Fracture Mechanics of Nonhomogeneous Materials 9811940622, 9789811940620

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Licheng Guo · Yu Hongjun · Wu Linzhi

Fracture Mechanics of Nonhomogeneous Materials

Fracture Mechanics of Nonhomogeneous Materials

Licheng Guo · Yu Hongjun · Wu Linzhi

Fracture Mechanics of Nonhomogeneous Materials

Licheng Guo Department of Astronautic Science and Mechanics Harbin Institute of Technology Harbin, Heilongjiang, China

Yu Hongjun Department of Astronautic Science and Mechanics Harbin Institute of Technology Harbin, Heilongjiang, China

Wu Linzhi Center for Composite Materials Harbin Institute of Technology Harbin, Heilongjiang, China

ISBN 978-981-19-4062-0 ISBN 978-981-19-4063-7 (eBook) https://doi.org/10.1007/978-981-19-4063-7 Jointly published with Science Press, Beijing, China The print edition is not for sale in China mainland. Customers from China mainland please order the print book from: Science Press ISBN of the Co-Publisher’s edition: 978-7-03-070071-1 © Science Press 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

In the past few decades, nonhomogeneous materials have been considerably designed and applied in the engineering fields. Typical nonhomogeneous materials (or functionally graded materials, FGMs) are designed with properties varying continuously from one side to another side with a specific gradient. The concept of FGMs was first proposed in 1984 by combining a variety of materials to serve the purpose of a thermal barrier capable of withstanding on a surface temperature of 2000 K and a temperature gradient of 1000 K. The FGMs have the advantages of both components, which makes them attractive in potential applications, including a potential reduction of in-plane and transverse through-the-thickness stresses, an improved residual stress distribution, enhanced thermal properties, higher fracture toughness, and reduced stress intensity factors. Due to the extreme load gradient and the existence of defects introduced during the manufacturing process, fracture becomes the key failure mode of FGMs. Initially, most of the theoretical researchers assumed that the transition between the two sides of the nonhomogeneous materials is exponential, so analytical solutions can be obtained. This assumption limits the development of theoretical study on nonhomogeneous materials with arbitrary changes in properties. On the other hand, nonhomogeneous materials are usually associated with particulate composites, where the volume fraction of particles varies in one or several directions. As the scale of research decreases to the particle size, nonhomogeneous materials need to be regarded as composite materials with complex interfaces. Fracture mechanics theories are applicable for homogeneous materials, and even for some nonhomogeneous materials with continuous characteristics, but there are few literatures on nonhomogeneous materials with complex interfaces. It hinders the development of fracture mechanics of nonhomogeneous materials with complex interfaces. Therefore, it is of significant academic value to establish and develop new theoretical models of fracture mechanics for nonhomogeneous materials with arbitrary properties and for nonhomogeneous materials with complex interfaces. This book titled Fracture Mechanics of Nonhomogeneous Materials mainly focuses on the fracture problems of two types of nonhomogeneous materials. One is the attribute with arbitrary properties and the other is the attribute with complex v

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interfaces. Chapter 1 provides the readers with a clear image of the crack-tip elastic fields of nonhomogeneous materials. In Chap. 2, the initial exponential models are introduced, which is the basic work for the next chapter. Then, in Chap. 3, the idea of a general fracture mechanics model (piecewise exponential model) of a nonhomogeneous material with general elasticity is described step by step. The further expansion of the piecewise- exponential model for thermomechanical and viscoelastic fracture problems is provided in Chap. 4. Then, this book introduces the fracture mechanics methods for nonhomogeneous materials with complex interfaces in Chaps. 5–7. The domain-independent interaction integral (DII-integral) method is introduced in Chap. 5, which is used to extract the fracture parameters of an internal crack in nonhomogeneous materials with complex interfaces. The DII-integral can decouple the mixed-mode stress intensity factors and extract the T-stress. For nonhomogeneous materials, the DII-integral has advantages over the classical methods, i.e., the integral domain is independent of nonhomogeneous material properties and interfaces. A region with arbitrary interfaces can be selected as the integral domain to facilitate the calculation of the fracture parameters for nonhomogeneous material with complex interfaces. Chapter 6 introduces the DII-integral for an interface crack in nonhomogeneous materials with complex interfaces. The thermal fracture problems of nonhomogeneous materials are discussed in Chap. 7 by introducing a frame for the design of a generalized DII-integral and establishing a new DII-integral for nonhomogeneous materials with complex interfaces. Compared with published books on the fracture of nonhomogeneous materials, this book gradually describes the fracture mechanics from homogeneous to nonhomogeneous materials with particular property assumptions, then to nonhomogeneous materials with arbitrarily continuous properties, and finally to nonhomogeneous materials with complex interfaces step by step. And research scale ranges from macro bulk material to micro constituents. This arrangement coincides with the research process, which will provide a revelation to new researchers. This book can be used as a reference book for undergraduate or graduate students. The book includes more than 40 research papers published by authors in recent two decades. The related research is funded by more than 6 National Natural Science Foundation of China (NSFC) projects. The first chapter is mainly written by Dr. Licheng Guo, Dr. Yu Hongjun, and Dr. Wu Linzhi. Chapters 2–4 were mainly written by Dr. Licheng Guo. Chapters 5–7 were mainly written by Dr. Yu Hongjun. Dr. Wang Zhihai has made many contributions to the contents of Chaps. 3 and 4 in preparing and editing this book. Due to the limited level of expertise, omissions and inadequate parts are unavoidable. Any kind of criticism and correction will be highly appreciated. Harbin, China January 2022

Licheng Guo Yu Hongjun Wu Linzhi

Contents

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Internal Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Basic Equations for Nonhomogeneous Materials . . . . . . . . . 1.1.2 Crack-Tip Fields for Homogeneous Materials . . . . . . . . . . . . 1.1.3 Crack-Tip Fields for Nonhomogeneous Materials . . . . . . . . . 1.1.4 Crack-Tip Fields for Nonhomogeneous Orthotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Interface Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Crack-Tip Fields of an Interface Crack . . . . . . . . . . . . . . . . . . 1.2.2 Crack-Tip Fields of an Interface Crack Between Two Nonhomogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Three-Dimensional Curved Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Internal Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Interface Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Exponential Models for Crack Problems in Nonhomogeneous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Crack Model for Nonhomogeneous Materials with an Arbitrarily Oriented Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Basic Equations and Boundary Conditions . . . . . . . . . . . . . . . 2.1.2 Full Field Solution for a Crack in the Nonhomogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Stress Intensity Factors (SIFs) and Strain Energy Release Rate (SERR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Crack Problems in Nonhomogeneous Coating-Substrate or Double-Layered Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Interface Crack in Nonhomogeneous Coating-Substrate Structures . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 5 9 11 11 14 18 18 19 20 23 23 23 25 29 31 31

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2.2.2 Cross -Interface Crack Parallel to the Gradient of Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Arbitrarily Oriented Crack in a Double-Layered Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Crack Problems in Orthotropic Nonhomogeneous Materials . . . . . . 2.3.1 Basic Equations and Boundary Conditions . . . . . . . . . . . . . . . 2.3.2 Solutions to Stress and Displacement Fields . . . . . . . . . . . . . 2.3.3 Crack-Tip SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Transient Crack Problem of a Coating-Substrate Structure . . . . . . . . 2.4.1 Basic Equations and Boundary Conditions . . . . . . . . . . . . . . . 2.4.2 Solutions to Stress and Displacement Fields . . . . . . . . . . . . . 2.4.3 Crack-Tip SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Representative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Example 1: Arbitrarily Oriented Crack in an Infinite Nonhomogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Example 2: Interface Crack Between the Coating and the Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Example 3: Cross-Interface Crack Perpendicular to the Interface in a Double-Layered Structure . . . . . . . . . . . . 2.5.4 Example 4: Inclined Crack Crossing the Interface . . . . . . . . . 2.5.5 Example 5: Vertical Crack in a Nonhomogeneous Coating-Substrate Structure Subjected to Impact Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 44 55 55 56 61 62 62 63 68 68 68 71 72 74

77 78 79

3 General Model for Nonhomogeneous Materials with General Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1 Piecewise-Exponential Model for the Mode I Crack Problem . . . . . 82 3.1.1 Piecewise-Exponential Model (PE Model) . . . . . . . . . . . . . . . 82 3.1.2 Solutions to Stress and Displacement Fields . . . . . . . . . . . . . 85 3.1.3 Crack-Tip SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.2 PE Model for Mixed-Mode Crack Problem . . . . . . . . . . . . . . . . . . . . . 90 3.2.1 Basic Equations and Boundary Conditions . . . . . . . . . . . . . . . 90 3.2.2 Solutions to Stress and Displacement Fields . . . . . . . . . . . . . 92 3.2.3 Crack-Tip SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3 PE Model for Dynamic Crack Problem . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3.1 Basic Equations and Boundary Conditions . . . . . . . . . . . . . . . 96 3.3.2 Solutions to Stress and Displacement Fields . . . . . . . . . . . . . 98 3.3.3 Crack-Tip SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.4 Representative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.4.1 Example 1: Mode I Crack Problem for Nonhomogeneous Materials with General Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Contents

3.4.2 Example 2: Mixed-Mode Crack Problem for Nonhomogeneous Materials with General Elastic Properties and an Arbitrarily Oriented Crack . . . . . . . . . . . . . 3.4.3 Example 3: Dynamic Mode I Crack Problem for Nonhomogeneous Materials with General Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Fracture Mechanics of Nonhomogeneous Materials Based on Piecewise-Exponential Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Thermomechanical Crack Models of Nonhomogeneous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Crack Model for Nonhomogeneous Materials Under Steady Thermal Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Crack Model for Nonhomogeneous Materials Under Thermal Shock Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Viscoelastic Crack Model of Nonhomogeneous Materials . . . . . . . . 4.2.1 The Correspondence Principle for Viscoelastic FGMs . . . . . 4.2.2 Viscoelastic Models for Nonhomogeneous Materials . . . . . . 4.2.3 PE Model for the Viscoelastic Nonhomogeneous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Crack Model for Nonhomogeneous Materials with Stochastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Stochastic Micromechanics-Based Model for Effective Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Probabilistic Characteristics of Effective Properties at Transition Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Crack in Nonhomogeneous Materials with Stochastic Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Example 1: Steady Thermomechanical Crack Problem . . . . 4.4.2 Example 2: Viscoelastic Crack Problem . . . . . . . . . . . . . . . . . 4.4.3 Example 3: Crack Problem in FGMs with Stochastic Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Fracture of Nonhomogeneous Materials with Complex Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Interaction Integral (I-Integral) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 J-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 I-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Auxiliary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Extraction of the SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Domain-Independent I-integral (DII-Integral) . . . . . . . . . . . . . . . . . . . 5.2.1 Domain Form of the I-Integral . . . . . . . . . . . . . . . . . . . . . . . . .

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112 117 121 123 123 123 126 137 137 140 141 143 143 147 149 152 152 157 159 163 165 166 166 167 168 169 169 169

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5.2.2 DII-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DII-Integral for Orthotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . Consideration of Dynamic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of the T-Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DII-Integral for 3D Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 I-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Auxiliary Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Extraction of the SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Domain Form of the I-Integral . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.5 DII-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Typical Fracture Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Stress Intensity Factor Evaluations . . . . . . . . . . . . . . . . . . . . . 5.7.2 T-Stress Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Stress Intensity Factors of a Penny-Shaped Crack . . . . . . . . . 5.7.4 Influences of the Material Continuity on the SIFs . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172 176 178 180 183 183 184 186 187 189 193 193 194 197 200 205

6 Interfacial Fracture of Nonhomogeneous Materials with Complex Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 I-integral for an Interface Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 J-integral and I-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Auxiliary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Relation Between the I-integral and the SIFs . . . . . . . . . . . . . 6.2 Domain-Independent I-integral (DII-Integral) . . . . . . . . . . . . . . . . . . . 6.2.1 DII-Integral for Materials with Continuous Properties . . . . . 6.2.2 DII-Integral for Materials with Complex Interfaces . . . . . . . 6.2.3 DII-Integral for a Curved Interface Crack . . . . . . . . . . . . . . . . 6.2.4 Consideration of Dynamic Fracture Process . . . . . . . . . . . . . . 6.3 T-stress Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Auxiliary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Extraction of the T-stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 DII-Integral for 3D Interface Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Definition of the I-integral on the Crack Front . . . . . . . . . . . . 6.4.2 Auxiliary Fields for 3D Interface Crack . . . . . . . . . . . . . . . . . 6.4.3 Extraction of the SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Domain Form of the I-integral . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 DII-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Representative Interfacial Fracture Problems . . . . . . . . . . . . . . . . . . . 6.5.1 Straight Interface Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 A Circular-Arc Shaped Interface Crack . . . . . . . . . . . . . . . . . . 6.5.3 T-stress Evaluation of Biomaterial Strips . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 207 208 209 210 211 211 212 215 216 217 217 218 218 218 219 220 221 223 224 224 228 230 234

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7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Internal Crack Under Thermal Loading . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Basic Equations of Thermoelasticity . . . . . . . . . . . . . . . . . . . . 7.1.2 I-integral for Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Auxiliary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Extraction of the SIFs for Thermoelastic Media . . . . . . . . . . 7.1.5 Domain Form of the I-integral . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Interface Crack Under Thermal Loading . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Definition of the I-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Auxiliary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Extraction of Thermal SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 I-integral for a Thermoelastic Solid with Complex Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 T-stress Evaluation for Nonhomogeneous Thermoelasticity . . . . . . . 7.3.1 Auxiliary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Extraction of T-stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Generalized DII-Integral for Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Requirements for the Establishment of the DII-Integral . . . . 7.4.2 Design of the Generalized Auxiliary Fields . . . . . . . . . . . . . . 7.4.3 Generalized DII-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 A New DII-Integral for Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 DII-Integral for an Internal Crack . . . . . . . . . . . . . . . . . . . . . . 7.5.2 DII-Integral for an Interface Crack . . . . . . . . . . . . . . . . . . . . . . 7.6 Typical Thermal Fracture Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Internal Crack Under Thermal Loading . . . . . . . . . . . . . . . . . . 7.6.2 Particulate Plate with an Internal Crack . . . . . . . . . . . . . . . . . 7.6.3 Multi-interface Plate with an Interface Crack . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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235 235 235 236 237 237 238 242 242 244 244 245 247 247 248 250 250 253 258 259 259 262 267 267 269 269 274

Chapter 1

Fundamental Theory of Fracture Mechanics of Nonhomogeneous Materials

Nonhomogeneous materials either exist naturally or are used intentionally to attain a required structural performance, such as bones, bamboo, shells, masonry composed of crushed stone, particulate composite materials, fiber-reinforced composite materials, etc. Nonhomogeneous materials are usually formed of two or more constituent phases with a variable composition, and they are prone to crack propagation during use which is due to the defects (holes, microcracks, debonding) introduced during manufacturing process. In recent decades, studies on fracture mechanics of nonhomogeneous materials have been carried out to ensure their reliable applications. To facilitate theoretical analysis, the interfaces between components are ignored, and the nonhomogeneous material is equivalent to a material with continuous macroscopic properties. This approach has been widely used in the fracture mechanics analysis of nonhomogeneous in the past few decades. Recently, due to the rapid development of computer technology, the real properties of components and interface conditions between components have been considered. This chapter will show the images of the near-tip field of a crack in nonhomogeneous material. Please note that throughout this book, an interface crack refers to the crack located along the interface between two neighboring materials while an internal crack refers to the crack with two surfaces located in the same materials. In addition, an embedded crack means that the crack is completely in a body, while an edge crack means that the crack intersects with the edge of the body. As a result, an embedded crack has two tips, while an edge crack has only one tip in the two-dimensional case.

1.1 Internal Crack 1.1.1 Basic Equations for Nonhomogeneous Materials A distinctive feature of nonhomogeneous materials is that the material parameters vary with coordinates. Taking an isotropic nonhomogeneous material as an example, © Science Press 2023 L. C. Guo et al., Fracture Mechanics of Nonhomogeneous Materials, https://doi.org/10.1007/978-981-19-4063-7_1

1

2

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous …

the Young’s modulus E and Poisson’s ratio ν need to be expressed respectively as E = E(x), ν = ν(x)

(1.1)

where x denotes the coordinate vector, which is given by x = {x1 , x2 , x3 }T or x = {x, y, z}T . The readers should remember the Eq. (1.1) firmly for nonhomogeneous materials. That is, even if only a single symbol E appears in some expression, it is usually not a constant but a function E(x) for nonhomogeneous materials. This is very important in the following study on nonhomogeneous materials. Then, the constitutive equation for the isotropic nonhomogeneous material is given in polar coordinate system by εαβ =

3 − κ(x) 1 σαβ − σγ γ δαβ (α, β, γ = r, θ ) 2μ(x) 1 + κ(x)

(1.2)

where εαβ and σαβ represent the components of strain and stress, respectively, μ(x) = E(x)/[2 + 2ν(x)] is the shear modulus, and κ(x) is the Kolosov constant defined by following formula { κ(x) =

(plane stress) 3 − 4ν(x) (plane strain) 3−ν(x) 1+ν(x)

The symbol δαβ is the Kronecker delta function and given by } δαβ =

1 (α = β) 0 (α /= β)

In addition, the elastic fields need to satisfy the strain–displacement relations εrr =

( ) ur ∂u θ uθ ∂u r 1 ∂u θ 1 1 ∂u r , εθθ = + , εr θ = + − ∂r r ∂θ r 2 r ∂θ ∂r r

(1.3)

and the equilibrium equations 1 ∂σr θ σrr − σθθ ∂σrr + + =0 ∂r r ∂r r 1 ∂σθθ ∂σr θ 2σr θ + + =0 r ∂θ ∂r r This chapter will discuss the crack-tip fields in this frame.

(1.4)

1.1 Internal Crack

3

1.1.2 Crack-Tip Fields for Homogeneous Materials For an internal crack in a homogeneous elastic solid, the stress has a singularity of r −1/2 , as shown in Fig. 1.1, where r represents the distance from the current point to the crack tip. As r tends to be zero, the stress tends to be infinite. Williams (1957) first provided the crack-tip fields by using the eigenfunction expansion technique. The near-tip displacements and stresses of a two-dimensional internal crack are given by KI I K II II σi j = √ gi j (θ ) + √ gi j (θ ) + T δ1i δ1 j + O(r 1/2 ) + . . . 2πr 2πr / / r I r II f i (θ ) + K II f (θ ) ui = KI 2π 2π i ) ( κ −3 κ +1 cos θ + δi2 sin θ + O(r 3/2 ) + . . . + T r δi1 8μ 8μ

(1.5)

(1.6)

√ √ The parameter K I = lim σ22 (r, 0) 2πr and K II = lim σ21 (r, 0) 2πr are moder →0

r →0

I and mode-II stress intensity factors (SIFs), showing tensile and shear effects near the crack tip. Generally, the stress intensity factor K N (N = I, II) depends on the material properties, geometry and loading conditions. T represents the constant term of the stress σ11 , known as the T-stress. The angular functions of the stress and displacement are given by (Gdoutos 2005) ) ) ( ( II I g11 = (1 − sin θ2 sin 3θ2 ) cos θ2 , g11 = − 2 + cos θ2 cos 3θ2 sin II I = (sin θ2 cos θ2 cos 3θ2) g22 = 1 + sin θ2 sin 3θ2 cos θ2 , g22 θ θ 3θ I II g12 = sin 2 cos 2 cos 2 , g12 = 1 − sin θ2 sin 3θ2 cos θ2

Fig. 1.1 Normal stress varying with the distance to the crack tip

θ 2

(1.7)

4

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous …

f 1I = f 2I =

κ−cos θ 2μ κ−cos θ 2μ

cos θ2 , f 1II = sin θ2 , f 2II =

2+κ+cos θ 2μ 2−κ−cos θ 2μ

sin θ2 cos θ2

(1.8)

In order to give readers an intuitive impression of stress distributions with respect to the angle θ , Fig. 1.2 provides √ the stress angular functions in Eq. (1.7). Here, the angular function giNj = σi j 2πr /K N can be regarded as the stresses on the circle of r = K N2 /(2πr ), where N = I, II denote opening and sliding crack modes, respectively.

