Anisotropic Elastic Plates

Anisotropic Elastic Plates Chyanbin Hwu Anisotropic Elastic Plates 123 Chyanbin Hwu Department of Aeronautics and...

Author: Chyanbin Hwu

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Anisotropic Elastic Plates

Chyanbin Hwu

Anisotropic Elastic Plates

123

Chyanbin Hwu Department of Aeronautics and Astronautics National Cheng Kung University No. 1 University Road Tainan 701 Taiwan R.O.C [email protected]

ISBN 978-1-4419-5914-0 e-ISBN 978-1-4419-5915-7 DOI 10.1007/978-1-4419-5915-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010922035 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Due to the nature of anisotropy, composite materials are usually modeled as anisotropic elastic solids. Therefore, the researchers and engineers interested in composite materials are usually advised to get acquainted with anisotropic elasticity. However, only three books related to anisotropic elasticity have been published in the literature. Two of them were written in Russian by Professor S.G. Lekhnitskii and were originally published in 1947 and 1950. Their English translations were published later in 1963 and 1968. These two books are the classical books of anisotropic elasticity and have great contributions for the follow-up research. In the present book, I arrange one chapter named Lekhnitskii formalism introducing the classical method presented in these two books. The third book about anisotropic elasticity was written by Professor T.C.T. Ting and was published in 1996. A great contribution of Ting’s book is the presentation of another systematic approach – Stroh formalism. Due to its importance, the Stroh formalism together with its related discussions introduced in Ting’s book is summarized in Chapter 3 of this book. Owing to the publication of Lekhnitskii’s and Ting’s books, during the last half century numerous new advances have been achieved. Therefore, I think it is a proper time to update this topic by publishing a new book entitled Anisotropic Elastic Plates. As structural elements, anisotropic elastic plates find wide applications in modern technology. The plates here are considered to be subjected to not only in-plane loads but also transverse loads. In other words, both plane problem and plate bending problem as well as stretching–bending coupling problem are all treated in this book. In addition to the introduction of the theory of anisotropic elasticity, several important subjects have also been discussed in this book such as interfaces, cracks, holes, inclusions, contact problems, piezoelectric materials, thermal stresses, and boundary element analysis. Most of the materials presented in this book can be found in the journal papers written by me and my co-workers, and some others are edited from the books and journal papers written by the other researchers. Even some notations have been unified in Lekhnitskii’s and Ting’s books, the notations used in the new advancements including my own works are still quite varied. Without a unified notation system, it is difficult for a beginner to study the subject. Therefore, in this book all the materials collected from the published results have been rewritten using a unified notation v

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system and some useful Appendices are provided for the symbols, sign convention, formalisms, and problem solutions. Elasticity and mechanics of composite materials are two important fundamental courses for senior undergraduate students and beginning graduate students in aerospace, civil, naval and mechanical engineering, applied mechanics, and engineering science. Several textbooks have been written for the studies of these two courses. I believe this book is helpful for engineers and scientists who want to have an advanced knowledge of the theory of elasticity and mechanics of composite materials. This book is appropriate to be a university textbook for the courses such as anisotropic elasticity, advanced elasticity, and advanced mechanics of composite materials. It is also a good reference book for the standard courses such as elasticity, mechanics of composite materials, fracture mechanics, plates and shells, and boundary element method and for the advanced courses such as micromechanics, contact mechanics, smart materials and structures, and thermal elasticity. Special features: 1. This book connects anisotropic elasticity and mechanics of composite materials. This book provides a systematic complex variable approach covering both plane problem and plate bending problem as well as stretching–bending coupling problem. The advancement of the stretching–bending coupling problem started nearly 15 years ago and hence has never been introduced in any book related to anisotropic elasticity or mechanics of composite materials. Most of the books related to anisotropic elasticity discuss only plane problem, whereas the books related to mechanics of composite materials discuss mainly the plate bending problem. Thus, we need a systematic approach to connect these two related topics. 2. This book connects anisotropic elasticity and fracture mechanics. Most of the crack problems are discussed in the books entitled Fracture Mechanics. Not too many books related to elasticity have special chapters named Cracks or Holes or Inclusions or Wedges & Interface Corners. I believe the arrangement of these chapters is helpful for the readers to understand the connection between elasticity and fracture mechanics. 3. This book connects theoretical treatment and numerical analysis. Most of the books related to elasticity introduce mainly the theoretical treatment of elastic deformable solids and leave the numerical analysis to special books such as finite element method or boundary element method. To let the readers see more clearly about the connection between theoretical treatment and numerical analysis, we arrange a chapter named boundary element analysis in this book. The boundary elements introduced in this chapter involve both two-dimensional problems and stretching–bending coupling problems. 4. Several special topics are discussed through one systematic approach In addition to cracks, holes, inclusions, wedges & interface corners, the topics such as contact problems, thermoelastic problems, piezoelectric materials, and

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holes/cracks/ inclusions in laminates, which are important in the engineering practice, are discussed separately in the specific chapters. Through these chapters the readers can understand how to apply the method introduced in this book to treat these special and interesting problems. 5. Collection of problem solutions In contrast to isotropic elastic materials that have only two elastic constants, anisotropic elastic materials may have as many as 21 elastic constants. Therefore, how to express the solutions for the problems of anisotropic elasticity in a simple and systematic way is really a big problem. Thus, even for one simple conventional problem, there may be several different kinds of mathematical expressions appeared in the literature. This also causes trouble for engineers to utilize the existing solutions, if they have difficulty in understanding the derivation details. In this book, more than 100 problem solutions are collected in Appendix D. To avoid the confusion caused by the symbols, Appendix A is provided in this book that describes the symbols, sign convention, and units. Moreover, to help the readers see clearly the unified expression used in this book, the summary of Stroh formalism is provided in Appendix C. Each problem collected in Appendix D is described with aid of a simple figure, and its solution is expressed in terms of the same symbol system. I believe through this collection most of the engineers and scientists can take advantage of these solutions freely and easily even they do not have enough time to understand their derivation details.

I wish to express my gratitude to my Ph.D. thesis adviser, Professor T.C.T. Ting. I am very fortunate to get into the field of anisotropic elasticity through his guidance. Several new advances of anisotropic elasticity have been achieved due to the publication of his book. Hope that the present book can also help the researchers go further. I also want to express my gratitude to my mentors, Professor C.S. Yeh of National Taiwan University and Professor W.H. Chen of National TsingHua University for their guidance during my studies for B.S. and M.S. degrees. In particular, since part of this book was written during my sabbatical leave, I am grateful to Professors K. Kishimoto (Tokyo Institute of Technology), M. Omiya (Keio University), N. Miyazaki (Kyoto University), T. Ikeda (Kyoto University), Y.W. Mai (Sydney University), T. Aoki (Tokyo University), and T. Yokozeki (Tokyo University), who have helped me during my staying in their departments. Special thanks also to my assistant H.E. Shen, my former student Y.C. Liang, and my present students C.Z. Tan, T.L. Kuo, Y.C. Chen, and H.Y. Huang who helped me draw part of figures presented in this book. I would also like to thank my friends C.C. Ma, K.C. Wu, T.T. Wu of National Taiwan University, C.K. Chao of National Taiwan University of Science and Technology, and T. Chen of National Cheng Kung University for their helpful discussions during my research on anisotropic elasticity. I acknowledge the National Science Council of Taiwan for the support of my research in the area of anisotropic elasticity.

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Finally, I would like to dedicate this book to my wife, Wenling, and my daughters, Frannie and Vevey, with thanks for their constant support and encouragement in everything. Tainan, Taiwan September, 2009

Chyanbin Hwu

Contents

1 Linear Anisotropic Elastic Materials . . . . . . 1.1 Theory of Elasticity for Anisotropic Bodies 1.1.1 State of Stress . . . . . . . . . . . 1.1.2 Deformation . . . . . . . . . . . . 1.1.3 Constitutive Laws . . . . . . . . . 1.1.4 Boundary Conditions . . . . . . . 1.2 Three-Dimensional Constitutive Relations . 1.2.1 Generalized Hooke’s Law . . . . . 1.2.2 Material Symmetry . . . . . . . . 1.2.3 Engineering Constants . . . . . . . 1.3 Two-Dimensional Constitutive Relations . . 1.3.1 Isotropic Materials . . . . . . . . . 1.3.2 Anisotropic Materials . . . . . . . 1.3.3 Monoclinic Materials . . . . . . . 1.3.4 Orthotropic Materials . . . . . . . 1.4 Laminate Constitutive Relations . . . . . . 1.4.1 Specially Orthotropic Lamina . . . 1.4.2 Generally Orthotropic Lamina . . . 1.4.3 Classical Lamination Theory . . .

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2 Lekhnitskii Formalism . . . . . . . . . . . . . . . . . . 2.1 Governing Differential Equations . . . . . . . . . . 2.2 General Solutions . . . . . . . . . . . . . . . . . . 2.3 Boundary Conditions . . . . . . . . . . . . . . . . 2.3.1 Lateral Surface Conditions . . . . . . . . 2.3.2 End Conditions . . . . . . . . . . . . . . 2.4 Special Cases . . . . . . . . . . . . . . . . . . . . 2.4.1 Generalized Plane Deformation . . . . . . 2.4.2 Plane Deformation . . . . . . . . . . . . . 2.4.3 Generalized Plane Stress . . . . . . . . . . 2.4.4 Anisotropic Rod by Bending and Twisting 2.5 Anisotropic Cantilever Under Transverse Force . .

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3 Stroh Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 General Solutions . . . . . . . . . . . . . . . . . . . . . . 3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 3.3 Material Eigenrelation . . . . . . . . . . . . . . . . . . . 3.3.1 Sextic Eigenrelation . . . . . . . . . . . . . . . . 3.3.2 Generalized Sextic Eigenrelation . . . . . . . . . 3.3.3 The Matrix Differential Equation . . . . . . . . . 3.4 Some Identities . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Explicit Expression of Fundamental Elasticity Matrix N . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Explicit Expressions of Barnett–Lothe Tensors S, H, and L . . . . . . . . . . . . . . . . 3.4.3 Identities Relating N, N (θ), S, H, L . . . . . . . . 3.4.4 Identities Converting Complex Form to Real Form 3.5 Degenerate Materials . . . . . . . . . . . . . . . . . . . .

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4 Infinite Space, Half-Space, and Bimaterials . . . . . . . . . . 4.1 Infinite Space . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Uniform Loading . . . . . . . . . . . . . . . . 4.1.2 Pure In-Plane Bending . . . . . . . . . . . . . . 4.1.3 Concentrated Forces . . . . . . . . . . . . . . . 4.1.4 Couple Moments . . . . . . . . . . . . . . . . . 4.1.5 Dislocations . . . . . . . . . . . . . . . . . . . 4.2 Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Green’s Function . . . . . . . . . . . . . . . . . 4.2.2 Surface Green’s Function . . . . . . . . . . . . 4.2.3 Distributed Load Along the Half-Space Surface 4.2.4 Couple Moments . . . . . . . . . . . . . . . . . 4.2.5 Dislocations . . . . . . . . . . . . . . . . . . . 4.3 Bimaterials . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Green’s Function . . . . . . . . . . . . . . . . . 4.3.2 Interfacial Green’s Function . . . . . . . . . . .

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87 87 87 91 92 95 99 100 100 105 106 107 107 108 108 111

5 Wedges and Interface Corners . . . . . . . . . . . . . . . . . 5.1 Uniform Tractions on the Wedge Sides . . . . . . . . . . 5.1.1 Non-critical Wedge Angles . . . . . . . . . . . 5.1.2 Critical Wedge Angles . . . . . . . . . . . . . . 5.1.3 Summary . . . . . . . . . . . . . . . . . . . . . 5.2 Forces at the Wedge Apex . . . . . . . . . . . . . . . . . 5.2.1 A Single Wedge – Under a Concentrated Force . 5.2.2 A Single Wedge – Under a Concentrated Couple 5.2.3 Multi-material Wedge Spaces . . . . . . . . . . 5.2.4 Multi-material Wedges . . . . . . . . . . . . . . 5.3 Stress Singularities . . . . . . . . . . . . . . . . . . . . 5.3.1 Multi-material Wedge Spaces . . . . . . . . . . 5.3.2 Multi-material Wedges . . . . . . . . . . . . . .

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5.3.3 Eigenfunctions . . . . . . . . . . . . . . . . . . . . 5.3.4 Special Cases . . . . . . . . . . . . . . . . . . . . Stress Intensity Factors of Interface Corners . . . . . . . . . 5.4.1 Near-Tip Field Solutions . . . . . . . . . . . . . . . 5.4.2 A Unified Definition . . . . . . . . . . . . . . . . . 5.4.3 H-Integral for Two-Dimensional Interface Corners . 5.4.4 H-Integral for Three-Dimensional Interface Corners 5.4.5 Numerical Examples . . . . . . . . . . . . . . . . .

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6 Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Elliptical Holes . . . . . . . . . . . . . . . . . . . . 6.1.1 Uniform Loading at Infinity . . . . . . . . . 6.1.2 In-Plane Bending at Infinity . . . . . . . . . 6.1.3 Arbitrary Loading Along the Hole Boundary 6.1.4 Concentrated Force at Arbitrary Location . . 6.1.5 Dislocation at Arbitrary Location . . . . . . 6.2 Polygon-Like Holes . . . . . . . . . . . . . . . . . . 6.2.1 Transformation Function . . . . . . . . . . . 6.2.2 Uniform Loading at Infinity . . . . . . . . . 6.2.3 Pure In-Plane Bending at Infinity . . . . . . 6.2.4 Discussions . . . . . . . . . . . . . . . . .

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7 Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Singular Characteristics of Cracks . . . . . . . . 7.1.1 Cracks in Homogeneous Materials . . . 7.1.2 Interfacial Cracks . . . . . . . . . . . . 7.1.3 Cracks Terminating at the Interfaces . . 7.2 A Finite Straight Crack . . . . . . . . . . . . . . 7.2.1 Uniform Loading at Infinity . . . . . . . 7.2.2 In-plane Bending at Infinity . . . . . . . 7.2.3 Arbitrary Loading on the Crack Surfaces 7.2.4 Concentrated Force at Arbitrary Location 7.2.5 Dislocation at Arbitrary Location . . . . 7.3 Collinear Cracks . . . . . . . . . . . . . . . . . . 7.3.1 General Solutions . . . . . . . . . . . . 7.3.2 Two Collinear Cracks . . . . . . . . . . 7.3.3 Collinear Periodic Cracks . . . . . . . . 7.3.4 Fracture Parameters . . . . . . . . . . . 7.4 Collinear Interface Cracks . . . . . . . . . . . . 7.4.1 General Solutions . . . . . . . . . . . . 7.4.2 A Semi-infinite Interface Crack . . . . . 7.4.3 A Finite Interface Crack . . . . . . . . . 7.4.4 Two Collinear Interface Cracks . . . . . 7.4.5 Fracture Parameters . . . . . . . . . . .

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7.5

Delamination Fracture Criteria . . . . . . . . . . . . . . . 7.5.1 Stress Intensity Factors and Energy Release Rates 7.5.2 Experimental Details . . . . . . . . . . . . . . . . 7.5.3 Delamination Fracture Toughness . . . . . . . . . 7.5.4 Mixed-Mode Fracture Criteria . . . . . . . . . . . 7.5.5 Prediction of Delamination Fracture . . . . . . . .

8 Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Elliptical Elastic Inclusions . . . . . . . . . . . . . . . 8.1.1 Uniform Loading at Infinity . . . . . . . . . . 8.1.2 Concentrated Forces at the Matrix . . . . . . . 8.2 Rigid Inclusions . . . . . . . . . . . . . . . . . . . . . 8.2.1 Elliptical Rigid Inclusions . . . . . . . . . . . 8.2.2 Rigid Line Inclusions . . . . . . . . . . . . . 8.2.3 Polygon-Like Rigid Inclusions . . . . . . . . 8.3 Interactions Between Inclusions and Dislocations . . . 8.3.1 Dislocations Outside the Inclusions . . . . . . 8.3.2 Dislocations Inside the Inclusions . . . . . . . 8.3.3 Dislocations on the Interfaces . . . . . . . . . 8.3.4 Interaction Energy . . . . . . . . . . . . . . . 8.4 Interactions Between Inclusions and Cracks . . . . . . 8.4.1 Cracks Outside the Inclusions . . . . . . . . . 8.4.2 Cracks Inside the Inclusions . . . . . . . . . . 8.4.3 Cracks Penetrating the Inclusions . . . . . . . 8.4.4 Curvilinear Cracks Lying Along the Interfaces

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9 Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Rigid Punches on a Half-Plane . . . . . . . . . . . . . . 9.1.1 General Solution . . . . . . . . . . . . . . . . . 9.1.2 Indentation by a Flat-Ended Punch . . . . . . . 9.1.3 A Flat-Ended Punch Tilted by a Couple . . . . . 9.1.4 Indentation by a Parabolic Punch . . . . . . . . 9.1.5 Analogy with the Interface Crack Problems . . . 9.2 Rigid Stamp Indentation on a Curvilinear Hole Boundary 9.2.1 General Solution . . . . . . . . . . . . . . . . . 9.2.2 Elliptical Hole Boundaries . . . . . . . . . . . . 9.2.3 Polygonal Hole Boundaries . . . . . . . . . . . 9.2.4 Numerical Calculation . . . . . . . . . . . . . . 9.3 Rigid Punches on a Perturbed Surface . . . . . . . . . . 9.3.1 Straight Boundary Perturbation . . . . . . . . . 9.3.2 Elliptical Boundary Perturbation . . . . . . . . 9.3.3 Illustrative Examples . . . . . . . . . . . . . . . 9.4 Sliding Punches With or Without Friction . . . . . . . . 9.4.1 General Solution . . . . . . . . . . . . . . . . . 9.4.2 A Sliding Wedge-Shaped Punch . . . . . . . . .

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10 Thermoelastic Problems . . . . . . . . . . . . . . . . . . . . 10.1 Extended Stroh Formalism . . . . . . . . . . . . . . . . 10.2 Holes and Cracks . . . . . . . . . . . . . . . . . . . . . 10.2.1 Elliptical Holes Under Uniform Heat Flow . . . 10.2.2 Cracks Under Uniform Heat Flow . . . . . . . . 10.3 Rigid Inclusions . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Elliptical Rigid Inclusions Under Uniform Heat Flow . . . . . . . . . . . . . . . . . . . . 10.3.2 Rigid Line Inclusions Under Uniform Heat Flow 10.4 Collinear Interface Cracks . . . . . . . . . . . . . . . . 10.4.1 General Solutions . . . . . . . . . . . . . . . . 10.4.2 Uniform Heat Flow . . . . . . . . . . . . . . . 10.5 Multi-material Wedges . . . . . . . . . . . . . . . . . . 10.5.1 Stress and Heat Flux Singularity . . . . . . . . . 10.5.2 Near-Tip Solutions . . . . . . . . . . . . . . . . 10.5.3 Special Cases . . . . . . . . . . . . . . . . . .

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11 Piezoelectric Materials . . . . . . . . . . . . . . . . . . . . . 11.1 Constitutive Laws . . . . . . . . . . . . . . . . . . . . . 11.1.1 Three-Dimensional State . . . . . . . . . . . . 11.1.2 Two-Dimensional State . . . . . . . . . . . . . 11.2 Expanded Stroh Formalism . . . . . . . . . . . . . . . . 11.2.1 General Solutions . . . . . . . . . . . . . . . . 11.2.2 Boundary Conditions . . . . . . . . . . . . . . 11.3 Explicit Expressions . . . . . . . . . . . . . . . . . . . . 11.3.1 Fundamental Matrix N . . . . . . . . . . . . . . 11.3.2 Material Eigenvector Matrices A and B . . . . . 11.3.3 Barnett–Lothe Tensors S, H, and L . . . . . . . 11.3.4 Bimaterial Matrices D and W . . . . . . . . . . 11.4 Multi-material Wedges . . . . . . . . . . . . . . . . . . 11.4.1 Orders of Stress/Electric Singularity . . . . . . 11.4.2 Near-Tip Solutions . . . . . . . . . . . . . . . . 11.4.3 Stress/Electric Intensity Factors . . . . . . . . . 11.4.4 Corner Opening Displacement/Electric Potential 11.5 Singular Characteristics of Cracks . . . . . . . . . . . . 11.5.1 Cracks in Homogeneous Piezoelectric Materials 11.5.2 Interface Cracks Between Two Dissimilar Piezoelectric Materials . . . . . . . . . . . . . .

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9.4.3 A Sliding Parabolic Punch . . . . . 9.4.4 Two Sliding Flat-Ended Punches . Contact Between Two Elastic Bodies . . . . 9.5.1 Contact in the Presence of Friction 9.5.2 Contact in the Absence of Friction 9.5.3 Contact in Complete Adhesion . .

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11.6

Some Crack Problems . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Cracks . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Interface Cracks . . . . . . . . . . . . . . . . . . . .

404 405 406

12 Plate Bending Analysis . . . . . . . . . . . . . . . . . . . . . . . 12.1 Bending Theory of Anisotropic Plates . . . . . . . . . . . . 12.2 Lekhnitskii Bending Formalism . . . . . . . . . . . . . . . 12.2.1 General Solutions . . . . . . . . . . . . . . . . . . 12.2.2 Boundary Conditions . . . . . . . . . . . . . . . . 12.2.3 Degenerate Materials . . . . . . . . . . . . . . . . 12.3 Stroh-Like Bending Formalism . . . . . . . . . . . . . . . . 12.3.1 General Solutions . . . . . . . . . . . . . . . . . . 12.3.2 Material Eigenrelation and Its Explicit Expressions . 12.3.3 Explicit Expressions of S, H, and L . . . . . . . . . 12.4 Holes/Inclusions/Cracks . . . . . . . . . . . . . . . . . . . 12.4.1 Elliptical Holes . . . . . . . . . . . . . . . . . . . . 12.4.2 Elliptical Rigid Inclusions . . . . . . . . . . . . . . 12.4.3 Cracks . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

411 412 415 415 416 420 420 421 424 426 428 428 431 433

13 Coupled Stretching–Bending Analysis . . . . . . . . . . 13.1 Coupled Stretching–Bending Theory of Laminates 13.2 Complex Variable Formulation . . . . . . . . . . . 13.2.1 Displacement Formalism . . . . . . . . . 13.2.2 Mixed Formalism . . . . . . . . . . . . . 13.2.3 Explicit Expressions of N, A, and B . . . . 13.2.4 Reduction to Symmetric Laminates . . . . 13.2.5 Comparison and Discussion . . . . . . . . 13.3 Stroh-Like Formalism . . . . . . . . . . . . . . . . 13.3.1 General Solutions . . . . . . . . . . . . . 13.3.2 Material Eigenrelation . . . . . . . . . . . 13.3.3 Stress Functions . . . . . . . . . . . . . . 13.3.4 Explicit Expressions of N . . . . . . . . . 13.3.5 Explicit Expressions of A and B . . . . . . 13.3.6 Explicit Expressions of N(ω) . . . . . . . 13.3.7 Explicit Expressions of S, H, and L . . . . 13.4 Hygrothermal Stresses . . . . . . . . . . . . . . . 13.4.1 Basic Equations . . . . . . . . . . . . . . 13.4.2 Extended Stroh-Like Formalism . . . . . . 13.5 Electro-elastic Composite Laminates . . . . . . . . 13.5.1 Basic Equations . . . . . . . . . . . . . . 13.5.2 Expanded Stroh-Like Formalism . . . . .

. . . . . . . . . . . . . . . . . . . . . .

435 436 440 440 446 451 454 457 458 458 459 461 465 467 469 472 474 474 476 481 481 484

14 Holes/Cracks/Inclusions in Laminates . . . . . . . . . . . . . . . . 14.1 Holes in Laminates Under Uniform Stretching and Bending Moments . . . . . . . . . . . . . . . . . . . . .

493

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494

Contents

xv

14.1.1 14.1.2

Field Solutions . . . . . . . . . . . . . . . . . . . . Stress Resultants and Moments Along the Hole Boundary . . . . . . . . . . . . . . . . . . . . 14.2 Holes in Laminates Under Uniform Heat Flow and Moisture Transfer . . . . . . . . . . . . . . . . . . . . . 14.2.1 Uniform Heat Flow and Moisture Transfer in x1 x2 -Plane . . . . . . . . . . . . . . . . . . . . . 14.2.2 Uniform Heat Flow and Moisture Transfer in x3 -Direction . . . . . . . . . . . . . . . . . . . . 14.3 Holes in Electro-Elastic Laminates . . . . . . . . . . . . . . 14.4 Green’s Functions for Laminates . . . . . . . . . . . . . . . 14.4.1 Concentrated In-Plane Forces and ˆ 1, m ˆ 2) . . . . . . . . Out-of-Plane Moments (fˆ1 , fˆ2 , m 14.4.2 Concentrated Transverse Force (fˆ3 ) . . . . . . . . . 14.4.3 Concentrated In-Plane Torsion (mˆ3 ) . . . . . . . . . 14.4.4 Explicit Real-Form Solutions . . . . . . . . . . . . 14.5 Green’s Functions for Laminates with Holes/Cracks . . . . . 14.5.1 Field Solutions . . . . . . . . . . . . . . . . . . . . 14.5.2 Stress Resultants and Moments Along the Hole Boundary . . . . . . . . . . . . . . . . . . . . 14.5.3 Verification and Discussions . . . . . . . . . . . . . 14.5.4 Cracks . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Green’s Functions for Laminates with Elastic Inclusions . . . 14.6.1 Concentrated Forces/Moments Outside the Inclusions . . . . . . . . . . . . . . . . . . . . 14.6.2 Concentrated Forces/Moments Inside the Inclusions 14.6.3 Verification and Discussions . . . . . . . . . . . . . 15 Boundary Element Analysis . . . . . . . . . . . . . . . . . . 15.1 Two-Dimensional Elastic Analysis . . . . . . . . . . . . 15.1.1 Boundary Integral Equations . . . . . . . . . . 15.1.2 Fundamental Solutions . . . . . . . . . . . . . 15.1.3 Boundary Element Formulation . . . . . . . . . 15.1.4 Stresses and Displacements at Internal Points . . 15.1.5 Stress Intensity Factors for Crack Problems . . . 15.1.6 Subregion Technique . . . . . . . . . . . . . . . 15.1.7 Numerical Implementation . . . . . . . . . . . 15.2 Two-Dimensional Electro-Elastic Analysis . . . . . . . . 15.2.1 Boundary Element Formulation . . . . . . . . . 15.2.2 Numerical Examples . . . . . . . . . . . . . . . 15.3 Coupled Stretching–Bending Analysis . . . . . . . . . . 15.3.1 Boundary Integral Equations – Internal Points . 15.3.2 Fundamental Solutions . . . . . . . . . . . . . 15.3.3 Boundary Integral Equations – Boundary Points 15.3.4 Free-Term Coefficients . . . . . . . . . . . . .

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506 508 510 512 515 516

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522 523 524 526

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529 534 539

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545 545 545 546 552 554 555 557 559 560 560 561 565 565 570 575 580

xvi

Appendix A: Symbols, Sign Convention and Units A.1 Common Symbols . . . . . . . . . . . . . A.2 Extended Symbols . . . . . . . . . . . . . A.3 Sign Convention . . . . . . . . . . . . . . A.4 Units . . . . . . . . . . . . . . . . . . . .

Contents

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589 589 594 599 600

Appendix B: Hilbert Problem . . . . . . . . . . . . . . B.1 Solution to the Hilbert Problem in Scalar Form B.2 Solution to the Hilbert Problem in Matrix form B.3 Evaluation of a Line Integral in Scalar Form . B.4 Evaluation of a Line Integral in Matrix Form .

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607 607 608 610 612

Appendix C: Summary of Stroh formalism . . . . . . . C.1 Two-dimensional Problems . . . . . . . . . . . C.2 Coupled Stretching-Bending Problems . . . . . C.3 Dimensions of Matrices used in Stroh Formalism

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615 615 618 624

Appendix D: Collection of Problem Solutions . . . . . . . . . . . . . . .

625

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

655

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

663

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

667

Chapter 1

Linear Anisotropic Elastic Materials

The mechanical properties of materials are described by constitutive laws. There are a wide variety of materials existing in the world. We are not surprised that there are a great many constitutive laws describing an almost infinite variety of materials. What is surprising is that a simple idealized stress–strain relationship gives a good description of the mechanical properties of many elastic materials around us. In this chapter, we present the relation between stresses and strains in a linear anisotropic elastic material. By this relation, we need 21 elastic constants to describe a linear anisotropic elastic material if the materials do not possess any symmetry properties. In engineering application, this number is somewhat higher than expected. Consideration of the material symmetry will then reduce the number of elastic constants. To provide these constants obvious physical interpretation, engineering constants such as the Young’s moduli, Poisson’s ratios and shear moduli as well as some other behavior constants will also be introduced in this chapter. If the problems considered can be treated as a two-dimensional problem, the elastic constants needed for the analysis of the mechanical behavior of anisotropic materials can be further reduced.

1.1 Theory of Elasticity for Anisotropic Bodies To study the behavior of an elastic continuous medium, the theory of elasticity is a generally accepted model. In this section we will briefly describe the general concept of elasticity for anisotropic bodies, such as state of stress, deformation, constitutive laws, and boundary conditions.

1.1.1 State of Stress In the study of elasticity, the body is generally considered to be a deformable continuous medium. Usually the objectives of analysis are the determination of stresses and strains induced by the external loads. The state of stress at a given point of a continuous body, which is either in equilibrium or in motion as a result of external C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_1, C Springer Science+Business Media, LLC 2010

1

2

1 Linear Anisotropic Elastic Materials σ 33

Fig. 1.1 Stress components

σ 32

σ 31

σ 23

σ 13

x3

σ 11

σ 12 σ 21

σ 22

x2 x1

forces, is known to be represented by the stress components on three mutually perpendicular planes passing through the point. If the infinitesimal volume element enclosing the point is taken in the shape of a rectangular parallelepiped, with faces parallel to the coordinate planes, the stress components σij are shown in Fig. 1.1. The first subscript of σ indicates the direction of normal to the plane, while the second indicates the direction of the stress component. The components σ11 , σ22 , and σ33 which are normal to the element surfaces are called normal stresses. The remaining components that are parallel to the surfaces are called shear stresses. The stress components are symmetric if the body moment is neglected. Knowing the stress components on the planes normal to the coordinate axes, we can always determine the stress vector t on any surface with unit outer normal vector n, which passes through the given point. This stress vector is determined by Cauchy’s formula as ti = σij nj ,

(1.1)

in which repeated indices imply summation through 1–3. This rule of summation convention will be applied in the whole text unless stated otherwise. The derivation of symmetry property and Cauchy’s formula can be found in any standard text of elasticity such as Sokolnikoff (1956). The stress components in the other coordinate systems are determined like a Cartesian tensor of rank 2, i.e., ∗ σpq = pi qj σij ,

(1.2)

where pi are the direction cosines between rotated (starred) and original (unstarred) axes. There exists a particular set of coordinate axes with respect to which all the shear stress components are zero. These coordinate axes are called the principal axes, and the corresponding stress components are called the principal stresses. By this definition, the principal axes and principal stresses can be determined by solving the following eigenvalue problem: (σij − σ δij )ni = 0,

(1.3)

1.1

Theory of Elasticity for Anisotropic Bodies

3

in which δij is defined as δij = 1 when i = j and δij = 0 when i = j, and is called the Kronecker delta. The stress components in a continuous body which is in equilibrium must satisfy the equilibrium equations, which in Cartesian coordinates are σij, j + fi = 0,

(1.4)

where fi designate the body forces referred to a unit volume in directions x1 , x2 , and x3 , and a comma stands for differentiation. Equations of motion of a continuous medium differ from equilibrium equations only by having inertia terms ρ u¨ i placed at the right hand side of (1.4) instead of zero, where ρ is the material density and ui is the displacement in the xi -direction and the double dot denotes twice differentiation with respect to time. Note that all the relations written above for the stresses are designated to every point of a continuous solid, which is nothing to do with the material properties, structure types and sizes. Therefore, it should be valid for the studies of isotropic elasticity or anisotropic elasticity, micromechanics or macromechanics, etc.

1.1.2 Deformation Forces applied to solids cause deformation. When the relative position of points in a continuous body is altered, the body is strained. The change in the relative position of points is a deformation. All material bodies are to some extent deformable. If there exists an ideal body which is nondeformable such that the distance between every pair of its points remains invariant throughout the history of the body, we call it a rigid body. The motion of a rigid body is usually described by translation and rotation. A deformable solid will experience an additional change in shape, i.e., deformation. One of the major objectives of elasticity theory is the determination of the deformation of the solid from some reference configuration. There are two modes of description of the deformation of a continuous medium, the Lagrangian and Eulerian. The Lagrangian description employs the coordinates of a typical particle in the initial state as the independent variables, while in the case of Eulerian coordinates the independent variables are the coordinates of a material point in the deformed state. When the deformation is infinitesimal, these two viewpoints coalesce and no distinction between them will be made. Let the variable (a1 , a2 , a3 ) refer to any particle in the original configuration of the body, and let (x1 , x2 , x3 ) be the coordinates of that particle when the body is deformed. The deformation of the body is known if x1 , x2 , x3 are known functions of a1 , a2 , a3 . The displacement vector u is defined by its components ui = xi − ai .

(1.5)

4

1 Linear Anisotropic Elastic Materials

Since the displacements defined in (1.5) may include the rigid body motion and deformation in which the former induces no stress. Thus, the displacements themselves are not directly related to the stress. To relate deformation with stress, we must consider the stretching and distortion of the body, which is related to the change in distance between any two points of the body. For this purpose, the Lagrangian and Eulerian strain tensors are defined as ∂uj 1 ∂ui ∂uk ∂uk , + + 2 ∂aj ∂ai ∂ai ∂aj ∂uj 1 ∂ui ∂uk ∂uk Eij = . + − 2 ∂xj ∂xi ∂xi ∂xj

Lij =

(1.6)

If the components of displacement are such that their first derivatives are so small that the squares and products of the partial derivatives of ui can be neglected, both of the Lagrangian and Eulerian strain tensors reduce to Cauchy’s infinitesimal strain tensor εij =

1 (ui, j + uj, i ). 2

(1.7)

Like the stresses, the components of the strains that reflect the stretching or shortening of the body, i.e., ε11 , ε22 and ε33 are called normal strains. The remaining components related to the distortion of the body are called shear strains. Sometimes γij = 2εij , i = j are used to represent engineering shear strains. Same as the stress components, according to the transformation law of the tensor of rank two the strains in the other coordinate systems can be calculated by ∗ = pi qj εij . εpq

(1.8)

The principal strains and the principal axes of strains can also be determined by solving the following eigenvalue problem: (εij − εδij )ni = 0.

(1.9)

With reference to the definition given in (1.7), it is clear that a sufficiently wellbehaved displacement field ui will generate an equally well-behaved strain field by differentiation. The converse, however, is not necessarily true. It is not always possible to find a continuous, single-valued displacement field for any set of six well-behaved scalar functions εij by integration of (1.7). For this reason we need to have the compatibility conditions for the strain fields to insure the existence of a single-valued, continuous displacement field for simply connected continuous body. The equations of compatibility obtained by St. Venant for infinitesimal strains are εij, kl + εkl, ij − εik, jl − εjl, ik = 0.

(1.10)

1.1

Theory of Elasticity for Anisotropic Bodies

5

The system of equations (1.10) consists of 34 = 81 equations, but some of these are identically satisfied, and some are repetitions. Considering the symmetry of the strains, there are only six strain components for three-dimensional problems. Since the six strains are defined in terms of three displacement functions, then only three independent compatibility equations within (1.10) are essential. In the case of two-dimensional problems, only three strain components and two displacement functions are necessary. Thus, only one independent compatibility equation should be satisfied for two-dimensional problems, which can be written as 2ε12, 12 = ε11, 22 + ε22, 11 .

(1.11)

Note that all the relations written above for the strains are true for any continuous body, both elastic and inelastic, and are nothing to do with the material properties, structure types and sizes. Derivation of these formulas can be found in textbooks of the theory of elasticity such as (Sokolnikoff, 1956).

1.1.3 Constitutive Laws In continuous media the state of stress is completely determined by the stress tensor σij , and the state of deformation by the strain tensor εij . If a material deforms as it is loaded and will return to its original dimensions during unloading, it is called an elastic material. In other words, an elastic material has a one-to-one analytical relation between the stresses and strains. If the materials obey a linear relationship between stresses and strains, which is usually called the generalized Hooke’s law, they are linear elastic materials. When a linear elastic material is maintained at a fixed temperature and the stresses vanish when the strains are all zero, i.e., the initial unstrained state of the solid is unstressed, the generalized Hooke’s law can be written as σij = Cijkl εkl ,

(1.12)

where Cijkl are elastic constants which characterize the elastic behavior of the solid. Since Cijkl is a fourth rank tensor, there are 81 elastic constants. Consideration of the symmetry properties of stresses and strains as well as the elastic and symmetric characteristics of the materials will reduce the number of elastic constants, which will be discussed detailedly in the following sections.

1.1.4 Boundary Conditions From the previous sections, we know that the basic equations for the anisotropic elasticity consist of equilibrium equations for the static loading conditions (1.4), strain–displacement relations for the small deformations (1.7) as well as the stress– strain laws for the linear anisotropic elastic solids (1.12). These three equation sets

6


constitute 15 partial differential equations with 15 unknown functions, ui , εij , σij , i, j = 1, 2, 3, in terms of three coordinate variables xi , i = 1, 2, 3. All these basic equations and unknown functions are designated to every point of the elastic body. A general solution for these 15 unknown functions satisfying 15 basic equations has been derived by complex variable formulation (Ting, 1996). Since all the equations stated are designated to points of the elastic body without considering the structure type and size, their associated general solutions can be applied to the studies of micromechanics or macromechanics, etc. If the structures constructed by the anisotropic elastic body are clearly defined and their associated loading and boundary conditions are well described, the undetermined functions of the general solutions and hence the 15 unknown functions will then be uniquely determined through the satisfaction of the boundary conditions. In other words, the state of stress and deformation will be determined by taking into account the boundary conditions. Depending on what is given at the boundary, there are several distinct problems. Generally, they are separated into the following three types: the first fundamental problem, the second fundamental problem, and the third fundamental problem. First fundamental problem is also called traction-prescribed problem. It is a problem that surface tractions and body forces are given on the elastic bodies. Surface tractions are force distributions which are applied to the surface of the solid, whereas body forces act on the internal matter of the solid. Examples of body forces are the action of gravity and magnetic attraction or repulsion. From (1.4) we see that the consideration of body forces will not induce too much difficulty if the solutions of the homogeneous parts have been found. Hence, in the whole text, the body forces will be neglected unless stated otherwise. The traction boundary conditions can usually be written as t = ˆt, along the body surface,

(1.13)

where t is the stress vector defined in (1.1), and the overhat denotes its prescribed value induced by the external forces. Second fundamental problem is also called displacement-prescribed problem. It is a problem that displacements are prescribed on the body surface. The boundary conditions in this case are ˆ along the body surface. u = u,

(1.14)

Third fundamental problem is also called mixed boundary value problem. In this case external forces are given on one part of the surface and displacements on another. This type of problem also includes problems where tangential forces and normal displacements are given on the surface, or normal forces and tangential displacements are given, etc. Mathematically, they can be written as t = ˆt, along part of the surface, ˆ along the remaining part of the surface; and u = u, or,

(1.15a)

1.2

Three-Dimensional Constitutive Relations

t1 = ˆt1 , u2 = uˆ 2 , u3 = uˆ 3 , or the other combinations.

7

(1.15b)

1.2 Three-Dimensional Constitutive Relations 1.2.1 Generalized Hooke’s Law As stated in Section 1.1.3, if the materials obey a linear relationship between stresses and strains, the constitutive law may be expressed by using the generalized Hooke’s law, i.e., σij = Cijkl εkl .

(1.16)

Since σij and εij are tensors of order two, Cijkl is a tensor of order four according to the quotient law. Consequently, the elastic constants transform according to the rule ∗ Cpqrs = pi qj rk sl Cijkl .

(1.17)

The elastic tensor Cijkl may vary from point to point of the medium. If the Cijkl are independent of the position of the point, the medium is called elastically homogeneous. In this book, most of our attention is confined to those media in which the Cijkl do not vary throughout the region under consideration. Since Cijkl is a fourth rank tensor, there are 34 = 81 elastic constants. Inasmuch as the stress components are symmetric, which is contingent upon the vanishing of the body moment, an interchange of the indices i and j in (1.16) does not alter these formulas, so that Cijkl = Cjikl .

(1.18)

Equation (1.18) reduces the number of independent elastic constants to 3 × 3 × 6 = 54. Moreover, through the symmetry of the strain tensor, which can be observed either from the Lagrangian description or from the Eulerian description, a further reduction of the elastic constants may be made. That is, εij = εji , and therefore σij = Cijkl εkl = Cijkl εlk = Cijlk εkl , which may lead to εkl (Cijkl − Cijlk ) = 0. Since this equality must hold for arbitrary values of εkl , it looks like we may conclude that Cijkl = Cijlk .

(1.19)

However, εkl are not actually arbitrary, they are restricted by the symmetry conditions. For example, expansion of εkl (Cijkl − Cijlk ) = 0 may lead to ε12 (Cij12 − Cij21 ) + ε21 (Cij21 − Cij12 ) + · · · = 0. Given that ε12 = ε21 , etc., this equation will automatically be satisfied and no conclusion such as (1.19) can be made. Hence, the above proof generally seen in the textbook is not correct. The correct proof can be

8


found in Sokolnikoff (1956), in which the elastic tensor is separated into two parts, + C where C = (C i.e., Cijkl = Cijkl ijkl + Cijlk )/2 and Cijkl = (Cijkl − Cijlk )/2. ijkl ijkl is symmetric and C is skew From the definition, it can easily be seen that Cijkl ijkl ε since the symmetric with respect to k and l. Thus, (1.16) may lead to σij = Cijkl kl ε will be identical to zero due to the skew symmetry of C and symsum of Cijkl kl ijkl , the linear metry of εkl . Hence, if we redefine the elastic constants by using Cijkl relationship still preserves and moreover the newly defined elastic constants will be symmetry with respect to k and l. Therefore, the symmetry property shown in (1.19) will be used in the theory of anisotropic elasticity. This symmetry relation now leads a further reduction of the number of the elastic constants to 6 × 6 = 36. Additional restrictions are possible if we consider elastic materials only. The εf strain energy density of a material can be calculated by W = 0ij σij dεij . If εij

a material is elastic, its strain energy should be independent of the loading and unloading path. Mathematically speaking, the integrand should be a total differential, i.e., σij dεij = dW. By using the stress–strain relation given in (1.16), we get Cijkl εkl dεij = dW. The elastic tensor is therefore related to the strain energy density through the differentiation, i.e., Cijkl = ∂ 2 W/∂εkl ∂εij . Since the differentiations with respect to εij and εkl are interchangeable, we may get an additional symmetry conditions for the elastic tensor, i.e., Cijkl = Cklij .

(1.20)

The symmetries (1.18), (1.19), and (1.20) result in (62 − 6)/2 + 6 = 21 independent elastic constants for the most general case of anisotropy. Due to the symmetry properties, the number of independent elastic constants has been drastically decreased from 81 to 21. To avoid dealing with double sums, a contracted notation has been introduced as σ11 = σ1 , σ22 = σ2 , σ33 = σ3 , σ23 = σ4 , σ31 = σ5 , σ12 = σ6 , ε11 = ε1 , ε22 = ε2 , ε33 = ε3 , 2ε23 = ε4 , 2ε31 = ε5 , 2ε12 = ε6 .

(1.21)

The generalized Hooke’s law (1.16) and the symmetry conditions of the elastic tensor Cijkl shown in (1.18), (1.19), and (1.20) can therefore be written as σp = Cpq εq , Cpq = Cqp , p, q = 1, 2, . . . , 6,

(1.22a)

or, in matrix notation, σ = Cε, C = CT .

(1.22b)

Note that σp , Cpq , εq are not tensor quantities and therefore cannot be transformed as tensors. Cpq is sometimes called stiffness matrix. The transformation between Cijkl and Cpq is accomplished by the replacement of the subscript according to the following rules for ij (or kl) ↔ p (or q):

1.2


9

11 ↔ 1, 22 ↔ 2, 33 ↔ 3, 23(or 32) ↔ 4, 31(or 13) ↔ 5, 12(or 21) ↔ 6. (1.23) The relations between stresses and strains written in (1.16) must be reversible, and we can write εij = Sijkl σkl ,

(1.24)

where Sijkl are the compliances which are components of a fourth rank tensor. They also possess the full symmetry conditions like (1.18), (1.19), and (1.20), i.e., Sijkl = Sjikl , Sijkl = Sijlk , Sijkl = Sklij .

(1.25)

Similar to the contracted notation introduced for the elastic tensor Cijkl , the compliance tensor Sijkl can also be contracted according to the rules shown in (1.23) except suitable factors should be added as (Ting, 1996) Sijkl = Spq , 2Sijkl = Spq , 4Sijkl = Spq ,

if both p, q ≤ 3, if either p or q ≤ 3, if both p, q > 3.

(1.26)

With (1.25) and (1.26), the stress–strain law (1.24) in contracted notation is εp = Spq σq , Spq = Sqp ,

(1.27a)

ε = Sσ, S = ST .

(1.27b)

or, in matrix notation,

Substitution of (1.27b) into (1.22b) yields CS = I = SC,

(1.28)

where I is the 6×6 unit matrix. The relation (1.28) indicates that the stiffness matrix C and the compliance matrix S are the inverses of each other.

1.2.2 Material Symmetry With the foregoing reduction from 81 to 21 independent constants, the stress–strain relations (1.22) are

10


⎧ ⎫ ⎡ C11 σ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪σ2 ⎪ ⎪ ⎢C12 ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ ⎢ ⎢C13 σ3 =⎢ ⎢C14 σ ⎪ ⎪ 4⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣C15 σ ⎪ ⎪ 5⎪ ⎪ ⎩ ⎭ σ6 C16

C12 C22 C23 C24 C25 C26

C13 C23 C33 C34 C35 C36

C14 C24 C34 C44 C45 C46

C15 C25 C35 C45 C55 C56

⎤⎧ ⎫ ε1 ⎪ C16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C26 ⎥ ε2 ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎨ ⎬ C36 ⎥ ε ⎥ 3 , C46 ⎥ ε4 ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ C56 ⎦ ⎪ ε5 ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ C66 ε6

(1.29)

which is the most general expression within the framework of linear elasticity. In engineering applications, σi and εi , i = 4, 5, 6 are usually replaced by the notation τij and γij to represent the engineering shear stresses and strains. Actually, the relations in (1.29) are referred to characterizing anisotropic materials since there are no planes of symmetry for the material properties. An alternative name for such an anisotropic material is a triclinic material. For most elastic solids, the number of independent elastic constants is far smaller than 21. The reduction is caused by the existence of material symmetry. If there is one plane of material symmetry such as the plane x3 = 0, the stress–strain relations reduce to ⎧ ⎫ ⎡ ⎤⎧ ⎫ C11 C12 C13 0 0 C16 ⎪ ε1 ⎪ σ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎥⎪ σ C C C 0 0 C ε2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 12 22 23 26 ⎪ ⎥⎪ ⎨ ⎪ ⎬ ⎢ ⎬ ⎨ ⎪ ⎢ ⎥ σ3 C C C 0 0 C ε 13 23 33 36 3 ⎥ =⎢ . (1.30) ⎢ ⎥ σ4 ⎪ 0 0 0 C44 C45 0 ⎥ ⎪ ε4 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ σ5 ⎪ ε5 ⎪ 0 0 0 C45 C55 0 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎭ ⎩ ⎪ σ6 ε6 C16 C26 C36 0 0 C66 Such a material is termed monoclinic or aelotropic, which has 13 independent elastic constants. To prove the relation (1.30), the symmetry with respect to the plane x3 = 0 is expressed by the statement that the Cij are invariant under the transformation x1 = x1∗ , x2 = x2∗ , x3 = −x3∗ .

(1.31)

The direction cosines pi of this transformation is ⎡

⎤ 10 0 = ⎣0 1 0 ⎦ . 0 0 −1

(1.32)

One may prove (1.30) by directly using the transformation given in (1.17) and let∗ ting Cijkl = Cijkl and using the contraction notation defined in (1.23). If one is not familiar with the transformation of fourth rank tensors, (1.30) can be proved indirectly through the transformation of stresses and strains. Since the stresses and strains are tensors of rank 2, they can be transformed according to the following transformation: ∗ = pi qj σij , σpq

∗ εpq = pi qj εij .

(1.33)

1.2


11

From (1.32), (1.33) and the contracted notation defined in (1.23), it is seen that σi∗ = σi , εi∗ = εi , i = 1, 2, 3, 6, σ4∗ = −σ4 , ε4∗ = −ε4 , σ5∗ = −σ5 ,

ε5∗ = −ε5 .

(1.34)

Consider the transformed coordinate (x1∗ , x2∗ , x3∗ ), the first equation of (1.29) can be written as ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ1∗ = C11 ε1 + C12 ε2 + C13 ε3 + C14 ε4 + C15 ε5 + C16 ε6 .

(1.35)

Since the elastic constants are invariant under the transformation with respect to x3 = 0, and the stresses and strains in the transformed coordinate are obtained in (1.34), (1.35) can now be rewritten as σ1 = C11 ε1 + C12 ε2 + C12 ε3 − C14 ε4 − C15 ε5 + C16 ε6 .

(1.36)

Comparison of this equation with the expression of σ1 given by (1.29) shows that C14 = C15 = 0.

(1.37)

Similarly, by considering σ2∗ , . . . , σ6∗ , we can get the results shown in (1.30) for a material with the symmetry property with respect to the plane x3 = 0. If a material has two orthogonal planes of material symmetry, it can be proved that the symmetry will exist relative to a third mutually orthogonal plane (Ting, 1996). Such materials are said to be orthotropic (or rhombic). By a way similar to that shown for the monoclinic materials, the elastic constants for the orthotropic materials in coordinates aligned with principal material directions can be proved to be ⎤ ⎡ C11 C12 C13 0 0 0 ⎢C12 C22 C23 0 0 0 ⎥ ⎥ ⎢ ⎢C13 C23 C33 0 0 0 ⎥ ⎥ ⎢ (1.38) ⎢ 0 0 0 C44 0 0 ⎥ , ⎥ ⎢ ⎣ 0 0 0 0 C55 0 ⎦ 0 0 0 0 0 C66 which has nine independent elastic constants. If at every point of a material there is one plane in which the mechanical properties are equal in all directions, then the material is termed transversely isotropic. If, for example, the x3 = 0 plane is the special plane of isotropy, the elastic constants of this kind of materials can be proved to be ⎡ ⎤ C11 C12 C13 0 0 0 ⎢C12 C11 C13 0 0 ⎥ 0 ⎢ ⎥ ⎢C13 C13 C33 0 0 ⎥ 0 ⎢ ⎥, (1.39) ⎢ 0 0 0 C44 0 ⎥ 0 ⎢ ⎥ ⎣ 0 0 0 0 C44 ⎦ 0 0 0 0 0 0 (C11 − C12 )/2 which have only five independent constants.

12


The greatest reduction in the number of elastic constants is obtained when the material is symmetric with respect to any plane and any axis, or say, the elastic properties are identical in all directions. Such materials are called isotropic materials. Although there are an infinite number of symmetry planes for isotropic materials, to determine the structure of the elastic constants Cij no more than three symmetry planes are required (Ting, 1996). For example (Sokolnikoff, 1956), we may determine the elastic constants by considering the invariance about the new coordinate axes obtained by rotating the x1 , x2 , x3 -system through a right angle about the x1 -axis, then another new coordinate by a rotation of axes through a right angle about the x3 -axis, finally by a rotation of 45◦ about the x3 -axis. It can be proved that for any isotropic material, exactly two independent elastic constants characterize the material and the structure of the elastic constants Cij is ⎡

C11 ⎢C12 ⎢ ⎢C12 ⎢ ⎢ 0 ⎢ ⎣ 0 0

C12 C11 C12 0 0 0

⎤ C12 0 0 0 ⎥ C12 0 0 0 ⎥ ⎥ C11 0 0 0 ⎥. ⎥ 0 (C11 − C12 )/2 0 0 ⎥ ⎦ 0 0 0 (C11 − C12 )/2 0 0 0 (C11 − C12 )/2

(1.40)

By letting C12 = λ and (C11 − C12 )/2 = μ,

(1.41)

where λ and μ are the Lame constants, the generalized Hooke’s law for an isotropic material can then be written in the following form: σij = λδij εkk + 2μεij , i, j = 1, 2, 3.

(1.42)

Restrictions on elastic constants are due to the positive definiteness of the strain energy which implies that the stiffness (or the compliances) matrices must be positive definite. The necessary and sufficient conditions for Cij (or Sij ) to be positive definite are that the eigenvalues of Cij (or Sij ) are positive, or alternatively all the leading principal minors of the stiffness (or compliance) matrix are positive. The obtained restrictions on elastic constants can then be used to examine experimental data to see if they are physically consistent within the framework of the mathematical elasticity model.

1.2.3 Engineering Constants Engineering constants (also known as technical constants) are generalized Young’s moduli, Poisson’s ratios, and shear moduli as well as some other behavior constants. These constants are measured in simple tests such as uniaxial tension or pure shear tests. Thus, these constants with their obvious physical interpretation have more direct meaning than the components of the relatively abstract compliance

1.2


13

and stiffness matrices discussed previously. Most simple tests are performed with a known load or stress. The resulting displacement or strain is then measured. Thus, the components of the compliance matrix are determined more directly than those of the stiffness matrix. For a general anisotropic material, the compliance matrix components in terms of the engineering constants are

⎡

1 E1

⎢ ⎢ −ν12 ⎢ E1 ⎢ ⎢ −ν13 ⎢ E ⎢ 1 S=⎢η ⎢ 23,1 ⎢ E1 ⎢ ⎢ η31,1 ⎢ E1 ⎣ η12,1 E1

−ν21 E2

−ν31 E3

1 E2

−ν32 E3

−ν23 E2 η23,2 E2 η31,2 E2 η12,2 E2

1 E3 η23,3 E3 η31,3 E3 η12,3 E3

η1,23 G23 η2,23 G23 η3,23 G23 1 G23 μ31,23 G23 μ12,23 G23

η1,31 G31 η2,31 G31 η3,31 G31 μ23,31 G31 1 G31 μ12,31 G31

⎤

η1,12 G12 ⎥ η2,12 ⎥ G12 ⎥ ⎥ η3,12 ⎥ ⎥ G12 ⎥ , μ23,12 ⎥ ⎥ ⎥ G12 ⎥ μ31,12 ⎥ G12 ⎥ 1 G12

(1.43)

⎦

where E1 , E2 , E3 are the Young’s moduli in x1 , x2 , and x3 directions, respectively; νij is the Poisson’s ratio for transverse strain in the xj -direction when stressed in the xi -direction, that is, νij = −εj /εi for σi = σ and all other stresses are zero; G23 , G31 , G12 are the shear moduli in the x2 x3 , x3 x1 , and x1 x2 planes, respectively; ηi,ij is the coefficient of mutual influence of the first kind which characterizes stretching in the xi -direction caused by shear in the xi xj -plane, that is, ηi,ij = εi /γij for τij = τ and all other stresses are zero; ηij,i is the coefficient of mutual influence of the second kind which characterizes shearing in the xi xj -plane caused by a normal stress in the xi -direction, that is, ηij,i = γij /εi for σi = σ and all other stresses are zero; ηij,kl is the Chentsov coefficient which characterizes the shearing strain in the xi xj -plane due to shearing stress in the xk xl -plane, that is, ηij,kl = γij /γkl for τkl = τ and all other stresses are zero. Due to the symmetry of the compliance matrix, the Poisson’s ratios, the coefficients of mutual influences, and the Chentsov coefficients are subject to the following reciprocal relations: νji ηi, jk ηjk,i μij,kl μkl,ij νij = , = , = . Ei Ej Gjk Ei Gkl Gij

(1.44)

Full matrix shown in (1.43) for the general anisotropic materials also indicates that application of a normal stress leads not only to extension in the direction of the stress and contraction perpendicular to it, but to shearing deformation. Conversely, shearing stress causes extension and contraction in addition to the distortion of shearing deformation. For example, the out-of-plane shearing strains of an anisotropic material due to in-plane shearing stress and normal stresses are γ13 =

η1,31 σ1 + η2,31 σ2 + μ12,31 τ12 η1,23 σ1 + η2,23 σ2 + μ12,23 τ12 , γ23 = , G31 G23 (1.45)

14


wherein both the Chentsov coefficients and the coefficients of mutual influence of the first kind are required. Note that neither of these shear strains arise in an orthotropic material unless it is stressed in directions other than the principal material directions. In such cases, the Chentsov coefficients and the coefficients of mutual influence would be obtained from the transformed compliances. From (1.28) we know by inversion of (1.43) the stiffness matrix components Cij in terms of the engineering constants can be obtained. However, since the compliance matrix shown in (1.43) is a 6 × 6 full symmetric matrix, it is not easy to get the analytical expression of its inverse matrix. Following we just list the stiffness matrix components Cij for the orthotropic materials. Orthotropic Materials ⎡

1 − ν23 ν32 ν12 + ν32 ν13 ν13 + ν12 ν23 ⎢ E2 E3 E1 E3 E1 E2 ⎢ ⎢ 1 − ν13 ν31 ν23 + ν21 ν13 ⎢ ⎢ E1 E3 E1 E2 ⎢ ⎢ 1 − ν12 ν21 C=⎢ ⎢ ⎢ E1 E2 ⎢ ⎢ symm. ⎢ ⎣

⎤ 0 ⎥ ⎥ ⎥ 0 0 0 ⎥ ⎥ ⎥ ⎥ ⎥, 0 0 0 ⎥ ⎥ ⎥ G23 0 0 ⎥ ⎥ G31 0 ⎦ G12 0

0

(1.46a)

where Δ=

1 − ν12 ν21 − ν23 ν32 − ν31 ν13 − 2ν21 ν32 ν13 , E1 E2 E3

(1.46b)

and the symmetry conditions give us ν12 + ν32 ν13 ν31 + ν21 ν32 ν13 + ν12 ν23 ν21 + ν31 ν23 = , = , E2 E3 E1 E3 Δ E2 E3 Δ E1 E2 Δ ν32 + ν12 ν31 ν23 + ν21 ν13 = . E1 E3 Δ E1 E2 Δ

(1.47)

1.3 Two-Dimensional Constitutive Relations 1.3.1 Isotropic Materials Two-dimensional problems usually considered in isotropic elasticity fall into two physical distinct types. One of these arises in the study of deformation of large cylindrical bodies acted by the external forces so distributed that the component of deformation in the direction of the axis of the cylinder vanishes and the remaining components do not vary along the length of the cylinder. This is the class of problems in plane deformation or plane strain. Take the cross section of the cylinder be a plane parallel to x1 x2 -plane, the state of plane deformation may be characterized by

1.3

Two-Dimensional Constitutive Relations

15

uα = uα (x1 , x2 ), u3 = 0, α = 1, 2,

(1.48)

ε13 = ε23 = ε33 = 0.

(1.49)

which will lead to

The other type appears in the study of the deformation of thin plates, in which the state of stress is characterized by the vanishing of the stress components in the direction of the thickness of the plate. These are the problems in plane stress and can be characterized by σ13 = σ23 = σ33 = 0.

(1.50)

In the plane strain problem, all the displacement and stress components are independent of the x3 -coordinate, whereas in the problem of plane stress these components may depend on x3 . Since the variable x3 may appear as a parameter in the elasticity equations (for example, if ε33 = 0 through integration we get u3 which will depend on x3 ), the problem of plane stress is not truly two dimensional. However, by dealing with the mean values of the displacements and stresses, and supposing that the faces of the thin plates are free of applied loads and all external surface forces acting on the edge of the plate lie in the plane parallel to the middle plane, a mathematical truly two-dimensional problem can be obtained for a thin plate (Sokolnikoff, 1956), which is called the problems of generalized plane stress. It can be proved that the mathematical formulations of the plane strain and generalized plane stress are identical. The relevant differential equations and boundary conditions differ only in the Lame constant λ as well as the use of average displacements and stresses in the generalized plane stress problems. For plane strain problems we use uα , σαβ , λ, etc., while for generalized plane stress problems we use u˜ α , σ˜ αβ , λ˜ where u˜ α (x1 , x2 ) =

1 h

1 σ˜ αβ (x1 , x2 ) = h

h/2

uα (x1 , x2 , x3 )dx3 ,

−h/2 h/2

−h/2

(1.51a) σαβ (x1 , x2 , x3 )dx3 ,

α, β = 1, 2,

and λ˜ =

2λμ . λ + 2μ

(1.51b)

In the above, h is the thickness of the plate. In practical applications, to describe the isotropic materials it is common to use the engineering constants E and ν instead of the Lame constants λ and μ. Thus, it is useful to know their relations which are shown below, E=

μ(3λ + 2μ) , λ+μ

ν=

λ , 2(λ + μ)

(1.52a)

16


or μ=

E , 2(1 + ν)

λ=

Eν . (1 + ν)(1 − 2ν)

(1.52b)

With these relations, the replacement of μ, λ by μ, λ˜ from plane strain problem to generalized plane stress problem can also be done with the replacement of E, ν by ˜ ν˜ where E, E˜ =

E(1 + 2ν) , (1 + ν)2

ν˜ =

ν . 1+ν

(1.53)

1.3.2 Anisotropic Materials Due to the mathematical equivalence, the plane strain and generalized plane stress problems are usually referred to be plane problems or two-dimensional problems. However, in a body with anisotropy of a general form, plane deformation is usually not possible except for some special forms because, assuming u3 = 0, it is generally impossible to satisfy the equations of equilibrium of an elastic body (Lekhnitskii, 1963). We can only assert that all components of stresses and displacements will not depend on x3 . The deformation of such a body (one of infinite length bounded by a cylindrical surface and possessing anisotropy of a general form) which corresponds to plane deformation in an isotropic body is called generalized plane deformation or generalized plane strain. In other words, a body is said to be in the state of generalized plane deformation (or generalized plane strain), parallel to the x1 x2 -plane, if all the displacement components u1 , u2 , and u3 are functions of the coordinates x1 and x2 , but not of x3 . Thus, the state of generalized plane deformation is characterized by ui = ui (x1 , x2 ),

i = 1, 2, 3

(1.54a)

which will lead to ε3 = 0.

(1.54b)

When ε3 = 0, the stress–strain relation written in (1.22a) becomes σp =

Cpq εq ,

p = 1, 2, . . . , 6,

(1.55)

p = 1, 2, 4, 5, 6,

(1.56a)

q=3

and its inverse relation (1.27a) becomes εp =

q=3

Sˆ pq σq ,

1.3

Two-Dimensional Constitutive Relations

17

where Sˆ pq are the reduced elastic compliances which are defined as Sp3 S3q Sˆ pq = Spq − = Sˆ qp . S33

(1.56b)

The relation (1.56) is derived by employing the requirement ε3 = 0 = S3q σq , which S3q σq /S33 . Substituting the result of σ3 into (1.27a), will lead to σ3 = − q=3

we obtain the relation (1.56a). Equations (1.55) and (1.56a) can also be written in matrix notation as (Ting, 1996) σ0 = C0 ε0 ,

ˆ 0, ε0 = Sσ

(1.57a)

⎧ ⎫ ε1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ε2 ⎪ 0 ε = ε4 , ⎪ ⎪ ⎪ ε5 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ ε6

(1.57b)

where ⎧ ⎫ σ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨σ2 ⎪ 0 σ = σ4 , ⎪ ⎪ ⎪ σ5 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ σ6 ⎡

C11 ⎢C12 ⎢ C0 = ⎢ ⎢C14 ⎣C15 C16

C12 C22 C24 C25 C26

C14 C24 C44 C45 C46

C15 C25 C45 C55 C56

⎤ C16 C26 ⎥ ⎥ C46 ⎥ ⎥, C56 ⎦ C66

⎡

Sˆ 11 ⎢Sˆ ⎢ 12 ⎢ Sˆ = ⎢Sˆ 14 ⎢ ⎣Sˆ 15 Sˆ 16

Sˆ 12 Sˆ 22 Sˆ 24 Sˆ 25 Sˆ 26

Sˆ 14 Sˆ 24 Sˆ 44 Sˆ 45 Sˆ 46

Sˆ 15 Sˆ 25 Sˆ 45 Sˆ 55 Sˆ 56

⎤ Sˆ 16 Sˆ 26 ⎥ ⎥ ⎥ Sˆ 46 ⎥ . ⎥ Sˆ 56 ⎦ Sˆ 66

(1.57c)

Similarly, when σ3 = 0 the stress–strain relation can be written as σp =

Cˆ pq εq ,

q=3

εp =

Spq σq ,

p = 1, 2, 4, 5, 6,

(1.58a)

q=3

or, in the matrix notation, ˆ 0 , ε0 = S0 σ0 , σ0 = Cε

(1.58b)

where S0 is a 5 × 5 compliance matrix whose components are Spq , p, q=1, 2, 4, 5, 6, ˆ is a 5 × 5 reduced stiffness matrix whose components Cˆ pq are the reduced and C elastic stiffnesses defined by Cp3 C3q Cˆ pq = Cpq − = Cˆ qp . C33

(1.59)

Similar to the inversion relation for C and S proved in (1.28), it can easily be proved that

18


ˆ 0, C0 Sˆ = I = SC

ˆ = I = CS ˆ 0. S0 C

(1.60)

By using the relations given in (1.40) and (1.41) for the isotropic materials, we see that the replacement given in (1.51b) is a special case of (1.59). Therefore, like the equivalence of the mathematical formulation for the plane strain and generalized plane stress in isotropic elasticity, we may conclude that when the general anisotropic materials are considered the elastic stiffnesses Cpq −1 ) should be employed for the or the reduced elastic compliances Sˆ pq (= Cpq generalized plane strain problems. While for the generalized plane stress problems, we employ the elastic compliances Spq or the reduced elastic stiffnesses −1 ). Cˆ pq (= Spq

1.3.3 Monoclinic Materials The elastic constants for the monoclinic materials with symmetry plane x3 = 0 have been given in (1.30). By deleting the third row and third column of the stiffness matrix for the general two-dimensional problems, we see that the in-plane and antiplane properties are decoupled. The stress–strain relation (1.57) for the problems of generalized plane deformation may therefore be split into two parts. One is the in-plane relation, the other is the anti-plane relation. They are ⎧ ⎫ ⎡ ⎤⎧ ⎫ C11 C12 C16 ⎨ε1 ⎬ ⎨σ1 ⎬ σ2 = ⎣C12 C22 C26 ⎦ ε2 , ⎩ ⎭ ⎩ ⎭ σ6 C16 C26 C66 ε6

C44 C45 ε4 σ4 = . σ5 C45 C55 ε5

(1.61)

For the problems of generalized plane stress, similar relations can be written in terms of Sij or Cˆ ij .

1.3.4 Orthotropic Materials The elastic constants for the orthotropic materials in coordinates aligned with principal material directions have been given in (1.38). Note that there is no interaction between normal stresses and shearing strains. Also, there is no interaction between shearing stresses and normal strains as well as none between shearing stresses and shearing strains in different planes. Similar to (1.61), the in-plane and antiplane relations for the problems of generalized plane deformation can then be written as ⎧ ⎫ ⎡ ⎤⎧ ⎫ C11 C12 0 ⎨ε1 ⎬ ⎨σ1 ⎬ 0 C ε4 σ4 σ2 = ⎣C12 C22 0 ⎦ ε2 , = 44 . (1.62) σ5 0 C55 ε5 ⎩ ⎭ ⎩ ⎭ σ6 ε6 0 0 C66 With (1.46), the above relation can be written in terms of the engineering constants. For generalized plane stress, we have

1.4

Laminate Constitutive Relations

⎧ ⎫ ⎡ Cˆ 11 ⎨σ1 ⎬ ⎣ σ2 = Cˆ 12 ⎩ ⎭ σ6 0

19

⎤⎧ ⎫ Cˆ 12 0 ⎨ε1 ⎬ Cˆ 22 0 ⎦ ε2 , ⎩ ⎭ ε6 0 Cˆ 66

Cˆ 44 0 ε4 σ4 = . ˆ σ5 ε 0 C55 5

(1.63)

By using the engineering constants given in (1.46) and the definitions for the reduced stiffness constants (1.59), we obtain E1 E2 , Cˆ 22 = , 1 − ν12 ν21 1 − ν12 ν21 ν12 E2 ν21 E1 = = , 1 − ν12 ν21 1 − ν12 ν21 = G12 , Cˆ 44 = G23 , Cˆ 55 = G31 .

Cˆ 11 = Cˆ 12 Cˆ 66

(1.64)

Note that from (1.60)2 we know that the reduced stiffness constants can also be determined by the inversion of the compliance matrix. That is, ⎡

Cˆ 11 ⎣Cˆ 12 0

⎤−1 ⎡ ⎤ Cˆ 12 0 1/E1 −ν21 /E2 0 0 ⎦, Cˆ 22 0 ⎦ = ⎣−ν12 /E1 1/E2 0 0 1/G12 0 Cˆ 66

(1.65a)

and −1 Cˆ 44 0 1/G23 0 . = 0 1/G31 0 Cˆ 55

(1.65b)

1.4 Laminate Constitutive Relations In engineering applications, one commonly used orthotropic material is unidirectional fiber-reinforced composite. The laminated composites are made by laying up various unidirectional fiber-reinforced composites (Fig. 1.2). A single layer of the laminated composites is generally referred to as a ply or lamina. A single lamina is T′ T

L

Transverse direction

Longitudinal direction (a)

(b)

Fig. 1.2 Laminated composites. (a) laminate, (b) lamina

20


generally too thin to be directly used in engineering applications. Several laminae are bonded together to form a structure termed a laminate. Properties of a lamina may be predicted by knowing the properties of its constituents, i.e., fibers, matrices, and their volume fractions. Properties and orientation of the laminae in a laminate are chosen to meet the laminate design requirements. Properties of a laminate may then be predicted by knowing the properties of its constituent laminae. Behavior of the laminate is governed by the behavior of individual laminae. Thus analysis or design of a laminate requires a complete knowledge of the behavior of the laminae. Therefore, in this section we will start from specially orthotropic lamina with principal material directions along with the fibers, then to generally orthotropic lamina and finally to the laminate constitutive relations.

1.4.1 Specially Orthotropic Lamina As shown in Fig. 1.2b, each lamina of the laminated composites is considered to be a unidirectional fiber-reinforced composite which consists of parallel fibers embedded in a matrix. The direction parallel to fibers is generally called the longitudinal direction and will be referred as L. The direction perpendicular to the fibers is called the transverse direction and will be referred as T. These axes are also referred to as the material axes of the ply. The ply depicted schematically in Fig. 1.2b shows only one fiber through the ply thickness. In practice, this may be true only for largediameter fibers such as boron. Plies formed by other fibers may have several fibers through the actual ply thickness. The fibers are randomly distributed throughout the cross section and may be in contact with each other in some locations. Because of this particular structure, the unidirectional fiber-reinforced composites show different properties in the longitudinal and transverse directions, and nearly identical properties in all directions of the cross section due to the random fiber distribution. Thus, by referring to the definitions given in Section 1.2.2, we know that a unidirectional composite can be considered to be orthotropic with xL xT , xL xT and xT xT as planes of symmetry and also transversely isotropic with xT xT plane as a plane of isotropy. If the reference coordinate axes coincide with the principal material directions, i.e., the lamina’s axes of symmetry L and T, the lamina is called specially orthotropic lamina whose two-dimensional stress–strain relationship has been shown in (1.62) for plane strain state and in (1.63) for generalized plane stress state or rewritten as σ∗ = Qε∗ ,

(1.66a)

where ⎧ ⎫ ⎨ σL ⎬ σ∗ = σT , ⎩ ⎭ τLT

⎤ Q11 Q12 0 Q = ⎣Q12 Q22 0 ⎦ , 0 0 Q66 ⎡

⎧ ⎫ ⎨ εL ⎬ ε∗ = εT . ⎩ ⎭ γLT

(1.66b)

1.4


21

In (1.66b) the subscripts L and T are used to denote the components related to the principal material directions, the engineering shear stress τLT and strain γLT are used instead of the contracted notations σ6 and ε6 , and the symbol Qij is used to stand for the elastic constants Cˆ ij of generalized plane stress state. Q is generally called stiffness matrix and its expressions in terms of engineering constants have been given in (1.64).

1.4.2 Generally Orthotropic Lamina If the principal material directions do not coincide with the reference coordinate directions that are geometrically natural to the solution of the problem, an orthotropic lamina appears to be anisotropic in reference coordinate direction, which is called a generally orthotropic lamina. Since a laminated composite is usually constructed by stacking several unidirectional laminae in a specified sequence of orientation, the principal directions of each lamina make a different angle with a common set of reference coordinate axes. Although each lamina is orthotropic and obeys the previously described stress–strain relation (1.66) referred to its principal material axes, for the purpose of analysis and synthesis of laminated structures it is necessary to refer the stress–strain relation to common reference coordinate axes. To know this relation, we now derive the stress–strain relation of a lamina with arbitrary orientation as follows. According to the transformation rule of tensor of rank 2, the stresses and strains in the transformed coordinates can be calculated by (1.2) and (1.8). This transformation rule can be expressed in terms of simple matrix multiplication if we consider the transformation from L–T axes to x–y axes as shown Fig. 1.3. If the angle θ between these two coordinate systems is taken positive when the angle of the L–T axes measured from x–y axes is in the counterclockwise direction, the transformation relations of the stresses and strains may be written as σ∗ = Tσ,

ε∗ = RTR−1 ε,

(1.67a)

y T

L

θ x

Fig. 1.3 Transformation between x–y and L–T coordinate systems

22


where ⎡

⎤ cos2 θ sin2 θ 2 sin θ cos θ T = ⎣ sin2 θ cos2 θ −2 sin θ cos θ ⎦ , − sin θ cos θ sin θ cos θ cos2 θ − sin2 θ ⎧ ⎫ ⎧ ⎫ ⎡ ⎤ 100 ⎨ σx ⎬ ⎨ εx ⎬ R = ⎣0 1 0⎦ , σ = σy , ε = εy . ⎩ ⎭ ⎩ ⎭ 002 τxy γxy

(1.67b)

Note that the appearance of the matrix R is because the transformation rule (1.8) is applied for the elastic strains while the constitutive relation (1.66) is written for the engineering strains, and the engineering shear strain is twice that of the elastic shear strain. Substituting (1.67) into (1.66), we obtain σ = Q∗ ε,

(1.68a)

Q∗ = T −1 QRTR−1 = T −1 QT −T .

(1.68b)

where

Usually, it will take time to calculate the inverse of a matrix. However, since the matrix T stands for the rotation about the coordinate axis, its inverse T −1 can be obtained directly from T by replacing the angle θ by −θ . By simple matrix multiplication shown in (1.68b), the relation between the transformed stiffness Q∗ and the stiffness Q can be written as Q∗11 = Q11 cos4 θ + 2(Q12 + 2Q66 ) sin2 θ cos2 θ + Q22 sin4 θ , Q∗12 = (Q11 + Q22 − 4Q66 ) sin2 θ cos2 θ + Q12 (sin4 θ + cos4 θ ), Q∗22 = Q11 sin4 θ + 2(Q12 + 2Q66 ) sin2 θ cos2 θ + Q22 cos4 θ , Q∗16 = (Q11 − Q12 − 2Q66 ) sin θ cos3 θ + (Q12 − Q22 + 2Q66 ) sin3 θ cos θ , Q∗26 = (Q11 − Q12 − 2Q66 ) sin3 θ cos θ + (Q12 − Q22 + 2Q66 ) sin θ cos3 θ , Q∗66 = (Q11 + Q22 − 2Q12 − 2Q66 ) sin2 θ cos2 θ + Q66 (sin4 θ + cos4 θ ).

(1.69)

∗

The matrix Q is now fully populated and similar in appearance to the Q matrix for an anisotropic material. Hence, there is coupling between shear strain and normal stresses and between shear stress and normal strains. However, because the lamina does have orthotropic characteristics in principal material directions, its mechanical behavior is still governed by only four independent material constants not six elastic constants fully populated in Q∗ . Similar to the transformation of the stiffness matrix Q, the transformed compliance matrix S∗ of the x–y coordinate axes can be found to be related to the compliance matrix S of the L–T coordinate axes by S∗ = T T ST.

(1.70)

1.4


23

Through simple matrix multiplication, expressions of the transformed compliance components similar to those shown in (1.69) for the transformed stiffness components can be obtained. By comparing the transformed compliances with the anisotropic compliances in terms of engineering constants in (1.43), the apparent engineering constants for an orthotropic lamina stressed in nonprincipal x–y coordinates can also be obtained as follows (Jones, 1974), 2νL T 1 1 1 4 sin2 θ cos2 θ + E1T sin4 θ , Ex = EL cos θ + GLT − EL νxy = Ex νELLT (sin4 θ + cos4 θ ) − E1L + E1T − G1L T sin2 θ cos2 θ , 2νLT 1 1 1 4θ + sin2 θ cos2 θ + E1L sin4 θ , = cos − Ey ET GLT EL 4νLT 1 2 2 1 1 2 4 2 4 = 2 + + − Gxy EL ET EL GL T sin θ cos θ + GL T (sin θ + cos θ ), (1.71) − G1LT sin θ cos3 θ ηxy,x = Ex E2L + 2νELT L 1 3 − E2T + 2νELT sin − θ cos θ , G L LT − G1LT sin3 θ cos θ ηxy,y = Ey E2L + 2νELT L 1 3θ . sin θ cos − − E2T + 2νELT G L LT

1.4.3 Classical Lamination Theory As stated in the beginning of this section, the overall properties of the laminates can be designed by changing the fiber orientation and the stacking sequence of laminae.To describe the overall properties and macromechanical behavior of a laminate, the most popular way is the classical lamination theory (Jones, 1974). According to the observation of actual mechanical behavior of laminates, following assumptions are made in this theory: (a) The laminate consists of perfectly bonded laminae and the bonds are infinitesimally thin as well as non-shear deformable. Thus, the displacements are continuous across lamina boundaries so that no lamina can slip relative to another. (b) A line originally straight and perpendicular to the middle surface of the laminate remains straight and perpendicular to the middle surface of the laminate when the laminate is deformed. In other words, the transverse shear strains are ignored, i.e., γxz = γyz = 0. (c) The normals have constant length so that the strain perpendicular to the middle surface is ignored, i.e., εz = 0. Based upon the above assumptions, the laminate displacements u, v, and w in the x, y, and z directions can be expressed as ∂w(x, y) , ∂x ∂w(x, y) v(x, y, z) = v0 (x, y) − z , ∂y w(x, y, z) = w0 (x, y), u(x, y, z) = u0 (x, y) − z

(1.72)

24


where u0 , v0 , and w0 are the middle surface displacements. If small deformations are considered, the laminate strains can be written in terms of the middle surface displacements as follows: ∂u ∂u0 ∂ 2w = −z 2, ∂x ∂x ∂x ∂v0 ∂ 2w ∂v = −z 2, εy = ∂y ∂y ∂y ∂u0 ∂v0 ∂ 2w ∂u ∂v γxy = + = + − 2z , ∂y ∂x ∂y ∂x ∂x∂y

εx =

(1.73)

or, in matrix notation, ε = ε0 + zκ,

(1.74)

where ε, ε0 , and κ denotes, respectively, strain vector, midsurface strain vector, and plate curvature vector, which are defined as

εx ε = εy , γxy

⎧ 0 ⎫ ⎧ ∂u0 ⎫ ⎪ ⎨ εx ⎬ ⎪ ⎬ ⎨ ∂x ∂v0 ε0 = εy0 = , ∂y ⎩ 0⎭ ⎪ ⎪ ∂v ∂u ⎭ ⎩ 0 0 γxy ∂y + ∂x

κx κ = κy κxy

=−

⎧ 2 ∂ w ⎪ ⎪ ⎪ ∂x2 ⎪ ⎨ 2

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

∂ w

∂y2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩2 ∂ 2 w ⎪ ⎭

.

∂x∂y

(1.75)

Substituting (1.74) into the stress–strain relation (1.68a) for each lamina, the stresses in the kth lamina can also be written in terms of the laminate middle surface strains ε0 and curvatures κ as ⎧ ⎫ ⎨ σx ⎬ σy = σ k = Q∗k (ε0 + zκ), ⎩ ⎭ τxy k

(1.76)

where Q∗k , as shown in (1.69) for a lamina with its principal material axes oriented at an angle θ from the reference coordinate axes, is the transformed stiffness matrix of the kth lamina. Note that since Q∗k may be different for each lamina, the stress variation through the laminate thickness is not necessarily linear even though the strain variation is linear. In other words, the stresses calculated from (1.76) may be discontinuous at the interface of two laminae due to the discontinuity of the material properties of laminae. Like the classical plate theory, the thickness of the laminate is considered to be small compared to its other dimensions. Therefore, instead of dealing the stress distribution across the laminate thickness, an integral equivalent system of forces and moments acting on the laminate cross section is used in the classical lamination theory. By integration of the stresses in each lamina through the laminate thickness,

1.4


25

the resultant forces N and moments M acting on a laminate cross section are defined as follows: ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ n hk ⎨σx ⎬ n hk ⎨ Nx ⎬ h/2 ⎨ σx ⎬ σy dz = σy N = Ny = dz = σk dz, ⎩ ⎭ ⎩ ⎭ h h −h/2 ⎩τ ⎭ k−1 k−1 Nxy τ k=1 k=1 xy xy k ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ n n hk M σ σ ⎨ x⎬ h/2 ⎨ x ⎬ hk ⎨ x ⎬ σy zdz = σy M = My = zdz = σk zdz, ⎩ ⎭ ⎩ ⎭ h h −h/2 ⎩τ ⎭ k−1 k−1 Mxy τ k=1 k=1 xy xy k (1.77) where hk and hk−1 are defined in Fig. 1.4. Substituting (1.76) into (1.77), the resultant forces N and moments M can be written in terms of the laminate middle surface strains ε0 and curvatures κ as N A B ε0 , = κ M BD

(1.78)

where A, B, and D are called the extensional, coupling, and bending stiffness matrices, respectively, and are determined by A= B= D=

n

hk−1

k=1 hk n hk−1 k=1 hk n hk−1 k=1 hk

Q∗k dz =

n

Q∗k (hk − hk−1 ),

k=1

Q∗k zdz =

1 ∗ 2 Qk (hk − h2k−1 ), 2

Q∗k z2 dz =

n

k=1 n

1 3

(1.79)

Q∗k (h3k − h3k−1 ).

k=1

From (1.78), we see that if the coupling matrix B is a zero matrix, the resultant forces N will induce only the midsurface strains while the resultant moments M will induce only the plate curvatures. The presence of the matrix B implies coupling between bending and extension of a laminate. Thus, when a laminate is subjected to an extensional force or a bending moment, it may suffer extensional as well as bending and/or twisting deformations at the same time. By (1.79)2 , we also know that the presence of a nonzero-coupling matrix is not attributable to the orthotropy or anisotropy of the layers but rather to the nonsymmetric stacking of laminae. The aim of the analysis of laminated composites is to determine the stresses and strains in each of the laminae forming the laminate. These stresses and strains can be used to predict the load at which failure initiates. If the resultant forces N and moments M are known at a particular cross section through the structural analysis, the midsurface strains and curvatures at this cross section may then be determined by the inversion of (1.78), i.e.,

26

1 Linear Anisotropic Elastic Materials dx Ny My M yx

Qy

dy

N yx Qx

x

Mx q

z Nx M xy

Mx + dMx

N xy Nxy + dNxy My + dMy

Qx + dQx

Nx + dNx Mxy + dMxy

Nyx + dNyx

My + dMyx

Qy + dQy Ny + dNy Lamina Number

y

1 2

h0

h1

Middle Plane

h2

x hk−1

h

hk h n−1

k

hn n

z

dx

Fig. 1.4 Laminate geometry, resultant forces, and moments

1.4


27

−1 ∗ ∗ A B ε0 N N AB = = ∗T ∗ , κ M M BD B D

(1.80a)

where A∗ = A−1 + A−1 BD∗ BA−1 , B∗ = −A−1 BD∗ , ∗

D = (D − BA

−1

(1.80b) B)

−1

.

The stresses and strains in each lamina can therefore be determined from (1.76) and (1.74).

Chapter 2

Lekhnitskii Formalism

From Section 1.1, we know that the basic equations for anisotropic elasticity consist of the equilibrium equations for static loading conditions (1.4), the strain– displacement relations for small deformations (1.7) as well as the stress–strain laws for linear anisotropic elastic solids (1.12). That is, σij, j + fi = 0,

εij =

1 (ui, j + uj, i ), 2

σij = Cijkl εkl ,

i, j, k, l = 1, 2, 3.

(2.1)

These three equation sets (2.1) constitute 15 partial differential equations with 15 unknown functions ui , εij , σij , i, j = 1, 2, 3, in terms of three coordinate variables xi , i = 1, 2, 3. If only the two-dimensional deformation is considered, the complex variable formulation can be used to establish the general solution for these 15 unknown functions satisfying 15 basic equations. In the literature, there are two different complex variable formulations for two-dimensional linear anisotropic elasticity. One is the Lekhnitskii formalism (Lekhnitskii, 1963, 1968) which starts with the equilibrated stress functions followed by compatibility equations, and the other is the Stroh formalism (Stroh, 1958, 1962; Ting, 1996) which starts with the compatible displacements followed by equilibrium equations. The Lekhnitskii formalism excerpted from the two well-known books (Lekhnitskii, 1963, 1968) will be presented in this chapter, whereas the Stroh formalism excerpted from the other well-known book (Ting, 1996) will be presented in the next chapter.

2.1 Governing Differential Equations In the Lekhnitskii formalism, the two-dimensional problem is considered as a body bounded by a cylindrical surface. The region of the cross section can be either finite or infinite. The body possesses rectilinear anisotropy of the form shown in (2.1)3 and is under the influence of body forces and tractions distributed along the surface. In order to have the fields in which the stresses depend only on two coordinates, the body forces and surface tractions are assumed to act in planes normal to the generator of the cylindrical surface and do not vary along the generator. In the case of a body of finite length and finite cross section, the stresses are assumed to reduce to an equivalent axial force and moment which act on the ends. C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_2, C Springer Science+Business Media, LLC 2010

29

30

2


Note that the rectilinear anisotropy stated above is different from curvilinear anisotropy. Since the curvilinear anisotropy is characterized by the fact that at different points of the body there exist directions which are not parallel but are equivalent in the sense of the elastic properties, a homogeneous curvilinear anisotropic body may be a nonhomogeneous rectilinear anisotropic body, and vice versa. In the following, we only consider the case of homogeneous rectilinear anisotropic body. For those who are interested in the case of curvilinear anisotropic body, refer to Lekhnitskii (1963) for further discussions. In order to conform with Lekhnitskii’s presentation (Lekhnitskii, 1963), in this chapter most of the equations in tensor notation will be written out one by one and the associated notation will also be changed slightly according to the conventional replacement and the contracted notation stated in Section 1.2 such as u1 → u, u2 → v, u3 → w, 2ε12 → γxy , σ12 → τxy ,C1112 → C16 ,4S2331 → S45 . With this understanding, the basic equations (2.1) can now be rewritten as follows. Strain–Displacement: ∂u ∂v ∂w , εy = , εz = , ∂x ∂y ∂z ∂w ∂v ∂w ∂u ∂u ∂v γyz = + , γxz = + , γxy = + . ∂y ∂z ∂x ∂z ∂y ∂x Stress–Strain: εx =

(2.2a)

εx = S11 σx + S12 σy + S13 σz + S14 τyz + S15 τxz + S16 τxy , εy = S12 σx + S22 σy + S23 σz + S24 τyz + S25 τxz + S26 τxy , εz = S13 σx + S23 σy + S33 σz + S34 τyz + S35 τxz + S36 τxy , γyz = S14 σx + S24 σy + S34 σz + S44 τyz + S45 τxz + S46 τxy ,

(2.2b)

γxz = S15 σx + S25 σy + S35 σz + S45 τyz + S55 τxz + S56 τxy , γxy = S16 σx + S26 σy + S36 σz + S46 τyz + S56 τxz + S66 τxy . Equilibrium Equations: ∂τxy ∂τxy ∂σy ∂τyz ∂σx ∂ Fˆ ∂ Fˆ ∂τxz + − = 0, + − = 0, + = 0, ∂x ∂y ∂x ∂x ∂y ∂y ∂x ∂y

(2.2c)

where Fˆ is the potential of the body forces fˆx , fˆy , i.e., ∂ Fˆ fˆx = − , ∂x

∂ Fˆ fˆy = − . ∂y

(2.2d)

Note that in (2.2c) we have employed the assumptions that the stresses depend only on two coordinates and the body forces act in planes normal to the generator of the cylindrical surface and do not vary along the generator. With the relations given in the third, fourth, and fifth equations of (2.2a), integration of the third, fourth, and fifth equations of (2.2b) with respect to z will then lead to

2.1

Governing Differential Equations

31

w = zD(x, y) + W 0 (x, y),

v = − z2 ∂D ∂y + z S14 σx + S24 σy + S34 σz + S44 τyz + S45 τxz + S46 τxy − +V0 (x, y), 2

u = − z2 ∂D ∂x + z S15 σx + S25 σy + S35 σz + S45 τyz + S55 τxz + S56 τxy − +U0 (x, y), 2

∂W0 ∂y ∂W0 ∂x

(2.3a) where D(x, y) = S13 σx + S23 σy + S33 σz + S34 τyz + S35 τxz + S36 τxy ,

(2.3b)

and U0 , V0 , W0 are arbitrary functions of x and y which appear as a result of integration with respect to z. Employing the results of (2.3a) into the first, second, and sixth equations of (2.2a) as well as (2.2b), and comparing the coefficients of z2 , z, and the free terms, we get ∂ 2D ∂ 2D ∂ 2D = 0, = 0, = 0, ∂x∂y ∂x2 ∂y2

(2.4a)

and ∂ ∂x ∂ ∂y ∂ ∂y

∂W0 ∂x ∂W0 ∂y ∂W0 ∂x

= 0, S14 σx + S24 σy + · · · + S46 τxy − = 0, S15 σx + S25 σy + · · · + S56 τxy − ∂ 0 S14 σx + S24 σy + · · · + S46 τxy − ∂W =0 + ∂x ∂y S15 σx + S25 σy + · · · + S56 τxy −

(2.4b)

and ∂U0 = S11 σx + S12 σy + · · · + S16 τxy , ∂x ∂V0 = S12 σx + S22 σy + · · · + S26 τxy , ∂y ∂V0 ∂U0 + = S16 σx + S26 σy + · · · + S66 τxy . ∂y ∂x

(2.4c)

From (2.4a), it follows that D is a linear function of x and y, D = S33 (Ax + By + C),

(2.5)

where A, B, and C are the arbitrary constants; then by (2.3b) we have σz = Ax + By + C −

1 (S13 σx + S23 σy + S34 τyz + S35 τxz + S36 τxy ). S33

(2.6)

32

2


Integration of (2.4b) gives ∂W0 = −αy + ω2 , ∂x ∂W0 = αx − ω1 , S14 σx + S24 σy + · · · + S46 τxy − ∂y

S15 σx + S25 σy + · · · + S56 τxy −

(2.7)

where α,ω1 ,ω2 are the new arbitrary constants. Substituting the results of (2.7) and (2.5) into (2.3a), the general expressions for the displacements can now be written as AS33 2 z − αyz + U(x, y) + ω2 z − ω3 y + u0 , 2 BS33 2 z + αxz + V(x, y) + ω3 x − ω1 z + v0 , v=− 2 w = (Ax + By + C)S33 z + W(x, y) + ω1 y − ω2 x + w0 , u=−

(2.8a)

where the new functions U, V, W are related to U0 , V0 , W0 by U0 = U − ω3 y + u0 , V0 = V + ω3 x + v0 , W0 = W + ω1 y − ω2 x + ω0 .

(2.8b)

In the general expression (2.8a), the constants u0 , v0 , w0 and ω1 , ω2 , ω3 obviously characterize the rigid body translations and rotations with respect to the x, y, and z axes; α is the relative angle of rotation about the z-axis associated with the torsion problems, i.e., the angle of twist per unit length; A and B characterize the bending of the body in the x–z and y–z planes. With the relations (2.8b), substitution of expression (2.6) for σz into (2.4c) and (2.7) will lead to the following equations for determining the unknown functions U, V, and W: ∂U = Sˆ 11 σx + Sˆ 12 σy + Sˆ 14 τyz + Sˆ 15 τxz + Sˆ 16 τxy + S13 (Ax + By + C), ∂x ∂V = Sˆ 12 σx + Sˆ 22 σy + Sˆ 24 τyz + Sˆ 25 τxz + Sˆ 26 τxy + S23 (Ax + By + C), ∂y ∂U ∂V + = Sˆ 16 σx + Sˆ 26 σy + Sˆ 46 τyz + Sˆ 56 τxz + Sˆ 66 τxy + S36 (Ax + By + C), ∂y ∂x (2.9a) ∂W = Sˆ 15 σx + Sˆ 25 σy + Sˆ 45 τyz + Sˆ 55 τxz + Sˆ 56 τxy + S35 (Ax + By + C) + αy, ∂x ∂W = Sˆ 14 σx + Sˆ 24 σy + Sˆ 44 τyz + Sˆ 45 τxz + Sˆ 46 τxy + S34 (Ax + By + C) − αx, ∂y (2.9b) where Sˆ ij are the reduced elastic compliance defined in (1.56).

2.1

Governing Differential Equations

33

Up to now only the strain–displacement and stress–strain relations, (2.2a) and (2.2b), and the assumption that σij = σij (x, y) have been utilized for getting (2.9a) and (2.9b). To determine the unknown functions U, V, and W through (2.9a) and (2.9b), we now consider the equilibrium equations (2.2c), which will be satisfied by introducing two Airy stress functions φ (x, y) and ψ(x, y), as σx =

∂ 2φ ∂ 2φ ∂ 2φ ∂ψ ∂ψ ˆ σy = ˆ τxy = − , τxz = , τyz = − . (2.10) + F, + F, 2 2 ∂x∂y ∂y ∂x ∂y ∂x

With relations (2.10), eliminating U and V from (2.9a) and W from (2.9b), by differentiating, we obtain the following system of differential equations which the stress functions must satisfy: ∂ 2 Fˆ ∂ 2 Fˆ ∂ 2 Fˆ − (Sˆ 11 + Sˆ 12 ) 2 , + (Sˆ 16 + Sˆ 26 ) 2 ∂x∂y ∂x ∂y (2.11a) ∂ Fˆ ∂ Fˆ ˆ ˆ ˆ ˆ L3 φ + L2 ψ = −2α + AS34 − BS35 + (S14 + S24 ) + (S15 + S25 ) , ∂x ∂y L4 φ + L3 ψ = −(Sˆ 12 + Sˆ 22 )

in which L2 , L3 , L4 are the differential operators of the second, third, and fourth order which have the form ∂2 ∂2 ∂2 + Sˆ 55 2 , L2 = Sˆ 44 2 − 2Sˆ 45 ∂x∂y ∂x ∂y 3 3 3 ∂ ∂ ∂3 ˆ 15 ∂ , L3 = −Sˆ 24 3 + (Sˆ 25 + Sˆ 46 ) 2 − (Sˆ 14 + Sˆ 56 ) + S ∂x ∂x ∂y ∂x∂y2 ∂y3 4 4 4 4 4 ∂ ∂ ∂ ∂ ˆ 11 ∂ . L4 = Sˆ 22 4 − 2Sˆ 26 3 + (2Sˆ 12 + Sˆ 66 ) 2 2 − 2Sˆ 16 + S ∂x ∂x ∂y ∂x ∂y ∂x∂y3 ∂y4 (2.11b) Solving the stress functions from (2.11), the stresses, strains, and displacements can then be determined from (2.10), (2.6), (2.2b), (2.9), and (2.8). To uniquely determine all these values, the boundary conditions and the requirement of the single-valued displacement should all be satisfied. Since the system of differential equations obtained in (2.11) is the combined result of the 15 basic equations shown in (2.2a–c), before discussing its associated general solutions we like to summarize briefly the derivation procedure stated between (2.2) and (2.11). Summary of the derivation for (2.11): 1. Introduce two Airy stress functions φ,ψ in (2.10). 2. Stresses σij in terms of φ,ψ will satisfy equilibrium (2.2c) automatically. 3. By employing the stress–strain relation (2.2b) and the strain–displacement relation (2.2a), the strain and displacement components can also be expressed in terms of the Airy stress functions. 4. During the integration the compatibility of displacements should be satisfied, which will then lead to the system of differential equations (2.11).

34

2


2.2 General Solutions The general solution of (2.11) can be written in the form φ = φ (h) + φ (p) ,

ψ = ψ (h) + ψ (p) ,

(2.12a)

where L4 φ (h) + L3 ψ (h) = 0, L3 φ (h) + L2 ψ (h) = 0,

(2.12b)

and φ (p) ,ψ (p) are particular solutions of the nonhomogeneous system (2.11a). The particular solutions depend on the form of the known functions of the right-hand sides of (2.11a) and are usually not difficult to find. Hence, in the following we only consider the general solutions of the homogeneous system (2.12b). Eliminating one of the functions, say ψ (h) , from (2.12b)1 and (2.12b)2 , we obtain an equation of the sixth order as (L4 L2 − L3 2 )φ (h) = 0.

(2.13)

The operator of the sixth homogeneous order L4 L2 − L3 2 can be decomposed into six linear operators of the first order. In other words, (2.13) can be represented in the following form: D6 D5 D4 D3 D2 D1 φ (h) = 0,

(2.14a)

where Dk =

∂ ∂ − μk , ∂y ∂x

(2.14b)

and μk are the roots of the algebraic equation associated with the differential equation (2.13), i.e., l4 (μ)l2 (μ) − l3 2 (μ) = 0,

(2.15a)

in which l2 (μ) = Sˆ 55 μ2 − 2Sˆ 45 μ + Sˆ 44 , l3 (μ) = Sˆ 15 μ3 − (Sˆ 14 + Sˆ 56 )μ2 + (Sˆ 25 + Sˆ 46 )μ − Sˆ 24 ,

(2.15b)

l4 (μ) = Sˆ 11 μ4 − 2Sˆ 16 μ3 + (2Sˆ 12 + Sˆ 66 )μ2 − 2Sˆ 26 μ + Sˆ 22 . One can prove that μk cannot be real if the strain energy is positive (Lekhnitskii, 1963). In other words, μk are always complex or purely imaginary and consist of three pairs of complex conjugates since the characteristic equation (2.15a) is a sixthorder algebraic equation with real coefficients. Let

2.2

General Solutions

35

μk+3 = μk ,

Im μk > 0, k = 1, 2, 3,

(2.16)

where Im denotes the imaginary part and the overbar denotes the complex conjugate. Assume that the roots μk are distinct; (2.14a) can be solved by considering the following six equations of the first order: D1 φ (h) = ϕ2 , D2 ϕ2 = ϕ3 , D3 ϕ3 = ϕ4 , D4 ϕ4 = ϕ5 , D5 ϕ5 = ϕ6 , D6 ϕ6 = 0. (2.17) Solving (2.17) successively in the order of ϕ6 , ϕ5 , ϕ4 , ϕ3 , ϕ2 , φ (h) , we obtain φ (h) = 2 Re

3

φk (zk ),

zk = x + μk y,

(2.18)

k=1

in which Re denotes the real part. Similarly, by expressing (2.13) in terms of ψ (h) and solving it in the same way as that shown in (2.14), (2.15), (2.16), (2.17), and (2.18), we can obtain ψ (h) = 2 Re

3

ψk (zk ).

(2.19)

k=1

Knowing that ∂φk ∂zk ∂φk ∂φk = = , ∂x ∂zk ∂x ∂zk

∂φk ∂zk ∂φk ∂φk = = μk , ∂y ∂zk ∂y ∂zk

(2.20)

and similarly for ψk , substitution of (2.18) and (2.19) into (2.12b) will then lead to 2 Re

3

l4 (μk )φk (zk ) + l3 (μk )ψk (zk ) = 0,

k=1

2 Re

3

(2.21) l3 (μk )φk (zk ) + l2 (μk )ψk (zk )

= 0,

k=1

in which the prime ( ) denotes differentiation with respect to zk . Integration of the first or second equation of (2.21) with respect to zk can now provide the relation between φk and ψk , i.e., ψk (zk ) = ηk φk (zk ) + ak zk + bk , k = 1, 2, 3,

(2.22a)

where ηk =

−l3 (μk ) −l4 (μk ) = , l2 (μk ) l3 (μk )

(2.22b)

36

2


and ak , bk are the arbitrary constants. For monoclinic materials with the symmetry plane at z = 0, the coefficients of the polynomial l3 (μ) all vanish and the sextic equation (2.15a) reduces to two equations, l2 (μ) = 0 and l4 (μ) = 0. Let μ1 and μ2 be the roots of l4 (μ) = 0 and μ3 be the roots of l2 (μ) = 0, i.e., l4 (μ1 ) = l4 (μ2 ) = l2 (μ3 ) = 0, which will then lead to the results that l2 (μ1 ), l2 (μ2 ), l4 (μ3 ) = 0 since μk are assumed to be distinct. With these values and l3 (μk ) = 0, we have η1 = η2 = 0 and η3 → ∞. To avoid using a coefficient that may approach infinity, we let λ 1 = η1 =

−l3 (μ1 ) −l3 (μ2 ) 1 −l3 (μ3 ) = , λ 2 = η2 = , λ3 = . l2 (μ1 ) l2 (μ2 ) η3 l4 (μ3 )

(2.23)

By choosing the arbitrary constants ak , bk to be zero, and using (2.23), (2.22a), (2.18), and (2.19), the general expressions for the stress functions now take the following form: φ = 2 Re{φ1 (z1 ) + φ2 (z2 ) + φ3 (z3 )} + φ (p) , 1 ψ = 2 Re λ1 φ1 (z1 ) + λ2 φ2 (z2 ) + φ3 (z3 ) + ψ (p) . λ3

(2.24)

Knowing that λ3 may approach zero for some materials, and φ3 (z3 ) is an arbitrary function at this stage, to avoid having the infinite coefficient we may absorb the coefficient into the function and introduce the new stress functions fk (zk ), k = 1, 2, 3, as follows: f1 (z1 ) = φ1 (z1 ),

f2 (z2 ) = φ2 (z2 ),

f3 (z3 ) =

1 φ (z3 ). λ3 3

(2.25)

With these new stress functions, substitution of (2.24) into (2.10) will then lead to ! " ∂ 2 φ (p) ˆ σx = 2 Re μ21 f1 (z1 ) + μ22 f2 (z2 ) + μ23 λ3 f3 (z3 ) + + F, ∂y2 ∂ 2 φ (p) ˆ σy = 2 Re f1 (z1 ) + f2 (z2 ) + λ3 f3 (z3 ) + + F, ∂x2 ∂ 2 φ (p) , τxy = −2 Re μ1 f1 (z1 ) + μ2 f2 (z2 ) + μ3 λ3 f3 (z3 ) − ∂x∂y ∂ψ (p) τxz = 2 Re μ1 λ1 f1 (z1 ) + μ2 λ2 f2 (z2 ) + μ3 f3 (z3 ) + , ∂y ∂ψ (p) τyz = −2 Re λ1 f1 (z1 ) + λ2 f2 (z2 ) + f3 (z3 ) − . ∂x

(2.26)

By substituting (2.26) into (2.9a) and (2.9b) and integrating the resulting equations, we can find the functions U, V, and W as

2.3

Boundary Conditions

37 3

U = 2 Re

a1k fk (zk ) + U (p) ,

k=1

V = 2 Re

3

a2k fk (zk ) + V (p) ,

(2.27a)

k=1

W = 2 Re

3

a3k fk (zk ) + W (p) ,

k=1

where a1k = p1 (μk ) + λk q1 (μk ), a2k = [p2 (μk ) + λk q2 (μk )]/μk , a3k = [p4 (μk ) + λk q4 (μk )]/μk , k = 1, 2, a13 = λ3 p1 (μ3 ) + q1 (μ3 ), a23 = [λ3 p2 (μ3 ) + q2 (μ3 )]/μ3 ,

(2.27b)

a33 = [λ3 p4 (μ3 ) + q4 (μ3 )]/μ3 , and pj (μk ) = μ2k Sˆ j1 + Sˆ j2 − μk Sˆ j6 , qj (μk ) = μk Sˆ j5 − Sˆ j4 ,

j = 1, 2, 4, 5, 6. (2.27c)

In (2.27a), U (p) ,V (p) ,W (p) are the solutions of (2.9a) and (2.9b) which correspond to the functions φ (p) ,ψ (p) ,Fˆ and to the linear functions Sij (Ax + By + C), αy, −αx which contain the constants α, A, B, C.

2.3 Boundary Conditions 2.3.1 Lateral Surface Conditions The general expressions for the stresses and displacements given in (2.26), (2.27), and (2.8a) contain the arbitrary complex functions fk (zk ) which should be determined through the satisfaction of the boundary conditions on the lateral surface. As stated in Chapter 1, the boundary conditions are usually described by prescribing the tractions or displacements. Because the general expressions for the stresses and displacements are written in terms of the unknown functions fk (zk ), it is now better to express the boundary conditions in terms of fk (zk ). First Fundamental Problem: σx n1 + τxy n2 = ˆtx , τxy n1 + σy n2 = ˆty , τxz n1 + τyz n2 = 0,

(2.28)

where (ˆtx ,ˆty , 0) are the tractions prescribed along the boundary. As shown in Fig. 2.1 the normal vector n of the boundary surface can be expressed by

38

2


n

Fig. 2.1 Tangent and normal directions of boundary surfaces

s

θ

ds

θ dx

dy

y

dx = n2 ds dy sin θ = = −n1 ds cos θ =

x

n1 = −

dy dx , n2 = , ds ds

(2.29)

where tangential direction s is chosen such that when one faces the direction of increasing s the material lies on the right side. Substituting (2.10) and (2.29) into (2.28) and integrating the resulting equations with respect to s, we obtain ∂φ = ˜tx (s) + c1 , ∂y

∂φ = ˜ty (s) + c2 , ∂x

ψ = c3 ,

(2.30a)

where c1 , c2 , c3 are the integration constants and s dy ˜tx (s) = − ˆtx + Fˆ ds, ds 0 s dx ˜ty (s) = ˆty − Fˆ ds. ds 0

(2.30b)

Employing (2.24) and (2.25) into (2.30a), we now have 2 Re {μ1 f1 + μ2 f2 + μ3 λ3 f3 } = ˜tx (s) −

∂φ (p) + c1 , ∂y

∂φ (p) + c2 , ∂x 2 Re {λ1 f1 + λ2 f2 + f3 } = −ψ (p) + c3 . 2 Re { f1 + f2 + λ3 f3 } = ˜ty (s) −

(2.31)

Second Fundamental Problem: u = uˆ , v = vˆ , w = w, ˆ

(2.32)

where (û,ˆv,w) ˆ are the displacements prescribed along the boundary. Substituting (2.8a), (2.27), into (2.32), we obtain

2.3

Boundary Conditions

2 Re

39 3

ˆ + ω3 y − u0 , a1k fk = −U (p) + U

k=1

2 Re

3

a2k fk = −V (p) + Vˆ − ω3 x − v0 ,

(2.33a)

k=1

2 Re

3

ˆ − w0 , a3k fk = −W (p) + W

k=1

ˆ V, ˆ W ˆ are given as where U, AS33 2 z + αyz − ω2 z, 2 BS33 2 Vˆ = vˆ + z − αxz + ω1 z, 2 ˆ = wˆ − (Ax + By + C)S33 z − ω1 y + ω2 x. W

ˆ = uˆ + U

(2.33b)

2.3.2 End Conditions As stated at the beginning of Section 2.1 that if a body of finite length and finite cross section is considered, the axial force and moment which act on the ends will be removed by imposing an equivalent stress distribution. Since the stresses do not depend on z, these conditions exist not only at the ends but also in any cross section. The conditions at the ends have the form:

τ xz dx dy = 0, σz y dx dy = M1 ,

τ yz dx dy = 0,

σz x dx dy = M2 ,

σz dx dy = Pz , (τ yz x − τ xz y)dx dy = Mt ,

(2.34) where Pz , M1 , M2 , and Mt are, respectively, the axial force, bending moments about x- and y-axes, and twisting moment. The integrals are taken over the entire area of the cross section. In the Lekhnitskii formulation, the net forces in the x and y directions should be zero under the assumption of two-dimensional equilibrated stress fields, and the first two conditions of (2.34) should then be satisfied identically. A direct proof is given as follows:

∂τ yz ∂τ xz τ xz + x + dx dy τ xz dx dy = ∂x ∂y ∂(xτ xz ) ∂(xτ yz ) dx dy = [xτ xz n1 + xτ yz n2 ]ds = 0. = + ∂x ∂y

(2.35)

40

2


In the above derivation, the first equality is obtained by adding a zero quantity from the equilibrium equation (2.2c) and the third equality is obtained by transforming the double integral into a contour integral (contour ), and the last equality is due to the boundary conditions set on the lateral surface (ˆtz = 0). It can be proved by the same way that the integral of τyz is also equal to zero. Substituting (2.6) into the remaining four equations of (2.34), we have 1 (S13 σx + S23 σy + S34 τ yz + S35 τ xz + S36 τ xy ) dx dy, Pz = CS − S33 1 M1 = BI1 − (S13 σx + S23 σy + S34 τ yz + S35 τ xz + S36 τ xy ) y dx dy, S33 1 (S13 σx + S23 σy + S34 τ yz + S35 τ xz + S36 τ xy )x dx dy, M2 = AI2 − S33 Mt = (τ yz x − τ xz y)dx dy, (2.36a) where S is the area of the cross section and I1 and I2 are the principal moments of inertia with respect to x- and y-axes defined by S=

dx dy,

I1 =

2

y dx dy,

I2 =

x2 dx dy.

(2.36b)

Note that the origin of the x–y coordinate passes through the centroid, and the x- and y-axes are the principal axes of inertia of the cross section, and hence x dx dy = 0, y dx dy = 0, and xy dx dy = 0. By employing similar techniques as (2.35), it can be shown that each term of the integrals in (2.36a) can be expressed in terms of the surface traction (ˆtx , ˆty , 0) and body force (fˆx , fˆy , 0) or the twisting moment Mt . For example,

∂τxy ∂σx σx + x + + xfˆx dx dy σx dx dy = ∂x ∂y ∂(xσx ) ∂(xτxy ) = dx dy + + xfˆx dx dy ∂x ∂y = x(σx n1 + τxy n2 )ds + xfˆx dx dy

=

xˆtx ds +

xfˆx dx dy,

(2.37a)

2.3

Boundary Conditions

41

∂τxy x ˆx y dx dy + x f σx + x ∂σ + ∂x ∂y ∂(xyσx ) ∂(xyτxy ) = dx dy − xτxy dx dy + + xyfˆx dx dy ∂x ∂y = xy(σx n1 + τxy n2 )ds + xyfˆx dx dy ∂σ ∂τ − x τxy + 12 ∂yy + ∂xxy x + 12 xfˆy dx dy 1 ∂(x2 τxy ) ∂(x2 σy ) dx dy xyfˆx dx dy − + ∂y = xyˆtx ds + 2 ∂x 1 2 ˆ − dy 2 x fy dx 1 2 = xyˆtx − 2 x ˆty ds + (xyfˆx − 12 x2 fˆy ) dx dy,

σxy dx dy =

(2.37b)

1 1 − (τyz x − τxz y) + (τyz x + τxz y) dx dy τxz y dx dy = 2 2 ∂τyz ∂τxz 1 1 + = − Mt + τyz x + τxz y + xy dx dy 2 2 ∂x ∂y ∂(xyτxz ) ∂(xyτyz ) 1 1 + dx dy = − Mt + 2 2 ∂x ∂y 1 1 = − Mt + xy(τxz n1 + τyz n2 ) ds 2 2

1 = − Mt , 2

(2.37c)

∂ψ ∂ψ x+ y dx dy (τyz x − τxz y) dx dy = − Mt = ∂x ∂y ∂(xψ) ∂(yψ) + dx dy + 2 ψ dx dy =− ∂x ∂y = − ψ(xn1 + yn2 ) ds + 2 ψ dx dy =2 ψ dx dy.

(2.37d)

Note that the last equality of (2.37d) comes from the fact that ψ = 0 along the boundary of the cross section, which can be proved by inserting (2.10)4,5 and (2.29) into (2.28)3 and knowing that if the region of the cross section is simply connected a constant value of ψ will not induce stresses and can be selected to be zero.

42

2


Similarly,

σy dx dy =

yˆty ds +

τxy dx dy =

xˆty ds +

σx x dx dy =

x2 2

yfˆy dx dy,

ˆtx ds +

xfˆy dx dy =

yˆtx ds +

yfˆx dx dy,

x2 2

fˆx dx dy,

y2 y2 ˆ ˆ xyfy − fx dx dy, xyˆty − ˆtx ds + σy x dx dy = 2 2

σy y dx dy =

y2 ˆty ds + 2

τxy x dx dy =

τxy y dx dy =

(2.37e)

(2.37f)

y2 ˆ fy dx dy, 2 x2 ˆty ds + 2 y2 ˆtx ds + 2

x2 ˆ fy dx dy, 2 y2 ˆ fy dx dy, 2

(2.37g)

τxz x dx dy = 0, τyz y dx dy = 0, τyz x dx dy =

(2.37h)

1 Mt . 2

With the results of (2.37a–h), the conditions (2.36a) take the form (Lekhnitskii, 1963) Pz = CS − S133 [(S13 x + S36 y)ˆtx + S23 yˆty ] ds − S133 [(S13 x + S36 y)fˆx + S23 yfˆy ] dx dy,

(2.38a)

S35 Mt 2S 33 1 S36 2 1 2 2 S13 xy + y ˆtx + (S23 y − S13 x )ˆty ds − S33 2 2 1 S36 2 ˆ 1 S13 xy + − y fx + (S23 y2 − S13 x2 )fˆy dx dy, S33 2 2

(2.38b)

M1 = BI1 +

2.4

Special Cases

43

S34 Mt 2S33 1 1 S36 2 − (S13 x2 − S23 y2 )ˆtx + S23 xy + x ˆty ds (2.38c) S33 2 2 1 1 S36 2 ˆ 2 2 ˆ − (S13 x − S23 y )fx + S23 xy + x fy dx dy, S33 2 2 Mt = 2 ψ dx dy. (2.38d)

M2 = AI2 −

The stress functions φ and ψ which satisfy (2.11a) and the boundary conditions on the cross section will contain the four arbitrary constants α, A, B, C. If the axial force Pz and the moments M1 , M2 , Mt are given, these constants can be determined from (2.38a–d).

2.4 Special Cases From the derivation shown in Section 2.1 we see that although the stresses depend only on two coordinates (x and y), through the integration the displacements may depend on the third coordinate (z) whose general expressions have been obtained in (2.8). In other words, the two-dimensional problems considered in Lekhnitskii formulation cover not only the pure 2D cases but also some special 3D cases. In this section, we like to discuss the special cases that all the physical responses such as the displacements and stresses depend only on two coordinates. In addition, the cases of finite length and finite cross section with equivalent axial forces and moments acting on the ends will also be discussed.

2.4.1 Generalized Plane Deformation Besides the assumptions stated at the beginning of this section, we assume that the length of the body is infinite and the region of the cross section is arbitrary (it can be finite or infinite, simply connected or multiply connected), see Fig. 2.2. An isotropic body under such conditions would experience plane deformation (or plane strain), i.e., w=0. In a body with anisotropy of a general form, plane deformation is usually not possible. We can only assert that all components of stresses and displacements will not depend on z. The deformation of such body is called generalized plane deformation or called generalized plane strain. The expressions for the displacements of generalized plane deformation are obtained as a special case by setting

A = B = C = α = 0, in (2.8a), and

ω1 = ω2 = 0

(2.39)

44

2


Fig. 2.2 Generalized plane deformation

y

tˆy tˆx x s

z

u = U(x, y) − ω3 y + u0 , v = V(x, y) + ω3 x + v0 ,

n

(2.40)

w = W(x, y) + w0 . The general solutions and boundary conditions for this kind of special case can also be expressed by (2.26), (2.27), (2.31), and (2.33).

2.4.2 Plane Deformation Let us assume that the body considered in the case of generalized plane deformation has at each point a plane of elastic symmetry normal to the generator, that is, normal to the z-axis. In other words, the body is composed of the monoclinic materials with the symmetry plane at z=0 for which C14 = C15 = C24 = C25 = C34 = C35 = C46 = C56 = 0. It is now possible to satisfy all the equations of the theory of elasticity by setting w = w0 Then, instead of (2.40), we have u = U(x, y) − ω3 y + u0 , v = V(x, y) + ω3 x + v0 , w = w0 .

(2.41)

Consequently, γyz = γxz = εz = 0, and on the basis of the equations of the generalized Hook’s law, τxz = τyz = 0 and σz = −

1 (S13 σx + S23 σy + S36 τxy ). S33

(2.42)

With the results of τxz = τyz = 0, by using the Airy stress function φ(x, y) introduced in (2.10) for the in-plane stresses σx , σy , τxy , the differential equation that the stress function must satisfy can then be reduced from (2.11) to

2.4

Special Cases

45

L4 φ = −(Sˆ 12 + Sˆ 22 )

∂ 2 Fˆ ∂ 2 Fˆ ∂ 2 Fˆ − (Sˆ 11 + Sˆ 12 ) 2 , + (Sˆ 16 + Sˆ 26 ) 2 ∂x∂y ∂x ∂y

(2.43a)

in which L4 is the differential operator of the fourth order which has the form 4 ∂4 ∂4 ∂4 ∂4 ˆ 11 ∂ . (2.43b) + S L4 = Sˆ 22 4 − 2Sˆ 26 3 + (2Sˆ 12 + Sˆ 66 ) 2 2 − 2Sˆ 16 ∂x ∂x ∂y ∂x ∂y ∂x∂y3 ∂y4

The general solution of the Airy stress function φ to the differential equation (2.43) can also be obtained by reducing the expression (2.24) to φ = 2 Re{φ1 (z1 ) + φ2 (z2 )} + φ (p) ,

(2.44a)

zk = x + μk y, k = 1, 2.

(2.44b)

in which

In (2.44b) the complex parameters μk are the roots of the algebraic equation l4 (μ) = Sˆ 11 μ4 − 2Sˆ 16 μ3 + (2Sˆ 12 + Sˆ 66 )μ2 − 2Sˆ 26 μ + Sˆ 22 = 0,

(2.45)

and are assumed to be distinct. In the case of repeated root such as an isotropic body whose μ1 = μ2 = i, the general solution should be expressed as φ = 2 Re{φ1 (z1 ) + z1 φ2 (z1 )} + φ (p) .

(2.46)

Detailed discussion of the general solutions for the bodies with repeated complex parameters μk will be given in the next chapter for Stroh formalism. With the result of (2.44), the general expressions for the stresses, displacements, and boundary conditions can then be obtained from (2.26), (2.27), (2.31), and (2.33) by deleting the terms associated with z3 . They are Stresses and Displacements ! " ∂ 2 φ (p) ˆ σx = 2 Re μ21 f1 (z1 ) + μ22 f2 (z2 ) + + F, ∂y2 ∂ 2 φ (p) ˆ σy = 2 Re f1 (z1 ) + f2 (z2 ) + + F, ∂x2 ∂ 2 φ (p) , τxy = −2 Re μ1 f1 (z1 ) + μ2 f2 (z2 ) − ∂x ∂y 2 u = 2 Re a1k fk (zk ) − ω3 y + u0 + u(p) , k=1

v = 2 Re

2 k=1

(2.47a)

(2.47b) (p)

a2k fk (zk ) − ω3 x + v0 + v ,

46

2


where a1k = μ2k Sˆ 11 + Sˆ 12 − μk Sˆ 16 , a2k = (μ2k Sˆ 21 + Sˆ 22 − μk Sˆ 26 )/μk .

(2.47c)

Boundary Conditions 2 Re {μ1 f1 + μ2 f2 } = ˜tx (s) −

∂φ (p) + c1 , ∂y

∂φ (p) + c2 , 2 Re { f1 + f2 } = ˜ty (s) − ∂x

(2.48a)

where s dy ˜tx (s) = − ˆtx + Fˆ ds, ds 0 s dx ˜ty (s) = ˆty − Fˆ ds, ds 0

(2.48b)

or

2 Re

2

a1k fk = −u(p) + uˆ + ω3 y − u0 ,

k=1

2 Re

2

(2.49) a2k fk = −v(p) + vˆ − ω3 x − v0 .

k=1

The problem then reduces to the determination of two complex functions f1 and f2 which should satisfy the boundary conditions on the contour of the cross-section region. Using St. Venant’s principle, we can apply all the formulas for an infinite cylinder to a body of finite length with fixed ends. In the case of free ends, the axial forces Pz and the bending moment M1 , M2 , which are given at the ends, can be removed by imposing the elementary distribution, M1 pz M2 − y− x, S I1 I2 σx = σy = τyz = τxz = τxy = 0, σz = −

on the distribution of the stresses in the finite cylinder.

(2.50)

2.4

Special Cases

47

2.4.3 Generalized Plane Stress Consider a thin plate of constant thickness, made from a homogeneous anisotropic material having at each point a plane of elastic symmetry parallel to the middle plane, i.e., monoclinic materials. Assume that the surface stresses are distributed along the edge symmetrically with respect to the middle plane, and that they vary negligibly with respect to the thickness of the plate; the body forces also are distributed symmetrically. Taking the middle plane as the coordinate plane, the x–y plane, we introduce the mean values of the stresses and displacements with respect to the thickness (see Fig. 2.3)

O

x

y z Fig. 2.3 Generalized plane stress

1 h/2 1 h/2 σx dz, σ˜ y = σy dz, h −h/2 h −h/2 1 h/2 1 h/2 u˜ = u dz, v˜ = v dz. h −h/2 h −h/2

σ˜ x =

τ˜xy =

1 h

h/2

−h/2

τxy dz, (2.51)

As we discussed in Section 1.3 for the stress–strain relation of two-dimensional problems, by replacing everywhere the reduced elastic compliances Sˆ ij to the elastic compliances Sij the basic equations that hold in the case of plane deformation can now be applied to the case of generalized plane stress for these average quantities. It is therefore the problems of plane deformation and generalized plane stress are usually combined and known as “the plane problem of the theory of elasticity”.

2.4.4 Anisotropic Rod by Bending and Twisting Consider certain cases of the equilibrium of an anisotropic rod where twisting and bending moments are applied simultaneously. Unlike the isotropic rods where the twisting moments will only induce twisting and the bending moments will only induce bending, due to the anisotropy of the rods both of the twisting and bending moments will induce bending as well as twisting. Figure 2.4 shows the case when

48

2


Fig. 2.4 Anisotropic rod by bending and twisting

z

x

M1

M2 Mt

y

the arbitrary twisting moment Mt and bending moments M1 , M2 act on the end of a rod where the x- and y-axes coincide with the principal axes of inertia of the cross section. Since the lateral surface of the rod is free from external loads and no body forces are considered in this case, application of the end moments Mt , M1 , M2 will then lead to ˆtx = ˆty = fˆx = fˆy = 0, Pz = 0, M1 , M2 , Mt = 0.

(2.52)

With (2.52), from (2.38a–c) we have A=

M2 S34 Mt + , I2 2S33 I2

B=

M1 S35 Mt − , I1 2S33 I1

C = 0.

(2.53)

The displacements for this kind of problems can then be expressed by substituting (2.53) into (2.8a). They are 1 (2S33 M2 + S34 Mt )z2 − αyz + U(x, y) + ω2 z − ω3 y + u0 , 4I2 1 v=− (2S33 M1 − S35 Mt )z2 + αxz + V(x, y) + ω3 x − ω1 z + v0 , 4I1 x y 1 (2S33 M2 + S34 Mt ) + (2S33 M1 − S35 Mt ) z + W(x, y) + ω1 y − ω2 x + w0 . w= 2 I2 I1 (2.54) u=−

The constants u0 , v0 , w0 and ω1 , ω2 , ω3 corresponding to the rigid body translation and rotation can be found from the conditions at the fixed end: u = v = w = 0,

∂u ∂v ∂u ∂v = = − = 0, ∂z ∂z ∂y ∂x

when z = l, x = y = 0.

(2.55)

2.4

Special Cases

49

Substituting (2.54) into (2.55), we get 1 (2S33 M2 + S34 Mt )l2 − U0 , 4I2 1 (2S33 M1 − S35 Mt )l2 − V0 , w0 = −W0 , v0 = − 4I1 1 (2S33 M1 − S35 Mt )l, ω1 = − 2I1 1 (2S33 M2 + S34 Mt )l, ω3 = −αl + ω30 , ω2 = 2I2 u0 = −

(2.56)

where U0 , V0 , W0 , and ω30 are the values of U, V, W, and (∂U/∂y − ∂V/∂x)/2 at the center of the cross section x = y = 0. In order to find U, V, W, and the associated stress fields, we can follow the steps described in (2.9)–(2.27) for the general cases. With the prior conditions (2.52) and the results of (2.53), the system of differential equations (2.11) which the stress functions must satisfy reduce to L4 φ + L3 ψ = 0, L3 φ + L2 ψ = −2α ∗ ,

(2.57a)

where Mt α =α− 4S33 ∗

#

2 S2 S34 + 35 I2 I1

$ +

S34 M2 S35 M1 − . 2I1 2I2

(2.57b)

A particular solution of the nonhomogeneous system (2.57) can be chosen as φ (p) = 0,

ψ (p) =

−α ∗ (Sˆ x2 + 2Sˆ 45 xy + Sˆ 44 y2 ). 2 ) 55 2(Sˆ 44 Sˆ 55 − Sˆ 45

(2.58)

With this particular solution, the unknowns remained in the general expressions for the stresses and displacements (2.26) and (2.27) are the complex functions f1 (z1 ), f2 (z2 ), f3 (z3 ) which should then be determined through the satisfaction of the lateral surface boundary conditions (2.31) or (2.33). In order to relate the twisting angle α and the twisting moment Mt we rewrite (2.38d) by letting ψ = α∗ψ ∗,

(2.59)

and hence Mt = 2α ∗

Mt ψ ∗ dxdy, α ∗ = α − 4S33

#

$ 2 2 S35 S34 S35 M1 S34 M2 + − . (2.60) + I2 I1 2I1 2I2

50

2


Re-organizing (2.60), we get α=

S35 M1 Mt S34 M2 − + , D 2I1 2I2

(2.61a)

where D is the generalized torsional rigidity defined by 1 1 1 = ∗+ D D 4S33

#

2 S2 S34 + 35 I2 I1

$ ,

or

D=

4S33 D∗ , 2 S2 S35 4S33 + I34 + D∗ I1 2

(2.61b)

and D∗ = 2

ψ ∗ dx dy.

(2.61c)

Based upon the derivation given in (2.52)–(2.61), several different rods deformed by twisting and bending moments have been analyzed and presented by Lekhnitskii (1963). The cross sections discussed in Lekhnitskii (1963) include elliptic, rectangular, aero-dynamic profile, elliptic ring, and elliptic sector. Following are some special cases of the joint action of twisting and bending moments. Twisting Moment Only (Mt = 0, Pz = M1 = M2 = 0) Substituting M1 = M2 = 0 into (2.53), (2.54), (2.55), (2.56), (2.57), and (2.58), we can get the results for a rod with anisotropy of a general kind under the influence of twisting moments. Since the constants A, B which are related to the bending deformation still exist, the twisting moments induce not only torsion but also bending. This situation is more complex than the usual pure torsion problem of isotropic rods and hence is called generalized torsion. If the rod has at each point a plane of elastic symmetry normal to its axis, which is one kind of monoclinic materials discussed in Section 1.2.2, S34 = S35 = 0 and hence Sˆ 34 = Sˆ 35 = 0, Sˆ 44 = S44 , Sˆ 45 = S45 , Sˆ 55 = S55 by (1.56). From (2.53) we see that A=B=C=0. In other words, under twisting moment pure torsion deformation can be obtained not only for the isotropic materials but also for the monoclinic materials. With M1 = M2 = 0, the relation between the twisting angle α and the twisting moment Mt , (2.61), now becomes 1 1 Mt 1 , = ∗+ α= D D D 4S33

#

2 S2 S34 + 35 I2 I1

$ .

(2.62)

Torsion Without Bending (α = 0, A = B = C = 0) From (2.53) we see that in some particular cases the bending deformation will not occur if the moments are selected in the following way:

2.4

Special Cases

51

M1 =

S35 S34 Mt , M2 = − Mt . 2S33 2S33

(2.63)

Substituting (2.63) into (2.61), we obtain α=

Mt , D

D = D∗ ,

(2.64)

which is the same as the case that only twisting moment is applied on a rod with S34 = S35 = 0. Obviously, the torsional rigidity D becomes more rigid than that of the generalized torsion problems because D∗ ≥ D in the general cases. Bending Without Torsion (B = 0, A = C = α = 0 or A = 0, B = C = α = 0) We seek moments such that the rod will be bent in the principal plane, the y–z plane or x–z plane, and the bending will not be accompanied by twisting. This situation can be made by letting A=0 or B=0 in (2.53) and then α = 0 in (2.61a), which will lead to M2 = −

S34 Mt , Mt = 2S33

S35 D∗ M1 , 2 S35 2I1 1 + 4S33 I1

(2.65a)

or M1 =

S35 Mt , Mt = 2S33

−S34 D∗ M2 . 2 S34 2I2 1 + 4S33 I1

(2.65b)

When we apply the moments given in (2.65a) the rod will be bent in the y–z plane without torsion. The bent axis along the center of the cross section may be found from (2.8a)2 with x=y=0. The bending rigidity may be represented by the inverse of BS33 /2, i.e., larger B means smaller bending rigidity which will induce larger deflection. When we substitute (2.65a) into (2.38b), we get B=

M1 4S33 I1 . 2 D∗ I 4S33 I1 + S35 1

(2.66)

For the same rod bent only by the moment M1 , from (2.38b) we have B=

M1 . I1

(2.67)

A comparison of (2.66) and (2.67) shows that the deflection of the rod undergoes bending without torsion due to the applied moments M1 , M2 and Mt given in (2.65a) will be less than the deflection induced by the single moment M1 . In other words, if the moments which prevent twisting are applied to the rod, the rigidity of its bending increases, or say, the constant B decreases. Similar situation can be made for the applied moments given in (2.65b) for the bending in the x–z plane without torsion.

52

2


2.5 Anisotropic Cantilever Under Transverse Force Consider an elastic body in the form of a cylinder or prism in equilibrium with one end fixed and a transverse force P acting on the other end (see Fig. 2.5). Assume that the cantilever is made from a homogeneous material with rectilinear anisotropy of the most general form. Under this condition, we may assume that the stresses σx , σy , τxy , τxz , τyz are functions of x and y only and that Fig. 2.5 Anisotropic cantilever under transverse force

z p

x

y

σz = −

M1 Pz y + σz0 = − y + σz0 , I1 I1

(2.68)

where σz0 is a function of x and y only. Then, the equilibrium equations stated in (2.2c) should be modified as ∂τxy ∂τxy ∂σy ∂τyz ∂σx ∂ Fˆ ∂ Fˆ ∂τxz P + − = 0, + − = 0, + − y = 0, (2.69) ∂x ∂y ∂x ∂x ∂y ∂y ∂x ∂y I1 in which the only difference from the two-dimensional problems discussed in the previous sections comes from the addition of the third term on the left-hand side of the third equation of (2.69). By following the procedure described in the previous sections, one can find the general solutions for the anisotropic cantilever under transverse force. Detailed discussion of this kind of problems can be found in Lekhnitskii (1963).

Chapter 3

Stroh Formalism

As stated at the beginning of Chapter 2, both Lekhnitskii and Stroh formalisms are for the analysis of two-dimensional deformation of an anisotropic linear elastic solid. The difference is that Lekhnitskii formalism begins with the two-dimensional stresses, while Stroh formalism starts with the two-dimensional displacements. Due to this difference, Lekhnitskii formalism is in terms of the reduced elastic compliances while Stroh formalism to be presented in this chapter will be in terms of the elastic stiffnesses. Moreover, the displacement fields as shown in (2.8) by Lekhnitskii formalism may be related to the third coordinate z due to integration of 2D strains, whereas all the physical quantities by Stroh formalism will depend on x and y only because stresses and strains are related to the derivatives of displacements not their integrations. In other words, Lekhnitskii formalism may cover more general fields than the pure two-dimensional problems discussed by Stroh formalism, such as the anisotropic rod by bending and twisting discussed in Section 2.4.4. However, for the pure two-dimensional problems, it will be observed in the rest of this book that Stroh formalism is indeed mathematically elegant and technically powerful in solving two-dimensional anisotropic elasticity problems. Stroh formalism can be traced to the work of Eshelby et al. (1953). Not all results for this well-known formalism are due to Stroh (1958, 1962). It was named after Stroh because he laid the foundations for researchers who followed him. The major contribution of Stroh formalism should be attributed to Ting (1996) for his thorough and detailed studies starting from the 1980s. Most of the materials presented in this chapter are therefore excerpted from Ting’s book (Ting, 1996).

3.1 General Solutions If we neglect the body forces fi , the basic equations for linear anisotropic elasticity shown in (2.1) are σij, j = 0,

εij =

1 (ui, j + uj, i ), 2

σij = Cijkl εkl ,

i, j, k, l = 1, 2, 3,

C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_3, C Springer Science+Business Media, LLC 2010

(3.1)

53

54

3

Stroh Formalism

where the elastic constants Cijkl are assumed to be fully symmetric and positive definite. Substituting (3.1)2 into (3.1)3 with Cijkl fully symmetric, we get σij = Cijkl uk, l .

(3.2)

The governing differential equations which the displacements must satisfy can then be obtained by substituting (3.2) into (3.1)1 , i.e., Cijkl uk, lj = 0.

(3.3)

Since the governing differential equations (3.3) are a set of homogeneous secondorder differential equations, for two-dimensional deformations in which uk , k = 1, 2, 3, depend on x1 and x2 only, a general solution for uk will depend on one composite variable which is a linear combination of x1 and x2 . Without loss in generality we choose uk = ak f (z),

or

u = a f (z),

(3.4a)

where z = x1 + μx2 .

(3.4b)

In the above a and μ are constants to be determined and f is an arbitrary function of z. By using chain rule, differentiation of (3.4a) with respect to xl gives uk, l = ak

df dz = ak (δl1 + μδl2 ) f (z), dz dxl

(3.5)

in which the prime denotes differentiation with respect to the argument z and δli is the Kronecker delta. Differentiating once more with respect to xj we find that the governing equation (3.3) can be satisfied if Cijkl (δl1 + μδl2 )(δj1 + μδj2 )ak = 0

(3.6a)

{Ci1k1 + μ(Ci1k2 + Ci2k1 ) + μ2 Ci2k2 }ak = 0.

(3.6b)

or

Equation (3.6b) can be rewritten in matrix form as {Q + μ(R + RT ) + μ2 T}a = 0,

(3.7)

where the superscript T stands for the transpose and Q, R, T are the 3 × 3 real matrices defined by Qik = Ci1k1 ,

Rik = Ci1k2 ,

Tik = Ci2k2 .

(3.8)

3.1

General Solutions

55

From definition (3.8) we see that Q and T are symmetric and positive definite if the strain energy is positive (Eshelby et al., 1953). In (3.7), a nontrivial solution of a exists if Q + μ(R + RT ) + μ2 T = 0,

(3.9)

which gives a sextic equation for μ. Since μ and a determined from (3.9) and (3.7) depend only on the elastic constants Cijkl , it is commonly called the material eigenvalues and material eigenvectors. The μ determined from (3.9) should also be equivalent to the one determined from (2.15a) by Lekhnitskii formulation since their corresponding governing equations (3.3) and (2.13) are all derived from the basic equations given in (3.1) for two-dimensional deformations. After obtaining μ and ak for the displacements given in (3.4), the stresses can be determined from (3.2) by employing (3.5). The results are σij = Cijkl (δl1 + μδl2 ) ak f (z) = (Cijk1 + μCijk2 ) ak f (z),

(3.10a)

which can also be written as σi1 = (Qik + μRik ) ak f (z),

σi2 = (Rki + μTik ) ak f (z).

(3.10b)

From (3.7) and (3.10b), we see that a simple expression for the stresses may be found by introducing a new vector b as 1 b = (RT + μT)a = − (Q + μR)a, μ

(3.11)

in which the second equality of (3.11) is obtained from (3.7). With (3.11), (3.10b) can now be written as σi1 = −μbi f (z),

σi2 = bi f (z).

(3.12)

Like the introduction of the Airy stress functions (2.10) in Lekhnitskii formulation, the equilibrium equation (3.1)1 can be satisfied automatically if we introduce the stress functions φi , i = 1, 2, 3, as σi1 = −φi, 2 ,

σi2 = φi, 1 .

(3.13)

Due to the stress symmetry property, σ12 = σ21 and hence φi are not independent of each other but are related by φ1, 1 + φ2, 2 = 0.

(3.14)

Comparing (3.12) and (3.13), we get φi = bi f (z),

or

φ = bf (z).

(3.15)

56

3

Stroh Formalism

Equation (3.9) is a sextic equation for material eigenvalue μ, which should give six roots for μ. Through the requirement that the strain energy should always be positive no matter which kind of deformations, we can prove that μ will not be real. If μ is real, (3.6a) multiplied by ai and summed with respect to i leads to Cijkl [(δj1 + μδj2 ) ai ][(δl1 + μδl2 ) ak ] = 0.

(3.16)

Since μ is assumed to be real and hence ai be real, we may choose an arbitrary strain as εij = (δj1 + μδj2 ) ai .

(3.17)

Cijkl εij εkl = 0,

(3.18)

With this strain, (3.16) gives us

which violates the condition that the strain energy is positive definite. Therefore, μ cannot be real. Since the coefficients of the sextic equation for μ arising from (3.9) are real, there are three pairs of complex conjugates for μ. If μk and ak , k = 1, 2, . . . , 6, are the material eigenvalues and eigenvectors, we may let Im μk > 0, μk+3 = μk , ak+3 = ak , bk+3 = bk , k = 1, 2, 3.

(3.19)

Assuming that μk are distinct, the general solution obtained by superposing six solutions of the form (3.4a) and (3.15) is u=

3

3 ! " {ak fk (zk ) + ak fk+3 (zk )} , φ = bk fk (zk ) + bk fk+3 (zk ) ,

k=1

(3.20)

k=1

where fk , k = 1, 2, . . . , 6 are arbitrary functions of their arguments and zk = x1 + μk x2 .

(3.21)

For the displacement u and the stress function φ to be real, we let fk+3 = f k ,

k = 1, 2, 3

(3.22)

and (3.20) becomes u = 2 Re

3

{ak fk (zk )} ,

k=1

φ = 2 Re

3

{bk fk (zk )} ,

(3.23)

k=1

which can also be written as u = 2 Re {Af (z)} ,

φ = 2 Re {Bf (z)} ,

(3.24a)

3.2

Boundary Conditions

57

where B = [b1 b2 b3 ], A = [a1 a2 a3 ], &T % f(z) = f1 (z1 ), f2 (z2 ), f3 (z3 ) .

(3.24b)

By combining the displacement and stress function vectors into one column vector, (3.24) can be further expressed as f(z) u AA . = φ BB f(z)

(3.25)

Through the use of orthogonality relation given later in (3.57a), we can get AT φ + BT u = f(z).

(3.26)

The matrix form general solutions (3.24) shown above are obtained by Stroh formalism, while the component form general solutions (2.26) and (2.27) are obtained by Lekhnitskii formulation. By comparing these two solutions we have ((3.24a)2 and (3.13) will be used to get the stress components from Stroh formalism) ⎤ c1 a11 c2 a12 c3 a13 A = ⎣c1 a21 c2 a22 c3 a23 ⎦, c1 a31 c2 a32 c3 a33 ⎡

⎡

⎤ −c1 μ1 −c2 μ2 −c3 μ3 λ3 c2 c3 λ3 ⎦, B = ⎣ c1 −c1 λ1 −c2 λ2 −c3

(3.27)

where aij and λi , i, j = 1, 2, 3 are given in (2.27b,c), (2.23), and (2.15b). The normalization factors ci , i = 1, 2, 3 can be determined from the orthogonality relation given later in (3.57b)1 , which requires that c2k =

1 , 2(a2k − μk a1k − λk a3k )

k = 1, 2,

c23 =

1 . 2(a23 λ3 − μ3 a13 λ3 − a33 ) (3.28)

3.2 Boundary Conditions The general solutions satisfying the 15 basic equations (3.1) have been shown in (3.24) for the displacement and stress functions vectors, in which A and B can be completely determined if the material properties are known and f(z) is the unknown to be determined through the satisfaction of boundary conditions. To find f(z), it is therefore better to express boundary conditions in terms of f(z) directly or u, φ indirectly. Since each component of the displacement vector u represents the displacement in the x1 , x2 , and x3 directions, like (2.32) the displacement prescribed boundary condition can be written directly by ˆ u = 2 Re {Af (z)} = u,

(3.29)

58

3

Stroh Formalism

where uˆ = (û1 , uˆ 2 , uˆ 3 ) are the displacements prescribed along the boundary. To state the traction prescribed boundary conditions, we need to know the relation between the stress function vector φ and the traction t. Like the derivation of (2.28), (2.29), (2.30), and (2.31), for plane problem n3 = 0 the surface traction ti can be written by Cauchy’s formula as ti = σij nj = σi1 n1 + σi2 n2 .

(3.30)

Substituting (3.13) and (2.29) into (3.30), we have ti =

dφi dx2 dφi dφi dx1 + = dx1 ds dx2 ds ds

(3.31)

dφ . ds

(3.32)

or in vector form t=

Note that since (2.29) has been used in deriving the relation (3.32), to keep this relation in correct sign the tangential direction s has to be chosen such that when one faces the direction of increasing s the material lies on the right side for the coordinate system shown in Fig. 2.1. Otherwise, a negative sign should be put in the right side of the equation. The stress resultants along the surface may then be expressed as

s2

t ds = φ(s2 ) − φ(s1 ).

(3.33)

s1

With the relation (3.32), if the boundary of a two-dimensional body is applied by the traction ˆt = (ˆt1, ˆt2 , ˆt3 ), the traction-prescribed boundary condition can be written as df (z) dφ = 2 Re B = ˆt ds ds

(3.34a)

or φ(s2 ) − φ(s1 ) = 2 Re {Bf (z)}]ss21 =

s2

ˆt ds.

(3.34b)

s1

When the direction of the boundary surface s is chosen to be x1 - or x2 -axis, (3.32) can be reduced to the relations shown in (3.13). It is known that the relations (3.13) are important for the calculation of the stresses from the stress function φ. To calculate the stress components along any other coordinate axes, one usually applies the transformation law of second-order tensors. Based upon the formula obtained in (3.32), an alternative approach to determine the stress components of the rotated coordinate axes has been introduced in Ting (1996). Let (s, n) be the unit vector

3.2

Boundary Conditions

59

Fig. 3.1 The original and rotated coordinate systems

x2

x2∗

x1∗ s

n

θ

x1

tangent and normal to a surface boundary. If tn and ts denote the traction vectors on the surfaces with normals n and s, respectively, by (3.32) we have (see Fig. 3.1) t n = φ, s ,

ts = −φ, n .

(3.35)

The stress components on s– n coordinate can therefore be calculated by σnn = nT tn = nT φ, s ,

σns = sT tn = sT φ, s ,

σss = sT ts = −sT φ, n ,

σsn = nT ts = −nT φ, n = σns ,

σn3 = iT3 tn = (φ, s )3 , σs3 = iT3 ts = −(φ, n )3 , (3.36a)

where sT = (cos θ , sin θ , 0),

nT = (− sin θ , cos θ , 0),

i3 = (0, 0, 1),

(3.36b)

and the angle θ is directed counterclockwise from the positive x1 -axis to the direction of s. For polar coordinate system (r, θ ), we may let s and n denote the directions of r and θ, and ∂s and ∂n be replaced by ∂r and r ∂θ . With this replacement, it can be proved that the traction vectors and stress components in the polar coordinate system are related to the stress functions by t θ = φ, r ,

tr = −φ, θ /r,

σθθ = n φ, r ,

σrr = −sT φ, θ /r,

σθ3 = (φ, r )3 ,

σr3 = −(φ, θ )3 /r.

T

σr θ = sT φ, r = −nT φ, θ /r,

(3.37)

60

3

Stroh Formalism

3.3 Material Eigenrelation From the general solutions shown in (3.24), we see that the material eigenvalues μk and their associated eigenvectors ak , bk play important roles in Stroh formalism. These values can be determined from (3.9), (3.7), and (3.11), which are not in the form of standard eigenvalue problems like Ax = λx. To know their determination and to reconstruct the standard eigenvalue problem, we now rewrite (3.8) in terms of the contracted notation introduced in Section 1.2.1, i.e., ⎤ C11 C16 C15 Q = ⎣ C16 C66 C56 ⎦, C15 C56 C55 ⎡

⎡

⎤ C16 C12 C14 R = ⎣C66 C26 C46 ⎦, C56 C25 C45

⎡

⎤ C66 C26 C46 T = ⎣C26 C22 C24 ⎦. C46 C24 C44

(3.38)

Note that Q and T are principal minors of Cij and hence symmetric and positive definite. Therefore, the matrices Q and T are non-singular. With (3.38), (3.7) becomes ⎡ ⎣

C11 + 2μC16 + μ2 C66

C16 + μ(C12 + C66 ) + μ2 C26 C66 + 2μC26 + μ2 C22

symm.

⎤ C15 + μ(C14 + C56 ) + μ2 C46 2 C56 + μ(C25 + C46 ) + μ C24 ⎦ a = 0. C55 + 2μC45 + μ2 C44

(3.39) A sextic equation for the eigenvalues μ is obtained by setting the determinant of the 3 × 3 matrix on the left of (3.39) to zero, like that shown in (3.9). This sextic equation can be simplified for special materials such as monoclinic, orthotropic, transversely isotropic, and isotropic materials discussed in Section 1.2.2. Detailed derivation for the eigenvalues of each special material can be found in Ting (1996). Here, we just show the results for the most commonly used materials – isotropic materials. By using the elastic constants for the isotropic materials given in (1.40) and (1.41), (3.39) can be simplified to ⎤ μ(λ + G) 0 (λ + 2G) + μ2 G ⎦ a = 0. ⎣ μ(λ + G) G + μ2 (λ + 2G) 0 2 0 0 G(1 + μ ) ⎡

(3.40)

Note that when we use the material constants given in (1.40) and (1.41), one of the Lame constants μ which is also the well-known shear modulus has been replaced by the symbol G to avoid confusion with the material eigenvalue μ. The sextic equation resulting from (3.40) for the material eigenvalues μ becomes G2 (λ + 2G)(1 + μ2 )3 = 0.

(3.41)

Since the Lame constants are always positive, the only roots to (3.41) are the triple roots μ = ±i.

(3.42)

3.3

Material Eigenrelation

61

The material eigenvectors a associated with the triple roots can then be found by substituting (3.42) into (3.40), which leads to ⎡

λ+G ⎣±i(λ + G) 0

±i(λ + G) −(λ + G) 0

⎤ 0 0⎦ a = 0. 0

(3.43)

The first two columns of the 3 × 3 matrix are proportional to each other, yielding only the following two independent eigenvectors a for μ = i, ⎧ ⎫ ⎧ ⎫ ⎨1⎬ ⎨0⎬ a1 = i , a3 = 0 . ⎩ ⎭ ⎩ ⎭ 0 1

(3.44)

With the results of (3.44), the eigenvector b can be found from (3.11) as ⎧ ⎫ ⎧ ⎫ ⎨ i ⎬ ⎨0⎬ b1 = 2G −1 , b3 = G 0 . ⎩ ⎭ ⎩ ⎭ 0 −i

(3.45)

The materials, which cannot yield a complete set of independent eigenvectors such as (3.44) and (3.45) discussed for the isotropic materials, are called the degenerate materials. Due to the degeneracy, the general solutions shown in (3.24) are not complete and should be modified. Detailed discussion of the degenerate materials can be found in Ting (1996) and also will be presented in Section 3.5. As we said at the beginning of this section that (3.7) is not in the form of the standard eigenvalue problems, by performing the determinant of (3.39) we will get a sixth-order polynomial equation for μ, which may look more complicate than the characteristic equation (2.15a) obtained by Lekhnitskii formulation written in terms of the reduced compliances instead of the elastic constants. From this viewpoint, we cannot see any particular advantages by using Stroh formalism instead of Lekhnitskii formulation. What makes Stroh formalism more powerful than the others is the reconstruction of the eigenrelation (3.7) into the form of standard eigenvalue problem like Ax = λx. With this standard form, many properties and identities that are important in applications can be obtained for the material eigenvalues and eigenvectors.

3.3.1 Sextic Eigenrelation The two equations (3.11) of which the second equality comes from the eigenrelation (3.7) can be recast in the form −Q 0 a RI a = μ , T0 b −RT I b

(3.46)

62

3

Stroh Formalism

where I is the 3 × 3 identity matrix. It can be shown that

0 T−1 I −RT−1

RI I0 = , T0 0I

(3.47)

where T−1 exists due to the positive definite property of T. With the relation (3.47), (3.46) can be reduced to the following standard eigenrelation: Nξ = μξ, where N=

' ( N1 N2 N3 NT1

,

(3.48a)

ξ=

a b

(3.48b)

and N1 = −T−1 RT ,

N2 = T−1 = NT2 ,

N3 = RT−1 RT − Q.

(3.48c)

From (3.48c) we see that N2 and N3 are symmetric and N2 is positive definite. It can be proved that −N3 is positive semi-definite (Ting, 1996). The explicit expressions for N1 , N2 , and N3 in terms of the reduced compliances have been obtained by Ting (1996) and will be listed in Section 3.4.1 for readers’ convenience. Due to the importance of the eigenrelation, the 6 × 6 real matrix N is called the fundamental elasticity matrix. It should be noted that N3 has the unit of stress (Nt/m2 ), N2 has the unit of compliance (m2 /Nt) and N1 is dimensionless. From (3.48b,c) it can also be seen that N is not symmetric but JN is symmetric where 0I J= . (3.49) I0 Since N is not symmetric, its associated right and left eigenvectors are different. ξ in (3.48a) is a right eigenvector, whereas the left eigenvector denoted by η satisfies the following eigenrelation: or NT η = μη. (3.50) ηT N = μηT ) T ) Note that N − μI = 0 and )N − μI) = 0 have the same eigenvalues, and the left and right eigenvectors are in the reverse order, i.e., b η = Jξ = . a

(3.51)

To prove (3.51), we pre-multiply both sides of (3.48a) by J, which leads to JNξ = μJξ.

(3.52)

3.3


63

Since JN is symmetric, we have JN = (JN)T = NT J. With this result, by comparing (3.52) with (3.50)2 we can get the relation between the left and right eigenvectors shown in (3.51). Besides the relation (3.51), we can further prove that the left and right eigenvectors associated with different eigenvalues are orthogonal to each other. To prove the orthogonality relation we consider (3.48a) associated with μi and ξi and multiplied by ηTj , and (3.50)2 associated with μj and ηj and multiplied by ξTi which lead to ηTj Nξi = μi ηTj ξi , ξTi NT ηj = μj ξTi ηj . (no sum for repeated i or j).

(3.53)

Since a scalar is equivalent to its own transpose, (3.53)2 can be rewritten as ηTj Nξi = μj ηTj ξi .

(3.54)

Subtracting (3.54) from (3.53)1 , we get (μi − μj )ηTj ξi = 0

(3.55a)

or, ηTj ξi = 0,

when μi = μj ,

= 0,

when μi = μj .

(3.55b)

Through normalization, (3.55b) can be combined into the following orthogonality relation ηTj ξi = δij , i, j = 1, 2, . . . , 6,

(3.56)

where δij is the Kronecker delta. With (3.19), (3.48b)2 , (3.51), and (3.24b)1,2 , the normalized orthogonality relation can be rewritten as '

( AA I0 = T T 0I BB B A BT AT

(3.57a)

or T

T

T

T

BT A + AT B = I = B A + A B, BT A + AT B = 0 = B A + A B.

(3.57b)

Equation (3.57a) implies that the two 6 × 6 matrices on the left-hand side are inverse of each other, and hence their product commutes, i.e., ' T T( B A AA I0 T T = 0 I BB B A

(3.58a)

64

3

Stroh Formalism

or T

T

T

T

ABT + A B = I = BAT + B A , AAT + A A = 0 = BBT + B B .

(3.58b)

Equations (3.58b) imply that AAT , BBT , and ABT − (1/2)I are purely imaginary. Hence, we may define three real matrices S, H, and L by H = 2iAAT ,

L = −2iBBT ,

S = i(2ABT − I),

(3.59)

where H and L are symmetric. Moreover, it can be shown that both H and L are positive definite if the strain energy is positive (Ting, 1996). Since these three real matrices occur frequently in the real-form expression for practical engineering problems, detailed discussions about their structures and explicit expressions have been shown in Ting (1996). For readers’ convenience, their explicit expressions in terms of the reduced compliances will be shown in Section 3.4.2. The matrix form orthogonality relations shown in (3.57) and (3.58) are based upon the fact that the eigenvectors of fundamental matrix N are independent. A matrix is called simple if all its eigenvalues are distinct. A matrix of order n that has repeated eigenvalues and possesses n independent eigenvectors is called semisimple. If the matrix has less than n independent eigenvectors, it is called nonsemisimple. When the fundamental matrix N is simple or semisimple, the eigenrelation (3.48) for μ1, μ2 , μ3 can be combined into one equation as 0 AA A A < μα > , N = 0 < μα > B B BB

(3.60)

where the angular bracket stands for the diagonal matrix in which each component is varied according to its subscript, e.g., < μα >= diag[μ1 , μ2 , μ3 ]. Use of the orthogonality relation (3.58a) will now lead (3.60) to

AA N= BB

0 < μα > 0 < μα >

' T T( B A T T , B A

(3.61)

which is the diagonalization of N. One may refer to Ting and Hwu (1988) or Ting (1996) for the diagonalization of nonsemisimple N.

3.3.2 Generalized Sextic Eigenrelation The problems and general solutions considered in Section 3.1 are under a fixed coordinate system xi , i = 1, 2, 3. If we consider a new coordinate system xi∗ obtained by rotating xi about the x3 -axis an angle θ (Fig. 3.1), we have

3.3


65

xi∗ = ij xj ,

(3.62a)

where ij is the rotation matrix defined as ⎡

cos θ sin θ = ⎣− sin θ cosθ 0 0

⎤ 0 0⎦ . 1

(3.62b)

∗ referred to the new coordinate x∗ are, as shown in (1.17), The elastic constants Cijks i ∗ Cijks = ip jq kr st Cpqrt .

(3.63)

All the other matrices related to the material properties such as Q, R, T in (3.8) and N in (3.48) may then be modified as ∗ , Q∗ik = Ci1k1

∗ R∗ik = Ci1k2 ,

∗ Tik∗ = Ci2k2 ,

N∗ ξ∗ = μ∗ ξ∗ , ∗ ∗ ∗ N1 N2 a ∗ ∗ , ξ = ∗ , N = b N∗3 N∗T 1 N∗1 = −T∗−1 R∗T ,

N∗2 = T∗−1 = N∗T 2 ,

N∗3 = R∗ T∗−1 R∗T − Q∗ .

(3.64) (3.65a) (3.65b) (3.65c)

The solution for the displacements and stress functions shown in (3.4a) and (3.15) becomes u∗ = a∗ f ∗ (z∗ ), φ∗ = b∗ f ∗ (z∗ ),

(3.66a)

z∗ = x1∗ + μ∗ x2∗ .

(3.66b)

where

Because the displacement and stress function vectors are tensors of rank one, they have the following relations: u∗ = u, φ∗ = φ.

(3.67)

Knowing the relations (3.67), by comparing (3.66) with (3.4) and (3.15), without loss in generality we may set a∗ = a, b∗ = b, f ∗ (z∗ ) = f (z).

(3.68)

Use of (3.62), z∗ defined in (3.66b) can be rewritten as sin θ + μ∗ cos θ z∗ = (cos θ − μ∗ sin θ ) x1 + x 2 . cos θ − μ∗ sin θ

(3.69)

66

3

Stroh Formalism

The equivalence of the functions shown in (3.68)3 means that their arguments can only be different by a multiplier. Hence, by comparing (3.69) with (3.4b), we obtain

μ=

sin θ + μ∗ cos θ μ cos θ − sin θ , or μ∗ = . ∗ cos θ − μ sin θ μ sin θ + cos θ

(3.70a)

Since μ∗ is a function of θ and μ∗ (0) = μ, to conform with the symbol used in the literature we delete the superscript∗ and keep the argument θ , i.e., we let μ∗ = μ(θ ).

(3.70b)

Instead of treating the problems in the xi - coordinate or xi∗ - coordinate, we may also consider a formalism in dual coordinate systems. That is, all the basic equations (3.1) including all the physical functions like the displacements, stresses, and strains are considered in the xi - coordinate, while the independent variables are referred to xi∗ - coordinate. Under this consideration, the displacement and stress function vectors may be written as u = af (z∗ ), φ = bf (z∗ ).

(3.71)

Based upon (3.71)1 and following the procedure described in (3.4)–(3.15), we can obtain (Ting, 1996) {Q(θ ) + μ(θ )(R(θ ) + RT (θ )) + μ2 (θ )T(θ )}a = 0, b = {RT (θ ) + μ(θ )T(θ )}a = −

1 {Q(θ ) + μ(θ )R(θ )} a, μ(θ )

(3.72) (3.73)

where Q(θ ), R(θ ), T(θ ) are the 3 × 3 real matrices defined by Qik (θ ) = Cijkl sj sl , s1 = cos θ ,

s2 = sin θ ,

Rik (θ ) = Cijkl sj nl , s3 = 0,

Tik (θ ) = Cijkl nj nl ,

n1 = − sin θ ,

n2 = cos θ ,

(3.74a) n3 = 0. (3.74b)

Expanding (3.74) and using the definition (3.8), we get the matrix form relations as Q(θ ) = Q cos2 θ + (R + RT ) sin θ cos θ + T sin2 θ , R(θ ) = R cos2 θ + (T − Q) sin θ cos θ − RT sin2 θ ,

(3.75)

T(θ ) = T cos2 θ − (R + RT ) sin θ cos θ + Q sin2 θ . The generalized sextic eigenrelation can then be obtained by following the procedure described in (3.46), (3.47), and (3.48), i.e., N(θ )ξ = μ(θ )ξ,

(3.76a)

3.3


67

where N(θ ) =

N1 (θ ) N2 (θ ) , N3 (θ ) NT1 (θ )

ξ=

a b

(3.76b)

and N1 (θ ) = −T−1 (θ )RT (θ ),

N2 (θ ) = T−1 (θ ) = NT2 (θ ),

N3 (θ ) = R(θ )T−1 (θ )RT (θ ) − Q(θ ).

(3.76c)

Note that Q(θ ), R(θ ), T(θ ) and N(θ ), N1 (θ ), N2 (θ ), N3 (θ ) defined in (3.74) and (3.76) are different from Q∗ (θ ), R∗ (θ ), T∗ (θ ) and N∗ (θ ), N∗1 (θ ), N∗2 (θ ), N∗3 (θ ) defined in (3.64) and (3.65). Their relation may be found directly from their definitions, which have been shown to be (Ting, 1996) Q∗ (θ ) = Q(θ )T , N∗1 (θ ) = N1 (θ )T ,

R∗ (θ ) = R(θ )T , N∗2 (θ ) = N2 (θ )T ,

T∗ (θ ) = T(θ )T , N∗3 (θ ) = N3 (θ )T .

(3.77)

3.3.3 The Matrix Differential Equation For the two-dimensional problems discussed in this chapter, the equilibrium equation (3.1)1 can be satisfied automatically if we use the stress functions defined in (3.13). By using (3.13) and the definition (3.8), the constitutive relations shown in (3.2) can be rewritten as −φi, 2 = σi1 = Ci1k1 uk, 1 + Ci1k2 uk, 2 = Qik uk, 1 + Rik uk, 2 , φi, 1 = σi2 = Ci2k1 uk, 1 + Ci2k2 uk, 2 = Rki uk, 1 + Tik uk, 2

(3.78a)

or in matrix form −φ, 2 = Qu, 1 + Ru, 2 , φ, 1 = RT u, 1 + Tu, 2 .

(3.78b)

By combining the displacement and stress function vectors into one column vector, (3.78b) can also be written as −Q 0 u,1 R I u, 2 = , T 0 φ, 2 −RT I φ, 1

(3.79)

which bears almost the same form as (3.46). By employing the identity (3.47), (3.79) now leads to the matrix differential equation for two-dimensional deformations of an anisotropic elastic solid as (Malen, 1971; Barnett and Lothe, 1973; Chadwick and Smith, 1977; Ting, 1996)

68

3

Nω, 1 = ω, 2 ,

Stroh Formalism

u ω= . φ

(3.80)

From the above matrix differential equation, we see that the fundamental matrix N really plays an important role in the two-dimensional anisotropic elasticity. The further application of (3.80) can be found in Ting (1996).

3.4 Some Identities 3.4.1 Explicit Expression of Fundamental Elasticity Matrix N The fundamental elasticity matrix N appears frequently in the solutions to twodimensional problems. If the explicit expression of the fundamental elasticity matrix is known, a lot of computing time may be saved and also a relatively elegant analytical expression may be obtained. Starting from the definitions given in (3.48c) and (3.8), and using the inverse relation between the elastic stiffness C0 and the reduced compliance Sˆ shown in (1.60), Ting (1996) provided the explicit expressions for N1 , N2 , and N3 as ⎡

⎤ ⎡ ⎤ r6 1 s6 Sˆ 55 0 −Sˆ 15 ⎣ 0 0 0 ⎦ N1 = − ⎣r2 0 s2 ⎦, N3 = −1 r4 0 s4 −Sˆ 15 0 Sˆ 11 ⎡ ⎤ Sˆ 66 Sˆ 26 Sˆ 46 N2 = ⎣Sˆ 26 Sˆ 22 Sˆ 24 ⎦ Sˆ 46 Sˆ 24 Sˆ 44 ⎡ ⎤⎡ ⎤⎡ ⎤ Sˆ 16 0 Sˆ 56 Sˆ 55 0 −Sˆ 15 Sˆ 16 Sˆ 12 Sˆ 14 − 1 ⎣Sˆ 12 0 Sˆ 25 ⎦ ⎣ 0 0 0 ⎦ ⎣ 0 0 0 ⎦ Sˆ 56 Sˆ 25 Sˆ 45 −Sˆ 15 0 Sˆ 11 Sˆ 14 0 Sˆ 45

(3.81a)

where = Sˆ 11 Sˆ 55 − Sˆ 15 Sˆ 15 , rk = (Sˆ 15 Sˆ 5 k − Sˆ 55 Sˆ 1 k )/,

(3.81b)

sk = (Sˆ 15 Sˆ 1 k − Sˆ 11 Sˆ 5 k )/, k = 2, 4, 6. By a similar approach as that for the fundamental elasticity matrices, the explicit expressions for the generalized fundamental elasticity matrices N1 (θ ), N2 (θ ), and N3 (θ ) should also be obtained directly. However, due to the complexity, no explicit expressions for the generalized fundamental elasticity matrices have been provided in the literature. Here, we list only their expressions for the isotropic materials.

3.4

Some Identities

69

Isotropic materials: ⎡ sin 2θ 1 ⎣1 − 2v − cos 2θ N1 (θ ) = 2(1 − ν) 0 ⎡ 3 − 4v + cos 2θ 1+ν ⎣ sin 2θ N2 (θ ) = 2E(1 − v) 0 ⎡ 1 + cos 2θ −E ⎣ sin 2θ N3 (θ ) = 2(1 − v2 ) 0

−(1 − 2v) − cos 2θ − sin 2θ 0 sin 2θ 3 − 4v − cos 2θ 0 sin 2θ 1 − cos 2θ 0

⎤ 0 0⎦, 0

(3.82a)

⎤ 0 0 ⎦, (3.82b) 4(1 − v)

⎤ 0 0 ⎦. 1−v

(3.82c)

Because Ni = Ni (0), i = 1, 2, 3, we have ⎡ ⎤ ⎡ ⎤ 0 −(1 − ν) 0 2(1 − ν) 0 0 1 ⎣ 1 + ν ⎣ 0 −ν 0 0⎦, N2 = 1 − 2ν 0 ⎦, N1 = (1 − ν) 0 E(1 − ν) 0 0 0 0 2(1 − ν) ⎡ ⎤ 2 0 0 −E ⎣0 0 0 ⎦. N3 = 2(1 − ν 2 ) 0 0 1 − ν (3.83) Note that if there is no special notification all the explicit expressions presented in this book are for the generalized plane strain condition. As we mentioned in Section 1.3.2., for the generalized plane stress condition the reduced compliances Sˆ ij should be replaced by the compliances Sij or the elastic stiffnesses Cij should be replaced by the reduced elastic stiffnesses Cˆ ij . For the isotropic materials, the replacement can be simplified by using (1.51b) or (1.53).

3.4.2 Explicit Expressions of Barnett–Lothe Tensors S, H, and L With explicit expressions of A and B presented in (3.27), the three real matrices S, H, and L defined in (3.59) can be determined. However, a direct substitution of A and B would lead to a very unwieldy algebraic calculation because of the presence of the normalization factors. An alternative approach using the dimensionless tensors AB−1 discussed later in Section 3.4.4 has been employed by Ting (1996) to get simple explicit expressions for S, H, and L. Following are the results for different types of materials (Ting, 1996; Dongye and Ting, 1989; Hwu, 1993b).

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Stroh Formalism

Monoclinic materials with the symmetry plane at x3 =0: ⎡ ⎤ ⎡ ⎤ bd 0 d −b 0 s2 ⎣ ⎦, 0 e −d 0⎦, H = Sˆ 11 (1 − s2 ) ⎣d e S= g 0 0 0 2 −1 ˆ 0 0 [hS11 (1 − s )] ⎡ ⎤ e −d 0 s2 ⎣ ⎦, −d b 0 L= Sˆ 11 g2 0 0 hSˆ g2 s−2 11

(3.84a)

where a, b, c, d, e, g, s, and h are all dimensionless real constants determined by μ1 + μ2 = a + ib,

μ1 μ2 = c + id,

g = (Sˆ 12 /Sˆ 11 ) − c > 0,

b > 0,

e = ad − bc > 0,

2 −1/2

1 > s = g(be − d )

> 0,

(3.84b)

2 −1/2 2 1/2 ) = (C44 C55 − C45 ) . h = (Sˆ 44 Sˆ 55 − Sˆ 45

Orthotropic materials: ⎡

0 −k1 e∗ ⎣ S = e∗ 0 0 0

⎤ 0 0⎦ , 0

⎤ k1 k2 e∗ 0 0 H = ⎣ 0 k2 e∗ 0 ⎦ , 0 0 h−1 ⎡

⎡

k3 e∗ 0 ⎣ 0 k1 k3 e∗ L= 0 0

⎤ 0 0⎦ , h (3.85a)

where √ 1/2 C66 ( C11 C22 − C12 ) , h = (C44 C55 )1/2 e = √ C22 (C12 + 2C66 + C11 C22 ) * (3.85b) √ + C22 C11 C22 + C66 , k3 = (C12 + C11 C22 ). k1 = , k2 = √ C11 C66 ( C11 C22 − C12 ) ∗

In terms of the engineering constants given in (1.46) for generalized plane strain and in (1.64) for generalized plane stress, (3.85b) can also be written as e∗ = α1 γ2 η1 ,

h = (G23 G31 )1/2 ,

k1 =

α2 γ1 , α1 γ2

k2 =

η2 , η1 G12

k3 =

γ1 E1 , γ2 η1 (3.85c)

where + γ1 = (E1 /G12 + 2η1 E1 /E2 )−1/2 , and for generalized plane strain

+ γ2 = (E2 /G12 + 2η1 E2 /E1 )−1/2 , (3.85d)

3.4

Some Identities

71

α1 = √ (1 − v13 v31 )−1/2 , α2 = (1 −√v23 v32 )−1/2 , η1 = √(1 − v13 v31 )(1 − v23 v32 ) − (v21 + v31 v23 )(v12 + v32 v13 ), η2 = (1 − v13 v31 )(1 − v23 v32 ) √ −(1 − v12 v21 − v23 v32 − v31 v13 − 2v21 v32 v13 )G12 / E1 E2 ,

(3.85e)

while for generalized plane stress, α1 = 1,

α2 = 1, √ η1 = 1 − v12 v21 ,

+ η2 = 1 + (1 − v12 v21 )G12 / E1 E2 .

(3.85f)

Substituting (3.85c) into (3.85a), the nonzero components of S, H, and L can now be written in terms of engineering constants as S12 = −α2 γ1 η1 ,

S21 = −α1 γ2 η1 ,

H11 = α2 γ1 η2 /G12 , L11 = α1 γ1 E1 ,

L22

+ H22 = α1 γ2 η2 /G12 , H33 = 1/ G23 G31 , + = α2 γ2 E2 , L33 = G23 G31 .

(3.85g)

Since αi , γi , and ηi defined in (3.85d–f) are dimensionless factors, the results shown in (3.85 g) reveal that Hij has inverse relation with shear modulus, while Lij is proportional to Young’s modulus or transverse shear modulus. Isotropic materials: ⎡

⎤ 0 −e∗ 0 S = ⎣e∗ 0 0⎦ , 0 0 0

⎤ k2 e∗ 0 0 H = ⎣ 0 k2 e∗ 0 ⎦ , 0 0 h−1 ⎡

⎡

⎤ k3 e∗ 0 0 L = ⎣ 0 k3 e∗ 0⎦ , 0 0 h

(3.86a)

where for plane strain E 1 − 2ν (3 − 4ν)(1 + ν) E , k2 e∗ = , k3 e∗ = = G, , h= 2(1 − ν) 2E(1 − ν) 2(1 + ν) 2(1 − ν 2 ) (3.86b) while for generalized plane stress

e∗ =

e∗ =

1−v , 2

k2 e∗ =

3−v , 4G

k3 e∗ =

E , 2

h=

E = G. 2(1 + ν)

(3.86c)

3.4.3 Identities Relating N, N(θ), S, H, L Before stating some identities relating the fundamental elasticity matrix N, N(θ ) and the Barnett–Lothe tensors S, H, and L, we like to introduce some notations as follows:

72

3

Stroh Formalism

μ(θ ˆ , α) = cos(θ − α) + sin(θ − α)μ(α), ˆ , α) = cos(θ − α)I + sin(θ − α)N(α), N(θ θ 1 θ ˜ , α) = 1 μ(θ ˜ , α) = μ(ω)dω, N(θ N(ω)dω π α π α

(3.87a)

and μ = μ(0),

N = N(0),

ˆ ) = N(θ ˆ , 0), μ(θ ˆ ) = μ(θ ˆ , 0), N(θ ˜ ) = N(θ ˜ , 0), μ(θ ˜ ) = μ(θ ˜ , 0), N(θ

(3.87b)

in which I is the identity matrix. With the notations (3.87), some special cases which appear frequently in applications are μ(θ ˆ , 0) = μ(θ ˆ ) = cos θ + μ sin θ ,

ˆ , 0) = N(θ ˆ ) = cos θ I + sin θ N, N(θ

μ(0, ˆ θ ) = μˆ −1 (θ ) = cos θ − μ(θ ) sin θ , π π = −μ θ + , μˆ θ , θ + 2 2 π μˆ θ + , θ = μ(θ ), 2 μ(θ ˆ , θ ) = 1,

ˆ θ) = N ˆ −1 (θ ) = cos θ I − sin θ N(θ ), N(0, ˆ θ , θ + π = −N θ + π , N 2 2 π ˆ N θ + , θ = N(θ ), 2 ˆ N(θ , θ ) = I. (3.88)

By directly employing the relation (3.70), it can be proved that the generalized material eigenvalue μ(θ ) and its derivative and integral have the following relations: μ(θ ) = tan(γ − θ ),

where μ = tan γ ,

μ (θ ) = −1 − μ (θ ), π μ(θ ˜ ) = ln(cos θ + μ sin θ ), 2

(3.89)

in which the particular value of the last equation of (3.89) for θ = π is (Ting, 1996) μ(π ˜ )=

i if Imμ > 0, −i if Imμ < 0.

(3.90)

The result of (3.90) tells us an interesting and sensible message that the average value of μ(θ ) over the interval 0 ≤ θ ≤ π is ±i which is the eigenvalue of isotropic materials. The relation (3.70) can also help us to prove that ˆ , θ0 ). μ(θ ˆ , α)μ(α, ˆ θ0 ) = μ(θ

(3.91)

3.4

Some Identities

73

Consider the special cases shown in (3.88), some particular relations can be obtained from (3.91) as follows. ˆ , θ0 ). When α = 0, ⇒ μ(θ ˆ )μˆ −1 (θ0 ) = μ(θ ˆ , α)μ(α, ˆ θ ) = 1. When θ = θ0 , ⇒ μ(θ When θ = θ0 and α = 0, (3.92) ⇒ μ(θ ˆ , 0)μ(0, ˆ θ ) = (cos θ + μ sin θ )(cos θ − μ(θ ) sin θ ) = 1. π When θ = θ0 and, α = θ +- 2 ,, , ⇒ μˆ θ , θ + π2 μˆ θ + π2 , θ = −μ θ + π2 μ(θ ) = 1. Based upon the relations given in (3.87), (3.88), (3.89), (3.90), (3.91), and (3.92), we may now obtain several identities which are useful for the application of Stroh formalism. Integral Formalism Similar to (3.61), when N(θ ) is simple or semisimple the diagonalization of the 6 × 6 matrix N(θ ) is ' T T( B A AA 0 < μα (θ ) > N(θ ) = . (3.93) 0 < μα (θ ) > BT AT BB Integrating (3.93) with respect to θ ranging from 0 to π and using (3.90) and (3.59), we get ˜ )= N(π

S H , −L ST

which can also be written as 1 π 1 π S= N1 (θ )dθ , H = N2 (θ )dθ , π 0 π 0

L=−

(3.94)

1 π

π

N3 (θ )dθ.

(3.95)

0

The relations shown in (3.95) indicate that S, H, and L are, respectively, the average values of N1 (θ ), N2 (θ ), and −N3 (θ ) over the interval 0 ≤ θ ≤ π . Since Ni (θ ), i = 1, 2, 3, is periodic in θ with periodicity π , which can be seen from the relations given in (3.76c) and (3.75), the range of integration (0, π ) can be replaced by (α, α + π ) where α can be any real constant. Equations (3.95) were first obtained by Barnett and Lothe (1973). The matrices S, H, and L which are tensors of rank two (Ting 1996) are thus called the Barnett–Lothe tensors. Equations (3.95) allow us to compute S, H, and L directly from the elastic constants Cijkl without computing the material eigenvalues and their associated eigenvectors. The problems associated with degenerate materials are therefore circumvented. Useful Relations From the standard form of the generalized eigenrelation (3.76), it is expected that the relations of the fundamental elasticity matrix may bear the same form as the

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Stroh Formalism

relations of the material eigenvalues. Thus, through the relations obtained in (3.89)2 , (3.90), (3.91), and (3.92) for the material eigenvalues, we may expect the following relations for the fundamental elasticity matrix: ˜ 2 (π ) = −I N

N (θ ) = −I − N2 (θ ),

(3.96a)

and ˆ , θ0 ), ˆ , α)N(α, ˆ N(θ θ0 ) = N(θ ˆ )N ˆ −1 (θ0 ) = N(θ ˆ , θ0 ), N(θ ˆ , α)N(α, ˆ N(θ θ ) = I, (cos θ I + sin θ N)(cos θ I − sin θ N(θ )) = I, π N θ+ N(θ ) = −I. 2

(3.96b)

Carrying out the matrix multiplication in (3.96a), we have − N1 (θ ) = I + N21 (θ ) + N2 (θ )N3 (θ ), − N2 (θ ) = N1 (θ )N2 (θ ) + N2 (θ )NT1 (θ ),

(3.97a)

− N3 (θ ) = N3 (θ )N1 (θ ) + NT1 (θ )N3 (θ ) and HL − SS = I = LH − ST ST , SH + HST = 0 = LS + ST L.

(3.97b)

Other identities that are related to the structures of the fundamental elasticity matrices N1 (θ ) and N3 (θ ) are N1 (θ )n(θ ) = −s(θ ),

N3 (θ )n(θ ) = 0,

(3.98a)

where sT = (cos θ , sin θ , 0),

nT = (− sin θ , cos θ , 0).

(3.98b)

Equation (3.98) can be proved as follows (Ting, 1996): N1 (θ )n(θ ) = −T−1 (θ )RT (θ )n(θ ) = −T−1 (θ )T(θ ) s(θ ) = −s(θ ), N3 (θ )n(θ ) = R(θ )T−1 (θ )RT (θ )n(θ ) − Q(θ )n(θ )

(3.99)

= R(θ )T−1 (θ )T(θ )s(θ ) − R(θ )s(θ ) = 0. Note that the first equality of the above two equations comes from the definitions of N1 (θ ) and N3 (θ ) given in (3.76c) whereas the second equality comes from the definitions Q(θ ), R(θ ), T(θ ) given in (3.74a), which shows that

3.4

Some Identities

75

(Cijkl sl nj )nk = (Cijlk nj nk )sl ,

(Cijkl sj sl )nk = (Cijlk sj nk )sl

(3.100a)

or RT (θ )n(θ ) = T(θ )s(θ ),

Q(θ )n(θ ) = R(θ )s(θ ).

(3.100b)

Based upon the identities (3.98a), one can further prove that GT1 (θ )n(θ ) = −s(θ ),

GT3 (θ )n(θ ) = 0,

(3.101a)

where G1 (θ ) = NT1 (θ ) − N3 (θ )SL−1 ,

G3 (θ ) = −N3 (θ )L−1 .

(3.101b)

In addition to the identities (3.96), (3.97), (3.98), (3.99), (3.100), and (3.101), several other useful identities which have been proved in Hwu and Ting (1990) and Ting (1996) are shown below. Through the rotation matrix defined in (3.62b), we know that

cos(θ − α) sin(θ − α) − sin(θ − α) cos(θ − α)

s(α) s(θ ) = n(α) n(θ )

(3.102a)

or by inversion cos(θ − α) − sin(θ − α) s(θ ) s(α) = . sin(θ − α) cos(θ − α) n(θ ) n(α)

(3.102b)

Multiplying (3.102a) and (3.102b) by N1 (θ ) or N3 (θ ) or N1 (α) or N3 (α) and using the relations (3.98a), we have N1 (θ )s(α) cos(θ − α) − sin(θ − α) N1 (θ )s(θ ) = , −s(θ ) N1 (θ )n(α) sin(θ − α) cos(θ − α) cos(θ − α) − sin(θ − α) N3 (θ )s(θ ) N3 (θ )s(α) , = sin(θ − α) cos(θ − α) 0 N3 (θ )n(α) cos(θ − α) sin(θ − α) N1 (α)s(α) N1 (α)s(θ ) , = − sin(θ − α) cos(θ − α) −s(α) N1 (α)n(θ ) cos(θ − α) sin(θ − α) N3 (α)s(α) N3 (α)s(θ ) . = 0 N3 (α)n(θ ) − sin(θ − α) cos(θ − α)

(3.103a) (3.103b) (3.103c) (3.103d)

Carrying out the matrix multiplication in (3.102a,b) and (3.103a–d), several identities can be obtained such as

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3

Stroh Formalism

cos(θ − α)s(α) − sin(θ − α)n(α) = s(θ ), − sin(θ − α)s(α) + cos(θ − α)n(α) = n(θ ), cos(θ − α)s(θ ) − sin(θ − α)n(θ ) = s(α), sin(θ − α)s(θ ) + cos(θ − α)n(θ ) = n(α), cos(θ − α)N1 (θ )s(θ ) = N1 (θ )s(α) − sin(θ − α)s(θ ), sin(θ − α)N1 (θ )s(θ ) = N1 (θ )n(α) + cos(θ − α)s(θ ), cos(θ − α)N3 (θ )s(θ ) = N3 (θ )s(α),

(3.104)

sin(θ − α)N3 (θ )s(θ ) = N3 (θ )n(α), ˆ 1 (θ , α)s(α) = −N1 (α)n(θ ), N ˆ 3 (θ , α)s(α) = −N3 (α)n(θ ), N ˆ 1 (θ , α)n(α) = n(θ ), N ˆ 3 (θ , α)n(α) = 0, N ˆ i (θ , α), i = 1, 2, 3, are the submatrices of N(θ ˆ , α). in which N T From (3.98) and (3.101a), we know that G1 (θ ) and GT3 (θ ) share certain properties with N1 (θ ) and N3 (θ ). Hence, by (3.104)5–8 we have cos(θ − α)GT1 (θ )s(θ ) = GT1 (θ )s(α) − sin(θ − α)s(θ ), sin(θ − α)GT1 (θ )s(θ ) = GT1 (θ )n(α) + cos(θ − α)s(θ ), cos(θ − α)GT3 (θ )s(θ ) = GT3 (θ )s(α),

(3.105)

sin(θ − α)GT3 (θ )s(θ ) = GT3 (θ )n(α). Commutative Properties From the eigenrelation (3.48) and the generalized eigenrelation (3.76), we can further prove that Nn (θ )ξ = μn (θ )ξ,

(3.106)

where n is an integer, positive or negative. When θ = 0, n = 1, (3.106) will reduce to the eigenrelation (3.48a), whereas n = 1 will lead (3.106) to the generalized eigenrelation (3.76). When n is a negative integer, (3.106) poses the eigenrelation of the nth power inversion of the fundamental elasticity matrix. The general eigenrelation shown in (3.106) reveals that for any n and θ all the matrices Nn (θ ) share the same set of eigenvectors. Thus, through the diagonalization process shown in (3.61) and (3.93) we can prove that the products of Nn (θ ) and Nm (ψ) commutes (Kirchner and Lothe, 1986; Ting, 1996), i.e., Nn (θ )Nm (ψ) = Nm (ψ)Nn (θ ). Integration of (3.107) with respect to θ and/or ψ yields

(3.107)

3.4

Some Identities

77

˜ n (θ ), ˜ n (θ )Nm (ψ) = Nm (ψ)N N ˜ m (ψ) = N ˜ m (ψ)N ˜ n (θ ). ˜ n (θ )N N

(3.108)

Specialization of (3.107) and (3.108) to some particular cases such as θ = 0 and/or ψ = π and/or n = ±1 and/or m = ±1, will then lead to N(θ )N(ψ) = N(ψ)N(θ ), ˜ )N(ψ) = N(ψ)N(θ ˜ ), N(θ ˜ )N ˜ =N ˜ N(θ ˜ ), N(θ ˜ = NN, ˜ NN

(3.109a)

˜ = NN ˜ −1 , N−1 N ˜ = NN ˜ n , . . . , etc., Nn N where N = N(0),

˜ = N(π ˜ ). N

(3.109b)

Carrying out the matrix multiplication, several identities related to the submatrices ˜ N(θ ), N(θ ˜ ) can then be obtained, for example, (3.109a)3 can be expanded of N, N, as ˜ 2 (θ )L = SN ˜ 1 (θ ) + HN ˜ 3 (θ ), ˜ 1 (θ )S − N N ˜ 1 (θ )H + N ˜ 2 (θ )ST = SN ˜ 2 (θ ) + HN ˜ T (θ ), N 1

˜ 3 (θ )S − N ˜ T (θ )L = −LN ˜ 1 (θ ) + ST N ˜ 3 (θ ), N 1 ˜ 3 (θ )H + N ˜ T (θ )ST = −LN ˜ 2 (θ ) + ST N ˜ T (θ ). N 1

(3.110)

1

Moreover, (3.96b) can also be generalized to ˆ , θ0 ) = N(α, ˆ ˆ , α), ˆ , α)N(α, ˆ θ0 )N(θ N(θ θ0 ) = N(θ ˆ ˆ ˆ ˆ N(θ , α)N(α, θ ) = I = N(α, θ )N(θ , α), (cos θ I + sin θ N)(cos θ I − sin θ N(θ )) θ I- + sin θ N), , =I= - (cos θ I − sin θ N(θ ))(cos , N θ + π2 N(θ ) = −I = N(θ )N θ + π2 .

(3.111)

3.4.4 Identities Converting Complex Form to Real Form The identities discussed in the previous section are related to the fundamental elasticity matrices N, N(θ ) and the Barnett–Lothe tensors S, H, and L, which are all real matrices related to the elastic constants Cijkl . In applications, by using Stroh formalism the final solutions to the anisotropic elasticity problems usually contain the matrices

78

3

A < f (zα ) > BT ,

A < f (zα ) > AT ,

Stroh Formalism

B < f (zα ) > BT ,

B < f (zα ) > BT , (3.112) in which A, B, and < f (zα ) > are all complex matrices. Since all the physical quantities such as the displacement u and stress function φ should be real, if possible people always like to find the real-form expressions for all physical quantities. To achieve this goal, in this section we will present some identities connecting A, B, and < f (zα ) > to the real matrices such as N, N(θ ), S, H, and L. Because the complex matrices A and B are the eigenvectors of the real matrix N (or the generalized matrix N(θ )), the best starting point for finding the identities converting complex form to real form is the eigenrelation (3.48) or (3.76). From the eigenrelation (3.48a), we have (x1 I + x2 N)ξ = (x1 + μx2 )ξ = zξ.

(3.113)

In the polar coordinate system r − θ , x1 = r cos θ ,

x2 = r sin θ ,

(3.114)

Equation (3.113) can be rewritten as ˆ )ξ = zξ, rN(θ

(3.115)

in which the notation defined in (3.87a)2 has been used. Repeated multiplication or ˆ ) on both sides of (3.115) will lead to inversion of N(θ ˆ n (θ )ξ = zn ξ, rn N

(3.116)

ˆ n (θ ) is simple or semisimple, the where n is an integer, positive or negative. When N n n n relation (3.116) for z1 , z2 , z3 can be combined into one equation as ˆ n (θ ) rn N

A A < znα > . = B < znα > B

(3.117)

Postmultiplying both sides of (3.117) by the 3 × 6 matrix [BT AT ] and using (3.59) and (3.95) leads to 1 ˆn A < znα > BT A < znα > AT ˜ = rn N (θ )[I − iN]. n T n T B < zα > B B < zα > A 2

(3.118)

Integrating both sides of (3.76a) and using the notations (3.87a)3,4 and the relation (3.89)3 , we obtain ˜ )ξ = π μ(θ π N(θ ˜ )ξ = (ln z − ln r)ξ.

(3.119)

3.4

Some Identities

79

Similar to the derivation of (3.116), from (3.119) we can get ˜ )]m ξ = (ln z)m ξ, [ln rI + π N(θ

(3.120)

where m is an integer, positive or negative. Again, similar to the derivation given in (3.116), (3.117), and (3.118), (3.120) will lead to 1 A < (ln zα )m > BT A < (ln zα )m > AT ˜ )]m [I − iN]. ˜ (3.121) = [ln rI + π N(θ m T m T B < (ln zα ) > B B < (ln zα ) > A 2 The identities obtained in (3.118) and (3.121) can be combined into one equation as 1 ˆn A < znα (ln zα )m > BT A < znα (ln zα )m > AT ˜ )]m [I−iN], ˜ (θ )[ln rI+π N(θ = rn N B < znα (ln zα )m > BT B < znα (ln zα )m > AT 2 (3.122) ˜ )]m , [I − iN] ˜ commute ˆ n (θ ), [ln rI + π N(θ in which the products of the matrices N since they share the same eigenvectors. Equation (3.122) is quite general because the integers m and n could be positive, negative, or zero, and θ could be variable or any fixed value. Specialization of (3.122) to some particular cases can be made as follows. When n = 1 and m = 0,

1 ˆ A < zα > BT A < zα > AT ˜ = rN(θ )[I − iN]. T T B < zα > B B < zα > A 2

(3.123a)

Carrying out the matrix multiplication and using (3.114), (3.87), and (3.94), we obtain 2A < zα > BT = x1 (I − iS) + x2 {N1 − i(N1 S − N2 L)}, 2A < zα > AT = −ix1 H + x2 {N2 − i(N1 H + N2 ST )}, 2B < zα > BT = ix1 L + x2 {N3 − i(N3 S − NT1 L)},

(3.123b)

2B < zα > AT = x1 (I − iST ) + x2 {NT1 − i(N3 H + NT1 ST )}. When n = 1, m = 0, and θ = π/2, 1 A < μα > BT A < μα > AT ˜ = N[I − iN]. B < μα > BT B < μα > AT 2

(3.124a)

Carrying out the matrix multiplication and using (3.94), we obtain 2A < μα > BT = N1 − i(N1 S − N2 L), 2A < μα > AT = N2 − i(N1 H + N2 ST ), 2B < μα > BT = N3 − i(N3 S − NT1 L), 2B < μα > AT = NT1 − i(N3 H + NT1 ST ).

(3.124b)

80

3

Stroh Formalism

T −1 T 1 −1 ˆ −1 A < z−1 α > B A < zα > A ˜ T B < z−1 > AT = r N (θ )[I − iN]. B < z−1 > B 2 α α

(3.125a)

When n = −1 and m = 0,

From the relation (3.96b)2 and the definition (3.87), we know ˆ θ ) = cos θ I − sin θ N(θ ). ˆ −1 (θ ) = N(0, N

(3.125b)

Carrying out the matrix multiplication of (3.125a) and using (3.94) and (3.125b), we obtain T 2rA < z−1 α > B = [cos θ I − sin θ N1 (θ )](I − iS) − i sin θ N2 (θ )L, T T 2rA < z−1 α > A = −i[cos θ I − sin θ N1 (θ )]H − sin θ N2 (θ )(I − iS ), T T 2rB < z−1 α > B = i[cos θ I − sin θ N1 (θ )]L − sin θ N3 (θ )(I − iS),

(3.125c)

T T T 2rB < z−1 α > A = [cos θ I − sin θ N1 (θ )](I − iS ) + i sin θ N3 (θ )H.

When n = −1, m = 0, and θ = π/2, T −1 T 1 −1 A < μ−1 α > B A < μα > A ˜ T B < μ−1 > AT = N [I − iN]. B < μ−1 > B 2 α α

(3.126a)

From the relation (3.96b)5 , we know N−1 = −N(π/2)

(3.126b)

Carrying out the matrix multiplication of (3.126a) and using (3.94) and (3.126b), we obtain T 2A < μ−1 α > B = −N1 (π/2) + i{N1 (π/2)S − N2 (π/2)L}, −1 2A < μα > AT = −N2 (π/2) + i{N1 (π/2)H + N2 (π/2)ST }, T T 2B < μ−1 α > B = −N3 (π/2) + i{N3 (π/2)S − N1 (π/2)L}, T −1 T 2B < μα > A = −N1 (π/2) + i{N3 (π/2)H + NT1 (π/2)ST }.

(3.126c)

When m = 0 and θ = π/2, 1 A < μnα > BT A < μnα > AT ˜ = Nn [I − iN]. n T n T B < μα > B B < μα > A 2

(3.127a)

3.4

Some Identities

81

Carrying out the matrix multiplication of (3.127a) and using (3.94), we obtain (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)T

2A < μnα > BT = N1 − i(N1 S − N2 L), 2A < μnα > AT = N2 − i(N1 H + N2 ST ), 2B < μnα > BT = N3 − i(N3 S − N1 (n)T

2B < μnα > AT = N1

(n)

(3.127b)

L),

(n)T T S ),

− i(N3 H + N1

(n)

in which Ni , i = 1, 2, 3 are the submatrices of Nn defined by ' N = n

(n)

(n)

(n)

(n)T

N1 N2

( .

(3.127c)

N3 N1

When n=0 and m=1, 1 A < ln zα > BT A < ln zα > AT ˜ )][I − iN]. ˜ = [ln rI + π N(θ T T B < ln zα > B B < ln zα > A 2

(3.128a)

Carrying out the matrix multiplication of (3.128a) and using (3.87) and (3.94), we obtain 2A < ln zα 2A < ln zα 2B < ln zα 2B < ln zα

˜ 1 (θ )](I − iS) + iπ N ˜ 2 (θ )L, > BT = [ln rI + π N T ˜ ˜ > A = −i[ln rI + π N1 (θ )]H + π N2 (θ )(I − iST ), ˜ T (θ )]L + π N ˜ 3 (θ )(I − iS), > BT = i[ln rI + π N 1 T ˜ T (θ )](I − iST ) − iπ N ˜ 3 (θ )H. > A = [ln rI + π N 1

(3.128b)

In some applications we need the expressions for A < znα (ln zα )m > B−1 A < znα (ln zα )m > A−1 . B < znα (ln zα )m > B−1 B < znα (ln zα )m > A−1

(3.129)

To convert the above complex form into real form, we follow the derivation described in (3.117), (3.118), (3.119), (3.120), (3.121), and (3.122). Postmultiplying both sides of (3.117) by the 3 ×6 matrix [B−1 A−1 ] leads to

A < znα > B−1 B < znα > B−1

A < znα > A−1 B < znα > A−1

−1 1 nˆn I AB = r N (θ ) , I BA−1 2

(3.130)

in which AB−1 = (ABT )(BBT )−1 = −iL−1 − SL−1 , BA−1 = (ABT )T (AAT )−1 = iH−1 + ST H−1 .

(3.131)

From (3.131), an impedance matrix M, which is commonly used in expressing the solutions of some practical problems, is defined as

82

3

Stroh Formalism

M = −iBA−1 = H−1 − iST H−1 = H−1 + iH−1 S, M−1 = iAB−1 = L−1 − iSL−1 = L−1 + iL−1 ST .

(3.132)

Note that the identities (3.97b)2 have been used in getting the last equality of (3.132)1 and (3.132)2 . Substituting (3.131) into (3.130) and carrying out the matrix multiplication, we can obtain the identities for A < znα > B−1 , A < znα > A−1 , B < znα > B−1 , B < znα > A−1 . Same approach as that described between (3.113) and (3.122) can be applied to obtain the identities for (3.129). An alternative approach to get the identities for (3.129) can be made by employing the following relations to the results stated in (3.122), (3.123), (3.124), (3.125), (3.126), (3.127), and (3.128): B−1 = BT (BBT )−1 = −2iBT L−1 , A−1 = AT (AAT )−1 = 2iAT H−1 .

(3.133)

For example, using (3.124b)1 and (3.133)1 we have A < μα > B−1 = −2i{A < μα > BT }L−1 = (N2 − N1 SL−1 ) − iN1 L−1 . (3.134) No matter which approach we employ, the general expressions of the identity for (3.129) have been obtained as A < znα (ln zα )m > B−1 A < znα (ln zα )m > A−1 B < znα (ln zα )m > B−1 B < znα (ln zα )m > A−1 −1 0 ˜ )]m [I − iN] ˜ −iL ˆ n (θ )[ln rI + π N(θ . = rn N 0 iH−1

(3.135)

Employing the techniques stated in this section, several identities converting the complex form to real form can be obtained such as A < zα > B−1 = −x1 (S + iI)L−1 + x2 {(N2 − N1 SL−1 ) − iN1 L−1 }, A < zα > A−1 = x1 I + x2 {(N1 + N2 ST H−1 ) + iN2 H−1 }, B < zα > B−1 = x1 I + x2 {(NT1 − N3 SL−1 ) − iN3 L−1 },

(3.136)

B < zα > A−1 = x1 (ST + iI)H−1 + x2 {(N3 + NT1 ST H−1 ) + iNT1 H−1 }, A < μα > B−1 = (N2 − N1 SL−1 ) − iN1 L−1 , A < μα > A−1 = (N1 + N2 ST H−1 ) + iN2 H−1 , B < μα > B−1 = (NT1 − N3 SL−1 ) − iN3 L−1 , B < μα > A−1 = (N3 + NT1 ST H−1 ) + iNT1 H−1 ,

(3.137)

3.4

Some Identities

83

−1 rA < z−1 = −[cos θ I − sin θ N1 (θ )](S + iI)L−1 − sin θ N2 (θ ), α >B −1 rA < z−1 = [cos θ I − sin θ N1 (θ )] − sin θ N2 (θ )(ST + iI)H−1 , α >A −1 rB < z−1 = [cos θ I − sin θ NT1 (θ )] + sin θ N3 (θ )(S + iI)L−1 , α >B

(3.138)

−1 rB < z−1 = [cos θ I − sin θ NT1 (θ )] (ST + iI)H−1 − sin θ N3 (θ ), α >A −1 A < μ−1 = −{N2 (π/2) − N1 (π/2)SL−1 } + iN1 (π/2)L−1 , α >B −1 A < μ−1 = −{N1 (π/2) + N2 (π/2)ST H−1 } − iN2 (π/2)H−1 , α >A −1 B < μ−1 = −{NT1 (π/2) − N3 (π/2)SL−1 } + iN3 (π/2)L−1 , α >B

(3.139)

−1 B < μ−1 = −{N3 (π/2) + NT1 (π/2)ST H−1 } + iNT1 (π/2)H−1 , α >A

˜ 1 (θ )](S + iI)L−1 + π N ˜ 2 (θ ), A < ln zα > B−1 = −[ln rI + π N ˜ 1 (θ )] + π N ˜ 2 (θ )(ST + iI)H−1 , A < ln zα > A−1 = [ln rI + π N ˜ T (θ )] − π N ˜ 3 (θ )(S + iI)L−1 , B < ln zα > B−1 = [ln rI + π N 1 ˜ T (θ )](ST + iI)H−1 + π N ˜ 3 (θ ). B < ln zα > A−1 = [ln rI + π N

(3.140)

1

In some applications, one uses the dual coordinate systems in which the displacements ui and stress functions φi are referred to the x1 − x2 coordinate system while the complex functions f (zα ) are referred to a rotated coordinate system x1∗ − x2∗ . In these cases, we may need the identities for

∗ m T A < z∗n α (ln zα ) > B ∗n ∗ m T B < zα (ln zα ) > B

∗ m T A < z∗n α (ln zα ) > A ∗ m T . B < z∗n α (ln zα ) > A

(3.141)

To convert the complex form (3.141) into a real-form expression, we may follow the derivation shown between (3.113) and (3.122). For the x1∗ − x2∗ coordinate, the eigenrelation (3.76a) may give us (x1∗ I + x2∗ N(θ0 ))ξ = (x1∗ + μ(θ0 )x2∗ )ξ = z∗ ξ,

(3.142)

where θ0 is the angle between x1 -axis and x1∗ -axis. In the polar coordinate system (r, θ ), x1∗ = r cos(θ − θ0 ),

x2∗ = r sin(θ − θ0 ).

(3.143)

Equation (3.142) can then be rewritten as ˆ , θ0 )ξ = z∗ ξ, rN(θ

(3.144)

in which the notation defined in (3.87a)2 has been used. Repeated multiplication or ˆ , θ0 ) on both sides of (3.144) will lead to inversion of N(θ

84

3

ˆ n (θ , θ0 )ξ = z∗n ξ, rn N

Stroh Formalism

(3.145)

where n is an integer, positive or negative. With the result of (3.145), by following the steps shown between (3.117) and (3.122) we obtain

∗ m T A < z∗n (ln z∗ )m > AT A < z∗n α (ln z α ) > B α α ∗n ∗ m T ∗ m T B < z α (ln z α ) > B B < z∗n α (ln z α ) > A 1 ˆn ˜ , θ0 )]m [I − iN], ˜ = rn N (θ , θ0 )[ln rI + π N(θ 2

(3.146a)

where ˜ ) − N(θ ˜ 0) = 1 ˜ , θ0 ) = N(θ N(θ π

θ

θ0

N(ω)dω,

(3.146b)

ˆ n (θ , θ0 ), [ln rI + π N(θ ˜ , θ0 )]m , [I − iN] ˜ comin which the products of the matrices N mute since they share the same eigenvectors. Like (3.122), specialization of (3.146) to some particular cases can be made by the way as those shown between (3.123) and (3.128).

3.5 Degenerate Materials Stroh formalism presented previously is based upon the assumption that the fundamental elasticity matrix N is simple, i.e., the material eigenvalues μk are distinct. The formalism remains valid when N is semisimple, i.e., repeated eigenvalues also yield independent eigenvectors. For degenerate materials N is nonsemisimple whose associated eigenvalues are repeated and cannot yield a complete set of independent eigenvectors. Thus, the solution shown in (3.20) or (3.23) or (3.24) is not general enough and should be modified for the degenerate materials. From (3.42), (3.44), and (3.45), we see that even the material eigenvalues of the isotropic materials are triple roots and they still yield two independent eigenvectors for each repeated eigenvalue. Hence without loss of generality, in the following we only consider the degenerate case that μ1 = μ2 and ξ1 = ξ2 (or say a1 = a2 and b1 = b2 ). When μk are distinct, any linear combination of the independent solutions can be used as a substitute for the solution. Therefore, the general solution (3.23) may also be written as a2 f2 (z2 ) − a1 f1 (z1 ) + a3 f3 (z1 ) , u = 2 Re a1 f1 (z1 ) + μ2 − μ1 b2 f2 (z2 ) − b1 f1 (z1 ) + b3 f3 (z1 ) . φ = 2 Re b1 f1 (z1 ) + μ2 − μ1

(3.147)

When μ2 and ξ2 approach to μ1 and ξ1 , the second term in the parenthesis becomes the differential with respect to μ1 , i.e.,

3.5

Degenerate Materials

85

d{a1 f1 (z1 )} u = 2 Re a1 f1 (z1 ) + + a3 f3 (z1 ) , dμ1 d{b1 f1 (z1 )} + b3 f3 (z1 ) , φ = 2 Re b1 f1 (z1 ) + dμ1

(3.148a)

df1 (z1 ) d{a1 f1 (z1 )} da1 = f1 (z1 ) + a1 , dμ1 dμ1 dμ1 d{b1 f1 (z1 )} df1 (z1 ) db1 = f1 (z1 ) + b1 . dμ1 dμ1 dμ1

(3.148b)

where

Note that da1 /dμ1 and db1 /dμ1 may be calculated by differentiating (3.7) or the standard eigenrelation (3.48a) with respect to μ1 , i.e., Nξ1

=

μ1 ξ1

ξ1

− ξ1 ,

da1 /dμ1 a1 . = = b1 db1 /dμ1

(3.149)

Whereas df1 (z1 )/dμ1 can be calculated by employing the chain rule for differentiation, i.e., ∂f1 (z1 ) ∂z1 ∂f1 (z1 ) ∂f1 (z1 ) df1 (z1 ) = + = f1 (z1 )x2 + . dμ1 ∂z1 ∂μ1 ∂μ1 ∂μ1

(3.150)

With the result of (3.148), we see that the matrix form general solution (3.24) should be modified as ∗ u = 2 Re A f (z) ,

φ = 2 Re B f∗ (z) ,

(3.151a)

where A = [a1 a1 a3 ],

%

B = b1

b1

&

b3 ,

⎧ ⎫ ⎨f1 (z1 ) + df1 (z1 )/dμ1 ⎬ f1 (z1 ) f∗ (z) = . ⎩ ⎭ f3 (z3 ) (3.151b)

As discussed in Section 3.3.1, after establishing the standard eigenrelation, the orthogonality relations (3.57) and (3.58) play important roles in Stroh formalism. By similar approach the orthogonality relation for the generalized eigenvector matrices A and B of the degenerate materials has been established as (Ting and Hwu, 1988; Ting, 1996) '

YBT YAT T T YB YA

( ' ( ' (' ( YBT YAT A A A A I 0 , T T = = 0I YB YA B B B B

(3.152a)

86

3

Stroh Formalism

where ⎡

⎤ 010 Y = ⎣1 δ 0⎦ , 001

δ = μ2 − μ1 .

(3.152b)

With the orthogonality relation (3.152), it has also been proved that the Barnett– Lothe tensors S, H, and L for the degenerate materials can be calculated by (Ting and Hwu, 1988) T

H = 2iA YA ,

T

L = −2iB YB ,

T

S = i(2A YB − I).

(3.153)

It should be noted that although the general solutions shown in (3.20) or (3.23) or (3.24) are valid only for nondegenerate materials and should be modified by using (3.148) or (3.151), if the final solution for a particular problem does not contain the material eigenvector matrices A and B explicitly they may be applicable to any kind of anisotropic materials including degenerate materials. In numerical calculation the general solutions shown in (3.20) or (3.23) or (3.24) may also be used for any kind of anisotropic materials because for degenerate materials a small perturbation of the material eigenvalues μk may be made and then the corresponding eigenvector matrices A and B can be approximately obtained. Therefore, in the following chapters of this book, without loss of applicability only the general solutions (3.20) or (3.23) or (3.24) for nondegenerate materials will be employed. The complex variable formulation for the isotropic materials has been developed long time ago (Muskhelishvili, 1954). Since the modified general solution (3.148) or (3.151) is valid for the degenerate materials including the isotropic materials, by embedding the explicit expressions of the generalized eigenvector matrices A and B of the isotropic materials into (3.151) we may recover the well-known Muskhelishvili formulation. By this way many useful correspondence relations between anisotropic and isotropic elasticity can be obtained (Hwu, 1996).

Chapter 4

Infinite Space, Half-Space, and Bimaterials

As stated in the last two chapters, the study of stresses and deformations in two-dimensional anisotropic elastic bodies can be done by using Lekhnitskii formalism or Stroh formalism. Most of the works published in the earlier literature were based on Lekhnitskii formalism (1968). Due to the mathematical difficulties involved for the generalized two-dimensional problems, when employing Lekhnitskii formalism one usually considers the state of generalized plane stress or plane strain. This implies that at each point of the anisotropic plate there should be a plane of elastic symmetry which is parallel to the middle plane of a thin plate or normal to the generator of a long cylinder. Therefore, most of the solutions obtained by using Lekhnitskii formalism are valid only for the monoclinic materials, and the inplane loading will not induce the anti-plane displacement or transverse shear. In the last two decades, many researchers started to study the problems of anisotropic elasticity by using Stroh formalism in order to cover more general anisotropic materials and to provide more elegant solutions. Usually the more complicated the boundaries are, the more difficult we solve it analytically. Therefore, in order to understand how to apply Stroh formalism for the general problems of anisotropic elasticity, in this chapter we begin with the cases with simple boundaries such as the infinite space, half-space, and bimaterials. For the infinite space we will consider the loading conditions such as uniform loading and pure in-plane bending at infinity, concentrated forces and moments, and dislocation at arbitrary locations. Also several different loading conditions are considered for the half-space problems such as the concentrated forces and moments on the edge or arbitrary point inside the bodies, the arbitrary loading along the edge. For the bimaterials, we will provide the solutions for the concentrated force and dislocation at arbitrary locations.

4.1 Infinite Space 4.1.1 Uniform Loading For an infinite homogeneous plate subjected to an equilibrated external loading at infinity, the stresses and strains must be distributed uniformly in the entire C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_4, C Springer Science+Business Media, LLC 2010

87

88

4


plate. Through the basic equations of elasticity such as the equilibrium equations, stress–strain relations, and strain–displacement relations, with the known stresses at infinity, the associated displacement and stress function vectors used in Stroh formalism can be determined by the following procedures. (1) The uniformly distributed stress field σij is obtained under the requirements that the infinity loading conditions σij∞ and the equilibrium equations should be satisfied. Moreover, for two-dimensional problems σij∞ have to be prescribed in such a way that ε33 = 0 or σ33 = 0. (2) By σi1 = −φi, 2 and σi2 = φi, 1 , integrations of the stresses with respect to x1 and x2 will lead to the stress function φ. (3) The associated strain field εij is calculated by using the constitutive law (1.56) or (1.58), i.e., ⎧ ˆ ⎪ ⎨ Sij σj , i = 3, plane strain, j=3 εi = ⎪ ⎩ Sij σj , i = 3, generalized plane stress,

(4.1a)

j=3

where Sij , Sˆ ij are, respectively, the compliances and reduced compliances and εi and σi ranging from 1 to 6 are the contracted notations of εij and σij . The correctness of εij and σij should be checked by the compatibility equations: ∂ε23 ∂ε31 = , ∂x1 ∂x2 ∂ 2 ε12 ∂ 2 ε11 ∂ 2 ε22 2 = + . ∂x1 ∂x2 ∂x22 ∂x12

(4.1b)

If the compatibility equations are not satisfied, one should return to step (1) and check the correctness of the given equilibrated stresses σij∞ . (4) The displacement u is then obtained by integration of the following reduced strain–displacement equations for two-dimensional problems, ∂u1 ∂u2 ∂u1 ∂u2 , ε22 = , 2ε12 = + , ∂x1 ∂x2 ∂x2 ∂x1 ∂u3 ∂u3 = , 2ε13 = . ∂x2 ∂x1

ε11 = 2ε23

(4.1c)

That is, u1 =

ε11 dx1 , u2 =

ε22 dx2 , u3 =

2ε23 dx2 ,

(4.1d)

4.1

Infinite Space

89

in which the integration constants can be determined by 2ε12 =

∂u1 ∂u2 ∂u3 + , 2ε13 = , ∂x2 ∂x1 ∂x1

(4.1e)

and the neglect of rigid body translation and rotation. By the procedures described above, we now list the solutions of u and φ for the two-dimensional anisotropic plate subjected to uniform loading at infinity: ∞ ∞ ∞ u = x1 ε∞ 1 + x2 ε2 , φ = x1 t2 − x2 t1 ,

(4.2a)

where

t∞ 1

⎧ ∞⎫ ⎧ ∞⎫ ⎧ ∞⎫ ⎧ ∞⎫ ⎪ ⎪ ⎪ ⎪ ⎨σ11 ⎪ ⎨σ21 ⎪ ⎨ ε11 ⎪ ⎨ ε21 ⎪ ⎬ ⎬ ⎬ ⎬ ∞ ∞ ∞ ∞ ∞ ∞ ∞ = σ12 , t2 = σ22 , ε1 = ε12 , ε2 = ε22 . ⎪ ⎪ ∞⎪ ⎪ ∞⎪ ⎪ ∞⎪ ⎩ ∞⎪ ⎩ ⎩ ⎩ ⎭ ⎭ ⎭ ⎭ σ13 σ23 2ε13 2ε23

(4.2b)

The displacement u shown in (4.2) is unique up to a rigid body translation and rotation. The uniform stress solution shown in (4.2) can also be represented by general solution (3.24) of Stroh formalism, in which the complex function vector f(z) is selected to be f(z) = < zα > q where q is a complex coefficient vector which may be replaced by two real constant vectors as q = AT g + BT h. That is, u = 2 Re{A < zα > (AT g + BT h)},

(4.3)

φ = 2 Re{B < zα > (AT g + BT h)}. By using the identity (3.123), (4.3) can be converted into a real form as u = x1 h + x2 (N1 h + N2 g),

(4.4)

φ = x1 g + x2 (N3 h + NT1 g). Comparison of (4.4) with (4.2a) yields h g

=

ε∞ 1

t∞ 2

,

ε∞ 2

−t∞ 1

=N

ε∞ 1

t∞ 2

.

(4.5)

In (4.2b), σij∞ are given and εij∞ can be determined by using the stress–strain relation (4.1a). The simplified solutions for some special loading conditions are listed below for plane strain condition. Replacing the reduced compliances by the compliances, the results are also valid for the case of generalized plane stress.

90

4


σ∞

Fig. 4.1 An infinite plate subjected to unidirectional tension and bending

x

∗ 2

M

α

x2

x1∗

α

x1

σ∞ M

Unidirectional Tension ∞ = σ ∞ cos2 α, σ ∞ = σ ∞ sin2 α, σ ∞ = σ ∞ cos α sin α) (Fig. 4.1) (σ11 22 12 ∞ ∞ ∞ t∞ 1 = σ cos αs(α), t2 = σ sin αs(α), ∞ 2 2 ε∞ 1 = σ {cos αS1 + sin αS2 + cos α sin αS6 },

ε∞ 2

∞

∗

= σ {cos αS1 + sin 2

2

αS∗2

+ cos α sin

(4.6a)

αS∗6 },

where ⎧ ⎫ ⎧ ⎧ ⎫ ⎫ ˆ ˆ ⎪ ⎪ ⎪ ⎨cos α ⎪ ⎬ ⎨ S1i ⎪ ⎨S6i /2⎪ ⎬ ⎬ s(α) = sin α , Si = Sˆ 6i /2 , S∗i = Sˆ 2i , i = 1, 2, 4, 5, 6. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎩ ⎭ ⎭ 0 Sˆ 4i Sˆ 5i

(4.6b)

∞ = σ ∞, σ ∞ = σ ∞) Biaxial Loading (σ11 1 22 2 ∞ ∞ t∞ t∞ 1 = σ1 i1 , 2 = σ2 i2 , ∞ ∞ ∞ ∞ ∗ ∞ ∗ ε∞ 1 = σ1 S1 + σ2 S2 , ε2 = σ1 S1 + σ2 S2 ,

(4.7a)

where ⎧ ⎫ ⎧ ⎫ ⎨1 ⎬ ⎨0⎬ i1 = 0 , i 2 = 1 . ⎩ ⎭ ⎩ ⎭ 0 0

(4.7b)

∞ = τ ∞) Pure Shear (σ12 ∞ ∞ ∞ t∞ 1 = τ i2 , t 2 = τ i1 , ∞ ∞ ∞ ∗ ε∞ 1 = τ S6 , ε2 = τ S6 .

(4.8)

4.1

Infinite Space

91

∞ = τ ∞ or σ ∞ = τ ∞ ) Anti-plane Shear (σ13 23 ∞ ∞ t∞ 1 = τ i3 , t2 = 0, ∞ ∞ ∞ ∗ ε∞ 1 = τ S5 , ε2 = τ S5

(4.9a)

or ∞ t∞ t∞ 1 = 0, 2 = τ i3 , ∞ ∞ ∞ ∗ ε∞ 1 = τ S4 , ε2 = τ S4 ,

(4.9b)

where ⎧ ⎫ ⎨0 ⎬ i3 = 0 . ⎩ ⎭ 1

(4.9c)

4.1.2 Pure In-Plane Bending Consider an infinite plate subjected to pure bending M in a direction at an angle α with the positive x1 -axis (Fig. 4.1). Since the bending moment M is equivalent to ∗ = Mx∗ /I where I is the moment of inertia of the the linear distributed stresses σ11 2 plate cross section, the stresses in the plate may be assumed to be σ11 =

Mx2∗ Mx2∗ Mx2∗ cos2 α, σ22 = sin2 α, σ12 = cos α sin α, I I I

(4.10a)

x2∗ = −x1 sin α + x2 cos α.

(4.10b)

where

By the procedure stated in Section 4.1.1, the displacement u and stress function φ can be obtained as u = −M 2I (sin αu1 + cos αu2 ) for γ = 0, (4.11a) 2 φ = −M 2I (x1 sin α − x2 cos α) s(α) where ⎧ ⎧ ⎫ ⎫ 2 2 ⎪ ⎪ ⎨ γ1 x1 − γ2 x2 ⎪ ⎨−(2γ1 x1 + γ6 x2 )x2 ⎪ ⎬ ⎬ u1 = (γ6 x1 + 2γ2 x2 )x1 , u2 = γ1 x12 − γ2 x22 ⎪ ⎪ ⎪ ⎩(γ x + 2γ x )x ⎪ ⎩ ⎭ ⎭ −γ4 x22 5 1 4 2 1

(4.11b)

and γi = Sˆ 1i cos2 α + Sˆ 2i sin2 α + Sˆ 6i cos α sin α, i = 1, 2, 4, 5, 6 γ = γ4 sin α + γ5 cos α.

(4.11c)

92

4


Note that the condition γ = 0 will always be satisfied for monoclinic materials. By a similar approach, one can obtain the solutions for γ = 0, in which the additional term such as (ax12 + bx1 x2 + cx22 )i3 should be considered in φ.

4.1.3 Concentrated Forces Consider an infinite homogeneous anisotropic elastic medium loaded by a concentrated force (or called point force, sometimes called line force for two-dimensional problems) pˆ = (ˆp1 , pˆ 2 , pˆ 3 ) applied at an internal point xˆ = (ˆx1 , xˆ 2 ) far from the boundary (Fig. 4.2). The elasticity solution of this problem is known to be Green’s function which can be used as a fundamental solution of the boundary element method. The boundary conditions of this problem can be written as . dφ = pˆ for any close curve C enclosing the point xˆ , .C

(4.12)

du = 0 for any closed curve C, C

σij → 0

at infinity.

The first equation of (4.12) comes from the force equilibrium around any closed curve enclosing the point xˆ and the relation for the stress resultants shown in (3.33). The second equation of (4.12) states the requirement of single-valued displacement, which is necessary when a multi-valued function is used and no discontinuous displacement is allowed in the whole field. The last one states that the stresses should approach zero when going away from the point of application of the force, i.e., the stresses should vanish at infinity. A solution satisfying these boundary conditions

x2

pˆ = ( pˆ 1 , pˆ 2 , pˆ 3 )

xˆ = ( x1 , x 2 )

x1 Fig. 4.2 An infinite plate subjected to a concentrated force

4.1

Infinite Space

93

has been found by many researchers, e.g., Barnett and Lothe (1975a), Chadwick and Smith (1977), Kirchner and Lothe (1987), and Ting (1996). The difference between these solutions is the form of their presentation. Some of them are presented in complex form, while the others are presented in real form. Some of them are written explicitly for the displacement gradient and the others are for the stresses. Some are organized in a simple form; some are complicated and are not easy to use. Because the detailed derivation and discussion of these solutions can be found in many papers and books, no further comments will be given. Following are the derivations similar to those presented in Ting (1996). To find a solution satisfying the conditions (4.12), the choice of fk (zk ) in (3.24) is very critical in the solution procedures. Equations (4.12) show that the stress functions φ1 , φ2 , and φ3 should be multi-valued functions if the concentrated force pˆ is nonzero. However, the physical quantities like the displacements should always be single-valued to confirm that the anisotropic medium will not break off when deformed. From the general solution shown in (3.24) we see that to get multi-valued functions for φi , we need to choose a multi-valued function for fk (zk ). It is known that the logarithmic function ln(zk − zˆk ) is holomorphic in the cut region and in an anticlockwise contour its value increases by 2π i. Hence, it is proper for us to choose f(z) as f(z) =< ln(zα − zˆα ) > q,

(4.13)

where q is a 3 × 1 unknown complex coefficient vector to be determined through the satisfaction of boundary conditions. With (4.13), the general solution (3.24) can then be written as , , u = 2 Re A < ln zα − zˆα > q , φ = 2 Re B < ln zα − zˆα > q .

(4.14)

Since . df(z) = 2π iq,

(4.15)

C

substitution of (4.14) into (4.12) we can get " ! 2π i Aq − Aq = 0, ! " ˆ 2π i Bq − Bq = p,

(4.16a)

which can also be re-organized as 0 AA q = . ˆ p/2π i −q BB

(4.16b)

94

4


The unknown coefficient vector q can then be obtained by using the orthogonality relation (3.57a), which will lead to ˆ i. q = AT p/2π

(4.17)

With the result of (4.17), the explicit closed-form solution for the infinite space loaded by a concentrated force pˆ can now be written as ˆ , u = Im{A < ln(zα − zˆα ) > AT }p/π ˆ . φ = Im{B < ln(zα − zˆα ) > AT }p/π

(4.18)

By using the identity given in (3.128), the complex form solution (4.18) can be converted into a real form as 1 ˜ 1 (θ )H + N ˜ 2 (θ )ST ]}p, ˆ {(ln r)H + π [N 2π 1 ˜ 3 (θ )H + N ˜ T (θ )ST ]}p. ˆ φ = − {(ln r)ST + π [N 1 2π u=−

(4.19)

Since the expressions (4.19) are written in terms of the polar coordinate system (r, θ ) with the origin located on (ˆx1 , xˆ 2 ), it may become cumbersome when calculating the Cartesian stresses and strains by (3.13) and (3.1)3 . Instead we can employ the relation given in (3.37) for the polar coordinate system. With that relation and the stress function obtained in (4.19), we get 1 T 1 T ˆ σrr = ˆ n (θ )ST p, s (θ ){N3 (θ )H + NT1 (θ )ST }p, 2π r 2π r 1 T 1 T T ˆ σθ3 = − ˆ s (θ )ST p, i S p, σrθ = − 2π r 2π r 3 1 T ˆ i {N3 (θ )H + NT1 (θ )ST }p. σr3 = 2π r 3 σθθ = −

(4.20)

From the solutions obtained in (4.20), we see that the stresses are inversely proportional to r and the stress components σθθ , σrθ , σθ3 of the surface traction tθ possess simpler forms than the stress components σrr , σr3 of the surface traction tr . Actually, tθ is independent of θ . The conclusion that the surface traction on any radial plane is invariant with θ has been proved (Ting, 1996) to be valid for any kind of materials: isotropic or anisotropic, homogeneous or nonhomogeneous, elastic or inelastic, and linear or nonlinear. In an isotropic medium, the stress distribution (4.20) can be further simplified by using the explicit expressions given in (3.82) and (3.86) for plane strain condition. The simplified results are

4.1

Infinite Space

95

1 − 2ν 2ν − 3 (ˆp1 cos θ + pˆ 2 sin θ ), σrr = (ˆp1 cos θ + pˆ 2 sin θ ) 4π (1 − ν)r 4π (1 − ν)r 1 − 2ν −1 σrθ = (ˆp1 sin θ − pˆ 2 cos θ ), σθ3 = 0, σr3 = pˆ 3 . 4π (1 − ν)r 2π r (4.21)

σθθ =

4.1.4 Couple Moments A couple is defined as two parallel forces that have the same magnitude, have opposite directions, and are separated by a perpendicular distance. Since the resultant force is zero, the only effect of a couple to a rigid body is to produce a rotation or tendency of rotation in a specified direction. However, as will be shown in this section that for an elastic body the effect of a couple may be influenced by the force direction and the separation distance even their resultant couple moment is identical. A concentrated moment can be produced by two perpendicular couples with approaching-zero distance as shown in Fig. 4.3 (Ting, 1996). Because a couple contains two concentrated forces, the stress fields induced by the couple moments may then be obtained by applying the superposition method to the solutions for the concentrated forces. Detailed discussions for several different kinds of couples can be found in Ting (1996). Consider two parallel and opposite concentrated forces pˆ and – pˆ applied at ˆ produced by this couple two arbitrary points A and B (Fig. 4.4). The moment m ˆ = ε × p, ˆ where ε is the vector directing from point A to B. For the is therefore m convenience of the following derivation, the center point of line AB is selected to be the origin of the coordinate system x1 − x2 . Moreover, a new coordinate system x1∗ − x2∗ is constructed by rotating the x1 − x2 coordinate angle θ0 to let x1∗ -axis

x2

pˆ pˆ

θ0 θ0

Fig. 4.3 Two perpendicular couples

x1

pˆ pˆ

96

4


x2

Fig. 4.4 Two parallel and opposite forces

x

∗ 2

pˆ

θ0

( xˆ1 , xˆ2 ) A

B

θ1

x1∗

θ0

x1

o

ε 2 ε 2

− pˆ

coincide with the direction of line AB. On these two coordinate systems, the general solutions may be expressed in the form like (3.71) where u, φ, a, b are quantities written in the x1 − x2 coordinate and z∗ is a variable in the x1∗ − x2∗ coordinate. With this dual coordinate system, the solutions for the concentrated forces pˆ and −pˆ acting at points B and A can be obtained from (4.18) as force pˆ acting at point B : ˆ , φ = Im{B < ln(z∗α − ε/2) > AT }p/π ˆ ; u = Im{A < ln(z∗α − ε/2) > AT } p/π force − pˆ acting at point A : ˆ , φ = −Im{B < ln(z∗α + ε/2) > AT }p/π ˆ . u = −Im{A < ln(z∗α + ε/2) > AT }p/π (4.22) The solution for the couple can then be obtained by superposing the two solutions shown in (4.22). Because the concentrated moment can be produced by two perpendicular couples with approaching-zero distance, we now like to find a solution for the couple of which ε → 0. In order to induce the couple moments for ε → 0, the magnitude of the applied concentrated forces pˆ and −pˆ should become infinite and the product εpˆ = p˘ remains bounded. With this understanding, the solution for the couple of which ε → 0 can be written as

1 Im A < ε→0 π 1 Im B < φ = lim ε→0 π u = lim

1 ∗ ln zα − ε 1 ∗ ln zα − ε

ε − ln z∗α + 2 ε − ln z∗α + 2

ε > AT p˘ , 2 ε > AT p˘ . 2

(4.23)

The limit of (4.23) can be evaluated by using L’Hospital’s rule which gives ˘ , φ = Im{B < z∗−1 ˘ . u = Im{A < z∗−1 > AT }p/π > AT }p/π α α

(4.24)

4.1

Infinite Space

97

To convert the complex form solution into real form, we may employ the identity given in (3.125). However, it should be noted that z−1 k of (3.125) is a function in ∗−1 x1 − x2 coordinate, whereas zk of (4.24) is a function in x1∗ − x2∗ coordinate. To get a useful identity for (4.24), we specialize the result of (3.146) to n = −1 and m = 0 and have '

> BT A < z∗−1 > AT A < z∗−1 α α B < z∗−1 > BT B < z∗−1 > AT α α

( =

1 ˆ ˜ N(θ0 , θ )[I − i N] 2r

(4.25)

Employing the identity (4.25), the complex form solution (4.24) can now be converted into the following real form: 1 ˆ u ˜ 0 = N(θ0 , θ ) N φ p˘ 2π r

(4.26)

Note that p˘ is not the moment applied on the elastic body, it is just a product of pˆ and ε. The equivalent moment induced by the couple can be determined from ˆ = ε × p. ˆ If we choose ε = ε(cos θ0 , sin θ0 ) and pˆ = pˆ (cos θ1 , sin θ1 ), then m ˆ =m ˆ = εpˆ sin(θ1 − θ0 ). From the solution (4.26), it can be shown that m ˆ i3 , where m ˆ the deformations even ε and pˆ are chosen such that they yield the same moment m; and stresses computed from (4.26) will still be different. This justifies that for an elastic body the effect of a couple may be influenced by the force direction and the separation distance even if their resultant couple moment is identical. Concentrated Moments ˆ = m Since a concentrated moment m ˆ i3 can be produced by two perpendicular couples with approaching-zero distance, we consider (see Fig. 4.3) ε = εs(θ0 ), pˆ = pˆ n(θ0 ), for the first couple, ε = εs(θ0 + π/2) = εn(θ0 ), pˆ = pˆ n(θ0 + π/2) = −ˆps(θ0 ), for the second couple, (4.27a) where ⎧ ⎧ ⎫ ⎫ ⎨cos θ0 ⎬ ⎨− sin θ0 ⎬ s(θ0 ) = sin θ0 , n(θ0 ) = cos θ0 . ⎩ ⎩ ⎭ ⎭ 0 0

(4.27b)

The total moment induced by these two couples is m ˆ = 2εpˆ and m ˆ n(θ0 ), for the first couple, 2 m ˆ p˘ = − s(θ0 ), for the second couple. 2 p˘ =

(4.28)

98

4


The displacement and stress function vectors for the infinite space subjected to a concentrated moment mi ˆ 3 can therefore be found by superposing two solutions obtained from (4.26) for these two couples. The result is π ˜ m ˆ ˆ m ˆ ˆ 0 u 0 ˜ − . N(θ0 , θ )N N θ0 + , θ N = n(θ0 ) φ s(θ0 ) 4π r 4π r 2

(4.29)

The solution (4.29) can be further simplified by expressing the 6 × 6 matrices in terms of their submatrices shown in the definitions (3.87)2 ,(3.76)1 ,(3.94), and (3.109b)2 , then performing the matrix multiplication and using the identities (3.103) and (3.109). The final simplified result is m ˆ Hn(θ ) − [N1 (θ )H + N2 (θ )ST ]s(θ ) u . = φ 4π r ST n(θ ) − [NT1 (θ )ST + N3 (θ )H]s(θ )

(4.30)

Note that unlike (4.26), the solution (4.30) is independent of θ0 . To calculate the stresses, we use the relations shown in (3.37), which give us −m ˆ {nT (θ )ST n(θ ) − nT (θ )[NT1 (θ )ST + N3 (θ )H]s(θ )}, 4π r2 −m ˆ T = s T φ, r = {s (θ )ST n(θ ) − sT (θ )[NT1 (θ )ST + N3 (θ )H]s(θ )}, 4π r2

σθθ = nT φ,r = σrθ

(4.31)

and the others. By employing the identities (3.103a,b), we know nT (θ )[NT1 (θ )ST + N3 (θ )H] = −sT (θ )ST .

(4.32)

Use of the relation (4.32) will lead (4.31) to −m ˆ {nT (θ )ST n(θ ) + sT (θ )ST s(θ )}, 4π r2 m ˆ = {nT (θ )W(θ )n(θ ) + sT (θ )W(θ )s(θ )}, 4π r2

σθθ = σrθ

(4.33a)

where W(θ ) = SN1 (θ ) + HN3 (θ ).

(4.33b)

By performing the matrix multiplication and employing the relation that trS = S11 + S22 + S33 = 0 (Ting, 1996), it can easily be proved that σθθ =

−m ˆ mS ˆ 33 m ˆ {S11 + S22 } = , σrθ = {W11 (θ ) + W22 (θ )}. 4π r2 4π r2 4π r2

(4.34)

Note that in this book same notation S has been used for the Barnett–Lothe tensor and the compliance matrix. To avoid confusion, Roman font is used to denote Barnett–Lothe tensor and italic font denotes compliance matrix.

4.1

Infinite Space

99

In addition to σθθ and σrθ , the other stress components can also be obtained by substituting (4.30) into (3.37), or by the following relations for the stress field to be proportional to r−2 (Ting, 1996) σrr = σrθ, θ − σθθ , σr3 = σθ3, θ , σθθ, θ = 0.

(4.35)

The above relations are obtained from the equilibrium equation alone, and hence are applicable to any kinds of materials. The last equation of (4.35) shows that σθθ is independent of θ , which agrees with the solution obtained in (4.34). By using the explicit expressions given in (3.82) and (3.86), a very simple stress distribution is obtained for an isotropic medium subjected to a concentrated moment: σrr = σθθ = σr3 = σθ3 = 0, σrθ =

−m ˆ . 2π r2

(4.36)

4.1.5 Dislocations The concentrated force will cause force discontinuity at the applied point like the first integral shown in (4.12). Likewise, the existence of dislocation means displacement discontinuity occurred at the internal point of a continuum. In the theory of linear elasticity, if the response of a body to a concentrated force is known, the deformation caused by any distribution of forces can be obtained by integration. Again likewise, if the effect of a dislocation to the body is known, the effects caused by any kind of cracks can be obtained by integration. In other words, a crack can be represented by a distribution of dislocations. Therefore, just like the Green’s function obtained from the problems of concentrated forces the solution of dislocation problem is frequently used as a kernel function of an integral equation to consider the crack problems. Consider a dislocation with Burgers vector bˆ = (bˆ 1 , bˆ 2 , bˆ 3 ) located on an internal point xˆ = (ˆx1 , xˆ 2 ) far from the boundary of an infinite space, the boundary conditions of this problem can be written as . dφ = 0, for any close curve C, C . (4.37) ˆ for any closed curve C enclosing the point xˆ , du = b, C

σij → 0,

at infinity.

From the general solutions shown in (3.24), we see that the mathematical forms of the displacement vector u and the stress function vector φ for the elasticity problems are different only by the eigenvector matrices A and B. If the mathematical forms of the boundary conditions for two problems are different by the shifting between u and φ, the solutions to these two problems may then be obtained by simple analogy. With

100

4


this understanding, by comparing the boundary conditions given in (4.12) and (4.37) for the point forces and dislocations, we see that the solutions to the dislocation problems can be obtained in a straightforward manner from the known solution to ˆ , i.e., ˆ replaced by BT b/π the point forces shown in (4.18) with AT p/π ˆ , u = Im{A < ln(zα − zˆα ) > BT }b/π ˆ . φ = Im{B < ln(zα − zˆα ) > BT }b/π

(4.38)

By a similar approach stated in (4.19) and (4.20), the real-form solutions for the displacement vector, stress function vector, and stresses can then be found as 1 ˜ 2 (θ )L]}b, ˆ ˜ 1 (θ )S − N {(ln r)S + π [N 2π 1 ˜ 3 (θ )S]}b, ˆ ˜ T (θ )L − N φ= {(ln r)L + π [N 1 2π u=−

1 T ˆ σrr = n (θ )Lb, 2π r 1 T ˆ σθ3 = = s (θ )Lb, 2π r

σθθ = σrθ

(4.39a)

1 T ˆ s (θ ){N3 (θ )S − NT1 (θ )L}b, 2π r 1 T ˆ 1 T ˆ i Lb, σr3 = i {N3 (θ )S − NT1 (θ )L}b. 2π r 3 2π r 3 (4.39b)

4.2 Half-Space 4.2.1 Green’s Function Consider a semi-infinite anisotropic half-space subjected to a concentrated force pˆ = (ˆp1 , pˆ 2 , pˆ 3 ) applied at point xˆ = (ˆx1 , xˆ 2 ), as shown in Fig. 4.5. The elasticity solution of this problem is known to be the Green’s function for half-space problems. If the surface of the half-space is assumed to be traction free, the boundary conditions of this problem can be written as t = φ = 0, along the surface x2 = 0, ˆ for anyclosed curve C enclosing the point xˆ , dφ = p, C du = 0, for any closed curve C,

(4.40)

C

σij → 0,

at infinity,

where t is the surface traction along the edge of the half-space. The equality between t and φ comes from (3.32), where the prime denotes the differentiation with respect to zk . The last three equations describing the force equilibrium around the action point, the single-valued displacement requirement, and the vanishing stresses at

4.2

Half-Space

101

x2

Fig. 4.5 A semi-infinite half-space subjected to concentrated force

x1 pˆ

( xˆ1 , xˆ2 )

infinity are the same as (4.12). To find a complex function vector f(z) satisfying the boundary conditions (4.40), several different methods have been used in the literature. Most of them utilize the Green’s function for the infinite space and then try to satisfy the surface traction-free condition through the superposition method. Although simple in concept, some of the solutions cannot be expressed in explicit form. The most recent explicit form solution was shown in Ting (1996) by a direct approach superimposing the Green’s functions of applied force and all image forces of the infinite space. Here we like to present the solution by using the method of analytical continuation (Muskhelishvili, 1954). When employing the analytical continuation method, the unknown complex function vector f(z) of the general solutions (3.24a) is usually written by f(z) = fu (z) + fp (z),

(4.41)

where fu is the function associated with the unperturbed elastic field which is analytic in the entire domain except some singular points such as the location of point forces and the points at zero or infinity; fp is the analytic function corresponding to the perturbed field of the problem and will be determined through satisfaction of the boundary conditions. By this way, the general solutions for the present problem can be written as u = 2 Re A[fu (z) + fp (z)] , φ = 2 Re B[fu (z) + fp (z)] ,

(4.42)

where fu (z) is selected to be the Green’s function for the infinite space, by (4.13) and (4.17) we choose ˆ i. fu (z) =< ln(zα − zˆα ) > AT p/2π

(4.43)

Since fu (z) has satisfied the last three boundary conditions shown in (4.40), to determine fp (z) we consider the first boundary condition of (4.40) which requires that the half-space surface be traction free. Substituting (4.42) into (4.40)1 , we get

102

4


B[f u (x1 ) + f p (x1 )] + B[f u (x1 ) + f p (x1 )] = 0.

(4.44)

From the choice of fu (z) given in (4.43), we know that f u (z) is analytic in the entire z-plane except the point zˆα which is located inside the half-space. Thus, we may say that f u (z) is analytic outside the half-space (S+ ). By a property used in the analytical continuation method, we know that f u (¯z) will be analytic inside the half-space (S− ). Because fp (z) is required to be analytic in the physical body, f p (z) should be analytic in S− and then f p (¯z) is analytic in S+ . With this background, (4.44) may be rewritten as θ(x1+ ) = θ (x1− ),

(4.45a)

where θ(z) =

−Bf p (z) − Bf u (¯z), z ∈ S− , Bf u (z) + Bf p (¯z), z ∈ S+ .

(4.45b)

Through the analytical conditions discussed previously, we conclude that this newly defined function θ(z) will be analytic in S+ and S− and is continuous across the half-space surface. This means that θ(z) is analytic in the entire z-plane including the points at infinity. By Liouville’s theorem we have θ(z) ≡ constant. Since θ(z) is related to f which is then related to the stresses, with the conditions that the stresses will vanish at infinite we may let this constant to be zero. Therefore, θ(z) ≡ 0.

(4.46)

With this result, (4.45b)1 now leads to f p (z) = −B−1 B fu (¯z).

(4.47)

Substituting (4.43) into (4.47) and integrating the results with respect to z, we have T ˆ i. fp (z) = B−1 B < ln(z − z¯ˆα ) > A p/2π

(4.48)

Note that during integration for (4.48) the constant term is neglected because the constant function f corresponds to rigid body motion which will not induce stresses. Moreover, the subscript α of the variable zα is dropped and the exact explicit solution of fp (z) can then be obtained by using the following translating technique. Translating Technique When we employ the method of analytical continuation, it is quite usual that a new analytical function will be introduced based upon the relations of the boundary conditions. For example, the new analytical function θ(z) given in (4.45b) is introduced

4.2

Half-Space

103

according to the relation (4.44) which comes from the traction-free boundary condition. If the function arguments zα , α = 1, 2, 3, have the same value on the boundary (e.g., z1 = z2 = z3 = x1 on x2 = 0), the arguments of the new analytical function can be any one of zk because their introduction is based upon the boundary conditions. Therefore, when we introduce the new analytical function it is better to represent its associated solutions by using the function vector f(z) defined as f(z) = (f1 (z), f2 (z), f3 (z))T ,

(4.49)

where the argument has the form z = x1 + μx2 without indicating the subscript of μ, such as (4.48). With this understanding, we know that the function vector f(z) obtained through the method of analytical continuation has the form of (4.49) which is not consistent with the solution form shown in (3.24b) and is valid only on the boundary. To get the explicit full domain solution, a mathematical operation based upon the following statement is needed: “Once the solution of f(z) is obtained from the condition of analytical continuation, a replacement of z1 , z2 or z3 should be made for each component function according to solution form required in the general solution (3.24b).” A technique translating the above mathematical operation can therefore be described below (Hwu, 1993). If an implicit solution with the form of (4.49) is written as f(z) = C < gα (z) > q,

(4.50)

with the understanding that the subscript of z is dropped before the matrix product and a replacement of z1 , z2 , or z3 should be made for each component function of f(z) after the multiplication of matrices, the explicit solution with the form shown in (3.24b) can be expressed as f(z) =

3

< gj (zα ) > CIj q,

(4.51a)

j=1

where ⎡

⎤ ⎡ ⎤ ⎡ ⎤ 100 000 000 I1 = ⎣ 0 0 0 ⎦ , I 2 = ⎣ 0 1 0 ⎦ , I 3 = ⎣ 0 0 0 ⎦ . 000 000 001

(4.51b)

According to the above translating technique, the solution obtained in (4.48) has been written with the subscript α of the variable zα dropped. (Note that zˆα is a definite value not a variable. Hence, its subscript is not dropped.) If we did not drop the subscript α of zα in (4.48), the solution obtained for fp (z) will not satisfy the basic form of f(z) in (3.24b). To translate the solution of fp (z) in (4.48) into a form required in (3.24b), we employ the technique introduced in (4.50) and (4.51). By comparing (4.48) with (4.50), we let C = B−1 B, gα (z) = ln(z − z¯ˆα ), and

104

4


T

ˆ i. Employing these corresponding values in (4.51) now gives us an q = A p/2π explicit solution as fp (z) =

3

ˆ i. < ln(zα − z¯ˆj ) > B−1 B Ij A p/2π T

(4.52)

j=1

With the results of (4.43) and (4.52), the explicit closed-form solution for the halfspace loaded by a concentrated force pˆ can now be written as u = Im{A[< ln(zα − zˆα ) > AT +

3

ˆ , < ln(zα − z¯ˆj ) > B−1 B Ij A ]}p/π T

j=1

φ = Im{B[< ln(zα − zˆα ) > A + T

3

(4.53) < ln(zα − z¯ˆj ) > B

−1

T

ˆ . B Ij A ]}p/π

j=1

Unlike the cases of infinite space discussed in the previous section, due to the lack of appropriate identities the real-form solutions associated with (4.53) have not been found. On the other hand, if we consider only the deformations and stresses along the half-space surface, real-form solutions may be obtained by the following way. Along the half-space surface, x2 = 0 and then zα = x1 . Substituting this value T into (4.53) and summing three terms, ln(x1 − z¯ˆj )B−1 BIj A , j =1, 2, 3, into one term, we get T ˆ , u(x1 , 0) = Im{A < ln(x1 − zˆα ) > AT + AB−1 B < ln(x1 − z¯ˆα ) > A }p/π T ˆ . φ(x1 , 0) = Im{B < ln(x1 − zˆα ) > AT + B < ln(x1 − z¯ˆα ) > A }p/π

(4.54)

By using the identities (3.128b) and (3.131), the real-form solutions of u and φ along the half-space surface x2 = 0 can be written as 1 ln r(H + L−1 + SL−1 ST )pˆ 2π 1 ˜ ˜ 2 (θ )ST + L−1 N ˜ T (θ )ST + N ˜ 3 (θ )H]}p, ˜ T (θ ) + SL−1 [N ˆ − {N 1 (θ )H + N 1 1 2 φ (x1 , 0) = 0, (4.55) u(x1 , 0) = −

in which (r, θ ) is the polar coordinate with origin at (ˆx1 , xˆ 2 ) and is related to (x1 , x2 ) by x1 − xˆ 1 = r cos θ , x2 − xˆ 2 = r sin θ .

(4.56)

In (4.55), φ (x1 , 0) = 0 means that the traction-free boundary condition is satisfied along the half-space surface, whereas the surface deformation u(x1 , 0) can be further

4.2

Half-Space

105

simplified through the use of the identities (3.97b) and (3.110). The final simplified result is ˜ T (θ )}p/π ˆ . u(x1 , 0) = −L−1 {ln rI + π N 1

(4.57)

The hoop stresses σ11 along the half-space surface can then be found by using the relations (3.13)1 in which we need to calculate φ, 2 . Differentiating φ given in (4.53)2 with respect to x2 and setting x2 = 0, we have ˆ φ, 2 (x1 , 0) = Im{B < μα (x1 − zˆα )−1 > AT + B < μα > B−1 B < (x1 − zˆα )−1 > A }p/π T

ˆ . = Im{B < μα > B−1 [B < (x1 − zˆα )−1 > AT + B < (x1 − zˆα )−1 > A ]}p/π T

(4.58) Using the identities (3.137) and (3.125), φ, 2 obtained in (4.58) can be further simplified as φ, 2 (x1 , 0) = −

1 ˆ N3 L−1 [cos θ I − sin θ NT1 (θ )]p. πr

(4.59)

The stresses σ11 , σ21 , and σ31 can then be calculated by substituting (4.59) into the first equation of (3.13). For isotropic materials, use of the explicit expressions given in (3.82), (3.83), and (3.86), we obtain ⎫ ⎧ ⎫ ⎧ sin 2θ ⎨σ11 ⎬ 1 ⎨2(ˆp1 cos θ − pˆ 2 sin θ ) − 1−ν (ˆp1 sin θ − pˆ 2 cos θ )⎬ σ21 = − . (4.60) 0 ⎭ ⎩ ⎭ πr ⎩ σ31 pˆ 3 cos θ

4.2.2 Surface Green’s Function If the point force considered in the above section is located on the half-space surface, the Green’s function obtained in (4.53) can be further simplified and the simplified function is called Surface Green’s function. When xˆ 2 = 0, we have zˆα = z¯ˆj = xˆ 1 . ¯ j AT , j=1, 2, 3, With this value, the last three terms of (4.53), < ln(zα − xˆ 1 ) > B−1 BI can be summed into one term and hence (4.53) can be simplified as ˆ , u = Im{A < ln (zα − xˆ 1 ) > B−1 }p/π ˆ . φ = Im{B < ln(zα − xˆ 1 ) > B−1 }p/π T

(4.61)

During the derivation of (4.61), the identity AT + B−1 BA = B−1 has been used, which can be proved from the orthogonality relation shown in (3.58b)1 . By using the identity (3.140) with the coordinate origin located on (ˆx1 , 0), the complex-form solution (4.61) can be converted into a real form as

106

4


˜ 1 (θ )]L−1 p/π ˆ , u = −[ln rI + π N ˜ 3 (θ )L−1 p. ˆ φ = −N

(4.62)

Substituting (4.62)2 into (3.37), we get σθθ = σrθ = σθ3 = 0, σrr =

1 T 1 T ˆ σr3 = ˆ n (θ )N3 (θ )L−1 p, i N3 (θ )L−1 p. πr πr 3 (4.63)

For isotropic materials, use of the explicit expressions given in (3.82) and (3.86), we obtain σrr =

2 1 (ˆp1 cos θ + pˆ 2 sin θ ), σr3 = pˆ 3 . πr πr

(4.64)

4.2.3 Distributed Load Along the Half-Space Surface Consider an elastic homogeneous anisotropic half-space, in which the distributed ˆ 1 ) = (ˆp1 (x1 ), pˆ 2 (x1 ), pˆ 3 (x1 ))T with normal and tangential components per load p(x unit length is applied on the half-space surface (Fig. 4.6). Since the solution associated with the concentrated forces has been obtained in (4.61), superposition may be used for distributed loads by considering an element of the distributed load as if it were a concentrated load and then integrating throughout the region of the load, (xa , xb ). By this procedure, the displacement and stress function vectors can be written as

x2

pˆ ( x1 )

xa

Fig. 4.6 Distributed loads applied on the half-space surface

xb

x1

4.2

Half-Space

107

1 xb ˆ x1 )dˆx1 , u= Im{A < ln(zα − xˆ 1 ) > B−1 }p(ˆ π xa xb 1 ˆ x1 )dˆx1 . φ= Im{B < ln(zα − xˆ 1 ) > B−1 }p(ˆ π xa

(4.65)

4.2.4 Couple Moments As discussed in Section 4.1.4 the solutions associated with the couple moments can be found by superposing the two solutions corresponding to two parallel and opposite concentrated forces applied at two arbitrary points just like that shown in Fig. 4.4, and the solution corresponding to each concentrated force for the half-space problems is provided in (4.53). To find the solution for the case of concentrated moment shown in Fig. 4.7, same approach as those described between (4.27) and (4.30) can also be employed, i.e., consider the concentrated moment be produced by two perpendicular couples with approaching-zero distance (see Fig. 4.3). Since the derivation is similar to that described in Section 4.1.4, no detailed derivation and results will be presented here. Fig. 4.7 A concentrated moments in the half-space

x2

x1 mˆ ( xˆ1 , xˆ2 )

4.2.5 Dislocations Consider a dislocation with Burgers vector bˆ = (bˆ 1 , bˆ 2 , bˆ 3 ) located on an internal point xˆ = (ˆx1 , xˆ 2 ) of the half-space (see Fig. 4.8). As that discussed in Section 4.1.5, the solution to this problem can be obtained from the corresponding point force ˆ . Hence ˆ replaced by BT b/π solution (4.53) with AT p/π

108

4


x2

Fig. 4.8 A dislocation in the half-space

x1

b ( xˆ1 , xˆ2 )

u = Im{A[< ln(zα − zˆα ) > BT +

3

ˆ , < ln(zα − zˆj ) > B−1 BIj B ]}b/π T

j=1

φ = Im{B[< ln(zα − zˆα ) > B + T

3

(4.66) −1

< ln(zα − zˆj ) > B

ˆ . BIj B ]}b/π T

j=1

By a similar approach stated between (4.54) and (4.59), the real-form solutions for the displacement and traction vectors along the half-space surface x2 = 0 can be obtained as follows (one may also refer to Ting (1996) for the detailed derivation): ˆ N3 (θ )b, u(x1 , 0) = −L−1 / sin θ ˆ N3 L−1 N3 (θ )b. φ, 2 (x1 , 0) = πr

(4.67)

4.3 Bimaterials Consider a bimaterial that consists of two dissimilar anisotropic elastic half-spaces. Let the upper half-space x2 > 0 be occupied by material 1 and the lower half-space x2 < 0 be occupied by material 2 (Fig. 4.9). Assume these two dissimilar materials are perfectly bonded along the interface x2 = 0.

4.3.1 Green’s Function The Green’s function for bimaterials considered in this section is the elasticity solution for a bimaterial subjected to a concentrated force pˆ and dislocation with Burgers vector bˆ applied at point xˆ = (ˆx1 , xˆ 2 ) of material 1. The boundary conditions of this problem can be expressed as

4.3

Bimaterials

109

Fig. 4.9 A concentrated force and a dislocation in bimaterials

pˆ bˆ

Material 1 ( xˆ1 , xˆ2 )

S1 S2

Material 2

u1 = u2 , φ1 = φ 2 , along the interface x2 = 0, ˆ for any closed curve C enclosing the point xˆ , dφ1 = p, C ˆ for any closed curve C enclosing the point xˆ , du1 = b,

(4.68)

C

σij → 0,

at infinity,

in which the subscripts 1 and 2 denote materials 1 and 2, respectively. To find a solution satisfying the above boundary conditions, we employ the method of analytical continuation introduced in Section 4.2.1 for the half-space problem. By that method, the general solution for the displacement and stress function vectors can be written as ! " ! " u1 = 2 Re A1 [f0 (z(1) ) + f1 (z(1) )] , φ1 = 2 Re B1 [f0 (z(1) ) + f1 (z(1) )] , ! " ! " (4.69) u2 = 2Re A2 f2 (z(2) ) , φ2 = 2Re B2 f2 (z(2) ) , where the subscripts 1 and 2 or the superscripts (1) and (2) denote materials 1 and 2, respectively. f0 (z(1) ) is selected to be the Green’s function for the infinite space, i.e., by (4.43) and (4.38), (1) T ˆ ˆ + BT1 b)/2π f0 (z(1) ) =< ln(z(1) i α − zˆ α ) > (A1 p

(4.70)

From (4.70), we know that f 0 (z(1) ) is analytic in the entire z-plane except the point (1) zˆα which is located inside the upper half-space. Thus, we may say that f 0 (z(1) ) is analytic in the lower half-space (S2 ). Moreover, f 1 (z(1) ) and f 2 (z(2) ) are required to be analytic in the upper and lower half-spaces, respectively. Since f0 (z(1) ) has satisfied the last three boundary conditions shown in (4.68), to determine f1 (z(1) ) and f2 (z(2) ) we consider the first two boundary conditions of (4.68) which require

110

4


that the traction and displacement be continuous across the interface. Substituting (4.69) into (4.68)1,2 , we get A1 [f 0 (x1 ) + f 1 (x1 )] + A1 [f 0 (x1 ) + f 1 (x1 )] = A2 f 2 (x1 ) + A2 f 2 (x1 ), B1 [f 0 (x1 ) + f 1 (x1 )] + B1 [f 0 (x1 ) + f 1 (x1 )] = B2 f 2 (x1 ) + B2 f 2 (x1 ).

(4.71)

From the relations (4.71) and the discussion stated between (4.44) and (4.46), we know that the following two functions −A1 f 1 (z) + A2 f 2 (¯z) − A1 f 0 (¯z), z ∈ S1 , θ1 (z) = A1 f 1 (¯z) − A2 f 2 (z) + A1 f 0 (z), z ∈ S2 , −B1 f 1 (z) + B2 f 2 (¯z) − B1 f 0 (¯z), z ∈ S1 , θ2 (z) = B1 f 1 (¯z) − B2 f 2 (z) + B1 f 0 (z), z ∈ S2 ,

(4.72a) (4.72b)

will be continuous across the interface and analytic in the entire z-plane including the points at infinity. By the Liouville’s theorem and the infinity condition (4.68)5 , we conclude that θ1 (z) ≡ 0, θ2 (z) ≡ 0,

(4.73)

which lead to A1 f 1 (z) = A2 f 2 (¯z) − A1 f 0 (¯z) , z ∈ S1 , B1 f 1 (z) = B2 f 2 (¯z) − B1 f 0 (¯z) A2 f 2 (z) = A1 f 1 (¯z) + A1 f 0 (z) , z ∈ S2 . B2 f 2 (z) = B1 f 1 (¯z) + B1 f 0 (z)

(4.74a) (4.74b)

From the first equation of (4.74a), we have −1

f 2 (¯z) = A2 [A1 f 1 (z) + A1 f 0 (¯z)].

(4.75)

Substituting (4.75) into the second equation of (4.74a), we obtain −1

−1

f 1 (z) = (B1 − B2 A2 A1 )−1 (B2 A2 A1 − B1 )f 0 (¯z),

(4.76)

which can also be expressed as −1 f 1 (z) = −A−1 1 (M2 + M1 ) (M2 − M1 )A1 f 0 (¯z),

(4.77)

where Mi is the impedance matrix defined in (3.132), i.e., Mi = −iBi A−1 i , i = 1, 2.

(4.78)

4.3

Bimaterials

111

Similarly, from (4.74b) we can obtain −1 −T f 2 (z) = −iA−1 2 (M2 + M1 ) A1 f 0 (z).

(4.79)

In deriving (4.79), the relations (3.131)2 and (3.59)1 have been used. With f0 (z) given in (4.70) integrating (4.77) and (4.79) with respect to z, and using the translating technique introduced in (4.50) and (4.51), we obtain f1 (z(1) ) =

3

(1)

−1 ˆ ˆ + B1 b)/2π < ln(zα − zˆj ) > A−1 i, 1 (M2 + M1 ) (M2 − M1 )A1 Ij (A1 p (1)

T

T

j=1

f2 (z(2) ) = −

3

−1 −T T ˆ + BT b)/2π ˆ < ln(zα − zˆj ) > A−1 . 1 2 (M2 + M1 ) A1 Ij (A1 p (2)

(1)

j=1

(4.80)

The explicit closed-form solution of the Green’s function for bimaterials can therefore be obtained by substituting (4.70) and (4.80) into (4.69). To get the explicit real-form expressions for the displacement and stress function along the interface, we may follow the steps shown between (4.53) and (4.59) by first letting x2 = 0. Detailed derivation can be found in Ting (1996) and the final simplified expressions are

u(x1 , 0) φ(x1 , 0)

ˆ ˜ 1 S˜ H b (1) ˜ ln rI + π N (θ ) =− , T ˜ ˜ pˆ − L S 2π

(4.81)

in which (r, θ ) is the polar coordinate with origin at (ˆx1 , xˆ 2 ) and is related to (x1 , x2 ) ˜ H, ˜ and L˜ are three 3 × 3 real tensors defined by by (4.56) and S,

˜ S˜ H ˜ ˜ −L ST

∗ ∗ ˜ S1 + S2 H1 + H2 S H S˜ H =2 ∗ S∗T = −2I. T ˜ ˜ −L −(L1 + L2 ) ST1 + ST2 −L S

(4.82)

Note that S, H, and L defined in (3.95) are the average values of N1 (θ ), N2 (θ ), and N3 (θ ) for homogeneous materials, now S∗ , H∗ , and L∗ are the average values of N1 (θ ), N2 (θ ), and N3 (θ ) for bimaterials. These definitions will be extended to multi-materials in the next chapter. Note also that in (4.81) the subscripts for u and φ have been deleted in view of the continuity conditions u1 = u2 , φ1 = φ2 along the interface x2 = 0.

4.3.2 Interfacial Green’s Function If the concentrated force pˆ and dislocation with Burgers vector bˆ are applied at the point of the interface, i.e., xˆ 2 = 0, the complex variable zˆα = xˆ 1 . With this value, by following the steps presented in Section 4.2.2 we may get simplified solutions for

112

4


f0 (z(1) ), f1 (z(1) ), and f2 (z(2) ) given in (4.70) and (4.80). The surface Green’s function can therefore be obtained by substituting these simplified solutions into (4.69). Following is an alternative approach for this problem (Ting, 1996). Consider a general case of bimaterials whose interface is inclined by angle α0 from x1 -axis (Fig. 4.10). This bimaterial is subjected to a concentrated force pˆ and dislocation with Burgers vector bˆ applied at point xˆ on the interface. The point xˆ is chosen to be the origin of the coordinate. For the displacements to be single valued with the form of logarithmic function, we introduce a branch cut at the interface θ = α0 ± π . With this choice, the boundary conditions shown in (4.68) may be rewritten as u1 (r, α0 ) = u2 (r, α0 ), φ1 (r, α0 ) = φ2 (r, α0 ), ˆ φ1 (r, α0 + π ) − φ2 (r, α0 − π ) = p,

(4.83)

ˆ ∀ r, u1 (r, α0 + π ) − u2 (r, α0 − π ) = b, σij → 0, at infinity. As that discussed in Section 4.1.3, to satisfy (4.83) we may assume (1) u1 = 2 Re{A1 < ln z(1) α > q1 }, φ1 = 2 Re{B1 < ln zα > q1 },

(4.84)

(2) u2 = 2 Re{A2 < ln z(2) α > q2 }, φ2 = 2 Re{B2 < ln zα > q2 },

where q1 and q2 are two complex coefficient vectors and each of them can be replaced by two real vectors gk and hk , k=1,2, as qk = ATk gk + BTk hk .

(4.85)

By substituting (4.85) into (4.84) and using the identities (3.128), we obtain the real-form expressions as

Material 1 pˆ

bˆ

α0 ( x1 , x2 ) Material 2 Fig. 4.10 A concentrated force and a dislocation on the bimaterial interface

4.3

Bimaterials

113

˜ (k) (θ )hk + N ˜ (k) (θ )gk ] + uˆ k , uk = ln rhk + π [N 1 2 ˜ (θ )hk + N ˜ φk = ln rgk + π [N 3 1 (k)

(k)T

(θ )gk ] + φˆ k , k = 1, 2,

(4.86)

in which uˆ k and φˆ k are constant vectors which do not alter the stresses. Substituting (4.86) into the continuity conditions (4.83)1, 2 , we get h1 = h2 = h, g1 = g2 = g, ˜ (k) (α0 )h − N ˜ (k) (α0 )g, uˆ k = −N 1 2

(4.87)

˜ (k) (α0 )h − N ˜ (k)T (α0 )g, k = 1, 2, φˆ k = −N 3 1 and hence ˜ (θ ) − N ˜ (α0 )]h + [N ˜ (θ ) − N ˜ (α0 )]g}, uk = ln rh + π {[N 1 1 2 2 (k)

(k)

(k)

(k)

˜ (k) (θ ) − N ˜ (k) (α0 )]h + [N ˜ (k)T (θ ) − N ˜ (k)T (α0 )]g}, k = 1, 2. φk = ln rg + π {[N 3 3 1 1 (4.88) ˜ (θ ), i = 1, 2, 3, are periodic in θ with periodicity π and that Noting that N i (k) ˜ (k) (π ) = Hk , N ˜ (k) (π ) = −Lk . Substitution of (4.88) into (4.83)3,4 ˜ N1 (π ) = Sk , N 2 3 leads to S1 + S2 H1 + H2 h bˆ π = . (4.89) g −(L1 + L2 ) ST1 + ST2 pˆ (k)

Use of the relation (4.82) may then give us ˜ p)/2π ˜ bˆ − S˜ T p)/2π ˆ ˆ . h = −(S˜ bˆ + H , g = (L

(4.90)

Chapter 5

Wedges and Interface Corners

Wedge is a piece of material with V-shaped edges (Fig. 5.1). Due to the sharp change at the apex of the wedge, stress singularity may occur. Hence, in analytical studies most of the wedge problems are considered asymptotically around the wedge apex, not for the entire region. These analytical results may provide useful information on the nature of stress singularities at the corner and may be applied to the problems containing local wedge geometries. With this concern, all the problems in this chapter consider only the boundary conditions near the wedge apex, i.e., no far-field conditions are prescribed in the wedge problems and hence all the problems are incomplete in the sense of structural geometry and force equilibrium. Their associated solutions are therefore necessarily non-unique. A unique solution can be expected when the boundary conditions at the far field are specified. With this understanding, Sections 5.1 and 5.2 will provide the particular solutions for the wedges subjected to forces on the edges and apex, and Section 5.3 will study the orders of stress singularity as well as homogeneous solutions for multi-material wedge spaces and multi-material wedges with several different boundary conditions. The homogeneous solutions are the field solutions near wedge apex and are also called the eigenfunctions of the wedges. In engineering applications, a unique solution for the complete problem may then be obtained by adding all the associated eigenfunctions to the particular solutions. Besides the orders of stress singularity and near-tip field solutions, the other important topics – stress intensity factors – are discussed in Section 5.4.

5.1 Uniform Tractions on the Wedge Sides Consider an anisotropic elastic wedge of wedge angle 2α subjected to uniform tractions, t+ and t− , on the sides of the wedge (Fig. 5.1). To make our solutions more generous and convenient for applications, the wedge is represented by θ − ≤ θ ≤ θ +,

r ≥ 0,

(5.1)

in which θ − and θ + are not required to be equal in magnitude, and the wedge angle 2α = θ + − θ − . From (3.32) and noting that s is the tangential direction when one C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_5, C Springer Science+Business Media, LLC 2010

115

116

5 Wedges and Interface Corners

Fig. 5.1 Wedges subjected to uniform tractions on the wedge sides

x2*

x2

x1* t−

θ θ

−

x1

+

t+

θ + − θ − = 2α

faces the direction of increasing s the material lies on the right side, the boundary conditions can be written as ∂φ/∂r = −t− , ∂φ/∂r = t+ ,

when θ = θ − , when θ = θ + .

(5.2)

Because the prescribed boundary conditions are uniform tractions (which may not equilibrate since we only consider the near-apex conditions, not the entire wedge fields), before finding any new solutions we first discuss the applicability of the uniform stress solution presented in (4.2), which can also be represented by (4.3). Using the solutions given in (4.2), the boundary conditions (5.2) give us − ∞ − cos θ − t∞ 2 −sin θ t1 = −t , + ∞ + cos θ + t∞ 2 −sin θ t1 = t ,

(5.3a)

which will then lead to − + + − t∞ 1 = −(cos θ t + cos θ t )/ sin 2α, − + + − t∞ 2 = −(sin θ t + sin θ t )/ sin 2α.

(5.3b)

Substituting (5.3b) into (4.2a)2 , we get φ=

r sin (θ − θ − )t+ + sin (θ − θ + )t− . sin 2α

(5.4)

In deriving the uniform stress solution (5.4), two requirements have been imposed implicitly, i.e.,

5.1

Uniform Tractions on the Wedge Sides

(1) sin 2α = 0, + −

117

or say,

2α = π , 2π ,

− +

(2) n (θ )t + n (θ )t = 0. T

T

(5.5)

The first requirement can be seen directly from the solution (5.4) because when sin 2α = 0 the denominator will be zero and the stresses will become infinite everywhere in the wedge. This requirement can also be seen from (5.3a) which leads to t+ = t− , when 2α = π , t+ = −t− , when 2α = 2π .

(5.6)

In other words, when sin 2α = 0 the two conditions on the wedge sides are dependent on each other in order to have uniform stress solution, which is not enough to solve ∞ the uniform stress vectors t∞ 1 and t2 . As to the second requirement, we consider T ∞ the symmetry of the stress components, i.e., σ12 = σ21 , which leads to iT2 t∞ 1 = i1 t 2 . From this symmetry relation and relation (5.3b), we know that to have uniform stress distribution, t+ and t− cannot be prescribed arbitrarily and should satisfy the relation given in (5.5)2 .

5.1.1 Non-critical Wedge Angles When the wedge angle and the prescribed traction cannot satisfy the requirements given in (5.5), the uniform stress solution (5.4) or (4.2) should be corrected. Because the tractions applied on the wedge sides are uniform and the solution obtained by choosing f(z) =< zα > q is invalid if (5.5) is violated, we now consider > q, f(z) =< z1+λ α

(5.7a)

where λ is any arbitrary constant and q is the complex coefficient vector which may be replaced by two real vectors g and h as q = AT g + BT h.

(5.7b)

For the convenience of applying the boundary conditions Ting (1996) suggested that the complex variable zα be replaced by the dual complex variable , z∗α = x1∗ + μα (θ − )x2∗ . Substituting (5.7a,b) into the general solution (3.24), we have u A < zα∗1+λ > BT A < zα∗1+λ > AT h = 2 Re . φ B < zα∗1+λ > BT B < zα∗1+λ > AT g

(5.8)

Employing the Taylor’s series expansion and letting g and h be a polynomial of λ, we have (Hwu and Ting, 1990)

118


, - λ2 , -2 zα∗1+λ = z∗α + λz∗α ln z∗α + z∗α ln z∗α + · · · , 2 g = g0 + λg1 + · · · , h=

(5.9)

hˆ 0 n(θ − ) + h0 + λh1 + · · ·, λ

where hˆ 0 is a real constant and hi , gi , i = 0, 1, 2, . . . are real constant vectors to be determined through the boundary conditions. The reason for adding the term hˆ 0 n(θ − )/λ is explained in Ting (1996). Substituting (5.9) into (5.8) and making use of the identities (3.146), the solution (5.8) can be written in real form as 1 uR u u0 u = + + λ 1 + · · ·, φ0 φ1 φ λ φR

(5.10a)

where − uR n(θ ) n(θ ) − ˆ ˆ ˆ = rh0 = rh0 N(θ , θ ) , 0 φR 0 u0 h0 n(θ ) − − ˆ / ˆ = rN(θ , θ ) + rh0 [ln rI + π N(θ , θ )] , g0 0 φ0 h u1 ˆ , θ − ) h1 + r[ln rI + π / = rN(θ N(θ , θ − )] 0 φ1 g1 g0 n(θ ) rhˆ 0 + 2 [ln rI + π / N(θ , θ − )]2 . 0

(5.10b)

Note that the second equality of (5.10b)1 , which can be proved by using the last two equations of (3.104), shows that φR = 0 and uR = rhˆ 0 n(θ ) and hence is an expression for rigid body rotation. Since λ can be any arbitrary constant, (u0 , φ0 ), (u1 , φ1 ),. . ., are all admissible solutions for the present problem. Written in full, we have ˆ 1 (θ , θ − )h0 + N ˆ 2 (θ , θ − )g0 + hˆ 0 [ln rI + π / N1 (θ , θ − )]n(θ ), u0 /r = N ˆ 3 (θ , θ − )h0 + N ˆ T (θ , θ − )g0 + π hˆ 0 / N3 (θ , θ − )n(θ ), φ0 /r = N 1

(5.11)

and ˆ 1 (θ , θ − )h1 + N ˆ 2 (θ , θ − )g1 + π / ˆ T (θ , θ − )g0 ˆ 3 (θ , θ − )h0 + N u1 /r = N N2 (θ , θ − ) N 1 % & − − − ˆ 2 (θ , θ )g0 ˆ 1 (θ , θ )h0 + N + ln rI + π / N1 (θ , θ ) N !% " &2 ˆ 3 (θ , θ − ) n(θ )/2, + hˆ 0 ln rI + π / N2 (θ , θ − )N N1 (θ , θ − ) + π 2 /

5.1


119

ˆ 3 (θ , θ − )h1 + N ˆ T (θ , θ − )g1 + π / ˆ 1 (θ, θ − )h0 + N ˆ 2 (θ, θ − )g0 ] φ1 /r = N N3 (θ , θ − )[N 1 ˆ T (θ , θ − )g0 ˆ 3 (θ , θ − )h0 + N + ln rI + π / NT1 (θ , θ − ) N 1 ! " N3 (θ , θ − ) + π / N3 (θ , θ − )/ N1 (θ , θ − ) + / NT1 (θ, θ − )/ N3 (θ, θ − ) n(θ)/2. + π hˆ 0 2 ln r/ (5.12)

By choosing u0 , φ0 in (5.11) to be the solution for the present problem, the boundary condition (5.2) leads to −t− = g0 , ˆ 3 (θ + , θ − )h0 + N ˆ T (θ + , θ − )g0 + π hˆ 0 / N3 (θ + , θ − )n(θ + ). t+ = N 1

(5.13)

ˆ 3 (θ + , θ − ), the Using the result of (5.13)1 and the definition given in (3.87a)2 for N second equation of (5.13) can now be written as ˆ T (θ + , θ − )t− − π hˆ 0 / N3 (θ + , θ − )n(θ + ). sin 2αN3 (θ − )h0 = t+ + N 1

(5.14)

In order to find the solution of h0 and hˆ 0 , we now consider the following problem N3 (ω)d = β.

(5.15)

From (3.98a)2 we see that N3 (ω) is singular and (5.15) has no solution unless nT (ω)β = 0.

(5.16)

When (5.16) is satisfied, d is not unique and has the extra term kn(ω) where k is an arbitrary constant. Like the explicit expression of N3 shown in (3.81a), the singular structure of N3 (ω) can be studied from its related matrix N∗3 (ω). Their relation has been shown in (3.77) as N3 (ω) = T (ω)N∗3 (ω)(ω), and N∗3 (ω) has the structure (Ting, 1996) ⎡

∗ N∗3 (ω) = ⎣ 0 ∗

(5.17)

⎤ 0 ∗ 0 0⎦. 0 ∗

(5.18)

The submatrix of N∗3 (ω) obtained by crossing out the second row and the second column is positive definite. This suggests that we may define the sub-inverse of N∗3 (ω) as

N∗3 (ω)

N∗3 (ω) = N∗3 (ω) N∗3 (ω)

= I2 ,

(5.19a)

120


where N∗3 (ω)

has the same structure as N∗3 (ω) and ⎡

1 I2 = ⎣ 0 0

0 0 0

⎤ 0 0⎦. 1

(5.19b)

{N∗3 (ω)} so defined exists and is unique. If we substitute (5.17) into (5.15) and make use of (5.19) we obtain d = {N3 (ω)} β + kn(ω),

(5.20)

where k is an arbitrary constant and {N3 (ω)} = T (ω) N∗3 (ω)

T (ω).

(5.21)

Thus when (5.16) is satisfied the solution to (5.15) is given by (5.20). The first term on the right-hand side is unique. The extra term kn(ω) on the right very often corresponds to a homogeneous solution or a rigid body motion and therefore can be ignored. From the discussion given between (5.15) and (5.21), we see that (5.14) has no solution unless ˆ T (θ + , θ − )t− − π hˆ 0 / N3 (θ + , θ − )n(θ + )} = 0. nT (θ − ){t+ + N 1

(5.22)

Using the second last identity of (3.104), (5.22) can be further simplified and lead to hˆ 0 = ˆt/π,

(5.23a)

where ˆt = nT (θ − )t+ + nT (θ + )t− ,

= nT (θ − )/ N3 (θ + , θ − )n(θ + ).

(5.23b)

When (5.22) is satisfied, i.e., hˆ 0 = ˆt/π , the solution for h0 is not unique and can be expressed as (Hwu and Ting, 1990) h0 =

1 N3 (θ − ) sin 2α

! " + − + ˆ T (θ + , θ − )t− −π hˆ 0 / N t+ + N (θ , θ )n(θ ) +kn(θ − ), 3 1 (5.24)

where k is an arbitrary constant and like φR and uR shown in (5.10b)1 the associated term kn(θ − ) represents rigid body rotation and can therefore be ignored. is the pseudo-inverse of N3 (θ − ) defined by (5.21). N3 (θ − ) Equations (5.13)1 , (5.23), and (5.24) provide the solutions to the unknown constants g0 , hˆ 0 , and h0 in (5.11). The solution is valid provided the wedge angle 2α is not one of the following:

5.1


2α = π , 2π , 2αc ,

121

(5.25)

where 2αc = θc+ − θc− is the wedge angle whose associated θc+ and θc− satisfy , - , + −- , + = nT θc− / N3 θc , θc n θc = 0.

(5.26)

The wedge angles given by (5.25) are three critical wedge angles of the wedge for which the solution given by (5.11) may not be valid. From the definition given in (3.146b), we know that , , , / N3 θc+ − / N3 θc− . N3 θc+ , θc− = /

(5.27)

By using the explicit expressions given in (3.82) for isotropic materials and the definition of n in (3.36b)2 , (5.26) can be further reduced to 2αc = tan 2αc ,

(5.28)

from which the critical wedge angle 2αc is obtained as 2αc = 257.45◦ . It should be pointed out that the solution given by (5.11), when it is valid, provides bounded stress everywhere in the wedge. If ˆt = 0, i.e., the second requirement of (5.5) is satisfied, from (5.23) we have hˆ 0 = 0 and hence the stress obtained from (5.11)2 become uniform in the wedge, which can be proved to agree with the solution shown in (5.4). For the general case hˆ 0 = 0, the stress depends on θ and is discontinuous but bounded at the wedge apex r = 0. We now discuss the validity of (5.11) for the three critical wedge angles separately. (i) 2α = π . In this case hˆ 0 exists and is determined from (5.23), but since sin 2α = 0, h0 does not exist unless the right-hand side of (5.14) vanishes. With 2α = π and hˆ 0 obtained from (5.23) and knowing that , ˆ T θ + , θ − = −I, / N3 (θ + , θ − ) = −L, n(θ + ) = −nT (θ − ), when 2α = π , N 1 (5.29) it can be shown that the vanishing of the right-hand side of (5.14) leads to t+ − t− =

nT (θ − )[t+ − t− ] Ln(θ − ). nT (θ − )Ln(θ − )

(5.30)

When (5.30) is satisfied, h0 is arbitrary. However, the h0 term produces no surface traction on θ = θ + and θ − and hence is a homogeneous solution which can be ignored. (ii) 2α = 2π . This case is similar to (i) in which hˆ 0 exists but h0 does not exist unless the right-hand side of (5.14) vanishes at 2α = 2π . Knowing that

122


ˆ T (θ + , θ − ) = I, / N3 (θ + , θ − ) = −2L, n(θ + ) = nT (θ − ), when 2α = 2π , N 1 (5.31) it can be shown that the vanishing of the right-hand side of (5.14) leads to t+ + t− =

nT (θ − )[t+ + t− ] Ln(θ − ). nT (θ − )Ln(θ − )

(5.32)

When (5.32) is satisfied, the h0 term is arbitrary which again corresponds to a homogeneous solution and hence can be ignored. (iii) 2α = 2αc . In this case, by (5.23), hˆ 0 exists and is arbitrary only if ˆt(αc ) = nT (θc− )t+ + nT (θc+ )t− = 0.

(5.33)

It is not difficult to show that the terms associated with hˆ 0 represent a homogeneous solution and hence can be ignored. Actually, (5.33) happens to be the requirement given in (5.5), which says that the uniform stress solution exists when (5.33) is satisfied.

5.1.2 Critical Wedge Angles If (5.30), (5.32), or (5.33) is not satisfied for the critical wedge angles 2α = π , 2π , 2αc , respectively, a bounded solution given by (5.11) for the wedge does not exist. We therefore employ u1 , φ1 in (5.12) as the solution for the wedges with critical angles. Satisfaction of the boundary conditions (5.2) now yields 0 = g0 , ˆ 3 (θ + , θ − )h0 + π hˆ 0 / 0=N N3 (θ + , θ − )n(θ + ),

(5.34)

and −t− = g1 , ˆ 3 (θ + , θ − )h1 − N ˆ T (θ + , θ − )t− + π / ˆ 1 (θ + , θ − )h0 t+ = N N3 (θ + , θ − )N 1 % & N3 (θ + , θ − )/ + π 2 hˆ 0 / N1 (θ + , θ − ) − / NT1 (θ + , θ − )/ N3 (θ + , θ − ) n(θ + )/2. (5.35) The right-hand side of (5.34) are the coefficients of the ln r terms of φ1 at θ = θ − and θ = θ + , which should vanish because the tractions applied on the sides of the wedge are independent of r. Equations (5.34) and (5.35)1 have been used in deriving (5.35)2 . Note that (5.34) is identical to (5.13) with t+ = t− = 0. By (5.23), we have

5.1


hˆ 0 = 0,

123

if 2α = 2αc .

(5.36)

We now discuss 2α = π , 2π , 2αc separately. (i) 2α = π or 2π . With hˆ 0 = 0 and sin 2α = 0, (5.34) is satisfied automatically and (5.35) will give us h0 = L−1 (t+ − t− )/π , −1 +

h0 = −L

−

(t + t )/2π ,

for 2α = π , for 2α = 2π .

(5.37)

The terms associated with h1 produce no surface tractions and hence represent a homogeneous solution. Ignoring the h1 terms, the final solution deduced from (5.12) is " & ˆ 3 (θ, θ − ) h0 , ˆ 1 (θ, θ − ) + π / ln rI + π / N1 (θ, θ − ) N N2 (θ, θ − )N !% " & ˆ T (θ, θ − )t− + ln rI + π / ˆ 3 (θ, θ − ) + π / ˆ 1 (θ, θ − ) h0 . φ1 /r = −N N3 (θ, θ − )N NT1 (θ, θ − ) N 1 (5.38)

ˆ 2 (θ, θ − )t− + u1 /r = −N

!%

The solution obtained in (5.38) shows that the stress has the lnr singularity at r=0 unless ˆ 3 (θ , θ − )h0 = sin(θ − θ − )N3 (θ − )h0 = 0. N

(5.39)

Equation (5.39) can be satisfied if we require (5.30) for 2α = π and (5.32) for 2α = 2π , which can be proved by substituting (5.37) into (5.39) and using the identity (3.98a)2 . This agree with the results obtained earlier that the stress is bounded everywhere when the applied tractions satisfy (5.30) or (5.32). (ii) 2α = 2αc . In this case hˆ 0 = 0. To find h0 from (5.34)2 , we first check the requirement shown in (5.16) for the existence of solutions. It leads to Δ = 0 which is the case for the present critical wedge angle. Therefore, h0 exists and its solution can be found by applying (5.20)–(5.34)2 . To find h1 , we consider (5.35)2 , which is also in the form of (5.15). The condition for h1 to exist gives us the solution for hˆ 0 when h0 from (5.34)2 is substituted into (5.35)2 . With hˆ 0 known, h0 can be found from (5.34)2 and h1 from (5.35)2 . Note that during the derivation all the extra terms related to kn(α − ) can be proved to correspond to homogeneous solution or rigid body motion and hence can be ignored.

5.1.3 Summary As discussed previously, several different conditions are considered in this section. Some provide uniform stress solutions such as (5.4), some provide nonuniform but bounded stress solutions such as (5.11), and some others provide solutions with

124


lnr stress singularity such as (5.12). All these solutions are particular solutions which satisfy the prescribed tractions on the wedge sides. As will be discussed in Section 5.3, there are infinitely many homogeneous solutions (or called eigenfunctions) that satisfy traction-free conditions at the wedge sides. A general solution can be obtained when all eigenfunctions are superimposed on the particular solution. A unique solution can then be expected when the complete boundary conditions for the wedge problems are specified. Table 5.1 is the summary for the solutions obtained in this section. Table 5.1 Solutions for the wedges subjected to uniform tractions on the wedge sides Wedge angle

Loading

Solution

Remark

2α = π , 2π 2α = π ,2π , 2αc 2α = π 2α = 2π 2α = π or 2π 2α = 2αc 2α = 2αc

ˆt = 0 ˆt = 0

(4.2) or (5.4) (5.11) with hˆ 0 = ˆt/π , h0 : (5.24) with k=0, g0 = −t− (5.11) with hˆ 0 = ˆt/π , h0 = 0, g0 = −t− (5.11) with hˆ 0 = ˆt/π , h0 = 0, g0 = −t− (5.38) with h0 : (5.37) (4.2) or (5.4) (5.12) with g0 = 0, g1 = −t− , h0 : (5.34) and (5.20), hˆ 0 : (5.35) and (5.16), h1 : (5.35) and (5.20)

Uniform stress solution Bounded solution

(5.30) (5.32) Any ˆt = 0 Any

Bounded solution Bounded solution lnr stress singularity Uniform stress solution lnr stress singularity

Note: ˆt = nT (θ − )t+ + nT (θ + )t− , = nT (θ − )/ N3 (θ + , θ − )n(θ + ).

5.2 Forces at the Wedge Apex 5.2.1 A Single Wedge – Under a Concentrated Force Consider an anisotropic elastic wedge of wedge angle 2α subjected to a concentrated force pˆ at the wedge apex (Fig. 5.2). If the wedge sides are traction-free, the boundary conditions of this problem can be written as tθ = φ, r = 0, when θ = θ − , θ + , ˆ φ(θ + ) − φ(θ − ) = p.

(5.40)

As that discussed in Section 4.1.3, to satisfy (5.40) we may assume u = 2 Re{A < ln zα > q}, φ = 2 Re{B < ln zα > q},

(5.41)

where q is a complex coefficient vector which can be replaced by two real vectors g and h as q = AT g + BT h.

(5.42)

5.2

Forces at the Wedge Apex

125

x2

Fig. 5.2 Wedges subjected to a concentrated force and moment

pˆ mˆ

θ−

θ+

x1

θ + − θ − = 2α

By substituting (5.42) into (5.41) and using the identities (3.128), we obtain the real-form expressions as & % N2 (θ )g , u = (ln r)h + π / N1 (θ )h + / % & φ = (ln r)g + π / N3 (θ )h + / NT (θ )g .

(5.43)

1

With the results of (5.43), the boundary conditions (5.40) lead to g = 0, h =

&−1 1 %/ + ˆ p. N3 (θ − ) N3 (θ ) − / π

(5.44)

A half-space is a special case of single wedge whose θ + = π and θ − = 0. With ˆ and the solutions obtained these values, h obtained in (5.44) becomes h = −L−1 p/π in (5.43) recover (4.62).

5.2.2 A Single Wedge – Under a Concentrated Couple If the wedge is under a concentrated couple m ˆ (counterclockwise) at the wedge apex and the wedge sides are traction free (Fig. 5.2), the boundary conditions of this problem can be written as tθ = 0, when θ = θ − , θ + , θ+ r2 σrθ dθ = −m, ˆ for r = constant, θ−

(5.45a)

126


in which, from (3.32) and (3.37), tθ = φ, r , σrθ = sT φ, r .

(5.45b)

From the last equation of (5.45a) we see that the stresses of this problem are proportional to r −2 . Therefore, like the problems discussed in Section 5.1.1 in order to satisfy the boundary conditions we may assume ! " ! " u = 2 Re A < zα−1+λ > q , φ = 2 Re B < zα−1+λ > q ,

(5.46)

where q is a complex coefficient vector which can be replaced by two real vectors g and h as q = AT g + BT h.

(5.47)

Following the procedure stated between (5.7) and (5.11) except that the terms associated with hˆ 0 are not needed, we obtain ˆ 1 (θ − , θ )h0 + N ˆ 2 (θ − , θ )g0 , ru0 = N ˆ 3 (θ − , θ )h0 + N ˆ T (θ − , θ )g0 , rφ0 = N 1

(5.48)

and ˆ 1 (θ − , θ )h1 + N ˆ 2 (θ − , θ )g1 + π / ˆ T (θ − , θ )g0 ˆ 3 (θ − , θ )h0 + N ru1 = N N2 (θ , θ − ) N 1 % & ˆ 2 (θ − , θ )g0 , ˆ 1 (θ − , θ )h0 + N + ln rI + π / N1 (θ , θ − ) N ˆ 3 (θ − , θ )h1 + N ˆ T (θ − , θ )g1 + π / ˆ 2 (θ − , θ )g0 ˆ 1 (θ − , θ )h0 + N rφ1 = N N3 (θ , θ − ) N 1 % & ˆ T (θ − , θ )g0 . ˆ 3 (θ − , θ )h0 + N + ln rI + π / NT1 (θ , θ − ) N 1 (5.49) ˆ i (θ , θ − ) Note that the above results are identical to (5.11) and (5.12) except that rN ˆ i (θ − , θ ) and the hˆ 0 terms have been eliminated. is replaced by r−1 N By choosing u0 , φ0 in (5.48) to be the solution for the present problem, the boundary condition (5.45a)1 leads to ˆ 3 (θ − , θ + )h0 = − sin 2α N3 (θ + )h0 = 0. g0 = 0, N

(5.50)

If sin 2α = 0, i.e., 2α = π or 2π , (5.50)2 will be satisfied automatically and hence h0 is arbitrary. Otherwise, from (5.15), (5.16), (5.17), (5.18), (5.19), and (5.20) we have h0 = kn(θ + ),

(5.51)

5.2


127

in which k is a constant to be determined. With the results of (5.50) and (5.51), imposition of the boundary condition (5.45a)2 and making use of the relations (5.45b)2 and (5.48)2 and the identity (3.104)10 and the definitions (3.146b) and (5.23b)2 , we get k = −m/π. ˆ

(5.52)

Combining the results of (5.50), (5.51), and (5.52), the solution for u and φ obtained in (5.48) now becomes u=

−m ˆ ˆ − −m ˆ ˆ − N1 (θ , θ )n(θ + ), φ = N3 (θ , θ )n(θ + ). πr π r

(5.53)

Critical Wedge Angle From (5.53), we see that the solution becomes unbounded everywhere in the wedge when = 0 , i.e., when 2α = 2αc where 2αc is the critical wedge angle satisfying the relation (5.26). To find a bounded solution, we now choose u1 , φ1 in (5.49) as the solution for the critical angle. The boundary condition (5.45a)1 leads to g0 = g1 = 0, h0 = k0 n(θ + ),

h1 =

(5.54) k0 π N3 (θ + , θ − )n(θ − ) + k1 n(θ + ), N3 (θ + ) / 2 sin 2αc

where k0 and k1 are arbitrary constants. Note that in deriving (5.54), the solution (5.20), the definition (3.87a)2 , and the identity (3.104)11 have been used. With the results of (5.54), substitution of (5.49)2 into the other boundary condition (5.45a)2 gives us k0 = −

m ˆ , c0 + c1

(5.55a)

where −

c0 = π n (θ ) T

θ+ θ−

%

& − − / / N1 (θ )N3 (θ , θ ) + N3 (θ )N1 (θ , θ ) dθ n(θ + ),

π2 N3 (θ + , θ − )n(θ − ). nT (θ − )/ N3 (θ + , θ − )N3 (θ + ) / c1 = 2 sin 2αc

(5.55b)

In deriving (5.55), the relations (5.45b)2 and (5.26), the commutative properties (3.109) and the identities (3.104)9,10 have been used. It should also be noted that the term associated with k1 corresponds to a homogeneous solution and can be ignored. Combining the results of (5.54) and (5.55), the solution for u and φ obtained in (5.49) now becomes

128


ˆ 1 (θ − , θ )n(θ + ) + N ˆ 1 (θ − , θ )h1 ru1 = k0 ln rN ˆ 3 (θ − , θ ) + / ˆ 1 (θ − , θ ) n(θ + ), + k0 π / N2 (θ , θ − )N N1 (θ , θ − )N ˆ 3 (θ − , θ )n(θ + ) + N ˆ 3 (θ − , θ )h1 rφ1 = k0 ln rN ˆ 1 (θ − , θ ) + / ˆ 3 (θ − , θ ) n(θ + ). + k0 π / NT1 (θ , θ − )N N3 (θ , θ − )N

(5.56)

5.2.3 Multi-material Wedge Spaces Consider a multi-material wedge that consists of n different anisotropic elastic wedges as shown in Fig. 5.3a and b. Assume perfect bond along each interface between two dissimilar wedges and each wedge occupies the region θk−1 ≤ θ ≤ θk ,

k = 1, 2, . . . , n.

(5.57)

If the entire space is filled up with the multi-material wedges and the nth wedge is bonded together with the first wedge, i.e., θn = θ0 , we call it a multi-material wedge space (Fig. 5.3a). If both of the first and the nth wedges have one side not bonded with any other wedges, we call it a multi-material wedge (Fig. 5.3b). The problem of a multi-material wedge space subjected to a concentrated force pˆ and a dislocation with Burgers vector bˆ applied at r=0 will be solved and discussed in this section. The problem for a multi-material wedge with free–free surfaces subjected to a concentrated force pˆ applied at r=0 will then be discussed next. Both of these two problems have been presented in Ting (1996). x2

x2

2 θ2

k

θ1

2

θ0 = θn

θk

θ2

1 x1

k

θk

θ n−1

θn n

n−1

(a)

θ1

(b)

Fig. 5.3 (a) Multi-material wedge space; (b) Multi-material wedge

1 θ0

x1

5.2


129

No matter which kinds of multi-material wedges are treated, all of them should consider the continuity conditions across the interface. For multi-material wedge spaces the displacement and traction continuity across each interface θ = θk , k = 1, 2, . . . , n, between two dissimilar wedges can be written as uk (θk ) = uk+1 (θk ), φk (θk ) = φk+1 (θk ), k = 1, 2, . . . , n − 1, un (θ0 ) = u1 (θ0 ), φn (θ0 ) = φ1 (θ0 ),

(5.58)

where the subscript k is used to denote the quantities pertaining to the kth wedge. Note that in (5.58) the traction continuity condition has been replaced by the stress function continuity because traction is related to stress function by (3.32) and along each interface only r varies and θ =constant. Like the bimaterials discussed in Section 4.3.2, to consider the concentrated force and dislocation a logarithmic function is selected for the complex function vector f(z). Moreover, to have a single-valued displacements and stresses, a branch cut at the interface of θ = θ0 and θ = θ0 + 2π is introduced. With this choice, the additional conditions that should be satisfied are ˆ φn (r, θ0 + 2π ) − φ1 (r, θ0 ) = p, ˆ un (r, θ0 + 2π ) − u1 (r, θ0 ) = b.

(5.59)

By following the steps described in (4.84), (4.85), and (4.86) for the displacement and traction continuity across the interface θ = θk−1 between the (k–1)th wedge and the kth wedge, the real-form expressions for the displacement and stress function vectors of the kth wedge can be written as ! " (k) (k) (k) (k) / N1 (θ ) − / N1 (θk−1 ) h + / N2 (θ ) − / N2 (θk−1 ) g + uˆ k , ! " (k) (k) (k)T (k)T φk = ln rg + π / N3 (θ ) − / N3 (θk−1 ) h + / N1 (θ ) − / N1 (θk−1 ) g + φˆ k , (5.60)

uk = ln rh + π

where uˆ k and φˆ k are constant vectors which do not alter the stresses. Employing (5.60) and successively checking the displacement continuity for each interface and letting uˆ 1 = 0, we get uˆ k = π

k−1 !

i=1 k−1 !

" (i) (i) (i) (i) / N1 (θi ) − / N1 (θi−1 ) h + / N2 (θi ) − / N2 (θi−1 ) g

" θi (i) (i) N1 (ω)dω h + θi−1 N2 (ω)dω g i=1 θ θ = θ0k−1 N1 (ω)dω h + θ0k−1 N2 (ω)dω g, k = 1, 2, . . . , n.

=

θi θi−1

(5.61)

Note that in the last equation of (5.61) the values of N1 (ω) and N2 (ω) are dictated by the values of ω, i.e., they are calculated by using the material properties where ω locates. Similarly, we have

130


φˆ k =

θk−1 θ0

N3 (ω)dω h +

θk−1 θ0

NT1 (ω)dω g,

k = 1, 2, . . . , n.

(5.62)

Substituting (5.61) and (5.62) into (5.60), we get u = ln rh +

N1 (ω)dω h +

θ θ0

φ = ln rg +

θ

θ0

N3 (ω)dω h +

N2 (ω)dω g,

θ θ0

θ θ0

NT1 (ω)dω

(5.63) g.

Note that in (5.63) the subscript k of u and φ have been omitted because (5.63) is valid for all wedges. The fundamental matrices N1 (ω), N2 (ω), and N3 (ω) are the generalization of their counterparts defined in (3.76c) for homogeneous materials. They are now defined for multi-materials, i.e., their values depend not only on ω but also on the associated material properties where ω locates. Substituting (5.63) into the concentrated force and dislocation conditions (5.59), we have 2π

S∗ H∗ −L∗ S∗T

h bˆ = , g pˆ

(5.64a)

Where S∗ =

1 2π

0

2π

N1 (ω)dω, H∗ =

1 2π

0

2π

N2 (ω)dω, L∗ = −

1 2π

2π

N3 (ω)dω. 0

(5.64b)

Note that in deriving (5.64a), we have used the properties that N1 (ω), N2 (ω), and N3 (ω) are periodic in ω with periodicity 2π , not π for homogeneous materials. Use of the relation (4.82) can then give us /p)/2π ˆ h = −(/ Sbˆ + H ,

ˆ . g = (/ Lbˆ − / ST p)/2π

(5.65)

/ and / It should be pointed out that the definitions of / S, H, L given in (4.82) have two times difference with those used in Ting (1996).

5.2.4 Multi-material Wedges Consider a multi-material wedge with free–free surfaces subjected to a concentrated force pˆ applied at r=0. The continuity, traction-free, and equilibrium conditions of this problem can be written as

5.3

Stress Singularities

uk (θk ) = uk+1 (θk ),

131

φk (θk ) = φk+1 (θk ),

k = 1, 2, . . . . . . , n − 1,

φn (θn ) = 0, φ1 (θ0 ) = 0, ˆ φn (r, θn ) − φ1 (r, θ0 ) = p.

(5.66)

As that described in the previous section, by employing the continuity conditions we can obtain the solutions (5.63) for the displacement and stress function vectors of all wedges. With the results of (5.63), the traction-free condition will then lead to g=0. Thus, u = ln rh + φ=

θ θ0

θ

θ0

N1 (ω)dω h,

(5.67)

N3 (ω)dω h.

Substitution of (5.67)2 into the equilibrium condition shown in the last equation of (5.66), we get h=

−1

θn θ0

N3 (ω)dω

ˆ p.

(5.68)

Equations (5.67) and (5.68) provide the solution for the multi-material wedge with free–free sides subjected to a concentrated force pˆ at the wedge apex. Because the results of (5.63) and (5.65) for the multi-material wedge spaces and (5.67) and (5.68) for the multi-material wedges are valid for any number of wedges and no restriction has been made for the wedge angles, it is reasonable to expect that these results are also applicable for angularly inhomogeneous spaces and wedges. A different approach provided by Ting (1989) proves that the solutions for the inhomogeneous spaces and wedges really have exactly the same form as those given in (5.63), (5.65), (5.67), and (5.68).

5.3 Stress Singularities The stress singularity generally occurs at the location of discontinuity. The discontinuity may come from geometries, materials, or loads, of which the typical examples are, respectively, cracks, multilayer media, or point forces. Due to the extremely high stresses near the points of discontinuity, failure is usually initiated at such locations. The study of stress singularity is generally helpful for the understanding of failure initiation. The nature of stress singularity for the above typical examples has been investigated by many researchers and is illustrated in standard texts such as Anderson (1991), Jones (1974), and Johnson (1985). Geometrical discontinuity other than cracks, which has also been studied vastly, is the singular stresses at wedge corners. They have been studied for single wedges by Williams

132


(1952), England (1971b), Bogy (1972), Stern and Soni (1976), Chen (1994), etc., and for bonded wedges by Bogy (1968, 1971), Dundurs (1969), Hein and Erdogan (1971), Theocaris (1974), Lin and Hartmann (1989), Reedy (1990, 1993), Chen and Nishitani (1992), Ding et al. (1994), Ting (1997), Berger et al. (1998), Chen (1998), and Desmorat and Leckie (1998), etc. However, due to the mathematical difficulties, most of the results leave the solutions to a system of simultaneous algebraic equations. Only Ting (1997) provides the explicit closed-form solutions by the introduction of a transfer matrix. Later, Hwu et al. (2003) and Hwu and Lee ˆ stated in (2004) introduced a key matrix for each wedge, which is the matrix N Section 3.4.3. This matrix contains the information of material properties and wedge angles and is related to the fundamental elasticity matrix N, and the transfer matrix is a power function of the key matrix. Therefore, no matter how many wedges are bonded together, the closed-form solutions for the orders of stress singularity can ˆ for bonded wedges. Because the easily be constructed by simple multiplication of N ˆ the results final expressions are quite simple and are dominated by the matrix N, may be useful for the understanding of failure initiation of bonded wedges and will be presented in Section 5.3.1 for the multi-material wedge spaces and in Section 5.3.2 for the multi-material wedges. In addition to the orders of stress singularity, its associated field deformation and stress distribution near the wedge apex, or called the eigenfunction of the wedges, can also be expressed in terms of the key matrix ˆ and will be presented in Section 5.3.3. Special cases such as a single wedge N (includes a semi-infinite crack in the homogeneous medium), bi-wedge (includes an interfacial crack along the bimaterial interface), and tri-wedge (includes a crack terminating at the bimaterial interface) will then be presented in Section 5.3.4. General Formulation To consider the stress singularity at wedge apex, the complex function vector f(z) of the general solution (3.24) is assumed as > g, f(z) =< z1−δ α

(5.69)

where δ is the singular order to be determined from the boundary conditions and g is its associated coefficient vector. Since the stress is proportional to the first derivative of the stress function f(z), if Re(δ) > 0 the stress at the wedge apex will be singular. However, when Re(δ) > 1 the strain energy of the elastic wedge may become unbounded. Thus, in the following derivation, we will only be interested in the region 0 < Re(δ) < 1.

(5.70)

If the singular order is a complex number, by using the representation given in (3.24) and (5.69) the singular order for the function with variable z¯α will be its complex conjugate. For example, the representation for u may be written as ¯

u = A < z1−δ > g + A < z¯α1−δ > g¯ . α

(5.71)

5.3


133

It means that to ensure the displacements to be real values the singular orders should come in pair of complex conjugate if they are complex. By superposition of this pair, the displacement u may be rewritten as ¯

¯

> g + A < z¯α1−δ > g¯ + A < zα1−δ > gc + A < z¯1−δ > g¯ c . (5.72) u = A < z1−δ α α With this understanding, for the convenience of latter derivation the displacement and stress function vectors given in (3.24) and (5.69) for each wedge may be expressed as u = A < z1−δ > g+A < z¯1−δ > h, α α φ = B < z1−δ > g+B < z¯1−δ > h, α α

(5.73)

where g and h are two complex coefficient vectors to be determined through the satisfaction of boundary conditions. If the singular order is a real value, g and h should be complex conjugate to keep the displacements and stresses to be real. Generally, they are not necessary to be complex conjugate. The final real values of the displacements and stresses will come from the superposition with another set of ¯ solution whose singular order is δ. To describe the boundary conditions of wedge problems it is appropriate by using the polar coordinate system (r, θ ) where the origin is located at the wedge apex and x1 = r cos θ ,

x2 = r sin θ .

(5.74)

μˆ α (θ ) = cos θ +μα sin θ .

(5.75)

Thus, from (3.21) we have zα = rμˆ α (θ ),

Substituting (5.75) into (5.73), the displacement and stress function vectors may be written as ! " ¯ˆ 1−δ (θ ) > h , u = r1−δ A < μˆ 1−δ α (θ ) > g + A < μ α (5.76a) ! " ¯ˆ 1−δ (θ ) > h , φ = r1−δ B < μˆ 1−δ (θ ) > g + B < μ α α or in matrix form, ( ' 0 < μˆ 1−δ g u α (θ ) > 1−δ A A . =r 1−δ φ BB 0 < μ¯ˆ α (θ ) > h

(5.76b)

Key Matrix We know that μα is the eigenvalue of the fundamental elasticity matrix N, and hence the diagonalization of N can be achieved by (3.61). Generalized relation for μα (θ ) ˆ introduced and N(θ ) is shown in (3.93). Similar to (3.61) and (3.93), the matrix N in (3.87) has the following diagonalization relation:

134


ˆ 1−δ

N

(' ( ' 1−δ 0 BT AT A A < μˆ α (θ , α) > (θ , α) = T T , 1−δ BB 0 < μ¯ˆ α (θ , α) > B A

(5.77)

in which 1 − δ may be integer (positive or negative or zero), real or complex. For ˆ we get the orthogonality relation (3.57a). As we will see the matrix zero power of N, ˆ plays an important role for the determination of the singular order of the multiN material wedges, and hence is called a key matrix for the wedge. Back to (5.76b) for the general expressions of displacement and stress function vectors of wedges. If we let ' T T( B A u g , = T T p, w = φ h B A

(5.78)

where p is a 6 × 1 complex coefficient vector and w is a 6 × 1 vector containing displacements and stress functions, from (5.77) and (5.78) we see that (5.76b) can ˆ as be rewritten in terms of the key matrix N ˆ 1−δ (θ )p. w = r1−δ N

(5.79)

5.3.1 Multi-material Wedge Spaces Consider a multi-material wedge space as shown in Fig. 5.3a. By using the newly introduced vector w defined in (5.78), the displacement and traction continuity conditions shown in (5.58) can now be written as wk (θk ) = wk+1 (θk ),

k = 1, 2, . . . , n − 1,

w1 (θ0 ) = wn (θn ),

(5.80)

in which θn = θ0 and the subscript k is used to denote the quantities pertaining to the kth wedge. To find a solution satisfying (5.80), we first consider the two boundaries of the kth wedge, from (5.79) we have ˆ 1−δ (θk−1 )pk , wk (θk−1 ) = r1−δ N k ˆ 1−δ (θk )pk , wk (θk ) = r1−δ N k

k = 1, 2, . . . , n.

(5.81)

Insertion of pk obtained from (5.81)1 into (5.81)2 and use of the relation (3.96b)2 lead to wk (θk ) = Ek wk (θk−1 ), where

k = 1, 2, . . . , n,

(5.82a)

5.3


135

ˆ 1−δ (θk , θk−1 ). Ek = N k

(5.82b)

Note that Ek defined in (5.82b) is the transfer matrix introduced by Ting (1997), and its submatrices can be obtained from the relation (5.77). With the relation (5.82a) for the two boundaries of each wedge, and the continuity conditions required in (5.80), it is possible for us to get the equation stating the relation between any two boundaries of the multi-material wedges. By setting k=n for (5.82) as the first equation of the follow-up iteration and repeated using (5.80)1 and (5.82), we can obtain wn (θn ) = Ke w1 (θ0 ),

(5.83a)

where n

Ke = En−k+1 = En En−1 . . . E1 ,

(5.83b)

k=1

By using the continuity condition across the interface θ = θ0 , (5.80)2 , equation (5.83) will now lead to [Ke − I]w1 (θ0 ) = 0.

(5.84)

Nontrivial solutions for w1 exist only when Ke − I = 0.

(5.85)

The roots of the determinant obtained from (5.85) provide the orders of stress singularity δ.

5.3.2 Multi-material Wedges We now consider a multi-material wedge for which the surfaces at θ = θ0 and θn are traction free or fixed. The Free–Free Wedge When both of the surfaces at θ = θ0 and θn are traction free, in addition to the continuity conditions stated in (5.80)1 , the boundary conditions on the outer edges of the wedge can be expressed as φ1 (θ0 ) = φn (θn ) = 0.

(5.86)

The results (5.83a) obtained from the continuity condition can be rewritten as

136

5 Wedges and Interface Corners (2) un (θn ) = K(1) e u1 (θ0 ) + Ke φ1 (θ0 ), (4) φn (θn ) = K(3) e u1 (θ0 ) + Ke φ1 (θ0 ),

(5.87)

(i)

where Ke , i = 1, 2, 3, 4, are the submatrices of Ke defined by ' ( (1) (2) Ke Ke Ke = (3) (4) . Ke Ke

(5.88)

Substituting (5.86) into (5.87)2 , we get (3) un (θn ) = K(1) e u1 (θ0 ), 0 = Ke u1 (θ0 ).

(5.89)

Nontrivial solutions for u1 (θ0 ) exist only when ) ) ) (3) ) )Ke ) = 0,

(5.90)

which will provide the singular orders for the free–free multi-material wedges. The Fixed-Fixed Wedge When both of the surfaces at θ = θ0 and θn are fixed, the boundary conditions can be expressed as u1 (θ0 ) = un (θn ) = 0.

(5.91)

Substituting (5.87)1 into (5.91) and following a similar argument as the previous case, we can get the singular order from ) ) ) (2) ) )Ke ) = 0.

(5.92)

The free–fixed and fixed–free wedges When one of the surfaces at θ = θ0 and θn is fixed and the other is free, their boundary conditions and corresponding results are free - fixed: φ1 (θ0 ) = un (θn ) = 0; fixed - free: u1 (θ0 ) = φn (θn ) = 0;

) ) ) ) ⇒ δ : )K(1) e ) = 0, ) ) ) ) ⇒ δ : )K(4) e ) = 0.

(5.93)

5.3.3 Eigenfunctions The general solutions for the displacements and stresses in the field near the wedge apex can be obtained from (5.79) for each different wedge. The unknown coefficient

5.3


137

vectors pk in (5.79) can be found by following the steps described in Sections 5.3.1 and 5.3.2. Through the use of (5.81)1 , the coefficient vectors pk can be expressed in terms of wk (θk−1 ). Similar to (5.83), repeated application of (5.80)1 and (5.82) may help us to express wk (θk−1 ) in terms of w1 (θ0 ). Thus, the field solutions (5.79) near the wedge apex, or called the eigenfunctions, can now be written as wk (r, θ ) = E∗k (θ )(Ke )k−1 w1 (θ0 ), k = 1, 2, 3, . . . , n,

(5.94a)

where ˆ 1−δ (θ , θk−1 ), E∗k (θ ) = N k k−1

(Ke )0 = I, (Ke )k−1 = Ek−i = Ek−1 Ek−2 . . . E1 for k = 2, 3, . . . , n.

(5.94b)

i=1

Note that from the definitions given in (5.82b), (5.83b), and (5.94b), we see that E∗k (θk ) = Ek , (Ke )n = Ke .

(5.95)

From (5.79) we know that w1 (θ0 ) can be expressed by w1 (θ0 ) = r1−δ w0 ,

(5.96)

where w0 is a 6 × 1 coefficient vector. With (5.96), the near-tip field solutions (5.94) can be rewritten as wk (r, θ ) = r1−δ E∗k (θ )(Ke )k−1 w0 , k = 1, 2, 3, . . . , n.

(5.97)

By solving the eigenvalue problems presented in (5.84) for the multi-material wedge spaces, or in (5.89) for the free–free multi-material wedges, the shapes of w1 (θ0 ) can be obtained as the eigenvectors of these problems. It should be noted that the expressions shown in (5.94) are the solutions corresponding to the singular order δ. If the singular order is a complex number, a conjugate part should be superimposed to get the real values for the displacements and stresses. If the singular order is a repeated root and no enough independent eigensolutions have been obtained, the logarithmic singularity should be considered (Ting, 1996). Moreover, if one concerns not only the near-apex field but also the entire domain of the multimaterial wedges, the solutions corresponding to all the singular and non-singular orders should be superimposed.

5.3.4 Special Cases A Single Wedge If only a single wedge is considered, the wedge space treated in Section 5.3.1 is a trivial problem because it is just a homogeneous space. No singularity will occur for

138


this trivial condition, and the solution shown in (5.85) will be satisfied automatically since Ke = I for any δ. For the single wedge with free–free surfaces, by using (5.88), (5.83b), (5.82b), and (5.77), we get 1−δ

T K(3) ˆ 1−δ ˆ α (θ1 , θ0 ) > B e =B B +B < μ

T

(5.98)

The equation for getting the singular orders can then be obtained by substituting (5.98) into (5.90). After finding the singular orders from (5.90), the displacements and stresses near the wedge apex obtained from (5.97) can also be simplified as u u(θ0 ) u 1−δ 1−δ ˆ 1−δ ˆ = r N (θ , θ0 ) 0 . = N (θ , θ0 ) 0 φ φ(θ0 )

(5.99)

In getting the second equality of (5.99), we have used the prescribed traction-free boundary condition (5.86)1 , and u0 is the eigenvector of eigenequation (5.89)2 . A semi-infinite crack in a homogeneous anisotropic medium can be represented by letting θ0 = −π and θ1 = π for the single wedge with free–free surfaces. With this special wedge angle, the solutions (5.98) and (5.99) can be further simplified. One may refer to Section 7.1.1 for detailed results of the simplification. Due to the similarity, the solutions for a single wedge with fixed–fixed, fixed–free, and free–fixed surfaces can also be obtained easily and will not be discussed here. Bi-wedge Consider a bi-wedge bonded together by two dissimilar wedges. Substituting n = 2 into (5.83b) and using (5.82b) we get ˆ 1−δ (θ2 , θ1 )N ˆ 1−δ (θ1 , θ0 ). Ke = N 2 1

(5.100)

By using this expression, the solutions for the singular order can be obtained directly from (5.85), (5.90), (5.92), and (5.93) for different boundary conditions. Similar to the derivation of (5.99), the displacements and stresses near the wedge apex obtained from (5.97) can be simplified as u1 ˆ 1−δ (θ , θ0 ) u0 , = r1−δ N 1 0 φ1 u2 u0 1−δ 1−δ ˆ ˆ = r1−δ N . (θ , θ ) N (θ , θ ) 1 1 1 0 2 φ2 0

(5.101)

Figure 5.4 shows the singular orders of an isotropic bi-wedge consisting of two wedges with the same wedge angles. The material properties of the wedges are EI = 100 GPa, EII = 10 GPa, and νI = νII = 0.3. A check point of this figure is α = 180◦ which corresponds to an interfacial crack and its solution 0.5 and 0.5 + iε

Stress Intensity Factors of Interface Corners

139 –0.20

Singular order δ (Real part, solid line)

0.5

0.4 –0.15 α

I

0.3 α

–0.10 II

0.2 –0.05 0.1

0.0 90

120

Wedge angle α

150

Singular order δ (Imagianary part, dot line)

5.4

0.00 180

Fig. 5.4 Order of stress singularity for an isotropic bi-wedge (Hwu et al., 2003)

(ε is the oscillation index of bimaterials) are exactly the same as those shown in (7.13). Tri-wedge Consider a tri-wedge bonded together by three dissimilar wedges. Substituting n = 3 into (5.83b) and using (5.82b) we get ˆ λ (θ3 , θ2 )N ˆ λ (θ2 , θ1 )N ˆ λ (θ1 , θ0 ). Ke = N 3 2 1

(5.102)

Similar to the bi-wedge problems, the solutions for singular orders and near-apex stress field can all be obtained by substituting (5.102) into (5.85), (5.90), (5.92), (5.93), and (5.97) for different boundary conditions. To see the numerical results for the singular orders, one may refer to Hwu et al. (2003) in which several different wedge combinations and its applications to the electric devices have been presented.

5.4 Stress Intensity Factors of Interface Corners If a structure, in meter, millimeter, or micrometer, or even smaller scale, is composed of many different parts and each part is made by different materials and may have different shapes, it is very possible that many interface corners exist in several local fields of the structure. Due to the mismatch of elastic properties, stress singularity usually occurs near the interface corners, which may initiate failure of structures. Therefore, it is important to design a proper joint shape to prevent the failure initiation and propagation. The singular order of stresses near the interface corners

140


is a good index for understanding failure initiation. However, in engineering applications one usually feels only the knowledge of singular orders is not enough for the prediction of failure initiation. The most apparent examples are homogeneous cracks whose singular order is –1/2 which is a constant value and is nothing to do with the surrounding environment and outside loading of cracks. These influential factors are reflected through another important parameter – stress intensity factor. Therefore, in addition to the singular orders one is always interested to know their associated stress intensity factors of interface corners. Although several detailed studies have been done about the determination of the singular orders and their associated stress intensity factors for interface corners, very few failure criteria were successfully established based upon these parameters. Even the cracks in homogeneous media or the cracks lying along the interface between two dissimilar materials are the special cases of the interface corners, the definitions of stress intensity factors proposed in the literature are usually not consistent with that of cracks. Therefore, to have a universal failure criterion for the homogeneous cracks, interface cracks, and interface corners, a unified definition for the stress intensity factors is indispensable. In the literature the stress intensity factors of interface corners are usually defined by the way similar to the homogeneous cracks, e.g., Sinclair et al. (1984) and Dunn et al. (1997), which may encounter trouble when the stress distributions near the interface corners exhibit the oscillatory characteristics like the interface cracks discussed in Rice (1988), Wu (1990), Suo (1990), Gao et al. (1992), and Hwu (1993). Thus, even for the interface cracks some definitions of their stress intensity factors proposed in the literature are not compatible with the conventional definitions for homogeneous cracks. To build a direct connection among the homogeneous cracks, interface cracks, and interface corners, Hwu and Kuo (2007) proposed a unified definition for the stress intensity factors, which is based upon the near-tip field solutions presented in Section 5.3.3. With the analytical solutions obtained in Section 5.3.3 for the multi-material anisotropic wedges, the near-tip solutions for the general interface corners can be divided into five different categories depending on whether the singular order is distinct or repeated, real or complex, and will be presented in Section 5.4.1. A unified definition for the stress intensity factors of interface corners will then be presented in Section 5.4.2. According to the experience of crack problems, finding a stable and accurate approach to calculate the stress intensity factors is also important. By the definition of stress intensity factors presented in Section 5.4.2, to calculate their values we need to know the stresses near the tip of interface corners. However, due to the singular and possibly oscillatory behaviors of the near-tip solutions, it is not easy to get convergent values for the stress intensity factors directly from the definition. To overcome this problem, a path-independent H-integral (Stern et al., 1976; Sinclair et al., 1984; Labossiere and Dunn, 1999) based on reciprocal theorem of Betti and Rayleigh is introduced in Section 5.4.3 to compute the possibly mixed-mode stress intensity factors, in which the complementary solutions needed for calculation can be obtained from the solutions presented in Section 5.3.3. By using the H-integral, the complexity of stresses around the tip of

5.4


141

interface corners can then be avoided. Modification of H-integral to suit for threedimensional interface corners will be presented in Section 5.4.4. Numerical examples of interface corners between two dissimilar materials will then be illustrated in Section 5.4.5.

5.4.1 Near-Tip Field Solutions To study the singular behavior of interface corners and to provide a proper definition for their associated stress intensity factors, like the concept of fracture mechanics it is important to know the near-tip field solutions. By employing Stroh formalism for anisotropic elasticity, the near-tip field solutions for multi-material anisotropic wedges have been obtained in (5.97), and can now be expanded as ! " ∗(1) (1) ∗(2) (3) uk (r, θ ) = r1−δ Ek (θ )(Ke )k−1 + Ek (θ )(Ke )k−1 u0 , ! " ∗(3) (1) ∗(4) (3) φk (r, θ ) = r1−δ Ek (θ )(Ke )k−1 + Ek (θ )(Ke )k−1 u0 , ∗(i)

(5.103a)

where Ek (θ ) and (Ke )k−1 are the submatrices of E∗k (θ ) and (Ke )k−1 defined by (i)

'

( ∗(1) ∗(2) E (θ ) E (θ ) k k E∗k (θ ) = , ∗(3) ∗(4) Ek (θ ) Ek (θ )

(Ke )k−1

' ( (1) (2) (Ke )k−1 (Ke )k−1 = . (3) (4) (Ke )k−1 (Ke )k−1

(5.103b)

From (5.103a)2 and the definitions given in (5.77) and (5.94), (5.95), and (5.96), we see that the singular orders are totally determined through the material properties and configurations of all wedges. Since we consider the singular fields and the strain energy cannot be unbounded, only the values located in the range of 0 < Re(δ) < 1 are considered. If more than one δ locate in this range, we select the one whose real part is maximum as δc , i.e., the one with the most critical singular order δc . If δ)c is a) complex number, it has been proved that its conjugate δ¯c is also a root of ) (3) ) )Ke ) = 0. When r → 0, i.e., the near-tip field, the terms associated with δc will dominate the stress behavior. Neglecting all the other singular and non-singular terms, and expanding the near-tip solution (5.103) for the terms associated with δc , we may express the displacement and stress function vectors in terms of the eigenvector u0 obtained from (5.89)2 for free–free edges. However, an eigenvalue δc may correspond to several linearly independent eigenvectors u0 . If δc is a non-repeated root, only one arbitrary scalar is needed to describe u0 . When δc is a double root, two arbitrary scalars are needed. While for a triple root δc , three arbitrary scalars are needed. If δc is complex, the arbitrary scalar associated with u0 is also complex which contains two real scalars. With the above understanding, the near-tip solutions (5.103a) may now be rewritten as follows:

142


Case 1: δc is distinct and real, δc = δR , u(r, θ ) = cr1−δR η(θ ), φ(r, θ ) = cr1−δR λ(θ ).

(5.104a)

Case 2: δc is double and real, δc = δR , u(r, θ ) = r1−δR {c1 η1 (θ ) + c2 η2 (θ )}, φ(r, θ ) = r1−δR {c1 λ1 (θ ) + c2 λ2 (θ )}.

(5.104b)

Case 3: δc is triple and real, δc = δR , u(r, θ ) = r1−δR {c1 η1 (θ ) + c2 η2 (θ ) + c3 η3 (θ )}, φ(r, θ ) = r1−δR {c1 λ1 (θ ) + c2 λ2 (θ ) + c3 λ3 (θ )}.

(5.104c)

Case 4: δc is distinct and complex, δc = δR ± iε, u(r, θ ) = r1−δR {criε η(θ ) + cr−iε η(θ )}, φ(r, θ ) = r1−δR {criε λ(θ ) + cr−iε λ(θ )}.

(5.104d)

Case 5: one is real δR and the others are complex δR ± iε, u(r, θ ) = r1−δR criε η1 (θ ) + cr−iε η1 (θ ) + c3 η3 (θ ) , φ(r, θ ) = r1−δR criε λ1 (θ ) + cr−iε λ1 (θ ) + c3 λ3 (θ ) .

(5.104e)

In the above, η(θ ) and λ(θ ) (or ηi (θ ) and λi (θ ), i=1,2,3) are functions related to ∗(i) (i) Ek (θ ), (Ke )k−1 , and u0 , in which the number of arbitrary scalars is dependent on the multiplicity of δc . Note that the solutions shown in (5.104a–e) are valid for any wedge of the multi-material wedges, and hence from now on unless special notification is needed the subscript k denoting the wedge has been neglected for simplicity. From (5.104a–e), we see that if we consider only the most critical singular order and disregard the possibility of logarithmic singularity, by superimposing all the eigenfunctions with the same real part of singular orders, without loss of generality the near-tip solutions can be further rewritten as u(r, θ ) = r1−δ R V(θ ) < riεα > c, where

φ(r, θ ) = r1−δ R (θ ) < riεα > c, V(θ ) = [η1 (θ ) η2 (θ ) η3 (θ )], (θ ) = [λ1 (θ ) λ2 (θ ) λ3 (θ )].

(5.105a)

(5.105b)

Note that the near-tip solutions shown above are not only valid for the interface corners between two dissimilar anisotropic materials but also valid for those among

5.4


143

more than two materials. The most critical singular order may be one real simple root, or one real double root with two independent eigenfunctions, or one real triple root with three independent eigenfunctions, or a pair of complex conjugate roots, or the combination of one real simple root and a pair of complex conjugate roots, etc. In other words, there are some cases where only one or two coefficients exist for the most critical singular order δ. For example, c1 = 0, ε1 = 0, c2 = c3 = 0 if δ is a real and simple root; c1 , c2 = 0, ε1 = ε2 = 0, c3 = 0 if δ is a real and double root; and c2 = c¯ 1 , ε2 = −ε1 , c3 = 0 if δ is a complex and simple root. In these cases, the missing eigenfunctions can be selected to be the one orthogonal with the existing one or simply taken to be zero. If they are taken to be zero, some of the matrices composed of them such as V(θ ) and (θ ) will contain one or two zero columns and become singular and hence their inverse will not exist in the common mathematical operation. With this understanding, if the inversion of such matrix is required, inverse it only in its submatrix to avoid the problem of missing eigenfunctions. For example, ⎡ ⎤ ⎤ ⎡ ⎤−1 ⎡ ⎤−1 ⎡ −1 d −b 0 ab0 a00 a 00 1 ⎣0 0 0⎦ = ⎣ 0 0 0⎦ , ⎣c d 0⎦ = ⎣−c a 0⎦ , = ad − bc, . . . . 0 0 0 000 000 0 00 Furthermore, sometimes it is not necessary to perform the inversion if the matrix can be multiplied by its own inverse, e.g., < (r/)−iεα > −1 = I when ε1 = ε2 = ε3 = 0. With the general expression given in (5.105), the near-tip stress distribution can then be obtained by substituting (5.105a)2 into (3.37), which leads to ⎧ ⎫ ⎨σrθ ⎬ σθθ = (θ )φ,r (r, θ ) = r−δR (θ ) (θ ) < (1 − δR + iεα )riεα > c, ⎩ ⎭ σθ3

(5.106a)

where ⎡

cos θ (θ ) = ⎣ − sin θ 0

sin θ cos θ 0

⎤ 0 0⎦. 1

(5.106b)

Along θ = 0, (0) = I and we let (0) = = [λ1 λ2 λ3 ]. The coefficient c can then be defined as c = lim rδR < (1 − δR + iεα )−1 r−iεα > −1 φ, r (r, 0). r→0

(5.107)

From (5.106a) with θ = 0, we see that the coefficient < (1 − δR + iεα ) > c can be thought as the intensity of singularity of the stresses (σrθ , σθθ , σθ3 ) in the direction of λ1 , λ2 , and λ3 . However, as a consequence of the peculiar singularity, c has an awkward physical unit. A remedy suggested by Rice (1988) is to appeal to

144


√ a fixed length and use the combination 2π < (1 − δR + iεα )iεα > c as the basic parameter which has the dimension and the scale of conventional stress intensity factors. Even so, the stress intensity factors still cannot be reduced to classical stress intensity factors for a crack tip in a homogeneous medium, because the directions of λ1 , λ2 , and λ3 are usually not the same as the direction of the crack. To have a comparable definition, transformation pre-multiplied by is suggested in Hwu (1993) for the interface crack problems. With this understanding, a proper definition of stress intensity factors for interface corners will be given in the next section.

5.4.2 A Unified Definition It is known that a semi-infinite crack in homogeneous materials can be represented by letting θ0 = −π and θ1 = π for a single wedge. Moreover, an interface crack can be represented by a bi-wedge with θ0 = −π , θ1 = 0, and θ2 = π . These two important special cases indicate that to propose a proper definition for the stress intensity factors of interface corners, it is better to review the corresponding definition for the crack problems. A conventional definition for the stress intensity factors k of a crack in homogeneous media is (Broek, 1974) ⎧ ⎫ ⎫ ⎧ ⎨σrθ ⎬ ⎨KII ⎬ √ √ (5.108) = lim 2πr σθθ = lim 2π rφ, r , k = KI ⎩ ⎭ r→0 ⎭ r→0 ⎩ KIII σ θ=0 θ=0 θz in which θ = 0 is a line along the crack. Due to the oscillatory behavior of the stresses near the tip of interface cracks, this definition cannot be applied to the cracks lying on the bimaterial interface. A proper definition for the bimaterial stress intensity factors has been given by Hwu (1993a,b) as ⎫ ⎧ ⎫ ⎧ ⎨σrθ ⎬ ⎨KII ⎬ √ KI = lim 2π r < (r/)−iεα > −1 σθθ , (5.109a) ⎭ r→0 ⎩ ⎭ ⎩ KIII σ θ=0 θz or in matrix form k = lim

r→0 θ=0

√

2π r < (r/)−iεα > −1 φ, r ,

(5.109b)

where = [λ1 λ2 λ3 ].

(5.110a)

εα , λα , α = 1, 2, 3, are the eigenvalues and eigenvectors of ∗

(M∗ − e2π ε M )λ = 0,

(5.110b)

5.4


145

in which M ∗ is the bimaterial matrix related to the material eigenvector matrices Ai , Bi , i = 1, 2 (see Appendix A.2). In (5.109), is a length parameter which may be chosen arbitrarily as long as it is held fixed when specimens of a given material pair are compared. Different values of will not alter the magnitude of k but will change its phase angle. In application, the reference length is usually selected to be the crack length. Considering the discussions at the end of the previous section and the consistency of the definitions between cracks, interface cracks, corners, and interface corners, a unified definition for the stress intensity factors has been proposed (Hwu and Kuo, 2007) as follows: ⎫ ⎧ ⎫ ⎧ ⎨σrθ ⎬ ⎨KII ⎬ √ KI = lim 2π rδR < (r/)−iεα > −1 σθθ , (5.111a) ⎭ r→0 ⎩ ⎭ ⎩ KIII σθz θ=0 or in matrix form, k = lim

√

r→0 θ=0

2π rδR < (r/)−iεα > −1 φ, r (r, θ ).

(5.111b)

Note that from the definition given in (5.111), we see that < (r/)−iεα > −1 should be real in order to have a real value of k, which has been proved analytically in Hwu (1993) for the interface cracks lying between two anisotropic elastic materials. For the general multi-wedges, only numerical check has been done. Comparison between (5.107) and (5.111b) leads to the following relation: k=

√

2π < (1 − δR + iεα )iεα > c.

(5.112)

Using the relation (5.112), the near-tip solution shown in (5.105) and (5.106) can now be rewritten in terms of the stress intensity factors k as u(r, θ ) = φ(r, θ ) = φ, r (r, θ ) =

√1 r 1−δR V(θ ) < (1 − δR + iεα )−1 (r/)iεα 2π √1 r 1−δR (θ ) < (1 − δR + iεα )−1 (r/)iεα 2π √1 r −δR (θ ) < (r/)iεα > −1 k. 2π

> −1 k, > −1 k,

(5.113)

In practical applications if only the most critical singular order of stresses is considered, certain modes of stress intensity factors will vanish (Kuo and Hwu, 2009). However, the lost of certain modes of stress intensity factors does not mean that it will not fracture by that mode since the stresses associated with the next critical singular order may dominate the failure behavior. With this understanding, sometimes it is necessary to compute the stress intensity factors for the lower singular orders. To provide a proper definition for the stress intensity factors associated with the lower singular orders, the stress field near the wedge apex may be expressed as φ, r (r, θ ) = φc,r (r, θ ) + φ2,r (r, θ ) + . . . ,

(5.114a)

146


where 1 φc,r (r, θ ) = √ r−δR (θ ) < (r/)iεα > −1 k, 2π (2) 1 −δ (2) φ2,r (r, θ ) = √ r R 2 (θ ) < (r/)iεα > −1 2 k2 , 2π

(5.114b)

and the subscript c denotes the value associated with the most critical singular order and the subscript 2 or the superscript (2) denotes the value associated with the second critical singular order. By the way similar to k, the stress intensity factors k2 associated with the second critical singular order can be defined as k2 = lim

r→0

√

(2)

(2)

2π rδR 2 < (r/)−iεα > −1 2 [φ, r (r, 0) − φc, r (r, 0)].

(5.115)

Similar definitions can also be applied to any other lower order terms. From the above discussion, we see that the definition for the stress intensity factors proposed in (5.111) is applicable not only to the interface corners but also to the cracks in homogeneous media or bimaterial interfaces. The conventional definition (5.108) is just a special case of (5.111) with δR = 1/2 and ε = 0, while the definition for the bimaterial stress intensity factor (5.109) is a special case of (5.111) with δR = 1/2. With this unified definition, it becomes possible that the failure criteria developed for the crack problems may be useful for the prediction of the failure of interface corners. Moreover, the fracture toughness measured from the standard crack specimen may also have a direct connection with the toughness of interface corners. In order to further unify the definitions of stress intensity factors associated with the most critical singular order and the lower singular orders shown in (5.111) and (5.115), we can generalize the near-tip solutions (5.105a) to include the complete singular orders as follows: u(r, θ ) = V(θ ) < r1−δα > c, φ(r, θ ) = (θ ) < r1−δα > c,

(5.116)

where δ1 , δ2 , and δ3 can be distinct or repeated, real or complex. The coefficient c defined in (5.107) can then be modified as c = lim < (1 − δα )−1 rδα > −1 φ, r (r, 0). r→0

(5.117)

Thus, it looks promising that the stress intensity factors associated with the complete singular orders can be defined as ⎧ ⎫ ⎧ ⎫ ⎨KII ⎬ ⎨σrθ ⎬ √ KI = lim 2π < (r/)δα > −1 σθθ , ⎩ ⎭ r→0 ⎩ ⎭ KIII σθz θ=0

(5.118a)

5.4


147

or in matrix form, k = lim

√

r→0 θ=0

2π < (r/)δα > −1 φ, r (r, θ ).

(5.118b)

Till now, no report according to the definition (5.118) has been presented. All the following discussions will then be based upon the definitions given in (5.111) and (5.115).

5.4.3 H-Integral for Two-Dimensional Interface Corners According to the definition of the stress intensity factors proposed in (5.111), to calculate their values we need to know the stresses near the tip of interface corners. However, due to the singular and possibly oscillatory behaviors of the near-tip solutions (5.103) for multi-material wedges, it is not easy to get convergent values for the stress intensity factors directly from the definition (5.111). To overcome this problem, several path-independent integrals have been proposed for the special cases of interface corners such as J-integral (Rice, 1968), L-Integral (Choi and Earmme, 1992), M-integral (Im and Kim, 2000), and H-integral (Sinclair et al., 1984) for crack problems. Since these integrals have a special feature that they are independent of paths, the complexity of stresses around the crack tip can then be avoided. The interface corners are usually in the status of mixed-mode intensity. Thus, employing H-integral to compute the stress intensity factors defined in (5.111) may be a good choice. The path-independent H-integral is based on the reciprocal theorem of Betti and Rayleigh (Sokolnikoff, 1956). It states that if an elastic body is subjected to two systems of body and surface forces, then the work that would be done by the first system in acting through the displacements due to the second system of forces is equal to the work that would be done by the second system in acting through the displacements due to the first system of forces. If we choose the first system to be the (actual) one we consider, and the second system to be the complementary (or called virtual) one. In the absence of body forces, this theorem can be written in the following form: .

(uT ˆt − uˆ T t)ds = 0,

(5.119)

C

where u and t are the displacement and traction vectors of the actual system, and uˆ and ˆt are those of the complementary system. C is any closed contour in a simply connected region, which is selected to be Cε + C1 + CR + C2 as shown in Fig. 5.5. Because the two outer surfaces of the multi-material wedges are considered to be free of tractions, t = ˆt = 0 along C1 and C2 , and hence,

148

5 Wedges and Interface Corners x2

k

n CR ′ C1 i

Traction-free C2

i–1

Cε

1

x1

2

C = Cε + C 1 + CR ′ + C2 : a closed contour C R = − CR ′ : in a counterclockwise direction

Fig. 5.5 Schematic diagram of H-integral contour

(uT ˆt − uˆ T t)ds = − Cε

(uT ˆt − uˆ T t)ds = CR

(uT ˆt − uˆ T t)ds,

(5.120)

CR

where both Cε and CR are the paths emanate from the lower wedge flank (θ = θ0 ) to the upper wedge flank (θ = θn ) counterclockwisely. In other words, the H-integral defined by H=

(uT ˆt − uˆ T t)ds,

(5.121)

is path-independent for free–free multi-material wedges when the path emanates from θ0 and terminates on θn in counterclockwise direction. By shrinking the inner path Cε inside the region dominated by the singular field and making a judicious choice for the complementary solution, we can get an analytical expression for the H-integral in terms of the coefficients ci (or simply c) which have a direct relation with the stress intensity factors as shown in (5.112). Thus, if one can evaluate the H-integral from the other path far from the tip, through the path-independent property shown in (5.120) we can calculate the stress intensity

5.4


149

factors. With this understanding, we will now try to find the suitable complementary solutions and then derive formulas for the coefficients ci of each case shown in (5.104a–e). Since the integral path can be selected arbitrarily from the lower wedge flank θ0 to upper wedge flank θn , for simplicity we choose a circular counterclockwise path through the region dominated by the singular field. Along this path, the traction t = φ, θ /r, which has been shown in (3.37), so (5.121) becomes H=

θn θ0

(uT φˆ , θ − uˆ T φ, θ )dθ.

(5.122)

Case 1: Substituting the near-tip solution (5.104a) into (5.122), we see that the suitable complementary solution, which will make the H-integral be independent of r, should be the one with eigenvalue -(1 − δR ), i.e., ˆ θ ) = cˆ rδR −1 λ(θ ˆ ), ˆ θ ) = cˆ rδR −1 η(θ ˆ ), φ(r, u(r,

(5.123)

ˆ ) can be obtained from (5.103a) with u0 determined by (5.89)2 ˆ ) and λ(θ where η(θ whose eigenvalue is δR − 1. Thus, it is important to know whether δ–1 is also an (3) eigenvalue of Ke u0 = 0 when 1-δ is. Since the explicit expression of the deter(3) minant of matrix Ke is quite complicated, it is not easy to perform rigorous proof. Instead, numerical check has been done in Hwu and Kuo (2007), which shows that (3) when 1-δ is a root of Ke u0 = 0, so is δ–1. Substituting (5.104a) and (5.123) into (5.122) with cˆ = 1, we get c = H ∗−1 H,

(5.124a)

where H∗ =

θn

θ0

{ηT (θ )λˆ (θ ) − ηˆ T (θ )λ (θ )}dθ.

(5.124b)

The prime • in (5.124b) denotes differentiation with respect to θ . Note that formula (5.124) is derived from the path through the singular field. By the path-independent property of H-integral, the value H appeared in (5.124a) can now be evaluated using (5.121) through any convenient path Γ far away from the tip. In (5.121), the displacement u and traction t of the actual state can be obtained from any other methods such as finite element or boundary element method, while uˆ and ˆt of the virtual state ˆ are from the complementary solution (5.123) whose cˆ = 1 and ˆt = ∂ φ/∂s. Case 2: Similar to the discussion of case 1, the suitable complementary solution will be the one associated with eigenvalue δR − 1, i.e.,

150


ˆ θ ) = rδR −1 {ˆc1 ηˆ 1 (θ ) + cˆ 2 ηˆ 2 (θ )}, u(r, ˆ θ ) = rδR −1 {ˆc1 λˆ 1 (θ ) + cˆ 2 λˆ 2 (θ )}. φ(r,

(5.125)

Substituting (5.104b) and (5.125) into (5.122) with cˆ 1 = 1, cˆ 2 = 0, and cˆ 1 = 0, cˆ 2 = 1, respectively, we get ∗ ∗ H1 = c1 H11 + c2 H12 , ∗ ∗ H2 = c1 H21 + c2 H22 ,

(5.126a)

where Hij∗ =

θn θ0

!

" ηTj (θ )λˆ i (θ ) − ηˆ Ti (θ )λj (θ ) dθ ,

i, j = 1, 2,

(5.126b)

and H1 is the value calculated from (5.121) with cˆ 1 = 1, cˆ 2 = 0 for the complementary solution, while H2 is the one associated with cˆ 1 = 0, cˆ 2 = 1. From (5.126), the coefficient ci can now be evaluated by matrix inversion as ∗ ∗ −1 H11 H12 H1 c1 = . ∗ H∗ c2 H21 H2 22

(5.127)

Case 3: Similar to case 2, the complementary solution can be written as ˆ θ ) = r−λR {ˆc1 ηˆ 1 (θ ) + cˆ 2 ηˆ 2 (θ ) + cˆ 3 ηˆ 3 (θ )}, u(r, ˆ θ ) = r−λR {ˆc1 λˆ 1 (θ ) + cˆ 2 λˆ 2 (θ ) + cˆ 3 λˆ 3 (θ )}. φ(r,

(5.128)

Substituting (5.104c) and (5.128) into (5.122) with cˆ 1 = 1, cˆ 2 = 0, cˆ 3 = 0, and cˆ 1 = 0, cˆ 2 = 1, cˆ 3 = 0, and cˆ 1 = 0, cˆ 2 = 0, cˆ 3 = 1, respectively, and then by matrix inversion, we get ⎧ ⎫ ⎡ ∗ H11 ⎨c1 ⎬ ∗ c2 = ⎣ H21 ⎩ ⎭ ∗ c3 H31

∗ H12 ∗ H22 ∗ H32

⎤−1 ⎧ ⎫ ∗ H13 ⎨H1 ⎬ ∗ ⎦ H23 H2 , ⎩ ⎭ ∗ H33 H3

(5.129)

in which Hij∗ has the same definition as (5.126b) for i, j = 1, 2, 3; H1 is the value calculated from (5.121) with cˆ 1 = 1, cˆ 2 = 0, cˆ 3 = 0 for the complementary solution, H2 is the one associated with cˆ 1 = 0, cˆ 2 = 1, cˆ 3 = 0, while H3 is the one associated with cˆ 1 = 0, cˆ 2 = 0, cˆ 3 = 1. Case 4: Each term of the solution shown in (5.104d), although its value is complex, satisfies all the basic equations and boundary conditions for the multi-material wedges. In this sense, the coefficient c, even is complex, can also be evaluated by the way of

5.4


151

case 1. That is, with the complementary solution ˆ θ ) = cˆ r−(1−δR +iε) λ(θ ˆ ), ˆ θ ) = cˆ r−(1−δR +iε) η(θ ˆ ), φ(r, u(r,

(5.130)

c is related to the H-integral by (5.124a), i.e., c = H ∗−1 H, where H ∗ and H are, respectively, calculated from (5.124b) and (5.121) with cˆ =1. Similarly, the coefficient c can also be calculated by (5.124a) with a complementary solution having exponent – (1–δR –iε) of r. Since c and c are complex conjugate, their results calculated separately by (5.124) can be used as a check for correctness. Case 5: As the explanation described for case 4, in this case 1–δR and 1–δR ±iε can be treated as three distinct roots. The complementary solutions associated with eigenvalues 1–δR and 1–δR + iε have been shown, respectively, in (5.123) and (5.130). Their associated coefficients c3 and c can therefore be evaluated separately by (5.124). The coefficient c associated with eigenvalue 1–δR − iε is then obtained by taking the conjugate of c. From the above discussion, we see that the coefficients ci (or simply c) can be evaluated from the relations (5.124), (5.127), (5.129), or (5.130) through the pathindependent H-integral. A general matrix form for these relations can then be shown as c = H∗−1 h,

(5.131a)

where ⎧ ⎫ ⎧ ⎫ ⎡ ∗ ∗ ∗⎤ H11 H12 H13 ⎨H1 ⎬ ⎨c1 ⎬ ∗ H∗ H∗ ⎦ , c = c2 , h = H2 , H∗ = ⎣H21 22 23 ⎩ ⎭ ⎩ ⎭ ∗ H∗ H∗ c3 H3 H31 32 33

(5.131b)

and the dimensions of vector c, matrix H∗ , and vector h depend on whether the singular order is distinct or repeated, real or complex as those described in (5.104). The component Hij∗ of H∗ is calculated through (5.126b), whereas the component Hi of h is calculated from the H-integral defined in (5.121) through any convenient path Γ emanating from θ0 and terminating on θn in counterclockwise direction. When calculating Hi through (5.121), u and t of the actual state can be obtained from any other methods, while uˆ and ˆt of the virtual state is from the complementary solution such as (5.123), (5.125), (5.128) with cˆ i = 1 and cˆ j = 0, j = i. Combining (5.112) and (5.131), the relation between the stress intensity factors k and the H-integral h is obtained as k=

√

2π < (1 − δR + iεα )iεα > H∗−1 h.

(5.132)

With this relation, the stress intensity factors defined in (5.108) can be computed in a stable and efficient way no matter what kind of interface corners is considered.

152


5.4.4 H-Integral for Three-Dimensional Interface Corners Same as the two-dimensional interface corner problems, in order to avoid the singular and possibly oscillatory behaviors of near corner tip stresses the most appropriate approach to calculate the stress intensity factors proposed in (5.111) is via the path-independent integral. In the previous section, we derived a pathindependent H-integral to compute the stress intensity factors for two-dimensional interface corner problems. Here, we like to extend our work to derive a threedimensional version of H-integral to deal with three-dimensional interface corner problems. With the Cauchy’s formula (1.1) for the tractions, the reciprocal theorem of Betti and Rayleigh shown in (5.119) can be rewritten as (σˆ ij ui − σij uˆ i )nj dS = 0, i, j = 1, 2, 3,

(5.133)

S

where nj is the surface normal, σij and ui are the stresses and displacements of the actual system, while σîj and uˆ i are those of the complementary system. S is any closed surface contour surrounding the corner front, which is selected to be S = So +Sc +Sl (Fig. 5.6). So is a region on the plane x3 = 0 bounded by Cε , C1 , C2 , and CR with an outward unit normal vector n = (0, 0, -1); Sl is a region on the plane x3 = l bounded by the contours similar to Cε , C1 , C2 , and CR with an outward unit normal vector n = (0, 0, 1); Sc = Sε + S1 + SR + S2 in which Sε , SR , S1 , and S2

x2 C1

SR

Cε

x3

′

C2

Sε

Sl

CR x1

l

S = S o + Sc +Sl : a closed surface contour

Sc = Sε + S1 +S R ′ + S 2 CR = − CR′ : in a counterclockwise direction

Fig. 5.6 Cornered body and its constitutive surface and contours

So

′

5.4


153

are, respectively, curved boundary surfaces extended by Cε , CR , C1 , and C2 from x3 = 0 to x3 = l. If we reverse the integration direction of SR and denote it by SR to conform with that of Sε , i.e., both of these two surface integrals are integrated from x3 = 0 to x3 = l and from the lower flank (θ = θ0 ) to the upper flank (θ = θn ), the outward unit normal vectors of SR and Sε will then be denoted by (n1 , n2 , 0) and (–n1 , –n2 , 0). With S = So + Sc + Sl , Sc = Sε + S1 + SR + S2 , the normal directions stated above, and the traction-free condition on surfaces S1 and S2 , (5.133) can now lead to (σˆ ij ui − σij uˆ i )nj dS = (σˆ ij ui − σij uˆ i )nj dS + (σˆ i3 ui − σi3 uˆ i )dS. (5.134) Sε

Sl −So

SR

Replacing dS by ldC for Sε and SR , dividing l on both sides of (5.134) and taking limit of l to zero, we get (σˆ ij ui − σij uˆ i )nj dC (5.135) = (σˆ ij ui − σij uˆ i )nj dC + (σˆ i3,3 ui + σˆ i3 ui,3 − σi3,3 uˆ i − σi3 uˆ i,3 )dS. Cε

CR

So

With the relations So = SoR − Soε and σij nj = ti in which ti stands for surface traction, (5.135) can be rewritten as

(ui ˆti − uˆ i ti )dC +

Cε

Soε

(ui ˆti − uˆ i ti )dC +

= CR

(σˆ i3, 3 ui + σˆ i3 ui, 3 − σi3, 3 uˆ i − σi3 uˆ i, 3 )dS

(5.136)

SoR

(σˆ i3, 3 ui + σˆ i3 ui, 3 − σi3, 3 uˆ i − σi3 uˆ i, 3 )dS,

which means that the following H-integral is domain independent: H=

(uT ˆt − uˆ T t)dC +

(σˆ i3, 3 ui + σˆ i3 ui, 3 − σi3, 3 uˆ i − σi3 uˆ i, 3 )dS.

(5.137)

S

In (5.137), Γ is the path emanating from θ0 and terminating on θn in counterclockwise direction, and S is the region enclosed by the path Γ and the two free edges of interface corners. Equation (5.137) is the three-dimensional version of H-integral for interface corner problems. In order to use the domain-independent property of H-integral to calculate the stress intensity factors, like the two-dimensional interface corner problems we first shrink the inner path Cε into the region dominated by the singular field and make a judicious choice for the integral path (a circular counterclockwise path) and the complementary solution (a near-tip solution associated with the singular order

154


−(1 − δR + iεα )). Along the circular path, the traction t = φ, θ /r, dC = rdθ , and dS = rdθ dr, so (5.137) becomes H=

θn

θ0

r (uT φˆ , θ − uˆ T φ, θ )dθ + 0

θn θ0

(σˆ i3, 3 ui + σˆ i3 ui, 3 − σi3, 3 uˆ i − σi3 uˆ i, 3 )rdθ dr. (5.138)

Substituting the near-tip solution (5.105) and its associated complementary solution into (5.138), we get H = cˆ T H∗ c,

(5.139a)

where H∗ =

θn θ0

T

ˆ T (θ ) (θ )]dθ . ˆ (θ )V(θ ) − V [

(5.139b)

Note that in (5.139) no contribution from the second term of (5.138) has been made since the near-tip solution and its associated complementary solution employed in (5.138) are those for two-dimensional problems whose differentiation with respect to x3 is equal to zero. Moreover, the terms related to the complex singular orders such as < r−iεα > and < riεα > pre- and postmultiplying H∗ are deleted since H∗ is a diagonal matrix in those cases (detailed later in (5.142)). If Hi , i = 1, 2, 3, denote the value of H with cˆ i = 1 and cˆ j = 0, j = i, a system of simultaneous algebraic equations with unknown ci , i = 1, 2, 3 can be constructed from (5.139a) and be solved as c = H∗−1 h, where

⎧ ⎫ ⎨H1 ⎬ h = H2 . ⎩ ⎭ H3

(5.140a)

(5.140b)

Substituting (5.140) into (5.112), same relation as that shown in (5.132) for twodimensional interface corner can be obtained as follows: k=

√

2π < (1 − δR + iεα )iεα > H∗−1 h.

(5.141)

Note that (5.140) and (5.141) are general expressions for the computation of the stress intensity factors concerning the most critical singular order, in which the singular order can be distinct or repeated, real or complex. The components Hij∗ of H∗ are calculated through (5.139b), which can be proved to be Case 1: when the most critical singular order is a real simple root, ∗ H11 = 0, all other Hij∗ = 0.

5.4


155

Case 2: when the most critical singular order is a real double root, ∗ ∗ = Hj3 = 0, j = 1, 2, 3. Hij∗ = 0, j = 1,2, in general; H3j

Case 3: when the most critical singular order is a real triple root, Hij∗ = 0, j = 1, 2, 3, in general.

(5.142)

Case 4: when the most critical singular order is a pair of complex root, ∗ ∗ H11 , H22 = 0, all other Hij∗ = 0.

Case 5: when the most critical singular order is a combination of one real simple root and one pair of complex root, ∗ ∗ ∗ , H22 , H33 = 0, all other Hij∗ = 0. H11

From (5.139b) we see that components Hij∗ of H∗ are calculated through the two-dimensional near-tip solutions provided in (5.105) or (5.113), whereas through the domain-independent property proved in (5.136) the components Hi of h can be calculated from the three-dimensional version of H-integral defined in (5.137) through any convenient path Γ emanating from θ0 and terminating on θn in counterclockwise direction. When calculating Hi through (5.137), u and t of the actual three-dimensional state can be obtained from any other methods, while uˆ and ˆt of the virtual state is from the two-dimensional complementary solution with cˆ i = 1 and cˆ j = 0, j = i. The relation (5.141) tells us that the stress intensity factors k is influenced locally by and H∗ , and globally by h. In other words, the corner angles and material properties of the interface corners which influence the singular orders will affect the stress intensity factors k locally through and H∗ , whereas the effects of external loads or structural geometry will be reflected through h. For example, for a center crack or edge crack in homogeneous materials, their and H∗ are the same since their local environments are the same, while their h are different because their external geometries are different.

5.4.5 Numerical Examples A unified definition for the stress intensity factors of cracks, interface cracks, and interface corners is proposed in (5.111). No matter what kind of stress singularity occurs for the general interface corners/cracks, (distinct or repeated, real or complex), an efficient and stable approach of calculating the stress intensity factors is suggested by using the relation (5.132) with the path-independent H-integral defined in (5.121) for two-dimensional problems or (5.137) for three-dimensional problems. Detailed studies about the convergency and efficiency of the H-integral as well as the

156


validity of the range and shape of the path have been done through several different kinds of examples (Hwu and Kuo, 2007). To illustrate the versatility of the unified definition, several different examples have also been implemented in Hwu and Kuo (2007) for two-dimensional interface corners and in Hwu and Kuo (2010) for threedimensional interface corners, such as two dimensional: (1) cracks in homogeneous isotropic or anisotropic materials, (2) central or edge notch in isotropic materials, (3) interface cracks between two dissimilar isotropic materials, and (4) interface corners between two dissimilar materials; and three dimensional: (1) an elliptical central crack in a homogeneous isotropic material, (2) a through thickness edge crack in a homogeneous isotropic material, (3) a through thickness edge notch in a homogeneous isotropic material, (4) a penny-shaped interface crack between two dissimilar isotropic materials, (5) a through thickness interface corner between two dissimilar materials, and (6) a representative block of electronic package subjected to tensile loading. To save the space of this section, only one representative example of interface corners is shown here. One may refer to Hwu and Kuo (2007) and Kuo and Hwu (2010) for the details of all the other examples. Example: Interface corners between two dissimilar materials, two dimensional or three dimensional Consider an interface corner between two dissimilar materials subjected to uniform tension σ = 10 MPa at far ends. Figure 5.7a is a schematic diagram of the present example for two-dimensional case, whereas Fig. 5.7b is for threedimensional case. In our example, d = 5mm, W/d = 3, h/W = 1/15, L/W = 6, and α = 30◦ for both two-dimensional and three-dimensional interface corners, and the thickness t is assumed to be infinitely large or small for plane strain or plane stress conditions. For three-dimensional case, x3 = ±t/2 are the free surfaces where σo

σ t

W L 2

Mat. 2 d h

α Mat. 1

(a)

L 2

d 1 = , W 3 1 h = , W 15 1 d = , L 18 α = 30o.

W

L/2

Mat. 2

x2 h

x1 x3 d

L/2

α Mat. 1

(b)

Fig. 5.7 Interface corner between two dissimilar materials (a) two dimensional; (b) three dimensional

5.4


157

thickness t = 2W. The material above the interface is isotropic whose elastic properties are E = 10GPa, ν = 0.2, while the other portion is orthotropic whose elastic properties are E11 = 134.45 GPa, E22 = E33 = 11.03 GPa, G12 = G13 = 5.84 GPa, G23 = 2.98 GPa, ν12 = ν13 = 0.301, ν23 = 0.49. For three-dimensional case, the boundary conditions used in the analysis are u1 = 0 on the surface x1 = d, u2 = 0 on the surface x2 = −L/2, and u3 = 0 on the surface x3 = 0. The singular orders of this example are obtained as 0.467 ± 0.0367i and 0.439 accompanied with one pair of complex conjugate eigenfunctions and one real eigenfunction. Because the real part of the complex singular orders is higher than the real singular order, 0.467 ± 0.0367i will be treated as the most critical singular order. The reference length used in the definition (5.111) is selected to be 10 mm. The numerical results of stress intensity factors calculated through H-integral show that if only the most critical singular order is considered mode-III stress intensity factor will be lost no matter which kind of loading is applied, which will then be picked up by considering the next higher singular order. Figure 5.8 shows the results

KI(MPa.mm 0.4667)

70

10

65

KI(present) KII(present)

5

KIII(present) KI=72.0(plane strain) KII=9.8(plane strain) KIII=0.0(plane strain)

60

0 0.0

0.1

0.2

0.3

0.4

KII(MPa.mm0.4667), KIII (MPa.mm0.4394)

15

75

0 0.5

x3/t Fig. 5.8 Stress intensity factor versus corner front location for a through thickness interface corner (three-dimensional case) (Kuo and Hwu, 2010)

158

5 Wedges and Interface Corners 100 KI KII

SIF. (MPa*mm 0.467)

80

60

40

20

0

–20 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Radius of H-integral path (mm).

Fig. 5.9 Stress intensity factors for two-dimensional interface corner (plane strain) (Hwu and Kuo, 2007)

of KI , KII , and KIII versus x3 /t for three-dimensional case. From this result we see that the stress intensity factors in the inner part of the interface corner will approach to certain constant values not far away from those of the corresponding plane strain problems shown in Fig. 5.9, which is acceptable from the engineering viewpoint.

Chapter 6

Holes

The problem of determining stress distributions induced by holes has aroused considerable interests for almost half a century. Of various holes the elliptic shape and its geometric limits such as circles and cracks have evoked the most interest among researchers. If the stress concentration is of more concern, the optimization in a class of hole shapes has shown that the optimized hole is not necessarily the elliptic type. However, due to the requirement of one-to-one mapping, most of the solutions found in the literature either by Lekhnitskii or Stroh formalism are valid only for the elliptical holes. Exact analytical solutions have been found for various shapes of holes if the materials are restricted to the isotropic ones (Savin, 1961). For general anisotropic media with holes of arbitrary curvilinear shapes, most of the solutions found in the literature are not exact but approximate (Hwu 1990a, 1992b). Because both of the exact and approximate solutions are useful for the engineering applications, in this chapter the elliptical holes and the general polygon-like holes will be discussed. Moreover, several different loading conditions will be considered such as uniform loading at infinity, pure in-plane bending at infinity, arbitrary loading along the hole boundary, point load and dislocation at arbitrary location.

6.1 Elliptical Holes Consider an elliptical hole in an infinite anisotropic medium. The contour of the hole boundary is represented by Fig. 6.1(a) x1 = a cos ψ,

x2 = b sin ψ,

(6.1)

where a, b are the half lengths of the major and minor axes of the ellipse and ψ is a real parameter. Since the elliptical hole boundary in the z-plane will map to three different slanted elliptical hole boundary in the zα -plane (see Fig. 6.1b), it is not convenient to solve the problems with elliptical holes by using the argument zα shown in the general solution (3.20) or (3.23) or (3.24). To transform all three different slanted elliptical hole boundary into the same hole boundary in the shape of a unit circle |ζ | = 1 (see Fig. 6.1c), we may assume the transformation function as C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_6, C Springer Science+Business Media, LLC 2010

159

160

6 Holes x2

k=1 k=2 k=3

b x1

a

x1=kacosψ x2=kbsinψ

(a) Im(ζk )

Im(zk)

k=1 k=2 k=3

k=1 k=2 k=3

1

a

2a

3a

–1

Re( zk )

1

Re(ζk )

–1

(c)

(b)

Fig. 6.1 Mapping of elliptical hole among (a) z-plane; (b) zα -plane; and (c) ζ α -plane (b/a = 0.6, μ1 = 2.13i, μ2 = 1.0698 + 2.1646i, μ3 = −1.0698 + 2.1646i)

zα = a1 ζα + a2 ζα−1 ,

α = 1, 2, 3.

(6.2a)

Substituting (6.1) into (6.2a) for the boundary, we have (a cos ψ + μα b sin ψ) = a1 (cos ψ + i sin ψ) + a2 (cos ψ − i sin ψ)

(6.2b)

and hence a1 = (a − iμα b)/2,

a2 = (a + iμα b)/2.

(6.2c)

Combining (6.2a) and (6.2c), and taking inverse we get 1 (a − ibμα )ζα + (a + ibμα ) , ζα + zα + z2α − a2 − b2 μ2α ζα = , α = 1, 2, 3. a − ibμα zα =

1 2

(6.3)

6.1

Elliptical Holes

161

To check whether the transformation given in (6.3) is single-valued, we calculate the roots of the equation, dzα /dζα = 0, which are at * ζαo

=±

a + ibμα √ = ± mα eiθα , a − ibμα

(6.4)

√ where mα and θα denote, respectively, the modulus and argument of the critical √ points ζαo . Since mα < 1, which can easily be proved if the imaginary part of μα has been set to be positive, the transformation is single-valued outside the elliptic hole. Although the inside region is double-valued, no particular treatment is required for the hole problems because no real body is inside the hole. When we consider the inclusion problems, special treatment becomes important for the transformation of the inside region, which will be discussed in detail in Chapter 8. From the above discussion, we know that for the problems with elliptical holes the complex functions fα (zα ) are better to be expressed in terms of the arguments ζα . Hence, the general solution (3.24) can be rewritten as u = 2 Re{Af(ζ )},

φ = 2 Re{Bf(ζ )},

(6.5a)

where f(ζ ) = [f1 (ζ1 ), f2 (ζ2 ), f3 (ζ3 )]T .

(6.5b)

Before solving the hole problems for each different loading conditions discussed in the following sections, we like to mention that all the solutions obtained in this section are also valid for their corresponding crack problems because a straight crack can be made by letting the minor axis of the ellipse be zero. Detailed discussions and solutions for the corresponding crack problems will be presented in Section 7.2.

6.1.1 Uniform Loading at Infinity Consider an infinite anisotropic elastic plate containing an elliptical hole subjected to uniform loading at infinity (see Fig. 6.2). The uniform load is applied such that the stresses at infinity are uniformly distributed. Thus, the displacement and stress function vectors at infinity, u∞ and φ∞ , which have been obtained in (4.2), can be expressed as ∞ u∞ = x1 ε∞ 1 + x2 ε2 ,

where t∞ 1

∞ φ∞ = x1 t∞ 2 − x2 t1 ,

⎧ ∞⎫ ⎧ ∞⎫ ⎧ ∞⎫ ⎧ ∞⎫ σ σ ε ε ⎪ ⎪ ⎪ ⎪ ⎨ 11 ⎪ ⎨ 21 ⎪ ⎨ 11 ⎪ ⎨ 21 ⎪ ⎬ ⎬ ⎬ ⎬ ∞ ∞ ∞ ∞ ∞ ∞ ∞ σ σ ε ε = , t2 = , ε1 = , ε2 = . 12 22 12 22 ⎪ ⎪ ⎪ ⎪ ⎩σ ∞ ⎪ ⎩σ ∞ ⎪ ⎩2ε∞ ⎪ ⎩2ε∞ ⎪ ⎭ ⎭ ⎭ ⎭ 13 23 13 23

(6.6a)

(6.6b)

162

6 Holes

σ 22∞

σ 13∞

•

•

•

•

•

σ 23∞

•

σ 12∞

σ 12∞

σ 12∞ σ 13∞

⊗

•

σ

∞ 11

x2

⊗

σ 11∞

•

⊗

b

θ

⊗

x1

s

•

⊗

•

n

a •

σ 23∞

⊗

⊗

⊗

⊗

⊗

σ 12∞

•

⊗

⊗

∞ σ 22

Fig. 6.2 An infinite anisotropic plate containing an elliptical hole subjected to uniform loading at infinity

In the above, σij∞ are given and εij∞ can be determined by using the stress–strain relation (4.1a). The simplified solutions for some special loading conditions such as the unidirectional tension, biaxial loading, pure shear, and anti-plane shear have been listed in (4.6), (4.7), (4.8), and (4.9) for the plane strain condition. For the case of generalized plane stress, all we need to do is just replace the reduced compliances Sˆ ij by the compliances Sij . If the hole boundary surface is traction free, the boundary conditions of the present problem can be expressed as φ = 0, along the hole boundary . φ → φ∞ , at infinity

(6.7)

Note that in getting (6.7), the relation between the traction and the stress function vector shown in (3.32) has been used and the integration constants denoting the rigid body motion have been neglected. In order to satisfy the boundary conditions (6.7), the solution can be assumed in the following form:

6.1

Elliptical Holes

163

! " u = u∞ + 2 Re A < ζα−1 > q , ! " φ = φ∞ + 2 Re B < ζα−1 > q ,

(6.8)

where q is the coefficient vector to be determined through the satisfaction of the boundary conditions. Note that in the selection of the solution form (6.8), we first consider the infinity condition (6.7)2 . To satisfy the infinity condition, we add u∞ and φ∞ into the general solution (6.5a) and let the complex function f(ζ ) of (6.5a) be a polynomial of ζα with a negative power. The power order can then be determined by the consideration of the traction-free boundary condition (6.7)1 with ζα = eiψ , x1 = a cos ψ, and x2 = b sin ψ. To determine the unknown complex coefficient vector q, we first write down the expressions for the displacement u and stress function vectors φ along the hole boundary. Substituting (6.6) into (6.8) using ζα = eiψ, x1 = a cos ψ, and x2 = b sin ψ, we get ∞ u = Re e−iψ (aε∞ 1 + ibε2 + 2Aq) , (6.9) ∞ φ = Re e−iψ (at∞ 2 − ibt1 + 2Bq) . With the result of (6.9)2 , the traction-free boundary condition (6.7)1 now gives us 1 ∞ q = − B−1 (at∞ 2 − ibt1 ). 2

(6.10)

Substituting (6.10) into (6.9) and using the identity (3.131)1 , the real-form solution for the displacement along the hole boundary can be obtained as −1 ∞ −1 ∞ ∞ −1 ∞ −1 ∞ u = cos ψ(aε∞ 1 + aSL t2 + bL t1 ) + sin ψ(bε2 − bSL t1 + aL t2 ). (6.11)

Hoop Stress Let s and n denote, respectively, the unit vector tangent and normal to the hole boundary (Fig. 6.1a). For the present problem, the prescribed condition along the hole boundary is traction free, i.e., tn = 0 (or say σnn = σns = σn3 = 0) for ζα = eiψ . Therefore, along the hole boundary the only stress component which should be concerned is the hoop stress σss . To calculate σss , we use the formula obtained in (3.36a), i.e., σss = sT ts = −sT φ, n ,

(6.12a)

sT = (cos θ , sin θ , 0),

(6.12b)

where

and the angle θ is directed counterclockwise from the positive x1 -axis to the direction of s, which is related to the contour parameter ψ by

164

6 Holes

ρ cos θ = −a sin ψ,

ρ sin θ = b cos ψ,

(6.13a)

where ρ 2 = a2 sin2 ψ + b2 cos2 ψ.

(6.13b)

The formula (6.12a) shows that to find the hoop stress we first need to calculate ts , which may be obtained by the direct differentiation of φ with respect to n and will be shown below. An alternative approach of calculating ts is through its relation with u , which can be found in Ting (1996). The calculation of φ, n can be performed by using the chain rule in which ∂ζ −1 ∂ζα ∂ψ ∂ζα−1 = α ∂n ∂ζα ∂ψ ∂zα

∂zα ∂x1 ∂zα ∂x2 + , ∂x1 ∂n ∂x2 ∂n

(6.14a)

where along the hole boundary, ζα = eiψ ,

∂ζα−1 ∂ζα = −ζα−2 = −e−2iψ , = ieiψ , ∂ζα ∂ψ

∂zα = −a sin ψ + μα b cos ψ = ρ (cos θ + μα sin θ ) , ∂ψ ∂x2 ∂zα ∂x1 ∂zα = − sin θ , = cos θ , = 1, = μα . ∂n ∂n ∂x1 ∂x2

(6.14b)

Substituting each term of (6.14b) into (6.14a) and using relation (3.70), we get ie−iψ ∂ζα−1 =− μα (θ ). ∂n ρ

(6.15)

By using the result of (6.15), the differentiation of φ with respect to n can then be performed on the stress function given in (6.8) with (6.6a) and (6.10). The result is " 1 ! −iψ ∞ Re ie B < μα (θ ) > B−1 (at∞ 2 − ibt1 ) . ρ (6.16) Using the generalized form of identity (3.137)3 and relation (6.13a), the real-form solution of φ,n along the hole boundary can be obtained as ∞ φ,n = − sin θ t∞ 2 − cos θ t1 +

φ, n = sin θ

!

G1 (θ )t∞ 1 −

a " b ∞ ∞ ∞ I+ G3 (θ ) t2 −cos θ I + G3 (θ ) t1 + G1 (θ )t2 , b a (6.17a)

where G1 (θ ) = NT1 (θ ) − N3 (θ )SL−1 ,

G3 (θ ) = −N3 (θ )L−1 .

(6.17b)

6.1

Elliptical Holes

165

Substituting (6.17a) into (6.12a) and using identities (3.105), a simplified real-form solution for the hoop stress can now be written as ! " b a ∞ T ∞ ∞ − i G , σss = iT1 G1 (θ )t∞ + (θ )t (θ )t − (θ )t G G 3 1 3 2 1 2 1 2 a b

(6.18a)

where iT1 = (1 0 0),

iT2 = (0 1 0).

(6.18b)

For the case of unidirectional tension shown in (4.6), the hoop stress can be further simplified as σss /σ ∞ = −nT (α)G1 (θ )s(α) + cT (α)G3 (θ )s(α),

(6.19a)

where cT (α) =

a b cos α, sin α, 0 . a b

(6.19b)

Employing the explicit expressions given in (3.82) and (3.86) for the isotropic materials, the hoop stress (6.19) is reduced to σss a b cos θ cos α + = −1 + 2 1 + sin θ sin α cos(θ − α), σ∞ a b

(6.20)

which is identical to the one shown in Muskhelishvili (1954). With this result, when the elliptic hole is subjected to a uniform tensile stress perpendicular to x1 -axis, α = π/2, the stress concentration at the end points of the major axis of the hole, θ = ±π/2, equals 1+2b/a which is a well-known factor for isotropic plates (Timoshenko and Goodier, 1970). For circular holes, the solutions can simply be obtained from the formulas given in (6.18), (6.19), and (6.20) with a = b. With this substitution, many equations % & can be simplified such as (6.20) which becomes σss = σ ∞ −1 + 4 cos2 (θ − α) .

6.1.2 In-Plane Bending at Infinity Consider an infinite anisotropic elastic plate containing an elliptical hole subjected to pure bending M in a direction at an angle α with the positive x1 -axis (Fig. 6.3). The in-plane bending is applied such that the stresses at infinity are distributed in the form as shown in (4.10). As discussed in Section 6.1.1, the boundary conditions of the present problem can be expressed by (6.7) and hence its associated solution can be assumed as

166

6 Holes x2

Fig. 6.3 An infinite anisotropic plate containing an elliptical hole subjected to pure bending M at infinity

M

α

b x1

M a

! " u = u∞ + 2 Re A < ζα−2 > q , ! " φ = φ∞ + 2 Re B < ζα−2 > q ,

(6.21a)

where u∞ and φ∞ are the displacement and stress function vectors at infinity, which have been shown in (4.11) as u∞ =

−M (sin αu1 + cos αu2 ), 2I

φ∞ =

−M (x1 sin α − x2 cos α)2 s(α), (6.21b) 2I

where ⎧ ⎨

⎫ γ1 x12 − γ2 x22 ⎬ u1 = (γ6 x1 + 2γ2 x2 )x1 , ⎩ ⎭ (γ5 x1 + 2γ4 x2 )x1

⎧ ⎫ ⎨−(2γ1 x1 + γ6 x2 )x2 ⎬ γ1 x12 − γ2 x22 u2 = ⎩ ⎭ −γ4 x22

(6.21c)

and γi = Sˆ 1i cos2 α + Sˆ 2i sin2 α + Sˆ 6i cos α sin α, γ = γ4 sin α + γ5 cos α.

i = 1, 2, 4, 5, 6

(6.21d)

To determine the unknown complex coefficient vector q, we first replace it by two real constant vectors g and h as q = AT g + BT h.

(6.22)

Substituting (6.21b) and (6.22) into (6.21a) with the boundary values ζα = cos ψ + i sin ψ, x1 = a cos ψ, and x2 = b cos ψ, and using identities (3.59), the traction-free boundary condition (6.7)1 will then give us

6.1

Elliptical Holes

g=

167

Ma2 c2 (α)s(α), 2I

h=

& Ma2 −1 % L s2 (α)I + c2 (α)ST s(α), 2I

(6.23a)

where c2 (α) =

1 [1 − c2 − (1 + c2 ) cos 2α], 4

s2 (α) =

−c sin 2α, 2

c=

b . a

(6.23b)

With this result and identities (3.59), the complex coefficient vector q can be further simplified as q=

Ma2 [c2 (α) + is2 (α)]B−1 s(α). 4I

(6.24)

With the results of (6.21) and (6.24), by following the same procedure as that discussed in Section 6.1.1, the full-field solution and the hoop stress along the hole boundary for the present problem can be expressed as (Hwu, 1990a, 1992b) " Ma2 ! Re [c2 (α) + is2 (α)]A < ζα−2 > B−1 s(α), 2I " Ma2 ! (6.25a) φ = φ∞ + Re [c2 (α) + is2 (α)]B < ζα−2 > B−1 s(α), 2I & σss a %∗ T = s∗o cos2 (θ − α) + s s (θ )G1 (θ )s(α) − c∗ sT (θ )G3 (θ )s(α) , Ma/I 2ρ u = u∞ +

where s∗o = sin(ψ − α) − (1 − c) cos α sin ψ, −1 [1 − c2 − (1 − c)2 cos 2α] sin 2ψ + c sin(2ψ + 2α), s∗ = 2 −1 [1 − c2 − (1 − c)2 cos 2α] cos 2ψ + c cos(2ψ − 2α), c∗ = 2

(6.25b) b c= , a

in which ψ is related to θ by (6.13a) and ρ is defined in (6.13b). For isotropic materials, by using (3.82) and (3.86) we have sT (θ )G1 (θ )s(α) = sin(θ − α),

sT (θ )G3 (θ )s(α) = 2 cos(θ − α).

(6.26)

The hoop stress shown in (6.25) can then be reduced to Ma ∗ a ∗ 2 ∗ σss = s0 cos (θ − α) + [s sin(θ − α) − 2c cos(θ − α)] . I 2ρ

(6.27)

168

6 Holes

6.1.3 Arbitrary Loading Along the Hole Boundary Consider an infinite anisotropic elastic plate containing an elliptical hole subjected to an arbitrarily prescribed traction ˆtn on the hole surface (Fig. 6.4). The boundary conditions of this problem can be expressed as

x2

Fig. 6.4 An infinite anisotropic plate containing an elliptical hole subjected to an arbitrarily prescribed traction ˆtn on the hole surface

b

tˆ n

x1

a

φ, s = ˆtn , along the hole boundary , φ → 0, at infinity

(6.28)

in which relation (3.35) has been used. Since any given function can be expressed by a Fourier series, we may assume ρ ˆtn = c0 +

∞

(ck cos kψ + dk sin kψ),

(6.29a)

k=1

where ρ is a length parameter defined in (6.13b) and c0 =

1 2π

π −π

ρ ˆtn dψ, ck =

1 π

π

−π

ρ ˆtn cos kψdψ, dk =

1 π

π

−π

ρ ˆtn sin kψdψ. (6.29b)

To satisfy the boundary conditions given in (6.28) with the applied traction ˆtn expressed as (6.29), the stress function vector φ and its associated displacement vector u can be assumed in the following form:

6.1

Elliptical Holes

169

u = 2 Re{A < ln ζα > q0 } + 2

∞

" ! Re A < ζα−k > qk ,

k=1

φ = 2 Re{B < ln ζα > q0 } + 2

∞

" ! Re B < ζα−k > qk ,

(6.30)

k=1

where qk , k = 0, 1, 2, . . . , ∞ are the unknown complex coefficient vectors to be determined. From (6.28)1 , we see that in order to find qk through the boundary conditions we need to calculate φ,s . By using the chain rule similar to that shown in (6.14a) and (6.14b), we can get ∂ζα−k ik = − e−ikψ , ∂s ρ

∂ ln ζα i = , ∂s ρ

along the hole boundary.

(6.31)

With the above results, we have ρφ, s = 2 Re{iBq0 } −

∞

" ! 2 k Re ie−ikψ Bqk ,

along the hole boundary. (6.32)

k=1

Similar to the technique we employ in (6.22), (6.23), and (6.24) for the bending case, we may now replace the complex vector qk by two real vectors gk and hk as qk = AT gk + BT hk .

(6.33)

Substituting (6.33) into (6.32) and using identities (3.59), the boundary condition (6.28)1 will then give us ST g0 − Lh0 = c0 ,

1 ST gk − Lhk = − ck , k

1 gk = − dk , k

k = 1, 2, . . . ∞. (6.34)

Because the logarithmic-function form has been introduced in (6.30), we need to consider the requirement of single-valued displacement, which gives 2 Re{iAq0 } = Hg0 + Sh0 = 0.

(6.35)

Combining (6.34) and (6.35), and using the identity (3.96a)2 , we obtain g0 = −ST c0 , h0 = −Hc0 , k = 1, 2, . . . ∞.

gk = − 1k dk ,

hk = − 1k L−1 (ST dk − ck ),

(6.36)

With the results of (6.36), by using identities (3.59) and (3.57b)1 , the complex constants qk defined in (6.33) can be further simplified as q0 = −iAT c0 ,

qk =

1 −1 B (ick − dk ), 2k

k = 1, 2, . . . ∞.