Fig. 1.2 Stress angular functions for an internal crack

1.1 Internal Crack

5

1.1.3 Crack-Tip Fields for Nonhomogeneous Materials Eischen (1987) extended the eigenfunction expansion technique to derive the cracktip fields of nonhomogeneous materials with continuously differentiable properties. For a traction-free crack, the stress equilibrium equations are satisfied identically by an Airy stress function φ = φ(r, θ ) as 1 ∂ 2φ 1 ∂φ ∂ 2φ ∂ + 2 2 , σθθ = 2 , σr θ = − σrr = r ∂r r ∂θ ∂r ∂r

(

1 ∂φ r ∂θ

) (1.9)

Substituting Eq. (1.9) into the compatibility equation 2 ∂εθθ 2 ∂ 2 εr θ 2 ∂εr θ 1 ∂εrr ∂εθθ 1 ∂εrr + = + 2 + − r 2 ∂θ 2 ∂r 2 r ∂r r ∂r r ∂r ∂θ r ∂r

(1.10)

one obtains the following equation governing the stress function for generalized plane stress condition (Eischen 1987) |

|| ( )| ) ( 2 ∂E 2 1 ∂ 2 E ∂ 2φ 1 ∂φ 1 ∂ 2φ ∇ φ+ − − ν + E 2 ∂r E ∂r 2 ∂r 2 r ∂r r 2 ∂θ 2 | ( |( ) ) 2 ∂E 2 1 ∂φ 1 ∂ 2φ 1 ∂2 E ν ∂ 2φ + + 4 2 − 2 2 − E 2 ∂θ E ∂θ 2 r 3 ∂r r ∂θ r ∂r ( 2 2 3 ) 1 ∂ φ 1 ∂ E 1 ∂φ 2∂ φ ∂ φ + 3 2 − + −2 3 2 2 E ∂r r ∂r r ∂θ r ∂r ∂r ) ( ν ∂ E 2 ∂ 3φ 1 ∂ 2φ 2 ∂ 2φ + + − 3 2 E ∂r r 2 ∂r ∂θ 2 r ∂r 2 r ∂θ ( ) 2 ∂ 3φ 1 ∂ E 2ν ∂ 3 φ 2 ∂ 2φ − + − E ∂θ r 2 ∂r 2 ∂θ r 4 ∂θ 3 r 3 ∂r ∂θ ( )( ) ∂2 E 2 ∂ 2φ 2 ∂φ 1 + ν 2 ∂E ∂E − − + E E ∂r ∂θ ∂r ∂θ r 2 ∂r ∂θ r 3 ∂θ ) | ( ( )| ∂ E 2 ∂ 2φ 2 ∂ 3φ 2 ∂φ 2 ∂ 3φ 1 + ν ∂ E 2 ∂ 2φ + − − − + E ∂r r 3 ∂θ 2 r 2 ∂r ∂θ 2 ∂θ r 3 ∂r ∂θ r 2 ∂r 2 ∂θ r 4 ∂θ )( ) ( ) ( 1 ∂φ 1 ∂ E ∂ν ∂ 2ν 1 ∂ 2φ ∂ 2ν 1 ∂ 2φ 2 ∂ E ∂ν − 2 + 2 2 + − 2 2 2 + E ∂r ∂r ∂r r ∂r r ∂θ E ∂θ ∂θ ∂θ r ∂r ) ) | ( 2 |( 2 ∂ E ∂ν ∂ ν 2 ∂ φ 2 ∂φ 1 ∂ E ∂ν + − − 2 + 3 E ∂r ∂θ ∂θ ∂r ∂r ∂θ r ∂θ r ∂r ∂θ ( ) 2 ∂ 2φ 2 ∂φ ∂ν 1 ∂ 2 φ ∂ν 1 ∂ 3 φ − 3 + 4 − + =0 (1.11) ∂θ r 2 ∂r 2 ∂θ r ∂r ∂θ r ∂θ ∂r r ∂r 2 4

6

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous …

As for homogeneous materials, the first term in the governing equation for φ involves the biharmonic operator. Applying the separation of variables method, one can express the stress function as φ(r, θ ) = r λ+1 F(θ ) + r λ+2 G(θ ) + O(r λ+3 )

(1.12)

where λ is an unspecified positive parameter. The functions F(θ ) and G(θ ) are unknown at this stage. The Young’s modulus E = E(r, θ ) is assumed to be a continuously differentiable function, and it can be expressed by a Maclaurin series expansion about the crack tip position as E(r, θ ) = E 0 + r E 1 (θ ) +

r2 E 2 (θ ) + O(r 3 ) + · · · 2

(1.13)

where E 0 = E(r = 0, θ ) is the Young’s modulus at r = 0, and E 1 (θ ), E 2 (θ ) are smooth, bounded function of θ . The Maclaurin series expansion for Poisson’s ratio can be written in a similar fashion to that of Young’s modulus. In order to reduce the algebraic burden, Poisson’s ratio is henceforth assumed to be constant. Substituting Eqs. (1.12) and (1.13) into the governing Eq. (1.11) yields { L 1λ+1 (F)} r λ−3 + {L 1λ+2 (G) − E 1'' (θ )L 3λ+1 (F) + E 1 (θ )L 4λ+1 (F) + E 1' (θ )L 5λ+1 (F) − (1 + ν)[E 1' (θ )L 6λ+1 (F) − E 1 (θ )L 7λ+1 (F) − E 1' (θ )L 8λ+1 (F)]}r λ−2 + O(r λ−1 ) + · · · = 0

(1.14)

where the symbol L m λ+n ( ) indicates differential operators, which acts on the functions F and G. The explicit forms of L m λ+n ( ) is given by L 1λ+n =

d4 d2 + [(λ + n)2 + (λ + n − 2)2 ] 2 + [(λ + n)2 (λ + n − 2)2 ] 4 dθ dθ L 2λ+n = −ν L 3λ+n =

d2 + (λ + n)[1 − ν(λ + n − 1)] d θ2

L 4λ+n = [1 + 2ν(λ + n − 1)] L 5λ+n = −2

d2 + (λ + n)(λ + n − 1 − ν) dθ 2

d2 + (λ + n)[(λ + n − 1)(2 − 2λ − 2n + ν) + 1] d θ2

d d3 + 2(λ + n)[ν(λ + n − 1) − 1] 3 dθ dθ L 6λ+n = 2(λ + n − 1)

d dθ

1.1 Internal Crack

7

L 7λ+n = 2(1 − λ − n)

d2 d θ2

L 8λ+n = 2[(λ + n)(2 − λ − n) − 1]

d dθ

Equation (1.14) is satisfied only if each of the function inside the braces {} are set equal to zero, which leads to L 1λ+1 (F) = 0

(1.15)

L 1λ+2 (G) − E 1'' L 3λ+1 (F) + E 1 L 4λ+1 (F) + E 1' L 5λ+1 (F) − (1 + ν)[E 1' L 6λ+1 (F) − E 1 L 7λ+1 (F) − E 1' L 8λ+1 (F)] = 0

(1.16)

Equations (1.15) and (1.16) are both the fourth order, linear, homogeneous ordinary differential equations for the unknown functions F(θ ) and G(θ ), respectively. Here, the general solution of F(θ ) is F(θ ) = a sin(λ + 1)θ + b cos(λ + 1)θ + c sin(λ − 1)θ + d cos(λ − 1)θ (1.17) where a, b, c and d are unspecified constants. The traction-free crack surface boundary conditions which must be imposed are σθθ (r, ±π ) = σr θ (r, ±π ) = 0

(1.18)

Imposition of these conditions leads to an eigenvalue problem requiring λn = n/2 (n = 1, 2, . . . , ∞). In addition, the constants a and b are expressible in terms of c and d. The eigenfunction Fn (θ ) is given by ) | |( n ) |} { |( n − 1 (θ + π ) − cos + 1 (θ + π ) Fn (θ ) = cn cos 2 2 } |( ) | n − 2 |( n ) |} n − 1 (θ + π ) − sin + 1 (θ + π ) + dn sin 2 n+2 2

(1.19)

Similarly, a general solution G n (θ ) can be obtained for a specified elastic modulus variation according to Eq. (1.16) (Eischen 1987). Fortunately, the explicit form of G(θ ) is not required. The stress components are expressed from linear superposition of the eigensolutions for F(θ ) and G(θ ) as σrr =

∞ { ) | ) | | | (n (n E + 1 Fn + r n/2 G ''n + + 2 Gn r n/2−1 Fn'' + 2 2 n=1,2... } +O(r n/2+1 ) + · · ·

8

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous …

σθθ = σr θ =

∞ { ) )( n ) (n } E n (n + 1 Fn + r n/2 +1 + 2 G n + O(r n/2+1 ) + · · · r n/2−1 2 2 2 2 n=1,2... ∞ { E n=1,2...

) (n } n + 2 G 'n + O(r n/2+1 ) + · · · −r n/2−1 Fn' − r n/2 2 2

(1.20)

Since n takes on the values 1, 2, . . ., the leading term in the expression of the stress components immediately following the term of O(r n/2−1 ) will have r raised to the powers 1/2, 0, 1/2, . . .. The second term in the expression that follows the term of O(r n/2 ) will have r raised to the powers 1/2, 1, 3/2, . . .. It is clear that all terms of O(r n/2 ) are non-singular. Only the leading term O(r n/2−1 ) (n = 1, 2) contribute to the stress components, which are singular as r → 0 (Eischen 1987; Jin and Noda 1994). The stress components can be re-written as 1 ∂φ 1 ∂ 2φ = r λ−1 [F '' (θ ) + (λ + 1)F(θ )] + O(r λ ) + · · · + r 2 ∂θ 2 r ∂r ∂ 2φ = 2 = r λ−1 λ(λ + 1)F(θ ) + O(r λ ) + · · · ∂r 1 ∂ 2φ 1 ∂φ =− (1.21) + 2 = −r λ−1 λF ' (θ ) + O(r λ ) + · · · r ∂r ∂θ r ∂θ

σrr = σθθ σr θ

The singularity parameter λ = n/2 and the leading term O(r n/2−1 ) (n = 1, 2) are identical to those given by Williams (1957) for homogeneous materials. Therefore, the nature of the stress singularity for cracks in nonhomogeneous materials is the same as the applicability of cracks in homogeneous materials. Applying the coordinate transform relationship, the crack-tip stress can be expressed as KI I K II II σi j = √ gi j (θ ) + √ gi j (θ ) + T δ1i δ1 j + O(r 1/2 ) + · · · 2πr 2πr

(1.22)

The angular functions giIj (θ ) and giIIj (θ ) are identical to those given in Eq. (1.7). The singular and constant terms in the stress expansion series are the same as those of a crack in a homogeneous material. The material nonhomogeneity only affects the term O(r 1/2 ) and higher-order terms. Similarly, the displacement is expressed by /

/ r I r II f i (θ ) + K II f (θ ) 2π 2π i ( ) κtip + 1 κtip − 3 + T r δi1 cos θ + δi2 sin θ + O(r 3/2 ) + · · · 8μtip 8μtip

u1 = KI

(1.23)

The terms O(r 1/2 ) and O(r ) in the expansion series of the crack-tip displacement are the same as those for a crack in the homogeneous solid with material properties

1.1 Internal Crack

9

μtip and κtip , which are the same to those evaluated at the crack-tip location in a nonhomogeneous material. The expressions of the angular functions f iI (θ ) and f iII (θ ) are identical to Eq. (1.8) just by replacing the parameters μ and κ with μtip and κtip , respectively.

1.1.4 Crack-Tip Fields for Nonhomogeneous Orthotropic Materials Many materials, such as fiber reinforced composite materials, piezoelectric ceramics, etc., exhibit orthotropic properties. Analogously, for a crack in nonhomogeneous orthotropic materials, the crack-tip stresses have the inverse square-root singularity, i.e., σi j ∼ r −1/2 . For the two-dimensional orthotropic material shown in Fig. 1.3, the crack-tip asymptotic stress and displacement can be expressed as | 1 | σi j = √ K I giIj (θ ) + K II giIIj (θ ) + T δ1i δ1 j + O(r 1/2 ) 2π r / | r | K I f iI (θ ) + K II f iII (θ ) + O(r ) + · · · ui = 2π

(1.24)

(1.25)

Here, the angular functions giIj , giIIj , f iI and f iII are given by (Sih et al. 1965)

I = g22 I = g12

(

tip tip

b1 b2 tip tip ( b1 −b2 1 Re tip tip ( b1 tip−btip2 b b Re tip1 2 tip b1 −b2

I g11 = Re

)

( II , g11 = Re tip 1 tip ( b1 −b2 tip tip ) Q 1 b1 −Q 2 b2 II , g22 = Re tip 1 tip Q1 Q2 ) ( b1 −b2 Q 1 −Q 2 II , g12 = Re tip 1 tip Q1 Q2 tip

tip

Q 1 b2 −Q 2 b1 Q1 Q2

Fig. 1.3 A two-dimensional nonhomogenous orthotropic solid with an edge crack

b1 −b2

tip

tip

Q 1 (b2 )2 −Q 2 (b1 )2 Q1 Q2 Q 1 −Q 2 Q1 Q2

)

)

tip tip Q 2 b1 −Q 1 b2

Q1 Q2

)

(1.26)

10

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous …

) ) ( tip tip ( tip tip tip tip b p Q −b p Q p Q −p Q f 1I = 2Re 1 2 tip2 2tip 1 1 , f 1II = 2Re 2 tip2 tip1 1 ( tip tipb1 −b2tip tip ) ( tipb1 −b2tip ) b1 q2 Q 2 −b2 q1 Q 1 q Q −q Q I II , f 2 = 2Re 2 tip2 1tip 1 f 2 = 2Re tip tip b1 −b2

where Q k =

/

(1.27)

b1 −b2

tip

cos θ + bk sin θ , and Re(∗) and Im(∗) denote the real and imaginary tip

tip

parts of the complex variable or function, respectively. The parameters bk , pk and tip qk (k = 1, 2) are the crack-tip constants, which can be determined from the crack-tip material constants by the following operations. By defining the strain vector {εm } and stress vector {σm } according to the following relations ε1 = ε11 , ε2 = ε22 , ε3 = ε33 , ε4 = 2ε23 , ε5 = 2ε13 , ε6 = 2ε12 σ1 = σ11 , σ2 = σ22 , σ3 = σ33 , σ4 = σ23 , σ5 = σ13 , σ6 = σ12

(1.28)

the constitutive equations for a homogenous orthotropic solid can be expressed in the compliance coefficients smn as {εm } = [smn ]{σn }, smn = snm (m, n = 1, 2, . . . , 6)

(1.29)

The components of the compliance matrix are defined as { amn =

smn

smn (plane stress) (m, n = 1, 2, 6) sm3 sn3 − s33 (plane strain)

(1.30)

The material parameters bk are determined from the following equation (Sih et al. 1965) a11 b4 − 2a16 b3 + (2a12 + a66 )b2 − 2a26 b + a22 = 0

(1.31)

The roots of Eq. (1.31) are always in conjugate pairs as b1 , b1 , b2 and b2 , where bk has positive imaginary parts (i.e., Im(bk ) > 0). The parameters pk and qk are given by pk = a11 bk2 − a16 bk + a12 , qk = a12 bk − a26 + a22 /bk tip

tip

tip

(1.32)

The crack-tip constants bk , pk and qk are determined by using Eqs. (1.31) and tip tip (1.32) through replacing ai j and bi by ai j and bi , respectively.

1.2 Interface Crack

11

1.2 Interface Crack In recent years, composite materials have been applied in more and more fields. The interfaces between the components often crack due to poor toughness and therefore, the interfacial fracture becomes a key failure mode for these composite materials.

1.2.1 Crack-Tip Fields of an Interface Crack Williams (1959) rigorously solved the problem of elastic interface cracks and deduced the characteristic oscillating stress singularity, thus laying a theoretical foundation for interface fracture mechanics. Rice (1988) gave the complete form of stress field and displacement field in the vicinity of the interface crack tip, and provided the interpretation of the complex stress intensity factor (SIF). Hutchinson and Suo (1992) gave a systemic illustration of the analytical and experimental work on the interface crack in layered materials. The near-tip in-plane stress of an interface crack is given by σi j =

√ Im(Kr iε ) II Re(Kr iε ) I σi j (θ ) + √ σi j (θ ) + Tm δi1 δ j1 + O( r ) + · · · √ 2πr 2πr

(1.33)

√ where i = −1. The complex SIF K = K 1 + iK 2 with real part K 1 and imaginary part K 2 imply that the tensile and shear effects near the crack tip are intrinsically inseparable. Tm denotes the T-stress of material m (m = 1, 2). The T-stress value of the upper-half plane is different from the T-stress value of the lower-half plane. This is why the T-stress is used to be denoted by Tm . The angular functions σiIj (θ ) and σiIIj (θ ) are (Sladek and Sladek 1997) I σ11 (θ )

=

I σ22 (θ ) = I (θ ) = σ12

II σ11 (θ ) = II σ22 (θ ) = II σ12 (θ ) =

| ( ) | θ 3θ 3θ 1 3A2 − 1 cos − A sin − 2ε cos sin θ 2B A 2 2 2 | ( ) | θ 3θ 3θ 1 A2 + 1 cos + A sin − 2ε cos sin θ 2B A 2 2 2 | ( ) | θ 3θ 3θ 1 A2 − 1 sin + A cos + 2ε sin sin θ 2B A 2 2 2 | ( ) | 3A2 + 1 θ 3θ 3θ 1 − sin − A cos + 2ε sin sin θ 2B A 2 2 2 | ( ) | A2 − 1 θ 3θ 3θ 1 − sin + A cos + 2ε sin sin θ 2B A 2 2 2 | ( ) | θ 3θ 3θ 1 A2 + 1 cos − A sin − 2ε cos sin θ 2B A 2 2 2

(1.34)

(1.35)

12

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous …

where the material constants take the values } ε(θ−π) (upper-half plane) e B = cosh(π ε), A = ε(θ+π) (lower-half plane) e

(1.36)

The parameter ε is expressed by the material constants as ε=

) ( 1 κ1 μ2 + μ1 ln 2π κ2 μ1 + μ2

(1.37)

where μm are the shear moduli and κm are defined as { κm =

3−νm 1+νm

(plane stress) 3 − 4νm (plane strain)

(1.38)

The stress angular functions are shown in Fig. 1.4 for μ2 = 2μ1 and κ1 = κ2 = 2.2. It can be clearly observed that the symmetry of stress regarding the crack face is broken. The Cartesian components of the crack-tip displacements are given by the following relation

Fig. 1.4 Stress angular functions for an interface crack (μ2 = 2μ1 and κ1 = κ2 = 2.2)

1.2 Interface Crack

1 uj = 2μm

/

13

) ( | r | ( iε ) I Re Kr u j (θ ) + Im Kr iε u IIj (θ ) + O(r ) + O(r 3/2 ) + . . . 2π (1.39)

where the angular functions are given by ( ) κm A 2 − 1 θ κm A 2 + 1 θ A θ 1 cos − 2ε sin + sin sin θ = B(1 + 4ε2 ) A 2 A 2 B 2 (1.40) ( ) κm A 2 + 1 A 1 θ κm A 2 − 1 θ θ + cos sin θ u II1 (θ ) = sin + 2ε cos 2 B(1 + 4ε ) A 2 A 2 B 2 (1.41) ( ) κm A 2 + 1 θ κm A 2 − 1 θ A θ 1 sin + 2ε cos − cos sin θ u I2 (θ ) = 2 B(1 + 4ε ) A 2 A 2 B 2 (1.42) ( ) κm A 2 − 1 θ κm A 2 + 1 θ A θ 1 − cos + 2ε sin + sin sin θ u II2 (θ ) = 2 B(1 + 4ε ) A 2 A 2 B 2 (1.43) u I1 (θ )

Taking θ = 0 into account, one can express the stress ahead of crack as Kr iε σ22 (r, 0) + i σ12 (r, 0) = √ 2π r

(1.44)

In order to measure the relative proportion of shear to normal tractions at a distance r ahead of the crack tip, the phase angle ψ is defined as (Rice 1988) tan ψ =