(6.37)

170

6 Holes

With the above results, by following the same procedure as that discussed in Section 6.1.1 the full-field solution and the hoop stress along the hole boundary for the elliptic hole subjected to arbitrary loading can now be obtained as ∞ ! " 1 Im A < ζα−k > B−1 (ck + idk ) , u = 2 Im A < ln ζα > AT c0 − k

φ = 2 Im B < ln ζα > AT c0 −

k=1 ∞ k=1

ρσss = sT (θ )NT1 (θ )c0 +

! " 1 Im B < ζα−k > B−1 (ck + idk ) , k (6.38a)

∞ cos kψsT (θ )[G1 (θ )ck − G3 (θ )dk ] k=1

(6.38b)

+ sin kψs (θ )[G3 (θ )ck + G1 (θ )dk ] . T

Uniform Loading Consider the plate with a traction-free hole subjected to a uniform loading at infinity. Due to the linear property, the principle of superposition can be used and the solution for this problem can be represented by the sum of an unnotched plate and corrective solution, for which the loading applied on the hole boundary are ˆ ˆtn = −Tn(θ ),

(6.39a)

where ∞ ∞ Tˆ = [t∞ 1 t2 t3 ],

nT (θ ) = (− sin θ

cos θ

0)

(6.39b)

and t∞ 1

⎧ ∞⎫ ⎧ ∞⎫ ⎧ ∞⎫ ⎨σ11 ⎬ ⎨σ21 ⎬ ⎨σ31 ⎬ ∞ , t∞ = σ ∞ , t∞ = σ ∞ , = σ12 3 ⎩ ∞⎭ 2 ⎩ 22 ⎩ 32 ∞⎭ ∞⎭ σ13 σ23 σ33

(6.39c)

in which σij∞ are the stresses at infinity. By using the relation of θ and ψ shown in (6.13a), substitution of (6.39) into (6.29b) gives the coefficients of Fourier series as c0 = ck = dk = 0, c1 = −bt∞ 1 ,

k = 1,

d1 = −at∞ 2 .

(6.40)

The displacement, stress function, and hoop stress given in (6.38) will then become ! , -" ∞ u = −Re A < ζα−1 > B−1 at∞ , 2 − ibt1 ! " , ∞ φ = −Re B < ζα−1 > B−1 at∞ , 2 − ibt1

(6.41a)

6.1

Elliptical Holes

171

" ! a b ∞ T ∞ ∞ + cos θ s σss = − sin θ sT (θ ) G1 (θ )t∞ − (θ )t (θ ) (θ )t + G (θ )t G G 3 3 1 1 2 1 2 b a ∞ = − sT (θ )(cos θ t∞ 1 + sin θ t2 ) ! " b a ∞ ∞ − iT2 G1 (θ )t∞ G , + (θ )t + iT1 G1 (θ )t∞ 3 2 1 1 − G3 (θ )t2 a b

(6.41b) which are equivalent to the solutions shown in (6.8), (6.10), and (6.18) when the solutions associated with u∞ and φ∞ are superimposed. In-Plane Bending In this case, we consider the plate subjected to in-plane bending M in the direction at an angle α with positive x1 -axis. By superposition with an unnotched plate, the hole boundary is subjected to the following equilibrated stress state, (4.10), σ11 = −

Mx2∗ cos2 α, I

σ22 = −

Mx2∗ sin2 α, I

σ12 = −

Mx2∗ cos α sin α, (6.42a) I

where x2∗ = −x1 sin α + x2 cos α.

(6.42b)

The applied traction on the hole boundary of which the normal is n can therefore be calculated by ˆtn = σij nj , and hence ˆ ), ˆtn = −Tn(θ

(6.43a)

−M (a cos ψ sin α − b sin ψ cos α)s(α)sT (α). Tˆ = I

(6.43b)

where

Substituting (6.43) into (6.29b), we get c0 = ck = dk = 0, k = 2, abM M sin 2αs(α), d2 = (a2 sin2 α − b2 cos2 α)s(α). c2 = 2I 2I

(6.44)

With the results of (6.44), the displacements, stress functions, and hoop stress can then be obtained from (6.38), which can be proved to be equivalent to those shown in (6.25) when the solutions associated with the unnotched plate are superimposed. Concentrated Force Consider a concentrated force pˆ acting at the point (a cos ψ0 , b sin ψ0 ) on the hole boundary. The surface loading ˆtn can be represented by

172

6 Holes

ˆ ρ ˆtn = −pδ(ψ − ψ0 ),

(6.45)

where δ(ψ − ψ0 ) is the Delta function with an impulse located at ψ = ψ0 . Substituting (6.45) into (6.29b), we obtain c0 = −

1 ˆ p, 2π

ck = −

cos kψ0 ˆ p, π

dk = −

sin kψ0 ˆ p. π

(6.46)

The displacement and stress function vectors shown in (6.38) will then become u=

∞ ! " 1 1 ˆ Im A < ln ζα > AT pˆ + Re ieikψ0 A < ζα−k > B−1 p, π πk k=1

1 φ = Im B < ln ζα > AT π

∞ ! " 1 ˆ Re ieikψ0 B < ζα−k > B−1 p. pˆ + πk

(6.47)

k=1

Knowing that −

k ∞ 1 eiψ0 k=1

k

ζα

eiψ0 , = ln 1 − ζα

(6.48)

where |ζα | > 1, (6.47) can be proved to be equivalent to " ! 1 Im A < ln ζα − eiψ0 > AT pˆ + π ! " 1 φ = Im B < ln ζα − eiψ0 > AT pˆ + π u=

" 1 ! T ˆ Im A < ln ζα−1 − e−iψ0 > B−1 BA p, π " 1 ! T ˆ Im B < ln ζα−1 − e−iψ0 > B−1 BA p. π (6.49)

During the derivation, a constant stress function which does not produce stresses has been neglected and identities (3.59) and (3.58b)1 have been used. By following the same procedure as that discussed between (6.12) and (6.18), the real-form solution of the hoop stress along the hole boundary can be obtained as σss = −

sin(ψ − ψo ) 1 T ˆ s (θ )N3 (θ )L−1 I + ST p. 2πρ 1 − cos(ψ − ψo )

(6.50)

Pin-Loaded Hole Consider a pin-loaded circular hole which is simulated by a cosine normal load distribution around the upper half of the hole boundary, i.e., σnn = pˆ cos θ ∗ ,

(6.51)

where θ ∗ is measured clockwise from x2 -axis, pˆ is the magnitude of pressure, and n is the direction normal to hole boundary. The prescribed surface traction ˆtn can then be described as

6.1

Elliptical Holes

173

⎧ ⎫ ⎨ sin θ ∗ ⎬ ˆtn = pˆ cos θ ∗ cos θ ∗ ⎩ ⎭ 0

(6.52a)

or ⎧ ⎫ ⎨sin ψ cos ψ ⎬ ˆtn = pˆ , sin2 ψ ⎩ ⎭ 0 = 0,

0 ≤ ψ ≤ π,

(6.52b)

π ≤ ψ ≤ 2π .

From (6.29b) and (6.52b) with a= b, we have c0 =

aˆp i2 , 4

and ⎧ 2aˆp 4aˆp ⎪ ⎪ ck = i1 , dk = i2 , ⎪ ⎪ 2 ⎪ π (4 − k ) π (4 − k2 ) ⎨ dk = 0, ck = 0, ⎪ ⎪ ⎪ ⎪ −aˆp aˆp ⎪ ⎩ ck = i2 , i1 , dk = 4 4

(6.53a)

when k = odd, when k = even except 2,

(6.53b)

when k = 2.

The displacements and stresses for the entire plate which contains a pin-loaded hole can then be calculated by substituting (6.53) into (6.38).

6.1.4 Concentrated Force at Arbitrary Location Consider an infinite anisotropic plate containing an elliptic hole under a concentrated force pˆ applied at point xˆ (Fig. 6.5). The elasticity solution of this problem is known to be the Green’s function for hole problems. If the hole is assumed to be traction free, the boundary conditions can be written as tn = φ, s = 0 along the hole boundary, dφ = pˆ for any closed curve C enclosing the point xˆ , C du = 0 for any closed curve C,

(6.54)

C

σij → 0 at infinity, where tn is the surface traction along the hole boundary of which the normal is n. The equality between tn and φ, s comes from (3.35), where s is tangent to the boundary (Fig. 6.5). The last three equations describing the point force and the vanishing

174

6 Holes x2

pˆ bˆ

xˆ = ( xˆ1 , xˆ2 )

b x1 n

s

θ a

Fig. 6.5 An infinite anisotropic plate containing an elliptic hole under a concentrated force pˆ applied at point xˆ

stresses at infinity are the same as (4.12). To find a complex function vector f(z) satisfying the boundary conditions shown in (6.54), several different methods have been used in the literature. One of them is the semi-inverse method by which the solution is found mainly due to the proper choice of the function form of f(z) (Hwu and Yen, 1991), which is then improved by Ting (1996) using the concept of image singularities. The other important and powerful method is the method of analytical continuation which was introduced by Hwu and Yen (1993) for general inclusion problem and has been employed in Section 4.2.1 for the half-space problem. By following the steps described in Section 4.2.1, the method of analytical continuation may now be employed again to find the solution for the present problem. Using the method of analytical continuation and understanding that the unknown complex function vector f(z) is better to be expressed in terms of the arguments ζα , the general solution for the present problem can now be written as u = 2 Re A[fu (ζ ) + fp (ζ )] ,

φ = 2 Re B[fu (ζ ) + fp (ζ )] ,

(6.55a)

where ˆ i, fu (ζ ) =< ln(ζα − ζˆα ) > AT p/2π

(6.55b)

and fp (ζ ) is an analytic function corresponding to the perturbed field of the problem and will be determined through satisfaction of the boundary conditions. Since fu (ζ ) given in (6.55b) is selected based upon the last three boundary conditions of (6.54), to determine fp (ζ ) we now consider the first boundary condition of (6.54) which requires that the hole is traction free. Substituting (6.55a) into (6.54)1 , we get

6.1

Elliptical Holes

175

B[fu (σ ) + fp (σ )] + B[fu (σ ) + fp (σ )] = 0,

(6.56)

where σ = eiψ is the point on the unit circle of the ζ -plane, which corresponds to the point along the hole boundary. Note that (6.56) is derived based upon φ = 0 which is obtained by integrating φ, s along the boundary. The integration constant which represents the rigid body translation (by (3.13), constant value of φ corresponds to zero value of stresses) has been neglected. From the choice of fu (ζ ) given in (6.55b), we know that fu (ζ ) is analytic in the whole ζ -plane except the points at ζˆα and infinity, which are all located outside the unit circle. Thus, we may say that fu (ζ ) is analytic in the region inside the unit circle (S− ). By a property used in the analytical continuation method, we know that fu (1/ζ ) will be analytic in the region outside the unit circle (S+ ). Because fp (ζ ) is required to be analytic in the physical body, fp (ζ ) is analytic in S+ and fp (1/ζ ) is analytic in S− . With this background, (6.56) may be rewritten as θ(σ + ) = θ (σ − ),

(6.57a)

where ⎧ ⎨ Bfp (ζ ) + Bfu (1/ζ ), ζ ∈ S+ , θ(ζ ) = ⎩ −Bf (ζ ) − Bf (1/ζ ), ζ ∈ S− . u p

(6.57b)

By the analytical conditions discussed before this equation, we conclude that this newly defined function θ(ζ ) will be analytic in S+ and S− and is continuous across the unit circle. This means that θ(ζ ) is analytic in the whole ζ -plane including the points at infinity. By Liouville’s theorem we have θ(ζ ) ≡ constant. However, the constant function corresponds to rigid body motion which may be neglected. Therefore, θ(ζ ) ≡ 0. With this result, (6.57b)1 now leads to fp (ζ ) = −B−1 Bfu (1/ζ ).

(6.58)

Substituting (6.55b) into (6.58), we have T ˆ i. fp (ζ ) = B−1 B < ln(ζ −1 − ζˆ α ) > A p/2π

(6.59)

As we have explained in the introduction of translating technique stated in Section 4.2.1, the subscript α of the variable ζ has been dropped in (6.59) and a replacement of ζ1 , ζ2 , ζ3 should be made for each component function of fp (ζ ). By comparing T ˆ (6.59) with (4.50), we let C = B−1 B, gα (z) = ln(ζ −1 − ζˆ α ), and q = A p/2π i. Employing these corresponding values in (4.51) now gives us an explicit solution as fp (ζ ) =

3 k=1

T

ˆ i. < ln(ζα−1 − ζˆ k ) > B−1 BIk A p/2π

(6.60)

176

6 Holes

Combining equations (6.55) and (6.60), the Green’s function for elliptical hole problems can be written as ( ' 3 1 T T −1 −1 ˆ ˆ ˆ p, < ln ζα − ζ k > B BIk A u = Im A < ln(ζα − ζα ) > A + π k=1 ( ' 3 1 T T −1 −1 ˆ φ = Im B < ln(ζα − ζˆα ) > A + p. < ln ζα − ζˆ k > B BIk A π k=1 (6.61) If one is interested in the hoop stress σss along the hole boundary, calculation may be performed by following the steps described in the “Hoop Stress” of Section 6.1.1 and the result is σss =

" ! 2 T ˆ s (θ )G3 (θ )Re B < ieiψ (eiψ − ζˆα )−1 > AT p. πρ

(6.62)

If the point force pˆ is applied at a point (ˆx1 , xˆ 2 ) on the surface of the elliptical hole, its corresponding solution is usually called the surface Green’s function. On the elliptical surface xˆ 1 = a cos ψo and xˆ 2 = b sin ψo , by the mapping given in (6.3) we get ζˆα = eiψo . Substituting these values into (6.61), the surface Green’s function can be obtained, which has been proved to be identical to that shown in (6.49) obtained by the other approach. The associated result for the hoop stress has been shown in (6.50).

6.1.5 Dislocation at Arbitrary Location Consider an infinite anisotropic plate containing an elliptic hole having a dislocation with Burgers vector bˆ = (bˆ 1 , bˆ 2 , bˆ 3 ) located on xˆ = (ˆx1 , xˆ 2 ) outside the hole (Fig. 6.5). As we discussed in Section 4.1.5, the solutions to the dislocation problems can be obtained in a straightforward manner from the corresponding solutions ˆ Thus, from (6.61) we can to the point force problems with AT pˆ replaced by BT b. get the solution for the present problem ' u = Im A < ln(ζα − ζˆα ) > B + T

3

T < ln ζα−1 − ζˆ k > B−1 BIk B

k=1

' φ = Im B < ln(ζα − ζˆα ) > B + T

3


( ˆ , b/π

( ˆ . b/π

k=1

(6.63)

6.2

Polygon-Like Holes

177

6.2 Polygon-Like Holes In this section various polygon-like holes such as the ellipse, circle, crack, triangle, oval and square are represented by a unified equation. Using this unified contour expression, the final results can then be written into one equation. The contour of the hole considered is represented by x1 = a(cos ψ + ε cos kψ), x2 = a(c sin ψ − ε sin kψ),

(6.64)

where 0 < c ≤ 1, and k is an integer. When ε = 0 we obtain an ellipse with semi-axes a and ac. By letting c be equal to zero, an elliptic hole can be made into a crack of length 2a. If c = 1, the ellipse becomes a circle. In the case that ε = 0, a polygon with k + 1 edges may be obtained by properly choosing the shape parameters a, c, ε, and k. For example, when c = 1 and k = 2 the hole has three symmetry axes and hence with an appropriate selection of the parameter ε the hole will differ from an equilateral triangle with rounded corners; when c = 1 and k = 3 there are four symmetry axes and at some values of ε the hole will differ little from a square with rounded corners; when c < 1 and k = 3 we will obtain ovals of a special type. Therefore, the polygon-like holes discussed throughout this section may include ellipse, crack, circle, triangle, oval, square, pentagon. Example is given in Fig. 6.6 for a pentagon with c = 1, k = 4 and ε = 0.1.

Fig. 6.6 An infinite anisotropic plate containing a polygon-like hole subjected to uniform loading and pure bending at infinity (a = 1, c = 1, ε = 0.1, k = 4)

178

6 Holes

6.2.1 Transformation Function In order to find the solution for the present problem, like the elliptical hole discussed in Section 6.1 we first consider the transformation of the polygon-like hole in the z-plane into a unit circular hole in the ζ -plane. The transformation function can be assumed as zα = a1 ζα + a2 ζα−1 + a3 ζαk + a4 ζα−k ,

α = 1, 2, 3.

(6.65a)

With zα defined in (3.21) and the boundary values given in (6.64) and ζα = cos ψ + i sin ψ, the transformation function assumed in (6.65a) requires a1 = a(1 − iμα c)/2, a3 = aε(1 + iμα )/2,

a2 = a(1 + iμα c)/2, a4 = aε(1 − iμα )/2.

(6.65b)

In order to make the transformation single-valued, it is necessary that all the roots of dzα /dζα = 0 be located inside the unit circle |ζα | = 1. However, in real application we find that there are many cases that some roots will locate outside the unit circle, i.e., the one-to-one mapping requirement cannot be satisfied. Because this requirement is important for the exactness of the solutions obtained by using the transformed variable ζα , before the detailed derivation of the solutions we like to discuss the validity of the transformation function (6.65). An Anisotropic Plate with Elliptic Holes (μα :complex, ε = 0) The transformation function for this condition has been given in (6.3) with b = ac. The roots of dzα /dζα = 0 are at ζα2 =

1 + iμα c . 1 − iμα c

(6.66)

If μαR and μαI are, respectively, the real and imaginary parts of μα , the absolute value of ζα2 is 0 0 (1 − cμ )2 + (cμ )2 αI αR 0 20 . 0ζα 0 = (1 + cμαI )2 + (cμαR )2

(6.67)

0 0 Since μαI > 0 and 0 < c ≤ 1, we have 0ζα2 0 < 1 which leads to |ζα | < 1. The roots are therefore located inside the unit circle |ζα | = 1 and the transformation function (6.3) is single-valued outside the elliptic hole. The solutions provided in Section 6.1 are therefore exact.

6.2

Polygon-Like Holes

179

An Isotropic Plate with Polygon-Like Holes (μα = i, ε :small) The transformation function given in (6.65) can be written as zα =

a 1 − c 2ε (1 + c)ζα + + k . 2 ζα ζα

(6.68)

Differentiating zα with respect to ζα , we have 1−c a 2εk dzα 1 + c − 2 − k+1 . = dζα 2 ζα ζα

(6.69)

If c = 1, the roots of dzα /dζα are at ζαk+1 = εk.

(6.70)

When the small number ε is chosen such that ε < 1/k, the critical point ζo will be located inside the unit circle |ζα | = 1 and the transformation function (6.65) is single-valued outside the hole. In the case that 0 ≤ c < 1 and ε is comparatively small such that 2εk/ζαk+1 can be neglected, the roots of dzα /dζα are at ζα2 =

1−c 1+c

(6.71)

which are located inside the unit circle and one-to-one mapping is obtained. For the other conditions, one can calculate the critical points ζ0 numerically and check whether |ζ0 | is smaller than one. From the discussions given above, we know that for isotropic plates the solutions are exact for most cases. An Anisotropic Plate with Polygon-Like Holes (μα :complex, ε :small) The critical points ζ0 for the most general conditions considered in (6.65) are determined by dzα /dζα = 0, i.e., kε(1 + iμα )ζα2 k + (1 − iμα c)ζαk+1 − (1 + iμα c)ζαk−1 − kε(1 − iμα ) = 0. (6.72) The product of all critical points, ζi , i = 1, 2, . . . , 2k, is equal to, if ε = 0 and μα = i, 12 k i=1

ζi = −

1 − iμα , 1 + iμα

(6.73)

of which the absolute value is greater than one because the imaginary part of μα is positive. Therefore, at least one of the critical points will be located outside the unit circle and (6.73) is not one-to-one outside the hole for general anisotropic materials. The occurrence of critical points outside the unit circle means that there will be multiple values of ζα , corresponding to one point zα , located outside the unit circle. If we designate the point nearest the unit circle to be the mapped point, we still

180

6 Holes

have one-to-one transformation. The hole is then mapped onto the unit circle and |ζα | → ∞ when |zα | → ∞ is also satisfied, which are the requirements for satisfaction of infinity boundary conditions. However, the ζα values may be discontinuous near the critical point, which may cause the discontinuity of displacements and stresses. To have a clear understanding of this phenomena, a typical example is shown below (Hwu and Wang, 1992). Example: k = 2, c = 1, a = 1, ε = 0.25, μα = 0.712i. The mapping function (6.65) becomes zα = 0.856ζ + 0.144

1 1 + 0.036ζ 2 + 0.214 2 . ζ ζ

(6.74)

The critical points ζ0 can then be calculated numerically by dzα /dζα = 0. The results are ζ0 = 0.84, −0.43 ± 0.64i, −11.88. The transformation from z-plane to ζα -plane is shown in Fig. 6.7. From this figure, we see that every point outside the hole on the z-plane maps to three points on the ζα -plane. Two of them locate on the right- and left-hand sides of ζ0 = −11.88, the other is inside the unit circle. The hole boundary is mapped to the unit circle and another triangular contour on the left side of ζ0 . The mapped plane (ζα -plane) looks symmetric with respective to the axis of ζ0 = −11.88, except that the mapped hole boundary on the right side is a unit circle not a triangle. To have a one-to-one transformation, the mapped plane is chosen to be the one nearest the unit circle, i.e., the plane on the right-hand side of ζ0 = −11.88. Discontinuity occurs at ζ0 = −11.88. The corresponding point in z-plane is z0 = −5.10. Figure 6.7 shows that the mapped points of z = x1∗ + i0± for x1∗ > −5.10 such as the points marked by (+) and (♦) will be continuous when they approach to x1 -axis. However, the mapped point of z = x1∗ + i0± for x1∗ < −5.10 such as the points marked by (∗), (×) and (◦), () will not be continuous when they approach to x1 -axis. The mapped points of z = x1∗ + i0± for x1∗ < −5.10 can be expressed as −11.88 ± iζI∗ of which the discontinuity is 2iζI∗ . Therefore, the corresponding displacements and stresses will be discontinuous near the critical point, which has been shown in Table 6.1. From Table 6.1 we see that the discontinuity really happens near the critical point, + − which has been shown by the nonvanished values of u2 /u+ 1 , u2 = u2 − u2 , + − at x1 = −5.2 and 20 with x2 − x2 = 0.0001, where the superscripts (+) and (–) denote the values at the near upper and lower points of x1 -axis. In other words, the displacements at the near upper and lower points of x1 -axis are not approaching to the same value when they approach to the same point on x1 -axis, which is not physically admissible for a continuum. However, when x1 approaches to infinity, the discontinuity will vanish due to the fact that the stress function has been chosen to be negative power of ζ . Moreover, for x1 > −5.1, the discontinuity also vanishes which can be seen in Fig. 6.7 and the data shown in Table 6.1. If the magnitude of discontinuity approaches to zero, the solution obtained will then approximate to the exact solution. Therefore, we may conclude that if the critical points are far away from the unit circle, the solutions may approximate to the exact solutions. The

6.2

Polygon-Like Holes

181

(a)

(b) Fig. 6.7 Mapping of triangular hole from z-plane to ζ α -plane: (a) z-plane; (b) ζ α -plane. (ε = 0.25, a = 1, c = 1, k = 2, μk = 0.712i)

182

6 Holes ± (Hwu and Wang, 1992) Table 6.1 u2 /u+ 1 at x2 = 0

x 2+ − x 2− x1

–5.0 –5.1 –5.2 –20 –2,000

0.1

0.01

0.001

0.000 1

0.365 0.387 0.457 0.355 2.10 ×10–3

0.036 0.041 0.151 0.209 4.40 ×10–4

3.65 ×10–3 4.13 ×10–3 0.120 0.194 2.75 ×10–4

3.65 ×10–4 4.13 ×10–4 0.117 0.193 2.58 ×10–4

numerical results presented in Section 6.2.4 show that the solutions are really well approximated for most cases.

6.2.2 Uniform Loading at Infinity As discussed in Section 6.2.1 the one-to-one transformation from z-plane to ζα -plane for the polygon-like hole may be established for general anisotropic plates. If the transformation is one-to-one, its associated solutions will be exact. Otherwise, only approximate solutions can be obtained. Knowing the establishment of the one-to-one transformation, the single-valued inverse of zα = zα (ζα ) expressed as ζα = ζα (zα ) can be obtained for all different cases discussed in Section 6.2.1. Therefore, like the elliptical hole discussed in Section 6.1 the general solution (3.24) can also be expressed in terms of ζα and be rewritten as shown in (6.5). Consider an infinite anisotropic elastic plate containing a polygon-like hole subjected to uniform loading at infinity. If the hole boundary surface is traction free, the boundary conditions of the present problem can be expressed as that shown in (6.7). To satisfy (6.7)2 , we may choose the complex function vector f(ζ ) be a polynomial with negative power which can make the stress function vector (6.5a)2 vanish at infinity. Along the hole boundary, (x1 , x2 ) is represented by (6.64) and its mapped point on ζα -plane is ζα = eiψ . With these boundary values and expressions (6.6) for the cases of uniform loading at infinity, the traction-free hole boundary condition (6.7)1 suggests that the function f(ζ ) be chosen as the combination of ζ −1 and ζ −k . Therefore, the general solution for the present problem can now be assumed as " ! ∞ −1 −k u = x1 ε∞ , 1 + x2 ε 2 + 2 Re A < ζα > q1 + < ζα > qk " ! ∞ −1 −k φ = x1 t∞ , 2 − x2 t1 + 2 Re B < ζα > q1 + < ζα > qk

(6.75)

∞ ∞ ∞ where ε∞ 1 , ε2 , t1 , t2 are defined in (6.6b); k is the shape parameter which has been described at the beginning of this section; q1 and qk are the coefficient vectors to be determined through the satisfaction of the boundary conditions. To determine q1 and qk , we first make the following replacement:

6.2

Polygon-Like Holes

183

q = AT g + BT h ,

= 1, k,

(6.76)

where g and h are real. Substituting (6.76) into (6.75) with the boundary values (6.64) and ζα = eiψ , and using identities (3.59), the traction-free boundary condition along the hole gives us gk = −aεt∞ g1 = −at∞ 2 , 2 , , , −1 ∞ T ∞ T ∞ h1 = aL ct1 − S t2 , hk = −aεL−1 t∞ 1 + S t2 .

(6.77)

With this result and identities (3.59), the complex coefficient vectors q1 and qk can be further simplified as 1 ∞ q1 = − aB−1 (t∞ 2 − ict1 ), 2

1 ∞ qk = − aεB−1 (t∞ 2 + it1 ). 2

(6.78)

The full-field solutions of the displacements and stresses for the anisotropic plate ∞ containing a polygon-like hole under uniform loading (t∞ 1 , t2 ) at infinity can therefore be obtained by substituting (6.78) into (6.75). By following the same procedure as that discussed in Section 6.1.1, the hoop stress along the hole boundary for the present problem can be obtained as (Hwu, 1990) σss = sT (θ )ts ,

∞ ∞ ∞ ts = cos θ t∞ 1 + sin θ t2 − G1 (θ ) sin θ t1 − cos θ t2 1 ∞ − G3 (θ ) [ρ cos θ + (1 + c)a sin ψ]t∞ 1 + [ρ sin θ − (1 + c)a cos ψ]t2 ρ (6.79) In the above, G1 (θ ) and G3 (θ ) are defined in (3.101b); s(θ ) is the unit vector tangent to the hole boundary (Fig. 6.6). The angle θ is directing counterclockwise from positive x1 -axis to the direction of s(θ ). For any point along the hole boundary, the unit tangent s(θ ) can also be expressed by dx where x is given in (6.64). Therefore, we have −a dx1 = (sin ψ + kε sin kψ) = cos θ , ρdψ ρ dx2 a = (c cos ψ − kε cos kψ) = sin θ , ρdψ ρ

(6.80a)

where ! " ρ 2 = a2 k2 ε2 + sin2 ψ + c2 cos2 ψ + 2 kε sin ψ sin kψ − 2ckε cos ψ cos kψ . (6.80b) For a plate under a unidirectional tension (or compression) σ ∞ directing at an ∞ angle α with the positive x1 -axis, by substituting t∞ 1 and t2 given in (4.6) into (6.79) we obtain

184

6 Holes

σss /σ ∞ = cos2 (θ − α) − sin(θ − α)sT (θ )G1 (θ )s(α) (1 + c)a − cos(θ − α) + sin(ψ − α) sT (θ )G3 (θ )s(α). ρ

(6.81)

The above result is valid for any kind of anisotropic materials. The shape of the opening can be ellipse, circle, crack, triangle, oval, or square. In this solution, the material properties are reflected by G1 and G3 . From the definitions given in (3.101b) and the integral formalism given in (3.95), we see that the effect of anisotropy on the hoop stress σss is totally determined through the fundamental elasticity matrices N1 and N3 . The effect of the loading direction is then reflected by α. To find the location of maximum or minimum hoop stress, all we have to do is to differentiate σss with respect to θ and set the result to zero. Note that ψ and ρ are dependent of θ . For isotropic materials, by using (6.26) the hoop stress (6.81) can now be reduced to 2(1 + c)a σss = σ ∞ −1 − sin(ψ − α) cos(θ − α) . (6.82) ρ For the hole subjected to a uniform tensile stress perpendicular to x1 -axis, α = π/2, the stress concentration at the end points of the major axis of the hole, ψ = 0, π (or θ = ±π/2) equals to σss /σ ∞ = −1 ±

2(1 + c)a , ρ

ρ = a |c ± kε| ,

(6.83)

where the ± sign depends on the shape of the hole and the position of the point, and can be determined by (6.80b) and (6.82). For elliptical hole ε = 0 and plus sign has been determined for both of the end points, so 2 σss /σ ∞ = 1 + , c

(6.84)

which is a well-known result for the isotropic plates (Timoshenko and Goodier, 1970).

6.2.3 Pure In-Plane Bending at Infinity Like the problem discussed in Section 6.1.2 for the case of elliptical hole, we now consider an infinite anisotropic elastic plate containing a polygon-like hole subjected to pure bending M in a direction at an angle α with the positive x1 -axis (Fig. 6.6). The in-plane bending is applied such that the stresses at infinity are distributed in the form as shown in (4.10). As discussed in Section 6.1.1, the boundary conditions of the present problem can also be expressed by (6.7). Similar to the procedure we use in the case of uniform loading the general solution for the plate under pure bending can be assumed as

6.2

Polygon-Like Holes

u = u∞ +2Re

! =2, k−1 k+1, 2 k

185

" A < ζα− > q ,

φ = φ∞ +2 Re

! =2, k−1 k+1, 2 k

" B < ζα− > q ,

(6.85) where u∞ and φ∞ are given in (4.11). By following the same steps as those described between (6.76) and (6.78) for the case of uniform loading, we obtain (Hwu, 1990a, 1992b) q =

Ma2 (c + is )B−1 s(α) 4I

(6.86a)

and 1 [1 − c2 − (1 + c2 ) cos 2α], 4 ε ck−1 = [(1 − c) − (1 + c) cos 2α], 2 ε ck+1 = [(1 + c) − (1 − c) cos 2α], 2 ε2 c2 k = − cos 2α, 2 c2 =

−c sin 2α, 2 ε sk−1 = (1 + c) sin 2α, 2 ε sk+1 = (1 − c) sin 2α, 2 ε2 s2 k = sin 2α. 2 s2 =

(6.86b)

By following the procedure described between (6.12) and (6.20), the hoop stress σss along the hole boundary can be obtained in real form as σss ac∗ T as∗ T = s∗0 cos2 (θ − α) − s (θ )G3 (θ )s(α) + s (θ )G1 (θ )s(α), Ma/I 2ρ 2ρ

(6.87a)

where s∗0 = sin(ψ − α) − ε sin(kψ + α) − (1 − c) cos α sin ψ, (c sin ψ + s cos ψ), s∗ = − =2, k−1 k+1, 2 k ∗

c =−

(6.87b)

(s sin ψ + c cos ψ).

=2, k−1 k+1, 2 k

For isotropic materials, by using (6.26) the hoop stress (6.87) can be reduced to Ma ∗ a ∗ 2 ∗ σss = s0 cos (θ − α) + [s sin(θ − α) − 2c cos(θ − α)] . I 2ρ

(6.88)

186

6 Holes

6.2.4 Discussions To confirm the discussions given in Section 6.2.1, a series of figures have been plotted in Hwu and Wang (1992) and Hwu (1992b) and compared with the solutions given in the literature (Lekhnitskii, 1968). In the figures shown in Hwu and Wang (1992), we cannot see any discontinuity due to the fact that the magnitude of discontinuity is very small compared to the total deformation, which also tells that the solution is really a good approximation to the exact solution. The hoop stresses along the triangular, oval, or square hole in orthotropic materials subjected to uniform loading or in-plane bending shown in Hwu (1992b) are compared with the approximate solutions provided by Lekhnitskii (1968). These figures show that in most cases the approximate solutions provided in this section are almost the same as those provided in Lekhnitskii (1968). The main difference is that the results provided by Lekhnitskii have different expressions for different holes whereas the solutions presented in this section have only one unified expression for various polygon-like holes in general anisotropic plates.

Chapter 7

Cracks

Although various types of material behaviors such as plasticity and viscoelasticity have been accounted for the study of material failure, the fundamental concept of fracture mechanics is derived from the analysis of a crack in linear elastic materials. The knowledge of stress singularity and stress distribution near the crack tip is indispensable in fracture mechanics. By specializing the work of multi-material wedges discussed in Chapter 5, the singular characteristics of cracks in homogenous materials, interface cracks between two dissimilar materials, and cracks terminating at the interfaces will be discussed in Section 7.1. To understand the effects of external loading and crack size, a finite straight crack subjected to various loading conditions will then be discussed in Section 7.2 by specializing the results of elliptical holes considered in Chapter 6. Following these two sections, some analytical solutions are derived in Sections 7.3 and 7.4 for the collinear cracks and collinear interfacial cracks. Detailed studies of the fracture parameters such as the stress intensity factors, crack opening displacements, and energy release rates will also be presented in these two sections. A delamination fracture criterion based upon the fracture parameters proposed for the interfacial cracks is then illustrated in Section 7.5.

7.1 Singular Characteristics of Cracks As discussed in Section 5.3 for the stress singularities of multi-material wedges, an infinite stress at the tip of a crack will occur if the material is considered to be linear elastic. This result caused concern when it was first discovered, because no material is capable of withstanding infinite stress. This paradox motivated Griffith (1920) to develop a fracture theory based upon energy rather than local stress. Later advancement connecting the local stress to stress intensity factors and energy release rates enhances the importance of understanding singular characteristics of cracks. Since the crack tip is a special case of wedge tip, in this section the singular order and near-tip stress distribution will be discussed based upon the results obtained in Section 5.3. C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_7, C Springer Science+Business Media, LLC 2010

187

188

7

Cracks

7.1.1 Cracks in Homogeneous Materials A semi-infinite crack in a homogeneous anisotropic medium can be represented by letting θ0 = −π and θ1 = π for the single wedge with free–free surfaces (Fig. 7.1). With this special wedge angle, the solutions for the singular order obtained in (5.98) can be simplified. From (5.75), (3.87), and (3.59), we have μˆ α (±π ) = e±iπ ,

μˆ α (π , −π ) = e2iπ and 2BBT = iL,

(7.1)

which will lead (5.98) to K(3) e = sin(2δπ )L,

(7.2)

and the singular order can be obtained by substituting (7.2) into (5.90). Knowing that the Barnett–Lothe tensor L is positive definite, L = 0, we get the singular order of the homogeneous crack as 1/2, which is a well-known result in fracture (3) mechanics. Since Ke is identical to a zero matrix when δ = 1/2, the eigenvector u0 of (5.89)2 will be totally arbitrary. In other words, there is no relation between three components of u0 . With this result, the displacements and stress functions near the crack tip can be further simplified from (5.99) to u u 1/2 ˆ 1/2 = r N (θ , −π ) 0 , 0 φ

(7.3)

where r is the distance ahead of the crack tip, θ is the angle directed from the positive x1 -axis, and u0 is a coefficient vector related to the external loading and crack geometry. For crack problems, it is custom to express the near-tip stress distribution in terms of the stress intensity factors which are usually defined as ⎧ ⎫ ⎧ ⎫ ⎨σ12 ⎬ ⎨KII ⎬ √ k = KI = lim 2π r σ22 . ⎩ ⎭ ⎩ ⎭ r→0 KIII σ23 θ=0

θ

θ Fig. 7.1 A semi-infinite crack in a homogeneous anisotropic material

π −π

(7.4)

7.1

Singular Characteristics of Cracks

189

Here, KII , KI , and KIII can be interpreted as the strength of stress singularity. For a homogeneous medium, KI , KII , and KIII represent the opening (or Mode I, tensile mode), the shearing (or Mode II, sliding mode), and the tearing (or Mode III, antiplane mode) stress intensity factors, respectively, which in turn are related to the stress components σ22 , σ12 , and σ32 for a horizontal straight crack. Hence, by the usual definition (7.4), k are determined by the near-tip solutions of σ22 , σ12 , and σ32 . However, this is not the only way to determine k; one may also use the other stress components or displacement components to determine k. From (3.13) or (3.37), we know that the stresses used in (7.4) can be obtained by differentiating the stress function φ given in (7.3)2 with respect to r and letting θ = 0, and hence k = lim

√

r→0 θ=0

2π rφ, r .

(7.5)

Substituting (7.3) into (7.5) we obtain 2 k=

π ˆ (1/2) (0, −π )u0 , N 2 3

(7.6)

ˆ ˆ 1/2 (0, −π ) and can be calculated where N (0, −π ) is one of the submatrices of N 3 from the relation given in (5.77), which leads to (1/2)

1/2 T T ˆ (1/2) (0, −π ) = B < μˆ 1/2 ˆ α (0, −π ) > B . N α (0, −π ) > B + B < μ 3

(7.7)

From (5.75) and (3.87), we know that μˆ α (0) = 1,

μˆ α (−π ) = e−iπ ,

μˆ α (0, −π ) = eiπ ,

iπ/2 μˆ 1/2 = i. α (0, −π ) = e (7.8a)

With the results of (7.7) and (7.8a) and the definition of Barnett–Lothe tensor given in (3.59), we have ˆ (1/2) (0, −π ) = −L. N 3

(7.8b)

Hence 2 u0 = −

2 −1 L k, π

(7.9)

and the stresses and displacements near the crack tip, (7.3), can now be rewritten in terms of the stress intensity factors k as 2 −1 2 1/2 ˆ 1/2 u L k r N (θ , −π ) . =− φ 0 π

(7.10)

190

7

Cracks

7.1.2 Interfacial Cracks If we set θ0 = −π , θ1 = 0, and θ2 = π , a bi-wedge with free–free surfaces can represent a bimaterial with a semi-infinite interface crack (Fig. 7.2). With these spe(3) cial angles and the results (5.100) for the bi-wedges, the matrix Ke used for the determination of the singular orders can be simplified to 1 ∗ (sin δπ )e−iδπ L2 (M∗ + e2iδπ M )L1 , 2 (7.11a) where M∗ is the bimaterial matrix defined as 2 K(3) e = − sin δπ L2 (W − cot δπ D)L1 =

−1

−1

−1 M∗ = M−1 1 + M2 = i(A1 B1 − A2 B2 ) = D − iW

(7.11b)

and −1 D = L−1 1 + L2 ,

−1 W = S1 L−1 1 − S2 L2 .

(7.11c)

The second and the third equalities of (7.11b) come from the identities given in (3.132). In deriving (7.11), the relations (5.77), (7.1), (3.59), and (3.97b) have been used. Because of the positive definiteness of L1 , L2 , and the nonzero of sin δπ for 0 < Re(δ) < 1, substitution of (7.11a) into (5.90) will further reduce to W − cot δπ D = 0,

(7.12)

of which the explicit solutions have been found by Ting (1986) as δ=

1 2

and

δ=

1 ± iε, 2

(7.13a)

ΙΙ

θ= π

θ = −π Fig. 7.2 A semi-infinite interfacial crack between two dissimilar anisotropic materials

Ι

7.1


191

where ε is called the oscillatory index since it characterizes the oscillatory behavior of the stresses near the crack tip, whose values can be determined by the following equation: 1+β 1 ln , ε= 2π 1 − β

1/2 1 −1 2 β = − tr(WD ) , 2

(7.13b)

in which tr stands for the trace of the matrix. Unlike the cracks in homogeneous materials, the eigenvector u0 determined from (5.89)2 will not be totally arbitrary. For each singular order δ, only one arbitrary scaling scalar is needed for its associated eigenvector u0 . Since we have three different singular orders, as shown in (7.13a), totally we still have three arbitrary constants. Like cracks in homogeneous materials, these three arbitrary constants will be related to three stress intensity factors. After finding the singular orders of interfacial cracks from (7.13), use of (5.101) will lead the displacement and stress function near the interfacial crack tip to −1 −r1−δ u1 L h 1−δ ˆ , N (θ , −π ) 1 = φ1 0 (1 − δ) sin π δ 1 −1 −r1−δ u2 ˆ 1−δ (0, −π ) L1 h , ˆ 1−δ (θ , 0)N = N 1 φ2 0 (1 − δ) sin π δ 2

(7.14)

in which h is a coefficient vector defined by h = lim rδ φ, r . r→0 θ=0

(7.15)

Note that as discussed previously for u0 , the three components of the coefficient vector h will be related to each singular order. Therefore, (7.15) is not a proper definition for the stress intensity factors. Moreover, the definition in (7.15) is not consistent with the stress intensity factor defined in (7.5) when materials I and II are chosen to be the same. Definition of the interfacial stress intensity factors will be given in Section 7.4 when we discuss the problems of interfacial cracks in detail.

7.1.3 Cracks Terminating at the Interfaces If one is interested in the stress behavior near the tip of a crack terminating at the bimaterial interface, the wedge angles can be selected to be θ0=−α, θ1=0, θ2 = π , θ3 = 2π − α and the material properties of wedge 3 can be chosen to be the same as those of wedge 1 (Fig. 7.3). By this selection, the formula shown in (5.102) can be further simplified and the singular order can be obtained from (5.90). The deformation and stress distribution near the crack tip can also be simplified from the solutions shown in (5.97).

192

7

Cracks

ΙΙ

Fig. 7.3 A crack terminating at the interface of the bimaterials

θ ＝π

α −

π =2

θ＝0

−α

θ=

θ

Ι

Ι

7.2 A Finite Straight Crack A finite straight crack is a special case of an elliptic hole where one of the axes becomes vanishingly small. Therefore, by letting the minor axis 2b of the elliptic hole to be equal to zero, a crack of length 2a can be made. With this understanding, all the solutions obtained in Section 6.1 for the elliptical hole problems can be used in this section with b = 0.

7.2.1 Uniform Loading at Infinity From (6.6), (6.8), and (6.10) with b = 0, the solution for a crack subjected to uniform loading at infinity can be obtained as " ! ∞ −1 −1 ∞ , u = x1 ε∞ 1 + x2 ε2 − a Re A < ζα > B t2 " ! ∞ −1 −1 ∞ φ = x1 t∞ , 2 − x2 t1 − a Re B < ζα > B t2

(7.16a)

where 3 1 2 2 zα + zα − a . ζα = a

(7.16b)

By using (7.16a)2 and σi2 = φi, 1 with x2 = 0, |x1 | > a, the stresses σi2 ahead of the crack tip along x1 -axis can be obtained as x1 σi2 = 3 t∞ 2 . 2 x1 − a2

(7.17)

7.2

A Finite Straight Crack

193

The above solution shows that the stresses are singular near the crack tip. From (7.4) or (7.5), the stress intensity factors k can be determined to be k=

√ π at∞ 2 .

(7.18)

Similarly, by using (7.16a)1 and setting x2 = 0± , |x1 | < a, where ± denotes the upper and lower surfaces of the crack, the crack opening displacement u can be obtained as 3 u = u(x1 , 0+ ) − u(x1 , 0− ) = 2 a2 − x12 L−1 t∞ 2 .

(7.19)

In deriving (7.19), the identity (3.131)1 has been used. By applying the virtual crack closure method (Irwin, 1957), the total strain energy release rate G can be calculated as follows: a 1 ui (s − a)σi2 (s)ds G = lim a→0 2a 0 a 1 = lim uT (s − a)φ (s)ds a→0 2a 0 π a ∞T −1 ∞ 1 = t L t2 = kT L−1 k, 2 2 2

(7.20)

where s is the distance ahead of the crack tip. For isotropic materials, by substituting the explicit expression of L given in (3.86) into (7.19) and (7.20) we get

u =

3 4(1 − v2 ) E

⎧ ∞ ⎫ σ ⎪ ⎨ 21 ⎪ ⎬ ∞ σ22 a2 − x12 , ⎪ ⎩ 1 σ ∞⎪ ⎭ 1−v 23

G=

(7.21)

1 − v2 2 1+ν 2 (KI + KII2 ) + KIII , E E

which is good for plane strain condition. For the generalized plane stress conditions, values given in (3.86c) should be used.

7.2.2 In-plane Bending at Infinity By following a similar procedure as that discussed in Section 7.2.1 for the case of uniform loading, the solutions for the cracks subjected to pure bending at infinity can now be obtained from (6.25) for hole problems. The solutions are

194

7

" M(a sin α)2 ! Re A < ζα−2 > B−1 s(α), 4I " M(a sin α)2 ! φ = φ∞ + Re B < ζα−2 > B−1 s(α), 4I √ −Ma π a sin2 α k= s(α), 2I 3 −M sin2 α x1 a2 − x12 L−1 s(α). u = I

Cracks

u = u∞ +

(7.22)

It can be seen that u2 will always be negative in either −a<x1 AT c0 − Im B < ζα−k > B−1 (ck + idk ) , (7.23) k k=1 $ # 2 ∞ π k= dk , ST c0 − a k=1

where ck and dk are defined in (6.29).

7.2.4 Concentrated Force at Arbitrary Location The field solution to an infinite anisotropic plate containing a crack of length 2a subjected to a concentrated force pˆ applied at xˆ can be obtained by letting the minor axis b = 0 in (6.61). If one is interested in the stresses σ2 = {σ21 , σ22 , σ23 }T ahead of the crack tip along the x1 -axis, calculation may be performed by differentiating

7.3

Collinear Cracks

195

the stress function φ with respect to x1 and letting x2 = 0, x1 > a. A simplified result is obtained as (Hwu and Yen, 1991) ⎞ ⎛ 1 ⎝ 1 1 x1 T ⎠ ˆ + > A p. Im B < 1+ 3 σ 2 = φ, 1 = πa ζ − ζˆα ζ − ζ 2 ζˆα x2 − a2 1

(7.24) With the result of (7.24), the stress intensity factors k defined in (7.4) or (7.5) can be calculated as " ! 2 ˆ k = √ Im B < (1 − ζˆα )−1 > AT p. πa

(7.25)

If the point force pˆ is applied on the upper crack surface x1 = c, (7.25) becomes 2 1 a + c ˆ k= √ I p, −ST + a−c 2 πa

(7.26)

which is identical to the solution given in Wu (1989) by using other approach.

7.2.5 Dislocation at Arbitrary Location Consider an infinite anisotropic plate containing a crack of length 2a having a dislocation with Burgers vector bˆ located on xˆ . As we discussed at the beginning of this section, the solutions can be obtained directly from (6.63) with b = 0, i.e. ' u = Im A < ln(ζα − ζˆα ) > B + T

3


k=1

' φ = Im B < ln(ζα − ζˆα ) > B + T

3


( ˆ , b/π ( ˆ . b/π

k=1

(7.27)

7.3 Collinear Cracks One of the standard crack problems is an infinite plate with an arbitrary number of collinear cracks. It is known (Sih et al., 1965) that the stress intensity factors of anisotropic problems are identical to those of the corresponding isotropic problems if the loads on the crack surface are self-equilibrated. Therefore, a proper stress function for collinear cracks in anisotropic bodies may be chosen by referring the corresponding isotropic problems. The solutions can then be obtained by satisfaction of the boundary conditions and the requirement of single valuedness of displacements. After getting the solutions for general collinear crack problems, two

196

7

Cracks

special collinear crack problems will be solved explicitly in this section. One is an infinite homogeneous anisotropic medium with two collinear cracks subjected to uniform loading at infinity and the other contains an infinite row of evenly spaced collinear cracks. For an arbitrary number of collinear cracks, similar approach can be applied. The closed-form solutions of the stresses and displacements in the entire domain are presented in matrix notation. Fracture parameters such as the stress intensity factors, crack opening displacements, and energy release rate are obtained in real form through the use of some identities provided in Chapter 3. Since the real-form solutions do not contain the complex eigenvalues and eigenvectors, the parameters presented here are also valid for degenerate materials such as isotropic materials. For the problem of two collinear cracks, the explicit form solutions given in this section cover equal and unequal crack lengths. Hence, the results are useful for the understanding of elastic interaction of a crack with another crack or microcrack. Like the isotropic cases, the results of an infinite row of evenly spaced collinear cracks are useful for the study of cracks in finite plates. All the above works have been presented in Hwu (1991) and will be stated in this section.

7.3.1 General Solutions Consider an infinite homogeneous anisotropic medium with a number of collinear cracks subjected to uniform loading at infinity (Fig. 7.4). The cracks lying at the x1 -axis are defined by the intervals ak ≤ x1 ≤ bk ,

k = 1, 2, . . . , n,

(7.28a)

with −∞ < a1 < b1 < a2 < b2 < · · · < an < bn < ∞,

(7.28b)

where n is the number of cracks. For isotropic medium, the solution of this problem has been given by Muskhelishvili (1954) in which the stress function φ (z) for traction-free cracks is obtained as φ (z) =

1 1 Pn (z)/X(z) − , 2 2

(7.29a)

where X(z) = nk=1 (z − ak )1/2 (z − bk )1/2 , Pn (z) = cn zn + cn−1 zn−1 + · · · + c0 , 1 = − (σ1 − σ2 )e2iα . 2

(7.29b)

Collinear Cracks

197 ∞ σ 22

∞ σ 22 •

σ 13∞

σ 12∞

( x1 , x2 )

r1 ∞

σ 13

σ 13∞

σ 11∞

r2 r3 r4

θ1 θ 2

θ3

a1 b1

θ4

a2

b2

σ 12∞

x1

ak

bk an

bn

σ 13∞

⊗

σ 12∞

∞ σ 23

σ 12∞

⊗

∞ σ 22

⊗

σ 12∞

•

σ 12∞

σ 11∞

σ 12∞

x2

⊗

σ 11∞

∞ σ 23

σ 12∞

•

σ 23∞

•

7.3

σ 11∞

∞ σ 23

∞ σ 22

Fig. 7.4 Collinear cracks in anisotropic plates subjected to uniform loading at infinity

σ1 , σ2 are the biaxial loading applied at infinity, α is the angle oriented from x1 -axis to the direction of σ1 . The coefficients c0 , c1 , . . . , cn are determined by the condition at infinity and the single valuedness of displacements, i.e. cn =

1 + , 2

1 (σ1 + σ2 ) 4

(7.30a)

k = 1, 2, · · ·, n,

(7.30b)

=

and Lk

Pn (t) dt = 0, X(t)

where Lk is the segment of crack ak bk . Based upon this solution and the general solutions for two-dimensional anisotropic elasticity given in (3.24), the possible solutions for collinear cracks in anisotropic bodies can be written as u = 2 Re {Af(z)} ,

φ = 2 Re {Bf(z)} ,

(7.31a)

where f(z) =

n k=0

ck + < zα > c.

(7.31b)

198

7

Cracks

Like the isotropic problems, the unknown coefficient vectors ck , k = 0, 1, 2, . . . , n and c can be determined from the condition at infinity and the single valuedness of displacements. Detailed derivation of these coefficient vectors will then be presented in Sections 7.3.2 and 7.3.3 for two special cases. One is two collinear cracks, and the other is periodic crack.

7.3.2 Two Collinear Cracks Consider an infinite homogeneous anisotropic medium with two collinear cracks subjected to uniform loading at infinity. Due to linear property, the principle of superposition can be used and the solution can be represented as the sum of a uniform stress in an uncracked solid and corrective solution, for which the boundary conditions are σi2 = −σi2∞ along the crack surface, σij → 0 at infinity.

(7.32)

Using the relation (3.13), the boundary condition (7.32) can be rewritten in terms of the stress function vector φ as φ = −t∞ , φ → 0,

when aj ≤ x1 ≤ bj and x2 = 0± , when |zα | → ∞,

j = 1, 2,

(7.33a)

where

t∞

⎧ ∞⎫ ⎪ ⎨σ12 ⎪ ⎬ ∞ . = σ22 ⎪ ⎩ ∞⎪ ⎭ σ32

(7.33b)

The prime • denotes differentiation with respect to the argument zα and ± represent the upper and lower surfaces of the crack, respectively. The possible general solution given in (7.31) is now specialized for two collinear cracks as

u=2

2 k=0

where

Re {A < fk (zα ) > ck },

φ=2

2 k=0

Re {B < fk (zα ) > ck },

(7.34a)

7.3

Collinear Cracks

199

1 dzα , X(zα ) zα f1 (zα ) = dzα , X(zα ) z2α f2 (zα ) = zα − dzα , X(zα ) f0 (zα ) =

(7.34b)

and X(zα ) = {(zα − a1 )(zα − b1 )(zα − a2 )(zα − b2 )}1/2 ,

α = 1, 2, 3.

(7.34c)

To calculate X(zα ) along the x1 -axis, it is convenient to introduce four sets of polar variables, ri , θi , i = 1, 2, 3, 4, as shown in Fig. 7.4. Thus X(zα ) =

√ r1 r2 r3 r4 ei(θ1 +θ2 +θ3 +θ4 )/2 .

(7.35)

With θ1 , θ2 , θ3 , θ4 ranging from −π to π , the values of X(zα ) along the crack surfaces are equal to X(z) since x2 = 0 and are obtained as X(zα ) = ∓iX0 (x1 ),

when

a1 ≤ x1 ≤ b1

a2 ≤ x1 ≤ b2

or

and

x2 = 0± , (7.36a)

where X0 (x1 ) =

+

(x1 − a1 )(b1 − x1 )(a2 − x1 )(b2 − x1 ), a1 ≤ x1 ≤ b1 , + X0 (x1 ) = − (x1 − a1 )(x1 − b1 )(x1 − a2 )(b2 − x1 ), a2 ≤ x1 ≤ b2 .

(7.36b)

We now replace the complex constant vector ck by ck = A−1 gk + B−1 hk ,

k = 0, 1, 2,

(7.37)

where gk and hk are real vectors. From the general solutions (7.34) as well as the replacements (7.37) and the values calculated in (7.36), the boundary condition on the crack surface (7.33a)1 will lead to g0 = g1 = g2 = 0,

1 h2 = − t∞ . 2

(7.38)

Since the displacements at the crack tips are single valued u+ (aj ) = u− (aj ),

u+ (bj ) = u− (bj ),

j = 1, 2,

(7.39a)

which can also be written as b

[u+ − u− ]ajj = 0,

j = 1, 2.

From (7.34a)1 , (7.36), (7.37), and (7.38), (7.39b) becomes

(7.39b)

200

7

2

2

" ! Re (fk (bj ) − fk (aj ))AB−1 hk = 0,

j = 1, 2,

Cracks

(7.40a)

k=0

where

bj

f0 (bj ) − f0 (aj ) = 2i

dx1 , X0 (x1 )

aj bj

f1 (bj ) − f1 (aj ) = 2i

x1 dx1 , X0 (x1 )

aj

f2 (bj ) − f2 (aj ) = −2i

bj aj

(7.40b)

x12 dx1 . X0 (x1 )

By using the identities (3.131) and the results (7.38), the two unknown vectors h0 and h1 can then be obtained from the two equations provided by (7.40). The results are hk =

λk ∞ t , 2λ

k = 0, 1, 2,

(7.41a)

where λ0 =

b1 a1 b1

x12 dx1 X0 (x1 )

b2

a2

x1 dx1 − X0 (x1 )

b1 a1

x1 dx1 X0 (x1 )

b2

a2

x12 dx1 , X0 (x1 )

b1 2 b2 2 x1 dx1 x1 dx1 b2 dx1 dx1 − , λ1 = a1 X0 (x1 ) a2 X0 (x1 ) a1 X0 (x1 ) a2 X0 (x1 ) b1 b1 b2 dx1 x1 dx1 x1 dx1 b2 dx1 λ2 = −λ = − . a1 X0 (x1 ) a2 X0 (x1 ) a1 X0 (x1 ) a2 X0 (x1 )

(7.41b)

Combining the results obtained in (7.37), (7.38), and (7.41), the field solutions for the displacements and stresses can now be written as u=2

2 λk k=0

φ=2

λ

2 λk k=0

λ

! " Re A < fk (zα ) > B−1 t∞ , ! " Re B < fk (zα ) > B−1 t∞ .

(7.42)

7.3.3 Collinear Periodic Cracks Consider an infinite sheet with an infinite row of evenly spaced collinear cracks. To find the solutions for this problem, one may apply the general solutions given in

7.3

Collinear Cracks

201

(7.31) with bj − aj = 2, aj+1 − aj = W and n approaching infinity, where 2 is the length of each crack and W is the distance between the crack centers. However, due to the fact that the crack is periodic with the periodicity W, the best way to solve this problem is treating it as a single-crack problem and representing the stress function by trigonometric functions. Referring the possible solutions (7.31) for general collinear cracks and letting n = 1 for a single crack and replacing zα and in the integrand by sin(π zα /W) and sin(π /W), the possible general solutions for collinear periodic cracks can be expressed as φ = 2 Re {B < f (zα ) > q} ,

(7.43a)

sin(π zα /W) 3 dzα , sin2 (π zα /W) − sin2 (π /W)

(7.43b)

u = 2 Re {A < f (zα ) > q} , where f (zα ) = zα −

and q is a complex coefficient vector. Like the replacement of ck introduced in (7.37), in the following derivation we may also replace q by two real vectors. Note that in (7.43b), the first term zα has not been replaced by sine function since it represents the uniform loading applied at infinity. Like (7.33), the boundary conditions for the present problem are φ = −t∞ ,

φ → 0,

when jW − ≤ x1 ≤ jW + and x2 = 0± , j = −∞, · · · , −2, −1, 0, 1, 2, · · · , ∞, when |zα | → ∞.

(7.44)

The infinity condition (7.44)2 is satisfied automatically when f (zα ) is chosen as (7.43b) since |sin(π zα /W)| → ∞ when zα → x1 + i∞. Similar to the calculation of X(zα ) in (7.36) sin(π x1 /W) , f (zα ) = 1 ± i 3 sin2 (π /W) − sin2 (π x1 /W)

along the crack surface.

(7.45)

The unknown coefficient vector q can then be determined by applying (7.43) and (7.45) to the boundary condition (7.44)1 . The results are 1 q = − B−1 t∞ . 2

(7.46)

7.3.4 Fracture Parameters The application of fracture mechanics bears largely upon the stress intensity factors k, crack opening displacements u, and strain energy release rate G. These fracture parameters have been defined in (7.4) or (7.5), (7.19), and (7.20), which are

202

7

k = lim

√

r→0 θ=0

Cracks

2π rφ ,

, , u = u x1 , 0+ − u x1 , 0− , aj ≤ x1 ≤ bj , a 1 G = lim uT (s − a)φ (s)ds. a→0 2a 0

(7.47)

In the above, (r, θ ) is the local coordinate with the origin at the crack tip and the direction θ = 0 opposite to the crack surface; (x1 , 0± ) denote the upper and lower surfaces of the crack. Using the solutions given in (7.42) for two collinear cracks and in (7.43) and (7.46) for collinear periodic cracks, we are now in a position to calculate k, u, and G by applying (7.47). 7.3.4.1 Two Collinear Cracks With the results of (7.34b), (7.34c), and (7.42), the stress intensity factors for each crack tip of two collinear cracks can now be calculated directly from the definition (7.47)1 as k = kt∞ ,

(7.48a)

or ∞ , KI = kσ22

∞ KII = kσ12 ,

∞ KIII = kσ32 ,

(7.48b)

where √ k = − 2π

−λ2 b21 + λ1 b1 + λ0

, when x1 = a1 λ2 (b1 − a1 )(a2 − b1 )(b2 − b1 ) √ −λ2 b21 + λ1 b1 + λ0 = 2π √ , when x1 = b1 λ2 (b1 − a1 )(a2 − b1 )(b2 − b1 ) √ −λ2 a22 + λ1 a2 + λ0 = 2π √ , when x1 = a2 λ2 (a2 − a1 )(a2 − b1 )(b2 − a2 ) √ −λ2 b22 + λ1 b2 + λ0 = − 2π √ , when x1 = b2 λ2 (b2 − a1 )(b2 − b1 )(b2 − a2 ) √

(7.48c)

and λ0 , λ1 , λ2 are given in (7.41b). It should be noted that these solutions are independent of material properties and are identical to the k values derived for isotropic materials. This is expected for balanced loads acting on the crack surfaces (Sih et al., 1965). As to the crack opening displacement defined in (7.47)2 , the use of (7.42)1 , (7.34b), (7.34c), and (7.36) and the identities (3.131) will then give us u = f (x1 )L−1 t∞ ,

aj ≤ x1 ≤ bj ,

j = 1, 2,

(7.49a)

7.3

Collinear Cracks

203

where

−λ2 x12 + λ1 x1 + λ0 dx1 λ2 X0 (x1 )

x1

f (x1 ) = −2

aj

(7.49b)

and X0 (x1 ) has been given in (7.36b). Unlike the stress intensity factors, the crack opening displacements are influenced by the material properties through Barnett– Lothe tensor L. The explicit expressions of L for orthotropic and isotropic materials have been shown in (3.85) and (3.86). With these expressions, the components of crack opening displacements in the x1 , x2 , and x3 directions can then be written explicitly for the orthotropic materials as ∞ u1 (x1 ) = f (x1 )σ12 /L11 , ∞ u2 (x1 ) = f (x1 )σ22 /L22 ,

u3 (x1 ) =

(7.50)

∞ f (x1 )σ32 /L33 .

For isotropic materials, we have ⎫ ∞ u1 (x1 ) = 2(1 − v2 )f (x1 )σ12 /E⎪ ⎬

∞ u2 (x1 ) = 2(1 − v2 )f (x1 )σ22 /E

u3 (x1 ) =

∞ 2(1 + v)f (x1 )σ32 /E

⎪ ⎭

for generalized plain strain

(7.51a)

for generalized plane stress.

(7.51b)

or ⎫ ⎪ ⎬

∞ u1 (x1 ) = 2f (x1 )σ12 /E ∞ u2 (x1 ) = 2f (x1 )σ22 /E

u3 (x1 ) =

⎪

⎭ ∞ 2(1 + v)f (x1 )σ32 /E

From (7.48) and (7.49), the stresses ahead of the crack tip and the crack opening displacement behind the crack tip can be obtained as ⎧ ⎫ ⎨σ12 ⎬ k ∞ 4k √ σ22 = φ (s) = √ a − sL−1 t∞ . (7.52) t , u(s − a) = √ ⎩ ⎭ 2π 2π s σ32 Substituting (7.52) into (7.47)3 , we obtain G=

k2 ∞T −1 ∞ t L t . 2

(7.53)

Since for orthotropic materials L is a diagonal matrix, the energy release rate G obtained in (7.53) can also be written in component form as k2 G= 2

#

∞2 σ ∞2 σ ∞2 σ12 + 22 + 32 L11 L22 L33

$

1 = 2

#

K2 K2 KII2 + I + III L11 L22 L33

$ .

(7.54)

204

7

Cracks

In the case when these two cracks have equal length 2 and the space between them is equal to d, the location of the crack tips can be denoted by a1 = −d/2 − 2, b1 = −d/2, a2 = d/2, b2 = d/2 + 2. λ0 , λ1 , λ2 calculated from (7.41b) are specialized to λ0 = −

π (4 + d) E(e), 2

λ1 = 0,

λ2 = −

2π K(e), 4 + d

√ 2 2(2 + d) e= . 4 + d

where

(7.55a)

(7.55b)

K(e) and E(e) are the complete elliptic integrals of the first and second kinds, respectively. With the values obtained in (7.55), the stress intensity factors (7.48), energy release rate (7.53), and crack opening displacements (7.49) can now be simplified to 1 G = k2 t∞T L−1 t∞ , 2 1 E(e) u = (4 + d) E(ψ, e) − F(ψ, e) L−1 t∞ , 2 K(e)

(7.56a)

√ k = FA π , x1 = a1 or x1 = b2 , √ = FB π , x1 = b1 or x1 = a2 , * (4 + d)2 − 4x12 , ψ = sin−1 8(2 + d)

(7.56b)

k = kt∞ ,

where

and 2 E(e) 4 + d 4 + d 1− , 4 2 + d K(e) 2 4 + d 2 E(e) d d −1 . FB = 4 2 + d d K(e)

FA =

(7.56c)

F(ψ, e) and E(ψ, e) are the elliptic integral of the first and second kinds, respectively. The stress intensity factors k shown in (7.56) are independent of material properties and are identical to those of the isotropic materials given in Murakami (1987). 7.3.4.2 Collinear Periodic Cracks The stresses σi2 ahead of the crack tips x1 = jW + can be obtained by differentiating (7.43)2 with respect to zα which gives

7.3

Collinear Cracks

205

⎫ ⎧ ⎧ ⎫ ⎬ ⎨ ⎨σ12 ⎬ /W sin π x 1 σ22 = φ = − 1 − 3 t∞ , ⎭ ⎩ ⎩ ⎭ 2 2 σ32 sin π x1 /W − sin π /W

jW + < x1 < (j + 1)W − . (7.57)

Considering x1 → jW + from the right-hand side of the crack tip and applying the definition (7.47)1 , the stress intensity factors for the collinear periodic cracks are obtained as 2 π ∞ , (7.58) k = kt , k = W tan W which are independent of material properties and identical to those of isotropic materials (Broek, 1974). By a similar procedure as that described in (7.49) for two collinear cracks, the crack opening displacements for collinear periodic cracks are obtained as u = f (x1 )L−1 t∞ , where

x1

f (x1 ) = −2

3

(7.59a)

sin πWx1

dx1 sin2 πW − sin2 πWx1 #2 $ 2 2W π 2 π x1 2 π 2 π x1 ln + sin − sin = 1 − sin / cos . π W W W W (7.59b) ak

It can also be proved that the relation between stress intensity factors and energy release rate shown in (7.53) is still valid for collinear periodic cracks. It is likely that the solution of (7.58) is approximately valid for the strip of width W, which is useful for the study of cracks in finite plates. In the case of the collinear cracks embedded in orthotropic plate, a strip of width W bears stresses along its edges (note that shear stresses are zero if symmetry conditions are considered), whereas the edges of a plate of finite size are stress free. Supposedly, the stresses parallel to the crack do not contribute much to k and consequently (7.58) can be used as an approximate solution for the strip of 3 finite size. It appears that (7.58) and (7.59) reduce to √ ∞ k = π t and u = 2 2 − x12 L−1 t∞ , respectively, if /W approaches to 0. This means that the finite strip behaves as an infinite plate if the cracks are small. 7.3.4.3 Discussions The explicit closed-form solutions of the fracture parameters obtained in this section show that the forms of the solutions are the same for different crack geometry, i.e., the solution forms of k = kt∞ , u = f (x1 )L−1 t∞ , and G = k2 t∞T L−1 t∞ /2 are valid for any collinear cracks subjected to uniform loading along the crack surfaces.

206

7

Cracks

The crack geometry is responsible for the factor k and the opening shape f (x1 ), while the effects of external loading and material anisotropy are reflected by t∞ and L−1 . The stress intensity factors are independent of the material properties and have linear relations with the external loading. The crack opening displacements and the energy release rate are influenced by the material anisotropy through the Barnett–Lothe tensor L. This tensor has been shown explicitly in terms of engineering constants for the orthotropic materials, which is a diagonal matrix. Hence, the solutions of crack opening displacement and energy release rate can be written in component form without any coupling terms between the external loadings. For the general anisotropic materials, L may be a full matrix and the coupling between the external loadings may not be avoided. From (3.85g) we see that L11 and L22 are proportional to E1 and E2 , respectively. To know the effects of crack geometry upon k (or G) and u and the interactions between the collinear cracks, a series of numerical results have been presented in Hwu (1991). The results show that when the crack spacing is infinitesimal, the inner tip stress intensity factors approach infinity, which means that the two cracks will be merged into one larger crack and the outer tip stress intensity factor will be equal to the one of the combined crack. When the crack spacing approaches infinity, the results approach those of single crack, which means that there are no interactions between cracks. This is expected since they are far away from each other. Moreover, the results also show that the stress intensity factors of collinear periodic cracks are larger than those of two collinear cracks if all the other conditions are the same, which means that the crack is influenced not only by the neighboring crack but also by the cracks next to its neighbors.

7.4 Collinear Interface Cracks One of the most frequently encountered problems in composite laminates is interface cracking, sometimes also known as delamination. Delaminations in layered composite materials may occur due to a variety of reasons, such as low energy impact, manufacturing defects, or high stress concentrations at geometry or material discontinuities. A quantitative assessment of the effect of realistic delaminations on the strength and lifetime of a laminate is difficult. Consequently, analytical efforts to date have only attempted to quantify the effect of idealized delaminations. Williams (1959) discovered the so-called oscillatory near-tip behavior for an interface crack between two isotropic materials. Since then many authors discuss the interface crack problems including isotropic and anisotropic materials. Although the problems of interface cracks between two dissimilar anisotropic materials have been widely studied by many researchers, most of the solutions are limited to the field of interface primarily because the fracture parameters such as stress intensity factors and energy release rates can be found by only knowing the near-tip solutions along the interface. If one is interested in applying the infinite domain solutions to the finite domain problems through some numerical approaches

7.4

Collinear Interface Cracks

207

such as the boundary element method or the finite element method, the field solutions other than those along the interface are also needed. With this intention, in this section we will derive the explicit full domain solutions which also include the interface solutions by following our previous works (Hwu, 1992a, 1993a,b). Unlike cracks in homogeneous bodies, the interface crack always induces opening, shearing, and tearing mode fracture simultaneously for a single mode loading. This coupling is due to the oscillatory characteristics of stresses near the crack tip. Because of this behavior, the conventional definition for Mode I, Mode II, and Mode III stress intensity factors and energy release rates should be modified. In the last four decades, several researchers have attempted to define a proper stress intensity factor which can represent the intensity of stress singularity near the tip of an interface crack for anisotropic bimaterials. Among them some are consistent with the classical definition for a crack in homogeneous media, some are not. Similarly, by the classical definition for a crack in homogeneous media (see Section 7.3), the Mode I, Mode II, and Mode III energy release rates may not exist for the interface cracks due to the oscillatory behavior. To have a unified definition of fracture parameters for any kind of anisotropic bimaterials and to be consistent with the classical definition for a crack in homogeneous media, at the end of this section a detailed discussion of fracture parameters will be provided.

7.4.1 General Solutions Consider an arbitrary number of collinear cracks lying along the interface of two dissimilar anisotropic materials. The materials are assumed to be perfectly bonded at all points of the interface x2 = 0 except those lying in the region of cracks L (see Fig. 7.5), which are defined by the intervals given in (7.28). On the upper and lower surfaces of the cracks, an arbitrary and self-equilibrated loading ˆt is specified. The continuity of displacement and traction across the bonded portion of the interface, as well as the prescribed traction on the crack portion, can be described by the following equation: x2

S1

x1 a1 b1

a2

b2

ak bk

an

bn S2

Fig. 7.5 Collinear interface cracks between two dissimilar anisotropic materials

208

7

u1 = u2 , φ1

=

φ2

φ1 = φ2 , = −ˆt,

when x1 ∈ / L, when x1 ∈ L.

Cracks

(7.60)

Here, the subscripts 1 and 2 are used to denote the quantities pertaining to the materials 1 and 2 which are located on x2 > 0(S1 ) and x2 < 0(S2 ), respectively. Note that the equality of traction continuity comes from the relation (3.32), i.e., ∂φ/∂s = t, where t is the surface traction on a curve boundary and s is the arc length measured along the curved boundary. Therefore, when t1 = ˆt and t2 = −ˆt, use the relation (3.32) and notice that the direction of s defined for (3.32) is opposite to the direction of increasing x1 on the upper crack surface and is the same as the direction of increasing x1 on the lower crack surface, we will get the last equation of (7.60), which has a negative sign before ˆt. To find a solution satisfying the boundary conditions (7.60), like (4.69) for the bimaterial problems the general solutions for the displacement and stress function vectors can be written as u1 = A1 f1 (z(1) ) + A1 f1 (z(1) ), u2 = A2 f2 (z(2) ) + A2 f2 (z(2) ),

φ1 = B1 f1 (z(1) ) + B1 f1 (z(1) ), φ2 = B2 f2 (z(2) ) + B2 f2 (z(2) ),

(7.61)

where the subscripts 1 and 2 or the superscripts (1) and (2) denote materials 1 and 2, respectively. From (7.60), we know that the traction continuity condition is valid not only for the bonded portion but also for the cracked portion (because self-equilibrated). In other words, the traction is continuous along the entire interface. By employing the method of analytical continuation as that discussed between (4.71) and (4.74) for the traction continuity along the entire interface, we get B1 f1 (z) = B2 f2 (z)

z ∈ S1 ,

B2 f2 (z) = B1 f1 (z)

z ∈ S2 .

(7.62)

With this result, the displacement continuity across the bonded portion of the interface stated in (7.60)1 now leads to ψ(x1+ ) = ψ(x1− ),

x1 ∈ / L,

(7.63)

where ψ(z) =

B1 f1 (z), z ∈ S1 , ∗ ∗−1 M M B2 f2 (z), z ∈ S2 .

(7.64)

In the above, M∗ is the bimaterial matrix defined in (7.11b) Using the results of (7.62), (7.63) and (7.64), the prescribed traction on the crack portion (7.60)2 leads to the following Hilbert problem (Muskhelishvili, 1954): , , ∗−1 ψ x1+ + M M∗ ψ x1− = −ˆt(x1 ),

x1 ∈ L.

(7.65)

7.4


209

The solution to this Hilbert problem is (see Appendix B.2) −1 ψ (z) = X0 (z) 2π i

L

1 % + &−1 ˆt(s)ds + X0 (z)pn (z), X (s) s−z 0

(7.66)

where pn (z) is a polynomial vector with degree not higher than n, and X0 (z) are the basic Plemelj functions defined as X0 (z) =
,

(7.67a)

j=1

where = [λ1

λ2

λ3 ]

(7.67b)

δα and λα , α = 1, 2, 3, of (7.67) are the eigenvalues and eigenvectors of ∗ M∗ + e2iπ δ M λ = 0.

(7.68)

The explicit solution for the eigenvalue δ has been given by Ting (1986) as δα =

1 + iεα , 2

α = 1, 2, 3,

(7.69a)

where ε1 = ε =

1+β 1 ln , 2π 1 − β

ε2 = −ε, ε3 = 0,

(7.69b)

in which β is related to the matrices W and D by (7.13b)2 . Again, as we discussed in the translating technique introduced in Section 4.2.1, once we get the solution of ψ (z) from (7.66), the complex function vectors f1 (z) and f2 (z) can be obtained from (7.64) with the understanding that the subscript of z is dropped since the analytical continuation is not affected by different arguments zα . That is f1 (z) = B−1 1 ψ(z), f2 (z) =

z ∈ S1 ,

∗−1 ∗ B−1 M ψ(z), 2 M

z ∈ S2 .

(7.70)

After the operation of matrices through (7.70), a replacement of z1 , z2 and z3 should be made for each function, because the complex function vectors f1 (z) and f2 (z) are required to have the form of (3.24b)3 . To get the explicit full field solution of f1 (z) and f2 (z), the translating technique (4.50) and (4.51) may be employed. The whole field solution of displacement and stress function vector can then be found by substituting the results of f1 (z) and f2 (z) into (7.61).

210

7

Cracks

If one is interested in the stresses σi2 along the interface and the crack opening displacements u, the following results show that they have a simple relation with ψ(z). By using the relation (3.13)2 , i.e., σi2 = φi, 1 , and the results given in (7.61), (7.62), (7.63), and (7.64), the stresses σi2 along the interface can be obtained as ⎧ ⎫ ⎨σ12 ⎬ ∗−1 σ22 = φ = (I + M M∗ )ψ (x1 ), ⎩ ⎭ σ32

x1 ∈ / L.

(7.71)

Similarly, the crack opening displacement u is related to ψ(z) by % , , -& u = −iM∗ ψ x1+ − ψ x1− ,

x1 ∈ L.

(7.72)

7.4.2 A Semi-infinite Interface Crack Let the semi-infinite planes of different materials be joined along the positive x1 axis. A line crack is situated along the negative x1 -axis extending from x1 = 0 to x1 = −∞ and is opened by a point force ˆt0 at x1 = −a on each side of the crack (Fig. 7.6). For this problem, the Plemelj function X0 (z) given in (7.67a) will be X0 (z) = < z−1/2+iεα > .

(7.73)

The point load ˆt(s) can be represented by a delta function, i.e. ⎧ ⎫ ⎨ˆt1 ⎬ ˆt0 = ˆt2 . ⎩ˆ ⎭ t3

ˆt(s) = δ(s + a)ˆt0 ,

(7.74)

x2

tˆ2

Fig. 7.6 A semi-infinite interface crack subjected to point loads

⊗

tˆ1

tˆ1

•

tˆ3

x1

tˆ3

a tˆ2

7.4


211

Substituting (7.73) and (7.74) into (7.66), we have % &−1 1 ˆt0 X0 (z) X+ (−a) 0 2π i(z + a)

(7.75a)

1 < eπ εα (a/z)1/2−iεα > −1 ˆt0 . 2π (z + a)

(7.75b)

ψ (z) = or ψ (z) =

With this solution, one can calculate the complex function vectors f1 (z) and f2 (z) by (7.70) with the understanding described in the translating technique (4.50) and (4.51). The results are (1)−1/2+iεk 3 1 π εk 1/2−iεk zα −1 −1 ˆ dz(1) f1 (z ) = e a α B1 Ik t0 , (1) 2π z + a α k=1 (1)

(2)−1/2+iεk 3 1 π εk 1/2−iεk zα −1 ∗−1 ∗ f2 (z ) = dz(2) e a M Ik −1 ˆt0 . α B2 M (2) 2π z + a α k=1 (7.76) (2)

Note that f2 (z(2) ) can be obtained from f1 (z(1) ) with zα and B−1 1 replaced by zα (1)

and B−1 2 M

∗−1

(2)

M∗ , respectively.

7.4.3 A Finite Interface Crack 7.4.3.1 Point Load Consider an interface crack located on a1 = −a, b1 = a subjected to a point load ˆt0 at z = c on each side of the crack (Fig. 7.7). For this problem, the Plemelj function X0 (z) is 8 9 z − a iεα 1 X0 (z) = √ , (7.77) z+a z2 − a2 while ˆt(s) is represented by ˆt(s) = δ(s − c)ˆt0 .

(7.78)

Substituting (7.77) and (7.78) into (7.66), we have 8

eπ εα ψ (z) = − 2π

a+c a−c

iεα

* 1 c−z

a2 − c2 z2 − a2

z−a z+a

iεα 9

−1 ˆt0 .

(7.79)

212

7

Cracks

x2

Fig. 7.7 A finite interface crack subjected to point loads

tˆ2

⊗

tˆ1

tˆ1

•

tˆ3

x1

tˆ3

tˆ2 c a

a

By (7.70), (4.50) and (4.51), the explicit full domain solution for f1 (z) is obtained as * 8 9 3 1 eπ εk a + c iεk a2 − c2 zα − a iεk −1 ˆ f1 (z) = − dzα B−1 1 Ik t0 , 2π a − c c − zα z2α − a2 zα + a k=1 (7.80) and f2 (z) can be obtained from f1 (z) with zα and B−1 replaced by zα 1

(2)

B−1 2 M

∗−1

M∗ ,

and

respectively.

7.4.3.2 Uniform Load By principle of superposition, the solution of point load problems may be used to attack all the general loading conditions with the same geometry. In the following, we will show the solutions for the case of uniform loading, i.e., ˆt(s) = T σˆ 12 σˆ 22 σˆ 23 (= constant). Instead of utilizing the results of point load problems, we evaluate the line integral (7.66) directly. By residue theory, the integral around a closed contour C shown in Fig. 7.8 can be calculated as .

1 [X0 (s)]−1 ˆt(s)ds = 2π i rk , s−z n

(7.81)

k=1

where rk are the residues of the integrand at its singular points within C. The closed contour C is the union of L+ , C0 , L− , L1 , C∞ , L1 , and C0 . The summation of the integrals along L1 and L1 vanishes since they have opposite directions and the integrand across this line is continuous. The integrals around the circle C0 and C0 can be proved to be zero when the radii of the circles C0 and C0 tend to zero. By replacing

7.4


213 x2

Uniform traction

(σˆ12 , σˆ 22 , σˆ 23 ) L+

C0'

C∞

C0

x1

−

L L'1

L1

Fig. 7.8 Integration contour for a finite interface crack subjected to uniform loads

the contour of C∞ by Reiψ and letting R → ∞, the integral around C∞ is obtained to be C∞

1 [X0 (s)]−1 ˆt(s)ds = 2π i < z + 2iεα a > −1 ˆt. s−z

(7.82)

If we let Y(z) =

L+

1 [X+ (s)]−1 ds, s−z 0

(7.83)

knowing that (see Appendix B.2) X+ 0 (x1 ) + M

∗−1

M∗ X− 0 (x1 ) = 0,

x1 ∈ L,

(7.84)

we have L+ +L−

1 ∗−1 [X0 (s)]−1 ˆt(s)ds = Y(z)[I − M M∗ ]ˆt. s−z

(7.85)

The only pole which has contribution to the residues is at s=z, and the residue at that point is [X0 (z)]−1 ˆt. With the above description, we are now in a position to evaluate the line integral and the final simplified result is 8

z + 2iεα a ψ (z) = − 1 − √ z2 − a2

z−a z+a

By (7.70), (4.50), and (4.51), we have

iεα 9

−1 (I + M

∗−1

M∗ )−1 ˆt.

(7.86)

214

7

f1 (z) = −

3

8 zα −

3 z2α

− a2

k=1

zα − a zα + a

Cracks

iεk 9 ∗−1 ∗ −1 −1 M ) ˆt. B−1 1 Ik (I + M (7.87)

Again, f2 (z) can be obtained from f1 (z) with zα and B−1 1 replaced by zα and (2)

B−1 2 M

∗−1

M∗ , respectively.

7.4.4 Two Collinear Interface Cracks Consider two collinear interface cracks located on (a1 , b1 ) and (a1 , b2 ) subjected to a uniform loading σij∞ at infinity (Fig. 7.9). If the crack surfaces are traction free, ˆt(s) = 0 and ψ (z) given in (7.66) can be simplified to ψ (z) = < Xα (z) > p2 (z),

(7.88a)

where Xα (z) = √

(z − b1 )(z − b2 ) iεα 1 , (z − a1 )(z − b1 )(z − a2 )(z − b2 ) (z − a1 )(z − a2 )

p2 (z) = c2 z + c1 z + c0 . 2

∞ σ 22

x2

∞ σ 23

σ 12∞

σ 13∞

σ 12∞

⊗

σ 13∞

σ 12∞

σ 11∞

•

σ 11∞

σ 12∞

•

σ 23∞

•

∞ σ 22

x1 σ 13

a1

b1

a2

b2

σ 12∞

σ 13∞

σ 12∞ ∞ σ 23

⊗

∞ σ 22

σ 12∞

⊗

σ 12∞

•

⊗

σ 11∞

∞

∞ σ 23

∞ σ 22

Fig. 7.9 Two collinear interface cracks subjected to uniform loads at infinity

σ 11∞

(7.88b)

7.4


215

c0 , c1 , and c2 are the coefficient vectors of polynomial p2 (z), which will be determined by the infinity condition and the single-valuedness requirement. From (7.71) and (7.88), the infinity condition provides c2 = −1 (I + M

∗−1

M∗ )−1 t∞ ,

(7.89a)

where ∞ t∞ = σ12

∞ σ22

∞ σ32

T

.

(7.89b)

The requirement of single-valuedness condition can be expressed by

bj

aj

[ψ (x1+ ) − ψ (x1− )]dx1 = 0,

j = 1, 2.

(7.90)

To calculate c0 and c1 from the above two equations, one should evaluate Xα (z) along the crack surfaces, which is in a way similar to those introduced in Section 7.3.2 Xα (z) = ± ie∓π εα χα (x1 ),

a1 ≤ x1 ≤ b1

or

x2 = 0± , (7.91a)

a2 ≤ x1 ≤ b2 ,

where (b1 − x1 )(b2 − x1 ) iεα 1 χα (x1 ) = √ , a1 ≤ x1 ≤ b1 , (x1 − a1 )(b1 − x1 )(a2 − x1 )(b2 − x1 ) (x1 − a1 )(a2 − x1 ) (x1 − b1 )(b2 − x1 ) iεα −1 , a2 ≤ x1 ≤ b2 . χα (x1 ) = √ (x1 − a1 )(x1 − b1 )(x1 − a2 )(b2 − x1 ) (x1 − a1 )(x1 − a2 ) (7.91b) Substituting (7.91) into (7.90) and solving a system of linear algebraic equations, we obtain c0 =< λ3α /λ1α > c2 ,

where λ1α = − λ2α = λ3α =

b1 a1

b1 a1 b1 a1

χα (x1 )dx1

χα (x1 )dx1

b2 a2

b2 a2

x12 χα (x1 )dx1

c1 =< λ2α /λ1α > c2 ,

x1 χα (x1 )dx1 +

x12 χα (x1 )dx1 −

b2 a2

b1 a1

b1 a1

x1 χα (x1 )dx1 −

x1 χα (x1 )dx1

x12 χα (x1 )dx1

b1 a1

(7.92a)

b2 a2

b2 a2

x1 χα (x1 )dx1

χα (x1 )dx1 ,

χα (x1 )dx1 ,

b2 a2

x12 χα (x1 )dx1 .

(7.92b)

Combining (7.88), (7.89), and (7.92), the final simplified result for ψ (z) is ψ (z) =
−1 (I + M M ) t . λ1α

(7.93)

216

7

Cracks

By (7.70), (4.50), and (4.51), the explicit full domain solutions for f1 (z) is obtained as f1 (z) =

3

B−1 M ) t , 1 Ik (I+M λ1 k

(7.94)

and f2 (z) can be obtained from f1 (z) with zα and B−1 replaced by zα 1

(2)

B−1 2 M

∗−1

M∗ ,

and

respectively.

7.4.5 Fracture Parameters Among all the near-tip solutions presented in the literature, important controversies seem to exist in the definition of stress intensity factors. Gao et al. (1992) tried to justify a proper definition from the mismatch analysis and found that only the solution proposed by Wu (1990), which conforms to the general definition suggested by Rice (1988), is consistent with the analysis of local interface mismatch near the crack tip. In this section, by the viewpoint of classical definition of stress intensity factors for the homogeneous media, which treats the factors as scalars measuring the intensity of singularities of the stresses near the tip, a relation between two seemingly different definitions given by Suo (1990) and Wu (1990) has been constructed. Through this relation, one can see more clearly about the physical meaning of stress intensity factors and understand that the controversy between those two definitions is just a vector transformation. 7.4.5.1 Proper Definition for Bimaterial Stress Intensity Factors As shown in Section 7.4.1, a general solution for the interface cracks between two dissimilar anisotropic media can be obtained from (7.61), (7.66), and (7.70). The stresses σi2 along the interface and the crack opening displacements u have simple relations with ψ(z), which are shown in (7.71) and (7.72). To define a proper fracture parameter, we now focus on a crack tip region which is small compared with the whole body. The physical problem considered is therefore a semi-infinite tractionfree interface crack. For this problem, the basic Plemelj function is given in (7.73), and hence the complex function ψ (z) shown in (7.66) for ˆt(s) = 0 is ψ (z) = X0 (z)p0 = < z−1/2+iεα > p0 .

(7.95a)

By integration, we have ψ(z) =
p0 . 1 + 2iεα

(7.95b)

Substituting (7.95a) into (7.71) with z = r ei0 , the stresses σi2 ahead of the crack tip are obtained as

7.4


217

⎧ ⎫ ⎨σ12 ⎬ ∗−1 σ22 = φ = r−1/2 (I + M M∗ ) < riεα > p0 . ⎩ ⎭ σ32

(7.96a)

Similarly, from (7.72), (7.95b), and z = r e±iπ for the upper and lower crack surfaces, we have (cosh π εα )riεα u = 4r1/2 M∗ < > p0 . (7.96b) 1 + 2iεα The matrix containing eigenvectors λα is fully determined by the eigenvalue problem given in (7.68) up to an arbitrary complex constant. From (7.95) and (7.96), we see that the constant can be absorbed into the coefficient vector p0 , which is determined by the external loading and geometry. Hence, the normalization of T eigenvectors will not affect the final results. Knowing that D is a diagonal matrix (Wu, 1990), normalization of can be performed by defining T

D = I

(7.97a)

1 T ∗ ∗ (M + M ) = I. 2

(7.97b)

or

Note that (7.97b) comes from (7.11b), which shows that D is the real part of bimaterial matrix M∗ . Using the explicit solution (7.69a) for δα , the eigenvalue problem (7.68) can also be written as ∗

M = M∗ < e2π εα > .

(7.98)

By applying the normalization defined in (7.97) and using (7.98), the following equalities, which are useful for the simplification of equations, can be obtained as T

M∗ =< 1 − tanh π εα >, T

∗

M =< 1 + tanh π εα >, 1 T ∗ T ∗ (M − M ) = −i W =< − tanh π εα >, 2 (I + M

∗−1

(7.99)

M∗ ) = 2 < e−π εα cosh π εα > .

Note that tanh π ε = β by (7.69b). Substituting (7.99) into (7.96), we have φ = 2r−1/2 < e−π εα (cosh π εα )riεα > p0 , −T

u = 4r1/2

p0 . 1 + 2iεα

(7.100)

218

7

Cracks

By using (7.100) and applying the virtual crack closure method (Irwin, 1957), the total strain energy release rate G can be calculated as 1 G = lim a→0 2a

a 0

uT (s − a)φ (s)ds = 2π pT0 < e−2π εα > p0 ,

(7.101)

where s is the distance ahead of the crack tip. In the derivation of (7.101), the fact that the displacement is real (hence u = u ) has been used and the integration is performed by knowing that a a − s 2 −iε

1

0

s

π a ds = cosh π ε

1 − iε . 2

(7.102)

To have a proper definition of the stress intensity factor, we recast the solution (7.100) into the form which looks like the classical near-tip solution. For this purpose, p0 is rescaled to the other coefficient vector k∗ by √ k∗ = 2 2π < e−π εα cosh π εα > p0 .

(7.103)

The near-tip solutions φ , u, and the energy release rate G can now be written as 1 φ = √ < riεα > k∗ , 2π r 2 2r −T riεα u = < > k∗ , π (1 + 2iεα ) cosh π εα 1 1 ∗T > k∗ , G= k < 4 cosh2 π εα

(7.104)

which are the same as those given by Suo (1990). From the eigenvalue problem (7.68) and the normalization defined in (7.97), one can show that λ2 = λ1 and λ3 is real. By (7.104)1 it can also be proved that the second component of k∗ is the conjugate of its first component and the third component of k∗ is real because the stress function vector φ is real. Hence, k∗ can be expressed as ⎧ ⎫ ⎨k⎬ k∗ = k , ⎩ ⎭ k3

k = k1 + ik2 ,

(7.105)

where k1 , k2 , and k3 are real parameters. With (7.105), the expression φ given in (7.104) can be written as φ = √

1 2π r

(kriε λ1 + kr−iε λ1 + k3 λ3 ),

(7.106)

7.4


219

√ √ √ from which kriε / 2π r, kr−iε / 2π r, and k3 / 2π r can be thought of as the components of the traction φ in the directions of λ1 , λ1 , and λ3 , respectively. The coefficients k∗ can therefore be explained as the intensity of singularity of the stresses σi2 in the direction of λ1 , λ1 , and λ3 . Evaluation of these components can be taken by the following matrix products: kriε T = λ1 Dφ , √ 2π r

k3 T = λ3 Dφ , √ 2π r

(7.107)

and the coefficient k∗ can be defined as k∗ = lim

√ T 2π r < r−iεα > Dφ

(7.108a)

k∗ = lim

√ 2π r < r−iεα > −1 φ .

(7.108b)

r→0

or, by (7.97a)

r→0

Since k1 , k2 , and k3 are the scalars measuring the intensity of singularity of the stresses σi2 , they may be treated as stress intensity factors. However, as a consequence of the peculiar singularity, k∗ has an awkward physical unit. A remedy suggested by Rice (1988) is to appeal to a fixed length and use the combination < iεα > k∗ as the basic parameter which has the unit of conventional stress intensity factors, i.e. kˆ ∗ =< iεα > k∗ = lim

r→0

√ 2π r < (r/)−iεα > −1 φ .

(7.109)

The stress intensity factors so defined cannot be reduced to the classical stress intensity factors for a crack tip in a homogeneous anisotropic medium when the two materials become the same, because the directions of λ1 , λ1 are usually not the same as the direction of the crack. To have a comparable definition, we may transform kˆ ∗ by k = kˆ ∗ ,

(7.110)

in order that it represents the intensity of singularity in the direction of x1 , x2 , and x3 . Hence, a proper definition for the bimaterial stress intensity factors may be given by k = lim

r→0 θ=0

where

√ 2π r < (r/)−iεα > −1 φ (r, θ ),

(7.111a)

220

7

⎧ ⎫ ⎨ KII ⎬ k = KI . ⎩ ⎭ KIII

Cracks

(7.111b)

It can be shown that < (r/)−iεα > −1 is a real matrix and can be calculated by the following equation (Wu, 1990): < cα > −1 = I +

1 − cR −1 2 cI −1 (D W) − D W, β β2

(7.112)

where c1 = c, c2 = c, c3 = 1, and cR , cI are real and imaginary parts of c. Therefore, KI , KII , KIII are real scalars and may be treated as the stress intensity factors which will be reduced to the classical stress intensity factors for a crack tip in homogeneous media. The near-tip solutions and energy release rate in terms of k can then be expressed as 1 φ = √ < (r/)iεα > −1 k, 2π r 2 2r −T (r/)iεα u = < > −1 k, π (1 + 2iεα )cosh π εα 1 1 −T > −1 k. G = kT < 4 cosh2 π εα

(7.113)

By (7.97) and (7.112), the relation between energy release rate G and stress intensity factors k shown in (7.113)3 can be simplified to G=

1 T k Ek, 4

E = D + WD−1 W,

(7.114)

which is equivalent to the one given by Wu (1990). Till now, there are three possible definitions for the bimaterial stress intensity factors which are given in (7.108b), (7.109), and (7.111a), respectively. k∗ defined in (7.108b) is similar to the one given by Suo (1990), while k defined in (7.111a) is similar to the one given by Wu (1990). The relations between these definitions have been given in (7.109) and (7.110) as k = < iεα > k∗ .

(7.115)

From these relations one may understand the controversy is just a vector transformation. To have a comparable definition with the classical stress intensity factors for a crack in homogeneous media, we will use k defined in (7.111) as the bimaterial stress intensity factors.

7.4


221

7.4.5.2 Some Explicit Expressions From the above discussion, we know that for interface cracks lying between two dissimilar anisotropic materials, the influence of material properties will be reflected through the eigenvalues δα (hence the oscillatory index ε) and the eigenvector matrix . The explicit solution for the eigenvalues δα and eigenvectors λα has been given in (7.69) and (7.13b). From these equations we see that the matrices W and D play important roles in the bimaterial problems. Therefore, before presenting the fracture parameters for each specific problem, here we like to provide the explicit expressions for W, D, and their related matrices such as M∗ and . With the explicit expressions of S, H, and L given in (3.85g) for orthotropic materials, by (7.11c) the matrices W and D can be expressed in terms of the engineering constants as −1 D11 = (α1 γ1 E1 )−1 (1) + (α1 γ1 E1 )(2) , −1 D22 = (α2 γ2 E2 )−1 (1) + (α2 γ2 E2 )(2) , −1/2

D33 = (G23 G31 )(1)

−1/2

+ (G23 G31 )(2) ,

(7.116)

W21 = −W12 = (γ2 η1 /γ1 E1 )(1) − (γ2 η1 /γ1 E1 )(2) , and all the other components of Dij and Wij are equal to 0. The subscripts (1) and (2) denote the properties of materials 1 and 2, respectively. The result of (7.116) shows that D is a diagonal matrix which is symmetric and positive definite, while W is antisymmetric with only one independent component W21 (= −W12 ), which has inverse relation with the Young’s modulus E1 . Furthermore, D11 , D22√, and D33 have inverse relation with the Young’s moduli E1 , E2 and shear modulus G23 G31 , respectively. For isotropic materials, by using (3.86), we have 2(1 − v21 ) 2(1 − v22 ) 2(1 + ν1 ) 2(1 + ν2 ) + , D33 = + , E1 E2 E1 E2 (1 − 2v1 )(1 + ν1 ) (1 − 2v2 )(1 + ν2 ) − , W21 = −W12 = E1 E2 (7.117a) for plane strain condition. While for generalized plane stress condition D11 = D22 =

D11 = D22 = W21 = −W12

2 2 2(1 + ν1 ) 2(1 + ν2 ) + , D33 = + , E1 E2 E1 E2 1 − v1 1 − v2 = − . E1 E2

(7.117b)

Note that in (7.117) the symbol for shear modulus is not used to avoid the confusion of the conventional symbol μ or G with the material eigenvalue or the energy release rate used in this book. The explicit expression of bimaterial matrix M∗ defined in (7.11b) can now be written as follows for orthotropic bimaterials:

222

7

⎤ D11 iW21 0 M∗ = ⎣−iW21 D22 0 ⎦ . 0 0 D33

Cracks

⎡

(7.118)

The oscillatory index ε given in (7.13b) is then simplified to ε=

1 1+β ln , 2π 1 − β

0 0 0 0 β = 0W21 (D11 D22 )−1/2 0 ,

(7.119)

where the positive value of β is chosen. By applying (7.118) and (7.119), the eigenvector matrix can now be obtained explicitly from the eigenvalue problem (7.68) as ⎡

iW21 − βD d1 11 = ⎣ d1 0

iW21 βD11 d1

d2 0

⎤ 0 0⎦, d3

(7.120)

where d1 , d2 , and d3 are arbitrary constants, which will be determined by the normalization defined in (7.97). After normalization, we have ⎤ √ √ −i/√ 2D11 i/√2D11 0 = ⎣ i/ 2D22 i/ 2D22 √0 ⎦ 0 0 1/ D33 ⎡

(7.121)

and ⎡

< cα > −1

√ cI D22 /D11 √ cR cR = ⎣−cI D11 /D22 0 0

⎤ 0 0⎦ , 1

(7.122)

where c1 = c, c2 = c, and c3 = 1 and cR and cI are real and imaginary parts of c, respectively. Equation (7.122) can also be derived from (7.112). It should be noted that after normalization, the eigenvector matrix is independent of the matrix W and is only related to the bimaterial constants D11 , D22 , and D33 . With the explicit expressions of ε and given in (7.119) and (7.121), the interface stresses σi2 ahead of the crack tip, the opening displacements u behind the tip, and the energy release rate G shown in (7.113) can therefore be written explicitly. The explicit relation between the stress intensity factors and the energy release rate shown in (7.114) can then be simplified to G=

! " 1 2 2 2 D K + D K 22 I 11 II + D33 KIII . 4 4 cosh2 π ε 1

(7.123)

By (7.116) and (7.119), we have W21 = 0 and ε = 0 for homogeneous orthotropic plates. With this result, the relation between energy release rates and stress intensity factors can be further simplified to

7.4


G=

223

" 1! 2 D22 KI2 + D11 KII2 + D33 KIII . 4

(7.124)

After knowing the proper definition for the bimaterial stress intensity factors and some explicit expressions for the related matrices, we are now in a position to provide the explicit closed-form solutions for the stress intensity factors discussed in Sections 7.4.2, 7.4.3, and 7.4.4.

7.4.5.3 A Semi-infinite Interfacial Crack Subjected to Point Load Substituting the solution obtained in (7.75b) for ψ into (7.71), we can get the stresses φ along the interface. The bimaterial stress intensity factors k defined in (7.111) can then be obtained as 2 k=

2 < (a/)−iεα cosh π εα > −1 ˆt0 . πa

(7.125)

By using (7.122), we now get the explicit solution of k for the orthotropic bimaterials as * 2 2 D11 cosh π ε[ˆt2 cos(ε ln a/) + ˆt1 KI = sin(ε ln a/)], πa D22 * 2 2 D22 (7.126) cosh π ε[ˆt1 cos(ε ln a/) − ˆt2 sin(ε ln a/)], KII = πa D11 2 2 ˆt3 . KIII = πa This result shows that the stress intensity factors for orthotropic bimaterial interface cracks are strongly similar to those of isotropic bimaterials given by Rice and √ Sih (1965). The only difference is that factor D11 /D22 may not be equal to unity since E1 and E2 are usually not the same for orthotropic materials. The results can also be reduced to the classical stress intensity factors for a crack tip in a homogeneous anisotropic medium in which the oscillatory index ε = 0 by (7.13b) with W = 0.

7.4.5.4 A Finite Interface Crack Subjected to Point Load By a similar approach as that for the semi-infinite interface crack discussed previously, the bimaterial stress intensity factors of the right tip for the problem discussed in Section 7.4.3.1 are obtained as 1 k= √ πa

2

(a + c) iεα a+c < cosh π εα > −1 ˆt0 . a−c 2a(a − c)

(7.127)

224

7

Cracks

For orthotropic bimaterials, we have * a+c D11 2a(a − c) 2a(a − c) ˆ ˆ + t1 KI sin ε ln cosh π ε t2 cos ε ln , a−c (a + c) D22 (a + c) * 2 a+c D22 1 2a(a − c) 2a(a − c) cosh π ε ˆt1 cos ε ln − ˆt2 sin ε ln KII = √ , (a + c) D11 (a + c) πa a − c 2 a+c 1 ˆt3 . KIII = √ πa a − c 1 = √ πa

2

(7.128)

7.4.5.5 A Finite Interface Crack Subjected to Uniform Load Similarly for the problem discussed in Section 7.4.3.2, we have

k=

√

π a < (1 + 2iεα )(2a/)−iεα > −1 ˆt.

(7.129)

For orthotropic bimaterials, we have % & π a σˆ 22 cos(ε ln 2a/) + 2ε sin(ε ln 2a/) + % &" +σˆ 12 D11 /D22 sin(ε ln 2a/) − 2ε cos(ε ln 2a/) , % & √ KII = π a σˆ 12 cos(ε ln 2a/) + 2ε sin(ε ln 2a/) " + −σˆ 22 D22 /D11 [sin(ε ln 2a/) − 2ε cos(ε ln 2a/)] , √ KIII = π aσˆ 32 . KI =

√

(7.130)

7.4.5.6 Two Collinear Interface Cracks Subjected to Uniform Load at Infinity Again, for the problem discussed in Section 7.4.4, by a similar approach we have

k= where

√

2π < kα > −1 t∞ ,

(7.131a)

7.5

Delamination Fracture Criteria

225

(b1 − a1 )(b2 − a1 ) iεα , (a2 − a1 ) λ1α (b1 − a1 )(a2 − a1 )(b2 − a1 ) iεα λ1α b21 + λ2α b1 + λ3α (b2 − b1 ) = , √ λ1α (b1 − a1 )(a2 − b1 )(b2 − b1 ) (b1 − a1 )(a2 − b1 ) λ1α a22 + λ2α a2 + λ3α (a2 − b1 )(b2 − a2 ) iεα = , √ (a2 − a1 ) λ1α (a2 − a1 )(a2 − b1 )(b2 − a2 ) iεα λ1α b22 + λ2α b2 + λ3α (b2 − b1 ) = , √ λ1α (b2 − a1 )(b2 − b1 )(b2 − a2 ) (b2 − a1 )(b2 − a2 )

kα =

√

λ1α a21 + λ2α a1 + λ3α

x1 = a1 , x1 = b1 , x1 = a2 , x1 = b2 . (7.131b)

It should be noted that when calculating the stress intensity factors of the left tip, i.e., x1 = a1 or a2 , the definition given in (7.111), which is defined based upon the horizontal right tip condition, should be modified as √ (7.132) k = lim 2π r < (r/)iεα > −1 φ . r→0

Otherwise, a coordinate transformation should be employed in order to be consistent with the condition of the original definition. By use of (7.122), the explicit solution of k for orthotropic bimaterials can be obtained, which is similar to the previous cases, and will not be presented here due to its complexity. The results shown in (7.131) with ε = 0 are identical to (7.48) for two collinear cracks in homogeneous anisotropic media. The elastic interaction between these two interface cracks can then be studied based upon this result.

7.5 Delamination Fracture Criteria Regardless of the nature of loading, delamination has been thought to be driven by the interlaminar stresses. Early analytical efforts have concentrated, therefore, on the determination of the interlaminar stresses which were subsequently used to predict the onset and growth of delamination (e.g., Pipes and Pagano, 1970; Pagano and Pipes, 1973; Zhou and Sun, 1990). Due to the singularity nature of delamination, the fundamental concept of classical fracture mechanics was applied later on. The laminate specimens were designed and loaded so as to induce delamination growth without significant interaction with other cracking modes. The commonly used test specimens for unidirectional composites are the double cantilever beam (DCB) (Wilkins et al., 1982) for Mode I fracture, the end-notched flexure (ENF) (Russell and Street, 1985) for Mode II fracture, and the cracked lap shear (CLS) (Wilkins et al., 1982) and the modified end-notched flexural (MENF) specimen (Yoon and Hong, 1990) for mixed-mode fracture. In the study of Hwu et al. (1995a), these four test specimens were applied to the cases that the delamination lies between two laminae with different fiber orientations. From the test data of these four different test specimens, the delamination fracture criterion can be established.

226

7

Cracks

In this section, the experimental details, the measurement of fracture toughness, and the establishment of fracture criteria will then be presented based upon the results of Hwu et al. (1995a).

7.5.1 Stress Intensity Factors and Energy Release Rates For a crack lying along the bimaterial interface, a single mode loading always induces opening, shearing, and tearing mode fracture simultaneously. This coupling is due to oscillatory characteristics of stresses near the interface crack tip, which was discussed in the previous section. Because of this behavior, the values from the conventional definition for Mode I, Mode II, and Mode III stress intensity factors and energy release rates may not exist for the interfacial cracks. To remedy this, the bimaterial stress intensity factors (7.111) and the energy release rate of each fracture mode proposed by Sun and his co-workers (Sun and Jih, 1987; Sun and Manoharan, 1989; Manoharan and Sun, 1990) are suggested to be the basic parameters for the establishment of delamination fracture criteria (Hwu et al., 1995a), i.e. ⎧ ⎫ ⎧ ⎫ ⎨σ12 ⎬ ⎨ KII ⎬ √ KI = lim 2π r < (r/)−iεα > −1 σ22 , ⎩ ⎭ ⎩ ⎭ r→0 KIII σ32 a 1 u2 (s − a)σ22 (s)ds, GI = 2a 0 a 1 GII = u1 (s − a)σ12 (s)ds, 2a 0 a 1 GIII = u3 (s − a)σ32 (s)ds. 2a 0

(7.133a)

(7.133b)

Note that different from the conventional definition where a → 0, in (7.133b) a is a small but finite characteristic length which may be material dependent; in (7.133a) is a length parameter which may be chosen arbitrarily as long as it is held fixed when specimens of a given material pair are compared. In this section, we choose = a, and according to actual calculation, a has been chosen to be 2h (h=ply thickness) since around this value the separate mode energy release rate remains relatively constant. εα and can be determined from (7.67b), (7.68), (7.69a), and (7.69b).

7.5.2 Experimental Details In order to measure the delamination fracture toughness and establish the mixedmode fracture criteria for delaminations, four different types of testing are implemented. They are DCB, ENF, CLS, and MENF. All four specimen geometries and loading conditions are indicated in Fig. 7.10. All these tests have been used

7.5


227

(a)

(b)

P

P B

2h

B

2h

a0

L

a0

P

(c)

(d) P

2

1 P

L

L

B h1 h2

a0 L

P

h1

2h

B

a0

L

L

Fig. 7.10 (a) DCB test specimen; (b) ENF test specimen; (c) CLS test specimen; (d) MENF test specimen

previously for unidirectional specimen to measure the pure and combined Mode I/ Mode II interlaminar fracture behavior. However, because of the asymmetry problems and the ambiguity of the definition for the fracture parameters of interfacial cracks, very few of them are applied to the delamination lying between two laminae with different fiber orientations. To understand the actual behavior of delamination, the study of the delamination lying between two laminae with different fiber orientations is necessary. The proper definition for the interfacial cracks has been provided in (7.133). The tests for unidirectional specimen are well documented, which may be modified for the present purpose. To avoid the asymmetry problem, two sets of 0◦ laminae have been put on the top and bottom of the composite laminates. Moreover, due to the asymmetry and the mixed-mode nature of interfacial cracks, the crack will not maintain its initial plane but shift interfaces as it grows. When this situation occurs, the definitions provided in (7.133) may not be applied since they are defined based upon the assumption of self-similarity. However, if we are only concerned about the crack initiation and not the route of crack propagation, we may discard the data when the crack goes slantingly. In addition to the problem of crack slanting, the fiber bridging may occur when the delamination propagates. This acts as a compressive force applied on the crack surface, which may increase the fracture toughness. To measure the delamination fracture toughness for the traction-free condition, and to establish a fracture criterion for delamination initiation, the data with fiber bridging that occurred should also be discarded. Because the crack slanting and the fiber bridging always occur after a few loops of cyclic loading, only the data from the second loop is suggested to be used for

228

7

Cracks

each specimen. Note that data from the first loop cannot be used since it measures the man-made crack and not the natural crack. Therefore, for one specimen, only one data has been picked up, which is not enough to interpolate the relation between the compliance and the crack length. To remedy this, several specimens (about five) with different initial delamination length are fabricated and tested, which is different from that for unidirectional specimen. With the above modification, these four types of testing originally for the unidirectional composite specimen may be used for the delamination lying between two laminae with different fiber orientations. It is known that DCB and ENF behave as pure Mode I and Mode II fracture for unidirectional composite specimen, respectively. Hence, one can directly obtain the critical Mode I and Mode II fracture toughness for unidirectional composite specimen by means of the general relationship between the energy release rate G and the compliance C: G=

P2 dC , 2B da

(7.134)

where P, B, and a denote applied load, specimen width, and crack length, respectively. However, due to the mixed-mode nature of the interfacial cracks, it is not guaranteed to get pure mode fracture from these tests for delaminated composites. Therefore, it is difficult to find the critical energy release rate of each fracture mode by just computing the total energy release rate from (7.134). In order to separate the total value of G into the sum of each mode, the finite element simulation is necessary. With the critical applied load Pcr measured from the tests as the input load of the finite element simulation, the stress intensity factors and energy release rates of each fracture mode are computed from (7.133) by using the numerical data of stresses and displacements obtained from the finite element simulation. 7.5.2.1 Materials and Specimen Fabrication The material used was Fiberite ICI glass/epoxy prepreg. The laminate layup used − 0 ◦ in all experiments was [004 /θ+ 4 /θ4 /04 ], where the 0 laminae were used to diminish the distortion caused by the unsymmetric arrangement and θ+ and θ− denote the fiber orientation of the laminae above and below the delamination, respectively. The thickness of each ply was 0.11 mm. A A4000RPS Teflon film with 0.025 mm thickness was imbedded at the mid-plane to simulate an initial delamination. After stacking layers of the prepreg, the vacuum pump was turned on, and the stack was placed on the hot press, and then the pressure was applied and the cure cycle was initiated as follows (Carlsson and Pipes, 1987). (1) The temperature was increased with the speed 2◦ C/min to 123◦ C without applying any pressure, then the temperature was maintained for 60 min. (2) The pressure of 0.55–0.65 MPa was applied and the temperature was increased to 176◦ C. (3) The pressure (0.55–0.65 MPa) and the temperature (176◦ C) were maintained for 2 hr, then the temperature was decreased. After the curing process, the specimen were cut into the following size: the width B=25.4 mm for all tests; the length L = 225 mm for DCB and ENF tests, L = 254 mm for CLS test, and L=120 mm for MENF test; the initial delamination

7.5


229

length ao =30, 35, 40, 45, and 50 mm for all tests. The material properties of the materials used in the experiment have been tested to be E1 = 44.6 GPa, E2 = 18.00 GPa, G12 = 9.69GPa, ν12 = 0.26, Xt = 1.65 GPa, Xc = 1.03 GPa, Yt = 137 MPa, Yc = 206 MPa, Vf = 0.55, where E, G, and ν are the Young’s modulus, the shear modulus, and the Poisson’s ratio, respectively. The subscripts 1 and 2 denote the fiber direction and the direction transverse to the fiber, respectively. X and Y are the longitudinal and transverse strengths. The subscripts t and c denote tension and compression. Vf stands for the fiber volume ratio. 7.5.2.2 Testing Procedure There are many factors which may influence the value of Gcr , such as the geometry of the specimen, loading speed, temperature, moisture, and the material properties. In this section, only the effect of fiber orientation θ + and θ − will be studied, and no attempt was made to examine the other effects. All the tests are conducted at room temperature, under 0.02 mm/s MTS crosshead speed for DCB and ENF, 0.008 mm/s for CLS and MENF. To achieve a stable delamination growth, most of the tests were conducted in a displacement-controlled mode. Prior to mounting the specimen, its free edges were coated with a typewriter correction fluid. After the fluid dried, fine marks were made on these edges, at 4 mm intervals on either side, to aid in the measurement of the extension of delamination. Since all these four types of specimen have been well documented, the detailed testing procedure will not be repeated here. The total energy release rate GC may be obtained by the compliance method shown in (7.134). The separation of the stress intensity factors and energy release rates of each fracture mode is then made with the aid of finite element simulation.

7.5.3 Delamination Fracture Toughness Unlike cracks in homogeneous bodies, the interface cracks always induce opening, shearing, and tearing mode fractures simultaneously for a single mode loading. The DCB/ENF test which belongs to pure Mode I/II test for unidirectional composite specimen now becomes the mixed-mode fracture test for delamination lying between two laminae with different fiber orientations. Hence, unlike the unidirectional composite specimen, it is not guaranteed that the DCB test may help us to measure KIC and GIC since the failure of DCB specimen may not only be determined by KIC or GIC but also be influenced by KIIC or GIIC . Before getting any experimental data from DCB test, one can only treat it like mixed-mode failure test and the criterion for mixed-mode fracture may look like

KI KIC

+

KII KIIC

m

+

KIII KIIIC

n =1

(7.135a)

230

7

Cracks

or

GI GIC

/2

+

GII GIIC

m/2

+

GIII GIIIC

n/2 = 1.

(7.135b)

Note that, as stated in Section 7.5.2, the stress intensity factors KI , KII , KIII and energy release rates GI , GII , GIII of (7.135a,b) are computed from (7.133) by using the numerical data of stresses and displacements obtained from the finite element simulation with the measured critical applied load Pcr as the input data. From the results for various specimens with different fiber orientations and different initiation delamination sizes (see Table 7.1 for two typical specimens), we observe that for DCB test the values of GII , GIII are far smaller than GI , and KII , KIII are far smaller than KI . If the critical fracture toughness of each separate mode is of the same order, (KII /KIIC )m , (KIII /KIIIC )n and (GII /GIIC )m/2 , (GIII /GIIIC )n/2 become much smaller than (KI /KIC ) and (GI /GIC )/2 . By neglecting the terms associated with Modes II and III, a conclusion of KI ∼ = KIC and GI ∼ = GIC can be made for DCB test of delaminated composites. With this conclusion, like the unidirectional composite specimen one can measure KIC and GIC for delaminated composites by just using the DCB test. Moreover, since the values of Modes II and III are much smaller than those of Mode I, the Mode I critical energy release rate may be approximated by the total value of the critical energy release rate. This approximation is very important for the present test since it means that the use of finite element simulation to separate the fracture mode may be unnecessary for obtaining the Mode I fracture toughness. One can approximate GIC just by computing the total critical energy release rate GC from (7.134) with the measured experimental data and calculate KIC by using the relation between G and k shown in (7.114) in which KII and KIII may also be approximated to 0 for DCB test. In addition to the evidence from the results shown in Table 7.1, the conclusion that GIC ∼ = GC may also be explained is as follows. (1) Usually, the oscillation index ε is very small, say of order 0.01–0.03, for very many material combinations Table 7.1 Results of DCB and ENF tests (Hwu et al., 1995a) Test Specimen √ KI (MPa √mm) KII (MPa √mm) KIII (MPa mm) GI (J/m2 ) GII (J/m2 ) GIII (J/m2 ) GT (J/m2 ) GC (J/m2 )

DCB

ENF

[0◦4 /30◦4 / − 30◦4 /0◦4 ]

[0◦4 /90◦4 / − 0◦4 /0◦4 ]

[0◦4 /45◦4 / − 45◦4 /0◦4 ]

[0◦4 /60◦4 / − 60◦4 /0◦4 ]

69.4 1.71 3.43 220 0.099 0.309 221 186

50.5 3.95 1.73 136.5 19.6 1.243 153.3 132

4.42 108.3 3.66 3.29 469.8 0.157 473.2 470.6

5.38 126.7 4.47 1.32 555.2 0.064 561.6 535

7.5


231

of interest (Rice, 1988). (2) Although the crack lies on the interface, it is still hardly distinguished whether the crack goes along the interface or the resin matrix. Now, it is interesting to see the effects of fiber orientation. Figure 7.11a shows the variation of GIC versus fiber orientation, in which the small circle (o) denotes the 500

[0o4 / 60o4 / θ 4 / 0o4 ]

GIC (J/m2)

400

[0 04 / + θ 4 / – θ 4 / 0 04 ]

300

200

100

0 –90 – 75 – 60 – 45 – 30 –15

0

15

30

45

60

75

90

θ (Degree)

(a)

1000

0

GII C (J/m 2)

o

o

[0 4 / 60 4 / θ 4 / 0 4 ]

800

600

400

200

0

–15

0

15

30

45

θ (Degree) (b)

60

75

Fig. 7.11 (a) Effects of fiber orientation on Mode I delamination fracture toughness(Hwu et al., 1995a); (b) effects of fiber orientation on Mode II delamination fracture toughness (Hwu et al., 1995a)

232

7

Cracks

average value of GIC from various tests, and the horizontal bars above and below the small circle stand for the variation of GIC from various tests. Due to the scatter in the data shown in Fig. 7.11a, there is no obvious way of describing the dependence of GIC upon θ . By viewing the distribution of the mean value of GIC , we see that GIC =206 ± 51 J/m2 , which is within 24.7% variation. Just by this variation, it is hard to say whether the fracture toughness is independent of the fiber orientation. However, there are many test results done by Chai (1984) supporting the conclusion that the fracture toughness is independent of the orientation of plies from both sides of the delaminating interface. For the purpose of establishing the mixed-mode fracture criterion for delamination in which the value of fracture toughness is necessary, we tentatively take the average value from various specimens shown in Fig. 7.11a to be our Mode I delamination fracture toughness of glass/epoxy for different types of interfaces. The values are √ KIC = 68.9MPa mm,

GIC = 206J/m2 .

The validity of this average value may then depend on the correctness of the prediction made by the fracture criterion proposed in the next section. Similar to the measurement of KIC and GIC , it is possible to measure KIIC and GIIC by just performing ENF test. In order to implement ENF test, a three-point flexure fixture is mounted in a properly aligned and calibrated test machine as shown in Fig. 7.10b. It has been observed that the crack generally propagates to the central loading point in an unstable manner even under displacement-controlled condition. Hence, to obtain the data from the second loop of loading, the crack should be carefully wedged open and extended about 2 mm beyond the insert in order to achieve a natural starter crack. Similar to DCB test, the conclusion of GIIC ∼ = GC can be made for ENF test by viewing the experimental data shown in Table 7.1. Moreover, from Fig. 7.11b, it can be seen that the mean value of GIIC =567.7 ± 95 J/m2 is within 16.7% variation. Similar to the reason stated previously for GIC , we take the average value from various specimens shown in Fig. 7.11b to be the Mode II delamination fracture toughness of glass/epoxy for different types of interfaces. The values are √ KIIC = 126MPa mm,

GIIC = 567.7J/m2 .

7.5.4 Mixed-Mode Fracture Criteria The measurement of delamination fracture toughness discussed in Section 7.5.3 tells us that the results of DCB and ENF tests for delamination are similar to those for unidirectional composite specimen. Moreover, the delamination fracture toughness is taken to be a single value for different types of interfaces. Therefore, one expects that the mixed-mode fracture criteria for delaminations may look the same

7.5


233

as the usual mixed-mode fracture criteria shown in (7.135), except that the fracture parameters used here are defined for the bimaterial interface cracks given in (7.133). If the form shown in (7.134) is accepted as the base to establish the fracture criteria for delaminations, the parameters that remain to be determined from the experiment are GIIIC (or KIIIC ) and , m, n since GIC and GIIC (or KIC and KIIC ) have been measured in the last section. Before finding these parameters, we assume GIIIC =GIIC (or KIIIC =KIIC ) and m=n by the experience of the cracks embedded in homogeneous materials. The validity of this assumption may then be checked in the next section when performing the prediction based upon the criterion established here. To determine the parameters and m, a series of mixed-mode fracture tests like CLS and MENF are performed. Tables 7.2 and 7.3 show the results of CLS and MENF tests, respectively, for various specimens from which we see that the contribution of Mode III fracture may be neglected. The range of the percentage of Mode I (or Mode II) energy release rate GI /GT (or GII /GT ) in CLS is 22.2–23.6%

Table 7.2 Results of CLS test (Hwu et al., 1995a) Specimen √ KI (MPa √mm) KII (MPa √mm) KIII (MPa mm) GI (J/m2 ) GII (J/m2 ) GIII (J/m2 ) GT (J/m2 ) GC (J/m2 ) GI /GT GII /GT

[0◦4 /0◦4 /0◦4 /0◦4 ]

[0◦4 /15◦4 / − 15◦4 /0◦4 ]

[0◦4 /30◦4 / − 30◦4 /0◦4 ]

[0◦4 /45◦4 / − 45◦4 /0◦4 ]

47.35 117.5 0 112.8 396.0 0 508.8 528.0 0.222 0.778

45.63 110.4 7.06 109.9 382.8 0.183 492.9 527.6 0.223 0.777

47.48 109.9 6.51 123.4 419.7 0.328 543.4 602.0 0.227 0.772

47.25 102.0 5.82 124.4 406.5 0.122 531.1 587.8 0.234 0.765

Table 7.3 Results of MENF test (Hwu et al., 1995a) Specimen √ KI (MPa √mm) KII (MPa √mm) KIII (MPa mm) GI (J/m2 ) GII (J/m2 ) GIII (J/m2 ) GT (J/m2 ) GC (J/m2 ) GI /GT GII /GT

[0◦4 /0◦4 /0◦4 /0◦4 ]

[0◦4 /15◦4 / − 15◦4 /0◦4 ]

[0◦4 /30◦4 / − 30◦4 /0◦4 ]

[0◦4 /45◦4 / − 45◦4 /0◦4 ]

68.69 47.98 0 189.1 127.8 0 316.9 331.2 0.597 0.403

69.50 47.07 0.74 190.9 131.4 0.76 323.0 326.3 0.591 0.407

68.30 46.97 2.43 188.4 132.5 1.40 322.0 333.4 0.585 0.411

69.98 46.98 5.28 193.2 138.4 1.35 333.0 324.1 0.580 0.416

234

7

Cracks

(or 76.3–77.8%), where GT =GI +GII +GIII denotes the total energy release rate calculated from the finite element simulation whose results may be checked by the GC value calculated from the compliance method (7.135). In MENF [004 /θ4+ /θ4− /004 ] test, we obtained 58.0–59.7% for GI /GT and 40.3–41.6% for GII/ GT . The range of percentage is very narrow, which means that the results are almost independent of fiber orientation. Since the mixed-mode percentage is almost unchanged in CLS or MENF, actually we only get two extra point groups (in addition to the data points from DCB and ENF) for the curve fitting of fracture criterion. In order to get more representative data points, we change the specimen thickness for MENF specimen, for example, MENF2: [003 /θ4+ /θ4− /005 ] and MENF3: [002 /θ4+ /θ4− /006 ]. Therefore, in Fig. 7.12a for the distribution of GI /GIC and GII /GIIC , five group points have been plotted for different test specimens. By least-squares approximation, we obtain =2.496 and m=3.103. Similarly, the distribution of KI /KIC and KII /KIIC is provided in Fig. 7.12b from which =2.512 and m=3.045. In applications, we choose =2.5 and m=3 since taking account of the experimental error the integer and halfinteger parameters seem to be more reasonable for a universal criterion. In summary, the mixed-mode fracture criterion for delamination can be written as

KI KIC

2.5

+

KII KIIC

3

+

KIII KIIIC

3 =1

(7.136a)

or

GI GIC

2.5/2

+

GII GIIC

3/2

+

GIII GIIIC

KIIC = KIIIC , GIIC = GIIIC .

3/2 = 1,

(7.136b)

(7.136c)

7.5.5 Prediction of Delamination Fracture Through a series of experiments, finite element simulation, and assumption, a mixed-mode delamination fracture criterion has been established in (7.136). With this criterion, most of the delamination fracture can be predicted correctly. In order to show its usefulness and preciseness, two commonly encountered problems are predicted by using this proposed criterion and verified by doing the test. One is the prediction of the most probable interface and the possible minimum load for the onset of delamination in a composite laminate which does not contain any delamination at the beginning of loading. The purpose of making this prediction is to show that the present fracture criterion can even be used for the cases that no cracks exist. The other prediction is for the test with delamination existing prior to loading.

7.5


235

(a) 1.0

l=3, m=4

GI /GIC

0.8

0.6

l=2, m=2

l=2.5, m=3

0.4

0.2

ENF DCB CLS MENF1 MENF2 MENF3

0.0 0.0

l=2, m=3

0.2

0.4

0.6

0.8

1.0

GII /GIIC l=3, m=4

(b) 1.0

l=2.5, m=3

KI /KIC

0.8

l=2, m=2

0.6

l=2, m=3

0.4

0.2

0.0 0.0

ENF DCB CLS MENF1 MENF2 MENF3 0.2

0.4

0.6

0.8

1.0

KII /KIIC Fig. 7.12 (a): The distribution of GI /GIC and GII /GIIC . (b) The distribution of KI /KIC and KII /KIIC (Hwu et al., 1995a)

7.5.5.1 The Onset of Delamination in a Perfect Composite Laminate The composite laminate to be predicted and tested does not contain any visible delamination. In order to predict the onset of delamination by using the proposed criterion, the concept of effective flaw (Wang et al., 1985) is introduced. It is generally

236

7

Cracks

assumed that small flaws exist in the material prior to loading. These flaws may be a result of residual stresses due to the curing process, external impact damage, environmental degradation, or the fabrication process. At some level of loading, these small flaws grow and join together to form a single dominant flaw of microscopic size. With the presence of a flaw, the application of fracture mechanics principles can be considered. However, the flaw size is a random value. Given a range of possible values for the edge flaw size, a corresponding range for the onset load can be determined. In particular, if for all probabilities the size of the edge flaw is on the order of the ply thickness or larger, then the predicted load range for the onset of delaminations is very narrow; in fact, it becomes practically a constant. Therefore, the predicted load based on the flaw size with the order of the ply thickness may represent the minimum possible load for the onset of delaminations. With the concept described above, a composite laminate with small edge flaws existing in every interface subjected to uniform tension is simulated by finite element program. By substituting the calculated stresses and displacements near the assumed crack tips into (7.133), the bimaterial fracture parameters for each interface can be found. Substituting these calculated fracture parameters into the criterion proposed in (7.136), one may judge the delamination onset for each interface subjected to the prescribed tension load. By this way, one may also predict the most probable interface and the minimum load for the onset of delamination. In order to verify the proposed criterion and the concept of effective flaw, a simple tension test specimen as shown in Fig. 7.13 is implemented. The stacking sequences are [0◦ / − 45◦ /90◦ /45◦ ]s , [90◦ /45◦ /0◦ / − 45◦ ]s , [−45◦ /90◦ /45◦ /0◦ ]s and [−30◦ / − 75◦ /60◦ /15◦ ]s . The geometry of the specimen is as follows: length P

t1

A

A L

t1 B

Fig. 7.13 Simple tension test specimen

h P

7.5


237

L=225 mm, width B=25.4 mm, and thickness h=3.4 mm. The small flaw size at each interface is assumed to be 0.425 mm, which is the lamina thickness. Since the specimen is long enough to be simulated by quasi-3D element, the cross section A–A shown in Fig. 7.13 will now be meshed to calculate the stresses and displacements near the assumed crack tips, then to calculate the bimaterial fracture parameters by (7.133). The prediction after substituting these parameters into the fracture criterion (7.136) is shown in Table 7.4. The results show that the prediction of the most probable interface for the onset of delamination is exactly correct, and the error for the prediction of the minimum load is within 10%. The success of this prediction shows that the present fracture criterion can even be used for the cases that no cracks exist. Table 7.4 Prediction for the onset of delamination (no cracks exist prior to loading) (Hwu et al., 1995a) Present criterion

Experiment

Specimen

Interface

Load (kN)

Interface

Load (kN)

Error (%)

[0◦ / −45◦ /90◦ /45◦ ]s [90◦ /45◦ /0◦ / −45◦ ]s [−45◦ /90◦ /45◦ /0◦ ]s [−30◦ / −75◦ /60◦ /15◦ ]s

90◦ /45◦ 0◦ / −45◦ 90◦ /45◦ −75◦ /60◦

57.3 56.9 51.2 46.1

90◦ /45◦ 0◦ / −45◦ 90◦ /45◦ −75◦ /60◦

52.6 53.9 50.3 42.2

8.9 5.6 1.8 9.2

Interface – the most probable interface for the onset of delamination. Load – the minimum possible load for the onset of delamination.

7.5.5.2 The Onset of Delamination in a Delaminated Composite Since the proposed criterion is established based upon the concept of fracture mechanics, it is better that we prove its preciseness by the delaminated test specimen. With this in mind, a 4 mm delamination lying along the entire length of the specimen was embedded into the composite. The location was chosen to be the most probable interface for the onset of delamination in a perfect composite laminate. Table 7.5 shows that the prediction and experimental results are well matched by using the proposed criterion. The error is within the range of 3.1–7.6%, which is better than the case that no delamination exists prior to loading. Table 7.5 Prediction of the minimum possible load for the onset of delamination (delamination exists prior to loading) (Hwu et al., 1995a) Specimen

Present criterion (kN)

Experiment (kN)

Error (%)

[0◦ / −45◦ /90◦ //45◦ ]s [90◦ /45◦ /0◦ // −45◦ ]s [−45◦ /90◦ //45◦ /0◦ ]s [−30◦ / −75◦ //60◦ /15◦ ]s

34.1 35.6 24.2 22.3

31.7 34.2 26.0 21.6

7.6 4.1 6.9 3.1

The location of delamination is denoted by the symbol //.

Chapter 8

Inclusions

Determination of the stress distributions induced by inclusions has aroused considerable interest for more than half a century. Several analytical solutions were presented in the literature. Some of them consider special loading conditions such as uniform loading or a concentrated couple (Chen, 1967b; Yang and Chou, 1976; Hwu and Ting, 1989), some others consider special matrices such as isotropic matrix (Eshelby, 1957; Jaswon and Bhargave, 1961; Sendeckyi, 1970; Stagni, 1982), or special inclusions such as rigid inclusions or holes (Santare and Keer, 1986; Hwu and Yen, 1991; Hwu and Wang, 1992), or special shapes such as lines or circles (Wang et al., 1985; Honein and Herrmann, 1990), or the uncoupling of in-plane and anti-plane deformations. A unified general analytical solution for the elliptical anisotropic elastic inclusions imbedded in an infinite anisotropic matrix subjected to an arbitrary loading was introduced by Hwu and Yen (1993). The presentation of this chapter will then follow mainly that of Hwu and Yen (1993) and their followup discussions about the interactions between inclusions and dislocations (Yen and Hwu, 1994; Yen et al., 1995) and the interactions between inclusions and cracks (Hwu et al., 1995b).

8.1 Elliptical Elastic Inclusions Consider an infinite plate containing an elliptical inclusion subjected to arbitrary loadings at the matrix. If both of the matrix and inclusion are made of anisotropic elastic materials and are assumed to be perfectly bonded along the interface, the displacements and surface tractions across the interface should be continuous. That is, u1 = u2 ,

φ1 = φ2 ,

along the interface,

(8.1)

where the subscripts 1 and 2 denote, respectively, the matrix and inclusion. The second equation of (8.1) comes from the relation t = ∂φ/∂s in which t is the surface traction and s is the arc length measured along the curved boundary. From the discussions given in Sections 4.2.1 and 6.1.4 for the method of analytical C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_8, C Springer Science+Business Media, LLC 2010

239

240

8 Inclusions

continuation, the general solutions (3.24) for the matrix and inclusion may be written in the following form: u1 = A1 [f0 (ζ ) + f1 (ζ )] + A1 [f0 (ζ ) + f1 (ζ )]

φ1 = B1 [f0 (ζ ) + f1 (ζ )] + B1 [f0 (ζ ) + f1 (ζ )] u2 = A2 f2 (ζ ∗ ) + A2 f2 (ζ ∗ ) ∗

φ2 = B2 f2 (ζ ) + B2 f2

(ζ ∗ )

,

,

ζ ∈ S1 ,

ζ ∗ ∈ S2 ,

(8.2a)

(8.2b)

where ζα =

zα +

+

z2α − a2 − b2 μ2α , a − ibμα

ζα∗

=

z∗α +

+ 2 2 ∗2 z∗2 α − a − b μα , a − ibμ∗α

(8.2c)

and 2a, 2b are the major and minor axes of the ellipse and ψ is a real parameter. The superscript ∗ denotes the value related to the inclusions. f0 represents the function associated with the unperturbed elastic field which is related to the solutions of homogeneous media and is holomorphic in the entire domain except some singular points such as the points under concentrated forces or dislocations and the points at zero or infinity. f1 (or f2 ) is the function corresponding to the perturbed field of matrix (or inclusion) and is holomorphic in region S1 (or S2 ) except some singular points. S1 and S2 denote, respectively, the regions occupied by the matrix and inclusion. The argument ζα related to zα (= x1 + μα x2 ) through the transformation function (8.2c) was first introduced in (6.3). The transformation (8.2c)1 is one-to-one outside the elliptical inclusion and has been discussed detailedly in Section 6.1. However, since the critical point ζα0 calculated at (6.4) is found to be located inside the elliptical inclusion, the transformation is double-valued in region S2 . Figure 8.1 shows the transformation among z-plane, zα -plane, and ζα -plane. It can be seen that there are two different ζα inside the unit circle corresponding to one zα inside the elliptic inclusion. To have a one-to-one transformation, we designate the point nearest the edge of the unit circle to be the mapped point. For this selection, the discontinuity occurs when two originally continuous points (zα )1 and (zα )2 are mapped √ onto (ζα )1 and (ζα )2 shown in Fig. 8.1. Actually the points (ζα )1 = mα σ and √ (ζα )2 = mα e2iθα / σ correspond to the same point in the zα -plane, where σ = eiψ denotes the points located on the edge of the unit circle. Hence, the transformation function (8.2c)1 now maps the whole zα -plane, cut along a slit, into the ζα -plane √ deprived of the circle of radius mα . To remedy this discontinuity, i.e., eliminating the slit which does not exist in the problem, the following restriction should be satisfied: f

√ ,√ mα σ = f mα e2iθ α / σ ,

(8.3a)

8.1

Elliptical Elastic Inclusions

241

b

a

a

zα(1) zα(2) 0

zα

z−plane

z−plane 1.0

ζα0 –1.0

mα

ζα(1) ζα

m1α 2

1.0

0

ζ(α2)

–1.0

ζα−plane

Fig. 8.1 Mapping between z-plane, zα -plane, and ζα -plane. (a) z-plane (b/a = 0.6,μα = 0.3 + 1.5i); (b) zα -plane (b/a = 0.6, μα = 0.3 + 1.5i); and (c)ζα -plane (b/a = 0.6, μα = 0.3 + 1.5i)

√ where mα and θα denote, respectively, the modulus and argument of the critical points ζαo , i.e., * ζαo = ±

a + ibμα √ √ = ± γα = ± mα eiθα . a − ibμα

(8.3b)

From the above discussions, we see that for the present problem in ζα -plane S1 is the region outside the unit circle, while+S2 is the region of the annual ring between the unit circle and the circle of radius m∗α . Since f2 is required to be holomorphic in the annual ring, it can be represented by Laurent’s expansion: f2 (ζ ∗ ) =

∞ k=−∞

Satisfaction of (8.3) gives

ck ζ ∗k .

(8.4a)

242

8 Inclusions

c−k =< γα∗k > ck ,

γα∗ =

a + ibμ∗α . a − ibμ∗α

(8.4b)

By using the general solution given in (8.2) and the expression given in (8.4), the traction boundary condition of (8.1) leads to ∞

B1 f1 (σ ) + B1 f0 (σ ) −

{B2 ck + B2 < γα∗k > ck }σ −k

k=1

∞

=−B1 f1 (σ ) − B1 f0 (σ ) +

{B2 ck + B2 < γ

k=1

∗k α

(8.5) > ck }σ k .

One of the important properties of holomorphic functions used in the method of analytic continuation is that if f(ζ ) is holomorphic in S1 (or S2 + S0 ), then f(1/ζ¯ ) is holomorphic in S2 + S0 (or S1 ). Here, S0 denotes the region inside the circle of √ radius mα . From this property and (8.5), we may introduce a function which is holomorphic in the entire domain including the interface boundary, i.e.,

θ(ζ ) =

⎧ ∞ ⎪ ⎪ {B2 ck + B2 < γα∗k > ck }ζ −k , ⎨B1 f1 (ζ ) + B1 f0 (1/ζ ) − k=1 ∞

⎪ ⎪ ⎩−B1 f1 (1/ζ¯ ) − B1 f0 (ζ ) +

{B2 ck + B2 < γ

k=1

∗k α

> ck }ζ k ,

ζ ∈ S1 , ζ ∈ S2 + S0 . (8.6)

In the above, the singular points of f0 are assumed to be located in the matrix only, and hence f0 can be expressed in Taylor’s expansion as

f0 (ζ ) =

∞

(k)

ek ζ k ,

ek =

k=0

f0 (0) 1 = k! 2π i

C

f0 (ζ ) dζ . ζ k+1

(8.7)

Since θ (ζ ) is now holomorphic and single-valued in the whole plane including the point at infinity, by Liouville’s theorem we have θ (ζ ) ≡ constant. However, constant function f corresponds to rigid body motion which may be neglected. Therefore, θ (ζ ) ≡ 0. With this result, (8.6) leads to ∞ k=1 ∞

{B2 ck + B2 < γα∗k > ck }ζ −k = B1 f1 (ζ ) + B1 f0 (1/ζ ),

ζ ∈ S1 , (8.8)

{B2 ck + B2
ck }ζ = B1 f1 (1/ζ ) + B1 f0 (ζ ), k

k=1

Similarly, the boundary condition u1 = u2 provides

ζ ∈ S2 + S0 .

8.1

Elliptical Elastic Inclusions ∞

243

{A2 ck + A2 < γα∗k > ck }ζ −k = A1 f1 (ζ ) + A1 f0 (1/ζ ),

ζ ∈ S1 ,

k=1 ∞

(8.9) {A2 ck + A2
ck }ζ = A1 f1 (1/ζ ) + A1 f0 (ζ ), k

ζ ∈ S2 + S0 .

k=1

Cancellation of f1 (ζ ) between (8.8) and (8.9) leads to f0 (ζ ) =

∞

iAT1 {(M1 + M2 )A2 ck + (M1 − M2 )A2 < γ¯α∗k > c¯ k }ζ k ,

(8.10)

k=1

where Mk is the impedance matrix defined in (3.132). By comparing the coefficients of corresponding terms of (8.7) and (8.10), the unknown constants ck are determined as −1 −1 ck = (Go − Gk G0 Gk )−1 (tk − Gk G0 ¯tk ),

k = 1, 2, · · · ∞,

(8.11a)

where G0 = (M1 + M2 )A2 ,

Gk = (M1 − M2 )A2 < γα∗k >,

tk = −iA−T 1 ek . (8.11b)

Note that the solutions associated with c0 are ignored because the constant stress function does not produce stress, which represents a rigid body motion. Having the solution of ck , function f1 (ζ ) can now be obtained from (8.8)1 or (8.9)1 , i.e., f1 (ζ ) = −B−1 1 B1 f0 (1/ζ ) +

∞

¯ k + B2 < γα∗k > ck }ζ −k B−1 1 {B2 c

(8.12a)

∗k −k A−1 1 {A2 ck + A2 < γα > ck }ζ .

(8.12b)

k=1

or f1 (ζ ) = −A−1 1 A1 f0 (1/ζ ) +

∞ k=1

When (8.7) is employed, we have f1 (ζ ) =

∞

¯ k + B2 < γα∗k > ck − B1 ek }ζ −k B−1 1 {B2 c

(8.12c)

k=1

or f1 (ζ ) =

∞ k=1

¯ k + A2 < γα∗k > ck − A1 ek }ζ −k . A−1 1 {A2 c

(8.12d)

244

8 Inclusions

Here the subscripts of ζ in (8.12a–d) have been dropped. To replace ζ by ζ1 , ζ2 , or ζ3 for each component function, the translating technique introduced in Section 4.2.1 should be employed in order to satisfy the solution form given in (3.24). The whole field solution can then be found by using (8.2). If one is interested in the interfacial stresses along the inclusion boundary, calculation may be performed by using the field solution of the matrix or inclusion. The stress components based upon the coordinate (s, n) tangent and normal to the interface boundary can be calculated by using the relations obtained in (3.36), from which the derivatives of the stress function vector φ with respect to the normal and tangent directions should be calculated first. The derivative of φ along the interface φ, s should be continuous across the interface since φ1 = φ2 along the interface. However, φ, n may be discontinuous. The evaluation of φ, n and φ, s can be performed by using chain rule as shown between (6.12) and (6.17). If the field solution of the inclusion given in (8.2b) with f2 (ζ ∗ ) obtained in (8.4) and (8.11) is used, we have φ2,n =

∞ 2k

ρ

k=1

! " Im B2 < μ∗α (θ ) >< e−ikψ γα∗k − eikψ > ck ,

φ1,s = φ2,s = φ, s =

∞ 2k k=1

ρ

! " Im B2 < e−ikψ γα∗k − eikψ > ck ,

(8.13a)

where μ∗α (θ ) =

μ∗α cos θ − sin θ , μ∗α sin θ + cos θ

3 ρ=

a2 sin2 ψ + b2 cos2 ψ.

(8.13b)

Similarly, φ1,n can be obtained by applying the field solution of the matrix given in (8.2a), or by (3.80) u u1,n = N(θ ) , s , φ1,n φ, s

(8.14)

in which u, s and φ, s can be obtained by using the field solution of the inclusion or the matrix since they are continuous across the interface. If the function f0 corresponding to the unperturbed elastic field is expressed in Laurent’s expansion, i.e., f0 (ζ ) =

∞ k=−∞

ek ζ k ,

e−k =< γαk > ek ,

ek =

1 2π i

C

f0 (ζ ) dζ , ζ k+1

(8.15)

by repeating the process given between (8.5) and (8.12) one may find that the solution of ck for this case has exactly the same expression as (8.11), and function f1 (ζ ) is obtained as

8.1


f1 (ζ ) = −

∞

245

! " B−1 B1 ek + B1 < γαk > ek − B2 c¯ k − B2 < γα∗k > ck ζ −k 1

k=1

(8.16a) or f1 (ζ ) = −

∞

! " k ∗k ¯ ζ −k . c A−1 A e + A < γ > e − A − A < γ > c 1 k 1 k 2 k 2 k α α 1

k=1

(8.16b) Notice again that a replacement of the argument ζ1 , ζ2 , or ζ3 should be made for each component function of the complex function vectors f0 (ζ ), f1 (ζ ), and f2 (ζ ).

8.1.1 Uniform Loading at Infinity In the case when the elastic inclusion in an infinite matrix is subjected to uniform loading at infinity, detail analysis has been given in Hwu and Ting (1989) and shown in Ting (1996) by using semi-inverse method, i.e., the function form of f(ζ ) is chosen before calculation. In this section, without any prior choices general solutions of f(ζ ) are obtained for arbitrary loading conditions. Here, we like to derive the solution for the case of uniform loading by using the results obtained in this section. To find the solution, first we need to know the function f0 associated with the unperturbed elastic field, i.e., the solution corresponding to a homogeneous medium subjected to uniform loading at infinity. From (4.3), (4.5)1 , and (6.3)1 , we know that the exact solution of f0 corresponding to homogeneous media is f0 (ζ ) =< zα > q =

1 < a − ibμα >< ζα + γα ζα−1 > q, 2

(8.17a)

where T ∞ q = AT1 t∞ 2 + B1 ε1 ,

t∞ 2

⎧ ∞⎫ ⎨σ21 ⎬ ∞ , = σ22 ⎩ ∞⎭ σ23

ε∞ 1

⎧ ∞ ⎫ ⎨ε11 ⎬ ∞ = ε12 ⎩ ∞⎭ 2ε13

(8.17b)

and σij∞ , εij∞ are the constant stresses and strains induced by the uniform loading applied at infinity. From the second equality of (8.17a), we see that the coefficient vector ek of f0 expressed by (8.15) is e1 =

1 < a − ibμα > q, 2

ek = 0,

k = 2, 3, . . . , ∞.

(8.18)

With the results of (8.18), the functions f1 (ζ ) and f2 (ζ ) corresponding to the perturbed fields of matrix and inclusion are obtained from (8.16), (8.4), and (8.11) as

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8 Inclusions

f1 (ζ ) =< ζα−1 > g1 ,

f2 (ζ ) =
c1 , a − ibμ∗α

(8.19a)

where ¯ 1 + B1 < γα > e1 − B2 c¯ 1 − B2 < γα∗ > c1 }, g1 = −B−1 1 {B1 e −1

−1

−T

c1 = −i{G0 − G1 G0 G1 }−1 {A−T 1 e1 + G1 G0 A1 e1 }

(8.19b)

and G0 , G1 , and e1 are defined in (8.11b) and (8.18). Note that f2 (ζ ) obtained in (8.19a)2 represents a state of uniform stress which has been observed by Eshelby (1957). Since the solutions shown in (8.17), (8.18), and (8.19) are somewhat complicated and so are those obtained in Hwu and Ting (1989) and Ting (1996) by the other approach, their outlooks are different and are not easy to be proved identical to each other. By numerical calculation (Yen, 1991), the solutions presented here have been proved to be identical to those given in Hwu and Ting (1989) and Ting (1996). Besides the numerical verification, the solution can also be checked by considering the special condition that the matrix and inclusion are composed of the same material. If A1 = A2 , B1 = B2 , M1 = M2 , we have, by (8.11) and (8.16), f1 (ζ ) = 0. The zero perturbed solution means that there is no inclusion effect for the homogeneous medium, which is expected since the selected f0 represents the exact solution of homogeneous medium subjected to uniform loading at infinity. Before closing this section, we want to emphasize that when employing the method of analytical continuation the unperturbed function f0 is not required to be the exact solution corresponding to homogeneous media. The selection of f0 depends on the satisfaction of loading condition and the convenience for the followup derivation. For example, in the present case an alternative choice may be provided by f0 (ζ ) =< ζα > q,

q=

1 T ∞ < a − ibμα > (AT1 t∞ 2 + B1 ε1 ). 2

(8.20)

This solution of f0 is not a solution for uniform stress distribution, which can be seen from the transformation function (8.2c)1 . However, in the follow-up derivation (8.20) may be more convenient than the choice of exact solution given in (8.17), because the singular points of (8.20) are at infinity while singularities occur at zero and infinity for (8.17). With the choice of (8.20), the solutions of the functions f1 (ζ ) and f2 (ζ ) corresponding to the perturbed fields of matrix and inclusion should be obtained from (8.12), (8.4), and (8.11). By careful derivation, one can prove that the final results of f0 + f1 are the same for different choices of f0 .

8.1.2 Concentrated Forces at the Matrix Consider an infinite anisotropic medium containing an elastic inclusion, subjected to a concentrated force pˆ = (ˆp1 , pˆ 2 , pˆ 3 ) applied on xˆ = (ˆx1 , xˆ 2 ) located in the

8.1


247

matrix. The elasticity solution of this problem can be used as a Green’s function for boundary element methods. Like the problem of uniform loading discussed in the previous section, to find a solution for the present problem first we need to choose a proper function f0 for the unperturbed elastic field. The Green’s function for the infinite homogeneous anisotropic plate has been found in Section 4.1.3 and its associated function f0 is ˆ f0 (ζ ) =< ln(zα − zˆα ) > AT1 p/2π i.

(8.21)

However, using (8.21) is inconvenient in calculation when our general solution is expressed in terms of the variable ζα not zα . An alternative choice for f0 is ˆ f0 (ζ ) =< ln(ζα − ζˆα ) > AT1 p/2π i.

(8.22)

This expression is more convenient than the one given in (8.21). Moreover, it also reflects the singularity characteristics of the original problems. Actually, this choice has ever been made in (6.55b) when we tried to find the Green’s function for hole problems. By using (8.22) as the solution for the unperturbed elastic field, the coefficient vector ek of f0 can be obtained from (8.7)2 as ek =
AT1 p, 2π ik α

k = 1, 2, 3, . . . , ∞.

(8.23)

With (8.22) and (8.23), the functions f1 (ζ ) and f2 (ζ ∗ ) corresponding to the perturbed fields of matrix and inclusion are obtained from (8.12a), (8.4), and (8.11) as −1 1 −1 − ζ¯ˆ ) > AT p α 1ˆ 2π i B1 B1 !< ln(ζ " ∞ −1 + B1 B2 c¯ k + B2 < γα∗k > ck ζ −k , k=1 ∞ ∗ f2 (ζ ) = {ck ζ ∗k + < γα∗k > ck ζ ∗−k }, k=1

f1 (ζ ) =

(8.24)

where ck , k = 1, 2, . . . , ∞, can be determined from (8.11) in which ek are given in (8.23). Replacing ζ in (8.24) by ζ1 , ζ2 , or ζ3 for each component function, the solutions of f1 (ζ ) and f2 (ζ ∗ ) for the whole field are f1 (ζ ) =

3

T ˆ < ln(ζα−1 − ζ¯ˆ k ) >B−1 1 B1 Ik A1 p k=1 ! " ∞ ∗k > c , ¯ + c < ζα−k > B−1 B + B < γ 2 k 2 k α 1 1 2π i

f2 (ζ ∗ ) =

(8.25)

k=1 ∞

k=1

< ζα∗k + (γα∗ /ζα∗ )k > ck .

To check the correctness of the present results, one may consider (1) the simplest condition that the matrix and inclusion are composed of the same material and (2)

248

8 Inclusions

the cases that the inclusions are very soft or hard, which can be compared with the results of holes or rigid inclusions. By setting A1 = A2 = A, M1 = M2 = M in (8.11), and evaluating ek from (8.23), one may obtain ck , f1 (ζ ), and f2 (ζ ) by (8.11) and (8.25). The infinite series representation of f1 (ζ ) and f2 (ζ ) can then be shown to be a Taylor’s expansion of logarithmic function. Combining the results, one can ˆ i, which is the solution prove that f0 (ζ ) + f1 (ζ ) = f2 (ζ ) =< ln(zα − zˆα ) > AT p/2π for a homogeneous medium under a concentrated force. In the case that the inclusion is elastic, numerical calculation has been performed (Yen, 1991) and the results show that the solutions for holes or rigid inclusions are really approximated by very soft or hard inclusions. To see the effect of elliptic shape and the singular behavior near the crack tips or the tips of rigid line inclusions, a series of numerical data for the hoop stress have been plotted (Yen, 1991). A nearly constant value of the hoop stress for b → 0 has been observed when the inclusion is not a hole or rigid medium, which means that no singular behavior occurs for the general elastic inclusions. For elliptic holes or rigid inclusions, singular behavior occurs when b → 0 which is expected for the cracks and rigid line inclusions.

8.2 Rigid Inclusions Holes are the extreme cases of elastic inclusions for which the inclusion is extraordinarily soft relative to the matrix. The other extreme case is rigid inclusion which means that the inclusion is absolutely rigid and cannot be deformed. However, a rigid body rotation ω relative to the matrix may occur. Hence, the boundary conditions for the case of rigid inclusions are u = u0 ,

along the inclusion boundary,

(8.26a)

where u0 = ω(x1 i2 − x2 i1 ),

⎧ ⎫ ⎨1⎬ i1 = 0 , ⎩ ⎭ 0

⎧ ⎫ ⎨0⎬ i2 = 1 . ⎩ ⎭ 0

(8.26b)

If no relative rigid body motion between the matrix and the inclusion occurs, the condition along the boundary of the rigid inclusion can be written as u = 0,

along the inclusion boundary

(8.27)

Therefore, the problems of rigid inclusions are counterparts of traction-free hole problems. The general solution given in (3.24) shows that the expressions of displacements and stress functions are distinguished only by the material eigenvector matrices A and B. Thus, the relevant boundary conditions of the displacementprescribed problems differ from those of the corresponding traction-prescribed

8.2

Rigid Inclusions

249

problems only in the appearance of the matrices A and B. Hence, it is very possible to find the solutions of displacement-prescribed problems from the solutions of their corresponding traction-prescribed problems through analogy technique, i.e., through the interchange of A and B, or vice versa. With this understanding, most of the solutions for the rigid inclusion problems can be obtained directly from their corresponding hole problems discussed in Chapter 6. A typical example of the analogy between hole and rigid inclusion is the Green’s function. Consider an infinite anisotropic plate containing an elliptic rigid inclusion under a concentrated force pˆ and a dislocation with Burgers vector bˆ applied at point xˆ . If no relative rotation occurs for the rigid inclusion, the boundary conditions can be written as u = 0, along the inclusion boundary, ˆ dφ = p, for any closed curve C enclosing the point xˆ , C ˆ du = b, for any closed curve C enclosing the point xˆ ,

(8.28)

C

σ ij → 0,

at infinity.

With reference to Green’s function for the problems with an elliptic hole, (6.61) and (6.63), we obtain the Green’s function for the problems with an elliptical rigid inclusion as 3 1 −1 ˆ ˆ A < ln(ζα − ζ k ) > q , u = Im A < ln(ζα − ζα ) > q0 + π k=1 3 1 −1 ˆ ˆ φ = Im B < ln(ζα − ζα ) > q0 + B < ln(ζα − ζ k ) > q , π

(8.29a)

k=1

where ˆ q0 = AT pˆ + BT b,

q = A−1 AIk q0 .

(8.29b)

Note that the solution form of (8.29a) is exactly the same as (6.61) and (6.63). q0 in (8.29b) is also exactly the same as those obtained in (6.61) and (6.63). The only difference between the hole and the rigid inclusion problems is the value of coefficient vector q in (8.29b) which is obtained from (6.61) and (6.63) by changing B to A. In the case that the relative rotation between matrix and inclusion occurs, the solutions for the rigid inclusion problems cannot be obtained directly from their corresponding hole problems. A small modification to include the relative rotation is necessary. In this section, three different shapes of rigid inclusions will be considered; they are elliptical rigid inclusions, rigid line inclusions, and polygon-like rigid inclusions. In the cases of elliptical rigid inclusions and rigid line inclusions,

250

8 Inclusions

by following the method of analytical continuation discussed previously for the elastic inclusions the general loading conditions will be considered. As to the case of polygon-like rigid inclusions, we consider only the cases of uniform loading at infinity.

8.2.1 Elliptical Rigid Inclusions Consider an infinite anisotropic elastic plate containing an elliptical rigid inclusion subjected to arbitrary loadings at the matrix. The contour of the inclusion boundary can be represented by (6.1), and hence the displacement along the inclusion boundary (8.26) can be written as u = ω(x1 i2 − x2 i1 ) = ω(a cos ψi2 − b sin ψi1 ) =

ω (σ k + σ −1 k), 2

(8.30a)

in which

σ = eiψ ,

⎧ ⎫ ⎨ib⎬ k= a . ⎩ ⎭ 0

(8.30b)

By using (8.2a)1 as the representation of the displacements, the boundary condition shown in (8.30a) can be rewritten as Af1 (σ ) + Af0 (σ ) −

ω −1 ω σ k = −Af1 (σ ) − Af0 (σ ) + σ k. 2 2

(8.31)

In (8.31), the singular points of f0 are assumed to be located in the matrix only and hence can be expressed in Taylor’s expansion as that shown in (8.7). Note that in this section the subscripts 1 and 2 used to distinguish matrix and inclusion are omitted for simplicity (except f1 which is related to f by f = f0 + f1 ) since only the deformation of the matrix is considered for the problems containing rigid inclusions. Similar to the elastic inclusion problems, by the method of analytic continuation equation (8.31) will lead to f1 (ζ ) = −A−1 Af0 (1/ζ ) +

ω −1 A k. 2ζ

(8.32)

To determine the rotation angle ω, we use the condition that the total moment about the x3 -axis due to the traction tn on the elliptic boundary vanishes, i.e.,

2π 0

[a cos ψ(tn )2 − b sin ψ(tn )1 ]ρdψ = 0,

(8.33)

8.2

Rigid Inclusions

251

in which (tn )1 and (tn )2 denote the first and second components of tn . From (3.35)1 , we know that the traction tn on the elliptic boundary can be calculated by differentiation of the stress function vectors φ with respect to the tangential direction s. Using the solutions given in (8.2a)2 and (8.32), (8.33) now leads to ω=−

2

2π o

yT Re{A−T f 0 (eiψ )}dψ , π kT Mk¯

(8.34a)

where yT = (−b sin ψ a cos ψ 0)

(8.34b)

and the prime (• ) denotes differentiation with respect to its argument. The matrix M in (8.34a) is the impendence matrix defined in (3.132) as M = −iBA−1 . The denominator of (8.34a) is real, positive, and nonzero because M is a positive definite Hermitian (Ting, 1996) and in component form is kT Mk = a2 (H−1 )22 + 2ab(H−1 S)21 + b2 (H−1 )11 .

(8.35)

Similar to the problems of elastic inclusions, the interfacial stresses can be determined by φ, n and φ, s , which are " 4 T ω ! N1 (θ )H−1 Re eiψ Af 0 (σ ) − Re ie−iψ B < μα (θ ) > A−1 k , ρ ρ 4 −1 iψ ω ! −iψ −1 " φ, s = H Re e Af 0 (σ ) − Re ie BA k ρ ρ (8.36a)

φ, n =

or " ω ! " ! 4 T N1 (θ )H−1 Re keikψ Aek − Re ie−iψ B < μα (θ ) > A−1 k , ρ ρ ∞

φ, n =

k=1

φ, s =

4 −1 H ρ

∞ k=1

" ω ! ! " Re keikψ Aek − Re ie−iψ BA−1 k . ρ

(8.36b) If the function f0 corresponding to the unperturbed elastic field is expressed in Laurent’s expansion as that shown in (8.15), we have f1 (ζ ) = −A−1

∞ ! " ω −1 A k, Aek + A < γαk > ek ζ −k + 2ζ

(8.37a)

k=1

where −T

ω=

−2 Im{kT A

kT Mk

e1 }

.

(8.37b)

252

8 Inclusions

The expressions of the derivatives φ, n and φ, s are exactly the same as (8.36b). Uniform Loading at Infinity When the elliptical rigid inclusion in an infinite matrix is subjected to uniform loading at infinity, like the choice made in (8.17) for the corresponding problems of elastic inclusions, f0 associated with the unperturbed elastic field can be selected as

where e1 =

f0 (ζ ) =< ζα + γα ζα−1 > e1 ,

(8.38a)

1 T ∞ < a − ibμα > (AT t∞ 2 + B ε1 ). 2

(8.38b)

Substitution of (8.38) into (8.37) will then provide the solutions for the rigid inclusions as f1 (ζ ) = −

1 ∞ ¯ < ζα−1 > A−1 (aε∞ 1 + ibε2 − ωk), 2

(8.39a)

where ω=

−1 ∞ −1 ∞ 2 −1 ∞ a2 (H−1 ε∞ 1 )2 − ab[(H Sε1 )1 + (H Sε2 )2 ] − b (H ε2 )1 . a2 (H−1 )22 + 2ab(H−1 S)21 + b2 (H−1 )11

(8.39b)

The solution (8.39) is obtained directly from the general solutions shown in (8.37), which is derived by the method of analytical continuation, whereas the solution of the corresponding hole problem discussed in Section 6.1.1 is obtained by the other way. To show that the solutions obtained by these two different ways are identical to each other, we now repeat this problem by following the approach presented in Section 6.1.1. By following the steps described between (6.8) and (6.10) for the corresponding hole problem, the solutions for the present problem are assumed in the form of ∞ −1 u = x1 ε∞ 1 + x2 ε2 + 2Re{A < ζα > q}, ∞ −1 φ = x1 t∞ 2 − x2 t1 + 2Re{B < ζα > q},

(8.40)

∞ ∞ ∞ where ε∞ 1 , ε2 , t1 , and t2 are defined in (6.6b) and ζα is defined in (8.2c). From (8.40), the displacement u and stress function vectors φ along the inclusion boundary have been obtained in (6.9) as ∞ u = Re{e−iψ (aε∞ 1 + ibε2 + 2Aq)}, ∞ φ = Re{e−iψ (at∞ 2 − ibt1 + 2Bq)}.

(8.41)

With the result of (8.41)1 , the inclusion boundary condition (8.30) now gives us 1 ∞ q = − A−1 (aε∞ 1 + ibε2 − ωk). 2

(8.42)

8.2

Rigid Inclusions

253

It can be seen clearly that q obtained in (8.42) is the same as the coefficient vector of (8.39a). To determine the rotation angle ω, we use the condition shown in (8.33) which will then give us ω=

∞ Re{kT M(aε∞ 1 + ibε2 )}

kT Mk

.

(8.43)

Written in component form, it can be proved that (8.43) is exactly the same as that shown in (8.39b). With the results of (8.40), (8.40), (8.41), (8.42), and (8.43), by using the relations (8.36) and following the steps described between (6.12) and (6.20), we can determine the stresses along the inclusion boundary. Concentrated Forces at the Matrix When a concentrated force pˆ = (ˆp1 , pˆ 2 , pˆ 3 ) applied on ˆx = (ˆx1 , xˆ 2 ) located in the matrix is considered, a proper function f0 for the unperturbed elastic field has been selected in (8.22) as ˆ i. f0 (ζ ) =< ln(ζα − ζˆα ) > AT p/2π

(8.44)

Substituting (8.44) into (8.32), and using the translating technique introduced in (4.50) and (4.51), we obtain ω 1 T < ln(ζα−1 − ζˆ k ) >A−1 AIk A pˆ + < ζα−1 > A−1 k, 2π i 2 3

f1 (ζ ) =

(8.45)

k=1

where the relative rotation ω can be evaluated by (8.34) with f0 given in (8.44). With the aid of residue theorem, we obtain T

ω=

Re{k A−T < ζˆα−1 > AT }pˆ π kT Mk

.

(8.46)

ˆ If the load is applied on the interface boundary, i.e., ζˆα = eiψ , we have

ω=

−ˆx2 pˆ 1 + xˆ 1 pˆ 2 π kT Mk

,

(8.47)

ˆ b sin ψ) ˆ is the location of the applied force p. ˆ This where (ˆx1 , xˆ 2 ) = (a cos ψ, solution is equivalent to the one given by Ting and Yan (1991) using different approach.

254

8 Inclusions

8.2.2 Rigid Line Inclusions Uniform Loading at Infinity A rigid line inclusion of length 2a can be made by letting the minor axis 2b of the ellipse equal to zero. From (8.40), (8.42), and (8.43) with b = 0, one obtains the solution for an infinite anisotropic elastic plate containing a rigid line inclusion subjected to uniform load at infinity, ∞ −1 −1 ∞ u = x1 ε∞ 1 + x2 ε2 − aRe{A < ζα > A }(ε1 − ωi2 ), ∞ −1 −1 ∞ φ = x1 t∞ 2 − x2 t1 − aRe{B < ζα > A }(ε 1 − ωi2 ),

(8.48a)

where ω=

a Re{kT Mε∞ 1 } kT Mk

,

ζα−1 =

1 (zα − a

3

z2α − a2 ).

(8.48b)

By using (8.48a)2 , (8.13), and the identities (3.131) and (3.137)4 with x2 = 0, |x1 | > a, the stresses ahead of the rigid line inclusion along x1 -axis can be obtained as ⎛

⎞

⎝1 − 3 σi2 = φ, 1 = t∞ 2 − ⎛

x1 x12

− a2

⎠ ST H−1 (ε∞ 1 − ωi2 ), ⎞

(8.49)

x1

⎝1 − 3 ⎠ (N3 + NT1 ST H−1 )(ε∞ σi1 = −φ, 2 = t∞ 1 + 1 − ωi2 ). 2 2 x1 − a In a similar way, one can obtain the stresses along the boundary of rigid line inclusion and the displacement along the x1 -axis. Concentrated Forces at the Matrix From (8.2a), (8.44), and (8.45) with b = 0, one obtains the Green’s function for an infinite anisotropic elastic plate containing a rigid line inclusion as 3 1 T T −1 −1 ˆ ˆ A < ln(ζα − ζ k ) > A AIk A pˆ u = Im A < ln(ζα − ζα ) > A + π k=1

ζα−1

−1

ζα−1

−1

+ Re{ωA < > A k}, 3 1 T T −1 −1 φ = Im B < ln(ζα − ζˆα ) > A + B < ln(ζα − ζˆ k ) > A AIk A pˆ π k=1

+ Re{ωB < where

>A

k}, (8.50a)

8.2

Rigid Inclusions

255

ω=

T Re{k A−T < ζˆα−1 > AT }pˆ

π kT Mk

(8.50b)

and 3 1 2 2 ζα = zα + zα − a , a

ζˆα =

1 α

3 2 2 zˆα + zˆα − a .

(8.50c)

3 If the load is applied on the rigid line, i.e., xˆ 2 = 0, we have ζˆα = (ˆx1 + i a2 − xˆ 12 )/a and ω=

xˆ 1 pˆ 2 π kT Mk

.

(8.51)

8.2.3 Polygon-Like Rigid Inclusions Consider an infinite anisotropic elastic plate containing a polygon-like rigid inclusion subjected to uniform loading at infinity. To find the solution of this problem we may follow the steps described in Section 6.2.2 for the corresponding hole problem. Thus, the general solution for the present problem can now be assumed as ∞ −1 −k u = x1 ε∞ 1 + x2 ε2 + 2 Re{A[< ζα > q1 + < ζα > qk ]}, ∞ −1 −k φ = x1 t∞ 2 − x2 t1 + 2 Re{B[< ζα > q1 + < ζα > qk ]},

(8.52a)

where ζα is related to zα by zα =

" a! (1 − iμα c)ζα + (1 + iμα c)ζα−1 + ε(1 + iμα )ζαk + ε(1 − iμα )ζα−k . 2 (8.52b)

∞ ∞ ∞ ε∞ 1 , ε2 , t1 , t2 are defined in (6.6b); a,c, k, and ε are shape parameters which have been described at the beginning of Section 6.2; q1 and qk are the coefficient vectors to be determined through satisfaction of the boundary conditions. To determine q1 and qk , we make the following replacement:

q = AT g + BT h ,

= 1, k,

(8.53)

where g and h are real. Substituting (8.53) into (8.52) with the boundary values (6.64) and ζα = eiψ , and using the identities (3.59), the boundary condition along the rigid inclusion (8.26) gives us (Hwu and Wang, 1992) the following:

256

8 Inclusions ∞ h1 = a(ωi2 − ε∞ 1 ) + x2 ε2 ,

hk = aε(ωi2 − ε∞ 1 ),

∞ g1 = −aH−1 {S(ωi2 − ε∞ 1 ) − c(ωi1 + ε2 )},

(8.54)

∞ gk = −aεH−1 {S(ωi2 − ε∞ 1 ) + (ωi1 + ε2 )},

where the rotation angle ω can be determined by the condition that the total moment about the x3 -axis due to the traction tn on the polygon-like boundary vanishes, i.e.,

2π

a[(cos ψ + ε cos kψ)(tn )2 − (c sin ψ − ε sin kψ)(tn )1 ]ρdψ = 0.

(8.55)

0

With the above results, by the way described between (6.79) and (6.84), we can determine the interfacial stresses along the inclusion boundary. The explicit expressions of ω and the interfacial stresses can be found in Wang (1990).

8.3 Interactions Between Inclusions and Dislocations Interactions between inclusions and dislocations have been a topic of considerable research. Greater understanding of material defects can be gained through the solution of suitable elasticity problems. The solutions of dislocations are frequently used as kernel function of integral equation to consider the interactions between inclusions and cracks (Erdogan et al., 1974; Patton and Santare, 1990). The problem of a circular elastic inclusion near an edge dislocation was solved in terms of Airys stress potentials by Dundurs and Mura (1964). Later, Dundurs and Sedeckyj (1965) solved the related problem where the dislocation is within the circular inclusion. The analytic solutions of isotropic elliptic inhomogeneity were obtained by Stagni and Lizzio (1983) for a dislocation located outside an elliptic inhomogeneity and by Warren (1983) for a dislocation inside an elastic elliptic inhomogeneity. A closedform solution is obtained by Santare and Keer (1986) for a dislocation near the rigid elliptical inclusion, in which particular attention is paid to the rigid body rotation of the inclusion relative to the dislocation. While these solutions are generally in isotropic materials, the concept of an elastically isotropic crystal is an idealization. All real crystals should be considered to be anisotropic. Isotropic theory may lead to useful results, but for some cases it is an inadequate approximation. Moreover, there are increasing numbers of observations of sufficient accuracy to warrant comparison with more precise anisotropic calculations. Also, some effects, such as the instability of some straight dislocations with respect to break up into zigzag shape, require anisotropic theory even for their qualitative explanation (Hirth and Lothe, 1982). With this concern, Hwu and his co-workers (Yen and Hwu, 1994; Yen et al., 1995) studied such interaction problems by using Stroh formalism (Stroh, 1958) for anisotropic elasticity to solve the general analytical solution for the

8.3

Interactions Between Inclusions and Dislocations

257

interactions between dislocations and anisotropic elastic elliptical inclusions. The presentation of this section will then follow that of Yen and Hwu (1994) and Yen et al. (1995). Because the transformation function introduced in (8.2c) is singlevalued outside the elliptical inclusion but nonsingle-valued inside the inclusion, in the following discussions we will separate the problems into three different categories: the dislocation outside, inside, and on the interface of the inclusion.

8.3.1 Dislocations Outside the Inclusions Consider an elliptical anisotropic elastic inclusion imbedded in an infinite matrix. A dislocation with Burgers vector bˆ = (bˆ 1 , bˆ 2 , bˆ 3 ) is located at the point xˆ = (ˆx1 , xˆ 2 ) which is outside the inclusion. As discussed in Section 4.1.5, the solutions of the dislocation problems can be obtained in a straightforward manner from the corresponding solutions of the concentrated force problems with AT pˆ replaced by ˆ Thus, from (8.2), (8.22), and (8.25), the solution for the present problem can BT b. be obtained as

u1 = A1 [f0 (ζ ) + f1 (ζ )] + A1 [f0 (ζ ) + f1 (ζ )] φ1 = B1 [f0 (ζ ) + f1 (ζ )] + B1 [f0 (ζ ) + f1 (ζ )] u2 = A2 f2 (ζ ∗ ) + A2 f2 (ζ ∗ ) φ2 = B2 f2 (ζ ∗ ) + B2 f2 (ζ ∗ )

,

,

ζ ∈ S1 ,

ζ ∗ ∈ S2 ,

(8.56a)

(8.56b)

where 1 ˆ < ln(ζα − ζˆα ) > BT1 b, 2π i 3 ∞ 1 T ˆ+ 1 ˆ < ln(ζα−1 − ζ¯ˆ k ) >B−1 B I B < ζα−k > Ek b, b f1 (ζ ) = 1 k 1 1 2π i 2π i f0 (ζ ) =

f2 (ζ ∗ ) =

k=1 ∞

1 2π i

k=1

< ζα∗k + γα∗k ζα∗−k > Ck bˆ

k=1

(8.56c) and

258

8 Inclusions −1 ∗k Ek = −B−1 1 B2 Ck + B1 B2 < γα > Ck , −1

−1

Ck = (Go − Gk G0 Gk )−1 (Tk + Gk G0 Tk ), i < ζˆα−k > BT1 , Tk = A−T k 1 G0 = (M1 + M2 )A2 , Gk = (M1 − M2 )A2 < γα∗k >,

k = 1, 2, . . . , ∞. (8.56d)

∗ Mk , k=1,2, are the impedance matrices defined as Mk = −iBk A−1 k ; and γα is defined in (8.4b). Note that the solutions associated with C0 are ignored because the constant stress function does not produce stress, which represents the rigid body motion. When the elastic inclusion is a traction-free hole, the solution can be obtained by considering the inclusion to be extraordinary soft or directly from that derived in (6.63) by considering the boundary condition to be φ = 0. Similarly, for a rigid inclusion interacted with the dislocation, the solution is found from (8.44), (8.45), ˆ The results are (8.46), and (8.47) with AT pˆ replaced by BT b.

u1 = A1 [f0 (ζ ) + f1 (ζ )] + A1 [f0 (ζ ) + f1 (ζ )]

,

φ1 = B1 [f0 (ζ ) + f1 (ζ )] + B1 [f0 (ζ ) + f1 (ζ )]

ζ ∈ S1 ,

(8.57a)

where f0 (ζ ) is given in (8.56c)1 and 1 T −1 −1 ˆ ω < ln(ζα−1 − ζˆ k ) >A−1 1 A1 Ik B1 b + 2 < ζα > A1 k, 2π i 3

f1 (ζ ) =

(8.57b)

k=1

in which T

ω=

Re{k

A−T 1

< ζˆα−1 > BT1 }bˆ

π kT M1 k

,

⎧ ⎫ ⎨ib⎬ k= a . ⎩ ⎭ 0

(8.57c)

8.3.2 Dislocations Inside the Inclusions When the dislocation is located inside the inclusion, the general solutions for the matrix and inclusion written in (8.2) should better be revised as u1 = A1 [f0 (ζ ) + f1 (ζ )] + A1 [f0 (ζ ) + f1 (ζ )] φ1 = B1 [f0 (ζ ) + f1 (ζ )] + B1 [f0 (ζ ) + f1 (ζ )]

,

ζ ∈ S1 ,

(8.58a)

8.3


259

u2 = A2 [f ∗0 (ζ ∗ ) + f2 (ζ ∗ )] + A2 [f ∗0 (ζ ∗ ) + f2 (ζ ∗ )] φ2 = B2 [f ∗0 (ζ ∗ ) + f2 (ζ ∗ )] + B2 [f ∗0 (ζ ∗ ) + f2 (ζ ∗ )]

ζ ∗ ∈ S2 ,

,

(8.58b)

in which the additional unperturbed function f ∗0 is added to consider the singularity behavior caused by the dislocation. Like the choices given in (8.4) and (8.56c)1 , to find an elasticity solution satisfying the dislocation singularity and the interface continuity condition we now make the following proper selection: f0 (ζ ) =< ln ζα > d, 1 ˆ f ∗0 (ζ ∗ ) = < ln(z∗α − zˆ∗α ) > BT2 b, 2π i ∞ ∗ < ζα∗k + γα∗k ζα∗−k > ck , f2 (ζ ) =

(8.59)

k=1

where d is the unknown coefficient vector to be determined by the satisfaction of the continuity condition described in (8.1). The main consideration for the above choice is the singularity characteristics and the single-valued transformation. Since f0 corresponds to the field of matrix (S1 ) and has the responsibility to reveal the singularity behavior at infinity caused by the dislocation, it becomes clear why we have the choice shown in (8.59)1 . As to f ∗0 , the singular point is the location of the dislocation, zˆ∗α . Therefore, the most appropriate choice is the exact solution for the dislocation in a homogeneous medium, which is the one shown in (8.59)2 . One should also note that unlike (8.56c)1 , the argument z∗α instead of ζα∗ is used in (8.59)2 . The reason for this difference is that the mapping from z∗α -plane to ζα∗ -plane is single-valued outside the inclusion and nonsingle-valued inside the inclusion, and the restriction (8.3) should be required for functions representing the inclusion to be single-valued. Hence, one may use the argument ζα or zα for the singularity located outside the inclusion, such as (8.21) or (8.22). However, for the present problem if the argument ζα∗ instead of z∗α is used in (8.59)2 for f ∗0 , the function will not be single-valued inside the inclusion. To avoid this trouble, the argument z∗α is used in (8.59)2 . By the relation given in (6.3), we have z∗α

− zˆ∗α

# $ 1 γα∗ ∗ ∗ ∗ = (a − ibμα )(ζα − ζˆα ) 1 − . 2 ζˆα∗ ζα∗

(8.60)

Substituting (8.60) into (8.59)2 and considering the points where |ζα∗ | > |ζˆα∗ |, by series expansion it can be shown that f ∗0 (ζ ) =

∞

1 ˆ < ln ζα∗ − e∗kα ζα∗−k > BT2 b, 2π i k=1

where

for

0 ∗0 0ζ 0 > |ζˆ ∗ |, α α

(8.61a)

260

8 Inclusions

e∗kα

$ # 1 γα∗k ∗k ζˆ + = . k α ζˆα∗k

(8.61b)

Note that in (8.61a), the constant term ln(a − ibμ∗α )/2 has been neglected since it corresponds to rigid body motion and has no contribution to the deformation. With a proper choice shown in (8.59) for f0 , f ∗0 , and f2 , the function remained to be determined is f1 which should be holomorphic in S1 . In addition to f1 , the unknown coefficients to be determined are d and ck in (8.59)1,3 . All these unknowns can now be found by the satisfaction of the continuity conditions given in (8.1) through the method of analytical continuation. Substituting (8.59)1,3 and (8.61) into (8.58), the stress continuity condition φ1 = φ2 along the interface ζα = ζα∗ = eiψ = σ leads to B1 [(ln σ )d+ f1 (σ )]∞+ B1 [(ln σ )d +f1 (σ )] ∞ < e∗kα > σ −k BT2 bˆ + (< γα∗k > σ −k + σ k )ck = B2 2π1 i ln σ − k=1 k=1 ∞ ∞ T −1 ∗ k + σ −k )c . + B2 2π i ln σ − < ekα > σ k B2 bˆ + (< γ ∗k > σ k α k=1

k=1

(8.62) Comparison of the coefficients of the ln terms on both sides of (8.62) provides that B1 d − B1 d = 0.

(8.63)

By deleting the ln terms through (8.63), we now re-arrange (8.62) according to the holomorphic property in S1 and S2 as B1 f1 (σ ) + θ1 (σ ) = −B1 f1 (σ ) + θ2 (σ ),

(8.64a)

where " ! 1 ˆ −k − {B2 < e∗kα > BT2 b}σ θ1 (σ ) = B2 c¯ k + B2 < γα∗k > ck σ −k , 2π i ∞

k=1 ∞

" ! 1 T k ˆ k+ θ2 (σ ) = {B2 < e∗kα > B2 b}σ B2 ck + B2 < γ ∗k > c k σ . α 2π i k=1 (8.64b) By (8.64) and the properties of holomorphic functions used in the method of analytic continuation, we may now introduce a function which is holomorphic in the entire domain including the interface boundary, i.e., B1 f1 (ζ ) + θ1 (ζ ), ζ ∈ S+ , θ(ζ ) = −B1 f1 (1/ζ ) + θ2 (ζ ), ζ ∈ S− .

(8.65)

8.3


261

Since θ(ζ ) is now holomorphic in the whole plane including the point at infinity, by Liouville’s theorem we have θ(ζ ) ≡ constant. However, constant function f corresponds to rigid body motion which can be neglected. Therefore, θ(ζ ) ≡ 0. With this result, (8.65) leads to B1 f1 (ζ ) =

−1 2π i

∞ k=1

∞

+

k=1

B1 f1 (1/ζ ) =

{B2 ck + B2 < γα∗k > ck }ζ −k , ζ ∈ S+ ,

1 2π i

+

ˆ −k {B2 < e∗kα > BT2 b}ζ

∞

T ˆ k {B2 < e∗kα > B2 b}ζ

k=1

∞

k=1

(8.66)

k − {B2 ck + B2 < γ ∗k α > ck }ζ , ζ ∈ S .

Similarly, the boundary condition u1 = u2 provides A1 d − A1 d =

1 ˆ b 2π i

(8.67)

and ∞ !

" A2 < e∗kα > BT2 bˆ ζ −k k=1 " ∞ ! + A2 ck + A2 < γα∗k > ck ζ −k , ζ ∈ S+ , k=1 " ∞ ! T A2 < e¯ ∗kα > B2 bˆ ζ k A1 f1 (1/ζ ) = 2π1 i k=1 " ∞ ! k − A2 ck + A2 < γ ∗k + α > ck ζ , ζ ∈ S . A1 f1 (ζ ) =

−1 2π i

(8.68)

k=1

From (8.63) and (8.67), we obtain d=

1 Tˆ B b. 2π i 1

(8.69)

By cancelling f1 (ζ ) between (8.66) and (8.68), and comparing the coefficients of corresponding terms, the unknown constants ck can be determined to have the same expression as (8.11a) except tk =

−1 T ˆ (M1 − M2 )A2 < e∗kα > B2 b. 2π i

(8.70)

Having the solution of ck , function f1 (ζ ) can now be obtained from (8.66)1 or (8.68)1 with the understanding that the subscripts of ζ in (8.66) or (8.68) are dropped. Once the solution of f1 (ζ ) is obtained from (8.66)1 or (8.68)1 , a replacement of ζ1 , ζ2 , or ζ3 should be made for each component function. The whole field solution can then be found by using (8.58) in which

262

8 Inclusions

1 ˆ < ln ζα > BT1 b, 2π i 1 ˆ < ln(z∗α − zˆ∗α ) > BT2 b, f ∗0 (ζ ∗ ) = 2π i ∞ ∞ −1 1 ∗ Tˆ ˆ < ζα−k >B−1 B < e > B < ζα−k > Ek b, b + f1 (ζ ) = 2 kα 2 1 2π i 2π i

f0 (ζ ) =

f2 (ζ ∗ ) =

k=1 ∞

1 2π i

k=1

ˆ < ζα∗k + γα∗k ζα∗−k > Ck b.

k=1

(8.71a) In the above, Ek and Ck have the same expressions as those given in (8.56d) except Tk in (8.56d)3 is now defined by T

Tk = −(M1 − M2 )A2 < e∗kα > B2 .

(8.71b)

To verify the solutions obtained in (8.71), some simplified cases are considered. First, we check the condition when the matrix and inclusion are composed of the same material. If A1 = A2 = A, B1 = B2 = B, M1 = M2 = M, by (8.56d)1,2,4,5 and (8.71b) we have G0 = −iA−T ,

Gk = 0,

Tk = 0,

Ck = 0,

Ek = 0,

k = 1, 2, . . . , ∞. (8.72)

With this result, the solutions (8.71a) become f0 (ζ ) =

1 ˆ < ln ζα > BT b, 2π i

−1 ˆ < ekα ζα−k > BT b, 2π i 3

f1 (ζ ) =

k=1

(8.73)

1 ˆ f2 (ζ ∗ ) = 0. < ln(zα − zˆα ) > BT b, f ∗0 (ζ ∗ ) = 2π i By the expansion shown in (8.61a), we prove that f0 (ζ ) + f1 (ζ ) = f ∗0 (ζ ) + f2 (ζ ∗ ) =

1 ˆ < ln(zα − zˆα ) > BT b, 2π i

(8.74)

which is the exact solution for the dislocation in a homogeneous medium. The next simplified case that has been checked is the condition when both of the matrix and inclusion are made by isotropic materials. This case is implemented based upon the introduction of a small perturbation in the values of material eigenvalues μα such as μ1 = 0.9952i, μ2 = i, μ3 = 1.0048i, because the material eigenvalues of isotropic materials are repeated which violates the assumption used in the general solutions (8.58). By implementing the numerical calculation it has been proved that the solutions (8.71) are identical to those presented by Dundurs and Sendeckyj (1965) for this special case. The difference is that the present solutions are suitable for general anisotropic media and the shape of the inclusion is

8.3


263

ellipse which includes circle and line; however, those presented by Dundurs and Sendeckyj (1965) are valid only for isotropic media and circular inclusion.

8.3.3 Dislocations on the Interfaces Before getting involved in the derivation for the present problem, it might be thought that the problem is trivial since it may be solved by using the results of “inside” or “outside” by just limiting the point to the interface. When actually employing the limiting procedure, we found that the convergent rate of the series form solutions shown in Sections 8.3.1 and 8.3.2 is very bad when we put the dislocation point on the interface since during derivation we have used the series expansions to represent the ln terms which may not be good for the dislocation located on the interface. Therefore, it becomes necessary to find the solution independently. Moreover, the continuity across the interface from the three different solution forms “inside,” “outside,” and “interface” should also be ensured. Following is the presentation for the solution of “interface.” When the dislocation is located on the interface, the choice of f0 and f ∗0 is not so clear as that shown in Sections 8.3.1 and 8.3.2 since the singular point is covered by both the inclusion and matrix. To find an appropriate function form for f0 and f ∗0 , we consider the problem as a combination of the hole problems and the elliptic plate problems. For a plate containing a hole with a singular point located on the hole boundary, a good reference is the solution for the elliptic hole subjected to a concentrated force on the hole boundary, (6.49). In that solution, the singularity is revealed by the function forms as < ln(ζα − ζˆα ) > and < ln(ζα−1 − ζ¯ˆ α ) > . The former also covers the singular behavior at infinity. In the case of the elliptic plate with a singular point on the elliptic boundary, a suitable reference may also be provided by the solution for the elliptic plate subjected to a concentrated force on the elliptic boundary. The singularity for this problem is revealed by < ln(zα − zˆα ) > . The reason that the argument zα is used instead of ζα is the same as that described in Section 8.3.2. From the above description, the general solutions for the present problem may be written by (8.58) with f0 (ζ ) =< ln(ζα − ζˆα ) > q1 + < ln(ζα−1 − ζ¯ˆ α ) > q1 ,

f ∗0 (ζ ∗ ) =< ln(z∗α − zˆ∗α ) > q2 , ∗

f2 (ζ ) =

∞

ck ,

k=1

where q1 , q1 , and q2 are the unknown coefficients to be determined. It should be noted that unlike those given in (8.56c)1 and (8.59)2 whose singular point is in a homogeneous medium (matrix or inclusion), the coefficients q1 , q1 , and q2 are still unknown in the present stage because they will be influenced by the interface condition. The two separate problems, holes and elliptic plates, are only used to help

264

8 Inclusions

us to choose the appropriate function forms like those shown in (8.75). Till now, the entire problem is reduced to finding the unknown coefficients q1 , q1 , q2 , and ck in (8.75) and the unknown function f1 in (8.58) through the use of continuity condition along the interface, the equilibrium, and the dislocation singularity, i.e., u1 = u2 , φ1 = φ2 , along the interface except the point xˆ , . . ˆ around the point xˆ . dφ = 0, du = b,

(8.76)

The mapped points around xˆ may be expressed as ζα = ζˆα + ρeiθ , ρ → 0, where θ starts from the line tangent : to the interface. Hence, the closed integrals may be expressed by dφ = φ1 (π ) − φ1 (0) + φ2 (2π ) − φ2 (π ) and of (8.76) 2 : du = u1 (π ) − u1 (0) + u2 (2π ) − u2 (π ). Substituting (8.75) into (8.58), the equilibrium and the dislocation singularity conditions shown in (8.76)2 now provide B1 q1 + B1 q1 + B2 q2 − B1 q1 − B1 q1 − B2 q2 = 0, 1 ˆ A1 q1 + A1 q1 + A2 q2 − A1 q1 − A1 q1 − A2 q2 = b. πi

(8.77)

As to the continuity conditions shown in (8.76)1 , we employ the method of analytical continuation as that described in Section 8.3.2. Like (8.63) and (8.67), comparison of the coefficients of the ln terms provides that B1 q1 + B1 q1 − B2 q2 + B1 q1 + B1 q1 − B2 q2 = 0, B1 q1 − B1 q1 − B2 q2 − B1 q1 + B1 q1 + B2 q2 = 0, A1 q1 + A1 q1 − A2 q2 + A1 q1 + A1 q1 − A2 q2 = 0,

(8.78)

A1 q1 − A1 q1 − A2 q2 − A1 q1 + A1 q1 + A2 q2 = 0. Like those described between (8.64) and (8.70), the holomorphic properties will then lead to " ∞ ! B2 < ∗kα > q2 ζ −k + B2 ck + B2 < γα∗k > ck ζ −k , ζ ∈ S+ , k=1 k=1 " " ∞ ! ∞ ! ∗ k − B2 ck + B2 < γ ∗k B1 f1 (1/ζ ) = − B2 < kα > q2 ζ k + α > ck ζ , ζ ∈ S , k=1 k=1 " ∞ ∞ ! A1 f1 (ζ ) = − A2 < ∗kα > q2 ζ −k + A2 ck + A2 < γα∗k > ck ζ −k , ζ ∈ S+ , k=1 k=1 " " ∞ ! ∞ ! ∗ k − A2 ck + A2 < γ ∗k A1 f1 (1/ζ ) = − A2 < kα > q2 ζ k + α > ck ζ , ζ ∈ S , B1 f1 (ζ ) = −

∞

k=1

k=1

(8.79a)

where

8.3


∗kα

1 = k

#

γα∗ ζˆα∗

265

$k .

(8.79b)

From (8.77) and (8.78), we obtain 1 Tˆ B b, 2π i 1 1 −1 −1 −1 −1 T ˆ (A2 A1 − B2 B1 )−1 (A2 A1 − B2 B1 )B1 b, q1 = 2π i 1 −1 −1 −1 −1 T ˆ (A A2 − B1 B2 )−1 (A1 A1 − B1 B1 )B1 b. q2 = 2π i 1 q1 =

(8.80)

By cancelling f1 (ζ ) between (8.79a)1 and (8.79a)3 , and comparing the coefficients of corresponding terms, the unknown constants ck can be determined to have the same expression as (8.11a) except ∗

tk = (M1 − M2 )A2 < kα > q2 .

(8.81)

Combining the results obtained in (8.75), (8.79), (8.80), and (8.81), and using the translating technique (4.50) and (4.51), the complex functions of the whole field solution (8.58) can now be written as 1 1 ˆ < ln(ζα − ζˆα ) > BT1 bˆ + < ln(ζα−1 − ζ¯ˆ α ) > Q1 b, 2π i 2π i 1 ˆ < ln(z∗α − zˆ∗α ) > Q2 b, f ∗0 (ζ ) = 2π i 3 ∞ −1 1 −1 −k ∗ Tˆ ˆ < ζα >B1 B2 < kα > Q2 b + < ζα−k > Ek b, f1 (ζ ) = 2π i 2π i f0 (ζ ) =

f2 (ζ ∗ ) =

k=1 ∞

1 2π i

k=1

ˆ < ζα∗k + γα∗k ζα∗−k > Ck b,

k=1

(8.82a) where −1

−1

−1

−1

T

−1

−1

−1

−1

T

Q1 = (A2 A1 − B2 B1 )−1 (A2 A1 − B2 B1 )B1 , Q2 = (A1 A2 − B1 B2 )−1 (A1 A1 − B1 B1 )B1

(8.82b)

and Ek and Ck have the same expressions as those given in (8.56d) except Tk in (8.56d)3 is now defined by ∗

T

Tk = −(M1 − M2 )A2 < kα > Q2 .

(8.82c)

The verification of this problem has also been done by checking (1) the simplest condition that the matrix and the inclusion are composed of the same material

266

8 Inclusions

and (2) the cases that both of the inclusion and matrix are isotropic (Dundurs and Sendeckyj, 1965). Discussions In order to show the continuity across the interface from the three different solution forms shown in Sections 8.3.1, 8.3.2, and 8.3.3, numerical comparison has been done in Yen et al. (1995) for the solutions of “interface,” “inside,” and “outside” by putting the dislocation on the interface. The numerical results show that the solutions from “outside,” “inside,” and “interface” will converge to the same value by different rate, and hence the continuity across the interface is ensured. The terms needed for the “interface” solution (usually below 10) are far less than the terms needed for “inside” or “outside” solution (usually over 1,000), which further explains why we need to derive the “interface” solution independently. When the dislocations are located inside or outside the inclusion and are not close to the interface, the series terms needed for a converged value of stress are usually below 20.

8.3.4 Interaction Energy In the absence of applied tractions, the interaction energy between the dislocation and the inclusion is equal to the elastic potential energy. Furthermore, all terms that are independent of the position of the dislocation, such as the divergent integrals corresponding to the energy of dislocation in the homogeneous materials, can be discarded in the computation of the interaction energy. Thus, the elastic potential energy for the dislocation with Burgers vector bˆ located on xˆ , set up as the work required to introduce the dislocation in the material, is (Eshelby, 1956, Dundurs and Gangadharan, 1969) Ei =

1 2

∞

xˆ 1

i i i (bˆ 1 σ21 + bˆ 2 σ22 + bˆ 3 σ23 )dx1 ,

(8.83)

where σiji = σij − σijd and σij is the stress field when the dislocation is located in an anisotropic matrix with an inclusion, which can be found by the results of Section 8.3.1. σijd is the self-stress of dislocation in an infinite homogeneous medium, which can be found by the results of Section 4.1.5. Since φi, 1 = σi2 and σij → 0 when x1 → ∞, (8.83) now leads to 1 Ei = − bˆ T φi (ˆx1 , xˆ 2 ) 2

(8.84)

If the dislocation moves along s direction, the generalized force Fi in the s direction on the dislocation is defined as the negative gradient of the interaction energy, i.e., Fi = −

∂Ei . ∂s

(8.85)

8.3


267

The total stress function φ(= φi + φd ) for the dislocation in a general position with respect to the inclusion has been provided in (8.56). To evaluate φi , a subtraction of the self-stress function φd , (4.38)2 , from the total stress function should be made. Substituting (8.56) and (4.38)2 into (8.84), the interaction energy can be obtained in an explicit form as 0 1 0 Ei = − bˆ T bˆ 0ζα =ζˆα , 2

(8.86a)

where ∞ −1 1 γα −1 ˆ T −k = > B1 + Im B1 Im B1 < ln 1 − < ζα > B1 Dk π π ζα ζˆα k=1 ∞ 1 T + Im B1 < ln(ζα−1 − ζ¯ˆ k ) > B−1 1 B1 Ik B1 π k=1 (8.86b) and k −T T ˆ k = i (B2 Dk − B2 < γα∗k > Dk Gk G−1 ˆ D 0 )A1 < ζ α > B1 k i −T T ˆ −k + (B2 < γα∗k > Dk − B2 Dk Gk G−1 0 )A1 < ζα > B1 , k −1

Dk = (G0 − Gk G0 Gk )−1 .

(8.86c)

(8.86d)

In a similar approach, the interaction energy between dislocations and holes can also be obtained in an explicit form as γα > BT bˆ 2π Ei = bˆ T Im B < ln 1 − ζα ζˆα 3 T ¯ T −1 −1 ˆ < ln(ζˆα − ζˆ k ) > B BIk B b. − bˆ Im B

(8.87)

k=1

For rigid inclusions, we have γα T T ˆ ˆ >B b 2π E = b Im B < ln 1 − ζα ζˆα 3 T ¯ T −1 −1 ˆ ˆ ˆ < ln(ζα − ζ k ) > A AIk B bˆ − b Im B i

k=1

! " − ωπ b Re B < ζˆα−1 > A−1 k . ˆT

(8.88)

268

8 Inclusions

From the solutions shown in (8.85), (8.86), (8.87), and (8.88), the interactions between dislocations and inclusions can then be studied by the contour of the glide component of the force which tends to move the dislocation toward or away from the inclusion. In regions where the glide component of the force is negative, the dislocation is attracted to the inclusion. A positive value indicates that the dislocation is repelled from the inclusion. When the glide force is zero, the position can be thought of as an equilibrium position for the dislocation. From the numerical results shown in Yen and Hwu (1994), we observe that the glide component of the force is positive for hard inclusions and is negative for soft inclusions. If the holes and rigid inclusions are treated as the limiting cases of the soft and hard inclusions, it has been shown that the absolute value of the glide component of the force increases when the hard inclusions become harder or the soft inclusions become softer. The numerical results also show that the outlooks of the force contour for the isotropic and anisotropic matrix are similar except that the latter looks like to be squeezed in the x2 -direction, which is due to the fact that the anisotropic matrix considered has stronger stiffness in x1 -direction than in x2 -direction, i.e., E1 > E2 .

8.4 Interactions Between Inclusions and Cracks The interaction between inclusions and cracks is a typical and important problem in fracture mechanics. Although several related works have been done in the literature, in order to connect this section with the previous sections we will follow the work presented in Hwu et al. (1995b) which mainly utilized the solutions of dislocation problems. Because the crack can be represented by a distribution of dislocation, by using the solutions of dislocation problems obtained in the previous section we can now study the interactions between inclusions and various types of cracks such as a crack located inside or outside the inclusions, a crack penetrating the inclusions, and a curvilinear crack lying along the interface between inclusion and matrix.

8.4.1 Cracks Outside the Inclusions Consider that a crack located outside an elliptical anisotropic elastic inclusion is subjected to uniform loading at infinity (see Fig. 8.2). Due to the linear property, the principle of superposition can be used and the problem is represented as the sum of the following two problems: (a) an elliptical anisotropic elastic inclusion embedded in an unbounded anisotropic matrix subjected to uniform loading at infinity and (b) same as the original problem except that no loading is applied at infinity and the crack surface is subjected to the loading which has opposite sense and equal magnitude as that obtained from problem (a) at the crack location. As to problem (a), the solution has been found in (8.17), (8.18), and (8.19), of which the stress functions φu1 and φu2 can be expressed as

8.4

Interactions Between Inclusions and Cracks

269

Fig. 8.2 A crack outside an elliptical inclusion

s

n

x2 d

α

( x10 , x20 )

b x1

a

−1 φu1 = 2 Re{B 1 < zα > q + B1 < ζα > g1 }, 2z α > c1 , φu2 = 2 Re B2 < a − ibμ∗α

(8.89)

where q, g1 , and c1 have been given in (8.17b) and (8.19b). The superscript u denotes that the solution is related to the uncracked problem. For problem (b), we represent the crack as a distribution of dislocation. By integrating the solution shown in (8.56) for the dislocation located outside the inclusion, we have φd1 (s) = −

1 π

−

Re iB1 < ln(zα − zˆα ) > BT1 β(t) dt

γα > BT1 β(t) dt Re iB1 < ln 1 − ζα ζˆα − 3 " ! 1 T − β(t) dt Re iB1 < ln(ζα−1 − ζˆj ) > B−1 B I B 1 j 1 1 π −

+

−

1 π

1 π

j=1 ∞

− k=1

(8.90)

Re{iB1 < ζα−k > Ek }β(t) dt,

where the superscript d denotes that the solution is obtained by integrating the dislocation solutions. β(t) stands for the dislocation density at point t and is an unknown function vector to be determined by the boundary conditions. The variables s and t are related to zα and zˆα by zα = (x1o + μα x2o ) + s(cos α + μα sin α), zˆα = (x1o + μα x2o ) + t(cos α + μα sin α),

(8.91)

where (x1o , x2o ) is the coordinate of the crack center and s (or t ) denotes the distance from the crack center to point zα (or zˆα ). The integration limits ± are the ends of the crack whose length is 2.

270

8 Inclusions

Through the use of superposition principle, we now obtain the stress function φ1 of the original problem as φ1 = φu1 + φd1 .

(8.92)

If the traction tn along the crack surface is considered to be zero, we have ∂φd1 = −tun , ∂s

along the crack surface,

(8.93)

where tun denotes the traction along the crack location induced by problem (a) and is related to φu1 by tun = ∂φu1 /∂s. Substituting (8.90) into (8.93) and reconstructing the results into a form of singular integral equation, we have −

1 2π

−

L1 β(t)

1 dt + t−s

−

ˆ 1 (t, s)β(t) dt = −tun (s), K

(8.94a)

where L1 is a real matrix defined by L1 = −2iB1 BT1 and γα ∂ζα > BT1 iB1 < ζα (ζα ζˆα − γα ) ∂s ∞ ∂ζα o 1 −k−1 +π > Ek Re iB1 < kζα ∂s k=1 3 ∂ζα −1 , > B−1 Re iB1 < ζα−1 (1 − ζα ζ¯ˆ j )−1 B I B + π1 1 j 1 1 ∂s j=1 ∂ζα 2ζα2 (cos α + μα sin α) = . ∂s (a − ibμα )ζα2 − (a + ibμα )

ˆ 1 (t, s) = K

1 π Re

(8.94b)

ˆ 1 is a kernel function of the singular integral equation and is Holder-continuous K along − ≤ s ≤ . Eok in (8.94b) is the matrix Ek defined in (8.56d)1,2 with Tk given in (8.56d)3 which is obtained for the dislocation outside the inclusion. If we nondimensionalize the variables s and t by letting ξ = s/ and η = t/, (8.94a) can be rewritten as 1 − 2π

1

1 dη + L1 β(η) η − ξ −1

1

−1

ˆ 1 (η, ξ )β(η) dη = −tun (ξ ). K

(8.95)

The requirement of crack tip continuity will lead to the following single-valued displacement condition:

1 −1

β(η) dη = 0.

Since the order of singularity at the crack tip is −1/2, it is convenient to let

(8.96)

8.4


β(η) = +

271

ˆ β(η) 1 − η2

,

(8.97)

ˆ where β(η) is Holder-continuous along − ≤ s ≤ . Up to now, the entire problem ˆ from the singular intehas been reduced to finding the unknown function vector β(η) gral equations (8.95), (8.96), and (8.97). Through the use of the numerical technique introduced by Gerasoulis (1982) and shown in Hwu et al. (1995b), the unknown dislocation density β(t) can be determined. With β(t) determined, the whole field solution for the stresses at point zα can be calculated by substituting β(t) into (8.89), (8.90), and (8.92). With the usual definition, the stress intensity factors may now be calculated by ⎧ ⎫ √ ⎨ KII ⎬ + π ˆ TL1 β(±1), k = KI = lim 2π (±ξ − 1)Ttn (ξ ) = ∓ ⎩ ⎭ ξ →±1 2 KIII where T is the transformation matrix defined as ⎡ − sin α cos α T = ⎣ cos α sin α 0 0

⎤ 0 0⎦ . 1

(8.98a)

(8.98b)

It should be noted that during the derivation of (8.98), the following relation has been used:

1 −1

f (η) dη π f (ξ ) + sgn(ξ ) + regular terms, =+ (η − ξ ) 1 − η2 ξ2 − 1

when |ξ | > 1, (8.99)

where the sign function sgn(ξ ) is defined as sgn(ξ ) = 1 if ξ > 0and sgn(ξ ) = −1 if ξ < 0. Moreover, the stresses tn (ξ ) near the crack tip (ξ → ±1) can be obtained by substituting (8.89), (8.90), (8.92), and (8.97) into the relation tn = φ1,s and using (8.99). The result is 1 ˆ )sgn(ξ ) + regular terms. tn (ξ ) = − + L1 β(ξ 2 ξ2 − 1

(8.100)

8.4.2 Cracks Inside the Inclusions Consider a crack located inside an elliptical anisotropic elastic inclusion subjected to uniform loading at infinity (Fig. 8.3). In a way similar to that described in the above section, one may set a singular integral equation as (8.94a) for the unknown dislocation density β(t), except that now L1 should be replaced by L2 = −2iB2 BT2

272

8 Inclusions x2

Fig. 8.3 A crack inside an elliptical inclusion

s d

n

b ∗

∗

( x1 , x2 )

α

x1

a

ˆ 1 should be replaced by K ˆ 2 as and K ˆ 2 (t, s) = − 1 K π

∞ k=1

∗ ∗k−1 ∗k ∗−(k+1) ∂ζα i > Ck . Re iB2 < k(ζα + γ α ζα ) ∂s

(8.101)

In the above equation Cik is the matrix Ck defined in (8.56d)2 with Tk given in (8.71b) which is obtained for the dislocation inside the inclusion. The formula for the stress intensity factors will also be the same as (8.98a) except that L1 is replaced by L2 .

8.4.3 Cracks Penetrating the Inclusions To consider the penetrating cracks, we first deal with the case that two cracks locate simultaneously inside and outside the inclusions (Fig. 8.4). In a way similar to that described in Section 8.4.1, we obtain the singular integral equation as

s1

n1

x2

1 1

d1

b

d2

n2

2

( x10 , x20 )

s2

2 ∗ 1

∗ α ( x , x2 )

Fig. 8.4 Two cracks located simultaneously outside and inside an elliptical inclusion

a

x1

8.4

Interactions Between Inclusions and Cracks 1 − 2π

1 − 2π

1

−1

1 −2 L2 β2 (t2 ) t2 −s2

1

ˆ 1 (t1 , s1 )β1 (t1 ) dt1 K ˆ 12 (t2 , s1 )β2 (t2 ) dt2 = −tu , + −2 K 1n 1 ˆ 21 (t1 , s2 )β1 (t1 ) dt1 dt2 + −1 K 2 ˆ 2 (t2 , s2 )β2 (t2 ) dt2 = −tu , K +

1 L1 β1 (t1 ) t1 −s dt1 + 1

2

273

1 − 2

−2

(8.102)

2n

ˆ 1 and K ˆ 2 are given in (8.94b) and (8.101) and where K " ! ˆ 12 (t2 , s1 ) = − 1 Re iB1 < 1 ∂ζα > BT K 1 π ζα ∂s1 " ! ∞ −(k+1) ∂ζα i Re iB1 < kζα > E + π1 k ∂s1 k=1 " (8.103) ! ∞ −(k+1) ∂ζα −1 ∗ > BT , − π1 Re iB1 < kζα > B B < e 2 kα 2 1 ∂s1 k=1 " ! ∞ ∗ ∗−(k+1) ∂ζα ˆ 21 (t1 , s2 ) = − 1 K Re iB2 < k(ζα∗k−1 + γα∗k ζα ) ∂s2 > Cok , π k=1

in which the superscripts i and o denote, respectively, the values related to “inside” and “outside.” In other words, Eik is the matrix Ek defined in (8.56d)1,2 with Tk given in (8.71b) which is obtained for the dislocation inside the inclusion, whereas Cok is the matrix Ck defined in (8.56d)2 with Tk given in (8.56d)3 which is obtained for the dislocation outside the inclusion. β1 (t) and β2 (t) denote, respectively, the dislocation density along the cracks outside and inside the inclusions. They will be determined by the set of singular integral equations shown in (8.102) and the singlevalued requirement for both of the cracks. The stress intensity factors for both of the cracks can be calculated by √ ki = ∓

π i TLi βˆ i (±1), 2

i = 1, 2,

(8.104)

where the subscripts 1 and 2 denote the values outside and inside the inclusions. With the above results, by letting the distance between these two cracks approach to zero, we may approximate the condition of penetrating cracks. The detail discussion of this approximation was shown in Hwu et al. (1995b).

8.4.4 Curvilinear Cracks Lying Along the Interfaces Consider a curvilinear crack lying along the interface between inclusion and matrix (see Fig. 8.5). Same as the problem discussed in Section 8.4.1 for a crack outside the inclusion, we represent the problem as the sum of two problems. One is an uncracked problem of which the solution is given in (8.89) and the other is a loaded cracked problem of which the crack is represented as a distribution of dislocations. The difference is that the dislocation is lying on the curvilinear interface. By integrating the solutions given in (8.82) along the elliptic interface, we have

274

8 Inclusions

x2

(a)

x2

(b)

s

2 2ψˆ

ψ 0 − ψˆ

ψˆ ψˆ

ψ0

ψ 0 − ψˆ

x1

b

x1

1

a

Fig. 8.5 A curvilinear crack lying along the interface of an elliptical inclusion: (a) z-plane; (b) ζα -plane

φd2 (s)

1 =− π 1 − π

ψˆ

Re{iB2 < ln(zα − zˆα ) > Q2 }β(t) dt

−ψˆ ψˆ ∞ −ψˆ k=1

(8.105) Re{iB2
Ck }β(t) dt.

In the above, we only show the stress function φ2 since φ1 = φ2 along the interface and φ2 is relatively simple than φ1 . After determining the dislocation density from the boundary conditions and the single-valued requirement, the calculation of the whole field solutions includes both φ1 and φ2 depending upon the point considered. In (8.105), s (or t) denotes the angle from the crack center to point zα (or zˆα ) in ζα -domain. The integration limits ±ψˆ are the ends of the cracks whose angle is 2ψˆ (in ζα -domain). By following the steps stated in (8.92), (8.93), and (8.94), and carefully differentiating φd2 along the curvilinear crack boundary (one may refer to (6.14) for detailed derivation about differentiation), we obtain 1 − 2π

ψˆ −ψˆ

1 2 Re(iB2 Q2 ) β(t) dt + ρ(s) t−s

ψˆ

−ψˆ

ˆ f (t, s)β(t) dt = −tun (s), K

(8.106a)

where ˆ f (t, s) = K

2 t−s 1 Re i − cot B2 Q2 2πρ(s) t−s 2 1 γα Re B2 < i(ψ +s) i(ψ +t) − > Q2 πρ(s) e o e o − γα ∞ " ! 1 − Re B2 < k(eik(ψo +s) − γα e−ik(ψo +s) ) > Ck πρ(s) k=1

and

(8.106b)

8.4


275

ρ 2 (s) = a2 sin2 (ψo + s) + b2 cos2 (ψo + s).

(8.106c)

Nondimensionalization of (8.106a) by ξ = s/ψˆ and η = t/ψˆ leads to 1 − 2π

1

1 dη + Lf (ξ )β(t) η − ξ −1

1 −1

ˆ f (η, ξ )β(η) dη = −tun (ξ ), K

(8.107a)

where Lf (ξ ) =

2 Re(iB2 Q2 ). ρ(ξ )

(8.107b)

The single-valued displacement requirement can also be written in the same form as (8.96). At first glance, the kernel function given in (8.106b) seems to be singular for the presence of the term 2/(t −s). However, by expanding the cot(t −s)/2 into the series expression as cot

2 t − s (t − s)3 t−s = − − − ······ , 2 t−s 0 6 0 360 0t − s0 0 < π, when 00 2 0

(8.108)

ˆ f will still be Holder-continuous along the curvilinear crack. we see that K Till now, we have set the traction-free boundary conditions by the singular integral (8.106) and the single-valued displacement requirement by (8.96). Like those stated in Section 8.4.1, the next thing should be finding a numerical technique to solve the unknown dislocation density vector β(t). The technique shown in Hwu et al. (1995b) is valid for β(t) to be singular in the order of −1/2. It is well known that the singularity order of the interface crack is −1/2 + iε where ε is the oscillation index depending on the material properties of matrix and inclusion. To overcome this problem, it looks like a new numerical technique should be developed. However, unlike singularity, oscillation will not cause numerical overflow. Moreover, by the experimental study of Hwu et al. (1995a), we see that the oscillation index ε is usually very small which means that the range of its influence is limited. The values of the bimaterial stress intensity factors considering the oscillation effects are also very close to those of the conventional stress intensity factors considering only the –1/2 singularity. With the above reasons, the numerical technique shown in Hwu et al. (1995b) can still be employed, and the stress intensity factors calculated by (8.98) ˆ One may can still be used except now L1 is replaced by Lf and is replaced by ψ. refer to (Hwu et al., 1995b) for numerical examples and detailed discussions.

Chapter 9

Contact Problems

Contact problem is one kind of mixed boundary value problems. Most of the analytical formula for contact problems can be found in the books written by Galin (1961), Gladwell (1980), and Johnson (1985). Combining the complex variable formulation with the method of analytical continuation, Muskhelishvili (1954) and England (1971a) provided solutions for several types of punch problems on isotropic elastic bodies. Usually, the mathematical model of elasticity can be divided into two parts. One is the basic equations which include equilibrium equations, constitutive laws, and kinematic relations. The other is the boundary conditions which can be distinguished into traction, displacement, and mixed boundary value problems. Once a problem is formulated based upon the basic equation, its solvability is usually dependent on the boundary conditions. As to the same boundary geometry, the mixed boundary value problems are more complicated than the traction or displacement boundary value problems. Therefore, the usual step to deal with the elasticity problems is from simple geometry to complicated geometry and then from traction (or displacement) boundary to mixed boundary. This is exactly the step we take in this book. In this chapter, several different kinds of contact problems are considered and are described by following the results obtained in Fan and Hwu (1996, 1998) and Hwu and Fan (1998a–d). First, we consider that the boundary geometries are simple like the straight boundary, and the main concern is the development of a systematic approach for solving the mixed boundary value problems. This is done in Section 9.1 for rigid punches on a half-plane by combining Stroh formalism for two-dimensional anisotropic elasticity and the method of analytical continuation for the manipulation of complex variables. In Section 9.2 we consider the problems with curvilinear boundaries and emphasize upon the introduction of conformal mapping functions. It is found that there are many practical boundary geometries that cannot be covered by the conformal mapping functions, such as coarse surfaces and polygonal holes. In order to handle this kind of boundaries, the perturbation technique is introduced in Section 9.3 to deal with the problems of rigid punches on perturbed surfaces. Punches with or without friction sliding along the surface of an anisotropic elastic half-plane is considered in Section 9.4. The word “sliding” used here has a generalized meaning which includes not only slowly sliding punches but


277

278

9 Contact Problems

also the punches that are in static equilibrium. The applied horizontal forces can be any value less than or equal to the maximum friction force. Different from the first four sections whose indenters are assumed to be rigid, in the last section the contact between two elastic bodies is considered. The shapes of the boundaries of these two elastic bodies are assumed to be approximately straight, but the contact region is not necessary to be small and the contact surface can be non-smooth. Based upon these assumptions, in Section 9.5 three different boundary conditions are considered and solved. They are the contact in the presence of friction, the contact in the absence of friction, and the contact in complete adhesion.

9.1 Rigid Punches on a Half-Plane The problem of punch indentation has been investigated for many years due to its broad application in engineering mechanics. This is one of the mixed boundary value problems and may be considered as a particular contact problem because of the line contact region. In this section by combining Stroh’s formalism and the method of analytical continuation, a set of rigid punches of arbitrary profiles indenting into the surface of an anisotropic elastic half-plane with no slip occurring is studied. Illustrations are presented for the normal and rotary indentation by a flatended punch and indentation by a parabolic punch. In the last section, some known analogical problems are discussed and the punch problems are resolved by analogy with the interface crack problems.

9.1.1 General Solution We examine the case that a set of rigid punches of given profiles are brought into contact with the surface of the half-plane and are allowed to indent the surface in such a way that the punches completely adhere to the half-plane on initial contact, and during the subsequent indentation no slip occurs and the contact region does not change. Let us suppose the contact region L is the union of a finite set of line segments Lk = (ak , bk ), k = 1, 2, . . . , n, where the ends of the segments are encountered in the order a1 , b1 , a2 , b2 , . . ., an , bn when moving in the positive x1 -direction. For this case the displacements of the surface of the half-plane are known at each point of the contact region, then the boundary conditions are ˆ 1 ), u(x1 ) = (uk (x1 ), vk (x1 ) + ck , 0)T = u(x t(x1 ) = (σ12 , σ22 , σ23 ) = 0, T

x1 ∈ / L,

x1 ∈ L,

(9.1)

in which uk (x1 ) and vk (x1 ) are related to the profile of the kth punch and ck is the relative depth of indentation. In addition to this condition, in this problem we suppose that the resultant forces applied to each punch are known. If qˆ k is the known resultant force vector on Lk , from the relation shown in (3.33) we have

9.1

Rigid Punches on a Half-Plane

279

φ ds = φ(bk ) − φ(ak ) = qˆ k .

tds = Lk

(9.2)

Lk

Considering the overall equilibrium of the elastic body, the total resultant force qˆ applied on the half-plane is the summation of all punch forces qˆ k , i.e., qˆ =

n

qˆ k .

(9.3)

k=1

To find a solution satisfying the boundary conditions stated in (9.1), (9.2) and (9.3), we start from the general solution provided in (3.24), which satisfies all the basic equations of anisotropic elasticity, i.e., u = Af(z) + Af(z),

φ = Bf(z) + Bf(z).

(9.4)

Concerning the employment of the analytical continuation method discussed in Section 4.2.1, we now introduce θ (z) such that

θ (z) =

Bf (z), −Bf (¯z),

z ∈ S− , z ∈ S+ ,

(9.5)

where S– is the region of the elastic half-plane, i.e., x2 < 0, and S+ denotes the upper half-plane (Fig. 9.1). Because f (z) is holomorphic in S– which also make f (¯z) holomorphic in S+ , θ (z) defined in (9.5) is holomorphic in S– and S+ , i.e., θ (z) is sectionally holomorphic in the whole plane except possibly on some segments of x1 -axis. With this newly defined function vector θ (z), using the general solutions (9.4) for the expressions of displacement and stress function vectors, and the relation (3.32) for the traction vector, the tractions and deformation along the half-plane surface can be expressed in terms of θ (z) as

x2

qˆ 1

qˆ n

qˆ k

S+

Fig. 9.1 Rigid punches on a half-plane

a1

b1

ak

bk

an bn

S−

x1

280

9 Contact Problems

t(x1 ) = θ (x1− ) − θ (x1+ ), iMuˆ (x1 ) = θ (x1+ ) + MM

(9.6a)

−1

θ (x1− ),

along the half plane surface,

where M is the impedance matrix defined in (3.132), and θ (x1− ) = lim θ (z),

θ (x1+ ) = lim θ (z).

x2 →0−

x2 →0+

(9.6b)

The boundary conditions (9.1) can then be written into the following Hilbert problem of vector form: θ (x1+ ) − θ (x1− ) = 0, θ (x1+ ) + MM

x1 ∈ / L,

−1

θ (x1− ) = iMuˆ (x1 ),

x1 ∈ L.

(9.7)

The solution to this Hilbert problem of vector form is (Appendix B.2) θ (z) =

1 X0 (z) 2π

L

1 [X+ (t)]−1 Muˆ (t)dt + X0 (z)pn (z), t−z 0

(9.8)

where pn (z) is an arbitrary polynomial vector with degree not higher than n and X0 (z) is the basic Plemelj function matrix satisfying the homogeneous part of the Hilbert problem, i.e., − X+ 0 (x1 ) = X0 (x1 ),

x1 ∈ / L,

−1 − X+ 0 (x1 ) + MM X0 (x1 )

= 0,

x1 ∈ L.

(9.9)

± Note that in the above, X± 0 (x1 ) which sometimes will be written as X0 is a simplified notation for X0 (x1± ). The solution X0 (z) to (9.9) is

X0 (z) = (z),

(9.10a)

where = [λ1

λ2

λ3 ],

n 1 (z) =< (z − aj )−δα (z − bj )δα −1 > .

(9.10b)

j=1

δα and λα , α=1, 2, 3, of (9.10b) are the eigenvalues and eigenvectors of (M−1 + e2π iδ M

−1

)λ = 0,

(9.11)

which is a special case of the eigenrelation shown in (7.68) for the interface crack problems. For the present eigenrelation (9.11) the solution given in (7.69) can be reduced to, by letting L−1 2 = 0 since the punch is rigid,

9.1


δα =

281

1 + iεα , 2

α = 1, 2, 3,

(9.12a)

where 1+β 1 ln , 2π 1 − β 1/2 1 β = − tr(S2 ) , 2 ε1 = ε =

ε2 = −ε,

ε3 = 0, (9.12b)

S = i(2ABT − I).

By referring to the normalization given in (7.97) for interface crack problems, the eigenvector matrix defined in (9.10b) should be normalized by T

(M−1 + M

−1

) = 2I

T

or L−1 = I.

(9.13)

The unknown in solution (9.8) that remains to be determined is the function pn (z) which is at most a polynomial of degree n–1 and can be expressed by pn (z) = d0 + d1 z + · · · · · · + dn−1 zn−1 .

(9.14)

To determine the unknown coefficient vectors dk , k = 0, 1, 2, . . . , n − 1, we use the force condition given in (9.2) and (9.3). With (9.4)2 and (9.5), the force equilibrium given in (9.2) can be rewritten as Lk

[θ (x1+ ) − θ (x1− )]dx1 = −qˆ k ,

k = 1, 2, 3, . . . . . . , n.

(9.15)

Substituting (9.8) into (9.15) yields n equations for the determination of the n coefficient vectors dk . The overall equilibrium condition (9.3) can be served as a check for the solutions obtained through (9.15). To apply the overall equilibrium condition, we consider the far-field solution. If the problem whose stresses tend to zero in the far field, for large |z| the complex function vector f(z) has the form f(z) =< ln zα > q∗ + O(1),

(9.16)

in which q∗ is a complex constant vector to be determined by the half-plane far-field condition. With the arc ab lying on the boundary of the half-plane, i.e., x1 -axis, we let a = R1 eiπ and b = R2 e2iπ . The resultant force qˆ applied on the surface of the half-plane can be calculated by substituting (9.16) and (9.4)2 into (9.2). The result is qˆ = (Bq∗ + Bq∗ ) ln(R2 /R1 ) + iπ (Bq∗ − B q∗ ). Since R1 , R2 tend to infinity independently, we conclude that

(9.17a)

282

9 Contact Problems

Bq∗ + B q∗ = 0,

qˆ = iπ (Bq∗ − B q∗ )

(9.17b)

and hence q∗ =

1 −1 ˆ B q. 2π i

(9.17c)

By (9.5), (9.16), and (9.17c), the half-plane far-field condition for θ (z) can now be written as [an alternative approach can be done by using (4.61)] θ (z) =

1 −1 ˆ B < z−1 α > B q, 2π i

as |z| → ∞.

(9.18)

−1 If x2 = 0− , the diagonal matrix < z−1 α > approximates to x1 I and hence

θ (x1 ) =

1 ˆ q, 2π ix1

as |x1 | → ∞ and

x2 = 0− .

(9.19)

Applying the far-field condition (9.18) to the solutions shown in (9.8) and (9.14), we get a relation for the coefficient dn−1 as dn−1 =

1 −1 ˆ q. 2π i

(9.20)

Contact Pressure To find a simplified expression for the contact pressure under the punches, we substitute (9.7)2 into (9.6a)1 and use the identity (3.132). The result is −1

t(x1 ) = 2ML

θ (x1− ) − iMuˆ (x1 ),

x1 ∈ L.

(9.21a)

Further reduction can also be done by substituting the definitions of M(= −iBA−1 ) and L(= −2iBBT ) into (9.21a) and using the identity AT B + BT A = 0 given in (3.57b). The result is t(x1 ) = −A−T f (x1− ) − iMuˆ (x1 ),

x1 ∈ L.

(9.21b)

Surface Deformation Similar to the contact pressure under the punches, the surface deformations outside the punches can also be obtained by using (9.6a) and (9.7). They are u (x1 ) = −2iL−1 θ (x1− ),

x1 ∈ / L,

(9.22a)

which can further be reduced to u (x1 ) = BT f (x1− ),

x1 ∈ / L.

(9.22b)

9.1


283

Now the problem is solved in principle. For illustrating the solutions derived above, some special cases will be investigated as follows.

9.1.2 Indentation by a Flat-Ended Punch We first examine the case of indentation by a single punch with a flat-ended profile which makes contact with S– over the region |x1 | ≤ a and the force qˆ applied on the punch is given. Then uˆ (x1 ) = 0 and from (9.8), (9.10a), and (9.20) we get θ (z) =

1 ˆ (z) −1 q, 2π i

(9.23a)

where (z) =< √

1 z2 − a2

z+a z−a

−iεα

>.

(9.23b)

Substituting (9.23) into (9.5)1 and using the translating technique (4.50) and (4.51), the complex function vector f(z) which is useful for the calculation of the full field solution for the stresses and displacements of the half-plane can be obtained as f (z) =

1 1 B−1 Ik −1 q.

(9.24)

To find the contact pressure and the surface deformation, we need to calculate (x1− ) for |x1 |≤ a and |x1 |> a. This can be evaluated by using a bipolar coordinate system with two origins located at the ends of the punch and introducing a cut along the punch region. The results are a+x1 ieπ εα −iε ln e α a−x1 >, (x1− ) =< 3 a2 − x12

(x1− )

=< ± 3

1 x12 − a2

0 0 0 x +a 0 −iεα ln0 x1 −a 0

e

1

for

|x1 | ≤ a, (9.25)

>,

for

x1 > a and

x1 < −a.

Substituting (9.23) and (9.25) into (9.21) and (9.22), we can get the solutions for the contact pressure under the punch and the surface deformation outside the punch, which is expressed in complex form. Since the stresses and deformations are real quantities, it is of interest to obtain the real-form solutions in order to have a better understanding of the physical behavior of the punch problems. To this end, the following equalities which are special cases of (7.99)4 and (7.112) are used for getting the real-form solutions, i.e.,

284

9 Contact Problems −1

) = 2 < e−π εα cosh πεα >, 1 − cR T 2 cI T (S ) + S . < cα > −1 = I + β β2

(I + MM

(9.26)

Using the procedure stated above, the real-form solution for the contact pressure under the punch and the surface deformation gradient outside the punch can be found to be 1 − cR T 2 cI T 1 ˆ I+ |x1 | ≤ a, t(x1 ) = 3 (S ) + S q, β β2 π a2 − x12 1 − c∗R T 2 c∗I T 1 −1 ˆ x1 > a and x1 < −a, I+ u (x1 ) = ∓ 3 L (S ) + S q, β β2 π x12 − a2 (9.27a) where a + x1 a + x1 cR = cosh(π ε) cos ε ln , cI = − cosh(π ε) sin ε ln , a − x1 a − x1 0 0 0 0 0 0 x1 + a 0 0 0 , c∗ = − sin ε ln 0 x1 + a 0 . c∗R = cos ε ln 00 I 0 0 x1 − a x1 − a 0 (9.27b) Solutions (9.24) and (9.27) are valid for general anisotropic materials. In principle the explicit full field solution shown in (9.24) is valid only for the nondegenerate materials, that is, the material eigenvalues μα , α = 1, 2, 3, are distinct or three independent material eigenvectors (aα , bα ), α = 1, 2, 3, can be found when μα are repeated. By using a correspondence relation between anisotropic and isotropic elasticity (Hwu, 1996), an analytical solution for isotropic materials deduced from (9.24) is obtained and is proved to be identical to that shown in Muskhelishvili (1954). To develop a unified computer program valid for any kind of anisotropic materials, the degenerate materials are generally treated by introducing a small perturbation in the material properties. As to the real-form solution shown in (9.27), no special numerical treatments or correspondence relations are needed for the degenerate materials since the solutions do not contain any material eigenvalues μα or eigenvector matrices A and B explicitly. Following is the presentation for the reduction to the orthotropic and isotropic half-planes. For orthotropic materials, (9.26)2 can be expressed as ⎡

< cα > −1

⎤ √ cI −S21 /S12 0 √ cR cR 0⎦, = ⎣ −cI −S12 /S21 0 0 1

(9.28)

where S12 and S21 are the {12} and {21} components of S. To avoid confusing with the other symbol used for representing the elastic compliance, the Sij here is typed in roman-face not italic-face. Then t(x1 ) under the punch becomes

9.1


285

⎧ 3 a+x1 S21 ⎪ cos ε ln q ˆ − − sin ε ln ⎪ 1 ⎪ a−x1 S12 ⎨ 1 3 3 t(x1 ) = + 1 ⎪ − SS12 sin ε ln a+x a−x1 qˆ 1 + cos ε ln π 1 − β 2 a2 − x12 ⎪ 21 ⎪ ⎩ qˆ 3

a+x1 a−x1 a+x1 a−x1

⎫ qˆ 2 ⎪ ⎪ ⎪ ⎬ qˆ 2 ⎪ ⎪ ⎪ ⎭

, |x1 | ≤ a,

(9.29) ˆ where qˆ 1 , qˆ 2 , and qˆ 3 are the components of the force vector q. Consider the special case of isotropic half-plane and the force qˆ applied on the punch is given as (0, −q0 , 0)T . For the isotropic materials, the explicit expression of S is shown in (3.86a)1 . With S given in (3.86a)1 , by (9.12b) we get β=

κ −1 , κ +1

ε=

1 ln κ, 2π

(9.30)

where κ = 3 − 4ν for plane strain condition and κ = (3 − ν)/(1 + ν) for generalized plane stress condition and ν is the Poisson’s ratio. With the values given in (3.86a)1 and (9.30) for the isotropic materials, the stresses under the punch shown in (9.29) can now be reduced to ⎫ ⎧ 1 ⎧ ⎫ sin ε ln a+x ⎪ ⎪ ⎪ ⎪ a−x 1 ⎨ ⎨σ12 ⎬ 1 + κ ⎬ q0 3 , (9.31) t(x1 ) = σ22 = √ a+x1 − cos ε ln a−x1 ⎪ ⎩ ⎭ κ 2π a2 − x2 ⎪ ⎪ ⎪ σ23 ⎩ ⎭ 1 0 which agree with those shown in Muskhelishvili (1954).

9.1.3 A Flat-Ended Punch Tilted by a Couple A second example illustrating the above general solution is the case of a flat-ended punch which adheres to the half-plane S– and is then tilted by the application of a couple m. ˆ Let us suppose the punch is of width 2a and is tilted through a small angle ω measured in the counterclockwise direction. Under this condition, ⎧ ⎫ ⎨0⎬ uˆ (x) = ω 1 = ωi2 , ⎩ ⎭ 0

|x1 | ≤ a,

(9.32)

and the solution of θ (z) shown in (9.8) can be reduced to ω X0 (z) θ (z) = 2π

a

−a

1 [X+ (t)]−1 dtMi2 . t−z 0

(9.33a)

Note that the last term in (9.8) vanishes since the resultant force is zero. In order to evaluate the above integral of matrix form, a special technique similar to that

286

9 Contact Problems

presented in England (1971a) for line integrals of scalar form has been developed in Appendix B.4. Applying that technique and using the relations given in (9.10) and (9.13), we obtain T

T

θ (z) = iω { − (z) < z + 2iaεα > }i2 /2,

(9.33b)

where (z) is the same as (9.23b). This result enables us to calculate the stresses over the contact region in terms of the tilt angle ω. In an alternative problem it may be assumed that the couple m ˆ acting on the punch is given and tilt angle ω is unknown. Hence it is necessary to evaluate the relation between the applied couple m ˆ and the tilt angle ω. For this purpose, we first calculate the stresses under the punch, by (9.21a), (9.32), and (9.33), we have −1

t(x1 ) = −iωML

T

(x1− ) < x1 + 2iaεα > i2 ,

|x1 | ≤ a.

(9.34)

With this result, the couple m ˆ can be calculated by m ˆ =

a

−a

x1 σ22 dx1 =

a −a

x1 iT2 t(x1 )dx1 ,

(9.35a)

in which the integral can be evaluated in a way similar to that presented in Appendix B. The result is m ˆ =

π 2T T ωa i2 < 1 + 4εα2 > i2 . 2

(9.35b)

With the above relation, the stresses under the punch obtained in (9.34) can be written in terms of the couple m ˆ applied on the punch. Using a similar approach as the above section, the explicit full field solution f (z) and the real-form solutions for contact pressure t(x1 ) under the punch and surface deformation gradients u (x1 ) outside the punch can be obtained as follows: 3 iω −1 −1 −1 < Γk (zα )(zα + 2iaεk ) > B Ik f (z) = Li2 , B − 2 k=1 1 − cR T 2 cI T ωx1 S I+ Li2 , |x1 | ≤ a, t(x1 ) = 3 (S ) + β β2 a2 − x12 ⎧ ⎫ ⎬ ⎨ ∗ ∗ 1 − cR 2 cI x1 I+ u (x1 ) = ω I ∓ 3 S − S i2 , x1 > a and x1 < −a, ⎩ β ⎭ β2 x12 − a2 (9.36a)

9.1


287

where 1

Γk (zα ) = + z2α − a2

zα + a zα − a

−iε ln

cR + icI = cosh(π ε)e

−iεk

a+x1 a−x1

, 2iaε 1+ , x1

0 0 0 x +a 0 −iε ln0 x1 −a 0

c∗R + ic∗I = e

1

1+

2iaε . x1

(9.36b) The real form relation between the applied couple m ˆ and the tilted angle ω is found to be m ˆ =

π 2 a ω 2

4ε2 I − 2 (ST )2 L . β 22

(9.36c)

9.1.4 Indentation by a Parabolic Punch Consider a symmetric punch whose end section can be expressed by a parabolic curve x2 = x12 /2R, |x1| ≤ , where R is the radius of curvature and 2 is the width of the punch. Let us suppose on indentation under the force qˆ the size of the contact region is 2a(≤ 2). By the assumption that the punch completely adhere to the half-plane, we have uˆ (x1 ) = (x1 /R)i2 , |x1 |≤ a. With this prescribed displacement gradient and the loading condition (9.20), the solution given in (9.8) can now be reduced to a t 1 1 −1 −1 ˆ X0 (z) [X+ (9.37a) θ (z) = 0 (t)] dtMi2 + 2π i X0 (z) q. 2π R t − z −a The line integral in (9.37a) can be evaluated in a way presented in Appendix B.4. The result is a t −1 [X+ 0 (t)] dtMi2 (9.37b) −a t − z −1 2 2 2 −1 = π i{z[X0 (z)] − < z + 2iaεα z − (1 + 4εα )a > }Li2 /2. Substituting (9.37b) and (9.10) into (9.37a), we have θ (z) =

" i ! zI − (z) < z2 + 2iaεα z − (1 + 4εα2 )a2 > −1 Li2 4R 1 ˆ (z) −1 q, + 2π i

(9.37c)

where (z) is the same as (9.23b). With the result of (9.37), by a similar approach as the previous two sections, we can get the explicit full field solution f (z) and

288

9 Contact Problems

the real-form solutions for contact pressure t(x1 ) under the punch and surface deformation gradients u (x1 ) outside the punch as follows: f (z) = −

i < Γk (zα )[z2α + 2iaεk zα − (1 + 4εk2 )a2 ] >B−1 Ik −1 Li2 4R 3

k=1

3 i 1 ˆ < zα > B−1 Li2 + < Γk (zα ) >B−1 Ik −1 q, 2R 2π i k=1 2x12 − a2 1 − (c/ c)R T 2 (c/ c)I T 3 I+ S Li2 t(x1 ) = (S ) + β β2 2R a2 − x12 1 − cR T 2 cI T 1 ˆ I+ q, |x1 | ≤ a, S + 3 (S ) + β β2 π a2 − x12 2x12 − a2 1 − (c∗/ c)R 2 (c∗/ c)I x1 I+ S i2 S − u (x1 ) = i2 ∓ 3 R β β2 2R x12 − a2 1 − c∗R T 2 c∗I T 1 −1 ˆ I+ x1 > a and x1 < −a, ∓ 3 L (S ) + S q, β β2 π x12 − a2 (9.38a) where Γk (zα ) is the same as (9.36b)1 and

+

−iε ln

a+x1 a−x1

c∗R

+ ic∗I

0 0 0 x +a 0 −iε ln0 x1 −a 0

cR + icI = cosh(π ε)e , =e 1 c˜ R + i˜cI = 2 [2x12 − (1 + 4ε2 )a2 + i4aεx1 ] 2x1 − a2

1

, (9.38b)

and cR − cI/ cI , (c/ c)I = cR/ cI + cI/ cR , (c/ c)R = cR/ ∗ ∗ ∗ ∗ ∗ (c / c)R = cR/ cR − cI / cI , (c / c)I = cR/ cI + c∗I / cR .

(9.38c)

9.1.5 Analogy with the Interface Crack Problems Solving problems by analogy techniques is not unusual. In the solution of torsional problems the membrane analogy has been proved very valuable (Timoshenko and Goodie, 1970). Their governing equations and boundary conditions are identical in the form of mathematical expressions, but their symbols have different physical meanings. Therefore, by replacing the symbols, they can communicate each other. Other kind of analogical problems known in the linear elasticity are point force and dislocation, crack and rigid line inclusion, hole and rigid inclusion, etc. Their governing equations are the same, but their boundary conditions are different in the sense that one is traction prescribed and the other is displacement prescribed.

9.1


289

Due to the fact that the traction-prescribed and displacement-prescribed boundary conditions have similar mathematical expressions, they can also benefit from analogy technique. By cutting through the contact/crack surface, the contact/crack problems can be treated as a half-plane problem with the displacements/tractions prescribed along a specified region. Through the conversion to half-plane problems, the analogy between contact problems and crack problems has been noticed before, e.g., Willis (1968) and Brock (1978). In this section, the analogy is discussed for the most general anisotropic linear elastic materials and the contact problems are solved directly by using the corresponding solutions of interface crack problems. The contact problems considered here are restricted to the two-dimensional cases in which the indenters are rigid punches. By this restriction, the materials above the interface cracks are also rigid. Through this study, it is hoped that the analogy technique can be extended to the general contact and interface crack problems. Moreover, it is also possible to explore the analogy between the three-dimensional contact and crack problems since both of these two problems were usually formulated as problems of Boussinesq type (Sneddon, 1946; Sneddon and Lowengrub, 1969; Willis, 1966, 1967, 1968, 1970; Gladwell, 1980). Known Analogical Problems One of the special features of Stroh formalism is that the solution form (3.24) is neat and elegant. Due to its elegancy, many important characteristics can be found at the first glance of the solution form. For example, the displacements and stress functions shown in (3.24a) are distinguished only by the material eigenvector matrices A and B. Thus, the relevant boundary conditions of the displacement-prescribed problems differ from those of the traction-prescribed problems only in the appearance of the symbols A and B. If the mathematical formulations for the displacementprescribed problems and the traction-prescribed problems are identical with A and B interchanged, their solutions should also be identical with A and B interchanged. Following are some examples for analogical problems. (1) The solution shown in (4.38) for dislocation problems can be obtained directly ˆ from (4.18) for point force problems with AT pˆ replaced by BT b. (2) The solution shown in (8.29) for problems with elliptical rigid inclusion can be obtained directly from (6.61) and (6.63) for elliptical hole problems with B−1 B replaced by A−1 A. (3) The solution shown in (8.48) for problems with rigid line inclusion can be replaced by obtained directly from (7.16) for crack problems with B−1 t∞ 2 (if ω is ignored). A−1 ε∞ 1 Solving the Punch Problems by Analogy with the Interface Crack Problems From the solutions obtained in Section 7.4 and this section, we see that the stress oscillatory characteristics near the interface crack tips and the ends of the flat-ended punches look similar. This similarity stimulates us to find the connection between these two problems. The boundary conditions for the interface crack problems and the punch problems have been shown in (7.60) and (9.1), respectively. Since the

290

9 Contact Problems

punches are assumed to be rigid, only the material of half-plane is considered in the boundary conditions (9.1). Whereas, for the interface crack problems two dissimilar materials above and below the interface are considered in the boundary conditions (7.60). If we consider the material above the interface to be rigid, the boundary conditions (7.60) will then be reduced to φ (x1 ) = −ˆt(x1 ) ,

x1 ∈ L,

and

u(x1 ) = 0,

x1 ∈ / L.

(9.39)

Comparing (9.1) and (9.39), we see that (9.39) is just a counterpart of (9.1) since the traction-prescribed condition φ= −ˆt of (9.39) corresponds to the displacementprescribed condition u=uˆ of (9.1) and the displacement-prescribed condition u = 0 of (9.39) corresponds to the traction-prescribed condition φ = 0 of (9.1). Therefore, we can solve the punch problems by using the solutions of the interface crack problems, (7.66), (7.67), (7.68), (7.69), and (7.70), with material 1 taken to be rigid, and interchanging the material eigenvector matrices A and B, or vice versa. Please refer to Hwu and Fan (1998a) for the detailed mathematical derivation by using the analogy technique.

9.2 Rigid Stamp Indentation on a Curvilinear Hole Boundary After discussing the mixed boundary value problems with straight edges, in this section we consider its counterparts with curvilinear boundaries. In engineering practice, the solutions to this kind of problems may be useful for the understanding of the mechanical behavior of a pin-loaded hole in composite materials.

9.2.1 General Solution Like the hole problems discussed in Chapter 6, to consider the curvilinear hole boundary a transformation function such as (6.3) for elliptical hole and (6.65) for polygon-like hole, which maps the points outside the curvilinear hole of the z-plane onto the points outside the unit circle of the ζα -plane, is introduced first for the problem formulation. With this transformation, the general solutions (3.24) can be written as u = Af(ζ ) + A f(ζ ),

φ = Bf(ζ ) + B f(ζ ).

(9.40)

Consider a set of rigid stamps of given profiles in contact with the curvilinear hole boundary of an anisotropic elastic body (Fig. 9.2). It is assumed that the stamps indent the hole boundary in such a way that the stamps completely adhere to the elastic body on initial contact, and during the subsequent indentation no slip occurs and the contact region does not change. Due to the assumption of rigid stamp indentation, over the contact region of the hole boundary the displacement u is known and

9.2

Rigid Stamp Indentation on a Curvilinear Hole Boundary

291

x2

Fig. 9.2 A curvilinear hole indented by a set of rigid stamps

R+

bn a1

Ln L1

b1 a2

an

R−

x1

s

L2

n

b2

θ

Lk

ak

bk

ˆ Over the remainder of the hole boundary, the surface traction is assumed set to be u. to be free. On transforming to the region |ζα | ≥ 1 of the ζα -plane, the boundary conditions can be expressed as ˆ u(s) = u(s), s ∈ L, / L, tn (s) = 0, s ∈

(9.41)

where s = eiψ denotes the points on the unit circle of the ζα -plane; L is the union of n arcs Lk = (ak , bk ), k = 1, 2, . . . , n, of the unit circle; tn is the surface traction along the boundary of which the normal is n; uˆ is the prescribed displacement which is related to the profile of the stamps. Like the straight edge problems discussed in Section 9.1, the resultant forces applied to each stamp are supposed to be known and denoted by qˆ k . The total resultant force qˆ applied on all stamps is the summation of all stamp forces qˆ k . To have a proper holomorphic function like θ (z) introduced in (9.5) for the half-plane problem, we first calculate the surface traction tn for the polygon-like boundary (6.64) whose mapped points s = eiψ approached from the region R+ outside the unit circle of ζα -plane. By performing the chain rule like that shown in (6.14) and (6.15), we obtain tn = lim

ζ →s+

i i ζ Bf (ζ ) − Bf (ζ ) , ρ ρζ

(9.42a)

where *0 0 0 0 0 ∂x1 02 0 ∂x2 02 0 0 . 0 0 +0 ρ= 0 ∂ψ 0 ∂ψ 0

(9.42b)

From the theory of complex variable function, we know that if f (ζ ) is holomorphic in R+ , then f (1/ζ¯ ) is holomorphic in R– . With this in mind, we introduce θ (ζ ) such

292

9 Contact Problems

that

θ (ζ ) =

ζ Bf (ζ ),

ζ ∈ R+ ,

1 ¯ ζ Bf (1/ζ ),

(9.43)

ζ ∈ R− .

Because f (ζ ) is holomorphic in the elastic body R+ which also make f (1/ζ¯ ) holomorphic in R– , θ (ζ ) defined in (9.43) is now holomorphic in R+ and R– , i.e., θ (ζ ) is sectionally holomorphic in the whole plane except possibly on some segments along the unit circle. With this newly defined function vector θ (ζ ), using the general solutions (9.40) for the expressions of displacement and stress function vectors, and the relation (9.42) for the traction vector, the tractions and displacement gradient along the hole boundary can be expressed in terms of θ (ζ ) as tn (s) = i[θ (s+ ) − θ (s− )]/ρ, − ρMu (s) = θ (s+ ) + MM

−1

θ (s− ),

(9.44) along the hole boundary.

The boundary conditions (9.41) can then be written into the following Hilbert problem of vector form: θ (s+ ) − θ (s− ) = 0, θ (s+ ) + MM

s∈ / L,

−1

θ (s− ) = −ρMuˆ (s),

s ∈ L.

(9.45)

The solution to this Hilbert problem of matrix form is (Appendix B.2) θ (ζ ) = −

1 Xc (ζ ) 2π i

L

ρ [X+ (s)]−1 Muˆ (s)ds + Xc (ζ )pc (ζ ), s−ζ c

(9.46a)

where pc (ζ ) is an arbitrary polynomial vector to be determined by the loading conditions and Xc (ζ ) is the basic Plemelj function matrix which has been given in (9.9), (9.10), (9.11), (9.12), and (9.13) with slight change made by interchange of M and M. The subscript c denotes that the quantities belong to the problems with curvilinear boundaries. Note that by comparing the results obtained in (9.44), (9.45), and (9.46) for the curvilinear boundary problems and those in (9.6), (9.7), and (9.8) for the straight edge problems, we see that they are different just by M and M, i and −ρ, which partly come from the definition for the regions + and − and partly come from the curvilinear geometry. With the above observation, it can easily be proved that the eigenvalues δ in the expressions of X0 (ζ ) and Xc (ζ ) are the same but their corresponding eigenvectors λ are conjugate each other. Hence, ¯ ), Xc (ζ ) = c c (ζ ) = (ζ

(9.46b)

where and (ζ ) are those given in (9.10b) calculated from (9.11) for half-plane problems.

9.2


293

Similar to pn (z) discussed in Section 9.1, the determination of pc (ζ ) can be done by considering the force conditions on each stamp. The difference is that in Section 9.1 only far-field condition is considered to decide the order of the polynomial pn (z) which is a degree not higher than the number of punches n, while in the present case to decide the order of pc (ζ ) both the infinity and zero conditions should be considered. By using (9.44)1 and (9.2), we have i + [θ (s ) − θ (s− )]ds, for k = 1, 2, . . . , n. qˆ k = (9.47) ρ Lk In addition to (9.47) which is the equilibrium equation for each stamp, the examination of the behavior of the stresses at infinity and zero points may also be helpful for the determination of pc (ζ ). If we consider that the stresses are bounded at infinity, like (9.16) it can be shown that for large |z| the complex function vector f(z) has the form f(z) =< ln zα > q∗ + < zα > q∞ + O(1)

(9.48a)

where q∗ is related to the resultant force qˆ applied on the entire body by (4.17) and q∞ is related to the stresses σij∞ at infinity by (4.5)1 . That is, q∗ =

1 T ˆ A q, 2π i

T ∞ q∞ = AT t∞ 2 + B ε1 .

(9.48b)

With the definition given in (9.43) and the transformation zα= mα (ζα ), (9.48) is then used to derive the infinity and zero conditions for θ (ζ ). The results are ζ mα (ζ ) ∗ ∞ −1 > q + < ζ mα (ζ ) > q + O(ζ ) , θ (ζ ) = B < mα (ζ )

|ζ | → ∞ (9.49a)

and m (ζ −1 ) θ (ζ ) = ζ −1 B < α −1 > q∗ + < mα (ζ −1 ) > q∞ + O(ζ ) , mα (ζ )

|ζ | → 0. (9.49b)

9.2.2 Elliptical Hole Boundaries Consider an infinite anisotropic body containing an elliptical hole of which the boundary is expressed by (6.1). A transformation which maps the points outside the elliptical hole to the points outside a unit circle is chosen to be (6.3). Suppose the hole is loaded by a rigid stamp along the segment between (a cos φ, −b sin φ) and (a cos φ, b sin φ), which is mapped onto an arc L = (e−iφ , eiφ ) in the ζα -plane. If the profile of the stamp is compatible with that of the hole boundary, uˆ (s) = 0

294

9 Contact Problems

when s ∈ L. With this specification, the general solution given in (9.46) can now be simplified to θ (ζ ) = c (ζ )pc (ζ ),

(ζ ) =< (ζ − e−iφ )−1/2−iεα (ζ − eiφ )−1/2+iεα >, (9.50)

in which the branch of (ζ ) is selected so that ζ (ζ ) → I as |ζ | → ∞. To determine the polynomial pc (ζ ), we first consider the infinity and zero conditions. Substituting (6.3) into (9.49), we obtain ζ θ (ζ ) = B q∗ + < a − ibμα > q∞ + O(ζ −1 ) , |ζ | → ∞, 2 1 θ (ζ ) = B q∗ + < a + ibμα > q∞ + O(ζ ) , |ζ | → 0. 2ζ

(9.51)

Moreover, (ζ ) given in (9.50)2 can also be expanded for large |ζ | as (ζ ) =

1 +< ζ

1 1 1 − iεα eiφ + + iεα e−iφ > 2 + O(ζ −3 ) 2 2 ζ

(9.52a)

and for small |ζ | as (ζ ) = − < e−2φεα > − < e−2φεα

1 1 + iεα eiφ + − iεα e−iφ > ζ + O ζ 2 . 2 2

(9.52b) With (9.50)1 , comparison of the infinity and zero conditions given in (9.51) and (9.52) now leads to pc (ζ ) = d2 ζ 2 + d1 ζ + d0 + d−1 ζ −1

(9.53a)

and B < a − ibμα > q∞ = 2 c d2 , 1 1 − iεα eiφ + + iεα e−iφ > d2 , Bq∗ = c d1 + c < 2 2 B < a + ibμα > q∞ = −2 c < e−2φεα > d−1 , 1 1 + iεα eiφ + − iεα e−iφ > d−1 . B q∗ = − c < e−2φεα > d0 + < e−2φεα 2 2

(9.53b) Once the loading conditions are given (i.e., q∗ and q∞ are known), the four unknowns d−1 , d0 , d1 , and d2 can be obtained by solving (9.53b). Thus, the problem is solved. Now consider a particular loading condition that all the stresses vanish at infinity and a resultant force qˆ = (ˆq1 qˆ 2 0)T is applied on the rigid stamp. In this case,

9.2


295

ˆ i. The four unknowns d−1 , d0 , d1 , and d2 can then be q∞ = 0 and q∗ = AT q/2π determined from (9.53b) as ∗

d0 = − < e2φεα > −1 c Bq ,

d−1 = 0,

∗ d1 = −1 c Bq ,

d2 = 0.

(9.54)

The solution of f (ζ ) can therefore be found by combining the results obtained in (9.43), (9.50), (9.53a), and (9.54), which is 1 −1 e2φεα T T −1 ˆ q. B c (ζ ) −1 > BA + < BA c c 2π i ζ

f (ζ ) =

(9.55)

Note that the function vector f (ζ ) of (9.55) which has the form of f (ζ ) = is not consistent with the solution form shown in (3.24b) and is valid only along the hole boundary ζ = s = eiψ . To get the explicit full field solution, a translating technique introduced in (4.50) and (4.51) is employed in (9.55), which leads to [f1 (ζ ) f2 (ζ ) f3 (ζ )]T

f (ζ ) =

3 1 e2φεk T T −1 −1 ˆ < Γk (ζα ) > B−1 c Ik −1 q, BA + < Γ (ζ ) > B I BA k α c k c c 2π i ζα k=1

(9.56a) where Γk (ζα ) =< (ζα − e−iφ )−1/2−iεk (ζα − eiφ )−1/2+iεk > .

(9.56b)

Using the result of (9.56) and applying a correspondence relation between isotropic and anisotropic elasticity proposed by Hwu (1996), the complex potential function ϕ (ζ ) (Muskhelishvili, 1954) of the present problem with isotropic media can be found to be κe−2φε qˆ 1 + iˆq2 1+ (ζ − e−iφ )−1/2+iε (ζ − eiφ )−1/2−iε , (9.57) ϕ (ζ ) = − 2π (1 + κ) ζ where κ = 3 − 4v for plane strain and κ = (3 − v)/(1 + v) for generalized plane stress and v is the Poisson’s ratio. The solution shown in (9.57) agrees with that presented in England (1971a).

9.2.3 Polygonal Hole Boundaries Consider a polygon-like hole, the contour of which is represented by (6.64) in which the symbol ε is replaced by ε∗ to avoid confusing with the oscillatory index ε. The transformation function is given in (6.65), which is one-to-one when the hole is in the shape of ellipse or the body is isotropic. For an anisotropic body containing a polygon-like hole, the transformation is generally not one-to-one. Detailed discussion about the validity of this transformation has been given in Section 6.2

296

9 Contact Problems

If we consider that the polygon-like hole is loaded by a resultant force qˆ = (ˆq1 , qˆ 2 , 0)T applied on a rigid stamp along the segment between a(cos φ + ε∗ cos kφ, −c sin φ + ε∗ sin kφ) and a(cos φ + ε∗ cos kφ, c sin φ − ε∗ sin kφ) which is mapped onto an arc L = (e−iφ , eiφ ) in the ζα -plane and uˆ (s) = 0 when s ∈ L, the solution form to this problem will then be exactly the same as that shown in (9.56) except that the relation between ζα and zα is now given in (6.65) instead of (6.1). Since there is no other analytical solution for the mixed boundary value problems with polygonal hole boundaries, for the purpose of verification another special case which can be reduced from the general solution (9.46) is studied as follows. Consider an anisotropic body containing a polygonal hole loaded by a uniform stress at infinity. If the entire boundary of the polygonal hole is traction-free, the displacement-prescribed segment L vanishes and the mixed boundary value problem reduces to a stress boundary value problem. Thus, the integral term of (9.46) vanishes and the Plemelj function Xc (ζ ) = c /ζ which is in the form of polynomials. The solution shown in (9.46) can therefore be reduced to θ (ζ ) = pc (ζ ),

(9.58)

in which the polynomial vector pc (ζ ) can be determined by substituting the transformation function (6.65) into the infinity and zero conditions shown in (9.49). The result is θ (ζ ) =

a {B < (1 − iμα c)ζ + kε∗ (1 + iμα )ζ k > q∞ 2 + B < (1 + i μα c)ζ −1 + kε∗ (1 − i μα )ζ −k > q∞ }.

(9.59)

The solution of f(ζ ) can then be found by applying (9.43)1 to (9.59) and integrating the results with respect to ζ . If we ignore the integration constant which corresponds to the rigid body motion, we get f(ζ ) =

a {< (1 − iμα c)ζ + ε∗ (1 + iμα )ζ k > q∞ 2 − B−1 B < (1 + i μα c)ζ −1 + ε∗ (1 − i μα )ζ −k > q∞ }.

(9.60)

Again, (9.60) is valid only along the hole boundary ζ = s = eiψ . To get an explicit full field solution, a translating technique is employed and the result is a f(ζ ) = {< (1 − iμα c)ζα + ε ∗ (1 + iμα )ζαk > q∞ 2 − < ζα−1 > B−1 B < (1 + iμ¯ α c) > q∞ + < ζα−k > B−1 B < ε ∗ (1 − iμ¯ α ) > q∞ }, (9.61)

which can be proved to be identical to that obtained in Section 6.2.2.

9.2


297

9.2.4 Numerical Calculation The solutions shown in this section have been checked analytically by their reduced forms such as (9.57) for isotropic body in the case of elliptical hole boundary and (9.61) for the stress boundary value problem in the case of polygonal hole boundary. For the general case of anisotropic media containing an elliptic or a polygonal hole indented by a rigid stamp, numerical calculation presenting the related hoop stress and stress contour has been done in Fan and Hwu (1998) to show its generality. In the following only the procedure for numerical calculation is stated, for those who are interested in the numerical results please refer to Fan and Hwu (1998). Calculation Procedure: (1) Calculate the material eigenvalues μα , α = 1,2,3, and material eigenvector matrices A and B from the elastic constants Cijkl. The eigenvector matrices A and B are normalized by AT B + BT A = I. (2) Calculate the eigenvalues δ and eigenvectors λ from (9.11) with M replaced by M and construct the matrices and (ζ ) by (9.10b) with the stamp location (aj , bj ) given. The eigenvector matrix is normalized by T

L−1 = I. (3) For any particular point (x1 , x2 ), calculate zα (= x1 + μα x2 ) and its corresponding ζα by (6.3)2 for elliptic hole boundary or by solving (6.65) and designating the point outside and nearest the unit circle to be the mapped point for polygonal hole boundary. (4) Calculate f(ζ ) by (9.56) for the case of elliptic or polygonal hole boundary. The solution form for the elliptic and polygonal hole boundary is the same except that the relations between ζα and zα are different. (5) Calculate the displacements, stresses, and strains from (3.24), (3.13), and (3.1)2 . The hoop stress σss can be calculated directly through the relations given in (3.36). During the numerical calculation, the determination of the complex variable function matrix (ζ ) should be very careful, because it is multi-valued if no special requirement was set. To deal with this problem, a special branch cut is introduced such that the complex variable function (ζ ) keeps continuous across the entire boundary C : |ζ | = 1 except the segment L in which (ζ ) induces a jump. For this purpose, we construct a bi-polar coordinate system (see Fig. 9.3), and let ζ − e−iϕ = r1 eiη1 , ζ − eiϕ = r2 eiη2 . (ζ ) can then be expressed as (ζ ) =< (r1 r2 )−1/2 (r2 /r1 )iεα e−i(η1 +η2 )/2 eεα (η1 −η2 ) > .

(9.62)

For a boundary point s = eiψ , the branch cut for the term (ζ − e−iϕ )−1/2−iεα will be set such that (see Fig. 9.3a) arg(s+ − e−iϕ ) = η1∗ ,

arg(s− − e−iϕ ) = η1∗ − 2π ,

(9.63a)

298

9 Contact Problems R+

R+

−

R

b

R

ζ

−

e

iϕ

η2*

η2 Δη 2

r1

Δη

ψ

η1*

r2

r

ζ

ψ

η1 r

a

e -iϕ

(a)

(b)

Fig. 9.3 A special polar coordinate system for the calculation of the complex variable function (ζ ) : (a) centered at point a (= e−iϕ ) ; (b) centered at point b (= eiϕ )

across the entire boundary C. On the other hand, the branch cut for (ζ −eiϕ )−1/2+iηα will be set such that (see Fig. 9.3b) arg(s+ − eiϕ ) = η2∗ , arg(s± − eiϕ ) = η2∗ ,

arg(s− − eiϕ ) = η2∗ − 2π , s ∈ L.

s ∈ C − L,

(9.63b)

By the definition given in (9.63a) and (9.63b), we find that when a point moves across the boundary from R+ to R− , the term (η1 +η2 ) in (9.62) undergoes a decrease 4π and 2π for s ∈ C − L and s ∈ L, respectively. Whereas, the term (η1 − η2 ) in (9.62) undergoes a decrease 0 and 2π for s ∈ C − L and s ∈ L, respectively. Thus, by (9.62), the relation between (s+ ) and (s− ) can be represented as (s+ ) = (s− ), +

2π εα

(s ) =< −e

s ∈ C − L, > (s− ),

(9.64)

s ∈ L.

The term < −e2π εα > representing the jump when (ζ ) moves across the segment L of the boundary is exactly the factor e2π iδ of the eigenrelation (9.11). This implies that the branch cut chosen in (9.63a) and (9.63b) will provide us the required values for the function (ζ ). For a point ζ not on the boundary and arg(ζ ) = ψ, we use η1∗ as a reference to calculate the argument η1 = arg(ζ − e−iϕ ), i.e., η1 =

η1∗

+ η1 ,

ζ − e−iϕ η1 = arg iψ e − e−iϕ

.

(9.65)

η2 = arg(ζ − e−iϕ ) can be obtained in a similar way. Thus, the complex variable function (ζ ) can be uniquely determined by (9.62) for all points of the anisotropic body.

9.3

Rigid Punches on a Perturbed Surface

299

9.3 Rigid Punches on a Perturbed Surface Of several physically distinct types of boundary conditions, there are three fundamental types of boundary conditions which seem to be of considerable physical interest. In the first it is supposed that the surface tractions are specified at all points along the boundary. Thus if ˆt is the prescribed traction value along the boundary C, the boundary conditions can be written as t(z) = ˆt(z), z ∈ C. Or, by ˆ integration, φ(z) = φ(z), z ∈ C. This kind of boundary value problem is referred to the stress boundary value problem. Alternatively, the displacement u may be specˆ ified at all points along the boundary, so that u(z) = u(z), z ∈ C, which is referred to the displacement boundary value problem. In some physical problems, the displacement boundary conditions hold over a part L of C and the stresses are defined over the remainder C − L of C. That is, ˆ u(z) = u(z), z ∈ L, and t(z) = ˆt(z), z ∈ C − L,

(9.66)

which constitutes the mixed boundary value problems. When L vanishes, the mixed boundary value problem reduces to the stress boundary value problem. When L=C, we get a displacement boundary value problem. Therefore, if we get a solution whose boundary conditions are set by (9.66), the corresponding solutions for the stress and displacement boundary value problems can also be obtained by letting L vanish or L=C. As we are dealing with the linear elasticity in which the superposition technique can be applied, without loss of generality, the following boundary conditions are usually considered: ˆ u(z) = u(z), z ∈ L, and t(z) = 0, z ∈ C − L,

(9.67)

instead of (9.66). Besides (9.66) or (9.67), there are other kinds of mixed boundary conditions such as ti (z) = ˆti (z), uj (z) = uˆ j (z),

i = 1, and/or 2, and/or 3, j = i,

z ∈ C,

z ∈ C,

(9.68)

which will be considered when we discuss the sliding punch problems in Section 9.4. In this section, two kinds of perturbed boundaries are considered. One is a boundary perturbed from a straight line and the other is a boundary perturbed from an ellipse. A general solution up to the first-order perturbation will be presented by using Stroh formalism, analytical continuation method, conformal mapping function, and perturbation technique. As to higher order perturbation solution, general procedure is depicted in this section. In order to illustrate the use of general solution, two typical examples are solved completely. One is a cosine wavy-shaped surface indented by a rigid flat-ended punch and the other is a triangular hole boundary indented by a rigid stamp.

300

9 Contact Problems

9.3.1 Straight Boundary Perturbation Consider an anisotropic elastic body occupying the lower half-plane whose boundary C is a wavy curve perturbed from the straight line x2 = 0 and can be expressed in terms of a small parameter ε˘ as (to avoid confusing with the oscillatory index, the symbol ε usually used for small number is now replaced by εˇ in this section) x2 = ε˘ ϕ(x1 ),

(9.69)

where ϕ(x1 ) is a wavy-shaped function such as cos x1 . Along this boundary the displacements and stresses are prescribed in (9.67) of which the displacementprescribed boundary L is a union of a set of segments Lk = (ak , bk ), k = 1, 2, . . . , n. In order to solve such a mixed boundary value problem, we let the complex function vector f(z) be expanded in the following perturbation form: f(z) = f0 (z) + ε˘ f1 (z) + ε˘ 2 f2 (z) + · · · .

(9.70)

If we introduce a new variable zˆ in place of z by z = zˆ + ε˘ μϕ(x1 ),

zˆ = x1 + μ[x2 − ε˘ ϕ(x1 )],

(9.71)

each term fk (z) can be expanded in terms of fk (ˆz) and their derivatives as 1 fk (z) = fk (ˆz) + ε˘ μϕ(x1 )fk (ˆz) + (˘ε μϕ(x1 ))2 fk (ˆz) + · · ·, 2

k = 0, 1, 2, . . .

(9.72)

and (9.70) becomes f(z) = f0 (ˆz) + ε˘ [f1 (ˆz) + μϕ(x1 )f0 (ˆz)] 1 + ε˘ 2 f2 (ˆz) + μϕ(x1 )f1 (ˆz) + μ2 ϕ 2 (x1 )f0 (ˆz) + · · · . 2

(9.73)

Note that in (9.73) the subscript α of μα is dropped because the μα term is always accompanied with the argument zα of which the subscript α has been dropped during the derivation by the method of analytical continuation as discussed previously in Section 4.2.1 . Surely, once the solution is obtained, a replacement of z1 , z2 , and z3 together with their corresponding μα should be made for each component function to evaluate the full field solution. With the general solution (9.4), traction relation (3.32), and the perturbation expression (9.73), the mixed boundary conditions (9.67) along the perturbed boundary zˆ = x1− can be expressed as ˆ 1 ), 2 Re{A[f0 (x1− ) + ε˘ (f1 (x1− ) + μϕ(x1− )f0 (x1− )) + · · ·]} = u(x 2 Re{B[f0 (x1− ) + ε˘ (f1 (x1− ) + μϕ(x1− )f0 (x1− )) + · · ·]}

= 0,

x1 ∈ L, x1 ∈ / L,

(9.74a)

9.3


301

ˆ 1 ) is assumed to depend on ε˘ and can be expanded into series of ε˘ as where u(x ˆ 1 ) = uˆ 0 (x1 ) + ε˘ uˆ 1 (x1 ) + ε˘ 2 uˆ 2 (x1 ) + · · · , u(x

(9.74b)

and a value with superscript – denotes that it is approaching from S− which stands for the region under the straight perturbed boundary. By comparing the coefficients of ε˘ k , k = 0, 1, 2, . . ., on both sides of (9.74), we obtain ⎧ ⎨2 Re{Af0 (x− )} = uˆ 0 (x1 ), 1

x1 ∈ L,

⎩2 Re{Bf (x− )} = 0, x1 ∈ / L, 0 1 ⎧ ⎨2 Re{A[f1 (x− ) + μϕ(x− )f (x− )]} = uˆ 1 (x1 ), 1 1 0 1

x1 ∈ L,

.

⎩2 Re{B[f (x− ) + μϕ(x− )f (x− )]} = 0, x∈ / L, 1 1 1 0 1 ⎧ ⎨2 Re{A[f2 (x− ) + μϕ(x− )f (x− ) + (μϕ(x− ))2 f (x− )/2]} = uˆ 2 (x1 ), 1 1 1 1 1 0 1 ⎩2 Re{B[f (x− ) + μϕ(x− )f (x− ) + (μϕ(x− ))2 f (x− )/2]} = 0, 2 1 0 1 1 1 1 1

x1 ∈ L, x1 ∈ / L. (9.75)

The first set equation of (9.75) for the zero-order perturbation is identical to that for the mixed boundary value problems with straight boundary x2 = 0, whose solution f0 (ˆz) has been obtained in (9.8) and (9.5). We can now use f0 (ˆz) as a base to solve the other unknown functions fk (ˆz) in sequence according to (9.75). For the purpose of illustration, the derivation for the first-order perturbation is carried out as follows. To employ the method of analytical continuation, we rewrite the second set of (9.75) as Af1 (x1− ) + μϕ(x1− )Af0 (x1− ) = −Af1 (x1− ) − μϕ(x1− )Af0 (x1− ) + uˆ 1 (x1 ), x1 ∈ L, / L. Bf1 (x1− ) + μϕ(x1− )Bf0 (x1− ) = −Bf1 (x1− ) − μϕ(x1− )Bf0 (x1− ), x1 ∈

(9.76)

Since both f1 (ˆz) and μϕ(ˆz)f0 (ˆz) are holomorphic in S− , by the theory of complex variable function f1 (zˆ¯) and μϕ(z¯ˆ)f0 (z¯ˆ) would also be holomorphic in S+ . If we introduce a new function θ1 (ˆz) such that

θ1 (ˆz) =

⎧ ⎨Bf1 (ˆz) + μϕ(ˆz)Bf0 (ˆz), ⎩−[Bf (z¯ˆ) + μϕ(z¯ˆ)Bf (z¯ˆ)], 1 0

zˆ ∈ S− , zˆ ∈ S+ ,

.

(9.77)

we can conclude that θ1 (ˆz) is sectionally holomorphic in the whole plane except possibly on some segments of x1 -axis, and (9.76) can be expressed in terms of θ1 (ˆz) as

302

9 Contact Problems

θ1 (x1+ ) + MM

−1

θ1 (x1+ ) − θ1 (x1− )

θ1 (x1− ) = iMuˆ 1 (x1 ), = 0,

x1 ∈ L,

x1 ∈ / L.

(9.78)

Equation (9.78) is in the form of Hilbert problem whose solution is (Appendix B.2)

1 θ1 (ˆz) = X0 (ˆz) 2π

L

1 [X+ (t)]−1 Muˆ 1 (t)dt + X0 (ˆz)pn (ˆz), t − zˆ 0

(9.79)

in which the polynomial vector pn (ˆz) can be determined by the loading condition. With this result and the definition given in (9.77), the first-order perturbation solution f1 (ˆz) can be obtained directly. In a similar way, the higher order perturbation solution can be solved step by step.

9.3.2 Elliptical Boundary Perturbation The exact solution for rigid stamp indentation on a curvilinear hole boundary is obtained in Section 9.2 only for the cases that the employed mapping function is single-valued and conformal. The most common examples are elliptic holes which include circles and cracks. For isotropic bodies, it can even be extended to polygonlike holes. If the mapping function is not single-valued, the solutions provided in Section 9.2 is not exact. In this section, we like to discuss the cases when the hole in an anisotropic body differs slightly from that of an ellipse or circle. By introduction of a small parameter ε˘ which characterizes the deviation of the hole from that of the ellipse or circle, we consider the hole whose contour is given by (Lekhnitskii, 1968) ' x1 = a cos ψ + ε˘ '

N

( (ck cos kψ + dk sin kψ) ,

k=1 N

x2 = a c sin ψ + ε˘

(

(9.80)

(−ck sin kψ + dk cos kψ) .

k=1

When ε˘ = 0 we obtain an ellipse with semiaxes a and ac. In the case of ε˘ = 0, c =1, and ε˘ = 0, c → 0, the contour represents, respectively, a circle and a crack. An infinite plane with a hole (9.80) can be conformally transformed to the ζ -plane with a hole in the shape of a unit circle | ζ |=1. The transformation function is z0 = ze0 + ε˘ ϕ0 (ζ ), where z0 = x1 + ix2 and

(9.81a)

9.3


303

ze0 = ω0 (ζ ) = a[(1 + c)ζ + (1 − c)ζ −1 ]/2, ϕ0 (ζ ) =

N

a(ck + idk )ζ −k .

(9.81b)

k=1

Note that the subscript “0” is used to denote that the transformation is related to the isotropic media whose material eigenvalues μα = ±i and their associated arguments zα (= x1 + μα x2 ) becomes z0 . Whereas the superscript “e” is used to denote that the transformation is related to the corresponding elliptical holes. In order to make the transformation (9.81) single-valued and conformal it is necessary that all the roots of equation, by differentiating (9.81) with respect to ζ , ω0 (ζ ) + ε˘ ϕ0 (ζ ) = 0,

(9.82)

located inside of the unit circle |ζ |=1. We will always consider that the coefficients ck , dk and parameter ε˘ are such that this condition is satisfied. Since the holomorphic function f(z) = {f1 (z1 ) f2 (z2 ) f3 (z3 )}T is expressed in terms of the arguments zα (= x1 + μα x2 ), α = 1, 2, 3, the transformation function between zα and ζ should be found. Assume zα is a polynomial of ζ and let the boundary point (x1 , x2 ) given in (9.80) be mapped to ζ = eiψ of the ζ -plane, we have zα = zeα + ε˘ ϕα (ζ ),

(9.83a)

where a 1 (1 − iμα c)ζ + (1 + iμα c) , = ωα (ζ ) = 2 ζ N a 1 k ϕα (ζ ) = (ck − idk )(1 + iμα )ζ + (ck + idk )(1 − iμα ) k , 2 ζ zeα

α = 1, 2, 3.

k=1

(9.83b) Note that although ω0 (ζ ), ω0 (ζ ) + ε˘ ϕ0 (ζ ), and ωα (ζ ) may be single-valued under a certain condition, e.g., (9.82), ωα (ζ ) + ε˘ ϕα (ζ ) is usually nonsingle-valued. Hence, it is possible to get the exact solutions for the isotropic body with an elliptical or a polygonal hole, and for the anisotropic body with an elliptical hole. To solve the problems of the anisotropic body with general curvilinear holes, we like to use the solutions for the anisotropic body with an elliptical hole as a reference. Then, by the perturbation technique an approximate solution for the anisotropic body with holes slightly different from ellipse can be found. To this end, the first step we need to take is to find the reference argument zeα (= x1e + μα x2e ) of the problem with an elliptical hole if a corresponding point z0 (= x1 + ix2 ) is given for the problem with a hole boundary of (9.80). The procedure is described as follows.

304

9 Contact Problems

(1) Given the position (x1 , x2 ) on the physical domain z which contains a hole described by (9.80). (2) Calculate z0 by using z0 = x1 + ix2 . (3) Calculate ζ by inverting z0 = ω0 (ζ ) + ε˘ ϕ0 (ζ ), which should be single-valued since the condition set in (9.82) is required. (4) Calculate zeα by zeα = ωα (ζ ). Like the problem of straight boundary perturbation, we now expand the complex function vector f(z) in the following perturbation form f(z) = f0 (z) + ε˘ f1 (z) + ε˘ 2 f2 (z) + · · · .

(9.84)

With zeα as the reference argument, each term fk (z), k = 0, 1, 2, . . . , can be rewritten as fk (z) = fk (ze + ε˘ ϕ(ζ )) = fk (ze ) + ε˘ ϕ(ζ )fk (ze ) + 12 (˘εϕ(ζ ))2 fk (ze ) + · · · , k = 0, 1, 2, . . .

(9.85)

Substituting (9.85) into (9.84), we have f(z) = f0 (ze ) + ε˘ [f1 (ze ) + ϕ(ζ )f0 (ze )] 1 + ε˘ 2 [f2 (ze ) + ϕ(ζ )f1 (ze ) + ϕ 2 (ζ )f0 (ze )] + · · · . 2

(9.86)

Note that as we discussed in the paragraphs following (9.73), the subscript α of zα , zeα , and ϕα (ζ ) is dropped in the expressions given in (9.84), (9.85), and (9.86). Since zeα = ωα (ζ ), for convenience we now express fk (ze ) by fk (ζ ) without changing the symbol fk . Thus, f(z) = f0 (ζ ) + ε˘ [f1 (ζ ) + ϕ(ζ )f0 (ζ )] 1 2 2 + ε˘ f2 (ζ ) + ϕ(ζ )f1 (ζ ) + ϕ (ζ )f0 (ζ ) + · · · . 2

(9.87)

Employing (9.87) into the general solution (9.40) and using the traction relation (3.32), the mixed boundary condition (9.67) along the elliptical perturbed boundary zα = ωα (σ ) + ε˘ ϕα (σ ) where σ = eiψ can then be expressed as ˆ ), 2 Re{A[f0 (σ + ) + ε˘ (f1 (σ + ) + ϕ(σ + )f0 (σ + )) + · · ·]} = u(σ +

2 Re{B[f0 (σ ) + ε˘ (f1 (σ

+

) + ϕ(σ + )f0 (σ + )) + · · ·]}

= 0,

σ ∈ L,

σ ∈ / L,

(9.88a)

where ˆ ) = uˆ 0 (σ ) + ε˘ uˆ 1 (σ ) + ε˘ 2 uˆ 2 (σ ) + · · · u(σ

(9.88b)

9.3


305

A value with superscript + denotes that it is approaching from S+ which stands for the region outside the unit circle of the mapped plane. Comparing the coefficients of ε˘ k (k=0,1,2,. . .) on both sides of (9.88) leads to ⎧ ⎨2 Re{Af0 (σ + )} = uˆ 0 (σ ), σ ∈ L, ⎩2 Re{Bf (σ + )} = 0, σ ∈ / L, 0 ⎧ ⎨2 Re{A[f1 (σ + ) + ϕ(σ + )f (σ + )]} = uˆ 1 (σ ), 0

σ ∈ L,

⎩2 Re{B[f (σ + ) + ϕ(σ + )f (σ + )]} = 0, σ ∈ / L, 1 0 ⎧ ⎨2 Re{A[f2 (σ + ) + ϕ(σ + )f (σ + ) + (ϕ(σ + ))2 f (σ + )/2]} = uˆ 2 (σ ), 1

0

⎩2 Re{B[f (σ + ) + ϕ(σ + )f (σ + ) + (ϕ(σ + ))2 f (σ + )/2]} = 0 2 1 0

σ ∈ L,

σ ∈ / L. (9.89)

Like the problem of straight perturbed boundary, the first equation set of (9.89) for the zero-order perturbation is identical to that for the mixed boundary value problems with elliptical boundary, whose solution f0 (ze ) has been obtained in (9.46a) and (9.43). Once f0 (ζ ) is obtained, it can be used as a base to solve the other unknown functions fk (ζ ) and we will demonstrate the derivation for the first-order perturbation solution as follows. The second equation of (9.89) can be written as Af1 (σ + ) + ϕ(σ + )Af0 (σ + ) + Af1 (σ + ) + μϕ(σ + )Af0 (σ + ) = uˆ 1 (σ ), +

Bf1 (σ ) + ϕ(σ

+

)Bf0 (σ + ) + Bf1 (σ + ) + μϕ(σ + )Bf0 (σ + )

σ ∈ L,

= 0, σ ∈ / L. (9.90)

Unlike the straight boundary perturbation, due to the form of ϕα (ζ ) given in (9.83b)2 ϕ(z)f0 (z) will not be holomorphic in S+ or S− . To find a sectionally holomorphic function, we need to separate ϕ(σ ) in (9.90) into two parts. One is holomorphic in S+ , and the other is holomorphic in S− . With this separation, using the following relation for the zero-order perturbation Af0 (σ + ) = σ −2 Af0 (1/σ¯ − ) + iρσ −1 uˆ 0 (σ ), Bf0 (σ + ) = σ −2 Bf0 (1/σ¯ − ),

σ ∈ / L,

σ ∈ L,

(9.91)

and introducing

θ1 (ζ ) =

⎧ ⎪ ⎪ ⎨Bf1 (ζ ) +

a 2

N k=1

⎪ ⎪ ⎩−Bf1 (1/ζ ) −

a 2

(ck + idk )[1 − iμ + (1 − iμ)ζ ¯ 2 ]ζ −k Bf0 (ζ ), ζ ∈ S+ , N k=1

(ck − idk )[1 + iμ¯ + (1 + iμ)ζ −2 ]ζ k Bf0 (1/ζ ), ζ ∈ S− , (9.92)

306

9 Contact Problems

(9.90) can be rewritten into the following Hilbert problem of matrix form: θ1 (σ + ) − θ1 (σ − ) = 0, θ1 (σ + ) + MM

−1

σ ∈ / L,

θ1 (σ − ) = Muˆ ∗1 (σ ),

σ ∈ L,

(9.93a)

where uˆ ∗1 (σ ) = i{uˆ 1 (σ ) −

N

Re[iaρ(ck − idk )(1 + iμ)σ k−1 ]uˆ 0 (σ )}.

(9.93b)

k=1

Since f0 (ζ ) and f1 (ζ ) are holomorphic in the elastic body S+ , from the theory of complex variable function f0 (1/ζ ) and f1 (1/ζ ) will be holomorphic in S− . Thus, when k ≥ 2, θ1 (ζ ) introduced in (9.92) will be holomorphic in the whole plane cut along L of the unit circle. The solution of θ1 (ζ ) can then be obtained by using the formula for the Hilbert problems of matrix form (Appendix B.2), which leads to 1 1 Xc (ζ ) [X+ (s)]−1 Muˆ ∗1 (s)ds + Xc (ζ )pc (ζ ), (9.94) θ1 (ζ ) = 2π i s − ζ c L where the polynomial vector pc (ζ ) should be determined by the loading conditions. Thus, θ1 (ζ ) is completely solved by (9.94) and hence f1 (ζ ) can be obtained from (9.92). The higher order perturbation solutions can also be obtained in a similar manner.

9.3.3 Illustrative Examples A Rigid Flat-Ended Punch on a Cosine Wavy-Shaped Boundary Consider a single rigid punch with a flat-ended profile indenting into the cosine wavy-shaped surface of an anisotropic elastic half-plane. If the wavy-shaped surface is slightly perturbed from a straight line by a small amount ε˘ in amplitude, it can be expressed as x2 = ε˘ ϕ(x1 ),

ϕ(x1 ) = cos x1 .

(9.95)

During the indentation, the punch is assumed to completely adhere to the half-plane over the contact region |x1 | ≤ a in such a way that no slip occurs. Thus, the displacement of the surface of the half-plane is known at each point of the contact region ˆ 1 ) = (−c + ε˘ ϕ(x1 ))i2 or and will be u(x uˆ 0 (x1 ) = −ci2 ,

uˆ 1 (x1 ) = ϕ(x1 )i2 ,

(9.96)

where i2 = (0, 1, 0)T and c is the relative depth of indentation. The resultant force applied to the punch is qˆ which will remain constant and will not be disturbed by the perturbed surface. From (9.15) we have

9.3


qˆ k = − Lk

307

{[θ 0 (x1+ )−θ 0 (x1− )]+˘ε[θ 1 (x1+ )−θ 1 (x1− )]+. . .}dx1 , k = 1, 2, 3, . . . . . . , n, (9.97)

and hence, for the present problem,

L

L

ˆ θ 0 (x1+ ) − θ 0 (x1− )dx1 = −q, (9.98)

θ 1 (x1+ ) − θ 1 (x1− )dx1 = 0.

To find the solution to this problem, we first derive the zero-order perturbation solution f 0 (ˆz) from (9.8) and (9.5) in which uˆ 0 (x) = 0 and X0 (ˆz) and pn (ˆz) can be obtained from (9.10), and (9.98)1 . Actually, the problem of zero-order perturbation has been discussed in Section 9.1.1 and the solution is given in (9.23a) for θ 0 (ˆz) in which (ˆz) is shown in (9.23b) and can be obtained from (9.10b)1 and (9.11). With the relation (9.5)1 , the zero-order perturbation solution f 0 (ˆz) can now be written as 1 1 −1 ˆ (ˆz) =< √ f 0 (ˆz) = B (ˆz) −1 q, 2 2π i zˆ − a2

zˆ + a zˆ − a

−iεα

>.

(9.99)

To find the first-order perturbation solution f 1 (ˆz), the sectionally holomorphic function θ1 (ˆz) derived in (9.79) should be evaluated first, in which the only unknown remained to be determined is the polynomial vector pn (ˆz). From (9.98) we see that the total applied force qˆ is considered when we solve the reference solution f 0 (ˆz) and no extra loading is applied on the perturbed problem. This implies that the polynomial vector pn (ˆz) associated with the higher order perturbation is identical to zero. With this result, θ1 (ˆz) is completely determined by the integral term of (9.79) with uˆ 1 (x1 ) = cos x1 i2 , which can be evaluated with the aid of the residue theory (Appendix B.4). The result is θ1 (ˆz) = i cos zˆ(M−1 + M

−1 −1

)

i2 = i cos zˆLi2 /2.

(9.100)

Substituting (9.99) and (9.100) into (9.77)1 , we obtain the first-order perturbation solution as f1 (ˆz) =

icosˆz −1 ˆ B [π Li2 + μ (ˆz) −1 q]. 2π

(9.101)

In a similar way, the higher order perturbation solution can also be obtained step by step. The complete solution to the present problem can then be expressed by (9.73). Note that a replacement of the function argument z1 , z2 , and z3 (including the material eigenvalues μ1 , μ2 and μ3 ) should be made for each component function of f(z) to calculate field quantities from the general solution (9.4).

308

9 Contact Problems

A Rigid Stamp on a Triangular Hole Boundary The previous example shows how to apply the results obtained in Section 9.3.1 for straight boundary perturbation to solve a real problem. To illustrate the application of the results obtained in Section 9.3.2 for the elliptical boundary perturbation, we now choose a hole whose contour differs slightly from that of an ellipse. To avoid tedious mathematical calculation, a relatively simple hole represented by (9.80) with N=2, c1 =d1 =d2 =0, c2 =1, ε˘ =0.25 is considered in this example. By actually plotting this contour, we see that it is a hole differing little from an equilateral triangle with rounded corners. If an anisotropic body contains this kind of hole and is indented by a rigid stamp under force qˆ over a segment, which is mapped onto an arc L=(e−iφ , eiφ ) in the ζ -domain, the stress and strain fields to this problem can be solved by applying the solutions obtained in Section 9.3.2. Since the stamp is assumed to completely adhere to the hole boundary, the displacement along the ˆ z)= constant along L. Note that this concontact region is a constant value. Hence, u(ˆ dition is different from that given in (9.96). In that case the punch profile (flat-ended) differs from the wavy-surface on initial contact, while in the present example, the surfaces of the stamp and the hole boundary are assumed to be perfectly matched on initial contact and during the subsequent indentation. To find the solution to this problem, we first calculate the zero-order perturbation solution f 0 (ζ ), which has been shown in (9.55) for rigid stamps on elliptical hole boundary. For the first-order perturbation solution f 1 (ζ ), we start from the solution θ1 (ζ ) obtained in (9.94). By (9.93b) with uˆ 0 (σ ) = uˆ 1 (σ ) = 0, we have uˆ ∗1 (σ ) = 0. Moreover, pc (ζ ) = 0 since no extra loading is applied on the perturbed problem. Hence, θ1 (ζ ) = 0 by (9.94). Substituting this result into (9.92)1 , we obtain f1 (ζ ) = −a[(1 − iμ)ζ −2 + (1 − iμ)]f 0 (ζ )/2.

(9.102)

Similarly, the higher order perturbation solution can be obtained step by step. The complete solution to the present problem can then be expressed by (9.87) and a replacement of the function argument ζ1 , ζ2 , and ζ3 (including the material eigenvalues μ1 , μ2 , and μ3 ) should be made for each component function of f(ζ ) to calculate field quantities from (9.40).

9.4 Sliding Punches With or Without Friction Sliding punch problem is one type of mixed boundary value problems of elasticity, which is similar to that shown in (9.68), and has been investigated for many years due to its broad application in engineering mechanics. Most of the works related to isotropic half-plane are collected in the books of Muskhelishvili (1954) and England (1971a). Punches with or without friction sliding along the surface of an anisotropic elastic half-plane is considered in this section. The word “sliding” used here has a generalized meaning which includes not only slowly sliding punches but also the punches that are in static equilibrium. The applied horizontal forces can be any value less than or equal to the maximum friction force.

9.4

Sliding Punches With or Without Friction

309

9.4.1 General Solution Consider a set of n punches sliding with or without friction along the surface of a half-plane. All the punches are assumed to be rigid, and the half-plane is considered to be an anisotropic elastic medium. If the kth punch is under the action of normal force Nk and is propelled simultaneously by a horizontal force Fk which is less than the maximum friction force Fk, max = γs Nk (γs : the coefficient of static friction) so that this punch is in equilibrium, we assume the ratio Fk /Nk = η is a constant along all the contact regions. If Fk = Fk, max , η = γs , the kth punch will then be on the verge of non-equilibrium which is an approximation to the case where the punches slide slowly along the surface of the half-plane. If the half-plane surface is frictionless, i.e., η = γs = 0, no horizontal force can be applied since the punch will never stop on a frictionless surface once it is started. The boundary conditions for this kind of problems can be expressed as ⎫ σ12 (x1 ) = ±ησ22 (x1 ) ⎬ σ32 (x1 ) = ±η∗ σ22 (x1 ) , on x1 ∈ Lk , ⎭ u2 (x1 ) = gk (x1 ) + constant σ12 (x1 ) = σ22 (x1 ) = σ32 (x1 ) = 0, on x1 ∈ / L,

(9.103)

where the upper plus sign corresponds to the case that the punch is propelled from right to left, while the lower minus sign corresponds to the case that the punch is propelled from left to right. gk (x1 ) is a given function for the profile of the kth punch, Lk = (ak , bk ) is the contact region between the kth punch and the half-plane surface, and L = L1 ∪ L2 ∪ · · · Ln . To ensure contact, the first two equations of (9.103) only holds provided σ22 < 0 which must be checked when the solution is obtained. Note that η∗ used in (9.103)2 stands for the ratio of the horizontal force in x3 direction to the normal force. Since only two-dimensional problems are considered in the following derivation, no motion or no horizontal force in x3 -direction is assumed in this section. The displacements in the x3 -direction under the punches are assumed to be deformed freely and smoothly without any constraint. Therefore, η∗ is taken to be zero in our following derivation. To find a solution satisfying all the boundary conditions set in (9.103), we start from the relations obtained in (9.6a) for the problems of rigid punches on a halfplane. By using (9.6a)1 , the relation between the pressure P(= −σ22 ) and the tangential stress T(= ∓σ12 or ∓ σ32 ) shown in (9.103)1,2 can be expressed as θ1 (x1+ ) − θ1 (x1− ) = ±η[θ2 (x1+ ) − θ2 (x1− )],

θ3 (x1+ ) − θ3 (x1− ) = 0,

x1 ∈ Lk , (9.104)

where θ1 ,θ2 , and θ3 are the three components of θ. Re-arrangement of (9.104) gives lim [θ1 (z) ∓ ηθ2 (z)] = lim [θ1 (z) ∓ ηθ2 (z)],

x2 →0+

x2 →0−

lim θ3 (z) = lim θ3 (z).

x2 →0+

x2 →0−

(9.105)

310

9 Contact Problems

Thus the functions θ1 (z) ∓ ηθ2 (z) and θ3 (z) are holomorphic in the whole plane including the point at infinity and it tends to zero as |z| → ∞, hence by Liouville’s theorem one can conclude that θ1 (z) ∓ ηθ2 (z) = θ3 (z) = 0.

(9.106)

The problem now reduces to determine a sectionally holomorphic scalar function θ2 (z) or θ1 (z) satisfying the displacement boundary condition of (9.103)3 . This condition can be expressed in terms of θ2 by employing (9.6a)2 and (9.106) into (9.103)3 , as θ2 (x1+ ) +

τ¯ − i θ2 (x1 ) = gk (x1 ), τ τ

on x1 ∈ Lk ,

(9.107a)

where τ = m22 ± ηm21 ,

(9.107b)

and m21 and m22 are the {21} and {22} components of the matrix M−1 . Equation (9.107a) is a standard Hilbert problem, the solution to it is (Appendix B.1) θ2 (z) =

χ0 (z) 2π τ n

k=1 L

g k (t) dt + χ0 (z)pn (z), χ0+ (t)(t − z)

k

(9.108a)

where χ0 (z) = δ=

n ;

(z − ak )−δ (z − bk )δ−1 ,

k=1 1 2π arg(−τ¯ /τ ),

(9.108b)

0 ≤ δ < 1,

and arg stands for the argument of a complex number; pn (z) is an arbitrary polynomial with degree not higher than the punch number n and can be expanded as

pn (z) = d0 + d1 z + · · · · · · + dn−1 zn−1 .

(9.109a)

The unknown coefficients of pn (z) can be determined by the infinity condition (9.20) dn−1

iN , = 2π

N=

n

Nk

k=1

and the force equilibrium condition of each punch, (9.15),

(9.109b)

9.4


Nk = Lk

[θ2 (x1+ ) − θ2 (x1− )]dx1 ,

311

for k = 1, 2, . . . , n.

(9.109c)

It is apparent that one of the equations of (9.109b) and (9.109c) is redundant as (9.109b) ensures the overall equilibrium of the elastic body. Note that δ is a real positive number and hence there are no oscillatory singularities in the solution which is different from the solution for the punch indentation with no slip. Moreover, 0 ≤ δ < 1 such that the total strain energy is bounded within the contact region. The solutions presented here is general in the following senses. (1) The horizontal force F is not necessary to be the value that causes the punch to slide. It may be any value less than or equal to γs N, since the solutions presented here are valid not only for the punches which slide slowly but also for the punches which are in equilibrium. When the punches slide slowly, η = γk (γk : the coefficient of kinetic friction). In the cases of equilibrium (i.e., F < γs N), η = F/N which is not a value of surface properties but a value of load ratio. (2) The profile of each punch gk (t) is not necessarily the same. (3) The punches can move freely in x3 -direction. Contact Pressure The normal stress σ22 and the vertical displacement u2 along the half-plane surface are related to θ2 (z) of (9.108) by (9.6a) as σ22 (x1 ) = θ2 (x1− ) − θ2 (x1+ ),

iu2 (x1 ) = τ θ2 (x1+ ) + τ¯ θ2 (x1− ).

(9.110)

To find a simplified expression for the contact pressure under the punches, we substitute (9.107a) into (9.110)1 , and obtain σ22 (x1 ) = {(τ + τ¯ )θ2 (x1− ) − igk (x1 )}/τ ,

x1 ∈ Lk .

(9.111)

The other components of the stresses under the punches can be obtained by employing (9.111) into (9.103)1,2 . Surface Deformation For the surface deformations outside the punches, we substitute the tractionprescribed condition (9.103)4 into (9.110)1 , which leads to θ2 (x1+ ) = θ2 (x1− ),

x1 ∈ / L.

(9.112)

Substituting (9.112) into (9.110)2 , we obtain u2 (x1 ) = −i(τ + τ¯ )θ2 (x1− ),

x1 ∈ / L.

(9.113)

The solution shown in (9.113) is only for the displacement in x2 -direction. To obtain the solutions for the displacements in x1 - and x3 -directions, similar approach can be applied. We will omit the detailed derivation and its corresponding solution for each displacement component. Alternatively, a compact matrix form solution has been

312

9 Contact Problems

given in (9.22) by using (9.6a)2 and (9.7)1 , which is u (x1 ) = −2iL−1 θ (x1− ),

⎧ ⎫ ⎨±ηθ2 (x1− )⎬ θ (x1− ) = , θ (x− ) ⎩ 2 1 ⎭ 0

/ L. x1 ∈

(9.114)

Note that the expressions given in (9.113) and (9.114) are for the displacement gradient not the displacement itself. To plot the deformation by using (9.113) or (9.114), an integration is necessary for finding u from u . However, the integration constant denoting the rigid body translation cannot be determined due to the infinite feature of our problem. To remedy this, we select the coordinate origin as a reference point whose displacement is set to be zero. By this way, the outlook of the surface deformation is preserved without affecting the understanding of physical behavior. Frictionless Surface If the surface between the half-plane and the punches is frictionless, η = 0 which leads to τ = m22 by (9.107b). The impedance matrix M, and hence its inverse M−1 , is a positive definite Hermitian (Ting, 1996). Therefore, the diagonal elements of M−1 are positive and nonzero. Thus, τ is a positive nonzero real number for the case of frictionless surface. By (9.108b)2 , we conclude that δ = 1/2 if η = 0, which means that the singular order of the punch ends sliding along the frictionless surface is the same as the traction-free crack tips.

9.4.2 A Sliding Wedge-Shaped Punch Consider a wedge-shaped punch under the action of normal force N and is propelled from left to right by a horizontal force F which is equal to the maximum friction force Fmax = γs N such that the punch is on the verge of equilibrium. Thus, η = γs , and the solution is an approximation to the case where the punch slides slowly along the surface of the half-plane. For the cases that F < Fmax , the punch will be in equilibrium and η = F/N. The profile of the punch can be expressed as g(x1 ) = ε∗ x1 where the origin is taken so that the contact region is −a ≤ x1 ≤ a in which a is unknown for the case of incomplete indentation and has to be determined by assuming the stresses vanish at the ends of the contact region. From (9.108) we see that the derivation of θ2 (z) depends on the following integral, which is evaluated by the way shown in Appendix B.3,

a

−a

dt 2π iτ = + τ +τ χ0 (t)(t − z)

1 − [z + (2δ − 1)a] . χ0 (z)

(9.115)

Substituting gk = ε∗ into (9.108), using the integral result of (9.115) and pn (z) = iN/2π from (9.109), we get

9.4


θ2 (z) =

313

iε∗ iN {1 − [z + (2δ − 1)a]χ0 (z)} + χ0 (z), τ +τ 2π

(9.116a)

χ0 (z) = (z + a)−δ (z − a)δ−1

(9.116b)

where

The contact pressure σ22 (x1 ) under the punch can then be calculated by (9.111). The final simplified results are − sin π δ 2π ε∗ σ22 (x1 ) = N− [x1 + (2δ − 1)a] . (τ + τ ) π (a + x1 )δ (a − x1 )1−δ

(9.117)

To have a complete indentation, the applied force N should be large enough that the end-face of the punch touches the half-plane, i.e., the pressure σ22 (x1 ) should be compressive under the punch. By letting σ22 (±a) < 0, we find the minimum requirement for the applied force N to reach complete indentation, which is N≥

4π ε∗ δ a. τ +τ

(9.118)

If N is not sufficiently large to satisfy the above inequality, a state of incomplete indentation will result. In this case the length of contact region will depend on the applied force N and is determined from the condition that the stress is bounded at the point x1 = a where the punch and the half-plane meet smoothly. For a bounded stress at x1 = a, from (9.117) a=

τ +τ N, 4π ε∗ δ

(9.119)

and hence σ22 (x1 ) = −

2ε∗ sin π δ τ +τ

a − x1 a + x1

δ .

(9.120)

The above expressions are valid for general anisotropic half-plane. By using the relations shown in (3.86) and (3.132), the components of M−1 for isotropic materials can be written explicitly. Then, from (9.107b) and (9.108b), we get τ and δ for isotropic half-plane, which has been proved to agree with that given in England (1971a). By letting ε∗ = 0, we get the results for the flat-ended punch. From (9.118) we see that no matter how small of the applied force N, the contact will always be complete indentation which is expected since the end is flat. Numerical Examples In the following, two numerical examples were done by considering an orthotropic half-plane whose material properties are E1 = 60.7 GPa, E2 = 24.8 GPa, G12 = 12 GPa,v12 = 0.23 where E,G, and v are, respectively, the Young’s

314

9 Contact Problems

N

N

F

F

σ 22 /( N / 2a)`

σ 22 /(N / 2a)

or

or v /( N / 2 a L22 )

v /(N / 2a L22)

(a)

(b)

Fig. 9.4 Contact pressure and surface deformation for a wedge-shaped punch sliding along an orthotropic half-plane (Hwu and Fan, 1998c): (a) incomplete indentation; (b) complete indentation

modulus, shear modulus, and Poisson’s ratio. The parameter ε∗ denoting the slope of the wedge shape is chosen to be 0.1. (Note that the slope of the wedge shown in Fig. 9.4a and b has been amplified. Otherwise, it will look like a flat-ended punch, because 0.1 is small.) The width of the punch 2 is set to be 2 m of which the size is just a reference for the infinite domain. The coefficient of friction is considered to be γs = 0.268. From (9.118) with a = , we obtain the minimum requirement for the applied normal force to reach complete indentation, which is 4.6 GNt. Based upon this value, we choose N = 3.6 GNt for the case of incomplete indentation, and N = 6.0 GNt for the case of complete indentation. The results of the contact pressure and surface deformation for these two different applied forces are shown in Fig. 9.4a and b. From these two figures, we observe that the behavior of the contact pressure at the right end of the punch is different. The singularity occurs only at the left end of the punch for the case of incomplete indentation. But for complete indentation, singularity occurs at both punch ends. Actually this result is expected since by applying the linear elasticity the singular behavior, which is physical unreal, usually occurs at a sharp corner or the boundary conditions change in type or change discontinuously. Moreover, due to the friction the singular orders are different at the two ends, which can be seen from (9.116b) and Fig. 9.4a and b.

9.4.3 A Sliding Parabolic Punch Consider a parabolic punch whose end section is expressed by x2 =

x12 , 2R

|x1 | ≤ ,

(9.121)

where R is the radius of curvature and 2 is the width of the punch. Let us suppose on indentation under the normal force N and the horizontal force F (≤ Fmax ), η = F/N and the contact region is the interval (–a,b) when the punch is moving to the right.

9.4


315

Same as the wedge-shaped punch, a and b are unknown for the case of incomplete indentation and has to be determined by assuming the stresses are bounded at the ends of the contact region. If the surface between the half-plane and the punch is assumed frictionless, η = 0 and no horizontal force can be applied to keep the punch in equilibrium. Since the results for the frictionless case can be obtained directly from the friction case by letting η = 0, in the following we first consider the case of sliding punch with friction. After that, we will present the simplified results for the frictionless case by inserting some known information obtained in Section 9.4.1 such as η = 0, δ = 1/2, and τ = τ = m22 Similar to the wedge-shaped punches, the following integral is useful for getting the closed-form solution of θ2 (z) (Appendix B.3),

b

−a

tdt 2π iτ = + τ +τ χ0 (t)(t − z)

z − j2 (z) , χ0 (z)

(9.122a)

where χ0 (z) = (z + a)−δ (z − b)δ−1 , (9.122b)

1 j2 (z) = z2 + [δ(a + b) − b]z − δ(1 − δ)(a + b)2 . 2

Substituting gk (t) = t/R into (9.108), using the integral result of (9.122) and pn (z) = iN/2π from (9.109), we get θ2 (z) =

i (τ + τ )RN z − χ0 (z) j2 (z) − . (τ + τ )R 2π

(9.123)

The contact pressure σ22 (x1 ) under the punch can then be calculated by (9.111). The result is −iχ0 (x1− ) (τ + τ )RN j2 (x1 ) − , σ22 (x1 ) = τR 2π

−a < x1 < b.

(9.124)

Substituting (9.123) into (9.113), the surface deformation outside the punches can be obtained as u2 (x1 )

1 (τ + τ )RN − x1 − χ0 (x1 ) j2 (x1 ) − , = R 2π

x1 < −a

or

x1 > b. (9.125)

Note that the expressions of χ0 (x1− ) for −a < x1 < b, x1 > b, and x1 < −a are different, which can be evaluated by using the bipolar coordinate with the origins located at the ends of the contact region and with the branch cut along the contact region. Their explicit expressions are

316

9 Contact Problems

⎧ −δ δ−1 −iπ δ , ⎪ −a < x1 < b, ⎨−(x1 + a) (b − x1 ) e − −δ δ−1 χ0 (x1 ) = (x1 + a) (x1 − b) , x1 > b, ⎪ ⎩ −(−a − x1 )−δ (b − x1 )δ−1 , x1 < −a.

(9.126)

Incomplete Indentation As in the case of wedge-shaped punch, when the applied force N is not sufficiently large, incomplete indentation occurs and the contact region (– a,b) should be determined by assuming the stresses vanish at the ends x1 = −a and x1 = b. From (9.124) with σ22 (−a) = σ22 (b) = 0, we obtain a2 =

δ(τ + τ )RN , π (1 − δ)

b2 =

(1 − δ)(τ + τ )RN . πδ

(9.127)

Corresponding to these values, (9.124) can be further simplified for the case of incomplete indentation, i.e., σ22 (x1 ) =

−ie−iπ δ (x1 + a)−δ+1 (b − x1 )δ , τR

−a < x1 < b.

(9.128)

Since the stresses are physical quantities which should be real, it is interesting to know the real form expression. From (9.108b)2 , we know −τ /τ = e2iπ δ , by which one can get 2 sin π δ ie−iπ δ = . τ τ +τ

(9.129)

Hence, the real-form solution of the contact pressure for incomplete indentation is σ22 (x1 ) = −

2 sin π δ (x1 + a)−δ+1 (b − x1 )δ , (τ + τ )R

−a < x1 < b.

(9.130)

Similarly, the displacement gradient outside the contact region for incomplete indentation can be found by substituting (9.127) into (9.125), which leads to u2 (x1 ) =

1 x1 ∓ |x1 + a|−δ+1 |x1 − b|δ , R R

x1 < −a

or

x1 > b,

(9.131)

where the upper minus sign is for the region of x1 > b while the lower plus sign is for the region of x1 < −a. From (9.127), we can find that the minimum requirement for the applied force N to reach complete indentation is N ≥ max

δ 1−δ , δ 1−δ

π 2 . (τ + τ )R

(9.132)

9.4


317

Frictionless Surface The solutions for the frictionless parabolic punch problems can be simplified by substituting η = 0, δ = 1/2, τ = τ = m22 into (9.123)–(9.132). The results are θ2 (z)

1 τ RN i a2 2 z− √ − . z − = 2τ R 2 π z2 − a2

(9.133)

When N ≥ π 2 /2τ R, complete indentation occurs and the solutions for the contact pressure and surface deformation are σ22 (x1 ) = u2 (x1 ) =

31 τ R 2 −x12 x1 R

∓

! x12 −

31 R x12 −2

2 2

! x12 −

τ RN π

− 2 2

−

" , τ RN π

|x1 | < , " ,

x1 >

or

x1 < −.

(9.134)

When N < π 2 /2τ R, incomplete indentation occurs and the solutions for the contact pressure and surface deformation are 3 σ22 (x1 ) = − τ1R a2 − x12 , |x1 | < a, 3 u2 (x1 ) = xR1 ∓ R1 x12 − a2 , x1 > a or

(9.135a) x1 < −a,

where a=

+

2τ RN/π .

(9.135b)

Discussions From the solutions shown in (9.130) and (9.135) for the case of incomplete indentation, we see that no singularity occurs at the ends of contact region. While for the complete indentation, the solutions obtained in (9.124) and (9.134) show that the singularity does occur at both ends (x1 = ±) of the punch. Moreover, the singular orders at these two ends are different for the friction case, but are identical to each other for the frictionless case. These phenomena are similar to what we see in Fig. 9.4a and b for the case of wedge-shaped punch. That is, when the punches pressed into the half-plane have no corners or when the force is not sufficiently large for the corners to come into contact with the half-plane, no singularity occurs. Otherwise, singularity will occur due to the change in type of the boundary conditions when applying the model of linear elasticity. By comparing the present solutions for anisotropic body with the solutions provided in England (1971a) for isotropic body, it is surprising to find that they are quite similar in solution form and the anisotropy is reflected only through the parameters

318

9 Contact Problems

τ and δ. Using the explicit expressions given in (3.86) for isotropic materials, it can be proved that the reduced result of the present solutions for the isotropic half-plane is equivalent to that given in England (1971a).

9.4.4 Two Sliding Flat-Ended Punches It should be noticed that the solutions presented in (9.108) are general in the following senses. (1) The half-plane can be any kind of anisotropic materials. (2) The number of rigid punches sliding along the surface is arbitrary. (3) The location of each punch on the surface is arbitrary. (4) The profile of each punch is arbitrary. (5) The surface between the punches and the half-plane can be rough or smooth. The features (1), (4), and (5) have been shown in the previous sections. For the purpose of illustrating features (2) and (3), a simplest case of two sliding flat-ended punches with/without friction is presented in this section. The location of these two punches is (a1 , b1 ) and (a2 , b2 ) where a1 , b1 , a2 , b2 can be arbitrary numbers in the order of a1 < b1 < a2 < b2 . Since the punches considered are flat-ended, g1 (t) = g2 (t) = 0 which make the first term of (9.108a) vanish. Thus, our major calculation will shift to the second term of (9.108a) which is relatively simpler for one punch problems since pn (z)(= do = iN/2π ) is a constant that can be obtained directly from (9.109b). If we have more than one punch, the unknown coefficients increase and more equations like (9.109c) should be employed to set a system of simultaneous algebraic equations for the unknown coefficients. Now by (9.108a), (9.108b)1 , and (9.109a) with n = 2 and g1 (t) = g2 (t) = 0, we have θ2 (z) = χ0 (z)(d0 + d1 z),

(9.136a)

where χ0 (z) = (z − a1 )−δ (z − b1 )δ−1 (z − a2 )−δ (z − b2 )δ−1 .

(9.136b)

The unknown coefficients d0 and d 1 can then be determined by substituting (9.136) into (9.109b) and (9.109c) and using the relation (9.107a), which leads to i (N1 + N2 ) , 2π bk τ χ0 (x1− )(d0 + d1 x1 )dx1 , Nk = − 1 + τ ak d1 =

(9.137) k = 1, 2.

There are three equations and two unknowns in (9.137). One of them is redundant and can be used as a check of the solution. Solving the unknowns from the last two equations of (9.137), we get

9.4


319

b1 b2 −τ − − d0 = x1 χ0 (x1 )dx1 − N2 x1 χ0 (x1 )dx1 , N1 (τ + τ ) a2 a1 b1 b2 −τ i − − χ0 (x1 )dx1 − N2 χ0 (x1 )dx1 = d1 = N1 (N1 + N2 ) , (τ + τ ) 2π a2 a1 (9.138a) where =

b1

a1

χ0 (x1− )dx1

b2 a2

x1 χ0 (x1− )dx1 −

b2

a2

χ0 (x1− )dx1

b1 a1

x1 χ0 (x1− )dx1 . (9.138b)

The normal pressure σ22 (x1 ) along the contact region can then be calculated by substituting (9.136) into (9.111). The result is τ χ0 (x1− )(d0 + d1 x1 ), σ22 (x1 ) = 1 + τ

a1 < x1 < b1 ,

a2 < x1 < b2 . (9.139)

Similarly, by substituting (9.136) into (9.113), the surface deformation can be found as u2 (x1 ) = −i(τ + τ )χ0 (x1− )(d0 + d1 x1 ),

x1 < a1 ,

b1 < x1 < a2 ,

x1 > b2 . (9.140)

Similar to the parabolic punch problem, the expressions of χ0 (x1− ) for x1 < a1 , a1 < x1 < b1 , b1 < x1 < a2 , a2 < x1 < b2 , and x1 > b2 are all different, which can be evaluated by introducing four sets of polar coordinates with each origin located at the ends of the two punches and with the branch cut along the two punches. Their explicit expressions are ⎧ 1, x1 < a1, ⎪ ⎪ ⎪ ⎪ ⎨ e−iπ δ , a1 < x1 < b1 , −1, b1 < x1 < a2 , χ0 (x1− ) = |χ0 (x1 )| ⎪ −iπ δ , a < x < b , ⎪ −e ⎪ 2 1 2 ⎪ ⎩ 1, b2 < x1 ,

(9.141a)

where |χ0 (x1 )| = |x1 − a1 |−δ |x1 − b1 |δ−1 |x1 − a2 |−δ |x1 − b2 |δ−1 .

(9.141b)

Frictionless Surface Based upon the solutions shown in (9.136), (9.137), (9.138), (9.139), (9.140), and (9.141) for the general two sliding rough flat-ended punches, we now consider a special problem with smooth sliding surface (η = 0) and two symmetric punches (a1 = −b, b1 = −a, a2 = a, b2 = b). With this consideration and the aid of the tables of integrals (Dwight, 1985), by (9.136) and (9.138) we obtain

320

9 Contact Problems

θ2 (z) =

π b(N1 − N2 ) − 2 K(k)(N1 + N2 )z + , 4π iK(k) (z2 − a2 )(z2 − b2 )

(9.142a)

where K(k) is the complete elliptic integrals defined as K(k) =

π/2

0

3

dφ 1 − k2 sin2 φ

, k=

+

b2 − a2 /b.

(9.142b)

The normal pressure σ22 (x1 ) and surface deformation u2 (x1 ) can be simplified from (9.139), (9.140), and (9.141). If N1 = N2 = N/2, the solutions can be further simplified to θ2 (z) =

+

iNz

, 2π − a2 )(z2 − b2 ) N σ22 (x1 ) = ± x1 (b2 − x12 )−1/2 (x12 − a2 )−1/2 , −b < x1 < −a, a < x1 < b, π τ N x1 |x12 − b2 |−1/2 |x12 − a2 |−1/2 , |x1 | > b, |x1 | < a, u2 (x1 ) = ± π (9.143) where the upper plus sign and the lower minus sign denote, respectively, the first and the second region given in (9.143). (z2

Numerical Examples Consider the same orthotropic half-plane as that given in Section 9.4.2. The results of the contact pressure and surface deformation for two symmetric smooth flatended punches calculated by (9.143) with a = 1, b = 2 are shown in Fig. 9.5. The interaction between these two punches can be observed from the different strengths of stress singularity at the inner and outer ends of the punches. To see the interaction effect more clearly, it is appropriate to define the strength of stress singularity in a way similar to the definition of stress intensity factor for the crack problems and express the results by using the spacing between two punches (s = a + b) and the width of the punches ( = b − a). Knowing that the singular order for all punch ends are equal to 1/2 for the frictionless surface, the strength of stress singularity S for the special solution shown in (9.143) can be defined and calculated as S(±a) = lim

√

N x ∓ a |σ22 (x1 )| = π

√

N x ∓ b |σ22 (x1 )| = π

x1 →±a

S(±b) = lim

x1 →±b

2 *

a N = 2 2 2π 2(b − a ) b N = 2 2 2π 2(b − a )

2 2

s− , s s+ . s

(9.144)

From (9.144), we see that the strength of stress singularity at the inner ends (x1 = ±a) is smaller than that at the outer ends (x1 = ±b), which agrees with Fig. 9.5.

9.5

Contact Between Two Elastic Bodies

321

N2

σ22/(N/2 ), v(N/2 L22)

N1

x1 / Fig. 9.5 Contact pressure and surface deformation for the indentation by two smooth flat-ended punches (N1 = N2 = N/2) (Hwu and Fan, 1998c)

Moreover, by (9.144) we see that S(±a) = S(±b) =

N √ , 2π

when s → ∞

(9.145)

which can be proved to be identical to that of the single flat-ended punch (using the results of Section 9.4.2 by letting δ = 1/2 and ε∗ = 0). This means that when the two punches are far away from each other, interaction effect can be ignored, which is reasonable and expected. To see the interaction of the two punches in depth, the contour diagram of nondimensionalized stress 2σ22 /N shown in Fig. 9.6 may be helpful, which is plotted by employing (9.143)1 , (9.4), and (9.5).

9.5 Contact Between Two Elastic Bodies Consider two dissimilar anisotropic elastic bodies S1 and S2 which are in contact along a segment L of their boundaries (Fig. 9.7). If these two elastic bodies satisfy the basic laws for two-dimensional linear anisotropic elasticity, by (3.24) the displacement vector u and stress function vector φ of these two bodies can be expressed

322

9 Contact Problems

–0.5

0 .5

–0 .2

––0 .7 0 .6

–0

.3

–0

.3

–0 .1

–0 .2

–1.5

–0

–1

–0.2

– – 0 .4

– 0– 0 .7 .6 – 0 .5 – 0.4

.1

x2/l

–2 –2.5 –3 –0.1

–3.5 –4 – 4.5 –5

–4

–3

–2

–1

0

1

2

3

4

5

x1/l Fig. 9.6 Contour plot of the dimensionless stress 2σ22 /N for the indentation by two smooth flat-ended punches (N1 = N2 = N/2)

as u1 = A1 f1 (z) + A1 f1 (z), , φ1 = B1 f1 (z) + B1 f1 (z),

z ∈ S1 ,

(9.146a)

u2 = A2 f2 (z) + A2 f2 (z), , φ2 = B2 f2 (z) + B2 f2 (z),

z ∈ S2 ,

(9.146b)

and

where the subscripts 1 and 2 are used to denote the quantities pertaining to the bodies S1 and S2 , respectively. Equation (9.146) does not consider the physical conditions of the contact problems. It is just a solution set for any two anisotropic elastic bodies. In this solution set, the material properties of these two elastic bodies are reflected by the eigenvalues μα and eigenvector matrices A and B. In order to have a complete knowledge about the contact behavior from (9.146), the unknown function vectors f1 (z) and f2 (z) should be determined through the satisfaction of the boundary conditions. For the contact problem, many different boundary conditions may be considered to suit for a real situation. In this section, three commonly encountered boundary conditions are formulated and solved. They are contact in the presence of friction, contact in the absence of friction, and contact in complete adhesion. Before we write

9.5


323

x2

x2 = g (1) ( x1 )

S1

*

x2 = g (1) ( x1 ) *

x2 = g (1) ( x1 )

b

a

x1 *

x2 = g ( 2 ) ( x1 )

x2 = g * ( x1 )

S2

*

x2 = g ( 2 ) ( x1 )

x2 = g ( 2 ) ( x1 )

Fig. 9.7 Contact of two anisotropic elastic bodies (solid line: before deformation; dot line: after deformation)

down the mathematical expressions for these three different boundary conditions, a few common physical assumptions are discussed and formulated as follows. If x2 = g(1) (x1 ) and x2 = g(2) (x1 ) are, respectively, the equations of the boundaries of bodies S1 and S2 before deformation, and x2 = g∗ (x1 ) is the equation of the line of contact after deformation (Fig. 9.7), the contact in the x2 -direction can be expressed as g(1) (x1 ) + u2 (x1 ) = g∗ (x1 + u1 ), (1)

(1)

g(2) (x1 ) + u2 (x1 ) = g∗ (x1 + u1 ), (2)

(2)

x1 ∈ L,

(9.147a)

where the superscripts (1) and (2) are used to denote the quantities pertaining to (1) bodies S1 and S2 , respectively. In the cases of small deformation, g∗ (x1 + u1 ) can (1) (2) be represented by its Taylor series g∗ (x1 )+u1 g∗ (x1 )+· · · Similarly, g∗ (x1 +u1 ) = (2) g∗ (x1 ) + u1 g∗ (x1 ) + · · · Subtracting (9.147a)2 from (9.147a)1 , we have g(1) (x1 ) + u2 (x1 ) − g(2) (x1 ) − u2 (x1 ) = (u1 − u1 )g∗ (x) + · · · . (1)

(2)

(1)

(2)

(9.147b)

If the line of contact after deformation is approximate to a straight line, i.e., g∗ (x1 ) ∼ = 0, (9.147) can then be replaced by a simple expression as (2)

(1)

u2 (x1 ) − u2 (x1 ) = g(x1 ),

x1 ∈ L,

(9.148a)

324

9 Contact Problems

where g(x1 ) = g(1) (x1 ) − g(2) (x1 ).

(9.148b)

In addition to the consideration of deformation in x2 -direction, the traction continuity across the contact region and traction-free condition along the uncontact region should also be considered. If the shapes of the boundaries are approximate to straight lines, the stresses normal and tangent to the boundaries can be approximated by σ22 , σ12 , and σ32 . The mathematical expressions for the traction conditions can then be written as (1)

(2)

(1) σ12

(2) σ12

(1)

σ12 = σ12 , =

(2)

(1)

σ22 = σ22 , =

(1) σ22

=

(2) σ22

=

(2)

σ32 = σ32 , (1) σ32

=

(2) σ32

= 0,

x1 ∈ L, x1 ∈ / L.

(9.149)

Equations (9.148) and (9.149) are written based upon the assumptions of small deformation and small boundary slopes and are common conditions for the three different boundary conditions described below. On the basis of these two conditions, we now write down the boundary conditions for three different cases as follows. Contact in the Presence of Friction ⎫ (1) (2) σ12 (x1 ) = σ12 (x1 ) = η1 σ22 (x1 )⎪ ⎪ ⎪ ⎪ ⎪ (1) (2) ⎪ σ (x ) = σ (x ) = η σ (x )⎬

3 22 1 32 1 32 1 , x1 ∈ L (1) (2) σ22 (x1 ) = σ22 (x1 ) = σ22 (x1 ) ⎪ ⎪ ⎪ ⎪ ⎪ (2) (1) ⎪ ⎭ u2 (x1 ) − u2 (x1 ) = g(x1 ) (1) (2) (1) (1) (1) σ12 (x1 ) = σ12 (x1 ) = σ22 (x1 ) = σ22 (x1 ) = σ32 (x1 )

(2)

= σ32 (x1 ) = 0,

x1 ∈ / L, (9.150)

where η1 and η3 are, respectively, the load ratios of the horizontal forces in x1 - and x3 -directions to the normal force. The limit values of these two load ratios will be the coefficients of static friction in the x1 - and x3 -directions. Contact in the Absence of Friction (1)

(2)

(1)

(2)

(1)

(2)

σ12 (x1 ) = σ12 (x1 ) = 0 σ32 (x1 ) = σ32 (x1 ) = 0 σ22 (x1 ) = σ22 (x1 ) (2)

(1)

(1)

(2)

u2 (x1 ) − u2 (x1 ) = g(x1 ) (1)

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

x1 ∈ L,

,

(2)

(1)

(2)

σ12 (x1 ) = σ12 (x1 ) = σ22 (x1 ) = σ22 (x1 ) = σ32 (x1 ) = σ32 (x1 ) = 0, x1 ∈ / L. (9.151)

9.5


325

Contact in Complete Adhesion (1)

(2)

σ12 (x1 ) = σ12 (x1 )

(1)

(2)

σ32 (x1 ) = σ32 (x1 )

u1 (x1 ) = u1 (x1 ), u3 (x1 ) = u3 (x1 ), (2) u2 (x1 ) (1)

=

(1) u2 (x1 ) (2)

(1)

(2)

(1)

(2)

= g(x1 ), (1)

(1) σ22 (x1 )

=

⎫ ⎪ ⎪ ⎪ ⎬ ⎪

⎪ (2) ⎭ σ22 (x1 )⎪

(2)

x1 ∈ L.

,

(1)

(2)

/ L. σ12 (x1 ) = σ12 (x1 ) = σ22 (x1 ) = σ22 (x1 ) = σ32 (x1 ) = σ32 (x1 ) = 0, x1 ∈ (9.152) The common conditions for the three different boundary conditions given in (9.150), (9.151), and (9.152) are the traction continuity across contact region and the traction free along uncontact region, which are expressed by (9.149). By using the relations (3.13), these common conditions can be rewritten in terms of φ1 and φ2 as x1 ∈ / L, (9.153) φ1 = φ2 , ∀x1 , and φ1 = φ2 = 0, which show that the traction is continuous on both the contact and the uncontact regions. This situation is the same as the interface cracks discussed in Section7.4 in which the method of analytical continuation provides the result of (7.62). That is, −1

f2 (z) = B2 B1 f1 (z), −1

f1 (z) = B1 B2 f2 (z),

z ∈ S1 , z ∈ S2 .

(9.154)

By employing (9.154) into the traction-free condition (9.153)2 , we get B1 f 1 (x1+ ) + B2 f 2 (x1− ) = 0. Based upon this relation, we introduce a new function vector −B1 f1 (z), z ∈ S1 , θ(z) = (9.155) B2 f2 (z), z ∈ S2 . Applying (9.154) and (9.155) to (9.146) for the points above and below the x1 -axis, we can get the following expressions: φ1 (x1+ ) = φ2 (x1− ) = θ (x1− ) − θ (x1+ ), −1

+ − u1 (x1+ ) = iM−1 1 θ(x1 ) + iM1 θ(x1 ),

u2 (x1− )

=

(9.156)

−1 − + −iM−1 2 θ(x1 ) − iM2 θ(x1 ),

where Mk , k = 1, 2, are the impedance matrix defined in (3.132). Subtracting (9.156)2 from (9.156)3 , we have ∗

u2 (x1− ) − u1 (x1+ ) = −i{M∗ θ(x1+ ) + M θ(x1− )}, where M∗ is a bimaterial matrix defined in (7.11b).

(9.157)

326

9 Contact Problems

9.5.1 Contact in the Presence of Friction By using the function vector θ(z) defined in (9.155), the boundary conditions given in (9.150) can be written as θ1 (x1− ) − θ1 (x1+ ) = η1 [θ2 (x1− ) − θ2 (x1+ )], θ3 (x1− ) − θ3 (x1+ ) = η3 [θ2 (x1− ) − θ2 (x1+ )], m∗21 θ1 (x1+ ) + m∗22 θ2 (x1+ ) + m∗23 θ3 (x1+ ) + m∗21 θ1 (x1− ) + m∗22 θ2 (x1− ) + m∗23 θ3 (x1− ) =ig(x1 ), (9.158) where θk , k = 1, 2, 3,are the components of the function vector θ and m∗ij , i, j = 1, 2, 3, are the components of the matrix M∗ . Re-arrangement of the first and second equations of (9.158) gives lim [θk (z) − ηk θ2 (z)] = lim [θk (z) − ηk θ2 (z)],

x2 →0+

x2 →0−

k = 1, 3.

(9.159)

Thus the function θk (z)−ηk θ2 (z), k = 1, 3, is holomorphic in the whole plane including the points at infinity and it tends to zero as |z| → ∞ since the stresses vanish at infinity; hence by Liouville’s theorem one can conclude that θk (z) − ηk θ2 (z) = 0,

k = 1, 3.

(9.160)

Employing this result into (9.158)3 , the displacement boundary conditions can now be expressed in terms of θ2 as θ2 (x1+ ) +

τ − i θ (x ) = g (x1 ), τ 2 1 τ

x1 ∈ L,

(9.161a)

where τ = m∗22 + η1 m∗21 + η3 m∗23 .

(9.161b)

Equation (9.161) is a standard Hilbert problem, which possesses exactly the same form as that shown in (9.107) for the sliding punch problems. The only difference is the definition of g(x1 ) and τ . Note that here m∗ij are the components of the bimaterial −1

matrix M∗ which is defined by M−1 1 + M2 , while for the sliding punches discussed in Section 9.4.1 mij stands for the components of M−1 . In Section 9.4.1, the punches are rigid and its elastic constant like Young’s modulus can be considered to be infinite. By (3.85) we know that the real matrix L is proportional to the Young’s modulus and hence L−1 vanishes, so does M−1 by (3.132) for punches. Therefore, by treating the punches as body 1 and the half-plane as body 2 the bimaterial matrix −1 M∗ discussed here will be reduced to M2 . Comparing (9.161) and (9.107), we see that one sliding punch is a special case of the present problem.

9.5


327

Since (9.161a) has the same form as (9.107a), the solution to the present problem can also be expressed by (9.108), i.e., χ0 (z) θ2 (z) = 2π

L

g (t) dt + iN , τ χ0+ (t)(t − z)

(9.162a)

where χ0 (z) = (z − a)−δ (z − b)δ−1 , 1 arg(−τ /τ ), 0 ≤ δ < 1, δ= 2π

(9.162b)

and N is the resultant normal force applied by the body S1 (or S2 ) to the body S2 (or S1 ). a and b are the ends of the contact region L, which should be determined by assuming the stresses vanish at x1 = a and x1 = b, i.e., σ22 (a) = σ22 (b) = 0. By following the way as described in Section 9.4.1, the contact pressure and surface deformation for the present problem can then be written as σ22 (x1 ) = {(τ + τ )θ2 (x1− ) − ig (x1 )}/τ , , , x1 ∈ L, g∗ (x1 ) = {τ (2) g(1) (x1 ) + τ (1) g(2) (x1 ) + i τ (1) τ (2) − τ (1) τ (2) θ2 (x1− )}/τ (9.163a) where ∗ g(1) (x1 ) = g(1) (x1 ) + i(τ (1) + τ (1) )θ2 (x1− ) / L, , x1 ∈ ∗ g(2) (x1 ) = g(2) (x1 ) + i(τ (2) + τ (2) )θ2 (x1− ) (9.163b) (k)

(k)

(k)

τ (k) = m22 + η1 m21 + η3 m23 ,

k = 1, 2,

and mij , i, j=1,2,3, are the components of the matrices M−1 k , k = 1, 2. (k)

Contact of Two Parabolic Elastic Bodies Consider the case of contact between two anisotropic elastic bodies bounded by surfaces g(1) (x1 ) = x12 /2R1 and g(2) (x1 ) = −x12 /2R2 where R1 and R2 are the radii of the curvatures and should be large enough to approximate to straight lines. The resultant vector of the external force applied by the body S1 (or S2 ) to the body S2 (or S1 ) is (N, F, 0), from which η1 =F/N and η3 = 0. By (9.148b), we have g(x1 ) =

x12 2

1 1 + R1 R2

.

(9.164)

Substituting (9.164) into (9.162), and evaluating the line integral with the aid of residue theory (Appendix B.3), we obtain θ2 (z) =

i(R1 + R2 ) iN χ0 (z), {z − χ0 (z)j2 (z)} + (τ + τ )R1 R2 2π

(9.165a)

328

9 Contact Problems

where 1 j2 (z) = z2 − [δ(a − b) + b]z − δ(1 − δ)(a − b)2 . 2

(9.165b)

The normal pressure σ22 (x) along the contact region can then be calculated by substituting (9.165) into (9.163a)1 . The result is −iχ0 (x1− )(R1 + R2 ) (τ + τ )R1 R2 N j2 (x1 ) − σ22 (x1 ) = , τ R1 R2 2π (R1 + R2 )

a < x1 < b, (9.166a)

where χ0 (x1− ) = −(x1 − a)−δ (b − x1 )δ−1 e−iπ δ ,

a < x1 < b.

(166b)

The contact region (a, b) can now be determined by assuming the stresses vanish at the ends x1 = a and x1 = b. Substituting (9.166) into σ22 (a) = σ22 (b) = 0, we obtain

a2 =

δ(τ + τ )R1 R2 N , π (1 − δ)(R1 + R2 )

b2 =

(1 − δ)(τ + τ )R1 R2 N π δ(R1 + R2 )

(9.167)

Corresponding to these values, (9.166) can be further simplified to

σ22 (x1 ) = −

2(R1 + R2 ) sin π δ (x1 − a)1−δ (b − x1 )δ , (τ + τ )R1 R2

a < x1 < b.

(9.168)

The solution (9.168) shows clearly how the parameters R1 , R2 , a, b,τ , δ influence the contact pressure. Numerical examples based upon the results of (9.163) and (9.168) for the contact of two orthotropic elastic bodies can be found in Hwu and Fan (1998d).

9.5.2 Contact in the Absence of Friction In the case that the surface between two contact bodies is frictionless, the load ratios η1 and η3 cannot exceed zero because no horizontal forces can be applied on these two bodies in order to have an equilibrium state. Thus, the contact problems in the

9.5


329

absence of friction can be solved by substituting η1 = η3 = 0 into the results obtained in Section 9.5.1. (k) From (9.161b) and (9.163b) with η1 = η3 = 0, we have τ = m∗22 and τ (k) = m22 which are positive nonzero real numbers by (7.116) and (7.118). Thus, δ defined in (9.162b)2 equals to 1/2 for the frictionless contact surface. Hence, by substituting τ = τ = m∗22 , δ = 1/2, and η1 = η3 = 0 into (9.162) and (9.163), the solutions to the contact problems in the absence of friction can be obtained. For the case of two parabolic elastic bodies, simplification of (9.165), (9.166), (9.167), and (9.168) leads to z2 − (b2 /2) iN i(R1 + R2 ) z − + , (9.169a) θ2 (z) = √ √ 2 2 2m∗22 R1 R2 z −b 2π z2 − b2 3 R1 + R2 σ22 (x1 ) = − ∗ b2 − x12 , |x1 | < b, (9.169b) m22 R1 R2 where * −a=b=

2m∗22 R1 R2 N . π (R1 + R2 )

(9.169c)

From (9.169b) and (9.169c), we see that the maximum contact pressure (σ22 )max occurs at the middle of the contact region. That is, when x1 = 0 we have |σ22 |max =

(R1 + R2 )b 2N , = m∗22 R1 R2 πb

(9.170)

in which the second equation is obtained by using the relation given in (9.169c). (1) (2) Knowing that τ (1) (= m22 ) and τ (2) (= m22 ) are real numbers for the bodies with frictionless surfaces, the surface deformation in the x2 -direction given in (9.163) can be simplified by integrating (9.169a) with respect to z and letting z = x1− . The result is g∗ (x1 ) = (1)∗

g

(2)

(1)

R2 m2 − R1 m22 2 x1 , 2m∗22 R1 R2

−b < x1 < b,

x2 (1) (x1 ) = 1 + 2im22 θ2 (x1− ), 2R1

(2)∗

g

x2 (2) (x1 ) = − 1 − 2im22 θ2 (x1− ), 2R2

(9.171a)

where

θ2 (x1− ) =

⎧ ⎪ ⎪ ⎨

iN 2π b2

⎪ ⎪ ⎩ 2πiNb2

3 3 2 2 2 2 2 2 x1 − x1 x1 − b + b ln(x1 + x1 − b ) , 3 3 2 2 2 2 2 2 x1 + x1 x1 − b + b ln(x1 − x1 − b ) ,

x1 > b, x1 < −b. (9.171b)

330

9 Contact Problems

By comparing the above solutions for anisotropic elastic bodies with the solutions provided in Rekach (1979) for isotropic elastic bodies, it is surprising to find that they are quite similar in solution form and the anisotropy is reflected only through the parameter m∗22 . When we substitute m∗22 by its corresponding isotropic value (given by (7.117)), we find that the present solution is identical to that given in Rekach (1979) in which the problem was solved by employing the concept of superposition of surface Green’s function. It is also possible to apply this concept to anisotropic media. In the following, the solution presented above will be checked by employing this concept. An Alternative Approach Because the problem we discuss in this section is contact in the absence of friction, we assume the state of surface tractions between the contact region have only σ22 component and σ12 = σ32 = 0. Let the compressive stress σ22 be expressed by the unknown distributed forces of intensity q(x) whose resultant N is given, i.e., q(x)dx = N. (9.172) L

The stress and displacement field of a half-plane subjected to this distributed forces can be found by integrating the solutions associated with the point forces. The solution to the anisotropic half-plane subjected to a point force applied on the half-plane surface has been given in (4.61). If the point force q(ξ ) is located on (ξ , 0), the surface deformation in the x2 -direction can be expressed by (4.61), i.e., du2 (x1 ) = −

1 −1 ln |x1 − ξ | L22 q(ξ )dξ , π

along the half plane surface.

(9.173)

Knowing that the forces applied on the bodies S1 and S2 have the same value but opposite direction, by integration of (9.173) for bodies S1 and S2 the boundary condition shown in (9.151)4 can now be written as m∗22 π

ln |x1 − ξ |q(ξ )dξ = g(x1 ).

(9.174)

L

For the case of two parabolic elastic bodies, substitution of (9.164) into (9.174) and differentiation of (9.174) with respect to x1 will lead to m∗22 π

a

−a

q(ξ ) dξ = x1 x1 − ξ

1 1 + R1 R2

.

(9.175)

The form of the integral equation (9.175) is identical to that of the corresponding isotropic integral equation (Rekach, 1979). Thus, by comparison one can prove that the solution of the unknown intensity q(x) is exactly the same as that presented in (9.169).

9.5


331

9.5.3 Contact in Complete Adhesion By the method of analytical continuation presented in this section, we show that the tractions and displacements can be expressed in terms of a single sectionally holomorphic function vector θ. Through the use of (9.156)1 and (9.157), the boundary conditions (9.152) for the contact in complete adhesion can be written as θ (x1− ) − θ (x1+ ) = 0, ∗

− i{M

x1 ∈ / L,

∗ θ(x1+ ) + M θ(x1− )}

= g(x1 ),

x1 ∈ L,

(9.176a)

where ⎧ ⎫ ⎨ 0 ⎬ g(x1 ) = g(x1 ) = g(x1 )i2 . ⎩ ⎭ 0

(9.176b)

To solve θ(z) from (9.176), we rewrite (9.176a)2 into a standard Hilbert problem of matrix form as ∗

θ (x1+ ) + M∗−1 M θ (x1− ) = ig (x1 )M∗−1 i2 ,

x1 ∈ L.

(9.177)

The solution to this Hilbert problem of matrix form is (Appendix B.2) 1 θ (z) = X0 (z) 2π

L

g (t) + −1 [X (t)] dtM∗−1 i2 + X0 (z)d0 . t−z 0

(9.178a)

X 0 (z) is the basic Plemelj function satisfying − X+ 0 (x1 ) = X0 (x1 ),

x1 ∈ / L,

∗−1 ∗ − X+ M X0 (x1 ) 0 (x1 ) + M

= 0,

x1 ∈ L,

(9.178b)

whose solution is 1 X0 (z) = < √ (z − a)(z − b)

z−a z−b

iεα

>,

(9.178c)

where = [λ1 λ2 λ3 ], εα , and λα , α = 1, 2, 3, are the eigenvalues and eigenvectors ∗ of (M − e2πεα M∗ )λα = 0 whose solution can be obtained by referring to (B.9) and (B.10) of Appendix B.2. To determine the constant vector d0 in (9.178a), we use the far-field condition given in (9.20), i.e., d0 =

1 −1 ˆ q, 2π i

(9.179)

332

9 Contact Problems

where qˆ is the resultant force vector applied on the elastic body S1 (or S2 ). The contact pressure can now be obtained by substituting the result of (9.178) into (9.156)1 . The contact region is then determined by assuming the stresses vanish at its ends. By letting M1 = 0 and g(2) (x1 ) = 0 one can prove that the present solutions are identical to those presented in Section 9.1 for the case of rigid punch indenting into a half-plane.

Chapter 10

Thermoelastic Problems

From the discussions of previous chapters, we see that Stroh formalism is a convenient and powerful complex variable method to deal with the plane deformation in general anisotropic elastic solids. This formalism was introduced by Stroh (1958), and the improvement and maturity of Stroh formulation relies on the latter development of identities among the elasticity constants, e.g., Barnett and Lothe (1973), Chadwick and Smith (1977), Ting (1988) and Ting and Hwu (1988), especially the works done by Ting (1996), which makes the mathematical formulation and manipulation simpler and easier. Otherwise, a complicated form solution will be obtained or an intractable mathematical problem will be left. The success of this improvement has been shown in various anisotropic elasticity problems presented in previous chapters. The extended version dealing with thermoelasticity has been given by Clements (1983), Wu (1984), and Hwu (1990). Based upon the work presented in Hwu (1990), the extended Stroh formalism for plane anisotropic thermoelasticity will be shown in Section 10.1. For a homogeneous body which deforms freely and is under the assumption of steady state linear anisotropic thermoelasticity, the linear distribution of temperature will not induce stresses inside the body. However, if the body contains cracks, holes, inclusions, interface, or corners, the temperature field will be disturbed and the thermal stresses will be induced. Severe thermal stresses even arise near the crack/corner/interface tip. With this concern, several different kinds of thermoelastic problems will be discussed in this chapter by following the works published in Hwu (1990, 1992), Lin and Hwu (1993), Hwu and Lee (2004), e.g., holes and cracks in Section 10.2, rigid inclusions in Section 10.3, interface cracks in Section 10.4, and multi-material wedges in Section 10.5.

10.1 Extended Stroh Formalism In a fixed rectangular coordinate system xi , i = 1, 2, 3, let ui , σij , εij , T, and hi be, respectively, the displacement, stress, strain, temperature, and heat flux. The heat conduction, energy equation, strain–displacement relation, constitutive law, and the equations of equilibrium for the uncoupled steady state thermoelastic problems are (Nowacki, 1962) C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_10, C Springer Science+Business Media, LLC 2010

333

334

10

hi = −kij T, j , hi, i = −kij T, ij = 0, 1 εij = (ui, j + uj, i ), σij = Cijkl εkl − βij T, 2

σij,j = 0,


i, j, k, l = 1, 2, 3,

(10.1)

where repeated indices imply summation, a comma stands for differentiation, and Cijkl , kij , βij are the elastic constants, heat conduction coefficients, and thermal moduli, respectively. In general, Cijkl and βij are assumed to be symmetric, while kij need not to be symmetric for anisotropic materials. Substituting (10.1)3 into (10.1)4 with Cijkl fully symmetric, we get σij = Cijkl uk, l − βij T.

(10.2)

The governing differential equations which the displacements must satisfy can then be obtained by substituting (10.2) into (10.1)5 , i.e., Cijkl uk, lj − βij T, j = 0.

(10.3)

Since (10.1)2 is a homogeneous second-order differential equation, for twodimensional deformations in which uk , k = 1, 2, 3 and T depend on x1 and x2 only, a general solution for T will depend on one composite variable which is a linear combination of x1 and x2 . Without loss in generality, the solution to (10.1)2 can be expressed as T = 2 Re g (zt ) ,

zt = x1 + τ x2 ,

(10.4)

where g(zt ) is an arbitrary function of complex variable zt and the prime • denotes differentiation with respect to its argument zt . τ is the thermal eigenvalue of k22 τ 2 + (k12 + k21 )τ + k11 = 0.

(10.5)

The fact that heat always flows from higher temperature to lower temperature tells us that the roots of (10.5) cannot be real (Ting, 1996). There is one pair of complex conjugates for τ , and we let τ in (10.4) is the one with positive imaginary part. Substituting (10.4) into (10.1)1 , the distribution of heat flux can be written as hi = −2 Re (ki1 + τ ki2 )g (zt ) ,

(10.6a)

h = −2 Re (k1 + τ k2 )g (zt ) ,

(10.6b)

or in vector form

where ⎧ ⎫ ⎨h1 ⎬ h = h2 , ⎩ ⎭ h3

⎧ ⎫ ⎨k11 ⎬ k1 = k21 , ⎩ ⎭ k31

⎧ ⎫ ⎨k12 ⎬ k2 = k22 . ⎩ ⎭ k32

(10.6c)

10.1

Extended Stroh Formalism

335

With (10.4), βij T, j in (10.3) is treated as known function, and the solution to uk in (10.3) can be found by adding the homogeneous solution obtained in (3.23) with a particular solution related to βij T, j . A general expression for the displacements can then be written in matrix notations as 3 u = 2 Re aα fα (zα ) + cg(zt ) , zα = x1 + μα x2 , (10.7) α=1

where fα (zα ) are arbitrary functions of their arguments, μα and aα are the elasticity eigenvalues with positive imaginary part and the associated eigenvectors of (3.7). c is determined by {Q + μ(R + RT ) + μ2 T}c = β1 + τ β2 ,

(10.8a)

where ⎧ ⎫ ⎨β11 ⎬ β1 = β21 , ⎩ ⎭ β31

⎧ ⎫ ⎨β12 ⎬ β2 = β22 . ⎩ ⎭ β32

(10.8b)

Like the elasticity formulation, the stresses σij can also be expressed by the stress functions φi introduced in (3.13). The stress function for thermoelasticity can be obtained by substituting (10.4) and (10.7) into (10.2). The result is φ = 2 Re

3

bα fα (zα ) + dg(zt ) ,

(10.9a)

α=1

in which b is related to a by (3.11) and d is related to c by 1 1 d = (RT + τ T)c − β2 = − (Q + τ R)c + β1 , τ τ

(10.9b)

where the second equality of (10.9b) comes from (10.8a). Like the general solution written in (3.24), the solution given in (10.7) and (10.9) can also be expressed in matrix form as u = 2 Re{Af(z) + cg(zt )}, φ = 2 Re {Bf(z) + dg(zt )} ,

(10.10)

where A, B, and f(z) have been defined in (3.24b). Like the eigenrelation established in (3.48) for the determination of the material eigenvectors a and b, (10.9b) can also be reconstructed into the following standard eigenrelation Nη = τ η + γ,

(10.11a)

where N is the fundamental elasticity matrix defined in (3.48b,c) and η and γ are defined by

336

10

c η= , d

0 γ=− I

N2 NT1


β1 . β2

(10.11b)

Like (3.76), the generalization of (10.11) for a rotated coordinate system can also be established by following the steps stated in Section 3.3.2 or referring to Hwu (1990). The result is N(θ )η(θ ) = τ (θ )η(θ ) + γ(θ ),

(10.12a)

where N1 (θ ) N2 (θ ) N(θ ) = , N3 (θ ) NT1 (θ ) 0 c(θ ) η(θ ) = , γ(θ ) = − d(θ ) I

N2 (θ ) β1 (θ ) NT1 (θ ) β2 (θ )

(10.12b)

and β1 (θ ) = βs(θ ), β2 (θ ) = βn(θ ), β = [β1 β2 β3 ], s(θ ) = (cos θ , sin θ , 0)T ,

n(θ ) = (− sin θ , cos θ , 0)T .

(10.12c)

As the relation for the eigenvalues μ(θ ) and μ shown in (3.70), it has been proved that τ (θ ) and τ also have the same relation, i.e., Hwu (1990) τ (θ ) =

τ cos θ − sin θ . τ sin θ + cos θ

(10.12d)

However, unlike a and b which are independent of θ , c(θ ) and d(θ ) are related to c and d by η(θ ) = τˆ (θ )η,

τˆ (θ ) = cos θ + τ sin θ ,

(10.13)

which can be proved as follows. Differentiating both sides of (10.12a) and using the identities (3.96a)1 , we obtain [τ (θ ) + 1 + τ 2 (θ )]η(θ ) + [N(θ )γ(θ ) + γ (θ )] + τ (θ )γ(θ ) =[N(θ ) − τ (θ )I]η (θ ).

(10.14a)

The first term of the left-hand side of (10.14a) vanishes through the use of (10.12d). Applying (10.12b)3 and (3.96a)1 , we get N(θ )γ(θ ) + γ (θ ) = 0. Equation (10.14a) now becomes N(θ )η (θ ) = τ (θ )η (θ ) + τ (θ )γ(θ ).

(10.14b)

Comparison between (10.14b) and (10.12a) suggests that η (θ ) = τ (θ )η(θ ).

(10.14c)

10.2

Holes and Cracks

337

Equation (10.14c) is a set of six uncoupled equations with unknown functions ηi (θ ), i = 1, 2, . . . , 6. Dividing both sides of the six equations by ηi (θ ) and integrating with τ (θ ) given in (10.12d), we can obtain the result shown in (10.13) where η = η(0). With the results of (10.13), (10.12a) can also be written as N(θ )η = τ (θ )η + τˆ −1 (θ )γ(θ ).

(10.15)

Integrating the generalized eigenrelation (10.12a) with respect to ω from 0 to 2π , and using the relations given in (3.87a)4 , (3.90), and (3.95), we can get another useful identities as Sc + Hd = ic + γ˜ ∗1 ,

−Lc + ST d = id + γ˜ ∗2 ,

(10.16a)

where 2π 2π 1 1 τˆ −1 (θ )γ1 (θ )dθ , γ˜ ∗2 = τˆ −1 (θ )γ2 (θ )dθ , 2π 0 2π 0 γ1 (θ ) = −N2 (θ )βn(θ ), γ2 (θ ) = −βs(θ ) − NT1 (θ )βn(θ ). γ˜ ∗1 =

(10.16b)

Note that the general solution given in (10.10) is obtained under the assumption that the thermal eigenvalue τ and the elasticity eigenvalues μα , α = 1, 2, 3, are distinct. For the cases that they are repeated, a small perturbation of the material constants can be employed to avoid the degenerate problem. Otherwise, a modified solution as discussed in Section 3.5 should be applied. However, if the final solutions do not contain the eigenvectors aα , bα , c, and d, the problems of repeated eigenvalues can be avoided, which can usually be done through the use of identities converting the complex form into real form as discussed in Section 3.4.4.

10.2 Holes and Cracks Thermal stress concentration occurs when a uniform heat flow is disturbed by a hole. As the hole reduces to a crack, the stresses induced by heat flow become singular at the tips of crack, which is the same as isothermal condition. Of various holes the elliptic shape has evoked the most interest among researchers for isothermal problems because of its flexibility to include the other special shapes such as circles or cracks. Several analytical solutions for thermal stresses disturbed by holes or cracks have been found in the literature for the isotropic plates (Florence and Goodier, 1960, 1963; Olesiak and Sneddon, 1960; Sih, 1962), orthotropic plates (Chen, 1967a; Tsai, 1984), or anisotropic plates (Atkinson and Clements 1977; Sturla and Barber, 1988; Hwu, 1990). In this section, we follow the work of Hwu (1990) to find the stresses, displacements, heat flux, and temperature in the anisotropic plates containing holes or cracks subjected to uniform heat flow.

338

10


10.2.1 Elliptical Holes Under Uniform Heat Flow Consider an infinite anisotropic plate containing an insulated elliptic hole and the heat h0 is flowing steady and uniformly in the direction of the positive x2 -axis. The contour of the hole boundary is represented by (6.1), i.e., x1 = a cos ψ,

x2 = b sin ψ,

(10.17)

where 2a, 2b are the major and minor axes of the ellipse and ψ is a real parameter. If the hole is assumed to be insulated and free of tractions, the boundary conditions for this problem can be written as hn = 0,

tn = 0,

h2 → h0 ,

along the hole boundary,

σij → 0,

at infinity,

(10.18)

where hn is the heat flux in the direction of n which is normal to the surface of the elliptic hole. tn is the surface traction along the hole boundary of which the normal is n. By using the coordinate transformation and applying (10.5) and (10.6), hn is obtained as hn = −h1 sin θ + h2 cos θ = 2k˜ Im (cos θ + τ sin θ )g (zt ) ,

(10.19a)

where k˜ is a real constant related to the heat conduction coefficients by k˜ = −ik22 (τ − τ¯ )/2 =

3 2 . k11 k22 − k12

(10.19b)

In (10.19a), the angle θ is directed counterclockwise from the positive x1 -axis to the direction of s (Fig. 10.1), which is related to the contour parameter ψ by ρ cos θ = −a sin ψ,

ρ sin θ = b cos ψ,

(10.20a)

where ρ 2 = a2 sin2 ψ + b2 cos2 ψ.

(10.20b)

Due to the linear property, the principle of superposition can be used and the solution can be represented as the sum of a uniform heat flux in an unnotched solid and corrective solution, for which the boundary conditions are hn = −h0 cos θ , tn = 0, along the hole boundary, h2 → 0, σij → 0, at infinity.

(10.21)

To satisfy the boundary conditions (10.21), the choice of functions f (zα ) and g(zt ) becomes critical in the solution procedure. A comparison with the isothermal problem under uniform loading, (6.8), may give us some idea to make a proper choice.

10.2

Holes and Cracks

339 h0 x2

x1

a n

b

s

θ x1 = a cos ψ x2 = b sin ψ

h0

Fig. 10.1 An elliptical hole under uniform heat flow

The boundary conditions of the heat flux shown in (10.21) are similar to those of the stresses in the isothermal condition. The general solutions for the heat flux shown in (10.6) and isothermal stresses in (10.10) with the last term neglected are proportional to g (zt ) and f (zα ), respectively. Therefore, by referring to the solutions for the isothermal problem (6.8), we may choose g(zt ) as g(zt ) = e1

ζ −1 (zt ) dzt ,

ζ (zt ) =

zt +

3

z2t − (a2 + τ 2 b2 ) a − iτ b

,

(10.22a)

where e1 is a complex constant to be determined. After taking the inverse and integrating, we obtain g(zt ) = e1 g0 (zt ), g0 (zt ) =

1 a + iτ b

3 1 2 1 zt − zt z2t − (a2 + τ 2 b2 ) 2 2 3 2 a + τ 2 b2 + ln zt + z2t − (a2 + τ 2 b2 ) . 2

(10.22b)

340

10


The only requirement for the choice of f (zα ) is the satisfaction of boundary conditions. Owing to the fact that f (zα ) and g(zt ) have the same order to affect the displacements and stresses in (10.10), possible function forms of f (zα ) come from the partition of g(zt ). That is, f1 (zα ) = f2 (zα ) =

1 a+iμα b a−iμα b 2

!

" + − 12 zα z2α − (a2 + μ2α b2 ) , + ln zα + z2α − (a2 + μ2α b2 ) , 1 2 2zα

(10.23)

where the subscripts 1 and 2 are the indices for the different possible functions. With this possible choice, the general solutions for the displacements and stress functions shown in (10.10) can then be written as u = 2 Re

2

φ = 2 Re

A < fi (zα ) > qi + cg(zt ) ,

i=1 2

(10.24a)

B < fi (zα ) > qi + dg(zt ) ,

i=1

where the complex constant vectors qi , i = 1, 2, can be replaced by two real constant vectors, qia and qib , as qi = AT qia + BT qib ,

i = 1, 2.

(10.24b)

The problem now reduces to the determination of the unknown constants e1 , qia , and qib which should satisfy the boundary conditions (10.21). Differentiating (10.22) and (10.23) with respect to their arguments, we obtain g (zt ) = e1 ζt−1 ,

g (zt ) = −e1 ζt−2

dζt , dzt

(10.25) a − iμα b a − iμα b f1 (zα ) = ζα−1 − + , f2 (zα ) = + , 2 z2α − (a2 + μ2α b2 ) 2 z2α − (a2 + μ2α b2 ) where ζt and ζα are the simplified notations of ζ (zt ) and ζ (zα ). Since g (zt ), g (zt ), and fi (zα ) → 0 when |zt | or |zα | → ∞, the infinity conditions (10.21)2 are satisfied. To satisfy the hole boundary condition (10.21)1 , one should evaluate g (zt ), g, s (zt ), and fi, s (zα ) along the hole boundary. Knowing that ζ = ζt = ζα = eiψ , dζt /dzt = ieiψ /ρ(cos θ + τ sin θ ), and the relation (10.20), on the hole boundary we have g (zt ) = −

ie1 (cos ψ − i sin ψ) , ρ(cos θ + τ sin θ )

g, s (zt ) = (cos θ + τ sin θ )g (zt ) =

ie1 {(a + iτ b)(cos 2ψ − i sin 2ψ) − a + iτ b}, 2ρ (10.26a)

10.2

Holes and Cracks

341

f1,s (zα ) = (cos θ + μα sin θ )f1 (zα ) = f2,s (zα ) =

a + iμα b (sin 2ψ + i cos 2ψ), 2ρ

−i (a − iμα b). 2ρ

(10.26b)

With the value calculated from (10.26a)1 and the relation given in (10.19a)2 , the heat flux boundary condition in (10.21)1 gives us e1 =

−iah0 . 2k˜

(10.27)

With the results obtained in (10.26a)2 , (10.26b), (10.27), and the identities given in Section 3.4.4 such as (3.124), the traction-free boundary condition of (10.21) will lead to , % , -& ˜ b N3 S − NT1 L q1b + aI + b NT1 ST + N3 H q1a = −ah0 Im{(a + ibτ )d}/k, , T ˜ − (aL + bN3 )q1b + aS − bNT1 q1a = −ah0 Re{(a + ibτ )d}/k, , T ˜ (aL − bN3 )q2b − aS + bNT1 q2a = −ah0 Re{(a − ibτ )d}/k. (10.28a) Due to the multi-valued characteristics of logrithmic functions contained in g(zt ) and f2 (zα ), the requirement of single-valuedness of displacements provides ˜ (aS + bN1 )q2b + (aH + bN2 )q2a = −ah0 Re{(a − ibτ )c}/k.

(10.28b)

Applying the anti-symmetric property of LS and the &identities (3.97b)1 and % (3.109a)3 , the manipulation L−1 ST (10.28a)2 + (10.28a)1 gives a simplified equation to replace (10.28a)1 , which is, ˜ (10.28c) (aS − bN1 )q1b + (aH − bN2 )q1a = −ah0 L−1 Re{(a + ibτ )(ST − iI)d}/k. Combining (10.28b,c) and (10.28a)2,3 , we obtain −1 T −ah0 ˜ q1b −1 L Re{(a + ibτ )(S − iI)d} = (aN − bN) , q1a Re{(a + ibτ )d} k˜ −ah0 ˜ q2b = (aN + bN)−1 Re{(a − ibτ )η}, q2a k˜

(10.29a)

˜ is the matrix defined in (3.94) and (3.109b)2 , i.e., in which N ˜ = N

S −L

H . ST

(10.29b)

˜ Note that the matrices (aN±bN) will become singular when one of their eigenvalues ai ± bμα equals to zero. This is the limiting case for the choice of f1 (zα ) in (10.23), which can be approached by applying L’Hospital rule and differentiating both of the

342

10


denominator and numerator with respect to μα . The real-form solutions for the hoop stress given later do not contain eigenvalues μα and any possible singular matrices, and hence that solutions will be valid for the limiting cases. Hoop Stress Using the stress function vector φ obtained in this section, the hoop stress σss can be determined by the relation given in (3.36a), i.e., σss = sT ts = −sT φ, n .

(10.30)

Similar to the calculation of tn , to get explicit real-form solution of σss one should evaluate g, n (zt ) and fi, n (zα )along the hole boundary. The tangential and normal directions of the hole boundary are, respectively, s(θ ) = (cos θ , sin θ , 0)T

n(θ ) = (− sin θ , cos θ , 0)T ;

and

(10.31)

thus, tn = φ, s = cos θ φ, 1 + sin θ φ, 2 ,

ts = −φ, n = sin θ φ, 1 − cos θ φ, 2 .

(10.32)

By comparing the two equations of (10.32) and using the relations given in (3.70) and (10.12d), we obtain g, n (zt ) = τ (θ )g, s (zt ),

fi, n (zα ) = μα (θ )fi, s (zα ),

(10.33)

where g, s (zt ) and fi, s (zα )have been obtained in (10.26). With these relations and the identities given in Section 3.4.4, by following the procedure stated in Section 6.1.1 one can find the hoop stress σss in real form. The final simplified expression is φ, n =

" ah0 ! {2ρ sin ψγ2 (θ ) − N3 (θ )L−1 Re e−2iψ (a + ibτ )γ˜ ∗2 . ˜ 2kρ

(10.34)

Using the identities (3.98), the above result has been checked by calculating σns = nT (θ )ts = −nT (θ )φ, n , which should be zero along the hole boundary. Note that the explicit closed-form solution for the hoop stress given in (10.30) and (10.34) does not contain the elasticity eigenvalues μα , elasticity eigenvectors A and B, and thermal eigenvectors c and d. Hence, the solution is valid for degenerate materials. Solution (10.34) shows that the hoop stress σss is influenced by the elastic and thermal anisotropy through N3 , L, γ2 , and γ˜ ∗2 . The shape of the elliptic opening is represented by a, b, and ρ. The effect of heat conductivity is reflected by k˜ and the thermal eigenvalue τ . The magnitude of uniform heat flow which is perpendicular to the x1 -axis is given by h0 . To find the location of maximum or minimum hoop stress, we differentiate σss with respect to θ and set the result to zero. Note that ψ and ρ are functions of θ , which can be seen in (10.20). ˜ ij where E, ν, α, and k˜ For isotropic materials, βij = (Eα/(1 − 2ν))δij , kij = kδ are the Young’s modulus, Poisson’s ratio, thermal expansion coefficient, and heat

10.2

Holes and Cracks

343

conductivity, respectively. δij is the Kronecker delta. The explicit expressions of the fundamental elasticity matrices N1 , N3 , and L for the isotropic materials have been given in (3.83) and (3.86). The thermal eigenvalue τ is determined by (10.5) and √ the root with positive imaginary part is obtained to be i = −1 , which is equal to the triple elasticity eigenvalues μα , α = 1, 2, 3, determined by (3.9) or (3.48). Therefore, the isotropic materials belong to the degenerate materials and the field solutions obtained in (10.24) and (10.29) cannot be applied directly. However, as stated above the hoop stress obtained in (10.30) and (10.34) can be applied directly. The determination of the hoop stress by (10.30) and (10.34) now relies on the calculation of γ2 (θ ) and γ˜ ∗2 , which are obtained by using the given values of βij , the explicit expression of N1 (θ ) in (3.82) and the relation in (10.16b)2 . The results are ⎧ ⎫ ⎧ ⎫ cos θ ⎬ ⎨−1⎬ −Eα ⎨ Eα sin θ , γ˜ ∗2 = i . γ2 (θ ) = 1−ν ⎩ 0 ⎭ 2 (1 − ν) ⎩ 0 ⎭

(10.35)

Substituting (10.35) and the explicit expressions of N3 (θ ), L given in (3.82) and (3.86) into (10.30) and (10.34), we obtain σss =

Eh0 αa2 (a + b) sin ψ, ˜ 2 (1 − ν) 2kρ

(10.36)

which is equivalent to the one given in Florence and Goodier (1960). Note that there is a typing error in (41) of Florence and Goodier (1960) which should be corrected to h = [ρ 4 − 2ρ 2 m cos 2θ + m2 ]2 .

10.2.2 Cracks Under Uniform Heat Flow An elliptic hole can be made into a crack of length 2a by letting b approach zero.The solutions for the uniform heat flux obstructed by a plane crack in a general anisotropic body can then be obtained from Section 10.2.1 by letting b=0. Using ˜ 2 = −I given in (3.96a)2 , (10.29) can be simplified to the identities N ˜ q1a = −ah0 Im{d}/k, q2a

˜ q1b = ah0 L−1 Re{(I + iST )d}/k, ˜ q2b = ah0 Re{Sc + Hd}/k. ˜ = ah0 Re{−Lc + ST d}/k,

(10.37)

With the results obtained in (10.27) and (10.37), the stresses σi2 ahead of the crack tip along the x1 -axis can be calculated by using the relations given in (3.13) and (10.24) with x2 = 0 and |x1 | > a. The solution can be further simplified through the use of the identities provided in (3.95) and (10.16), which is obtained as σi2 =

a2 h0 3 Re γ˜ ∗2 . 2k˜ x2 − a2 1

(10.38)

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10


Same as the isothermal problems, the above solution shows that the stresses are singular near the crack tip. With the usual definition, the stress intensity factors are given by ⎧ ⎫ ⎫ ⎧ ⎨σ12 ⎬ √π h ⎨ KII ⎬ + 0 3/2 KI = lim 2π (x1 − a) σ22 = a Re γ˜ ∗2 . (10.39) ⎩ ⎭ ⎭ x1 →a ⎩ 2k˜ KIII σ32 Similarly, by using (10.24a)1 and setting x2 = 0± and |x1 | < a where ± denotes the upper and lower surface of the crack, the crack opening displacements u are obtained as 3 ˜ (10.40) u = u(x1 , 0+ ) − u(x1 , 0− ) = h0 x1 a2 − x12 L−1 Re γ˜ ∗2 /k. It can be seen that u2 will always benegative in either −a < x1 < 0 or 0 < x1 < a if the second component of L−1 Re γ˜ ∗2 is not equal to zero, which violates the assumption of fully open crack and the solution is invalid. Sturla and Barber (1988) discussed this situationand gave a solution with partial contact crack. If the second component of L−1 Re γ˜ ∗2 equals to zero, u2 = 0 for all x1 within the crack, and hence there is no tendency for the crack to open or close. However, a relative tangential displacement may exist between the crack faces. Therefore, (10.40) is valid if u2 = 0 or the negative u2 is increased to a positive value by an applied tensile load. By applying the virtual crack closure method (Irwin, 1957), the total energy release rate G can be calculated as 1 a→0 2a

a

G = lim

ui (x − a)σi2 (x) dx =

0

π h20 a3 ∗ Re γ˜ 2 8k˜ 2

T

L−1 Re γ˜ ∗2 .

(10.41) For isotropic materials, substituting (10.35)2 and (3.86a)3 into (10.40), we have u2 = 0 which validates the following solutions. From (10.39) and (10.41), √ KII = −

π Eαh0 a3/2 , KI = KIII = 0, 4(1 − ν)k˜

π Eα 2 h20 a3 (1 + ν) , G= 16(1 − ν)k˜ 2

(10.42)

which is identical to the solutions given in the literature for isotropic plates (Sih, 1962).

10.3 Rigid Inclusions The thermal stress concentration induced by holes or cracks has been studied in the previous section. In Section 10.3.1 we will consider the thermal stress induced by

10.3

Rigid Inclusions

345

elliptical rigid inclusions, which are important for many practical applications such as bolted joints and fiber reinforced composites. By letting the minor axis of the ellipse approach to zero, the solutions for rigid line inclusion are provided in Section 10.3.2. The strength of thermal stress singularity is defined in the way similar to the stress intensity factor of crack problems and is found to be dependent on the material properties.

10.3.1 Elliptical Rigid Inclusions Under Uniform Heat Flow Consider a rigid inclusion embedded in an infinite anisotropic plate under a uniform heat flow h0 in the direction at an angle ϕ with the positive x1 -axis. The contour of the inclusion is considered to be an ellipse of which the boundary is represented by (10.17). If the inclusion is considered to be perfectly bonded with the matrix and is insulated, the boundary conditions for this problem can be written as hn = 0, u = uo , along the inclusion boundary, h1 → h0 cos ϕ, h2 → h0 sin ϕ, σij → 0, at infinity,

(10.43)

where hn is the heat flux in the normal direction n of inclusion surface. uo is the rigid body displacement caused by the relative rotation ω between matrix and inclusion, which can be written as uo = x1 εo1 + x2 εo2 ,

where

εo1 = ωn(0),

εo2 = −ωs(0).

(10.44)

The expressions for n and s are shown in (10.12c) where the angle θ is directed counterclockwise from the positive x1 -axis to the direction of s. The relation between θ and ψ is given in (10.20). If the rigid inclusion is considered to have a good thermal conductivity, the temperature and heat flux across the boundary should be continuous. The boundary conditions along the interface between inclusion and matrix, (10.43)1 , are then replaced by T = T o , hn = hon ,

u = uo ,

along the inclusion boundary,

(10.45)

where the superscript o refers to the rigid inclusion. In this section, only the insulated rigid inclusion will be studied, i.e., we only consider the boundary conditions given in (10.43). Similar approach can be applied for the conductive rigid inclusion (10.45). Like the hole problems discussed in Section 10.2.1, to find a solution satisfying the boundary conditions (10.43), the choice of functions g(zt ) and f (zα ) becomes critical in the solution procedures. Since a certain analogy exists between the hole and the rigid inclusion problems, one can now use the results for hole problems as a reference. By referring to the solutions given in (10.24), (10.27), and (10.29) for hole problems, the general solutions for rigid inclusion problems can be written as

346

10

u = 2 Re A
q0 +

φ = 2 Re B < z2α > q0 +

2

A < fi (zα ) > qi + c

e0 z2t


+ e1 g0 (zt )

i=1 2

B < fi (zα ) > qi + d e0 z2t + e1 g0 (zt )

,

(10.46) ,

i=1

where f1 (zα ), f2 (zα ), and g0 (zt ) are given in (10.23) and (10.22b), the complex constant vectors qi , i = 0, 1, 2 are replaced by two real constant vectors, qia and qib , as that shown in (10.24b). Note that the choice of g(zt ) and fi (zα ) given in (10.46) is almost the same as that of the hole problem shown in (10.24) except that the terms associated with z2t and z2α are now included to consider the uniform heat flux at infinity. In hole problems due to the application of superposition principle, a negative uniform heat flux is considered to be applied on the hole boundary. However, it is not appropriate in rigid inclusion problems since the rigid body motion conditions of inclusions cannot be satisfied. The problem now reduces to the determination of the unknown complex constants ω, e0 , e1 and real constants qia , qib , i = 0, 1, 2, which should satisfy the boundary conditions (10.43). By following the same procedure as that described between (10.25) and (10.29), we have (Lin and Hwu, 1993) −h0 −h0 (b cos ϕ + ia sin ϕ), (cos ϕ + τ¯ sin ϕ), e1 = ˜ 4kτI 2k˜ ! " q0a = −2 Re{e0 d}, q0b = 2 Re e0 N3 (NT1 − τ I)d + v0 n(0), (10.47a) q∗d q1b ˜ − bN)−1 , = 2(aN , ∗ −1 ∗ q1a qc + Sqd −H q2b ˜ + bN)−1 Re{ie1 (a − ibτ )η}, = −2(aN q2a

ω = 0,

e0 =

where q∗c = −Re{e1 (a + ibτ )c} + q∗b ,

q∗d = −Re{e1 (a + ibτ )c} + q∗a

(10.47b)

and ! " q∗b = −2 Re e0 (a2 − τ 2 b2 )c , + b2 N1 N2 + N2 NT1 q0a − a2 I − b2 (N1 N1 + N2 N3 ) q0b ,

(10.47c)

q∗a = 4Re{abτ e0 c} + 2ab(N2 q0a + N1 q0b ). In (10.47a) k˜ is defined in (10.19b); the superscript < −1 > denotes sub-inverse defined in (5.19) and v0 is a constant determined by substituting (10.47a)3,4 into the following equation:

10.3

Rigid Inclusions

347

! " , , N3 N2 + NT1 NT1 q0a + N3 N1 + NT1 N3 q0b + 2 Re e0 τ 2 d = 0.

(10.47d)

Interfacial Stresses If one is interested in the interfacial stresses along the inclusion boundary, calculation can be performed by using the solutions provided in (10.46)2 and (10.47) and the formulas given in (3.36) for the stress components on s-n coordinate system. The results are (Lin and Hwu, 1993) σnn = nT φ, s , σn3 = (φ, s )3 ,

σss = −sT φ, n , σns = sT φ, s = −nT φ, n , σs3 = −(φ, n )3 ,

(10.48a)

in which ! &" % ρφ, s = H−1 Re e−2iψ ie1 (a + ibτ )γ˜ ∗1 − (S − iI)(q∗a + iq∗b ) , " ! ρφ, n = NT1 (θ )H−1 Re ie1 (a + ibτ )e−2iψ γ˜ ∗1 − 2ρ Re e1 e−iψ γ2 (θ ) ! " − Re e−2iψ NT1 (θ )H−1 S − N3 (θ ) − iNT1 (θ )H−1 (q∗a + iq∗b ) (10.48b) and q∗a and q∗b are given in (10.47c).

10.3.2 Rigid Line Inclusions Under Uniform Heat Flow With the results of elliptical rigid inclusions discussed in the previous section, the problems of insulated rigid line inclusion can now be studied by letting the minor axis 2b of the ellipse approach to zero. With b=0, (10.47) can be reduced to e0 =

−h0 (cos ϕ + τ¯ sin ϕ), ˜ I 4kτ

q0a = −2 Re{e0 d},

q0b

−iah0 sin ϕ, e1 = 2k˜ ! , - " = 2 Re e0 N3 NT1 − τ I d + v0 n(0),

q1b = −2 Re{(2ae0 + e1 )c} − 2aq0b , ! " q1a = 2 Re H−1 [(2ae0 + e1 )S − ie1 I]c + 2aH−1 Sq0b ,

(10.49)

q2b = 2 Re{ie1 (Sc + Hd)}, q2a = 2 Re ie1 (−Lc + ST d) . Differentiating φi with respect to x1 with x2 = 0 and |x1 | > a, the stresses σi2 ahead of the tip of rigid line inclusion along the x1 -axis are obtained as

348

10


⎧ ⎫ ⎛ ⎞ ⎨σ12 ⎬ a a ⎠ q1a σ22 = ⎝ 3 − 3 ⎩ ⎭ x1 + x12 − a2 2 x12 − a2 σ32

(10.50)

a

2a 3 + 3 q2a + Re{e1 d}. 2 x12 − a2 x1 + x12 − a2 Similar to the crack problems, the above solution shows that the stresses are singular near the tip of rigid line inclusion. Hence, like the stress intensity factors for crack problems, we define the strength of thermal stress singularity as ⎧ ⎫ ⎧ ⎫ ⎨ FII ⎬ ⎨σ12 ⎬ + FI = lim 2π (x1 − a) σ22 . ⎩ ⎭ x1 →a ⎩ ⎭ FIII σ32

(10.51)

Using the results obtained in (10.49) and (10.50) and the identities (10.16), the strength of thermal stress singularity for rigid line inclusion can be obtained as ⎧ ⎫ ⎨ FII ⎬ √π a ! " FI = {q2a −q1a } = 2 Re ie1 γ˜ ∗2 − 2ae0 H−1 Sc −2aH−1 Sq0b . (10.52) ⎩ ⎭ 2 FIII Based upon the results of (10.52), the numerical calculation for the strength of thermal stress singularity has been presented in Lin and Hwu (1993) to study the effect of flow direction. Their results show that FII = 0 when the flow is parallel to the line inclusion, i.e., ϕ = 0o ; FI = 0 when the flow is perpendicular to the line inclusion, i.e., ϕ = 90o . Same results have been obtained for the isotropic materials. Opposite phenomena was observed for the mechanical loading, i.e., KI = 0 for the load parallel to line inclusion and KII = 0 for the load perpendicular to the line inclusion. For the other directions, FI decreases and FII increases when ϕ increases.

10.4 Collinear Interface Cracks The collinear interface crack problems have been discussed in Section 7.4 for mechanical loading condition. By using the extended Stroh formalism introduced at the beginning of this chapter, a general thermoelastic collinear interface crack problem will be discussed in this section. The steady state thermoelastic problems of interface cracks between two dissimilar isotropic media have been studied by Erdogan (1965), Barber and Comninou (1982, 1983), Martin-Moran et al. (1983), and Sumi and Ueda (1990). As to the cracks between two dissimilar anisotropic media, solutions were presented by Clements (1983) and Hwu (1992a). Due to the lack of identities among the thermoelastic constants developed in the later years, the solutions provided by Clements (1983) is complicated and without further notification those solutions cannot be applied to the full field domain except the interface. In Section 10.4.1, we follow the work of Hwu (1992a) to consider an arbitrary

10.4


349

number of collinear cracks lying along the interface subjected to an arbitrary and self-equilibrated loading and heat flux on the upper and lower surfaces of the cracks. The materials are assumed to be perfectly bonded at all points except those lying in the region of cracks. With the general solutions presented in Section 10.4.1, two simplest cases subjected to uniform heat flux and loading are discussed detailedly in Section 10.4.2.

10.4.1 General Solutions Consider an arbitrary number of collinear cracks lying along the interface of two dissimilar anisotropic materials. The materials are assumed to be perfectly bonded at all points of the interface x2 = 0 except those lying in the region of cracks L (see Fig. 7.5), which are defined by the intervals given in (7.28). On the upper and lower surfaces of the cracks, an arbitrary and self-equilibrated loading ˆt(x1 ) and heat flux ˆ 1 ) are specified. The continuity of displacement, traction, temperature, and heat h(x flux across the bonded portion of the interface, as well as the prescribed traction and heat flux conditions on the crack portion can be described by u1 = u2 ,

φ1 = φ2 ,

φ1 = φ2 = −ˆt,

T1 = T2 ,

(1) (2) ˆ h2 = h2 = h,

(1)

(2)

h2 = h2 ,

x1 ∈ / L,

x1 ∈ L.