Im(Kr iε ) Re(Kr iε )

(1.45)

And the displacement jumps ∆u i = u i (r, π ) − u i (r, −π ) across the crack faces can be expressed as ∆u 2 (r ) + i∆u 1 (r ) =

Kr iε 8 1 + 2iε E ∗ cosh(π ε)

/

r 2π

(1.46)

Here, E ∗ is the averaged elastic modulus and defined by 2 1 1 = ' + ' ∗ E E1 E2

(1.47)

14

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous …

where the in-plane modulus is E m' = E m for plane stress and E m' = E m /(1 − νm2 ) for plane strain. Please note that K 1 and K 2 play the same role as to the conventional mode I and mode II SIFs only at the crack-tip location, while they are not the familiar tensile and shear modes at other locations except for ε = 0. Due to the existence of r iε , the crack opening modes I and II always occur in couples. As a result, it is no longer possible to associate with the crack-face displacement u 2 and u 1 or the stress σ22 and σ21 ahead of the crack in a unique way to the opening and sliding modes anymore. Considering r iε = cos(ε ln r ) + i sin(ε ln r ), Rice (1988) suggested the relation K I + iK II = K rˆ iε = (K 1 + iK 2 )ˆr iε

(1.48)

to convert the relations in Eqs. (1.44) and (1.46) into the classical form, if the stress is coupled to a certain distance rˆ . This means that the mode ratio at rˆ is just the material-specific size of the crack process zone which should be smaller than the crack length and larger than the oscillation region. The energy release rage G for the interface crack is G=

K 12 + K 22 E ∗ cosh2 (π ε)

(1.49)

1.2.2 Crack-Tip Fields of an Interface Crack Between Two Nonhomogeneous Media First, the singularity of the interface cracks between two nonhomogeneous bonded half-planes will be discussed. The half plane above the crack is marked by Material 1 with Young’s modulus E 1 (x) and Poisson’s ratio ν1 (x), and the half plane below the crack is marked by Material 2 with Young’s modulus E 2 (x) and Poisson’s ratio ν2 (x), respectively. In this section, it is assumed that the properties in Material 1 (or Material 2) vary continuously. For the interface crack shown in Fig. 1.5, the stress function φm = φm (r, θ ) for generalized plane stress conditions should meet the governing equation given by ∇ 4 φm + Φm (φm , r, θ, E m , νm ) = 0 (m = 1, 2)

(1.50)

where ∇ 4 = ∇ 2 ∇ 2 and ∇ 2 is the Laplacian operator. The function Φm is given by (Eischen 1987) | Φm =

2 E m2

(

∂ Em ∂r

)2

1 ∂ 2 Em − E m ∂r 2

||

( )| ∂ 2 φm 1 ∂φm 1 ∂ 2 φm + 2 − νm ∂r 2 r ∂r r ∂θ 2

1.2 Interface Crack

15

Fig. 1.5 An interface crack located between two nonhomogeneous materials

|

|( ) ) 1 ∂φm ∂ Em 2 1 ∂ 2 φm 1 ∂ 2 Em νm ∂ 2 φm + + − − ∂θ E m ∂θ 2 r 3 ∂r r 4 ∂θ 2 r 2 ∂r 2 ) ( 1 ∂ 2 φm 1 ∂ E m 1 ∂φm 2 ∂ 2 φm ∂ 3 φm + + − − 2 E m ∂r r 2 ∂r r 3 ∂θ 2 r ∂r 2 ∂r 3 ) ( νm ∂ E m 2 ∂ 3 φm 1 ∂ 2 φm 2 ∂ 2 φm + + − 3 2 2 2 E m ∂r r ∂r ∂θ r ∂r r ∂θ 2 ( ) 2 ∂ 3 φm 1 ∂ E m 2νm ∂ 3 φm 2 ∂ 2 φm − + − E m ∂θ r 2 ∂r 2 ∂θ r 4 ∂θ 3 r 3 ∂r ∂θ ( )( ) 2 ∂ 2 φm 1 + νm 1 + νm 2 ∂ E m ∂ E m ∂ 2 Em 2 ∂φm + + − − 3 Em E m ∂r ∂θ ∂r ∂θ r 2 ∂r ∂θ r ∂θ Em | ) ( ( )| 2 3 2 3 ∂ E m 2 ∂ φm ∂ E m 2 ∂ φm 2 ∂ φm 2 ∂φm 2 ∂ φm + − 2 2 − 4 − 2 ∂r r 3 ∂θ 2 r ∂r ∂θ 2 ∂θ r 3 ∂r ∂θ r ∂r ∂θ r ∂θ )( ) ( 1 ∂φm ∂ 2 νm 1 ∂ 2 φm 2 ∂ E m ∂νm − + 2 + E m ∂r ∂r ∂r 2 r ∂r r ∂θ 2 ) ( 2 2 ∂ νm 1 ∂ φm 1 ∂ E m ∂νm − + E m ∂θ ∂θ ∂θ 2 r 2 ∂r 2 ) |( ) ( | ∂ E m ∂νm ∂ 2 νm 2 ∂φm 2 ∂ 2 φm 1 ∂ E m ∂νm + − − + E m ∂r ∂θ ∂θ ∂r ∂r ∂θ r 3 ∂θ r 2 ∂r ∂θ ( ) 2 ∂ 2 φm 2 ∂φm ∂νm 1 ∂ 2 φm ∂νm 1 ∂ 3 φm − 3 + 4 (1.51) − + 2 2 2 ∂r r ∂r ∂θ r ∂r ∂θ r ∂r ∂θ r ∂θ 2 E m2

(

For plane strain, E m and νm should be replaced with E m /(1−νm2 ) and νm /(1−νm ), respectively. In order to obtain the solution of the stress function φm , a separation of variables method is applied, and an appropriate expression of φm is selected, as

16

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous …

follows φm (r, θ ) = r λ+1 Fm (θ ) + r λ+2 G m (θ ) + r λ+3 Hm (θ ) + O(r λ+4 ) + · · ·

(1.52)

where Fm , G m and Hm are unknown functions; λ is an unspecified parameter. The elastic moduli and Poisson’s ratio can be represented by the Maclaurin series expansion about the crack-tip position (r = 0) as follows | | r2 3 1 + r E m1 (θ ) + E m2 (θ ) + O(r ) + · · · E m (r, θ ) = 2 | | 2 r 3 tip νm (r, θ ) = νm 1 + r νm1 (θ ) + νm2 (θ ) + O(r ) + · · · 2 E mtip

(1.53)

where the superscript “tip” indicates the corresponding variable located at the crack tip. By substituting Eqs. (1.52) and (1.53) into Eq. (1.50), it can be found that only the first term in Eq. (1.52) (r λ+1 Fm (θ )) contributes to the stress components, which are singular as r → 0 (Eischen 1987; Jin and Noda 1994). Here, the general solution of Fm (θ ) is Fm (θ ) = am sin(λ + 1)θ + bm cos(λ + 1)θ + cm sin(λ − 1)θ + dm cos(λ − 1)θ (1.54) where am , bm , cm and dm are unspecified constants. Therefore, the stresses can be written as (Timoshenko and Goodier 1987) 1 ∂φm 1 ∂ 2 φm = r λ−1 [Fm'' (θ ) + (λ + 1)Fm (θ )] + r 2 ∂θ 2 r ∂r + O(r λ ) + · · ·

σrr =

∂ 2 φm = r λ−1 λ(λ + 1)Fm (θ ) + O(r λ ) + · · · ∂r 2 1 ∂φm 1 ∂ 2 φm + 2 = −r λ−1 λFm' (θ ) =− r ∂r ∂θ r ∂θ + O(r λ ) + · · ·

σθθ = σr θ

(1.55)

The stress–strain relationship and the series expansion of material parameters [Eq. (1.53)] are employed to derive the strain components, and then the displacement components can be obtained as } } 4 −(λ + 1)F (θ ) + sin(λ − 1)θ + d cos(λ − 1)θ [c ] m m m tip tip 2μm 1 + νm + O(r λ+1 ) + · · ·

ur =

rλ

1.2 Interface Crack

17

} } 4 ' −Fm (θ ) − uθ = [c sin(λ − 1)θ − dm cos(λ − 1)θ ] tip tip m 2μm 1 + νm + O(r λ+1 ) + · · · rλ

(1.56)

Since the crack faces are assumed to be the traction-free, we have σθθ (r, ±π ) = σr θ (r, ±π ) = 0

(1.57)

The stress and displacement boundary conditions along θ = 0 can be expressed as σθθ (r, 0+ ) = σθθ (r, 0− ), σr θ (r, 0+ ) = σr θ (r, 0− ) u r (r, 0+ ) = u r (r, 0− ), u θ (r, 0+ ) = u θ (r, 0− )

(1.58)

After some algebraic substitutions, the nontrivial solution for the constants (am , bm , cm and dm ) exists if the parameter λ meets the following equation (Yu et al. 2010) | cot (λ π ) + 2

tip

tip

tip

2k tip (1 − p2 ) − 2(1 − p1 ) − (k tip − 1) tip

tip

2k tip (1 − p2 ) + 2(1 − p1 ) tip

tip

tip

|2 =0

(1.59)

tip

where k tip = μ1 /μ2 and pm = νm /(1 + νm ) are the material constants of the nonhomogeneous materials at the crack tip. It can be observed that Eq. (1.59) on λ is of the same form as that given by Williams (1959) for the interface crack between two homogeneous materials. In addition, the terms proportional to r λ−1 of stress components in Eq. (1.55) and those proportional to r λ of displacement components in Eq. (1.56) are of the same form as the corresponding terms of an interface crack between two homogeneous materials. Therefore, the nature of the stress singularity of the interface crack between two nonhomogeneous materials is the same as that applicable to the interface crack between two homogeneous materials. Namely, the lower-order terms (O(r −1/2 ) and O(r 0 ) for stress; O(r 1/2 ) and O(r ) for displacement) are not affected by material nonhomogeneity. The crack-tip asymptotic stress and displacement can be expressed as follows √ Re(Kr iε ) I Im(Kr iε ) II σi j = √ σi j (θ ) + √ σi j (θ ) + Tm δi1 δ j1 + O( r ) + · · · 2π r 2π r / | r | 1 Re(Kr iε )u Ij (θ ) + Im(Kr iε )u IIj (θ ) + O(r ) + · · · uj = 2μm 2π

(1.60)

(1.61)

The terms O(r −1/2 ) and O(r 0 ) in the stress and terms O(r 1/2 ) and O(r ) in the displacement are identical to those for a crack in the homogeneous solid with material tip tip properties μm and νm .

18

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous …

1.3 Three-Dimensional Curved Crack 1.3.1 Internal Crack In principle, as we have discovered in the two-dimensional case, the same near field exists at three-dimensional crack fronts. The asymptotic solution now only applies locally with respect to a point on the crack front. Therefore, an associated Cartesian coordinate system perpendicular to the crack front is introduced. As r → 0, the near-tip solution of the two-dimensional crack problem under the plane strain state can be found (Kuna 2013), as shown in Fig. 1.6. Therefore, for a curved crack front in a three-dimensional nonhomogenous material, the asymptotic stress near the crack front is given by K II (s) II K III (s) III K I (s) I σi j (r, θ, s) = √ gi j (θ ) + √ gi j (θ ) + √ gi j (θ ) 2π r 2π r 2π r + Ti j (s) + O(r 1/2 ) + · · ·

(1.62)

I In this section, the subscripts i, j = 1, 2, 3. The angular functions gαβ (θ ) and II gαβ (θ ) (α, β = 1, 2) are identical to those given in Eq. (1.7). The angular functions are given by

) ( I g11 = (1 − sin θ2 sin 3θ2 ) cos θ2 , I g22 = 1 + sin θ2 sin 3θ2 cos θ2 , I I g12 = g21 = sin θ2 cos θ2 cos 3θ2 , I(II) I(II) I(II) g33 = ν(s)(g11 + g22 ),

) ( II g11 = − 2 + cos θ2 cos 3θ2 sin θ2 θ 3θ II g22 = sin θ2 cos ) ( 2 cos θ2 II II g12 = g21 = 1 − sin 2 sin 3θ2 cos θ2 I(II) I(II) I(II) I(II) g13 = g31 = g23 = g32 =0

θ θ III III III III g13 = g31 = − sin , g23 = g32 = cos 2 2

Fig. 1.6 A closed contour in the ξ1 − ξ2 plane normal to the front of a curved crack

(1.63)

1.3 Three-Dimensional Curved Crack

19

III III III III III g11 = g22 = g33 = g12 = g21 =0

(1.64)

And the term Ti j includes all stress components of the 2nd order, representing constant finite values in the plane perpendicular to the crack front. ⎤ T11 0 T13 [Ti j ] = ⎣ 0 0 0 ⎦ T13 0 T33 ⎡

(1.65)

For the point where the crack front crosses the surface, special considerations are required, since a situation similar to the plane stress state exists, where the singularity vanishes. The asymptotic displacement near the crack front is given by / u i (r, θ, s) =

| r | K I (s) f iI (θ ) + K II (s) f iII (θ ) + K III (s) f iIII (θ ) + O(r ) + . . . 2π (1.66)

The angular functions are given by θ κ(s) − cos θ cos , 2μ(s) 2

f 2I =

θ κ(s) − cos θ sin , 2μ(s) 2

θ 2 + κ(s) + cos θ sin , 2μ(s) 2

f 2II =

θ 2 − κ(s) − cos θ cos , 2μ(s) 2

f 1I = f 1II =

f 1III = f 2III = 0,

f 3III =

f 3I = 0

(1.67)

f 3II = 0 (1.68)

θ 2 sin μ(s) 2

(1.69)

The angular functions f αI (θ ) and f αII (θ ) (α = 1, 2) are the same as given in Eq. (1.8) by replacing the material parameters with those evaluated at the point s.

1.3.2 Interface Crack For an interface crack located between two three-dimensional nonhomogenous materials, the asymptotic stress near the crack front is given by Re(Kr iε ) I Im(Kr iε ) II K III III σi j = √ σi j (θ ) + √ σi j (θ ) + √ σi j (θ ) 2π r 2π r 2π r √ + (Ti j )m + O( r ) + · · ·

(1.70)

I II The angular functions σαβ (θ ) and σαβ (θ ) (α, β = 1, 2) are the same as those given in Eqs. (1.34) and (1.35), respectively, by replacing the material parameters with those evaluated at the point s, and the other angular functions are given as

20

1 Fundamental Theory of Fracture Mechanics of Nonhomogeneous … I(II) I(II) I(II) σ33 = −νm (s)(σ11 + σ22 ) I(II) I(II) I(II) I(II) σ13 = σ31 = σ23 = σ32 =0 θ III III σ13 (θ ) = σ31 (θ ) = − sin 2 θ III III σ23 (θ ) = σ32 (θ ) = cos 2 III III III III III = σ22 = σ33 = σ12 = σ21 =0 σ11

(1.71)

The non-zero T-stress components T11 , T13 , T31 and T33 possess different values in different material regions, i.e., (Ti j )1 /= (Ti j )2 . The Cartesian components of the crack-tip displacements are /

} } r Re[K (s)r iε ] I Im[K (s)r iε ] II K III (s) III u j (θ ) + u j (θ ) + u j (θ ) 2π 2μm (s) 2μm (s) μm (s) + O(r ) + · · · (1.72)

uj =

The angular functions u Iα (θ ) and u IIα (θ ) (α = 1, 2) are the same as those given in Eqs. (1.40)–(1.43) by replacing the material parameters with those evaluated at the point s, and the other angular functions are given as III III u I3 = u II3 = u III 1 = u 2 = 0, u 3 = 2 sin

θ 2

(1.73)

References Eischen, J.W. 1987. Fracture of nonhomogeneous materials. International Journal of Fracture 34: 3–22. Gdoutos, E.E. 2005. Fracture mechanics—An introduction, 2nd ed. Dordrecht: Springer. Hutchinson, J.W., and Z. Suo. 1992. Mixed mode cracking in layered materials. Advances in Applied Mechanics 29: 63–191. Jin, Z.H., and N. Noda. 1994. Crack-tip singular fields in nonhomogeneous materials. Journal of Applied Mechanics—Transactions of the ASME 61: 738–740. Kuna, M. 2013. Finite elements in fracture mechanics: Theory-numerics-applications. In Solid mechanics and its applications, vol. 201. Springer. Rice, J.R. 1988. Elastic fracture mechanics concepts for interfacial cracks. Journal of Applied Mechanics 55: 98–103. Sih, G.C., P.C. Paris, and G.R. Irwin. 1965. On cracks in rectilinearly anisotropic bodies. International Journal of Fracture Mechanics 1: 189–203. Sladek, J., and V. Sladek. 1997. Evaluations of the T-stress for interface cracks by the boundary element method. Engineering Fracture Mechanics 56 (6): 813–825. Timoshenko, S.P., and J.N. Goodier. 1987. Theory of elasticity, 3rd ed. New York: McGraw-Hill.

References

21

Williams, M.L. 1957. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics 24: 109–114. Williams, M.L. 1959. The stresses around a fault or crack in dissimilar media. Bulletin of the Seismological Society of America 49(2): 199–204. Yu, H.J., L.Z. Wu, L.C. Guo, et al. 2010. Interaction integral method for the interfacial fracture problems of two nonhomogeneous materials. Mechanics of Materials 42: 435–450.

Chapter 2

Exponential Models for Crack Problems in Nonhomogeneous Materials

Nonhomogeneous materials, or functionally Graded Materials (FGMs), are substantially more efficient than homogeneous materials, depending on the specific application they are designed for. In particular, FGMs can greatly reduce stress concentration in the composition with those with abrupt transitions in materials composition and properties. In order to make the crack problems solvable analytically, most theoretical investigations assumed the material properties to be an exponential function. Since, when the properties are assumed to be an exponential function, the governing equations can be transformed into the common partially differential equations with constant coefficients, so that it is easy to determine analytical solution. This chapter focuses on the crack problems of FGMs by assuming the properties to be an exponential function. It provides models for nonhomogeneous materials with an arbitrarily oriented crack (Sect. 2.1), nonhomogeneous coating-substrate or double-layered structures (Sect. 2.2), cracked orthotropic nonhomogeneous plates (Sect. 2.3), and transient crack problem of a coating-substrate structure (Sect. 2.4).

2.1 Crack Model for Nonhomogeneous Materials with an Arbitrarily Oriented Crack 2.1.1 Basic Equations and Boundary Conditions Figure 2.1 shows the geometry of the infinite nonhomogeneous material plane with a generally oriented crack. According to previous work (Delale and Erdogan 1983; Erdogan and Wu 1997), the effect of the Poisson’s ratio on the SIFs is not very significant for the present problem. Therefore, the Poisson’s ratio ν is assumed a constant throughout without specificaiton. In the global coordinate (x0 O y0 ), the shear modulus of the nonhomogeneous material is defined as follows

© Science Press 2023 L. C. Guo et al., Fracture Mechanics of Nonhomogeneous Materials, https://doi.org/10.1007/978-981-19-4063-7_2

23

24

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

Fig. 2.1 The FGMs model with a generally oriented crack

μ(y0 ) = μ0 eγ y0

(2.1)

where μ0 denotes shear modulus at y0 = 0, and γ is the exponential parameter. Thus, in the local (xOy) coordinate system, the shear modulus can be expressed as μ(x, y) = μ0 eδx+βy = μ0 eγ x cos θ +γ y sin θ

(2.2)

If assuming the displacement components in the x-direction and y-direction as u(x, y) and v(x, y), the constitutive equations can be written as | | ⎧ ∂u ∂v μ(x, y) ⎪ ⎪ σx x = (1 + κ) + (3 − κ) ⎪ ⎪ κ −1 ∂x ∂y ⎪ ⎪ ⎪ | | ⎨ ∂v ∂u μ(x, y) (1 + κ) + (3 − κ) σ yy = ⎪ κ −1 ∂y ∂x ⎪ ⎪ ) ( ⎪ ⎪ ∂v ∂u ⎪ ⎪ ⎩ τx y = μ(x, y) + ∂y ∂x

(2.3)

where the Kolosov constant κ = 3 − 4ν for plane strain and κ = (3 − ν)/(1 + ν) for plane stress. The equilibrium equations in terms of the stresses are given by ⎧ ∂σx x (x, y) ∂σx y (x, y) ⎪ ⎪ + =0 ⎨ ∂x ∂y ∂σ (x, y) ∂σx y (x, y) ⎪ ⎪ ⎩ yy + =0 ∂y ∂x

(2.4)

The continuity conditions of the structure are {

σ yy (x, 0+ ) = σ yy (x, 0− ) σx y (x, 0+ ) = σx y (x, 0− )

(2.5)

2.1 Crack Model for Nonhomogeneous Materials with an Arbitrarily …

25

The following auxiliary functions defined at crack faces will be used to deal with crack problems ⎧ ∂ ⎪ [u(x, 0+ ) − u(x, 0− )] ⎨ f 1 (x) = ∂x ⎪ ⎩ f (x) = ∂ [v(x, 0+ ) − v(x, 0− )] 2 ∂x

(2.6)

2.1.2 Full Field Solution for a Crack in the Nonhomogeneous Medium Substituting Eq. (2.3) into the equilibrium Eq. (2.4) and on applying the Fourier transform about x leads to the following expressions. ⎧ ∂v(s, y) ∂ 2 u(s, y) ∂u(s, y) ⎪ ⎪ + a14 + a15 =0 ⎨ a11 u(s, y) + a12 v(s, y) + a13 ∂ y ∂y ∂ y2 2 ⎪ ⎪ ⎩ a21 u(s, y) + a22 v(s, y) + a23 ∂u(s, y) + a24 ∂v(s, y) + a25 ∂ v(s, y) = 0 ∂y ∂y ∂ y2 (2.7) where s is the Fourier variable, the coefficients ai j are given in the following forms a11 a12 a13 a14 a15

= −(1 + κ)s(s − iδ), a21 = i(κ − 1)sβ, a22 = (κ − 1)β, a23 = 2is − (κ − 3)δ, a24 = κ − 1, a25

= −i(κ − 3)sβ = −(κ − 1)s(s − iδ) = 2is + (κ − 1)δ = (κ + 1)β =κ +1

Here, Eq. (2.7) can be written as {

d11 u(s, y) + d12 v(s, y) = 0 d21 u(s, y) + d22 v(s, y) = 0

where the differential factors d11 and d22 can be expressed as

(2.8)

26

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

⎧ ∂ ∂2 ⎪ ⎪ d + a = a + a 15 11 11 13 ⎪ ⎪ ∂y ∂2 y ⎪ ⎪ ⎪ ⎪ ∂ ⎪ ⎪ ⎪ ⎨ d12 = a12 + a14 ∂ y ∂ ⎪ ⎪ ⎪ d21 = a21 + a23 ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ d22 = a22 + a24 ∂ + a25 ∂ ∂y ∂2 y

(2.9)

If we introduce a function f (s, y) with (d11 d22 − d12 d21 ) f (s, y) = 0

(2.10)

The solution of Eq. (2.8) can be written as {

u(s, y) = d22 f (s, y) v(s, y) = −d21 f (s, y)

(2.11)

From Eqs. (2.10) and (2.11), the solution of Eq. (2.7) can be written as ⎧ ( ( 4 ( 2 ) E ⎪ ∂ ∂ ⎪ ⎪ + a25 2 Ai (s)eλi y ⎪ u(s, y) = a22 + a24 ⎪ ⎨ ∂y ∂y i=1 ( )(E ( 4 ⎪ ⎪ ∂ ⎪ λ y ⎪ Ai (s)e i ⎪ v(s, y) = − a21 + a23 ⎩ ∂y i=1

(2.12)

( ( / 3−κ 2 1 δ = −β − 2 1+κ / / 1 3−κ 2 3−κ 2 ∓ δ (2is + δ) + (4s 2 + β 2 − 4isδ) + δ 2β 1+κ 1+κ 2 ( ( / 3−κ 2 1 δ = −β + 2 1+κ / / 1 3−κ 2 3−κ 2 ∓ δ (2is + δ) + (4s 2 + β 2 − 4isδ) + δ −2β 1+κ 1+κ 2

(2.13)

Here, λ1,3

λ2,4

are the roots of the following characteristic equation of Eq. (2.10) b11 + b12 λ + b13 λ2 + b14 λ3 + b15 λ4 = 0

(2.14)

2.1 Crack Model for Nonhomogeneous Materials with an Arbitrarily …

27

Here, the coefficients bi j are given in the following forms b11 = −a12 a21 + a11 a22 b12 = −a14 a21 + a13 a22 − a12 a23 + a11 a24 b13 = a15 a22 − a14 a23 + a13 a24 + a11 a25 b14 = a15 a24 + a13 a25 b15 = a15 a25

(2.15)

From Eqs. (2.12)–(2.14), the displacement and the stress can be expressed by ⎧ { ∞E 4 ⎪ 1 ⎪ ⎪ u(x, y) = E j (s)A j (s)eλ j y+isx ds ⎪ ⎪ 2π −∞ j=1 ⎨

(2.16)

{ ∞E 4 ⎪ ⎪ 1 ⎪ ⎪ G j (s)A j (s)eλ j y+isx ds ⎪ ⎩ v(x, y) = 2π −∞ j=1 ⎧ { ∞E 4 ⎪ μ0 1 ⎪ ⎪ σ B j (s)A j e(λ j +β)y+(δ+is)x ds = x x ⎪ ⎪ κ − 1 2π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ { ∞E 4 ⎨ μ0 1 C j (s)A j e(λ j +β)y+(δ+is)x ds σ yy = ⎪ κ − 1 2π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ { ∞E 4 ⎪ ⎪ 1 ⎪ ⎪ = μ D j (s)A j e(λ j +β)y+(δ+is)x ds τ ⎪ x y 0 ⎩ 2π

(2.17)

−∞ j=1

where the coefficients B j , C j , D j , E j and G j are given by E j = a22 + a24 λ j + a25 λ2j ,

G j = −a21 − a23 λ j

B j = (a22 + a24 λ j + a25 λ2j )(κ + 1)is + (−a21 − a23 λ j )λ j (3 C j = (a22 + a24 λ j + a25 λ2j )(3 − κ)is + (−a21 − a23 λ j )λ j (κ D j = (a22 + a24 λ j + a25 λ2j )λ j + (−a21 − a23 λ j )is

− κ) + 1)

(2.18)

Consider that the displacement components vanish when y approaches to infinity, and on applying the Fourier transform to Eq. (2.6), one gets ⎞ ⎛ ⎧ 2 4 E E ⎪ ⎪ ⎪ ⎪ F1 (s) = is ⎝ Ej Aj − E j Aj⎠ ⎪ ⎪ ⎨ j=1 j=3 ⎛ ⎞ ⎪ 2 4 ⎪ E E ⎪ ⎪ ⎝ ⎪ G j Aj − G j Aj⎠ ⎪ ⎩ F2 (s) = is j=1

j=3

(2.19)

28

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

where Fk (s) =

{b a

f k (x)e−isx dx (k = 1, 2)

(2.20)

Combining Eq. (2.19) with the stress continuity conditions on y = 0 in Eq. (2.5) yields ⎡

C1 ⎢ D1 ⎢ ⎣ E1 G1

C2 D2 E2 G2

−C3 −D3 −E 3 −G 3

⎧ ⎫ ⎤⎧ ⎫ 0 ⎪ A1 ⎪ −C4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎬ 0 −D4 ⎥ ⎥ A2 = 1 ⎪ F1 (s) ⎪ −E 4 ⎦⎪ A ⎪ is ⎪ ⎪ ⎪ ⎩ ⎩ 3⎪ ⎭ ⎭ −G 4 A4 F2 (s)

(2.21)

Substituting Eq. (2.21) into Eq. (2.17), the required stress can be expressed by the following two auxiliary functions {

σ yy τx y

}

μ0 1 = κ − 1 2π

{

|

∞

e −∞

isx+xδ

h 11 (s) h 12 (s) h 21 (s) h 22 (s)

|{

} F1 (s) ds F2 (s)

(2.22)

where h i j are the known functions of the characteristic roots λ j , and their expressions are determined in the derivations of Eq. (2.22). Thus, the stress can be written as ⎧ { b { b { 1 μ0 eδx ∞ ⎪ −ist ⎪ ⎪ f 1 (t)e dt + h 12 f 2 (t)e−ist dt]eisx ds [h 11 ⎨ σ yy (x, 0) = 2π κ − 1 −∞ a a { b { b { ⎪ 1 μ0 eδx ∞ ⎪ −ist ⎪ f 1 (t)e dt + h 22 f 2 (t)e−ist dt]eisx ds [h 21 ⎩ σx y (x, 0) = 2π κ − 1 −∞ a a (2.23) Considering the singularity of the existing problem, the singular integral equations can be obtained by referring to the method of Shbeeb et al. (1999a, b). As s → ∞, h i j can be expanded in asymptotic form 1κ −1 2γ sin θ (i = j ) s κ +1 4(κ − 1) 1 κ − 1 hi j → + 2γ cos θ (i /= j ) κ +1 s κ +1

hi j →

(2.24)

Now let’s introduce the stress boundary conditions on the crack faces {

σ yy (x, 0) = − p1 (x) σx y (x, 0) = − p2 (x)

(a < x < b)

the singular integral equations can be rewritten as

(2.25)

2.1 Crack Model for Nonhomogeneous Materials with an Arbitrarily …

⎧ κ + 1 −δx ⎪ ⎪ ⎪ ⎨ − 4μ e p1 (x) = 0 ⎪ κ + 1 −δx ⎪ ⎪ e p2 (x) = ⎩− 4μ0

29

| { | 1 b f 2 (t) + f 1 (t)ζ11 (x, t) + f 2 (t)ζ12 (x, t) dt π a t−x | { b| f 1 (t) 1 + f 1 (t)ζ21 (x, t) + f 2 (t)ζ22 (x, t) dt π a t−x (2.26)

where {{ { U ∞ 1 2i[−Im(h 1 j ) + id1 j ] sin[s(t − x)]ds + 2Re(h 1 j ) cos[s(t − x)]ds 2d12 0 0 } { ∞ { ∞ e1 j e1 j + 2 2[Re(h 1 j ) − ] cos[s(t − x)]ds + cos[s(t − x)]ds s s U { U { U { ∞ 1 2[−Im(h 2 j ) + id2 j ]i sin[s(t − x)]ds + 2Re(h 2 j ) cos[s(t − x)]ds ζ2 j (x, t) = 2d21 0 0 } | { ∞ | { ∞ e2 j e2 j + 2 Re(h 2 j ) − 2 cos[s(t − x)]ds + cos[s(t − x)]ds s s U U ζ1 j (x, t) =

(2.27)

(2.28)

Here, U is an arbitrary positive value. The unknown auxiliary functions f 1 (t) and f 2 (t) in Eq. (2.26) can be solved numerically.

2.1.3 Stress Intensity Factors (SIFs) and Strain Energy Release Rate (SERR) The singular integral Eq. (2.26) can be solved by the Lobatto-Chebyshev method (Theocaris and Ioakimidis 1977). The auxiliary function can be expressed as f i (t) = gi (t)w(t) (i = 1, 2)

(2.29)

where gi (t) is an unknown bounded function and w(t) is a known weight function / w(t) = 1/ 1 − t 2

(2.30)

More details can be found in Theocaris and Ioakimidis (1977). After solving the equations numerically, the SIFs K I (a), K II (a), K I (b) and K II (b) can be obtained as

30

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

√ 4 2μ0 δa g2 (a) K I (a) = lim 2(a − x)σ yy (x, 0) = e √ x→a κ +1 b−a √ / 4 2μ0 δb g2 (b) K I (b) = lim 2(x − b)σ yy (x, 0) = − e √ x→b κ +1 b−a √ / 4 2μ0 δa g1 (a) e √ K II (a) = lim 2(a − x)τx y (x, 0) = x→a κ +1 b−a √ / 4 2μ0 δb g1 (b) e √ K II (b) = lim 2(x − b)τx y (x, 0) = − x→b κ +1 b−a /

(2.31)

Furthermore, the SERRs can be written as ⎧ π(κ + 1) 2 ⎪ ⎪ K (a) ⎨ G N (a) = 8μ(a, 0) N (N = I, II) π(κ + 1) 2 ⎪ ⎪ ⎩ G N (b) = K N (b) 8μ(b, 0)

(2.32)

where G I (a) and G II (a) represent the mode I and mode II SERRs, respectively, at the crack tip x = a, G I (b) and G II (b) correspond those at the crack tip x = b. The total SERR is given by G = G I + G II . In order to transform the singular integral equations into standard form, the following normalized variables are introduced (t, x) =

b+a b−a (s, z) + 2 2

(2.33)

Then Eq. (2.26) can be rewritten as ⎧ | n | κ + 1 −δ(c z +d ) 1 E g2 (sk )wk ⎪ ⎪ 0 i 0 [− p1 (z i )] = ⎪ e + c0 g1 (sk )wk k11 (z i , sk ) + c0 g2 (sk )wk k12 (z i , sk ) ⎪ ⎪ 4μ π s − z ⎨ 0 k i k=1 | n | ⎪ κ +1 E ⎪ (s )w 1 g ⎪ 1 k k ⎪ e−δ(c0 zi +d0 ) [− p2 (z i )] = + c0 g1 (sk )wk k21 (z i , sk ) + c0 g2 (sk )wk k22 (z i , sk ) ⎪ ⎩ 4μ0 π sk − z i k=1

(2.34) where | sk = cos

| (κ − 1)π , k = 1, . . . , n n−1

π π , wj = , j = 2, . . . , n − 1 2(n − 1) n−1 | | (2i − 1)π , i = 1, . . . , n − 1 z i = cos 2n − 2

w1 = wn =

(2.35) (2.36) (2.37)

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

31

c0 =

b−a 2

(2.38)

d0 =

b+a 2

(2.39)

2.2 Crack Problems in Nonhomogeneous Coating-Substrate or Double-Layered Structures In fact, in FGMs the fracture-related failures can occur in various ways (Erdogan 1996). For the nonhomogeneous coatings bonded to homogeneous substrates, a widely observed failure mode is known to be cracking along the interface between the coating and the substrate. Therefore, it is important to study such crack problems. Therefore, firstly, the interface crack problem in nonhomogeneous coating-substrate structure is discussed in this section. On the other hand, very limited studies have been conducted on the crack problems in nonhomogeneous structures with a crack crossing the interface. Regarding this issue, mainly some numerical analyses have been conducted, please refer to Bleeck et al. (1998) and Chi and Chung (2003). Therefore, the nonhomogeneous double-layered structure with cracks crossing the interface is studied by analysis. These problems are solved by using the integral transform, singular integral equation method and residual theory. The influence of the nonhomogeneity constants and geometric parameters on the SIFs will be investigated. Especially, the variation of SIFs with the nonhomogeneity constant will be discussed when cracks move from one layer to another.

2.2.1 Interface Crack in Nonhomogeneous Coating-Substrate Structures Figure 2.2 schematically illustrates the coating-substrate structure under concentrated load and crack surface loads. The crack of length 2a is located along the interface between the homogeneous substrate (material 1) and the nonhomogeneous coating (material 2). The thicknesses of the substrate and the coating are h1 and h2 , respectively. Concentrated load p(x) is applied to the top and the bottom of the structure. Poisson’s ratio is assumed constant because it has variation affect on SIFs slightly. The shear modulus of the FGMs is defined as follows μ2 (y) = μ1 eβy

(2.40)

32

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

Fig. 2.2 Geometry of a crack between the FGMs coating and the homogeneous substrate

where μ1 is the shear modulus of the substrate (material 1), and β denotes the nonhomogeneity constant of the coating (material 2). Firstly, for the FGM coating, substituting the constitutive relation ⎧ ∂u 2 ∂v2 μ2 (x, y) ⎪ σ2x x = [(1 + κ) + (3 − κ) ] ⎪ ⎪ ⎪ κ − 1 ∂ x ∂y ⎪ ⎪ ⎨ ∂v2 ∂u 2 μ2 (x, y) [(1 + κ) + (3 − κ) ] σ2yy = ⎪ κ −1 ∂y ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ τ2x y = μ2 (x, y)( ∂u 2 + ∂v2 ) ∂y ∂x

(2.41)

into the equilibrium equation ⎧ ∂σ2x x (x, y) ∂σ2x y (x, y) ⎪ ⎪ + =0 ⎨ ∂x ∂y ∂σ (x, y) ∂σ2x y (x, y) ⎪ ⎪ ⎩ 2yy + =0 ∂y ∂x

(2.42)

and using the Fourier transform about x, then the following equations can be obtained ⎧ ∂u 2 (s, y) ∂v 2 (s, y) ∂ 2 u 2 (s, y) ⎪ ⎪ + a14 + a15 =0 ⎨ a11 u 2 (s, y) + a12 v 2 (s, y) + a13 ∂y ∂y ∂ y2 2 ⎪ ⎪ ⎩ a21 u 2 (s, y) + a22 v 2 (s, y) + a23 ∂u 2 (s, y) + a24 ∂v 2 (s, y) + a25 ∂ u 2 (s, y) = 0 ∂y ∂y ∂ y2 (2.43)

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

33

where the coefficients ai j are given by a11 a12 a13 a14 a15

= −(1 + κ)s 2 , a21 = i(−1 + κ)sβ, a22 = (κ − 1)β, a23 = 2is, a24 = κ − 1, a25

= −i(κ − 3)sβ = −(κ − 1)s 2 = 2is = (κ + 1)β =κ +1

Solving Eq. (2.43) yields the expressions of the displacement components ⎧ { ∞E 4 ⎪ 1 ⎪ ⎪ u 2 (x, y) = E 2 j (s)A2 j (s)eλ2 j y+isx ds ⎪ ⎪ 2π −∞ j=1 ⎨ { ∞E 4 ⎪ ⎪ 1 ⎪ ⎪ G 2 j (s) A2 j (s)eλ2 j y+isx ds (x, y) = v ⎪ 2 ⎩ 2π −∞ j=1

(2.44)

and the stress components ⎧ { ∞E 4 ⎪ μ1 1 ⎪ ⎪ σ2x x = B2 j (s)A2 j e(λ2 j +β)y+isx ds ⎪ ⎪ κ − 1 2π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ { ∞E 4 ⎨ μ1 1 C2 j (s)A2 j e(λ2 j +β)y+isx ds σ2yy = ⎪ κ − 1 2π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ { 4 ⎪ ⎪ μ1 ∞ E ⎪ ⎪ = D2 j (s) A2 j e(λ2 j +β)y+isx ds τ ⎪ 2x y ⎩ 2π

(2.45)

−∞ j=1

where the characteristic roots are given as /

/ 1 3−κ 1 is + β 2 λ21,23 = − β ∓ 4s 2 + 4β 1+κ 2 2 / / 1 3−κ 1 2 is + β 2 λ22,24 = − β ∓ 4s − 4β 2 2 1+κ

(2.46)

The expressions of B2 j , C2 j , D2 j , E 2 j and G 2 j are identical to those of B j , C j , D j , E j and G j in Eq. (2.18), respectively, just by replacing the parameters λ j with λ2 j . Similarly, the displacement and stress of material 1 can be obtained as follows

34

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

⎧ { ∞E 4 ⎪ 1 ⎪ ⎪ u (x, y) = E 1 j (s)A1 j (s)eλ1 j y+isx ds ⎪ 1 ⎪ 2π −∞ j=1 ⎨ { ∞E 4 ⎪ ⎪ 1 ⎪ ⎪ (x, y) = G 1 j (s)A1 j (s)eλ1 j y+isx ds v ⎪ 1 ⎩ 2π −∞ j=1 ⎧ { ∞E 4 ⎪ μ1 1 ⎪ ⎪ σ1yy = C1 j (s) A1 j eλ1 j y+isx ds ⎪ ⎪ κ − 1 2π −∞ j=1 ⎨ { ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎩ τ1x y = μ1 2π

∞

4 E

−∞ j=1

(2.47)

(2.48)