(10.53)

Here and after the subscripts 1 and 2 (if they are not subscripts denoting the components of a vector) or the superscripts (1) and (2) denote, respectively, the quantities pertaining to the materials 1 and 2 which are located on x2 > 0 (S1 ) and x2 < 0 (S2 ). Note that the last second equation of (10.53) has a negative sign before ˆt, which has been explained in the paragraph after (7.60). From the general solutions provided in Section 10.1, i.e., (10.4), (10.6), and (10.10), we know that to find a solution for a problem in two-dimensional anisotropic thermoelasticity, all we need to do is finding the complex functions f(z) and g(zt ), which should satisfy the boundary conditions of that problem. For the problems involving two different materials, a complete solution to the problem requires the knowledge of two complex function vectors f1 (z), f2 (z) and two complex scalar functions g1 (zt ), g2 (zt ). The functions f1 (z), g1 (zt ) and f2 (z), g2 (zt ) are holomorphic in the regions S1 and S2 , respectively. For the present problem, they are sought to satisfy the boundary conditions set in (10.53). For the convenience of later derivation, we now rewrite the general solutions shown in (10.4), (10.6), and (10.10) for the two dissimilar materials discussed in this section, i.e., ! " (i) , Ti = 2 Re gi zt " ! (i) (i) (i) hi = −2 Re k1 + τi k2 gi zt , " ! (i) ui = 2 Re Ai fi (z(i) ) + ci gi zt , ! " (i) φ i = 2 Re Bi fi (z(i) ) + di g zt , i = 1, 2.

(10.54)

350

10


Another form which drops all the subscripts related to zα , α = 1, 2, 3, and zt and is appropriate for the method of analytical continuation is Ti = gi (z) + g i (z), h∗i = −iki gi (z) + iki gi (z),

(10.55)

ui = Ai fi (z) + ci gi (z) + Ai fi (z) + c¯ i gi (z), φi = Bi fi (z) + di gi (z) + Bi fi (z) + di gi (z), i = 1, 2,

in which h∗i is a symbol representing the heat flux in the surface normal direction (i) of material i, i.e., hn given in (10.19a) where ki is a real constant related to the heat conduction coefficients of material i and is defined in (10.19b) with the tilde dropped for simplicity. From (10.53) we see that the traction and heat flux are continuous along the entire interface, i.e., h∗1 = h∗2 , along the entirem interface.

φ 1 = φ2 ,

(10.56)

Note that here h∗i = h2 , i = 1, 2, because the normal direction of the interface coincides with the direction of x2 . Using (10.55)2 , the heat flux continuity condition (10.56)2 leads to (i)

, , , , − ik1 g1 x1+ − ik2 g2 x1− = −ik2 g2 x1− − ik1 g1 x1+ ,

(10.57)

where x1± denote, respectively, the points above and below the interface. With the results of (10.57), by the method of analytical continuation introduced in Section 4.2.1, we can now introduce a function which is holomorphic in the entire domain including the interface, i.e., ∗

g (z) =

−ik1 g1 (z) − ik2 g2 (¯z), −ik2 g2 (z) − ik1 g1 (¯z),

z ∈ S1 , z ∈ S2 .

(10.58)

Since g∗ (z) is holomorphic and single-valued in the whole plane including the points at infinity, by Liouville’s theorem we have g∗ (z) ≡ constant. If the heat flux tends to zero when |z| → ∞, g∗ (z) is then identically zero, i.e., g∗ (z) ≡ 0. If the temperature field also tends to zero as |z| → ∞ and the terms corresponding to the rigid body motion are neglected, (10.58) can further lead to g2 (z) = −

k1 g1 (z), k2

z ∈ S1 ,

and

g1 (z) = −

k2 g2 (z), k1

z ∈ S2 .

(10.59)

Similarly, with the results of (10.59) by using (10.55)4 , the traction continuity (10.56)1 along the entire interface leads to

10.4


351

k1 f2 (z) = B1 f1 (z) + d1 + d2 g1 (z) , z ∈ S1 , k2 k2 −1 B2 f2 (z) + d2 + d1 g2 (z) , z ∈ S2 . f1 (z) = B1 k1 −1 B2

(10.60)

By employing (10.55)1,3 and the results of (10.59) and (10.60), the temperature and displacement continuity along the bonded portion of the interface, shown in (10.53), now provide θ (x1+ ) = θ (x1− ),

ψ(x1+ ) = ψ(x1− ),

when

x1 ∈ / L,

(10.61a)

where ⎧ ⎨ 1 + kk1 g1 (z), z ∈ S1 , 2 θ (z) = ⎩ 1 + k2 g (z), z ∈ S , 2 2 k1 2 d1 + e1 , z ∈ S1 , B1 f1 (z) + k1k+k 2 ψ(z) = ∗ ∗−1 1 M M B2 f2 (z) + k1k+k d2 + e2 , z ∈ S2 , 2

(10.61b)

and M∗ is the bimaterial matrix defined in (7.11); the complex vectors e1 and e2 are related to the thermal eigenvectors c and d and are defined as 1 k1 + k2 1 e2 = k1 + k2

e1 =

−1 iM∗−1 c∗ − iM2 d∗ − k2 d1 , ∗−1 −1 ∗ iM c ∗ − iM2 d − k1 d2 ,

(10.61c)

where c∗ = k2 c1 + k1 c2 ,

d∗ = k2 d1 + k1 d2 .

(10.61d)

Using the results of (10.59), (10.60), and (10.61b), the prescribed traction and heat flux conditions on the crack portion (10.53)2 lead to the following Hilbert problems: θ (x1+ ) + θ (x1− ) = ψ (x1+ ) + M

∗−1

i(k1 + k2 ) ˆ h(x1 ), k1 k2

(10.62)

M∗ ψ (x1− ) = −ˆt(x1 ) + θ (x1+ )e1 + θ (x1− )e2 ,

x1 ∈ L.

The solutions to these Hilbert problems are (Appendix B) θ (z) = ψ (z) =

ˆ ds h(s) k1 +k2 2π k1 k2 χ0 (z) L χ + (s)(s−z) + χ0 (z)pn (z), 0 1 % + &−1 1 −ˆt(s) + θ (s+ )e1 X (z) L s−z X0 (s) 2π i 0

+ θ (s− )e2 ds + X0 (z)pn (z), (10.63a)

352

10


where pn (z) and pn (z) are arbitrary polynomials with the degree not higher than n and χ0 (z) and X0 (z) are the basic Plemelj functions defined as χ0 (z) =

n 1

(z − aj )−1/2 (z − bj )−1/2 ,

X0 (z) = (z),

(10.63b)

j=1

where = [λ2 λ2 λ3 ],

(z) =
.

(10.63c)

j=1

δα and λα , α = 1, 2, 3, of (10.63c) are the eigenvalues and eigenvectors of (7.68) whose explicit solution has been given in (7.69). Note that the order of singularity δα is independent of the heat conduction coefficients kij and thermal moduli βij and is the same as those of the isothermal interface crack problems. The order of singularity related to the heat flux is –1/2 as shown in (10.63b). Once we get the solution of θ (z) and ψ (z) from (10.63), the complex functions g1 (z), g2 (z) and f1 (z), f2 (z) can be obtained from (10.61b) with the understanding that the subscript of z is dropped since the analytical continuation is not affected (i) (i) by different arguments zt or zα . After the operation of matrices, a replacement of (i) (i) (i) (i) zt or z1 , z2 , z3 , i=1, 2, should be made for each function, because the functions g1 (z), g2 (z) and f1 (z), f2 (z) are required to have the form (i) gi (z) = gi zt ,

T (i) (i) (i) fi (z) = f1 z1 f2 z2 f3 z3 ,

i = 1, 2,

(10.64)

which can be seen from the general solution given in (10.54). The whole field solution can then be found by using the translating technique introduced in (4.50) and (4.51). If one is interested in the stresses σi2 along the interface and the crack opening displacements u, the following results show that they have a simple relation with functions ψ(z) and θ (z). By applying σi2 = φi, 1 , (10.55)4 , (10.59), (10.60), and (10.61b), the stresses σi2 along the interface are calculated as ⎧ ⎫ ⎨σ12 ⎬ ∗−1 σ22 = φ1 = φ2 = (I+M M∗ )ψ (x1 )−θ (x1 )(e1 +e2 ), ⎩ ⎭ σ32

x1 ∈ / L.

(10.65)

From (10.55)3 , (10.59), (10.60), and (10.61b), the crack opening displacements u can also be calculated and simplified as , % , , -& , u = u x1 , 0+ − u x1 , 0− = −iM∗ ψ x1+ − ψ x1− ,

x1 ∈ L.

(10.66)

10.4


353

10.4.2 Uniform Heat Flow Cracks in Homogeneous Media The simplest case of the interface cracks is when the two media are composed of the same materials. The results for the thermoelastic interface crack problems should therefore be checked by this simplest case. Consider an infinite homogeneous anisotropic plate containing an insulated crack of which the heat is flowing uniformly in the direction of the positive x2 -axis (which has been discussed in Section 10.2.2 by another approach). Due to the linear property, the principle of superposition can be used and the problem can be represented as the sum of a uniform heat flux in an uncracked solid and a corrective problem which is described by ˆ 1 ) = −h0 = constant, h(x A1 = A2 = A,

ˆt(x1 ) = 0,

B1 = B2 = B,

n = 1,

a1 = −a,

c1 = c2 = c,

b1 = a,

d1 = d2 = d,

˜ k1 = k2 = k. (10.67)

To find the solution for this corrective problem, the line integral of (10.63a)1 should be evaluated first. Through the aid of residue theory, we have (Appendix B.3)

h0 ds + χ (s)(s − z) L 0

+ = −iπ h0 z − z2 − a .

(10.68a)

After evaluating the line integral, the polynomial pn (z) of (10.63a) with n = 1, i.e., p1 (z) = c0 + c1 z,

(10.68b)

could be determined by the infinity condition and the single-valuedness requirement. If θ (z) → 0 as |z| → ∞ , we have c1 = 0. The requirement of single-valuedness condition can be expressed by

a

−a

% , + , -& θ x1 − θ x1− dx1 = 0.

(10.68c)

With the function θ (z) shown in (10.63a)1 and its related3results obtained between √ (10.68a) and (10.68c), and knowing that z2 − a2 = ±i a2 − x12 for |x1 | < a and x2 = ±0, the requirement stated in (10.68c) gives us c0 = 0. Combining all these results into (10.63a)1 , the final simplified solution for θ (z) is ih0 θ (z) = − k˜

1− √

z z2 − a2

.

(10.69)

To find the solution for ψ (z) from (10.63a)2 , we first calculate the terms related to θ (z). Integrating θ (z) obtained in (10.69) with the assumption that the temperature field tends to zero as |z| → ∞ , and substituting the identities (3.132) and (10.16a)

354

10


into (10.61c) with the condition of homogeneous media (10.67)2 , we have θ (s+ )e1 + θ (s− )e2 = −

h0 s ∗ Re γ˜ 2 . k˜

(10.70)

The evaluation of the line integral and the determination of the polynomial pn (z) are similar to those described in (10.68) and (10.69). The result is ψ (z) = −

∗ 2z2 − a2 h0 Re / γ2 . 2z − √ 4k˜ z2 − a2

(10.71)

With the specialized condition given in (10.67) and the identity (10.16a)2 , the solutions of the complex functions g(z) and f(z) can be obtained from (10.61b) and be simplified as 1 θ (z), 2

g(z) =

f(z) =

% , -& 1 −1 B 2ψ(z) − θ (z) d − iRe γ˜ ∗2 . 2

(10.72)

Note again that the subscript of z is dropped before the multiplication of matrices and a replacement of zt or zα should be made for each component function after the matrix product. By this calculation procedure, the explicit expressions for the complex functions g(z) and f(z) can be obtained from (10.69), (10.71), and (10.72) as g(z) =

− ih˜0 4k

f(z) =

ih0 4k˜

+ ih0ã 4k

2

3 3 2 2 2 2 2 2 zt − zt zt − a + a ln zt + zt − a ,

+ < z2α − zα z2α − a2 > B−1 d + % , -& < ln zα + zα − a2 > B−1 d − iRe γ˜ ∗2 ,

(10.73)

which can be proved to be identical to those shown in Section 10.2.2, i.e., a combination of (10.22), (10.23), (10.24), (10.27), and (10.37), through different approach. The whole field solutions for the temperature, heat flux, displacements, and stresses can then be found by using (10.54). The stresses σi2 ahead of the crack tip along the x1 -axis can be calculated by (10.65) with θ (z) and ψ(z) given in (10.69) and (10.71). Similarly, the crack opening displacements u can be obtained from (10.66) and (10.71). All these results together with the stress intensity factors and energy release rate have been proved to be exactly the same as those shown in (10.38), (10.39), (10.40), and (10.41) of Section 10.2.2. Interface Cracks Lying Between Two Dissimilar Materials Consider an interface crack located on a1 = −a, b1 = a subjected to uniform heat ˆ 1 ) = h0 and uniform loading ˆt(x1 ) = t0 . To find the solution of θ (z) from flux h(x (10.63a), similar approach as previous example for cracks in homogeneous media can be employed and the result is

10.4


355

+ θ (z) = −ih∗0 z − z2 − a2 ,

where h∗0 =

h0 (k1 + k2 ) . 2k1 k2

(10.74)

With the aid of residue theory, the following line integrals which are useful for the calculation of ψ (z) can be obtained (see Appendix B.4): 1 2π i L 1 2π i L 1 2π i L

1 + −1 1 − (z + 2iεα a) > t∗0 , t0 ds =< X0 (s) s−z χα (z) ( ' s + −1 z a2 ek ds =< > e∗k , X0 (s) − z(z + 2iεα a) − 1 + 4εα2 s−z χα (z) 2 + + i a2 − s2 + −1 z2 − a2 ek ds =< X0 (s) − z(z + 2iεα a) − a2 1 + 2εα2 > e∗k , s−z χα (z) (10.75a)

where t∗0 = −1 (I + M

∗−1

M∗ )−1 t0 , e∗k = −1 (I + M z − a iεα 1 χα (z) = √ . z2 − a2 z + a

∗−1

M∗ )−1 ek ,

k = 1, 2,

(10.75b) Applying the results of (10.74) and (10.75), the polynomial pn (z) in (10.63a)2 can now be determined by the ainfinity % , condition, , ψ (z) & → 0 as |z| → ∞, and the single+ − valuedness requirement −a ψ x1 − ψ x1 ) dx1 = 0 . The result is ! " pn (z) = −ih∗0 a2 e∗1 + < 16 iaεα z − a2 1 + 16εα2 > e∗2 /4.

(10.76)

In the derivation of (10.76) the following integrals calculated by the residue theory have been used: a

−a

a

−a

a

−a

√

1 a2 −t2

√

t

a2 −t2

√t

2

a2 −t2

a−t a+t a−t a+t a−t a+t

iεα iεα iεα

dt =

π cosh π εα ,

dt =

−2iπ aεα cosh π εα ,

dt =

π a2 2 cosh π εα .

(10.77)

Combining the results of (10.74), (10.75), and (10.76), the final simplified solution for ψ (z) of (10.63a)2 is obtained as , ψ (z) = − J0 (z)t∗0 + ih∗0 J1 (z)e∗1 + J2 (z)e∗2 , where

(10.78a)

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10


J0 (z) =< 1 − (z + 2iaεα )χα (z) >, + 1 J1 (z) =< z − z2 − a2 − a2 χα (z) >, 4 + 5 2 2 2 2 J2 (z) =< z + z − a − 2z − a χα (z) > . 4

(10.78b)

The solutions for the isothermal interface crack problems can be found by letting h∗0 = 0, which is identical to those presented in Section 7.4.3. The fracture parameters such as stress intensity factors, crack opening displacements, and energy release rate can then be obtained from this solution and the proper definition for the interface crack problems.

10.5 Multi-material Wedges The multi-material wedge problems have been discussed in Chapter 5 for mechanˆ that includes the information ical loading conditions. In that study a key matrix N of material properties and wedge angles is introduced to express the closed-form solutions for the orders of stress singularity and their associated near-tip solutions. ˆ to extend the disIn this section, we like to take advantage of the key matrix N cussions to another important topic – thermal effects on the singular behavior of multi-material wedges.

10.5.1 Stress and Heat Flux Singularity General Formulation Like the discussions in Section 5.3, to consider the stress and heat flux singularities at the wedge apex, the unknown functions f(z) and g(zt ) in the general solution (10.4), (10.6), and (10.10) are assumed as > p, f(z) =< z1−δ α

g(zt ) = qz1−δ , t

(10.79)

where δ is the singular order to be determined from the boundary conditions and p and q are the associated coefficients. Since the stress is proportional to the first derivative of the stress function, if Re(δ) > 0 the stress at the wedge apex will be singular. However, when Re(δ) > 1 the strain energy of the elastic wedge may become unbounded. Hence, when one is concerned about the stress singularity, it is usual to consider the region 0 < Re(δ) < 1.

(10.80a)

However, it should be noticed that in this region, not only the stresses will be singular but also the temperature and heat flux will be singular. Since the singularity

10.5

Multi-material Wedges

357

of the temperature field is generally not permissible, when we discuss the heat flux singularity we will restrict our region to − 1 < Re(δ) < 0.

(10.80b)

Substituting (10.79) into (10.4), (10.6), and (10.10), and considering that the singular order δ may come in pair if they are complex (Hwu et al., 2003), the temperature, heat flux, displacement, and stress function vectors of each wedge can be expressed as " ! −δ , T = (1 − δ) q1 z−δ + q z ¯ 2 t t " ! h = δ(1 − δ) q1 (k1 + τ k2 )zt−1−δ + q2 (k1 + τ¯ k2 )¯zt−1−δ ,

(10.81)

u = A < z1−δ > p1 + A < z¯1−δ > p2 + q1 cz1−δ + q2 c¯ z¯1−δ , t t α α ¯ z1−δ φ = B < z1−δ > p1 + B < z¯1−δ > p2 + q1 dz1−δ + q2 d¯ , t t α α where p1 , p2 , q1 , and q2 are complex coefficients to be determined through the satisfaction of the boundary conditions. If the singular order δ is a real value, q1 and q2 should be complex conjugate to keep the temperature and heat flux to be real, and so are p1 and p2 to keep the displacements and stresses to be real. Generally, they are not necessary to be complex conjugate. The final real values of the temperature, heat flux, displacements, and stresses will come from the ¯ superposition with another set of solution whose singular order is δ. For the description of boundary conditions of the wedge problems it is better to use the polar coordinate system (r, θ ) where the origin is located at the wedge apex and x1 = r cos θ ,

x2 = r sin θ .

(10.82)

Substituting (10.82) into (10.4)2 and (10.7)2 and using the notations defined in ˆ ) = cos θ + μ sin θ , τˆ (θ ) = cos θ + τ sin θ , we have (3.88)1 , i.e., μ(θ zα = rμˆ α (θ ),

zt = rτˆ (θ ),

α = 1, 2, 3.

(10.83)

Substituting (10.83) into (10.81), we have ! " −δ T = (1 − δ)r−δ q1 τˆ −δ (θ) + q2 τˆ (θ) , ! " −1−δ (θ) , h = δ(1 − δ)r−1−δ q1 (k1 + τ k2 )τˆ −1−δ (θ) + q2 (k1 + τ k2 )τˆ ! " 1−δ 1−δ ˆ α (θ) > p2 + q1 cτˆ 1−δ (θ) + q2 cτˆ (θ) , u = r1−δ A < μˆ 1−δ α (θ) > p1 + A < μ ! " 1−δ 1−δ ˆ α (θ) > p2 + q1 dτˆ 1−δ (θ) + q2 d τˆ (θ) . φ = r1−δ B < μˆ 1−δ α (θ) > p1 + B < μ (10.84)

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10


To consider the continuity of the heat flux, only the heat flux h∗ in the direction normal to the interface is required to be continuous. Therefore, it is more convenient to use h∗ instead of h to state the general solutions. If the interface is located on a radial line with angle θ measured counterclockwise from x1 -axis, the heat flux h∗ normal to the interface can be calculated from the vector coordinate transformation as that shown in (10.19). Substituting (10.84)2 into (10.19) and using the characteristic equation for τ given in (10.5), we get ! ˜ − δ)r−1−δ q1 τˆ −δ (θ ) − q2 τˆ h∗ = ikδ(1

−δ

" (θ ) ,

(10.85)

where k˜ is a real constant related to the heat conduction coefficients defined in (10.19b). Observing (10.84) from the viewpoint of singular function, we see that u and φ are in the same order, whereas T and h are not in the same order. To have a function related to the temperature T and consistent with h∗ , we consider ∂T/∂r. From (10.84)1 , we have ! " −δ T, r = −δ(1 − δ)r−1−δ q1 τˆ −δ (θ ) + q2 τ¯ˆ (θ ) .

(10.86)

By introducing two new vectors v and w as T v = ∗, r , h

u w= , φ

(10.87)

the four equations (10.84)3,4 , (10.85), and (10.86) can now be written as ( q1 , v = −δ(1 − δ)r−1−δ −δ ¯ 0 τˆ (θ ) q2 ( ' 1−δ (θ ) > 0 < μ ˆ p1 A A α w = r1−δ 1−δ ¯ B B 0 < μˆ α (θ ) > p2 ( ' 1−δ τ ˆ (θ ) 0 c c q1 . +r1−δ 1−δ ¯ d d 0 τˆ (θ ) q2

1 −ik˜

1 ik˜

'

τˆ −δ (θ )

0

(10.88)

ˆ if From (5.77), we see that (10.88)2 can be written in terms of the key matrix N we let ' T T( B A p1 = T T p, p2 B A where p is a 6 × 1 complex vector. In addition, we let

(10.89)

10.5


359

c c¯ U= , d d¯

1 1 = , −ik˜ ik˜

q q= 1 . q2

(10.90)

By using (5.77), (10.89), and (10.90), (10.88) can now be written in a compact matrix form as v = −δ(1 − δ)r−1−δ < τˆα−δ (θ ) > q, ! " ˆ 1−δ (θ )p + U < τˆα1−δ (θ ) > q . w = r1−δ N

(10.91)

Note that in (10.91) the diagonal matrix related to τˆα is a 2 × 2 diagonal matrix with τ1 = τ and τ2 = τ¯ , which will be used throughout this section. Interface Continuity Conditions Consider the multi-material wedges as shown in Fig. 5.3a,b. The temperature, heat flux, traction, and displacement continuity across each interface θ = θk , k = 1, 2, . . . , n − 1, between two dissimilar wedges can be written as Tk (θk ) = Tk+1 (θk ),

h∗k (θk ) = h∗k+1 (θk ),

uk (θk ) = uk+1 (θk ),

tk (θk ) = tk+1 (θk ),

k = 1, 2, . . . , n − 1,

(10.92)

where the subscript k is used to denote the quantities pertaining to the kth wedge. Along each interface only r varies and θ = constant. Therefore, from (3.32) we see that the traction continuity conditions can be replaced by the stress function continuity, i.e., φk (θk ) = φk+1 (θk ). Moreover, the temperature continuity will lead to ∂Tk (θk )/∂r = ∂Tk+1 (θk )/∂r. With this understanding and using the two new vectors w and v introduced in (10.87), the boundary conditions (10.92) can be replaced by vk (θk ) = vk+1 (θk ),

wk (θk ) = wk+1 (θk ),

k = 1, 2, . . . , n − 1.

(10.93)

The temperature, heat flux, displacement, and stress in each wedge can be expressed by the equations shown in (10.91). The only unknowns in (10.91) are the coefficient vectors p, q and the singular order δ , which should then be determined through the satisfaction of the continuity conditions given in (10.93). To solve this problem, we first consider the two boundaries of the kth wedge; from (10.91) we have vk (θk−1 ) = −δ(1 − δ)r−1−δ k < τˆα−δ (θk−1 ) >k qk , vk (θk ) = −δ(1 − δ)r−1−δ k < τˆα−δ (θk ) >k qk , ! " (10.94) ˆ 1−δ (θk−1 )pk + Uk < τˆα1−δ (θk−1 ) >k qk , wk (θk−1 ) = r1−δ N k ! " ˆ 1−δ (θk )pk + Uk < τˆα1−δ (θk ) >k qk , k = 1, 2, . . . , n. wk (θk ) = r1−δ N k In order to get a relation between the two boundaries of each wedge, we insert the result of qk obtained from (10.94)1 into (10.94)2 , which leads to vk (θk ) = k vk (θk−1 ),

k = 1, 2, . . . , n,

(10.95a)

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10


where k = k < τˆα−δ (θk , θk−1 ) >k −1 k

(10.95b)

and τˆα (θk , θk−1 ) = cos(θk − θk−1 ) + sin(θk − θk−1 )τα (θk−1 ).

(10.95c)

Similar to the relation for the temperature/heat flux obtained in (10.95), the relation for the displacement/stress function can be also obtained by using (10.94). From (10.94)3 , we can express pk in terms of wk (θk−1 ) and qk . This result together with the insertion of qk obtained from (10.94)1 into (10.94)4 gives wk (θk ) = Ek wk (θk−1 ) + r2 Fk vk (θk−1 ),

k = 1, 2, . . . , n,

(10.96a)

where ˆ 1−δ (θk , θk−1 ), Ek = N k 1 Fk = δ(1−δ) Ek Uk − Uk < τˆα1−δ (θk , θk−1 ) >k < τˆα (θk−1 ) >k −1 k .

(10.96b)

Note that Ek defined in (10.96b) is the transfer matrix introduced by Ting (1997), and its submatrices can be obtained from the identity (5.77). With the relations obtained in (10.95) and (10.96) for the two boundaries of each wedge, and the continuity conditions required in (10.93), it is possible for us to get the equations stating the relations between the two outer boundaries of the multi-material wedges. To get this relation, by repeated application of (10.95a) and (10.93)1 we have vn (θn ) = KT v1 (θ0 ),

(10.97a)

where KT =

n 1

n−k+1 = n n−1 . . . 1 .

(10.97b)

k=1

Similarly but more complicatedly, by setting k=n for (10.96a) as the first equation of the next iteration and repeated using (10.93), (10.95a), and (10.96a), we can obtain wn (θn ) = Ke w1 (θ0 ) + r2 Kc v1 (θ0 ),

(10.98a)

where Ke =

n 1 k=1

En−k+1 = En En−1 · · · E1 ,

(10.98b)

10.5


Kc =

n

#

j=1

n−1 ;

361

$

#

En−k+j Fj

k=j

j−1 ;

$ j−i

= En En−1 · · · E2 F1 + En En−1 · · · E3 F2 1

i=1

+En En−1 · · · E4 F3 2 1 + · · · + En Fn−1 n−2 · · · 2 1 + Fn n−1 n−2 · · · 2 1 . (10.98c)

Note in (10.98c), the results after the multiplication operator is enforced to be unity if the starting index is larger than the end index, for example, 0 1

(· · · ) =

n−1 1

i=1

(· · · ) = I.

(10.98d)

k=n

Combining the results of (10.97a) and (10.98a), we get K 0 vn (θn ) v1 (θ0 ) = 2T . wn (θn ) r Kc Ke w1 (θ0 )

(10.99)

For the purpose of the following discussions, we divide the 2 × 2 matrix KT , 6 × 2 matrix Kc , and 6 × 6 matrix Ke as '

( (1) (2) KT KT KT = (3) (4) , KT KT (i)

(i)

'

( (1) (2) kc kc Kc = (3) (4) , kc kc

' ( (1) (2) Ke Ke Ke = (3) (4) , Ke Ke

(10.100)

(i)

in which KT , kc , and Ke , i = 1, 2, 3, 4, are, respectively, scalars, 3 × 1 vectors, and 3 × 3 matrices. By using the submatrices defined in (10.100) and (10.87), (10.99) can be rewritten as Tn,r (θn ) = KT T1,r (θ0 ) + KT h∗1 (θ0 ), (1)

(2)

h∗n (θn ) = KT T1,r (θ0 ) + KT h∗1 (θ0 ), (3)

(4)

(2) 2 (1) 2 (1) ∗ un (θn ) = K(1) e u1 (θ0 ) + Ke φ1 (θ0 ) + r kc T1,r (θ0 ) + r kc h1 (θ0 ),

(10.101)

(4) 2 (3) 2 (4) ∗ φn (θn ) = K(3) e u1 (θ0 ) + Ke φ1 (θ0 ) + r kc T1,r (θ0 ) + r kc h1 (θ0 ).

Multi-material Wedge Space For a multi-material wedge space, the entire space is filled up with wedges and the nth wedge is bonded together with the first wedge, i.e., θn = θ0

and

v1 (θ0 ) = vn (θn ),

w1 (θ0 ) = wn (θn ).

Substituting (10.99) into (10.102), we obtain KT − I 0 v1 (θ0 ) = 0. r2 Kc Ke − I w1 (θ0 )

(10.102)

(10.103)

Nontrivial solutions for v1 and w1 exist only when KT − I = 0 or

Ke − I = 0.

(10.104)

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10


Either of the roots of the determinant obtained from (10.104) provides the stress singularities δ. The former is related to the heat conduction coefficients, whereas the latter is related to the elastic constants and is identical to that obtained in (5.85) for the cases without considering the thermal effects. It should also be noted that the results are independent of the thermal moduli βij . Moreover, consideration of the thermal effects will not change the original singular orders obtained from the elastic constants but will add additional singular orders to the multi-material wedge space. However, if we restrict our consideration to the region that −1 < Re{δ} < 0, i.e., the temperature field should not be singular, then from (10.81) we see that no stress singularity will occur. In other words, under the assumption of uncoupled static thermoelasticity, consideration of the thermal effects (e.g., the mismatch of thermal properties of multi-material wedges) will not influence the stress singularity but will induce the heat flux singularity. This is reasonable and expectable for the uncoupled static thermoelasticity considered in this chapter. Multi-material Wedges We now consider a multi-material wedge for which the surfaces at θ = θ0 and θn are insulated or isothermal, traction free, or fixed. Case I: The Insulated and Free–Free Wedge. When both of the surfaces at θ = θ0 and θn are insulated and traction-free, the boundary conditions can be expressed as h∗1 (θ0 ) = h∗n (θn ) = 0,

φ1 (θ0 ) = φn (θn ) = 0.

(10.105)

Substituting (10.101) into (10.105), we get (1)

Tn,r (θn ) = KT T1,r (θ0 ), (3)

0 = KT T1,r (θ0 ), 2 (1) un (θn ) = K(1) e u1 (θ0 ) + r kc T1,r (θ0 ),

(10.106)

2 (3) 0 = K(3) e u1 (θ0 ) + r kc T1,r (θ0 ).

Nontrivial solutions for T1,r (θ0 ) and u1 (θ0 ) exist only when ) ) ) ) (3) KT = 0 or )K(3) e ) = 0,

(10.107)

which will provide the singular orders. Same as the multi-material wedge space, the former roots are related to the heat conduction coefficients, whereas the latter roots are related to the elastic constants and are identical to those obtained in (5.90) for the cases without considering the thermal effects. Again, these results are independent of the thermal moduli βij . Case II: The Insulated and Fixed–Fixed Wedge When both of the surfaces at θ = θ0 and θn are insulated and fixed, the boundary conditions can be expressed as h∗1 (θ0 ) = h∗n (θn ) = 0,

u1 (θ0 ) = un (θn ) = 0.

(10.108)

10.5


363

Substituting (10.101) into (10.108) and following a similar argument, we can get the singular order from (3)

KT = 0

or

) ) ) (2) ) )Ke ) = 0.

(10.109)

Case III: The Insulated and Free–Fixed Wedge For this case, the boundary conditions can be expressed as h∗1 (θ0 ) = h∗n (θn ) = 0,

φ1 (θ0 ) = un (θn ) = 0.

(10.110)

The results are (3)

KT = 0

or

) ) ) (1) ) )Ke ) = 0.

(10.111)

Case IV: The Insulated and Fixed–Free Wedge For this case, the boundary conditions can be expressed as h∗1 (θ0 ) = h∗n (θn ) = 0,

u1 (θ0 ) = φn (θn ) = 0.

(10.112)

The results are (3)

KT = 0

or

) ) ) (4) ) )Ke ) = 0.

(10.113)

Case V: The Isothermal Wedge When the surfaces at θ = θ0 and θn are isothermal and free–free or fixed–fixed, free–fixed, or fixed–free, their boundary conditions and corresponding results are ) ) ) ) (2) free-free : T1 (θ0 ) = Tn (θn ) = 0, φ1 (θ0 ) = φn (θn ) = 0; δ : KT = 0 or )K(3) e ) = 0, ) ) ) ) (2) fixed-fixed : T1 (θ0 ) = Tn (θn ) = 0, u1 (θ0 ) = un (θn ) = 0; δ : KT = 0 or )K(2) e ) = 0, ) ) ) ) (2) free-fixed : T1 (θ0 ) = Tn (θn ) = 0, φ1 (θ0 ) = un (θn ) = 0; δ : KT = 0 or )K(1) e ) = 0, ) ) ) ) (2) fixed-free : T1 (θ0 ) = Tn (θn ) = 0, u1 (θ0 ) = φn (θn ) = 0; δ : KT = 0 or )K(4) e ) = 0. (10.114)

Case VI: One Insulated and One Isothermal Wedge. When one of the surfaces at θ = θ0 and θn is insulated and the other is isothermal, no matter they are free–free or fixed–fixed or free–fixed or fixed–free, the singular orders for the stresses will still be the same those obtained in cases I–V, while the thermal singularities can be found from (1)

KT = 0, (4) KT = 0,

for insulated-isothermal wedges, for isothermal-insulated wedges.

(10.115)

364

10


10.5.2 Near-Tip Solutions The general solutions for the temperature, heat flux, displacement, and stress in the field near the wedge apex can be obtained from (10.91) for each different wedge. Through the use of (10.94)1 and (10.94)3 , the coefficient vectors pk and qk can be expressed in terms of vk (θk−1 ) and wk (θk−1 ). Similar to (10.97) and (10.98), repeated application of (10.93), (10.95), and (10.96) may help us to express vk (θk−1 ) and wk (θk−1 ) in terms of v1 (θ0 ) and w1 (θ0 ). Thus, the field solutions (10.91) near the wedge tip can be written as ∗ 0 (KT )k−1 (θ ) vk (r, θ ) = 2 k∗ wk (r, θ ) r Fk (θ ) E∗k (θ ) r2 (Kc )k−1 ∗ 0 1 (θ ) v1 (θ0 ) , = 2 ∗ r F1 (θ ) E∗1 (θ ) w1 (θ0 )

0 (Ke )k−1

v1 (θ0 ) , k = 2, 3, . . . , n, w1 (θ0 )

k = 1, (10.116a)

where

∗k (θ ) = k < τˆα−δ (θ , θk−1 ) >k −1 k , ˆ 1−δ (θ , θk−1 ), E∗k (θ ) = N k ! " 1 ˆ 1−δ (θ , θk−1 )Uk − Uk < τˆα1−δ (θ , θk−1 ) >k < τˆα (θk−1 ) >k −1 F∗k (θ ) = δ(1−δ) N k k (10.116b) and (KT )k−1 =

k−1 ;

k−i = k−1 k−2 . . . 1 ,

i=1

(Ke )k−1 =

k−1 ; i=1

(Kc )k−1 =

k−1 j=1

Ek−i = Ek−1 Ek−2 . . . E1 , #

k−2 ; l=j

$ Ek−1−l+j Fj

#

j−1 ;

$ j−i

= Ek−1 Ek−2 · · · E2 F1

i=1

+Ek−1 Ek−2 · · · E3 F2 1 + Ek−1 Ek−2 · · · E4 F3 2 1 + · · · · · · +Ek−1 Fk−2 k−3 · · · 2 1 + Fk−1 k−2 k−3 · · · 2 1 . (10.116c) Note that from the definitions given in (10.116b), (10.116c), (10.95b), (10.96b), (10.97b), and (10.98b,c), we see that ∗k (θk ) = k , E∗k (θk ) = Ek , F∗k (θk ) = Fk , (KT )n = KT , (Ke )n = Ke , (Kc )n = Kc .

(10.117)

10.5


365

It can be proved that if the thermal effects are ignored the solutions shown in (10.116) can be reduced to those shown in (5.97) for the pure mechanical loading conditions. By solving the eigenvalue problems presented in (10.103) for the multi-material wedge space, or in (10.106) for the insulated free–free multi-material wedges, or in (10.108), (10.109), (10.110), (10.111), (10.112), (10.113), (10.114), and (10.115) for the other boundary conditions, the shapes of v1 (θ0 ) and w1 (θ0 ) can be obtained as the eigenvectors of these problems. It should be noted that the expressions shown in (10.116) are the solutions corresponding to the singular order δ. If the singular order is a complex number, a conjugate part should be superimposed to get the real values for the temperature, heat flux, displacement, and stress. If the singular order is a repeated root and no enough independent eigensolutions have been obtained, the logarithmic singularity should be considered (Ting and Yan, 1992). Moreover, if one concerns not only the near-tip field but also the entire domain of the multi-material wedges, the solutions corresponding to all the singular and non-singular orders should be superimposed.

10.5.3 Special Cases The explicit closed-form solutions for the orders of stress and heat flux singularities near the wedge apex have been obtained in simple compact matrix forms in (10.104) for multi-material wedge space and in (10.107), (10.109), (10.111), (10.113), (10.114), and (10.115) for the multi-material wedges with different boundary conditions. Their associated near-tip field solutions for the temperature, heat flux, displacement, and stress distribution have also been presented in (10.116). As we discussed in Section 10.5.1, these solutions show that if we restrict our consideration of singular orders to the region that −1 < Re{δ} < 0, i.e., the temperature field should not be singular, the mismatch of the thermal properties of multi-material wedges will not influence the stress singularity but will induce the heat flux singularity. Because the conclusion that the thermal properties will not influence the stress singularity has been observed and the cases without considering the thermal effects have been discussed in Section 5.3, in this section the studies will be focused on the heat flux singularities. Because the solutions presented in this section are not only simple but also general in the sense of thermal effect, wedge number, wedge angle, and material properties, now we like to consider some special cases that are usually discussed in the literature. From the solutions obtained in Section 10.5.2, we know that the singularities induced by the thermal properties of wedges are dominated by the matrix KT defined in (10.97b) which is then determined by k of each wedge. From (10.95b) and (10.90)2 , we have τ˘ −τ˘I /k˜ k = ˜ R , kτ˘I τ˘R k

(10.118a)

366

10


where ! " (τ˘R )k = Re τˆk−δ (θk , θk−1 ) ,

! " (τ˘I )k = Im τˆk−δ (θk , θk−1 ) .

(10.118b)

A Single Wedge (Including Crack Problems) If only a single wedge is considered, the multi-material wedge space treated in the previous section is a trivial problem because it is just a homogeneous space. No singularity will occur for this trivial condition, and the solution shown in (10.104) will be satisfied automatically. Substituting (10.118) and (10.97b) into the solutions shown between (10.107) and (10.115), we can obtain the singular orders from (3) KT = k˜ τ˘I = 0, (2) KT = −τ˘I /k˜ = 0, (1) (4) KT = KT = τ˘R = 0,

for the insulated wedge boundaries, for the isothermal wedge boundaries, for one insulated and one isothermal wedge boundaries. (10.119)

A semi-infinite crack in a homogeneous anisotropic medium can be represented by letting θ0 = −π and θ1 = π for the single wedge. With this special wedge angle, τˆ can be calculated as τˆ −δ (π , −π ) = e−2iπ δ , and hence the solutions (10.119), in the range −1 < Re{δ} < 0, will lead to the following results: δ = 0, −1/2, δ = 0, −1/2, δ = −1/4, −3/4,

for the insulated crack surfaces, for the isothermal crack surfaces, for one insulated and one isothermal crack surfaces. (10.120)

Bi-wedges (Including Interfacial Crack Problems) Consider a bi-wedge bonded together by two dissimilar wedges. Substituting n=2 into (10.97b) and using (10.118) we get KT = 2 1 '1 ˜ ˜ ˜ {k2 (τ˘R )2 (τ˘R )1 − k1 (τ˘I )2 (τ˘I )1 } = ˜k2 k2 (τ˘I )2 (τ˘R )1 + k˜ 1 (τ˘R )2 (τ˘I )1

−

1 ˜ {k2 (τ˘R )2 (τ˘I )1 k˜ 1 k˜ 2

1 ˜ {k1 (τ˘R )2 (τ˘R )1 k˜ 1

+ k˜ 1 (τ˘I )2 (τ˘R )1 }

− k˜ 2 (τ˘I )2 (τ˘I )1 }

( .

(10.121)

By using (10.121), the solutions shown between (10.107) and (10.115) for the singular orders can then be obtained explicitly. If we set θ0 = −π , θ1 = 0, and θ2 = π , the bi-wedge can represent the interface crack problem. With these special angles, τˆ −δ (0, −π ) = τˆ −δ (π , 0) = e−iπ δ , and hence the solutions (10.121), in the range −1 < Re{δ} < 0, will lead to the following results:

10.5


367

δ = 0, −1/2,

for the insulated interfacial crack surfaces,

δ = 0, −1/2,

for the isothermal interfacial crack surfaces,

k˜ 2 cos2 δπ − k˜ 1 sin2 δπ = 0, for lower surface insulated and upper surface isothermal, k˜ 1 cos2 δπ − k˜ 2 sin2 δπ = 0, for lower surface isothermal and upper surface insulated. (10.122)

To know the influence of thermal properties and wedge angles on the heat flux singularity, several numerical examples have been illustrated in Hwu and Lee (2004) such as a single wedge, bi-wedge, tri-wedge, and fully bonded junction and disbonded junction. Most of the results show that the influence of material properties and wedge angles on the heat flux singularity bears the same trends as that on the stress singularity shown in Section 5.3. Whereas consideration of boundary conditions shows that higher values of singular orders are usually obtained for the wedges with one insulated and one isothermal wedge boundaries.

Chapter 11

Piezoelectric Materials

Due to the rapid development of intelligent space structure and mechanical system, advanced structures with integrated self-monitoring and control capabilities are increasingly becoming important. It is well known that piezoelectric materials produce an electric field when deformed and undergo deformation when subjected to an electric field. Because of this intrinsic coupling phenomenon, piezoelectric materials are widely used as the sensors and actuators in intelligent advanced structure design. When subjected to mechanical and electric stresses in service, these piezoelectric materials can fail due to defects such as cracks and holes arisen during their manufacturing process. Therefore, it is interesting to study the electromechanical behavior for the piezoelectric materials with defects. To study their electromechanical behaviors, suitable mathematical modeling becomes important. Since the expanded Stroh formalism for piezoelectric materials preserves most essential features of Stroh formalism, it becomes a popular tool for the study of piezoelectric anisotropic elasticity. Most of the analytical solutions presented in the literature such as Barnett and Lothe (1975b), Pak (1990), Sosa (1991), Kuo and Barnett (1991), Suo et al. (1992), Park and Sun (1995), and Liang and Hwu (1996) show that the solutions for the problems of piezoelectric anisotropic materials can be purposely organized to have the same mathematical forms as those of the corresponding anisotropic elastic materials. This observation tells us the importance of getting the corresponding explicit expressions of the fundamental matrix N, the material eigenvector matrices A and B, the Barnett–Lothe tensors L, S, and H, and the bimaterial matrices D and W for piezoelectric materials, which has been discussed by Suo et al. (1992), Soh et al. (2001), and Hwu (2008). With the above understanding, the following contents based upon the works of Hwu (2008) and Hwu and Ikeda (2008) will be discussed in this chapter. In Section 11.1, we discuss the general three-dimensional constitutive laws for piezoelectric materials and further reduce the laws to two-dimensional state. By using the reduced two-dimensional constitutive laws, the expanded Stroh formalism for two-dimensional piezoelectric materials will be introduced in Section 11.2. And some explicit expressions which are useful for detailing the matrix form solutions into component form solutions will be introduced in Section 11.3. With this background, several important topics will be discussed in the following sections such as


369

370

11


multi-material wedges in Section 11.4, singular characteristics of cracks in Section 11.5, and some cracks/interface crack problems in Section 11.6.

11.1 Constitutive Laws 11.1.1 Three-Dimensional State For an anisotropic and linearly electro-elastic solid, the constitutive relation between elastic field tensors (stresses σij and strains εij ) and electric field vectors (electric displacements or called induction Dj and electric field Ej ) can be represented by four equally important systems of piezoeffect equations. In tensor notation, they can be written as (Rogacheva, 1994)

E ε −e E , σij = Cijkl kl kij k εE , Dj = ejkl εkl + ωjk k D ε −h D , σij = Cijkl kl kij k Ej = −hjkl εkl + βjkε Dk ,

E σ +d E , εij = Sijkl kl kij k σE , Dj = djkl σkl + ωjk k D εij = Sijkl σkl + gkij Dk , Ej = −gjkl σkl + βjkσ Dk ,

i, j, k, l = 1, 2, 3,

(11.1) E and SD are elastic compliances at constant electric field and inducwhere Sijkl ijkl E and CD are elastic stiffnesses at constant electric field and induction, tion, Cijkl ijkl ε , ωσ and β ε , β σ are dielectric permittivities and non-permittivities at conωjk jk jk jk stant strains and stresses, and dkij , ekij , gkij , and hkij are piezoelectric strain/charge, stress/charge, strain/voltage, stress/voltage tensors, respectively. The SI units used E , CD : Nt/m2 , D , e : Coul/m2 or Nt/(m-volt), for the above symbols are σij , Cijkl j kij ijkl ε σ E , SD : m2 /Nt, d : Ek , hijk : Nt/Coul or volt/m, ωjk , ωjk : Coul2 /(m2 Nt) or Nt/volt2 , Sijkl ijk ijkl Coul/Nt or m/volt, gijk : m2 /Coul or m-volt/Nt, and βjkε , βjkσ : m2 Nt/Coul2 or volt2 /Nt. Considering the symmetry of stresses and strains, and the path-independency of elastic strain energy, these constants have the following symmetry properties: E E E ε ε = Cjikl = Cklij , ekij = ekji , ωjk = ωkj , Cijkl E E E σ σ = Sjikl = Sklij , dkij = dkji , ωjk = ωkj , Sijkl D D D Cijkl = Cjikl = Cklij , hkij = hkji , βjkε = βkjε ,

(11.2)

D D D = Sjikl = Sklij , gkij = gkji , βjkσ = βkjσ . Sijkl

To express the constitutive laws in matrix form, the contracted notation assigning 11 to 1, 22 to 2, 33 to 3, 23 or 32 to 4, 13 or 31 to 5, 12 or 21 to 6, 14 or 41 to 7, 24 or 42 to 8, and 34 or 43 to 9 are usually used in engineering expressions. With this assignment and the symmetry properties (11.2), certain transformations need to add a factor of 2 or 4. They are (see (1.26) for reference)

11.1

Constitutive Laws

2Sijkl = Spq ,

371

if either p or q > 3,

4Sijkl = Spq , if both p and q > 3, 2εij = εp , 2dkij = dkp , 2gkij = gkp ,

(11.3)

if p > 3.

no factors are needed for all other transformations. By using the contracted notation, the constitutive laws (11.1) can be written in matrix form as σ ε ε σ CE eT SE −dT = , = , D D e −ωε −E d −ωσ −E σ ε σ CD −hT ε S gT = , = D , −E D −E h −βε g −βσ D

(11.4a)

where ⎡ E E E E E E⎤ C11 C12 C13 C14 C15 C16 ⎧ ⎫ ⎧ ⎧ ⎫ ⎧ ⎫ ⎫ σ ε ε σ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ 1 11 1 11 E CE CE CE CE CE ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢C12 ⎪ ⎪ ⎪ ⎪ 22 23 24 25 26 ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ σ ε ε ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎥ 2 22 2 22 ⎪ ⎪ ⎪ ⎨ ⎪ ⎨ ⎪ ⎨ ⎨ ⎪ ⎢CE CE CE CE CE CE ⎥ ⎬ ⎪ ⎬ ⎬ ⎪ ⎬ σ3 σ33 ε3 ε33 ⎢ 13 23 33 34 35 36 ⎥ = ,ε = = , CE = ⎢ E E E E E E ⎥ , σ= σ4 ⎪ σ23 ⎪ ε4 ⎪ 2ε23 ⎪ ⎪ ⎪ ⎪ ⎪ ⎢C14 C24 C34 C44 C45 C46 ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ σ ε 2ε ⎪ ⎪ ⎪ ⎪ ⎢ E E E E E E⎥ ⎪ ⎪ ⎪ ⎪ 5 13 5 13 ⎪ ⎪ ⎪ C15 C25 C35 C45 C55 C56 ⎦ ⎩ ⎪ ⎩ ⎪ ⎩ ⎩ ⎪ ⎭ ⎪ ⎭ ⎭ ⎪ ⎭ ⎣ σ6 σ12 ε6 2ε12 E CE CE CE CE CE C16 26 36 46 56 66 ⎡ ε ε ε ⎤ ⎧ ⎫ ⎧ ⎫ ⎤ ⎡ ω11 ω12 ω13 e11 e12 e13 e14 e15 e16 ⎨E1 ⎬ ⎨D1 ⎬ ⎢ ε ωε ωε ⎥ E = E2 , D = D2 , e = ⎣e21 e22 e23 e24 e25 e26 ⎦, ωε = ⎣ω12 22 23 ⎦ ⎩ ⎭ ⎩ ⎭ E3 D3 e31 e32 e33 e34 e35 e36 ωε ωε ωε 13

23

33

(11.4b)

and similar expressions for SE , SD , CD , d, g, h, ωσ , βσ , βε . In the above, the superscript T of the matrix denotes the transpose. Since the four equation sets shown in (11.4a) describe the same materials from different bases, the matrices in different set of (11.4a) should have some relations. In other words, from any one of the four equation sets, one can obtain the other three sets by simple mathematical operation. For example, starting from the first set of (11.4a) we can obtain the following relations: SE = C−1 E ,

d = eC−1 E ,

T ωσ = eC−1 E e + ωε ,

CD = CE + eT ω−1 ε e,

h = ω−1 ε e,

βε = ω−1 ε ,

SD =

C−1 E

−1 T −1 − C−1 E e ωσ eCE ,

g=

−1 ω−1 σ eCE ,

βσ =

ω−1 σ .

Generally, their relations can be expressed by the following equations:

(11.5a)

372

11

CE = S−1 E ,

CD = S−1 D ,

d = eSE = ωσ g,

e = dCE = ωε h,


βε = ω−1 ε ,

βσ = ω−1 σ ,

g = hSD = βσ d,

h = gCD = βε e,

ωσ − ωε = dCE d = eSE e = de , T

T

T

βε − βσ = hSD hT = gCD gT = hgT , CD − CE = eT βε e = hT ωε h = hT e, SE − SD = gT ωσ g = dT βσ d = dT g.

(11.5b)

11.1.2 Two-Dimensional State If we consider the most general anisotropic materials, the in-plane and anti-plane deformations will not be decoupled. Under this condition, the two-dimensional states are usually described by generalized plane strain (ε3 = 0) or generalized plane stress (σ3 = 0) without requiring the transverse shear strain or transverse shear stress to be zero. While for electric fields, open circuit condition (D3 = 0) is considered when the faces of piezoelectric materials are in contact with nonconducting media and the top and bottom surfaces are free of charge, or short circuit condition (E3 = 0) is considered if the top and bottom surfaces of the piezoelectric materials are held at the same electric potential. With the above consideration, the two-dimensional states will be divided into four different situations, i.e.,

I. II. III. IV.

Generalized plane strain and short circuit: ε3 = 0 and E3 = 0. Generalized plane strain and open circuit: ε3 = 0 and D3 = 0. Generalized plane stress and short circuit: σ3 = 0 and E3 = 0. Generalized plane stress and open circuit: σ3 = 0 and D3 = 0.

(11.6)

Under the above four different states, the constitutive laws (11.4) can be further reduced by eliminating the terms associated with zero values of ε3 (or σ3 ) and E3 (or D3 ) and replacing σ3 (or ε3 ) and D3 (or E3 ) by the other two-dimensional terms. By this way, the constitutive laws for piezoelectric materials in two-dimensional states can be written in matrix form as follows: State I: ε3 = 0 and E3 = 0. ( 0 ' 0 0T ( 0 0 ' CE e Sˆ E −dˆ T σ ε σ0 ε = , = , 0 D0 D0 −E0 ˆσ dˆ −ω e0 −ω0ε −E ( 0 0 ' ˆ SD gˆ T σ σ0 ε CD −hT ε0 = , = , 0 0 0 h −βε −E D −E D0 gˆ −βˆ σ

(11.7a)

11.1

Constitutive Laws

373

where ⎡ E E E E E⎤ ⎧ ⎫ ⎧ ⎫ C11 C12 C14 C15 C16 σ1 ⎪ ε1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢CE CE CE CE CE ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ 12 22 24 25 26 ⎥ ⎬ ⎬ ⎨σ2 ⎪ ⎨ε2 ⎪ ⎢ E E E E E⎥ 0 0 0 ⎥ σ = σ4 , ε = ε4 , CE = ⎢ ⎢C14 C24 C44 C45 C46 ⎥, ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ε5 ⎪ E CE CE CE CE ⎥ ⎪ ⎪ ⎪ ⎪σ5 ⎪ ⎪ ⎣C15 ⎭ ⎭ ⎩ ⎩ ⎪ 25 45 55 56 ⎦ σ6 ε6 E E CE CE CE C16 C26 46 56 66 ' ' ( ( ε ωε e e e15 e16 e ω11 11 12 14 E1 D1 12 0 0 0 0 E = , D = , e = , ωε = ε ωε , E2 D2 ω12 e21 e22 e24 e25 e26 22 (11.7b) ⎡ ˆ D S11 ⎢ ˆ D ⎢S12 ⎢ ⎢ D Sˆ D = ⎢Sˆ 14 ⎢ ⎢ ˆ D ⎣S15 D Sˆ 16

D S ˆ D Sˆ D Sˆ D ⎤ Sˆ 12 14 15 16 D S ˆ D Sˆ D Sˆ D ⎥ ⎥ ' ' ( ( Sˆ 22 24 25 26 ⎥ σ βˆ σ gˆ 11 gˆ 12 gˆ 14 gˆ 15 gˆ 16 βˆ11 ⎥ 12 D D D D Sˆ 24 Sˆ 44 Sˆ 45 Sˆ 46 ⎥, gˆ = , βˆ σ = σ βˆ σ ⎥ gˆ 21 gˆ 22 gˆ 24 gˆ 25 gˆ 26 βˆ12 22 ˆSD Sˆ D Sˆ D Sˆ D ⎥ 25 45 55 56 ⎦ D S ˆ D Sˆ D Sˆ D Sˆ 26 46 56 66 (11.7c)

and gˆ 3i gˆ 3j Sˆ ijD = Sˆ ijD + σ = Sˆ jiD , βˆ33

gˆ ij

βˆ σ gˆ 3j = gˆ ij − 3iσ , βˆ

βîjσ = βîjσ −

33

σ βˆ σ βˆ3i 3j σ βˆ33

= βˆjiσ , (11.7d)

in which Sˆ ijD = SijD −

D SD S3i 3j D S33

= Sˆ jiD ,

gˆ ij = gij −

D gi3 S3j D S33

,

βîjσ = βijσ +

gi3 gj3 = βˆjiσ . (11.7e) D S33

ˆ ω ˆ σ , CD , h , and βε , which can To save the space, the expressions for Sˆ E , d, be obtained by the same way, are not shown here. Similarly, only two types of constitutive laws are shown below for the other three states. State II: ε3 = 0 and D3 = 0 ( 0 0 0 ' Sˆ D gˆ T σ0 σ ε ε CE eT = , = . (11.8) 0 0 0 ˆ e −ω D −E −E D0 gˆ −βσ ε State III: σ3 = 0 and E3 = 0 0 0 ˆ E eˆ T σ ε C = , D0 ê −ω ˆ ε −E0 State IV: σ3 = 0 and D3 = 0

ε0 SD gT σ0 = . g −βσ D0 −E0

(11.9)

374

11

( 0 ' T ˆ ˆ C e ε0 σ E = ˆ , D0 ˆ ε −E0 e −ω

'S0 g0T ( 0 D σ ε0 = . 0 0 −E0 D0 g −βσ

In the above, CijE = CijE +

Cˆ ijE = CijE −

e3i e3j = CjiE , ε ω33

E CE C3i 3j

eˆ ij = eij −

eˆ 3i eˆ 3j Cˆ ijE = Cˆ ijE + ε = Cˆ jiE , ωˆ 33

eˆ ij = eˆ ij −

g3i g3j = SjiD , σ β33

gij = gij −

SijD = SijD +

(11.10)

ε ωε εe ω3i ω3i 3j 3j ε ε , ω = ω − = ωjiε , ij ij ε ε ω33 ω33 (11.11a)

eij = eij −

= Cˆ jiE ,

E C33


E ei3 C3j E C33

,

ωˆ ijε = ωijε +

ei3 ej3 , E C33 εω ε ωˆ 3i ˆ 3j

εe ˆ 3j ωˆ 3i , ε ωˆ 33

ωˆ ijε = ωˆ ijε −

σg β3i 3j , σ β33

βijσ = βijσ −

ε ωˆ 33

σ βσ β3i 3j σ β33

(11.11b) = ωˆ jiε , (11.11c) = βjiσ (11.11d)

and Sˆ ijD , gˆ ij , βîjσ , Sˆ ijD , gˆ ij , βîjσ are given in (11.7d) and (11.7e). Similar to the three-dimensional states, some relations between the material constants can be obtained through simple inversion such as (' ( ' SˆD gˆ T C0E e0T e0 −ω0ε and ' CE

e T

e −ωε

(' Sˆ D

gˆ −βˆ σ

gˆ T

gˆ −βˆ σ

( =

' = I,

' Ê C eˆ

eˆ T ˆε −ω

CD −hT

('

h −βε ('

SD

g T

g −βσ

Sˆ E −dˆ T

(

ˆσ dˆ −ω (

' =

=I

Cˆ E eˆ T ˆ ε eˆ −ω

(11.12a)

(' ( S0D g0T

= I. g0 −β0σ (11.12b)

11.2 Expanded Stroh Formalism 11.2.1 General Solutions For two-dimensional linear anisotropic elasticity, there are two major complex variable formalisms in the literature. One is Lekhnitskii formalism (Lekhnitskii, 1963) which starts with the equilibrated stress functions followed by constitutive laws, strain–displacement relations, and compatibility equations; the other is Stroh formalism (Stroh, 1958) which starts with the compatible displacements followed by

11.2

Expanded Stroh Formalism

375

strain–displacement relations, constitutive laws, and equilibrium equations. With this understanding, to develop the expanded Stroh formalism for piezoelectric anisotropic elasticity the most appropriate constitutive relation is the first equation set of (11.1). While for the extended Lekhnitksii formalism the most appropriate constitutive relation is the last equation set of (11.1). Thus, to describe the expanded Stroh formalism for piezoelectric anisotropic elasticity, it is better to write the basic equations as E ε −e E , σij = Cijkl kl kij k εE , Dj = ejkl εkl + ωjk k

1 εij = (ui, j +uj, i ), 2

σij, j = 0, Di, i = 0,

i, j, k, l = 1, 2, 3, (11.13)

where repeated indices imply summation, a comma stands for differentiation, and ui is the displacement in xi -axis. By letting Dj = σ4j , Cijkl =

−Ej = u4,j = 2ε4j ,

E Cijkl ,

Cij4l = elij , C4jkl = ejkl , C4j4l = −ωjlε ,

j = 1, 2, 3

i, j, k, l = 1, 2, 3 i, j, l = 1, 2, 3 j, k, l = 1, 2, 3

(11.14)

j, l = 1, 2, 3

the basic equations (11.13) can be rewritten in an expanded tensor notation as σIJ = CIJKL εKL ,

εIJ =

1 (uI,J + uJ,I ), 2

σIJ,J = 0,

I, J, K, L = 1, 2, 3, 4, (11.15a)

where expanded elastic stiffness tensor CIJKL has the following symmetry property: CIJKL = CJIKL = CKLIJ = CIJLK .

(11.15b)

Since the mathematical form of expanded expression (11.15) for piezoelectric anisotropic elasticity is exactly the same as that of pure anisotropic elasticity shown in (3.1), the general solutions satisfying all basic equations (11.13) under twodimensional deformation can therefore be written in the form of Stroh formalism and is usually called extended Stroh formalism. However, in this book to have a simple distinction with the extended Stroh formalism for thermoelasticity, we like to call it expanded Stroh formalism since it really expands all the related matrices. Following is the general solutions satisfying (11.13), which possesses exactly the same matrix form as that shown in (3.24) for pure elastic materials. The only differences between (11.16) and (3.24) are the size and contents of the matrices. u = 2 Re{Af(z)}, φ = 2 Re{Bf(z)}, where

(11.16a)

376

11

⎧ ⎫ u1 ⎪ ⎪ ⎪ ⎬ ⎨ ⎪ u2 , u= u3 ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ u4

⎧ ⎫ φ1 ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ φ2 φ= , φ3 ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ φ4


⎧ ⎫ f1 (z1 )⎪ ⎪ ⎪ ⎪ ⎨ ⎬ f2 (z2 ) f(z) = , f3 (z3 )⎪ ⎪ ⎪ ⎪ ⎩ ⎭ f4 (z4 )

A = [a1 a2 a3 a4 ], B = [b1 zk = x1 + μk x2 , k = 1, 2, 3, 4.

b2

b3

(11.16b) b4 ],

The generalized stress function φi is related to the stresses and electric displacements by σi1 = −φi, 2 , σi2 = φi, 1 , i = 1, 2, 3,

and D1 = σ41 = −φ4,2 , D2 = σ42 = φ4,1 . (11.17) fk (zk ), k = 1, 2, 3, 4 are four holomorphic functions of complex variables zk , which will be determined by the boundary conditions set for each particular problem. μk and (ak , bk ) are the material eigenvalues and eigenvectors which can be determined by the following eigenrelations: Nξ = μξ,

(11.18a)

where N is a 8×8 fundamental matrix and ξ is a 8×1 column vector defined by N=

N1 N2 , N3 NT1

ξ=

a b

(11.18b)

and N1 = −T−1 RT ,

N2 = T−1 = NT2 ,

N3 = RT−1 RT − Q = NT3 .

(11.18c)

Q, R, and T are three 4×4 real matrices defined by the elastic constants as Qik = Ci1k1 ,

Rik = Ci1k2 ,

Tik = Ci2k2 ,

i, k = 1, 2, 3, 4.

(11.18d)

Note that the general solutions (11.16) are obtained by considering the twodimensional deformation in which ui , i = 1, 2, 3, 4, depend on x1 and x2 only. Through the strain–displacement relation (11.13)2 and the relation for the electric field (11.14)2 , we know that the two-dimensional state considered in the expanded Stroh formalism is state I: generalized plane strain and short circuit (ε3 = 0 and E3 = 0). From (11.7) we see that the material constants used in this state are CijE , eij , ωijε or Sˆ ijD , gˆ ij , βîjσ or Sˆ ijE , dˆ ij , ωˆ ijσ or CijD , hij , βijε . For the other two-dimensional states, to employ the general solution (11.16) the material constants should be replaced according to the relations shown in (11.8), (11.9), (11.10), and (11.11). For example, CijE , eij , ωijε should be replaced by CijE , eij , ωijε for state II, and replaced by Cˆ ijE , eˆ ij , ωˆ ijε for state III, and replaced by Cˆ ijE , eˆ ij , ωˆ ijε for state IV. For the convenience of readers’ reference, we now show the matrix components of Q, R, and T defined in (11.18d) for state I.

11.2

Expanded Stroh Formalism

⎡

E CE CE C11 16 15

⎢ E E E ⎢C16 C66 C56 Q=⎢ ⎢C E C E C E ⎣ 15 56 55 e11 e16 e15

e11

⎤

377

⎡

E CE CE C16 12 14

e21

⎤

⎡

E CE CE C66 26 46

e26

⎤

⎢ E E E ⎢ E E E ⎥ ⎥ ⎥ ⎢C C C ⎢C C C e16 ⎥ e26 ⎥ e22 ⎥ ⎥, R = ⎢ 66 26 46 ⎥, T = ⎢ 26 22 24 ⎥. ⎢C E C E C E ⎢C E C E C E ⎥ ⎥ e15 ⎥ ⎣ 56 25 45 e25 ⎦ ⎣ 46 24 44 e24 ⎦ ⎦ ε ε ε −ω11 e16 e12 e14 −ω12 e26 e22 e24 −ω22 (11.19)

Note that because the material eigenvalues μk obtained from the eigenrelation (11.18) cannot be real if the strain energy is positive (Suo et al., 1992; Ting, 1996), μk occurs as four pairs of complex conjugates. In the general solution (11.16), the material eigenvalues μk and material eigenvectors ak , bk have been arranged to be μk+4 = μk , Im(μk ) > 0, and ak+4 = ak , bk+4 = bk , k=1,2,3,4. Moreover, in the general solution (11.16), the material eigenvalues are assumed to be distinct and their associated eigenvectors are independent of each other. For the cases that the material eigenvalues are repeated so that their associated eigenvectors are not independent of each other, the general solution (11.16) should be modified (Ting, 1996) or one may introduce a small perturbation in the values of material properties to avoid the problem of degeneracy (Hwu and Yen, 1991). From the above discussions we know that the fundamental matrix N and its associated eigenvector ξ = (a, b) play important roles in Stroh formalism. Due to their importance, several works have been done to get their explicit expressions for pure anisotropic elastic materials (as shown in Chapter 3). Since ak , bk are the right eigenvectors of the fundamental matrix N, to have unique values of ak , bk normalization is necessary. The orthogonality relation for the material eigenvector matrices A and B of anisotropic materials is shown in (3.57) and (3.58) and will also be extended to the piezoelectric anisotropic materials. From this relation, it has been observed that the three matrices S, H, and L defined in (3.59) are real and have been proved to be the average values of N1 (θ ), N2 (θ ), and −N3 (θ ) over the interval θ = (0, π ) and are usually called Barnett–Lothe tensors.

11.2.2 Boundary Conditions The generalized surface traction vector t can be calculated by using Cauchy’s formula (Sokolnikoff, 1956), i.e., ti = σij nj , i, j = 1, 2, 3, and

t4 = σ4j nj = Dj nj = Dn ,

(11.20)

where nj is the unit normal to the surface boundary; t1 , t2 , t3 are the components of surface traction vector and t4 (= Dn ) is the electric displacement on the normal direction of the surface. If s is the arc length measured along a curved boundary, using the Cauchy’s formula and the relation given in (11.17), we obtain a useful formula:

378

11

t=


∂φ . ∂s

(11.21)

To keep this relation in correct sign the tangential direction s has to be chosen such that when one faces the direction of increasing s the material lies on the right side. Otherwise, a negative sign should be put on either side of the equation. From (11.21), we have t = φ, r for a radial line surface and t = −φ, θ /r for a circular surface, and hence the stresses and electric displacement in polar coordinate can also be calculated from the generalized stress function vector φ by t θ = φ, r ,

tr = −φ, θ /r,

σθθ = n φ, r , T

σrr = −sT φ, θ /r,

σθ3 = iT3 φ, r ,

σr3 = −iT3 φ, θ /r,

Dθ = iT4 φ, r ,

Dr = −iT4 φ, θ /r,

σrθ = sT φ, r = −nT φ, θ /r,

(11.22a)

where sT = (cos θ , sin θ , 0, 0), iT3 = (0, 0, 1, 0),

nT = (− sin θ , cos θ , 0, 0),

iT4 = (0, 0, 0, 1)

(11.22b)

and the angle θ is directed counterclockwise from the positive x1 -axis to the direction of s. The counterpart of t is the generalized displacement vector u shown in (11.16b)1 in which u1 ,u2 ,u3 are the displacement components and u4 defined in (11.14)2 is the electric potential. Understanding the physical meaning of the generalized surface traction vector t and generalized displacement vector u, and using the relation (11.21), several commonly encountered boundary conditions can be written as follows: Continuity condition: u1 = u2 ,

φ1 = φ2 ,

along the interface.

(11.23a)

Mechanically free and electrically open: φ = 0, along the free surface.

(11.23b)

Mechanically clamped and electrically closed: u = 0, along the clamped surface.

(11.23c)

Point force (ˆp1 , pˆ 2 , pˆ 3 ) and point charge pˆ 4 = qˆ : (11.23d) . . ˆ dφ = p, du = 0, for any close curve C enclosing the loading point. C

C

11.3

Explicit Expressions

379

11.3 Explicit Expressions 11.3.1 Fundamental Matrix N Although the fundamental matrix N is defined clearly in (11.18b) and (11.18c), their calculation involves the matrix inversion. Therefore, if we do not pay special attention to get their explicit expressions, their results from pure numerical calculation can only provide their numerical values which are not appropriate for the understanding of the physical meaning of the analytical solutions obtained by using the expanded Stroh formalism. Even it is possible to find the explicit expressions by using the symbolic computational software such as Mathematica, ignorant of the relations among the components may lead to complicated expressions. Although it has been indicated by Ting (1996) that certain elements of N1 and N3 are zero for piezoelectric materials, the explicit expressions for Ni were only presented in Hwu (2008). In this section, the work of Hwu (2008) by following the steps described in Ting (1996) for pure anisotropic materials to get the explicit expressions of Ni for piezoelectric materials is described below. In order to find the explicit expressions of Ni , we first re-organize (11.18c) into the following compact matrix form: −N3 0 I NT1 Q R = . −N1 I 0 N2 RT T

(11.24)

Re-arrangement of (11.12a)1 and knowing the matrix expressions of Q, R, and T given in (11.19), we can get Q R Q∗ R∗ I−2 I21 = , 0 I RT T R∗T T∗

(11.25a)

where ⎡ ˆ D D S11 0 Sˆ 15 ⎢ ⎢ 0 0 0 Q∗ = ⎢ ⎢Sˆ D 0 Sˆ D ⎣ 15 55 gˆ 11 0 gˆ 15

gˆ 11

⎤

⎥ 0 ⎥ ⎥, gˆ 15 ⎥ ⎦ −βˆ σ 11

⎡ ˆ D ˆ D ˆ D S16 S12 S14 ⎢ ⎢ 0 0 0 R∗ = ⎢ ⎢Sˆ D Sˆ D Sˆ D ⎣ 56 25 45 gˆ 16 gˆ 12 gˆ 14

⎡ D D Sˆ 66 Sˆ 26 ⎢ ⎥ D Sˆ D Sˆ 26 0 ⎥ ∗ ⎢ 22 ⎥, T = ⎢ ⎢ ⎥ ⎢Sˆ D Sˆ D gˆ 25 ⎦ ⎣ 46 24 −βˆ σ gˆ gˆ gˆ 21

⎤

12

26

D Sˆ 46

gˆ 26

⎤

⎥ gˆ 22 ⎥ ⎥ ⎥ ˆSD gˆ ⎥ 44 24 ⎦ ˆ σ 22 gˆ 24 −β22 (11.25b) D Sˆ 24

and ⎡

I−2

1 ⎢0 =⎢ ⎣0 0

0 0 0 0

0 0 1 0

⎡ ⎤ 0 0 ⎢1 0⎥ ⎥,I = ⎢ 0⎦ 21 ⎣0 1 0

0 0 0 0

0 0 0 0

⎤ ⎡ ⎤ 0 1000 ⎢ ⎥ 0⎥ ⎥ , I = ⎢0 1 0 0⎥ . ⎣0 0 1 0⎦ 0⎦ 0 0001

Employing the relation (11.24), equation (11.25a) becomes

(11.25c)

380

11


−N3 0 Q∗ R∗ I NT1 I−2 I21 = 0 I −N1 I R∗T T∗ 0 N2

(11.26a)

or − N3 Q∗ = I−2 , −N3 R∗ = I21 + NT1 , −N1 Q∗ + R∗T = 0, −N1 R∗ + T∗ = N2 . (11.26b) From the above results, we see that the explicit expressions of N3 can be obtained directly from the first equation of (11.26b) since Q∗ shown in (11.25b) is a matrix of rank 3, not a full matrix of rank 4. Substituting the result of N3 into (11.26b)2 we can get N1 and then obtain N2 through (11.26b)4 . Through this procedure, the explicit expressions of N1 ,N2 and N3 have been obtained as ⎡

X6 ⎢X2 N1 = ⎢ ⎣X4 X8

−1 0 0 0

Y6 Y2 Y4 Y8

⎤ ⎡ ∗ Z6 S11 ⎢ −1 Z2 ⎥ ⎥ , N = −N1 R∗ + T∗ , N3 = ⎢ 0∗ Z4 ⎦ 2 ⎣S15 Z8 g∗11

∗ 0 S15 0 0 ∗ 0 S55 ∗ 0 g15

⎤ g∗11 0 ⎥ ⎥, g∗15 ⎦ ∗ β11 (11.27a)

Where D ˆ D ˆ D D 2 D 2 ˆ D 2 = −Sˆ 11 S55 β11 + 2Sˆ 15 gˆ 11 gˆ 15 − Sˆ 55 gˆ 11 − Sˆ 11 gˆ 15 + βˆ11 (S15 ) , ∗ D ˆ S11 β11 − gˆ 2 = −Sˆ 55 15 ,

∗ D ˆ β11 + gˆ 11 gˆ 15 , S15 = Sˆ 15

D D gˆ 15 − Sˆ 55 gˆ 11 , g∗11 = Sˆ 15

∗ D ˆ β11 − gˆ 2 S55 = −Sˆ 11 11 ,

D D g∗15 = −Sˆ 11 gˆ 15 + Sˆ 15 gˆ 11 ,

D ∗ D ∗ D ∗ S11 + Sˆ 5α S15 + Sˆ 7α g11 )/, Xα = (Sˆ 1α D ∗ D ∗ D ∗ g11 + Sˆ 5α g15 + Sˆ 7α β11 )/, Zα = (Sˆ 1α

∗ D ˆ D D 2 S55 − (Sˆ 15 β11 = Sˆ 11 ) ,

D ∗ D ∗ D ∗ Yα = (Sˆ 1α S15 + Sˆ 5α S55 + Sˆ 7α g15 )/,

α = 2, 4, 6, 8. (11.27b)

D , S ˆ D , which are Note that in (11.27) some expanded notations are used such as Sˆ 7j 8j σ related to gˆ and βˆ by ij

ij

D D = Sˆ j7 = gˆ 1j , Sˆ 7j D Sˆ 77 = −βˆ11 ,

D D Sˆ 8j = Sˆ j8 = gˆ 2j ,

D Sˆ 78 = −βˆ12 ,

j = 1, 2, 4, 5, 6,

D Sˆ 88 = −βˆ22 .

(11.28)

11.3.2 Material Eigenvector Matrices A and B From the general solutions shown in (11.16) we see that it would be of much benefit if we can get the explicit expressions of the material eigenvector matrices A and B. For pure anisotropic materials, the explicit expressions of material eigenvectors are obtained through comparison with the stress-based Lekhnitskii formalism (Ting, 1996). With this understanding, to obtain the material eigenvectors for piezoelectric materials we start from the fourth type constitutive laws of state I, i.e., (11.7a)4 ,

11.3


381

εp = Sp σp ,

(11.29a)

where 0 ε0 σ , σ , = εp = p −E0 D0

'

( Sˆ D gˆ T Sp = . gˆ −βˆ σ

(11.29b)

Knowing that stresses are related to stress functions by (11.17) and strains are related to displacements by (11.15a)2 , (11.29a) can be further written as Dε u = Sp Dσ φ,

(11.30a)

where Dε and Dσ are, respectively, the matrix of differential operators related to the strain and stress calculation as ⎡

∂ ∂x1

⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ Dε = ⎢ ⎢ 0 ⎢ ∂ ⎢ ∂x2 ⎢ ⎣ 0 0

0 ∂ ∂x2

0 0 ∂ ∂x1

0 0

0 0

0 0 0 0 0

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ 0 ⎥ ⎥ 0 ∂x∂ 1 ⎦ 0 ∂x∂ 2

∂ ∂x2 ∂ ∂x1

⎡

− ∂x∂ 2 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ Dσ = ⎢ ⎢ 0 ⎢ ∂ ⎢ ∂x1 ⎢ ⎣ 0 0

0 ∂ ∂x1

0 0 0 0 0

0 0 ∂ ∂x1 − ∂x∂ 2

0 0 0

0 0 0

⎤

⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥. ⎥ 0 ⎥ ⎥ − ∂x∂ 2 ⎦

(11.30b)

∂ ∂x1

By using the relation obtained in (11.30), each displacement component can be expressed in terms of the stress functions through integration. With this relation, the following compatibility equations for two-dimensional problems ε11,22 + ε22,11 − 2ε12,12 = 0,

−ε23,1 + ε13,2 = 0,

E1,2 − E2,1 = 0

(11.31)

will give us Dc Sp Dσ φ = 0,

(11.32a)

where ⎡ ∂2

∂x22

⎢ Dc = ⎣ 0 0

∂2 ∂x12

0

0 − ∂x∂1 ∂x2 0

0 − ∂x∂ 1 ∂x∂ 2 0 0 0

2

0 0

0 ∂ ∂x2

0

⎤

⎥ 0 ⎦.

(11.32b)

− ∂x∂ 1

Since (11.32) is a system of homogeneous partial differential equations in two independent variables x1 and x2 . A general solution for φi depends on one composite variable that is a linear combination of x1 and x2 , which is also applicable for ui through (11.30a). Without loss of generality the coefficient of x1 is usually selected to be unity, i.e., z = x1 + μx2 . By comparison with the general solutions shown in

382

11


(11.16a), we now let ui = ai f (z),

φi = bi f (z),

(11.33)

in which ai and bi are the material eigenvectors to be determined in this section. Substituting (11.33)2 into (11.32a) we obtain c Sp σ b = 0,

(11.34a)

where ⎡

−μ ⎢ 0 ⎢ ⎡ 2 ⎤ ⎢ 0 μ 1 0 0 −μ 0 0 ⎢ c = ⎣ 0 0 −1 μ 0 0 0 ⎦ , σ = ⎢ ⎢ 0 ⎢ 1 0 0 0 0 0 μ −1 ⎢ ⎣ 0 0

0 0 1 0 0 1 0 −μ 0 0 0 0 0 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ −μ⎦ 1

(11.34b)

Note that in the above c and σ can be obtained directly from the matrices of differential operator, Dc and Dσ , with ∂/∂x1 replaced by 1 and ∂/∂x2 replaced by μ. Equation (11.34a) is a linear algebraic system of equations with four unknowns and three equations. To solve the unknown vector b, we need one more relation. From (11.17) we see that φi , i = 1, 2, 3, 4, are not independent of each other because of the symmetry of stress σ12 = σ21 , which will lead to − φ2,2 = φ1,1 .

(11.35)

Substituting (11.33)2 into (11.35) we obtain − μb2 = b1 .

(11.36)

With the relation (11.36), the system of equations (11.34a) can be rewritten as c Sp − σ b−1 = 0,

(11.37a)

where ⎡

⎤ μ2 0 0 ⎢ 1 0 0⎥ ⎢ ⎥ ⎢ 0 1 0⎥ ⎢ ⎥ ⎢ − 0⎥ σ = ⎢ 0 −μ ⎥, ⎢−μ 0 0⎥ ⎢ ⎥ ⎣ 0 0 −μ⎦ 0 0 1

b−1

⎧ ⎫ ⎨b2 ⎬ = b3 . ⎩ ⎭ b4

(11.37b)

11.3


383

After operating the matrix multiplication for (11.37), we get ⎤⎧ ⎫ −4 3 m3 ⎨b2 ⎬ ⎣−3 2 m2 ⎦ b3 = 0, ⎩ ⎭ −m3 m2 ρ2 b4 ⎡

(11.38a)

where D 2 D D μ − 2Sˆ 45 μ + Sˆ 44 , 2 = Sˆ 55 D 3 D D 2 D D D 3 = Sˆ 15 μ − (Sˆ 14 + Sˆ 56 )μ + (Sˆ 25 + Sˆ 46 )μ − Sˆ 24 , D 4 D 3 D D 2 D D μ − 2Sˆ 16 μ + (2Sˆ 12 + Sˆ 66 )μ − 2Sˆ 26 μ + Sˆ 22 , 4 = Sˆ 11

m2 = gˆ 15 μ2 − (ˆg14 + gˆ 25 )μ + gˆ 24 ,

(11.38b)

m3 = gˆ 11 μ3 − (ˆg21 + gˆ 16 )μ2 + (ˆg12 + gˆ 26 )μ − gˆ 22 , σ 2 σ σ μ + 2βˆ12 μ − βˆ22 . ρ2 = −βˆ11

Nontrivial solutions of b2 , b3 , b4 exist only when the determinant of the coefficient matrix equal to zero, which will lead to the following characteristic equation for the determination of the material eigenvalues μ, 2 4 ρ2 + 23 m2 m3 − 2 m23 − 4 m22 − ρ2 23 = 0.

(11.39)

Equation (11.39) is an eighth-order polynomial which should lead to the same eigenvalues as those obtained from the eigenrelation (11.18). Furthermore, after obtaining the eigenvalues from (11.39), its associated eigenvectors can be obtained from (11.38a) and (11.36). The results are ⎧ ∗⎫ ⎧ ∗⎫ ⎧ ⎫ ⎧ ⎫ ∗ ⎪ ⎪ ⎪ ⎪ ⎪ μ4∗ ⎪ ⎪μ3∗ ⎪ ⎪ μm3∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b1 ⎪ ⎨ ⎨ ⎨ ⎬ ⎬ ⎬ ⎬ ⎨ b2 −4 −3 −m3 =c , or c , or c , (11.40a) b= b3 ⎪ ∗ ⎪ ∗ ⎪ m∗2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 3∗ ⎪ ⎩ 2∗ ⎪ ⎩ ⎭ ⎭ ⎭ ⎭ ⎩ ⎪ b4 m3 m2 ρ2∗ where ∗2 = 4 ρ2 − m23 , m∗2 = 3 m3 − 4 m2 ,

∗3 = m2 m3 − 3 ρ2 , m∗3 = 3 m2 − 2 m3 ,

∗4 = 2 ρ2 − m22 , ρ2∗ = 2 4 − 23 .

(11.40b)

In the above c is the scaling factor. Generally, the three different expressions shown in (11.40a) should be the same if they are nontrivial. If one or two of them is a trivial solution, i.e., zero, just take the nontrivial one as the eigenvector b. If all of them are trivial, one may take any three independent vectors of b-1 as the eigenvectors and employ the relation (11.36) to complete the eigenvector b. The solutions shown in (11.40) cover all the possible eigenvectors in which one of them agrees with that presented by Soh et al. (2001) whose solution will fail if its denominator equals to zero for some piezoelectric materials.

384

11


When b is determined from (11.40), the other eigenvector a can be obtained by the following relation, which is derived by substituting (11.33) into (11.30a): ε a = Sp σ b,

(11.41)

where σ is given in (11.34b)2 and ε can be obtained from Dε of (11.30b)1 with ∂/∂x1 replaced by 1 and ∂/∂x2 replaced by μ. By choosing an appropriate matrix − − ε making ε ε = I, the eigenvector a can then be determined by − − a = − ε Sp σ b = ε Sp σ b−1 ,

(11.42a)

where the second equality of (11.42a) comes from (11.36) and ⎡

1 0 ⎢0 1/μ − ε = ⎢ ⎣0 0 0 0

0 0 1/μ 0

⎤ 0000 0 0 0 0⎥ ⎥. 0 0 0 0⎦ 0010

(11.42b)

After operating the matrix multiplication for (11.42a), we get ⎫ ⎧ ⎫ ⎧ p1 b2 + q1 b3 + r1 b4 ⎪ a1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎬ ⎨ a2 [p2 b2 + q2 b3 + r2 b4 ]/μ = , a= a3 ⎪ ⎪ [p4 b2 + q4 b3 + r4 b4 ]/μ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎭ ⎩ ⎪ a4 p7 b2 + q7 b3 + r7 b4

(11.43a)

where D D D pj = μ2 Sˆ j1 + Sˆ j2 − μSˆ j6 , D D − μSˆ j5 , qj = Sˆ j4

rj =

D Sˆ j8

D − μSˆ j7 ,

(11.43b) j = 1, 2, 4, 7.

Note that although the choice of − ε in (11.42b) may not be unique, different will lead to the same a. For example, if the third row of − choice of − ε ε is selected to be (0 0 0 1 0 0 0), the third component of a will be p5 b2 + q5 b3 + r5 b4 which can be proved to be identical to the one shown in (11.43a). After getting the explicit expressions of ak and bk through (11.40) and (11.43) for each material eigenvalue μk , k = 1, 2, 3, 4, the material eigenvector matrices A and B can be constructed as that shown in (11.16b)4,5 . To have a unique value for the eigenvectors, the scaling factors ck should be normalized. The normalization has been defined through the orthogonality relation (3.57), which shows that c2k =

1 , 2(a1k b1k + a2k b2k + a3k b3k + a4k b4k )

k = 1, 2, 3, 4,

(11.44)

11.3


385

where ajk and bjk are the components of material eigenvector matrices A and B before scaling. In order to let readers have a clear picture about the explicit expressions of A and B, two typical examples for the piezoelectric materials are shown below. Example 1: Piezoelectric ceramics poling in x3 -axis The constitutive relations for piezoelectric ceramics with poling direction parallel to x3 -axis can be written as ⎡ E C11 ⎧ ⎫ σ ⎢ ⎪ ⎪ 1 E ⎪ ⎪ ⎪ ⎪ ⎢C12 ⎪ σ2 ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎢C E σ3 ⎢ = ⎢ 13 σ ⎢ 0 ⎪ ⎪ 4 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ σ ⎢ ⎪ ⎪ 5 ⎪ ⎩ ⎪ ⎭ ⎣ 0 σ6 0

E CE 0 C12 13

0

0

E CE 0 C11 13

0

0

E CE 0 C13 33

0 0 0

0 0 0

0

0

E C44

0

0

0

E C44

0

0

⎤

0 E (C11

E )/2 − C12

⎧ ⎫ ⎡ ε1 ⎪ 0 ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢0 ⎪ ε ⎪ ⎪ ⎥⎪ 2 ⎨ ⎪ ⎬ ⎢ ⎥⎪ ⎢0 ⎥ ε3 −⎢ ⎥ ⎢0 ε4 ⎪ ⎥⎪ ⎢ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎣e15 ε5 ⎪ ⎥⎪ ⎪ ⎪ ⎦ ⎩ε ⎭ 0 6

0 0 0 e15 0 0

⎤ e31 ⎧ ⎫ e31 ⎥ ⎥ ⎨E1 ⎬ ⎥ e33 ⎥ E2 , 0⎥ ⎥ ⎩E3 ⎭ 0⎦ 0 (11.45a)

⎧ ⎫ ⎪ ⎪ ⎪ε1 ⎪ ⎪ ⎡ ε ⎧ ⎫ ⎡ ⎪ε2 ⎪ ⎤⎧ ⎫ ⎤⎪ ⎪ ⎪ ⎪ ω11 0 0 ⎨E1 ⎬ 0 0 0 0 e15 0 ⎨ ⎪ ⎬ ⎨D1 ⎬ ε ε D2 = ⎣ 0 0 0 e15 0 0⎦ 3 + ⎣ 0 ω11 0 ⎦ E2 . ε4 ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ε ⎪ ⎪ D3 E3 e31 e31 e33 0 0 0 ⎪ 0 0 ω33 ⎪ ⎪ε5 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ ε6

(11.45b)

If we consider the two-dimensional state of generalized plane strain and short circuit (ε3 = 0 and E3 = 0), the explicit expressions of the eigenvectors a and b shown in (11.40) and (11.43) are written in terms of Sˆ ijD , gˆ ij , βîjσ . Therefore, to get the explicit expressions for the material eigenvector matrices A and B, the first thing we need to do is finding the inverse relation of (11.45a) and (11.45b), which gives us D D = Sˆ 22 = Sˆ 11 D D Sˆ 44 = Sˆ 55

E C11

E )2 − (CE ) (C11 12 ε ω11 = 2 , ε CE e15 + ω11 44

D D = gˆ 24 = Sˆ 57 = gˆ 15 = Sˆ 48

D D Sˆ 12 = Sˆ 21 =

, 2

D Sˆ 66 =

E C11

E C44

ε CE e215 + ω11 44

and all the other constants are zero, i.e.,

E )2 − (CE )2 (C11 12

2 , E − C12

e15 , ε CE e215 + ω11 44

D σ D − Sˆ 77 = βˆ11 = −Sˆ 88 = βˆ22 =

E −C12

,

(11.46a)

386

11


D D D D D D D D D D Sˆ 14 = Sˆ 15 = Sˆ 16 = Sˆ 17 = Sˆ 18 = Sˆ 24 = Sˆ 25 = Sˆ 26 = Sˆ 27 = Sˆ 28 D D D D D D D D = Sˆ 45 = Sˆ 46 = Sˆ 47 = Sˆ 56 = Sˆ 58 = Sˆ 67 = Sˆ 68 = Sˆ 78 = 0.

(11.46b)

Substituting (11.46a) and (11.46b) into (11.38b), we get D 2 D 2 (μ + 1), 4 = Sˆ 11 (μ + 1)2 , m2 = gˆ 24 (μ2 + 1), 2 = Sˆ 44 σ (μ2 + 1), 3 = m3 = 0. ρ2 = βˆ11

(11.47)

With this result, the characteristic equation (11.39) gives us the material eigenvalues with positive imaginary part, μk , k = 1, 2, 3, 4, as μ1 = μ2 = μ3 = μ4 = i.

(11.48)

With the repeated eigenvalues obtained in (11.48), we see that the coefficient matrix of b-1 of (11.38a) is identical to zero and all the explicit expressions shown in (11.40a) provide trivial solution, i.e., b=0. As we explain in the paragraph following (11.40b), under this condition we may take any three independent vectors of b-1 as the eigenvectors and the relation −μb2 = b1 should be employed to complete the eigenvector b. With this understanding, the eigenvector matrix B can be written as ⎡

−c1 μ1 −c2 μ2 ⎢ c1 c2 B=⎢ ⎣ 0 0 0 0

0 0 c3 0

⎤ 0 0⎥ ⎥. 0⎦ c4

(11.49)

Note that in the above all four eigenvectors are independent of each other depending on the assumption that μ1 = μ2 . However, from (11.48) we know that it is not true since all the eigenvalues are the same. Therefore, the material eigenvector matrix B for this special case does not exist owing to the fact that no enough independent eigenvectors exist for the repeated eigenvalues. This is the so-called degenerate materials. For this special kind of materials, the general solution shown in (11.16) is not valid and should be modified. A modified formalism for degenerate materials has been proposed in Section 3.5 for two-dimensional anisotropic elasticity, which may also be applied to the piezoelectric materials. Even (11.49) is not valid for the present case when μ1 = μ2 = i. In many applications, it is very useful by treating μ1 = i and μ2 = i + ε where ε is a small perturbed value. Successful application of (11.49) can be seen in the next section when we derive the explicit expressions of Barnett–Lothe tensors L, S, and H. After getting the eigenvectors bk , k = 1, 2, 3, 4, their associated eigenvector ak can be obtained from (11.43). By using the constant values given in (11.46) and the results obtained in (11.48) and (11.49) as well as the assumption that μ1 = μ2 , we can now write down the explicit expression for the material eigenvector matrix A as

11.3


⎡

387

D + S ˆ D ) c1 (μ21 Sˆ 11 12

D + S ˆ D ) c2 (μ22 Sˆ 11 12

0

0

⎤

⎥ ⎢ ⎢c (μ2 Sˆ D + Sˆ D )/μ c (μ2 Sˆ D + Sˆ D )/μ 0 0 ⎥ 1 2 2 12 2 ⎥ ⎢ 1 1 12 11 11 A=⎢ ⎥. ⎢ D ˆ 0 0 −ic3 S44 −ic4 gˆ 24 ⎥ ⎦ ⎣ σ ˆ 0 0 −ic3 gˆ 24 ic4 β11

(11.50)

The scaling factors ck , k = 1, 2, 3, 4, shown in (11.49) and (11.50) can then be determined by the orthogonality relation (3.57), or obtained directly from (11.44) with ajk and bjk given in (11.49) and (11.50), e.g., b11 = −μ1 , b21 = 1, b31 = σ . b41 = 0, . . . .., a44 = iβˆ11 Example 2: Piezoelectric ceramics poling in x2 -axis The constitutive relations for piezoelectric ceramics with poling direction parallel to x2 -axis can be written as ⎡ E C11 ⎧ ⎫ σ1 ⎪ ⎢ E ⎪ ⎪ ⎪ ⎪ ⎢C12 ⎪ ⎪ σ2 ⎪ ⎪ ⎢ ⎪ ⎪ ⎬ ⎢C E ⎨ ⎪ σ3 ⎢ = ⎢ 13 ⎪ ⎢ 0 ⎪σ4 ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎢ ⎪σ5 ⎪ ⎪ ⎭ ⎣ 0 ⎩ ⎪ σ6 0

E CE 0 C12 13

0

E C22 E C12

0

E C12 E C11

0

0

0

0

0

0

0

E C44

0

0

0

0

0

0

0

0

E (C11

E )/2 − C12

0

⎤

0

0 E C44

⎧ ⎫ ⎡ 0 ε1 ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎥⎪ ⎢ ⎪ ⎪ ε 0 ⎥⎪ 2⎪ ⎨ ⎪ ⎬ ⎢ ⎥⎪ ⎢ ε 0 ⎥ 3 −⎢ ⎥ ⎢ ε ⎥⎪ 4⎪ ⎢0 ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎣0 ε ⎥⎪ 5⎪ ⎩ ⎪ ⎭ ⎦⎪ ε6 e16

e21 e22 e21 0 0 0

⎤ 0 ⎧ ⎫ 0⎥ ⎥ ⎨E1 ⎬ 0⎥ ⎥ E2 , e16 ⎥ ⎥ ⎩E3 ⎭ 0⎦ 0 (11.51a)

⎧ ⎫ ⎪ ⎪ ⎪ε1 ⎪ ⎪ ⎡ ⎪ ⎧ ⎫ ⎡ ⎤⎧ ⎫ ⎤⎪ ε2 ⎪ ⎪ ⎪ ε ⎪ ω11 0 0 0 0 0 e16 ⎨ ⎪ 0 0 ⎨E1 ⎬ ⎬ ⎨D1 ⎬ ε ε D2 = ⎣e21 e22 e21 0 0 0 ⎦ 3 + ⎣ 0 ω22 0 ⎦ E2 . ε4 ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ε ⎪ ⎪ D3 E3 0 0 0 e16 0 0 ⎪ 0 0 ω11 ⎪ ⎪ ε5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ε6

(11.51b)

Similar to the previous case, we first find the inverse relation of (11.51a) and (11.51b) under the condition that ε3 = 0 and E3 = 0, which gives us D Sˆ 11 =

E C22 E CE − (CE )2 C11 22 12

D D Sˆ 12 = Sˆ 21 =−

E − e C E )2 (e21 C11 22 12 E CE − (CE )2 ] C∗ [C11 22 12

E )2 − (CE )2 ] e21 e22 [(C11 12 E CE − (CE )2 ] C∗ [C11 22 12

+

,

ε CE ω22 12 , C∗

ε CE ω22 11 + , E CE − (CE )2 ] C∗ C∗ [C11 22 12 ε ω11 1 2 D ˆ D = = E , Sˆ 55 = E , S , 66 E ε CE C44 C11 − C13 e216 + ω11 44

D = Sˆ 22 D Sˆ 44

−

E )2 − (CE )2 ] e221 [(C11 12

(11.52a)

388

11

D E E Sˆ 18 = gˆ 21 = (e21 C11 − e22 C12 )/C∗ , D Sˆ 67 = gˆ 16 =

e216

e16 , ε CE + ω11 44


D E E Sˆ 28 = gˆ 22 = (e22 C11 − e21 C12 )/C∗ , D σ −Sˆ 77 = βˆ11 =

E C44

e216

ε CE + ω11 44

D σ E E E 2 − Sˆ 88 = βˆ22 = (C11 C22 − (C12 ) )/C∗ ,

,

(11.52b)

where E E ε E E E 2 C∗ = (e221 + e222 )C11 − 2e21 e22 C12 + ω22 [C11 C22 − (C12 ) ]

(11.52c)

and all the other constants are zero, i.e., D D D D D D D D = Sˆ 15 = Sˆ 16 = Sˆ 17 = Sˆ 24 = Sˆ 25 = Sˆ 26 = Sˆ 27 Sˆ 14 D D D D D D D D D = Sˆ 45 = Sˆ 46 = Sˆ 47 = Sˆ 48 = Sˆ 56 = Sˆ 57 = Sˆ 58 = Sˆ 68 = Sˆ 78 = 0.

(11.52d)

Substituting (11.52a), (11.52b), (11.52c), and (11.52d) into (11.38b), we get D 2 D D 4 D D 2 D μ + Sˆ 44 , 4 = Sˆ 11 μ + (2Sˆ 12 + Sˆ 66 )μ + Sˆ 22 , 2 = Sˆ 55 σ 2 σ m3 = −(ˆg21 + gˆ 16 )μ2 − gˆ 22 , ρ2 = −(βˆ11 μ + βˆ22 ), 3 = m2 = 0.

(11.53)

With the results of (11.53), the characteristic equation (11.39) can be reduced to 2 (4 ρ2 − m23 ) = 0.

(11.54)

Let μ1 be the root of 2 = 0 and μ2 , μ3 , μ4 be the roots of 4 ρ2 − m23 = 0, whose imaginary parts are positive. Since 4 ρ2 − m23 = 0 is a third-order polynomial of μ2 whose coefficients are all real and μ cannot be real, the most general expressions for the roots of μ2 are one pair of complex conjugates and one real. And hence, we may let 3 D /S ˆ D , μ2 = α2 + iβ2 , μ3 = −α2 + iβ2 , μ4 = iβ4 . μ1 = i Sˆ 44 55

(11.55)

The material eigenvector matrix B can then be constructed through (11.40) with the values given in (11.53). Its final simplified expression is ⎤ 0 −c2 μ2 −c3 μ3 −c4 μ4 ⎢ 0 c2 c3 c4 ⎥ ⎥, B=⎢ ⎣c1 0 0 0 ⎦ 0 c2 η2 c3 η3 c4 η4 ⎡

(11.56a)

11.3


389

where ηk =

4 (μk ) m3 (μk ) = , m3 (μk ) ρ2 (μk )

k = 2, 3, 4.

(11.56b)

Note that through (11.53), (11.55) and (11.56b) we see that some relations exist for μk and ηk : μ3 = −μ2 , η3 = η2 , μ4 = −μ4 , η4 = η4 .

(11.57)

By the way similar to Example 1, the material eigenvector matrix A can then be written as ⎤ c2 a12 c3 a13 c4 a14 ⎢ c2 a22 c3 a23 c4 a24 ⎥ ⎥, A=⎢ ⎣c1 Sˆ D /μ1 0 0 0 ⎦ 44 0 c2 a42 c3 a43 c4 a44 ⎡

0 0

(11.58a)

where D 2 D a1 k = Sˆ 11 μk + Sˆ 12 + gˆ 21 ηk ,

D 2 D a2 k = (Sˆ 12 μk + Sˆ 22 + gˆ 22 ηk )/μk ,

σ a4 k = −(ˆg16 − βˆ11 ηk )μk ,

k = 2, 3, 4.

(11.58b)

The scaling factors ck , k = 1, 2, 3, 4 shown in (11.56) and (11.58) can be determined by the orthogonality relation (3.57), or obtained directly from (11.44).

11.3.3 Barnett–Lothe Tensors S, H, and L In two-dimensional problems, three real matrices S, H, and L defined in (3.59) appear frequently in the final real-form solutions. Although this definition is not valid for degenerate materials whose material eigenvector matrices A and B may not exist, such as Example 1 shown in the previous section, Barnett and Lothe (1973) devised an integral formalism to compute these matrices directly from the elastic stiffnesses. Their integral formalism (3.95) shows that S, H, and L are, respectively, the average values of N1 (θ ), N2 (θ ), and −N3 (θ ). By this integral formalism, the problems associated with degenerate materials disappear. Hence, S, H, and L sometimes are called Barnett–Lothe tensors. Due to the importance of S, H, and L, it is always desirable to have their explicit expressions. Although the integral formalism (3.95) has its advantage to avoid the degenerate problems, it is not convenient for the calculation. If we have the explicit expressions of A and B, it seems that a direct substitution by the definition (3.59) is a good approach. However, the presence of the normalization factors ck for the eigenvector matrices A and B would lead the direct substitution to unwieldy algebraic calculation. An alternative approach by employing AB-1 is suggested by Ting (1996). In the following, we will follow his

390

11


steps to find the explicit expressions of S, H, and L for the two examples discussed in the last section. Knowing that AB−1 = (ABT )(BBT )−1 and using the definitions given in (3.59), we have AB−1 = −(SL−1 + iL−1 ).

(11.59a)

When A is multiplied by B–1 , it is seen that the normalization factors cancel each other, which may prevent the unwieldy results by direct substitution into the definitions given in (3.59). Equation (11.59a) shows that L and S can be obtained, respectively, from the imaginary part and real part of AB–1 by L = (L−1 )−1 ,

S = (SL−1 )L.

(11.59b)

Once L and S are determined, H can be obtained by using the following identity (3.97b)1 : H = L−1 + S(SL−1 ).

(11.59c)

Example 1: Piezoelectric ceramics poling in x3 -axis Using the procedure outlined above, we first calculate AB−1 by (11.49) and (11.50). The result is ⎡ AB

−1

D 2iSˆ 11

⎢ ˆ D ˆ D ⎢S11 + S12 = −⎢ ⎢ 0 ⎣ 0

D + S ˆ D ) 0 −(Sˆ 11 12 D 2iSˆ 22

0

0

D iSˆ 44

0

iˆg24

⎤

0

⎥ 0 ⎥ ⎥, ⎥ iˆg24 ⎦ −iβˆ σ

(11.60)

11

which can be proved to be identical to that presented in Suo et al. (1992). Note that during the derivation of (11.60), μ1 = μ2 was assumed for the calculation of B−1 . After getting AB−1 in terms of μ1 and μ2 , we insert their actual values, i.e., μ1 = μ2 = i to get (11.60). By this way the problem of degeneracy disappears, which means that although B−1 does not exist for the degenerate materials, AB−1 and hence L, S, and H exist. With the result obtained in (11.60) and following the procedure described between (11.59a) and (11.59c), we now get

11.3


−1

L

391

⎡ ˆ D 2S11 0 0 0 ⎢ D ˆ ⎢ 0 2S11 0 0 =⎢ ⎢ D 0 Sˆ 44 gˆ 24 ⎣ 0 0 ⎡

0 ⎢1 S = k2 ⎢ ⎣0 0

−1 0 0 0

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

⎡ ˆ D −1 (2S11 ) 0 0 0 ⎢ D −1 ˆ ⎢ 0 0 0 (2S11 ) L=⎢ ⎢ σ 0 0 k1 βˆ11 k1 gˆ 24 ⎣

σ 0 gˆ 24 −βˆ11

⎤ 00 0 0⎥ ⎥, 0 0⎦ 00

0

⎡ ˆ D k3 S11 0 0 0 ⎢ D ˆ ⎢ 0 k3 S11 0 0 H=⎢ ⎢ D 0 Sˆ 44 gˆ 24 ⎣ 0 0

0

0 ⎤

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

D k1 gˆ 24 −k1 Sˆ 44

⎥ ⎥ ⎥, ⎥ ⎦

σ gˆ 24 −βˆ11

(11.61a)

where ε E ˆ σ ˆ D −1 = e215 + ω11 k1 = (ˆg2 C44 , 24 + β11 S44 )

k2 = (1 + k0 )/2,

D ˆ D k3 = (3 + k0 )(1 − k0 )/2, and k0 = Sˆ 12 /S11 .

(11.61b)

Example 2: Piezoelectric ceramics poling in x2 -axis By a similar approach as Example 1, we first calculate AB−1 by using the results given in (11.55), (11.56), (11.57), and (11.58). The result is ⎡

AB−1

∗ −χ iL11 21 ⎢ χ21 iL∗ 22 ⎢ = −⎣ 0 0 ∗ χ41 iL24

⎤ 0 −χ41 ∗ ⎥ 0 iL24 ⎥, ∗ iL33 0 ⎦ ∗ 0 iL44

(11.62a)

in which Lij∗ and χij are all real values and are related to the material constants by ! " ∗ D = 2Sˆ 11 Im μ22 η2 + (μ24 η2 − μ22 η4 ) /λ, L11 " ! ∗ L22 = 2 Im γ2 μ2 μ4 η4 + (γ2 μ24 − γ4 μ22 )η2 /μ2 μ4 /λ, 3 ∗ D S ˆ D , L33 = Sˆ 44 55 " ! ∗ L24 = −2 Im γ2 μ2 μ4 + (γ2 μ24 − γ4 μ22 ) /μ2 μ4 /λ σ Im μ2 μ2 η2 η4 + μ2 μ4 η2 (η2 − η4 ) /λ, = 2βˆ11 ∗ σ L44 = −2βˆ11 Im μ2 μ2 η2 + μ2 μ4 (η2 − η4 ) /λ, ! " D D χ21 = Sˆ 12 − 2Sˆ 11 Re μ22 μ2 η4 + μ2 μ4 η2 (μ2 − μ4 ) /λ % & D = Sˆ 12 + 2 Re γ2 μ4 η2 + (γ4 μ2 η2 − γ2 μ4 η4 ) /μ2 μ4 /λ, ! " D χ41 = gˆ 21 + 2Sˆ 11 Re μ22 μ2 + μ2 μ4 (μ2 − μ4 ) /λ σ Re μ2 η2 η2 − η2 η4 (μ2 − μ4 ) /λ = −ˆg16 + 2βˆ11

(11.62b)

392

11


and λ = 2 Re μ2 η2 + (μ4 η2 − μ2 η4 ) ,

D γk = Sˆ 22 + gˆ 22 ηk ,

k = 2, 4.

(11.62c)

∗ , χ , and χ , which come Note that in (11.62b) two equalities are given for L24 21 41 −1 −1 from the fact that L is symmetric and SL is skew-symmetric. With the result of (11.62) and the procedure described in (11.59), we get

⎡ ∗ −1 ⎤ ⎤ ∗ (L11 ) L11 0 0 0 0 0 0 ⎢ 0 ⎢ 0 L∗ 0 L∗ ⎥ ∗ ∗ ⎥ k L44 0 −k L24 ⎢ ⎢ ⎥ 22 24 ⎥ , L = L−1 = ⎢ ⎢ ⎥ ⎥, ∗ ∗ −1 ⎣ 0 ⎣ 0 0 L33 0 ⎦ 0 (L33 ) 0 ⎦ ∗ ∗ ∗ ∗ 0 L24 0 L44 0 −k L24 0 k L22 ⎡ ⎡ ⎤ ⎤ 0 S12 0 S14 H11 0 0 0 ⎢S21 0 0 0 ⎥ ⎢ 0 H22 0 H24 ⎥ ⎢ ⎥ ⎥ S=⎢ ⎣ 0 0 0 0 ⎦, H = ⎣ 0 0 H33 0 ⎦, 0 H24 0 H44 S41 0 0 0 (11.63a) where ⎡

∗ ∗ S12 = −k (χ21 L44 − χ41 L24 ), ∗ S21 = χ21 /L11 ,

∗ ∗ S14 = k (χ21 L24 − χ41 L22 ), ∗ S41 = χ41 /L11 ,

∗ 2 ∗ ∗ 2 ∗ H11 = L11 − k (χ21 L44 − 2χ21 χ41 L24 + χ41 L22 ),

∗ 2 ∗ H22 = L22 − (χ21 /L11 ),

∗ ∗ H24 = L24 − (χ21 χ41 /L11 ),

∗ 2 ∗ H44 = L44 − (χ41 /L11 ) (11.63b)

∗ H33 = L33 ,

and ∗ ∗ ∗ 2 −1 L44 − (L24 ) ] . k = [L22

(11.63c)

11.3.4 Bimaterial Matrices D and W Example 1: Piezoelectric ceramics poling in x3 -axis Using the expression of AB−1 given in (11.60) for two dissimilar piezoelectric ceramics poling in x3 -axis, matrices D and W defined in (7.11b) and (7.11c) can then be expressed as follows: ⎡

D1 ⎢0 D=⎢ ⎣0 0

0 D22 0 0

0 0 D33 D34

⎤ ⎡ 0 0 −W21 ⎢W21 0 0 ⎥ ⎥, W = ⎢ ⎣ 0 D34 ⎦ 0 D44 0 0

⎤ 00 0 0⎥ ⎥, 0 0⎦ 00

(11.64a)

11.4


393

in which the nonzero components of Dij and Wij are obtained directly from (11.60) by using the relations (11.59a), (7.11b), and (7.11c). For example, D D )1 + (2Sˆ 11 )2 , D11 = (2Sˆ 11

D D D D W21 = (Sˆ 11 + Sˆ 12 )1 − (Sˆ 11 + Sˆ 12 )2 , . . . , etc. (11.64b)

Example 2: Piezoelectric ceramics poling in x2 -axis Similar to the previous piezoelectric ceramics, D and W for this case can be expressed by ⎡

D11 ⎢0 D=⎢ ⎣0 0

0 D22 0 D24

0 0 D33 0

⎤ 0 D24 ⎥ ⎥, 0 ⎦ D44

⎡

0 −W21 ⎢W21 0 W=⎢ ⎣0 0 W41 0

⎤ 0 −W41 0 0 ⎥ ⎥, 0 0 ⎦ 0 0

(11.65a)

in which the nonzero components of Dij and Wij are obtained directly from (11.62) by using the relations (11.59a), (7.11b), and (7.11c). For example, ∗ ∗ )1 + (L11 )2 , D11 = (L11

W21 = (χ21 )1 − (χ21 )2 , . . . , etc.

(11.65b)

11.4 Multi-material Wedges Consider a piezoelectric multi-wedge that consists of n different piezoelectric anisotropic elastic wedges as that shown in Fig. 5.3. Assume perfect bond along each interface between two dissimilar wedges and each wedge occupies the region θk−1 ≤ θ ≤ θk , k = 1, 2, . . . , n. As stated in (11.23a), the traction/induction and displacement/electric potential continuity conditions across each interface θ = θk , k = 1, 2, . . ., n − 1, between two dissimilar wedges can be written as uk (θk ) = uk+1 (θk ),

φk (θk ) = φk+1 (θk ),

k = 1, 2, . . . . . . , n − 1,

(11.66)

of which the mathematical form is exactly the same as those given in (5.58)1 . The two outer surfaces of the multi-wedges may be bonded together or free–free, fixed– fixed, free–fixed, or fixed–free (here “free” means mechanically free and electrically open and “fixed” means mechanically clamped and electrically closed). By (11.23a), (11.23b), and (11.23c), these boundary conditions can also be expressed in the forms which are exactly the same as those given in (5.58)2 , (5.86), (5.91), and (5.93), i.e., bonded :

u1 (θ0 ) = un (θn ),

φ1 (θ0 ) = φn (θn ),

free - free: φ1 (θ0 ) = φn (θn ) = 0, free - fixed: φ1 (θ0 ) = un (θn ) = 0,

fixed - fixed: u1 (θ0 ) = un (θn ) = 0, (11.67) fixed - free: u1 (θ0 ) = φn (θn ) = 0.

Since the mathematical forms of the general solution (11.16), the boundary conditions (11.66) and (11.67), and all the related equations of the present problems are

394

11


exactly the same as the corresponding pure anisotropic multi-material wedge problems discussed in Sections 5.3.1 and 5.3.2, the solutions for the present problem should also preserve the same mathematical form as the solutions for the corresponding pure anisotropic multi-material wedge problems. The only difference is the contents and dimensions of the matrices/vectors, which have all been clearly defined in the expanded Stroh formalism discussed in Section 11.2. With this understanding, we can now write down the solutions for the present problem by just referring to those given in Section 5.3 for the pure anisotropic multi-material wedge problems.

11.4.1 Orders of Stress/Electric Singularity The orders of stress/electric singularity of the piezoelectric multi-wedges can be determined by the eigenrelations mentioned later in (11.69c), which provide the following relations for the singular orders: Ke − I = 0, ) ) ) ) ) (2) ) ) ) K = 0, fixed fixed: free - free: )K(3) ) ) e e ) = 0, ) ) ) ) ) ) (4) ) ) free - fixed: )K(1) e ) = 0, fixed - free: )Ke ) = 0,

bonded :

(11.68)

(i)

where Ke , i = 1, 2, 3, 4, are the submatrices of Ke defined by (5.83b) and (5.88).

11.4.2 Near-Tip Solutions The displacements, stresses, electric fields, and electric displacements near the apex of piezoelectric multi-wedges can be written in the same form as that shown in (5.94). For the convenience of following discussions, (5.94) is now rewritten as

uk (r, θ ) φk (r, θ )

=r

1−δ

ˆ 1−δ (θ , θk−1 )(Ke )k−1 N k

u0 , φ0

k = 1, 2, 3, . . . ., n, (11.69a)

in which (r, θ ) is the polar coordinate with origin located on the wedge apex, uk (r, θ ) is a 4×1 vector containing three displacement components and one electric potential of the kth wedge, φk (r, θ ) is a 4 × 1 vector containing three stress functions and one potential of electric displacement of the kth wedge, and (Ke )k−1 is a 8 × 8 matrix defined by

(Ke )k−1

⎧ k = 1, ⎨ I, k−1 ; = Ek−i = Ek−1 Ek−2 . . . E1 , k = 2, 3, . . . , n, ⎩ i=1

(11.69b)

11.4


395

in which Ek is defined in (5.82b) and u0 and φ0 are two 4×1 coefficient vectors which can be determined through the eigenrelation of singular orders, i.e., bonded :

(Ke − I)w0 = 0,

w0 = (u0 φ0 )T ,

K(3) e u0 = 0,

φ0 = 0,

fixed - fixed: K(2) e φ0 = 0, (1) free - fixed: Ke u0 = 0, fixed - free: K(4) e φ0 = 0,

u0 = 0,

free - free:

(11.69c)

φ0 = 0, u0 = 0.

It should be noted that if the singular order δ is a simple root, the associated eigenvector (u0 , φ0 ) is unique up to a constant multiplier. If δ is a root of multiplicity m, the eigenrelation (11.69c) may or may not provide m independent eigenvectors (u0 , φ0 ). When the eigenrelation cannot provide enough independent eigenvectors, the missing eigenfunctions can be found by differentiating the eigenfunction (11.69a) with respect to δ, which will provide the supplemented solutions with logarithmic singularity in the form of (ln r)r1−δ . If the singular order is a complex number, a conjugate part should be superimposed to get the real values for the generalized stresses and displacements. Moreover, if one concerns not only the near apex field but also the entire domain of the multi-material wedges, the solutions corresponding to all the singular and non-singular orders should be superimposed. If we consider only the most critical singular order and disregard the possibility of logarithmic singularity, by superimposing all the eigenfunctions with the same real part of singular orders, without loss of generality the near-tip solutions shown in (11.69a) can be further written as u(r, θ ) = r1−δR V(θ ) < riεα > c, φ(r, θ ) = r1−δR (θ ) < riεα > c,

(11.70a)

where V(θ ) = [η1 (θ ) η2 (θ ) η3 (θ ) η4 (θ )], (θ ) = [λ1 (θ ) λ2 (θ ) λ3 (θ ) λ4 (θ )]

(11.70b)

ˆ 1−δ (θ , θk−1 ), (Ke )k−1 and and ηi (θ ) and λi (θ ), i=1,2,3,4, are functions related to N k the eigenvectors (u0 , φ0 ) determined from (11.69c). In (11.70a), c is a coefficient vector containing the constant multiplier cα of each eigenvector (u0 , φ0 ) associated with δ = δR + iεα , α = 1, 2, 3, 4. For example, if δ is a real and simple root, c1 = 0, ε1 = 0, c2 = c3 = c4 = 0, if δ is a real and double root, c1 , c2 = 0, ε1 = ε2 = 0, c3 = c4 = 0, if δ is a real and triple root, c1 , c2 , c3 = 0, ε1 = ε2 = ε3 = 0, c4 = 0; if δ is a complex and simple root, c2 = c1 , ε2 = −ε1 , c3 = c4 = 0, if the most critical singular order happens to contain a simple real root and a simple complex root that have the same real part, we may take c2 = c1 , ε2 = −ε1 , c3 = 0, ε3 = 0, c4 = 0; if the most critical singular order happens to contain a double real root and a simple

396

11


complex root that have the same real part, we may take c2 = c1 , ε2 = −ε1 , c3 = 0, c4 = 0,ε3 = ε4 = 0, etc. Note that the solution shown in (11.70) covers all five different cases discussed in (5.104) for pure anisotropic multi-material wedges. For piezoelectric multi-wedges, it may contain more than five different cases; however, by using the expression shown in (11.70) all possible different cases are included, which is useful for the further discussion. Since (11.70) is valid for any wedge of the piezoelectric multiwedges, from now on unless special notification is needed the subscript k denoting the wedge is neglected for simplicity.

11.4.3 Stress/Electric Intensity Factors A proper definition for the interface corner has been proposed in Section 5.4.2, which can be reduced to the conventional definition for cracks in homogeneous anisotropic materials or for cracks along the interfaces between two dissimilar anisotropic materials. We now try to extend this definition to the present cases of piezoelectric multi-wedges which include corners without any interfaces. For the problems of cracks or interface cracks or interface corners, it is quite apparent to choose θ = 0 along the crack or interface to study the intensity factors of stresses on the surface parallel to the crack or interface. For the general wedge problems, if we choose θ = 0 on different directions, different stress intensity factors are studied. Just like the stresses are second-order tensors, the stress intensity factors for different directions may also be second-order tensors. In other words, if we know the stress intensity factors on the surfaces parallel to the x1 x3 or x2 x3 or x1 x2 planes, the stress intensity factors on the other surfaces may be obtained through the transformation of second-order tensors. Through this way, one may study whether the principal stress intensity factor and its associated principal direction are useful for the prediction of propagation of wedge apex. Several works are needed to make sure the above statements. To focus on the piezoelectric effects, in this section we disregard the difference coming up from the different choices of the coordinates. The direction θ = 0 is chosen arbitrarily for the following discussions. From (11.22) and (11.70), we have ⎧ ⎫ σrθ ⎪ ⎪ ⎪ ⎬ ⎨ ⎪ σθθ = (θ )φ, r (r, θ ) = r−δR (θ ) (θ ) < (1 − δR + iεα )riεα > c, σ ⎪ ⎪ θ3 ⎪ ⎪ ⎭ ⎩ Dθ

(11.71a)

where ⎡

⎤ cos θ sin θ 0 0 ⎢− sin θ cos θ 0 0⎥ ⎥. (θ ) = ⎢ ⎣ 0 0 1 0⎦ 0 0 01

(11.71b)

11.4


397

Along θ = 0, (0) = I, and we let (0) = = [λ1 λ2 λ3 λ4 ]. The coefficient c can then be defined as c = lim rδR < (1 − δR + iεα )−1 r−iεα > −1 φ, r (r, 0). r→0

(11.72)

Similar to the discussions stated in Section 7.4.5, from (11.71a) with θ = 0 we see that the coefficient < (1 − δR + iεα ) > c can be thought as the intensity of singularity of the stresses (σrθ , σθθ , σθ3 ) and electric displacement Dθ in the direction of λ1 , λ2 , λ3 , and λ4 . However, as a consequence of the peculiar singularity, c has an awkward physical unit. A remedy suggested by Rice (1988) is to appeal to a √ fixed length and use the combination 2π < (1 − δR + iεα )iεα > c as the basic parameter which has the dimension and the scale of conventional stress intensity factors. Even so, the stress intensity factors still cannot be reduced to classical stress intensity factors for a crack tip in a homogeneous medium, because the directions of λ1 , λ2 , λ3 , and λ4 are usually not the same as the direction of the crack. To have a comparable definition, transformation pre-multiplied by is suggested in (7.115) for the interface crack problems. With this understanding, a proper definition of stress/electric intensity factors for the piezoelectric multi-wedges can be given as ⎫ ⎧ ⎫ ⎧ σrθ ⎪ KII ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎪ ⎬ ⎨ √ KI δR −iεα −1 σθθ = lim 2π r < (r/) > σ3θ ⎪ ⎪ ⎪ ⎪ r→0 ⎪ ⎪KIII ⎪ ⎭ ⎩ ⎪ ⎭ ⎩ θ=0 KIV Dθ

(11.73a)

or in matrix form k = lim

√

r→0 θ=0

2π rδR < (r/)−iεα > −1 φ, r (r, θ ).

(11.73b)

From (11.72) and (11.73b), we see that the relation between the coefficient vector c and the stress intensity factors k are k=

√

2π < (1 − δR + iεα )iεα > c.

(11.74)

Note that from the definition given in (11.73), we see that < (r/)−iεα > −1 should be real in order to have a real value of k, which has been proved analytically in (7.112) for the interface cracks lying between two anisotropic elastic materials. For the general piezoelectric multi-wedges, only numerical check has been done. Using the relation (11.74), the near-tip solution shown in (11.70) and (11.71) can now be rewritten in terms of the stress/electric intensity factors k as

398

11


1 u(r, θ ) = √ r1−δR V(θ ) < (1 − δR + iεα )−1 (r/)iεα > −1 k, 2π 1 φ(r, θ ) = √ r1−δR (θ ) < (1 − δR + iεα )−1 (r/)iεα > −1 k, 2π 1 φ, r (r, θ ) = √ r−δR (θ ) < (r/)iεα > −1 k. 2π

(11.75)

If we consider the intensity factors associated with lower singular orders, the stress field near the wedge apex may be expressed as φ, r (r, θ ) = φc,r (r, θ ) + φ2,r (r, θ ) + · · · ,

(11.76a)

where 1 φc,r (r, θ ) = √ r−δR (θ ) < (r/)iεα > −1 k, 2π (2) 1 −δ (2) φ2,r (r, θ ) = √ r R 2 (θ ) < (r/)iεα > −1 2 k2 . 2π

(11.76b)

The subscript c denotes the value associated with the most critical singular order and the subscript 2 or the superscript (2) denotes the value associated with the second critical singular order. By the way similar to k, the stress intensity factors k2 associated with the second critical singular order can be defined as k = lim

r→0

√

(2)

(2)

2π rδR 2 < (r/)−iεα > −1 2 [φ, r (r, 0) − φc,r (r, 0)].

(11.77)

Similar definitions can also be applied to any other lower order terms.

11.4.4 Corner Opening Displacement/Electric Potential The difference of the displacements and electric potential between two outer wedge surfaces can be calculated through the use of (11.75)1 as u(r, θ ) = u(r, θn ) − u(r, θ0 ) 1 = √ r1−δR [V(θn ) − V(θ0 )] < (1 − δR + iεα )−1 (r/)iεα > −1 k, 2π (11.78) in which the first three components of u denote the corner opening displacements and the last component of u denotes the difference of electric potential on the two outer wedge surfaces.

11.5


399

11.5 Singular Characteristics of Cracks 11.5.1 Cracks in Homogeneous Piezoelectric Materials As described in Section 7.1.1, a semi-infinite crack in piezoelectric materials can be represented by letting θ0 = −π and θ1 = π for the single wedge with free-free surfaces. With this special wedge angle, the solutions for the singular order obtained in (11.68a)2 can be further reduced. Since the mathematical form of (11.68a) is exactly the same as that for the corresponding anisotropic problems, without any further derivation we can now write down the results obtained in (7.2), i.e., K(3) e = sin(2δπ )L.

(11.79)

From the explicit expression of L shown in Section 11.3.3, we know that L = 0, and hence the singular order of the crack in piezoelectric materials is 1/2, which is (3) a well-known result in fracture mechanics. Since Ke is identical to a zero matrix when δ = 1/2, the eigenvector u0 of (11.69c)2 will be totally arbitrary. In other words, no relation between four components of u0 . With this result, the near-tip solution can be further reduced from (11.69a) and written in the form of (11.70a), i.e., u(r, θ ) = r1/2 V(θ )c,

φ(r, θ ) = r1/2 (θ )c,

(11.80a)

where 1/2

T

1/2

T

T V(θ ) = A < μˆ 1/2 ˆ α (θ , −π ) > B , α (θ , −π ) > B + A < μ

(11.80b)

T ˆ α (θ , −π ) > B . (θ ) = B < μˆ 1/2 α (θ , −π ) > B + B < μ 1/2

Knowing that μˆ α (0, −π ) = eiπ/2 = i, from (11.80b)2 and the definition of L given in (3.59) we have (0) = −L, and hence, 1 φ, r (r, 0) = − r−1/2 Lc, 2

(11.81)

which will lead (11.73b) to + k = − π/2Lc.

(11.82)

With the above relation (11.82), the near-tip solution (11.80) can now be rewritten in terms of the stress/electric intensity factors k as + u(r, θ ) = − 2/π r1/2 V(θ )L−1 k,

+ φ(r, θ ) = − 2/π r1/2 (θ )L−1 k,

(11.83)

which can also be expressed in the form obtained in (7.10). Through this simple solution (11.83), the crack opening displacement/electric potential u, the energy

400

11


release rate G, and the mechanical energy release rate GM introduced by Park and Sun (1995) can all be written in terms of the stress/electric intensity factors as 2 u = u(r, π ) − u(r, −π ) = 2

2 1/2 −1 r L k, π

a 1 ui (s − a)σi2 (s) ds, i = 1, 2, 3, 4, a→0 2a 0 a 1 1 = lim uT (s − a)φ (s) ds = kT L−1 k, a→0 2a 0 2 a 1 GM = lim ui (s − a)σi2 (s) ds, i = 1, 2, 3, a→0 2a 0 a 1 1 = lim uTM (s − a)φM (s) ds = kT L−T M kM . a→0 2a 0 2

(11.84a)

G = lim

(11.84b)

(11.84c)

In (11.84b) and (11.84c), the superscript • denotes the differentiation with respect to the argument of the function; the subscript M denotes the values associated with the mechanical responses, e.g., uM , φM , and kM are 3×1 vectors containing the first three components of u, φ , and k, L−T M is a 4×3 matrix containing the first three columns of L−1 . By using the explicit expressions shown in (11.61a) for L−1 of the piezoelectric ceramics poling in x3 -axis, the crack opening displacement/electric potential u, energy release rate G, and mechanical energy release rate GM obtained in (11.84a), (11.84b), and (11.84c) can be further expressed in the following explicit component forms: 2 u = 2

2 1/2 ˆ D D D σ r (2S11 KII , 2Sˆ 11 KI , Sˆ 44 KIII + gˆ 24 KIV , gˆ 24 KIII − βˆ11 KIV )T , (11.85a) π

D 2 D 2 D 2 σ 2 KI + 2Sˆ 11 KII + Sˆ 44 KIII − βˆ11 KIV + 2ˆg24 KIII KIV , 2G = 2Sˆ 11

(11.85b)

D 2 D 2 D 2 KI + 2Sˆ 11 KII + Sˆ 44 KIII + 2ˆg24 KIII KIV . 2GM = 2Sˆ 11

(11.85c)

Similarly, for the piezoelectric ceramics poling in x2 -axis, use of (11.63) gives us 2 u = 2

2 1/2 ∗ ∗ ∗ ∗ ∗ ∗ r (L11 KII , L22 KI + L24 KIV , L33 KIV , L24 KI + L44 KIV )T , (11.86a) π

∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ KI + L11 KII + L33 KIII + L44 KIV + 2L24 KI KIV , 2G = L22

(11.86b)

∗ 2 ∗ 2 ∗ 2 ∗ KI + L11 KII + L33 KIII + 2L24 KI KIV . 2GM = L22

(11.86c)

11.5


401

11.5.2 Interface Cracks Between Two Dissimilar Piezoelectric Materials If we set θ0 = −π , θ1 = 0, and θ2 = π , a bi-wedge with free–free surfaces can represent a bimaterial with a semi-infinite interface crack. With these special angles, the near-tip solution shown in (11.69a) can be further reduced to ⎧ ⎨u1 (r, θ ) = r1−δ E∗(1) 1 (θ )u0 ,

⎧ (1) ∗(2) (3) ⎨u2 (r, θ ) = r1−δ [E∗(1) 2 (θ )E1 + E2 (θ )E1 ]u0 ,

⎩φ (r, θ ) = r1−δ E∗(3) (θ )u , 0 1 1

⎩φ (r, θ ) = r1−δ [E∗(3) (θ )E(1) + (E∗(1) (θ ))T E(3) ]u , 0 2 2 1 2 1 (11.87a)

in which ⎡ ⎣

∗(1)

⎤

∗(2)

Ek (θ ) Ek (θ ) ∗(3) Ek (θ )

∗(4) Ek (θ )

⎡ ⎣

ˆ 1−δ (θ , θk−1 ), k = 1, 2. ⎦ = E∗k (θ ) = N k

(1)

(2)

(3) Ek

(4) Ek

Ek Ek

(11.87b)

⎤ ˆ 1−δ (θk , θk−1 ) ⎦ = Ek = E∗k (θk ) = N k

Sk Hk = −(cos δπ )I + sin δπ Lk STk

(11.87c)

k = 1, 2.

(3)

The matrix Ke used for the determination of the singular orders can then be calculated from (5.88) in which Ke = E2 E1 for the present problem. The final simplified (3) expression for Ke has been shown in (7.11a). For the convenience of following discussions, we rewrite (7.11a) as follows: K(3) e =

1 ∗ (sin δπ )e−iδπ L2 (M∗ + e2iδπ M )L1 . 2

(11.88)

It is known that matrix L is negative definite for piezoelectric materials (Kuo and Barnett, 1991), and hence L1 = 0, L2 = 0. Also, sin δπ = 0 in the region of 0 < Re(δ) < 1. With these nonzero values in (11.88), the determination of the singular order δ and its associated eigenvector u0 from (11.68)2 and (11.69c)2 can now be reduced to ∗

(M∗ + e2iδπ M )λ = 0, where λ = − sin δπ L1 u0 .

(11.89)

In the above a factor − sin δπ is multiplied on L1 u0 for the purpose of adjusting the eigenvector λ to λ(0) defined later in (11.96d). By using the last equality of (7.11b), (11.89) can be further reduced to (W − cot δπ D)λ = 0.

(11.90)

402

11


The explicit solution to (11.90) has been given by Ting (1986) for the pure anisotropic materials and by Kuo and Barnett (1991) for piezoelectric materials. A simple derivation for readers to quickly understand how the solution was derived is presented as follows. To solve the singular order and its associated eigenvector from (11.90), we first make a replacement of δ by letting δ=

1 + iε. 2

(11.91)

Note that this replacement has no indication that ε is real; actually it will be shown that two of ε are real, while the other two are pure imaginary. With this replacement, (11.90) can be rewritten as (W + iβD)λ = D(D−1 W + iβI)λ = 0

or

(D−1 W + iβI)λ = 0,

(11.92a)

where β = tanh πε and cot δπ = −i tanh πε = −iβ.

(11.92b)

Inversely, we have ε=

1 1+β 1 tanh−1 β = ln . π 2π 1 − β

(11.92c)

Since the second equation of (11.92a) is a standard eigenvalue problem of matrix D−1 W, its eigenvalue can be determined by the following polynomial equation (−iβ)4 + I1 (−iβ)3 + I2 (−iβ)2 + I3 (−iβ) + I4 = 0,

(11.93)

in which I1 , I2 , I3 , and I4 are four invariants of the matrix D−1 W. Knowing that D is real and symmetric, and W is real and antisymmetric, by taking the complex conjugate or transpose of (11.92a)2 we observe that if β is a root of (11.92a)2 , so are −β, β and −β (Suo et al. 1992). Therefore, in (11.93) I1 = I3 = 0, and the solution to (11.93) can then be obtained as β1 = βε , β2 = −βε , β3 = iβτ , β4 = −iβτ ,

(11.94a)

where 3 3 βε = {[ I22 − 4I4 + I2 ]/2}1/2 , βτ = {[ I22 − 4I4 − I2 ]/2}1/2 , ) ) 1 ) ) I4 = )D−1 W) , I2 = − tr(D−1 W)2 , 2

(11.94b)

in which tr stands for the trace of the matrix. Note that since I4 ≤ 0, βε and βτ are real. From (11.91) and (11.92c), we have

11.5


δα =

403

1 + iεα , α = 1, 2, 3, 4, and ε1 = ε, ε2 = −ε, ε3 = iτ , ε4 = −iτ , (11.95a) 2

where ε is called the oscillatory index since it characterizes the oscillatory behavior of the stresses near the crack tip and τ is the difference of the singular order affected by the piezoelectric properties, whose values can be determined by ε=

1 1 1 + βε = tanh−1 βε , ln 2π 1 − βε π

τ=

1 tan−1 βτ . π

(11.95b)

Since βε and βτ are real, so are ε and τ by (11.95b). Let cα , α = 1, 2, 3, 4, be the constant multiplier of λα for each singular order δα = 1/2 + iεα , the near-tip solution (11.87a) can now be written in the form of (11.70a), i.e., uk (r, θ ) = r1/2 Vk (θ ) < riεα > c, φk (r, θ ) = r1/2 k (θ ) < riεα > c,

(11.96a)

k = 1, 2,

where Vk (θ ) = [η1 (θ ) η2 (θ ) η3 (θ ) η4 (θ )]k , k (θ ) = [λ1 (θ ) λ2 (θ ) λ3 (θ ) λ4 (θ )]k

(11.96b)

and ηα (θ ) =

λα (θ ) =

∗(1)

−(E1 (θ ))α L−1 for material 1, 1 λα / sin δα π , ∗(1) (1) ∗(2) (3) −1 −[E2 (θ )E1 + E2 (θ )E1 ]α L1 λα / sin δα π , for material 2, (11.96c)

⎧ −1 ⎨−(E∗(3) 1 (θ ))α L1 λα / sin δα π ,

for material 1,

⎩−[E∗(3) (θ )E(1) + E∗(1) (θ )E(3) ] L−1 λ / sin δ π , α α 2 1 2 1 α 1

for material 2. (11.96d)

In the above, the subscript α = 1, 2, 3, 4 denotes the values related to δα . From (11.87b), (11.87c), and (5.77), we know that E∗1 (0) = E1 , E∗2 (π ) = E2 , E∗1 (π ) = E∗2 (0) = I.

(11.96e)

With the values obtained in (11.96e) and the relation given in (11.74), the near-tip solution (11.96a), the stresses/electric displacement along the interface φ, r (r, 0), the crack opening displacement/electric potential u, and the energy release rate G can all be written in terms of stress/electric intensity factors k as

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11


2

2r Vk (θ ) < (1 + 2iεα )−1 (r/)iεα > −1 k, k = 1, 2, π 2 2r φk (r, θ ) = k (θ ) < (1 + 2iεα )−1 (r/)iεα > −1 k, k = 1, 2, π 1 φ1,r (r, 0) = φ2,r (r, 0) = √ < (r/)iεα > −1 k, 2π r 2 2r −T (r/)iεα u(r) = u2 (r, π ) − u1 (r, −π ) = < > −1 k, π (1 + 2iεα ) cosh π εα 1 G = kT (D + WD−1 W)k. 4 (11.97)

uk (r, θ ) =

Note that the mathematical form of (11.97) is exactly the same as that of the corresponding pure anisotropic interface crack problems shown in Section 7.4.5 which is derived by a different approach. By using the explicit expressions shown in (11.64) for D and W of the piezoelectric ceramics poling in x3 -axis, the relation between energy release rate G and stress/electric intensity factors shown in the last equation of (11.97) can be written in component form as 4G =

2 2 D11 D22 − W21 D11 D22 − W21 2 2 KI2 + KII2 + D33 KIII + D44 KIV + 2D34 KIII KIV . D11 D22 (11.98)

Similarly, for the piezoelectric ceramics poling in x2 -axis, use of (11.65) will give us # $ 2 2 − 2D W W + D W 2 D44 W21 D11 D22 − W21 24 21 41 22 41 2 KI + D11 − 4G = KII2 D11 D22 D44 − D224 2 + D33 KIII

+

2 D11 D44 − W41 D11 D24 − W21 W41 2 KIV +2 KI KIV . D11 D11

(11.99)

11.6 Some Crack Problems As stated at the beginning of Section 11.4, if the mathematical forms of general solution (11.16) and boundary conditions of the problems in piezoelectric materials are exactly the same as those of the corresponding pure anisotropic elastic problems, the solutions for the cracks in piezoelectric materials can be obtained directly by referring to their counterparts for pure anisotropic materials. By this way, most of the solutions for cracks in piezoelectric materials can be obtained directly without

11.6

Some Crack Problems

405

any further derivation. With this understanding, we will not list all the full-field solutions for the crack problems. For the convenience of readers’ reference, we only list the solutions related to the stress/electric intensity factors and energy release rates with focus on the explicit expressions in component form. For the other kinds of problems such as holes, inclusions, and contacts, which will not be discussed in this chapter, similar approach can be applied.

11.6.1 Cracks Collinear Cracks Subjected to Uniform Load/Induction at Infinity The full-field solutions u(r, θ ) and φ(r, θ ) for displacements, electric potential, stress functions, and induction potential can be obtained by referring to the solutions presented in Section 7.3 for pure anisotropic problems. By reducing the full-field solutions to the near tip, the stress/electric intensity factors k, crack opening displacements u, and energy release rate G for several different crack numbers all have the following relations: k = kt∞ 2 ,

u = f (x1 )L−1 t∞ 2 ,

G=

1 2 ∞ T −1 ∞ k (t2 ) L t2 , 2

(11.100a)

.

(11.100b)

in which

t∞ 2 =

⎧ ∞⎫ σ12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨σ ∞ ⎪ ⎬ 22 ∞⎪ ⎪ σ23 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∞⎪ ⎭ D2

k and f (x1 ) will be different for different crack geometries. In other words, the crack geometry is responsible for the factor k and the opening shape f (x1 ), while the effects of external loading/induction and material properties are reflected by t∞ 2 and L−1 . The stress/electric intensity factors are independent of the material properties and have linear relations with the external loading/induction. The crack opening displacements and energy release rate are influenced by the material properties through the real matrix L−1 whose explicit expressions have been shown in (11.61a) and (11.63a) for two specific piezoelectric materials. The detailed expressions of k and f (x1 ) for different crack geometries are shown below. I. A crack located on −a ≤ x1 ≤ a: see (7.18) and (7.19) k=

√ π a,

3 f (x1 ) = 2 a2 − x12 .

II. Two collinear cracks located on a1 ≤ x1 ≤ b1 and a2 ≤ x1 ≤ b2 :

(11.101a)

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11

k : (7.48c),


f (x1 ) : (7.49b).

(11.101b)

III. Evenly spaced collinear periodic cracks. (The length of each crack is 2 and the distance between the crack centers is W.) k : (7.58),

f (x1 ) : (7.59b).

(11.101c)

A Crack Subjected to Arbitrary Load/Induction on the Crack Surfaces By referring to the corresponding problems for pure anisotropic materials shown in Section 7.2.3, the stress/electric intensity factor for a crack subjected to arbitrary load/induction on the crack surfaces can be expressed by (7.23)3 . A Crack Subjected to Point Force/Charge at Arbitrary Location The stress/electric intensity factor for a crack subjected to point force/charge applied at arbitrary location can be written as that shown in (7.25). If the point force/charge is applied on the upper crack surface, the simplified solution is shown in (7.26).

11.6.2 Interface Cracks As the discussion stated following (11.100), the effects of material properties on k, u, and G are reflected only by L−1 , whose explicit expressions have been given in (11.61a) and (11.63a) for two piezoelectric ceramics. All the others are the same as those shown in Sections 7.2 and 7.3 for anisotropic materials. Therefore, no further detailed expressions are provided in the last section. Unlike the cracks in homogeneous materials, the piezoeffects will influence the results of interface cracks through the combination of material properties and crack geometries. Since most of the expressions shown below contain < cα > −1 , before presenting the results for each different interface crack problem we like to show the explicit expressions of < cα > −1 for two different piezoelectric ceramics discussed in Section 11.3. Explicit Expressions With βα obtained in (11.94), the matrix D−1 W of (11.92a)2 can be diagonalized by its eigenvalues βα , α = 1, 2, 3, 4, as iD−1 W = < βα > −1 , = [λ1 λ2 λ3 λ4 ].

(11.102)

Followings are the determination of βα , their associated eigenvector matrix , and the expression < cα > −1 for two typical piezoelectric materials. (i) Piezoelectric ceramics poling in x3 -axis With the bimaterial properties given in (11.64), βα can be determined from (11.94b) as

11.6

Some Crack Problems

407

+ β1 = βε = W21 / D11 D22 , β2 = −βε , β3 = β4 = 0.

(11.103)

Substituting (11.103) into (11.92a)2 , the associated eigenvector matrix can be constructed as ⎤ ⎡ −ik1 W21 /βε D11 ik2 W21 /βε D11 0 0 ⎢ k2 0 0⎥ k1 ⎥, (11.104) =⎢ ⎣ 0 0 k3 0 ⎦ 0 0 0 k4 in which k1 , k2 , k3 , and k4 are scaling factors. With (11.104), we can get the following relation, which is useful for the calculation of the component expressions of the stress/electric intensity factors shown in this section: ⎡ −1

< cα >

cR

√ ⎢ ⎢−cI D11 /D22 =⎢ ⎢ 0 ⎣

√ ⎤ cI D22 /D11 0 0 ⎥ cR 0 0⎥ ⎥, 0 1 0⎥ ⎦

0

0

(11.105a)

0 1

in which c1 = cR + icI ,

c2 = c1 = cR − icI ,

c3 = c4 = 1.

(11.105b)

(ii) Piezoelectric ceramics poling in x2 -axis Similar to material (i), with the properties given in (11.65), βα can be obtained from (11.94b) as * β1 = βε =

2 − 2D W W + D W 2 D44 W21 24 21 41 22 41

D11 (D22 D44 − D224 )

, β2 = −βε , β3 = β4 = 0. (11.106)

Substituting (11.106) into (11.92a)2 , we have ⎡

k1

k2

0

0

⎤

⎢ ⎥ ⎢ik1 d1 /βε −ik2 d1 /βε 0 k4 W41 ⎥ ⎢ ⎥, =⎢ ⎥ 0 0 k 0 3 ⎣ ⎦ ik1 d2 /βε −ik2 d2 /βε 0 −k4 W21

(11.107a)

in which k1 , k2 , k3 , and k4 are scaling factors and d1 =

D44 W21 − D24 W41 D22 W41 − D24 W21 , d2 = . 2 D22 D44 − D24 D22 D44 − D224

With (11.107) and (11.105b), we get

(11.107b)

408

11


⎤ cR cI W21 /βε 0 cI W41 /βε 2 2 ⎢ −cI d1 /βε (cR d1 W21 + d2 W41 )/βε 0 (cR − 1)d1 W41 /βε ⎥ ⎥. < cα > −1 = ⎢ ⎦ ⎣ 0 0 1 0 2 2 (cR − 1)d2 W21 /βε 0 (cR d2 W41 + d1 W21 )/βε −cI d2 /βε (11.108) ⎡

A Semi-infinite Interface Crack Subjected to Point Force/Charge on Crack Surfaces Let the semi-infinite planes of different materials be joined along the positive x1 axis. A line crack is situated along the negative x1 -axis extending from x1 = 0 to x1 = −∞ and is opened by a point force/charge pˆ = (ˆp1 , pˆ 2 , pˆ 3 , qˆ ) at x1 = −a on each side of the crack. The stress/electric intensity factors for this and the following interface crack problems can be found by referring to the results obtained in (7.125), i.e., 2 k=

2 ˆ < (a/)−iεα cos hπεα > −1 p. πa

(11.109)

To write down (11.109) in explicit component form, we use the relations given in (11.105) and (11.108) for two typical piezoelectric materials. The results are as follows: (i) Piezoelectric ceramics poling in x3 -axis: + 2/πa hˆ 1 (ε), KII = 2/πa hˆ 2 (ε), + + = 2/πa pˆ 3 , KIV = 2/πa qˆ ,

KI = KIII

+

(11.110a)

where + hˆ 1 (ε) = cosh(πε)[ˆp2 cos(εln(a/)) + pˆ 1 D11 /D22 sin(εln(a/))], (11.110b) + hˆ 2 (ε) = cosh(πε)[ˆp1 cos(εln(a/)) − pˆ 2 D22 /D11 sin(εln(a/))]. (ii) Piezoelectric ceramics poling in x2 -axis: + + 2/πa{d1 hˆ 1 (ε) + W41 (d2 pˆ 2 − d1 qˆ )}/βε2 , KII = 2/πa hˆ 2 (ε)/βε , + + KIII = 2/πa pˆ 3 , KIV = 2/πa{d2 hˆ 1 (ε) − W21 (d2 pˆ 2 − d1 qˆ )}/βε2 , (11.111a) where d1 and d2 are given in (11.107b) and KI =

hˆ 1 (ε) = cosh(π ε)[βε pˆ 1 sin(ε ln(a/)) + (W21 pˆ 2 + W41 qˆ ) cos(ε ln(a/))], hˆ 2 (ε) = cosh(π ε)[βε pˆ 1 cos(ε ln(a/)) − (W21 pˆ 2 + W41 qˆ ) sin(ε ln(a/))]. (11.111b)

11.6

Some Crack Problems

409

An Interface Crack Subjected to Point Force/Charge on the Crack Surfaces An interface crack located on −a ≤ x1 ≤ a subjected to a point force/charge pˆ at x1 = c on each side of the crack. The stress intensity factor for this problem is given in (7.127), i.e., 1 k= √ πa

2

(a + c) iεα a+c ˆ < cos hπεα > −1 p. a−c 2a(a − c)

(11.112)

Same as the procedure to get (11.110) and (11.111), we now have the following: (i) Piezoelectric ceramics poling in x3 -axis: + 1/πa∗ hˆ 1 (ε), KII = 1/πa∗ hˆ 2 (ε), + + = 1/πa∗ pˆ 3 , KIV = 1/πa∗ qˆ ,

KI = KIII

+

(11.113a)

where + hˆ 1 (ε) = cosh(πε)[ˆp2 cos(ε ln(2a∗ /)) + pˆ 1 D11 /D22 sin(ε ln(2a∗ /))], + hˆ 2 (ε) = cosh(πε)[ˆp1 cos(ε ln(2a∗ /)) − pˆ 2 D22 /D11 sin(ε ln(2a∗ /))] (11.113b) and a∗ = a(a − c)/(a + c).

(11.113c)

(ii) Piezoelectric ceramics poling in x2 -axis: + 1/πa∗ {d1 hˆ 1 (ε) + W41 (d2 pˆ 2 − d1 qˆ )}/βε2 , KII = 1/πa∗ hˆ 2 (ε)/βε , + + = 1/πa∗ pˆ 3 , KIV = 1/πa∗ {d2 hˆ 1 (ε) − W21 (d2 pˆ 2 − d1 qˆ )}/βε2 , (11.114a)

KI = KIII

+

where d1 and d2 are given in (11.107b) and a∗ is defined in (11.113c) and hˆ 1 (ε) = cosh(π ε)[βε pˆ 1 sin(ε ln(2a∗ /)) + (W21 pˆ 2 + W41 qˆ ) cos(ε ln(2a∗ /))], hˆ 2 (ε) = cosh(π ε)[βε pˆ 1 cos(ε ln(2a∗ /)) − (W21 pˆ 2 + W41 qˆ ) sin(ε ln(2a∗ /))]. (11.114b) An Interface Crack Subjected to Uniform Load/Induction on the Crack Surfaces An interface crack located on −a ≤ x1 ≤ a subjected to uniform load/induction ˆt = ˆ 2 ) on both sides of the crack surfaces. The stress/electric intensity (σˆ 12 , σˆ 22 , σˆ 23 , D factor for this problem is given in (7.129), i.e.,

410

11

k=

√


πa < (1 + 2iεα )(2a/)−iεα > −1 ˆt.

(11.115)

Similar to the previous two problems, the explicit expressions of the stress/electric intensity factor of each mode can be written as follows:. (i) Piezoelectric ceramics poling in x3 -axis: √ √ πa hˆ 1 (ε), KII = πa hˆ 2 (ε), √ √ ˆ 2, = πa σˆ 23 , KIV = πa D

(11.116a)

+ hˆ 1 (ε) = σˆ 22 c(ε) + σˆ 12 D11 /D22 s(ε), + hˆ 2 (ε) = σˆ 12 c(ε) − σˆ 22 D22 /D11 s(ε)

(11.116b)

KI = KIII where

and c(ε) = cos(ε ln(2a/)) + 2ε sin(ε ln(2a/)), s(ε) = sin(ε ln(2a/)) − 2ε cos(ε ln(2a/)).

(11.116c)

(ii) Piezoelectric ceramics poling in x2 -axis: √ √ ˆ 2 )}/βε2 , KII = πa hˆ 2 (ε)/βε , πa{d1 hˆ 1 (ε) + W41 (d2 σˆ 22 − d1 D √ √ ˆ 2 )}/βε2 , = πaσˆ 23 , KIV = πa{d2 hˆ 1 (ε) − W21 (d2 σˆ 22 − d1 D (11.117a)

KI = KIII where

ˆ 2 )c(ε) + βε σˆ 12 s(ε), hˆ 1 (ε) = (W21 σˆ 22 + W41 D ˆ 2 )s(ε) hˆ 2 (ε) = βε σˆ 12 c(ε) − (W21 σˆ 22 + W41 D

(11.117b)

and c(ε) and s(ε) are defined in (11.116c). Two Collinear Interface Cracks Subjected to Uniform Load/Induction at Infinity Two collinear interface cracks located on (a1 , b1 ) and (a2 , b2 ) subjected to uniform load/induction t∞ 2 at infinity. The stress intensity factor for this problem is obtain in (7.131a), i.e., k=

√

2π < kα > −1 t∞ 2 ,

(11.118)

where kα is given in (7.131b). By following the same procedure as the above problems, the component form can also be derived. Since the results are relatively complicated, detailed expressions are not shown.

Chapter 12

Plate Bending Analysis

Plate bending analysis concerns a plate subjected to transverse loadings and/or bending moments. Under this kind of loading conditions, the assumptions made in two-dimensional problems are not valid for plate bending problems which involve all three coordinate variables. Besides the two-dimensional problems, around 60 years ago Lekhnitskii also developed a complex variable formalism for the plate bending analysis (Lekhnitskii, 1938) and used his formalism to solve the problems of orthotropic plates containing circular holes or rigid inclusions (Lekhnitskii, 1968). After that, very few contributions can be found in the literature for the improvement of complex variable formulation in plate bending analysis. Through the connection between Stroh formalism and Lekhnitskii formalism for the twodimensional problems, Hwu (2003a) developed a Stroh-like complex variable formalism for the bending theory of anisotropic plates, which can be applied directly to the symmetric laminates. Because the Stroh-like complex variable formalism developed by Hwu (2003a) possesses almost the same matrix form as Stroh formalism for two-dimensional linear anisotropic elasticity, almost all the mathematical techniques developed for two-dimensional problems can be employed to the plate bending analysis. By simple analogy, many problems that cannot be solved previously have the possibility to be solved even without detailed derivation if their counterparts in two-dimensional problems have been solved. Although the Stroh-like complex variable formalism has been developed (Hwu, 2003a) and applied to get some analytical solutions for holes/cracks/inclusions problems (Hsieh and Hwu, 2002a), due to the peculiar selection of the stress function vector and slope vector, that formalism will cause confusion when comparing with Stroh formalism for two-dimensional problem introduced in Chapter 2 and Stroh-like formalism for coupled stretching–bending analysis introduced later in Chapter 13. For example, the slope vector of the formalism introduced in Hwu (2003a) is arranged in the order of (β2 , −β1 ) instead of (β1 , β2 ), where βi is the slope defined by the negative derivative of the deflection w, i.e., βi = −w, i . Similar situation occurs for the selection of the stress function vector. In order to have comparable formalism for different types of problems, in this chapter the Strohlike bending formalism will be introduced in Section 12.3 by an approach different from that shown in Hwu (2003a). In order to present Stroh-like bending formalism, the bending theory of anisotropic plates and Lekhnitskii bending formalism will C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_12, C Springer Science+Business Media, LLC 2010

411

412

12 Plate Bending Analysis

be introduced in Sections 12.1 and 12.2. With the Stroh-like bending formalism, some analytical solutions for anisotropic plates with holes/cracks/inclusions subjected to out-of-plane bending moments will then be presented in the last section of this chapter.

12.1 Bending Theory of Anisotropic Plates By following the description given in Section 1.4.3 for classical lamination theory and considering the anisotropic materials have one plane of elastic symmetry located at the middle surface of the plate (or say, the material is monoclinic not the most general anisotropic), the coupling stiffness Bij will be identical to zero. With Bij = 0 in (1.78), the bending moments Mx , My , and Mxy (see Fig. 12.1) can be expressed in terms of the deflection w as

q

h x

n

θ

Mˆ n

x

θ

y s

y Vˆn

z dx

Qy

dy

M yx My x

M Qx x

q

z

M x + dM x

M xy

M xy + dM xy

Qx + dQx M y + dM y M yx + dM yx

y

Qy + dQy

Fig. 12.1 Plate geometry, resultant forces, and moments

12.1

Bending Theory of Anisotropic Plates

413

∂ 2w ∂ 2w ∂ 2w , Mx = − D11 2 + D12 2 + 2D16 ∂x∂y ∂x ∂y ∂ 2w ∂ 2w ∂ 2w My = − D12 2 + D22 2 + 2D26 , ∂x∂y ∂x ∂y ∂ 2w ∂ 2w ∂ 2w Mxy = − D16 2 + D26 2 + 2D66 , ∂x∂y ∂x ∂y

(12.1)

where Dij , i, j = 1, 2, 6, are the bending stiffness defined in (1.79). Neglecting the body forces and the tractions on the top and bottom surfaces of the plate except the lateral load q(x, y), the force and moment equilibrium equations of the plate can be expressed as ∂Qx ∂Qy + + q = 0, ∂x ∂y

∂Mx ∂Mxy + − Qx = 0, ∂x ∂y

∂Mxy ∂My + − Qy = 0, (12.2) ∂x ∂y

where Qx and Qy are the transverse shear forces defined by Q=

h/2 τxz Qx = dz. Qy −h/2 τyz

(12.3)

Substitution of (12.2)2 and (12.2)3 into (12.2)1 will further lead the force equilibrium equation in the z direction to ∂ 2 My ∂ 2 Mxy ∂ 2 Mx + + 2 + q = 0. ∂x∂y ∂x2 ∂y2

(12.4)

With the moment–curvature relation (12.1) and the equilibrium equations (12.2)2 and (12.2)3 , the transverse shear forces and the equilibrium equation (12.4) can also be expressed in terms of the lateral deflection w as ∂ 3w ∂ 3w ∂ 3w ∂ 3w Qx = − D11 3 + 3D16 2 + (D12 + 2D66 ) + D26 3 , ∂x ∂x ∂y ∂x∂y2 ∂y 3 3 3 ∂ w ∂ w ∂ 3w ∂ w Qy = − D16 3 + (D12 + 2D66 ) 2 + 3D26 + D22 3 , ∂x ∂x ∂y ∂x∂y2 ∂y

(12.5)

and D11

∂ 4w ∂ 4w ∂ 4w ∂ 4w ∂ 4w + 2(D + 4D + 2D ) + 4D + D = q. 16 12 66 26 22 ∂x4 ∂x3 ∂y ∂x2 ∂y2 ∂x∂y3 ∂y4 (12.6)

Equation (12.6) is the governing differential equation for deflection of anisotropic thin plates. To determine w through this equation, appropriate boundary conditions should be set properly. For a fourth-order differential equation, only two boundary

414


conditions are required at each edge. These may be a given deflection and slope, or force and moment, or some combinations. Mathematically, they can be written as ∂ wˆ ∂w ˆ n or Mn = km ∂w , = or Mn = M ∂n ∂n ∂n and

w=w ˆ

(12.7a)

or Vn = Vˆ n or Vn = kv w,

where Vn is the well-known Kirchhoff force of classical plate theory, or effective transverse shear force, defined by Vn = Qn +

∂Mns . ∂s

(12.7b)

The subscripts n and s denote, respectively, the directions normal and tangent to the boundary. The overhat •ˆ denotes the prescribed value. km and kv are given spring constants. Note that to have n be the external normal and to conform with the sign convention of the plate bending analysis that the positive direction of x3 (or x2 ) is downward, the direction of s is chosen such that when one faces the direction of increasing s the material lies on the left side (see Fig. 12.1), which is opposite to the direction of s for two-dimensional problems. With the definition given in (12.7b) and the expressions in (12.1) and (12.5), the effective transverse shear force in x–y coordinate can also be expressed in terms of the lateral deflection as ∂Mxy ∂ 3w ∂ 3w ∂ 3w ∂ 3w , = − D11 3 + 4D16 2 + (D12 + 4D66 ) + 2D 26 ∂y ∂x ∂x ∂y ∂x∂y2 ∂y3 ∂Mxy ∂ 3w ∂ 3w ∂ 3w ∂ 3w + D Vy = Qy + = − 2D16 3 + (D12 + 4D66 ) 2 + 4D26 . 22 ∂x ∂x ∂x ∂y ∂x∂y2 ∂y3 (12.8) Vx = Qx +

If θ denotes the angle directed clockwise from the positive x-axis to the tangent direction of s (Fig. 12.1), the values in the n–s coordinate can be calculated from the values in the x–y coordinate according to the following transformation laws: ∂w ∂w ∂w = − sin θ + cos θ , ∂n ∂x ∂y

∂w ∂w ∂w = cos θ + sin θ , ∂s ∂x ∂y

(12.9a)

Mn = sin2 θ Mx + cos2 θ My − 2 sin θ cos θ Mxy , Ms = cos2 θ Mx + sin2 θ My + 2 sin θ cos θ Mxy ,

(12.9b)

Mns = sin θ cos θ (My − Mx ) + (cos2 θ − sin2 θ )Mxy , Qn = − sin θ Qx + cos θ Qy ,

Qs = cos θ Qx + sin θ Qy .

(12.9c)

12.2

Lekhnitskii Bending Formalism

415

Among all the possible boundary conditions, the commonly used conditions like simply supported, clamped, free, forced, displaced, and elastic-supported edge boundary conditions can then be expressed as simply supported edge : w = 0, clamped edge : w = 0, free edge : Vn = 0, forced edge : Vn = Vˆ n , displaced edge : w = w, ˆ elastic-supported edge : Vn = kv w,

Mn = 0; w, n = 0; Mn = 0; ˆ n; Mn = M ˆ , n; w, n = w Mn = km w, n .

(12.10)

12.2 Lekhnitskii Bending Formalism 12.2.1 General Solutions To solve the governing differential equation (12.6) together with the boundary conditions (12.10) for deflection of anisotropic thin plate, Lekhnitskii (1968) rewrote (12.6) symbolically with the use of four linear differential operators of the first order: ∂ ∂ − μk , ∂y ∂x

k = 1, 2, 3, 4,

(12.11a)

D22 μ4 + 4D26 μ3 + 2(D12 + 2D66 )μ2 + 4D16 μ + D11 = 0.

(12.11b)

d1 d2 d3 d4 w = q,

where dk =

and μk are the roots of the characteristic equation

It has been proved (Lekhnitskii, 1938) that (12.11b) has no real roots for any elastic homogeneous material. Since the coefficients of the fourth-order equation for μ are real, there are two pairs of complex conjugates for μ. If we let Im μk > 0, μk+2 = μk , k = 1, 2, and assume that μ are distinct, the general solution for deflection w can be expressed as (Lekhnitskii, 1968) w = w0 + 2 Re {w1 (z1 ) + w2 (z2 )},

(12.12a)

where w0 is a particular solution of (12.6) whose form depends on the load distribution q(x, y) on the plate surface; w1 (z1 ) and w2 (z2 ) are holomorphic functions of complex variables z1 = x + μ1 y and z2 = x + μ2 y.

(12.12b)

Note that the complex parameters μk of bending differ in general from the complex parameters of plane stress or plane strain for the same plate. On the basis of (12.1),

416


(12.5), and (12.8), general expressions for the moments and shear forces can be obtained as (for the case μ1 = μ2 ) (Lekhnitskii, 1968): Mx = Mx0 − 2 Re {g1 (μ1 )w1 (z1 ) + g1 (μ2 )w2 (z2 )}, My = My0 − 2 Re {g2 (μ1 )w1 (z1 ) + g2 (μ2 )w2 (z2 )}, 0 − 2 Re {g (μ )w (z ) + g (μ )w (z )}, Mxy = Mxy 6 1 1 1 6 2 2 2 Qx = Q0x − 2 Re {μ1 s(μ1 )w 1 (z1 ) + μ2 s(μ2 )w2 (z2 )},

(12.13)

Qy = Q0y + 2 Re {s(μ1 )w 1 (z1 ) + s(μ2 )w2 (z2 )}, 2 Vx = Vx0 + 2 Re {μ21 g2 (μ1 )w 1 (z1 ) + μ2 g2 (μ2 )w2 (z2 )}, Vy = Vy0 + 2 Re {g1 (μ1 )w 1 (z1 )/μ1 + g1 (μ2 )w2 (z2 )/μ2 }.

Here Mx0 , . . . , Q0x , . . . , Vy0 are the bending moments, transverse shear forces, and effective transverse shear forces corresponding to the particular solution of deflection w0 , which can be found by the relations given in (12.1), (12.5), and (12.8). The prime • denotes differentiation with respect to the function argument zk . The coefficients gj (μα ) and s(μα ), j = 1, 2, 6, α = 1, 2, are defined as gj (μα ) = Dj1 + Dj2 μ2α + 2Dj6 μα , g1 (μα ) s(μα ) = g6 (μα ) + , j = 1, 2, 6, μα

α = 1, 2.

(12.14)

12.2.2 Boundary Conditions With the general solutions given in (12.12) and (12.13), to get the complete solutions the only functions remained to be found are the complex functions w1 (z1 ) and w2 (z2 ), which should be determined through the satisfaction of boundary conditions. If the plate is subjected to bending only by forces and moments distributed along the edge and no transverse load is applied on the top or bottom surfaces (i.e., q=0), the boundary conditions of forced and displaced edges shown in (12.10) can be replaced by (Lekhnitskii, 1968) s ˆ n dy − Vˆ˜ n dx) − cx + c1 , (M 2 Re {g1 (μ1 )w 1 (z1 )/μ1 + g1 (μ2 )w 2 (z2 )/μ2 } = − 0 s ˆ n dx − Vˆ˜ n dy) + cy + c2 2 Re {g2 (μ1 )w 1 (z1 ) + g2 (μ2 )w 2 (z2 )} = (−M 0

(12.15a)

and 2 Re {w 1 (z1 ) + w 2 (z2 )} = wˆ , s cos θ − wˆ , n sin θ , 2 Re {μ1 w 1 (z1 ) + μ2 w 2 (z2 )} = wˆ , s sin θ + wˆ , n cos θ ,

(12.15b)

12.2


417

where Vˆ˜ n =

s

Vˆ n ds,

(12.15c)

0

and s is the arc length measured along a curved boundary; θ is the angle between the tangent s and x-axis (see Fig. 12.1). Although (12.15a), (12.15b), and (12.15c) are given in Lekhnitskii’ book (1968), only a reference written in Russian (Lekhnitskii, 1938) is cited and no detailed explanation is given. Since their associated physical meaning is important for the following development, we now try to relate (12.15a) and (12.15b) to the original boundary conditions (12.10)4 and (12.10)5 . Moments and Transverse Shear Forces ˆn Consider the force boundary condition (12.10)4 for which Vn = Vˆ n and Mn = M along the edges. To calculate the moments and transverse shear forces of (12.10)4 in n–s coordinate system, the conventional way is to calculate the moments (Mx , My , Mxy ) and transverse shear forces (Qx , Qy ) or effective transverse shear forces (Vx , Vy ) by using their relations with the deflection w such as those shown in (12.1), (12.5), and (12.8) or the complex variable formulation (12.13), then use the transformation law (12.9) and the definition (12.7b) to find their values in the normal and tangent coordinate system, i.e., Mn , Ms , Mns , Qn , Qs , Vn , Vs . For elasticity problems, the force conditions are usually called traction boundary conditions in which the tractions ti are prescribed along the boundaries. Using Cauchy’s formula ti = σij nj , and introducing the stress functions φi such that σi1 = −φi, 2 and σi2 = φi, 1 for two-dimensional problems, a very simple relation ti = ∂φi /∂s has been found. This simple relation is very helpful for the description of traction conditions especially along the curved boundaries. Therefore, we now try to express the force boundary conditions by employing the concept of traction boundary conditions used in 2D elasticity. By following the concept of 2D elasticity and considering the non-lateral load condition (q = 0), the force and moment equilibrium equations of the plate shown in (12.2) can be rewritten as ∂Qy ∂Qx + = 0, ∂x ∂y

∂Mxy ∂Mx + = Qx , ∂x ∂y

∂My ∂Mxy + = Qy . ∂x ∂y

(12.16)

The first equation of (12.16) will be satisfied automatically if we let Qx = −

∂η , ∂y

Qy =

∂η . ∂x

(12.17)

∂ψ2 . ∂x

(12.18)

Introduce two stress functions φ1 and φ2 such that Mx = −

∂ψ1 , ∂y

My =

418


Substituting (12.17) and (12.18) into (12.16)2,3 , we obtain Mxy =

∂ψ2 ∂ψ1 −η =− + η. ∂x ∂y

(12.19)

The second equality of (12.19) will then lead to η=

1 2

∂ψ1 ∂ψ2 + ∂x ∂y

.

(12.20)

Like the surface traction for two-dimensional problems, we define the surface (n) (n) moments Mj , j = 1, 2, along the surface with normal n by Mj = Mij nj . If n = (− sin θ , cos θ ), where θ denotes the angle between the tangent s and x-axis (see Fig. 12.1), we have (n)

M1 = −Mx sin θ + Mxy cos θ , (s)

(n)

M2 = −Mxy sin θ + My cos θ .

(12.21a)

(s)

The surface moments M1 and M2 along the surface with normal s = (cos θ , sin θ ) perpendicular to n can then be written as (s)

M1 = Mx cos θ + Mxy sin θ ,

(s)

M2 = Mxy cos θ + My sin θ .

(12.21b)

Substituting (12.18) and (12.19) into (12.21a) and (12.21b) and using the following relations − sin θ = n1 = − (n)

the surface moment Mi (n)

∂x ∂y ∂y ∂x = and cos θ = n2 = = , ∂s ∂n ∂s ∂n (s)

and Mi , i = 1, 2 can then be expressed as

∂x ∂y ∂ψ1 ∂ψ2 (n) − η , M2 = −η , ∂s ∂s ∂s ∂s ∂x ∂y ∂ψ1 ∂ψ 2 (s) + η , M2 = − +η . =− ∂n ∂n ∂n ∂n

M1 = (s) M1

(12.21c)

(12.22)

The values in the n–s coordinate can be calculated from the values in the x–y coordinate according to the following transformation laws: (n)

(n)

Mn = −M1 sin θ + M2 cos θ , (n)

(n)

(s)

(s)

Ms = M1 cos θ + M2 sin θ , (s)

(s)

Mns = M1 cos θ + M2 sin θ = −M1 sin θ + M2 cos θ .

(12.23a)

Substituting (12.22) into (12.23a) and using (12.21c), the moments Mn , Mns , Ms can be written as

12.2


419

∂ψ2 ∂ψ2 ∂ψ1 ∂ψ1 cos θ − sin θ , Ms = − sin θ − cos θ , ∂s ∂s ∂n ∂n ∂ψ2 ∂ψ1 ∂ψ2 ∂ψ1 = sin θ + cos θ − η = − cos θ + sin θ + η, ∂s ∂s ∂n ∂n

Mn = Mns

(12.23b)

or in matrix notation, Mn = nT ψ, s ,

Ms = −sT ψ, n ,

Mns = sT ψ, s − η = −nT ψ, n + η.

(12.23c)

From the second equality of (12.23c)3 , we obtain η=

1 T (s ψ, s + nT ψ, n ), 2

(12.24)

so that the moment Mns can also be expressed as Mns =

1 T (s ψ, s − nT ψ, n ). 2

(12.25)

Substituting (12.17) into (12.9c) and using (12.21c) and the chain rule for differentiation, the transverse shear forces Qn and Qs can now be expressed as Qn = η, s ,

Qs = −η, n .

(12.26a)

With (12.24), (12.26) can be rewritten as Qn =

1 T (s ψ, s + nT ψ, n ), s , 2

1 Qs = − (sT ψ, s + nT ψ, n ), n . 2

(12.26b)

With the results of (12.25) and (12.26b), the effective transverse shear forces Vn and Vs can be found to be Vn = Qn +

∂Mns = (sT ψ, s ), s , ∂s

Vs = Qs +

∂Mns = −(nT ψ, n ), n . ∂n

(12.27)

In summary, from (12.23c), (12.24), (12.25), (12.26), and (12.27), we see that all of the moments and transverse shear forces and effective transverse shear forces in the normal–tangent coordinate system have simple relations with the stress function vector ψ. Using the relations obtained in (12.23b)1 and (12.27)1 , and knowing that dy = sin θ ds, dx = cos θ ds, we now prove that

s 0

(Mn dy − V˜ n dx) = −ψ1 (s) + ψ1 (0),

0

s

(−Mn dx − V˜ n dy) = −ψ2 (s) + ψ2 (0), (12.28a)

420


where V˜ n =

s

Vn ds.

(12.28b)

0

This result (12.28) together with the definition (12.18) and the expressions (12.13)1,2 can then help us to prove that the expressions given in (12.15a) are really the equivalent expressions of the force boundary condition (12.10)4 . From this derivation, we know that (12.15a) stands for the prescribed conditions of the stress functions ψ1 and ψ2 defined in (12.18). Deflections and Slopes After proving the replacement of force boundary condition, we consider the displacement boundary condition (12.10)5 for which w = wˆ and w, n = wˆ , n along the edges. Since w = wˆ along the edges, it leads to w, s = wˆ , s along the edges. ˆ , s and Hence, the displacement boundary condition can be replaced by w, s = w w, n = wˆ , n , which means that both of the slopes in the tangential and normal directions of the boundary are prescribed. According to the transformation law of vectors, this condition can be further replaced by w, x = −wˆ , n sin θ + wˆ , s cos θ ,

w, y = wˆ , n cos θ + wˆ , s sin θ .

(12.29)

Substituting (12.12a) with w0 = 0 into (12.29), we obtain (12.15b). From this derivation, we know that (12.15b) stands for the prescribed conditions of slopes in the x- and y-directions.

12.2.3 Degenerate Materials In the above the complex parameters of bending μ1, μ2 are assumed distinct. If they are repeated (μ1 = μ2 ), the general solution for deflection w should be modified as (Lekhnitskii, 1968) w = w0 + 2 Re {w1 (z1 ) + z¯1 w2 (z1 )},

(12.30)

and all the following expressions (12.13), (12.14), and (12.15) should also be modified.

12.3 Stroh-Like Bending Formalism Two different complex variable formalisms are introduced in Chapters 2 and 3 for two-dimensional linear anisotropic elasticity. One is the Lekhnitskii formalism which starts with the equilibrated stress functions followed by compatibility equations, and the other is the Stroh formalism which starts with the compatible displacements followed by equilibrium equations. For plate bending analysis, opposite process is employed for Lekhnitskii bending formalism introduced in Section 12.2,

12.3

Stroh-Like Bending Formalism

421

which is developed with deflection as the basic unknown variable. Therefore, in this section to develop Stroh-like bending formalism we will start from the equilibrated stress functions and then consider the compatibility equations.

12.3.1 General Solutions Consider the homogeneous case that no lateral load is applied on the plate, i.e., q=0. The force equilibrium equation (12.4) can be satisfied automatically if we introduce two stress functions ψ1 (x, y) and ψ2 (x, y) such that Mx = −

∂ψ1 , ∂y

My =

∂ψ2 , ∂x

Mxy =

1 2

∂ψ1 ∂ψ2 − ∂x ∂y

.

(12.31)

If the anisotropic plate is symmetric with respect to the mid-plane of the plate, the coupling stiffness Bij will be identical to zero. Substituting (12.31) into (1.80) with B=0, the curvature κx , κy , and κxy can be expressed in terms of the stress functions as ⎧ 2 ⎫ ⎧ ⎫ ∂ w ⎪ ⎪ ∂ψ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ∗ ∗ ∗ ⎤⎪ ∂x ∂y ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D D D ⎪ ⎪ ⎪ ⎪ 11 12 16 ⎨ ⎬ ⎨ ∂ 2w ⎬ ⎨κx ⎬ ∂ψ2 ⎢ ∗ ∗ ∗ ⎥ κy = − D D D = , (12.32) ⎣ ⎦ 12 22 26 ⎪ ⎪ ⎪ ⎩ ⎭ ∂y2 ⎪ ∂x ⎪ ⎪ ⎪ κxy ∗ D∗ D∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D ⎪ ⎪ ⎪ 16 26 66 ⎪ ⎪ ⎪ ⎪ 1 ∂ψ1 − ∂ψ2 ⎪ ⎪ ⎪ ⎪ ∂ 2w ⎪ ⎩ ⎪ ⎭ ⎪ ⎭ ⎩2 2 ∂x ∂y ∂x∂y in which D∗ij are the components of the inverse of bending stiffness, D−1 . Knowing that the curvatures are not independent of each other, the following compatibility relations should be satisfied for the curvatures expressed by the stress functions, ∂κxy ∂κx = , ∂y 2∂x

∂κy ∂κxy = . ∂x 2∂y

(12.33)

The governing equations for the plate bending problem can then be obtained by substituting (12.32) into (12.33), which are 2 2 D∗66 ∂ 2 ψ1 D∗26 ∂ 2 ψ2 D∗66 ∂ 2 ψ2 D∗ ∂ 2 ψ2 ∗ ∂ ψ1 ∗ ∂ ψ1 ∗ + D − + D + = 0, − D − 16 16 11 12 2 2 2 4 ∂x ∂x∂y 2 ∂x 4 ∂x∂y 2 ∂y2 ∂y 2 2 D∗ ∂ 2 ψ1 D∗16 ∂ 2 ψ1 D∗66 ∂ 2 ψ2 D∗66 ∂ 2 ψ1 ∗ ∗ ∂ ψ2 ∗ ∂ ψ2 − 26 − − + D + − D + D = 0. 12 22 26 2 ∂x2 4 ∂x∂y 2 ∂y2 ∂x∂y 4 ∂y2 ∂x2 −

(12.34) Since the governing equations (12.34) are a set of homogeneous second-order differential equations, a general solution for ψ1 (x, y) and ψ2 (x, y) will depend on one composite variable which is a linear combination of x1 and x2 . Without loss in generality we choose

422


ψk = bk f (z) or

ψ = b f (z),

(12.35a)

where z = x + μy.

(12.35b)

Similar to relation (3.7) for two-dimensional problems, through the use of (12.35) the governing equations (12.34) will lead to {Qb + μ(Rb + RTb ) + μ2 Tb }b = 0,

(12.36a)

where ' Qb =

− 14 D∗66 − 12 D∗26 − 12 D∗26 −D∗22

'1

( , Rb =

∗ 2 D16 D∗12

1 ∗ 4 D66 1 ∗ 2 D26

'

( , Tb =

−D∗11 − 12 D∗16

(

, − 12 D∗16 − 14 D∗66 (12.36b)

and the subscript b denotes the value related to the bending properties. In (12.36a), a nontrivial solution of b exists if ) ) ) ) (12.37a) )Qb + μ(Rb + RTb ) + μ2 Tb ) = 0, which gives a fourth-order polynomial equation for μ. It can be proved that (12.37a) is equivalent to the characteristic equation (12.11b) of Lekhnitskii bending formalism. Similar to (3.11), through (12.36a) we can now introduce a new vector a as 1 a = (RTb + μTb )b = − (Qb + μRb )b. μ

(12.37b)

Substituting (12.35a) into the first two equations of (12.32) and integrating with respect to x and y, respectively, we get βk = ak f (z) or

β = a f (z),

(12.38a)

where β1 = −

∂w , ∂x

β2 = −

∂w . ∂y

(12.38b)

Like the general solution shown in (3.24) for two-dimensional problem, if the material eigenvalues are distinct and are arranged such that Im μk > 0, μk+2 = μ¯ k , k = 1, 2, the general solutions (12.35) and (12.38) for the plate bending problems can now be written as β = 2 Re {Af(z)},

ψ = 2 Re {Bf(z)},

(12.39a)

12.3


423

where β=

β1 , β2

A = [a1

ψ=

a2 ],

ψ1 , ψ2

B = [b1

f(z) = b2 ],

f1 (z1 ) , f2 (z2 )

zα = x + μα y,

(12.39b) α = 1, 2.

By comparing the solution given in (12.39), (12.12), and (12.13), and using relations (12.31) and (12.38b), we can write down the explicit expressions for the material eigenvector matrices A and B as follows: c1 c2 , A= c1 μ1 c2 μ2

−c1 g1 (μ1 )/μ1 −c2 g1 (μ2 )/μ2 B= , c1 g2 (μ1 ) c2 g2 (μ2 )

(12.40a)

where c1 and c2 are the normalization factors determined from the orthogonality relation (3.57), which are c2k =

μk , 2(μ2k g2 (μk ) − g1 (μk ))

k = 1, 2.

(12.40b)

Because the general solutions (12.39) are expressed in terms of the slope vector β and stress function vector ψ, for the convenience of problem solving it is better that the boundary conditions (12.10) generally used in plate bending problems are also expressed in terms of β and ψ. With this consideration, use of (12.28) and (12.15) will lead the displacement and force boundary conditions to ⎧ ⎫ ⎨− s (M ˆ n dy − Vˆ˜ n dx) − cx + c1 ⎬ −wˆ , s cos θ + wˆ , n sin θ β= , ψ= 0 . ⎩ s (M −wˆ , s sin θ − wˆ , n cos θ ˆ n dx + Vˆ˜ n dy) − cy − c2 ⎭ 0 (12.41) Thus, the commonly encountered boundary conditions (12.10)2,3 can be written as clamped edge :

β = 0;

and

free edge :

ψ = 0.

(12.42)

As to the mixed boundary conditions like the simply supported edge (12.10)1 and the elastic-supported edge (12.10)6 , no simple vector form expressions can be obtained. Under this condition, the problems can be solved by using the component expressions. In this sense, relations (12.23b)1 and (12.27)1 can be utilized to express the moment and effective transverse shear force, while the deflection and slope can be expressed with the assist of (12.9a) and (12.12). For example, the boundary condition of simply supported edges can be written as simply supported edge: w = 0 and Mn =

∂ψ1 ∂ψ2 cos θ − sin θ = 0. ∂s ∂s

(12.43)

424


Note that the Stroh-like bending formalism introduced in this section has been purposely arranged to have the same form as those of the corresponding Stroh formalism for two-dimensional problems. Due to the similarity, all the mathematical techniques developed for two-dimensional problems can be transferred to the plate bending problems. By simple analogy, many problems which cannot be solved previously, now have the possibility to be solved analytically.

12.3.2 Material Eigenrelation and Its Explicit Expressions The key feature that makes Stroh formalism more attractive than Lekhnitskii formalism is that it possesses the eigenrelation which relates the eigenmodes of stress functions and displacements to the material properties. Without the eigenrelation, one cannot feel the benefit of Stroh formalism. Therefore, it is important for us to establish the eigenrelation for Stroh-like bending formalism developed in this section. Otherwise, the formalism is just a skeleton without any spirit inside its body. In order to establish the eigenrelation for Stroh-like bending formalism, it is better to know the eigenrelation (3.48) for Stroh formalism of two-dimensional problems and its corresponding characteristic equation (2.15) of Lekhnitskii formalism. In general, the two-dimensional problems considered in anisotropic plates include not only in-plane but also anti-plane problems and the problems where in-plane and anti-plane deformations couple each other. For this general case, the characteristic equation associated with anisotropic materials is a sixth-order algebraic equation. In this section, we only consider the anisotropic plates having one plane of elastic symmetry located at the middle surface, i.e., the monoclinic plates. For this case, the in-plane and anti-plane problems will decouple. The characteristic equation for the in-plane problems is a fourth-order algebraic equation, while that for the anti-plane problems is a second-order equation. For the bending problems considered in this section, a fourth-order characteristic equation is also obtained in (12.11b). Therefore, the comparison is better to be performed through in-plane problems for monoclinic plates not through general two-dimensional problems for anisotropic plates. In order to derive the material eigenrelation for plate bending problem, we follow the steps stated in Section 3.3.1. By reconstruction of (12.37b) and use of (3.47), we can get the following eigenrelation: Nb η = μη, where

(Nb )1 Nb = (Nb )3

(Nb )2 , (Nb )T1

(12.44a) b η= a

(12.44b)

and T (Nb )1 = −T−1 b Rb ,

T (Nb )2 = T−1 b = (Nb )2 ,

T T (Nb )3 = Rb T−1 b Rb − Qb = (Nb )3 . (12.44c)

12.3


425

To get exactly the same material eigenrelation as that of two-dimensional problems, substituting relation (3.51) into (12.44a), we get Nξ = μξ,

(12.45a)

where N=

N1 N3

N2 , NT1

ξ=

a b

(12.45b)

and N1 = (Nb )T1 ,

N2 = (Nb )3 = NT2 ,

N3 = (Nb )2 = NT3 .

(12.45c)

Substituting (12.36b) into (12.44c) and using relation (12.45c), we obtain the explicit expressions of the fundamental elasticity matrix N as ⎡

⎤ 0 1 0 0 ⎣ ⎦ D12 2D26 , N2 = , N1 = 0 D122 − − D22 D22 ⎤ ⎡ D212 2D12 D26 −2D16 + ⎢ −D11 + D D22 ⎥ 22 ⎥ ⎢ N3 = ⎢ ⎥. 2 ⎣ 4D26 ⎦ 2D12 D26 −2D16 + −4D66 + D22 D22

(12.46)

Substituting (12.46) into (12.45), we can prove that N − μI = 0 is equivalent to (12.11b) and its associated eigenvector is the one written in (12.40). Like the generalized eigenrelation for two-dimensional problems, the eigenrelation (12.45) can also be generalized to have the same form as that shown in (3.76). By a similar approach as that described in Ting (1996) for the derivation of Ni , j = 1, 2, 3 for two-dimensional problems, the explicit expressions of the generalized Ni (θ ), i = 1, 2, 3 for the plate bending problems can be obtained as follows: 1 Y1 Y3 N1 (θ ) = , Y2 Y4

1 −s2 cs N2 (θ ) = , cs −c2

1 Y5 Y6 N3 (θ ) = , Y6 Y7 (12.47a)

where Y1 = −2D16 s2 + cs[(D22 − D12 )c2 − (4D26 − 4D16 )cs + (D12 + 4D66 − D11 )s2 ], Y2 = D12 c2 − 2D16 cs + D11 s2 , Y3 = −D22 c2 + 2D26 cs − D12 s2 , Y4 = 2D26 c2 − cs[(D12 + 4D66 − D22 )c2 + (4D26 − 4D16 )cs + (D11 − D12 )s2 ],

426


Y5 = (D11 D22 − D212 )c2 − 4(D11 D26 − D12 D16 )cs + 4(D11 D66 − D216 )s2 , Y6 = 2(D16 D22 − D12 D26 )c2 + (D212 − D11 D22 + 4D12 D66 − 4D16 D26 )cs + 2(D11 D26 − D12 D16 )s2 , Y7 = 4(D22 D66 − D226 )c2 − 4(D16 D22 − D12 D26 )cs + (D11 D22 − D212 )s2 , (12.47b) and = −[D22 c4 − 4D26 c3 s + 2(D12 + 2D66 )c2 s2 − 4D16 cs3 + D11 s4 ], c = cos θ , s = sin θ .

(12.47c)

12.3.3 Explicit Expressions of S, H, and L To get the explicit expressions of the three real matrices S, H, and L defined in (3.59) for Stroh-like bending formalism, we follow the steps described in Section 11.3.3. With the explicit expressions of A and B obtained in (12.40), we have

AB−1

⎡ ⎤ D11 −D22 (μ1 + μ2 ) − 2D26 − D12 ⎥ μ1 μ2 1 ⎢ ⎢ ⎥, = ⎣ ⎦ B 1 1 −D22 μ1 μ2 + D12 D11 ( + ) + 2D16 μ1 μ2

(12.48a)

D11 [D12 + (μ21 + μ1 μ2 + μ22 )D22 + 2(μ1 + μ2 )D26 ] μ1 μ2 + D12 [μ1 μ2 D22 − D12 ] + 2D16 [(μ1 + μ2 )D22 + 2D26 ].

(12.48b)

where B =

Note that the roots μ1 , μ2 , μ1 , and μ2 are related to the coefficients of the characteristic equation (12.11b), from which we have Re(μ1 + μ2 ) = −2D26 /D22 , (μ1 + μ2 )(μ1 + μ2 ) + (μ1 μ2 + μ1 μ2 ) = 2(D12 + 2D66 )/D22 , Re{μ1 μ2 (μ1 + μ2 )} = −2D16 /D22 ,

(12.49)

μ1 μ2 μ1 μ2 = D11 /D22 , where the overbar denotes complex conjugate. By (12.49), AB−1 shown in (12.48a) can be rewritten as ( ' Im {μ + μ } D μ μ − D −iD 22 1 2 22 12 1 1 2 AB−1 = . (12.50) B −D22 μ1 μ2 + D12 −iD22 Im {μ1 μ2 (μ1 + μ2 )}

12.3


427

With the result of (12.50), by following the steps described in (11.59a), (11.59b), and (11.59c) we can get the explicit expressions of S, H, and L as S=

s2 −d b , g −e d

H=

s2 b D22 g2 d

d , e

L = D22 (1 − s2 )

e −d , (12.51a) −d b

where a, b, d, e, g, and s are defined by μ1 + μ2 = a + ib, μ1 μ2 = c + id, b > 0, g D12 − c > 0, 0 < s = √ < 1. g= D22 be − d2

e = ad − bc > 0, (12.51b)

Although the explicit expressions of S, H, and L have been obtained in (12.51) for the general monoclinic plates, the results show that when using (12.51) we still need to calculate the material eigenvalues from the characteristic equation (12.11b). Following are the special cases for which the material eigenvalues can be expressed in terms of the bending stiffness explicitly. Consider an orthotropic plate or regular symmetric cross-ply laminate whose bending–twisting response is uncoupled, i.e., D16 = D26 = 0. With this special arrangement, use of (12.11b) will lead (12.51b) to < ' * ( = 2 = D12 + 2D66 D11 D11 > a = d = 0, b = 2 + , , c=− D22 D22 D22 (12.52) * D12 D11 g e = −bc > 0, g = > 0. + , s= √ D22 D22 −b2 c Substituting (12.52) into (12.51), for orthotropic plates the explicit expressions of S, H, and L can be further simplified to 0 t3 /t1 , −t3 /t2 0

S=

H=

1/t2 0 , 0 1/t1

L=

−t4 /t2 0 , 0 −t4 /t1

(12.53a)

where 3

3 + + 2D11 (D12 + 2D66 + D11 D22 ), t2 = 2D22 (D12 + 2D66 + D11 D22 ), + + t3 = D12 + D11 D22 , t4 = D212 − D11 D22 − 4D66 D11 D22 . (12.53b) t1 =

For a single-layer isotropic plate, D11 = D22 = D12 + 2D66 = D = Eh3 /12 (1−ν 2 ), D16 = D26 = 0, and hence, μ1 = μ2 = i, where E is the Young’s modulus; ν is the Poisson’s ratio; and h is the thickness of the plate. With this information, we get

428


a = d = 0,

c = −1,

s = (1 + ν)/2,

b = e = 2, g = 1 + ν,

D22 = D = Eh3 /12(1 − ν 2 ),

(12.54a)

and hence, S=

1+ν 0 1 , −1 0 2

H=

1 10 , 2D 0 1

L=

(3 + ν)(1 − ν)D 1 0 . (12.54b) 01 2

12.4 Holes/Inclusions/Cracks 12.4.1 Elliptical Holes Consider an unbounded anisotropic plate weakened by an elliptical hole subjected ˆ y and twisting moment M ˆ xy at infinity (see ˆ x, M to out-of-plane bending moments M ˆ Fig. 12.2; note that to avoid confusion the twisting moment Mxy is not shown in this figure). There is no load around the edge of the elliptical hole. The contour of the elliptical hole is represented by x = a cos ϕ,

y = b sin ϕ,

(12.55)

where 2a, 2b are the major and minor axes of the ellipse and ϕ is a real parameter. The boundary conditions of this problem can be expressed as ˆ x , My = M ˆ y , Mxy = M ˆ xy , at infinity, Mx = M Mn = Vn = 0, along the hole boundary.