D1 j (s)A1 j eλ1 j y+isx ds

where the characteristic roots are given by λ11,12 = −|s|, λ13,14 = |s|

(2.49)

The expressions of C1 j , D1 j , E 1 j and G 1 j are as follows C11 = C13 = −4i(κ − 1)s 3 , C12,14 = 2i(κ − 1)s[±(κ + 1)|s| − 2s 2 y] D11,13 = ∓|s|3 , D12,14 = 2(3 + κ)s 2 ∓ 4|s|3 y E 11 = E 13 = 2s 2 , E 12,14 = 2[∓(κ + 1)|s| + s 2 y] G 11,13 = ±2is|s|, G 12,14 = 2is(−1 ± |s|y) The boundary and continuity conditions are given by σ1yy (x, −h 1 ) = p0 δ(x − r0 ), τ1x y (x, −h 1 ) = 0 (−∞ < x < ∞)

(2.50)

σ2yy (x, h 2 ) = p0 δ(x − r0 ), τ2x y (x, h 2 ) = 0 (−∞ < x < ∞)

(2.51)

σ1yy (x, 0− ) = σ2yy (x, 0+ ), τ1x y (x, 0− ) = τ2x y (x, 0+ )(x < −a or x > a) (2.52) σ1yy (x, 0− ) = q1 (x), τ1x y (x, 0− ) = q2 (x)(−a < x > a) u 1 (x, 0− ) = u 2 (x, 0+ ), v1 (x, 0− ) = v2 (x, 0+ )(x < −a or x > a)

(2.53) (2.54)

To obtain the integral equations, let us introduce the following auxiliary functions

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

35

⎧ ∂ ⎪ [u 2 (x, 0+ ) − u 1 (x, 0− )] ⎨ f 1 (x) = ∂x ⎪ ⎩ f (x) = ∂ [v (x, 0+ ) − v (x, 0− )] 2 1 2 ∂x

(2.55)

By solving Eqs. (2.50)–(2.55), the integral equations can be written as follows {

} |{ } { a{ ∞| p1 (x) 1 μ0 h 11 (s) h 12 (s) f 1 (t) is(x−t) e dsdt = f 2 (t) 2π κ − 1 −a −∞ h 21 (s) h 22 (s) p2 (x)

(2.56)

The expression at the left side of Eq. (2.56) is related to the load, and detailed information is given in Appendix 2A. The functions h i j (s) are the known functions determined in the derivations from Eq. (2.48). Considering the singularity of the present problem, the singular integral equations can be simplified as {

{{ } b p1 (x) 2 μ0 {ab = πκ +1 a p2 (x)

f 2 (t) dt t−x f 1 (t) dt t−x

} {b {b + a f 1 (t)ζ11 (x, t)dt + a f 2 (t)ζ12 (x, t)dt {b {b + a f 1 (t)ζ21 (x, t)dt + a f 2 (t)ζ22 (x, t)dt (2.57)

where ζ11 (x, t) =

1+κ 4(κ − 1)

{{ ∞ 0

−2iIm(h 11 ) sin[s(t − x)]ds +

{ U 0

2Re(h 11 ) cos[s(t − x)]ds

} { ∞ (κ − 1)β (κ − 1)β + [2Re(h 11 ) − ] cos[s(t − x)]ds + cos[s(t − x)]ds (1 + κ)s (1 + κ)s U U {{ ∞ κ −1 1+κ (κ − 1)(5 + κ)β [−2iIm(h 12 ) − 4 ζ12 (x, t) = ] sin[s(t − x)]ds − 4(κ − 1) 0 1+κ (1 + κ)2 s { U + 2Re(h 12 ) cos[s(t − x)]ds { ∞

{ ∞

0

(2.59)

}

t − x (κ − 1)(5 + κ)β 2Re(h 12 ) cos[s(t − x)]ds + |t − x| 2(1 + κ)2 {{ ∞ 4 1+κ β [−2iIm(h 21 ) − ζ21 (x, t) = − ] sin[s(t − x)]ds 4 1+κ (1 + κ)s 0 } { U { ∞ t−x β + 2Re(h 21 ) cos[s(t − x)]ds + 2Re(h 21 ) cos[s(t − x)]ds + 2(1 + κ) |t − x| 0 U {{ { U ∞ 1+κ −2iIm(h 22 ) sin[s(t − x)]ds + 2Re(h 22 ) cos[s(t − x)]ds ζ22 (x, t) = 4 0 0 } { ∞ { ∞ β β + [2Re(h 22 ) + ] cos[s(t − x)]ds − cos[s(t − x)]ds (1 + κ)s U (1 + κ)s U +

(2.58)

U

(2.60)

(2.61)

Applying the Lobatto-Chebyshev method (Theocaris and Ioakimidis 1977), the unknown auxiliary functions f 1 (x) and f 2 (x) in Eq. (2.57) can be solved numerically. The auxiliary functions can be expressed by an unknown bounded function gi (t) as gi (t) f i (t) = √ (i = 1, 2) 1 − t2

(2.62)

36

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

After solving the equations numerically, the SIFs can be obtained as ⎧ / 2μ0 g2 (−a) ⎪ ⎪ lim 2(−a − x)σ yy (x, 0) = √ ⎨ K I (−a) = x→−a κ +1 a / 2μ0 g2 (a) ⎪ ⎪ ⎩ K I (a) = lim 2(x − a)σ yy (x, 0) = − √ x→a κ +1 a ⎧ / 2μ0 g1 (−a) ⎪ ⎪ lim 2(−a − x)τx y (x, 0) = √ ⎨ K II (−a) = x→−a κ +1 a / g (a) 2μ ⎪ 1 0 ⎪ ⎩ K II (a) = lim 2(x − a)τx y (x, 0) = − √ x→a κ +1 a

(2.63)

(2.64)

Correspondingly, the SERRs can be written as ⎧ | π(κ + 1) | 2 ⎪ ⎪ K I (a) + K II2 (a) ⎨ G(a) = 8μ(a, 0) | π(κ + 1) | 2 ⎪ ⎪ ⎩ G(−a) = K I (−a) + K II2 (−a) 8μ(−a, 0)

(2.65)

2.2.2 Cross -Interface Crack Parallel to the Gradient of Material Properties Figure 2.3 shows a layered structure with a crack crossing the interface. Both layers may be nonhomogeneous, and the mechanical properties are continuous across the interface. The layers are infinite in the y direction. The thicknesses of two layers are h 1 and h 2 , respectively. The crack is located in either of the two layers. The shear modulus of the both FGMs layers are defined as μ1 (x) = μ01 eδ1 x (0 ≤ x ≤ h 1 )

(2.66)

μ2 (x) = μ02 eδ2 x (h 1 ≤ x ≤ h 1 + h 2 )

(2.67)

where μ01 and μ02 are shear moduli at x = 0. The parameters δ1 and δ2 are gradient constants. The shear modulus across the interface is assumed to be continuous, i.e., μ2 (h 1 ) = μ1 (h 1 )

(2.68)

μ2 (x) = μ01 e(δ1 −δ2 )h 1 eδ2 x

(2.69)

Therefore,

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

37

Fig. 2.3 Geometry of functionally graded materials with a crack crossing the interface

When δ1 = δ2 , μ1 (x) and μ2 (x) are different parts of the same function. The constitutive equations can be expressed in the displacement components u(x, y) and v(x, y) as | | ⎧ ∂u n (x, y) ∂vn (x, y) μn (x) ⎪ ⎪ σnx x (x, y) = (1 + κn ) + (3 − κn ) ⎪ ⎪ κn − 1 ∂x ∂y ⎪ ⎪ ⎪ | | ⎨ ∂vn (x, y) ∂u n (x, y) μn (x) (1 + κn ) σnyy (x, y) = (n = 1, 2) + (3 − κn ) ⎪ κn − 1 ∂y ∂x ⎪ ⎪ | | ⎪ ⎪ ∂u n (x, y) ∂vn (x, y) ⎪ ⎪ ⎩ σnx y (x, y) = μn (x) + ∂y ∂x (2.70) The equilibrium equations are ⎧ ∂σnx x (x, y) ∂σnx y (x, y) ⎪ ⎪ + =0 ⎨ ∂x ∂y ⎪ ∂σnyy (x, y) ∂σnx y (x, y) ⎪ ⎩ + =0 ∂y ∂x

(2.71)

The boundary and continuity conditions are given by σ1x x (0, y) = 0, σ1x y (0, y) = 0

(2.72)

σ1x x (h 1 , y) = σ2x x (h 1 , y), σ1x y (h 1 , y) = σ2x y (h 1 , y)

(2.73)

v1 (h 1 , y) = v2 (h 1 , y), u 1 (h 1 , y) = u 2 (h 1 , y)

(2.74)

σ2x x (h 1 + h 2 , y) = 0, σ2x y (h 1 + h 2 , y) = 0

(2.75)

38

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

v1 (x, 0) = 0, when 0 ≤ x ≤ h 1 and x ∈ / (a, b)

(2.76)

v2 (x, 0) = 0, when h 1 ≤ x ≤ h 1 + h 2 and x ∈ / (a, b)

(2.77)

σ1x y (x, 0) = 0, 0 ≤ x ≤ h 1

(2.78)

σ2x y (x, 0) = 0, h 1 ≤ x ≤ h 1 + h 2

(2.79)

σnyy (x, 0) = −σ0 (x), when a < x < b

(2.80)

Equation (2.72) is a consequence of the assumption that the crack faces are traction-free. Equations (2.73) and (2.74) are the continuity conditions of stress and displacement components. Equations (2.76)–(2.79) follow the symmetry of the geometry, and the loading in Fig. 2.3. Equation (2.80) is the loading condition on the crack face. The function σ0 (x) is the crack face traction. Please note that any layer may contain part of the crack, please refer to Eq. (2.80), n = 1 for the cracked part located in the Layer 1 and n = 2 for the cracked part located in the Layer 2. For the symmetry of the structure depicted in Fig. 2.3, it is enough to only analyze the left half part (y < 0). Considering this point, substituting the constitutive Eq. (2.70) into the governing Eq. (2.71) and using Fourier transform method leads to the following expressions for the displacement components. ⎧ { ∞E 2 ⎪ 1 ⎪ ⎪ ⎪ u n (x, y) = E n1 j (s) An1 j (s)eλn1 j (s)y−isx ds ⎪ ⎪ 2π ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ { ∞E ⎪ 4 ⎪ ⎪ 2 ⎪ ⎪ E n2 j (α)An2 j (α)eλn2 j (α)y cos(αy)dα + ⎪ ⎪ π 0 j=1 ⎨ { ∞E 2 ⎪ ⎪ 1 ⎪ ⎪ (x, y) = Fn1 j (s)An1 j (s)eλn1 j (s)y−isx ds v ⎪ n ⎪ ⎪ 2π −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ { 4 ⎪ ⎪ 2 ∞E ⎪ ⎪ ⎪ Fn2 j (α)An2 j (α)eλn2 j (α)y sin(αy)dα + ⎪ ⎩ π 0

(2.81)

j=1

where E n1 j (s), Fn1 j (s), E n2 j (α) and Fn2 j (α) are intermediate parameters depending on the Fourier variables s and α (Guo and Noda 2008). An1 j (s) and An2 j (α) are unknown quantities to be solved. At first, substituting the constitutive Eq. (2.70) into Eq. (2.71), we can obtain two partially differential equations (PDEs) regarding u n (x, y) and vn (x, y). In order to solve these PDEs, Fourier and inverse Fourier transforms are applied. The solutions to u n (x, y) and vn (x, y) include two parts. The procedures are: ➀ Applying Fourier transform to the x variable, the two PDEs

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

39

become two ordinary differential equations (ODEs) regarding y in the Fourier field s. It is easy to solve the ODEs and obtain the displacement components u n (s, y) and v n (s, y) in the Fourier field s. An1 j (s) are unknowns in u n (s, y) and v n (s, y). Applying inverse Fourier transform to u n (s, y) and v n (s, y), we have the first part (the first integral) of the displacement expression (2.81); ➁ Similarly, applying the Fourier transform to y variable in the PDEs, the corresponding transform variable is denoted by α. It is easy to obtain the displacement components u n (x, α) and v n (x, α) in the Fourier field α. Then applying inverse Fourier cosine transform to u n (x, α) and v n (x, α), we have the second part (the second integral) of the expression (2.81). λn1 j (s) and λn2 j (α) are the roots of the characteristic equation of the ODEs. Then, substituting Eq. (2.81) into the constitutive Eq. (2.70), the stress components are obtained as ⎡ ⎧ { ∞E 2 ⎪ ⎪ ⎪ δn x ⎣ 1 ⎪ σ (x, y) = e Bn1 j (s)An1 j (s)eλn1 j (s)y−isx ds ⎪ nx x ⎪ 2π ⎪ −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎤ ⎪ ⎪ { ∞E 4 ⎪ ⎪ 2 ⎪ ⎪ + Bn2 j (α)An2 j (α)eλn2 j (α)x cos(αy)dα ⎦ ⎪ ⎪ ⎪ π 0 ⎪ j=1 ⎪ ⎪ ⎪ ⎡ ⎪ ⎪ { ∞E 2 ⎪ ⎪ ⎪ δn x ⎣ 1 ⎪ ⎪ σ Cn1 j (s)An1 j (s)eλn1 j (s)y−isx ds (x, y) = e ⎪ ⎪ nyy 2π −∞ j=1 ⎨ ⎤ (2.82) { ∞E ⎪ 4 ⎪ ⎪ 2 ⎪ ⎪ + Cn2 j (α) An2 j (α)eλn2 j (α)x cos(αy)dα ⎦ ⎪ ⎪ π ⎪ 0 ⎪ j=1 ⎪ ⎪ ⎪ ⎡ ⎪ ⎪ { ∞E 2 ⎪ ⎪ ⎪ ⎪ σnx y (x, y) = eδn x ⎣ 1 Dn1 j (s) An1 j (s)eλn1 j (s)y−isx ds ⎪ ⎪ ⎪ 2π −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎤ ⎪ ⎪ { ∞E 4 ⎪ ⎪ 2 ⎪ ⎪ ⎪ + Dn2 j (α)An2 j (α)eλn2 j (α)x sin(αy)dα ⎦ ⎪ ⎩ π 0 j=1 where the known expressions Bn1 j (s), Cn1 j (s), Dn1 j (s), Bn2 j (α), Cn2 j (α) and Dn2 j (α) are intermediate quantities, and λn1 j (s) and λn2 j (α) are the characteristic roots of the ODEs (Guo and Noda 2008). Now, we introduce the auxiliary function of the following form f (x) =

{ ∂v1 (x,0)

∂x ∂v2 (x,0) ∂x

(0 < x < h 1 ) (h 1 ≤ x < h 1 + h 2 )

(2.83)

According to the crack-face boundary conditions in Eqs. (2.76) and (2.77), we have

40

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

f (x) = 0, x ∈ / (a, b) {

b

(2.84)

f (x)dx = 0

(2.85)

a

Substituting the displacement components of Eq. (2.81) into Eq. (2.83), and considering the crack-face boundary conditions in Eqs. (2.84) and (2.85), and applying Fourier transform to Eq. (2.83) yields −is

2 E

{ Fn1 j (s)An1 j (s) =

b

f (u)eisu du

(2.86)

a

j=1

Using the conditions (2.78) and (2.79), we have 2 E

Dn1 j (s)An1 j (s) = 0

(2.87)

j=1

Then, the parameter An1 j (s) is obtained as { An1 j (s) = qn j (s)

b

f (u)eisu du(n = 1, 2; j = 1, 2)

(2.88)

a

where qn j (s) is the solved coefficients from Eqs. (2.86) and (2.87). According to the boundary conditions as given in Eqs. (2.72) and (2.75), we have {

b

[X 1 (α)]{ A12 (α)} =

{ P 0 (u, α)} f (u)du

(2.89)

{ P 1 (u, α)} f (u)du

(2.90)

a

{ [X 2 (α)]{ A22 (α)} =

b

a

where | [X 1 (α)] = { A12 (α)} =

|

| X 2 (α) =

|

{

B121 (α) B122 (α) B123 (α) B124 (α) D121 (α) D122 (α) D123 (α) D124 (α) A121 (α) A122 (α) A123 (α) A124 (α)

}T { { P 0 (u, α)} = R01 (u, α) R02 (u, α)

|

}T

(2.91) (2.92) (2.93)

B221 (α)eλ221 (h 1 +h 2 ) B222 (α)eλ222 (h 1 +h 2 ) B223 (α)eλ223 (h 1 +h 2 ) B224 (α)eλ224 (h 1 +h 2 ) D221 (α)eλ221 (h 1 +h 2 ) D222 (α)eλ222 (h 1 +h 2 ) D223 (α)eλ223 (h 1 +h 2 ) D224 (α)eλ224 (h 1 +h 2 )

|

(2.94)

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

{ A22 (α)} =

{

A221 (α) A222 (α) A223 (α) A224 (α)

41

}T

(2.95)

}T { { P 1 (u, α)} = R03 (u, α) R04 (u, α)

(2.96)

Also from the stress continuity conditions (2.73), we have the following equations [Y 1 (α)]{ A12 (α)} − e(δ2 −δ1 )h 1 [Y 2 (α)]{ A22 (α)} } { b{ R11 (u, α) − e(δ2 −δ1 )h 1 R21 (u, α) = g(u)du R12 (u, α) − e(δ2 −δ1 )h 1 R22 (u, α) a

(2.97)

in which | [Y n (α)] =

Bn21 (α)eλn21 h 1 Bn22 (α)eλn22 h 1 Bn23 (α)eλn23 h 1 Bn24 (α)eλn24 h 1 Dn21 (α)eλn21 h 1 Dn22 (α)eλn22 h 1 Dn23 (α)eλn23 h 1 Dn24 (α)eλn24 h 1

| (2.98)

Using the displacement continuity conditions (2.74), the following equation can be obtained } { b{ R13 (u, α) − R23 (u, α) f (u)du [Z 1 (α)]{ A12 (α)} − [Z 2 (α)]{ A22 (α)} = R14 (u, α) − R24 (u, α) a (2.99) where | [Z n (α)] =

E n21 (α)eλn21 h 1 E n22 (α)eλn22 h 1 E n23 (α)eλn23 h 1 E n24 (α)eλn24 h 1 Fn21 (α)eλn21 h 1 Fn22 (α)eλn22 h 1 Fn23 (α)eλn23 h 1 Fn24 (α)eλn24 h 1

|

On applying the theory of residues, Rni (u, α) (n = 0, 1, 2; i = 1, 2, 3, 4) listed in Guo and Noda (2008). Through the above procedures, then Eqs. (2.97) and (2.99) can be written as | |{ } { b −Z 2 (α) Z 1 (α) A12 (α) { P 2 (u, α)} f (u)du = (2.100) Y 1 (α) −e(δ2 −δ1 )h 1 Y 2 (α) A22 (α) a Here, ⎧ ⎪ ⎪ ⎨

⎫ R13 (u, α) − R23 (u, α) ⎪ ⎪ ⎬ R14 (u, α) − R24 (u, α) { P 2 (u, α)} = ⎪ R (u, α) − e(δ2 −δ1 )h 1 R21 (u, α) ⎪ ⎪ ⎪ ⎩ 11 ⎭ R12 (u, α) − e(δ2 −δ1 )h 1 R22 (u, α) Thus, relating Eqs. (2.89), (2.100) and (2.90), we have

(2.101)

42

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

⎤ ⎧ ⎫ 0 X 1 (α) { } { b ⎨ P 0 (u, α) ⎬ ⎥ ⎢ Z 1 (α) −Z (α) (α) A 2 12 ⎥ ⎢ = f (u)du P 2 (u, α) ⎣ Y 1 (α) −e(δ2 −δ1 )h 1 Y 2 (α) ⎦ A22 (α) 8×1 a ⎩ P (u, α) ⎭ 1 8×1 0 X 2 (α) 8×8 (2.102) ⎡

where the unknown An2 j (α) can be obtained by solving the linear Eq. (2.102). Next, substituting An1 j (s) and An2 j (α) into the crack-face boundary condition in Eq. (2.80) that yields a singular integral equation regarding the unknown auxiliary function f (u). First, using the stress expression in Eq. (2.82), and the crack-face boundary condition in Eq. (2.80), we obtain the following relation on the crack face 1 2π 2 + π

{

∞

2 E

−∞ j=1

{

∞ 0

4 E

Cn1 j (s)An1 j (s)eλn1 j (s)y−isx ds (2.103)

Cn2 j (α)An2 j (α)eλn2 j (α)x cos(αy)dα = −σ0 (x)e−δn x

j=1

Substituting An1 j (s) in Eq. (2.88) and An2 j (α) in Eq. (2.102) into Eq. (2.103) and then defining |

| P˜n (α, x) = [ Cn21 (α)eλn21 x Cn22 (α)eλn22 x Cn23 (α)eλn23 x Cn24 (α)eλn24 x ] (2.104) 2 E

Cn1 j (s)qn j (s)eλn1 j (s)y (n = 1, 2)

(2.105)

| | K n2 (u, x, α) = P˜n (α, x) {Sn (u, α)} (n = 1, 2)

(2.106)

K n1 (y, s) =

j=1

{

S1 (u, α) S2 (u, α)

}

⎤−1 ⎧ ⎫ 0 X 1 (α) ⎨ P 0 (u, α) ⎬ ⎥ ⎢ Z 1 (α) −Z (α) 2 ⎥ =⎢ ⎣ Y 1 (α) −e(δ2 −δ1 )h 1 Y 2 (α) ⎦ ⎩ P 2 (u, α) ⎭ P 1 (u, α) 0 X 2 (α) ⎡

(2.107)

Thus using the relation in Eq. (2.103) reduces to {

b

[h n1 (u, x) + h n2 (u, x)] f (u)du = −eδn x σ0 (x) (a < x < b; n = 1, 2)

a

(2.108) where

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

43

{ ∞ ⎧ 1 ⎪ ⎪ h (u, x) = lim K n1 (y, s)eis(u−x) ds ⎨ n1 y→0 2π −∞ { ⎪ 2 ∞ ⎪ ⎩ h n2 (u, x) = lim K n2 (u, x, α) cos(αy)dα y→0 π 0

(2.109)

Considering the singularity of Eq. (2.108), an asymptotic analysis is conducted. As s → ∞, the asymptotic form of K n1 (y = 0, s) can be obtained as K n1∞ (0, s) = wn11 i + wn12

1 (n = 1, 2) s

(2.110)

where wn11 =

4μ0n |s| −2μ0n δn , wn12 = 1 + κn s 1 + κn

(2.111)

Finally, the singular integral equation is obtained as 1 π

{ a

b

|

| wn11 + kn1 (u, x) + πh n2 (u, x) f (u)du = −eδn x σ0 (x) x −u

(2.112)

where { kn1 (u, x) =

+∞

−∞

[K n1 (0, s) − K n1∞ (0, s)]eis(u−x) ds

(2.113)

The singular integral Eq. (2.112) will be solved by the numerical methods of Erdogan and Gupta (1972) and Shbeeb et al. (1999a, b). In order to simplify Eq. (2.112) into a standard form, the intervals are normalized by setting u=

b+a b−a b+a b−a r+ ,x= s+ 2 2 2 2

(2.114)

The solution of Eq. (2.112) may be written as g(r ) f (r ) = √ 1 − r2

(2.115)

The SIF for the crack can be defined as K I (a) = lim

/

x→a

K I (b) = lim

x→b

4μ0n δn a 2(a − x)σ yy (x, 0) = − e 1 + κn

/

4μ0n δn b 2(x − b)σ yy (x, 0) = e 1 + κn

/ /

b−a g(−1) 2

(2.116)

b−a g(1) 2

(2.117)

44

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

2.2.3 Arbitrarily Oriented Crack in a Double-Layered Structure The geometry of the arbitrarily oriented crack problem in a double-layered structure is shown in Fig. 2.4. The crack may cross the interface, or it may be completely located in either Layer 1 or Layer 2. h 1 and h 2 denote the thicknesses of both layers, respectively. The origin of the global coordinate system x O y is located on the upper surface of Layer 1. The total thickness is represented by h = h 1 + h 2 . The origin of the local coordinate system x0 O0 y0 is denoted by O0 with the coordinates (xc , yc ) in the global coordinate system x O y. Without loss of generality, the x0 axis of the coordinate system x0 O0 y0 can be arranged along the crack line. In the local coordinate system x0 O0 y0 , the x0 coordinates of both crack tips are a0 and b0 , respectively. As shown in Fig. 2.4, both the layers may be nonhomogeneous, and the mechanical properties remain continuous at the interface, but their derivatives may be discontinuous. Therefore, the interface can correspond to the kink line of the mechanical property distributions. Applying the superposition principle, as shown in Fig. 2.4, the stress field can be expressed as the sum of two states [corresponding to Fig. 2.5b, c] of stress sets σni(1)j (x0 , y0 ) and σni(2)j (x, y) (n = 1, 2; i, j = x, y or x0 , y0 ). The first state is associated with the FGMs structure with one crack as shown in Fig. 2.5b, and the stress is denoted by σni(1)j (x0 , y0 )(n = 1, 2). The second state is associated with the FGMs structure without the crack as shown in Fig. 2.5c, and the stress is denoted by σni(2)j (x, y) (n = 1, 2). Thus, the total stresses σni(1+2) (x, y) in the global coordinate j system xOy can be written as

Fig. 2.4 Geometry of a functionally graded layered structure with an arbitrarily oriented crack crossing the interface

2.2 Crack Problems in Nonhomogeneous Coating-Substrate … ⎫ ⎧ (2) ⎫ ⎡ ⎧ (1+2) ⎪ cos2 θ ⎨ σnx x (x, y) ⎪ ⎬ ⎪ ⎬ ⎨ σnx x (x, y) ⎪ ⎢ (1+2) (2) σnyy (x, y) = σnyy (x, y) + ⎣ sin2 θ ⎪ ⎪ ⎪ ⎩ (2) ⎭ ⎪ ⎭ ⎩ (1+2) sin θ cos θ σnx y (x, y) σnx y (x, y)

45

⎫ ⎤⎧ (1) ⎪ ⎪ σnx0 x0 (x0 , y0 ) ⎪ ⎪ sin2 θ −2 sin θ cos θ ⎨ ⎬ ⎥ (1) 2 sin θ cos θ ⎦ σny0 y0 (x0 , y0 ) (n = 1, 2) cos2 θ ⎪ ⎪ ⎪ (1) ⎪ ⎭ − sin θ cos θ cos2 θ − sin2 θ ⎩ σnx y (x 0 , y0 ) 0 0

(2.118)

Similarly, in the local coordinate system x0 O0 y0 , the total stress σni(1+2) (x0 , y0 ) of j the general problem can be expressed as ⎫ ⎡ ⎧ (1+2) ⎪ ⎪ ⎪ cos2 θ ⎬ ⎨ σnx0 x0 (x0 , y0 ) ⎪ ⎢ (1+2) σny0 y0 (x0 , y0 ) = ⎣ sin2 θ ⎪ ⎪ ⎪ ⎭ ⎩ σ (1+2) (x , y ) ⎪ − sin θ cos θ nx y 0 0 0 0

⎫ ⎫ ⎧ (1) ⎤⎧ (2) ⎪ σnx x (x0 , y0 ) ⎪ ⎪ ⎪ sin2 θ 2 sin θ cos θ ⎨ ⎬ ⎬ ⎪ ⎨ σnx x (x, y) ⎪ 0 0 ⎥ (1) (2) −2 sin θ cos θ ⎦ σnyy cos2 θ (x, y) + σny0 y0 (x0 , y0 ) (n = 1, 2) ⎪ ⎪ ⎪ ⎪ (2) ⎪ ⎭ ⎩ ⎩ σ (1) (x , y ) ⎪ ⎭ sin θ cos θ cos2 θ − sin2 θ σnx y (x, y) nx0 y0 0 0

(2.119)

where {

x − xc y − yc

}

| =

cos θ − sin θ sin θ cos θ

|{

x0 y0

} (2.120)

Let u (1+2) (x, y) and vn(1+2) (x, y) (n = 1, 2) be the total displacement component n in x and y-direction, respectively. The boundary and continuity conditions can be expressed as

Fig. 2.5 Schematic of the principle of superposition for the present problem

46

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

{

{

{

{

(1+2) σ1x x (0, y) = 0 (1+2) σ1x y (0, y) = 0

(−∞ < y < ∞)

(1+2) σ2x x (h 1 + h 2 , y) = 0 (1+2) σ2x y (h 1 + h 2 , y) = 0

(−∞ < y < ∞)

(1+2) (1+2) σ1x x (h 1 , y) = σ2x x (h 1 , y) (1+2) (1+2) σ1x y (h 1 , y) = σ2x y (h 1 , y)

u (1+2) (h 1 , y) = u (1+2) (h 1 , y) 1 2 v1(1+2) (h 1 , y) = v2(1+2) (h 1 , y) { (1+2) σny0 y0 (x0 , 0) = p1 (x0 ) (1+2) (x0 , 0) = p2 (x0 ) σnx 0 y0

(2.121)

(2.122)

(−∞ < y < ∞)

(2.123)

(−∞ < y < ∞)

(2.124)

(a0 < x < b0 )

(2.125)

Equations (2.121) and (2.122) indicate that both surfaces of the structure are traction-free. Equations (2.123) and (2.124) denote the stress continuity conditions and the displacement continuity conditions at the interface, Eq. (2.125) represents the load conditions of both crack faces. In the global coordinate system x O y, the shear moduli of both nonhomogeneous layers are defined by μ1 (x) = μ01 eδ1 x 0 ≤ x ≤ h 1

(2.126)

μ2 (x) = μ02 eδ2 x h 1 ≤ x ≤ h 1 + h 2

(2.127)

Thus, in the local coordinate system x0 O0 y0 , the shear moduli of both layers can be written as μ1 (x0 , y0 ) = μ01 eδ1 x = μ01 eβ1 x0 +γ1 y0 +δ1 xc

(2.128)

μ2 (x0 , y0 ) = μ02 eδ2 x = μ02 eβ2 x0 +γ2 y0 +δ2 xc

(2.129)

where β1 = δ1 cos θ , γ1 = −δ1 sin θ , β2 = δ2 cos θ , and γ2 = −δ2 sin θ . If the material properties at the interface are continuous but their derivatives are discontinuous, then we have { μ2 (h 1 ) = μ1 (h 1 ) (2.130) δ1 /= δ 2 Applying Eqs. (2.126) and (2.127) to Eq. (2.130) yields

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

47

μ02 = μ01 e(δ1 −δ2 )h 1 (1)

(2.131)

Stress and Displacement Fields for the First Stress State

In Fig. 2.5b, the displacement components in the x 0 and y0 -direction are defined as (1) u (1) n (x 0 , y0 ) and vn (x 0 , y0 ) (n = 1, 2). By using Hooke’s law, the relation of strain and stress can be expressed as | | ⎧ ∂u (1) ∂vn(1) (x0 , y0 ) μn (x0 , y0 ) n (x 0 , y0 ) (1) ⎪ ⎪ σ (1 + κ) + (3 − κ) (x , y ) = ⎪ nx0 x0 0 0 ⎪ κ −1 ∂ x0 ∂ y0 ⎪ ⎪ ⎪ | | ⎨ (1) (1) ∂vn (x0 , y0 ) μn (x0 , y0 ) ∂u n (x0 , y0 ) (1) (1 + κ) (x , y ) = + (3 − κ) σny 0 0 0 y0 ⎪ κ −1 ∂ y0 ∂x0 ⎪ ⎪ | | (1) ⎪ ⎪ (1) ⎪ ∂u n (x0 , y0 ) ∂vn (x0 , y0 ) ⎪ (1) ⎩ σnx (x0 , y0 ) = μn (x0 , y0 ) + 0 y0 ∂ y0 ∂ x0 (2.132) And the governing equations are ⎧ (1) (1) (x0 , y0 ) ∂σnx0 x0 (x0 , y 0 ) ∂σnx ⎪ 0 y0 ⎪ ⎪ + =0 ⎨ ∂ x0 ∂ y0 (n = 1, 2) (1) (1) ⎪ ∂σ (x , y ) (x , y ) ∂σ 0 0 0 0 ⎪ y y ny nx 0 0 0 0 ⎪ ⎩ + =0 ∂ y0 ∂ x0

(2.133)

Substituting the stress-displacement relation [Eq. (2.132)] into the governing Eq. (2.133), two partial differential equations (PDEs) about u (1) n (x 0 , y0 ) and vn(1) (x0 , y0 ) can be derived. The Fourier transform is applied to variable x0 , and the two PDEs are transformed into two ordinary differential equations (ODEs) regarding y0 in the Fourier domain. It is easy to solve the ODEs and obtain the displacement components in the Fourier domain. Finally, by using inverse Fourier transform, the expressions of the displacement components can be obtained as ⎧ { ∞E 4 ⎪ 1 ⎪ λ(1) (1) n j (s)y0 −isx 0 ds ⎪ u E n(1)j (s) A(1) (x , y ) = ⎪ 0 0 n n j (s)e ⎪ 2π −∞ j=1 ⎨ { ∞E 4 ⎪ ⎪ 1 ⎪ (1) λ(1) (1) ⎪ n j (s)y0 −isx 0 ds Fn(1) (x , y ) = v ⎪ 0 0 j (s) An j (s)e ⎩ n 2π −∞ j=1

(n = 1, 2)

(2.134)

where s is the Fourier variable, λ(1) n j (s) (n = 1, 2; j = 1, 2, 3, 4) is characteristic root (1) (1) of ODE, and E n j (s) and Fn j (s) are intermediate parameters, which can be solved directly from the above equations (Guo et al. 2014). Without loss of generality, the (1) above-mentioned characteristic roots can be arranged as Re(λ(1) n1 ) < 0, Re(λn2 ) < 0, (1) (1) (1) Re(λn3 ) > 0 and Re(λn4 ) > 0. Thus, the unknown functions An j (s) (n = 1, 2;

48

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

j = 1, 2) are equal to zero when y0 < 0, and A(1) n j (s) (n = 1, 2; j = 3, 4) are equal to zero when y0 > 0. Substituting Eq. (2.134) into the constitutive Eq. (2.132), the stress components σni(1)j (x0 , y0 ) can be derived as ⎧ { ∞E 4 ⎪ (1) λ(1) (1) βn x0 +γn y0 +δn xc 1 ⎪ n j (s)y0 −isx 0 ds ⎪ σ (x , y ) = e Bn(1) 0 0 ⎪ nx0 x0 j (s)An j (s)e ⎪ 2π ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ { ∞E 4 ⎨ (1) λ(1) (1) βn x0 +γn y0 +δn xc 1 n j (s)y0 −isx 0 ds Cn(1) (x , y ) = e σny (n = 1, 2) 0 0 j (s)An j (s)e 0 y0 ⎪ 2π −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ { ∞E 4 ⎪ ⎪ ⎪ λ(1) (1) βn x0 +γn y0 +δn xc 1 ⎪ n j (s)y0 −isx 0 ds Dn(1)j (s)A(1) (x , y ) = e σ ⎪ n j (s)e ⎩ nx0 y0 0 0 2π −∞ j=1

(2.135) (1) (1) The coefficients Bn(1) j (s), C n j (s) and Dn j (s) (n = 1, 2; j = 1, 2, 3, 4) are the intermediate parameters that can be solved by the constitutive equation (Guo et al. 2014). Considering the stress continuity condition at y0 = 0 in Fig. 2.5b, we have

⎧ (1) (1) ⎪ σny (x0 , y0 ) = lim− σny (x0 , y0 ) ⎨ y lim 0 y0 0 y0 →0+ y →0 0

0

(1) (1) ⎪ ⎩ lim+ σnx0 y0 (x0 , y0 ) = lim− σnx0 y0 (x0 , y0 ) y0 →0

(2.136)

y0 →0

Substituting Eq. (2.135) into Eq. (2.136) gives {

(1) (1) (1) (1) (1) (1) (1) (1) Cn1 An1 + Cn2 An2 = Cn3 An3 + Cn4 An4 (1) (1) (1) (1) (1) (1) (1) (1) Dn1 An1 + Dn2 An2 = Dn3 An3 + Dn4 An4

(2.137)

In order to determine the unknown functions A(1) n j (s) (n = 1, 2; j = 1, 2, 3, 4), the following new auxiliary functions are introduced: ⎧ ∂ ⎪ − ⎪ [u (1) (x0 , y0 = 0+ ) − u (1) ⎨ f 1 (x0 ) = n (x 0 , y0 = 0 )] ∂ x0 n ∂ ⎪ ⎪ ⎩ f 2 (x0 ) = [v (1) (x0 , y0 = 0+ ) − vn(1) (x0 , y0 = 0− )] ∂ x0 n

(2.138)

From the definitions of f 1 (x0 ) and f 2 (x0 ), we can easily conclude that f 1 (x0 ) = f 2 (x0 ) = 0 when x0 < a0 or x0 > b0 , and {

b0

a0

f i (x0 )dx0 = 0 (i = 1, 2)

(2.139)

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

49

Applying Eqs. (2.139) and (2.138), then the following equations can be obtained {

(1) (1) (1) (1) (1) (1) (1) (1) −is(E n1 An1 + E n2 An2 − E n3 An3 − E n4 An4 ) = F1 (s) (1) (1) (1) (1) (1) (1) (1) (1) An1 + Fn2 An2 − Fn3 An3 − Fn4 An4 ) = F2 (s) −is(Fn1

(n = 1, 2)

(2.140)

where { F1 (s) =

b0

{ f 1 (x0 )e

isx0

dx0 , F2 (s) =

a0

b0

f 2 (x0 )eisx0 dx0

a0

Solving Eqs. (2.137) and (2.140), the unknown functions A(1) n j (s) (n = 1, 2; j = 1, . . . , 4) can be easily determined as ⎧ ⎫ ⎪ ⎪ ⎡ A(1) ⎪ n1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ A(1) ⎪ ⎬ ⎢ ⎢ n2 =⎢ (1) ⎪ ⎪ ⎪ An3 ⎪ ⎪ ⎣ ⎪ ⎪ ⎪ ⎪ ⎩ (1) ⎪ ⎭ An4

Wn11 Wn Wn21 Wn Wn31 Wn Wn41 Wn

Wn12 Wn Wn22 Wn Wn32 Wn Wn42 Wn

⎤

{ } ⎥ F (s) ⎥ 1 (n = 1, 2) ⎥ ⎦ F2 (s)

(2.141)

The expressions of Wn (n = 1, 2) and Wn jk (n = 1, 2; k = 1, 2; j = 1, 2, 3, 4) can be obtained by solving Eqs. (2.137) and (2.140) that are described by Guo et al. (2014). (2) Stress and Displacement Fields for the Second Stress State If the displacement components in the x-direction and y-direction are defined as (2) u (2) n (x, y) and vn (x, y), the stress–strain relationship of the uncracked elastic medium in Fig. 2.5c can be expressed as { } ⎧ ∂u (2) ∂vn(2) (x, y) μn (x) n (x, y) (2) ⎪ ⎪ σ (1 + κ) + (3 − κ) (x, y) = ⎪ nx x ⎪ κ −1 ∂x ∂y ⎪ ⎪ ⎪ { } ⎨ (2) (2) ∂vn (x, y) ∂u n (x, y) μn (x) (2) (n = 1, 2) (1 + κ) + (3 − κ) σnyy (x, y) = ⎪ κ − 1 ∂ y ∂ x ⎪ ⎪ | | (2) ⎪ ⎪ ⎪ ∂u n (x, y) ∂vn(2) (x, y) ⎪ (2) ⎩ σnx + y (x, y) = μn (x) ∂y ∂x (2.142) The governing equation of the structure shown in Fig. 2.5c is similar to the Eq. (2.133). Substituting the constitutive Eq. (2.142) into the governing equation and applying the Fourier transform, the displacement components can be obtained as