(12.56)

Mˆ y

2a

Mˆ x

Mˆ x x

2b

Fig. 12.2 An anisotropic plate weakened by an elliptical hole subjected to out-of-plane bending ˆ x and M ˆy moments M

Mˆ y

y

z

12.4

Holes/Inclusions/Cracks

429

Since the boundary considered in this problem is an elliptical boundary, it is not easy to find a solution satisfying (12.56) due to the complexity of describing the normal direction n in the conventional way. The substitutes of the force boundary conditions have been given in (12.15a). From the discussion presented in (12.16)– (12.28) we know the stress functions ψ1 and ψ2 introduced in (12.18) have simple relations with the bending moments and effective transverse shear forces in n–s coordinate system, which are suitable for Stroh-like bending formalism to describe the curvilinear force boundary conditions such as (12.41)2 . With this understanding, the boundary conditions (12.56) can now be expressed in terms of the stress function vector ψ as ∞ ψ = ψ∞ = x1 m∞ 2 − x2 m1 , at infinity, ψ = 0, along the hole boundary,

(12.57a)

where m∞ 1 =

ˆx M ˆ xy , M

m∞ 2 =

ˆ xy M ˆy . M

(12.57b)

The slope vector β∞ associated with ψ ∞ for a homogeneous plate can be obtained from the constitutive laws given in (12.32), which leads to ⎧ ∞⎫ ⎡ ∗ ∗ ∗ ⎤ ⎧ ⎫ ˆx ⎬ D11 D12 D16 ⎨ M ⎨ κx ⎬ ˆ κ ∞ = ⎣D∗12 D∗22 D∗26 ⎦ M . ⎩ˆ y ⎭ ⎩ y∞ ⎭ κxy D∗16 D∗26 D∗66 Mxy

(12.58)

From (12.58), we can get the following relation: ∞ T ∞ κ∞ 1 = −Tb m1 + Rb m2 ,

∞ ∞ κ∞ 2 = Rb m1 − Qb m2 ,

(12.59a)

∞ where m∞ 1 , m2 and Qb , Rb , Tb are defined in (12.57b) and (12.36b), and

κ∞ 1

κx∞ , = ∞ κxy /2

κ∞ 2

∞ κ /2 = xy∞ . κy

(12.59b)

Integration of (12.59a) with respect to x1 and x2 , respectively, leads to the results of β∞ as ∞ β∞ = x1 κ∞ 1 + x2 κ2 .

(12.60)

In order to satisfy the infinity boundary condition (12.57a)1 , the slope vector β∞ and the stress function vector ψ∞ are added to the general solution (12.39a). Whereas for the satisfaction of traction-free hole boundary condition (12.57a)2 , by referring to the solutions of the corresponding two-dimensional problems, (6.8), the complex function vector f(z) of (12.39a) is selected to be

430


f(z) =
k,

ζα =

zα +

+

z2α − a2 − μ2α b2 , a − iμα b

(12.61)

where k is the unknown coefficient to be determined through the satisfaction of the hole boundary condition. Therefore, the solution for the present problem can be expressed as β = β∞ + 2 Re {A < ζα−1 > k},

ψ = ψ∞ + 2 Re {B < ζα−1 > k},

(12.62)

where β∞ , ψ∞ are given in (12.60) and (12.57a), respectively. Knowing the value ζα = eiϕ along the hole boundary and the solutions selected in (12.62), the boundary condition (12.57a)2 will then provide us 1 ∞ k = − B−1 (am∞ 2 − ibm1 ). 2

(12.63)

The explicit solutions can therefore be expressed as ∞ β = β∞ − Re {A < ζα−1 > B−1 (am∞ 2 − ibm1 )}, ∞ ψ = ψ∞ − Re {B < ζα−1 > B−1 (am∞ 2 − ibm1 )},

(12.64)

which has the same form as those of the corresponding two-dimensional problems, (6.8) and (6.10). The only difference is that the symbols like A, B,. . ., etc., have different dimensions and different contents for different types of problems. With the explicit solution found in (12.64), all the physical responses such as the deflection, bending moments, and transverse shear forces of the plate can be obtained by using their relations with the slopes and stress functions given in (12.38b), (12.31), and (12.17). Moments Around the Hole Boundary According to the relations obtained in (12.23), (12.25), (12.26), and (12.27), we know that to obtain the explicit solutions for the moments around the elliptical hole boundary, the first step we need to do is calculating ψ, n (note that ψ, s should be zero along the traction-free hole boundary, which can be used as a check in calculation). Following the way described between (6.14) and (6.17), we obtain " ! a ∞ G − I + (θ ) m ψ, n = sin θ G1 (θ )m∞ 3 1 2 b b ∞ ∞ − cos θ I + G3 (θ ) m1 + G1 (θ )m2 , a

(12.65a)

where G1 (θ ) = NT1 (θ ) − N3 (θ )SL−1 ,

G3 (θ ) = −N3 (θ )L−1 .

(12.65b)

12.4


431

With the results of (12.65), the moments around the hole boundary can then be calculated through the relations given in (12.23c) and (12.25). Circular Holes For an anisotropic plate weakened by a circular hole with radius a subjected to ˆ My = Mxy = 0, the explicit solution (12.64) can be reduced to Mx = M, ˆ Im{A < ζα−1 > B−1 }i1 , β = β∞ − aM ˆ Im{B < ζα−1 > B−1 }i1 . ψ = ψ∞ − aM

(12.66)

If the plate is composed of orthotropic materials with thickness h, the bending stiffness Dij will be D11 =

E1 h3 , 12(1 − ν1 ν2 )

D16 = D26 = 0,

D66

D22 =

E2 h3 , 12(1 − ν1 ν2 )

D12 = ν1 D22 = ν2 D11 ,

Gh3 , = 12

(12.67)

where E1 and E2 are the Young’s moduli in x1 and x2 directions, respectively; G is the shear modulus in the x1 x2 plane; ν1 and ν2 are the major and minor Poisson’s ratios, respectively, and satisfy ν1 /E1 = ν2 /E2 . With the plate properties (12.67) and the expressions of A and B given in (12.40), the explicit solution obtained in (12.66)1 can lead to the following results for w1 (z1 ) and w2 (z2 ) of (12.12). They are , ˆ i ν1 + μ22 Ma , 1 (z1 ) = ζ1 ν1 − n2 + 1 + ν1 ν2 + kν2 2D11 (μ2 − μ1 ) k , ˆ −i ν1 + μ21 Ma w 2 (z2 ) = , 2 ζ2 ν1 − n + 1 + ν1 ν2 + kν2 2D11 (μ2 − μ1 ) k

w

(12.68a)

where * k = −μ1 μ2 =

D11 , D22

n = −i(μ1 + μ2 ),

(12.68b)

which can be proved to be identical to that shown in Lekhnitskii (1968).

12.4.2 Elliptical Rigid Inclusions Consider an unbounded anisotropic plate embedded with an elliptical rigid incluˆ y and twisting moment M ˆ xy ˆ x, M sion subjected to out-of-plane bending moments M

432


at infinity. Since the rigid inclusion cannot undergo deformation, the boundary conditions of this problem can be expressed as ˆ x, Mx = M

ˆ y, My = M

w = ∂w/∂n = 0,

ˆ xy , at infinity, Mxy = M

along the inclusion boundary.

(12.69)

From the discussions given in Section 12.2.2, we know that the boundary condition (12.69) can also be expressed in terms of the stress function vector ψ and slope vector β as ∞ ψ = ψ∞ = x1 m∞ 2 − x2 m1 , at infinity, β = 0, along the hole boundary.

(12.70)

By referring to the corresponding two-dimensional solutions (8.40) and using the approach similar to the previous section, the explicit solution for the problem with elliptical rigid inclusion can be obtained as ∞ β = β∞ − Re {A < ζα−1 > A−1 (aκ∞ 1 + ibκ2 )}, ∞ ψ = ψ∞ − Re {B < ζα−1 > A−1 (aκ∞ 1 + ibκ2 )},

(12.71)

∞ where β∞ and ψ∞ are given in (12.60) and (12.70), and κ∞ 1 and κ2 are related to ˆ ˆ ˆ the applied moments Mx , My , Mxy by (12.58) and (12.59).

Moments Around the Inclusion Boundary Similar to Section 12.4.1, to know the moments around the inclusion boundary we need to calculate ψ, s and ψ, n , which can be obtained by the same way stated in that section. The results are b ∞ ∞ ∞ m1 + E1 (θ)κ1 − E3 (θ)κ2 , a " ! a T H−1 κ∞ + H−1 κ∞ + cos θ m∞ − ST H−1 κ∞ + b H−1 κ∞ , ψ, s = − sin θ m∞ + S 1 2 1 2 1 2 b a !

"

a ∞ ∞ ψ, n = − sin θ m∞ 2 + E1 (θ )κ2 + b E3 (θ )κ1 − cos θ

(12.72a) where E1 (θ ) = N3 (θ ) + NT1 (θ )ST H−1 ,

E3 (θ ) = NT1 (θ )H−1 .

(12.72b)

With the results of (12.72), the moments around the inclusion boundary can then be calculated through the relations given in (12.23c) and (12.25). Circular Inclusions For an anisotropic plate embedded with a rigid circular inclusion with radius a ˆ My = Mxy = 0, the explicit solution (12.71) can be subjected to Mx = M, reduced to

12.4


433

ˆ Re {A < ζα−1 > A−1 (Tb − iRb )}i1 , β = β∞ + aM ˆ Re {B < ζα−1 > A−1 (Tb − iRb )}i1 , ψ = ψ∞ + aM

(12.73)

If the plate is orthotropic, then D∗11 =

D22 , D11 D22 − D212

D∗16 = D∗26 = 0,

D∗22 =

D∗66 =

D11 , D11 D22 − D212

D∗12 = −

1 , D66

D12 , D11 D22 − D212

(12.74)

where Dij are given in (12.67). Again, like the derivation of (12.68), w1 (z1 ) and w2 (z2 ) of (12.12) can be obtained as w 1 (z1 ) = − w

ˆ Ma μ2 + iν1 , 2D11 (μ1 − μ2 )(1 − ν1 ν2 ) ζ1

ˆ Ma μ1 + iν1 , 2 (z2 ) = 2D11 (μ1 − μ2 )(1 − ν1 ν2 ) ζ2

(12.75)

which is exactly the same as that shown in Lekhnitskii (1968).

12.4.3 Cracks An elliptic opening can be made into a crack of length 2a by letting the minor axis b be zero. The explicit solution obtained in (12.64) is then applicable to crack problem ˆ y, ˆ x, M with b = 0. Consider an infinite plate containing a center crack subjected to M ˆ and Mxy at infinity. The stress function vector ψ and slope vector β of this problem can therefore be obtained from (12.64) with b = 0, i.e., β = β∞ − a Re {A < ζα−1 > B−1 }m∞ 2 , ψ = ψ∞ − a Re {B < ζα−1 > B−1 }m∞ 2 ,

(12.76a)

where 3 1 2 2 ζα = zα + zα − a . a

(12.76b)

For the crack problems, it is interesting to know the moment intensity factors defined by K=

√ 6 KI My (r, θ ) = 2 lim 2π r , KII Mxy (r, θ ) h r→0 θ=0

(12.77)

434


where r is the distance ahead of the crack tip, h is the thickness of the plate, and θ = 0 is the direction along the crack. Thus, to find the moment intensity factors, we need to calculate My , Mxy ahead of the crack tip. To calculate the moments ahead of the crack tip, we can use the relations given in (12.18)2 and (12.19), and let x = a + r, y = 0. The differentiation of ζα−1 with respect to x and y can be obtained as 0 0 ∂ζα−1 00 1 μα ∂ζα−1 00 x x = = , . 1 − 1 − √ √ ∂x 0y=0 a ∂y 0y=0 a x2 − a2 x2 − a2 (12.78) Using the result of (12.78) and identity (3.137), the differentiation of the stress function vector ψ (12.76a) with respect to x and y will lead to 2 ψ, x →

a ∞ m , 2r 2

2 ψ, y →

a G1 m∞ 2 , 2r

when r → 0.

(12.79)

With (12.79), the moments My and Mxy near the crack tip can then be obtained from (12.18)2 and (12.19). Substituting their near-tip values into the definition (12.77), the moment intensity factor can be obtained as KI =

6√ ˆ y, π aM h2

KII =

" 3√ ! ˆ xy − [G1 m∞ πa M 2 ](2) , 2 h

(12.80a)

where G1 = NT1 − N3 SL−1 .

(12.80b)

In (12.80a), the subscript (2) used in the expression of KII denote the second component of the vector. From (12.80), we see that the moment intensity factor KII depends on the loading, crack length as well as the material properties of the plate. ˆ y only, from the explicit If the plate is made of orthotropic materials subjected to M expressions of N1 , N3 , S, and L shown in (12.46) and (12.53) we find that mode II moment intensity factor of (12.80a) equals to zero for cracks in orthotropic plates ˆ y at infinity. subjected to bending moments M According to the solutions presented in Section 12.4, several numerical examples have been done to show the effects of material properties and hole/inclusion geometries on the moment distributions around the hole/inclusion boundary. For those who are interested in the numerical results and their related discussions, please refer to Hsieh and Hwu (2002a).

Chapter 13

Coupled Stretching–Bending Analysis

Although the classical lamination theory was developed long time ago, it is not easy to apply this theory to find an analytical solution for the problem with curvilinear boundaries, especially when the laminates are composed of the laminae that will make the in-plane and plate bending deformations couple each other. Even under inplane forces if the laminates are unsymmetric they will be stretched as well as bent, and the two-dimensional analysis cannot be applied. Because this kind of problems was solved by the complex variable formalism in two-dimensional deformation, it is hoped that similar formalism can be developed for the classical lamination theory. For two-dimensional linear anisotropic elasticity, there are two major complex variable formalisms in the literature. One is Lekhnitskii formalism introduced in Chapter 2, and the other is Stroh formalism introduced in Chapter 3. Because Stroh formalism is expressed in matrix form, it possesses a special feature that many different kinds of problems may have the same matrix form solutions and the only difference among them is the contents of the matrices. With this special feature, by Stroh formalism it becomes much easier to solve a new problem whose solution has not been obtained but its counterparts have been solved. Due to this reason, the efforts of this chapter will be focused upon the establishment of a counterpart of Stroh formalism for the lamination theory. If the formulas developed for the lamination theory can be purposely arranged into the form of Stroh formalism for two-dimensional linear anisotropic elasticity, almost all the mathematical techniques developed for two-dimensional problems can be transferred to the coupled stretching–bending problems. Thus, by simple analogy, many unsolved lamination problems can be solved if their corresponding two-dimensional problems have been solved successfully. In the previous chapter, through the understanding of the connection between Stroh formalism and Lekhnitskii formalism for the two-dimensional problems, the Stroh-like bending formalism is developed for the plate bending analysis of anisotropic plates, which can be applied directly to the symmetric laminates. However, it is difficult to apply the same approach to the unsymmetric laminates with in-plane and plate bending coupling. Instead of using the advantages of Stroh–Lekhnitskii’s connection, a displacement-based derivation has been introduced by Lu and Mahrenholtz (1994) and modified by Cheng and Reddy (2002) and improved by Hwu (2003b). In addition, some researchers devoted their efforts to the C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_13, C Springer Science+Business Media, LLC 2010

435

436

13


development and application of the complex variable method on the laminates with bending extension coupling such as Becker (1991) and Zakharov (1992). Due to the complexity, the resemblance between Stroh formalism and displacement formalism as well as the published complex variable methods is not perfect enough to employ most of the key features of Stroh formalism. By comparing the formalisms developed by Hwu (2003a), Lu and Mahrenholtz (1994), and Cheng and Reddy (2002), Hwu (2003b) found that there exists an alternative formalism that is more alike to the Stroh formalism for two-dimensional problems from the viewpoint of material eigenrelation, which is called mixed formalism because during the derivation the basic functions are not pure displacements or pure stresses but in-plane displacements plus plate bending moments. The associated eigenrelation of the mixed formalism shows that it is really more alike to Stroh formalism. Moreover, by the mixed formalism, the explicit expressions of the fundamental elasticity matrix and material eigenvectors have also been obtained in Hwu and his co-worker’s papers (Hsieh and Hwu, 2002b; Hwu, 2003b). In this chapter, to have a deep insight of the complex variable formalism for lamination theory, both displacement and mixed formalisms will be introduced. In Section 13.1, we introduce the coupled stretching–bending theory of laminates. The displacement and mixed formalisms introduced by Hwu (2003b) are presented in Section 13.2. The Stroh-like formalism presented in Section 13.3 will then be a combination of these two formalisms by using the general solutions of displacement formalism and the eigenrelation of mixed formalism. Some useful relations such as the relations between the stress functions and stress resultants/moments, and the explicit expressions for A, B, N, S, H, and L, will also be presented in Section 13.3. Same as Stroh formalism for two-dimensional problems, the Strohlike formalism for stretching–bending coupling analysis can also be extended to hygrothermal problems and electro-elastic composite laminates, which will then be discussed in Sections 13.4 and 13.5 by following the works of Hwu and Hsieh (2005) and Hsieh and Hwu (2006).

13.1 Coupled Stretching–Bending Theory of Laminates To describe the overall properties and mechanical behavior of a laminate, the most popular way is the classical lamination theory presented in Section 1.4.3. According to the observation of actual mechanical behavior of laminates, Kirchhoff assumptions are usually made in this theory. Based upon the Kirchhoff assumptions, the displacement fields, the strain–displacement relations, the constitutive laws, and the equilibrium equations can be written as follows: Displacement Fields u(x, y, z) = u0 (x, y) + zβx (x, y), v(x, y, z) = v0 (x, y) + zβy (x, y), w(x, y, z) = w0 (x, y),

(13.1a)

13.1

Coupled Stretching–Bending Theory of Laminates

437

where βx (x, y) = −

∂w(x, y) , ∂x

βy (x, y) = −

∂w(x, y) . ∂y

(13.1b)

u, v, and w are the displacements in x-, y-, and z-directions; u0 , v0 , and w0 are the middle surface displacements; and βx and βy are the negative slopes in x and y directions. Strain–Displacement Relation 0 + zκxy , εx = εx0 + zκx , εy = εy0 + zκy , γxy = γxy

(13.2a)

∂v0 ∂u0 ∂v0 ∂u0 0 , εy0 = , γxy + , = ∂x ∂y ∂y ∂x ∂ 2w ∂ 2w ∂ 2w κx = − 2 , κy = − 2 , κxy = −2 . ∂x∂y ∂x ∂y

(13.2b)

where εx0 =

0 ) are the mid-plane strains, and (εx , εy , γxy ) are the strains, (εx0 , εy0 , γxy (κx , κy , κxy ) are the plate curvatures.

Constitutive Laws ⎧ ⎫ ⎡ A11 Nx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ Ny ⎪ ⎢A12 ⎪ ⎪ ⎪ ⎢ ⎪ ⎨N ⎪ ⎬ ⎢A xy 16 =⎢ ⎢B ⎪ ⎪ M x ⎪ ⎪ ⎢ 11 ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪My ⎪ ⎪ ⎣B12 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ Mxy B16

A12

A16

B11

B12

A22 A26

A26 A66

B12 B16

B22 B26

B12 B22 B26

B16 B26 B66

D11 D12 D16

D12 D22 D26

⎤⎧ 0 ⎫ ⎪ εx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎥ ⎪ ⎪ ε B26 ⎥ ⎪ y ⎪ ⎪ ⎪ 0⎪ ⎪ ⎥⎪ ⎨ B66 ⎥ γxy ⎬ ⎥ , D16 ⎥ κx ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ D26 ⎦ ⎪ κy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ D66 κxy B16

(13.3)

where Nx , Ny , Nxy and Mx , My , Mxy are the stress resultants and bending moments defined in (1.77) (Fig. 13.1). Aij , Bij , and Dij are, respectively, the extensional, coupling, and bending stiffnesses defined in (1.79). Equilibrium Equations ∂Nyx ∂Nxy ∂Ny ∂Nx + = 0, + = 0, ∂x ∂y ∂x ∂y ∂ 2 My ∂ 2 Mxy ∂ 2 Mx + + 2 + q = 0, ∂x∂y ∂x2 ∂y2

(13.4)

where q is the lateral distributed load applied on the laminates. Note that (13.4)3 represents the forces equilibrium in the thickness direction, which is usually written in terms of the transverse forces Qx and Qy as that shown in (12.2)1 . The moment

438

13


q

h A

s

B

fˆ3

h mˆ 2

x1

θ n

mˆ 3

x2

θ

Vˆn s

x2

x1

fˆ2 fˆ1

xˆ

mˆ 1

Mˆ n

x3

dx1

Lamina Number

N22

Q2

1

M 22 M 21

dx2 N21 Q1

h1

x1

M11 N11 M12

2

h0

M11 + dM11

N12

hk hk

N11 + dN11

N12 + dN12

M 22 + dM 22 M 21 + dM 21

Q1+ dQ1

Middle Plane

h2

q

x3

M12 + dM12

hn

x1

k

1

hn

N21 + dN21 Q 2 + dQ2

N22 + dN22

x2

h

1

n x3 dx1

Fig. 13.1 Laminate geometry, stress resultants, and bending moments

equilibrium in the x- and y-directions shows that the transverse shear forces are related to the bending moments by (12.2)2,3 . Substituting (12.2)2,3 into (12.2)1 , we get (13.4)3 . Governing Equations To get governing equations satisfying all the basic equations, we first use (13.2b) 0 and curvatures κ , κ , κ in terms of to express the mid-plane strains εx0 , εy0 , γxy x y xy the mid-plane displacements u0 , v0 , and w, then use (13.3a) to express the stress resultants Nx , Ny , Nxy and bending moments Mx , My , Mxy in terms of the mid-plane displacements. After these direct substitutions, the three equilibrium equations (13.4) can now be written in terms of three unknown displacement functions u0 , v0 , and w as

A11

∂ 2 u0 ∂ 2 u0 ∂ 2 u0 ∂ 2 v0 ∂ 2 v0 ∂ 2 v0 + A66 2 + A16 2 + (A12 + A66 ) + A26 2 + 2A16 2 ∂x∂y ∂x∂y ∂x ∂y ∂x ∂y 3 3 3 3 ∂ w ∂ w ∂ w ∂ w − B26 3 = 0, − B11 3 − 3B16 2 − (B12 + 2B66 ) ∂x ∂x ∂y ∂x∂ 2 y ∂y (13.5a)

13.1

Coupled Stretching–Bending Theory of Laminates

A16

439

∂ 2 u0 ∂ 2 u0 ∂ 2 v0 ∂ 2 v0 ∂ 2 v0 ∂ 2 u0 + A + A + (A + A ) + A + 2A 12 66 26 66 26 22 ∂x∂y ∂x∂y ∂x2 ∂y2 ∂x2 ∂y2 ∂ 3w ∂ 3w ∂ 3w ∂ 3w − B16 3 − (B12 + 2B66 ) 2 − 3B26 − B22 3 = 0, 2 ∂x ∂x ∂y ∂x∂y ∂y (13.5b)

D11

∂ 4w ∂ 4w ∂ 4w ∂ 4w ∂ 4w + 4D16 3 + 2(D12 + 2D66 ) 2 2 + 4D26 + D22 4 4 3 ∂x ∂x ∂y ∂x ∂y ∂x∂y ∂y 3 3 3 3 ∂ u0 ∂ u0 ∂ u0 ∂ u0 − B11 3 − 3B16 2 − (B12 + 2B66 ) − B26 3 ∂x ∂x ∂y ∂x∂y2 ∂y 3 3 3 ∂ v0 ∂ v0 ∂ 3 v0 ∂ v0 − B16 3 − (B12 + 2B66 ) 2 − 3B26 − B = q, 22 ∂x ∂x ∂y ∂x∂y2 ∂y3 (13.5c)

which are the governing equations for the laminated plates. The governing equations shown in (13.5a), (13.5b), and (13.5c) are system of partial differential equations with three unknown functions u0 , v0 , and w. Due to the mathematical complexity of these equations, it is not easy to get solutions by solving these partial differential equations. In practical engineering applications, it is common to have a symmetric laminate or to construct a balanced laminate. In those cases the coupling stiffness components like Bij and/or A16 , A26 and/or D16 , D26 will be zero, and (13.5a), (13.5b), and (13.5c) will be drastically simplified. Boundary Conditions For the general cases of laminated plates, the in-plane and bending problems will couple each other. Hence, every boundary of the plates should be described by four prescribed values. Two of them correspond to the in-plane problems and the other two correspond to the bending problems. Generally, they may be expressed as un = uˆ n

or

Nn = Nˆ n

or

Nn = kn un ,

us = uˆ s

or

Nns = Nˆ ns

or

Nns = ks us ,

w, n = wˆ , n

or

ˆn Mn = M

or

Mn = km w, n ,

w = wˆ

or

Vn = Vˆ n

or

Vn = kv w,

(13.6)

where Vn is the well-known Kirchhoff force of classical plate theory, or called effective shear force, defined in (12.7b). The subscripts n and s denote, respectively, the directions normal and tangent to the boundary. The overhat denotes the prescribed value. The values in the n–s coordinate can be calculated from the values in the x–y coordinate according to the transformation laws (12.9).

440

13


13.2 Complex Variable Formulation The governing equations (13.5a), (13.5b), and (13.5c) involve both in-plane and plate bending problems, i.e., these two problems are coupled to each other if the coupling stiffnesses Bij are not equal to zero. Due to the mathematical complexity, very few systematic approaches can be found in the literature. Because of the two-dimensional nature of the thin laminates, it is hoped that the complex variable method which is powerful and elegant for the two-dimensional problems can also be applied to the stretching–bending coupling problems. As discussed in Chapters 2 and 3, for two-dimensional linear anisotropic elasticity, there are two important formalisms. One is Lekhnitskii formalism and the other is Stroh formalism. The former is a stress formalism, whereas the latter is a displacement formalism. In Section 13.2.1, we will use the mid-plane displacements and slopes as the basic functions for our formulation, which is therefore called displacement formalism. In addition to the displacement formalism, by using mid-plane displacements and bending moments as basic functions, another formalism called the mixed formalism will be introduced in Section 13.2.2.

13.2.1 Displacement Formalism Although several different kinds of displacement formalisms have been developed in the literature, some fail in their eigenrelation and some fail in their complexity. To have a correct and clear formalism, in this section we follow the work of Hwu (2003b) to derive the displacement formalism in a more Stroh-like approach. For the convenience of later derivation, first we rewrite all the basic equations (13.1), (13.2), (13.3), and (13.4) in terms of tensor notation as follows:

Ui = ui + x3 βi ,

β1 = −w, 1 , β2 = −w, 2 , 1 1 1 ξij = εij + x3 κij = (Ui, j + Uj, i ), εij = (ui, j + uj, i ), κij = (βi, j + βj, i ), 2 2 2 Nij = Aijkl εkl + Bijkl κkl , Mij = Bijkl εkl + Dijkl κkl ,

Nij, j = 0,

Mij, ij + q = 0,

Qi = Mij, j ,

i, j, k, l = 1, 2. (13.7)

Note that in the above tensor notation, we have made a slight change for some symbols and employed the following conventional replacements:

x ↔ x1 , y ↔ x2 , z ↔ x3 , 11 ↔ 1, 22 ↔ 2, 12 or 21 ↔ 6,

(13.8a)

13.2

Complex Variable Formulation

441

for example, u ↔ U1 , εx ↔ ξ11 ,

v ↔ U2 ,

u0 ↔ u1 ,

γxy ↔ 2ξ12 ,

εx0

v0 ↔ u2 ,

↔ ε11 ,

0 γxy ↔ 2ε12 ,

κx ↔ κ11 ,

κxy ↔ 2κ12 ,

Ny ↔ N22 , Mxy ↔ M12 , Qx ↔ Q1 , A16 ↔ A1121 , B21 ↔ B2211 , D66 ↔ D1212 , . . . (13.8b) It should be noted that the replacements of shear strain γxy and curvature κxy are not only symbol change but also two times difference, which is the same as the conventional contracted notation, (1.21). Substituting the strains/curvatures and displacements/slopes relations into the constitutive laws, i.e., substituting (13.7)2 into (13.7)3 , the stress resultants and bending moments can be expressed in terms of mid-plane displacements ui and slopes βi as Nij = Aijkl uk, l + Bijkl βk, l ,

Mij = Bijkl uk, l + Dijkl βk, l .

(13.9)

By employing the results of (13.9) to the equilibrium equations (13.7)4 , the governing equations can also be expressed in terms of the mid-plane displacements ui and slopes βi as Aijkl uk, lj + Bijkl βk, lj = 0,

Bijkl uk, lij + Dijkl βk, lij + q = 0.

(13.10)

Consider the homogeneous case that no lateral load is applied on the laminates, i.e., q=0. Because the mid-plane displacements ui and slopes βi depend only on two variables, x1 and x2 , and (13.10) are homogeneous partial differential equations, we may let uk = auk f (z),

β

βk = ak f (z),

z = x1 + μx2 ,

k = 1, 2.

(13.11)

βk, l = ak (δl1 + μδl2 ) f (z),

(13.12)

Differentiation of (13.11) with respect to xl gives uk, l = auk (δl1 + μδl2 ) f (z),

β

in which the prime • denotes differentiation with respect to the argument z and δij is Kronecker delta. Further differentiating with respect to xj and xi , we find that (13.10) with q=0 will be satisfied if ! " " ! QA + μ(RA + RTA ) + μ2 TA au + QB + μ(RB + RTB ) + μ2 TB aβ = 0, ! " ! " μ∗T QB + μ(RB + RTB ) + μ2 TB au + μ∗T QD + μ(RD + RTD ) + μ2 TD aβ = 0, (13.13a)

442

13


where QA = Ai1k1 , RA = Ai1k2 ,

QB = Bi1k1 , RB = Bi1k2 ,

QD = Di1k1 , RD = Di1k2 ,

TA = Ai2k2 , TB = Bi2k2 , TD = Di2k2 , β au1 a 1 au = u , aβ = 1β , μ∗ = . μ a2 a2

(13.13b)

From the second and third equations of (13.7)1 and the assumption of the slope βk given in the second equation of (13.11), we get β

β

a2 = μa1 .

(13.14)

Equations (13.13a) and (13.14) constitute four equations with four unknowns β β au1 , au2 , a1 , a2 . Thus, the problem is solved in principle. Substituting (13.12) into (13.9), we have Ni1 = −μbu f (z), Ni2 = bu f (z), Mi1 = −μd∗ f (z), Mi2 = df (z),

(13.15a)

bu = (RTA + μTA )au + (RTB + μTB )aβ 1 = − {(QA + μRA )au + (QB + μRB )aβ }, μ d = (RTB + μTB )au + (RTD + μTD )aβ , 1 d∗ = − {(QB + μRB )au + (QD + μRD )aβ }. μ

(13.15b)

where

Note that the second equality of (13.15b)1 comes from (13.13a)1 . Using the relation for the bending moments and transverse shear forces given in the third equation of (13.7)4 , the definition for the effective transverse shear force given in (12.7b), and the results for the bending moments obtained in (13.15a)2 , we obtain Q1 = μiT1 (d − d∗ )f (z), V1 = μiT1 (2d − d∗ )f (z),

Q2 = μiT2 (d − d∗ )f (z), V2 = μiT2 (d − 2d∗ )f (z),

(13.16a)

where 1 i1 = , 0

0 i2 = . 1

(13.16b)

With the definitions of d and d∗ given in (13.15b), the second equation of (13.13a) leads to

13.2


443

μ∗T (d − d∗ ) = 0.

(13.17)

Substituting (13.15a)2 into the symmetry condition of the twist moments, i.e., M12 = M21 , we have d1 = −μd2∗ .

(13.18)

Combining (13.17) and (13.18), we can express d∗ in terms of d or vice versa. Through their relation, we now introduce a new vector bβ as bβ = d + b0 i1 = d∗ +

b0 i2 , μ

(13.19a)

where 1 ∗T 1 β β μ bβ = (b1 + μb2 ) = d1 + μd2 . (13.19b) 2 2 By the relation given in (13.19), the expressions for the bending moments and transverse shear forces obtained in (13.15a)2 and (13.16a) can now be written as b0 =

M11 M12 = (−μbβ + b0 i2 )f (z), = (bβ − b0 i1 )f (z), M21 M22 2 β −μ b2 Q1 −μ V1 = b0 = f (z). f (z), β Q2 V2 1 b

(13.20)

1

Observing the results obtained in (13.15a) and (13.20), we introduce two stress functions φi = bui f (z),

β

ψi = bi f (z).

(13.21)

With the use of these two stress functions, the moments, transverse shear forces, and effective transverse shear forces can be expressed as Ni1 = −φi, 2 ,

Ni2 = φi,1 , 1 1 Mi1 = −ψi, 2 − λi1 ψk, k , Mi2 = ψi,1 − λi2 ψk, k , 2 2 1 1 Q1 = − ψk, k2 , Q2 = ψk, k1 , 2 2 V1 = −ψ2, 22 , V2 = ψ1, 11 ,

(13.22a)

where λij is the permutation tensor defined as λ11 = λ22 = 0,

λ12 = −λ21 = 1.

(13.22b)

Up to now, the formalism is almost complete because the displacements, slopes, moments, and transverse shear forces have all been expressed elegantly in (13.11),

444

13


(13.21), and (13.22). The eigenvalues μ and the displacement eigenvectors au , aβ can be obtained from (13.13a) and (13.14), and the stress function eigenvectors bu , bβ can be obtained from (13.15b) and (13.19). From (13.13) and (13.14), the determination of the eigenvalues μ will lead to an equation of eighth order polynomial, which can be proved to have eight roots with four pairs of complex conjugates (Cheng and Reddy, 2002). By arranging the complex eigenvalues whose imaginary parts are positive to be the first four eigenvalues, and superimposing all their corresponding solutions, the solutions shown in (13.11) and (13.21) can now be written in a compact matrix form as ud = 2Re{Ad f(z)},

φd = 2Re{Bd f(z)},

(13.23a)

φ φd = , ψ

(13.23b)

where u ud = , β

Ad = [a1 a2 a3 a4 ], Bd = [b1 b2 b3 b4 ], ⎧ ⎫ f1 (z1 )⎪ ⎪ ⎪ ⎪ ⎨ ⎬ f2 (z2 ) , zk = x1 + μk x2 , k = 1, 2, 3, 4 f(z) = f3 (z3 )⎪ ⎪ ⎪ ⎪ ⎩ ⎭ f4 (z4 )

(13.23c)

β φ ψ1 u1 , β= 1 , φ= 1 , ψ= , u2 β2 φ2 ψ2 a b ak = u , bk = u , k = 1, 2, 3, 4. aβ k bβ k

(13.23d)

and u=

In order to establish an eigenrelation like the Stroh formalism for two-dimensional problems, we recast (13.15b)1 and (13.19a) with the assist of (13.15b)2,3 into ( ' ( −R −I + 12 I44 a a =μ , b RT −I + 12 I33 b −T − 12 I34

' Q − 12 I43

(13.24a)

where Q=

QA QB , QB QD

R=

a=

RA RB , RB RD

au , aβ

b=

T=

TA TB , TB TD

bu . bβ

(13.24b)

(13.24c)

13.2


445

In the above, I denotes the identity matrix and Imn stands for a matrix with all zero components except the mn component, for example, ⎡

I34

0 ⎢0 =⎢ ⎣0 0

0 0 0 0

0 0 0 0

⎤ 0 0⎥ ⎥, 1⎦ 0

⎡

I44

0 ⎢0 =⎢ ⎣0 0

0 0 0 0

0 0 0 0

⎤ 0 0⎥ ⎥,... 0⎦ 1

(13.25)

In order to see more clearly the eigenrelation shown in (13.24a), we now write down the expressions of QA , QB , . . . , TD defined in (13.13b) as A11 A16 , A16 A66 A A RA = 16 12 , A66 A26 A A TA = 66 26 , A26 A22

QA =

B11 B16 D11 D16 , QD = , B16 B66 D16 D66 B B D16 D12 RB = 16 12 , RD = , B66 B26 D66 D26 B B D66 D26 TB = 66 26 , TD = . B26 B22 D26 D22 QB =

(13.26)

By expanding (13.24a) with the assist of (13.26), we observe that the second and fifth equations of (13.24a) will ensure the equality bu1 = −μbu2 , which is also the consequence of the symmetry of in-plane forces, i.e., N12 = N21 by (13.15a)1 . Moreover, it is observed that the fourth and seventh equations of (13.24a) are identical, which has also been noticed by Cheng and Reddy (2002). Due to the equivalence of the fourth and seventh equations, only seven independent equations remain in (13.24a). The extra independent equation may come from the equality of (13.14), which is a result of thin plate Kirchhoff assumption because the slopes βx and βy are not independent in the classical lamination theory both of them are related to the deflection w. According to the suggestion of Cheng and Reddy (2002), the complete eigenrelation is given by adding (13.14) with two arbitrarily different multipliers, respectively, to the fourth and seventh equations of (13.24a). To have a definite expression, we select these two multipliers to be –1/2 and 1/2, and the final complete eigenrelation can then be expressed as Nd ξ = μξ,

(13.27a)

−1 1 1 L1 + J1 Nd = L2 + J2 2 2

(13.27b)

Q 0 RI a , L = − , ξ = , 2 T0 b RT −I −I44 −I43 −I43 I44 , J2 = . J1 = I34 I33 I33 −I34

(13.27c)

where

and L1 =

446

13


13.2.2 Mixed Formalism In (13.3a), the constitutive laws are written by expressing the stress resultants and bending moments in terms of mid-plane strains and curvatures, which is similar to the use of elastic constants Cijkl for the elastic solids. In applications, sometimes it is convenient by using the compliances Sijkl , i.e., expressing the mid-plane strains and curvatures in terms of stress resultants and bending moments. In this section, the mixed expression will be used with mid-plane strains and moments as basic functions. In order to get a clear relation about these expressions, we rewrite (13.3) in the matrix form as N A B ε0 , (13.28) = M BD κ which may lead to the following mixed expression: ' / /( A B ε0 N = , T κ −/ B / D M

(13.29a)

where / A = A − BD−1 B,

/ B = BD−1 ,

/ D = D−1 .

(13.29b)

The inversion of (13.29a), which will also be used in the following derivation, is now written as ' / A ε0 = T M −/ B

(−1 ' ∗ / / B A N = ∗T / κ D −/ B

( ∗ / B N , ∗ / κ D

(13.30a)

where ∗ / A = A−1 ,

∗ / B = −A−1 B,

∗ / D = D − BA−1 B.

(13.30b)

∗ ∗ Because A, B, and D defined in (1.79) are symmetric matrices, / A, / D, / A , and / D ∗ defined in (13.29b) and (13.30b) will also be symmetric, whereas / B and / B are not symmetric. Note that in the above matrix expressions, the symbols A, B, and N have different representations from the eigenvector matrices Ad , Bd defined in (13.23c) and the fundamental elasticity matrix Nd defined in (13.27). The former is the traditional notation used in the community of mechanics of composite materials, while the latter is the notation generally used in the community of anisotropic elasticity. To let the readers from both communities see clearly what we express in this chapter, we just use the italic bold-faced fonts to denote the extensional, coupling matrices and the vector of stress resultants, and use the roman bold-faced fonts to denote the material eigenvector matrices and the fundamental elasticity matrix.

13.2


447

Similar to the displacement formalism, we rewrite the mixed constitutive laws (13.29) in terms of tensor notation as Aijkl εkl + / Bijkl Mkl , κij = −/ Bklij εkl + / Dijkl Mkl , Nij = /

i, j, k, l = 1, 2.

(13.31)

Note that due to the two times difference for the tensor notation and contracted notation of shear strain and twist curvature denoted in (13.8b), the following rule should be followed: / Aij , Apqrs ↔ /

for all i and j, i, j = 1, 2, 6,

/ Bpqrs ↔ / Bij ,

if j = 6,

/ Dpqrs ↔ / Dij , if i, j = 6, 1 / Dpqrs ↔ / Dij , if either i or j = 6, 2

1 / Bpqrs ↔ / Bij , 2 / Dpqrs ↔

if j = 6,

1 / Dij , 4

if both i and j = 6. (13.32) Because the basic functions we use in the mixed constitutive laws are the strains εij and moments Mij , the kinematic relations shown in (13.7)2 and the equilibrium equations shown in (13.7)4 are better replaced by 1 1 (ui, j + uj, i ), Mij = (ψi, j∗ + ψj, i∗ ), 2 2 = 0, κij, j∗ = 0,

εij = Nij, j where the superscript (x1 , x2 ) by

∗

(13.33a)

denotes the coordinate system (x1∗ , x2∗ ) which is related to x1∗ = −x2 ,

x2∗ = x1 .

(13.33b)

By (13.33b), we have ∂/∂x1∗ = −∂/∂x2 and ∂/∂x2∗ = ∂/∂x1 . Thus, the expressions in (13.33a) for Mij and κij are equivalent to M11 = −ψ1, 2 , M22 = ψ2, 1 , M12 = (ψ1, 1 − ψ2, 2 )/2, κ11, 2 − κ12, 1 = 0, κ12, 2 − κ22, 1 = 0.

(13.34)

The three equations of (13.34)1 show that the moments defined in the second equation of (13.33a)1 will automatically satisfy the equilibrium equation Mij, ij = 0

(13.35)

On the other hand, the kinematic relations for the curvatures shown in the second and third equations of (13.7)1 and the third equation of (13.7)2 will lead to κ11 = −w, 11 ,

κ22 = −w, 22 ,

κ12 = −w, 12 ,

(13.36)

448

13


which will then automatically satisfy the two equations of (13.34)2 , i.e., the second equation of (13.33a)2 . In other words, we may call the second equation of (13.33a)1 as the kinematic relation for the moments, and the second equation of (13.33a)2 as the compatibility equation for the curvatures. Substituting (13.33a)1 into (13.31), the stress resultants and curvatures can be expressed in terms of mid-plane displacements ui and stress function ψi as Nij = / Aijkl uk, l + / Bijkl ψk, l∗ , κij = −/ Bklij uk, l + / Dijkl ψk, l∗ .

(13.37)

With this result, the equilibrium equations and compatibility equations shown in (13.33a)2 can also be expressed in terms of mid-plane displacements ui and stress function ψi as / Aijkl uk, lj + / Bijkl ψk, l∗ j = 0, −/ Bklij uk, lj∗ + / Dijkl ψk, l∗ j∗ = 0.

(13.38)

Like the derivation for the displacement formalism, we now let uk = auk f (z),

ψ

ψk = ak f (z),

z = x1 + μx2 ,

k = 1, 2.

(13.39)

Substituting (13.39) into (13.38) with the use of (13.33b), we obtain !

" " ! T 2 2/ Q/ a + μ(R + R ) + μ T + R + μ(T − Q ) − μ R / / / / / / u / B B B B aψ = 0, A A A A " " ! ! T T T 2 /T T 2 R a aψ = 0, −R/ + μ(Q − T ) + μ + T − μ(R + R ) + μ Q / / / u D D D / / / / B B B B D (13.40a) where au =

au1 au2

,

aψ =

ψ a1 ψ

,

(13.40b)

a2

and the definition of Q, R, and Tare the same as those given in (13.13b). Concerning the unsymmetry of / B, a new matrix / R/ B is defined by / / R/ B = Bi2k1 .

(13.41)

With the understanding of the transformation rules given in (13.8) and (13.32), we /B defined in (13.13b) and can write down the expressions of Q/ B , . . . T/ D , R/ A , Q/ (13.41) as A˜ 11 A˜ 16 QA˜ = ˜ ˜ , A16 A66

B˜ 11 B˜ 16 /2 , QB˜ = ˜ ˜ B61 B66 /2

˜ 16 /2 ˜ 11 D D QD˜ = ˜ ˜ 66 /4 , D16 /2 D

(13.42a)

13.2


449

˜ 12 ˜ /2 D ˜ ˜ A˜ A˜ B˜ /2 B˜ D ˜ ˜ = B61 B66 /2 , , R RA˜ = ˜ 16 ˜ 12 , RB˜ = ˜ 16 ˜ 12 , RD˜ = ˜ 16 B ˜ 26 /2 B˜ 21 B˜ 26 /2 B66 /2 B62 D66 /4 D A66 A26 (13.42b) ˜ 26 /2 ˜ 66 /4 D A˜ 66 A˜ 26 B˜ 66 /2 B˜ 62 D TA˜ = ˜ ˜ , TB˜ = ˜ , TD˜ = ˜ . ˜ 22 D26 /2 D B26 /2 B˜ 22 A26 A22

(13.42c)

Substituting (13.39) into (13.37), we have Ni1 = −μbu f (z), Ni2 = bu f (z), κi1 = bψ f (z), κi2 = μbψ f (z),

(13.43a)

T /B )aψ bu = (R/ + μT/ B − μR/ A )au + (T/ A 1 (Q/ =− B − μQ/ B )aψ , A + μR/ A )au + (R/ μ T T bψ = −(Q/ + μ/ R/ )a + (R/ D − μQ/ D )aψ B u A 1 T T T (R/ =− + μT/ )a − (T/ )a . D − μR/ B B u D ψ μ

(13.43b)

where

Note that the second equalities of (13.43b)1 and (13.43b)2 come from the two equations of (13.40a). Observing the results obtained in (13.43), we introduce one stress function φi and one slope function βi as φi = bui f (z),

ψ

βi = bi f (z).

(13.44)

With the use of these two functions, the stress resultants and curvatures can be expressed as Ni1 = −φi, 2 ,

Ni2 = φi,1 ,

and

κi1 = βi,1 ,

κi2 = βi, 2 .

(13.45)

By (13.44) and (13.45), the symmetry requirements N12 = N21 and κ12 = κ21 lead to − μbu2 = bu1 ,

ψ

ψ

b2 = μb1 .

(13.46)

Similar to the displacement formalism, by superimposing all the associated solutions shown in (13.39) and (13.44), the general solutions to the basic equations (13.31) and (13.33) can be written in a compact matrix form as um = 2Re{Am f(z)}, where

φm = 2Re{Bm f(z)},

(13.47a)

450

13

u um = , ψ


φ φm = , β

(13.47b)

Bm = [b˜ 1 b˜ 2 b˜ 3 b˜ 4 ]

Am = [ã1 a˜ 2 a˜ 3 a˜ 4 ], and

au b ˜ , b = u , a˜ k = aψ k k bψ k

k = 1, 2, 3, 4.

(13.47c)

Like the displacement formalism, (13.43b) can be re-organized into the following eigenrelation: ' ( / a −Rm =μ −Tm −I / b

' Qm

( / a , / 0 b

−I

0

RTm

(13.48a)

where Qm =

' Q/ A T R/ B

(

R/ B −T/ D

,

Rm =

' R/ A T T/ B

−Q/ B T R/ D

'

( ,

T/ A Tm = −/ RT

/ B

−/ R/ B

(

. −Q/ D (13.48b)

Better than the eigenrelation of the displacement formalism shown in (13.24a), (13.48a) has exactly the same form as that of Stroh formalism for two-dimensional problems. Therefore, all the relations originated from the eigenrelation for the twodimensional problems can automatically be transferred to the present eigenrelation of mixed formalism. By using the inverse relation −1 0 −T−1 −Rm −I m , = −Tm 0 −I Rm T−1 m

(13.49)

the eigenrelation (13.48a) can then be written into the following standard eigenrelation as Nm/ ξ = μ/ ξ,

(13.50a)

where Nm =

(Nm )1 (Nm )2 , (Nm )3 (Nm )T1

/ a / ξ= / b

(13.50b)

and T (Nm )1 = −T−1 m Rm ,

T (Nm )2 = T−1 m = (Nm )2 ,

T T (Nm )3 = Rm T−1 m Rm − Qm = (Nm )3 .

(13.50c)

13.2


451

13.2.3 Explicit Expressions of N, A, and B As shown in the general solution (13.23) or (13.47) and the eigenrelation (13.27) or (13.50), the material eigenvector matrices (Ad , Bd for displacement formalism and Am , Bm for the mixed formalism) and the fundamental elasticity matrix (Nd for displacement formalism and Nm for the mixed formalism) play important roles in the Stroh-like formalism for coupled stretching–bending analysis. Like the Stroh formalism for two-dimensional problems, it would be of much benefit if we can get the explicit expressions of the material eigenvectors and the fundamental elasticity matrix. As described in Chapter 3, the explicit expressions of material eigenvector matrices A and B for two-dimensional problems are obtained from comparison with the stress-based Lekhnitskii formalism. And, the explicit expression of the fundamental elasticity matrix N is obtained from definitions same as those given in (13.50) for the mixed formalism not those given in (13.27) for the displacement formalism. With this understanding, we see that to obtain the explicit expressions for the material eigenvectors and fundamental elasticity matrix, the mixed formalism is more suitable than the displacement formalism. Material Eigenvectors To find the explicit expressions of material eigenvectors, we first consider the constitutive laws shown in (13.30). By using (13.33a)1 , (13.39), (13.44), (13.45), and (13.46), we get ⎫ ⎫ ⎧ ⎧ au1 ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ 11 ⎪ ⎪ ⎪ ⎪ u ⎪ ⎪ ⎪ ⎪ μa ⎪ ⎪ ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ 2 22 ⎪ ⎪ ⎪ ⎪ u u ⎬ ⎬ ⎨ μa + a ⎨ 2ε12 1 2 ψ f (z), = −μa M ⎪ ⎪ ⎪ ⎪ 11 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ψ ⎪ ⎪ ⎪ M22 ⎪ ⎪ ⎪ ⎪ ⎪ a2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎩ ψ ψ ⎩ ⎭ M12 (a − μa )/2 1

2

⎧ ⎫ ⎧ 2 u⎫ μ b2 ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ 11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ bu2 ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ 22 ⎪ ⎪ ⎪ u ⎬ ⎨ ⎬ ⎨−μb ⎪ N12 2 ψ f (z). = b κ ⎪ ⎪ ⎪ ⎪ 11 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ψ⎪ ⎪ ⎪ κ22 ⎪ ⎪ ⎪ ⎪ ⎪ μ2 b1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎩ ⎭ ψ⎪ 2κ12 2μb1

(13.51)

Substituting (13.51) into (13.30), and expanding its results into six equations, we obtain ψ

ψ

ψ

au1 = p1 bu2 + q1 b1 , μau2 = p2 bu2 + q2 b1 , μau1 + au2 = p6 bu2 + q6 b1 , ψ ψ ψ ψ ψ ψ ψ −μa1 = −h1 bu2 + g1 b1 , a2 = −h2 bu2 + g2 b1 , (a1 − μa2 )/2 = −h6 bu2 + g6 b1 , (13.52a) where A∗j2 − μ/ A∗j6 , qj = / B∗j1 + μ2/ B∗j6 , A∗j1 + / pj = μ2/ B∗j2 + 2μ/ hj = μ2/ B∗2j − μ/ B∗6j , gj = / D∗j1 + μ2 / D∗j6 . B∗1j + / D∗j2 + 2μ/

(13.52b)

Since all the three equations of (13.52a)1 and (13.52a)2 are not independent of each other, by standard elimination procedure with proper multiplication, addition, and subtraction we obtain

452

13 ψ

Coupled Stretching–Bending Analysis ψ

4 bu2 + 3 b1 = 0, 2 bu2 − 1 b1 = 0,

(13.53a)

g1 h1 μg2 μh2 + + g6 , 2 = + + h6 , 2μ 2 2μ 2 q2 p2 − q6 , 4 = μp1 + − p6 . 3 = μq1 + μ μ

(13.53b)

where 1 =

ψ

Equations (13.53) show that nontrivial solutions of bu2 and b1 exist only when 1 (μ)4 (μ) + 2 (μ)3 (μ) = 0.

(13.54)

By viewing (13.52b) and (13.53b), we know that (13.54) is an eighth order polynomial which should lead to the same eigenvalues as those obtained from the eigenvalue relation (13.50). Furthermore, after obtaining the eigenvalues from (13.54), (13.53) give us ψ

b1 = λbu2 ,

λ=−

4 2 = , 3 1

if 1 and/or 3 = 0

(13.55a)

or ψ

bu2 = λ−1 b1 ,

λ−1 = −

3 1 = , 4 2

if 2 and/or 4 = 0.

(13.55b)

With the results of (13.46), (13.55), and (13.52), the explicit expressions for the eigenvectors / a and / b can then be written as a˜ =

au aψ

⎧ ⎧ ⎫ ⎫ p1 + λq1 ⎪ −μ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎪ ⎬ ⎬ 1 b (p2 + λq2 )/μ u , , b˜ = = = λ ⎪ bψ ⎪ ⎪ ⎪(h1 − λg1 )/μ⎪ ⎪ ⎪ ⎩ ⎩ ⎪ ⎭ ⎭ λμ −h2 + λg2

if 1 and/or 3 = 0 (13.56a)

or ⎧ −1 ⎧ −1 ⎫ ⎫ λ p1 + q1 ⎪ −λ μ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −1 ⎨ −1 ⎪ ⎬ ⎬ b (λ p2 + q2 )/μ au λ u ˜ , b = a˜ = = = , if 2 and/or 4 = 0. aψ bψ (λ−1 h1 − g1 )/μ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ μ −λ−1 h2 + g2 (13.56b) Fundamental Elasticity Matrix

From the definition given in (13.50), we know that the fundamental elasticity matrix Nm of the mixed formalism is a 8×8 matrix which is related to the extensional, bending, and coupling stiffness matrices. Although it looks complicated, it is not

13.2


453

difficult to get the explicit expression because the definition of the fundamental elasticity matrix given in (13.50) has been purposely arranged to be the same form as that of the two-dimensional problems. With this understanding, by following the steps described in Ting’s book (1996) for Stroh formalism we can find the explicit expressions of Nm . This benefit for the mixed formalism cannot be applied to the displacement formalism because the definition given in (13.27) is not perfectly matched with the Stroh formalism for two-dimensional problems. In the following section, we will try to find the relation between Nm and Nd through the proper comparison of these two formalisms. With that relation, the explicit expression of Nd can be found via Nm . To find the explicit expressions for the fundamental elasticity matrix Nm , one can follow the steps described in Section 11.3.1 for piezoelectric materials or refer to Hsieh and Hwu (2002b) for detailed derivation. Following are the explicit expressions of (Nm )i , i = 1,2,3, which are the sub-matrices of Nm : ⎡

⎡ ⎤ X11 X12 0 X14 Y11 ⎢Y12 ⎥ 1 ⎢ 1 X 0 0 X 21 24 ⎥ , (Nm )2 = ⎢ (Nm )1 = ⎢ / ⎣X31 0 0 X34 ⎦ / ⎣Y13 X41 0 X43 X44 Y14 ⎡ ⎤ ∗ ∗ −/ D22 0 0 / B12 1 ⎢ 0 0 0 0 ⎥ ⎥, (Nm )3 = ⎢ /⎣ 0 00 0 ⎦ / B∗12 0 0 / A∗11

Y12 Y22 Y23 Y24

Y13 Y23 Y33 Y34

⎤ Y14 Y24 ⎥ ⎥, Y34 ⎦ Y44

(13.57a)

where / =/ B∗12/ A∗11 / B∗12 + / D∗22 , A∗16 / B∗12/ A∗11 / B∗12/ A∗11/ A∗16/ X11 = / D∗22 + / B∗62 , X12 = −(/ D∗22 + / B∗12 ), X14 = / B∗62 − / B∗12 , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ A12 / B12/ A11/ A12/ X21 = / D22 + / B22 , X24 = / B22 − / B12 , ∗ ∗ ∗ ∗ ∗ ∗ B11 / B12 / A11 / B∗11/ X31 = / D22 − / D12 , X34 = −(/ D12 + / B∗12 ), B∗16 / B∗12 / A∗11 / B∗12/ A∗11 / B∗16/ X41 = 2(/ D∗22 − / D∗26 ), X43 = / D∗22 + / B∗12 , X44 = −2(/ D∗26 + / B∗12 ), (13.57b) /∗ /∗2 /∗ /∗ /∗ /∗2 /∗ /∗ /∗ /∗ A∗11/ Y11 = / B∗2 62 + A66 B12 + A11 A66 D22 − A16 D22 − 2A16 B12 B62 , /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ Y12 = / A∗11/ A∗26/ B∗22/ B∗62 + / B∗2 12 + A11 A26 D22 − A12 A16 D22 − A12 B12 B62 − A16 B12 B22 , /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ Y13 = / A∗16/ B∗2 B∗12 / D∗12 + / 12 B61 + A11 B61 D22 − A16 B11 D22 − A11 B62 D12 − B11 B12 B62 , /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ Y14 = 2(/ A∗16/ B∗2 B∗12 / D∗26 + / 12 B66 + A11 B66 D22 − A16 B16 D22 − A11 B62 D26 − B12 B16 B62 ), (13.57c) /∗ /∗2 /∗ /∗ /∗ /∗2 /∗ /∗ /∗ /∗ Y22 = / A∗11/ B∗2 22 + A22 B12 + A11 A22 D22 − A12 D22 − 2A12 B12 B22 , /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ A∗12/ B∗2 Y23 = / B∗12 / D∗12 + / 12 B21 + A11 B21 D22 − A12 B11 D22 − A11 B22 D12 − B11 B12 B22 , /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ /∗ Y24 = 2(/ A∗12/ B∗2 B∗12 / D∗26 + / 12 B26 + A11 B26 D22 − A12 B16 D22 − A11 B22 D26 − B12 B16 B22 ), (13.57d)

454

13


/∗ /∗ /∗ /∗ /∗ /∗ /∗2 /∗ /∗2 /∗ Y33 = / A∗11 / D∗2 12 + 2B11 B12 D12 − B11 D22 − B12 D11 − A11 D11 D22 , /∗ /∗ /∗ /∗ Y34 = 2(/ B∗12/ A∗11 / B∗11/ A∗11 / B∗2 B∗16 / D∗12 + / D∗12 / D∗26 + / B∗12 / D∗26 − / D∗16 / D∗22 − / 12 D16 − B11 B16 D22 ), /∗ /∗ /∗ /∗ /∗ /∗ /∗2 /∗ /∗2 /∗ Y44 = 4(/ A∗11 / D∗2 26 + 2B12 B16 D26 − B16 D22 − B12 D66 − A11 D22 D66 ),

(13.57e)

∗ ∗ ∗ in which / A ,/ B , and / D are defined in (13.30b).

13.2.4 Reduction to Symmetric Laminates In our previous derivation, displacement or mixed formalism, no symmetry condition is required on the laminates. In practical engineering applications, it is common to design a symmetric laminate whose coupling stiffnesses Bij are zero. For this kind of composite structures, the in-plane problem and plate bending problem decouple. In Chapter 12, without involving the coupling conditions, a Stroh-like bending formalism for the anisotropic plate is developed through the Stroh–Lekhnitskii’s connection. Because that formalism only considers the plate bending analysis, to avoid coupling effects the anisotropic plates should have one plane of elastic symmetry located at the mid-plane of the plate. Therefore, it can only be applied directly to the symmetric laminates not the general composite laminates with coupling. In this section, we will use the Stroh-like bending formalism as a check of the displacement and mixed formalisms by reducing the results of Sections 13.2.1 and 13.2.2 to the symmetric laminates. Displacement Formalism Substituting Bij = 0 into (13.26), we obtain QB = RB = TB = 0. With this result and the definitions of Q, R, and T given in (13.24b), the eigenrelation (13.27) can now be separated into two parts as

−1 QA 0 au au −RA −I = μ bu −TA 0 RTA −I bu

(13.58a)

and ' −RD − 12 I21 −TD + 12 I11

−I + 12 I22 − 12 I12

(−1 ' QD − 12 I22 RTD + 12 I12

− 12 I21 −I + 12 I11

(

aβ bβ

aβ =μ . bβ (13.58b)

13.2


455

In the above, (13.58a) corresponds to the in-plane problems, while (13.58b) corresponds to the plate bending problems. By careful comparison, we see that (13.58a) is identical to that of two-dimensional problems, (3.46), but (13.58b) looks different. Substituting (13.26) into (13.58b) and performing 4×4 matrix inversion and multiplication carefully, we obtain Nβ ξβ = μξβ ,

ξβ =

aβ , bβ

1 26 −2 D D22

0 0

(13.59a)

where ⎡

0

0

⎤

1 12 ⎥ ⎢ −D ⎢ D22 D22 ⎥ ⎥ ⎢ 2 D Nβ = ⎢ D12 D26 D12 ⎥ , −2(D − ) 0 − ⎢ −D11 + D12 16 D22 D22 ⎥ 22 ⎦ ⎣ 2 D D12 D26 D26 26 −2(D16 − D22 ) −4(D66 − D22 ) 1 −2 D22

(13.59b)

which is equivalent to the explicit form given in (12.46) for plate bending analysis. Because the eigenvalues and eigenvectors for the symmetric laminates can be separated into two parts, in-plane problem and plate bending problem, the general solutions shown in (13.23) can also be separated into these two parts. They are u = 2Re{Au f(z)}, φ = 2 Re{Bu f(z)}, β = 2Re{Aβ f(z)}, ψ = 2 Re{Bβ f(z)},

(13.60a)

Au = [(au )1 (au )2 ], Bu = [(bu )1 (bu )2 ], Aβ = [(aβ )1 (aβ )2 ], Bβ = [(bβ )1 (bβ )2 ]

(13.60b)

f1 (z1 ) f(z) = . f2 (z2 )

(13.60c)

where

and

From the solutions shown in (13.60) for the in-plane and plate bending problems, we see that they have exactly the same form as that of the Stroh formalism shown in (3.24) for two-dimensional problems and in (12.39) for plate bending problem. Mixed Formalism Substituting Bij = 0 into (13.29), we obtain / A = A, / B = 0, / D = D−1 , which / will then lead to Q/ B = R/ B = R/ B = T/ B = 0 by (13.42). With this result and the definitions of Qm , Rm , and Tm given in (13.48b), the eigenrelation (13.50) can now be separated into two parts as

456

13

'

T −T−1 A RA

T−1 A


( au

−1 T RA T−1 A RA − QA −RA TA

bu

=μ

au bu

(13.61a)

and '

−1 R/ Q/ D D

−1 −Q/ D

(

T Q−1 R + T T Q−1 −R/ / / D D R/ D D D / D /

aψ bψ

aψ =μ . bψ

(13.61b)

Similar to the displacement formalism, eigenrelations (13.61a) and (13.61b) correspond to the in-plane and plate bending problems, respectively. By simple inversion and multiplication, it can be proved that (13.58a) and (13.61a) are equivalent. To prove that (13.61b) is identical to (13.58b), we first note that aψ = bβ ,

bψ = aβ ,

(13.62)

which can be observed from (13.11), (13.21), (13.39), and (13.44). With this information, through the use of (13.42) we can prove that (13.61b) is identical to (13.58b). Similar to the displacement formalism, the general solutions shown in (13.47) can also be separated into two parts, which are exactly the same as those shown in (13.60). Because the explicit expressions for the eigenvectors have been obtained in the mixed formalism, the eigenvectors can also be checked by the results given in (12.40a) for the symmetric laminates. Substituting Bij = 0 into (13.30b), we ∗ ∗ ∗ B = 0, / D = D. If we use A∗ij to denote the components of A−1 , have / A = A−1 , / (13.52b) gives us pj = μ2 A∗j1 + A∗j2 − μA∗j6 ,

qj = hj = 0,

gj = Dj1 + μ2 Dj2 + 2μDj6 , (13.63)

which will then lead to, by the use of (13.53b), 2 = 3 = 0.

(13.64)

With this result, the characteristic equation for the eigenvalues shown in (13.54) becomes 1 (μ)4 (μ) = 0.

(13.65)

In the above, 4 (μ) = 0 will provide the eigenvalues for the in-plane problems, whereas 1 (μ) = 0 will provide the eigenvalues for the plate bending problems. The explicit expressions for the eigenvectors can therefore be separated into two parts. One is from (13.56a) and the other is from (13.56b). They are p1 −μ , bu = , p2 /μ 1

au =

for in-plane problems,

(13.66a)

13.2


aψ =

−g1 /μ 1 , , bψ = μ g2

457

for plate bending problems,

(13.66b)

which are identical to those shown in (3.27) for the in-plane problems and in (12.40a) for the plate bending problems. ∗ ∗ ˜ ∗ = D into (13.57) for the symmetric lamiSubstituting A˜ = A−1 , B˜ = 0, D nates, we can also prove that the two separate parts of the explicit expressions of the fundamental elasticity matrices corresponding to the in-plane and bending problems are exactly the same as those presented in (3.81) and (12.46).

13.2.5 Comparison and Discussion From the identities presented in Section 3.4, we observe that through the use of the eigenrelation many useful identities relating the material properties to the eigenmodes of stress functions and displacements can be established. With the assist of these identities, many problems that are left with unsolved linear algebraic system can be solved explicitly. Moreover, many complex variable form solutions may be transformed to real-form solutions. With this understanding, in this section the comparison will be emphasized upon the resemblance of the general solutions and their associated eigenrelations, because the more alike to the Stroh formalism the more possible we can benefit from the experience of two-dimensional problems. By comparing the general solutions of Stroh formalism for two-dimensional problem (3.24), Stroh-like bending formalism for plate bending analysis (12.39), displacement formalism (13.23), and mixed formalism (13.47), we see that the solution form of displacement formalism is exactly the same as that of Stroh formalism for two-dimensional problem. While for the mixed formalism, its in-plane part is still exactly the same as Stroh formalism for two-dimensional problem, but its plate bending part conforms in the opposite way with A corresponding to the stress function and B corresponding to the slope. Purely from the comparison of general solutions, one may conclude that displacement formalism should be the one most alike to Stroh formalism for twodimensional problem. However, comparison of eigenrelation shows the opposite. From (13.50), we see that the eigenrelation of mixed formalism has exactly the same form as that for two-dimensional problem shown in (3.48). However, the eigenrelation for displacement formalism (13.27) is different by the addition of the matrices J1 and J2 . Because the matrices N1 , N2 , and N3 play important roles in Stroh formalism for two-dimensional problems, they are called fundamental elasticity matrices. Therefore, although the displacement formalism has exactly the same form in the general solution, sometimes it may not be a good choice for the lamination theory due to the lack of resemblance of its eigenrelation. Thus, from the viewpoint of the eigenrelation, the mixed formalism is a better choice than the displacement formalism for solving the practical lamination problems. However, because the generalized displacement vector um and the generalized stress function vector φm in

458

13


mixed formalism have their mix nature, it may become inconvenient when one deals with the pure stress or displacement boundary-valued problems. On the other hand, if a mixed boundary-valued problem (prescribed in-plane displacements and out-of-plane bending moments/effective transverse shear forces, or prescribed inplane forces and out-of-plane deflections/slopes) is considered, mixed formalism may be a good choice. From the above discussion, we know that both the displacement and mixed formalisms are not perfectly alike to the Stroh formalism for two-dimensional problems. One is alike in general solution and the other is alike in eigenrelation. To combine the merits from both formalisms, we may use the general solutions formed by the displacement formalism and when there is a need to count on the eigenrelation we may use the eigenrelation from the mixed formalism. From this viewpoint, we need to know the relation between the fundamental elasticity matrices Nd and Nm . By (13.62), we get the following relation: ⎧ ⎫ ⎡ ⎤⎧ ⎫ I 000 ⎪ au ⎪ ⎪ ⎪ ⎪au ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ ⎥ ⎨aβ ⎬ / a 0 0 0 I I1 I2 a a ψ ⎥ =⎢ = = It ξ, ξ˜ = / = (13.67) ⎣0 0 I 0⎦ ⎪bu ⎪ bu ⎪ I2 I1 b b ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0I 00 bψ bβ in which I1 , I2 , and It are defined through the equalities. Substituting (13.67) into (13.50) and comparing its results with (13.27), we obtain Nd = It Nm It ,

(13.68)

N1 N2 , Nd = N3 NT1

(13.69a)

or in the sub-matrix form,

where N1 = I1 (Nm )1 I1 + I1 (Nm )2 I2 + I2 (Nm )3 I1 + I2 (Nm )T1 I2 , N2 = I1 (Nm )1 I2 + I1 (Nm )2 I1 + I2 (Nm )3 I2 + I2 (Nm )T1 I1 , N3 =

(13.69b)

I2 (Nm )1 I1 + I2 (Nm )2 I2 + I1 (Nm )3 I1 + I1 (Nm )T1 I2 .

13.3 Stroh-Like Formalism 13.3.1 General Solutions From the comparison and discussion presented in Section 13.2.5, we know that the general solution obtained from the displacement formalism is more convenient to be utilized if the boundary conditions are purely displacement prescribed or force/moment prescribed. However, the eigenrelation of the mixed formalism is

13.3

Stroh-Like Formalism

459

more alike to the Stroh formalism for two-dimensional problems. Therefore, in the following we list the general solutions of displacement formalism and employ the eigenrelation of mixed formalism. Note that for the convenience of later employment, the subscript d of Ad , Bd , and Nd has been dropped if no confusion occurs. ud = 2 Re{Af(z)},

φd = 2 Re{Bf(z)},

(13.70a)

where ud =

u , β

φd =

φ , ψ

A = [a1 a2 a3 a4 ], B = [b1 b2 b3 b4 ], ⎫ ⎧ f1 (z1 )⎪ ⎪ ⎪ ⎪ ⎬ ⎨ f (z ) f(z) = 2 2 , zk = x1 + μk x2 , k = 1, 2, 3, 4, f3 (z3 )⎪ ⎪ ⎪ ⎪ ⎭ ⎩ f4 (z4 )

(13.70b)

and u1 β1 −w, 1 φ1 ψ1 u= , β= = , φ= , ψ= , u2 β2 −w, 2 φ2 ψ2

(13.70c)

in which the subscript comma denotes differentiation. In (13.70c), φi , i = 1, 2, are the stress functions related to the in-plane forces Nij , and ψi , i = 1, 2, are the stress functions related to the bending moments Mij , transverse shear forces Qi , and effective transverse shear forces Vi . Their relations are given in (13.22) and can be rewritten as N11 = −φ1, 2 ,

N22 = φ2, 1 ,

N12 = φ1, 1 = −φ2, 2 = N21 ,

M11 = −ψ1, 2 , M22 = ψ2, 1 , M12 = ψ1, 1 − η = −ψ2, 2 + η = M21 , (13.71a) Q1 = −η, 2 , Q2 = η, 1 , V1 = −ψ2, 22 ,

V2 = ψ1, 11 ,

where η=

1 ψk, k . 2

(13.71b)

13.3.2 Material Eigenrelation Combining the relations shown in (13.27a) and (13.68), the material eigenvalues μk and their associated eigenvectors ak and bk can be determined from the following eigenrelation:

460

13


Nξ = μξ,

(13.72a)

where ξ=

N = It Nm It ,

a b

(13.72b)

and (Nm )1 (Nm )2 N1 N2 I1 I2 , N , I , = = m t I2 I1 N3 NT1 (Nm )3 (Nm )T1 T T −1 (Nm )1 = −T−1 m Rm , (Nm )2 = Tm = (Nm )2 , I0 00 T −1 T , I2 = . (Nm )3 = Rm Tm Rm − Qm = (Nm )3 , I1 = 00 0I

N=

(13.72c)

Note that the material eigenvalues μk have been assumed to be distinct in the general solution (13.70). Moreover, the four pairs of material eigenvectors (ak , bk ), k = 1, 2, 3, 4, are assumed to be those corresponding to the eigenvalues with positive imaginary parts. For the materials whose eigenvalues are repeated, a small perturbation in their values may be introduced to avoid the degenerate problems (Hwu and Yen, 1991) or a modification on the general solution can be done (see Section 13.5, or refer to Yin (2003a, b)). In (13.72b), Nm is the fundamental elasticity matrix of mixed formalism whose explicit expressions have been given in (13.57). In (13.72c), the three 4×4 real matrices Qm , Rm , and Tm are defined in (13.48b) and can be rewritten as ⎡ / A11 / A16 ⎢ / / ⎢ A16 A66 Qm = ⎢ ⎣/ B66 /2 B16 /2 / / B12 / B62

/ B16 /2 / B66 /2

/ B12 / B62

⎤

⎥ ⎥ ⎥, −/ D66 /4 −/ D26 /2 ⎦ −/ D26 /2 −/ D22 ⎡ / ⎤ A16 / B11 −/ B16 /2 A12 −/ ⎢ / B61 −/ B66 /2⎥ A26 −/ ⎢ A66 / ⎥ Rm = ⎢ ⎥, ⎣/ B66 /2 / B26 /2 / D16 /2 / D66 /4 ⎦ / B62 / B22 / D12 / D26 /2

⎡ / / A66 A26 ⎢ / / A22 ⎢ A26 Tm = ⎢ ⎣ −/ B61 −/ B21 / / −B66 /2 −B26 /2

⎤ −/ B61 −/ B66 /2 −/ B21 −/ B26 /2 ⎥ ⎥ ⎥. −/ D11 −/ D16 /2⎦ −/ D16 /2 −/ D66 /4

(13.73a)

(13.73b)

(13.73c)

/ Aij , / Bij , and / Dij are components of the matrices / A, / B, and / D which are related to the extensional, bending, and coupling stiffness matrices A, B, and D by

13.3


461

/ A = A–BD−1 B,

/ B = BD−1 ,

/ D = D−1 .

(13.74)

Like the generalized eigenrelation for the two-dimensional problems (3.76), the eigenrelation (13.72) can be generalized as N(ω)ξ = μ(ω)ξ,

(13.75a)

where a N(ω) = It Nm (ω)It , ξ = , b ' ( ( ' N1 (ω) N2 (ω) (Nm (ω))1 (Nm (ω))2 , Nm (ω) = , N(ω) = (Nm (ω))3 (Nm (ω))T1 N3 (ω) NT1 (ω) T (Nm (ω))1 = −T−1 m (ω)Rm (ω),

(Nm (ω))2 = T−1 m (ω),

(13.75b)

(13.75c)

T (Nm (ω))3 = Rm (ω)T−1 m (ω)Rm (ω) − Qm (ω)

and μ(ω) =

− sin ω + μ cos ω . cos ω + μ sin ω

(13.75d)

In (13.75c), Qm (ω), Rm (ω), and Tm (ω) are related to the matrices Qm , Rm , and Tm defined in (13.73) by Qm (ω) = Qm cos2 ω + (Rm + RTm ) sin ω cos ω + Tm sin2 ω, Rm (ω) = Rm cos2 ω + (Tm − Qm ) sin ω cos ω − RTm sin2 ω, Tm (ω) = Tm cos ω 2

− (Rm + RTm ) sin ω cos ω

(13.75e)

+ Qm sin ω, 2

in which ω denotes the angle between the transformed and original coordinates.

13.3.3 Stress Functions Because in Stroh-like formalism the solution fields are expressed in terms of generalized displacement and stress function vectors, ud and φd , in order to employ this formalism to a practical engineering problem we must transform the related displacement and force boundary conditions into the form of ud and φd . From the displacement fields assumed in (13.7)1 and the generalized displacement vector ud defined in (13.70b) and (13.70c) we see that the displacement boundary values usually described by using u1 , u2 , w and β1 , β2 have direct relation with ud , and hence no further discussion about the displacement is needed. However, for prescribed force boundaries, it is not clear on how to describe the boundary conditions in terms of stress function vector φd . In this section some useful relations about the stress

462

13


resultants, bending moments, transverse shear forces, and their resultants will be derived, which are helpful for the description of force boundary conditions in terms of stress function vector φd . Stress Resultants, Bending Moments, and Transverse Shear Forces Since in Stroh-like formalism for coupled stretching–bending analysis the general solutions are expressed in terms of the displacement vector ud and the stress function vector φd , it is hoped that the stress resultants, bending moments, and transverse shear forces can also be calculated directly from φd . Like the relations derived for two-dimensional problems (3.35), (3.36), and (3.37) and plate bending problems (12.17)–(12.27), we can now define the surface traction tn and surface moment mn along the surface with normal n as (n)

ti

= Nij nj ,

(n)

mi

= Mij nj ,

(13.76a)

and the surface traction ts and surface moment ms along the surface with normal s which is perpendicular to the direction n as (s)

(s)

ti = Nij sj ,

mi = Mij sj ,

∂x1 ∂x2 = , ∂s ∂n

n2 = s1 = cos θ =

(13.76b)

in which n1 = −s2 = − sin θ = −

∂x2 ∂x1 = , (13.77) ∂s ∂n

where θ denotes the angle from the positive x1 -axis to the direction s in clockwise direction (see Fig. 13.1). Substituting (13.71) into (13.76a) and (13.76b), we get t n = φ, s , ts = −φ, n ,

mn = ψ, s − ηs, ms = −ψ, n + ηn,

(13.78a)

where η=

1 2

∂ψ1 ∂ψ2 + ∂x1 ∂x2

,

sT = (cos θ , sin θ ),

nT = (− sin θ , cos θ ). (13.78b)

The stress resultants and bending moments in s–n coordinates can therefore be calculated by Nn = nT tn = nT φ, s ,

Nns = sT tn = sT φ, s ,

Ns = sT ts = −sT φ, n ,

Nsn = nT ts = −nT φ, n = Nns ,

Mn = nT mn = nT ψ, s ,

Mns = sT mn = sT ψ, s − η,

Ms = sT ms = −sT ψ, n ,

Msn = nT ms = −nT ψ, n + η = Mns .

(13.79)

13.3


463

By the equalities of Nns = Nsn and Mns = Msn , we have sT φ, s + nT φ, n = 0,

and

η=

1 T (s ψ, s + nT ψ, n ). 2

(13.80)

The transverse shear forces and effective transverse shear forces in s–n coordinates can be obtained by utilizing the transformation laws (12.9c), the definition (12.7b), and relations (13.71)3, 4 , (13.78), and (13.79). The results are Qn = η, s , Qs = −η, n , Vn = (sT ψ, s ), s , Vs = −(nT ψ, n ), n .

(13.81)

From the above equations, we see that all the forces and moments in s–n coordinates have simple relations with the stress function vector φd which contains subvectors φ and ψ. Through the relations given in (13.79), (13.80), and (13.81), it is clear on how to describe the force/ moment boundary conditions in terms of φd . For example, a traction-free boundary condition usually written by Nn = Nns = Mn = Vn = 0,

(13.82a)

can now be written in terms of the generalized stress function vector φd as nT φ, s = sT φ, s = nT ψ, s = (sT ψ, s ), s = 0,

or

φd = 0.

(13.82b)

In addition to the s–n coordinate, in engineering applications sometimes it is interested to present results in terms of polar coordinate system instead of the rectangular coordinate system. For polar coordinate system (r, θ ), we may let s and n denote the directions of r and θ , and ∂s and ∂n be replaced by ∂r and r∂θ . With this replacement, it can be proved that the stress resultants/bending moments in the polar coordinate system are related to the stress functions by 1 1 Nrθ = sT φ, r = − nT φ, θ , Nr = − sT φ, θ , r r 1 1 Mθ = nT ψ, r , Mrθ = sT ψ, r − η = − nT ψ, θ + η, Mr = − sT ψ, θ , r r 1 1 T 1 T s ψ, r + n ψ, θ , Qθ = η, r , Qr = − η, θ , η = r 2 r 1 1 1 nT ψ, θ Vθ = (sT ψ, r ), r = sT ψ, rr , Vr = − = − 2 (nT ψ, θθ − sT ψ, θ ), r r r ,θ (13.83) where s and n denoting the directions of r and θ can still be expressed by those given in (13.78b). Nθ = nT φ, r ,

Resultant Forces and Moments Consider an edge surface bounded by two arcs (lie on the edge of the top and bottom surfaces of the laminates) and two straight lines (normal to the laminate surface) as

464

13


shown in Fig. 13.1. The arcs have the positive direction from A to B and the thickness of the laminate is h. The surface traction ti on each point of the boundary surface can be calculated by Cauchy’s formula as ti = σij nj ,

(13.84)

where nj is the unit normal to the surface boundary (Fig. 13.1). Knowing that Nij , Qi , Mij are defined as the stress resultants across the laminate thickness, (1.77) and (12.3), by using relations (13.84) and (13.77) the resultant forces/ ti and moments / mi about the coordinate origin along the boundary surface can then be integrated as / t1 = / t2 = / t3 =

B h/2

A −h/2 B h/2 A −h/2 B h/2 A

B h/2

/ m1 =

A

/ m3 =

−h/2 B h/2

=

B

t2 dx3 ds =

A B

t3 dx3 ds =

−N11 dx2 + N12 dx1 , −N12 dx2 + N22 dx1 ,

(13.85a)

−Q1 dx2 + Q2 dx1 ,

A

(x2 t3 − x3 t2 )dx3 ds =

(x3 t1 − x1 t3 )dx3 ds =

−h/2

A

B

A B

A −h/2 B h/2

/ m2 =

−h/2

t1 dx3 ds =

B

M12 dx2 − M22 dx1 − x2 (Q1 dx2 − Q2 dx1 ),

A B

−M11 dx2 + M12 dx1 + x1 (Q1 dx2 − Q2 dx1 ),

A

(x1 t2 − x2 t1 )dx3 ds

x1 (−N12 dx2 + N22 dx1 )+x2 (N11 dx2 − N12 dx1 ).

A

(13.85b) By using the relation given in (13.71), the resultant forces and moments can then be expressed in terms of the stress functions as / t2 = φ2 ]BA , / t3 = η]BA , t1 = φ1 ]BA , / / m1 = −(ψ2 − x2 η)]BA , / m2 = (ψ1 − x1 η)]BA , B x1 dφ2 − x2 dφ1 = (x1 φ2 − x2 φ1 − / m3 = A

(13.86a) )]BA ,

where φ1 = −

, 2,

φ2 =

,1

(13.86b)

Note that the introduction of is based upon the relation given in (13.71)1 that N12 = φ1, 1 = N21 = −φ2, 2 , which is usually called Airy stress function.

13.3


465

With the relations given in (13.86), the force and moment equilibrium around a ˆ = (m ˆ 1, m ˆ 2, m ˆ 3 ) applied at point concentrated force ˆf = (fˆ1 , fˆ2 , fˆ3 ) and moment m xˆ = (ˆx1 , xˆ 2 ) can then be written as . . . dφ1 = fˆ1 , dφ2 = fˆ2 , dη = fˆ3 , C

.

C

C

dψ1 = m ˆ 2 + (x1 − xˆ 1 )fˆ3 ,

C

.

dψ2 = −m ˆ 1 + (x2 − xˆ 2 )fˆ3 ,

(13.87)

C

.

d((x1 − xˆ 1 )φ2 − (x2 − xˆ 2 )φ1 −

)=m ˆ 3,

C

where C is any closed contour enclosing the point xˆ anticlockwise.

13.3.4 Explicit Expressions of N As we said previously, in eigenrelation the mixed formalism is more alike to Stroh formalism. Therefore, it is easier to find the explicit expressions (13.57) for Nm by following the steps employed in Stroh formalism. After getting Nm , we can then use the relation provided in (13.72b)1 to obtain the explicit expression of the fundamental elasticity matrix N for the displacement formalism which are now listed below: ⎡

X11 X12 Y13 1 ⎢ X 21 0 Y23 N1 = ⎢ /⎣ 0 0 0 / B∗ 0 X34 ⎡ 12 ∗ −/ D22 0 X31 ⎢ 1 ⎢ 0 0 0 N3 = ⎣ / X31 0 Y33 X41 0 Y34

⎡ ⎤ Y11 Y14 ⎢Y12 1 Y24 ⎥ ⎥ , N2 = ⎢ /⎣ 0 X43 ⎦ X44 X14 ⎤ X41 0 ⎥ ⎥, Y34 ⎦ Y44

Y12 Y22 0 X24

⎤ 0 X14 0 X24 ⎥ ⎥, 0 0 ⎦ 0/ A∗11

(13.88)

/ are given in (13.57b), (13.57c), (13.57d), and (13.57e). where Xij , Yij , and Symmetric Laminates If the plate is symmetric with respect to the mid-plane, such as symmetric laminates, the coupling stiffness B will be identical to zero and the stretching and bending deformation will be uncoupled. Substituting B=0 into (13.30b), we have ∗ / A = A−1 ,

∗ / B = 0,

∗ / D = D.

(13.89)

With this result, the fundamental elasticity matrix (13.57) can be further reduced to

466

13

⎤

⎡

A∗16 ⎢ A∗ ⎢ 11 ⎢ ∗ ⎢ A12 ⎢ ⎢ ∗ N1 = ⎢ A11 ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0

−1

0

0

0

0

0

D12 0 − D22

0

⎡ ∗ ⎢A66


A∗2 − 16 A∗11

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A∗ A∗ ⎢ ∗ ⎥ 0 ⎥ ⎢A26 − 12∗ 16 A11 ⎥ , N2 = ⎢ ⎢ ⎥ ⎢ ⎥ 1 ⎥ 0 ⎢ ⎢ ⎢ D26 ⎥ ⎦ ⎣ −2 0 D22

⎤

⎡

1 ⎢− A∗ ⎢ 11 ⎢ 0 ⎢ ⎢ N3 = ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎣ 0

0

0

0

0

0

0

⎤ A∗12 A∗16 − 0 0 ⎥ A∗11 ⎥ ⎥ ⎥ ∗2 A12 ⎥ ∗ A22 − ∗ 0 0 ⎥ ⎥, A11 ⎥ ⎥ 0 0 0 ⎥ ⎥ 1 ⎥ ⎦ 0 0 D22

A∗26

⎥ ⎥ ⎥ ⎥ ⎥ D212 D12 D26 ⎥ , 0 −(D11 − ) −2(D16 − )⎥ D22 D22 ⎥ ⎥ ⎥ D226 ⎦ D12 D26 ) −4(D66 − ) 0 −2(D16 − D22 D22

(13.90)

where A∗ij , i, j = 1, 2, 6, denote the components of A−1 . Isotropic Plates For a single layer isotropic plate with Young’s modulus E and Poisson’s ratio ν, its elastic stiffness for generalized plane stress condition can be expressed as E νE E , C16 = C26 = 0. , C12 = C21 = , C66 = 2(1 + ν) 1 − ν2 1 − ν2 (13.91) Substituting (13.91) into (1.79), the coupling stiffness B will be identical to zero and the extensional and bending stiffnesses can be obtained as C11 = C22 =

A11 = A22 = A, A12 = A21 = νA, A66 = (1 − ν)A/2, A16 = A26 = 0, D11 = D22 = D, D12 = D21 = νD, D66 = (1 − ν)D/2, D16 = D26 = 0, (13.92a) where

A=

Eh , 1 − ν2

D=

Eh3 , 12(1 − ν 2 )

(13.92b)

and h is the thickness of the plate. The fundamental elasticity matrix N for the isotropic plates can therefore be obtained by substituting (13.92) into (13.90), which is

13.3


467

⎡ 2(1 + ν) ⎡

0 ⎢−ν N1 = ⎢ ⎣ 0 0

−1 0 0 0

0 0 0 −ν

⎢ ⎤ ⎢ 0 ⎢ ⎢ ⎥ 0⎥ ⎢ , N2 = ⎢ ⎦ 1 ⎢ ⎢ 0 ⎢ ⎣

⎡

−Eh 0 0 ⎢ 0 0 0 ⎢ ⎢ 3 ⎢ N3 = ⎢ 0 0 − Eh ⎢ 12 ⎢ ⎣ 0 0 0 −

0 0 0 Eh3 6(1 + ν)

⎤ 0

Eh

1 − ν2

0

0

0

0

0

Eh 0

0

0

0

0

0

12(1 − ν 2 ) Eh3

0

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦ (13.93)

⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

13.3.5 Explicit Expressions of A and B Like the fundamental elasticity matrices, the explicit expressions of the material eigenvectors have also been obtained through the use of mixed formalism and the results are shown in (13.56). Through the relation shown in (13.67) for the mixbased eigenvectors and the displacement-based eigenvectors, i.e., / ξ = It ξ which can also be written inversely as ξ = It/ ξ, we can get the explicit expressions for the material eigenvector matrices A and B as follows: ⎡

c1 a11 ⎢c1 a21 A=⎢ ⎣c1 a31 c1 a41

c2 a12 c2 a22 c2 a32 c2 a42

c3 a13 c3 a23 c3 a33 c3 a43

⎤ c4 a14 c4 a24 ⎥ ⎥, c3 a34 ⎦ c3 a44

⎡

c1 b11 ⎢c1 b21 B=⎢ ⎣c1 b31 c1 b41

c2 b12 c2 b22 c2 b32 c2 b42

c3 b13 c3 b23 c3 b33 c3 b43

⎤ c4 b14 c4 b24 ⎥ ⎥, c3 b34 ⎦ c3 b44

(13.94)

where aij and bij are determined by the following relations. If 1 and/or 3 = 0: a1 k = p1 (μk ) + λk q1 (μk ), b1 k = −μk , a2 k = [p2 (μk ) + λk q2 (μk )]/μk , b2 k = 1, a3 k = λk , a4 k = λk μk ,

b3 k = [h1 (μk ) − λk g1 (μk )]/μk , b4 k = −h2 (μk ) + λk g2 (μk ),

λk = −4 (μk )/3 (μk ) = 2 (μk )/1 (μk ),

k = 1, 2, 3, 4.

(13.95)

468

13


If 2 and/or 4 = 0: a1 k = λ−1 k p1 (μk ) + q1 (μk ),

b1 k = −λ−1 k μk ,

−1 a2 k = [λ−1 k p2 (μk ) + q2 (μk )]/μk , b2 k = λk ,

a3 k = 1,

b3 k = [λ−1 k h1 (μk ) − g1 (μk )]/μk ,

a4 k = μk ,

b4 k = −λ−1 k h2 (μk ) + g2 (μk ),

λ−1 k = −3 (μk )/4 (μk ) = 1 (μk )/2 (μk ),

(13.96)

k = 1, 2, 3, 4.

In the above, pj (μk ), qj (μk ), gj (μk ), hj (μk ) and 1 (μk ), 2 (μk ), 3 (μk ), 4 (μk ) are defined in (13.52b) and (13.53b), respectively. The scaling factor ck are determined by c2k =

1 , k = 1, 2, 3, 4. 2(a1 k b1 k + a2 k b2 k + a3 k b3 k + a4 k b4 k )

(13.97)

Symmetric Laminates To get the simplified expressions of the material eigenvectors, we first substitute (13.89) into (13.52b) and (13.53b), which will lead to qj = hj = 0,

j = 1, 2, 6,

and

2 = 3 = 0.

(13.98)

Thus, by the characteristic equation (13.54) we know that the material eigenvalues may be obtained from 1 (μ) = 0

or

4 (μ) = 0.

(13.99)

If we let the eigenvalues with positive imaginary parts obtained from 4 (μ) = 0 be μ1 and μ2 , and those obtained from 1 (μ) = 0 be μ3 and μ4 , by (13.94), (13.95), and (13.96) we have ⎡

c1 p1 (μ1 )

c2 p1 (μ2 )

0

⎢c p (μ )/μ c p (μ )/μ 0 1 2 2 2 2 ⎢ 1 2 1 A=⎢ ⎣ 0 0 c3 0

0

0 0 c4

⎤ ⎥ ⎥ ⎥, ⎦

(13.100a)

c3 μ3 c4 μ4

⎡ ⎤ −c1 μ1 −c2 μ2 0 0 ⎢ c ⎥ c2 0 0 ⎢ 1 ⎥ B=⎢ ⎥. ⎣ 0 0 −c3 g1 (μ3 )/μ3 −c4 g1 (μ4 )/μ4 ⎦ 0 0 c3 g2 (μ3 ) c4 g2 (μ4 )

(13.100b)

13.3


469

Isotropic Plates With the coupling stiffness being zero and the extensional and bending stiffnesses given in (13.92) for isotropic plates, from (13.99) we get μ1 = μ2 = μ3 = μ4 = i,

(13.101)

which are repeated roots. From (13.100a) and (13.100b) we see that the first and second eigenvectors are dependent on each other, and so are the third and fourth eigenvectors. In other words, the material eigenvector matrices A and B do not exist for the isotropic plates, which are classified as plates of degenerate materials. In this situation, a small perturbation of the material properties is suggested in the numerical calculation to avoid the degenerate problems (Hwu and Yen, 1991), or a generalized eigenvector is introduced in the modification of the general solution (Ting and Hwu, 1988).

13.3.6 Explicit Expressions of N(ω) The explicit expressions for the fundamental elasticity matrix shown in Section 13.3.4 were derived by following the steps for the corresponding two-dimensional problems (Hsieh and Hwu, 2002b). It seems that same approach can be applied to obtain the explicit expressions for the generalized fundamental elasticity matrix. However, the difficulty we face is getting the explicit solutions from a complicated system of linear algebraic equations. Although it is possible to get the solutions through the symbolic computer program, such as mathematica, their solutions are too complicated and do not deserve to be written down explicitly. With this understanding, in the following only the explicit expressions for symmetric laminates and isotropic plates are presented. Symmetric Laminates For symmetric laminates, the coupling stiffness B=0 which will lead to (13.89). Substitution of (13.89) into (13.73a), (13.7b), (13.73c) and then to (13.75e), we get ⎤ ⎡ Q11 Q12 0 0 R11 ⎢Q12 Q22 0 0 ⎥ ⎢R21 ⎥ ⎢ Qm (ω) = ⎢ ⎣ 0 0 Q33 Q34 ⎦ , Rm (ω) = ⎣ 0 0 0 Q34 Q44 0 ⎤ ⎡ T11 T12 0 0 ⎢T12 T22 0 0 ⎥ ⎥ Tm (ω) = ⎢ ⎣ 0 0 T33 T34 ⎦ , 0 0 T34 T44 ⎡

R12 R22 0 0

0 0 R33 R43

⎤ 0 0 ⎥ ⎥, R34 ⎦ R44

(13.102)

470

13


where Q11 = c2 A11 + 2csA16 + s2 A66 ,

Q22 = c2 A66 + 2csA26 + s2 A22 ,

Q12 = c2 A16 + cs(A12 + A66 ) + s2 A26 , c2 ∗ s2 Q44 = −c2 D∗22 + csD∗26 − D∗66 , D66 + csD∗16 − s2 D∗11 , 4 4 s2 ∗ c2 ∗ 1 ∗ ∗ Q34 = − D26 + cs D12 + D66 − D16 , 2 4 2

(13.103a)

Q33 = −

R11 = (c2 − s2 )A16 − cs(A11 − A66 ),

R22 = (c2 − s2 )A26 + cs(A22 − A66 ),

R12 = c2 A12 + cs(A26 − A16 ) − s2 A66 , R21 = c2 A66 + cs(A26 − A16 ) − s2 A12 , 1 1 1 1 R33 = (c2 − s2 )D∗16 − cs D∗11 − D∗66 , R44 = (c2 − s2 )D∗26 + cs D∗22 − D∗66 , 2 4 2 4 R34 =

c2 ∗ cs D − (D∗16 − D∗26 ) − s2 D∗12 , 4 66 2

T11 = c2 A66 − 2csA16 + s2 A11 ,

R43 = c2 D∗12 −

cs ∗ s2 (D16 − D∗26 ) − D∗66 , 2 4 (13.103b)

T22 = c2 A22 − 2csA26 + s2 A66 ,

T12 = c2 A26 − cs(A12 + A66 ) + s2 A16 , s2 c2 T33 = −c2 D∗11 − csD∗16 − D∗66 , T44 = − D∗66 − csD∗26 − s2 D∗22 ,. 4 4 s2 ∗ c2 ∗ 1 ∗ ∗ T34 = − D16 − cs D12 + D66 − D26 . 2 4 2

(13.103c)

In the above, D∗ij , i, j = 1, 2, 6, are the components of the inversion of the bending stiffness, D−1 , and c = cos ω,

s = sin ω.

(13.104)

Substituting (13.102) directly into (13.75c), the explicit expressions for the generalized fundamental elasticity matrix N(ω) can then be obtained as ⎤ P11 P12 0 0 ⎢P21 P22 0 0 ⎥ ⎥ N1 (ω) = − ⎢ ⎣ 0 0 P33 P43 ⎦ , 0 0 P34 P44 ⎤ ⎡ V11 V12 0 0 ⎢V12 V22 0 0 ⎥ ⎥ N3 (ω) = ⎢ ⎣ 0 0 T∗ T∗ ⎦ , 33 34 ∗ T∗ 0 0 T34 44 ⎡

⎡

∗ T11 ⎢T ∗ 12 N2 (ω) = ⎢ ⎣ 0 0

∗ T12 ∗ T22 0 0

0 0 V33 V34

⎤ 0 0 ⎥ ⎥, V34 ⎦ V44

(13.105)

where ∗ T11 = T22 /A ,

∗ T22 = T11 /A ,

= T44 /D ,

= T33 /D ,

∗ T33

∗ T44

∗ T12 = −T12 /A ,

∗ T34 = −T34 /D ,

(13.106a)

13.3


471

∗ ∗ P11 = T11 R11 + T12 R12 , ∗ ∗ P21 = T12 R11 + T22 R12 , ∗ ∗ P33 = T33 R33 + T34 R34 , ∗ ∗ P43 = T34 R33 + T44 R34 ,

V11 = R11 P11 + R12 P21 − Q11 ,

∗ ∗ P12 = T11 R21 + T12 R22 , ∗ ∗ P22 = T12 R21 + T22 R22 ,

(13.106b)

∗ ∗ P34 = T33 R43 + T34 R44 , ∗ ∗ P44 = T34 R43 + T44 R44 ,

V22 = R21 P12 + R22 P22 − Q22 ,

V12 = R11 P12 + R12 P22 − Q12 = R21 P11 + R22 P21 − Q12 , V33 = R33 P33 + R34 P43 − Q33 , V44 = R43 P34 + R44 P44 − Q44 ,

(13.106c)

V34 = R33 P34 + R34 P44 − Q34 = R43 P33 + R44 P43 − Q34 , and 2 , A = T11 T22 − T12

2 D = T33 T44 − T34 .

(13.106d)

By letting ω = 0, we can prove that the results of N1 (0), N2 (0), and N3 (0) obtained from (13.105) are identical to those obtained from (13.90) for N1 , N2 , and N3 . Isotropic Plates The explicit expressions for the generalized fundamental elasticity matrix N(ω) of isotropic plates can be obtained by substituting the extensional and bending stiffnesses (13.92) into (13.105). The results are ⎡

⎤ 0 0 sin 2ω −k1 − cos 2ω ⎥ 1+ν ⎢ 0 0 ⎢k1 − cos 2ω − sin 2ω ⎥, N1 (ω) = ⎣ ⎦ sin 2ω 1 + k cos 2ω 0 0 −k 2 1 1 0 0 −1 + k1 cos 2ω k1 sin 2ω ⎡ ⎤ 0 0 k2 + cos 2ω sin 2ω ⎥ (1 + ν)2 ⎢ 0 0 ⎢ sin 2ω k2 − cos 2ω ⎥, N2 (ω) = 0 0 d1 (1 − cos 2ω) −d1 sin 2ω ⎦ 2Eh ⎣ 0 0 −d1 sin 2ω d1 (1 + cos 2ω) ⎡ ⎤ 1 + cos 2ω sin 2ω 0 0 ⎥ −Eh ⎢ 0 0 ⎢ sin 2ω 1 − cos 2ω ⎥, N3 (ω) = 0 0 d2 (k3 − cos 2ω) −d2 sin 2ω ⎦ 2 ⎣ 0 0 −d2 sin 2ω d2 (k3 + cos 2ω) (13.107a) where k1 =

1−ν , 1+ν

k2 =

3−ν , 1+ν

k3 =

3+ν , 1−ν

d1 =

12 k1 , h2

d2 =

h2 k1 . 12 (13.107b)

472

13


13.3.7 Explicit Expressions of S, H, and L To get the explicit expressions for the Barnett–Lothe tensors S, H, and L, we may follow the steps described in Section 11.3.3. Like the problem we discussed for the generalized fundamental elasticity matrix, due to the complexity occurred from the inversion of B, the results of S, H, and L for the general unsymmetric laminates will become awkward, that may be more inconvenient than direct numerical calculation. In this sense, here we also ignore the presentation for the general laminates and start from the symmetric laminates. Symmetric Laminates Substituting the explicit expressions of A and B shown in (13.100) into (11.59a), and using the relations obtained from the characteristic equations (13.99) (one may refer to Ting (1996) for similar simplification technique), we get ⎡ ⎤ 0 cA A∗11 − A∗12 0 0 ⎢−cA A∗ − A∗ ⎥ 0 0 0 11 12 ⎥, SL−1 = ⎢ ⎣ 0 0 0 kD (D12 − cD D22 )⎦ 0 0 0 −kD (D12 − cD D22 ) ⎤ 0 0 bA A∗11 dA A∗11 ⎥ ⎢dA A∗ eA A∗ 0 0 11 11 ⎥, =⎢ ⎣ 0 0 kD bD D22 kD dD D22 ⎦ 0 0 kD dD D22 kD eD D22 ⎡

L−1

(13.108) where bA , cA , dA , eA and bD , cD , dD , eD , kD are real coefficients related to the material eigenvalues by μ1 + μ2 = aA + ibA , μ1 μ2 = cA + idA , eA = aA dA − bA cA , μ3 + μ4 = aD + ibD , μ3 μ4 = cD + idD , eD = aD dD − bD cD , kD =

{bD eD − c2D

2 − dD )D222

(13.109)

+ 2cD D12 D22 − D212 }−1 .

With the results of (13.108), L, S, and H can then be obtained by using the relations given in (11.59b) and (11.59c). Their final simplified results are ⎤ ⎡ ⎤ ⎡ 0 0 0 0 −dA Sa bA Sa eA La −dA La ⎢ ⎢−dA La bA La 0 0 ⎥ 0 0 ⎥ ⎥ , S = ⎢−eA Sa dA Sa ⎥, L=⎢ ⎣ 0 ⎣ 0 0 eD Ld −dD Ld ⎦ 0 −dD Sd bD Sd ⎦ 0 0 −dD Ld bD Ld 0 0 −eD Sd dD Sd (13.110a) ⎡

⎤ bA Ha dA Ha 0 0 ⎢dA Ha eA Ha 0 0 ⎥ ⎥, H=⎢ ⎣ 0 0 bD Hd dD Hd ⎦ 0 0 dD Hd eD Hd

(13.110b)

13.3


473

where La = Sa = Ha =

A∗11

kD D22 , Ld = , 2 A0 D20 A∗11 (cA A∗11 − A∗12 ), Sd = A20 A∗ A∗11 − 11 (cA A∗11 − A∗12 )2 , A20

2D kD 22

D20

(D12 − cD D22 ),

Hd = kD D22 −

3D kD 22

D20

(D12 − cD D22 )2 , (13.111a)

and A20 = (bA eA − dA2 )A∗2 11 ,

2 2 2 D20 = (bD eD − dD )kD D22 .

(13.111b)

Isotropic Plates With the extensional and bending stiffnesses given in (13.92) and the material eigenvalues obtained in (13.101), the real constants defined in (13.109), (13.111a) and (13.111b) for the isotropic plates can be obtained as aA = aD = dA = dD = 0, bA = bD = eA = eD = 2, cA = cD = −1, 1 2 , D0 = 2kD D, , A0 = kD = Eh (3 + ν)(1 − ν)D2 Eh (3 + ν)(1 − ν)D 1−ν 1+ν , Ld = , Sa = − , Sd = , La = 4 4 4 4 (3 − ν)(1 + ν) 1 , Hd = . Ha = 4Eh 4D (13.112) Substituting (13.112) into (13.110), we get L11 = L22 = Eh/2, L33 = L44 = (3 + ν)(1 − ν)D/2, all other Lij = 0, S12 = −S21 = −(1 − ν)/2, S34 = −S43 = (1 + ν)/2, all other Sij = 0, H11 = H22 = (3 − ν)(1 + ν)/(2Eh),

H33 = H44 = 1/(2D), all other Hij = 0. (13.113)

Note that all the solutions presented in this section are for the coupled stretching– bending analysis of thin laminated plates. When the solutions are reduced to the special cases such that the stretching and bending deformation uncouple, the results of stretching part can be proved to agree with the known results obtained in (3.86c) for generalized plane stress condition. Moreover, (13.113) also agrees with the results obtained by (3.95) with the integrals of Ni (ω), i = 1, 2, 3, calculated from (13.107).

474

13


13.4 Hygrothermal Stresses Composite laminates are increasingly being used not only in traditional areas like aerospace, but also in many engineering applications. Some of these applications are the structures under hygrothermal environment. For a unidirectional lamina the coefficients of thermal and moisture expansion, like its other properties, change with direction. Thus, the hygrothermal changes result in unequal strains in the longitudinal and transverse directions. Hygrothermal strains do not produce a resultant force or moment when the body is completely free to expand, bend, and twist. However, for a composite laminate each individual lamina is not completely free to deform. The lamina stresses are therefore induced by the constraints placed on its deformation by adjacent lamina. Moreover, the unsymmetry of laminates will cause coupling between stretching and bending, which may complicate the analysis. Due to the designable characteristics of composite laminates, sometimes the engineering designers want to utilize the coupling effects to do something that cannot be achieved by using metallic or symmetric laminates. Thus, the study of hygrothermal stresses in unsymmertic laminates becomes important for practical engineering design. Like the extension of Stroh formalism to anisotropic thermoelasticity discussed in Chapter 10, in this section we extend the Stroh-like formalism to the hygrothermal stress analysis of laminates.

13.4.1 Basic Equations In a fixed rectangular coordinate system xi , i = 1, 2, 3, let Ui , σij , ξij , T, H, qi , and mi be, respectively, displacement, stress, strain, change in temperature, change in moisture content, heat flux, and moisture transfer. If the coupling terms between the elastic deformation, heat conduction, and moisture transport are neglected, the heat conduction, the moisture diffusion, the strain–displacement relation, the constitutive law, the force, heat and moisture equilibrium equations for linear anisotropic elastic materials under static loading and small deformation conditions can be written as (Nowacki, 1962) qi = −kijt T, j ,

mi = −kijh H, j ,

ξij =

t h σij = Cijks ξks − Cijks αks T − Cijks αks H, σij, j = 0, qi,i = 0, mi,i = 0,

1 (Ui, j + Uj, i ), 2

(13.114)

i, j, k, s = 1, 2, 3,

where repeated indices imply summation, a subscript comma stands for differentiation, and Cijks , kijt , kijh and αijt ,αijh are, respectively, the elastic constants, heat conduction coefficients, moisture diffusion coefficients, and the coefficients of thermal and moisture expansion. Cijks are assumed to be fully symmetric, i.e., Cijks = Cjiks = Cijsk = Cksij and are required to be positive definite due to the positiveness

13.4

Hygrothermal Stresses

475

of strain energy. kijt , kijh , αijt , and αijh are also assumed to be symmetric, i.e., kijt = kjit , kijh = kjih , αijt = αjit , and αijh = αjih . Equation (13.114) constitutes 23 partial differential equations in terms of three coordinate variables xi , i = 1, 2, 3. If the deformations are considered to be dependent upon two coordinate variables x1 and x2 only, a general solution satisfying these 23 equations can be found by following Stroh formalism for two-dimensional linear anisotropic thermoelasticity. In this section, we consider a composite laminate composed of layers of various materials. Each layer is assumed to be made of anisotropic materials. If the laminate thickness is smaller than its other dimensions, according to Kirchhoff assumptions the displacement Ui , temperature T, and moisture content H may be assumed to vary linearly through laminate thickness as Ui (x1 , x2 , x3 ) = ui (x1 , x2 ) + x3 βi (x1 , x2 ), U3 (x1 , x2 , x3 ) = w(x1 , x2 ),

i = 1, 2,

T(x1 , x2 , x3 ) = T 0 (x1 , x2 ) + x3 T ∗ (x1 , x2 ),

(13.115a)

H(x1 , x2 , x3 ) = H 0 (x1 , x2 ) + x3 H ∗ (x1 , x2 ), where β1 = −w, 1 ,

β2 = −w, 2 .

(13.115b)

(u1 , u2 , w), T 0 , and H 0 are the middle surface displacements, temperature, and moisture content, and βi , i = 1, 2, are the negative of the slope of the middle surface in the x1 -and x2 -directions. T ∗ and H ∗ are the rates of changes in temperature and moisture content. Based upon the assumptions given in (13.115) and the 23 basic equations (13.114) for anisotropic materials under hygrothermal condition, we can now write down the kinematic relations, constitutive laws, and equilibrium equations for hygrothermal stress analysis of composite laminates as follows: t ∗ T , q˘ i = −Kijt T,0j − Kij∗t T,∗j − Ki3

h ∗ m ˘ i = −Kijh H,0j − Kij∗h H,∗j − Ki3 H ,

1 1 (ui, j + uj, i ), κij = (βi, j + βj, i ), 2 2 Nij = Aijkl εkl + Bijkl κkl − Atij T 0 − Ahij H 0 − Btij T ∗ − Bhij H ∗ , εij =

Mij = Bijkl εkl + Dijkl κkl − Btij T 0 − Bhij H 0 − Dtij T ∗ − Dhij H ∗ , Nij, j = 0, Mij, ij + p = 0, Qi = Mij, j , q˘ i,i + q = 0, m ˘ i,i + m = 0, i, j, k, l = 1, 2, (13.116a) where εij and κij denote the mid-plane strain and plate curvature; Nij , Mij , and Qi denote the stress resultants, bending moments, and shear forces; q˘ i and m ˘ i denote the heat flux resultant and moisture transfer resultant, Aijkl , Bijkl , and Dijkl are, respectively, the extensional, coupling, and bending stiffness tensors; Atij , Btij , Dtij and

476

13


Ahij , Bhij , Dhij are the corresponding tensors for the thermal and moisture expansion coefficients; Kijt , Kijh and Kij∗t , Kij∗h are the coefficients related to the heat conduction and moisture diffusion coefficients; p, q , and m are the lateral distributed load, heat flux, and moisture concentration transfer applied on the laminates. Their definitions are Nij =

−h/2 h/2

q˘ i =

=

σij dx3 ,

−h/2 h/2 −h/2 h/2

−h/2 h/2 −h/2

Mij =

−h/2 h/2

Atij =

Kijt

qi dx3 ,

Aijks =

Ahij =

h/2

m ˘i =

t Cijks αks dx3 ,

−h/2

Btij =

kijt dx3 ,

Kij∗t

Bhij =

=

h/2 −h/2

Qi =

h/2 −h/2

σi3 dx3 ,

mi dx3 ,

h/2

−h/2 h/2

h Cijks αks dx3 ,

σij x3 dx3 ,

−h/2 h/2

Bijks =

Cijks dx3 ,

h/2

−h/2 h/2 −h/2

Cijks x3 dx3 ,

Dijks =

t Cijks αks x3 dx3 ,

Dtij =

h/2

Cijks x32 dx3 ,

−h/2 h/2

−h/2 h/2

t 2 Cijks αks x3 dx3 ,

h Cijks αks x3 dx3 ,

kijt x3 dx3 ,

Kijh

=

h/2 −h/2

Dhij = kijh dx3 ,

−h/2

Kij∗h

h 2 Cijks αks x3 dx3 ,

=

h/2 −h/2

kijh x3 dx3 ,

(13.116b)

in which h is laminate thickness. Note that like Cijks ,kijt , kijh , αijt , and αijh , according to the definitions given in (13.116b) Aijkl , Bijkl , Dijkl , Atij , Btij , Dtij , Ahij , Bhij , Dhij , Kijt , Kijh ,Kij∗t , and Kij∗h still preserve the symmetry property.

13.4.2 Extended Stroh-Like Formalism Since the basic equations stated in (13.115) and (13.116) are quite general, it is not easy to find a solution satisfying all these basic equations. In the following, we consider two special cases that occur frequently in engineering applications. One is the case that temperature and moisture distributions depend on x1 and x2 only, i.e., T ∗ = H ∗ = 0, and the other is the case that temperature and moisture distributions depend on x3 only, i.e., T = T 0 + x3 T ∗ and H = H 0 + x3 H ∗ in which T 0 , T ∗ , H 0 , and H ∗ are constants independent of x1 and x2 . Case 1: temperature and moisture content depend on x1 and x2 only If the temperature and moisture content are assumed to depend on x1 and x2 only and the lateral distributed load, heat flux, and moisture concentration transfer applied on the laminates are neglected, i.e., T ∗ = H ∗ = p = q = m = 0, the basic equations stated in (13.116a) can be simplified as

13.4


q˘ i,i = −Kijt T, ij = 0,

477

m ˘ i,i = −Kijh H, ij = 0,

Nij, j = Aijkl uk, lj + Bijkl βk, lj − Atij T, j − Ahij H, j = 0, Mij, ij =

Bijkl uk, lij + Dijkl βk, lij − Btij T, ij

− Bhij H, ij

(13.117)

= 0, i, j, k, l = 1, 2.

By following the steps described in Chapter 10 for Stroh formalism of twodimensional thermoelasticity and in Section 13.3 for Stroh-like formalism of coupled stretching–bending analysis of composite laminates, we can find a general solution satisfying the basic equations (13.117) and call it the extended Stroh-like formalism. The solution is T = 2 Re{gt (zt )},

H = 2Re{gh (zh )},

t t h h q˘ i = −2 Re{(Ki1 + τt Ki2 )gt (zt )}, m ˘ i = −2 Re{(Ki1 + τh Ki2 )gh (zh )}, (13.118a) ud = 2 Re{Af(z) + ct gt (zt ) + ch gh (zh )},

φd = 2 Re{Bf(z) + dt gt (zt ) + dh gh (zh )}, where ud =

u φ u β φ ψ1 , φd = , u= 1 , β= 1 , φ= 1 , ψ= , (13.118b) β ψ u2 β2 φ2 ψ2

A = [a1 a2 a3 a4 ] ,

B = [b1 b2 b3 b4 ] ,

⎧ ⎫ ⎪ ⎪f1 (z1 )⎪ ⎪ ⎨ ⎬ f (z ) f(z) = 2 2 , zk = x1 + μk x2 , ⎪f3 (z3 )⎪ ⎪ ⎪ ⎩ ⎭ f4 (z4 ) zt = x1 + τt x2 ,

k = 1, 2, 3, 4,

zh = x1 + τh x2 .

(13.118c)

(13.118d)

(13.118e)

In the above, Re stands for the real part of a complex number and the prime (• ) denotes differentiation with respect to its argument. ud and φd are the generalized displacement and stress function vectors. φ1 , φ2 and ψ1 , ψ2 are the stress functions related to the stress resultants Nij , shear forces Qi , effective shear forces Vi , and bending moments Mij by (13.71). fk (zk ), k = 1, 2, 3, 4, gt (zt ) and gh (zh ) are six holomorphic functions of complex variables zk , zt and zh , which will be determined by the boundary conditions set for each particular problem. μk , τt , τh and (ak , bk ), (ct , dt ), (ch , dh ) are, respectively, the material eigenvalues and eigenvectors, which can be determined by the following eigenrelation: t t t + 2τt K12 + τt2 K22 = 0, K11

Nξ = μξ,

h h h K11 + 2τh K12 + τh2 K22 = 0,

Nηt = τt ηt + γt ,

Nηh = τh ηh + γh ,

(13.119a) (13.119b)

478

13


where N is a 8×8 real matrix which is the fundamental elasticity matrix for coupled stretching–bending analysis; ξ, ηt and ηh are three 8×1 complex vectors which are composed of the material eigenvectors. These matrices and vectors are composed of some well-defined submatrices and vectors and are defined by N=

N1 N2 , N3 NT1

ξ=

a b

ηt =

ct c , ηh = h , dt dh

(13.119c)

in which the superscript T denotes the transpose of a matrix. Detailed definition of the submatrices N1 , N2 , and N3 have been given in (13.72), (13.73), and (13.74). γt and γh are two 8×1 complex vectors related to the elastic constants and the coefficients of thermal and moisture expansion, whose detailed expressions can be obtained by following either the displacement formalism or mixed formalism described in Section 13.2. They are −1 t 1 0 (Nm )2 α˜ t1 α1 = −It , γt = L2 + J2 αt2 I (Nm )T1 α˜ t2 2

(13.120a)

t αA1 , αtB1

(13.120b)

where αt1 =

αt2 =

t αA2 , αtB2

α˜ t1 =

t α˜ A1 , α˜ tB2

α˜ t2 =

t α˜ A2 −α˜ tB1

and αtAi

t t t t / / A1i B1i A1i B t t t , αBi = ,/ αAi = /t , / αBi = /1i , i = 1, 2, = At2i Bt2i Bt2i A2i / Atij = Atij − / Bijkl Btkl ,

/ Btij = / Dijkl Btkl ,

(13.120c)

(13.120d)

in which / Bijkl and / Dijkl are the tensor notations of / B and / D defined in (13.74). Same expressions as (13.120) are defined for γh only by replacing the subscript or superscript from t to h. Detailed definitions of L2 , J2 , It , (Nm )1 and (Nm )2 have also been given in (13.27c) and (13.72c). The second equality of (13.120a) comes from the equivalence between the displacement formalism and mixed formalism discussed in (13.27), (13.68), and (13.69). By following the steps of two-dimensional thermoelasticity discussed between (10.12) and (10.16), the eigenrelations for the thermal and moisture properties shown in (13.119b) can be generalized as N(ω)ηt (ω) = τt (ω)ηt (ω) + γt (ω), N(ω)ηh (ω) = τh (ω)ηh (ω) + γh (ω), where

(13.121a)

13.4


479

τt cos ω − sin ω τh cos ω − sin ω , τh (ω) = , τt sin ω + cos ω τh sin ω + cos ω ηt (ω) = τˆt (ω)ηt , ηh (ω) = τˆh (ω)ηh , τˆt (ω) = cos ω + τt sin ω, τˆh (ω) = cos ω + τh sin ω,

τt (ω) =

t I1 γ1 (ω) = − I2 γt2 (ω) h γ (ω) I γh (ω) = 1h =− 1 I2 γ2 (ω) γt (ω) =

t / α1 (ω) , / αt2 (ω) h I2 0 (Nm (ω))2 / α1 (ω) , I1 I (Nm (ω))T1 / αh2 (ω) I2 I1

(13.121b)

0 (Nm (ω))2 I (Nm (ω))T1

αt1 + sin ω/ αt2 , / αt1 (ω) = cos ω/

/ αt2 (ω) = − sin ω/ αt1 + cos ω/ αt2 ,

/ αh1 (ω) = cos ω/ αh1 + sin ω/ αh2 ,

/ αt2 (ω) = − sin ω/ αh1 + cos ω/ αh2 .

(13.121c)

(Nm (ω))1 , (Nm (ω))2 , and (Nm (ω))3 defined in (13.75) are the submatrices of the generalized fundamental elasticity matrices Nm (ω). Integrating the generalized eigenrelations (13.121a) from 0 to π and using the relation (3.95), we get

S −L S −L

t∗ ct c / γ = i t + 1t∗ , dt dt / γ2 h∗ H ch c / γ = i h + 1h∗ , dh ST dh / γ2

H ST

(13.122a)

where

/ γt∗ i =

1 π

0

π

τˆt−1 (ω)γti (ω)dω,

/ γh∗ i =

1 π

π 0

τˆh−1 (ω)γhi (ω)dω,

i = 1, 2. (13.122b)

Case 2: temperature and moisture content depend on x3 only If the temperature and moisture content depend on x3 only, the distribution assumed in (13.115a)3,4 can be rewritten as T = T 0 + x3 T ∗ ,

H = H 0 + x3 H ∗ ,

(13.123)

where T 0 , T ∗ , H 0 , and H ∗ are real constants independent of x1 and x2 . With this assumption, the basic equations (13.116) can be simplified as

480

13 t ∗ q˘ i = −Ki3 T ,


h ∗ m ˘ i = −Ki3 H ,

Nij = Aijkl uk, l + Bijkl βk, l − Atij T 0 − Ahij H 0 − Btij T ∗ − Bhij H ∗ , Mij = Bijkl uk, l + Dijkl βk, l − Btij T 0 − Bhij H 0 − Dtij T ∗ − Dhij H ∗ , Nij, j = Aijkl uk, lj + Bijkl βk, lj = 0, Mij, ij = Bijkl uk, lij + Dijkl βk, lij = 0, Qi = Mij, j ,

(13.124)

i, j, k, l = 1, 2,

and q˘ i,i = 0 and m ˘ i,i = 0 are satisfied automatically. Note that the mathematical expressions of the governing equations written in (13.124)4, 5 , i.e., Nij, j = 0 and Mij, ij = 0 in terms of uk and βk , are exactly the same as those of the nonhygrothermal problems discussed in Section 13.2. Therefore, the general solution for uk and βk should be exactly the same as that shown in (13.70). Substituting this solution into (13.124)2,3 and following the steps of Section 13.2, we can find the solution for Nij and Mij . The final solution expressed in the form of Stroh-like formalism can then be written as ud = 2 Re{Af(z)}, φd = 2 Re{Bf(z)} − x1 ϑ2 + x2 ϑ1 ,

(13.125a)

where ∗ ∗h ∗ ϑi = αti T 0 + αhi H 0 + α∗t i T + αi H ,

i = 1, 2.

(13.125b)

In the above, αti and αhi have been defined in (13.120b), whereas α∗t i are defined by α∗t i =

t αBi , αtDi

αtBi =

t B1i , Bt2i

αtDi =

t D1i , Dt2i

i = 1, 2,

(13.125c)

and same expressions as (13.125c) are defined for α∗h i only by replacing superscript from t to h. With the general solution shown in (13.125) for the generalized displacement and stress function vectors, the middle surface displacements ui and slopes βi can be obtained directly from the components of the generalized displacement vector ud . As to the stress resultants Nij , shear forces Qi , effective shear forces Vi and bending moments Mij , we can utilize the relations shown in (13.71). Moreover, all the relations for the non-hygrothermal problems such as eigenrelation (13.72) and force relation (13.79)–(13.87) are all valid for the present case. Note that unlike the general solution shown in (13.118) for case 1, the general solution shown in (13.125) for case 2 does not include the expressions for temperature, moisture content, heat flux resultant, and moisture transfer since they have been given in (13.123) and (13.124)1 as known linear distributions and constant ˘i flows. In other words, for case 1 to find the plane distributions of T, H, q˘ i , and m we need to prescribe their associated boundary conditions, while for case 2 their

13.5

Electro-elastic Composite Laminates

481

distributions are assumed to be known and hence the boundary conditions for these physical quantities will not be stated in the related problems.

13.5 Electro-elastic Composite Laminates If a multilayered composite is made up of different layers such as fiber-reinforced composite layers and composite layers consisting of the piezoelectric materials, it may exhibit electric effects that are more complicated than those of single-phase piezoelectric materials. According to the intrinsic coupling phenomenon, piezoelectric materials are widely used as sensors and actuators in intelligent advanced structure design. Like the extension of Stroh formalism to the anisotropic piezoelectric materials discussed in Chapter 11, in this section the Stroh-like formalism will be extended to the coupled mechanical–electrical analysis for the electro-elastic composite laminates.

13.5.1 Basic Equations Because the thickness of laminate is smaller than its other dimensions, the thin laminate keeps the Kirchhoff hypotheses that the normals to the mid-plane of the undeformed laminate remain straight and normal to the mid-plane of the deformed laminate. Based upon Kirchhoff assumptions, the displacement and electric fields can be written as U1 (x1 , x2 , x3 ) = u1 (x1 , x2 ) + x3 β1 (x1 , x2 ), U2 (x1 , x2 , x3 ) = u2 (x1 , x2 ) + x3 β2 (x1 , x2 ), (0)

(1)

(0)

(1)

∂w(x1 , x2 ) , ∂x1

β2 (x, y) = −

E1 (x1 , x2 , x3 ) = E1 (x1 , x2 ) + x3 E1 (x1 , x2 ), E2 (x1 , x2 , x3 ) = E2 (x1 , x2 ) + x3 E2 (x1 , x2 ),

(13.126a)

(13.126b)

where β1 (x, y) = −

∂w(x1 , x2 ) . ∂x2

(13.126c)

Ui , Ei , i = 1, 2 are the displacements and electric fields in xi -direction; u1 , u2 , and w are, respectively, the mid-plane displacements in the x1 -, x2 - and x3 -directions; β1 (0) and β2 are the negative slopes in x1 and x2 directions; Ei are the mid-plane electric (1) fields and Ei are the rate change of the electric fields in the thickness direction. Like the expansion given in (11.14), when the notations of stresses and strains are expanded to the electro-state, the electric potential and the electric displacement have been treated, respectively, as the fourth component of displacements and the additional stress components, i.e.,

482

13

Ej = −U4, j ,


Dj = σ4j ,

j = 1, 2.

(13.127)

Using the assumption (13.126b) for Ek and integrating (13.127)1 with respect to x1 and x2 , we obtain U4 (x1 , x2 , x3 ) = u4 (x1 , x2 ) + x3 β4 (x1 , x2 ),

(13.128a)

where u4 = −

(0) E1 dx1

=−

(0) E2 dx2 ,

β4 = −

(1) E1 dx1

=−

(1)

E2 dx2 . (13.128b)

With (13.126a) and (13.128), the generalized displacements Up and generalized strains εpj for the electro-elastic composite laminates can be expressed in tensor notation as Up = up + x3 βp ,

0 εpj = εpj + x3 κpj ,

p = 1, 2, 4,

j = 1, 2,

(13.129a)

where β1 = −w, 1 , β2 = −w, 2 , 1 1 0 εpj = (up, j + uj, p ), κpj = (βp, j + βj, p ), 2 2 (0) (1) u4, j = −Ej , β4, j = −Ej .

(13.129b)

To write down the constitutive law for the electro-elastic composite laminates, we first define an integral equivalent system of forces and moments for the stress distribution across the laminate thickness h, i.e., define Npj =

h/2 −h/2

σpj dx3 ,

Mpj =

h/2

−h/2

σpj x3 dx3 ,

p = 1, 2, 4, j = 1, 2.

(13.130)

From the definitions given in (13.127) and (13.130), we know that the generalized stress resultants related to the electric displacement are N4i = / Di =

h/2

−h/2

Di dx3 , M4i = / D∗i =

h/2

−h/2

Di x3 dx3 , i = 1, 2.

(13.131)

By the way similar to the classical lamination theory, we can now write down the constitutive relation in terms of the generalized stress resultants/moments (Npj , Mpj ) 0 , κ ), i.e., and generalized mid-plane strains/curvatures (εpj pj

13.5


⎧ ⎫ ⎡ A11 N ⎪ ⎪ 11 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ N22 ⎪ ⎢ A12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ N ⎢ ⎪ ⎪ 12 ⎪ ⎪ ⎪ ⎪ ⎢ A16 ⎪ ⎪ ⎢ ⎪ ⎪ N41 ⎪ ⎪ ⎢ A17 ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ N ⎨ 42 ⎬ ⎢ A18 ⎢ M11 ⎪ = ⎢ ⎢ B11 ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ M ⎢ B12 ⎪ ⎪ 22 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪M12 ⎪ ⎪ ⎢ B16 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪M41 ⎪ ⎪ ⎢ B17 ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎩M42 ⎪ ⎭ B18

483

⎤⎧ 0 ⎫ ⎪ ⎪ε11 ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎥⎪ ⎪ B28 ⎥ ⎪ ⎪ ⎪ ε ⎪ 22 ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎥ 0 B68 ⎥ ⎪ ⎪2ε12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ 0 ⎪ B78 ⎥ ⎪ ⎪ ⎪ 2ε ⎪ 41 ⎪ ⎪ ⎥⎪ ⎪ ⎨ 0 ⎪ ⎬ B88 ⎥ ⎥ 2ε42 . ⎥ D18 ⎥ ⎪ ⎪ κ ⎪ ⎪ 11 ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D28 ⎥ κ22 ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ D68 ⎥ ⎪ ⎪ ⎪ 2κ ⎪ ⎪ 12 ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D78 ⎥ ⎪ ⎪ 2κ ⎪ ⎦⎪ 41 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ D88 2κ42

A12 A16 A17 A18 B11 B12 B16 B17 B18 A22 A26 A27 A28 B12 B22 B26 B27 A26 A66 A67 A68 B16 B26 B66 B67 A27 A67 A77 A78 B17 B27 B67 B77 A28 A68 A78 A88 B18 B28 B68 B78 B12 B16 B17 B18 D11 D12 D16 D17 B22 B26 B27 B28 D12 D22 D26 D27 B26 B66 B67 B68 D16 D26 D66 D67 B27 B67 B77 B78 D17 D27 D67 D77 B28 B68 B78 B88 D18 D28 D68 D78

(13.132)

In the above Aαβ , Bαβ , Dαβ , α, β = 1, 2, 6, 7, 8, denote, respectively, the generalized extensional, coupling, and bending stiffness, which are defined as

Aαβ =

n

(Cαβ )k (hk − hk−1 ),

k=1

1 (Cαβ )k (h2k − h2k−1 ), 2 n

Bαβ = Dαβ =

1 3

k=1 n

(13.133a)

(Cαβ )k (h3k − h3k−1 ),

k=1

where E Cαβ = Cαβ ,

= e1α , = e2α , S = −ω(α−6)(β−6) ,

α, β = 1, 2, 6, α = 1, 2, 6, α = 1, 2, 6,

β = 7, β = 8,

(13.133b)

α, β = 7, 8

and hk and hk−1 denote, respectively, the location of the bottom and top surface of the kth lamina (see Fig. 13.1). Like the constitutive laws, for the electro-elastic composite laminates the equilibrium equations and the electrostatic equations are better expressed in terms of the generalized stress resultants and moments. If we neglect the body forces and shear stresses on the top and bottom surfaces of the laminates, by the way similar to the classical plate we can express the equilibrium and electrostatic conditions as

484

13


∂N11 ∂N21 + = 0, ∂x1 ∂x2 ∂ 2 M11 ∂ 2 M12 + 2 + ∂x1 ∂x2 ∂x12

∂N12 ∂N22 ∂N41 ∂N42 h/2 + = 0, + + D3 |−h/2 = 0, ∂x1 ∂x2 ∂x1 ∂x2 ∂ 2 M22 ∂M41 ∂M42 h/2 = q, + − N43 + (x3 D3 )|−h/2 = 0, 2 ∂x ∂x ∂x2 1 2 (13.134) where q is the lateral distributed load applied on the laminates.

13.5.2 Expanded Stroh-Like Formalism As described in Section 13.5.1, the displacement fields, the strain–displacement relations, the constitutive relations, and the equilibrium equations for the electroelastic composite laminates can be expressed by the equations given in (13.129), (13.132), and (13.134). Consider the homogeneous case that no lateral load and electric charge are applied on the laminates and the entire laminates are in the open circuit condition, i.e., q = 0, D3 (±h/2) = 0, and N43 = 0, the generalized constitutive law (13.132) and equilibrium equations (13.134) can be expressed in tensor notation as Npq = Apqrs ur, s + Bpqrs βr, s , Mpq = Bpqrs ur, s + Dpqrs βr, s , (13.135) Npj, j = 0, Mij, ij = 0, M4j, j = 0, p, q, r, s = 1, 2, 4; i, j = 1, 2. In getting (13.135), relation (13.129) and the symmetry property of Aαβ , Bαβ , Dαβ have been used. The tensors Apqrs , Bpqrs , and Dpqrs are related to their associated contracted notation by the following rules: 11 ↔ 1,

22 ↔ 2,

12 or 21 ↔ 6,

41 ↔ 7, 42 ↔ 8.

(13.136)

To solve the system of partial differential equations (13.135), the Stroh-like formalism presented in Section 13.2 for the coupled stretching–bending analysis of composite laminates will be used to treat the present homogeneous case. When q = 0 and/or D3 (±h/2) = 0 and/or N43 = 0, the complete solutions can be found by adding a particular solution to the homogeneous solutions derived in this section. By employing the constitutive relations (13.135)1 to the equilibrium equations (13.135)2 , the governing equations for the electro-elastic laminates can be expressed in terms of the mid-plane displacements ur and slopes βr as Apjrl ur, lj + Bpjrl βr, lj = 0, Bijrl ur, lij + Dijrl βr, lij = 0, B4jrl ur, lj + D4jrl βr, lj = 0,

(13.137) p, r = 1, 2, 4; i, j, l = 1, 2.

Because the mid-plane displacements ur and slopes βr depend only on two variables, x1 and x2 , and (13.137) are homogeneous partial differential equations, we may let

13.5


ur = aur f (z),

485

βr = aβr f (z),

z = x1 + μx2 ,

r = 1, 2, 4.

(13.138)

Substituting (13.138) into (13.137), we find that the governing equation (13.137) will be satisfied if " " ! ! QA + μ(RA + RTA ) + μ2 TA au + QB + μ(RB + RTB ) + μ2 TB aβ = 0, ! " ! " μ∗T QB + μ(RB + RTB ) + μ2 TB au + μ∗T QD + μ(RD + RTD ) + μ2 TD aβ = 0, ! " ! " iT3 QB + μ(RB + RTB ) + μ2 TB au + iT3 QD + μ(RD + RTD ) + μ2 TD aβ = 0, (13.139a) where ⎧ u⎫ a ⎪ ⎨ 1⎪ ⎬ u au = a2 , ⎪ ⎩au ⎪ ⎭ 4

⎧ ⎫ β⎪ ⎪ ⎪ ⎬ ⎨a1 ⎪ aβ = aβ2 , ⎪ ⎪ ⎪ ⎭ ⎩ β⎪ a4

⎧ ⎫ ⎨1 ⎬ μ∗ = μ , ⎩ ⎭ 0

⎧ ⎫ ⎨0⎬ i3 = 0 ⎩ ⎭ 1

(13.139b)

In (13.139a), QX , RX , TX , X = A, B, or D, are the 3 × 3 matrices defined by QX = Xp1r1 , RX = Xp1r2 , TX = Xp2r2 , p, r = 1, 2, 4,

(13.140a)

or in matrix notation, ⎡

⎡ ⎤ ⎤ X11 X16 X17 X16 X12 X18 QX = ⎣X61 X66 X67 ⎦ , RX = ⎣X66 X62 X68 ⎦ , X71 X76 X77 X76 X72 X78 ⎡ ⎤ X66 X62 X68 TX = ⎣X26 X22 X28 ⎦ , X = A, B or D. X86 X82 X88

(13.140b)

From (13.129b)1 we know β1, 2 = β2, 1 . With this relation, (13.138) gives us β

β

a2 = μa1 .

(13.141)

Equations (13.139a) and (13.141) constitute six equations with six unknowns β β β au1 , au2 , au4 , a1 , a2 , a4 . Thus, the problem is solved in principle. The non-vanishing values of au and aβ exist only when the determinant of their coefficient matrix becomes zero, which leads to an equation of 12th order polynomial of μ and can be proved to have 12 roots with six pairs of complex conjugates. To find the generalized stress resultants, we substitute (13.138) into (13.135)1 and follow the steps described between (13.15) and (13.22). By this approach, we obtain

486

13


Nr1 = −φr, 2 ,

Nr2 = φr, 1 , 1 Mr1 = −ψr, 2 − λr1 ψk, k , Mr2 = ψr, 1 − 2 1 1 Q1 = − ψk, k2 , Q2 = ψk, k1 , 2 2 V1 = −ψ2, 22 , V2 = ψ1, 11 , r = 1, 2, 4;

1 λr2 ψk, k , 2

(13.142a)

k = 1, 2,

where λrk is the tensor defined as λ11 = λ22 = λ41 = λ42 = 0,

λ12 = −λ21 = 1.

(13.142b)

In (13.142a) we see that the calculation of the generalized stress resultants depends on the six stress functions φr and ψr , r=1,2,4, which have been assumed in the following form:

φr = bur f (z),

ψr = bβr f (z),

r = 1, 2, 4.

(13.143)

β

bur and br are the components of the vectors bu and bβ , which are related to au and aβ by the following relations: ⎧ u⎫ ⎪ ⎨b1 ⎪ ⎬ bu = bu2 , ⎪ ⎩ u⎪ ⎭ b4

⎧ ⎫ β⎪ ⎪ ⎪ ⎬ ⎨b1 ⎪ bβ = bβ2 ⎪ ⎪ ⎪ ⎭ ⎩ β⎪ b4

(13.144a)

and bu = (RTA + μTA )au + (RTB + μTB )aβ 1 = − {(QA + μRA )au + (QB + μRB )aβ }, μ bβ = (RTB + μTB )au + (RTD + μTD )aβ + b0 i1 , 1 b0 = − {(QB + μRB )au + (QD + μRD )aβ } + i2 , μ μ

(13.144b)

13.5


487

where 1 ∗T 1 β β μ bβ = (b1 + μb2 ), 2 2 ⎧ ⎫ ⎧ ⎫ ⎨1 ⎬ ⎨0⎬ i1 = 0 , i2 = 1 . ⎩ ⎭ ⎩ ⎭ 0 0

b0 =

(13.144c) (13.144d)

Knowing that (13.139) and (13.141) will provide six pairs of eigenvalues μk , μ¯ k , k = 1, 2, . . . , 6, the results of (13.138) and (13.143) can now be combined into ud = 2 Re{Af(z)},

φd = 2 Re{Bf(z)},

(13.145a)

where Re stands for the real part and ud =

u , β

φd =

φ , ψ

A = [a1 a2 a3 a4 a5 a6 ], B = [b1 b2 b3 b4 b5 b6 ], ⎧ ⎫ f1 (z1 )⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f2 (z2 )⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ f3 (z3 ) , zk = x1 + μk x2 , k = 1, 2, 3, 4, 5, 6, f(z) = f4 (z4 )⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f5 (z5 )⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ f6 (z6 )

(13.145b)

and ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎨β1 ⎬ ⎨φ1 ⎬ ⎨ψ1 ⎬ ⎨u1 ⎬ u = u2 , β = β2 , φ = φ2 , ψ = ψ2 , ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ u4 β4 φ4 ψ4 au bu , bk = , k = 1, 2, 3, 4, 5, 6. ak = aβ k bβ k

(13.145c)

Note that in (13.145a) we put a subscript d for the augmented displacement and stress function vectors to emphasize that this solution is obtained by following the displacement formalism introduced in Section 13.2.1. The eigenvalues μk and their associated eigenvectors ak and bk can be determined from (13.139), (13.140), (13.141) and (13.144). Re-organizing these relations, one can formulate a standard eigenrelation by following the steps described in Section 13.2.1. However, as discussed in Section 13.2.5, the eigenrelation constructed by the displacement formalism is not fully compatible with that of Stroh formalism for two-dimensional problems. To get a compatible eigenrelation, the mixed formalism was introduced in Section 13.2.2. By following the steps described in Section 13.2.2 for the mixed formalism and the discussions in Section 13.2.5, a compatible eigenrelation substituting (13.139), (13.140), (13.141) and (13.144) can be written as

488

13


Nξ = μξ

(13.146a)

where N = It Nm It ,

ξ=

a b

(13.146b)

and N1 N2 N= , N3 NT1

(Nm )1 (Nm )2 Nm = , (Nm )3 (Nm )T1

I1 I2 It = , I2 I1

T (Nm )1 = −T−1 m Rm ,

T (Nm )2 = T−1 m = (Nm )2 , I 0 −1 T T (Nm )3 = Rm Tm Rm − Qm = (Nm )3 , I1 = , 00

I2 =

(13.146c)

00 . 0I

The subscript m here denotes that its related values are constructed by the mixed formalism. In (13.146c), I is the 6 × 6 unit matrix, whereas Qm , Rm , and Tm in (13.146c) are three 6 × 6 matrices defined by Qm =

' ( Q/ B A R/ T −T R/ / D B

,

Rm =

' ( R/ B A −Q/ T RT T/ / B D

'

,

( /B T/ A −R/ Tm = , T −Q −/ R/ / D B

(13.147a)

where QX , RX , TX , X = / A, / B, or / D, have the same definition as those given in (13.140). Concerning the unsymmetry of / B, a new matrix / R/ B is defined by / / R/ B = Bp2r1 ,

p, r = 1, 2, 4.

(13.147b)

In matrix notation, we have ⎡ ⎡ ⎤ ⎤ / / A11 / A16 / A17 B17 /2 B11 / B16 /2 / ⎣/ / / ⎦ ⎣/ / ⎦ / Q/ Q/ B = B61 B66 /2 B67 /2 , A = A16 A66 A67 , / / B77 /2 B17 / B67 /2 / A17 / A67 / A77 ⎡ ⎤ / D17 /2 D16 /2 / D11 / ⎣/ ⎦ / / Q/ D = D16 /2 D66 /4 D67 /4 , / D67 /4 / D77 /4 D17 /2 /

(13.147c)

⎡ ⎤ / A16 / A12 / A18 ⎣/ / / ⎦ R/ A = A66 A26 A68 , / A67 / A27 / A78

(13.147d)

⎡ ⎤ / B67 /2 B66 /2 / B61 / / ⎣/ / ⎦ / R/ B = B21 B26 /2 B27 /2 , / / / B18 B68 /2 B87 /2

⎡ ⎤ / B12 / B18 /2 B16 /2 / ⎣/ ⎦ / / R/ B = B66 /2 B62 B68 /2 , / B27 / B67 /2 / B78 /2 ⎡ ⎤ / D12 / D18 /2 D16 /2 / ⎣/ ⎦ / / R/ D = D66 /4 D26 /2 D68 /4 , / / / D67 /4 D27 /2 D78 /4

(13.147e)

13.5


489

⎡ ⎤ / A66 / A26 / A68 ⎣/ / / ⎦ T/ A = A26 A22 A28 , / A68 / A28 / A88 ⎡ ⎤ / B62 / B68 /2 B66 /2 / ⎣/ ⎦ / / T/ B = B26 /2 B22 B28 /2 , / B28 / B68 /2 / B88 /2

(13.147f)

⎡ ⎤ / D26 /2 / D68 /4 D66 /4 / ⎣/ ⎦ / / T/ D = D26 /2 D22 D28 /2 . / D28 /2 / D88 /4 D68 /4 /

(13.147g)

Bαβ , and / Dαβ , α, β = 1, 2, 6, 7, 8, are components of the matrices In the above / Aαβ , / / A, / B, and / D which are related to the generalized extensional, coupling, and bending stiffness matrices A, B, and D by / A = ABD−1 B,

/ B = BD−1 ,

/ D = D−1 .

(13.148)

Material Eigenvalues and Eigenvectors By using the mixed formalism, ur , ψr , φr and βr , r =1,2,4, are related to the material eigenvectors by ur = aur f (z), ψr = aψ r f (z), z = x1 + μx2 , r = 1, 2, 4.

φr = bur f (z),

βr = bψ r f (z),

⎧ ⎫ ψ ⎪ ⎨a1 ⎪ ⎬ , aψ = aψ 2 ⎪ ⎩ ψ⎪ ⎭ a4

⎧ u⎫ ⎨b1 ⎬ bu = bu2 , ⎩ u⎭ b4

⎧ ⎫ ψ ⎪ ⎨b1 ⎪ ⎬ , bψ = bψ 2 ⎪ ⎩ ψ⎪ ⎭ b4

(13.149)

Let ⎧ u⎫ ⎨a1 ⎬ au = au2 , ⎩ u⎭ a4

(13.150)

comparing (13.149) with (13.138) and (13.143), we have bβ = aψ ,

aβ = bψ .

(13.151)

By the symmetry requirement N12 = N21 and κ12 = κ21 , we get − μbu2 = bu1 ,

ψ

ψ

b2 = μb1 .

(13.152)

With the relations given in (13.152), (13.129b)2 , and (13.145a)1,2 , substituting (13.149) into (13.30) we can get

⎧ u⎫ b2 ⎪ ⎪ ' ∗ ( ⎪ ∗ ⎨ bu ⎪ ⎬ / A / B XN 0 Xε 0 au 4 f (z) = ψ f (z), ∗T ∗ b1 ⎪ 0 XM aψ 0 Xκ ⎪ D −/ B / ⎪ ⎪ ⎩ ψ⎭ b4

(13.153a)

490

13


where ⎡

⎤ ⎡ 0 −μ 0 ⎢ 0 1 0⎥ ⎥ ⎢ ⎢ 0⎥ ⎥ , XM = ⎢ 1 −μ ⎦ ⎣ 0 0 1 μ 0 0

⎤ ⎡ 2 0 μ ⎢ 1 0 ⎥ ⎥ ⎢ ⎢ 0 ⎥ ⎥ , XN = ⎢−μ ⎦ ⎣ 0 −μ 1 0

⎡ ⎤ ⎤ 1 0 0 ⎢ μ2 0 ⎥ 0 ⎥ ⎢ ⎥ ⎥ ⎥ ⎥ 0 ⎥ , Xκ = ⎢ ⎢2μ 0 ⎥ . ⎣ ⎦ 0 1⎦ −μ 0 μ 1 (13.153b) Multiplying both sides of (13.153a) by the following 4 × 10 matrix,

1 0 ⎢0 μ ⎢ Xε = ⎢ ⎢μ 1 ⎣0 0 0 0

Yε 0 , 0 YM

(13.154a)

where

μ 1/μ −1 0 0 1/μ μ 2 0 0 Yε = , YM = , 0 0 0 −μ 1 0 0 01μ

(13.154b)

the left-hand side vanishes and (13.153a) becomes ⎧ u⎫ b2 ⎪ ( ⎪ ⎪ ' ∗ ⎪ ∗ u⎬ ⎨ / / b A B XN 0 Yε 0 4 = 0, ψ ∗T ∗ b ⎪ 0 YM −/ 0 Xκ ⎪ D B / ⎪ ⎩ 1ψ ⎪ ⎭ b4

(13.155)

which can be written explicitly as ⎡

11 ⎢21 ⎢ ⎣31 41

12 22 32 42

13 23 33 43

⎤⎧ u ⎫ b2 ⎪ 14 ⎪ ⎪ ⎨bu ⎪ ⎬ 24 ⎥ ⎥ 4ψ = 0. ⎦ 34 ⎪ b ⎪ ⎪ ⎩ 1ψ ⎪ ⎭ 44 b4

(13.156a)

where 1 1 pA2 − pA6 , 12 = μrA1 + rA2 − rA6 , μ μ 1 1 = μqB1 + qB2 − qB6 , 14 = μsB1 + sB2 − sB6 , μ μ = −μpA7 + pA8 , 22 = −μrA7 + rA8 , 23 = −μqB7 + qB8 , 24 = −μsB7 + sB8 , 1 1 = − pBT 1 − μpBT 2 − 2pBT 6 , 32 = − rBT 1 − μrBT 2 − 2rBT 6 , μ μ 1 1 = qD1 + μqD2 + 2qD6 , 34 = sD1 + μsD2 + 2sD6 , μ μ = −pBT 7 − μpBT 8 , 42 = −rBT 7 − μrBT 8 , 43 = qD7 + μqD8 , 44 = sD7 + μsD8 (13.156b)

11 = μpA1 + 13 21 31 33 41

13.5


491

and ∗ ∗ ∗ + X˜ j2 − μX˜ j6 , pXj = μ2 X˜ j1

∗ ∗ rXj = −μX˜ j7 + X˜ j8 ,

∗ ∗ ∗ qXj = X˜ j1 + μ2 X˜ j2 + 2μX˜ j6 ,

∗ ∗ sXj = X˜ j7 + μX˜ j8 , ψ

ψ

X = A, B, or D.

(13.156c)

Nontrivial solutions of bu2 , bu4 , b1 , and b4 exist when the determinant of the coefficient matrix of (13.156a) vanishes, which will lead to an equation of 12th order polynomial of μ. The material eigenvalues obtained from this characteristic equation should be the same as those calculated by the eigenrelation (13.146). From (13.152) and (13.156a), one can write down the explicit expressions for bu and bψ , whereas those of au and aψ are obtained by substituting bu and bψ into (13.153a). With these results and relations (13.151), the expressions for the augmented material eigenvectors a and b defined in the last two equations of (13.145c) can then be obtained explicitly.

Chapter 14

Holes/Cracks/Inclusions in Laminates

The problems of holes, cracks, and inclusions are important not only in macromechanics but also in micromechanics. From the viewpoint of macromechanics, holes and inclusions are usually parts of the structure design. Due to the stress concentration induced by the existence of holes/inclusions and pre-existing flaws, the cracks may initiate, propagate, and fracture. From micromechanical viewpoint of composite materials, fibers can be treated as inclusions, and micro-cracks and voids always exist in the materials due to imperfect composite fabrication. Thus, understanding holes/cracks/inclusions is of importance due to the increased utilization of composites in recent aerospace and commercial applications. As discussed in Chapters 6, 7, 8 the problems of anisotropic plates containing holes/cracks/inclusions have been studied extensively for two-dimensional problems. If the in-plane and bending deformations are uncoupled, the problems of anisotropic plates containing holes/cracks/inclusions subjected to bending moments were also solved in Section 12.4 by using Stroh-like bending formalism. In this chapter we like to apply the Stroh-like formalism introduced in Chapter 13 to deal with the coupled stretching–bending problems of holes, cracks, and inclusions in general composite laminates. In Section 14.1, an unbounded laminate with an elliptical hole subjected to uniform in-plane loads and out-of-plane bending moments will be considered. Besides the mechanical loading, by using the extended Strohlike formalism for hygrothermal analysis introduced in Section 13.4, the problems of the general laminates, symmetric or unsymmetric, disturbed by an elliptical hole subjected to uniform heat flow and moisture transfer in x1 x2 -plane and x3 -direction will be solved analytically and presented in Section 14.2. The mechanical–electrical analysis of holes in the electro-elastic composite laminates is then solved in Section 14.3 by applying the expanded Stroh-like formalism introduced in Section 13.5. In addition to the uniform loadings (mechanical, thermal, and electrical) discussed in Sections 14.1, 14.2, and 14.3, the Green’s functions will be presented in Sections 14.4 for infinite laminates, and in Sections 14.5 and 14.6 for laminates with holes/cracks and inclusions. The Green’s functions here are the solutions for an infinite laminate with holes/cracks/inclusions subjected to concentrated forces and moments at any arbitrary points. The complete loading cases such as transverse loading, in-plane loading, out-of-plane bending moment, and in-plane torsion


493

494

14


will all be considered in the study of Green’s functions discussed in Sections 14.4, 14.5, and 14.6. These solutions are important because analytically they can provide solutions for arbitrary loading through superposition and numerically they can be employed as the fundamental solutions for boundary element method and as the kernel functions of integral equations to consider interactions between holes/inclusions and cracks.

14.1 Holes in Laminates Under Uniform Stretching and Bending Moments Consider an unbounded composite laminate with an elliptical hole subjected to ∞, N ∞ ∞ in-plane forces N11 = N11 22 = N22 , N12 = N12 and out-of-plane bending ∞ ∞ ∞ moments M11 = M11 , M22 = M22 , M12 = M12 at infinity (Fig. 14.1). There is no load around the edge of the elliptical hole. The contour of the elliptical hole is represented by x1 = a cos ϕ, x2 = b sin ϕ,

(14.1)

where 2a, 2b are the major and minor axes of the ellipse and ϕ is a real parameter. The boundary conditions of this problem can be expressed as ∞ ∞ ∞ N11 = N11 , N22 = N22 , N12 = N12 , ∞ ∞ ∞ M11 = M11 , M22 = M22 , M12 = M12 ,

Nn = Nns = 0, Mn = Vn = 0,

at infinity; at infinity;

(14.2)


∞ N 22

M 12∞

N12∞ ∞ M 22

N12∞ N11∞

2a

M 11∞ 2b

M 12∞

M 11∞

N12∞ M 22∞

M 12∞

x2

N11∞ M 12∞

x1

N12∞

∞ N 22

x3

Fig. 14.1 A composite laminate weakened by an elliptical hole subjected to in-plane forces and out-of-plane bending moments

14.1

Holes in Laminates Under Uniform Stretching and Bending Moments

495

From the relations given in (13.71) and (13.82), we know that the boundary condition (14.2) can be expressed in terms of the stress function as ∞ ∞ at infinity, φ d = φ∞ d = x1 m2 − x2 m1 , φd = 0, along the hole boundary,

(14.3a)

where

m∞ 1

⎧ ∞⎫ ⎧ ∞⎫ N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪N11 ⎪ ⎬ ⎨ ⎨ 12 ∞ ∞⎬ N12 N22 ∞ , m = = ∞ ∞ . 2 M12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪M11 ⎪ ⎭ ⎩ ⎩ ∞ ∞⎭ M12 M22

(14.3b)

∞ The displacement vector u∞ d associated with φd for a homogeneous composite laminate can be obtained from the inverse of constitutive laws given in (13.28), i.e.,

∗ ∗ ε0 N A B = ∗ ∗ , κ B D M

(14.4a)

where ⎧ 0 ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎨ε11 ⎬ ⎨N11 ⎬ ⎨M11 ⎬ ⎨κ11 ⎬ 0 N = N22 , M = M22 , ε0 = ε22 , κ = κ22 , ⎩ 0⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ N12 M12 κ12 γ12

(14.4b)

and A∗ , B∗ , D∗ are related to the extensional, bending, and coupling stiffness matrices A, B and D by A∗ = A−1 + A−1 BD∗ BA−1 , B∗ = −A−1 BD∗ , D∗ = (D − BA−1 B)−1 . (14.4c) From (14.4a), we get the following relation:

d∞ 1 d∞ 2

=

∗ ∗ ∞ Q R m1 , R∗T T∗ m∞ 2

(14.5a)

∞ where m∞ 1 , m2 are defined in (14.3b) and

d∞ 1

⎧ ∞ ⎫ ⎧ ∞ ⎫ ε11 ⎪ γ12 /2⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎬ ⎨γ ∞ /2⎪ ⎨ ε∞ ⎪ 12 22 ∞ , d = = 2 ∞ ⎪ ∞ /2 ⎪ . ⎪ ⎪ κ11 κ12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ∞ ⎭ ⎩ ∞ ⎪ κ22 κ12 /2

(14.5b)

The three matrices Q∗ , R∗ , and T∗ are related to Q, R, and T in (13.24b) and are defined by

496

14

⎡

A∗11

A∗16 /2


B∗11

B∗16 /2

⎤

⎢ A∗ /2 A∗ /4 B∗ /2 B∗ /4 ⎥ ⎢ ⎥ 66 61 66 Q∗ = ⎢ 16∗ ⎥, ∗ ∗ ∗ ⎣ B11 B61 /2 D11 D16 /2 ⎦ ⎡

B∗16 /2 B∗66 /4 D∗16 /2 D∗66 /4

A∗16 /2

A∗12

B∗16 /2

B∗12

⎤

⎢ A∗ /4 A∗ /2 B∗ /4 B∗ /2 ⎥ ⎢ ⎥ 26 66 62 R∗ = ⎢ 66 ⎥, ⎣ B∗61 /2 B∗21 D∗16 /2 D∗12 ⎦ ⎡

B∗66 /4 B∗26 /2 D∗66 /4 D∗26 /2

A∗66 /4 A∗26 /2 B∗66 /4 B∗62 /2

(14.5c)

(14.5d)

⎤

⎢ A∗ /2 A∗ B∗26 /2 B∗22 ⎥ ⎢ ⎥ 22 T∗ = ⎢ 26 ⎥, ⎣ B∗66 /4 B∗26 /2 D∗66 /4 D∗26 /2 ⎦ B∗62 /2 B∗22 D∗26 /2 D∗22

(14.5e)

in which A∗ij , B∗ij , and D∗ij are components of the matrices A∗ , B∗ , and D∗ given in (14.4c). Integration of (14.5a) with respect to x1 and x2 , respectively, leads to the results of u∞ d as ∞ ∞ u∞ d = x1 d1 + x2 d2 .

(14.6)

14.1.1 Field Solutions In order to satisfy the infinity boundary condition (14.3a)1 , the displacement vector ∞ u∞ d and the stress function vector φd are added to the general solution (13.70a). Whereas for the satisfaction of hole boundary condition (14.3a)2 , by referring to the solutions of the corresponding two-dimensional problems, (6.8), the complex function vector f(z) of (13.70a) is selected to be f(z) =
k, ζα =

zα +

+ z2α − a2 − μ2α b2 , a − iμα b

(14.7)

where k is the unknown coefficient vector to be determined through the satisfaction of the hole boundary condition. Therefore, the solution for the present problem can be expressed as ! " ! " −1 ∞ −1 ud = u∞ + 2Re A < ζ > k , φ = φ + 2Re B < ζ > k , d α α d d

(14.8)

∞ where u∞ d , φd are given in (14.6) and (14.3), respectively. Knowing the value ζα = eiϕ along the hole boundary and the solutions selected in (14.8), the boundary condition (14.3a)2 will then provide us

14.1

Holes in Laminates Under Uniform Stretching and Bending Moments

1 ∞ k = − B−1 (am∞ 2 − ibm1 ). 2

497

(14.9)

The explicit solutions can therefore be expressed as ! " −1 −1 ∞ ∞ ud = u∞ d − Re A < ζα > B (am2 − ibm1 ) , ! " −1 −1 ∞ ∞ φd = φ∞ − Re B < ζ > B (am − ibm ) , α d 2 1

(14.10)

which has the same form as those of the corresponding two-dimensional problems, (6.8) and (6.10), and pure bending problems, (12.64). The only difference is that the symbols like u, φ, A, B, . . . have different dimensions and different contents for different types of problems.

14.1.2 Stress Resultants and Moments Along the Hole Boundary According to the relations obtained in (13.79), (13.80), and (13.81) we know that the calculation of the stress resultants and bending moments relies upon the calculation of the differentials φd, s and φd, n . Again, because our final solutions for φd obtained in (14.10) have exactly the same form as those of the corresponding twodimensional problems, just by copying our previous corresponding results, (6.17), without any further detailed derivation, we can get φd, s = 0,

" ! a ∞ φd, n = sin θ G1 (θ )m∞ 1 − I + G3 (θ ) m2 b b ∞ , − cos θ I + G3 (θ ) m∞ + G (θ )m 1 1 2 a

(14.11a)

where G1 (θ ) = NT1 (θ ) − N3 (θ )SL−1 ,

G3 (θ ) = −N3 (θ )L−1 .

(14.11b)

Substituting (14.11) into (13.79), we obtain the stress resultants and bending moments around the hole boundary as Nn = Nns = Mn = Vn = 0, " ! 1 (3) (0) (3) (1) (0) (1) Ns = cos θ σˆ 1 − cσˆ 1 + σˆ 2 + sin θ σˆ 2 − σˆ 2 − σˆ 1 , c " ! 1 (3) (0) (3) (1) (0) (1) Ms = cos θ σ1 − cσ1 + σ2 + sin θ σ2 − σ2 − σ1 , c " ! 1 (3) (0) (3) (1) (0) (1) σ2 − / σ2 − / Mns = cos θ / σ1 − c/ σ1 + / σ2 + sin θ / σ1 . c

(14.12a)

498

14


In the above, c = b/a and (0)

σˆ i

(0)

σi

(1) (3) = sT (θ )Iˆn m∞ ˆ i = sT (θ )Iˆn G1 (θ )m∞ ˆ i = −sT (θ )Iˆn G3 (θ )m∞ i , σ i , σ i , (14.12b)

= sT (θ )Iˆm m∞ i , σi

(1)

= sT (θ )Iˆm G1 (θ )m∞ i , σi

(3)

(0)

= nT (θ )Iˆm m∞ σi i /2, /

(3)

= −nT (θ )Iˆm G3 (θ )m∞ i /2, i = 1, 2,

/ σi / σi

(1)

= −sT (θ )Iˆm G3 (θ )m∞ i , (14.12c)

= nT (θ )Iˆm G1 (θ )m∞ i /2,

(14.12d)

where G1 (θ ) and G3 (θ ) are defined in (14.11b) and 1000 Iˆn = [I 0] = , 0100

0010 Iˆm = [0 I] = , 0001

sT = (cos θ sin θ ), nT = (− sin θ cos θ ).

(14.12e) (14.12f)

Based upon the solutions obtained in (14.12) for the stress resultants and bending moments around the hole boundary, several examples have been done in Hsieh and Hwu (2003) to discuss the laminate coupling effect and hole shape effect.

14.2 Holes in Laminates Under Uniform Heat Flow and Moisture Transfer By using the extended Stroh-like formalism introduced in Section 13.4, the hygrothermal stresses in composite laminates disturbed by an elliptical hole subjected to uniform heat flow and moisture transfer in x1 x2 -plane or x3 -direction are now considered in this section.

14.2.1 Uniform Heat Flow and Moisture Transfer in x1 x2 -Plane In an infinite composite laminate, heat qˆ and moisture m ˆ are flowing uniformly in the direction of angle θˆ clockwise from the positive x1 -axis (Fig. 14.2). The uniform steady heat and moisture flow is disturbed by the presence of an insulated elliptic hole whose boundary is given by (14.1). If the hole is assumed to be free of tractions, the boundary conditions for this problem can be written as ˘i → m ˆ i , Nij → 0, Mij → 0, i, j = 1, 2, at infinity, q˘ i → qˆ i , m (14.13) q˘ n = m ˘ n = 0, Nn = Nns = 0, Mn = Vn = 0, along the hole boundary, where q˘ n and m ˘ n are the heat flux and moisture transfer in the direction of n which is normal to the surface of the elliptic hole. From relations (13.79), (13.80), and (13.81)

14.2

Holes in Laminates Under Uniform Heat Flow and Moisture Transfer

499

θˆ

qˆ , mˆ Lamina Number

θ

1

2b

2

h0

h1

n

h2

x1

2a

hk-1

h

hk

hn-1

s

x1

k

hn

x2

n

x3

qˆ , mˆ

x3 dx1

Fig. 14.2 Laminates weakened by an elliptical hole subjected to uniform heat flow and moisture transfer in x1 – x2 -plane

we know that the traction boundary condition given in (14.13) can be expressed in terms of the generalized stress function vector φd as φd → 0, φd, s = 0,

at infinity, along the hole boundary.

(14.14)

To consider the heat flux and moisture transfer boundary conditions given in (14.13)2 , we use the coordinate transformation and employ (13.118a)2 and (13.119a) to get q˘ n = −˘q1 sin θ + q˘ 2 cos θ = 2Kt Im (cos θ + τt sin θ )gt (zt ) , m ˘ n = −m ˘ 1 sin θ + m ˘ 2 cos θ = 2Kh Im (cos θ + τh sin θ )gh (zh ) ,

(14.15a)

where Kt and Kh are real constants defined by t Kt = K22 (τt − τ t )/2i,

h Kh = K22 (τh − τ h )/2i.

(14.15b)

θ is the angle between x1 -axis and the tangent direction s (Fig. 14.2) and is related to ϕ by ρ cos θ = a sin ϕ,

ρ sin θ = −b cos ϕ,

(14.16a)

in which 3 ρ=

a2 sin2 ϕ + b2 cos2 ϕ.

(14.16b)

500

14


Because the mathematical forms of the general solutions (13.118) and boundary conditions (14.13) together with relations (14.14), (14.15), and (14.16) are the same as those of the corresponding two-dimensional problems, by following the steps described in Section 10.2.1, the unknown functions in the general solution (13.118) can be assumed to be f(z) =

2

< fk (zα , μα ) > AT qka + BT qkb ,

k=0

gt (zt ) =

2

gˆ tk fk (zt , τt ),

gh (zh ) =

k=0

2

(14.17a) gˆ hk fk (zh , τh ),

k=0

where 1 2 z , 2 3 1 2 1 1 f1 (z, μ) = z − z z2 − (a2 + μ2 b2 ) , a + iμb 2 2 3 a − iμb ln z + z2 − (a2 + μ2 b2 ) f2 (z, μ) = 2 f0 (z, μ) =

(14.17b)

and gˆ t1 = gˆ t2 ,

gˆ h1 = gˆ h2 .

(14.17c)

The problem now reduces to the determination of the unknown constants qka , qkb , gˆ tk and gˆ hk , k = 0, 1, 2, which should satisfy the boundary conditions set in (14.13) with useful relations given in (14.14) and (14.15). Substituting the assumed functions (14.17) into the general solutions (13.118) then to the boundary conditions (14.13), and following the steps described in Section 10.2.1, we get " 1 t 1 ! h t h h (K , + iK )ˆ q − K q ˆ = + iK ) m ˆ − K m ˆ (K , g ˆ t 2 1 h 2 1 12 22 0 12 22 2Kt2 2Kh2 1 1 gˆ t1 = − bˆq1 + iaˆq2 , gˆ h1 = − bm ˆ 1 + iam ˆ2 , 2Kt 2Kh (14.18a) " ! q0a = −2Re gˆ t0 dt + gˆ h0 dh , " ! (14.18b) q0b = 2N3 Re gˆ t0 (NT1 − τt I)dt + gˆ h0 (NT1 − τh I)dh + k∗ i2 ,

gˆ t0 =

−1 T q1b L (S − iI)d∗1 = 2[a/ N − bN]−1 Im , q1a d∗1 " ! q2b = 2[a/ N + bN]−1 Im gˆ t1 (a − ibτt )ηt + gˆ h1 (a − ibτh )ηh , q2a

(14.18c)

14.2

Holes in Laminates Under Uniform Heat Flow and Moisture Transfer

501

where d∗1 = gˆ t1 (a + ibτt )dt + gˆ h1 (a + ibτh )dh .

(14.18d)

In the above, S and L are Barnett–Lothe tensors defined in (3.59) and are related ˜ by (3.95); the superscript to the average of the fundamental elasticity matrix N < −1 > of N3 denotes sub-inverse defined in (5.19); k∗ is a constant determined by substituting (14.18b) into the following equation which comes from the satisfaction of the infinity condition (14.14)1 T(2)

N1

" ! (2) q0a + N3 q0b + 2Re gˆ t0 τt2 dt + gˆ h0 τh2 dh = 0,

(14.19a)

where T(2)

N1

= N3 N2 + NT1 NT1 ,

(2)

N3 = N3 N1 + NT1 N3 ,

(14.19b)

and i2 = (0 1 0 0)T . The general solution (13.118) together with (14.17), (14.18), and (14.19) now provide us the complete full-field solution for the present problem. To calculate the stress resultants and bending moments along the hole boundary, the relations provided in (13.79), (13.80), and (13.81) can be employed, from which we see that the determination of stress resultants/moments depends on the derivative of stress function, φd, n . By following the steps described in Section 10.2.1, the explicit form solution of φd, n for the present case is obtained as φd, n =

1 1 (bˆq1 cos ϕ + aˆq2 sin ϕ)γt2 (θ ) + (bm ˆ 1 cos ϕ + am ˆ 2 sin ϕ)γh2 (θ ) Kt Kh "" ! 1! G3 (θ )Re ie−2iϕ gˆ t1 (a + ibτt )/ , + γt∗ ˆ h1 (a + ibτh )/ γh∗ 2 +g 2 ρ (14.20)

where G3 (θ ) is defined in (14.11b); γt2 (θ ), γh2 (θ ),/ γt∗ γh∗ 2 , and / 2 are vectors related to thermal and moisture moduli and their detailed definitions are given in (13.121) and (13.122).

14.2.2 Uniform Heat Flow and Moisture Transfer in x3 -Direction All the conditions are the same as Section 14.2.1 except that the heat qˆ and moisture m ˆ is now flowing uniformly in the thickness direction instead of plane direction. This may occur when the temperature and moisture content on the top and bottom surfaces of the laminate are different, for example, Tu and Hu on the top surface and Tl and Hl on the bottom surface. If the temperature and moisture content are assumed to vary linearly as shown by (13.123), we have

502

14

T0 =


Tl + Tu Hl + Hu Tl − Tu Hl − Hu , H0 = , T∗ = , H∗ = . 2 2 h h

(14.21)

By the first and second equations of (13.124)1 , the heat flux qˆ and moisture transfer m ˆ in the thickness direction are related to T ∗ and H ∗ by t T ∗, qˆ = −K33

h m ˆ = −K33 H∗,

(14.22)

which are constant throughout the entire plate. If the hole is assumed to be free of traction, the boundary conditions for this problem can be written as ˆ ij , i, j = 1, 2, Nij → Nˆ ij , Mij → M Nn = Nns = 0, Mn = Vn = 0,

at infinity, along the hole boundary.

(14.23)

As stated in the last paragraph of Section 13.4.2, in (14.23) there is no need to describe the boundary conditions of heat and moisture flow as those shown in (14.13) for the case discussed in Section 14.2.1. The prescribed values Nˆ ij and ˆ ij are the stress resultants and bending moments induced by the temperature and M moisture which are related to T 0 , H 0 , T ∗ , and H ∗ by ⎫ ⎧ Nˆ 11 ⎪ ⎪ ⎪ ⎬ ⎨ˆ ⎪ N12 ˆ ⎪ = −ϑ1 , ⎪ M ⎪ ⎭ ⎩ 11 ⎪ ˆ 12 M

⎫ ⎧ Nˆ 12 ⎪ ⎪ ⎪ ⎬ ⎨ˆ ⎪ N22 ˆ ⎪ = −ϑ2 , ⎪ M ⎪ ⎭ ⎩ 12 ⎪ ˆ 22 M

(14.24)

where ϑ1 and ϑ2 are given in (13.125b). Note that (14.24) is obtained by using (13.71) and (13.125) and knowing that no mechanical loading is applied in this problem. From relations (13.71), (13.79), (13.80), (13.81) and (14.24), the boundary conditions (14.23) can now be expressed in terms of the generalized stress function vector φd as φd → −x1 ϑ2 + x2 ϑ1 , at infinity, φd, s = 0, along the hole boundary.

(14.25)

By comparing the general solution (13.125) and boundary condition (14.25) of this problem with those of non-hygrothermal problem solved in Section 14.1, without any further detailed derivation we can obtain directly the solution for this problem, by referring to (14.10), as ! " ud = Re A < ζα−1 > B−1 (aϑ2 − ibϑ1 ) , ! " φd = Re B < ζα−1 > B−1 (aϑ2 − ibϑ1 ) − x1 ϑ2 + x2 ϑ1 ,

(14.26)

14.3

Holes in Electro-Elastic Laminates

503

where ζα , α = 1, 2, 3, are defined in (14.7). Again, by referring to the corresponding non-hygrothermal solutions obtained in (14.11), without any further detailed derivation we get φd, n

b a = cos θ ϑ1 + G1 (θ )ϑ2 + G3 (θ )ϑ1 + sin θ ϑ2 − G1 (θ )ϑ1 + G3 (θ )ϑ2 . a b (14.27)

Based upon the solutions presented in this section, several numerical examples discussing the hygrothermal effects on the holes in unsymmetric laminates have been shown in Hsieh and Hwu (2006).

14.3 Holes in Electro-Elastic Laminates Consider an unbounded electro-elastic composite laminate with an elliptical hole ∞ , N ∞ , N ∞ , M ∞ , M ∞ , M ∞ and subjected to the generalized forces and moments N11 22 12 11 22 12 ∞ ∞ ∞ ∞ N41 , N42 , M41 , M42 at infinity. The generalized forces and moments are defined in (13.130) in which N4i and M4i , i =1, 2 are related to the electric displacement by (13.131). The contour of the elliptical hole is represented by (14.1). If the hole edge and the upper and lower surfaces of the laminate are free of traction and electric charge, no particular solution is required and the boundary conditions of this problem can be expressed in terms of the augmented stress function φd as ∞ ∞ φd = φ ∞ at infinity, d = x1 m2 − x2 m1 , φd = 0, around the hole boundary,

(14.28a)

where

m∞ 1

⎧ ∞⎫ ⎧ ∞⎫ N11 ⎪ N12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ∞⎪ ⎪ ⎪ ⎪ N12 ⎪ N22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨N ∞ ⎬ ⎨N ∞ ⎪ 41 42 ∞ , m2 = . = ∞ ∞ ⎪ ⎪ M11 ⎪ M12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∞⎪ ∞⎪ ⎪ ⎪ ⎪ ⎪ M12 M22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ∞⎭ ⎩ ∞⎪ M41 M42

(14.28b)

By inversion of the constitutive laws shown in (13.132) and integrating the midplane strains and curvatures with respect to x1 and x2 , the displacement vector u∞ d associated with φ∞ d can be obtained as ∞ ∞ u∞ d = x1 d1 + x2 d2 ,

(14.29a)

504

14


where

d∞ 1

⎧ ∞⎫ ⎧ ∞⎫ ε11 ⎪ ε12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ε12 ⎪ ε22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎬ ⎨ 2ε∞ ⎪ ⎨ 2ε∞ ⎪ 41 42 ∞ , d = = 2 ∞ ∞ ⎪, ⎪ ⎪ κ12 ⎪ ⎪ ⎪ κ11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∞⎪ ∞⎪ ⎪ ⎪ ⎪ κ12 ⎪ κ22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎭ ⎩ ∞⎪ ⎩ ∞⎪ 2κ41 2κ42

(14.29b)

∞ and κ ∞ , p=1, 2, 4, j=1, 2, are the generalized mid-plane strains and and εpj pj ∞ and M ∞ . curvatures at infinity, which are induced by the application of Npj pj Note that the general solution (13.145) which satisfies all the basic equations for the electro-elastic laminates has been purposely organized to have the same form as that of Stroh formalism for two-dimensional problems (3.24), and Stroh-like bending formalism for pure bending problems (12.39), and coupled stretching– bending problems (13.70). The only difference is that the symbols like u, φ, A, B, . . . have different dimensions and different contents for different types of problems. If both of general solution and boundary condition keep the same form for different types of problems, their associated solutions should also have the same form. With this understanding, to find a solution from the general solution (13.145) satisfying the boundary conditions shown in (14.28), all we need to do is reproducing the corresponding solutions found in the literature, e.g., (6.8) for two-dimensional hole problems, or (12.64) for pure bending hole problems, or (14.10) for coupled stretching–bending hole problems. If one is interested in the comparison and discussions of these solutions, one can refer to Hwu (2003c). Thus, without any further derivation we can write down the solution for the present problem as

! , ∞ -" −1 −1 ∞ am , − Re A < ζ > B − ibm ud = u∞ α d 2 1 ! " , −1 −1 ∞ φd = φ∞ am∞ , d − Re B < ζα > B 2 − ibm1

(14.30)

where ζα , α = 1, 2, 3, are defined in (14.7). The angular bracket stands for the diagonal matrix whose components vary according to the subscripts α = 1, 2, 3, 4, 5, 6, i.e., < fα >= diag[f1 , f2 , f3 , f4 , f5 , f6 ]. The real-form solutions for the generalized stress resultants and moments around the hole boundary are Dn = 0, Nn = Nns = Mn = / (14.31a) 1 Ds = −i∗T Ns = −sT1 φd, n , Ms = −sT2 φd, n , Mns = − nT2 φd, n , / 3 φd, n , 2

14.4

Green’s Functions for Laminates

505

where T 0), sT2 = (0 sT ), nT2 = (0 nT ), i∗T 3 = (i3 " ! a ∞ = sin θ G1 (θ )m∞ 1 − I + G3 (θ ) m2 b b ∞ , − cos θ I + G3 (θ ) m∞ + G (θ )m 1 1 2 a

sT1 = (sT φd, n

0), (14.31b)

and G1 (θ ) and G3 (θ ) are defined in (14.11b), and sT = (cos θ sin θ 0), nT = (− sin θ cos θ 0), iT3 = (0 0 1).

(14.31c)

Note that the values of stress resultants and bending moments Nij , Mij associated with the s–n and x1 –x2 coordinates are related by the second-order transformation laws, while those of the resultant electric displacements / Di (defined in (13.131)) are related by the first-order transformation laws. Although the explicit real-form solutions for the generalized stress resultants have been obtained in (14.31), usually in stress analysis one is also interested in the internal stresses at every point of the laminates. To know the internal stresses we first calculate the strains by (13.129a)2 in which the generalized mid-plane strains 0 and curvatures κ can be obtained by differentiating the solution u shown in εpj pj d (14.30)1 with respect to x1 and x2 . With the results of strains, the stresses of each lamina are calculated by using the constitutive relation (11.7a)1 for generalized plane strain and short circuit condition, and (11.8)1 , (11.9)1, and (11.10)1 for the other two-dimensional conditions. According to the solutions presented in (14.30) and (14.31), several numerical examples have been done in Hwu and Hsieh (2005) to study the hole effects on the electro-elastic laminates.

14.4 Green’s Functions for Laminates On account of the linear character of the related equations, the principle of superposition is applicable to most of the fundamental problems of elasticity. Thus, the solutions associated with the concentrated forces and moments applied on any arbitrary points, generally called Green’s functions, become important in constructing general solutions through superposition. Many analytical closed-form solutions of Green’s functions have been obtained for several different problems such as the twodimensional infinite spaces, half-spaces, bimaterials, and the infinite spaces with the presence of cracks, holes, or inclusions discussed in Chapters 4 and 6, 7, and 8. In this section, problems with stretching–bending coupling will be discussed for infinite laminates by following the work presented in Hwu (2004), and Green’s functions for laminates with holes/cracks/inclusions will be discussed in the next two sections by following the works presented in Hwu (2005) and Hwu and Tan (2007).

506

14


Consider an infinite laminate subjected to a concentrated force ˆf = (fˆ1 , fˆ2 , fˆ3 ) and ˆ = (m ˆ 2, m ˆ 3 ) at point xˆ = (ˆx1 , xˆ 2 ) (see Fig. 13.1). Using the relamoment m ˆ 1, m tions in (13.86) for the resultant forces/moments, the force and moment equilibrium conditions can be written as . C

. C

.

dφ1 = fˆ1 ,

.

dφ2 = fˆ2 ,

C

.

dη = fˆ3 ,

C

dψ1 = m ˆ 2 + (x1 − xˆ 1 )fˆ3 ,

.

dψ2 = −m ˆ 1 + (x2 − xˆ 2 )fˆ3 ,

(14.32)

C

d((x1 − xˆ 1 )φ2 − (x2 − xˆ 2 )φ1 −

)=m ˆ 3,

C

where C is any closed contour enclosing the point xˆ anticlockwise. To find a solution satisfying the conditions (14.32), the choice of fk (zk ) in the general solutions (13.70b) is very critical in the solution procedures. Equations (14.32) show that the stress functions φ1 , φ2 , ψ1 , and ψ2 should be multi-valued ˆ 1, m ˆ 2 are applied, whereas functions if the concentrated forces fˆ1 , fˆ2 and moments m η(=ψk, k /2) and (= φ2 dx1 ) should be multi-valued functions if the concentrated forces fˆ3 and moments m ˆ 3 are applied. However, no matter which kind of force conditions are considered, the physical quantities such as the displacements and slopes should always be single-valued to confirm that the laminates will not break off when deformed. Since φi (or ψi ), η(= ψk, k /2), and (= φ2 dx1 ) stand for three different function status, in the following we like to separate the force conditions into three ˆ 1, m ˆ 2 ; (2) fˆ3 ; (3) m ˆ 3. cases, i.e., (1) fˆ1 , fˆ2 , m

14.4.1 Concentrated In-Plane Forces and Out-of-Plane Moments ˆ 1, m ˆ 2) (fˆ1 , fˆ2 , m In this case, the stress functions φ1 , φ2 , ψ1 , and ψ2 should be multi-valued. From the general solution shown in (13.70) we see that to get multi-valued functions for φi and ψi , we need to choose a multi-valued function for fα (zα ). It is known that ln(zα − zˆα ) is holomorphic in the cut region and in an anticlockwise contour its value is increased by 2π i. Hence, it is proper for us to choose f(z) as f(z) =< ln(zα − zˆα ) > q1 ,

(14.33)

where q1 is a 4×1 unknown complex coefficient vector to be determined through the satisfaction of the boundary conditions. The integral of the differential of f(z) can then be evaluated as

14.4


507

. df(z) = 2π iq1 .

(14.34)

C

By combining the four equations given in the first, second, fourth, and fifth equations of (14.32) and the single-valued displacement conditions, the boundary conditions for this case can be written as . . ˆ dφd = p, dud = 0, (14.35a) C

C

where pˆ = (fˆ1 fˆ2 m ˆ2 −m ˆ 1 )T .

(14.35b)

With the general solution and the function vector given in (13.70) and (14.33), and the results of (14.34), the boundary conditions (14.35) can be evaluated as ! " ˆ 2π i Bq1 − Bq1 = p, ! " 2π i Aq1 − Aq1 = 0,

(14.36a)

where the overbar denotes the conjugate of a complex number. Equation (14.36a) can also be reorganized as

AA BB

q1 −q1

=

0 . ˆ p/2π i

(14.36b)

Because the general solution (13.70) and the eigenrelation (13.72) of Stroh-like formalism have been purposely arranged into the form of Stroh formalism for two-dimensional problems, all the identities developed for two-dimensional problems can be employed in Stroh-like formalism. With this understanding, use of the orthogonality relation (3.57a) will lead (14.36b) to ˆ i. q1 = AT p/2π

(14.37)

Combining the results of (13.70), (14.33), and (14.37), the explicit closed-form solution for an infinite laminate subjected to concentrated forces fˆ1 , fˆ2 and moments m ˆ 1, m ˆ 2 at xˆ = (ˆx1 , xˆ 2 ) can then be written as ˆ , ud = Im A < ln(zα − zˆα ) > AT p/π T ˆ . φd = Im B < ln(zα − zˆα ) > A p/π

(14.38)

Moreover, by using the identity given in (3.128), the complex form solution (14.38) can be converted into a real form as

508

14

1 2π 1 φd = − 2π

ud = −


% & ˆ (ln r)H + π / N1 (θ )H + / N2 (θ )ST p,

% & ˆ (ln r) S + π / N3 (θ )H + / NT1 (θ )ST p.

(14.39)

T

In the above the detailed definition of the real matrices / Ni (θ ), S, and H can be found in Chapter 3. It should be noted that the solution forms obtained in (14.38) and (14.39) are exactly the same as those for two-dimensional problems, i.e., (4.18) and (4.19). Actually, this is what we expect because the general solution (13.70) and the boundary conditions (14.35) have been purposely organized into exactly the same forms as those for the two-dimensional problems.

14.4.2 Concentrated Transverse Force (fˆ3 ) From the third, fourth, and fifth equations of (14.32) we see that in this case η(= ψk, k /2) should be a multi-valued function. Since ψk is proportional to fα (zα ), to have a multi-valued ψk, k we may choose f(z) as f(z) =
q2 =< (zα −ˆzα ) ln(zα −ˆzα )−(zα −ˆzα ) > q2 . (14.40)

With this choice, the first derivatives of the displacements and slopes become multivalued. Hence, the unknown coefficient vector q2 should be determined not only by satisfying (14.32)3,4,5 but also by the single-valued requirement of the displacement’s first derivatives. The boundary conditions for this case can therefore be written as . C

.

dη = fˆ3 ,

. C

dψ1 = (x1 − xˆ 1 )fˆ3 ,

C

dψ2 = (x2 − xˆ 2 )fˆ3 ,

C

.

dud,1 = 0,

.

(14.41)

dud,2 = 0. C

It should be noted that the conditions written in (14.41) are not all independent. For example, by the second equation of (13.70c) we have iT1 β, 2 = iT2 β, 1 .

(14.42)

Moreover, the satisfaction of the first equation of (14.41) will make its second and third equations be satisfied automatically. With this understanding, use of the general solution (13.70) and the function vector given in (14.40) will lead the boundary conditions (14.41) to

14.4


509

, Re 2π i iT3 Bq2 + iT4 B < μα > q2 = fˆ3 , 2Re{2π iAq2 } = 0, 2Re {2π iA < μα > q2 } = 0.

(14.43)

Because of relation (14.42), the fourth component of (14.43)2 and the third component of (14.43)3 are dependent. Thus, we may add (14.43)1 to the third component of (14.43)3 to form a system of eight linear algebraic equations for the four complex numbers contained in the 4×1 unknown complex coefficient vector q2 . By this way, the 8×8 system of equations can be written as

A A < μα > +I33 B + I34 B < μα > 0 , = ˆ f3 i3 /π i

A q2 A < μα > +I33 B + I34 B < μα > −q2

(14.44a) where ⎡

I33

⎤ 0000 ⎢0 0 0 0⎥ ⎥ =⎢ ⎣0 0 1 0⎦ , 0000

⎡

I34

0 ⎢0 =⎢ ⎣0 0

0 0 0 0

0 0 0 0

⎤ 0 0⎥ ⎥, 1⎦ 0

⎧ ⎫ 0⎪ ⎪ ⎪ ⎨ ⎪ ⎬ 0 . i3 = ⎪ ⎪1⎪ ⎪ ⎩ ⎭ 0

(14.44b)

Since q2 is a complex vector, for the convenience of derivation it may be replaced by two real vectors g and h as ' T T( B A ig q2 = . T T −q2 ih B A

(14.45)

Substituting (14.45) into (14.44a) and using the identities given in (3.124), we obtain g = 0, (N2 + I33 + I34 NT1 )h = −fˆ3 i3 /π .

(14.46)

By using the explicit expressions of Ni given in (13.88) and knowing that the submatrix of N2 by deleting the third row and third column is nonsingular, we can find that h = −fˆ3 i3 /2π ,

or

q2 = fˆ3 AT i3 /2π i.

(14.47)

With the result of (14.47), by using identity (3.122) and the explicit expression given in (13.88) we can prove that the conditions required in the second and third equations of (14.41) are satisfied automatically. Combining the results of (13.70), (14.40), and (14.47)2 , the explicit closed-form solution for an infinite laminate subjected to the concentrated force fˆ3 can now be written as

510

14


ud = Im A < (zα − zˆα ) ln(zα − zˆα ) − (zα − zˆα ) > AT φd = Im B < (zα − zˆα ) ln(zα − zˆα ) − (zα − zˆα ) > AT

fˆ3 i3 /π , fˆ3 i3 /π .

(14.48)

Use of the identity given in (3.122) will then convert (14.48) into the following real form: " r ! ˆ ∗ (θ ) + π N ˜ ∗ (θ ) + N ˆ 2 (θ )N ˜ ∗T (θ ) fˆ3 i3 , ˆ 1 (θ )N (ln r − 1)N ud = − 2 2 1 2π ! " (14.49a) r ˆ ∗T (θ ) + π N ˜ ∗ (θ ) + N ˆ T (θ )N ˜ ∗T (θ ) fˆ3 i3 , ˆ 3 (θ )N (ln r − 1)N φd = − 1 2 1 1 2π where ˜ 1 (θ )H + N ˜ 2 (θ )ST , ˜ ∗ (θ ) = N N 2

˜ ∗T (θ ) = N ˜ 3 (θ )H + N ˜ T (θ )ST , N 1 1

ˆ ∗ (θ ) = N ˆ 1 (θ )H + N ˆ 2 (θ )ST , N 2

ˆ ∗T (θ ) = N ˆ 3 (θ )H + N ˆ T (θ )ST . N 1 1

(14.49b)

14.4.3 Concentrated In-Plane Torsion (m ˆ 3) When only the concentrated moment m ˆ 3 is applied, the last equation of (14.32) shows that (= φ2 dx1 ) should be a multi-valued function. Since φ2 is pro portional to fα (zα ), to have a multi-valued φ2 dx1 we may choose f(z) as f(z) =
q3 =< > q3 . dzα zα − zˆα

(14.50)

With this choice, the requirement of single-valued displacement ud will be satisfied automatically. However, the integration of β will let the transverse deflection w become multi-valued and hence should be requested in the boundary conditions. Moreover, by applying the concentrated moment m ˆ 3 , only pure shear stress will be induced on the laminates, i.e., Nns = 0, Nn = Mn = Vn = 0. With this understanding, by use of (13.79), (13.80), and (13.81) and the last equation of (14.32), the boundary conditions for this case can be written as .

. d((x1 − xˆ 1 )φ2 − (x2 − xˆ 2 )φ1 − C T

)=m ˆ 3,

dw = 0, C

n φ, s = n ψ, s = (s ψ, s ), s = 0, T

T

(14.51)

along any arbitray surface boundary.

Note that in (14.51), (x1 − xˆ 1 )φ2 − (x2 − xˆ 2 )φ1 is a single-valued function when we choose f(z) in the form of (14.50). Moreover, the differentiation with respect to s can be dropped when all the points along the surface boundary are required to be satisfied. Thus, (14.51) can be further simplified as

14.4


. C T

.

dφ˜ 2 = −m ˆ 3,

511

dβ˜1 = 0, (14.52a)

C

n φ = n ψ = s ψ, s = 0, along any arbitray surface boundary. T

T

In getting the above equations, the relations given in the second equation of (13.70c) and (13.86b) have been used and φ˜ i and β˜i are defined by φ˜ i =

φi dzα ,

β˜i =

βi dzα ,

i = 1, 2.

(14.52b)

To employ the pure shear condition described in the last equation of (14.52a), we choose two particular boundaries. One is a line parallel to x1 -axis and passing through point xˆ , and the other is a line parallel to x2 -axis and passing through point xˆ . With this choice, use of the general solution (13.70) and the function vector given in (14.50) will lead the boundary conditions (14.52) to ˆ 3 , 2Re 2π iiT3 Aq3 = 0, 2Re 2π iiT2 Bq3 = −m 2Re iT2 Bq3 = 0, 2Re iT4 Bq3 = 0, 2Re iT3 Bq3 = 0, ! " ! " −1 T 2Re iT1 B < μ−1 k > q3 = 0, 2Re i3 B < μk > q3 = 0, ! " 2Re iT4 B < μ−1 = 0. > q 3 k

(14.53)

In order to utilize the benefit of the identities provided in Stroh formalism, we reorganize (14.53) into

(I12 + I−1 )B (I12 − I−1 )B −1 > I A I23 A + I−2 B < μ−1 23 − I−2 B < μα > α

im ˆ 3 i1 /2π q3 = , −q3 0 (14.54a)

where ⎡

⎡ ⎤ ⎤ 0100 0000 ⎢0 0 0 0⎥ ⎢0 0 1 0⎥ ⎢ ⎥ ⎥ I12 = ⎢ ⎣0 0 0 0⎦ , I23 = ⎣0 0 0 0⎦ , 0000 0000 ⎧ ⎫ ⎡ ⎡ ⎤ ⎤ 1⎪ 0000 1000 ⎪ ⎪ ⎨ ⎪ ⎬ ⎢0 1 0 0⎥ ⎢0 0 0 0⎥ ⎥ , I−2 = ⎢ ⎥ , i1 = 0 . I−1 = ⎢ ⎣0 0 1 0⎦ ⎣0 0 1 0⎦ 0⎪ ⎪ ⎪ ⎩ ⎪ ⎭ 0 0001 0001

(14.54b)

By employing the procedure described between (14.45) and (14.47) into (14.54), we can obtain q3 = −m ˆ 3 AT i2 /2π i.

(14.55)

512

14


Combining the results of (13.70), (14.50), and (14.55), the explicit closed-form solution for an infinite laminate subjected to the concentrated moment m ˆ 3 can then be written as " ! ˆ 3 i2 /π , ud = −Im A < (zα − zˆα )−1 > AT m " ! (14.56) φd = −Im B < (zα − zˆα )−1 > AT m ˆ 3 i2 /π . Use of the identity given in (3.122) will then convert (14.56) into the following real form: " 1 !˘ ˘ 2 (θ )ST m ˆ 3 i2 , N1 (θ )H + N 2π r " 1 !˘ ˘ T (θ )ST m ˆ 3 i2 . φd = N3 (θ )H + N 1 2π r ud =

(14.57)

˘ i (θ ), i = 1, 2, 3 are the submatrices of N ˆ −1 (θ ) which is Note that in the above N ˆ ˆ the inverse of N(θ ) and N(θ ) = cos θ I + sin θ N.

14.4.4 Explicit Real-Form Solutions Combining the solutions obtained in (14.38), (14.48) and (14.56), the Green’s function for an infinite laminate subjected to a concentrated force ˆf = (fˆ1 , fˆ2 , fˆ3 ) and ˆ = (m ˆ 2, m ˆ 3 ) at point xˆ = (ˆx1 , xˆ 2 ) can be written in complex matrix moment m ˆ 1, m form as ud = 2 Re {Af(z)},

φd = 2 Re {Bf(z)},

(14.58a)

where f(z) =

1 fˆ3 < ln(zα − zˆα ) > AT pˆ + < (zα − zˆα )[ln(zα − zˆα ) − 1] > AT i3 2π i 2π i 1 m ˆ3 < + > AT i2 2π i zα − zˆα (14.58b)

and ⎧ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎫ fˆ1 ⎪ 1⎪ 0⎪ 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ˆ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎪ ⎬ 0 1 0 f 2 , i2 = , i3 = . , i1 = pˆ = 0⎪ 0⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ m ˆ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 ⎪ ⎩ ⎪ ⎭ ⎩ ⎪ ⎭ ⎩ ⎪ ⎭ ⎭ 0 0 0 −m ˆ1

(14.58c)

Furthermore, by combining the solutions obtained in (14.39), (14.49), and (14.57), the real matrix form solution of Green’s function can be rewritten as

14.4


513

" 1 rfˆ3 ! ˆ ∗ (θ ) i ˆ ∗ (θ ) + π / N (ln r)H + π / N∗2 (θ ) pˆ − (ln r − 1)N 3 2 2 2π 2π m ˆ3 ˘∗ + N (θ )i2 , 2π r 2 ∗T 1 rfˆ3 ˆ T ∗T ∗T ˆ / / (ln r)S + π N1 (θ ) pˆ − (ln r − 1)N1 (θ ) + π N1 (θ ) i3 φd = − 2π 2π m ˆ 3 ˘ ∗T + N (θ )i2 , 2π r 1 (14.59a) where ud = −

˜ 3 (θ )H + N ˜ T (θ )ST , N ˜ ∗ (θ ) = N ˜ 1 (θ )H + N ˜ 2 (θ )ST , ˜ ∗T (θ ) = N N 1 1 2 ˆ ∗T (θ ) = N ˆ 3 (θ )H + N ˆ T (θ )ST , N ˆ ∗ (θ ) = N ˆ 1 (θ )H + N ˆ 2 (θ )ST , N 1 1 2 ˘ ∗T (θ ) = N ˘ 3 (θ )H + N ˘ T (θ )ST , N ˘ ∗ (θ ) = N ˘ 1 (θ )H + N ˘ 2 (θ )ST , N 1 1 2 ∗ ˆ˜ ∗T (θ ) = N ˆ 3 (θ )N ˜ ∗ (θ ) + N ˆ T (θ )N ˜ ∗T (θ ), N ˆ 1 (θ )N ˜ ∗ (θ ) + N ˆ 2 (θ )N ˜ ∗T (θ ) ˜ˆ 2 (θ ) = N N 1 2 1 1 2 1 (14.59b) and

θ θ θ ˜ 2 (θ ) = 1 ˜ 3 (θ) = 1 ˜ 1 (θ ) = 1 N1 (ω)dω, N N2 (ω)dω, N N3 (ω)dω, N π 0 π 0 π 0 ˆ 2 (θ ) = sin θN1 , N ˆ 3 (θ ) = sin θN3 , ˆ 1 (θ ) = cos θ I + sin θ N1 , N N ˘ 2 (θ ) = − sin θ N2 (θ ), N ˘ 3 (θ) = cos θI − sin θNT (θ). ˘ 1 (θ ) = cos θ I − sin θ N1 (θ ), N N 1 (14.59c)

ˆ i (θ ) and N ˘ i (θ ), i = 1, 2, 3, are submatrices of / ˆ ) and N(θ ˘ ) which ˜ i (θ ), N N(θ ), N(θ N are integral, z-associate, and inversion of the fundamental elasticity matrix N defined by θ ˆ ) = cos θ I + sin θ N, ˜ )= 1 N(ω)dω, N(θ N(θ π 0 ˘ )=N ˆ −1 (θ ) = (cos θ I + sin θ N)−1 = cos θ I − sin θ N(θ ). N(θ

(14.60)

The second equality of the last equation of (14.60) has been shown in (3.96b)4 . Isotropic Plates Substituting the results obtained in (13.93), (13.107), and (13.113) for isotropic plates into (14.59a), (14.59b), and (14.59c), and using the relations given in (13.83), we can obtain the following explicit real-form solutions:

514

14


1+ν ! 2(3 − ν)fˆ1 ln r − (1 + ν) fˆ1 (cos 2θ − 1) + fˆ2 sin 2θ 8π Eh 2(1 + ν) m ˆ 3 sin θ cos 2θ , − r (14.61a) 1+ν ! ˆ ˆ ˆ 2(3 − ν)f2 ln r − (1 + ν) f1 sin 2θ − f2 (cos 2θ − 1) u2 = − 8π Eh 2 ˆ 3 cos θ [4 − (1 + ν) cos 2θ ] , − m r " 1 ! β1 = − m ˆ 2 (2 ln r + cos 2θ − 1) − m ˆ 1 sin 2θ + 2fˆ3 r(ln r − 1) cos θ , 8π D " 1 ! m ˆ 1 (−2 ln r + cos 2θ − 1) + m ˆ 2 sin 2θ + 2fˆ3 r ln r sin θ , β2 = − 8π D (14.61b) u1 = −

w=

" 1 ! 4m ˆ 2 r(ln r − 1) cos θ − 4m ˆ 1 r ln r sin θ + fˆ3 r2 [2(ln r − 1) − cos 2θ ] , 16π D (14.61c) " 1 − ν !ˆ f1 cos θ + fˆ2 sin θ , 4π r " 3 + ν !ˆ m ˆ3 f1 cos θ + fˆ2 sin θ − Nr = − (1 + ν) sin 2θ , 4π r 2π r2 ! " 1−ν ˆ m ˆ3 {2 − (1 + ν) cos 2θ } , f1 sin θ − fˆ2 cos θ − Nrθ = 4π r 4π r2 Nθ =

(14.61d)

" 1+ν fˆ3 ! m ˆ 1 sin θ − m (1 + ν) ln r − (1 − ν) sin2 θ , ˆ 2 cos θ − 4π r 4π " 1+ν fˆ3 ! m ˆ 1 sin θ − m (1 + ν) ln r + (1 − ν) sin2 θ , Mr = ˆ 2 cos θ − 4π r 4π 1−ν fˆ3 Mrθ = m ˆ 1 cos θ + m (1 − ν) sin 2θ , ˆ 2 sin θ − 4π r 8π (14.61e) Mθ =

1 m ˆ 1 cos θ + m ˆ 2 sin θ , 2π r2 −1 fˆ3 , m ˆ sin θ − m ˆ cos θ − Qr = 1 2 2π r 2π r2 Qθ =

1+ν m ˆ 1 cos θ + m ˆ 2 sin θ , 4π r2 3−ν fˆ3 {2 + (1 − ν) cos 2θ } . m ˆ 1 sin θ − m ˆ 2 cos θ − Vr = − 2 4π r 4π r

(14.61f)

Vθ =

(14.61g)

The generalized displacement vector ud defined in (13.70b) and (13.70c) contains only the in-plane displacements u1 , u2 and slopes β1 , β2 . The deflection w should be calculated through the integration of β1 and β2 . However, since the

14.5

Green’s Functions for Laminates with Holes/Cracks

515

explicit solutions shown in (14.61) are written in terms of polar coordinates r and θ , it is more convenient to use βr and βθ instead of β1 and β2 . They are related by βθ = −

∂w ∂w = −β1 sin θ + β2 cos θ , βr = − = β1 cos θ + β2 sin θ . (14.62) r∂θ ∂r

The deflection w can, therefore, be calculated through the integration of slopes βr and βθ , i.e., w=−

βr dr = −r

βθ dθ .

(14.63)

During integration, both the first and second equality of (14.63) are needed because both of them will leave the unknown integration functions such as f (θ ) for the first equality and g(r) for the second equality. These two integration constant functions should be determined through the equality relation. For the readers’ reference, the explicit expressions of βr and βθ are shown in the following: " 1 ! 2(m ˆ 1 cos θ + m ˆ 2 sin θ ) ln r − 2m ˆ 2 sin θ − fˆ3 r sin 2θ , 8π D " 1 ! 2(m ˆ 1 sin θ − m ˆ 2 cos θ ) ln r + 2m ˆ 1 sin θ − fˆ3 r(2 ln r − 1 − cos 2θ ) . βr = 8π D (14.64)

βθ =

Note that the Green’s functions shown above may differ from those shown in the literature, e.g., Stern (1979) and Hartmann and Zotemantel (1986), by nonlogarithmic terms, which will not influence the boundary conditions set for the Green’s function and therefore can be treated as one of the Green’s functions.

14.5 Green’s Functions for Laminates with Holes/Cracks In this section we will employ the method of analytical continuation to find Green’s functions for hole/crack problems. By this method, we need to know the unperturbed solutions f0 for non-hole problems which are the solutions we get in Section 14.4. With this concern, in this section we first rewrite the solutions obtained in (14.38), (14.48), and (14.56) as follows: Case 1 : f0 (z) =< ln(zα − zˆα ) > q1 , Case 2 : f0 (z) =< (zα − zˆα )[ln(zα − zˆα ) − 1] > q2 , 1 Case 3 : f0 (z) =< > q3 , zα − zˆα

(14.65a)

where q1 =

1 T fˆ3 T m ˆ3 T ˆ q2 = A p, A i3 , q3 = A i2 . 2π i 2π i 2π i

(14.65b)

516

14


xˆ

2b

n

θ

fˆ1 mˆ 1

fˆ2

fˆ3

mˆ 2

mˆ 3

s

x1

2a

x2

x3

Fig. 14.3 An elliptic hole in laminates subjected to concentrated forces and moments

Consider an infinite composite laminate containing an elliptical hole under a concentrated force and moment at point xˆ (Fig. 14.3). The contour of the hole boundary is represented by (14.1). The force equilibrium and single-valued requirement of this problem are the same as those shown in (14.35), (14.41), (14.42), and (14.51) for each different loading case. If the hole is assumed to be traction free, the additional boundary conditions are Nn = Nns = Mn = Vn = 0, along the hole boundary.

(14.66)

From (13.82b), we know that the traction-free boundary conditions (14.66) can be rewritten in terms of the augmented stress function vector φd as φd = 0,


(14.67)

14.5.1 Field Solutions Since the elliptical hole boundary in the z-plane will map to four different slanted elliptical hole boundary in the zα -plane, it is not convenient to solve problems with elliptical boundary by using the argument zα defined in (13.70b). Therefore, as discussed in Chapter 6, to treat the problems with elliptical boundary, most of the solutions shown in the literature are expressed in terms of the transformed complex variable ζα , which can transform all four different slanted elliptical hole boundary into the same hole boundary in the shape of a unit circle |ζ | = 1. The relation between zα and ζα is given in (14.7), and ζα = cos ϕ + i sin ϕ = eiϕ = σ along the hole boundary. Using the method of analytical continuation and understanding that the unknown complex function vector f(z) is better expressed in terms of the arguments ζα , the

14.5


517

general solution (13.70) for the present problem can now be written as ud = 2Re A[f0 (ζ ) + fp (ζ )] ,

φd = 2Re B[f0 (ζ ) + fp (ζ )] ,

(14.68)

where f0 is the function associated with the unperturbed elastic field and fp is the holomorphic function corresponding to the perturbed field of the problem and will be determined through satisfaction of the boundary conditions. To solve fp , we first need to have a proper choice for f0 . If some parts of f0 are holomorphic outside the hole (S+ ) while others are holomorphic inside the hole (S− ), we may split f0 into − two functions f+ 0 and f0 , i.e., f0 (ζ ) = f0+ (ζ ) + f0− (ζ ),

(14.69)

− where f0+ is holomorphic in S + and f− 0 is holomorphic in S . By employing general solution (14.68) and (14.69), the traction-free boundary condition (14.67) now becomes

& % B f0+ (σ ) + f0− (σ ) + fp (σ ) + B f0+ (σ ) + f0− (σ ) + fp (σ ) = 0,

(14.70)

which can also be written as Bf0+ (σ ) + Bfp (σ ) + Bf0− (σ ) = −Bf0− (σ ) − Bf0+ (σ ) − Bfp (σ ).

(14.71)

One of the important properties of holomorphic functions used in the analytical continuation method is that if f (ζ ) is holomorphic outside the unit circle S+ then f (1/ζ ) will be holomorphic inside the unit circle S− and vice versa. With this knowledge, we now rewrite (14.71) as θ(σ + ) = θ(σ − ),

(14.72a)

where θ(ζ ) =

⎧ ⎨ Bf + (ζ ) + Bfp (ζ ) + Bf − (1/ζ ), ζ ∈ S+ , 0 0 ⎩−Bf − (ζ ) − Bf + (1/ζ ) − Bf (1/ζ ), ζ ∈ S− . p 0 0

(14.72b)

By the holomorphic conditions discussed before this equation, we conclude that this newly defined function θ(ζ ) will be holomorphic in S+ and S− and is continuous across the unit circle. This means that θ (ζ ) is holomorphic in the whole ζ -plane including the points at infinity. By Liouville’s theorem we have θ(ζ ) ≡ constant. However, the constant function corresponds to rigid body motion which may be neglected. Therefore, θ (ζ ) ≡ 0. With this result, (14.72b)1 leads to fp (ζ ) = −f0+ (ζ ) − B−1 Bf0− (1/ζ ).

(14.73)

518

14


Note that when employing the method of analytical continuation the subscript α of ζα has been dropped in (14.73) and a replacement of ζ1 , ζ2 , ζ3 , ζ4 should be made for each component function of fp (ζ ) after the multiplication of matrices. The problem now becomes how to select an appropriate unperturbed solution − f0 and split it into f+ 0 and f0 , and then use (14.73) to get fp . Following are the discussions based upon the Green’s functions of non-hole problems provided in (14.65), which are presented in three different loading cases. ˆ 1, m ˆ2 Case 1: fˆ1 , fˆ2 , m In order to save the effort of considering the force equilibrium and single-valued requirement caused by the concentrated forces and moments, it is appropriate to select f0 to be the solution for the non-hole problems, i.e., the solution given in (14.65a)1 . From (6.3)1 we know that ! " γα −1 −1 ˆ ˆ ˆ , (14.74a) zα − zˆα = cα ζα − ζα + γα (ζα − ζα ) = cα (ζα − ζα ) 1 − ζˆα ζα where cα =

1 a + ibμα (a − ibμα ), γα = . 2 a − ibμα

(14.74b)

In order to express f0 in terms of ζα , we substitute (14.74a) into (14.65a)1 and get γα f0 (ζ ) =< ln(ζα − ζˆα ) + ln 1 − + ln cα > q1 . ζˆα ζα

(14.75)

Knowing that |γα /(ζˆα ζα )| < 1 when ζα ∈ S+ and |ζα /ζˆα | < 1 when ζα ∈ S− , we may split f0 of (14.75) into the following two parts: γα + ln cα > q1 , f0+ (ζ ) =< ln 1 − ζˆα ζα

(14.76)

f0− (ζ ) =< ln(ζα − ζˆα ) > q1 .

Substituting (14.76) into (14.73) and understanding that the subscript of ζα should be dropped before the matrix multiplication, we get γα + ln cα > q1 − B−1 B < ln(ζ −1 − ζˆ α ) > q1 . (14.77) fp (ζ ) = − < ln 1 − ζˆα ζ Using the translating technique (4.50) and (4.51), the explicit full-field solution of fp (ζ ) can now be written as

14.5


# fp (ζ ) = − < ln 1 −

γα ζˆ α ζα

$ + ln cα > q1 −

4

519

< ln(ζα−1 − ζˆ k ) > B−1 BIk q1 .

k=1

(14.78)

Adding f0 and fp obtained in (14.75) and (14.78) together and using the first equation of (14.65b), we have f(ζ ) = f0 (ζ ) + fp (ζ ) 4 1 T ˆ = < ln ζα−1 − ζˆ k > B−1 BIk A p, < ln(ζα − ζˆα ) > AT + 2π i k=1 (14.79) which is exactly the same as that obtained in (6.61) for the Green’s function of hole in two-dimensional problems. In Hwu and Yen (1991) the solution was found by proper selection of the function form of f(ζ ), which is then improved by Ting (1996) using the concept of image singularities. The analytical continuation method was first introduced by Hwu and Yen (1993) for general elastic inclusion problems. In + Hwu and Yen (1993) f0 was selected to be f− 0 of (14.76), which means that f0 = 0. By that selection, fp will be different from the one obtained in (14.78), while the final result of f = f0 + fp still keeps the same. Note that the selections of f0 given in Hwu and Yen (1993) and (14.75) are different only by their arguments. One is < ln(ζα − ζˆα ) > q1 , and the other is < ln(zα − zˆα ) > q1 . It looks that f0 can be selected directly from the non-hole problems, i.e., (14.75), or just by replacing the argument of the non-hole problems from zα to ζα . However, in general the latter way may not be correct, which should depend on the satisfaction of the force equilibrium and single-valued requirement described in (14.35). Unlike (14.65a)1 , when zα is replaced by ζα the unperturbed solutions (14.65a)2,3 cannot satisfy their associated boundary conditions (14.41), (14.42), and (14.51). Therefore, in the following two cases one should be very careful about the selection of f0 . Case 2: fˆ3 As stated above f0 cannot be chosen to be the non-hole solution (14.65a)2 with zα replaced by ζα since it does not satisfy the boundary conditions given in (14.41) and (14.42). Similar to case 1, to save the effort of considering the force equilibrium and single-valued requirement (14.41) and (14.42), it is appropriate to select f0 directly from (14.65a)2 without making replacement. In order to express f0 in terms of ζα , we substitute (14.74a) into (14.65a)2 and get γα γα ˆ ˆ f0 (ζ ) =< cα (ζα − ζα ) 1 − ln(ζα − ζα ) + ln 1 − + ln cα − 1 > q2 . ζˆα ζα ζˆα ζα (14.80)

520

14


Knowing that

ln(ζα − ζˆα ) = ln(−ζˆα ) −

∞ 1 ζα k

k ζˆα k=1 ∞ γα γα k 1 ln 1 − =− , k ζˆα ζα ζˆα ζα k=1

,

0 0 0 ζα 0 for 00 00 < 1, ζˆα 0 0 0 γα 0 0 < 1, 0 for 0 0 ζˆα ζα

(14.81)

and carrying out the multiplication of (14.80) into series expansion and then check the holomorphic condition of each term, we may split f0 of (14.80) into two − + − + functions f+ 0 and f0 where f0 is holomorphic in S (outside the hole) and f0 is holomorphic in S− (inside the hole). They are γα γα ˆ ln 1 − > qc =< (ζα − ζα ) 1 − ζˆα ζα ζˆα ζα + < γα (ζα−1 − ζˆα−1 ) > q∗c + < (ζα−1 − ζˆα−1 ) > q∗∗ c , γ α ln(ζα − ζˆα ) > qc f0− (ζ ) =< (ζα − ζˆα ) 1 − ζˆα ζα + < (ζα − ζˆα ) > q∗c − < (ζα−1 − ζˆα−1 ) > q∗∗ c ,

f0+ (ζ )

(14.82a)

where ˆ qc =< cα > q2 , q∗c =< cα (ln cα − 1) > q2 , q∗∗ c =< cα γα ln(−ζα ) > q2 . (14.82b) Substituting (14.82a) into (14.73) and understanding that the subscript of ζα should be dropped before the matrix multiplication, we get γα γα ˆ ln 1 − > qc fp (ζ ) = − < (ζ − ζα ) 1 − ζˆα ζ ζˆα ζ − < γα (ζ −1 − ζˆα−1 ) > q∗c − < (ζ −1 − ζˆα−1 ) > q∗∗ c −1

(14.83)

− B−1 B < (ζ −1 − ζˆ α )(1 − γ α ζˆ α ζ ) ln(ζ −1 − ζˆ α ) > qc −1

− B−1 B < (ζ −1 − ζˆ α ) > q∗c + B−1 B < (ζ − ζˆ α ) > q∗∗ c . Using the translating technique (4.50) and (4.51), the explicit full-field solution of fp (ζ ) can then be written as

14.5


521

γα γα ˆ fp (ζ ) = − < (ζα − ζα ) 1 − ln 1 − > qc ζˆα ζα ζˆα ζα − < γα (ζα−1 − ζˆα−1 ) > q∗c − < (ζα−1 − ζˆα−1 ) > q∗∗ c −

4

−1

< (ζα−1 − ζˆ k )(1 − γ k ζˆ k ζα ) ln(ζα−1 − ζˆ k ) > B−1 BIk qc

k=1

−

4

< (ζα−1 − ζˆ k ) > B−1 BIk q∗c +

k=1

4

−1

< (ζα − ζˆ k ) > B−1 BIk q∗∗ c .

k=1

(14.84) Adding f0 and fp obtained in (14.80) and (14.84) together and using (14.74) we get f(ζ ) = f0 (ζ ) + fp (ζ ) =< (zα − zˆα ) ln(ζα − ζˆα ) > q2 −

4

−1

< (ζα−1 − ζˆ k )(1 − γ k ζˆ k ζα ) ln(ζα−1 − ζˆ k ) > B−1 BIk < cα > q2

k=1

+ < (ζα − ζˆα ) > q∗c −

4

< (ζα−1 − ζˆ k ) > B−1 BIk q∗c

k=1

− < (ζα−1 − ζˆα−1 ) > q∗∗ c +

4

−1

< (ζα − ζˆ k ) > B−1 BIk q∗∗ c .

k=1

(14.85) Case 3: m ˆ3 Similar to case 2, f0 is selected to be the unperturbed solution (14.65a)3 in which zα − zˆα is related to ζα and ζˆα by (14.74), and hence f0 (ζ ) =
q3 .

ζα − γα /ζˆα

(14.86)

Since ζˆα and γα /ζˆα are located in S+ and S− , respectively, f0 can be split into the following two parts, f0+ (ζ ) =
q∗∗ 3 , f0 (ζ ) =
q∗3 ,

(14.87a)

where q∗3 =
q3 , q∗∗ 3 =
q3 .

(14.87b)

522

14


Substituting (14.87) into (14.73) and using the translating technique (4.50) and (4.51), the full-field solution of fp (ζ ) can be obtained as fp (ζ ) =
q∗∗ 3 −

4

B−1 BIk q∗3 .

(14.88)

Adding f0 and fp obtained in (14.86) and (14.87) together we get f(ζ ) = f0 (ζ ) + fp (ζ ) =

q∗3

−

4 k=1

B−1 BIk q∗3 . (14.89)

14.5.2 Stress Resultants and Moments Along the Hole Boundary In engineering applications, one is usually interested in the stress resultants and moments along the hole boundary. Since the hole considered in this section is in the shape of ellipse, it is better to calculate the stress resultants and bending moments in the tangent–normal (s–n) coordinate instead of the Cartesian (x1 –x2 ) coordinate. By the relations given in (13.79), (13.80), and (13.81) we know that the calculation of stress resultants and moments replies upon the calculation of the differentials φd, s and φd, n . Moreover, due to the traction-free boundary condition prescribed in (14.67), along the hole boundary φd, s should be zero, which will then be used as a check of the solutions. From (13.70a)2 , we have φd, n = 2Re{Bf, n (ζ )},

(14.90)

in which each term of f, n (ζ ) can be obtained by using chain rule for differentiation, such as those shown in (6.14) and (6.15). Substituting the results of f(ζ ) obtained in (14.79), (14.85), and (14.89) for three different loading cases into (14.90) and carefully performing the differentiation and summation for each case, we get (Hwu, 2005) φd, n =

−2 G3 (θ )Im B < hi (ϕ) > AT pˆ i , i = 1, 2, 3, πρ

(14.91a)

where h1 (ϕ) = eiϕ (eiϕ − ζˆα )−1 , h2 (ϕ) = cα [eiϕ ln(cα (eiϕ − ζˆα )) − γα e−iϕ ln(1 − ζˆα−1 eiϕ ) − γα ζˆα−1 ], eiϕ ζˆα h3 (ϕ) = − cα (ζˆα − γα /ζˆα )(eiϕ − ζˆα )2

(14.91b)

14.5


523

and ˆ pˆ 1 = p,

pˆ 2 = fˆ3 i3 ,

pˆ 3 = m ˆ 3 i2 .

(14.91c)

The subscripts i = 1,2,3, denote the loading cases discussed in this section. Note that in deriving (14.91) an identity converting complex form into real form has been used, which is the generalization of (3.137)3 , i.e., B < μα (θ ) > B−1 = G1 (θ ) + iG3 (θ ),

(14.91d)

where G1 (θ ) and G3 (θ ) are two real matrices defined in (14.11b).

14.5.3 Verification and Discussions When we employed the general solution (13.70), all the basic equations for the laminates with stretching–bending coupling have been satisfied. Thus, for the purpose of verification all we need to do is checking the satisfaction of the boundary conditions described in (14.35), (14.41), (14.42), (14.51), and (14.67). By the method of analytical continuation, when we selected the non-hole solutions (14.65) as the unperturbed solutions f0 , the boundary conditions (14.35), (14.41), (14.42), and (14.51) have been satisfied. Moreover, the perturbed solutions fp obtained in (14.78), (14.84), and (14.88) are all holomorphic in the region outside the hole, and hence will make all the contour integrals shown in (14.35), (14.41), (14.42), and (14.51) vanish and let the boundary conditions (14.35), (14.41), (14.42), and (14.51) are satisfied not only by f0 but also by f0 + fp . After verifying the force equilibrium and single-valued requirement shown in (14.35), (14.41), (14.42), and (14.51), we now check the traction-free condition (14.67). Similar to the derivation shown in (14.90) and (14.91), the differentiation with respect to the tangential direction can be calculated by ieiϕ ∂f ∂f = , ∂s ρ ∂ζα


(14.92)

With this relation by following the steps described in (6.14) and (6.15) we get φd, s = 0,

along the hole boundary,

(14.93)

which shows that the traction-free boundary conditions are also satisfied by the solutions obtained in (14.79), (14.85), and (14.89).

524

14


14.5.4 Cracks An elliptic hole can be made into a crack of length 2a by letting the minor axis 2b be equal to zero. The Green’s functions for crack problems can therefore be obtained from (14.79), (14.85), and (14.89) by letting b = 0 and the results are Case 1 : f(ζ ) =< ln(ζα − ζˆα ) > q1 −

4

< ln(ζα−1 − ζˆ k ) > B−1 BIk q1 , (14.94a)

k=1

Case 2 : f(ζ ) =< (zα − zˆα ) ln(ζα − ζˆα ) > q2 4 −1 a −1 ˆ ˆ − < ζα − ζ k 1 − ζ k ζα ln ζα−1 − ζˆ k > B−1 BIk q2 2 k=1

+ < (ζα − ζˆα ) > q∗c −

4

< ζα−1 − ζˆ k > B−1 BIk q∗c

k=1

4 −1 −1 −1 ˆ > q∗∗ − < ζα − ζ α < (ζα − ζˆ k ) > B−1 BIk q∗∗ c + c , k=1

(14.94b) Case 3 : f(ζ ) =
q∗3 −

4 k=1

B−1 BIk q∗3 ,

(14.94c)

where 3 3 1 1 2 2 2 2 ˆ z α + z α − a , ζα = zˆα + zˆα − a ζα = a a

(14.94d)

and 1 T fˆ3 T ˆ q2 = A p, A i3 , q3 = 2π i 2π i a q∗c =< (ln(a/2) − 1) > q2 , q∗∗ c =< 2 2ζˆα q∗3 = q3 , a(ζˆα − ζˆα−1 ) q1 =

m ˆ3 T A i2 , 2π i a ln(−ζˆα ) > q2 , 2 (14.94e)

I1 = diag[1, 0, 0, 0], I2 = diag[0, 1, 0, 0], I3 = diag[0, 0, 1, 0], I4 = diag[0, 0, 0, 1]. Stress Intensity Factors With the complex function vectors obtained in (14.94), the field solutions of displacements and stresses for the cracks in laminates subjected to concentrated forces and moments can be calculated through the use of general solution (13.70) and

14.5


525

relations (13.71). To know the stresses ahead of the crack tip, we consider x2 = 0, |x1 | > a. When x1 is approaching to a, the stresses will be singular. To deal with the stress singularity, the stress intensity factors are usually defined as ⎧ ⎫ ⎫ ⎧ KII ⎪ N12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎬ ⎨ √ KI N22 = lim 2π r , k= M12 ⎪ ⎪ ⎪ ⎪KIIB ⎪ ⎪ r→0 ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ KIB M22

(14.95)

where r is the distance ahead of the crack tip. By using the relations given in (13.71), the definition (14.95) can now be rewritten in terms of the augmented stress function vector as k = lim

r→0

√

2π r(φd,1 − ηi3 ), η = (iT3 φd,1 + iT4 φd,2 )/2.

(14.96)

From this relation we know that to obtain the stress intensity factors we need to calculate φd, 1 and φd, 2 . By the approach similar to those described in Section 14.5.2 for the calculation of stress resultants and moments along the hole boundary, and letting x2 = 0, x1 > a, x1 − a = r, and r → 0, which will lead to ζα → 1, we obtain √ 2 2π rφd,1 = √ Im B < hi > AT pˆ i , r→0 πa √ 2 lim 2π rφd,2 = √ G1 Im B < hi > AT pˆ i , i = 1, 2, 3, r→0 πa

(14.97a)

ˆ pˆ 2 = fˆ3 i3 , pˆ 3 = m G1 = NT1 − N3 SL−1 , pˆ 1 = p, ˆ 3 i2 ,

(14.97b)

ˆ ) a 1 a(1 − ζ α h1 = (1 − ζˆα )−1 , h2 = − , ln 2 2(1 − ζˆα−1 ) ζˆα

(14.97c)

lim

where

and

h3 = −

2ζˆα

a(ζˆα − ζˆα−1 )(1 − ζˆα )2

.

For case 1, if the force pˆ is applied on the upper crack surface x1 = c where 0 < c < a, the solution (14.97a) can be further reduced to

526

14

lim

√

r→0

lim

√

r→0

2π rφd, 1 2π rφd, 2


2

a+c T ˆ I − S p, a−c 2 a+c 1 T ˆ = √ G1 I − S p. a−c 2 πa 1 = √ 2 πa

(14.98)

The modes I and II stress intensity factors, KI and KII , calculated from (14.98) are identical to the solutions obtained in (7.26).

14.6 Green’s Functions for Laminates with Elastic Inclusions Consider an infinite composite laminate containing an elliptical inclusion subjected ˆ = (m ˆ 1, m ˆ 2, m ˆ 3 ) at point xˆ = to a concentrated force ˆf = (fˆ1 , fˆ2 , fˆ3 ) and moment m (ˆx1 , xˆ 2 ) (Fig. 14.4). The contour of the inclusion boundary is represented by (14.1). The inclusion and the matrix (the laminates) are assumed to be perfectly bonded along the interface. The boundary conditions related to the concentrated forces and moments, i.e., the force equilibrium and single-valued requirement of this problem are the same as those shown in (14.35), (14.41), (14.42), and (14.51) for each different loading case. For the convenience of following discussions, these conditions are rewritten below by deleting some dependent and unnecessary relations. .

ˆ 1, m ˆ2 : Case 1. fˆ1 , fˆ2 , m

. ˆ dφd = p,

C

around the point xˆ . (14.99a)

dud = 0, C

fˆ1

xˆ

fˆ2 mˆ 2 n

2b

mˆ 1

fˆ3 mˆ 3

θ

s

x1

2a

x2 x3

Fig. 14.4 An elliptic inclusion in laminates subjected to in-plane/out-of-plane concentrated forces and moments

14.6

Green’s Functions for Laminates with Elastic Inclusions

Case 2. fˆ3 :

.

dη = fˆ3 ,

C

.

527

. dud,1 = 0,

C

dud,2 = 0,

around the point xˆ .

C

(14.99b) .

. Case 3. m ˆ3

= −m ˆ 3,

d C

dw = 0,

around the point xˆ ,

C

nT φ = nT ψ = sT ψ, s = 0,

along any arbitray surface boundary. (14.99c)

In (14.99a) and (14.99c), ˆ2 −m ˆ 1 )T , pˆ = (fˆ1 fˆ2 m

,1

= φ2 ,

,2

= −φ1 .

(14.99d)

Note that through the definitions of rotation angles βi and generalized displacement vector ud given in (13.70), we see that the conditions stated in (14.99b) are still not all independent. They are related by iT3 ud,2 = iT4 ud,1 ,

(14.99e)

where ik , k = 1, 2, 3, 4, are the unit base vectors defined as ⎧ ⎫ 1⎪ ⎪ ⎪ ⎨ ⎪ ⎬ 0 , i1 = 0⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0

⎧ ⎫ 0⎪ ⎪ ⎪ ⎨ ⎪ ⎬ 1 i2 = , 0⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0

⎧ ⎫ 0⎪ ⎪ ⎪ ⎨ ⎪ ⎬ 0 i3 = , 1⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0

⎧ ⎫ 0⎪ ⎪ ⎪ ⎨ ⎪ ⎬ 0 i4 = . 0⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 1

(14.99f)

If the inclusion and the matrix are assumed to be perfectly bonded along the interface, the displacements and surface tractions across the interface should be continuous and the associated boundary conditions can be written as (2) (1) (2) (1) (2) (1) (2) u(1) n = un , us = us , βn = βn , wn = wn , (1) (2) Nn(1) = Nn(2) , Nns = Nns , Mn(1) = Mn(2) , Vn(1) = Vn(2) ,

along the interface, (14.100a)

where the superscripts (1) and (2) denote, respectively, the quantities of the matrix and the inclusion. By relations (13.70), (13.79), (13.80), and (13.81) the displacement and traction continuity conditions can be rewritten as (1)

(2)

ud = ud ,

(1)

(2)

φd = φd ,

along the interface.

(14.100b)

To satisfy the boundary conditions stated in (14.99) and (14.100), by referring to the corresponding two-dimensional problems discussed in Sections 8.1 and 8.3, the general solution (13.70a) for the matrix and inclusion can now be written as

528

14 (1)


(1)

ud = 2Re {A1 [f0 (ζ ) + f1 (ζ )]} , (2) ud = 2Re A2 [f∗0 (ζ ∗ ) + f2 (ζ ∗ )] ,

φd = 2Re {B1 [f0 (ζ ) + f1 (ζ )]} , (2) φd = 2Re B2 [f∗0 (ζ ∗ ) + f2 (ζ ∗ )] , (14.101a)

where ζα =

zα +

+

z2α − a2 − μ2α b2 , a − iμα b

ζα∗

=

z∗α +

+ 2 ∗2 2 z∗2 α − a − μα b . a − iμ∗α b

(14.101b)

The subscripts 1 and 2 or the superscripts (1) and (2) denote, respectively, the matrix and the inclusion; μα and μ∗α are the material eigenvalues of the matrix and inclusion. f0 is a function associated with the unperturbed elastic field and is holomorphic in the entire domain except some singular points. f1 and f2 are functions corresponding to the perturbed field of the matrix and the inclusion, and are holomorphic in the regions of matrix (S1 ) and inclusion (S2 ), respectively (see Fig. 14.5). As described in Section 8.1 and shown in Fig. 14.5 that the transformation (14.101b) will map the points outside the elliptic inclusion into the points outside the unit circle in ζα -domain. Whereas the points inside the elliptic inclusion will be √ mapped into an annular ring of mα ≤ |ζα | ≤ 1 and is a one-to-one transformation only when the following restriction is satisfied f

+ + γα∗ σ = f γα∗ /σ ,

(14.102a)

where σ = eiϕ along the elliptical interface, and γα∗ =

a + ibμ∗α = mα e2iθα . a − ibμ∗α

(14.102b)

1

x2

S1 b

S1 S0

−1

a

−a

1

x1 mα

S2 −b (a)

Fig. 14.5 Mapping from (a) z-plane to (b) ζα -plane

S2

−1 (b)

14.6


529

Upon the requirement of (14.102), the function vector f2 (ζ ∗ ) which is holomorphic in S2 can be written as − ∗ ∗ f2 (ζ ∗ ) = f+ 2 (ζ ) + f2 (ζ ),

(14.103a)

− ∗ ∗ where f+ 2 (ζ ) and f2 (ζ ) are holomorphic in S1 and S2 + S0 , respectively, and are related by

+ ∗ ∗ ∗ f+ 2 (ζ ) = g2 ( γα /ζ ),

+ ∗ ∗ ∗ f− 2 (ζ ) = g2 (ζ / γα ),

(14.103b)

in which g2 is a function vector to be determined through the satisfaction of the boundary conditions. As discussed in Section 14.5.1, the function vectors f0 (ζ ) and f∗0 (ζ ∗ ) are usually chosen to be the one corresponding to the unperturbed elastic field, i.e., the one for the laminates without inclusions. According to the holomorphic characteristics, they may be split into three parts, i.e., f0 (ζ ) = f0+ (ζ ) + f0− (ζ ) + fs (ζ ),

(14.104a)

∗− ∗ ∗ ∗ ∗ f∗0 (ζ ∗ ) = f∗+ 0 (ζ ) + f0 (ζ ) + fs (ζ ),

(14.104b)

∗+ ∗ − ∗− ∗ where f+ 0 (ζ ) and f0 (ζ ) are holomorphic in S1 , f0 (ζ ) and f0 (ζ ) are holomorphic in S2 + S0 , and fs (ζ ) and f∗s (ζ ∗ ) are singular function vectors and are not holomorphic in either S1 or S2 + S0 . From some problems discussed in previous chapters, we know that the method of analytical continuation is a powerful method for finding solutions satisfying the boundary conditions. To employ this method, the following property is usually used, i.e., if f (ζ ) is holomorphic outside the unit circle S+ then f (1/ζ ) will be holomorphic inside the unit circle S− and vice versa. Using this property and the holomorphic conditions discussed following (14.101), we now list the holomorphic condition of each function vector in Table 14.1.

Table 14.1 Holomorphic condition of each function vector Region

Holomorphic function

S1

f1 (ζ )

S2 + S0 S2

f1 (1/ζ ) f2 (ζ ∗ )

∗ f+ 2 (ζ )

∗

∗

f+ 2 (1/ζ )

f− 2 (1/ζ )

∗ f− 2 (ζ )

f0+ (ζ )

f0+ (1/ζ )

f0− (1/ζ ) f0− (ζ )

∗ f∗+ 0 (ζ )

∗

∗

f∗+ 0 (1/ζ )

f∗− 0 (1/ζ )

∗ f∗− 0 (ζ )

14.6.1 Concentrated Forces/Moments Outside the Inclusions If the in-plane/out-of-plane concentrated forces and moments are located outside the inclusion,

530

14


f∗0 (ζ ∗ ) = 0,

(14.105)

and f0 (ζ ) can be selected to be the one associated with the solutions for the laminates without inclusions, i.e., (14.65), and will satisfy the boundary conditions shown in (14.99a) or (14.99b) or (14.99c) depending the loading type. The solutions shown in (14.65) are expressed in terms of zα instead of ζα , which are not convenient for considering the continuity conditions along the elliptic interface. Moreover, to employ the method of analytical continuation, it is necessary for us to split f0 (z) into three parts as those shown in (14.104a). This work has been done in the previous section when deriving the Green’s functions for holes/cracks in laminates. Thus, no detailed derivation will be presented here and only the results of f0− (ζ ) are shown below: Case 1. f0− (ζ ) = Case 2. f0− (ζ ) =

Case 3. f0− (ζ ) =

1 ˆ < ln(ζα − ζˆα ) > AT1 p, 2π i

(14.106a)

fˆ3 < (zα − zˆα ) ln(ζα − ζˆα ) 2π i + c2α (ζα − ζˆα ) − c3α (ζα−1 − ζˆα−1 ) > AT1 i3 ,

(14.106b)

c4α ζˆα m ˆ3 < > AT1 i2 , 2π i ζα − ζˆα

(14.106c)

where cα =

1 (a − ibμα ), 2

c3α = cα γα ln(−ζˆα ),

c2α = cα (ln cα − 1), c4α =

1 cα (ζˆα − γα /ζˆα )

(14.106d) .

Note that in the above fs (ζ ) and f0+ (ζ ) are not shown because no singular function vector remained for the present problem, i.e., fs (ζ ) = 0,

(14.107)

and f0+ (ζ ) has the same holomorphic condition as f1 (ζ )(see Table 14.1) and can be combined together, i.e., we may let ˘f1 (ζ ) = f1 (ζ ) + f + (ζ ). 0

(14.108)

Substituting (14.103a), (14.104a), (14.105), (14.107), and (14.108) into (14.101a), the generalized displacement and stress function vectors can now be rewritten as

14.6


531

ud = A1 [f0− (ζ ) + ˘f1 (ζ )] + A1 [f0− (ζ ) + ˘f1 (ζ )], (1)

(1) φd = B1 [f0− (ζ ) + ˘f1 (ζ )] + B1 [f0− (ζ ) + ˘f1 (ζ )], − ∗ + ∗ − ∗ ∗ ud = A2 [f+ 2 (ζ ) + f2 (ζ )] + A2 [f2 (ζ ) + f2 (ζ )], (2)

(14.109)

− ∗ + ∗ − ∗ ∗ φd = B2 [f+ 2 (ζ ) + f2 (ζ )] + B2 [f2 (ζ ) + f2 (ζ )]. (2)

Since f0 (z) shown in (14.65) have satisfied the boundary conditions for the concen∗ trated forces and moments, to determine the unknown function vectors ˘f1 (ζ ), f+ 2 (ζ ), − ∗ and f2 (ζ ) we consider the displacement and traction continuity conditions shown in (14.100b). Substituting (14.109) into (14.100b) with ζα = σ along the interface, and following the standard approach of analytical continuation with the knowledge of the holomorphic conditions shown in Table 14.1, we can obtain + − A1 ˘f1 (ζ ) + A1 f− 0 (1/ζ ) = A2 f2 (ζ ) + A2 f2 (1/ζ ) , − B1 ˘f1 (ζ ) + B1 f0− (1/ζ ) = B2 f+ 2 (ζ ) + B2 f2 (1/ζ ) + A1 f0− (ζ ) + A1 ˘f1 (1/ζ ) = A2 f− (ζ ) + A f (1/ζ ) 2 2 2 , B1 f − (ζ ) + B1 ˘f1 (1/ζ ) = B2 f− (ζ ) + B2 f+ (1/ζ ) 0

2

ζ ∈ S1 , (14.110) ζ ∈ S0 + S2 .

2

Note that in (14.110) the superscript∗ denoting the argument of the inclusion has been dropped, which is due to the use of analytical continuation. When employing the method of analytical continuation, the results come up from the continuation across the boundary in which all the arguments ζα , ζα∗ , α = 1, 2, 3, 4, have the same value σ = eiϕ . To make the final results have more flexibility to suit for the solution form shown in (13.70b), at this stage the superscript ∗ and/or the subscript α will be dropped, and a replacement of ζα or ζα∗ , α = 1, 2, 3, 4, should be made for each component function of f(ζ ) at the final stage. To get the complete full-field solution possessing the standard form shown in (13.70), a translating technique stated in (4.50) and (4.51) can be employed. Although the solutions of f− 0 (ζ ) have been given explicitly in (14.106), it is diffi− ∗ ∗ cult to get the explicit solutions of ˘f1 (ζ ), f+ 2 (ζ ), and f2 (ζ ) directly from relations (14.110). Instead they can be obtained in series form. With this consideration, by Taylor’s expansion f0− (ζ ) of (14.106) is now expressed in series form as f0− (ζ ) =

∞

k e− k ζ ,

(14.111a)

−1 −k 1 ˆ < ζˆ > AT1 p; 2π i k α

(14.111b)

k=1

where Case 1. e− k =

532

14


cα fˆ3 γα ζˆα ln(−cα ζˆα ) + > AT1 i3 , < 2π i ζˆα 2ζˆα ' ( −ζˆα cα γα fˆ3 − < + ek = > AT1 i3 , 2π i kζˆαk k − 1 (k + 1)ζˆα

Case 2. e− 1 =

Case 3. e− k =−

(14.111c) k = 1;

m ˆ3 < c4α ζˆα−k > AT1 i2 . 2π i

(14.111d)

− From the holomorphic condition of f+ 2 (ζ ) and f2 (ζ ), the function g2 in (14.103b) ∞ can also be assumed in series form such as g2 (ζ ) = dk ζ k . With this series k=1

− form, the relations given in (14.103b) will provide us the series of f+ 2 (ζ ) and f2 (ζ ). To get a comparable solution with those of two-dimensional problem discussed in Section 8.1, the unknown coefficient dk is replaced by ck =< γα∗−k > dk . With this replacement, we have

f+ 2 (ζ ) =

∞

< γα∗k > ck ζ −k ,

f− 2 (ζ ) =

k=1

∞

ck ζ k .

(14.112a)

k=1

− Adding f+ 2 (ζ ) and f2 (ζ ) together, we get ∞

f2 (ζ ) =

ck ζ k ,

(14.112b)

c−k =< γα∗k > ck .

(14.112c)

k=−∞

where

Note that in the above equations, (14.111) and (14.112), the constant term associated with rigid body motion has been neglected. Substituting (14.111a) and (14.112a) into (14.110), and following the standard approach of analytical continuation, the coefficient vector ck can be obtained as −1

−1

ck = {G0 − Gk G0 Gk }−1 {tk − Gk G0 tk },

k = 1, 2, · · · , ∞,

(14.113a)

where G0 = {M1 + M2 }A2 ,

Gk = {M1 − M2 }A2 < γα∗k >,

− tk = −iA−T 1 ek , (14.113b)

and Mk , k = 1, 2, are the impedance matrices defined as M1 = −iB1 A−1 1 ,

M2 = −iB2 A−1 2 .

(14.113c)

14.6


533

− Having the solution of ck , function vectors f+ 2 (ζ ), f2 (ζ ) and f2 (ζ ) can be determined by (14.112). Function vector ˘f1 (ζ ) can then be obtained either from the first or the second equation of (14.110). The results are

˘f1 (ζ ) = −

∞

− ∗k −k A−1 1 {A1 ek − A2 < γα > ck − A2 ck }ζ ,

(14.114a)

− ∗k −k B−1 1 {B1 ek − B2 < γα > ck − B2 ck }ζ .

(14.114b)

k=1

or ˘f1 (ζ ) = −

∞ k=1

From the above derivation we see that to obtain the solutions of generalized displacement and stress function vectors from (14.109), we need to combine the results of (14.106) for f0− (ζ ), (14.112b) and (14.113) for f2 (ζ ∗ ), and (14.114) for ˘f1 (ζ ). − ∗ ∗ There is no need to calculate the separate solutions for f0+ (ζ ), f+ 2 (ζ ), and f2 (ζ ), unless special purpose is requested. Moreover, as stated in the paragraph following (14.110), to get the complete full-field solutions the translating technique should be employed. Comparison of the present solutions with those of the corresponding two-dimensional problems presented in Section 8.1.2 shows that the solutions obtained here for case 1 possess exactly the same mathematical form as those for the corresponding two-dimensional problems. The only difference is the contents of the symbols. While for the other two cases, one more difference comes from f0− (ζ ). Instead of the full-field solutions, in engineering application one is usually interested in the stress resultants and moments along the interface boundary. From the relations shown in (13.79), (13.80), and (13.81), we know that the calculation of stress resultants and moments replies upon the calculation of the differentials φd, s and φd, n . Due to the continuity conditions (14.100b), the derivative of φd along the (1) (2) interface should be continuous across the interface, i.e., φd, s = φd, s . On the other (1)

(2)

hand, φd, n may not equal to φd, n . Thus, to calculate φd, s along the interface one may use the field solutions of the inclusion or the matrix. While for the calculation of φd, n , we may differentiate φd directly with respect to the normal direction n for both the inclusion and the matrix, or calculate indirectly from φd, s through the following relation, which is generalized from (3.80):

ud, n φd, n

= N(θ )

ud, s , φd, s

(14.115)

where N(θ ) is the generalized matrix of the fundamental matrix N. By using chain rule, two relations useful for the calculation of φd, s and φd, n have been obtained as ∂f ieiϕ μα (θ ) ∂f , = ∂n ρ ∂ζα

∂f ieiϕ ∂f , along the interface boundary, = ∂s ρ ∂ζα

(14.116)

534

14


where μα (θ ) is the generalized material eigenvalue related to μα by (3.70). Using the second relation of (14.116) and the solutions for the inclusion shown in (14.109)4 , (14.112), and (14.113) we obtain ⎧ ⎫ ∞ ⎨ ⎬ ikeikϕ (1) (2) B2 ck , along the interface boundary. φd, s = φd, s = φd, s = 2 Re ⎩ ⎭ ρ k=−∞

(14.117) With solution (14.117) and relation (14.115), φd, n can be obtained for both the inclusion and the matrix. Or alternatively, we can obtain φd, n directly through the use of (14.116)1 and the solutions for the matrix and inclusion obtained in (14.109), (14.106), (14.112), and (14.113).

14.6.2 Concentrated Forces/Moments Inside the Inclusions The holomorphic condition discussed previously will change if the singular point ζˆα is located inside the inclusion. For example, function ln(ζα − ζˆα ) of (14.106a) which is holomorphic inside the unit circle (S2 + S0 ) when |ζˆα | > 1, will not be holomorphic in S2 + S0 because ζˆα is now located in this region. For this reason, when the concentrated forces and moments are applied on the points inside the inclusion, the solutions obtained in Section 14.6.1 become invalid and new solutions should be found. Since the forces/moments are applied on the point ζˆα∗ inside the inclusion, instead of f0 (ζ ) we now select f∗0 (ζ ∗ ) to be the one associated with the solutions for the laminates without inclusions. That is, Case 1. f∗0 (ζ ∗ ) =

1 ˆ < ln(z∗α − zˆ∗α ) > AT2 p; 2π i

(14.118a)

Case 2. f∗0 (ζ ∗ ) =

fˆ3 < (z∗α − zˆ∗α )[ln(z∗α − zˆ∗α ) − 1] > AT2 i3 ; 2π i

(14.118b)

Case 3. f∗0 (ζ ∗ ) =

m ˆ3 1 > AT2 i2 . < ∗ 2π i zα − zˆ∗α

(14.118c)

Note that the functions shown in (14.118) are written in terms of z∗α , which can be re-written in terms of ζα∗ by using the relation given in (14.74). Also because f∗0 (ζ ∗ ) is selected to be a function of argument z∗α , the restriction (14.102a) for one-to-one mapping from z∗α to ζα∗ will be satisfied automatically. Substituting (14.74) into (14.118a), (14.118b), and (14.118c) and considering the holomorphic condition of each term, we may now split f∗0 (ζ ∗ ) into three parts as those shown in (14.104b). They are

14.6


535

Case 1. 1 ˆ < ln(z∗α − zˆ∗α ) − ln ζα∗ > AT2 p, 2π i 1 ∗ ˆ < ln ζα∗ > AT2 p. f∗− f∗s (ζ ∗ ) = 0 (ζ ) = 0, 2π i

∗ f∗+ 0 (ζ ) =

(14.119a)

Case 2. % & fˆ3 < (z∗α − zˆ∗α ) ln(z∗α − zˆ∗α ) − ln ζα∗ − 1 − c∗α ζα∗ (ln c∗α − 1) > AT2 i3 , 2π i fˆ3 ∗ ∗ ∗ ∗ T f∗− 0 (ζ ) = 2π i < cα ζα (ln cα − 1) > A2 i3 , fˆ3 < (z∗α − zˆ∗α ) ln ζα∗ > AT2 i3 . f∗s (ζ ∗ ) = 2π i (14.119b) Case 3. ∗ f∗+ 0 (ζ ) =

∗ f∗+ 0 (ζ ) =

1 m ˆ3 < ∗ > AT2 i2 , 2π i zα − zˆ∗α

∗ f∗− 0 (ζ ) = 0,

f∗s (ζ ∗ ) = 0.

(14.119c)

Unlike the problems considered in Section 14.6.1, here the function f0 (ζ ) cannot be set to be zero like that of (14.105) because in region S1 we need corresponding term fs (ζ ) for the singular function f∗s (ζ ∗ ) in region S2 + S0 . Moreover, f+ 0 (ζ ) and ∗− f− (ζ ) are also needed to have comparable terms with f (ζ ) which is holomorphic 0 0 in S2 + S0 but not holomorphic in S1 . With this consideration, f0 (ζ ) is also split into three parts as that shown in (14.104a), in which each term is assumed as follows: Case 1. fs (ζ ) =< ln ζα > d,

− f+ 0 (ζ ) = f0 (ζ ) = 0.

Case 2. fs (ζ ) =< ln ζα > {< ζα > d1 + < ζα−1 > d−1 + d0 }, f0+ (ζ ) =< ζα−1 > k−1 ,

f0− (ζ ) =< ζα > k1 .

− Case 3. fs (ζ ) = f+ 0 (ζ ) = f0 (ζ ) = 0.

(14.120a)

(14.120b) (14.120c)

Like (14.109), we now substitute (14.103a), (14.104a), and (14.104b) into (14.101a) and rewrite the general solutions for the generalized displacement and stress function vectors as

536

14


− + − ud = A1 [fs (ζ ) + f+ 0 (ζ ) + f0 (ζ ) + f1 (ζ )] + A1 [fs (ζ ) + f0 (ζ ) + f0 (ζ ) + f1 (ζ )], (1)

(1)

= B1 [fs (ζ ) + f0+ (ζ ) + f0− (ζ ) + f1 (ζ )] + B1 [fs (ζ ) + f0+ (ζ ) + f0− (ζ ) + f1 (ζ )],

(2)

∗− ∗ + ∗ − ∗ ∗ = A2 [f∗s (ζ ∗ ) + f∗+ 0 (ζ ) + f0 (ζ ) + f2 (ζ ) + f2 (ζ )]

φd ud

∗− ∗ + ∗ − ∗ ∗ + A2 [f∗s (ζ ∗ ) + f∗+ 0 (ζ ) + f0 (ζ ) + f2 (ζ ) + f2 (ζ )], (2)

φd

∗− ∗ + ∗ − ∗ ∗ = B2 [f∗s (ζ ∗ ) + f∗+ 0 (ζ ) + f0 (ζ ) + f2 (ζ ) + f2 (ζ )] ∗− ∗ + ∗ − ∗ ∗ + B2 [f∗s (ζ ∗ ) + f∗+ 0 (ζ ) + f0 (ζ ) + f2 (ζ ) + f2 (ζ )].

(14.121)

Again, as described in the statement following (14.109), to determine the − + ∗ − ∗ unknown function vectors fs (ζ ),f+ 0 (ζ ), f0 (ζ ), f1 (ζ ), f2 (ζ ), and f2 (ζ ), only the displacement and traction continuity conditions will be considered because f∗0 (ζ ∗ ) = ∗− ∗ ∗ ∗ ∗ f∗+ 0 (ζ ) + f0 (ζ ) + fs (ζ ) shown in (14.118) and (14.119) have satisfied the boundary conditions for the concentrated forces and moments. We now substitute (14.121) into (14.100b) with ζα = σ along the interface and separate its results into three parts: (1) the terms related to f∗s (ζ ∗ ) which has singular points in both regions S1 and S2 + S0 , (2) the terms related to f∗− 0 (ζ ) which is not holomorphic in S1 , and (3) all the others. By comparison of corresponding terms we obtain A1 fs (σ ) + A1 fs (σ ) = A2 f∗s (σ ) + A2 f∗s (σ ), B1 fs (σ ) + B1 fs (σ ) = B2 f∗s (σ ) + B2 f∗s (σ ), − + − ∗− ∗− A1 f+ 0 (σ ) + A1 f0 (σ ) + A1 f0 (σ ) + A1 f0 (σ ) = A2 f0 (σ ) + A2 f0 (σ ), − + − ∗− ∗− B1 f+ 0 (σ ) + B1 f0 (σ ) + B1 f0 (σ ) + B1 f0 (σ ) = B2 f0 (σ ) + B2 f0 (σ )

(14.122a)

(14.122b)

and + − A1 f1 (σ ) − A2 [f∗+ 0 (σ ) + f2 (σ )] − A2 f2 (σ ) + − = −A1 f1 (σ ) + A2 [f∗+ 0 (σ ) + f2 (σ )] + A2 f2 (σ ), + − B1 f1 (σ ) − B2 [f∗+ 0 (σ ) + f2 (σ )] − B2 f2 (σ )

(14.122c)

+ − = −B1 f1 (σ ) + B2 [f∗+ 0 (σ ) + f2 (σ )] + B2 f2 (σ ). + − ∗ With f∗s (ζ ∗ ) and f∗− 0 (ζ ) given in (14.119) and fs (ζ ), f0 (ζ ), and f0 (ζ ) assumed in (14.120), equations (14.122a) and (14.122b) can now provide us the solutions of the unknown coefficients d, d0 , d−1 , d1 , k−1 , and k1 of functions fs (ζ ), f0+ (ζ ), and f0− (ζ ). They are (Hwu and Tan, 2007)

Case 1. d =

1 T ˆ A p. 2π i 1

(14.123a)

14.6


537

−fˆ3 T {B J2R + AT1 JT1R }i3 , πi 1 d1 = {(BT1 g2 + AT1 h2 ) + i(BT1 g1 + AT1 h1 )},

Case 2. d0 =

d−1 = {−(BT1 g2 + AT1 h2 ) + i(BT1 g1 + AT1 h1 )},

(14.123b) fˆ3 T (B1 Q2R + AT1 QT1R ) + i(BT1 Q2I + AT1 QT1I ) i3 , k1 = 2π i fˆ3 −(BT1 Q2R + AT1 QT1R ) + i(BT1 Q2I + AT1 QT1I ) i3 . k−1 = 2π i Case 3. None. (14.123c) In the above, −fˆ3 (E2R + F2R )i3 g1 = , h1 2π (ET1R + FT1R )i3

fˆ3 (E2I − F2I )i3 g2 = , h2 2π (ET1I − FT1I )i3

(14.124)

where EkR , FkR , JkR , QkR and EkI , FkI , JkI , QkI , k=1,2,3, are real and imaginary parts of the matrices Ek , Fk , Jk , and Qk , which is defined as ' A2 < c∗α > BT2 E1 E2 = E3 ET1 B2 < c∗α > BT2 ' A2 < c∗α γα∗ > BT2 F1 F2 = F3 FT1 B2 < c∗α γα∗ > BT2 ζˆα∗ + A < ⎢ 2 J1 J2 =⎢ ⎣ J3 JT1 ∗ B2 < cα ζˆα∗ + ⎡

c∗α

γα∗ ζˆα∗ γα∗ ζˆα∗

A2 < c∗α > AT2 B2 < c∗α > AT2

,

A2 < c∗α γα∗ > AT2

BT2

> BT2

(14.125a)

(

B2 < c∗α γα∗ > AT2

>

(

,

ζˆα∗ + A2 < ∗ B2 < cα ζˆα∗ + c∗α

(14.125b)

γα∗ ζˆα∗ γα∗ ζˆα∗

> AT2 > AT2

⎤ ⎥ ⎥, ⎦

(14.125c)

( ' A2 < c∗α (ln c∗α − 1) > BT2 A2 < c∗α (ln c∗α − 1) > AT2 Q1 Q2 = , (14.125d) Q3 QT1 B2 < c∗α (ln c∗α − 1) > BT2 B2 < c∗α (ln c∗α − 1) > AT2 Ek = EkR + iEkI ,

Fk = FkR + iFkI ,

Jk = JkR + iJkI ,

Qk = QkR + iQkI ,

i = 1, 2, 3.

(14.125e)

From (14.122c) and following the standard approach of analytical continuation as that shown in Section 14.6.1, the unknown function vectors f1 (ζ ) and f2 (ζ ∗ ) can be determined as f2 (ζ ∗ ) =

∞ k=−∞

ck ζ ∗k ,

c−k =< γα∗k > ck

(14.126a)

538

14


and f1 (ζ ) =

∞

+ ∗k −k A−1 1 {A2 ek + A2 < γα > ck + A2 ck }ζ

(14.126b)

+ ∗k −k B−1 1 {B2 ek + B2 < γα > ck + B2 ck }ζ

(14.126c)

k=1

or f1 (ζ ) =

∞ k=1

where −1

−1

ck = {G0 − Gk G0 Gk }−1 {tk − Gk G0 tk },

k = 1, 2, · · · , ∞

(14.126d)

and tk = −(M1 − M2 )A2 e+ k .

(14.126e)

∗+ ∗ In the above, e+ k are the coefficients of the Taylor’s expansions of f0 (ζ ), i.e.,

f∗+ 0 (ζ ) =

∞

−k e+ k ζ .

(14.127a)

k=1

From (14.119a), (14.119b), and (14.119c), we have Case 1. e+ k =

−1 ˆ < ζˆα∗k + (γα∗ /ζˆα∗ )k > AT2 p. 2π ki

(14.127b)

Case 2. , 1 ∗2 fˆ3 < c∗α ζˆα + (γα∗ /ζˆα∗ )2 + γα∗ ln c∗α + 1 > AT2 i3 , 2π i 2 ∗ ˆ cα 1 ∗k+1 f3 ζˆα + (γα∗ /ζˆα∗ )k+1 e+ k = 2π i < k k+1 ∗ γα ∗k−1 ∗ ˆ ∗ k−1 ˆ ζ > AT2 i3 , k = 1. + (γα /ζα ) − k−1 α e+ 1 =

(14.127c)

Case 3. e+ k =

ζˆ ∗k − (γα∗ /ζˆα∗ )k m ˆ3 > AT2 i2 . < α 2π i c∗α [ζˆα∗ − (γα∗ /ζˆα∗ )]

(14.127d)

14.6


539

Same as Section 14.6.1, the solutions obtained here for case 1 possess exactly the same mathematical form as those for the corresponding two-dimensional problems. The only difference is the contents of the symbols. Similarly, the stress resultants and moments along the interface boundary can also be obtained through the explicit solution of φd, s , which is

(1)

(2)

φd, s = φd, s = φd, s

⎧ ⎨

⎫ ∞ ⎬ ikϕ ike B2 ck , = 2Re B2 f∗0,s + ⎩ ⎭ ρ

along the interface,

k=−∞

(14.128a) where

f∗0,s

⎧ 1 T ⎪ ⎪ ˆ < (cos θ + μ∗α sin θ )−1 ⎪ α > A2 p, ⎪ 2π i ⎪ ⎪ ⎨ ˆ f3 = < (cos θ + μ∗α sin θ ) ln α > AT2 i3 , ⎪ 2π i ⎪ ⎪ ⎪ ⎪ ˆ3 ⎪ T ⎩ −m < (cos θ + μ∗α sin θ )−2 α > A2 i2 , 2π i

case 1, case 2,

(14.128b)

case 3

and α = a cos ϕ + μα b sin ϕ − zˆα .

(14.128c)

14.6.3 Verification and Discussions From the statements and the solutions given in the previous sections, we know that Green’s function is a solution to the incomplete problem whose domain is infinite and whose loading is an unbalanced point load. Based upon this solution, we may construct the solutions for any complete problem whose boundary is prescribed by tractions or displacements and all applied forces and reactive forces are in equilibrium condition. In other words, the complete problem may be solved through superposition of several Green’s function with different intensities located at different points. That is why the Green’s function is sometimes called the fundamental solution for the boundary element method. Therefore, in this stage it is not appropriate to compare the solutions with those obtained from the other numerical methods such as the finite element method since they can only deal with complete problems. Due to the inappropriateness of numerical check by finite element method, in this section we first consider the analytical verification with some special cases reduced from the general solutions, and then present the numerical results for various elastic inclusions. To check the correctness of the solutions we consider (1) the simplest condition that the matrix and the inclusion are composed of the same materials, i.e., no inclusions are embedded in the laminates; (2) the case that the inclusion is very soft, which can be checked by the results of corresponding hole problems. The

540

14


analytical solutions of these two special cases have been presented in Sections 14.4 and 14.5. Without Inclusions: Specialization of the Solutions from Section 14.6.1 If the matrix and the inclusion are composed of the same material, the material eigenvector matrices should be identical, and hence we can set A1 = A2 = A and B1 = B2 = B. Thus, from (14.113c) we have M1 = M2 = M. From the identities shown in (3.59) and (3.132), we know that the impedance matrix M is related to the Barnett–Lothe tensors H, S , and L, which are real matrices, by M = H−1 (I + iS),

where

H = 2iAAT ,

S = i(2ABT − I).

(14.129)

By using the above relations, the coefficient vector ck and the function vector ˘f1 (ζ ) obtained in (14.113) and (14.114) can be reduced to ˘ ck = e− k , f1 (ζ ) =

∞

< (γα /ζα )k > e− k .

(14.130)

k=1

Note that in deriving the second equation of (14.130), the translating technique (4.50) and (4.51) has been employed. The coefficient vector e− k for different loading cases has been given in (14.111b), (14.111c), and (14.111d). From − (14.130)1 , (14.111a), and (14.112a)2 , we see that f− 2 (ζ ) = f0 (ζ ) whose solutions for three different loading cases have been shown in (14.106). With the results of (14.111b), (14.111c), (14.111d), and (14.130), the function vectors ˘f1 (ζ ) and f+ 2 (ζ ) can be obtained from (14.130)2 and (14.112a)1 . Their final simplified results are Case 1. ˘f1 (ζ ) = f+ (ζ ) = 1 < ln 1 − γα ˆ > AT p. 2 ˆζα ζα 2π i

(14.131a)

ˆ3 ˆα c γ − ζ f ζ α α α ˘f1 (ζ ) = < = ln(−cα ζˆα ) + 2π i ζα ζˆα γα > AT i3 . +(zα − zˆα ) ln 1 − ζˆα ζα

(14.131b)

ˆ3 c4α (γα /ζˆα ζα ) ˘f1 (ζ ) = f+ (ζ ) = −m > AT i2 . < 2 2π i 1 − (γα /ζˆα ζα )

(14.131c)

Case 2. f+ 2 (ζ )

Case 3.

14.6


541

Note that in deriving solutions of (14.131), relations (14.74) and (14.81)2 have been used. Combining the results of (14.106) and (14.131), it can be shown that − f0− (ζ ) + ˘f1 (ζ ) = f+ 2 (ζ ) + f2 (ζ ) = f0 (z),

(14.132)

in which f0 (z) is given in (14.65) for three different loading cases. In other words, the present solutions have been successfully reduced to the cases of homogeneous laminates. Without Inclusions: Specialization of the Solutions from Section 14.6.2 Similar to the reduction process stated above, by setting A1 = A2 = A and B1 = B2 = B the solutions obtained in (14.126b), (14.126c), (14.126d), and (14.126e) can be reduced to ck = 0,

f1 (ζ ) =

∞

−k e+ = f∗+ k ζ 0 (ζ ).

(14.133)

k=1

Combining the results of (14.133), (14.120), and (14.123), and using the orthogonality relations given in (3.57) and (3.58), we can prove that fs (ζ ) + f0+ (ζ ) + f0− (ζ ) + f1 (ζ ) = f0 (z),

(14.134)

in which f0 (z) is given in (14.65) for three different loading cases. To prove that the other part of the general solution (14.121) also reduces to the solution of homogeneous laminates, we combine the results of (14.133)1 , (14.126), and (14.119), which also give us ∗− ∗ + ∗ − ∗ ∗ f∗s (ζ ∗ ) + f∗+ 0 (ζ ) + f0 (ζ ) + f2 (ζ ) + f2 (ζ ) = f0 (z).

(14.135)

Holes If the inclusion is replaced by a hole, the elastic constants Cijkl of the inclusion can be set to be zero, and then the extensional, coupling, and bending stiffness Aijkl , Bijkl , Dijkl will all be equal to zero. Substituting these zero values into the explicit solutions of the material eigenvector matrices A and B given in Chapter 13, we get B2 = 0 whereas A2 may not be a zero matrix. With B2 = 0, the solution ˘f1 (ζ ) obtained in (14.114b) can be reduced to ˘f1 (ζ ) = −

∞

− −k B−1 1 B1 ek ζ .

(14.136)

k=1

Substituting e− k of (14.111b), (14.111c), and (14.111d) into (14.136), using the following Taylor series relations, and employing the translating technique (4.50) and (4.51),

542

14

−

∞ k x k=1

k

= ln(1 − x),

∞

xk =

k=1


1 , 1−x

for|x| < 1,

(14.137)

we can prove that the solution (14.136) is identical to those shown in Section 14.5.1 for all three loading cases of the coupled stretching–bending hole problems. Elastic Inclusions After the analytical check through the above two special cases, we now perform the numerical calculation for the cases of general elastic inclusions to see whether the elastic responses shown by the Green’s function are conformable to the engineering intuition. Although the solutions are valid for the general composite laminates, symmetric or unsymmetric, to save the space only the simplest orthotropic lamina is considered in the following examples. Consider both of the matrix and inclusion are orthotropic materials. The material properties of the matrix are E1 = 144.8 GPa, E2 = E3 = 10.7 GPa, ν12 = ν13 = ν23 = 0.31, G12 = G13 = G23 = 4.5 GPa, while the properties of the inclusion are assumed to be proportional to the matrix as (Gij )2 (Ei )2 = , i, j = 1, 2, 3, ν12 = ν13 = ν23 = 0.31, k= (Ei )1 (Gij )1

0.2

0.0

aN s pˆ

Index of softness

Hole k=1

-0.2

k = 0.5 k = 10-1 k = 10-3 k = 10-5

Fig. 14.6 Hoop stress resultant along the elliptical inclusion boundary (Hwu and Tan, 2007)

-0.4 -90

-60

-30

30 0 ϕ (degree)

60

90

14.6


543

where k is the index of softness (or hardness). When k < 1 the inclusion is softer than the matrix, while for k > 1 means hard. A hole or rigid inclusion can therefore be approximated by letting k → 0 or k → ∞. Consider a concentrated force pˆ (Nt) directed in the x2 -axis applied on the point (ˆx1 /a, xˆ 2 /a) = (0, 5). The elliptical inclusion is represented by b/a=0.75. The hoop stress resultant Ns of elastic inclusion for various k are calculated by (14.115), (14.116), and (14.117). The results for the normalized hoop stress resultant aNs /ˆp are shown in Fig. 14.6, from which we see that the solutions for holes are approximated by k = 10−5 . The correctness of the results in the limiting cases is therefore verified and the trend from soft inclusions to holes is also reasonable. Note that in numerical calculation, the infinite series are truncated into finite terms which are determined by truncating error defined by Tr = [(Ns )i+1 − (Ns )i ]/(Ns )i . In the present example, if Tr = 10−6 , ϕ = 0o , k = 2, the term needed is 14. When the applied load pˆ is located on (ˆx1 /a, xˆ 2 /a) = (0, 1.1), the term needed is 27. To see the effect of elliptic shape and the singular behavior near the crack tip, a series of numerical data for the hoop stress resultant at ϕ = 0o are shown in Fig. 14.7. A nearly constant value of the hoop stress resultant for b → 0 is achieved when the inclusion is not a hole, which means that no singular behavior occurs for the general elastic inclusions. For elliptical holes, singular behavior occurs when b → 0 which is expected for cracks. 108

107

106

105 104 aN s pˆ

Hole

k = 10−5

103 102

k = 10−3

101 100

k = 10 −1

10−1

k = 0.5

10−2

k=1

10−3 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100

b/a Fig. 14.7 Hoop stress resultant at ϕ = 0o for the soft inclusion (Hwu and Tan, 2007)

Chapter 15

Boundary Element Analysis

Finite element and boundary element methods are two important and popular techniques for practical problems in engineering. Many commercial softwares have been developed based upon these two methods. It’s known that the core function of the boundary element method is the fundamental solutions which are derived from Green’s functions – the solutions for point load problems. If the Green’s functions for particular problems such as half-space, interface, hole, crack, and inclusion are obtained analytically, their associated boundary elements will possess several advantages than the other boundary elements or finite elements. With this understanding, in addition to the conventional boundary element which employs the Green’s function for infinite space, several special boundary elements considering half-space/interface/hole/crack/inclusion will be discussed. Based upon the works presented in Hwu and Liao (1994), Liang and Hwu (1996), and Hwu (1999, 2010), the boundary element analysis will be presented in this chapter for the twodimensional elastic problems, the two-dimensional electro-elastic problems and the coupled stretching–bending problems.

15.1 Two-Dimensional Elastic Analysis 15.1.1 Boundary Integral Equations If body forces are omitted, the boundary integral equations for the boundary value problem in solid mechanics can be written as (Brebbia et al., 1984) u∗ij (ξ, x) tj (x)d(x), i, j = 1, 2, 3, (15.1) cij (ξ)uj (ξ) + tij∗ (ξ, x) uj (x)d(x) =

where denotes the boundary of the elastic solid; uj (x) and tj (x) are the displacements and surface tractions along the boundaries; u∗ij (ξ, x) and tij∗ (ξ, x) are, respectively, the displacements and tractions in the xj -direction at point x = (x1 , x2 ) corresponding to a unit point force acting in the xi -direction applied at point ξ = (ˆx1 , xˆ 2 ); cij (ξ) is a coefficient dependent on the location of ξ, which equals to δij /2 for a smooth boundary and cij = δij for an internal point. The symbol δij C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7_15, C Springer Science+Business Media, LLC 2010

545

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15


is the Kronecker delta, i.e., δij = 1 when i = j and δij = 0 when i = j. In practical applications, cij (ξ) can be computed by considering rigid body motion. In other words, if we let a unit rigid body movement in the direction of xj , uj = 1 which will not induce any stresses and hence tj = 0. Substituting this condition into (15.1), we get cij (ξ) = −

tij∗ (ξ, x) d(x).

(15.2)

The boundary integral equations given in (15.1) have now three unknown functions, i.e., uj or tj , j = 1, 2, 3, and three equations, which constitute the basis of the boundary element formulation. For the convenience of boundary element formulation, the boundary integral equations (15.1) are usually written in matrix form as C(ξ)u(ξ) +

T∗ (ξ, x)u(x)d(x) =

U∗ (ξ, x)t(x)d(x),

(15.3a)

where ⎧ ⎫ ⎧ ⎫ ⎤ c11 (ξ) c12 (ξ) c13 (ξ) ⎨u1 (x)⎬ ⎨t1 (x)⎬ C(ξ) = ⎣c21 (ξ) c22 (ξ) c23 (ξ)⎦ , u(x) = u2 (x) , t(x) = t2 (x) , ⎩ ⎭ ⎩ ⎭ c31 (ξ) c32 (ξ) c33 (ξ) u3 (x) t3 (x) ⎡∗ ⎤ ∗ (ξ, x) t∗ (ξ, x) t11 (ξ, x) t12 13 ∗ (ξ, x) t∗ (ξ, x) t∗ (ξ, x)⎦ , T∗ (ξ, x) = ⎣t21 22 23 ∗ (ξ, x) t∗ (ξ, x) t∗ (ξ, x) t31 32 33 ⎡ ∗ ⎤ ∗ u11 (ξ, x) u12 (ξ, x) u∗13 (ξ, x) U∗ (ξ, x) = ⎣u∗21 (ξ , x) u∗22 (ξ, x) u∗23 (ξ, x)⎦ . u∗31 (ξ, x) u∗32 (ξ, x) u∗33 (ξ, x) ⎡

(15.3b)

To make (15.3) work for the programming of boundary element codes, we need to find the fundamental solutions tij∗ (ξ, x) and u∗ij (ξ, x) which are related to the Green’s functions obtained in previous chapters.

15.1.2 Fundamental Solutions In order to find the fundamental solutions tij∗ (ξ, x) and u∗ij (ξ, x) for infinite space, half-space, bimaterials, holes, cracks, rigid inclusions, or elastic inclusions, we first collect the Green’s functions obtained in (4.18) for infinite space, (4.53) for halfspace, (4.70) and (4.80) for bimaterials, (6.61) for holes and cracks, (8.44), (8.45), and (8.46) for rigid inclusions, and (8.22) and (8.25) for elastic inclusions. All these Green’s functions can be expressed by ˆ u = 2 Re{AF(z)}p,

ˆ φ = 2 Re{BF(z)}p,

(15.4)

15.1

Two-Dimensional Elastic Analysis

547

for problems with one elastic material, whereas for problems with two dissimilar elastic materials, they can be expressed by ˆ ˆ u1 = 2 Re {A1 F1 (z)} p, φ1 = 2 Re {B1 F1 (z)} p, z ∈ S1 , ∗ ∗ ˆ ˆ u2 = 2 Re A2 F2 (z ) p, φ2 = 2 Re B2 F2 (z ) p, z∗ ∈ S2 ,

(15.5)

in which pˆ is the vector of concentrated force applied at point ξ = (ˆx1 , xˆ 2 ). Note that the symbols with or without subscript 1 and the arguments without superscript ∗ denote the values associated with material 1, whereas the symbols with subscript 2 and the arguments with superscript ∗ denote the values related to material 2. The complex function vectors for each different problem are as follows: (i) Infinite space: F(z) = FI (z).

(15.6a)

F(z) = FI (z) + FS (z).

(15.6b)

(ii) Half-space:

(iii) Bimaterial: F1 (z) = FI (z) + FB (z),

F2 (z∗ ) = FB¯ (z∗ ).

(15.6c)

(iv) Hole or crack: (b = 0 for crack): F(z) = FI (ζ ) + FH (ζ ).

(15.6d)

F(z) = FI (ζ ) + FR (ζ ).

(15.6e)

(v) Rigid inclusion:

(vi) Elastic inclusion: F1 (z) = FI (ζ ) + FE (ζ ),

F2 (z∗ ) = FE¯ (ζ ∗ ).

(15.6f)

In the above, for problems (i)–(iii) we have 1 < ln(zα − zˆα ) > AT , 2π i 3 1 ¯ k AT , < ln(zα − z¯ˆk ) > B−1 BI FS (z) = 2π i FI (z) =

k=1

(15.7a)

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1 T −1 ¯ ¯ ¯ ¯ < ln(zα − z¯ˆj ) > A−1 1 (M2 + M1 ) (M2 − M1 )A1 Ij A1 , 2π i 3

FB (z) =

j=1

FB¯ (z∗ ) = −

3 1 ¯ −1 −T T < ln(z∗α − zˆj ) > A−1 2 (M2 + M1 ) A1 Ij A1 , 2π j=1

(15.7b) whereas for problems (iv)–(vi), 1 < ln(ζα − ζˆα ) > AT , 2π i 3 1 T < ln(ζα−1 − ζˆ k ) > B−1 BIk A , FH (ζ ) = 2π i FI (ζ ) =

(15.8a)

k=1

3 1 −1 −1 T −1 −1 T ¯ kA ¯ + < ζα > A kw , < ln(ζα − ζˆ k ) >A AI FR (ζ ) = 0 2π i k=1 (15.8b) FE (ζ ) = FH (ζ ) +

∞

! " ¯ k + B2 < γα∗k > Ck , < ζα−k > B−1 B2 C 1

k=1

(15.8c)

∞ ! " FE¯ (ζ ∗ ) = < ζα∗k + (γα∗ /ζα∗ )k > Ck . k=1

The components of the applied concentrated force pˆ are pˆ 1 , pˆ 2 , and pˆ 3 , and Ik , k = 1, 2, 3, are the diagonal matrices with unit value at the kk component and all other components are zero, i.e., ⎧ ⎫ ⎨pˆ 1 ⎬ pˆ = pˆ 2 , ⎩ ⎭ pˆ 3

⎡

⎤ 100 I1 = ⎣0 0 0⎦ , 000

⎡

⎤ 000 I2 = ⎣0 1 0⎦ , 000

⎡

⎤ 000 I3 = ⎣0 0 0⎦ . 001

(15.9)

w0 of (15.8b) is a vector related to the rotation angle ω of rigid inclusion by ˆ i. Using (8.46) we have ω = wT0 p/π T

wT0 =

i Re{k A−T < ζˆα−1 > AT } kT Mk

,

⎧ ⎫ ⎨ib⎬ k= a . ⎩ ⎭ 0

(15.10)

M is the impedance matrix defined in (3.132), i.e., M = −iBA−1 = H−1 − iST H−1 = H−1 + iH−1 S.

(15.11)

15.1


549

ζα is the transformed variable mapping an ellipse to a unit circle, which is related to complex variable zα by (6.3), i.e., ζα =

zα +

+ z2α − a2 − b2 μ2α , a − ibμα

α = 1, 2, 3.

(15.12a)

If a polygon-like hole or inclusion is considered, the transformation relation (15.12a) should be replaced by the one given in (6.65) or (8.52b), i.e., zα =

" a! (1 − iμα c)ζα + (1 + iμα c)ζα−1 + ε(1 + iμα )ζαk + ε(1 − iμα )ζα−k . 2 (15.12b)

Ck , k = 1, 2, 3, . . . of (15.8c) are the coefficient matrices for the elastic inclusion problems, which can be obtained from the relations given in (8.11) and (8.23). They are −1 −1 Ck = (G0 − Gk G0 Gk )−1 (Tk − Gk G0 T¯ k ), ∗

G0 = (M1 + M2 )A2 , Gk = (M1 − M2 )A2 < γα k >, 1 −T A < ζˆα−k > AT1 , k = 1, 2, 3 · · · ∞, Tk = 2π k 1

(15.13a)

where γα∗ =

a + ibμ∗α . a − ibμ∗α

(15.13b)

From (3.32) we know that the traction t is related to the stress function φ by t = ∂φ/∂s. With this relation and the Green’s functions collected in (15.6), (15.7), and (15.8), the fundamental solutions tij∗ (ξ, x) and u∗ij (ξ, x) can be obtained by the following way: (1) Employ a unit point force at ξ acting in the x1 -direction, i.e., ∗ , j = 1, 2, 3. With this point load apply pˆ = (1, 0, 0)T at ξ = (ˆx1 , xˆ 2 ) to get u∗1j and t1j ∗ u1j can be obtained directly from the Green’s functions u listed in (15.6), (15.7), and ∗ are obtained by differentiating the stress function φ of (15.6), (15.8), whereas t1j (15.7), and (15.8) with respect to s which is the tangential direction of the body at point x, i.e., t(x) =

∂φ(x) ∂x1 ∂φ(x) ∂x2 ∂φ(x) = + = (s1 + μα s2 )φ , ∂s ∂x1 ∂s ∂x2 ∂s

(15.14)

where φ is the differential of φ with respect to its argument zα , and s1 and s2 are the components of the tangential direction defined by s1 =

∂x1 ∂x2 = n2 = = cos θ , ∂s ∂n

s2 =

∂x2 ∂x1 = −n1 = − = sin θ , ∂s ∂n

(15.15)

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15


in which θ denotes the angle from the positive x1 -axis to the direction s in counterclockwise direction. (2) Employ pˆ = (0, 1, 0)T at ξ = (ˆx1 , xˆ 2 ) to get u∗2j and ∗ , j = 1, 2, 3 through the same way as (1). (3) Similarly, u∗ and t∗ , j = 1, 2, 3, t2j 3j 3j can be obtained by applying pˆ = (0, 0, 1)T . (4) Combining the results for the above three different unit point forces, we can obtain a full matrix form solution U∗ and T∗ for u∗ij and tij∗ . By following the steps (1)–(4) described above, the results of U∗ (ξ, x) and ∗ T (ξ, x) for problems with one elastic material, and the results of U∗i (ξ, x) and T∗i (ξ, x), i = 1, 2 for problems with two elastic materials can be written as follows: U∗ = 2 Re{[AF(z)]T },

T∗ = 2 Re{[BF, s (z)]T },

U∗1 = 2 Re{[A1 F1 (z)]T },

T∗1 = 2 Re{[B1 F1, s (z)]T },

U∗2

T∗2

= 2 Re{[A2 F2

(z∗ )]T },

= 2 Re{[B2 F2, s

(15.16)

(z∗ )]T },

in which the complex function matrices F, F1 , F2 have been given in (15.6), (15.7), and (15.8), whereas their derivatives F, s , F1, s , F2, s for each different problems are as follows: (i) Infinite space: F, s (z) = FI, s (z).

(15.17a)

F, s (z) = FI, s (z) + FS, s (z).

(15.17b)

(ii) Half-space:

(iii) Bimaterial: F1, s (z) = FI, s (z) + FB, s (z),

∗ F2, s (z∗ ) = FB, ¯ s (z ).

(15.17c)

(iv) Hole or crack: (b = 0 for crack): F, s (z) = FI, s (ζ ) + FH, s (ζ ).

(15.17d)

F, s (z) = FI, s (ζ ) + FR, s (ζ ).

(15.17e)

(v) Rigid inclusion:

(vi) Elastic inclusion: F1, s (z) = FI, s (ζ ) + FE, s (ζ ), In the above, for problems (i)–(iii) we have

∗ F2, s (z∗ ) = FE, ¯ s (ζ ).

(15.17f)

15.1


551

@ ? 1 s1 + μα s2 T FI, s (z) = A , 2π i zα − zˆα (15.18a) @ 3 ? 1 s1 + μα s2 −1 ¯ T B BIk A , FS, s (z) = 2π i zα − z¯ˆk k=1 8 9 3 1 s1 + μα s2 ¯ 2 + M1 )−1 (M ¯ 2−M ¯ 1 )A ¯ 1 Ij AT1 , FB, s (z) = (M A−1 1 ¯ 2π i zα − zˆj j=1 (15.18b) @ 3 ? 1 s1 + μ∗α s2 −1 −T ∗ −1 T ¯ 1 ) A Ij A , A2 (M2 + M FB, ¯ s (z ) = − 1 1 2π z∗α − zˆj j=1

whereas for problems (iv)–(vi), @ ? ζα, s 1 AT , FI, s (ζ ) = 2π i ζα − ζˆα 8 9 3 1 ζα, s T FH, s (ζ ) = − B−1 BIk A , 2π i 2 ˆ k=1 ζα − ζ k ζα

(15.19a)

9 3 8 1 ζα, s −1 ¯ ¯ T −2 −1 T FR, s (ζ ) = − A AIk A + < ζα, s ζα > A kw0 , 2π i 2 k=1 ζα − ζˆ k ζα (15.19b) FE, s (ζ ) = FH, s (ζ ) −

@ ∞ ? kζα, s ζαk+1 8

! " ∗k B−1 , B C + B < γ > C 2 k 2 k α 1

k=1

FE, ¯ s (ζ ) =

∞

< ζα, s >

k=1

∗ k−1

kζα

∗

−

kγα k ∗

ζα k+1

9

(15.19c)

Ck ,

in which ζα, s =

∂ζα ∂ζα ∂zα = (s1 + μα s2 ), ∂zα ∂s ∂zα

(15.20a)

and ∂ζα ζα ζα 2ζα2 . =+ = = ∂zα (a − ibμα )ζα − zα (a − ibμα )ζα2 − (a + ibμα ) z2α − a2 − b2 μ2α (15.20b)

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15


15.1.3 Boundary Element Formulation After getting the fundamental solutions in (15.16)–(15.20), the unknowns remained in the boundary integral equations (15.3) are u and t over the boundary Γ . In boundary element formulation, the boundary Γ is approximated by a series of elements, and the points x, displacements u, and tractions t on the boundary are approximated by the nodal points xn , nodal displacement un , and nodal traction tn through different interpolation functions. In this section, we assume the same linear variation within each element for the boundary points x, displacements u and tractions t. Thus, the values of x, u, and t at any point on the mth element can be defined in terms of their nodal values and two linear interpolation functions !1 and !2 of the dimensionless coordinate ς , such that (1) (2) (1) (1) (2) x = !1 xm +!2 xm , u = !1 um +!2 u(2) m , t = !1 tm +!2 tm , on the mth element, (15.21a)

where a symbol with subscript m and superscript (1) or (2) denotes the value of node 1 or 2 of the mth element. The interpolation functions !1 and !2 are given by !1 =

1 (1 − ς ), 2

!2 =

1 (1 + ς ), 2

(15.21b)

where ς is the dimensionless coordinate defined by ς = 2s/m in which m is the length of the mth element and s is the coordinate lying along the linear element and directed from the first node to the second node of element m. If the boundary Γ is discretized into M segments with N nodes, substitution of (15.21) into (15.3a) yields C(ξ)u(ξ) +

M ! M ! " " (1) (2) (2) (1) (1) (2) (2) ˆ ˆ (1) = G (ξ)u + Y (ξ)u (ξ)t + G (ξ)t Y m m m m m m m m , m=1

m=1

(15.22) (i) (i) ˆm in which Y (ξ) and Gm (ξ), i = 1, 2, are the matrices of influence coefficients defining the interaction between the point ξ and the particular node (1 or 2) on element m, and are defined as (i) (2) ˆm (ξ) = T∗ (ξ, x(1) Y m , xm , ς )!i (ς )dm (ς ), m (15.23) (i) (2) Gm (ξ) = U∗ (ξ, x(1) , x , ς )! (ς )d (ς ), i = 1, 2. i m m m m

m denotes the mth segment of the discretized boundary. To evaluate the integrals along m , T∗ , and U∗ are expressed in terms of the dimensionless coordinate ς and the differential dm (x) (or written as dm (ς ) since x can be expressed in terms of ς in each element through (15.21a)) is transformed to dς multiplied by the Jacobian |Jm | as

15.1


553

* dm (ς ) = |Jm | dς ,

|Jm | =

dx1 dς

2

+

dx2 dς

2 = m /2.

(15.24)

Substituting (15.24) into (15.23) and employing a numerical integration scheme, (i) ˆ (i) Y m (ξ) and Gm (ξ) can be evaluated numerically. Here, all the integrals including the one with singular terms such as ln(zα − zˆα ) and (zα − zˆα )−1 in (15.7a) and (15.18a) are evaluated by using Gaussian quadrature rules. The required Gauss points for integration depend on the distance between the point under consideration and the midpoint of each element because the smaller the distance the larger the variation of tractions and displacements. For the point located on the boundary, 16 Gauss points are used to have a convergence solution for the singular integrals. If the connecting elements, for example, the (m − 1)th and the mth element are continuous at the connecting nodal points, the second node of the (m − 1)th element will be the first node of the mth element and can be named as the nth node of the whole boundary element. We let (2)

(2)

um−1 = u(1) m = un ,

(1) tm−1 = tm = tn , . . . , etc.

(15.25)

To write (15.3) corresponding to point ξ in discrete form, we need to add the contribution from two adjoining elements, m and m–1, into one term. Hence, we let ˆ (1) ˆ ˆ (2) + Y Y m = Yn , m−1

(2)

Gm−1 + G(1) m = Gn , . . . , etc.

(15.26)

Equation (15.3) can then be rewritten as C(ξ)u(ξ) +

N

ˆ n (ξ)un = Y

n=1

N Gn (ξ)tn .

(15.27)

n=1

ˆ in , Gin to denote the values Consider ξ to be the location of node i and use Ci , ui , Y ˆ of C, u, Yn , Gn at node i.Equation (15.27) can now be expressed as N n=1

Yin u n =

N

Gin t n ,

i = 1, 2, . . . , N,

(15.28a)

n=1

in which ˆ in , Yin = Y

for i = n,

ˆ in + Ci , Yin = Y

for i = n.

(15.28b)

When all the nodes are taken into consideration, (15.28) produces a 3 N×3 N system of equations which can be represented in matrix form as

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15

⎡

Y11 ⎢ Y21 ⎢ ⎢ · ⎢ ⎢ · ⎢ ⎢ Yi1 ⎢ ⎢ · ⎢ ⎣ · YN1

Y12 Y22 · · Yi2 · · YN2

··· ··· ··· ··· ··· ··· ··· ···

⎤⎧ ⎫ ⎡ u1 ⎪ G11 Y1 N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ Y2 N ⎥ u ⎪ ⎪ 2⎪ ⎢ G21 ⎥⎪ ⎪ ⎪ ⎪ ⎢ · ⎪ · ⎥ · ⎪ ⎪ ⎥⎪ ⎨ ⎪ ⎬ ⎢ ⎢ · · ⎥ · ⎥ =⎢ ⎢ Gi1 YiN ⎥ u ⎪ ⎪ i⎪ ⎢ ⎥⎪ ⎪ ⎪ · ⎪ ⎪ ⎢ · · ⎥ ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪ · ⎪ ⎪ ⎣ · · ⎦⎪ ⎪ ⎪ ⎪ ⎩ ⎭ YNN uN GN1

G12 G22 · · Gi2 · · GN2

··· ··· ··· ··· ··· ··· ··· ···


⎤⎧ ⎫ t1 ⎪ G1 N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G2 N ⎥ t2 ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ · ⎥ · ⎪ ⎪ ⎪ ⎥⎪ ⎨ ⎬ · ⎥ · ⎥ . GiN ⎥ ti ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪·⎪ ⎪ · ⎥ ⎪ ⎥⎪ ⎪ ⎪ ⎪·⎪ ⎪ · ⎦⎪ ⎪ ⎪ ⎩ ⎪ ⎭ GNN tN

(15.29)

By applying the boundary condition such that either ui or ti at each node is prescribed, the system of equations (15.29) can be reordered in such a way that the final system of equations can be expressed as Kv=p where K is a fully populated matrix, v is a vector containing all the boundary unknowns, and p is a vector containing all the prescribed values given on the boundary. Note that through the system of equations shown in (15.29), a rigid body movement represented by u1 = u2 = · · · = uN = ik , t1 = t2 = · · · = tN = 0, k=1,2,3, where i1 = (1, 0, 0)T , i2 = (0, 1, 0)T , and i3 = (0, 0, 1)T , can give us the following relation: Yii = − Yij , (15.30) i=j

which can be used to check the correctness of the coefficient matrix Ci and the ˆ in calculated from (15.2) and (15.23). matrices of influence coefficients Y Once (15.29) has been solved, all the values of tractions and displacements on the boundary are determined. With this result, the values of stresses and displacements at any interior point can be calculated as follows.

15.1.4 Stresses and Displacements at Internal Points If ξ is an internal point, cij (ξ) in (15.1) becomes δij , i.e., C(ξ) in (15.3a) becomes the unit matrix I. The displacement at any point inside the body can therefore be obtained from (15.27) as u(ξ) =

N ! " ˆ n (ξ)un . Gn (ξ)tn − Y

(15.31)

n=1

The internal strains at point ξ can be found by differentiating (15.31) with respect to xˆ i , i = 1, 2, 3 and using the strain–displacement relation εij = (ui, j + uj, i )/2. For two-dimensional problems considered in this section, we have ε11 = u1, 1 , ε22 = u2, 2 , ε33 = 0, 2ε12 = 2ε21 = u1, 2 + u2, 1 , 2ε13 = 2ε31 = u3, 1 ,

2ε23 = 2ε32 = u3, 2 . (15.32)

15.1


555

From (15.32), we see that all the strain components can easily be calculated if we know u, 1 and u, 2 . Their relations are ⎧ ⎫ ⎨ ε11 ⎬ (1) = u, 1 , ε1 = 2ε12 ⎩ ⎭ 2ε13

⎧ (2) ⎫ ⎨2ε12 ⎬ ε2 = ε22 = u, 2 ⎩ ⎭ 2ε23

(15.33a)

and (1)

(2)

2ε12 = 2ε12 + 2ε12 = iT2 u, 1 + iT1 u, 2 .

(15.33b)

By differentiating (15.31) with respect to xˆ 1 and xˆ 2 , we obtain u, i (ξ) =

N !

" ˆ n, i (ξ)un , Gn, i (ξ)tn − Y

i = 1, 2.

(15.34)

n=1

ˆ n, i are obtained by using (15.23) and (15.26) in which the Note that Gn, i and Y ∗ differentiation of T and U∗ with respect to xˆ 1 and xˆ 2 can be done analytically from the expressions given in (15.16), (15.17), (5.18), (5.19), and (15.20) for different kinds of problems. To find the internal stresses at point ξ, the stress–strain law σij = Cijks εks should be employed. If we let ⎧ ⎫ ⎨σ11 ⎬ σ1 = σ12 , ⎩ ⎭ σ13

⎧ ⎫ ⎨σ21 ⎬ σ2 = σ22 , ⎩ ⎭ σ23

(15.35)

it can be proved that σ1 (ξ) = Qu, 1 (ξ) + Ru, 2 (ξ), σ2 (ξ) = RT u, 1 (ξ) + Tu, 2 (ξ),

(15.36)

where Q, R, and T have been defined in (3.8) and u, 1 and u, 2 are given in (15.34).

15.1.5 Stress Intensity Factors for Crack Problems For crack problems, it is always useful to know the stress intensity factors, which have been defined in (7.4), i.e., ⎧ ⎧ ⎫ ⎫ ⎨ KII ⎬ ⎨σ12 ⎬ + + k = KI = lim 2π (ˆx1 − a) σ22 = lim 2π (ˆx1 − a)σ2 , ⎩ ⎩ ⎭ xˆ 1 →a ⎭ xˆ 1 →a KIII σ23 xˆ 2 =0 xˆ 2 =0

(15.37)

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15


where a is the half-length of the crack and (ˆx1 , xˆ 2 ) is the point ahead of the crack tip. From (15.36)2 and (15.34), we have N ! " ˆ n, 1 (ξ) + TYˆ n, 2 (ξ)]un , (15.38a) σ2 (ξ) = [RT Gn, 1 (ξ) + TGn, 2 (ξ)]tn − [RT Y n=1

where ˆ n, i (ξ) = 1 Y 2 1 Gn, i (ξ) = 2

1

−1 1 −1

T∗, i (ξ)(!1 m + !2 m−1 )dς , (15.38b) U∗, i (ξ)(!1 m

+ !2 m−1 )dς ,

i = 1, 2.

In (15.38b),!1 and !2 are the interpolation functions defined in (15.21b); m and m−1 are the lengths of the mth and (m–1)th element connecting at the nth node; and T∗ and U∗ are the fundamental solutions given in (15.16) as well as (15.6d) and (15.17d) with b = 0 for crack problems. Substituting (15.38) into (15.37), and using proper differentiation and limitation, a closed-form solution for the stress intensity factors k can then be obtained as

k=

N ! " ˆ cn2 ]un , ˆ c + TY [RT Gcn1 + TGcn2 ]tn − [RT Y n1

(15.39a)

n=1

where ˆ cni = lim Y

xˆ 1 →a xˆ 2 =0

Gcni = lim

xˆ 1 →a xˆ 2 =0

+

ˆ n, i (ξ) = 1 2π (ˆx1 − a)Y 2

+

2π (ˆx1 − a)Gn, i (ξ) =

1 2

1

−1

1

−1

Tci (ξ)(!1 m + !2 m−1 )dς , Uci (ξ)(!1 m + !2 m−1 )dς,

i = 1, 2 (15.39b)

and @ ? 1 1 T T −1 A − < ζα > B BA , = √ Im A 1 − ζα πa ? ( @' 3 1 1 T c T T −1 (U2 ) = √ Im A < μ¯ k ζα >B BIk A , < μα > A − 1 − ζα πa k=1 (15.39c) (Uc1 )T

15.1


557

@ ? 2 ζα (s1 + μα s2 ) 2 T T −1 (A (Tc1 )T = √ Im B − B BA ) , (ζα2 − 1)(1 − ζα )2 a πa ? ( @' 3 2 ζα2 (s1 + μα s2 ) T c T T −1 μ¯ k B BIk A . (T2 ) = √ Im B < μα > A − (ζα2 − 1)(1 − ζα )2 a πa k=1 (15.39d) In (15.39d), s1 and s2 are the components of the tangential direction defined in (15.15) and ζα is defined in (15.12a) with b = 0. A matrix with superscript c means that it is a matrix related to crack problems, and the superscript T denotes the transpose of matrix. Note that (15.39) provide a direct method for evaluating the stress intensity factors if the remote displacements and tractions are known on some closed contour containing the crack. The data on the remote closed contour may also be supplied by any other methods like the finite element method or experimental measurement. This is very different to the conventional computation of the stress intensity factors since it is usually obtained from data near the crack tip. In the case of the well-known path-independent J-integral or H-integral, the path is still needed to start and end at the crack surfaces. In (15.39), however, all the data used to calculate k are from the boundary. It is known that very fine meshes near the crack tips are usually needed when using the conventional finite element method to determine the stress intensity factors. This is not only time-consuming but also inaccurate. All these defects have been overcome by the present BEM since the crack does not need to be meshed, and the stress intensity factors are obtained using only the data from the boundary.

15.1.6 Subregion Technique The system of simultaneous linear algebraic equations, (15.29), may also be written in the following simple matrix form: Yu = Gt,

(15.40)

where Y and G are 3N × 3N matrices and u and t are, respectively, 3 N × 1 vectors for the nodal displacements and tractions. With the fundamental solutions obtained from the problems of infinite space, half-space, bimaterial, hole, crack, or inclusion, (15.40) is valid for the analysis of two-dimensional anisotropic elastic solids containing a continuous media, a free surface, an interface, a single hole, crack, or inclusion. To consider anisotropic solids containing multiple free surfaces, interfaces, holes, cracks, and inclusions, the technique of subregions (Brebbia et al., 1984) may be applied, where each subregion contains only one free surface, interface, hole, crack, or inclusion. The final system of equations for the whole region

558

15


is obtained by adding the set of equations (15.40) for each subregion together with compatibility and equilibrium conditions between their interfaces. Consider for simplicity a region consisting of two subregions 1 and 2 (Fig. 15.1). Over subregion 1 , (15.40) may be written as u1 t1 1 1 Y1 Y1I = G , G I u1I t1I

(15.41a)

where the subscript I denotes the interface between these two subregions and the superscript 1 (or 2) denotes the subregion. For subregion 2 , we have u2 t2 2 2 2 2 Y YI = G GI . u2I t2I

(15.41b)

It should be noted that in order to be compatible with the fundamental solution the system of equations for each subregion is obtained with respect to the local coordinate (x(i) , y(i) ) shown in Fig. 15.1. In order to combine the system of equations for the whole region, a commonly based global coordinate (x, y) should be used. The relation between the local and global coordinates can be written as (i)

(i)

x(i) = (x − x0 ) cos θ (i) + (y − y0 ) sin θ (i) , (i)

(15.42)

(i)

y(i) = −(x − x0 ) sin θ (i) + (y − y0 ) cos θ (i) ,

in which the superscript θ (i) denotes the angle between the global and local (i) (i) coordinates, and (x0 , y0 ) is the center of the local coordinate of subregion i (Fig. 15.1).

Γ1

Γ2

y

(1)

y ( 2)

x (1)

θ (1) ( x0( 2) , y0( 2 ) )

(1) 0

( x , y0(1) )

θ ( 2) x ( 2)

y

o

Ω1

x

Ω2

ΓI (interface)

Fig. 15.1 A domain divided into two subregions

15.1


559

Each nodal point is first transformed to the local coordinate and the matrices Y and G are then obtained using the local coordinate system. Before the whole region is assembled from each subregion, the matrices Y and G should be transformed to the matrices Y and G in the global coordinate system. Since the dimension of Y and G of each subregion may be very large, it is suggested that the transformation be performed at the nodal level, not the subregion level. That is, instead of transforming a 3N × 3N matrix, we perform N’s 3 × 3matrix transformations according to Yij = Y ij ,

Gij = G ij ,

Ci = C i ,

(15.43a)

where ⎤ cos θ sin θ 0 = ⎣− sin θ cos θ 0⎦ . 0 0 1 ⎡

(15.43b)

If now (15.41a) and (15.41b) are expressed with respect to the global coordinate system, the compatibility and equilibrium conditions to be applied at the interface I between 1 and 2 are u1I = u2I = uI ,

t1I = −t2I = tI .

(15.44)

The combination of (15.41a,b) and (15.44) leads to the system of equations for the whole region as

Y1 0

Y1I Y2I

⎧ ⎫ ⎧ ⎫ ⎨u1 ⎬ 1 ⎨t1 ⎬ 1 0 G GI 0 u = t . 0 −G2I G2 ⎩ 2I ⎭ Y2 ⎩ 2I ⎭ u t

(15.45)

By imposing the boundary conditions, (15.45) can be reordered in such a way that all the unknowns are written on the left-hand side in a v vector. The final result can be written as Kv = p and then be solved numerically.

15.1.7 Numerical Implementation To help the reader understand the implementation of the numerical technique developed in this section, the computational process is briefly described as follows. (1) Read the input data. The input data are divided into three parts: coordinates of the element nodes, prescribed boundary values, and material properties. Note that all the input data are based upon the global coordinate system. For a corner node connecting two different surfaces which have different tractions, two nodes with the same coordinate are used to represent each corner node. (2) Eigenrelation of Stroh formalism. Calculate the material eigenvalues μα and eigenvectors aα , bα by using the material eigenrelation (3.48). Establish the

560

(3) (4)

(5)

(6)

(7)

(8)

15


material eigenvector matrices A and B by (3.24b)1, 2 which should be normalized using the orthogonality relation (3.57). For degenerate materials such as isotropic materials, a small perturbation of material eigenvalues μα should be made, and then their corresponding eigenvector matrices A and B are obtained approximately. Coordinate transformation. Transform the global coordinate system to the local coordinate system of each subregion according to (15.42). Programming the fundamental solution. Set a subroutine for the evaluation of the fundamental solution T∗ and U∗ provided in (15.16), (15.17), (15.18), and (15.20). Evaluation of the interaction coefficient matrices. Evaluate the interaction coefficient matrices Yij and Gij by (15.2), (15.23), (15.26), and (15.28b). The integrals, including the one with singular terms, are calculated using Gaussian quadrature rules. Coordinate transformation and subregion combination. Transform back to the global coordinate system according to (15.43). Combine all the subregion formulation and generate a system of simultaneous linear algebraic equations for the entire body by using (15.45). Solutions of the simultaneous linear algebraic equations. The Gauss elimination technique is employed to solve the system of equations. All the values of un and tn on the boundary are determined. Calculation of the internal stresses and the stress intensity factors. The internal displacements, strains, and stresses can be calculated by using (15.31), (15.33), (15.34), and (15.36). For the crack problems, the stress intensity factors are calculated by using (15.39).

A computer code based upon the above computational process has been designed and several numerical examples about multiple holes, cracks, and inclusions have been done to illustrate its accuracy, efficiency, and versatility. Detailed descriptions of these numerical examples can be found in Hwu and Liao (1994).

15.2 Two-Dimensional Electro-Elastic Analysis 15.2.1 Boundary Element Formulation From the discussion presented in Chapter 11, we know that by treating electric displacement Di and electric field Ei to be generalized stresses and strains, and expanding elastic tensor Cijkl to include piezoelectric tensors and dielectric permittivities and non-permittivities such as that shown in (11.14), the complete basic equations (11.13) for electromechanical analysis can be rewritten in an expanded tensor notation as (11.15) which possesses exactly the same mathematical form as (3.1) for pure elastic problems. With this understanding, the boundary integral equations for two-dimensional electro-elastic analysis can also be written in the same mathematical form as (15.1), i.e.,

15.2

Two-Dimensional Electro-Elastic Analysis

cij (ξ)uj (ξ) +

tij∗ (ξ, x) uj (x)d(x)

561

=

u∗ij (ξ, x) tj (x)d(x), i, j = 1, 2, 3, 4, (15.46)

in which the first three components of uj and tj have the same physical meaning as that given in (15.1) for pure elastic problems, whereas u4 and t4 are related to the electric displacement and electric field by u4, j = 2ε4 j = −Ej ,

t4 = σ4 j n j = Dj n j = Dn .

(15.47)

u∗ij (ξ, x) and tij∗ (ξ, x) are, therefore, the generalized displacements and generalized tractions in the xj -direction at point x = (x1 , x2 ) corresponding to a unit point force or point charge acting in the xi -direction applied at point ξ = (ˆx1 , xˆ 2 ). Same as the boundary integral equation (15.46), the fundamental solutions u∗ij (ξ, x) and tij∗ (ξ, x) have also exactly the same mathematical forms as those shown in (15.15)-(15.20) for different kinds of problems. The only difference is the content and dimension of the matrices used in (15.15)-(15.20) whose details can be found in Chapter 11. Due to the same mathematical formulation for the pure elastic and electroelastic analysis, all the follow-up works are also the same as those presented in Section 15.1 and hence, will not be repeated here. In the following only two representative examples published by Liang and Hwu (1996) are presented to show its applicability to piezoelectric materials.

15.2.2 Numerical Examples In the following examples, the piezoelectric materials are chosen to be PZT4, whose properties are S11 = 10.9 × 10−12 , S12 = S21 = −2.1 × 10−12 , S13 = S31 = −5.42 × 10−12 , S22 = 7.9 × 10−12 , S33 = 10.9 × 10−12 , S23 = S32 = S12 , S44 = 19.3 × 10−12 , S55 = 32.64 × 10−12 , S66 = S44 ,

(unit : m2 /Nt).

g12 = g21 = −11.1 × 10−3 , g22 = 2.61 × 10−2 , g23 = g32 = g12 , g43 = g34 = g61 = g16 = 39.4 × 10−3 ,

(unit: Volt-m/Nt).

σ β11

σ = β11 ,

= 7.66 × 10 , 7

σ β22

= 8.69 × 10 , 7

σ β33

(unit: Volt2 /Nt), others zero.

Example 1: Holes in Piezoelectric Plates To check the correctness of boundary element modeling, we first consider a square plate containing a circular hole whose a/W = 0.01 in order to compare with the analytical solution obtained in (6.12) and (6.16) or (6.18) for an infinite piezoelectric

562

15


plate containing a hole. The plate is subjected to uniform tension σ0 at two opposite edges. Figure 15.2 is a schematic diagram of the boundary meshes and loading conditions for the present problem. Figures 15.3 and 15.4 show the results of the hoop stress and electric displacement along the circular hole by the present boundary element method, which agree with the analytical solution obtained in (6.12) and (6.16) by expanding the contents of the vectors and matrices to include the piezoelectric properties according to the discussions of Chapter 11. From these results we see that the maximum hoop stress and the maximum electric displacement occur at ψ = 0◦ , 180◦ . The applied stresses of order 107 Nt/m induce only an order of 10−3 Coul/m2 electric displacement and an order of 105 Volt/m electric field. Example 2: Cracks in Piezoelectric Plates Consider a uniformly loaded cracked plate as shown in Fig. 15.5. The applied tensile stress σ = 1 Pa; the plate width W and length L are W = 10 cm, L = 30 cm. The variation of the crack is represented by its center location (d1 , d2 ) and length 2a with unit centimeter. From Fig. 15.5 we see that the larger the crack length a the higher the stress intensity factor KI and KIV . The variation of the crack center location (d1 , d2 ) also influences the stress intensity factors. However, when the crack size is under a certain value such as 2a = 2.0 cm in Fig. 15.5, it is very difficult to distinguish the difference from different locations with the same small crack size. Moreover, the variation of d2 has almost no influence on the value of the stress intensity factor. It is also noted that the values of KIV are in the order of 10−11 Coul/cm3/2 , which is close to zero. KII and KIII are not shown for their relative

Fig. 15.2 Holes in piezoelectric plates subjected to uniform tension

15.2

Two-Dimensional Electro-Elastic Analysis

563

Fig. 15.3 Hoop stress along the hole boundary

Fig. 15.4 Electric displacement along the hole boundary

small values compared to KI . The results that KII , KIII , and KIV are all near to zero are expected by referring to the analytical solution shown in (11.100) and (11.101a) for an infinite plate, which shows that these four different modes of stress intensity factors are uncoupled. Besides the results shown in Fig. 15.5, we would like to mention that if one used the conventional boundary element with fundamental solution given by (15.6a) and (15.17a), the crack boundary should be meshed. Moreover, very fine meshes are usually required near the crack tips in order to achieve high accuracy. On the other hand, by the special boundary element with fundamental solution given by (15.6d) and (15.17d), the discretization around the crack (or hole) is avoided. This results in a saving of computer time and storage. Furthermore, since the fundamental solution (15.16d) and (15.17d) has satisfied the traction-free crack (or hole) boundary condition, discretization along the outer boundary with relatively coarse meshes can achieve high accuracy. To see more clearly about this advantage, a convergence study is done for the case of 2a = 2.0 cm and (d1 , d2 ) = (0, 0). The results presented in Fig. 15.6 show that even the roughest mesh of four elements can achieve good value of KI , and the convergence is made by only 24 elements. This is a very coarse mesh for the crack problems by comparing with any other boundary elements and finite elements.

564

15


Fig. 15.5 Variation of the right tip stress intensity factor versus crack size and location

Fig. 15.6 Convergence of the stress intensity factor

15.3


565

15.3 Coupled Stretching–Bending Analysis 15.3.1 Boundary Integral Equations – Internal Points Unlike the boundary integral equation (15.1) for two-dimensional elastic analysis, which can be found in many textbooks of boundary element method, most of the books or journal papers did not mention about the boundary integral equation for the coupled stretching–bending problems. In order to develop the boundary element for the stress analysis of coupled stretching–bending problems, in this section their associated boundary integral equations derived by Hwu (2010) will be presented as follows. The reciprocal theorem of Betti and Raleigh in terms of stresses and strains can be expressed as

σij εij∗ d =

σij∗ εij d

(15.48)

where σij , εij and σij∗ , εij∗ , i, j = 1, 2, 3, are the stresses and strains induced by two different loading systems on the same elastic body whose region is denoted by . If the elastic body is a thin laminated plate with plane region A, according to the Kirchhoff assumptions the integration of (15.48) with respect to the thickness may leadto A = A

∗ ∗ ∗ ∗ (Nx εx0 + Ny εy0 + Nxy γxy0 + Mx κx∗ + My κy∗ + Mxy κxy + Qx γxz∗ + Qy γyz∗ )dA

∗ ∗ (Nx∗ εx0 + Ny∗ εy0 + Nxy γxy0 + Mx∗ κx + My∗ κy + Mxy κxy + Q∗x γxz + Q∗y γyz )dA,

(15.49)

where (Nx , Ny , Nxy ), (Mx , My , Mxy ), and (Qx , Qy ) are the stress resultants, bending moments, and transverse shear forces; (εx0 , εy0 , γxy0 ), (κx , κy , κxy ), and (γxz , γyz ) are the mid-plane strains, curvatures, and transverse shear strains, which are related to the mid-plane displacements u0 , v0 , and w by ∂v0 ∂u0 ∂v0 ∂u0 , εy0 = , γxy0 = + , ∂x ∂y ∂y ∂x ∂βy ∂βy ∂βx ∂βx κx = , κy = , κxy = + , ∂x ∂y ∂y ∂x ∂w ∂w βx = γxz − , βy = γyz − . ∂x ∂y

εx0 =

(15.50)

With relations (15.50), the surface integral of (15.49) can be reduced to line integral by taking integration by parts, for example, A

∗ Nx εx0 dA

∂u∗ = Nx 0 dA = ∂x A

Nx u∗0 n1 d

− A

∂Nx ∗ u dA, . . . ∂x 0

(15.51)

566

15


in which Γ is the boundary of area Awhose normal direction is denoted by (n1 , n2 ). After taking integration by parts term by term as that shown in (15.51), (15.49) can be separated into two parts. One is a surface integral and the other is a line integral. To have a further reduction for the part of surface integral, we now consider the equilibrium equations of the plates, which are ∂Nxy ∂Nxy ∂Ny ∂Qy ∂Nx ∂Qx + + qx = 0, + + qy = 0, + + q = 0, ∂x ∂y ∂x ∂y ∂x ∂y ∂Mxy ∂My ∂Mxy ∂Mx + = Qx − mx , + = Qy − my , . . . etc., ∂x ∂y ∂x ∂y (15.52a) where qx = mx =

h/2 −h/2 h/2 −h/2

h/2

fx dz + σxz |−h/2 ,

qy =

h/2

fx zdz + (z σxz )|−h/2 ,

h/2 −h/2

my =

0h/2 fy dz + σyz 0−h/2 ,

h/2 −h/2

q=

h/2 −h/2

h/2

fz dz + σz |−h/2 ,

0h/2 fy dz + (z σyz )0−h/2 ,

(15.52b)

and fx , fy , and f are body forces in x-, y- and z-directions. Thus, qx , qy , q and mx , my represent the total distributed loads and moments applied on the upper and bottom surfaces of the plates including the forces/moments induced by the body forces (Fig. 15.7). By employing the equilibrium equations (15.52a), (15.49) can now be reduced to

z or x3

y or x2

Γ

fˆ3

fˆ2

mˆ 1

(+)

corner point

xˆ

mˆ 2

(−) q

my mx

fˆ1

mˆ 3

x or x1

qy

qx

Fig. 15.7 Positive directions of coordinates, integration routes, point forces/moments, and distributed forces/moments

15.3


567

(qx u∗0 + qy v∗0 + qw∗ + mx βx∗ + my βy∗ )dA A + [(Nx n1 + Nxy n2 )u∗0 + (Ny n2 + Nxy n1 )v∗0 ]d ∗ ∗ + [(Mx n1 + Mxy n2 )βx + (My n2 + Mxy n1 )βy ]d + (Qx n1 + Qy n2 )w∗ d ∗ ∗ ∗ ∗ ∗ = (qx u0 + qy v0 + q w + mx βx + my βy )dA A ∗ ∗ + [(Nx∗ n1 + Nxy n2 )u0 + (Ny∗ n2 + Nxy n1 )v0 ]d ∗ ∗ ∗ ∗ + [(Mx n1 + Mxy n2 )βx + (My n2 + Mxy n1 )βy ]d + (Q∗x n1 + Q∗y n2 )wd.

(15.53)

Using the following transformation relations, u0 = un n1 + us n2 ,

v0 = un n2 − us n1 ,

βx = βn n1 + βs n2 , βy = βn n2 − βs n1 , Qn = Qx n1 + Qy n2 , mn = mx n1 + my n2 , Nn = Nx n21 + 2Nxy n1 n2 + Ny n22 , Mn = Mx n21 + 2Mxy n1 n2 + My n22 ,

ms = mx n2 − my n1 ,

Nns = (Nx − Ny )n1 n2 + Nxy (n22 − n21 ), Mns = (Mx − My )n1 n2 + Mxy (n22 − n21 ), (15.54)

Fig. 15.8 Positive directions for Cartesian coordinate system and tangent–normal coordinate system: (x, y) − (s, n), (u0 , v0 ) − (us , un ), (βx , βy ) − (βs , βn ), (mx , my ) − (ms , mn )

y,v0

n, un

mx , β x

ms , β s

s, u s

θ θ

my , β y

x,u0

mn , β n

where the subscripts n and s denote the values in normal and tangential directions (Fig. 15.8), (15.53) can be further reduced to

568

15


(qx u∗0 + qy v∗0 + qw∗ + mn βn∗ + ms βs∗ )dA A + (Nn u∗n + Nns u∗s + Mn βn∗ + Mns βs∗ + Qn w∗ )d = (q∗x u0 + q∗y v0 + q∗ w + m∗n βn + m∗s βs )dA A ∗ ∗ + (Nn∗ un + Nns us + Mn∗ βn + Mns βs + Q∗n w)d.

(15.55)

For thin plates, transverse shear deformations are usually ignored, i.e., γxz = γyz = 0.

(15.56)

With this assumption and the last two equations of (15.50), the rotation angles βx and βy will not be independent but be related to the deflection w by βx = −

∂w , ∂x

βy = −

∂w . ∂y

(15.57)

Using the transformation relations shown in (15.54)3,4 , we have βn = −

∂w , ∂n

βs = −

∂w . ∂s

(15.58)

Substituting (15.58)2 into (15.55) and taking integration by parts as that shown in (15.51), we get

(qx u∗0 + qy v∗0 + qw∗ + mn βn∗ + ms βs∗ )dA 0 + (Nn u∗n + Nns u∗s + Mn βn∗ + Vn w∗ )d − (Mns w∗ )0f 0 = (q∗x u0 + q∗y v0 + q∗ w + m∗n βn + m∗s βs )dA A 0 ∗ ∗ + (Nn∗ un + Nns us + Mn∗ βn + Vn∗ w)d − (Mns w)0f , A

(15.59)

0

where Vn is the effective shear force defined as Vn = Qn +

∂Mns , ∂s

(15.60)

and 0 and f represent, respectively, the starting and final points of the boundary ∗ w value is continuous, the last terms of both sides of (15.59) . If the Mns w∗ or Mns will vanish. Otherwise, the addition of these two terms becomes necessary, which may occur on corners where the Mns value is discontinuous. When the boundary has

15.3


569

many corners, the last terms of both sides of (15.59) may be represented as Nc 0 + − (Mns w∗ )0f = − (Mns − Mns )k w∗k , 0

Nc 0 ∗ ∗+ ∗− (Mns w)0f = − (Mns − Mns )k wk , 0

k=1

k=1

(15.61) where the subscript k stands for the value in the kth corner, the superscripts + and – denote, respectively, the value ahead of and behind the corner and Nc is the number of corners (Fig. 15.7). Through the above derivation, it should be noted that the integral equation (15.55) is applicable for the relatively thick plates whose transverse shear deformation is taken into account, whereas (15.59) is applicable for the thin plates in which the effects of transverse shear deformation is ignored. Since the fundamental solutions to be presented in the next section is obtained for thin laminated plates, in the following derivation we will start from (15.59) instead of (15.55). Consider four independent unit point loads or moment applied at the point ξ inside the body, e.g., the load applied in each of three orthogonal directions ei , i = 1, 2, 3, and the moment applied on the surface with normal n, i.e., q∗x (x) = δ(ξ, x),

q∗y = q∗ = m∗n = m∗s = 0,

or,

q∗y (x) = δ(ξ, x),

q∗x = q∗ = m∗n = m∗s = 0,

or,

q∗ (x) = δ(ξ, x),

q∗x = q∗y = m∗n = m∗s = 0,

or,

m∗n (x) = δ(ξ, x),

q∗x = q∗y = q∗ = m∗s = 0,

(15.62)

where δ(ξ, x) represents the Dirac delta function, ξ is the singular load point, and x ∈ A is the field point. Substituting each point load of (15.62) independently into (15.59) and using (15.61), we get ui (ξ) + =

tij∗ (ξ, x)uj (x)d(x) +

u∗ij (ξ, x)tj (x)d(x) +

Nc

∗ tic (ξ, xk )u3 (xk )

k=1

A

u∗ij (ξ, x)qj (x)dA(x) +

Nc

u∗i3 (ξ, xk )tc (xk ),

(15.63)

k=1

i, j = 1, 2, 3, 4, in which new notations are used for the convenience of later presentation. They are u1 = u0 , u2 = v0 , u3 = w, u4 = βn , t1 = Tx = Nx n1 + Nxy n2 , t2 = Ty = Nxy n1 + Ny n2 , t3 = Vn , t4 = Mn , (15.64) q1 = qx , q2 = qy , q3 = q, q4 = mn , + − tc = Mns − Mns ,

570

15


∗ (ξ, x) , i = 1, 2, 3, j = 1, 2, 3, 4, represent, respectively, and u∗ij (ξ, x) , tij∗ (ξ, x), and tic uj , tj , and tc at point x corresponding to a unit point force acting in the ei direction ∗ (ξ, x), and t∗ (ξ, x) , j = 1, 2, 3, 4, represent applied at point ξ, whereas u∗4j (ξ, x), t4j 4c uj , tj , and tc at point x corresponding to a unit point moment acting on the surface with normal n applied at point ξ. The normal n associated with the boundary (x) is the normal of the surface boundary at point x, while the normal n associated with the interior domain A(x) is the normal compatible with the direction of the distributed moment q4 (x)(= mn (x)) at point x. For example, if a plate is subjected to a uniform moment expressed by mx and my , an equivalent expression with ms = 0, mn = 0 should be found to determine the direction n of the applied moment. Note that in derivation of (15.63), the following relations have been used:

Nn u∗n + Nns u∗s = Tx u∗0 + Ty v∗0 ,

∗ Nn∗ un + Nns us = Tx∗ u0 + Ty∗ v0 .

(15.65)

Since tij∗ (ξ, x) are the stress resultants corresponding to the unit point load applied at ξ directing in ei , when x approaches to ξ they will become singular. Therefore, if ξ goes to the boundary Γ , the integral given in (15.63) should be modified. Detailed derivation of the modified boundary integral equations for smooth or non-smooth boundaries will be given in Section 15.3.3. It should also be noted that the surface integral appears in the second term of the right-hand side of (15.63) will not influence the discretization of boundary element since its integrand contains only known functions. One is the fundamental solution u∗ij provided in the next section, and the other is the given surface load qj . The benefit of boundary element over finite element is that all the unknown basic variables appear on the boundary instead of the entire body. The unknown basic variables of the present chapter are uj , tj , tc , j = 1, 2, 3, 4, and they do not appear in the surface integral of (15.63).

15.3.2 Fundamental Solutions (i) Infinite Laminates Consider an infinite laminate subjected to a concentrated force fˆ = (fˆ1 , fˆ2 , fˆ3 ) and ˆ = (m ˆ 2, m ˆ 3 ) at point xˆ = (ˆx1 , xˆ 2 ). The elasticity solution of this moment m ˆ 1, m problem is generally called Green’s function and has been obtained in explicit closed-form in Section 14.4 through the use of Stroh-like formalism. For the convenience of following discussions, the Green’s function presented in Section 14.4 is now listed below: ud = 2 Re{Af(z)}, where the complex function vector f(z) is

φd = 2 Re{Bf(z)},

(15.66a)

15.3


f(z) =

571

fˆ3 1 < ln(zα − zˆα ) > AT pˆ + < (zα − zˆα )[ln(zα − zˆα ) − 1] > AT i3 2π i 2π i ? @ 1 m ˆ3 AT i2 , + 2π i zα − zˆα (15.66b)

in which ⎫ ⎧ fˆ1 ⎪ ⎪ ⎪ ⎬ ⎨ ˆ ⎪ f2 , pˆ = ⎪ m ˆ ⎪ ⎪ ⎭ ⎩ 2 ⎪ −m ˆ1

⎧ ⎫ 1⎪ ⎪ ⎪ ⎨ ⎪ ⎬ 0 , i1 = 0⎪ ⎪ ⎪ ⎩ ⎪ ⎭ 0

⎧ ⎫ 0⎪ ⎪ ⎪ ⎨ ⎪ ⎬ 1 i2 = , 0⎪ ⎪ ⎪ ⎩ ⎪ ⎭ 0

⎧ ⎫ 0⎪ ⎪ ⎪ ⎨ ⎪ ⎬ 0 i3 = . 1⎪ ⎪ ⎪ ⎩ ⎪ ⎭ 0

(15.66c)

From (15.63) and (15.64) we know that the basic functions for the boundary integral equations are u0 , v0 , w and βn which are different from the generalized displacement vector ud defined in (13.70b) and (13.70c). Hence, to get the fundamental solutions u∗ij (ξ, x), the following two relations are needed: u3 = w = −

β1 dx1 = −

β2 dx2 ,

u4 = βn = β1 n1 + β2 n2 .

(15.67)

∗ (ξ, x ), the following Furthermore, to get the fundamental solutions tij∗ (ξ, x) and tic k relations obtained from (15.64), (13.71), (13.79), (13.80), and (13.81) are needed:

t1 = Tx = N11 n1 + N12 n2 = −φ1, 2 n1 − φ2, 2 n2 , t2 = Ty = N21 n1 + N22 n2 = φ1, 1 n1 + φ2, 1 n2 , t3 = Vn = (sT ψ, s ), s = sT ψ, s + sT ψ, ss

= 2s1 s1 ψ1, 1 + 2s2 s2 ψ2, 2 + (s1 s2 + s1 s2 )(ψ1, 2 + ψ2, 1 ) + s1 (n22 ψ1, 11 − 2n1 n2 ψ1, 12 + n21 ψ1, 22 ) + s2 (n22 ψ2, 11 − 2n1 n2 ψ2, 12 + n21 ψ2, 22 ),

t4 =Mn = nT ψ, s = n1 ψ1, s + n2 ψ2, s = n1 (n2 ψ1, 1 − n1 ψ1, 2 ) + n2 (n2 ψ2, 1 − n1 ψ2, 2 ), 1 + − tc = Mns − Mns = [sT ψ, s − nT ψ, n ]+ − 2 1 1 2 2 ψ1, 1 − n1 n2 (ψ1, 2 + ψ2, 1 ) + n1 − ψ2, 2 ]+ = [ n2 − −. 2 2 (15.68) In the above, s1 and s2 are the first and second component of tangent vector s, and n1 and n2 are the first and second component of normal vector n. They are related to the direction angle θ by that shown in (15.15), and have the following relations: s1 = n2 ,

s2 = −n1 ,

n21 + n22 = s21 + s22 = 1.

(15.69)

572

15


As to s1 and s2 , they can be calculated by s1 =

∂(cos θ ) sin θ n1 ∂s1 = =− = , ∂s R∂θ R R

s2 =

∂(sin θ ) cos θ n2 ∂s2 = = = , ∂s R∂θ R R (15.70)

where R is the radius of curvature of the considered point. If the point is located on a straight boundary, R → ∞ and s1 = s2 = 0. With relations (15.67) and (15.68), and the Green’s functions (15.66), the ∗ (ξ, x ) can now be obtained as fundamental solutions tij∗ (ξ, x), u∗ij (ξ, x), and tic k u∗i1 (ξ, x) = 2Re{iT1 Afi (ξ, x)}, u∗i2 (ξ, x) = 2Re{iT2 Afi (ξ, x)}, u∗ (ξ, x) = −2Re{iT A˜fi (ξ, x)},

i3 3 ∗ T ui4 (ξ, x) = 2Re{nb (x)Afi (ξ, x)}, ∗ ti1 (ξ, x) = −2Re{nTp (x)B < μα > f i (ξ, x)}, ∗ (ξ, x) = 2Re{nTp (x)Bf i (ξ, x)}, ti2 ∗ ti3 (ξ, x) = 2Re{sTb (x)B < s1 + s2 μα > f i (ξ, x)} + 2Re{sT b (x)B < n2 − n1 μα > f i (ξ, x)} + 2Re{sTb (x)B < (n2 − n1 μα )2 > f i (ξ, x)}, ∗ ti4 (ξ, x) = 2Re{nTb (x)B < n2 − n1 μα > f i (ξ, x)}, ∗ tic (ξ, xk ) = 2Re{nTc (xk )Bf i (ξ, xk ) + sTc (xk )B < μα > f i (ξ, xk )},

i = 1, 2, 3, 4, (15.71a)

where 1 2π i 1 f2 (ξ, x) = 2π i 1 f3 (ξ, x) = 2π i 1 f4 (ξ, x) = 2π i f1 (ξ, x) =

< ln(zα − zˆα ) > AT i1 , < ln(zα − zˆα ) > AT i2 , (15.71b) < (zα − zˆα )[ln(zα − zˆα ) − 1] > AT i3 , < ln(zα − zˆα ) > AT nb (ξ)

and ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎧ ⎫ ⎫ 0 ⎪ 0 ⎪ n1 ⎪ 0 ⎪ 0⎪ 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎨ ⎪ ⎨ ⎪ ⎨ ⎪ ⎨ ⎨ ⎬ ⎬ ⎬ ⎬ ⎬ ⎬ 0 0 0 0 0 n2 , nb = , sb = , sb = , nt = 1 , s , np = = t ⎪0 ⎪ ⎪n 1 ⎪ ⎪s1 ⎪ ⎪s1 ⎪ ⎪ 2 − n21 ⎪ ⎪ −n1 n2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎪ ⎩ ⎪ ⎩ ⎪ ⎩ ⎩ 2 1⎪ ⎭ ⎭ ⎭ ⎭ ⎭ 0 n2 s2 s2 −n1 n2 n1 − 2 − − n c = n+ sc = s+ t − nt , t − st , zα = x1 + μα x2 , zˆα = ξ1 + μα ξ2 .

(15.71c)

15.3


573

In the above, the subscripts p, b, t, and c denote, respectively, the values related to the in-plane, bending, twisting, and corner responses. The superscript T, +, and – stand for the transpose of a matrix or vector, and the values ahead of and behind the corner. The normal and tangential vectors, n and s, will depend on the location of x or ξ. The prime (• ), double prime (• ), and the overtilde (˜•) denote, respectively, the first derivative, second derivative, and the integral, for example, 1 < (zα − zˆα )−1 > AT i1 , 2π i −1 < (zα − zˆα )−2 > AT i1 , f1 (ξ, x) = 2π i ˜f1 (ξ, x) = 1 < (zα − zˆα )[ln(zα − zˆα ) − 1] > AT i1 . 2π i

f1 (ξ, x) =

(15.72)

(ii) Laminates with Holes or Cracks Similar to the case discussed previously for the infinite laminates, the fundamental solutions for laminates with holes or cracks can be found through the use of the Green’s functions obtained in Section 14.5, i.e., the solutions shown in (14.79) for ˆ 1, m ˆ 2 , and in (14.85) for case 2 loaded by fˆ3 , and in (14.89) case 1 loaded by fˆ1 , fˆ2 , m for case 3 loaded by m ˆ 3 . By following the steps described between (15.66) and (15.72), we obtain the fundamental solutions for laminates with holes or cracks, which can also be expressed by (15.71a) in which fi (ξ, x), i = 1, 2, 3, 4, are as follows: f1 (ξ, x) = Fi1 , f2 (ξ, x) = Fi2 , f3 (ξ, x) = (F31 + F32 + F33 + F34 + F35 + F36 )i3 , f4 (ξ, x) = Fnb (ξ),

(15.73a)

where 4 1 ¯ T −1 −1 ¯ T ¯ ˆ ˆ F= < ln(ζα − ζ k ) > B BIk A , < ln(ζα − ζα ) > A + 2π i k=1

F31 F32

1 < (zα − zˆα ) ln(ζα − ζˆα ) > AT , = 2π i 4 −1 1 ¯ T, ¯ k < c¯ α > A = < (ζα−1 − ζ¯ˆ k )(1 − γ¯k ζ¯ˆ k ζα ) ln(ζα−1 − ζ¯ˆ k ) > B−1 BI 2π i k=1

F33 F34

1 < cα (ln cα − 1)(ζα − ζˆα ) > AT , = 2π i 4 1 ¯ k < cα (ln cα − 1) > A ¯ T, = < (ζα−1 − ζ¯ˆ k ) > B−1 BI 2π i k=1

F35

1 < cα γα ln(−ζˆα )(ζα−1 − ζˆα−1 ) > AT , =− 2π i

574

15


−1 1 ¯ k < cα γα ln(−ζˆα ) > A ¯ T, < (ζα − ζ¯ˆ k ) > B−1 BI 2π i 4

F36 = −

(15.73b)

k=1

and cα =

1 (a − ibμα ), 2

γα =

a + ibμα . a − ibμα

(15.73c)

(iii) Laminates with Elastic Inclusions Similar to the previous two cases, with the Green’s functions obtained in (14.106)– (14.114), the fundamental solutions for laminates with elastic inclusions can also be expressed by (15.71a) in which A and B are the material eigenvector matrices of the matrix, i.e., A = A1 and B = B1 , and fi (ξ, x) , i = 1, 2, 3, 4, are as follows: f1 (ξ, x) = [FI (ζ ) + FE (ζ )]i1 , f2 (ξ, x) = [FI (ζ ) + FE (ζ )]i2 , f3 (ξ, x) = F3 (ζ )i3 ,

(15.74a)

f4 (ξ, x) = [FI (ζ ) + FE (ζ )]nb (ξ), where FI and FE are identical to those given in (15.8a) and (15.8c) for twodimensional analysis and

F3 (ζ ) =

∞ !

−1 ¯ ¯ − −k ∗k ∗ ¯ ¯∗ < ζαk > E− k + < ζα > B1 (B1 Ek + B2 < γα > Ck − B2 Ck )

"

k=1

(15.74b) in which @ cα γα ζˆα ln(−cα ζˆα ) + AT1 , ζˆα 2ζˆα 8 ' (9 cα −ζˆα γα − + AT1 , Ek = kζˆαk k − 1 (k + 1)ζˆα E− 1 =

?

−1

−1 ∗

C∗k = {G0 − Gk G0 Gk }−1 {T∗k − Gk G0 Tk }, G0 = {M1 + M2 }A2 , − T∗k = −iA−T 1 Ek

(15.74c) k = 1 k = 1, 2, · · · , ∞

Gk = {M1 − M2 }A2 < γα∗k >,

and cα and γα are defined in (15.73c).

15.3


575

15.3.3 Boundary Integral Equations – Boundary Points The boundary integral equation shown in (15.63) is valid only when the unit point force or moment is applied at the point ξ inside the body. In order to derive a general boundary integral equation valid for any location, such as internal, external, boundary, or corner points, we now place the source point ξ on the boundary and for generality, suppose that ξ is a corner point with internal angle ϕ(= ϕ + − ϕ − ) (see Fig. 15.9) and should ξ be on a smooth boundary, we merely set ϕ = π . When the source point ξ is placed on the boundary, the fundamental solutions tij∗ (ξ, x) will become singular when x = ξ. However, even tij∗ (ξ, x) → ∞ when x → ξ, its associated integral tij∗ (ξ, x)uj (x)d(x) in (15.63) may not be infinite in the sense of Cauchy principal value and can be written as

tij∗ (ξ, x)uj (x)d(x)

= lim

ρ→0 ¯ ρ


lim

ρ→0 −ρ

tij∗ (ξ, x)uj (x)d(x). (15.75)

In (15.75) the integral path (= ρ + ∗ ) of the left-hand side is replaced by (= ρ + ∗ ) of the right-hand side, where the paths ρ , ∗ = − ρ = ∗+ + ∗− , and ρ = ρ+ + ρ− are shown in Fig. 15.9. Through the new path ρ + ∗ , the source x2∗ Γ∗+

ξ+ +

Γρ

ϕ+

ξ

n

s

r

θ

ϕ ρ

ϕ− −

Γρ

x1∗ Γρ

ξ−

Γ∗− Fig. 15.9 Coordinates and integration routes for corner point: local Cartesian coordinate system x1∗ − x2∗ (x1∗ = x1 − xˆ 1 , x2∗ = x2 − xˆ 2 ), local polar coordinate system r − ϕ (x1∗ = r cos ϕ, x2∗ = r sin ϕ), and local tangent–normal coordinate system s − n

576

15


point ξ is located outside the body, and hence the first term of the left-hand side of (15.63) vanishes. Moreover, the corner point ξ also disappears and is replaced by two new corners ξ+ and ξ− . Thus, the corner number shown in the boundary integral equation (15.63) should be modified to Nc∗ which excludes the corner point ξ if it happens to be the source point, whereas the contribution of the new corners ξ+ and ξ− will be assigned to the coefficient of free term ui (ξ). Therefore, through the replacement of the integral path shown in (15.75), the boundary integral equation (15.63) can now be modified as i (ξ) +

∗

∗


Nc


k=1

Nc∗ ∗ ∗ = uij (ξ, x)tj (x)d(x) + uij (ξ, x)qj (x)dA(x) + u∗i3 (ξ, xk )tc (xk ), i, j = 1, 2, 3, 4,

A

k=1

(15.76a) where i (ξ) = lim

ρ→0 ρ

∗ ∗ tij∗ (ξ, x)uj (x)d(x) + tic (ξ, ξ+ )u3 (ξ+ ) + tic (ξ, ξ− )u3 (ξ− ). (15.76b)

Note that the first integral on the left-hand side of (15.76a) which comes from the second integral on the right-hand side ∗ of (15.75) is taken in the sense of Cauchy principal value and represented by . The remaining integrals in (15.76a) present no special singularities and can be interpreted in the normal sense of integration. ∗ is Furthermore, from the fundamental solutions provided in (15.71) we see that t43 proportional to 1/ρ 2 when ρ → 0, where ρ is the distance of the integration point x to the source point ξ, and hence 4 (ξ) will not converge. To remedy this situation, like the plate bending problems discussed in Stern (1979), the fourth equations of (15.76a) and (15.76b) should be modified by the replacement of u3 (x) with u3 (x) − u3 (ξ), i.e., ∗

4 (ξ) +

Nc

∗ (ξ, x )[u (x ) − u (ξ)] t4c k 3 k 3

k=1

∗ ∗ ∗ (ξ, x)u (x) + t∗ (ξ, x)[u (x) − u (ξ)] + t∗ (ξ, x)u (x) d(x) + t41 (ξ, x)u1 (x) + t42 2 3 3 4 43 44

=

u∗4j (ξ, x)tj (x)d(x) +

A

∗

u∗4j (ξ, x)qj (x)dA(x) +

Nc

u∗43 (ξ, xk )tc (xk ), j = 1, 2, 3, 4,

k=1

(15.77a)

where 4 (ξ) = lim

ρ→0 ¯ ρ

∗ ∗ ∗ t41 (ξ, x)u1 (x) + t42 (ξ, x)u2 (x) + t43 (ξ, x)[u3 (x) − u3 (ξ)]

15.3


577

∗ ∗ + t44 (ξ, x)u4 (x) d(x) + t4c (ξ, ξ+ )[u3 (ξ+ ) − u3 (ξ)] ∗ + t4c (ξ, ξ− )[u3 (ξ− ) − u3 (ξ)].

(15.77b)

In order to evaluate i (ξ), i = 1, 2, 3, 4 from (15.76b) and (15.77b), we consider ρ be a circular path starting from ξ− and ending at ξ+ . Along this path, x1 − xˆ 1 = ρ cos ϕ, n1 = − cos ϕ,

x2 − xˆ 2 = ρ sin ϕ,

n2 = − sin ϕ,

s1 = − sin ϕ,

s2 = cos ϕ

(15.78)

and ui (x) → ui (ξ) − ρui, n (x) + · · · · · · , i = 1, 2, 3, − u3, n (x) = βn (x) → −[cos(ϕ − α)βs (ξ) + sin(ϕ − α)βn (ξ)], d(x) = ρdϕ,

−

(15.79)

+

ϕ ≤ϕ≤ϕ ,

where α is the value of direction angle θ of the source point ξ. If the source point is a corner, α = (α + + α − )/2 where α + and α − are, respectively, the direction angles of the boundary ahead of and behind the corner point ξ. In Fig. 15.9, the polar angle ϕ of point x on route ρ is related to its direction angle θ by ϕ + (π/2) = θ . Substituting (15.78) into (15.71) we get tij∗ (ξ, x) =

1 tij (ϕ), ρ

1 ti3 (ϕ), ρ2

i, j = 1, 2, 4,

∗ t33 (ξ, x) =

1 t33 (ϕ), ρ

j = 1, 2, 4,

∗ 0 t3c (ξ, x) = t3c (ϕ) ln ρ + t3c (ϕ),

i = 1, 2, 4,

∗ ti3 (ξ, x) =

∗ 0 t3j (ξ, x) = t3j (ϕ) ln ρ + t3j (ϕ), ∗ tic (ξ, x) =

1 tic (ϕ), ρ

(15.80)

where 1 T ti1 (ϕ) = − nTp (x)Im{B < μα μˆ −1 α (ϕ) > A }hi , π 1 T ti2 (ϕ) = nTp (x)Im{B < μˆ −1 α (ϕ) > A }hi , π 1 T ti3 (ϕ) = − sTb (x)Im{B < 1 + (n2 − n1 μα )2 μˆ −2 α (ϕ) > A }hi π 1 T + nTb (x)Im{B < (n2 − n1 μα )μˆ −1 α (ϕ) > A }hi , π 1 T ti4 (ϕ) = nTb (x)Im{B < (n2 − n1 μα )μˆ −1 α (ϕ) > A }hi , π 1 T T T ˆ −1 tic (ϕ) = Im{nTc (xk )B < μˆ −1 α (ϕ) > A + sc (xk )B < μα μ α (ϕ) > A }hi , π i = 1, 2, 4,

578

15


1 t31 (ϕ) = − nTp (x)Im{B < μα > AT }i3 , π 1 t32 (ϕ) = nTp (x)Im{BAT }i3 , π 1 ˆ α (ϕ) > AT }i3 t33 (ϕ) = sTb (x)Im{B < (n2 − n1 μα )2 μˆ −1 α (ϕ) + (n1 + n2 μα ) ln μ π 1 + nTb (x)Im{B < (n2 − n1 μα ) ln μˆ α (ϕ) > AT }i3 , π (15.81a) 1 T n (x)Im{B < (n2 − n1 μα ) > AT }i3 , π b 1 t3c (ϕ) = Im{nTc (xk )BAT + sTc (xk )B < μα > AT }i3 , π t34 (ϕ) =

(15.81b)

1 0 t31 (ϕ) = − nTp (x)Im{B < μα ln μˆ α (ϕ) > AT }i3 , π 1 0 (ϕ) = nTp (x)Im{B < ln μˆ α (ϕ) > AT }i3 , t32 π 1 0 t34 (ϕ) = nTb (x)Im{B < (n2 − n1 μα ) ln μˆ α (ϕ) > AT }i3 , π 1 0 t3c (ϕ) = Im{nTc (xk )B < ln μˆ α (ϕ) > AT + sTc (xk )B < μα ln μˆ α (ϕ) > AT }i3 π (15.81c) and μˆ α (ϕ) = cos ϕ + μα sin ϕ,

h1 = i1 ,

h2 = i2 ,

h4 = nb (ξ).

(15.81d)

Substituting (15.80) into (15.76b) and (15.77b), and using (15.79) with ρ → 0, we get s c 1 (ξ) = k11 u1 (ξ) + k12 u2 (ξ) + k14 βn (ξ) + k14 βs (ξ), s c 2 (ξ) = k21 u1 (ξ) + k22 u2 (ξ) + k24 βn (ξ) + k24 βs (ξ),

3 (ξ) = g33 u3 (ξ), 4 (ξ) = k41 u1 (ξ) + k42 u2 (ξ) + gs44 βn (ξ) + gc44 βs (ξ),

(15.82a)

where 0 g33 = k33 + k3c ,

and

s s s gs44 = k43 + k44 + k4c ,

c c c gc44 = k43 + k44 + k4c

(15.82b)

15.3


kij =

579

ϕ+

tij (ϕ)dϕ,

ϕ−

kijs (ξ) = − kijc (ξ) = −

ϕ+ ϕ− ϕ+

ϕ− +

tij (ϕ) sin(ϕ − α)dϕ,

(15.82c)

tij (ϕ) cos(ϕ − α)dϕ,

i, j = 1, 2, 3, 4,

0 0 0 = t3c (ϕ ) + t3c (ϕ − ), k3c s k4c (ξ) = −t4c (ϕ + ) sin(ϕ + − α) − t4c (ϕ − ) sin(ϕ − − α), c k4c (ξ)

+

+

−

(15.82d)

−

= −t4c (ϕ ) cos(ϕ − α) − t4c (ϕ ) cos(ϕ − α).

By employing the results of (15.82) to (15.76) and (15.77), the modified boundary integral equations can now be written as cij (ξ)uj (ξ) + ci5 (ξ)βs (ξ) + =

u∗ij (ξ, x)tj (x)d(x) +

∗

A

∗


Nc


k=1

u∗ij (ξ, x)qj (x)dA(x) +

Nc∗

u∗i3 (ξ, xk )tc (xk ), i, j = 1, 2, 3, 4,

k=1

(15.83)

in which the first two terms of the left-hand side of (15.83) can be expressed in matrix form as ⎧ ⎫ u1 (ξ) ⎪ ⎤⎪ ⎡ ⎪ ⎪ s c ⎪ k11 k12 0 k14 k14 ⎪ ⎪ ⎪ ⎪ ⎪ u (ξ) ⎪ ⎪ 2 ⎨ ⎬ ⎥ ⎢k s c ⎢ 21 k22 0 k24 k24 ⎥ (15.84) cij (ξ)uj (ξ) + ci5 (ξ)βs (ξ) = ⎢ ⎥ u3 (ξ) . ⎪ ⎣0 0 g33 0 0 ⎦⎪ ⎪ ⎪βn (ξ)⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k41 k42 g43 gs44 gc44 ⎪ ⎩ ⎭ βs (ξ) Note that the extra term g43 related to 4 (ξ), which comes from the terms associated with u3 (ξ) in (15.77a), is g43 = −

∗

∗

∗ t43 (ξ, x)d(x) −

Nc

∗ t4c (ξ, xk ).

(15.85)

k=1

In the next section we will prove that ci5 (ξ) = 0, i = 1, 2, 3, 4 only when the source point ξ is a corner point. In other words, the corner points possess extra degree of freedom βs (ξ) than all the other points, which is not convenient for boundary element formulation. Since in boundary element formulation one corner is usually represented by two nodes, it is suggested that βs (ξ) and βn (ξ) of the corner points be transformed to the normal slopes of the boundaries ahead of and behind

580

15


the corner point ξ, βn+ (ξ), and βn− (ξ). Their transformation can be done by using the following relations: βn+ (ξ) = − sin(α + − α)βs (ξ) + cos(α + − α)βn (ξ), βn− (ξ) = − sin(α − − α)βs (ξ) + cos(α − − α)βn (ξ),

(15.86)

which leads to 1 cos(α + − α)βn− (ξ) − cos(α − − α)βn+ (ξ) , − −α ) 1 sin α + − α)βn− (ξ) − sin(α − − α)βn+ (ξ) . βn (ξ) = + − sin(α − α )

βs (ξ) =

sin(α +

(15.87)

Alternatively, one may also consider the transformation of βs (ξ) and βn (ξ) to the global x–y coordinate, βx (ξ) and βy (ξ), which can be done by the following transformation law:

cos α sin α βx (ξ) βs (ξ) = . − sin α cos α βy (ξ) βn (ξ)

(15.88)

15.3.4 Free-Term Coefficients The free-term coefficients cij (ξ) of the modified boundary integral equations have been provided in (15.84) when the nodal degrees of freedom are set to be (u1 , u2 , u3 , βn , βs ). Substituting (15.87) or (15.88) into (15.82a), the coefficients cij (ξ) for the alternative degrees of freedom can be obtained as follows: cij (ξ)uj (ξ) + ci5 (ξ)βs (ξ) ⎧ ⎫ ⎡ − n+ ⎤ ⎪u1 (ξ) ⎪ ⎡ n ⎪ ⎪ ⎪ k11 k12 0 k14 k14 ⎪ k11 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪u2 (ξ) ⎪ ⎬ ⎢k ⎢k21 k22 0 kn− kn+ ⎥ ⎨ ⎢ 21 24 24 ⎥ u (ξ) =⎢ =⎢ 3 ⎢0 0 g ⎥ ⎪ ⎪ 0 ⎦⎪ ⎪ ⎣ 0 33 0 ⎣ ⎪ βn− (ξ)⎪ ⎪ ⎪ − + ⎪ ⎪ ⎪ k41 ⎩ ⎭ k41 k42 g43 gn44 gn44 ⎪ βn+ (ξ)

⎤ ⎧u (ξ)⎫ ⎪ 1 ⎪ k12 0 g14 g15 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ξ) u ⎬ ⎨ ⎥ 2 k22 0 g24 g25 ⎥ ⎥ u3 (ξ) , ⎪ 0 g33 0 0 ⎦ ⎪ ⎪ βx (ξ)⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ k42 g43 g44 g45 βy (ξ) (15.89a)

where

15.3

Coupled Stretching–Bending Analysis −

581

c 1 s k cos(α + − α) + ki4 sin(α + − α) , i = 1, 2, sin(α + − α − ) i4 c −1 s k cos(α − − α) + ki4 = sin(α − − α) , i = 1, 2, + − sin(α − α ) i4 (15.89b) c 1 + s + g = cos(α − α) + g sin(α − α) , 44 sin(α + − α − ) 44 c −1 g cos(α − − α) + gs44 sin(α − − α) = sin(α + − α − ) 44

n ki4 = +

n ki4

−

gn44

+

gn44 and

c s c s gi4 = ki4 cos α − ki4 sin α, gi5 = ki4 sin α + ki4 cos α, i = 1, 2, c s c g44 = g44 cos α − g44 sin α, g45 = g44 sin α + gs44 cos α.

(15.89c)

Explicit Real-Form Solutions Although the closed-form solutions for the coefficients cij (ξ) have been obtained in (15.84), their calculation depends on the integrals of complex functions given in (15.81). For isotropic plates either in-plane problems or pure bending problems, it is well known that cij = δij /2 for a smooth boundary, cij = δij for an internal point, in which δij is the Kronecker delta, and cij = 0 for a point outside the body (Brebbia et al., 1984). Therefore, with the knowledge of the results for the special cases – isotropic plates, it is interesting to derive the explicit real-form solutions of the coefficients cij (ξ) for the general coupled stretching–bending problems. In order to get the explicit real-form solutions, the identities converting the complex form to real form become important. Because the Stroh-like formalism introduced in Section 13.3 was purposely organized into the form of Stroh formalism, all the identities established for two-dimensional problems can be applied to the present coupling case without any further rigorous proof. With this understanding, for the convenience of readers’ reference several identities which are useful for the present derivation are collected below: 2 Im{BAT } = −ST , 2 Im{B < μα (ϕ) > AT } = −X, 2 Im{B < n2 − n1 μα > AT } = sin ϕST − cos ϕX, T T 2 Im{B < μˆ −1 α (ϕ) > A } = − cos ϕS + sin ϕX(ϕ), T 2 Im{B < μα μˆ −1 α (ϕ) > A } = − cos ϕX + sin ϕY(ϕ), T T 2 Im{B < (n2 − n1 μα )μˆ −1 α (ϕ) > A } = 2 Im{B < μα (ϕ) > A } = −X(ϕ), T 2 T ∗ 2 Im{B < (n2 − n1 μα )2 μˆ −2 α (ϕ) > A } = 2 Im{B < μα (ϕ) > A } = −X (ϕ), T 2 Im{B < (n2 − n1 μα )2 μˆ −1 α (ϕ) > A } = sin ϕX(ϕ) − cos ϕY(ϕ), ˜ 2 Im{B < ln μˆ α (ϕ) > AT } = −π X(ϕ),

˜ 2 Im{B < μα ln μˆ α (ϕ) > AT } = −π Y(ϕ),

582

15


˜ ˜ 2 Im{B < (n1 + n2 μα ) ln μˆ α (ϕ) > AT } = π cos ϕ X(ϕ) + π sin ϕ Y(ϕ), (15.90a) ˜ ˜ − π cos ϕ Y(ϕ), 2 Im{B < (n2 − n1 μα ) ln μˆ α (ϕ) > AT } = π sin ϕ X(ϕ) where n1 = − cos ϕ,

n2 = − sin ϕ,

μˆ α (ϕ) = cos ϕ + μα sin ϕ,

μα (ϕ) =

− sin ϕ + μα cos ϕ , cos ϕ + μα sin ϕ

(15.90b)

X = NT1 ST + N3 H = ST NT1 − LN2 , X3 = N3 S − NT1 L = ST N3 − LN1 , X(ϕ) = NT1 (ϕ)ST + N3 (ϕ)H = ST NT1 (ϕ) − LN2 (ϕ), ˜ ˜ T (ϕ)ST + N ˜ T (ϕ) − LN ˜ 3 (ϕ)H = ST N ˜ 2 (ϕ), X(ϕ) =N 1 1 ∗

T ∗ X (ϕ) = N∗T 1 (ϕ)S + N3 (ϕ)H Y(ϕ) = XNT1 (ϕ) + X3 N2 (ϕ), ˜ 2 (ϕ), ˜ ˜ T1 (ϕ) + X3 N Y(ϕ) = XN

=S

T

(15.90c)

∗ N∗T 1 (ϕ) − LN2 (ϕ),

and T T N∗T 1 (ϕ) = N3 (ϕ)N2 (ϕ) + N1 (ϕ)N1 (ϕ),

N∗2 (ϕ) = N1 (ϕ)N2 (ϕ) + N2 (ϕ)NT1 (ϕ), N∗3 (ϕ) = N3 (ϕ)N1 (ϕ) + NT1 (ϕ)N3 (ϕ), ϕ ˜ i (ϕ) = 1 Ni (ω)dω, i = 1, 2, 3. N π 0

(15.90d)

In the above, Ni and Ni (ϕ), i = 1, 2, 3, are the fundamental elasticity matrices and the generalized fundamental elasticity matrices, respectively, which are related by Ni (0) = Ni ; S, H, and Lare Barnett–Lothe tensors. Using the identities shown in (15.90), the real-form solutions of 0 (ϕ) of (15.81a), (15.81b), and (15.81c) for general composite tij (ϕ), tic (ϕ), t3c ∗ , can now be laminates, which are related to the fundamental solutions tij∗ , tic obtained as tij (ϕ) = pij (ϕ), i = 1, 2, j = 1, 2, 3, 4, t4j (ϕ) = −p3j (ϕ) sin α + p4j (ϕ) cos α, j = 1, 2, 3, 4, tic (ϕ ± ) = ±pic (ϕ ± ), i = 1, 2, ±

±

(15.91a) ±

t4c (ϕ ) = ∓p3c (ϕ ) sin α ± p4c (ϕ ) cos α,

15.3


583

1 {cos ϕX13 + sin ϕX23 } , 2π 1 {cos ϕS31 + sin ϕS32 } , t32 (ϕ) = 2π 1 X33 (ϕ) + Y43 (ϕ) − cos 2ϕ[X33 (ϕ) − Y43 (ϕ) + 2π (X˜ 43 (ϕ) + Y˜ 33 (ϕ))] t33 (ϕ) = − 4π − sin 2ϕ[X43 (ϕ) + Y33 (ϕ) − 2π (X˜ 33 (ϕ) − Y˜ 43 (ϕ))] , 1 {S34 − X33 − cos 2ϕ[S34 + X33 ] + sin 2ϕ[S33 − X43 ]} , t34 (ϕ) = − 4π 1 cos 2ϕ ± [S33 − X43 ] + sin 2ϕ ± [S34 + X33 ] , t3c (ϕ ± ) = ∓ 2π 1 0 (ϕ ± ) = ∓ cos 2ϕ ± [X˜ 33 (ϕ) − Y˜ 43 (ϕ)] + sin 2ϕ ± [X˜ 43 (ϕ) + Y˜ 33 (ϕ)] , t3c 2 (15.91b)

t31 (ϕ) = −

where 1 {X1i − Y2i (ϕ) + cos 2ϕ[X1i + Y2i (ϕ)] + sin 2ϕ[X2i − Y1i (ϕ)]} , 4π 1 {Si1 − X2i (ϕ) + cos 2ϕ[Si1 + X2i (ϕ)] + sin 2ϕ[Si2 − X1i (ϕ)]} , pi2 (ϕ) = 4π 1 ∗ ∗ cos ϕ[Si4 + X3i (ϕ) + X4i (ϕ)] − sin ϕ[Si3 − X4i (ϕ) + X3i (ϕ)] , pi3 (ϕ) = 2π 1 {cos ϕX3i (ϕ) + sin ϕX4i (ϕ)} , pi4 (ϕ) = 2π 1 {cos 2ϕ cos ϕ[Si3 − X4i ] + sin 2ϕ cos ϕ[Si4 + X3i ] pic (ϕ) = − 2π − cos 2ϕ sin ϕ[X3i (ϕ) − Y4i (ϕ)] − sin 2ϕ sin ϕ[X4i (ϕ) + Y3i (ϕ)]} , pi1 (ϕ) = −

i = 1, 2, 3, 4. (15.91c) In (15.91b) and (15.91c), Sij , Xij , Xij (ϕ), X˜ ij (ϕ), Xij∗ (ϕ), Yij (ϕ), Y˜ ij (ϕ), i, j = 1, 2, 3, 4, ˜ ˜ defined are the components of the real matrices S, X, X(ϕ), X(ϕ), X∗ (ϕ), Y(ϕ), Y(ϕ) in (15.90c) and (15.90d). Note that the complex-form solutions shown in (15.81) are valid for the laminates whose material eigenvector matrices A and B exist, i.e., the material eigenvalues μα are distinct or a complete set of independent eigenvectors exists when μα are repeated. For the degenerate laminates whose material eigenvalues are repeated and their associated eigenvectors are dependent on each other, a small perturbation of the material constants is usually made for the numerical calculation to yield a complete set of independent material eigenvectors. On the other hand, when we apply the real-form solutions obtained in (15.91), the calculation of the material eigenvalues and eigenvectors has been avoided. Hence, the explicit real-form solutions (15.91)

584

15


are valid for any kind of laminated plates including the degenerate laminates such as the isotropic plates discussed in next section. Isotropic Plates By using the explicit expressions shown in Sections 13.3.4, 13.3.5, 13.3.6 and 13.3.7 for the isotropic plates, the results of (15.91a), (15.91b), and (15.91c) can be further reduced to 1 1+ν {2 + (1 + ν) cos 2ϕ} , t12 (ϕ) = t11 (ϕ) = sin 2ϕ, 4π 4π 1+ν 1 {2 − (1 + ν) cos 2ϕ} , sin 2ϕ, t22 (ϕ) = t21 (ϕ) = 4π 4π 1 1+ν {2 + (1 − ν) cos 2ϕ} , t34 (ϕ) = − , t33 (ϕ) = 4π 4π 3−ν 1+ν sin(ϕ − α), t44 (ϕ) = − sin(ϕ − α), t43 (ϕ) = − 4π 4π t13 (ϕ) = t14 (ϕ) = t23 (ϕ) = t24 (ϕ) = t31 (ϕ) = t32 (ϕ) = t41 (ϕ) = t42 (ϕ) = 0, (15.92a) t3c (ϕ + ) = t3c (ϕ − ) = 0, 1−ν 1−ν cos(ϕ + − α), t4c (ϕ − ) = cos(ϕ − − α), t4c (ϕ + ) = − 2π 2π 1−ν 1−ν 0 0 sin 2ϕ + , t3c sin 2ϕ − . (ϕ + ) = − (ϕ − ) = t3c 4π 4π

(15.92b)

Substituting (15.92) into (15.82c) and (15.82d), we get 1+ν 1+ν ϕ ϕ + (sin 2ϕ + − sin 2ϕ − ), k22 = − (sin 2ϕ + − sin 2ϕ − ), 2π 8π 2π 8π 1+ν = k21 = − (cos 2ϕ + − cos 2ϕ − ), 8π 1−ν ϕ 1+ν + (sin 2ϕ + − sin 2ϕ − ), k34 = − ϕ, = 2π 8π 4π (15.93a)

k11 = k12 k33

3−ν 2ϕ − sin 2(ϕ + − α) + sin 2(ϕ − − α) , 16π 3−ν cos 2(ϕ + − α) − cos 2(ϕ − − α) , =− 16π 1+ν 2ϕ − sin 2(ϕ + − α) + sin 2(ϕ − − α) , = 16π 1+ν cos 2(ϕ + − α) − cos 2(ϕ − − α) , =− 16π 1+ν 2ϕ − sin 2(ϕ + − α) + sin 2(ϕ − − α) , = 16π

s k43 = c k43 s k44 c k44 s k44

15.3


1+ν cos 2(ϕ + − α) − cos 2(ϕ − − α) , 16π 1−ν s sin 2(ϕ + − α) − sin 2(ϕ − − α) , = k4c 4π 1−ν c cos 2(ϕ + − α) − cos 2(ϕ − − α) , = k4c 4π k13 = k14 = k23 = k24 = k31 = k32 = k41 = k42 = 0,

585

c k44 =−

(15.93b)

s c s c 0 k14 = k14 = k24 = k24 = k3c = 0,

in which k11 , k12 , k21 , and k22 agree with the coefficients for plane stress problems (Hartmann, 1983). Note that Hartmann’s solution was presented for the plane strain case, which should be converted to the plane stress case with ν replaced by ν/(1+ν). By (15.82b), we have 1−ν ϕ + (sin 2ϕ + − sin 2ϕ − ), 2π 8π ν ϕ − sin 2(ϕ + − α) − sin 2(ϕ − − α) , = 2π 4π ν cos 2(ϕ + − α) − cos 2(ϕ − − α) , =− 4π

g33 = gs44 gc44

(15.94)

which conceptually agree with the solutions presented by Stern (1979) for isotropic plates. Here, the word “conceptually” is used because the Green’s functions used in our problem are not exactly the same as those used by Stern. From the explicit realform solutions shown in (14.61) for the isotropic plates, we see that the deflection w associated with the transverse load fˆ3 is w=

fˆ3 r2 {2(ln r − 1) − cos 2ϕ} , 16π D

(15.95)

which is different from w = fˆ3 r2 ln r/8π D given by Stern (1979). Moreover, the fundamental solution of the deflection w associated with the bending moment m ˆ n, ˆ n cos α, m ˆ 2 = −m ˆ n sin α, is or say m ˆ 1 = −m w=

m ˆ nr {sin(ϕ − α) ln r + cos ϕ sin α} , 4π D

(15.96)

which is different from w = sin(ϕ − α)r ln r/2π D given by Stern (1979). Knowing that the non-logarithmic terms will not influence the effectiveness of the Green’s functions, the fundamental solutions used here as well as those used by Stern (1979) should all be applicable to their corresponding problems. From Fig. 15.9 we see that at point ξ+ the polar angle ϕ + and the direction angle α + are related by ϕ + = α + (some other cases they may be related by ϕ + = α + − 2π ), whereas at point ξ− , ϕ − = α − − π . If the point moment is applied in the direction α = (α + + α − )/2, it can be proved that gc44 = 0. Together with the results

586

15


obtained in the last equation of (15.93b), we see that all the coefficients related to the extra degree of freedom βs (ξ) are zero for isotropic plates. For a smooth boundary, we have ϕ + = ϕ − +π . With this relation, the coefficients obtained in (15.93) and (15.94) can be further reduced, which leads to the wellknown result that cij = δij /2. Similarly, cij = δij for an internal point and cij = 0 for an external point can also be proved by letting ϕ + = ϕ − for a closed contour. Knowing the coefficients cij (ξ) for the nodal degrees of freedom (u1 , u2 , u3 , βn , βs ), the coefficients cij (ξ) for the alternative degrees of freedom (u1 , u2 , u3 , βn− , βn+ ) and (u1 , u2 , u3 , βx , βy ) can then be obtained by substituting (15.93) and (15.94) into (15.89b) and (15.89c). Their results are −

+

−

+

n = kn = kn = kn = 0, k14 14 24 24 − 1 n 2ϕ sin(α + − α) − ν[cos(α + − α) − cos(2α − − α + − α)] , g44 = + − 4π sin(α − α ) + −1 2ϕ sin(α − − α) + ν[cos(α − − α) − cos(2α + − α − − α)] , gn44 = + − 4π sin(α − α )

(15.97a) g14 = g15 = g24 = g25 = 0, ν ϕ sin α − cos(2ϕ + − α) − cos(2ϕ − − α) , g44 = − 2π 4π ν ϕ cos α − sin(2ϕ + − α) − sin(2ϕ − − α) . g45 = 2π 4π

(15.97b)

If the point moment is applied in the direction α = (α + +α − )/2, it can be proved that ϕ + ν sin ϕ − + , (15.98) gn44 = gn44 = 4π sin(ϕ/2) which also conceptually agree with the solutions presented by Stern (1979). Calculated from Rigid Body Motion Usually, in boundary element formulation even no explicit closed-form solutions are provided for the coefficients cij (ξ), they still can be computed by applying the boundary integral equations to represent rigid body movements. With this understanding, we now consider the following five different rigid body movements: i. u1 = 1, u2 = u3 = βn = βs = 0,

for all x,

ii. u2 = 1, u1 = u3 = βn = βs = 0, for all x, iii. u3 = 1, u1 = u2 = βn = βs = 0, for all x, iv. βn = 1, u1 = u2 = βs = 0, for all x, u3 (ξ) = 0, v. βs = 1, u1 = u2 = βn = 0, for all x, u3 (ξ) = 0.

(15.99)

15.3


587

Fig. 15.10 Transformation between two different local tangent–normal coordinates

n

s* n

s

y

α

ξ

s*

n *

n

s

*

θ x

α θ

x

Since βn and βs defined in (15.58) are, respectively, the negative slopes in the normal and tangent directions, they are dependent on the direction of the boundary. Through the vector transformation, we have (see Fig. 15.10) cos α − sin α βs (ξ) βx (ξ) = , sin α cos α βy (ξ) βn (ξ) βs∗ (ξ) cos(α − θ ) − sin(α − θ ) βs (ξ) = . βn∗ (ξ) βn (ξ) sin(α − θ ) cos(α − θ )

(15.100)

Substituting the conditions set in the last two rigid body movements of (15.99) into the first relation of (15.100), and integrating with respect to x1 and x2 , we can get u3 (x) for the whole plate. Also, substituting (15.99)4,5 into (15.100)2 , we can get u4 (x), i.e., βn∗ (x) for all x along the boundary Γ . Their final expressions are iv. u3 = u04 (x), u4 = βn∗ = cos(α − θ ), for all x, v. u3 = u05 (x), u4 = βn∗ = sin(α − θ ), for all x,

(15.101a)

u04 (x) = (x1 − ξ1 ) sin α − (x2 − ξ2 ) cos α, u05 (x) = −(x1 − ξ1 ) cos α − (x2 − ξ2 ) sin α.

(15.101b)

where

Knowing that tj = qj = tc = 0, j = 1, 2, 3, 4 for rigid body movements, substitution of (15.99) and (15.101) into the modified boundary integral equations (15.83) will then give us

588

15

ci1 (ξ) = − ci2 (ξ) = −

∗

∗

ci3 (ξ) = −

∗

ci4 (ξ) = − ci5 (ξ) = −

∗ ti1 (ξ, x)d(x), ∗ ti2 (ξ, x)d(x), ∗

∗ ti3 (ξ, x)d(x) −

Nc

∗ tic (ξ, xk ),

k=1

∗ ∗


∗

∗ ∗ [ti3 (ξ, x)u04 (x) + ti4 (ξ, x) cos(α

− θ )]d(x) −

Nc

∗ tic (ξ, xk )u04 (xk ),

k=1 ∗ ∗ [ti3 (ξ, x)u05 (x) + ti4 (ξ, x) sin(α − θ )]d(x)

∗

−

Nc

∗ tic (ξ, xk )u05 (xk ), i = 1, 2, 3, 4.

k=1

(15.102) After getting the coefficients cij (ξ), i = 1, 2, 3, 4, j = 1, 2, 3, 4, 5 for the modified boundary integral equations (15.83), the alternative coefficients for different degrees of freedom shown in (15.89a) can then be calculated through the relations shown in (15.89b) and (15.89c). From the results shown above, we see that the free-term coefficients cij (ξ) can be calculated from (15.82) or (15.102). The former is an integral around the source point ξ whose integrands tij (ϕ) have been obtained explicitly in (15.81) with complex form and in (15.91) with real-form, whereas the latter is an integral around the whole boundary of the body and is obtained from the application of five different rigid body movements. From the computational viewpoint, (15.82) is more efficient and accurate than (15.102). While the relations shown in (15.102) are the good checking points for the boundary element coding especially when the fundamental solutions obtained in Section 15.3.2 are expressed in terms of the multivalued logarithmic functions of complex variables. With the boundary integral equations and the explicit closed-form fundamental solutions derived in this section, the boundary element for the coupled stretching– bending analysis of composite laminates can then be established and coded by following the steps described in Section 15.1.3.