50

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

⎧ { ∞E 4 ⎪ 1 ⎪ λ(2) (2) n j (α)x−iαy dα ⎪ u (x, y) = E n(2)j (α)A(2) ⎪ n n j (α)e ⎪ 2π −∞ j=1 ⎨ { ∞E 4 ⎪ ⎪ 1 ⎪ λ(2) (2) ⎪ n j (α)x−iαy dα (x, y) = F (2) (α)A(2) v ⎪ n n j (α)e ⎩ 2π −∞ j=1 n j

(n = 1, 2)

(2.143)

where’ α is the Fourier variable, λ(2) n j (α) (n = 1, 2; j = 1, 2, 3, 4) is the characteristic root of the ODE, and E n(2)j (α) and Fn(2) j (α) are intermediate parameters that can be solved directly from the above equations. According to Eqs. (2.142) and (2.143), the stress components can be expressed as ⎧ { ∞E 4 ⎪ 1 (2) λ(2) (2) δ x ⎪ n n j (α)x−iαy dα ⎪ σ (x, y) = e Bn(2) ⎪ nx x j (α)An j (α)e ⎪ 2π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ { ∞E 4 ⎨ 1 (2) λ(2) (2) n j (α)x−iαy dα (x, y) = eδn x Cn(2) σnyy (n = 1, 2) (2.144) j (α)An j (α)e ⎪ 2π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ { ∞E 4 ⎪ ⎪ ⎪ (2) λ(2) δn x 1 ⎪ n j (α)x−iαy dα Dn(2)j (α)A(2) (x, y) = e σ ⎪ n j (α)e ⎩ nx y 2π −∞ j=1

(3) Derivation of SIFs Applying Eqs. (2.134) and (2.143), the total displacement components u (1+2) (x, y) n and vn(1+2) (x, y) in the global coordinate system x O y can be written as {

(x, y) u (1+2) n (1+2) (x, y) vn

}

{ =

} | } |{ (1) u (2) cos θ − sin θ u n (x0 , y0 ) n (x, y) + vn(2) (x, y) vn(1) (x0 , y0 ) sin θ − cos θ

(2.145)

By using the boundary conditions (2.121), the following expressions can be obtained { } { } { b0 f 1 (u) (2) du (2.146) [X 1 (α, x)]|x=0 A1 (α) = [ P 01 (u, α, x)]|x=0 f 2 (u) a0 where |

(2)

(2)

(2)

(2)

(2) (2) (2) (2) B11 (α)eλ11 x B12 (α)eλ12 x B13 (α)eλ13 x B14 (α)eλ14 x [X 1 (α, x)] = (2) (2) (2) (2) (2) (2) (2) (2) D11 (α)eλ11 x D12 (α)eλ12 x D13 (α)eλ13 x D14 (α)eλ14 x { } { }T (2) (2) (2) (2) A(2) 1 (α) = A11 (α) A12 (α) A13 (α) A14 (α)

| (2.147) (2.148)

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

|

51

R01 (u, α, x) R02 (u, α, x) R03 (u, α, x) R04 (u, α, x)

[ P 01 (u, α, x)] =

| (2.149)

Here, R0 j (u, α, x) ( j = 1, 2, 3, 4) can be determined from the boundary conditions. Applying the boundary conditions (2.122) yields [X 2 (α, x)]|x=h 1 +h 2

{

A(2) 2 (α)

}

{ =

b0

a0

{ [ P 02 (u, α, x)]|x=h 1 +h 2

} f 1 (u) du (2.150) f 2 (u)

where |

| (2) (2) (2) (2) (2) (2) (2) (2) B21 (α)eλ21 (α)x B22 (α)eλ22 (α)x B23 (α)eλ23 (α)x B24 (α)eλ24 (α)x [X 2 (α, x)] = (2) (2) (2) (2) (2) (2) (2) (2) D21 (α)eλ21 (α)x D22 (α)eλ22 (α)x D23 (α)eλ23 (α)x D24 (α)eλ24 (α)x (2.151) } { { } T (2) (2) (2) (2) (2.152) A(2) 2 (α) = A21 (α) A22 (α) A23 (α) A24 (α) | [ P 02 (u, α, x)] =

R05 (u, α, x) R06 (u, α, x) R07 (u, α, x) R08 (u, α, x)

| (2.153)

Here, R0 j (u, α, x) ( j = 5, 6, 7, 8) can be determined from the boundary conditions. Using the stress continuity conditions (2.123), we have the following equations { } { } (δ2 −δ1 )h 1 [X 1 (α, h 1 )] A(2) [X 2 (α, h 1 )] A(2) 1 (α) − e 2 (α) } { { b0 { } f (u) 1 = du P 1 (u, α, h 1 ) − e(δ2 −δ1 )h 1 P 2 (u, α, h 1 ) f 2 (u) a0

(2.154)

Here, | [ P n (u, α, x)] =

| Rn1 (u, α, x) Rn2 (u, α, x) (n = 1, 2) Rn3 (u, α, x) Rn4 (u, α, x)

(2.155)

where Rn j (u, α, x) ( j = 1, 2, 3, 4) can be determined from the continuity conditions (Guo et al. 2014). Applying the displacement continuity conditions (2.124) and Eq. (2.145), the following equations can be obtained |

{

{ b |{ (2) } | |{ (2) } } 0{ ' Z 1 (α, h 1 ) A1 (α) − Z 2 (α, h 1 ) A2 (α) = P 1 (u, α, h 1 ) − P ' 2 (u, α, h 1 ) a0

where

f 1 (u)

}

f 2 (u)

du

(2.156)

52

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

|

| (2) (2) (2) (2) (2) (2) (2) (2) E n1 (α)eλn1 h 1 E n2 (α)eλn2 h 1 E n3 (α)eλn3 h 1 E n4 (α)eλn4 h 1 [Z n (α, h 1 )] = (2) (2) (2) (2) (2) (2) (2) (2) Fn1 (α)eλn1 h 1 Fn2 (α)eλn2 h 1 Fn3 (α)eλn3 h 1 Fn4 (α)eλn4 h 1 (2.157) | | | ' | Rn5 (u, α, x) Rn6 (u, α, x) P n (u, α, x) = (2.158) Rn7 (u, α, x) Rn8 (u, α, x) Here, Rn j (u, α, x) ( j = 5, 6, 7, 8) can be determined from the continuity condition (Guo et al. 2014). Through the above procedure, applying Eqs. (2.146), (2.150), (2.154) and (2.156), the following equations can be obtained ⎤ 0 X 1 (α, 0) { (2) } ⎢ X 1 (α, h 1 ) −e(δ2 −δ1 )h 1 X 2 (α, h 1 ) ⎥ A1 (α) ⎥ ⎢ ⎦ ⎣ Z 1 (α) −Z 2 (α) A(2) 2 (α) 8×1 0 Z 2 (α, 0) 8×8 ⎡ ⎤ P 01 (u, α, 0) { } { b0 ⎢ ⎥ (δ −δ )h f 1 (u) ⎢ P 1 (u, α, h 1 ) − e 2 1 1 P 2 (u, α, h 1 )⎥ = du ⎢ ' ⎥ ' ⎦ f 2 (u) a0 ⎣ P 1 (u, α, h 1 ) − P 2 (u, α, h 1 ) ⎡

P 02 (u, α, h 1 + h 2 )

(2.159)

8×2

According to Eq. (2.159), the unknown A(2) n j (α) (n = 1, 2; j = 1, . . . , 4) can be expressed by the unknown auxiliary functions f 1 (u) and f 2 (u). Using the total stress expression (2.105) and the crack faces conditions (2.125) can result in the following equations ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

| (1) (2) (2) (2) σny0 y0 + σnx x sin2 θ + σnyy cos2 θ − 2σnx y sin θ cos θ = p1 (x0 )

| lim

y0 →0+

lim

y0 →0+

| (1) (2) (2) (2) σnx0 y0 − σnx x sin θ cos θ + σnyy cos θ sin θ − 2σnx y (cos2 θ − sin2 θ) = p2 (x0 )

|

(2.160)

According to Eqs. (2.135), (2.141), and (2.144), then Eq. (2.160) can be written as ⎧ ⎡ ( { 4 ⎪ Wn j1 eβn x0 +γn y0 +δn xc ∞ E (1) ⎪ ⎪ Cn j (s) F + ⎪ ⎢ ⎪ ⎪ ⎢ 2π Wn 1 −∞ j=1 ⎪ ⎪ ⎢ ⎪ ⎪ ⎢ lim ⎢ ⎪ ⎪ ⎪ 4 (1) ⎪ y0 →0+ ⎢ eδn x { ∞ E ⎪ λ (α)x−iαy (1) ⎪ ⎣+ ⎪ Tn5 j An j (α)e n j dα ⎪ ⎪ ⎪ 2π −∞ j=1 ⎨ ⎡ ( { ⎪ 4 ⎪ Wn j1 ⎪ eβn x0 +γn y0 +δn xc ∞ E (1) ⎪ ⎪ Dn j (s) F + ⎢ ⎪ ⎪ ⎢ 2π Wn 1 ⎪ −∞ ⎪ ⎢ j=1 ⎪ ⎪ ⎪ lim ⎢ ⎪ ⎢ ⎪ ⎪ 4 (2) ⎪ y0 →0+ ⎢ eδn x { ∞ E λ (α)x−iαy ⎪ (2) ⎣+ ⎪ ⎪ Tn6 j An j (α)e n j dα ⎪ ⎩ 2π −∞ j=1

⎤ ) (1) λ (s)y0 −isx0 F2 e n j ds ⎥ ⎥ Wn ⎥ ⎥ = p1 (x0 ) ⎥ ⎥ ⎦

Wn j2

⎤ ) (1) λ (s)y0 −isx0 F2 e n j ds ⎥ ⎥ Wn ⎥ ⎥ = p2 (x0 ) ⎥ ⎥ ⎦

Wn j2

(2.161)

2.2 Crack Problems in Nonhomogeneous Coating-Substrate …

53

where (2) (2) 2 2 Tn5 j = Bn(2) j sin θ + C n j cos θ − 2Dn j sin θ cos θ (2) (2) 2 2 Tn6 j = −Bn(2) j sin θ cos θ + C n j cos θ sin θ − 2Dn j (cos θ − sin θ )

(2.162) (2.163)

In order to derive the final singular integral equation, the second terms of the lefthand side of the Eq. (2.161) need to be written in the known integrals Hn11 (x0 , u), Hn12 (x0 , u), Hn21 (x0 , u) and Hn22 (x0 , u) as ⎧ { b0 { ∞E 4 ⎪ 1 ⎪ (2) λ(2) (α)x−iαy n j ⎪ lim Tn5 An j (α)e dα = Hn11 (x0 , u) f 1 (u)du ⎪ ⎪ y0 →0+ 2π −∞ ⎪ a0 ⎪ j=1 ⎪ ⎪ ⎪ { b0 ⎪ ⎪ ⎪ ⎪ + Hn12 (x0 , u) f 2 (u)du ⎪ ⎨ a0

{ b0 { ∞E 4 ⎪ ⎪ 1 ⎪ (2) λ(2) ⎪ n j (α)x−iαy dα = lim T A (α)e Hn21 (x0 , u) f 1 (u)du ⎪ n6 n j ⎪ ⎪ y0 →0+ 2π −∞ a0 ⎪ j=1 ⎪ ⎪ ⎪ { b0 ⎪ ⎪ ⎪ ⎪ ⎩ + Hn22 (x0 , u) f 2 (u)du a0

(2.164) When s → ∞, the limit of the first terms of the left-hand side in Eq. (2.161) can be obtained as lim

s→∞

2 E

Cn(1) j (s)

j=1

Wn j1 ds = 0 Wn (2.165)

2 E

Wn j2 Cn(1) ds lim j (s) s→∞ Wn j=1 lim

s→∞

lim

s→∞

2 E j=1 2 E j=1

Dn(1)j (s)

2sμ0n i = (1 + κn )|s|

Wn j1 2sμ0n i ds = Wn (1 + κn )|s| (2.166)

Wn j2 Dn(1)j (s) ds = 0 Wn

Adding and subtracting the limit of Eqs. (2.165)–(2.166) from the integrands, Eq. (2.161) can be reduced to

54

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

⎧ { b0 { b0 ⎪ ⎪ 2π p1 (x0 ) = ⎪ (K n11 + Hn11 ) f 1 (u)du + (K n12 + Hn12 ) f 2 (u)du ⎪ ⎪ ⎪ eβn x0 +δn xc a0 a0 ⎪ ⎪ ) ({ { b0 ⎪ +∞ ⎪ 2sμ0n i −|s|y0 −is(x0 −u) ⎪ ⎪ ds f 2 (u)du + lim+ e ⎪ ⎨ y0 →0 −∞ (1 + κn )|s| a0 { b0 { b0 ⎪ 2π p2 (x0 ) ⎪ ⎪ = (K + H ) f (u)du + (K n22 + Hn22 ) f 2 (u)du ⎪ n21 n21 1 ⎪ ⎪ eβn x0 +δn xc a0 a0 ⎪ ⎪ ⎪ ) { b0 ({ +∞ ⎪ ⎪ 2sμ0n i −|s|y0 −is(x0 −u) ⎪ ⎪ ds f 1 (u)du + lim+ e ⎩ y0 →0 a0 −∞ (1 + κn )|s| (2.167) where { K n11 (x0 , u) = ⎡

∞

2 E

−∞ j=1

Cn(1) j (s)

Wn j1 −is(x0 +u) e ds Wn

⎤ W i 2sμ n j2 0n ⎣ ⎦e−is(x0 +u) ds K n12 (x0 , u) = Cn(1) − j (s) W (1 + κ )|s| n n −∞ j=1 {

{ K n21 (x0 , u) =

∞

∞ −∞

⎡ ⎣

2 E

2 E j=1

⎤ W i 2sμ n j1 0n ⎦e−is(x0 +u) ds Dn(1)j (s) − Wn (1 + κn )|s| {

K n22 (x0 , u) =

∞

2 E

−∞ j=1

Dn(1)j (s)

Wn j2 −is(x0 +u) e ds Wn

(2.168)

(2.169)

(2.170)

(2.171)

Through simplification of Eq. (2.167), the final singular integral equations can be obtained as ⎧{ | { b | b0 0 2π p (x ) 4μ0n ⎪ ⎪ ⎪ f (u)du = β x 1+δ 0x K n12 + Hn12 − (K n11 + Hn11 ) f 1 (u)du + ⎪ ⎨ a (1 + κ)(u − x0 ) 2 e n 0 n c a0 0 | { b | { b ⎪ 0 0 ⎪ 4μ0n 2π p (x ) ⎪ ⎪ K n21 + Hn21 − (K n22 + Hn22 ) f 2 (u)du = β x 2+δ 0x f 1 (u)du + ⎩ (1 + κ )(u − x ) n e n 0 n c a0 a0 0

(2.172)

The singular integral equation obtained above can be solved numerically using the method adopted by Shbeeb et al. (1999a, b). The singular unknown functions f i (u) in Eq. (2.172) can be written as (b0 − a0 )gi (u) f i (u) = √ (i = 1, 2) 2 (u − a0 )(b0 − u) Finally, the mixed-mode SIFs can be expressed as

(2.173)

2.3 Crack Problems in Orthotropic Nonhomogeneous Materials

K I (a0 ) = lim

x0 →a0

/

2(a0 −

(1+2) x0 )σny (x0 , 0) 0 y0

55

2μ0n βn a0 +δn xc e =− 1 + κn

/ 2μ0n βn b0 +δn xc (1+2) e 2(x0 − b0 )σny (x0 , 0) = 0 y0 x0 →b0 1 + κn

/

b0 − a0 g2 (a0 ) 2 (2.174)

/

b0 − a0 g2 (b0 ) 2 (2.175) / / 2μ0n βn a0 +δn xc b0 − a0 (1+2) g1 (a0 ) e K II (a0 ) = lim 2(a0 − x0 )σnx (x0 , 0) = − y 0 0 x0 →a0 1 + κn 2 (2.176) / / 2μ0n βn b0 +δn xc b0 − a0 (1+2) g1 (b0 ) e K II (b0 ) = lim 2(x0 − b0 )σnx (x, 0) = 0 y0 x0 →b0 1 + κn 2 (2.177) K I (b0 ) = lim

2.3 Crack Problems in Orthotropic Nonhomogeneous Materials 2.3.1 Basic Equations and Boundary Conditions Consider the plane elasticity problem of a nonhomogeneous orthotropic strip with properties varying in the x-direction. The strip is infinite along y-axis and has a thickness h along x-axis. The principal direction of orthotropy is parallel and perpendicular to the boundary of the strip. The cracks perpendicular to the boundary can be internal (embedded) or edge cracks, as shown in Fig. 2.6. The material properties are defined as

Fig. 2.6 Geometry of a crack between the FGMs coating and the homogeneous substrate

56

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

c11 (x) = c110 eδx , c12 (x) = c120 eδx , c22 (x) = c220 eδx , c66 (x) = c660 eδx

(2.178)

where c110 ,c120 ,c220 ,c660 and δ are constants. The general constitutive relation can be written as ⎧ ∂u ∂v ⎪ σx x = c11 (x) + c12 (x) ⎪ ⎪ ⎪ ∂ x ∂y ⎪ ⎪ ⎨ ∂u ∂v + c22 (x) σ yy = c12 (x) ⎪ ∂x ∂y ⎪ ⎪ ⎪ ⎪ ∂v ∂u ⎪ ⎩ σx y = c66 (x)( + ) ∂y ∂x

(2.179)

The boundary conditions of the mode-I crack problem described in Fig. 2.6 can be written as σx x (0, y) = 0, σx y (0, y) = 0 (−∞ < y < ∞)

(2.180)

σx x (h, y) = 0, σx y (h, y) = 0 (−∞ < y < ∞)

(2.181)

σx y (x, 0) = 0 (0 < x < h)

(2.182)

σ yy (x, 0) = −σ0 (x) (a < x < b)

(2.183)

v(x, 0) = 0 (0 < x < a or b < x < h)

(2.184)

The equilibrium equation in terms of the displacement can be given as ⎧ ∂2u ∂2u ∂2v ∂u ∂v ⎪ ⎪ + c11 (x)δ + c12 (x)δ =0 ⎪ c11 (x) 2 + c66 (x) 2 + [c12 (x) + c66 (x)] ⎨ ∂ x∂ y ∂x ∂y ∂x ∂y ) ( ⎪ ⎪ ∂2v ∂2v ∂2u ∂u ∂v ⎪ ⎩ c22 (x) =0 + c66 (x) 2 + [c12 (x) + c66 (x)] + c66 (x)δ + ∂ x∂ y ∂y ∂x ∂ y2 ∂x

(2.185)

2.3.2 Solutions to Stress and Displacement Fields Similar to the solving process in Sect. 2.1, applying the Fourier transform method to Eq. (2.185) yields

2.3 Crack Problems in Orthotropic Nonhomogeneous Materials

57

⎧ { ∞ E { 4 4 ⎪ 1 2 ∞E λ (s)y−isx λ (α)x ⎪ ⎪ u(x, y) = E 1 j (s)A1 j e 1 j ds + E 2 j (α)A2 j e 2 j cos(αy)dα ⎪ ⎪ ⎪ 2π π −∞ j=1 0 j=1 ⎨

(2.186)

{ ∞ E { ⎪ 4 4 ⎪ ⎪ 2 ∞E 1 λ (s)y−isx λ (α)x ⎪ ⎪ A1 j e 1 j ds + A2 j e 2 j sin(αy)dα ⎪ ⎩ v(x, y) = 2π −∞ π 0 j=1 j=1

According to the constitutive relations, the stress expressions can be written as ⎡ ⎧ ⎪ ⎪ 1 ⎪ ⎪ σx x = eδx ⎣ ⎪ ⎪ 2π ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎪ ⎪ ⎨ 1 σ yy = eδx ⎣ ⎪ 2π ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎪ ⎪ ⎪ ⎪ 1 ⎪ δx ⎪ ⎪ σx y = e ⎣ ⎪ ⎩ 2π