Appendix A Symbols, Sign Convention, and Units

A.1 Common Symbols Aij , Bij , Dij or Aijkl , Bijkl , Dijkl or A, B, D: extensional, coupling, bending stiffnesses ~ ~ ~ Aij , Bij , Dij : detailed in extended symbols 2 Aijt , Bijt , Dijt : detailed in extended symbols 2 Aijh , Bijh , Dijh : detailed in extended symbols 2 ~ ~ Aijt , Bijt : detailed in extended symbols 2

~ ~ ~ A, B, D : detailed in extended symbols 2

A,B : material eigenvector matrices a k , b k : material eigenvectors

Cij or Cijkl or C: elastic stiffness E CijE or Cijkl : elastic stiffness at constant electric field

c, d or ct , d t : heat eigenvectors c h , d h : moisture eigenvectors

D j : electric displacement Dij : bending stiffness (used in coupling problem) D : detailed in extended symbols 8 (used in 2D problem) E : Young’s modulus of isotropic materials Ek : electric field, sometimes used as Young’s modulus of k-direction Ek( 0 ) : mid-plane electric field Ek(1) : rate of electric field change C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7, C Springer Science+Business Media, LLC 2010

589

590

Appendix A

eij or ekij : piezoelectric stress tensors fî : prescribed point force fα ( zα ) : elasticity complex functions f ( z ) : complex function vector G: energy release rate, sometimes used as shear modulus

G1 (θ ), G 3 (θ ) : detailed in extended symbols 7 g ( zt ) or g t ( zt ) : thermal complex function g h ( z h ) : hygrocomplex function g ij or g ijk : piezoelectric strain/voltage tensor H : moisture content

H 0 : mid-plane moisture content

H * : rate of moisture content change hi : heat flux (used in 2D problem) h* : heat flux in surface normal direction (2D problem) I : unit matrix

I1 , I 2 , I 3 : detailed in extended symbols 3 i1 , i 2 , i 3 : base vectors, detailed in extended symbols 3

i = − 1 : pure imaginary number

Im: imaginary part of a complex number K I , K II , K III : stress intensity factors K ijt , K ijh , K ij*t , K ij*h : detailed in extended symbols 2

kij or kijt : heat conduction coefficient kijh : moisture diffusion coefficients

k = ( K II , K I , K III )T : vector of stress intensity factors, sometimes used as coefficient vector

M ij : bending moments (used in coupling problems)

591

Appendix A

M : impedance matrix, detailed in extended symbols 8

M * : bimaterial matrix, detailed in extended symbols 8 ( mi : moisture transfer resultant (used in hygroproblem)

mˆ i : prescribed moisture transfer resultant (used in hygroproblems) or prescribed bending moment (used in coupling problems)

~ : surface resultant moments m i

mˆ : prescribed bending moment m n , m s : moment on the surface with normal n and s

N ij : stress resultants N : fundamental elasticity matrix N1 , N 2 , N 3 : submatrices of fundamental elasticity matrix N m : fundamental elasticity matrix of mixed formalism N(θ ) : detailed in extended symbols 6 ˆ (θ ), N ˆ (θ , α ) : detailed in extended symbols 6 N ~ ~ N (θ ), N (θ , α ) : detailed in extended symbols 6 ( N (θ ) : detailed in extended symbols 6

{N 3 (θ )} , N 3

< −1>

: detailed in extended symbols 6

n : normal direction, detailed in extended symbols 3 Qi : transverse shear force ( qi : heat flux resultant (used in coupling problem, same as hi in 2D problem) qî : prescribed heat flux resultant

Qij or Q: elastic stiffness (used for lamina) Q, R, T : matrices related to material properties Q m , R m , Tm : Q, R, T of mixed formalism r , θ : polar coordinate system

Re: real part of a complex number Sîj : reduced elastic compliances

592

Sij or S: elastic compliances S, H, L: Barnett–Lothe tensors S* , H * , L* : detailed in extended symbols 7 ~ ~ ~ S, H, L : detailed in extended symbols 7 s : tangential direction, detailed in extended symbols 3 T : temperature T 0 : mid-plane temperature T * : rate of temperature change

ti : surface traction

~ ti : surface resultant force t : surface traction vector t n , t s : traction on the surface with normal n and s t1∞ 　 , t ∞2 : detailed in extended symbols 9

tr : trace of the matrix U i : displacement (used in coupling problem) ui : mid-plane displacement or displacement (2D problem)

u : displacement vector u d : generalized displacement vector (coupling problem) Δu : crack opening displacement

Vi : effective transverse shear force

W : detailed in extended symbols 8 w : deflection, displacement in x3-direction

x1 , x2 , x3 or x,y,z : rectangular coordinate system zα = x1 + μα x2 : anisotropic complex variables zt = x1 + τx2 : heat complex variables zˆα : prescribed value of zα

α : thermal expansion coefficient of isotropic material

Appendix A

Appendix A

α ijt : coefficients of thermal expansion α ijh : coefficients of moisture expansion α ih , α ti , α *i h , α *i t : detailed in extended symbols 5

β i : negative slope in xi -direction (coupling problem) β ij : thermal moduli (used in 2D thermal problems) β ijσ : dielectric non-permittivities at constant stress (used in piezoelectric problems) β : slope vector (used in coupling problem)

β, β1 , β 2 , β3 : vector of thermal moduli, detailed in extended symbols 5 (used in 2D problem)

γ , γ (θ ), γ t , γ h , ~ γ i* : detailed in extended symbols 5

ε ij : mid-plane strains or strains (2D problem) ε1∞ , ε ∞2 : detailed in extended symbols 9

ζ α : detailed in extended symbols 10 ζˆα : prescribed value of ζ α η or ηt : heat eigenvector ηh : moisture eigenvector

κ ij : curvature μ or μα : material eigenvalue μ (θ ) : detailed in extended symbols 4 μˆ (θ ), μˆ (θ , α ) : detailed in extended symbols 4 μ~ (θ ), μ~ (θ , α ) : detailed in extended symbols 4

ν : Poisson’s ratio ξ ij : strains (used in coupling problems) ξ = (a, b)T : material eigenvector

σ = eiψ : point on the circular boundary

593

594

Appendix A

σ ij : stresses τ or τ t : heat eigenvalue τ h : moisture eigenvalue τ (θ ),τˆ(θ ),τˆ(θ , α ) : detailed in extended symbols 5 ϕ : hole boundary parameter (used in coupling problem) φi : stress function φ d : generalized stress function vector (coupling problem) φ : stress function vector

ψ : hole boundary parameter (used in 2D problem) ψ i : bending stress function ψ : bending stress function vector

ω εjk : dielectric permittivities at constant strains

A.2 Extended Symbols 1. Superscripts, subscripts, etc. •T : transpose of vector or matrix • −1 : inverse of matrix • k : kth power of variable or function or matrix •( k ) or • k : variable or function or matrix of the kth material or sometimes used as the kth sub-matrix

•′ : differentiation of function

• : complex conjugate •ˆ : usually used as a prescribed value < •α >: diagonal matrix whose component is varied according to its subscript α

2. Extension, coupling, bending stiffnesses, etc.: ~ ~ ~ ~ ~ Aij , Bij , Dij , Aijt , Bijt , Dijt , K ijt , K ij*t , Aijt , Bijt , K t , K h

595

Appendix A

~ ~ ~ ~ ~ ~ Aij , Bij , Dij : related to Aij , Bij , Dij by A = A − BD −1 B, B = BD −1 , D = D −1 ~ ~ Aijt , Bijt , Dijt , Aijh , Bijh , Dijh , K ijt , K ijh , K ij*t , K ij*h , Aijt , Bijt , K t , K h : defined as follows h/2

Aijks = ∫−h / 2 Cijks dx3 ,

h/2

Bijks = ∫−h / 2 Cijks x3 dx3 ,

h/2

Bijt = ∫−h / 2 Cijksα kst x3 dx3 ,

h/2

Bijh = ∫−h / 2 Cijksα ksh x3 dx3 ,

Aijt = ∫−h / 2 Cijksα kst dx3 , Aijh = ∫− h / 2 Cijksα ksh dx3 , h/2

h/2

Dijt = ∫−h / 2 Cijksα kst x32 dx3

h/2

Dijh = ∫−h / 2 Cijksα ksh x32 dx3

h/2

K ijt = ∫−h / 2 k ijt dx3 , K ij*t = ∫−h / 2 k ijt x3 dx3 ,

~ ~ Aijt = Aijt − Bijkl Bklt ,

~ ~ Bijt = Dijkl Bklt ,

t K t = K 22 (τ t − τ t ) / 2i,

h/2

Dijks = ∫−h / 2 Cijks x32 dx3 h/2

h/2

h/2

h/2

K ijh = ∫−h / 2 k ijh dx3 , K ij*h = ∫− h / 2 k ijh x3 dx3

h K h = K 22 (τ h − τ h ) / 2i

3. Base vectors, unit matrix, and their extensions: i k , s, n, I , I k ⎧− sin θ ⎫ ⎧1⎫ ⎧0⎫ ⎧0⎫ ⎧cos θ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i1 = ⎨0⎬　 , i 2 = ⎨1⎬, i 3 = ⎨0⎬　 , s(θ ) = ⎨ sin θ ⎬, n(θ ) = ⎨ cosθ ⎬ ⎪ 0 ⎪ ⎪0⎪ ⎪0⎪ ⎪1⎪ ⎪ 0 ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎡1 0 0⎤ ⎡1 0 0⎤ ⎡0 0 0 ⎤ ⎡0 0 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ I = ⎢0 1 0⎥, I1 = ⎢0 0 0⎥ , I 2 = ⎢0 1 0⎥ , I 3 = ⎢⎢0 0 0⎥⎥ ⎢⎣0 0 1⎥⎦ ⎢⎣0 0 0⎥⎦ ⎢⎣0 0 0⎥⎦ ⎢⎣0 0 1⎥⎦

Note: the dimensions of the above vectors or matrices may change depending on the problems, for example,

sT = (cosθ , sin θ ), nT = (− sin θ , cos θ ) for inplane problem sT = (cosθ , sin θ , 0), nT = (− sin θ , cos θ , 0) for in-plane anti-plane coupling sT = (cosθ , sin θ , 0, 0), nT = (− sin θ , cos θ , 0, 0) for electromechanical problem where the angle θ is directed counterclockwise from the positive x1-axis to the direction of s

596

Appendix A

4. Material eigenvalue: μ , μ (θ ), μˆ (θ ), μ~ (θ ), μˆ (θ , α ), μ~ (θ , α ) μ (θ ) =

μ cos θ − sin θ μ sin θ + cos θ

μˆ (θ , α ) = cos(θ − α ) + sin(θ − α ) μ ( α ), μ = μ (0 ), μˆ

μˆ (θ ) = μˆ ( θ , 0 ),

μ% ( θ , α ) =

1 θ ∫ μ (ω ) dω

π α

μ% (θ ) = μ% ( θ , 0 ),

−1 (θ ) = μˆ (0, θ ) = cos θ − μ ( θ ) sin θ

5. Heat eigenvalue and its extension: τ ,τ (θ ),τˆ(θ ),τˆ(θ , α ) , β1 , β 2 , β , ~ γ1* , ~ γ 2* , etc.

Two-dimensional problems: τ cos θ − sin θ τ (θ ) = τ sin θ + cos θ τˆ(θ , α ) = cos(θ − α ) + sin(θ − α )τ (α ),

τ = τ (0),

τˆ(θ ) = τˆ(θ ,0) = cos θ + τ sin θ

⎡0 N 2 ⎤ ⎧ β1 ⎫ ⎧ γ1 (θ ) ⎫ ⎡0 N 2 (θ ) ⎤ ⎧β1 (θ ) ⎫ ⎧c ⎫ η = ⎨ ⎬, γ = − ⎢ ⎬, γ (θ ) = ⎨ ⎬ = −⎢ ⎬, ⎥⎨ T ⎥⎨ T I N β γ θ ( ) d 1 ⎦⎩ 2 ⎭ ⎩ ⎭ ⎣ ⎩ 2 ⎭ ⎣ I N1 (θ )⎦ ⎩β 2 (θ )⎭ ⎧ β11 ⎫ ⎧ β12 ⎫ ⎪ ⎪ ⎪ ⎪ β1 = ⎨β 21 ⎬, β 2 = ⎨β 22 ⎬ ⎪β ⎪ ⎪β ⎪ ⎩ 31 ⎭ ⎩ 32 ⎭ β1 (θ ) = βs(θ ), β 2 (θ ) = βn(θ ), β = [β1 β 2 β 3 ] 1 2π −1 1 2π −1 ~ γ1* = τˆ (θ ) γ1 (θ )dθ , ~γ 2* = τˆ (θ ) γ 2 (θ )dθ 2π ∫0 2π ∫0

Coupled stretching–bending problems: τ cos θ − sin θ τ cos θ − sin θ , τ h (θ ) = h , τ t (θ ) = t τ t sin θ + cos θ τ h sin θ + cos θ τˆ t (θ ) = cos θ + τ t sin θ , τˆ h (θ ) = cos θ + τ h sin θ ⎧c t ⎫ ⎧c h ⎫ ηt = ⎨ ⎬, ηh = ⎨ ⎬ ⎩d t ⎭ ⎩d h ⎭ ~t ⎫ ~h ⎫ ⎡0 (N m ) 2 ⎤ ⎧⎪α ⎡0 (N m ) 2 ⎤ ⎧⎪α 1⎪ 1⎪ γ t = −I t ⎢ γ , I , = − h t⎢ T ⎥ ⎨~ t ⎬ T ⎥ ⎨~ h ⎬ I N ( ) I ( N ) m 1 ⎦⎪ m 1 ⎦⎪ ⎣ ⎣ ⎩α 2 ⎪⎭ ⎩α 2 ⎪⎭ ⎡I I2 ⎤ ⎡ I 0⎤ ⎡0 0 ⎤ It = ⎢ 1 , I1 = ⎢ , I2 = ⎢ ⎥ ⎥ ⎥ ⎣0 0 ⎦ ⎣0 I ⎦ ⎣I 2 I 1 ⎦

597

Appendix A

⎧⎪ γ1t (θ ) ⎫⎪ γ t (θ ) = ⎨ t ⎬ = −It ⎩⎪ γ 2 (θ ) ⎭⎪

t ⎡ 0 (N m (θ )) 2 ⎤ ⎧⎪α% 1 (θ ) ⎫⎪ ⎬, ⎢ I (N (θ ))T ⎥ ⎨ t ⎪α% 2 (θ ) ⎭⎪ 1 ⎦⎩ m ⎣

h h ⎡ 0 (N m (θ )) 2 ⎤ ⎪⎧α% 1 (θ ) ⎪⎫ ⎪⎧ γ1 (θ ) ⎪⎫ γ h (θ ) = ⎨ h ⎬ = − I t ⎢ ⎨ ⎬ ⎥ T h ⎪⎩ γ 2 (θ ) ⎪⎭ ⎣ I (N m (θ ))1 ⎦ ⎪⎩α% 2 (θ ) ⎪⎭

~ t (θ ) = cos θα ~ t + sin θα ~t , α ~ t (θ ) = − sin θα ~ t + cos θα ~t α 1 1 2 2 1 2 h h h t h ~ ~ ~ ~ ~ ~ α1 (θ ) = cos θα1 + sin θα 2 , α 2 (θ ) = − sin θα1 + cos θα 2h ~t ~t ~t ~t ~ t = ⎧⎪ α A2 ⎫⎪, α ~ t = ⎧⎪α A1 ⎫⎪, α ~ t = ⎧⎪ A1i ⎫⎪, α ~ t = ⎧⎪ B1i ⎫⎪ α ⎨ ~t ⎬ ⎨ ~t ⎬ ⎨~ t ⎬ ⎨ ~t ⎬ 2 1 Bi Ai ⎪⎩α B 2 ⎪⎭ ⎪⎩− α B1 ⎪⎭ ⎪⎩ B2i ⎪⎭ ⎪⎩ A2i ⎪⎭ ~ ~h h h h ~ ~ ~ h = ⎧⎪α A1 ⎫⎪, α ~ h = ⎧⎪ α A2 ⎫⎪, α ~ h = ⎧⎪ A1i ⎫⎪, α ~ h = ⎧⎪ B1i ⎫⎪ α ⎨ ~h ⎬ ⎨ ~h ⎬ ⎨~ h ⎬ ⎨ ~h ⎬ 1 2 Bi Ai ⎪⎩α B 2 ⎪⎭ ⎪⎩− α B1 ⎪⎭ ⎪⎩ B2i ⎪⎭ ⎪⎩ A2i ⎪⎭ ⎧⎪α tAi ⎫⎪ ⎧⎪α hAi ⎫⎪ ⎧⎪α tBi ⎫⎪ ⎧⎪α hBi ⎫⎪ α ti = ⎨ t ⎬ , α ih = ⎨ h ⎬ , α*i t = ⎨ t ⎬ , α*i h = ⎨ h ⎬ ⎪⎩α Bi ⎪⎭ ⎪⎩α Bi ⎪⎭ ⎪⎩α Di ⎪⎭ ⎪⎩α Di ⎪⎭ t t t h ⎪⎧ A1i ⎪⎫ ⎪⎧ B1i ⎪⎫ ⎪⎧ D1i ⎪⎫ ⎪⎧ A1i ⎫⎪ α tAi = ⎨ t ⎬ , α tBi = ⎨ t ⎬ , α tDi = ⎨ t ⎬ , α hAi = ⎨ h ⎬ , ⎪⎩ A2i ⎪⎭ ⎪⎩ B2i ⎪⎭ ⎪⎩ D2i ⎪⎭ ⎪⎩ A2i ⎪⎭ h h ⎧⎪ B1i ⎫⎪ ⎧⎪ D1i ⎫⎪ α hBi = ⎨ h ⎬ , α hDi = ⎨ h ⎬ , i = 1, 2 ⎩⎪ B2i ⎭⎪ ⎩⎪ D2i ⎭⎪ 1 π ~ γ it* = ∫ τˆt−1 (ω )γ ti (ω )dω ,

π

0

~γ h* = 1 π τˆ −1 (ω )γ h (ω )dω , i = 1,2 i i ∫ h

π

0

6. Elasticity matrix: ~ ~ ~ ~ ~ ˆ (θ ), N ˆ (θ , α ), N N, N (θ ), N (θ ), N (θ , α ) , N1( k ) (θ ), N (2k ) (θ ), N 3( k ) (θ ) , {N 3 (θ )} Nξ = μ ξ,

N(θ )ξ = μ (θ )ξ

ˆ (θ , α ) = cos(θ − α )I + sin(θ − α )N (α ), N

% (θ , α ) = 1 N

% (θ ) = N % (θ , 0), ˆ (θ ) = N ˆ (θ , 0), N N = N(0), N ( ˆ −1 (θ ) = N ˆ (0, θ ) = cos θ I − sin θ N(θ ) N(θ ) = N

θ

N (ω )d ω π ∫α

% =N % (π ), N

~ ~ ~ ~ ~ ~ ~ N1 (θ ), N 2 (θ ), N 3 (θ ) & N1 (θ , α ), N 2 (θ , α ), N 3 (θ , α ) : submatrices of N (θ ) & ~ N (θ , α ) ( ( ( ( N1 (θ ), N 2 (θ ), N 3 (θ ) : sub-matrices of N (θ )

598

Appendix A

~ ~ ~ ~ ~ ~ N1( k ) (θ ), N (2k ) (θ ), N 3( k ) (θ ) : N1 (θ ) , N 2 (θ ) , N 3 (θ ) of kth material {N 3 (θ )} : sub-inverse of N 3 (θ ) defined in (5.21); N3

< −1>

: sub-inverse of N 3

7. Barnett–Lothe tensors and their extensions: ~ ~ ~ S, H, L, S* , H * , L* , S, H, L , G1 (θ ), G 3 (θ ) S = i (2 ABT − I ) = L = −2iBBT = −

1

π∫

1

π∫

π 0

π 0

N1 (θ )d θ ,

H = 2iAAT =

1

π∫

π 0

N 2 (θ )d θ ,

N 3 (θ )d θ

⎡ S H ⎤ ~ ~ or N = N(π ) = ⎢ T⎥ ⎣− L S ⎦ S* , H * , L* : S, H, L defined for multi-materials, for example, ⎡ S * H * ⎤ 1 ⎡ S1 + S 2 H1 + H 2 ⎤ , for bimaterials ⎢ * ⎥= ⎢ T T ⎥ *T ⎢⎣− L S ⎥⎦ 2 ⎣− (L1 + L 2 ) S1 + S 2 ⎦ 1 2π 1 2π 1 2π S* = N1 (ω )dω , H* = N 2 (ω )dω , L* = − N 3 (ω )dω , for ∫ ∫ 0 0 2π 2π 2π ∫0 general multi-materials ~ ~ ~ S, H, L : S, H and L defined for multi-materials: ~ ~ ⎤ ⎡ S H ⎡ S* H * ⎤ ⎢ ~ ~T ⎥ ⎢ * ⎥ = −I . *T ⎢⎣− L S ⎥⎦ ⎢⎣− L S ⎥⎦

⎡ S H ⎤ ~ where N(π ) = ⎢ T⎥ ⎣− L S ⎦ G1 (θ ) = NT1 (θ ) − N 3 (θ )SL−1 , G 3 (θ ) = −N 3 (θ )L−1 ~ Note: N 2 (π ) = −I,

8. Impedance matrix, bimaterial matrix, and their extensions: M, M* , D, W M = − iBA −1 = H −1 − iST H −1 = H −1 + iH −1S, M −1 = iAB −1 = L−1 − iSL−1 = L−1 + iL−1ST M* = M1−1 + M 2−1 = i ( A1B1−1 − A 2 B 2−1 ) = D − iW, D = L−11 + L−21 ,

W = S1L−11 − S 2 L−21

599

Appendix A

9. Uniform stresses and strains at infinity: ε1∞ , ε ∞2 , t1∞ 　 , t ∞2 ∞ ∞ ⎧σ 11∞ ⎫ ⎧σ 21 ⎫ ⎧ ε 11∞ ⎫ ⎧ ε 21 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ⎪ t1 = ⎨σ 12 ⎬　 , t 2 = ⎨σ 22 ⎬, ε1 = ⎨ ε 12 ⎬　 , ε 2 = ⎨ ε 22 ⎬ ⎪σ ∞ ⎪ ⎪σ ∞ ⎪ ⎪2ε ∞ ⎪ ⎪2ε ∞ ⎪ ⎩ 13 ⎭ ⎩ 23 ⎭ ⎩ 13 ⎭ ⎩ 23 ⎭

10. Mapped variable: ζ α ,ψ , θ , ρ

Elliptical hole or inclusion:

ζα =

zα + zα2 − a 2 − b 2 μα2 a − ibμα

, zα =

1⎧ 1 ⎫ ⎨(a − ibμα )ζ α + (a + ibμα ) ⎬ 2⎩ ζα ⎭

ρ cos θ = a sin ψ , ρ sin θ = −b cosψ , ρ = a 2 sin 2 ψ + b 2 cos 2 ψ use ϕ to replace ψ in the stretching–bending problem Polygon-like hole or inclusion: zα = a{(1 − iμα c)ζ α + (1 + iμα c)ζ α−1 + ε (1 + iμα )ζ αk + ε (1 − iμα )ζ α− k } / 2 ρ cos θ = a(sinψ + kε sin kψ ), ρ sin θ = − a(c cosψ − kε cos kψ )

ρ 2 = a 2 {k 2ε 2 + sin 2 ψ + c 2 cos 2 ψ + 2kε sinψ sin kψ − 2ckε cosψ cos kψ } Crack or line inclusion: 1 a

ζ α = {zα + zα2 − a 2 },

zα =

a⎧ 1 ⎫ ⎬ ⎨ζ α + ζα ⎭ 2⎩

A.3 Sign Convention Since the positive directions of stresses and displacements and their related physical responses are defined according to their directions along the coordinate axes, to consider the sign conventions for these symbols two different right-handed Cartesian coordinate systems are shown in Figs. A.1 and A.2. The former contains five sub-figures showing the positive directions of several physical quantities based upon the coordinate system with upward positive x3 , whereas the latter is based upon the coordinate with downward positive x 3 . These two figures show the positive directions of (a) the coordinate system ( x1 , x2 , x3 ) , displacements (u , u , u ) , tractions (t , t , t ) , resultant forces ( ~ t ,~ t ,~ t ) , moments 1

2

3

1

2

3

1

2

3

~ ,m ~ ,m ~ ) , and slopes ( β , β ) ; (b) the stresses σ ; (mx , m y ) , resultant moments (m 1 2 3 x y ij

(c) the stress resultants N ij , bending moments M ij , transverse shear forces Qi , effective transverse shear forces Vi , corner forces tc , and transverse distributed load q; (d) the quantities in the Cartesian coordinate ( x , y ) and the tangent–normal coordinate ( s, n) : (ux , u y ) – (us , un ) , ( β x , β y ) – ( β s , β n ) , and (mx , m y ) – (ms , mn ) .

600

Appendix A

Note that in Figs. A.1 and A.2 the moment or slope is represented by a vector in the form of a double-headed arrow. The direction of the moment or slope is indicated by the right-hand rule – namely, using your right hand, let your fingers curl in the direction of the moment or slope, and then your thumb will point in the direction of the vector.

A.4 Units The International System of Units, abbreviated SI units, is a worldly accepted version of the metric system. In SI units for mechanics, mass in kilograms (kg), length in meters (m), and time in seconds (s) are selected as the base units, and force in newton (N) is derived from the preceding three by Newton’s second law. Thus, force (N) = mass (kg) × acceleration (m/s2) is one of the derived units used in mechanics. For the convenience of readers’ reference, Table A.1 shows some physical quantities and their SI units as well as SI symbols and relations used in this book. The SI unit prefixes used for the multiplication factors of the SI symbols are then shown in Table A.2.

Scaling Factor Table A.3 is an example of material properties of graphite/epoxy and left-hand quartz by using SI units. From this table we see that some values of the properties such as elastic stiffness constants are in the order above 9, while some others such as dielectric permittivity are in the order of −11. Thus, it is very possible that the fundamental elasticity matrix N constructed through (11.18) and (11.19) for piezoelectric materials will contain the components ranging from − 11

9

to 10 , which is numerical ill-condition. To avoid any trouble caused by 10 the ill-conditioned matrix, a proper scale adjustment is necessary. For example, we may rewrite the constitutive relation (11.13)1 into σ ij / E0 = (Cijkl / E0 )ε kl − ekij ( Ek / E0 ), E

(A.1)

D j = e jkl ε kl + ( E0ω εjk )( Ek / E0 ), 9

2

in which E 0 is a reference number, such as 10 N/m , used for scale adjustment. Equation (A.1) means that in numerical calculation the material E

ε

E

properties C ijkl and ω jk are replaced by C ijkl /E

0

and

ε

E 0 ω jk . By such

replacement the output values of stresses and electric fields are σ ij / E 0 and E k / E 0 , which should be multiplied by E 0 to return to their original units.

Appendix A

601

Elastic Materials From the definition given in (3.48) we know that N 1 is dimensionless and N 2 and N3 are, respectively, proportional to 1/E and E where E stands for the

Young’s modulus. Thus, the fundamental elasticity matrix N will become ill-conditioned even for the pure elastic materials. Therefore, the scaling factor E 0 introduced in (A.1) to nondimensionalize the elastic constants Cijkl is suggested to be used in the computer programming for elastic materials, i.e., every Cijkl will be divided by E 0 in numerical calculation. To trace the effects of scaling factor, it is better to know the proportional relation with Young’s modulus for some commonly used matrices, which is shown in Table A.4. Take some examples from this table such as S, H, and L; we know that their values will become S, E 0 H , and L / E 0 when C ijkl is nondimensionalized by using Cijkl

/ E 0 . To return their original values, the latter two should be divided or

multiplied by E 0 .

602

Appendix A

x3 , u3 , t3 , t%3

x2 , u2 , t2 , t%2

β x , mx , m% 2 m% 3

β y , my

x1 , u1 , t1 , t%1

m% 1 (a) σ 33

σ 32

σ 31 σ 13

σ 12 σ 11

σ 21

σ 22

σ 23

(b)

M 21 N 22

M 22 M 11

M 12

N11

q

N12

M 12

Q1 ,V1 M 22

N 21

N 22 M 21 Q2 ,V2

tc

tc

Q2 ,V2

tc

N12

N12

M 11

N11

Q1 ,V1

tc

(c )

y, u y

n, un

β x , mx

β s , ms

β y , my β n , mn

θ

s, u s

θ

x, u x

body(on the right hand side of the route)

θ :counterclockwise

(d)

Fig. A.1 Sign convention for a right-handed Cartesian coordinate system with % i , β i ; (b) σ ij ; (c) N ij , M ij ,Qi ,Vi ,tc ,q ; upward positive x3 : (a) ui , ti , t%i , mi , m (d) s − n coordinate.

603

Appendix A

β y , my

m% 1

x1 , u1 , t1 , t%1

β x , mx , m% 2 m% 3

x2 , u2 , t2 , t%2

x3 , u3 , t3 , t%3 (a) σ 33 σ 32

σ 31

σ 12 σ 13

σ 21

σ 22

σ 23

(b)

N 22 M 21 M 22

N 21

N12

M 11

q

N11

M 12 M 11

σ 11

Q2 ,V2

tc

M 12

Q1 ,V1 M 22 M 21

N 22

tc

tc

N11

N12 Q1 ,V1

N 21 tc

Q2 ,V2

(c )

β n , mn β y , my β s , ms n, u n

x, u x

θ

θ

β x , mx

θ

s, u s

y, u y

(d)

Fig. A.2 Sign convention for a right-handed Cartesian coordinate system with % i , β i ; (b) σ ij ; (c) N ij , M ij ,Qi ,Vi ,tc , downward positive x3 : (a) ui , ti , t%i , mi , m q ; (d) s − n coordinate.

604

Appendix A

Table A.1 SI units Quantity

SI unit

SI symbol

Meter Kilogram Second Kelvin Ampere

m kg s °K A

Newton

N

Stress, pressure Energy, work

Pascal Joule

Pa J

Power Electric charge

Watt Coulomb

W C

N = kg ⋅ m/s 2 Pa=N/m2 J= N ⋅ m (= volt ⋅ coul) W=J/s C= A ⋅ s (= joule/volt )

Electric potential

Volt

V

V=J/C ( = amp ⋅ ohm)

Electric resistance Capacitance Magnetic flux Magnetic field

Ohm Farad Weber Tesla

Ω F Wb T

Ω = V/A F=C/V Wb = V ⋅ s T=Wb/m2 (= N/(amp⋅ m))

Celsius temperature Electric field Electric displacement Permittivity

Degree

°C

° C = ° K - 273

(Base units) Length Mass Time Temperature Electric current (Derived units) Force

Thermal conductivity Thermal expansion Thermal modulus

Relation

Volt/m Coul/m2

Volt/m=N/coul Coul/m2=N/ (m ⋅ volt)

Farad/m

Farad/m=N/V2= C /(m ⋅ N)

Watt/ (m° K)

W/ (m° K) = W/ (m° C)

1/ ° K

1/ ° K = 1/ ° C

2

2

2

N/m ° K

N/m ° K

Table A.2 SI prefixes Prefix Pico Nano Micro Milli Centi Kilo Mega Giga Tera

Symbol p n μ

m c k M G T

Factor 10−12 10−9 10−6 10−3 102 103 106 109 1012

= N/m2 ° C

2

605

Appendix A

Table A.3 Material properties of graphite/epoxy and left-hand quartz Graphite/epoxy

Left-hand quartz

C11E (GPa)

183.71

86.74

C12E (GPa)

4.63

6.99

C13E (GPa)

4.22

11.91

C 22E (GPa)

11.29

86.74

C 23E (GPa)

3.24

11.91

C 33E (GPa)

11.27

107.2

C 44E , C 55E (GPa)

7.17

57.94

C 66E (GPa)

2.87

39.88

− C14E , C24E ,−C56E (GPa)

–

17.91

ω11ε

(F/m)

–

39.21×10−12

ε ω22

(F/m)

–

39.21×10−12

ω33ε

(F/m)

–

41.03×10−12

e11 ,−e12 ,−e26 (C/m2)

–

0.171

− e25 , e14 (C/m2)

–

0.0406

Table A.4 Proportional relation with Young’s modulus of elastic materials ~E ~1 ~ 1/E Fundamental elasticity N , N (θ ) N 2 , N 2 (θ ) N 3 , N 3 (θ ) 1 1 matrices Barnett–Lothe tensors S H L * * Green’s functions U T Influence matrices

Yin

Material matrices

A ~ 1/ E , B ~ E

eigenvector

Gin

⇒ AB −1 ~ 1 / E , ABT ~ 1, AAT ~ 1 / E , BBT ~ E

Appendix B Hilbert Problem

B.1 Solution to the Hilbert Problem in Scalar Form The Hilbert problem is usually expressed in the form of scalar functions, like

F + ( s ) − gF − ( s ) = f ( s ) , s on L except at the ends,

(B.1)

where g in general is a complex constant; L is the union of a finite number of arcs L1 , L 2 ,..., Ln where the ends of the arcs L k , k = 1, 2 ,..., n, are a k and b k ; f ( s) is a complex function which satisfies the Holder conditions on L, i.e., for

any two points s1 , s2 on L,

| f ( s1 ) − f ( s2 ) |≤ c | s1 − s2 |γ ,

(B.2)

where c and γ are positive constants and 0 < γ ≤ 1 ; F ( z) is a sectionally +

−

holomorphic function in the plane cut along L and F ( s) and F ( s ) are, respectively, the limiting values of F ( z ) as z → s from the left and right. The solution to (B. 1) with g ≠ 1 has been shown in Muskhelishvili (1954) as

F ( z) =

χ0 ( z) f ( s)ds + χ 0 ( z ) pn ( z ) , 2π i ∫L χ 0+ ( s)( s − z )

(B.3a)

where pn (z ) is an arbitrary polynomial with the degree not higher than n and χ 0 ( z ) is the basic Plemelj function satisfying the relation

χ 0+ ( s ) = g χ 0− ( s ),

s on L,

(B.3b)

whose solution can be expressed as n

χ 0 ( z ) = ∏ ( z − a j ) −δ ( z − b j )δ −1 , j =1

δ=

1 ln( g ). 2π i

(B.3c)

607

608

Appendix B

When g=1, the solution to (B.1) is

F ( z) =

1 f ( s )ds + pn ( z ). 2π i ∫L s − z

(B.4)

B.2 Solution to the Hilbert Problem in Matrix Form The Hilbert problem in matrix form can be expressed as ψ + ( s ) + Gψ − ( s ) = tˆ ( s ), s on L except at the ends,

(B.5)

where G is a complex constant matrix, L is the union of arcs, and tˆ ( s ) is a given complex function vector. To find the solution ψ ( z ) to problem (B.5), a similar approach as that of scalar form can be employed. First, we consider the homogeneous problem (B.6) ψ + ( s ) + Gψ − ( s ) = 0 , 　　s ∈ L　 . With reference to (B.3c), a particular solution ψ 0 to the problem (B.6) will be sought in the form n

ψ 0 ( z ) = ∏ ( z − a j ) −δ ( z − b j )δ −1 λ ,

(B.7)

j =1

where δ is a complex constant and λ is a complex constant vector. The function ψ 0 ( z ) is holomorphic in the entire plane cut along L, if a definite branch of this function is selected. It is readily verified by an investigation of the variation in the argument of z − a j or z − b j , when z describes a closed path beginning at a point s of the arc a j b j and leading, without intersecting L, from the left side of a j b j around the end a j to the right side of the arc or around the end b j , that (B.8)

ψ 0+ ( s ) = e 2iπδ ψ 0− ( s ). Hence, ψ 0 ( z ) will satisfy the condition (B.6), provided

(B.9)

(e 2iπδ I + G )λ = 0.

If G = M*−1M* where M* = D − iW is the bimaterial matrix, the explicit solution for the eigenvalue δ has been given by Ting (1986) as

1 2

1 2

1 2

δ 1 = + iε , 　　δ 2 = − iε , 　　δ 3 = , where

　

(B.10a)

Appendix B

609

ε=

[

1 1+ β 1 ln , β = − tr( WD−1 ) 2 2π 1 − β 2

]

1/2

,

(B.10b)

and tr stands for the trace of the matrix. Thus, a particular solution ψ 0 ( z ) of the homogeneous problem has been found; it is given by (B.7) with δ and λ determined by (B.9). Since there are three eigenvalues to (B.9), a linear combination of these particular solutions will still be one of the particular solutions, i.e., (B.11a) ψ 0 ( z ) = X0 ( z )p 0 , where (B.11b)

X0 ( z ) = ΛΓ( z ) and

Λ = [λ 1　λ 2 λ 3 ],

n

Γ( z ) =< ∏ ( z − a j ) −δα ( z − b j )δα −1 > .

(B.11c)

j =1

p 0 　is a coefficient vector. This particular solution does not vanish anywhere in the finite part of the plane and it is unbounded like | z − a j |−1/ 2 and | z − b j |−1 / 2

near the ends a j and b j , respectively. We now look for the most general solution to the homogeneous Hilbert problem. For this purpose it will be noted that ψ 0 ( z ) = X 0 ( z )p 0 , being a solution of the homogeneous problem, satisfies the condition

X0+ ( s )p 0 + GX0− ( s )p 0 = 0,

s ∈ L,

(B.12)

and hence,

G = − X0+ ( s )[ X0− ( s )]−1.

(B.13)

By applying (B.13), (B.6) becomes

[ X0+ ( s )]−1 ψ + ( s) − [ X0− ( s )]−1 ψ − ( s ) = 0 , 　　　s ∈ L　 ,

(B.14a)

ψ *+ ( s ) − ψ *− ( s ) = 0 , 　　　s ∈ L　 ,

(B.14b)

or

where ψ * ( z ) denotes the sectionally holomorphic function [ X 0 ( z )]−1 ψ ( z ) . It follows from (B.14) that ψ * ( z ) is holomorphic in the entire plane, except at the point z = ∞ , provided it is given suitable values on L. Further, since ψ * ( z ) can only have a pole at infinity, it must, by the generalized Liouville theorem, be a polynomial. Thus, the most general solution of the homogeneous problem is given by

610

Appendix B

(B.15)

ψ ( h ) ( z ) = X0 ( z )p n ( z ),

where p n (z ) is an arbitrary polynomial vector. If it is desired to obtain a solution which is also holomorphic at infinity, it must be assumed that the degree of the polynomial p n (z ) does not exceed n. This follows from the behavior of X 0 ( z ) at infinity as given in (B.11b) and (B.11c). Next consider the non-homogeneous problem. Using (B.13), the boundary condition (B.5) may be written as

ψ *+ ( s ) − ψ *− ( s ) = [ X0+ ( s )]−1 tˆ( s ),

(B.16a)

s ∈ L,

where

ψ * ( z ) = [ X0 ( z )]−1 ψ ( z ) .

(B.16b)

Each component of equation (B.16) is in the form of (B.1) with g = 1; hence, by (B.4) we have 1 1 (B.17) [X0+ (s)]−1 tˆ(s)ds + X0 ( z )pn ( z ), ψ( z ) = X0 ( z ) ∫ L 2πi s−z where p n (z ) is an arbitrary polynomial vector with the degree not higher than n. n. B.3 Evaluation of a Line Integral in Scalar Form L1

L1+

C1' L1−

C1

L1−

L1''

L1' C∞

Fig. B.1 Integration contour Consider the line integral

j( z) = ∫

L

h( s )ds

χ 0+ ( s)( s − z )

,

(B.18)

where L is the union of a finite number of arcs L1 , L2 ,..., Ln and χ 0 ( z ) is the basic Plemelj function. Suppose that h(s ) is a polynomial, a situation which often occurs in practice. By residue theory, the integral around a closed contour C can be calculated as

Appendix B

611

∫

C

h( s )ds = 2π iη , χ 0 ( s)( s − z )

(B.19)

where η = h( z ) / χ 0 ( z ) is the sum of residues of the poles of the integrand within C. The closed contour C is the union of ( L+k , L−k , Ck , Ck′ , Lk′ , Lk′′), k = 1,2,..., n and C∞ (see Fig. B.1 for example with one arc L1 ). The summation of the integrals along Lk′ and Lk′′ vanishes since they have opposite directions and the integrand

across this line is continuous. If h( s ) is bounded near the ends s = ak and s = bk , the integrals around the circles Ck and Ck′ can be proved to be zero when the radii of the circles Ck and Ck′ tend to zero. Knowing that χ o ( z ) satisfying (B.3b), we have

∫

h( s )ds

L+k + L−k

χ o ( s)( s − z )

= (1 − g ) ∫

Lk

h( s )ds

χ o+ ( s )( s − z )

.

(B.20)

Therefore, the integral around a closed contour C can now be written as

∫

h( s )ds C

χ 0 ( s )( s − z )

= 2π iη = (1 − g ) ∫

L

h( s)ds

χ o+ ( s )( s − z )

+∫

C∞

h( s )ds

χ 0 ( s )( s − z )

.

(B.21) By replacing the contour of C ∞ as ρeiθ and letting ρ → ∞ , the line integral j ( z ) of (B.18) can now be evaluated by iθ iθ 2π h ( ρ e )i ρ e dθ ⎞ h( s )ds 1 ⎛ h( z ) 2π i = − lim ∫ ⎜ ⎟. i θ i θ L χ ( s )( s − z ) 1− g ⎝ χ 0 ( z ) ρ →∞ 0 χ 0 ( ρ e )( ρ e − z ) ⎠

∫

+ o

(B.22) As an example, we now consider a straight arc L located on (− a, a ) and h( s ) = h0 , g = −1. From (B.3c), we have δ = 1 / 2 and

χ0 ( z) =

1 z − a2 2

.

(B.23)

With the values given between (B.22) and (B.23), the line integral j ( z ) can be evaluated from (B.22) as h ds 2 (B.24) ∫L χ o+ (s)(o s − z ) = −iπho ( z − z − a ) .

612

Appendix B

B.4 Evaluation of a Line Integral in Matrix Form Consider the line integral in matrix form

1 [ X 0+ ( s )]−1 h( s )ds, s z − L

j( z ) = ∫

(B.25)

where L is the union of a finite number of arcs L1 , L2 ,..., Ln and X0(z) is the matrix of basic Plemelj function satisfying the relation

X0+ ( s ) + GX0− ( s ) = 0,

s ∈ L.

(B.26)

Suppose that h( s) is a polynomial and consider the closed contour shown in Fig. B.1. Following the procedure stated between and (B.22), we can obtain

j( z ) = 2πiη −

1

∫ s − z [X (s)]

−1

0

(I + G ) −1 h( s )ds,

(B.27)

C∞

where η is the sum of residues of the poles of the integrand within a closed contour C, i.e., 1 −1 −1 (B.28) ∫C s − z [X0 (s)] (I + G ) h(s)ds = 2π iη . The second term of (B.27) has the form

lim ∫

2π

ρ →∞ 0

i ρ eiθ [ X ( ρ eiθ )]−1 (I + G ) −1 h( ρ eiθ )dθ , ρ eiθ − z 0

(B.29)

where ρ is the radius of the contour C ∞ . It can be shown that only terms independent of ρeiθ can contribute to the above integral. Then with a given function h(s), the integral j(z) can be explicitly evaluated from (B.27), (B.28), and (B.29). For example, consider an integral along a single line L = ( −a, a ) and let h(t)=h, where h is a given constant vector. The sum of the residues is η = [ X0 ( z )]−1 (I + G ) −1 h. (B.30) To calculate the integral shown in (B.29), by (B.11c) we express Γα−1 (ζ ) for large | ζ | as

Γα−1 (ζ ) = (ζ + a)δ α (ζ − a ) −δ α +1 = ζ + 2iaε α + O(ζ −1 ), α = 1,2,3. With (B.31), (B.29) becomes

(B.31)

Appendix B

lim ∫

ρ→∞

613 2π 0

(1 + ρez

iθ

)

+ L < ρ eiθ + 2iaε α + L > id θ Λ −1 (I + G ) −1 h

(B.32)

= 2π i < z + 2iaε α > Λ (I + G ) g −1

−1

From (B.27), (B.30), and (B.32) we obtain the final result of j(z) as j( z ) = 2πi{[ X0 ( z )]−1 − < z + 2iaε α > Λ −1}(I + G ) −1 h.

(B.33)

Appendix C Summary of Stroh Formalism C.1 Two-Dimensional Problems: ui = ui ( x1 , x2 ), i = 1,2,3 Basic equations I.

II.

Anisotropic bodies under mechanical loadings (Section 3.1):

ε ij = (ui , j + u j ,i ) / 2, σ ij = Cijkl ε kl , σ ij , j = 0, i, j , k , l = 1,2,3.

(C.1)

Governing equations: Cijkl u k ,lj = 0, i, j , k , l = 1,2,3.

(C.2)

Anisotropic bodies under thermal loadings (Section 10.1): hi = −kij T, j ,

hi ,i = 0,

(C.3a)

ε ij = (ui , j + u j ,i ) / 2, σ ij = Cijkl ε kl − β ij T , σ ij , j = 0,

i, j , k , l = 1,2,3.

Governing equations: kijT,ij = 0, Cijkl u k ,lj − β ijT, j = 0, i, j , k , l = 1,2,3.

III.

(C.3b) (C.4)

Piezoelectric bodies under electromechanical loadings (Section 11.2):

ε ij = (ui , j + u j ,i ) / 2, σ ij , j = 0,

D j = e jkl ε kl + ω εjk Ek ,

E σ ij = Cijkl ε kl − ekij Ek ,

Di ,i = 0, i, j , k , l = 1, 2, 3.

(C.5a) (C.5b)

Governing equations: C pjql u q ,lj = 0, j , l = 1, 2, 3, p, q = 1,2,3,4 ,

(C.6a)

Cijkl = C , 　　i, j , k , l = 1,2,3;

(C.6b)

E ijkl

C 4 jkl = e jkl , 　　j , k , l = 1,2,3;

Cij 4l = elij , 　　i, j , l = 1,2,3, ε

C 4 j 4l = −ω jl , 　 j , l = 1,2,3,

u 4, j = − E j , σ 4 j = D j , j = 1,2,3 .

(C.6c) (C.6d)

General solution

I.

Anisotropic bodies under mechanical loadings (Section 3.1): u = 2 Re{Af ( z )}, φ = 2 Re{Bf ( z )} ,

(C.7a)

A = [a1 a 2 a 3 ] , B = [b1 b 2 b 3 ] ,

(C.7b)

⎧u1 ⎫ ⎧φ1 ⎫ ⎧ f1 ( z1 ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u = ⎨u 2 ⎬, φ = ⎨φ2 ⎬, f ( z ) = ⎨ f 2 ( z 2 )⎬ , zα = x1 + μα x2 , α = 1,2,3 . ⎪u ⎪ ⎪φ ⎪ ⎪ f (z ) ⎪ ⎩ 3⎭ ⎩ 3⎭ ⎩ 3 3 ⎭

(C.7c)

Anti-plane:

u = 2 Re{af ( z )},

φ = 2 Re{bf ( z )} .

In-plane:

u = 2 Re{Af ( z )}, φ = 2 Re{Bf ( z )} ,

(C.9a)

A = [a1 a 2 ] , B = [b1 b 2 ] ,

(C.9b)

⎧ f1 ( z1 ) ⎫ ⎧u ⎫ ⎧φ ⎫ u = ⎨ 1 ⎬, φ = ⎨ 1 ⎬, f ( z ) = ⎨ ⎬. ⎩ f 2 ( z 2 )⎭ ⎩u2 ⎭ ⎩φ2 ⎭

(C.8)

(C.9c)

615

616

II.

Appendix C

Anisotropic bodies under thermal loadings (Section 10.1): T = 2 Re{g ′( z t )}, hi = −2 Re{(ki1 + τki 2 ) g ′′( zt )},

III.

zt = x1 + τx2 ,

(C.10a)

u = 2 Re{Af ( z ) + cg ( zt )}, 　φ = 2 Re{Bf ( z ) + dg ( zt )},

(C.10b)

A = [a1 a 2 a 3 ] , B = [b1 b 2 b 3 ] ,

(C.10c)

⎧u1 ⎫ ⎧φ1 ⎫ ⎧ f1 ( z1 ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u = ⎨u 2 ⎬, φ = ⎨φ2 ⎬, f ( z ) = ⎨ f 2 ( z 2 )⎬ , zα = x1 + μα x2 , α = 1,2,3 . ⎪u ⎪ ⎪φ ⎪ ⎪ f (z ) ⎪ ⎩ 3⎭ ⎩ 3⎭ ⎩ 3 3 ⎭

(C.10d)

Piezoelectric bodies under electromechanical loadings (Section 11.2): u = 2 Re{Af ( z )}, φ = 2 Re{Bf ( z )} ,

(C.11a)

A = [a1 a 2 a 3 a 4 ] , B = [b1 b 2 b 3 b 4 ] ,

(C.11b)

⎧u1 ⎫ ⎧φ1 ⎫ ⎧ f1 ( z1 ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u φ ⎪ 2⎪ ⎪ 2⎪ ⎪ f 2 ( z 2 )⎪ u = ⎨ ⎬, φ = ⎨ ⎬, f ( z ) = ⎨ ⎬ , zα = x1 + μα x2 , α = 1,2,3,4 . ⎪u3 ⎪ ⎪φ3 ⎪ ⎪ f 3 ( z3 ) ⎪ ⎪⎩u 4 ⎪⎭ ⎪⎩φ4 ⎪⎭ ⎪⎩ f 4 ( z 4 )⎪⎭

(C.11c)

Material eigen-relation ⎡ N1 N 2 ⎤ ⎧a ⎫ Nξ = μξ , N = ⎢ , ξ = ⎨ ⎬ , N1 = −T −1 R T , N 2 = T −1 = N T2 , T⎥ ⎩b ⎭ ⎣ N 3 N1 ⎦ N 3 = RT −1 R T − Q = N T3 .

I.

Anisotropic bodies under mechanical loadings (Section 3.1): Qik = Ci1k1 , 　Rik = Ci1k 2 , 　Tik = Ci 2 k 2 , ⎡C11 C16 or Q = ⎢⎢C16 C66 ⎢⎣C15 C56

II.

(C.12)

C15 ⎤ ⎡C16 C56 ⎥⎥, R = ⎢⎢C66 ⎢⎣C56 C55 ⎥⎦

C12 C26 C25

i, k = 1,2,3 , C14 ⎤ ⎡C66 C46 ⎥⎥, T = ⎢⎢C26 ⎢⎣C46 C45 ⎥⎦

(C.13a) C26 C22 C24

C46 ⎤ C24 ⎥⎥. C44 ⎥⎦

(C.13b)

Anisotropic bodies under thermal loadings (Section 10.1): k22τ 2 + (k12 + k21 )τ + k11 = 0.

(C.14a)

⎡0 N 2 ⎤ ⎧ β1 ⎫ ⎧c ⎫ Nη = τη + γ , η = ⎨ ⎬, γ = − ⎢ ⎬ , β1 , β 2 : extended symbols 5, T ⎥⎨ d ⎩ ⎭ ⎣ I N1 ⎦ ⎩β 2 ⎭

(C.14b)

Q, R, T: same as Problem I, Eq. (C.13).

617

Appendix C

III.

Piezoelectric bodies under electromechanical loadings (Section 11.2): Qik = Ci1k1 , 　Rik = Ci1k 2 , 　Tik = Ci 2 k 2 , ⎡C ⎢ E or Q = ⎢C16 ⎢ E ⎢C15 ⎢e ⎣ 11

E 11

E 16

C

C

E 15

C

E 66

C

E 56

C56E C55E e16

⎡C e11 ⎤ ⎥ ⎢ E e16 ⎥ ⎢C66 R = , ⎥ ⎢ E e15 ⎥ ⎢C56 ε ⎥ ⎢e − ω11 ⎦ ⎣ 16 E 16

e15

E 12

C

C

E 26

i, k = 1,2,3,4, E 14

C

C

E 46

C 25E C 45E e12

e14

(C.15a)

⎡C e21 ⎤ ⎢ E ⎥ e26 ⎥ ⎢C 26 T = , ⎢ E ⎥ e25 ⎥ ⎢C 46 ε ⎥ ⎢e − ω12 ⎦ ⎣ 26 E 66

C

E 26

C

E 22

C

E 46

C

E 24

C 24E C 44E e22

e24

e26 ⎤ ⎥ e22 ⎥ (C.15b) . ⎥ e24 ⎥ ε ⎥ − ω 22 ⎦

Generalized eigen relation

Elasticity relation (Section 3.3): N (θ )ξ = μ (θ )ξ ,

⎡N1 (θ ) N 2 (θ ) ⎤ N(θ ) = ⎢ ⎥, T ⎣N 3 (θ ) N1 (θ )⎦

(C.16a)

⎧a ⎫ ξ = ⎨ ⎬, ⎩b ⎭

N1 (θ ) = −T −1 (θ ) R T (θ ), N 2 (θ ) = T −1 (θ ) = N T2 (θ ), N 3 (θ ) = R (θ )T −1 (θ ) R T(θ ) − Q(θ ),

(C.16b)

Q(θ ) = Q cos 2 θ + (R + R T ) sin θ cos θ + T sin 2 θ ,

(C.16c)

R (θ ) = R cos 2 θ + (T − Q) sin θ cos θ − R T sin 2 θ ,

(C.16d)

T(θ ) = T cos 2 θ − (R + R T ) sin θ cos θ + Q sin 2 θ .

(C.16d)

Thermal relation (Section 10.1): N (θ ) η = τ (θ ) η + τˆ −1 (θ ) γ (θ ),

Sc + Hd = ic + γ% 1* ,

⎡0 N 2 (θ ) ⎤ ⎧ β1 (θ ) ⎫ ⎧c ⎫ η = ⎨ ⎬, γ (θ ) = − ⎢ ⎬, ⎥⎨ T d ⎩ ⎭ ⎣ I N1 (θ )⎦ ⎩β 2 (θ )⎭

− Lc + ST d = id + γ% *2 ,

τˆ(θ ), β1 (θ ), β 2 (θ ), γ% 1* , γ% *2 :

(C.17a) (C.17b)

extended symbols 5. Stresses (Sections 3.2 and 11.2) Stresses in xi -coordinate system:

σ i1 = −φi ,2 , σ i 2 = φi ,1 , i = 1, 2,3 for Problems I&II, σ i1 = −φi ,2 , σ i 2 = φi ,1 ,

(C.18a)

i = 1, 2,3, 4 for Problem III ( D1 = σ 41 , D2 = σ 42 ). (C.18b)

Stresses in s-n coordinate system: (s: tangential direction, n: normal direction) t n = φ , s , t s = −φ , n ,

(C.19a)

σ nn = n φ, s , σ ss = −s φ ,n , σ ns = s φ ,s = −n φ ,n = σ sn , σ n3 = i φ ,s ,

(C.19b)

Dn = i T4 φ ,s ,

(C.19c)

T

T

T

T

T 3

σ s 3 = −i T3 φ,n ,

Ds = −i T4 φ ,n .

618

Appendix C

Stresses in polar coordinate system:

t r = −φ ,θ / r ,

tθ = φ ,r ,

(C.20a)

σ θθ = n φ , r , σ rr = −s φ ,θ / r , T

σ rθ = s φ ,r = −n φ ,θ / r , σ θ 3 = i φ ,r ,

T

T

T

T 3

(C.20b)

σ r 3 = −i φ ,θ / r , T 3

Dθ = i T4 φ ,r ,

Dr = −i T4 φ ,θ / r ,

(C.20c)

C.2 Coupled Stretching–Bending Problems U i = ui ( x1 , x2 ) + x3 β i ( x1 , x2 ), U 3 = w( x1 , x2 ) β1 = − w,1 , β 2 = − w, 2 , ξ ij = ε ij + x3κ ij = (U i , j + U j ,i ) / 2, i = 1,2 . Basic equations I.

Laminates under mechanical loadings (Section 13.2):

ε ij = (ui , j + u j ,i ) / 2, κ ij = ( β i , j + β j ,i ) / 2, N ij = Aijkl ε kl + Bijkl κ kl , M ij = Bijkl ε kl + Dijkl κ kl ,

(C.21a)

N ij , j = 0, M ij ,ij = 0, Qi = M ij , j , i, j , k , l = 1,2.

(C.21b)

Governing equations: Aijkl u k ,lj + Bijkl β k,lj = 0, Bijkl u k ,lij + Dijkl β k ,lij = 0,

II.

β1,2 = β 2,1 , i,j,k,l = 1,2 .

(C.22)

Laminates under hygrothermal loadings (Section 13.4): T = T 0 ( x1 , x2 ) + x3T * ( x1 , x2 ),

H = H 0 ( x1 , x2 ) + x3 H * ( x1 , x2 ),

( ( qi = − K ijt T, 0j − K ij*t T, *j − K it3T * , mi = − K ijh H ,0j − K ij*h H ,*j − K ih3 H * ,

(C.23a) (C.23b)

1 ε ij = (ui , j + u j ,i ) / 2, κ ij = ( β i , j + β j ,i ) / 2, 2 N ij = Aijkl ε kl + Bijkl κ kl − Aijt T 0 − Aijh H 0 − Bijt T * − Bijh H * ,

(C.23c)

M ij = Bijkl ε kl + Dijkl κ kl − B T − B H − D T − D H , ( ( N ij , j = 0, M ij ,ij = 0, Qi = M ij , j , qi ,i = 0, mi ,i = 0, i, j , k , l = 1,2.

(C.23e)

t ij

0

h ij

0

t ij

*

h ij

(C.23d)

*

(C.23f)

Case 1: temperature and moisture content depend on x1 and x2 only Governing equations: ( ( qi ,i = − K ijt T,ij = 0, mi ,i = − K ijh H ,ij = 0,

(C.24a)

N ij , j = Aijkl uk ,lj + Bijkl β k ,lj − A T − A H , j = 0,

(C.24b)

t ij , j

h ij

M ij ,ij = Bijkl uk , lij + Dijkl β k ,lij − B T − B H , ij = 0, t ij , ij

h ij

i, j , k , l = 1,2.

(C.24c)

Appendix C

619

Case 2: temperature and moisture content depend on x3 only ( ( T = T 0 + x3T * , H = H 0 + x3 H * , qi = − K it3T * , mi = − K ih3 H * ,

(C.25a)

N ij = Aijkl u k ,l + Bijkl β k ,l − A T − A H − B T − B H ,

(C.25b)

M ij = Bijkl u k ,l + Dijkl β k ,l − B T − B H − D T − D H , Qi = M ij , j .

(C.25c)

t ij

t ij

0

0

h ij

0

0

h ij

*

t ij

t ij

*

h ij

*

h ij

*

Governing equations: N ij , j = Aijkl u k ,lj + Bijkl β k ,lj = 0, M ij ,ij = Bijkl u k ,lij + Dijkl β k ,lij = 0, i, j , k , l = 1,2 . (C.26)

III.

Electro-Elastic Composite Laminates (Section 13.5): E1 = E1( 0 ) ( x1 , x2 ) + x3 E1(1) ( x1 , x2 ),

E2 = E2( 0 ) ( x1 , x2 ) + x3 E2(1) ( x1 , x2 ),

(C.27a)

ε rs = (u r ,s + u s ,r ) / 2, κ rs = ( β r ,s + β s ,r ) / 2,

(C.27b)

N pq = Apqrsl ε rs + B pqrsκ rs , M pq = B pqrs ε rs + D pqrsκ rs ,

(C.27c)

N pj , j = 0, M ij ,ij = 0, M 4 j , j = 0, p, q, r , s = 1,2,4; i,j = 1,2.

(C.27d)

Governing equations:

Apjrl u r ,lj + B pjrl β r ,lj = 0, Bijrl u r ,lij + Dijrl β r ,lij = 0, B4 jrl u r ,lj + D4 jrl β r ,lj = 0,

(C.28a)

β1, 2 = β 2,1 ,

(C.28b)

p, r = 1,2,4; i, j , l = 1,2.

General solution I.

Laminates under mechanical loadings (Section 13.2): u d = 2 Re{Af ( z )},

φ d = 2 Re{Bf ( z )},

(C.29a)

⎧u1 ⎫ ⎧ β1 ⎫ ⎧φ1 ⎫ ⎧ψ 1 ⎫ ⎧φ ⎫ ⎧u ⎫ u d = ⎨ ⎬, φ d = ⎨ ⎬, u = ⎨ ⎬, β = ⎨ ⎬, φ = ⎨ ⎬, ψ = ⎨ ⎬ , u β φ β ψ ⎩ ⎭ ⎩ ⎭ ⎩ 2⎭ ⎩ 2⎭ ⎩ 2⎭ ⎩ψ 2 ⎭

(C.29b)

A = [a1 a 2 a 3 a 4 ], B = [b1 b 2 b 3 b 4 ],

(C.29c)

⎧ f1 ( z1 ) ⎫ ⎪ ⎪ ⎪ f 2 ( z 2 )⎪ f ( z) = ⎨ ⎬, zα = x1 + μα x2 , ⎪ f 3 ( z3 ) ⎪ ⎪⎩ f 4 ( z 4 )⎪⎭

α = 1,2,3,4.

(C.29d)

Appendix C

620

IIa.

Laminates under hygrothermal loadings – temperature and moisture content depend on x1 and x2 only (Section 13.4): T = 2 Re{g t′ ( zt )}, H = 2 Re{g ′h ( z h )}, ( ( qi = −2 Re{( K it1 + τ t K it2 ) g t′′( zt )}, mi = −2 Re{( K ih1 + τ h K ih2 ) g h′′ ( z h )},

(C.30a)

u d = 2 Re{Af ( z ) + c t g t ( zt ) + c h g h ( z h )},

(C.30c)

(C.30b)

φ d = 2 Re{Bf ( z ) + d t g t ( zt ) + d h g h ( z h )},

IIb.

⎧u1 ⎫ ⎧β1 ⎫ ⎧φ1 ⎫ ⎧ψ 1 ⎫ ⎧u ⎫ ⎧φ ⎫ u d = ⎨ ⎬, φ d = ⎨ ⎬, u = ⎨ ⎬, β = ⎨ ⎬, φ = ⎨ ⎬, ψ = ⎨ ⎬, β ψ u β φ ⎩ ⎭ ⎩ ⎭ ⎩ 2⎭ ⎩ 2⎭ ⎩ 2⎭ ⎩ψ 2 ⎭

(C.30d)

A, B, f ( z ) : same as Problem I, Eqs.(C.29c) and (C.29d). zt = x1 + τ t x2 , zh = x1 + τ h x2 .

(C.30e)

Laminates under hygrothermal loadings – temperature and moisture content depend on x3 only (Section 13.4) : ( ( T = T 0 + x3T * , H = H 0 + x3 H * , qi = − K it3T * , mi = − K ih3 H * ,

(C.31a)

u d = 2 Re{Af ( z )},

(C.31b)

φ d = 2 Re{Bf ( z )} − x1ϑ 2 + x2 ϑ1 ,

ϑi = α T + α H + α T + α H , t i

0

h i

0

*t i

*

*h i

*

i = 1,2.

(C.31c)

A, B, f ( z ) : same as Problem I, Eqs. (C.29c) and (C.29d). α ti , α ih , α *i t , α *i h : extended symbols 5.

III.

Electro Elastic Composite Laminates (Section 13.5): u d = 2 Re{Af ( z )},

φ d = 2 Re{Bf ( z )},

(C.32a) ⎧ f1 ( z1 ) ⎫ ⎪ ⎪ ⎪ f 2 ( z 2 )⎪ ⎧ψ 1 ⎫ ⎧φ1 ⎫ ⎧u1 ⎫ ⎧ β1 ⎫ ⎪⎪ f 3 ( z3 ) ⎪⎪ ⎧u ⎫ ⎧φ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u d = ⎨ ⎬, φ d = ⎨ ⎬, u = ⎨u 2 ⎬, β = ⎨β 2 ⎬, φ = ⎨φ2 ⎬, ψ = ⎨ψ 2 ⎬, f ( z ) = ⎨ ⎬, (C.32b) ⎩β ⎭ ⎩ψ ⎭ ⎪ f 4 ( z 4 )⎪ ⎪ψ ⎪ ⎪φ ⎪ ⎪u ⎪ ⎪β ⎪ ⎩ 4⎭ ⎩ 4⎭ ⎩ 4⎭ ⎩ 4⎭ ⎪ f 5 ( z5 ) ⎪ ⎪ ⎪ ⎪⎩ f 6 ( z 6 ) ⎪⎭ A = [a1 a 2 a 3 a 4 a 5 a 6 ], B = [b1 b 2 b 3 b 4 b 5 b 6 ], z k = x1 + μ k x2 , (C.32c) k = 1,2,3,4,5,6.

621

Appendix C

Material eigen relation Nξ = μξ ,

⎡N N=⎢ 1 ⎣N 3

N = I t N mI t , N2 ⎤ ⎡ (N ) , Nm = ⎢ m 1 N1T ⎥⎦ ⎣( N m ) 3 ⎧a ⎫ ξ = ⎨ ⎬, ⎩b ⎭

(C.33a) (N m ) 2 ⎤ ⎡I , It = ⎢ 1 (N m )1T ⎥⎦ ⎣I 2

I2 ⎤ ⎡ I 0⎤ ⎡0 0 ⎤ , I1 = ⎢ ⎥ , I 2 = ⎢0 I ⎥ , I1 ⎥⎦ 0 0 ⎣ ⎦ ⎣ ⎦

(N m )1 = −Tm−1R Tm , (N m ) 2 = Tm−1 = (N m )T2 , (N m ) 3 = R m Tm−1R Tm − Q m = (N m )T3 .

I.

(C.33b)

(C.33c)

Laminates under mechanical loadings (Section 13.2): ~ ⎡ T~ R B~ ⎤ ⎡Q ~ ⎡R A~ − Q B~ ⎤ (C.34a) − R B~ ⎤ A Q m = ⎢ TA R = , , T = ⎥, ⎢ ~T ⎢ T~T ⎥ m T ⎥ m ~ ~ ~ R − T R ~ ~ R Q − − ⎥ D ⎦ D⎦ ⎣ B ⎣ B B D⎦ ⎣⎢ ~ ~ ~ ~ ~ R B~ = Bi 2 k1 , Q X = X i1k1 , R X = X i1k 2 , TX = X i 2 k 2 , X = A, B or D, i, k = 1,2 , (C.34b)

or ~ ~ ~ ~ ⎡ A A16 B16 / 2 B12 ⎤ 11 ⎢ ~ ⎥ ~ ~ ~ A A66 B66 / 2 B62 ⎥ Q m = ⎢ ~ 16 , ~ ~ ~ ⎢B / 2 B − D66 / 4 − D26 / 2⎥ 16 66 / 2 ⎢ ~ ~ ~ ~ ⎥ B62 − D26 / 2 − D22 ⎥⎦ ⎢⎣ B12 ~ ⎡ A 16 ⎢ ~ A R m = ⎢ ~ 66 ⎢B / 2 ⎢ 66 ~ ⎢⎣ B62 ~ ⎡ A 66 ⎢ ~ A Tm = ⎢ ~26 ⎢ −B ⎢ ~ 61 ⎢⎣− B66 / 2

II.

~ ~ ~ A12 − B11 − B16 / 2 ⎤ ⎥ ~ ~ ~ A26 − B61 − B66 / 2⎥ , ~ ~ ~ B26 / 2 D16 / 2 D66 / 4 ⎥ ⎥ ~ ~ ~ B22 D12 D26 / 2 ⎥⎦ ~ ~ ~ A26 − B61 − B66 / 2 ⎤ ⎥ ~ ~ ~ A22 − B21 − B26 / 2 ⎥ . ~ ~ ~ − B21 − D11 − D16 / 2 ⎥ ⎥ ~ ~ ~ − B26 / 2 − D16 / 2 − D66 / 4⎥⎦

(C.34c)

(C.34d)

Laminates under hygrothermal loadings (Section 13.4): t K11t + 2τ t K12t + τ t2 K 22 = 0,

Nξ = μξ,

Nηt = τ t ηt + γ t ,

h K11h + 2τ h K12h + τ h2 K 22 = 0,

Nηh = τ h ηh + γ h ,

⎡N1 N 2 ⎤ ⎧c t ⎫ ⎧c h ⎫ ⎧a ⎫ , ξ = ⎨ ⎬, ηt = ⎨ ⎬, ηh = ⎨ ⎬, N=⎢ T⎥ b d N N ⎩ ⎭ ⎩ t⎭ ⎩d h ⎭ 1 ⎦ ⎣ 3 γ t , γ h : extended symbols 5. Q m , R m , Tm : same as Problem I, (C.34a), (C.34b), (C.34c), and (C.34d).

(C.35a) (C.35b) (C.35c)

Appendix C

622

III.

Electro Elastic Composite Laminates (Section 13.5): ~ ⎡ T~ R B~ ⎤ (C.36a) ⎡R ~ − Q B~ ⎤ ⎡Q ~ − R B~ ⎤ Q m = ⎢ TA Tm = ⎢ ~A T , R m = ⎢ TA ⎥, ⎥ T ⎥, ~ ~ ~ ~ − R T T R ⎢⎣− R B~ − Q D~ ⎥⎦ D ⎦ D⎦ ⎣ B ⎣ B ~ ~ ~ ~ ~ R B~ = B p 2 r1 , Q X = X p1r1 , R X = X p1r 2 , TX = X p 2 r 2 , X = A, B or D, p, r = 1,2,4, (C.36b)

or ~ ⎡ A 11 ⎢ ~ ⎢ A16 ~ ⎢ A 17 Qm = ⎢ ⎢ B16 / 2 ~ ⎢ B ⎢ ~ 12 ⎣⎢ B18 / 2

~ A16 ~ A66 ~ A67 ~ B66 / 2 ~ B62 ~ B68 / 2

~ ⎡ A16 ⎢ ~ ⎢ A66 ⎢ ~ R m = ⎢ ~ A67 ⎢ B 66 / 2 ⎢ B~ ⎢ ~ 62 ⎢⎣ B 68 / 2 ~ ⎡ A 66 ⎢ ~ ⎢ A26 ⎢ ~ Tm = ⎢ A~68 ⎢ − B61 ⎢ ~ ⎢ − B66 / 2 ~ ⎢− B ⎣ 67 / 2

~ A12 ~ A26 ~ A ~ 27 B 26 / 2 ~ B 22 ~ B 28 / 2

~ A17 ~ A67 ~ A77 ~ B67 / 2 ~ B 27 ~ B78 / 2

~ ~ B16 / 2 B12 B18 / 2 ⎤ ⎥ ~ ~ ~ B66 / 2 B62 B68 / 2 ⎥ ~ ~ ~ B67 / 2 B 27 B78 / 2 ⎥ , ⎥ ~ ~ ~ − D66 / 4 − D26 / 4 − D68 / 4⎥ ~ ~ ~ − D26 / 2 − D22 − D28 / 2⎥⎥ ~ ~ ~ − D68 / 4 − D28 / 2 − D88 / 4 ⎥⎦

~ ~ ~ A18 − B11 − B16 / 2 ~ ~ ~ A68 − B 61 − B 66 / 2 ~ ~ ~ − B17 − B 67 / 2 A78 ~ ~ ~ B 68 / 2 D16 / 2 D 66 / 4 ~ ~ ~ B 28 D12 D 26 / 4 ~ ~ ~ B88 / 2 D18 / 2 D 68 / 4

~ − B17 / 2 ⎤ ⎥ ~ − B 67 / 2⎥ ~ − B 77 / 2⎥ ⎥, ~ D 67 / 4 ⎥ ~ D 27 / 2 ⎥ ⎥ ~ D 78 / 4 ⎥⎦ ~ ~ ~ ~ ~ A26 A68 − B61 − B66 / 2 − B67 / 2 ⎤ ⎥ ~ ~ ~ ~ ~ − B21 − B26 / 2 − B27 / 2 ⎥ A22 A28 ~ ~ ~ ⎥ ~ ~ − B18 − B68 / 2 − B87 / 2 ⎥ . A28 A88 ~ ~ ~ ~ ~ − B21 − B18 − D11 − D16 / 2 − D17 / 2 ⎥ ⎥ ~ ~ ~ ~ ~ − B26 / 2 − B68 / 2 − D16 / 2 − D66 / 4 − D67 / 4⎥ ~ ~ ~ ~ ~ − B27 / 2 − B87 / 2 − D17 / 2 − D67 / 4 − D77 / 4⎥⎦

(C.36c)

(C.36d)

(C.36e)

Generalized eigen relation Elasticity relation (Section 13.2): ⎧a ⎫ N(θ )ξ = μ (θ )ξ, N(θ ) = I t N m (θ )I t , ξ = ⎨ ⎬, ⎩b ⎭

(C.37a)

⎡N1 (θ ) N 2 (θ ) ⎤ ⎡ (N m (θ ))1 (N m (θ )) 2 ⎤ N(θ ) = ⎢ ⎥, N m (θ ) = ⎢ T ⎥, T N ( ) N ( ) θ θ ⎣(N m (θ )) 3 (N m (θ ))1 ⎦ 1 ⎣ 3 ⎦ (N m (θ ))1 = −Tm−1 (θ )R Tm (θ ), (N m (θ )) 2 = Tm−1 (θ ), (N m (ς )) 3 = R m (θ )Tm−1 (θ )R Tm (θ ) − Q m (θ ),

(C.37b) (C.37c)

Q m (θ ) = Q m cos 2 θ + (R m + R Tm ) sin θ cos θ + Tm sin 2 θ ,

(C.37d)

R m (θ ) = R m cos θ + (Tm − Q m ) sin θ cos θ − R sin θ ,

(C.37e)

Tm (θ ) = Tm cos θ − (R m + R ) sin θ cos θ + Q m sin θ .

(C.37f)

2

2

T m

T m

2

2

Appendix C

623

Thermal relation: N (θ )ηt = τ t (θ )ηt + τˆt−1 (θ ) γ t (θ ),

N (θ )ηh = τ h (θ )ηh + τˆh−1 (θ ) γ h (θ ),

(C.38a)

ηt , ηh , τ t (θ ),τ h (θ ),τˆt (θ ),τˆh (θ ), γ t (θ ) , γ h (θ ) : extended symbols 5.

(C.38b)

γ ⎫⎪ ⎡ S H ⎤ ⎧c h ⎫ ⎧c h ⎫ ⎧⎪~ γ ⎫⎪ ⎡ S H ⎤ ⎧c t ⎫ ⎧c t ⎫ ⎧⎪~ ⎬ = i ⎨ ⎬ + ⎨~ t* ⎬, ⎢ ⎬ = i ⎨ ⎬ + ⎨~ h* ⎬, ⎢ T ⎥⎨ T ⎥⎨ ⎣− L S ⎦ ⎩d t ⎭ ⎩d t ⎭ ⎪⎩ γ 2 ⎪⎭ ⎣− L S ⎦ ⎩d h ⎭ ⎩d h ⎭ ⎪⎩ γ 2 ⎪⎭ ~ γ t* , ~γ h* , i = 1,2 : extended symbols 5.

(C.38c)

h* 1

t* 1

i

i

Stress resultants (Section 13.3) Stresses in xi -coordinate system: N i1 = −φi , 2 ,

N i 2 = φi ,1 ,

M i1 = −ψ i , 2 −

1 λi1ψ k ,k , 2

1 1 Q1 = − ψ k ,k 2 , Q2 = ψ k ,k1 , V1 = −ψ 2, 22 , 2 2 λ11 = λ22 = 0, λ12 = −λ21 = 1.

M i 2 = ψ i ,1 −

1 λi 2ψ k ,k , (C.39a) 2

V2 = ψ 1,11 ,

(C.39b)

Stresses in s-n coordinate system (s: tangential direction, n: normal direction): t n = φ ,s ,

t s = − φ , n , m n = ψ , s − η s,

N n = n φ ,s , T

M n = n ψ ,s , T

N s = −s φ , n , T

M s = −s ψ ,n , T

m s = − ψ , n + ηn ,

(C.40a)

N ns = s φ ,s = −n φ ,n = N sn , T

T

(C.40b)

M ns = s ψ , s − η = −n ψ ,n + η = M sn , T

T

(C.40c)

Qn = η,s , Qs = −η,n , Vn = (s ψ ,s ) ,s , Vs = −(n ψ ,n ) ,n , η = (s ψ ,s + n ψ ,n ) / 2. (C.40d) T

T

T

T

Stresses in polar coordinate system: 1 1 Nθ = nT φ, r , N rθ = sT φ, r = − nT φ,θ , N r = − sT φ,θ , r r 1 T 1 T T M θ = n ψ , r , M rθ = s ψ , r − η = − n ψ ,θ + η , M r = − sT ψ ,θ , r r 1 1 T 1 T Qθ = η, r , Qr = − η,θ , η = (s ψ , r + n ψ ,θ ), 2 r r 1 1 T 1 T T Vθ = (s ψ , r ), r = s ψ , r r , Vr = − ( n ψ ,θ ),θ = − 2 (nT ψ ,θθ − sT ψ ,θ ), r r r

Resultant forces and bending moments: ~ ~ ~ B B B t1 = φ1 ]A , t2 = φ2 ]A , t3 = η ]A ,

(C.41a) (C.41b) (C.41c) (C.41d)

(C.42a)

~ = − (ψ − x η )]B , m ~ = (ψ − x η )]B , m 1 2 2 2 1 1 A A ~ = B x dφ − x dφ = ( x φ − x φ − Φ )]B , m 3 A ∫A 1 2 2 1 1 2 2 1

(C.42b)

φ1 = −Φ , 2 ,

(C.42c)

φ2 = Φ ,1 .

624

Appendix C

C.3 Dimensions of Matrices Used in Stroh Formalism Two-dimensional problems

2×1

scalar

In-plane anti-plane Coupling 3×1

A, B

2×2

scalar

3×3

4×4

c,d (thermal)

2×1

scalar

3×1

4×1

f (z )

2×1

function

3×1

4×1

g ( zt ) (thermal)

Scalar function 2×2

–



6×6

8×8

In-plane

aα , bα

Ni

Anti-plane

Electro elastic coupling 4×1

N Q, R , T

2×2

scalar 2×2 C55 , C 45 , C 44

3×3

4×4

u ( z ), φ( z )

2×1

scalar

3×1

4×1

μα

2 pairs

1 pairs

3 pairs

4 pairs

1 pairs

1 pairs

4×4

τ (thermal) 1 pairs – Coupled stretching–bending problems 3×3 ( Aij )

3×3 ( Dij )

Stretching bending coupling 3×3

aα , bα

2×1

2×1

4×1

6×1

A, B

2×2

2×2

4×4

6×6

ct , dt (thermal)

2×1

2×1

4×1

6×1

c h , d h (hygro)

2×1

2×1

4×1

6×1

f (z )

2×1

2×1

4×1

6×1

g t ( zt ) (thermal)

Scalar function

–

Scalar function

Scalar function

g h ( z h ) (hygro)


– 2×2



Stretching

Aij , Bij , Dij

Ni

Bending

Electro elastic coupling 5×5

N Q m , R m , Tm

4×4

4×4

8×8

12×12

2×2

2×2

4×4

6×6

u d ( z ), φ d ( z )

2×1 (u, φ)

2×1 (β, ψ )

4×1

6×1

μα τ t (thermal) τ h (hygro)

2 pairs

2 pairs

4 pairs

6 pairs

1 pairs

–

1 pairs

1 pairs

1 pairs

–

1 pairs

1 pairs

Note: aα , bα : complex vectors, material properties A, B : complex matrices, material properties Aij , Bij , Dij : real tensors, material properties c, d : complex vectors, material properties c t , d t : complex vectors, material properties c h , d h : complex vectors, material properties f ( z ) : complex function vector, problem dependent

g ( z t ) : complex function, problem dependent g t ( zt ), g h ( z h ) : complex functions, problem dependent N, N i , Q, R, T : real matrices, material properties u( z ), φ( z ) : real vectors, physical quantities u d ( z ), φ d ( z ) : real vectors, physical quantities μ α , τ : complex numbers, material properties τ t ,τ h : complex numbers, material properties

Appendix D Collection of the Problem Solutions Note 1: Two-dimensional problems: u = 2 Re{Af ( z )}, φ = 2 Re{Bf ( z )} . Note 2: Stretching–bending coupling problems: u d = 2 Re{Af ( z )}, φ d = 2 Re{Bf ( z )} . Note 3: Mechanical properties are given in all the following problems. 4.1.1 Infinite space: uniform loading (given σ ij∞ ) ∞ σ 22

σ 13∞ σ 12∞

⊗

•

•

σ 11∞

•

•

•

σ 23∞

•

σ 12∞

σ 13∞

⊗

•

x2

⊗

f ( z ) =< zα > ( AT t ∞2 + BT ε1∞ ) .

σ 12∞

σ

∞ 11

Real-form solution:

•

⊗

ε1∞ , ε ∞2 , t1∞ 　 , t ∞2 : (4.2b). Special cases: (4.6)–(4.9).

•

x1

⊗

•

⊗

u = x1ε1∞ + x2ε ∞2 　 , φ = x1t ∞2 − x2t1∞ .

• ∞ σ 23

⊗

⊗

⊗

⊗

⊗

σ 12∞

•

⊗

⊗

∞ σ 22

4.1.2 Infinite space: pure in-plane bending (given M , α , I ) −M −M u= (sin αu1 + cos αu 2 ), φ = ( x1 sin α − x2 cos α ) 2 s(α ), 2I 2I for γ = 0. u1 , u 2 , γ : (4.11b,c). I: moment of inertia of the plate cross section. 4.1.3 Infinite space: point load (given pˆ , zˆα ) x2

M

α

x1

M

x2

1 < ln( zα − zˆα ) > A T pˆ . 2πi Real-form solution: ~ ~ 1 u=− {(ln r )H + π [N1 (θ )H + N 2 (θ )ST ]}pˆ , 2π ~ ~ 1 φ=− {(ln r )ST + π [N 3 (θ )H + N1T (θ )S T ]}pˆ . 2π Stresses: (4.20); isotropic medium: (4.21). 4.1.4 Infinite space: point moment (given mˆ ) mˆ {Hn(θ ) − [N1T (θ )H + N 2 (θ )ST ]s(θ )}, u= 4πr mˆ {S T n(θ ) − [N1T (θ )ST + N 3 (θ )H ]s(θ )}. φ= 4πr Stresses: (4.33a,b) or (4.34); isotropic medium: (4.36). f ( z) =

pˆ = ( pˆ 1 , pˆ 2 , pˆ 3 )

xˆ = ( x1 , x 2 )

x1

x2

mˆ

x1

625

626

Appendix D

4.1.5 Infinite space: dislocation (given bˆ , zˆα ) f ( z ) =< ln( zα − zˆα ) > BT bˆ / 2πi. Real-form solution: ~ ~ 1 u=− {(ln r )S + π [N1 (θ )S − N 2 (θ )L]}bˆ , π 2 x ~ ~ 1 φ= {(ln r )L + π [N1T (θ )L − N 3 (θ )S]}bˆ . 2π Stresses: (4.39b). 4.2.1 Half space: point load (given pˆ , zˆα ) x2

bˆ = (bˆ1 , bˆ2 , bˆ3 )

xˆ = ( x1 , x 2 )

1

x2

f ( z) =

Along the half-space surface x2 = 0 : (4.57) and (4.59). Isotropic medium: (4.60).

x1

pˆ

3 T⎫ 1 ⎧ T −1 ⎨< ln( zα − zˆα ) > A + ∑ < ln ( zα − zˆ j ) > B B I j A ⎬pˆ . 2πi ⎩ j =1 ⎭

( xˆ1, xˆ2 )

4.2.2 Half space: surface point force (given pˆ , xˆ 1) x2

1 < ln( zα − xˆ1 ) > B −1pˆ . 2πi ~ ~ Real-form solution: u = −[ln rI + πN1 (θ )]L−1pˆ /π , φ = − N 3(θ )L−1pˆ . Stresses: (4.63); isotropic medium: (4.64). f ( z) =

pˆ

x1

( xˆ1 ,0)

4.2.3 Half space: distributed load (given pˆ ( xˆ1 ) ) x2

f ( z) =

pˆ ( x1 )

xa

1 xb < ln( zα − xˆ1 ) > B −1pˆ ( xˆ1 )dxˆ1 . 2πi ∫x a

x1

xb

4.2.5 Half space: dislocation (given bˆ , zˆα ) x2

f ( z) = x1 b

( xˆ1 , xˆ2 )

3 T ⎫ 1 ⎧ T −1 ⎨< ln( zα − zˆα ) > B + ∑ < ln ( zα − zˆ j ) > B B I j B ]⎬bˆ . 2πi ⎩ j =1 ⎭

Along the half-space surface x2 = 0 : (4.67).

627

Appendix D

4.3.1 Bimaterials: point load and dislocation (given pˆ , bˆ , zˆα(1) ) u1 = 2 Re{A1[f 0 ( z (1) ) + f1 ( z (1) )]}, 　 φ1 = 2 Re{B1[f 0 ( z (1) ) + f1 ( z (1) )]},

pˆ

u 2 = 2 Re{A 2f 2 ( z ( 2) )}, 　φ 2 = 2 Re{B 2f 2 ( z ( 2) )}.

bˆ ( xˆ1 , xˆ2 )

S2

1 < ln( zα(1) − zˆα(1) ) > ( A1T pˆ + B1T bˆ ), 2πi 1 3 f1 ( z (1) ) = ∑ < 〈ln( zα(1) − zˆ j(1) )〉 > p1, 2π i j =1

f 0 ( z (1) ) =

S1

f2 ( z (2) ) = −

1 2π

3

∑ 〈ln( zα j =1

(2)

− zˆ (1) j )〉 p 2 ,

T p1 = A1−1 ( M 2 + M1 ) −1 ( M 2 − M1 ) A1I j ( A1 pˆ + B1T bˆ ),

p 2 = A 2−1 ( M 2 + M1 ) −1 A1−T I j ( A1T pˆ + B1T bˆ ).

Along the interface x2 = 0 : (4.81). 4.3.2 Bimaterials: point load and dislocation on the interface (given pˆ , bˆ , zˆα(1) , α 0 ) ~ ~ ~ ~ u k = ln rh + π {[N1( k ) (θ ) − N1( k ) (α 0 )]h + [N (2k ) (θ ) − N (2k ) (α 0 )]g}, ~ ~ ~ ~ φ k = ln rg + π {[N 3( k ) (θ ) − N 3( k ) (α 0 )]h + [N1( k )T (θ ) − N1( k )T (α 0 )]g}, α k = 1,2. ~ ~ ~ ~ h = −(Sbˆ + Hpˆ ) / 2π , g = (Lbˆ − ST pˆ ) / 2π . Material 1

pˆ

bˆ

0

( x1 , x2 )

Material 2

5.1 Wedge: uniform traction (given t + , t − , θ + , θ − ) Wedge angle Loading Solution 2α ≠ π , 2π (4.2) or (5.4) tˆ = 0 x2

x2*

x1*

t−

θ−

θ+

x1

2α ≠ π , 2π , 2α c tˆ ≠ 0

t+

θ + − θ − = 2α

tˆ = n T (θ − )t + + n T (θ + )t − , ~ Δ = n T (θ − ) N 3 (θ + ,θ − )n(θ + ), 2α = θ + − θ − Δ(α c )

~ = n T (θ c− )N 3 (θ c+ , θ c− )n(θ c+ ) = 0.

where 2α c = θ c+ − θ c−

(5.11) with hˆ0 = tˆ / πΔ, h 0 : (5.24) with k=0,

g0 = −t −

2α = π

(5.30)

(5.11) with hˆ0 = tˆ / πΔ, h 0 = 0, g 0 = − t −

2α = 2π

(5.32)

(5.11) with hˆ0 = tˆ / πΔ, h 0 = 0, g 0 = − t −

2α = π or 2π

any

(5.38) with h 0 : (5.37)

2α = 2α c

tˆ = 0

2α = 2α c

any

(4.2) or (5.4) (5.12) with g 0 = 0, g1 = − t − , h 0 : (5.34) and (5.20), hˆ0 : (5.35) and (5.16), h1 : (5.35) and (5.20)

628

Appendix D

5.2.1 Wedge: point force (given pˆ , θ + , θ − ) x2

f ( z) =

θ−

~ ~ < lnzα > B T [N 3 (θ + ) − N 3 (θ − )]−1 pˆ .

real-form solution: ~ ~ ~ u = [(ln r )I + πN1 (θ )][N 3 (θ + ) − N 3 (θ − )]−1 pˆ / π , ~ ~ ~ φ = N 3 (θ )[N 3 (θ + ) − N 3 (θ − )]−1 pˆ .

pˆ

x1

θ+

1

π

θ + − θ − = 2α

5.2.2 Wedge: point moment (given mˆ , θ + , θ − ) x2

2α ≠ π , 2π , 2α c : u = φ=

mˆ

θ−

x1

θ+

θ − θ = 2α +

−

− mˆ ˆ N1 (θ − ,θ )n(θ + ),

π rΔ

− mˆ ˆ N (θ − ,θ )n(θ + ) . πrΔ 3

ˆ (θ − ,θ )h , r φ = N ˆ (θ − ,θ )h , 2α = π or 2α = 2π : ru = N 1 0 3 0 h 0 : arbitrary.

2α = 2α c : (5.49), (5.54) and (5.55).

5.2.3 Multi-material wedge space: point force and dislocation (given pˆ , bˆ , θ 0 ,..., θ n ) x ~ ~ u = [(ln r )I + π N1 (θ , θ 0 )]h + π N 2 (θ , θ 0 )g, ~ ~ bˆ φ = [(ln r )I + π N1T (θ , θ 0 )]g + π N 3 (θ , θ 0 )h, ~ ~ ~ ~ θ θ pˆ h = −(Sbˆ + Hpˆ ) / 2π , g = (Lbˆ − S T pˆ ) / 2π . 2

2

2

k

1

1

θ0 = θn

θk

x1

θ n−1 n −1

5.2.4 Multi-material wedge: point force (given pˆ , θ 0 ,..., θ n ) x ~ ~ u = [(ln r )I + πN1 (θ , θ 0 )]h, φ = πN 3 (θ , θ 0 )h, ~ h = N 3−1 (θ n , θ 0 )pˆ / π . pˆ 2

2

θ2 k

θ1

θk

1

θ0

x1

θn n

5.3 Multi-material wedge and wedge space: near-tip solutions (given θ 0 ,..., θ n ) x2

⎧u k (r ,θ )⎫ ⎧u 0 ⎫ 1−δ ˆ 1−δ ⎨ ⎬=r N ⎬, k = 1,2,3,...., n, k (θ ,θ k −1 )(K e ) k −1 ⎨ ⎩φ k (r ,θ ) ⎭ ⎩φ 0 ⎭

2

θ2 k

θ1

θk

1

θ0 = θn

x1

θ n −1 n −1

x2

2

θ2 k

θ1

θk θn n

1

θ0

x1

Bonded : (K e − I )w 0 = 0, w 0 = (u 0 φ 0 )T , Free –free : K(e3)u 0 = 0, φ 0 = 0, Fixed –fixed: K(e2 ) φ 0 = 0, u 0 = 0, Free –fixed : K (e1)u 0 = 0, φ 0 = 0, Fixed –free: K (e4 ) φ 0 = 0, u 0 = 0. K e , (K e ) k −1 : (5.82b), (5.83b), and (5.94); K e(1) , K e( 2 ) , K e(3) , K e( 4 ) : (5.88).

629

Appendix D

6.1.1 Elliptical hole: uniform loading (given σ ij∞ , a, b ) ∞ σ 22

⊗

•

•

σ 13∞

σ 12∞

•

•

•

∞ σ 23

•

σ 12∞

u = x1ε1∞ + x2ε ∞2 − Re{A < ζ α−1 > B −1 (at ∞2 − ibt1∞ )},

σ 12∞

σ 13∞

⊗

σ

•

σ 11∞

x2

⊗

•

⊗

b

⊗

x1

s

•

θ

•

n

⊗

a •

σ 23∞

⊗

⊗

⊗

⊗

⊗

σ 12∞

φ = x1t ∞2 − x2 t1∞ − Re{B < ζ α−1 > B −1 (at ∞2 − ibt1∞ )}, Deformation of the hole boundary: u: (6.11). Hoop stress σ ss : (6.18a,b); unidirectional tension: (6.19a,b); isotropic materials: (6.20)

•

⊗

⊗

∞ 11

σ 22∞

x1 = a cosψ x2 = b cosψ 6.1.2 Elliptical hole: in-plane bending (given M , I , α , a, b ) M

x2

Ma 2 Re{[c2 (α ) + is 2 (α )]A < ζ α−2 > B −1}s(α ), 2I Ma 2 Re{[c2 (α ) + is 2 (α )]B < ζ α− 2 > B −1}s(α ), φ = φ∞ + 2I

u = u∞ +

α

b

x1

M

u ∞ , φ ∞ : (6.21b,c); Hoop stress: (6.25a,b);

a

c2 (α ), s2 (α ) : (6.23b) isotropic materials: (6.27)

6.1.3 Elliptical hole: arbitrary loading (given tˆ n , a, b ) ∞

i {< ζ α− k > B−1 (ck + idk )}, k 2 k =1 1 π ρ tˆ n dψ , c k = ∫ ρ tˆ n cos kψ dψ ,

f ( z ) = i < ln ζ α > AT c0 − ∑

x2

c0 =

b tˆ n

x1

1 2π

π

∫π −

π

dk = a

1

−π

π

ρ tˆ π ∫π −

n

sin kψ dψ

Hoop stress: (6.38) Point force on the hole surface: (6.49) and (6.50) 6.1.4 Elliptical hole: point force (given pˆ , ζˆ ) ρ 2 = a 2 sin 2 ψ + b2 cos2 ψ

α

x2

3 T ⎫ 1 ⎧ T −1 −1 f ( z) = ⎨[< ln(ζ α − ζˆα ) > A + ∑ < ln(ζ α − ζˆ k ) > B BI k A ]⎬pˆ , 2πi ⎩ k =1 ⎭

pˆ

( x1 , x2 )

b

x1

Hoop stress: σ ss =

2

πρ

sT (θ )G 3 (θ ) Re{B < ie iψ (e iψ − ζˆα )−1> A T }pˆ .

a

6.1.5 Elliptical hole: dislocation (given bˆ , ζˆα ) x2

f ( z) =

bˆ

( x1 , x2 )

b x1

a

3 T ⎫ 1 ⎧ T −1 −1 ⎨[< ln(ζ α − ζˆα ) > B + ∑ < ln(ζ α − ζˆ k ) > B BI k B ]⎬bˆ 2πi ⎩ k =1 ⎭

630

Appendix D

6.2.2 Polygon-like hole: uniform loading (given σ ij∞ , a, c, ε ) σ∞ α

x2

x2*

x1*

α

x1

n

s θ

σ∞ α

x1 = a (cosψ + ε cos kψ ),

u = x1ε1∞ + x2ε ∞2 + 2 Re{A[< ζ α−1 > q1 + < ζ α− k > q k ]}, φ = x1t ∞2 − x2t1∞ + 2 Re{B[< ζ α−1 > q1 + < ζ α− k > q k ]}, 1 1 q1 = − aB −1 (t ∞2 − ict1∞ ) 　 , q k = − aεB −1 (t ∞2 + ict1∞ ) . 2 2 Hoop stress: (6.79); unidirectional tension: (6.81); isotropic materials: (6.82); end points: (6.83)

x2 = a ( c sinψ − ε sin kψ ), 6.2.3 Polygon-like hole: in-plane bending (given M , I , α , a, c, ε ) M α

x2*

x2

x1*

θ

α

∑{A < ζ α−l > q l},　φ = φ ∞ + 2 Re ∑ {B < ζ α− l > q l },　

l = 2 ,k −1 k +1, 2 k

α x 1 s n

M

u = u ∞ + 2Re

l = 2 ,k −1 k +1, 2 k

Ma 2 (cl + isl )B −1s(α ), 4I s2 = −c sin 2α/ 2, c2 = [1 − c 2 − (1 + c 2 ) cos 2α ] / 4, 　　　

ql =

, ck −1 = ε [(1 − c) − (1 + c) cos 2α ] / 2, 　　　 sk −1 = ε (1 + c) sin 2α / 2 　 s k +1 = ε (1 − c ) sin 2α / 2　 ck +1 = ε [(1 + c) − (1 − c) cos 2α ] / 2, 　　　 , s2 k = ε 2 sin 2α / 2　 c2 k = −ε 2 cos 2α/2, 　　　　　　　　 . Hoop stress: (6.87a,b); isotropic materials: (6.88) 7.1.1 Crack: near-tip solution and fracture parameters ⎧L−1k ⎫ ⎧u ⎫ 2 1/ 2 ˆ 1/ 2 r N (θ ,−π )⎨ ⎨ ⎬=− ⎬, π ⎩φ ⎭ ⎩ 0 ⎭ θ ＝π θ ＝- π 1 Energy release rate: G = k T L−1k , 2 k: vector of stress intensity factors

Appendix D

631

7.1.2 and 7.4.5 Interface crack: near-tip solution and fracture parameters ⎧L−1h ⎫ ⎧u1 ⎫ − r 1−δ N11−δ (θ ,−π )⎨ 1 ⎬, ⎨ ⎬= ⎩φ1 ⎭ (1 − δ ) sin πδ ⎩ 0 ⎭ θ ＝π ΙΙ

⎧L−1h ⎫ ⎧u 2 ⎫ − r 1−δ N12−δ (θ ,0)N11−δ (0,−π )⎨ 1 ⎬, ⎨ ⎬= ⎩φ 2 ⎭ (1 − δ ) sin πδ ⎩ 0 ⎭

θ ＝- π

Ι

1 + iε : (7.13a,b) and (7.11c); 2 Along the interface (θ = 0) :

δ=

1

φ' =

2πr

Δu =

2r

π

h: arbitrary

Λ < (r / l) iεα > Λ −1k , Λ −T
Λ −1k (1 + 2iε α ) cos h (πε α )

Λ : (7.67b) and (7.68); l : reference length

ε α : (7.69b) and (7.13b);

1 T k Ek , E = D + WD −1 W 4 Orthotropic bimaterials: (7.123) and (7.116); Isotropic bimaterials: (7.117a,b)

Energy release rate: G =

7.2.1 Crack: uniform loading (given σ ij∞ , a ) ∞ σ 22

σ 13∞

⊗ ∞

σ 12

•

•

σ 11∞

•

•

•

σ 23∞

•

σ 12∞

− a Re{A < ζ α−1 > B −1t ∞2 }, u = x1ε1∞ + x2ε ∞2 　

σ 12∞

σ 13∞

⊗

σ 11∞

•

x2

⊗

φ = x1t ∞2 − x2 t1∞ − a Re{B < ζ α−1 > B −1t ∞2 },

•

Fracture parameters: k = πa t ∞2 , Δu = 2 a 2 − x12 L−1t ∞2 , πa ∞T −1 ∞ G= t2 L t2 2 Isotropic materials: (7.21) 7.2.2 Crack: in-plane bending (given M , I , α , a ) ⊗

x1

•

a

⊗

a

•

⊗

•

∞ σ 23

⊗

⊗

⊗

⊗

⊗

σ 12∞

•

⊗

⊗

σ 22∞

M

x2

x1

a

M

a

M (a sin α ) 2 Re{A < ζ α−2 > B −1}s(α ), 4I M (a sin α ) 2 φ = φ∞ + Re{B < ζ α−2 > B −1}s(α ), 4I

u = u∞ +

k=

− Ma πa sin 2 α s(α ), 2I

Δu =

− M sin 2 α x1 a 2 − x12 L−1s(α ) I

632

Appendix D

7.2.3 Crack: arbitrary loading on the crack surface (given tˆ n , a )

⊗

⊗

⊗

a

x1

⊗

{

}

{

}

{

}

{

}

∞ 1 u = 2 Im A < ln ζ α > A T c 0 − ∑ Im A < ζ α− k > B −1 (c k + id k ) , k =1 k ∞ 1 φ = 2 Im B < ln ζ α > A T c 0 − ∑ Im B < ζ α− k > B −1 (c k + id k ) , k =1 k

x2

a

c k , d k : (6.29);

k=

π⎛

∞ ⎞ ⎜ ST c 0 − ∑ d k ⎟ a⎝ k =1 ⎠

7.2.4 Crack: point force (given pˆ , xˆ1 , xˆ 2 ) f ( z) =

x2

pˆ

( xˆ1 , xˆ 2 ) a

k=

x1

a

3 T ⎫ 1 ⎧ T −1 −1 ⎨[< ln(ζ α − ζˆα ) > A + ∑ < ln(ζ α − ζˆ k ) > B BI k A ]⎬pˆ . 2πi ⎩ k =1 ⎭

2

πa

Im{B < (1 − ζˆα ) −1 > A T }pˆ .

When ( xˆ1 , xˆ 2 ) = (c,0) :

k=

⎧⎪ T a + c ⎫⎪ I ⎬pˆ . ⎨− S + a − c ⎪⎭ 2 πa ⎪⎩

1

7.2.5 Crack: dislocation (given bˆ , xˆ1 , xˆ 2 ) x2

f ( z) =

bˆ ( xˆ1 , xˆ 2 ) a

x1

a

3 T ⎫ 1 ⎧ T −1 −1 ⎨[< ln(ζ α − ζˆα ) > B + ∑ < ln(ζ α − ζˆ k ) > B BI k B ]⎬bˆ 2πi ⎩ k =1 ⎭

7.3.1 Collinear cracks: general solutions (given a1 , b1 ,...., an , bn ) n

f ( z ) = ∑ < ∫ zαk / X ( zα )dzα > c k + < zα 　 >c

x2

k =0

a1 b1

a2

bk an

ak

b2

X ( zα ) = Π nk =1 ( zα − ak )1/ 2 ( zα − bk )1/ 2

x1

bn

7.3.2 Two collinear cracks: uniform loading (given a1 , b1 , a2 , b2 , σ ij∞ )

λk < f k ( zα ) > B −1t ∞ , k =0 λ 2

σ 12∞

x2

⊗

σ 13∞

σ 12∞ σ 13∞

⊗

a2

b2

a1

⊗

σ 12∞

σ 23∞

∞ σ 22

σ 12∞

σ 13∞

σ 23∞

∞ σ 22

σ 11∞

σ 12∞

x1 b1

σ 12∞

⊗

σ 11∞

•

σ 13∞

σ 12∞

•

•

σ 11∞

σ 23∞

σ 12∞

•

∞ σ 23

f ( z) = ∑

σ 22∞

σ 22∞

σ 11∞

t ∞ : (7.33b); f 0 ( zα ), f1 ( zα ), f 2 ( zα ) : (7.34b,c);

λ , λ0 , λ1 , λ2 : (7.41b) and (7.36b) Fracture parameters: k = kt ∞ , Δu = Δf (x1 )L−1t ∞ , 1 G = k 2t ∞T L−1t ∞ 　 2 k: (7.48c); Δf ( x1 ) : (7.49b) Orthotropic materials: (7.50) and (7.54); Isotropic materials: (7.51a,b) Two equal cracks: λ , λ0 , λ1 , λ2 : (7.55a,b); k , Δu, G : (7.56a,b,c)

Appendix D

633

7.3.3 Collinear periodic cracks: uniform loading (given l, W , σ ij∞ ) σ 22∞

σ 12∞ σ 13∞

σ

w

⊗

σ 11∞

σ 12∞

σ 13∞

2l

2l

σ 23∞

σ 12∞

⊗

∞ σ 22

sin(πzα / W ) 1 < zα − ∫ dzα 　 > B −1t ∞ 2 2 sin (πzα / W ) − sin 2 (πl / W )

Fracture parameters: k = kt ∞ , Δu = Δf ( x1 )L−1t ∞ , G =

∞ 12

x1 2l

σ 12∞

⊗

σ 11∞

σ 13∞

•

•

x2

σ 12∞

σ 11∞

•

σ

f ( z) = −

σ 12∞

∞ 13

⊗

σ 11∞

σ 23∞

σ 12∞

•

σ 22∞ ∞ σ 23

Δf ( x1 ) : (7.59b)

k: (7.58);

σ 23∞

∞ σ 22

1 2 ∞T −1 ∞ k t L t 　 2

7.4.1 Collinear interface cracks: general solution (given a1 , b1 ,...., an , bn , tˆ( s ) ) x2

u1 = A1f1 ( z (1) ) + A1 f1 ( z (1) ), 　φ1 = B1f1 ( z (1) ) + B1 f1 ( z (1) ), u 2 = A 2f 2 ( z ( 2) ) + A 2 f 2 ( z ( 2) ), 　φ 2 = B 2f 2 ( z ( 2) ) + B 2 f 2 ( z ( 2) ), S1

x1 a1 b1

a2

b2

ak bk

an

, 　z ∈ S2 ; f 2 (z) = B −21 M *−1M * ψ ( z ) 　 = B1−1ψ ( z ), 　 z ∈ S1 　 f1 ( z ) 　

bn

ψ' ( z ) =

S2

−1 1 X0 ( z ) ∫ [X0+ (s)]−1 tˆ(s)ds + X0 ( z )p n ( z ) L s−z 2πi n

X 0 ( z ) = Λ < ∏ ( z − a j ) −δα ( z − b j ) δα −1 > ; j =1

1 + iε α : (7.69a,b) and (7.13b) 2 7.4.2 A semi-infinite interface crack: point force on the crack surface (given a, tˆ 0 )

δα =

Λ : (7.67b) and (7.68);

x2 tˆ2

tˆ1

•

⊗

tˆ3 tˆ1

x1

tˆ3

1 2π

∑ < eπε a1/ 2−iε ∫ k =1

f 2 ( z ( 2) ) =

1 2π

∑ < eπε a1/ 2−iε ∫ k =1

3

k

k

3

k

k

a

tˆ2

−1/ 2+iε k

zα(1) dzα(1) > B1−1I k Λ −1tˆ 0 , zα(1) + a

f1 ( z (1) ) =

k=

−1 zα( 2)−1/ 2+iε k ( 2) dzα > B −21M* M* I k Λ −1tˆ 0 , zα( 2 ) + a

2 Λ < (a / l) −iε α cos hπε α > Λ −1tˆ 0 ; πa

Orthotropic bimaterials: (7.126) 7.4.3a A finite interface crack: point force on the crack surface (given a, c, tˆ 0 ) x2

eπε k ⎛ a + c ⎞ f1 ( z ) = −∑ < ⎜ ⎟ 2π ⎝ a − c ⎠ k =1 3

tˆ2

tˆ1

•

tˆ3

⊗

tˆ1

tˆ3

iε k

1 ∫ c − zα

iε k

a 2 − c 2 ⎛ zα − a ⎞ ⎟ dzα ⎜ zα2 − a 2 ⎜⎝ zα + a ⎟⎠

> B1−1ΛI k Λ −1tˆ 0 ,

x1

−1

tˆ2

f 2 ( z ) : from f1 ( z ) with zα and B1−1 replaced by zα( 2 ) and B −21M* M*.

c a

a

k=

1

⎡ l(a + c) ⎤ a+c Λ Λ −1tˆ 0 .

634

Appendix D

7.4.3b A finite interface crack: uniform loading on the crack surface (given a, tˆ ) x2

3 ⎛ z −a⎞ ⎟ f1 ( z ) = −∑ < zα − zα2 − a 2 ⎜⎜ α ⎟ k =1 ⎝ zα + a ⎠

•

•

•

•

−1

>B1−1ΛI k Λ −1 (I + M* M* ) −1 tˆ. −1

f 2 ( z ): from f1 ( z ) with zα and B1−1 replaced by zα( 2 ) and B −21M* M*.

x1

⊗

⊗

•

tˆ

iε k

⊗

⊗

⊗

k = πa Λ < (1 + 2iε α )(2a / l) −iε α > Λ −1tˆ.

orthotropic bimaterials: (7.130) a

a

7.4.4 Two collinear interface cracks: uniform loading at infinity (given a1 , b1 , a2 , b2 , σ ij∞ ) ∞ σ 22

σ

σ 11∞

x2

•

σ 12∞

σ 23∞

3

f1 ( z ) = ∑ < ∫

σ 12∞

•

∞ σ 22

σ 23∞ ∞ 13

σ

⊗

σ 11∞

•

σ 13∞

σ 12∞

k =1

∞ 12

x1 a1

σ 13∞

σ 11∞

b1

a2

b2

σ 12∞

σ 23∞

σ 12∞

⊗

⊗

σ 12∞

σ 22∞

σ 11∞

•

⊗

σ

σ 13∞

∞ 12

1

λ1k

(λ1k zα2 + λ2 k zα + λ3k )X k ( zα ) *−1

> B1−1 ΛI k Λ −1 (I + M M * ) −1 t ∞

λ1k , λ2 k , λ3k : (7.92b) and (7.91b);

t ∞ : (7.89b) −1

σ 23∞

f 2 ( z ): from f1 ( z ) with zα and B1−1 replaced by zα( 2 ) and B −21M* M*.

σ 22∞

k = 2π Λ < kα > Λ −1t ∞ ,

kα : (7.131b)

8.1 Elliptical elastic inclusion: general solution (given a,b) u1 = A1[f0 (ζ ) + f1 (ζ )] + A1[f0 (ζ ) + f1 (ζ )]⎫⎪ ⎬, ζ ∈ S1 , ζ α : (8.2c) φ1 = B1[f0 (ζ ) + f1 (ζ )] + B1[f0 (ζ ) + f1 (ζ )] ⎪⎭ b x u 2 = A 2f 2 (ζ * ) + A 2 f 2 (ζ * )⎫⎪ * ζ α* : (8.2c) ⎬, ζ ∈ S 2 , a φ 2 = B 2f 2 (ζ * ) + B 2 f 2 (ζ * ) ⎪⎭ f 0 (ζ ) : (8.7) or (8.15); f1 (ζ ) : (8.12c,d) or (8.16a,b) f 2 (ζ ) : (8.4a,b) in which c k : (8.11a,b) Interfacial stresses: (8.13a,b) and (8.14) Superscript •* : quantity related to the inclusion. 8.1.1 Elliptical elastic inclusion: uniform loading at infinity (given a, b, σ ij∞ ) x2

1

f 0 (ζ ) =< zα > ( A1T t ∞2 + B1T ε1∞ ) , ∞ σ 23

•

•

•

•

σ 12∞

σ

x2

⊗

σ 12∞ •

⊗

•

σ 13∞

•

⊗ ⊗

•

b

x1 •

⊗ a

•

⊗

σ 12∞

∞ σ 23

⊗

⊗

⊗

⊗

•

⊗

⊗

σ 12∞

⊗

σ 11∞

•

σ 22∞

σ 22∞

∞ 13

σ 11∞

f1 (ζ ) =< ζ α−1 > g1 ,

f 2 (ζ ) =
c1 , a − ibμα*

g1 = −B1−1{B1 e1 + B1 < γ α > e1 − B 2 c1 − B 2 < γ α* > c1}, γ α : (8.4b) −1

−1

c1 = −i{G 0 − G 1 G 0 G 1}−1{A1−T e1 + G 1 G 0 A1−T e1}, G 0 ,G 1:(8.11b) 1 e1 = < a − ibμα > ( A1T t ∞2 + B1T ε1∞ ). 2

635

Appendix D

8.1.2 Elliptical elastic inclusion: point force at the matrix (given a, b, pˆ , ζˆα ) f 0 (ζ ) =< ln(ζ α − ζˆα ) > A1T pˆ / 2πi f1 (ζ ) =

∞ 1 3 < ln(ζ α−1 − ζˆk ) >B1−1 B1I k A1T pˆ + ∑ < ζ α−k ∑ 2πi k =1 k =1

{

> B1−1 B 2 c k + B 2 < γ α*k > c k ∞

{

}

}

f 2 (ζ * ) = ∑ < ζ α*k + γ α*k ζ α*− k > c k , k =1

γ α* : (8.4b)

c k : (8.11a,b) in which e k : (8.23)

8.2.1a Elliptical rigid inclusion: general loading condition (given a,b) u = A[f0 (ζ ) + f1 (ζ )] + A[f0 (ζ ) + f1 (ζ )]⎫⎪ ⎬ , ζ ∈ S1 , φ = B[f0 (ζ ) + f1 (ζ )] + B[f0 (ζ ) + f1 (ζ )] ⎪⎭

ω −1 A k, 2ζ ω : (8.34a,b) and (8.35)

f1 (ζ ) = − A −1 Af 0 (1 / ζ ) +

k: (8.30b);

∞

∑ e k ζ k , e -k

When f 0 (ζ ) = ek =

k = −∞

=< γ αk > e k ,

f 0 (ζ ) 1 dζ , γ α : (8.4b) ∫ C 2π i ζ k +1

∞

{

}

f1 (ζ ) = − A −1 ∑ Ae k + A < γ αk > e k ζ −k + k =1

ω −1 A k, 2ζ

−T

ω=

− 2 Im{k T A e1}

. k T Mk Interfacial stresses: (8.36a,b)

8.2.1b Elliptical rigid inclusion: uniform loading at infinity (given a, b, σ ij∞ ) f0 (ζ ) =< ζ α + γ α ζ α−1 > e1 ,

e1 =

1 < a − ibμα > ( A T t ∞2 + BT ε1∞ ) , γ α : (8.4b) 2

1 < ζ α−1 > A −1 (a ε1∞ + ibε ∞2 − ω k ), k: (8.30b) 2 a 2 (H −1ε1∞ ) 2 − ab (H −1Sε1∞ )1 + (H −1Sε ∞2 ) 2 − b 2 (H −1ε ∞2 )1 ω= a 2 (H −1 ) 22 + 2ab(H −1S) 21 + b 2 (H −1 )11 f1 (ζ ) = −

[

]

636

Appendix D

8.2.1c Elliptical rigid inclusion: point load at the matrix (given a, b, pˆ , ζˆα ) f 0 (ζ ) =< ln(ζ α − ζˆα ) > A T pˆ / 2πi f1 (ζ ) =

ω=

1 3 ω ∑ < ln(ζ α−1 − ζˆ k ) >A −1AI k AT pˆ + 2 < ζ α−1 > A −1 k 2πi k =1

T Re{k A −T < ζˆα−1 > A T }pˆ , πk T Mk

k: (8.30b)

If the load is applied on the interface boundary, ˆ i.e., ζˆα = eiψ , ω : (8.47) 8.2.2a Rigid line inclusion: uniform loading at infinity (given a, σ ij∞ ) σ σ

σ

σ 11∞ σ 12∞

σ 12∞

σ σ

x2

∞ 12

∞ 13

σ

∞ 11

φ = x1t ∞2 − x2t1∞ − a Re{B < ζ α−1 > A −1}(ε1∞ − ωi 2 ),

ω=

x1 a

a

u = x1ε1∞ + x2ε ∞2 　 − a Re{A < ζ α−1 > A −1}(ε1∞ − ωi 2 ),

∞ 22

∞ 23

∞ 13

σ

σ 12∞ σ

a Re{k T Mε1∞ } , k T Mk

k = ai 2

∞ 23

∞ 22

8.2.2b Rigid line inclusion: point load at the matrix (given a, pˆ , ζˆα ) pˆ

x2

f ( z) =

(xˆ , xˆ ) 1

+ {ω < ζ α−1 > A −1k} / 2,

2

T

x1 a

a

3 1 ⎧ T T⎫ −1 −1 ⎨< ln(ζ α − ζˆα ) > A + ∑ < ln(ζ α − ζˆ k ) > A AI k A ⎬pˆ 2πi ⎩ k =1 ⎭

ω=

Re{k A −T < ζˆα−1 > AT }pˆ

π k T Mk

.

k = ai 2

If the load is applied on the rigid line, i.e., xˆ2 = 0 , ω =

xˆ1 pˆ 2 π kT M k

8.2.3 Polygon-like rigid inclusion: uniform loading at infinity (given a, c, ε , σ ij∞ ) u = x1ε1∞ + x2ε ∞2 + 2 Re{A[< ζ α−1 > q1 + < ζ α− k > q k ]}, φ = x1t ∞2 − x2 t1∞ + 2 Re{B[< ζ α−1 > q1 + < ζ α−k > q k ]}, q l = A T g l + B T h l , l = 1, k h1 = a (ω i2 − ε1∞ ) + x2ε ∞2 , g1 = −aH −1{S(ω i2 − ε1∞ ) − c(ω i1 + ε ∞2 )}　 h k = aε (ω i2 − ε1∞ ) 　 , g k = −aεH −1{S(ω i2 − ε1∞ ) + (ω i1 + ε ∞2 )}　 ω : (8.55)

637

Appendix D

8.3.1 Elliptical elastic inclusion: dislocations outside the inclusions (given a, b, bˆ , ζˆα ) 1 < ln(ζ α − ζˆα ) > B1T bˆ , 2πi 1 3 1 ∞ f1 (ζ ) = < ln(ζ α−1 − ζˆk )>B1−1 B1I k B1T bˆ + ∑ ∑ < ζ α−k > E k bˆ , 2πi k =1 2πi k =1 1 ∞ f 2 (ζ * ) = ∑ < ζ α*k + γ α*k ζ α*− k > Ck bˆ , 2πi k =1 f 0 (ζ ) =

E k , C k : (8.56d); γ α* : (8.4b) Rigid inclusion: 1 f 0 (ζ ) = < ln(ζ α − ζˆα ) > B1T bˆ , 2πi 1 3 ω f1 (ζ ) = ∑ < ln(ζ α−1 − ζˆ k ) >A1−1A1I k B1T bˆ + 2 < ζ α−1 > A1−1 k , 2πi k =1 ω : (8.57c); k: (8.30b) 8.3.2 Elliptical elastic inclusion: dislocations inside the inclusions (given a, b, bˆ , zˆ * ) α

u1 = A1[f0 (ζ ) + f1 (ζ )] + A1[f0 (ζ ) + f1 (ζ )]⎫⎪ ⎬ , ζ ∈ S1 , φ1 = B1[f0 (ζ ) + f1 (ζ )] + B1[f0 (ζ ) + f1 (ζ )] ⎪⎭ u 2 = A 2 [f0* (ζ * ) + f 2 (ζ * )] + A 2 [f0* (ζ * ) + f 2 (ζ * )]⎫⎪ * ⎬ , ζ ∈ S2 , * * * * * * φ2 = B 2 [f0 (ζ ) + f 2 (ζ )] + B 2 [f0 (ζ ) + f 2 (ζ )] ⎪⎭ 1 f 0 (ζ ) = < ln ζ α > B1T bˆ , , 2πi 1 f 0* (ζ ) = < ln( zα* − zˆα* ) > B T2 bˆ , 2πi −1 ∞ 1 ∞ f1 (ζ ) = < ζ α−k >B1−1B 2 < ek*α > B T2 bˆ + ∑ ∑ < ζ α−k > E k bˆ , 2πi k =1 2πi k =1 1 ∞ f 2 (ζ * ) = ∑ < ζ α*k + γ α*k ζ α*− k > C k bˆ . 2πi k =1 ek*α : (8.61b); E k , C k : (8.56d) in which Tk : (8.71b); γ α* : (8.4b)

638

Appendix D

8.3.3 Elliptical elastic inclusions: dislocation on the interface (given a, b, bˆ , zˆα* ) u1 , φ1 and u 2 , φ2 : same expressions as Section 8.3.2. 1 1 f 0 (ζ ) = < ln(ζ α − ζˆα ) > B1T bˆ + < ln(ζ α−1 − ζˆα ) > Q1bˆ , 2πi 2πi 1 f 0* (ζ ) = < ln( zα* − zˆα* ) > Q 2bˆ , 2πi −1 3 1 ∞ f1 (ζ ) = < ζ α− k >B1−1B 2 < l*kα > QT2 bˆ + ∑ ∑ < ζ α− k > Ek bˆ , 2πi k =1 2πi k =1 1 ∞ f 2 (ζ * ) = ∑ < ζ α*k + γ α*kζ α*− k > Ck bˆ , 2πi k =1 l*kα : (8.79b); γ α* : (8.4b) Q1 , Q 2 , E k , C k : (8.82b) and (8.56d) in which Tk : (8.82c) 9.1 Rigid punches on a half-plane: general solution (given uˆ , qˆ k ) f ' ( z ) = B −1θ' ( z ) , θ' ( z ) =

X 0 ( z ) = ΛΓ( z ),

1 1 X 0 ( z ) ∫L [ X 0+ (t )]−1 M uˆ ' (t )dt 2π t−z + X 0 ( z )p n ( z ),

Λ = [λ1 λ 2 λ 3 ],

n

Γ( z ) =< ∏ ( z − a j ) − (1+ δ α ) ( z − b j )δ α > . j =1

p n ( z ) = d 0 + d1 z + LL + d n −1 z n −1. λ α , δ α : (9.11) and (9.12a,b); d k , k = 0,1,2,..., n − 1 : (9.15) and (9.20) t ( x1 ) = − A −T f ′( x1− ) − iM uˆ ′( x1 ), x1 ∈ L.

Contact pressure:

Surface deformation: u′( x1 ) = BT f ′( x1− ),

x1 ∉ L.

9.1.2 Indentation by a flat-ended punch (given a, qˆ ) ⎛ zα + a ⎞ ⎜ ⎟ 2 2 ⎜ z −a⎟ zα − a ⎝ α ⎠

1 3 f ' ( z) = ∑< 2πi k =1 t ( x1 ) =

1

π a −x

u ′( x1 ) = m

2

2 1

1

[I +

1 − cR

β

2

(S T ) 2 +

− iε k

>B −1ΛI k Λ −1qˆ .

cI

β

S T ]qˆ ,

| x1 |≤ a,

⎡ 1 − c *R T 2 c *I T ⎤ L−1 ⎢I + (S ) + S ⎥qˆ , β β2 π x12 − a 2 ⎣ ⎦ 1

x1 > a and x1 < −a,

cR , c I , c*R , c*I : (9.27b);

ε k , β : (9.12b)

Orthotropic half-plane: (9.29);

Isotropic half-plane: (9.31) and (9.30)

639

Appendix D

9.1.3 A flat-ended punch tilted by a couple (given a, ω or mˆ ) iω ⎧ −1 3

−1 −1 ⎫ ⎨B − ∑ < Γ k ( zα )( zα + 2iaε k ) > B ΛI k Λ ⎬Li 2 , 2 ⎩ k =1 ⎭

f ′( z ) =

ωx1

⎡ 1 − cR T 2 cI T ⎤ S + S ⎥ Li 2 , | x1 |≤ a, ⎢I + β ⎦ β2 a −x ⎣

t ( x1 ) =

2

2 1

⎧⎪ u′( x1 ) = ω ⎨I m ⎪⎩

mˆ =

⎡ 1 − c*R 2 c*I ⎤ ⎫⎪ S − S ⎥ i 2 ⎬, ⎢I + β ⎦ ⎪ β2 x −a ⎣ ⎭ x1

2 1

2

x1 > a and x1 < −a,

⎧⎪⎡ 4ε 2 ⎤ ⎫⎪ a 2ω ⎨⎢I − 2 (S T ) 2 ⎥ L ⎬ 2 β ⎪⎩⎣ ⎦ ⎪⎭ 22

π

Γ k ( zα ), c R , c I , c R* , c *I : (9.36b) ;

ε , ε k , β : (9.12b)

9.1.4 Indentation by a parabolic punch (given a, R, qˆ ) i 3 ∑ < Γ k ( zα )[ zα2 + 2iaε k zα − (1 + 4ε k2 )a 2 ] >B −1Λ I k Λ −1Li 2 4 R k =1 1 3 i + < zα > B −1Li 2 + ∑ < Γ k ( zα ) >B −1Λ I k Λ −1qˆ , 2R 2πi k =1 2 x12 − a 2 ⎡ 1 − (cc~ ) R T 2 (cc~ ) I T ⎤ t ( x1 ) = S + S ⎥ Li 2 ⎢I + β β2 ⎦ 2 R a 2 − x12 ⎣

f ′( z ) = −

⎡ 1 − cR T 2 cI T ⎤ (S ) + S ⎥qˆ , ⎢I + β ⎦ β2 π a −x ⎣ 2 x12 − a 2 ⎡ 1 − (c *c~ ) R 2 (c *c~ ) I x u′( x1 ) = 1 i 2 m S − ⎢I + β R β2 2 R x12 − a 2 ⎣ +

m

1 2

2 1

| x1 |≤ a, ⎤ S⎥i 2 ⎦

⎡ 1 − c *R T 2 c I* T ⎤ (S ) + S ⎥qˆ , x1 > a and x1 < − a, L−1 ⎢I + β ⎦ β2 π x12 − a 2 ⎣ 1

ε k , β : (9.12b) Γ k ( zα ) : (9.36b)1; * * ~ ~ c R , c I , c R , c I , (cc ) R , (cc ) I , (c *c~ ) R , (c *c~ ) I : (9.38b,c) 9.2 Rigid stamp indentation on a curvilinear hole boundary: general solution (given a, c, ε , uˆ , qˆ k ) f ′(ζ ) = B −1θ′(ζ )/ζ 1 ρ θ' (ζ ) = − X c (ζ ) ∫L [ X c+ ( s )]−1 Muˆ ' ( s )ds + X c (ζ )p c (ζ ), 2πi s −ζ X c (ζ ) = Λ c Γ c (ζ ) = ΛΓ(ζ ), Λ = [λ1 λ 2 λ 3 ],

n

Γ( z ) =< ∏ ( z − a j ) − (1+δ α ) ( z − b j )δ α > . j =1

ρ : (9.42b) and (6.64); λ α , δ α : (9.11) and (9.12a,b); p c : (9.47), (9.48b) and (9.49a,b)

640

Appendix D

9.2.2 Rigid stamp indentation on an elliptical hole boundary (given a, b, φ , qˆ ) f ′(ζ ) =

e 2φε k 1 3 ⎡ −1 −1 T ∑ ⎢< Γ k (ζ α ) > B Λ c I k Λ c BA + < Γ k (ζ α ) ζα 2πi k =1 ⎣ ⎤ > B −1 Λ c I k Λ c−1 B A T ⎥qˆ ⎦

Γ k (ζ α ) =< (ζ α − e −iφ ) −1/ 2−iε k (ζ α − e iφ )1/ 2+iε k > ;

ε k : (9.12b)

Isotropic medium: (9.57) 9.2.3 Rigid stamp indentation on a polygon-like hole boundary (given a, c, ε , φ , qˆ ) f ′(ζ ) =

e 2φε k 1 3 ⎡ −1 −1 T ∑ ⎢< Γ k (ζ α ) > B Λ c I k Λ c BA + < Γ k (ζ α ) 2πi k =1 ⎣ ζα ⎤ > B −1 Λ c I k Λ c−1 B A T ⎥qˆ ⎦

Γ k (ζ α ) =< (ζ α − e −iφ ) −1/ 2−iε k (ζ α − e iφ )1/ 2+iε k > ;

ε k : (9.12b)

Note: the solution form of this problem is the same as that of 9.2.2 except ζ α . 9.3.1 Rigid punch on a boundary perturbed by a straight line: general solution (given ( ε ϕ ( x1 ), uˆ , uˆ 0 , uˆ 1 ) ( ( f ( z ) = f 0 ( z ) + ε f1 ( z ) + ε 2f 2 ( z ) + L. ( ( uˆ ( x1 ) = uˆ 0 ( x1 ) + ε uˆ 1 ( x1 ) + ε 2uˆ 2 ( x1 ) + L f 0′ ( z ) = B −1θ′0 ( z ) , 1 1 θ′0 ( z ) = X0 ( z)∫ [ X 0+ (t )]−1 M uˆ ′0 (t )dt + X 0 ( z )p n ( z ), Lt − z 2π f1 ( zˆ ) = B −1{θ1 ( zˆ ) − μϕ ( zˆ )Bf 0′ ( zˆ )},

θ1 ( zˆ ) =

1 1 X 0 ( zˆ ) ∫L [ X 0+ (t )]−1 M uˆ 1 (t )dt + X 0 ( zˆ )p (n1) ( zˆ ). 2π t − zˆ

641

Appendix D

( 9.3.2 Rigid stamp on an elliptical perturbed hole: general solution (given a, ck , d k , ε , ϕ (ζ ), uˆ , uˆ 0 , uˆ 1 ) ( ( f ( z ) = f 0 (ζ ) + ε [f1 (ζ ) + ϕ (ζ )f 0′ (ζ )] + ε 2 [f 2 (ζ ) + ϕ (ζ )f1′(ζ ) 1 + ϕ 2 (ζ )f 0′′(ζ )] + L 2 ( ( uˆ (σ ) = uˆ 0 (σ ) + ε uˆ 1 (σ ) + ε 2 uˆ 2 (σ ) + L

f 0′ (ζ ) = B −1θ′0 (ζ )/ζ 1 ρ θ′0 (ζ ) = − X c (ζ ) ∫L [ X c+ ( s )]−1 Muˆ ′0 ( s )ds + X c (ζ )p c (ζ ), 2πi s −ζ a N ⎫ ⎧ f1 (ζ ) = B −1 ⎨θ1 (ζ ) − ∑ (c k + id k )[1 − iμ + (1− iμ )ζ 2 ]ζ − k Bf 0′ (ζ )⎬ 2 k =1 ⎭ ⎩ 1 1 θ1 (ζ ) = X c (ζ ) ∫ [ X c+ ( s )]−1 Muˆ 1* ( s )ds + X c (ζ )p (c1) (ζ ), L s −ζ 2πi N

uˆ 1* ( s ) = i{uˆ 1 ( s ) − ∑ Re[iaρ (ck − id k )(1 + iμ ) s k −1 ]uˆ ′0 ( s )}. k =1

ρ : (9.42b) and (9.80) 9.3.3a A rigid flat-ended punch on a cosine wavy-shaped boundary (given x2 = ε(ϕ ( x1 ), ϕ ( x1 ) = cos x1 , qˆ ) ( f ( z ) = f 0 ( zˆ ) + ε [f1 ( zˆ ) + μϕ ( x1 )f 0′ ( zˆ )]

⎛ zˆ + a ⎞ ⎜ ⎟ 2 2 zˆ − a ⎝ zˆ − a ⎠

f0′ ( zˆ ) =

1 −1 B ΛΓ( zˆ ) Λ −1qˆ , 2πi

1

f1 ( zˆ ) =

icoszˆ −1 B (πLi 2 + μΛΓ( zˆ ) Λ −1qˆ ). 2π

Γ( zˆ) =
Λ c−1B A T ⎥qˆ ⎦

>.

642

Appendix D

9.4.1 Sliding punches with or without friction: general solution (given ak , bk , Fk , N k , g k (t ) ) ⎧± ηθ 2′ ( x1− )⎫ ⎪ ⎪ f ' ( z ) = B −1θ' ( z ) , θ′( x1− ) = ⎨ θ 2′ ( x1− ) ⎬ , η = Fk / N k ⎪ ⎪ 0 ⎭ ⎩ n ′ χ ( z) g (t ) θ 2′ ( z ) = 0 ∑ ∫ + k dt + χ 0 ( z ) pn ( z ), 2πτ k =1 Lk χ 0 (t )(t − z ) n

χ 0 ( z ) = ∏ ( z − ak ) −δ ( z − bk )δ −1 , δ = k =1

1 arg(−τ / τ ), 0 ≤ δ < 1 ; 2π

τ : (9.107b) pn ( z ) = d 0 + d1 z + LL + d n−1 z n−1 ,

d k : (9.109b,c)

Contact pressure: σ 22 ( x1 ) = {(τ + τ )θ 2′ ( x ) − ig ′k ( x1 )} / τ , x1 ∈ Lk . − 1

Surface deformation: u ′2 ( x1 ) = −i (τ + τ )θ 2′ ( x1− ),

u′( x1 ) = −2iL−1θ′( x1− ), x1 ∉ L. Frictionless surface: η = 0 , τ = m22 , m22 : (22) components of M −1

9.4.2 A sliding wedge-shaped punch (given a, F , N , ε * ; a: to be determined for incomplete indentation)

θ 2′ ( z ) =

iε * {1 − [z + (2δ − 1)a ]χ 0 ( z )} + iN χ 0 ( z ), τ +τ 2π

χ 0 ( z ) = ( z + a) −δ ( z − a)δ −1 Contact pressure:

− sin πδ 2πε * {N − [ x1 + (2δ − 1)a]}. δ 1−δ π (a + x1 ) (a − x1 ) (τ + τ ) 4πε *δ a. Complete indentation: N ≥ τ +τ τ +τ Incomplete indentation: a = N, 4πε *δ δ 2ε * sin πδ ⎛ a − x1 ⎞ ⎟ . σ 22 ( x1 ) = − `⎜ τ + τ ⎜⎝ a + x1 ⎟⎠

σ 22 ( x1 ) =

643

Appendix D

9.4.3 A sliding parabolic punch (given l, a, b, F , N , R ; a,b: to be determined for incomplete indentation )

θ 2′ ( z ) =

⎧ i (τ + τ ) RN ⎤ ⎫ ⎡ ⎨ z − χ 0 ( z ) ⎢ j2 ( z ) − ⎥⎬ , (τ + τ ) R ⎩ 2π ⎣ ⎦⎭

χ 0 ( z ) = ( z + a ) −δ ( z − b)δ −1 , 1 j2 ( z ) = z 2 + [δ (a + b) − b]z − δ (1 − δ )(a + b) 2 . 2 Contact pressure and surface deformation: − iχ 0 ( x1− ) ⎧ (τ + τ ) RN ⎫ σ 22 ( x1 ) = ⎨ j2 ( x1 ) − ⎬, − a < x1 < b. τR 2π ⎩ ⎭ 1⎧ (τ + τ ) RN ⎤ ⎫ ⎡ u2′ ( x1 ) = ⎨ x1 − χ 0 ( x1− ) ⎢ j2 ( x1 ) − ⎥ ⎬, x1 < −a or x1 > b. R⎩ 2π ⎣ ⎦⎭ χ 0 ( x1− ) : (9.126) 2 ⎧1 − δ δ ⎫ πl , . Complete indentation: N ≥ max ⎨ ⎬ ⎩ δ 1 − δ ⎭ (τ + τ ) R δ (τ + τ ) RN 2 (1 − δ )(τ + τ ) RN Incomplete Indentation: a 2 = ,b = , π (1 − δ ) πδ 2 sin πδ σ 22 ( x1 ) = − ( x1 + a ) −δ +1 (b − x1 )δ , − a < x1 < b , (τ + τ ) R x 1 u ′2 ( x1 ) = 1 m | x1 + a |−δ +1 | x1 − b |δ , x1 < −a or x1 > b . R R Frictionless surface: (9.133), (9.134) and (9.135a,b) 9.4.4 Two sliding flat-ended punches (given a1 , b1 , a2 , b2 , F1 , F2 , N1 , N 2 ) θ 2′ ( z ) = χ 0 ( z )(d 0 + d1 z ), d 0 , d1 : (9.138a,b) and (9.141a,b)

χ 0 ( z ) = ( z − a1 ) −δ ( z − b1 )δ −1 ( z − a2 ) −δ ( z − b2 )δ −1 Contact pressure and surface deformation: ⎛ τ ⎞ σ 22 ( x1 ) = ⎜1 + ⎟ χ 0 ( x1− )(d 0 + d1 x1 ), a1 < x1 < b1 , a2 < x1 < b2 . ⎝ τ⎠ u ′2 ( x1 ) = −i (τ + τ ) χ 0 ( x1− )(d 0 + d1 x1 ),

x1 < a1 , b1 < x1 < a2 ,

χ 0 ( x ) : (9.141a,b) Frictionless surface: (9.142a,b); special case: (9.143) 9.5 Contact between two elastic bodies: general expressions u1 = A1f1 ( z ) + A1 f1 ( z ),⎫⎪ z ∈ S1 , ⎬, φ1 = B1f1 ( z ) + B1 f1 ( z ), ⎪⎭ u 2 = A 2f 2 ( z ) + A 2 f 2 ( z ),⎪⎫ z ∈ S2 , ⎬, φ 2 = B 2f 2 ( z ) + B 2 f 2 ( z ), ⎪⎭ ⎧− B1f1 ( z ), 　z ∈ S1 , θ( z ) = ⎨ ⎩B 2f 2 ( z ), 　z ∈ S 2 . − 1

x1 > b2 .

644

Appendix D

9.5.1a Contact in the presence of friction: general solution (given η1 ,η 3 , N , g (1) ( x1 ), g ( 2) ( x1 ) )

θ1′( z ) = η1θ 2′ ( z ), θ 3′ ( z ) = η 3θ 2′ ( z ),

θ 2′ ( z ) =

⎫⎪ χ 0 ( z ) ⎧⎪ n g ′(t ) dt + iN ⎬ ⎨∑ ∫ + 2π ⎪⎩ k =1 Lk τχ 0 (t )(t − z ) ⎪⎭

χ 0 ( z ) = ( z − a) −δ ( z − b)δ −1 , δ = g ( x1 ) = g (1) ( x1 ) − g (2) ( x1 ).

1 arg(−τ / τ ), 0 ≤ δ < 1, 2π

τ : (9.161b) contact pressure and surface deformation: ⎪⎫ ⎬, x1 ∈ L g * ( x1 ) = {τ ( 2 ) g (1) ( x1 ) + τ (1) g ( 2 ) ( x1 ) + i (τ (1)τ ( 2 ) − τ (1)τ ( 2 ) )θ 2 ( x1− )} / τ ⎪⎭

σ 22 ( x1 ) = {(τ + τ )θ 2′ ( x1− ) − ig ′( x1 )} / τ ,

g (1)* ( x1 ) = g (1) ( x1 ) + i (τ (1) + τ (1) )θ 2 ( x1− ) ⎫⎪ ⎬, 　x1 ∉ L, g ( 2 )* ( x1 ) = g ( 2 ) ( x1 ) − i (τ ( 2 ) + τ ( 2 ) )θ 2 ( x1− )⎪⎭ τ (1) ,τ ( 2 ) : (9.163b) 9.5.1b Contact of two parabolic elastic bodies with friction (given F , N , R1 , R2 ) i ( R1 + R2 ) iN {z − χ 0 ( z ) j2 ( z )} + χ 0 ( z ). (τ + τ ) R1R2 2π 1 j2 ( z ) = z 2 − [δ (a − b) + b]z − δ (1 − δ )(a − b) 2 ; 2 δ : (9.162b) and (9.161b)

θ 2′ ( z ) =

x2 1 1 g ( x1 ) = 1 ( + ). 2 R1 R2

τ : (9.161b); χ 0 ( z ) : (9.162b) and (9.161b) δ τ + τ R R N ( ) (1 − δ )(τ + τ ) R1R2 N 1 2 a2 = , 　　b 2 = . π (1 − δ )( R1 + R2 ) πδ ( R1 + R2 ) Contact pressure: 2( R1 + R2 ) sin πδ σ 22 ( x1 ) = − ( x1 − a )1−δ (b − x1 )δ , (τ + τ ) R1 R2

a < x1 < b

9.5.2 Contact of two parabolic elastic bodies without friction (given N , R1 , R2 ) i ( R1 + R2 ) ⎧⎪ z 2 − (b 2 / 2) ⎫⎪ iN , z − ⎨ ⎬+ * 2 2 2m22 R1 R2 ⎪⎩ z − b ⎪⎭ 2π z 2 − b 2 Contact pressure and surface deformation:

θ 2′ ( z ) =

σ 22 ( x1 ) = − g ( x1 ) =

x12 1 1 ( + ). 2 R1 R2

g * ( x1 ) =

R1 + R2 b 2 − x12 , 　　 | x1 |< b, * m22 R1R2

b=

* R1 R2 N 2m22 . π ( R1 + R2 )

( 2) (1) − R1m22 R2 m22 − b < x1 < b, x12 , 　 * 2m22 R1 R2

g (1)* ( x1 ) =

x2 x12 (1) ( 2) + 2im22 θ 2 ( x1− ), 　g ( 2 )* ( x1 ) = − 1 − 2im22 θ 2 ( x1− ), 2 R1 2 R2

(1) (2) * , m22 , m22 : (22) components of M1−1 , M −2 1, M * θ 2 ( x1− ) : (9.171b); m22

645

Appendix D

9.5.3 Contact of two elastic bodies in complete adhesion (given a, b, g ( x1 ), qˆ ) g ′(t ) + −1 1 θ′( z ) = X 0 ( z ) ∫L [ X 0 (t )] dtM *−1i 2 + X 0 ( z )d 0 t−z 2π 1 z − a iε α 1 −1 X0 ( z) = Λ < ( ) > ; d0 = Λ qˆ − z b 2 πi ( z − a)( z − b)

ε 1 = ε , ε 2 = −ε , ε 3 = 0 in which ε : (B.10b)

Λ : (B.9);

10.2 Elliptical holes: uniform heat flow at infinity (given a, b, h0 ) h0

2

u = 2 Re{∑ A < f i ( zα ) > q i + cg ( zt )},

x2

i =1 2

φ = 2 Re{∑ B < f i ( zα ) > q i + dg ( zt )} i =1

x1

a b x1 = a cosψ x2 = b sin ψ

h0

e1 ⎧ 1 2 1 2 2 2 2 ⎨ zt − zt zt − ( a + τ b ) a + iτb ⎩ 2 2 a 2 + τ 2b 2 ln( z + z 2 − (a 2 + τ 2b 2 ) )⎫ ⎬ + t t 2 ⎭ g ( zt ) =

f1 ( zα ) =

1 ⎧1 2 1 2 2 2 2 ⎫ ⎨ zα − zα zα − (a + μα b ) ⎬, a + iμα b ⎩ 2 2 ⎭

a − iμ α b ln( zα + zα2 − (a 2 + μα2 b 2 ) ), 2 e1 : (10.27) and (10.19b); q1 , q 2 : (10.24b) and (10.29a) Hoop stress: f 2 ( zα ) =

σ ss = −sT φ ,n , ah φ ,n = ~0 2 ρ sin ψγ 2 (θ ) − N 3 (θ )L−1 Re[e −2iψ (a + ibτ )~ γ 2* ] 2 k ρ ~ k : (10.19b); θ -ψ relation: (10.20a); ρ : (10.20b);

{

}

γ 2* : (10.16b) and (10.13) γ 2 (θ ) : (10.16b), (10.12c), and (10.8b); ~

Isotropic materials:

σ ss =

Eh0αa 2 (a + b) sinψ , ~ 2k ρ 2 (1 −ν )

646

Appendix D

10.2.2 Cracks: uniform heat flow at infinity (given a, h0 ) u = 2 Re{Af ( z ) + cg ( zt )}, h0

ih g ( z ) = − ~0 {zt2 − zt zt2 − a 2 + a 2 ln( zt + zt2 − a 2 )}　 , 4k ih f ( z ) = ~0 < zα2 − zα zα2 − a 2 > B −1d 4k

x2

a

+

x1

a

φ = 2 Re{Bf ( z ) + dg ( zt )}

ih0 a 2 2 −1 ~* ~ < ln( zα + zα − a ) > B [d − iRe( γ 2 )] , 4k

~ k : (10.19b);

~ γ 2* : (10.16b) and (10.13)

π h0

3/ 2 ~* ~ a Re{γ 2 } , 2k πh02 a 3 G= ~ Re{~ γ 2* }T L−1 Re{~ γ 2* } , 8k 2 ~ ~ Δu = h0 x1 a 2 − x12 L−1Re{　 γ 2* } / k ,

h0

Fracture parameters: k =

Isotropic materials: (10.42) 10.3.1 Insulated elliptical rigid inclusions: uniform heat flow at infinity (given a, b, h0 , ϕ ) 2

h0

u = 2 Re{A < zα2 > q 0 + ∑ A < f i ( zα ) > q i + c[e0 zt2 + e1 g 0 ( zt )]},

x2

i =1 2

φ = 2 Re{B < zα > q 0 + ∑ B < f i ( zα ) > q i + d[e0 zt2 + e1 g 0 ( zt )]}, 2

i =1

~ e0 , e1 : (10.47a) in which τ I : imaginary part of τ ; k : (10.19b) q 0 , q1 , q 2 : (10.24b) and (10.47a,b,c,d) h Interfacial stresses: (10.48a,b) 10.3.2 Insulated rigid line inclusions: uniform heat flow at infinity (given a, h0 , ϕ ) x1

b

a

0

2

h0

u = 2 Re{A < zα2 > q 0 + ∑ A < f i ( zα ) > q i + c[e0 zt2 + e1 g 0 ( zt )]}, i =1 2

φ = 2 Re{B < zα > q 0 + ∑ B < f i ( zα ) > q i + d[e0 zt2 + e1 g 0 ( zt )]},

x2

2

i =1

~ e0 , e1 : (10.49) in which τ I : imaginary part of τ ; k : (10.19b)

x1

a

q 0 , q1 , q 2 : (10.24b), (10.49), and (10.47d) Strength of thermal stress singularity: (10.51) and (10.52)

a

h0

647

Appendix D

10.4.1 Collinear interface cracks: general thermal loading (given a1 , b1 ,...., an , bn , hˆ( s ), tˆ( s ) ) Ti = 2 Re{ g i′ ( z t( i ) )},

hˆ x2

h i = −2 Re{(k 1( i ) + τ i k (2i ) ) g i′′( z t( i ) )},

u i = 2 Re{A i f i ( z ( i ) ) + c i g i ( z t( i ) )}, S1

x1 a1 b1

a2

ak bk

b2

an

bn S2

φ i = 2 Re{B i f i ( z ( i ) ) + d i g ( z t( i ) )},

i = 1,2.

h i , k , k : h, k 1 , k 2 (defined in (10.6c)) of ith material g1 ( z ), g 2 ( z ), f1 ( z ), f 2 ( z ) : (10.61b,c,d) in which (i ) 1

(i ) 2

θ ′′( z ) =

k1 + k 2 hˆ( s )ds χ 0 ( z ) ∫L + + χ 0 ( z ) pn ( z ), 2πk1k 2 χ 0 ( s)(s − z )

ψ′( z ) =

1 1 X 0 ( z) ∫ [ X 0+ ( s)]−1[−tˆ( s ) + θ ′( s + )e1 + θ ′( s − )e 2 ]ds + L s−z 2πi X 0 ( z )p n (z ),

~ ki : k (defined in (10.19b)) of ith material with the tilde dropped. e1 , e 2 : (10.61c,d) χ 0 ( z ), X 0 ( z ) : (10.63b,c), (7.68), (7.69a,b), and (7.13b) Stresses σ i 2 along the interface: (10.65) Crack opening displacement: (10.66)

10.4.2 Interface crack: uniform heat flow (given a, h0 , t 0 )

θ ′( z ) = −ih0* ( z − z 2 − a 2 ),

where h0* =

x2 h0 t0 x1 h0

a

a

h0 (k1 + k 2 ) 2k1k 2

ψ ′( z ) = − Λ{J 0 ( z )t *0 + ih0* (J1 ( z )e1* + J 2 ( z )e*2 )}, Λ : (10.63c), (7.68) J 0 ( z ), J1 ( z ), J 2 ( z ) : (10.78b), (10.75b), (7.69b), and (7.13b) t *0 , e1* , e*2 : (10.75b), (10.63c), and (7.68)

648

Appendix D

10.5 Multi-material wedges under thermal loading: near-tip solution (given θ 0 ,..., θ n ) x2

* ⎧ v1 (r , θ ) ⎫ ⎡ Γ1 (θ ) 0 ⎤ ⎧ v1 (θ 0 ) ⎫ ⎥⎨ ⎬= ⎢ 2 * ⎬ ⎨ * ⎩w1 (r , θ )⎭ ⎢⎣r F1 (θ ) E1 (θ )⎥⎦ ⎩w1 (θ 0 )⎭

2

θ2 k

θ1

θk

1

θ0

x1

θn n

* ⎤ ⎧ v1 (θ 0 ) ⎫ ⎧ v k (r ,θ ) ⎫ ⎡ Γ k (θ ) 0 ⎤ ⎡ (K T ) k −1 0 = ⎢ ⎥⎢ 2 ⎥⎨ ⎬, ⎨ ⎬ 2 * * ⎩w k (r ,θ )⎭ ⎣⎢r Fk (θ ) E k (θ )⎦⎥ ⎣r (K c ) k −1 (K e ) k −1 ⎦ ⎩w1 (θ 0 )⎭

k = 2,3,...., n

⎧⎪T,r (r , θ )⎫⎪ v k (r , θ ) = ⎨ * ⎬ , ⎪⎩h (r , θ ) ⎪⎭ k

⎧u k (r , θ )⎫ w k (r , θ ) = ⎨ ⎬ ⎩φ k (r , θ ) ⎭

Γ *k (θ ), E*k (θ ), Fk* (θ ) : (10.116b) in which Λ k , U k : Λ, U (defined in (10.90)) of kth material (K T ) k −1 , (K c ) k −1 , (K e ) k −1 : (10.116c) in which Γ k , E k , Fk : (10.95b), (10.96b) v1 (θ 0 ), w1 (θ 0 ) and singular order δ : (10.103)–(10.115) Special cases (including single wedge, bi-wedge): (10.119)–(10.122)

11.4.2 Piezoelectric multi-material wedges: near-tip solutions (given θ 0 ,..., θ n ) x2

1

u( r , θ ) =

2π 1

2

θ2 k

θ1

θk

1

θ0

x1

φ( r , θ ) =

θn n

2π 1

r 1−δ R V (θ ) < (1 − δ R + iε α ) −1 (r / l) iεα > Λ −1k , r 1−δ R Λ (θ ) < (1 − δ R + iε α ) −1 (r / l) iεα > Λ −1k ,

r −δ R Λ (θ ) < (r / l) iεα > Λ −1k. 2π V (θ ), Λ (θ ) : (11.70b) and (11.69a,b,c); Λ = Λ (0) δ α = δ R + iε α : (11.69c), (5.83b), (5.82b), and (5.88) 11.5.1 Cracks in piezoelectric materials: near-tip solution φ ,r ( r , θ ) =

u(r ,θ ) = − 2 / π r 1/ 2 V (θ )L−1k , θ ＝π

θ ＝- π

φ(r ,θ ) = − 2 / π r 1/ 2 Λ (θ )L−1k ,

V (θ ) = A < μˆ α1 / 2 (θ ,−π ) > B T + A < μˆ α1/ 2 (θ ,−π ) > B T Λ (θ ) = B < μˆ α1 / 2 (θ ,−π ) > BT + B < μˆ α1 / 2 (θ ,−π ) > B T Δu = 2

2

π

r 1/ 2 L−1k ,

G=

Piezoelectric ceramics poling in Piezoelectric ceramics poling in (11.56b) and (11.53)

1 T −1 k L k 2

x3-axis: (11.85a,b,c) and (11.7d,e) x2 -axis: (11.86a,b,c), (11.62b,c),

649

Appendix D

11.5.2 Interface cracks between two dissimilar piezoelectric materials: near-tip solution 2 2r u k ( r ,θ ) = Vk (θ ) < (1 + 2iε α ) −1 (r / l) iεα > Λ −1k , k = 1,2, π φ k ( r ,θ ) =

2r

π

Λ k (θ ) < (1 + 2iε α ) −1 (r / l) iεα > Λ −1k , k = 1,2,

φ1,r (r ,0) = φ 2,r (r ,0) =

1

Δu(r ) =

2r

π

Λ −T
Λ −1k ,

(r / l) iεα > Λ −1k , (1 + 2iε α ) cosh πε α

1 G = k T (D + WD −1W )k. 4 Vk (θ ), Λ k (θ ) : (11.96b,c,d), (11.87b,c), (11.95a,b), and

(11.94b) in which λ : (11.89) Λ = Λ (0) ; ε α : (11.95a,b) and (11.94b); l : reference length Piezoelectric ceramics poling in x3 -axis: (11.98) Piezoelectric ceramics poling in x2 -axis: (11.99) 11.6.1 Collinear cracks: uniform load/induction at infinity (given a1 , b1 ,...., an , bn , σ ij∞ , D2∞ ) σ 22∞

σ 22∞ σ 23∞

σ 12∞

k = kt ∞2 , Δu = Δf ( x1 )L−1t ∞2 , G =

σ 12∞

•

•

∞ σ 23

x2

σ 13∞

σ 12∞

⊗

σ 13∞

σ 12∞ a1 b1

σ 13∞

a2

ak

b2

bk an

σ 12∞

⊗

σ 13∞

σ 12∞

σ 23∞

σ 12∞

⊗

σ 12∞

⊗

∞ σ 22

σ 11∞

x1

bn

•

σ 11∞

•

σ

∞ 11

σ 11∞

1 2 ∞ T −1 ∞ ∞ k (t 2 ) L t 2 , t 2 : (11.100b) 2

A single crack: k = π a , Δf ( x1 ) = 2 a 2 − x12 . Two collinear cracks: k : (7.48c), Δf ( x1 ) : (7.49b) in which

λk : (7.41b) and (7.36b)

σ 23∞

Evenly spaced collinear periodic cracks: k :(7.58), Δf ( x1 ) : (7.59b) 　 .

∞ σ 22

11.6.2a A semi-infinite interface crack: point force/charge on crack surfaces (given a, pˆ ) x2

k=

pˆ 2

Piezoelectric ceramics poling in x3 -axis: (11.110a,b)

pˆ 1

•

pˆ 1

⊗

pˆ 3

2 Λ < (a / l) −iε α cos hπε α > Λ −1pˆ , Λ, ε α , l : §11.5.2 πa

x1

pˆ 3


a pˆ 2

11.6.2b A finite interface crack: point force/charge on the crack surfaces (given a, c, pˆ ) x2

k=

pˆ 2

⊗ pˆ 2

c a

a

pˆ 1

•

pˆ 3 pˆ 1

pˆ 3

x1

1

πa

⎡ l( a + c ) ⎤ a+c Λ Λ −1pˆ ,

Λ, ε α , l : §11.5.2

Piezoelectric ceramics poling in x3 -axis: (11.113a,b,c) Piezoelectric ceramics poling in x2 -axis: (11.114a,b)

650

Appendix D

11.6.2c A finite interface crack: uniform load/induction on the crack surfaces (given a, tˆ ) x2

k = πa Λ < (1 + 2iε α )(2a / l) −iε α > Λ −1tˆ, Λ, ε α , l : §11.5.2

Piezoelectric ceramics poling in x3 -axis: (11.116a,b,c)

tˆ •

⊗

•

•

⊗

⊗

•

•

⊗

⊗ a

x1


a

11.6.2d Two collinear interface cracks: uniform load/induction at infinity (given a1 , b1 , a2 , b2 , σ ij∞ ) ∞ σ 22

σ 23∞

σ 12∞

•

x2

σ 12∞

σ 13∞

σ 12∞

σ 13∞

σ 12∞

σ 11∞

•

⊗

σ

∞ 11

•

∞ σ 22

σ 23∞

k = 2π Λ < kα > Λ −1t ∞2 , Λ, ε α , l : Section 11.5.2

kα : (7.131b),

x1

σ

b1

a2

b2

σ

σ 13∞

σ 12∞

σ 23∞

σ 12∞

⊗

σ 22∞

⊗

σ 12∞

∞ 12

•

⊗

σ 11∞

a1

∞ 13

σ 11∞

σ 23∞

σ 22∞

12.4.1 Elliptical hole: uniform bending at infinity (given a, b, Mˆ x , Mˆ y , Mˆ xy ) Mˆ y

Mˆ x

β = β∞ − Re{A < ζ α−1 > B −1 (am ∞2 − ibm1∞ )},

2a

Mˆ x

2b

ψ = ψ ∞ − Re{B < ζ α−1 > B −1 (am ∞2 − ibm1∞ )}, β, ψ : (12.39b); A,B: (12.39b), (12.40), and (12.14);ζ α : (12.61) β ∞ , ψ ∞ : (12.57) − (12.60) in which D ij*: components of D −1 ,

Mˆ y

D: bending stiffness Moments around the hole boundary: (12.65) Circular holes with M x = Mˆ , M y = M xy = 0 : β = β ∞ − aMˆ Im{A < ζ −1 > B −1}i , α

1

ψ = ψ ∞ − aMˆ Im{B < ζ α−1 > B −1}i1 .

Orthotropic plate: (12.68a,b) 12.4.2 Elliptical rigid inclusion: uniform bending at infinity (given a, b, Mˆ x , Mˆ y , Mˆ xy ) Mˆ y

Mˆ x

β = β ∞ − Re{A < ζ α−1 > A −1 (aβ1∞ + ibβ ∞2 )},

2a

Mˆ x

2b

ψ = ψ ∞ − Re{B < ζ α−1 > A −1 (aβ1∞ + ibβ ∞2 )}, β, ψ,A,B, ζ k : Section12.4.1; β1∞, β ∞2 : (12.58) and (12.59) in which

Mˆ y

Dij* : Section 12.4.1

Moments around the inclusion boundary: (12.72) Circular inclusions with M x = Mˆ , M y = M xy = 0 : (12.73) Orthotropic plate: (12.75)

651

Appendix D

12.4.3 Crack: uniform bending at infinity (given a, Mˆ x , Mˆ y , Mˆ xy ) Mˆ y

β = β ∞ − a Re{A < ζ α−1 > B −1}m ∞2 , ψ = ψ ∞ − a Re{B < ζ α−1 > B −1}m ∞2 ,

2a

ˆx

Mˆ x

β, ψ ,A,B: Section 12.4.1; β ∞ , ψ ∞ , ζ α : (12.57)–(12.60), (12.76b) x

{

}

6 3 π aMˆ y , K II = 2 π a Mˆ xy − [G1m ∞2 ](2) h2 h G1 = N1T − N 3SL−1 , m ∞2 : (12.57b) The subscript (2): the second component of the vector. 14.1 Holes in laminates: uniform stretching and bending moments at infinity (given a, b, N ij∞ , M ij∞ ) KI =

Mˆ y

z

y

N 22∞

φ d = φ ∞d − Re{B < ζ α−1 > B −1 (am ∞2 − ibm1∞ )},

∞ M 22

N12∞ N11∞

u d = u ∞d − Re{A < ζ α−1 > B −1 (am ∞2 − ibm1∞ )},

M 12∞

N12∞

2a

M 11∞

M 11∞

N11∞

2b

M 12∞

M

N12∞ ∞ M 22

M 12∞

u ∞d , φ ∞d : (14.3a,b), (14.6), (14.5a,b,c,d,e) and (14.4c);ζ α : (14.7);

∞ 12

N12∞

m1∞ , m ∞2 : (14.3b) Stress resultants and bending moments along the hole boundary: (14.12a–f)

N 22∞

14.2.1 Holes in laminates: uniform heat flow and moisture transfer in x1 x2 plane (given a, b, qî , mˆ i ) T = 2 Re{g t′ ( zt )}, H = 2 Re{g h′ ( z h )}, ( ( qi = −2 Re{( K it1 + τ t K it2 ) g t′′( zt )}, mi = −2 Re{( K ih1 + τ h K ih2 ) g h′′ ( z h )},

θˆ

qˆ , mˆ

n

θ

s

u d = 2 Re{Af ( z ) + ct g t ( zt ) + c h g h ( z h )},

x1

φ d = 2 Re{Bf ( z ) + d t g t ( zt ) + d h g h ( z h )},

f ( z ), g t ( zt ), g h ( z h ) : (14.17a,b,c), (14.18a,b,c,d), (14.19a,b) Stress resultants and bending moments along the hole boundary: (14.20), (13.79)–(13.81), and (13.70b) 14.2.2 Holes in laminates: uniform heat flow and moisture transfer in x3 direction (given a, b, h, Tl , Tu , H l , H u ) x2

qˆ , mˆ

x3

T = T 0 + x3T * , H = H 0 + x3 H * , n

x2

x3 qˆ , mˆ

θ

s

x1

( ( qi = − K it3T * , mi = − K ih3 H *

u d = Re{A < ζ α−1 > B −1 (aϑ2 − ibϑ1 )},

φ d = Re{B < ζ α−1 > B −1 (aϑ2 − ibϑ1 )} − x1ϑ2 + x2ϑ1 , ϑ1 , ϑ 2 : (13.125b,c); ζ α : (14.7); T 0 , H 0 , T * , H *: (14.21) Stress resultants and bending moments along the hole boundary: (14.27), (13.79)–(13.81), and (13.70b)

652

Appendix D

14.3 Holes in electroelastic laminates (given a, b, N N

∞ 22

∞ M 22

N12∞ N11∞

2a

M 11∞

M 11∞

2b

M 12∞

N12∞ ∞ M 22

N12∞

, M ij∞ )

u d = u ∞d − Re{A < ζ α−1 > B −1 (am ∞2 − ibm1∞ )},

M 12∞

N12∞

∞ ij

N11∞ M 12∞

φ d = φ ∞d − Re{B < ζ α−1 > B −1 (am ∞2 − ibm1∞ )},

ζ α : (14.7);

u ∞d , φ ∞d : (14.28a,b), (14.29a,b) and (13.145); ∞ 1

∞ 2

m , m : (14.28b) Generalized stress resultants and bending moments along the hole boundary: (14.31a,b,c) 14.4.1 Concentrated inplane forces and out-of-plane moments (given zˆ , fˆ , fˆ , mˆ , mˆ ) M 12∞

N 22∞

α

u d = Im{A < ln( zα − zˆα ) > A T }pˆ / π ,

1

2

1

2

pˆ = ( fˆ1 fˆ2 mˆ 2 − mˆ 1 )T

φ d = Im{B < ln( zα − zˆα ) > A T }pˆ / π , Real form solution: (14.39)

14.4.2 Concentrated transverse force (given zˆα , fˆ3 ) u d = Im{A < ( zα − zˆα ) ln( zα − zˆα ) − ( zα − zˆα ) > A T } fˆ3i 3 / π , φ = Im{B < ( z − zˆ ) ln( z − zˆ ) − ( z − zˆ ) > A T } fˆ i / π . d

α

α

α

α

α

α

3 3

Real-form solution: (14.49a,b)

14.4.3 Concentrated in-plane torsion (given zˆα , mˆ 3 ) u d = − Im{A < ( z k − zˆk ) −1 > AT }mˆ 3i 2 / π ,

φ d = − Im{B < ( zk − zˆk ) −1 > AT }mˆ 3i 2 / π . Real-form solution: (14.57)

14.5 Holes in laminates: point forces and moments (given a, b, zˆα , fˆ1 , fˆ2 , fˆ3 , mˆ 1 , mˆ 2 , mˆ 3 ) u d = 2 Re{A[f0 (ζ ) + f p (ζ )]},

Case 1 fˆ1 , fˆ2 , mˆ 1 , mˆ 2 :

φd = 2 Re{B[f0 (ζ ) + f p (ζ )]}

f 0 (ζ ) + f p (ζ ) : (14.79)

Case 2 fˆ3 : f 0 (ζ ) + f p (ζ ) : (14.85), (14.74b), (14.65b) and (14.82b) Case 3 mˆ 3: f 0 (ζ ) + f p (ζ ) : (14.89), (14.87b), (14.65b) and (14.74b) Stress resultants and moments along the hole boundary: (14.91a–c) and (14.74b) in which ρ : (14.16b); ζ α : (14.7)

653

Appendix D

14.5.4 Cracks in laminates: point forces and moments (given a, zˆα , fˆ1 , fˆ2 , fˆ3 , mˆ 1 , mˆ 2 , mˆ 3 ) u d = 2 Re{Af (ζ )}, φ d = 2 Re{Af (ζ )} , f (ζ ) : (14.94a–e) Stress intensity factors: (14.95), (14.96) and (14.97a,b,c), in which ζ α : (14.94d)

When the force is applied on the crack surface: (14.98)

14.6.1 Point forces/moments outside the Inclusions (given a, b, zˆα , fˆ1 , fˆ2 , fˆ3 , mˆ 1 , mˆ 2 , mˆ 3 ) ( ( u (d1) = A1[f 0− (ζ ) + f1 (ζ )] + A1[f 0− (ζ ) + f1 (ζ )], ( ( φ (d1) = B1[f 0− (ζ ) + f1 (ζ )] + B1[f 0− (ζ ) + f1 (ζ )], u (d2) = A 2 [f 2+ (ζ * ) + f 2− (ζ * )] + A 2 [f 2+ (ζ * ) + f 2− (ζ * )],

φ (d2) = B 2 [f 2+ (ζ * ) + f 2− (ζ * )] + B2 [f 2+ (ζ * ) + f 2− (ζ * )]. f 0− (ζ ) : (14.106a–d) and (14.74b); ( f1 (ζ ) : (14.114a,b), (14.111b,c,d) and (14.113a,b,c); f 2+ (ζ * ) + f 2− (ζ * ) = f 2 (ζ * ) : (14.112a–c)

and (14.113a–c) in which cε , γ α : (14.74b), ζ α : (14.7) and the superscript * denotes the quantities of the inclusion Stress resultants and moments along the interface: (14.115) and (14.117) 14.6.2 Point forces/moment inside the Inclusion (given a, b, zˆ , fˆ , fˆ , fˆ , mˆ , mˆ , mˆ ) α

u

(1) d

φ

(1) d

u

( 2) d

1

2

3

1

2

3

= A1[f s (ζ ) + f (ζ ) + f (ζ ) + f1 (ζ )] + A1[f s (ζ ) + f (ζ ) + f 0− (ζ ) + 0

− 0

+ 0

+ f1 (ζ )], = B1[f s (ζ ) + f 0+ (ζ ) + f 0− (ζ ) + f1 (ζ )] + B1[f s (ζ ) + f 0+ (ζ ) + f 0− (ζ ) + f1 (ζ )], = A 2 [f s* (ζ * ) + f 0*+ (ζ * ) + f 0*− (ζ * ) + f 2+ (ζ * ) + f 2− (ζ * )] + A 2 [f s* (ζ * ) + f 0*+ (ζ * ) + f 0*− (ζ * ) + f 2+ (ζ * ) + f 2− (ζ * )], φ (d2 ) = B 2 [f s* (ζ * ) + f 0*+ (ζ * ) + f 0*− (ζ * ) + f 2+ (ζ * ) + f 2− (ζ * )] + B 2 [f s* (ζ * ) + f 0*+ (ζ * ) + f 0*− (ζ * ) + f 2+ (ζ * ) + f 2− (ζ * )].

f s (ζ ), f 0+ (ζ ), f 0− (ζ ) : (14.120a–c), (14.123a–c), (14.124) and (14.125a–e) f1 (ζ ) : (14.126b,c,d,e) and (14.127b,c,d); f s* (ζ * ) + f 0*+ (ζ * ) + f 0*− (ζ *) = f 0* (ζ * ) : (14.118a–c) and (14.119a–c) f 2+ (ζ * ) + f 2− (ζ * ) = f 2 (ζ * ) : (14.126a,d,e) and (14.127a–d)

in which cε , γ α : (14.74b), ζ α : (14.7) and the superscript * denotes the quantities of the inclusion Stress resultants and moments along the interface: (14.128a–c)

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Author Index

A Anderson, T. L., 131 Atkinson, C., 337 B Barber, J. R., 348 Barnett, D. M., 333, 389 Becker, W., 436 Berger, J. R., 132 Bhargave, R. D., 239 Bogy, D. B., 132 Brebbia, C. A., 545, 567, 581 Brock, L. M., 289 Broek, D., 144, 205 C Carlsson, L. A., 228 Chadwick, P., 67, 93, 333 Chai, H., 232 Chen, D. H., 132 Chen, H. P., 132 Chen, W. T., 239 Cheng, Z. Q., 435–436, 444–445 Choi, N. Y., 147 Chou, Y. T., 239 Clements, D. L., 337 Comninou, M., 348 D Desmorat, R., 132 Ding, S., 132 Dongye, C., 69 Dundurs, J., 132, 256, 262–263, 266 Dunn, M. L., 140 Dwight, H. B., 319 E Earmme, Y. Y., 147 England, A. H., 277, 286, 295, 308, 317–318

Erdogan, F., 256 Eshelby, J. D., 53, 55, 239, 246, 266 F Fan, C. W., 277, 297 Florence, A. L., 337, 343 G Galin, L. A., 277 Gangadharan, A. C., 266 Gao, H., 140, 216 Gerasoulis, A., 271 Gladwell, G. M. L., 277, 289 Goodier, J. N., 165, 184, 337, 343 Griffith, A. A., 187 H Hartmann, F., 515, 585 Hartmann, H. H., 132 Hein, V. L., 132 Herrmann, G., 239 Hirth, J. P., 256 Honein, T., 239 Hong, C. S., 225 Hsieh, M. C., 411, 434, 436, 453, 469, 498, 503, 505 Hwu, C., 64, 69, 75, 85–86, 103, 120, 132, 139–140, 144–145, 149, 156–158, 159, 167, 174, 180, 182, 185–186, 195–196, 206–207, 225–226, 230–231, 233, 235, 237, 245–246, 257, 268, 271, 273, 275, 277, 284, 290, 295, 297, 314, 321, 328, 333, 336–337, 346–348, 357, 367, 369, 377, 379, 411, 434–436, 440, 453, 460, 469, 498, 503–505, 519, 522, 536, 542–543, 545, 560–561, 565 I Ikeda, T., 369 Im, S., 147 Irwin, G. R., 193, 218, 344

663

664 J Jaswon, M. A., 239 Jih, C. J., 226 Johnson, K. L., 131, 277 Jones, R. M., 23, 131 K Keer, L. M., 239, 256 Kim, K. S., 147 Kirchner, H. O. K., 76, 93 Kuo, C. M., 369, 401, 402 Kuo, T. L., 140, 145, 149, 156–158 L Labossiere, P. E. W., 140 Leckie, F. A., 132 Lee, W. J., 132, 333, 367 Lekhnitskii, S. G., 16, 29–30, 34, 42, 50, 87, 186, 302, 374, 411, 415–417, 420, 431, 433 Liang, Y. C., 369, 545, 561 Liao, C. Y., 545, 560 Lin, C. C., 333, 346–348 Lin, K. Y., 132 Lizzio, R., 256 Lothe, J., 67, 73, 93, 256, 369, 389 Lowengrub, M., 289 Lu, P., 435–436 M Mahrenholtz, O., 435–436 Malen, K., 67 Manoharan, M. G., 226 Martin-Moran, C. J., 348 Mura, T., 256 Murakami, Y., 204 Muskhelishvili, N. I., 86, 101, 165, 196, 208, 277, 284–285, 295, 308 N Nishitani, H., 132 Nowacki, W., 333, 474 O Olesiak, Z., 337 P Pagano, N. J., 225 Pak, Y. E., 369 Park, S. B., 369, 400 Patton, E. M., 256 Pipes, R. B., 225, 228

Author Index R Reddy, J. N., 435–436, 444–445 Reedy, J. E. D., 132 Rekach, V. G., 330 Rice, J. R., 140, 143, 147, 216, 219, 223, 231, 397 Rogacheva, N. N., 370 Russell, A. J., 225 S Santare, M. H., 239, 256 Savin, G. N., 159 Sendeckyi, G. P., 239 Sih, G. C., 195, 202, 223, 337, 344 Sinclair, G. B., 140, 147 Smith, G. D., 67, 93, 333 Sneddon, I. N., 289, 337 Soh, A. K., 369, 383 Sokolnikoff, I. S., 2, 5, 8, 12, 15, 147, 377 Soni, M. L., 132 Sosa, H., 369 Stagni, L., 239, 256 Stern, M., 132, 140, 515, 576, 585–586 Street, K. N., 225 Stroh, A. N., 29, 53, 256, 333, 374 Sturla, F. A., 337, 344 Sumi, N., 348 Sun, C. T., 225–226, 369, 400 Suo, Z., 140, 216, 218, 220, 369, 377, 390, 402 T Tan, C. Z., 536, 542–543 Theocaris, P. S., 132 Timoshenko, S. P., 165, 184, 288 Ting, T. C. T., 6, 9, 11–12, 17, 53, 58, 60–62, 64, 66–67, 74–76, 95, 98–99, 101, 108, 117, 120, 128, 130, 137, 174, 190, 209, 245–246, 253, 334, 360, 365, 377, 379–380, 402 Tsai, Y. M., 337 U Ueda, S., 348 W Wang, A. S. D., 235 Wang, W. Y., 180, 182, 186, 239, 255–256 Wang, Z. Y., 239 Warren, W. E., 256 Wilkins, D. J., 225 Williams, M. L., 131, 206 Willis, J. R., 289

Author Index

665

Wu, C. H., 333 Wu, K. C., 140, 195, 216–217, 220

Yin, W. L., 460 Yoon, S. H., 225

Y Yan, G., 253, 365 Yang, H. C., 239 Yen, W. J., 174, 195, 239, 246, 248, 256–257, 266, 377, 460, 469, 519

Z Zakharov, D. D., 436 Zhou, S. G., 225 Zotemantel, R., 515

Subject Index

A Airy stress function, 33, 44–45, 55, 444 Analogy technique, 249, 288–290 Analytical continuation, 101–103, 109, 174, 246, 264, 279, 325, 350, 517–519, 529–532, 537 Analytic function, 101, 174 Angular bracket, 64, 504 Anisotropic elasticity, 3, 5, 29, 87, 197, 295, 374–375, 386, 420, 440, 446 elastic solid, 5–6, 29, 67, 333, 557 matrix, 239, 266, 268 plates, 182, 186, 337, 411–415, 424, 454 Antiplane deformation, 18 Antiplane shear, 18, 189 Arbitrary loading, 87, 159, 168–173, 194, 239, 245, 250, 494 Axial force, 29, 39, 43, 46 B Barnett–Lothe tensor, 69–71, 73, 77, 86, 98, 188–189, 203, 206, 377, 389–392, 540, 582 Bending moment, 25, 39, 46–48, 50, 412, 416, 428–431, 434, 437–438, 440–443, 477, 493–498, 505, 585 rigidity, 51 theory, 411–415, 436–439 Biaxial loading, 90, 162, 197 Bimaterial matrix, 145, 190, 208, 217, 221, 325–326 stress intensity factors, 144, 216–220, 223, 226, 275 Body force, 3, 6, 29–30, 40, 47–48, 147, 413, 483, 566 Boundary element, 92, 149, 207, 247, 494, 545–588

Boundary integral equation, 545–546, 552, 561, 565–571, 573–580, 586–588 Boundary value problem first (traction-prescribed), 6 second (displacement-prescribed), 6 third (mixed), 6 Branch cut, 112, 129, 297–298, 315, 319 Burgers vector, 99, 107–108, 111–112, 128, 176, 195, 249, 257, 266 C Cauchy principal value, 575–576 Cauchy’s formula, 2, 58, 152, 377, 417, 464 Chain rule, 54, 85, 164, 169, 244, 291, 522, 533 Characteristic equation, 34, 61, 358, 383, 386, 388, 422, 424, 427, 468 Chentsov coefficient, 13–14 Classical lamination theory, 23–27, 412, 445, 482 Coefficient matrix, 383, 386, 485, 491, 554 of moisture expansion, 474, 476, 478 of mutual influence, 13 of thermal expansion, 342–343 Commutative properties, 76–77, 127 Compatibility equation, 5, 29, 88, 374, 381, 420–421, 448 Complementary solution, 140, 148–151, 154–155 Complete adhesion, 278, 322, 325, 331–332 Complex conjugate, 34–35, 56, 133, 143, 151, 157, 357, 388, 415, 444, 485 method, 229, 234 parameter, 45, 415, 420 reduced elastic, 17–18, 32, 47 variable formulation, 6, 29, 86, 417, 440–458

667

668 Compliance method, 229, 234 reduced elastic, 17–18, 32, 47, 53, 591 Concentrated force, 87, 92, 95–96, 99, 111, 131, 173–176, 194–195, 246–248, 263, 505–507, 526, 529–539, 570 Concentrated moment, 95–99, 107, 510, 512 Conformal mapping function, 277, 299 Constitutive law, 1, 5, 7, 277, 369–375, 436–437, 451, 482–484, 503 Contact pressure, 282–284, 286, 311, 313–317, 327–329 surface, 278, 329 Continuity condition, 111, 113, 129, 134–135, 208, 259–260, 359–360, 527, 530–531, 536 Contracted notation, 8–9, 11, 21, 30, 60, 88, 370–371, 441, 484 Coordinate bipolar, 283, 315 cartesian, 3, 567, 575 global, 558–560 local, 202, 558–560 polar, 59, 78, 83, 94, 104, 111, 283, 297–298, 315, 319, 378, 463, 575 s–n, 59, 347, 462–463, 522 tangent–normal, 567, 575, 587 transformation, 225, 338, 358, 499, 560 Correspondence relation, 86, 284, 295 Coupled stretching–bending analysis, 411, 435–491, 565–588 Couple moment, 95–99, 107 Coupling effect, 454, 474, 498 Crack collinear, 187, 195–207, 225, 349, 405 collinear interfacial, 187 collinear periodic, 200–202, 204–206, 406 curvilinear, 268, 273–275 edge, 156 elliptical central, 156 insulated, 353, 366 interface, 140, 144–145, 190, 206–226, 229, 278, 288–290, 333, 348–356, 370, 396–397, 408–410 isothermal, 366 opening displacement, 187, 193, 196, 201–206, 210, 216, 344, 354, 405 penetrating, 272–273 penny-shaped interface, 156 semi-infinite, 132, 138, 144, 188, 366, 399

Subject Index tip, 144, 147, 187–189, 191–194, 199, 202–205, 207, 216, 237, 289, 344, 403, 563 Cracked lap shear (CLS), 225–229, 233–235 Critical point, 161, 179–180, 240–241 Cross section, 14–15, 20, 24–25, 39–41, 48–51, 237 Curvilinear boundaries, 277, 290, 435 Curvilinear interface, 273 D Degenerate material, 61, 73, 84–86, 196, 284, 342–343, 386, 389–390, 420, 560 Degrees of freedom, 580, 586, 588 Delaminated composite, 228, 230, 237 Delamination, 187, 206, 225–237 Delamination fracture criterion, 187, 225, 234 Delta function, 172, 210, 569 Diagonalization, 64, 73, 76, 133 Dielectric permittivities, 370, 560 Direction cosine, 2, 10 Dislocation, 87, 99–100, 107–108, 111–112, 176, 239–240, 256–275, 288–289 Dislocation density, 269, 271, 273–275 Displacement formalism, 436, 440–445, 447–451, 454–459, 465, 478, 487 Displacement gradient, 93, 287, 292, 312, 316 Double cantilever beam (DCB), 225–230, 232, 234–235 Dual coordinate systems, 66, 83 E Edge notch, 156 Effective transverse shear force, 414, 416–417, 419, 423, 429, 442–443, 458–459, 463 Eigenfunction, 115, 124, 132, 136–137, 143, 157, 395 Eigen-relation generalized, 73–74, 76, 85–86, 337, 425, 461, 469, 479 material, 60–68, 424–425, 459–461, 560 sextic, 61–67 Eigenvalue elasticity, 335, 337, 342–343 material, 55–56, 60–61, 72–74, 84, 86, 221, 262, 303, 307–308, 376–377, 384, 386, 422, 427, 459–460, 468, 534 repeated, 64, 84, 337, 386 thermal, 334, 337, 342–343 Eigenvector elasticity, 342 left, 62

Subject Index material, 55, 61, 86, 145, 248, 284, 289–290, 380–389, 436, 451, 467–469, 491, 541, 583 right, 62–63, 377 thermal, 342, 351 Elastically homogeneous, 7 Elastic constant, 1, 5, 7–8, 10–12, 18, 21–22, 54–55, 60–61, 73, 297, 326, 362, 446, 474, 478, 541 Elastic material anisotropic (or triclinic), 1–27 isotropic, 239 monoclinic (or aelotropic), 10 orthotropic (or rhombic), 11 transversely isotropic, 11, 20, 60 Elastic stiffness, 17–18, 68–69, 370, 375, 389, 466 Electric displacement, 370, 376–378, 394, 397, 403, 481–482, 505, 560–563 field, 369–370, 372, 376, 394, 481, 560–562 intensity factor, 396–400, 403–410 potential, 372, 378, 393–394, 398–400, 403, 481 singularity, 394 Electro-elastic analysis, 560–565 Electromechanical analysis, 560 Electronic package, 156 Elliptic integral, 204, 320 End-notched flexure (ENF), 225–230, 232, 234–235 Energy equation, 333 Energy release rate, 193, 196, 201, 203–206, 218, 220–222, 226, 228–230, 400, 405 Engineering constant, 1, 12–15, 18–19, 21, 23, 70–71, 221 Equilibrium equation, 3, 5, 29–30, 33, 52, 293, 413, 417, 420–421, 436–438, 447–448, 474–475, 566 F Failure initiation, 132, 140 Fiber bridging, 227 Fiber orientation, 23, 225, 227–232, 234 Fiber-reinforced composite, 19–20, 481 Field solution, 115, 137, 141–144, 183, 194, 200, 207, 209, 283, 352, 405, 496–497, 516–522, 533 Finite element, 149, 207, 228–230, 234, 539, 557, 570 Fracture mechanics, 141, 187–188, 225, 236–237, 268, 399

669 Fracture parameter, 187, 196, 201–207, 216–225, 236, 356 Fracture toughness, 146, 226–232 Free term coefficient, 580–588 Friction coefficient of, 314 force, 278, 308–309, 312 surface, 309, 312, 319–320, 329 Fundamental elasticity matrix, 62, 68–69, 71–73, 84, 132, 133, 436, 446, 469–472, 501, 513 Fundamental solution, 92, 494, 539, 545–551, 556–558, 560–561, 569–576, 588 G Gauss elimination, 560 Gaussian quadrature rule, 553, 560 Gauss point, 553 Generalized Hooke’s law, 5, 7–9, 12 Glide force, 268 Green’s function interfacial, 111–113 surface, 105–106, 112, 176, 330 H Half-plane, 277–292, 300, 306, 308–309, 311–315, 317–318, 326, 330, 332 Half-space, 87–113, 174, 545–547, 550, 557 Heat conduction, 333–334, 338, 358, 362, 474, 476 flux, 333–334, 337–339, 343, 345–346, 349–354, 356–365, 474–476, 498–499, 502 Hilbert problem, 208–209, 280, 292, 302, 306, 310, 326, 331, 351 H-integral, 141, 147–158, 557 Holder-continuous, 270–271, 275 Hole circular, 165, 172, 178, 411, 431, 562 curvilinear, 290–298, 302–303 elliptical, 159–176, 178, 182, 289–290, 293–295, 338–343, 428, 516, 543 oval, 177, 184, 186 pin-loaded, 172–173, 290 polygon-like, 159, 177–186, 290, 295–296, 549 square, 186 triangular, 181, 299, 308 Holomorphic function, 242, 260, 291, 303, 305, 307, 331, 376, 415, 477, 517, 529 Homogeneous material, 52, 111, 130, 144, 155, 188, 191, 266, 406, 415

670 Homogeneous solution, 115, 120, 122–124, 335, 484 Hoop stress, 105, 163–165, 167, 170–172, 176, 183–186, 342–343, 542–543, 562–563 Hygrothermal stress, 474–481, 498 I Identities, 61, 68–84, 104–105, 112, 127, 165–167, 183, 200, 336–337, 341–343, 457, 511, 581, 583 Image force, 101 Image singularities, 174, 519 Impedance matrix, 81, 110, 243, 280, 312, 325, 540, 548 Inclusion circular, 256, 263, 432 elastic, 239–248, 250–252, 256–258, 526–543, 546–547, 574 elliptical, 239–240, 256–257, 272, 274, 526, 542–543 hard, 248, 268 polygon-like rigid, 249–250, 255–256 rigid, 239, 248–256, 268, 288–289, 344–348, 431–433, 546–548, 550 rigid line, 248–249, 254–255, 288–289, 347–348 soft, 268, 543 Indentation complete, 312–317 depth of, 278, 306 incomplete, 312–317 normal, 278, 311–312, 314 punch, 278, 311 rigid stamp, 290–298, 302 rotary, 278 Indenter, 278, 289 Induction, 370, 393, 405–406, 409–410 Infinite space, 87–113, 505, 545–547, 550, 557 Integral formalism, 73, 184, 389 Interaction coefficient matrices, 560 Interaction effect, 320–321 Interaction energy, 266–268 Interface corner, 115–158, 396 Interfacial stress, 191, 244, 251, 256, 347 Interlaminar stress, 225 Internal point, 92, 99, 107, 545, 554–555, 565–570, 581, 586 strain, 554 stress, 505, 555, 560 Interpolation function, 552, 556 Isothermal condition, 337, 339

Subject Index Isotropic plate, 87, 89, 162, 165–166, 168, 173–174, 176–179, 182–184, 186, 194–195, 338, 344–345, 412–415, 424, 427–428, 469, 473, 581, 584–586 J J-integral, 147, 557 K Kernel function, 99, 256, 270, 275, 494 Key matrix, 132–134, 356, 358 Kinematic relation, 277, 447–448, 475 Kirchhoff force, 414, 439 Kirchhoff’s assumption, 436, 445, 475, 481, 565 Kronecker delta, 3, 54, 63, 441, 546, 581 L Lame constant, 12, 15, 60 Lamina generally orthotropic, 20–23 specially orthotropic, 20–21 Laminate balanced, 439 composite, 206, 227, 234–237, 474–475, 481–491, 493–495, 498, 526 cross-ply, 427 degenerate, 583–584 electro-elastic, 484, 503–505 symmetric, 411, 435, 454–457, 465, 468–469, 472–474, 503 unsymmetric, 435, 472, 503 Laurent’s expansion, 241, 244, 251 Lekhnitskii bending formalism, 411, 415–420, 422 Lekhnitskii formalism, 29–52, 87, 380, 411, 420, 424, 440, 451 L’Hospital’s rule, 96 Linear algebraic equation, 215, 469, 509, 557, 560 Line force, 92 Line integral, 212–213, 286–287, 327, 353–355, 566 Liouville’s theorem, 102, 110, 175, 242, 261, 310, 326, 350, 517 Logarithmic function, 93, 112, 129, 169, 248, 588 Logarithmic singularity, 137, 142, 365, 395 Longitudinal direction, 19–20 M Macromechanics, 3, 6, 493 Mapping function, 180, 277, 299, 302 Material axes, 20–21, 24

Subject Index Material symmetry, 1, 9–12 Matrices of influence coefficients, 552, 554 Matrix diagonal, 64, 154, 203, 206, 217, 221, 282, 359, 504 identity, 62, 72, 445 nonsemisimple, 64, 84 rotation, 65, 75 semisimple, 64, 73, 78 simple, 21–23, 557 Matrix differential equation, 67–68 Mechanical energy release rate, 400 Membrane analogy, 288 Micromechanics, 3, 6, 493 Mid-plane strain, 437–438, 446, 475, 482, 504–505, 565 Mixed formalism, 436, 440, 446–460, 465, 467, 487–489 Mixed-mode fracture, 225, 229, 232–234 Mode I, II, III, 189, 207, 225–233 Modified end-notched flexural (MENF), 225–229, 233–235 Moisture content, 474–476, 479–480, 501 diffusion, 474, 476 transfer, 474–475, 493, 498–503 Moment–curvature relation, 413 Moment of inertia, 91 Moment intensity factor, 433–434 Multi-valued function, 92–93, 506, 508, 510 N Near-tip solution, 140–142, 146–147, 149, 154–155, 189, 206, 216, 356, 364–365, 394–397, 399, 401, 403 Nodal displacement, 552, 557 point, 552–553, 559 traction, 552 Non-permittivities, 370, 560 Normalization factor, 57, 69, 389–390, 423 O Open circuit, 372, 484 Opening mode (or tensile mode), 189 Orthogonality relation, 57, 63–64, 85–86, 94, 105, 377, 384, 387, 389, 507, 541, 560 Oscillatory characteristics, 140, 207, 226, 289 Oscillatory index, 191, 221–223, 295, 300, 403 P Parabolic elastic bodies, 327, 329–330 Particular solution, 34, 49, 115, 124, 335, 415–416, 484, 503

671 Path-independent property, 148–149 Perturbation elliptical perturbed boundary, 304 higher order, 299, 302, 306–308 straight perturbed boundary, 301, 305 surface, 277, 299–308 technique, 277, 299, 303 zero-order, 301, 305, 307–308 Piezoeffect equation, 370 Piezoelectric ceramics, 385, 387, 390–393, 400, 404, 406–410 material, 369–410, 453, 481, 561 tensor, 560 Plane generalized plane strain, 16, 18, 43–44, 69–70, 372, 376, 385, 505 generalized plane stress, 15–16, 18, 20–21, 47, 69–71, 88–89, 203, 221, 285, 372–373 strain (plane deformation), 14, 43–44 stress, 15–16, 18, 20–21, 44, 47, 69–71, 87–89, 156, 162, 193, 203, 221, 295, 372, 466, 585 Plate bending analysis, 411–434, 454, 457 Plate curvature, 24–25, 475 Plemelj function, 209–211, 216, 280, 292, 296, 331, 352 Point charge, 378, 561 Point force, 92, 100–101, 105, 107, 131, 173, 176, 195, 210, 288–289, 408–409, 549–550, 570, 575 Poisson’s ratio, 1, 12–13, 285, 295, 314, 342, 427, 431, 466 Poling direction, 385, 387 Positive definite, 12, 54–56, 60, 62, 64, 119, 188, 190, 221, 251, 312, 474 Positive semi-definite, 62 Potential energy, 266 Principal axes of inertia, 40, 48 Principal material direction, 11, 14, 18, 20–22 Principal moments of inertia, 40 Pseudo-inverse, 120 Punch flat-ended, 283–287, 289, 299, 306, 314, 318–322 parabolic, 278, 287–288, 314–319 rigid, 277–290, 299–309, 318, 332 sliding, 277–278, 299, 308–321, 326 wedge-shaped, 312–317 Pure in-plane bending, 87, 91–92, 159, 184–185 Pure shear, 12–13, 90, 162, 510–511

672 R Reciprocal theorem of Betti and Rayleigh, 140, 147, 152 Reference length, 145, 157 Residue theorem, 253 Rigid body motion, 4, 102, 120, 123, 162, 175, 258, 260–261, 350, 517, 532, 586 rotation, 118, 120, 248, 256 translation, 32, 48, 89, 175, 312 Root double, 141, 143, 155, 395 non-repeated, 141 triple, 60–61, 84, 141, 143, 155, 395 Rotated coordinate system, 59, 83, 336 S Scaling factor, 383–384, 387, 389, 407, 468 Sectionally holomorphic, 279, 292, 301, 305, 307, 310, 331 Semi-inverse method, 174 Sensors and actuators, 369, 481 Shearing mode (or sliding mode), 189 Shear modulus, 60, 71, 221, 229, 314, 431 Short circuit, 372, 376, 385, 505 Sign convention, 414 Sign function, 271 Single-valued displacement, 4, 33, 92, 100, 129, 169, 270, 275, 507, 510 Singular characteristics, 187–192, 370, 399–404 field, 141, 148–149, 153 integral, 270–273, 275, 553 integral equation, 270–273 order, 133–134, 137–143, 145–146, 154–155, 187–188, 190–191, 356–357, 366–367, 394–395, 402–403 point, 101, 212, 240, 242, 246, 250, 259, 263, 528, 534, 536 Smooth boundary, 545, 575, 581, 586 Source point, 575, 577, 579, 588 Specimen fabrication, 228–229 Stacking sequence, 23, 236 Stiffness bending, 25, 413, 421, 427, 469–471, 489, 541 coupling, 412, 421, 439–440, 452, 454, 460, 465–466, 469, 495 extensional, 25, 437, 452, 473, 495, 541 reduced, 17, 19 Strain Cauchy’s infinitesimal, 4

Subject Index engineering shear, 4, 22 Eulerian, 4 Lagrangian, 4, 7 normal, 4, 18, 22 shear, 4, 14, 22–23, 372, 441, 447, 565 Strain energy, 8, 12, 34, 55–56, 64, 132, 141, 193, 201, 218, 356, 377, 475 Strength of stress singularity, 189, 320 Stress concentration, 159, 165, 184, 206, 337, 344, 493 contour, 297 expanded, 369, 374–379, 394 expanded Stroh-like formalism, 484–491, 493 extended Stroh formalism, 333–337, 348, 375 extended Stroh-like formalism, 476–481, 498 function, 29, 33, 43–45, 55–59, 65–67, 141, 172, 189, 270, 374, 457, 501, 530 intensity factor, 115, 139–147, 152–158, 196, 201–207, 226, 273, 345, 396–398, 555–557, 562–564 normal, 2, 13, 18, 22, 311 principal, 2, 396 resultant, 58, 92, 436–438, 446, 462–464, 485, 497–498, 522–523, 570 shear, 2, 10, 21–22, 205, 372, 483, 510 singularity, 115, 124, 135, 155, 189, 207, 320, 348, 362, 365, 525 –strain relation, 1, 8–10, 16–18, 20–21, 47, 88–89, 162 Stroh formalism Stroh-like bending formalism, 411–412, 420–429, 435, 454, 457, 504 Stroh-like formalism, 411, 436, 451, 458–474, 476–481, 484–491, 507, 581 St. Venant’s principle, 46 Sub-inverse, 119, 346, 501 Subregion technique, 557–559 Superposition method, 95, 101 Surface deformation, 104, 282–284, 286, 288, 311–312, 315, 317, 319–321, 329–330 integral, 153, 565–566, 570 moment, 418, 462 traction, 6, 29, 40, 58, 94, 100–101, 121, 123, 172–173, 208, 239, 291, 377–378, 462, 545 System of equation, 5, 382, 509, 553–554, 558–560

Subject Index T Taylor’s series expansion, 117 Tearing mode (or antiplane mode), 189, 207, 226, 229 Technical constant, 12 Temperature field, 333, 350, 353, 357, 362, 365 Thermal eigenvalue, 334, 337, 342–343 eigenvector, 342, 351 expansion coefficient, 342 moduli, 334, 352, 362 properties, 362, 365, 367 stress, 337, 344–345, 348 Thermoelasticity, 333, 335, 349, 362, 375, 474–475, 477–478 Tilted angle, 287 Torsion generalized, 50–51 generalized torsional rigidity, 50 pure torsion deformation, 50 Transfer matrix, 132, 135, 360 Transformation function, 159–160, 178–182, 240, 246, 257, 295–296, 302–303 law, 4, 58, 414, 417–418, 420, 463, 505, 580 matrix, 271 Translating technique, 102–103, 111, 175, 209, 211, 244, 253, 265, 283, 295–296, 352, 518, 520, 522, 540–541 Transverse direction, 19–20, 474 Transverse loading, 411, 493 Transverse shear, 23, 71, 87, 372, 413–414, 416–417, 419, 442–443, 462–463, 569 Twisting moment, 39–40, 47–51, 428, 431 Two-dimensional deformation, 29, 53–55, 67, 435

673 U Unidirectional tension, 90, 162, 165, 183 Uniform heat flow, 337–339, 342–348, 493, 498–503 loading, 87–91, 161–165, 170, 177, 182–186, 245–247, 255, 268, 338, 354, 493 stress solution, 89, 116–117, 122, 124 traction, 115–124, 213 Unperturbed elastic field, 101, 240, 244–245, 247, 251–253, 517, 528–529 V Virtual crack closure method, 193, 218, 344 W Wavy-shaped surface, 299, 306 Wedge angle, 115, 117–125, 127, 132, 188, 191, 356, 365–367, 399 apex, 115, 121, 125–134, 137–138, 145, 356–357, 394, 396, 398 bi-, 132, 138–139, 144, 190, 366–367, 401 critical wedge angle, 117–123, 127–128 insulated, 363, 366 isothermal, 363, 366–367 multi-material, 115, 128–132, 134–137, 356–367, 393–398 multi-material wedge space, 115, 128–132, 134–137, 361–362, 365–366 non-critical wedge angle, 117–122 tri-, 132, 139, 367 Y Young’s modulus, 71, 221, 229, 326, 342, 427, 466

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