⎤ { 4 2 ∞E λ (s)y−isx λ (α)x B1 j (s) A1 j e 1 j ds + B2 j (α)A2 j e 2 j cos(αy)dα ⎦ π 0 −∞ j=1 j=1 ⎤ { ∞ E { ∞E 4 4 2 λ (s)y−isx λ (α)x C1 j (s)A1 j e 1 j ds + C2 j (α)A2 j e 2 j cos(αy)dα ⎦ π 0 −∞ { ∞ E 4

j=1

{ ∞ E 4 −∞ j=1

j=1

2 λ (s)y−isx D1 j (s) A1 j e 1 j ds +

{ ∞E 4

π 0

(2.187)

⎤ λ (α)x D2 j (α)A2 j e 2 j sin(αy)dα ⎦

j=1

where the coefficients Bm j ,Cm j ,Dm j and E m j (m = 1, 2) can be determined from the constitutive equations, and the characteristic roots λm j (m = 1, 2) can be obtained by solving the ODEs (Guo et al. 2004a, b, c). If we order the characteristic roots to make Re(λm j ) > 0 and note that the stress vanishes at the time when y → ∞, we have A1 j = 0 ( j = 3, 4)

(2.188)

The other six unknowns A11 , A12 and A2 j ( j = 1, 2, 3, 4) can be obtained from the conditions of Eqs. (2.180)–(2.184). To obtain the integral equations, let us introduce the auxiliary function ∂v(x, 0) ∂x

f (x) =

(2.189)

Using the boundary conditions in Eqs. (2.182), (2.184), (2.186) and (2.187) and by applying Fourier transform to (2.189), one obtains the following result { A 1 j = q1 j

b

f (u)eisu du

(2.190)

a

where q11 =

s + iλ12 E 12 s + iλ11 E 11 , q12 = − (λ12 E 12 − λ11 E 11 )s (λ12 E 12 − λ11 E 11 )s

(2.191)

Using the homogeneous conditions given in Eqs. (2.180) and (2.181), and then by solving the linear algebraic equations we have

58

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

{ { A2 } = a

b

{

} X 2 f (u)du

(2.192)

where { A2 } =

{

A21 A22 A23 A24

}T

(2.193)

{X 2 } = [X 1 ]−1 { P(u, α)} ⎡

B21

B22

B23

(2.194) B24

⎤

⎥ ⎢ D21 D22 D23 D24 ⎥ ⎢ [X 1 ] = ⎢ c660λ21 h c660λ22 h c660λ23 h c660λ24 h ⎥ B22 e B23 e B24 e ⎦ ⎣ B21 e D21 λ21 h D22 λ22 h D23 λ23 h D24 λ24 h e e e e c660 c660 c660 c660

(2.195)

}T { { P(u, α)} = R1 (u, α) R2 (u, α) R3 (u, α) R4 (u, α)

(2.196)

Here, Ri (u, α) can be determined according to the boundary conditions (Guo et al. 2004a, b, c). Now let us substitute Eqs. (2.188), (2.190) and (2.192) into (2.187) and make use of the boundary condition of Eq. (2.183), then the following equation can be obtained {

b

[h 1 (u, x) + h 2 (u, x)] f (u)du = −eδx σ0 (x)

(2.197)

a

where { +∞ 1 H1 (y, s)eis(x−u) ds h 1 (u, x) = lim y→0 2π −∞ { 2 +∞ H2 (u, x, α) cos(αy)dα h 2 (u, x) = lim y→0 π 0 H1 (y, s) =

2 E

C1 j q1 j eλ1 j y

(2.198) (2.199)

(2.200)

j=1

H2 (u, x, α) = X 3 X 2 eλ2 j x

(2.201)

{X 3 } = { C21 eλ21 x C22 eλ22 x C23 eλ23 x C24 eλ24 x }T

(2.202)

To derive the singular integral equation, one of the most important tasks is to perform an asymptotic analysis. When s → ∞, after lengthy manipulation, the asymptotic form of the intermediate parameter H1 (y, s) can be obtained as

2.3 Crack Problems in Orthotropic Nonhomogeneous Materials

59

H1∞ (0, s) = w11 + w12 /s

(2.203)

w11 = w12 =

2 2 2 2 2 2 + 16c220 p11 p12 + 4c120 c220 ( p11 + p12 )] ic660 [c120 8c220 (c120 + c660 ) p11 p12 ( p11 + p12 ) ⎧

(2.204) ⎫ ⎬

⎨ 3c2 c660 − 16c2 (c660 + 3c120 ) p 2 p 2 1 c660 δ 120 220 11 12 16c220 (c120 + c660 )2 p11 p12 ( p11 + p12 ) ⎩ +c3 + 4c120 c220 (c660 − c120 )( p 2 + p 2 )⎭ 120 11 12

(2.205)

where p11 and p12 are intermediate quantities (Guo et al. 2004a, b, c, d). For isotropic nonhomogeneous materials with shear modulus μ(x) = μ0 eδx , then Eq. (2.203) reduces to H1∞ (0, s) =

−2μ0 δ 1 4μ0 i + 1+κ 1+κ s

(2.206)

Erdogan and Wu (1996, 1997) and Chen et al. (2002) considered the first constant term in Eq. (2.206). A second-order asymptotic term w12 /s is kept here, which can handle the oscillations in the integrand (Shbeeb et al., 1999a, 1999b). By adding and subtracting H1∞ from the integrand, h 1 (u, x) can be obtained as h 1 (u, x) = { k1 (u, x) =

+∞

−∞

| | 1 −Im(w11 ) + k1 (u, x) π u−x

(2.207)

[H1 (0, s) − H1∞ (0, s)]eis(u−x) ds

(2.208)

Based on the analysis of Erdogan and Wu (1997) and Kadioglu et al. (1998), for isotropic FGMs, h 2 (u, x) and k1 (u, x) for an internal crack (0 < a < b < h) are bounded in the interval a ≤ (u, x) ≤ b, but h 2 (u, x) for an edge crack (a = 0; b = h) will include singular terms. For the functionally graded orthotropic strip with an edge crack, we’ll analyze the singularity of h 2 (u, x) and give the asymptotic behavior of H2 when a = 0. As α → ∞, the asymptotic form of H2 can be written as H2∞ (u, x, α) = w21 e−uλ21 +xλ23 + w22 e−uλ21 +xλ24 + w23 e−uλ22 +xλ23 + w24 e−uλ22 +xλ24 (2.209) where w21 =

(B21 D22 − B22 D21 )C23 D24 R11 + (B22 D21 − B21 D22 )B24 C23 R21 (2.210) −(B22 D21 − B21 D22 )(B24 D23 − B23 D24 )

w22 =

(B22 D21 − B21 D22 )C24 D23 R11 + (B21 D22 − B22 D21 )B23 C24 R21 (2.211) −(B22 D21 − B21 D22 )(B24 D23 − B23 D24 )

w23 =

(B21 D22 − B22 D21 )C23 D24 R12 + (B22 D21 − B21 D22 )B24 C23 R22 (2.212) −(B22 D21 − B21 D22 )(B24 D23 − B23 D24 )

60

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

w24 =

(B22 D21 − B21 D22 )C24 D23 R12 + (B21 D22 − B22 D21 )B23 C24 R22 (2.213) −(B22 D21 − B21 D22 )(B24 D23 − B23 D24 )

As α → ∞, λ2 j ( j = 1, 2, 3, 4) need to be expanded in the asymptotic forms λ21 = p21 α + p23 , λ22 = p22 α + p24 , λ23 = − p21 α + p24 , λ24 = − p22 α + p23 (2.214) where p21 , . . . , p24 are constants p21 =

1 / ξ21 M21 + M22 2

(2.215)

p22 =

1 / ξ22 M21 − M22 2

(2.216)

p23 = lim (λ21 − p21 α)

(2.217)

p24 = lim (λ22 − p22 α)

(2.218)

/ ξ21 = sgn[Re( M21 + M22 )]

(2.219)

/ ξ22 = sgn[Re( M21 − M22 )]

(2.220)

α→∞

α→∞

M21 = M22 =

1 c110 c660

c110 c220 − c120 (c120 + 2c660 ) c110 c660

/ 2 (c120 − c110 c220 )[(c120 + 2c660 )2 − c110 c220 ]

(2.221) (2.222)

Here, √the values –1 or 1, which depend on whether the real parts √ ξ21 and ξ22 take of M21 + M22 and M21 − M22 are negative or positive. Thus Eq. (2.209) can be expressed as H2∞ (u, x, α) =

e p24 x− p23 u e p23 x− p23 u e p24 x− p24 u e p23 x− p24 u w21 + ( p u+ p x)α w22 + ( p u+ p x)α w23 + p (u+x)α w24 p (u+x)α e 21 e 21 22 e 22 21 e 22 (2.223)

where { kb (u, x) = 0

+∞

[H2 (u, x, α) − H2∞ (u, x, α)]dα

(2.224)

2.3 Crack Problems in Orthotropic Nonhomogeneous Materials

{ ks (u, x) =

+∞

61

H2∞ (u, x, α)dα

(2.225)

0

To verify the expression (2.223), let us consider the isotropic nonhomogeneous materials with shear modulus μ(x) = μ0 eδx . It can be obtained that ⎛ p21 = p22 = 1, p23,24 =

/

1⎝ −δ ± 2

⎞ (κ − 1+κ

3)δ 2

⎠

(2.226)

Then, H2∞ (u, x, α) can be simplified to the form of isotropic FGMs given by Erdogan and Wu (1997). The above formula is valid for isotropic FGMs, homogeneous orthotropic materials and even homogeneous isotropic materials. Finally, substituting Eq. (2.207) into Eq. (2.197), we have 1 π

{

b

a

|

| −Im(w11 ) + k1 (u, x) + 2ks (u, x) + 2kb (u, x) g(u)du = −eδx σ0 (x) u−x (2.227)

Since a second-order asymptotic term w12 /s remains in the Eq. (2.203), used to solve Eq. (2.227) numerically, some manipulations are required for k1 (u, x), referencing the procedure outlined by Shbeeb et al. (1999a, b).

2.3.3 Crack-Tip SIFs The singular integral Eq. (2.227) can be solved by the method given by Kadioglu et al. (1998) and Erdogan and Gupta (1972). For an internal crack, a bounded function g(x) can be defined, and then the singular nature of the unknown function f (x) can be expressed as f (x) = √

g(x) (a < x < b) (x − a)(b − x)

(2.228)

The SIFs of the internal crack tips can be expressed as K I (a) = lim

/

x→a

K I (b) = lim

y→b

/ 2(a − x)σ yy (x, 0) = −Im(w11 )e

/

δa

/ 2(x − b)σ yy (x, 0) = Im(w11 )e

δb

2 g(a) b−a

2 g(b) b−a

For the edge crack, the solution of Eq. (2.227) may be written as

(2.229)

(2.230)

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2 Exponential Models for Crack Problems in Nonhomogeneous Materials

g(x) f (x) = √ (0 < x < b) b−x

(2.231)

The SIF of the edge crack tip can be expressed as K I (b) = lim

y→b

/

g(b) 2(x − b)σ yy (x, 0) = Im(w11 )eδb √ b

(2.232)

2.4 Transient Crack Problem of a Coating-Substrate Structure 2.4.1 Basic Equations and Boundary Conditions As shown in Fig. 2.7, a nonhomogeneous coating with an internal crack perpendicular to the interface and an infinite homogeneous substrate are bonded together. The coating is of thickness h 1 . The shear modulus and density of the FGM layer is defined as follows μ1 (x) = μ0 eδx , ρ1 (x) = ρ0 eδx

(2.233)

The material properties are assumed to vary continuously across the interface, so that the shear modulus and density of the substrate can be written as

a −σ 0 (x) H (t)

μ 1(x), ρ1(x) μ 2 , ρ2

Fig. 2.7 Geometry of the dynamic crack problem in a nonhomogeneous coating-substrate system

2.4 Transient Crack Problem of a Coating-Substrate Structure

μ2 = μ1 (h 1 ) = μ0 eδh 1 , ρ2 = ρ2 (h 1 ) = ρ0 eδh 1

63

(2.234)

where μ0 , ρ0 and δ are material constants. The displacement and stress components of transient problems are dependent on the coordinates and the time t, i.e., u n = u n (x, y, t), vn = vn (x, y, t) and σni j = σni j (x, y, t), where the subscript n = 1 corresponds to the FGM coating and n = 2 corresponds to the substrate. Please note that the symbol t in this section denotes the time. The constitutive relation are identical with those given in Eq. (2.70). The equilibrium equations are given by ⎧ ∂σnx y ∂σnx x ∂ 2un ⎪ ⎪ ⎨ ∂ x + ∂ y = ρn (x) ∂t 2 (n = 1, 2) ⎪ ∂σnyy ∂σnx y ∂ 2 vn ⎪ ⎩ + = ρn (x) 2 ∂y ∂x ∂t

(2.235)

The boundary and continuity conditions of the structure are σ1x x (0, y, t) = 0, σ1x y (0, y, t) = 0 (−∞ < y < ∞)

(2.236)

σ1x x (h 1 , y, t) = σ2x x (h 1 , y, t), σ1x y (h 1 , y, t) = σ2x y (h 1 , y, t)

(2.237)

v1 (h 1 , y, t) = v2 (h 1 , y, t), u 1 (h 1 , y, t) = u 2 (h 1 , y, t)

(2.238)

σ1x y (x, 0, t) = 0 (0 < x < h 1 )

(2.239)

σ1yy (x, 0, t) = −σ0 (x)H (t) (a < x < b)

(2.240)

v1 (x, 0, t) = 0 (0 < x < a, b < x < h 1 )

(2.241)

where σ0 (x) is known and H (t) denotes the Heaviside function.

2.4.2 Solutions to Stress and Displacement Fields Substituting Eq. (2.70) into the equilibrium Eq. (2.235) and upon use of Laplace transform and Fourier transform, the following displacement forms can be obtained

64

2 Exponential Models for Crack Problems in Nonhomogeneous Materials ⎧ { ∞ E { 2 4 ⎪ 2 ∞ E 1 λ y−isx ⎪ ⎪ u ∗1 (x, y, p) = E 1 j (s, p)A1 j e 1 j ds + E 2 j (s, p) A2 j cos(sy)ds ⎪ ⎪ ⎪ 2π π −∞ j=1 −∞ j=1 ⎨ { ∞ E { ⎪ 2 4 ⎪ ⎪ 2 ∞ E 1 λ y−isx ⎪ ∗ ⎪ A1 j e 1 j ds + A2 j sin(sy)ds ⎪ ⎩ v1 (x, y, p) = 2π −∞ π −∞ j=1 j=1

⎧ { 4 ⎪ 2 ∞E ⎪ ∗ ⎪ u (x, y, p) = E 3 j (s, p) A3 j eλ3 j x cos(sy)ds ⎪ 2 ⎪ π −∞ j=1 ⎨ { 4 ⎪ ⎪ 2 ∞E ⎪ ∗ ⎪ A3 j eλ3 j x sin(sy)ds ⎪ ⎩ v2 (x, y, p) = π −∞ j=1

(2.242)

(2.243)

where the variables marked with the superscript * denote those obtained by the Laplace transform, p is Laplace variable in the time transform domain, E m j (m = 1, 2, 3) is intermediate quantity (Guo et al. 2004a, b, c). According to the constitutive relation, the stress components are written as ⎡ ⎧ { ∞E 2 ⎪ ⎪ ⎪ ∗ δx ⎣ 1 ⎪ σ (x, y, p) = e B1 j (s, p)A1 j eλ1 j y−isx ds ⎪ 1x x ⎪ 2π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎤ ⎪ ⎪ { ∞E 4 ⎪ ⎪ 2 ⎪ ⎪ + B2 j (s, p) A2 j eλ2 j x cos(sy)ds ⎦ ⎪ ⎪ ⎪ π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎡ ⎪ ⎪ { ∞E 2 ⎪ ⎪ ⎪ ∗ δx ⎣ 1 ⎪ ⎪ σ1yy (x, y, p) = e C1 j (s, p) A1 j eλ1 j y−isx ds ⎪ ⎪ 2π −∞ j=1 ⎨ ⎤ { ∞E ⎪ 4 ⎪ ⎪ 2 ⎪ ⎪ + C2 j (s, p)A2 j eλ2 j x cos(sy)ds ⎦ ⎪ ⎪ π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎪ ⎪ { ∞E 2 ⎪ ⎪ ⎪ ∗ δx ⎣ 1 ⎪ σ D1 j (s, p)A1 j eλ1 j y−isx ds (x, y, p) = e ⎪ 1x y ⎪ ⎪ 2π −∞ j=1 ⎪ ⎪ ⎪ ⎪ ⎤ ⎪ ⎪ { ∞E 4 ⎪ ⎪ 2 ⎪ ⎪ ⎪ + D2 j (s, p)A2 j eλ2 j x sin(sy)ds ⎦ ⎪ ⎩ π −∞ j=1

(2.244)

2.4 Transient Crack Problem of a Coating-Substrate Structure

⎧ ⎪ ∗ ⎪ ⎪ ⎪ σ2x x (x, y, p) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∗ (x, y, p) = σ2yy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ ⎩ σ2x y (x, y, p) =

2 π 2 π 2 π

{

4 E

∞

−∞ j=1

{

4 E

∞

−∞ j=1

{

4 E

∞

−∞ j=1

65

B3 j (s, p) A3 j eλ3 j x cos(sy)ds C3 j (s, p) A3 j eλ3 j x cos(sy)ds

(2.245)

D3 j (s, p) A3 j eλ3 j x sin(sy)ds

where the coefficients Bm j ,Cm j ,Dm j (m = 1, 2, 3) are intermediate quantities (Guo et al. 2004a, b, c). For the homogeneous substrate, considering that the stress in Eq. (2.245) vanish as x → ∞, we have A3 j = 0 ( j = 1, 2)

(2.246)

To obtain the integral equation, let us introduce the following auxiliary function f (x, p) =

∂v1∗ (x, 0, p) ∂x

(2.247)

Substituting Eq. (2.242) into Eq. (2.247), and upon using the boundary conditions given in Eqs. (2.239) and (2.241) and then applying Fourier transform, we obtain the following result {

b

A 1 j = q1 j

f (u, p)eisu du ( j = 1, 2)

(2.248)

a

where q11 =

1 s + iλ12 E 12 1 s + iλ11 E 11 , q12 = − s λ12 E 12 − λ11 E 11 s λ12 E 12 − λ11 E 11

(2.249)

From the boundary conditions (2.236) and (2.237) on solving the linear equations, the following relation can be obtained { [X 1 ]{ A2 } = a

b

| | { P 1 }g(u, p)du + X 2 { A3 }

(2.250)

where { A2 } =

{

A21 A22 A23 A24

{ A3 } =

{

A33 A34

}T

}T

(2.251) (2.252)

66

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

⎤ B22 B23 B24 B21 ⎢ D21 /μ0 D22 /μ0 D23 /μ0 D24 /μ0 ⎥ ⎥ [X 1 ] = ⎢ λ22 h λ23 h ⎣ B21 eλ21 h B22 e B23 e B24 eλ24 h ⎦ D21 eλ21 h /μ0 D22 eλ22 h /μ0 D23 eλ23 h /μ0 D24 eλ24 h /μ0 ⎤ ⎡ 0 0 ⎥ ⎢ 0 0 ⎥ [X 2 ] = ⎢ (λ34 −δ)h 1 ⎦ ⎣ B33 e(λ33 −δ)h 1 B34 e (λ31 −δ)h 1 (λ31 −δ)h 1 D33 e /μ0 D34 e /μ0 }T { { P 1 (u, s, p)} = R1 (u, s, p) R2 (u, s, p) R3 (u, s, p) R4 (u, s, p) ⎡

(2.253)

(2.254)

(2.255)

where Ri (u, s, p) is an intermediate quantity. From the displacement continuity condition (2.238), we can obtain the following expression |

| X 3 { A3 } =

{

b

a

| | { P 2 } f (u, p)du + X 4 { A2 }

(2.256)

where | [X 3 ] = |

E 33 eλ33 h 1 E 34 eλ34 h 1 eλ33 h 1 eλ34 h 1

|

E 21 eλ21 h 1 E 22 eλ22 h 1 E 23 eλ23 h 1 E 24 eλ24 h 1 eλ21 h 1 eλ22 h 1 eλ23 h 1 eλ24 h 1

[X 4 ] =

(2.257) |

}T { { P 2 (u, s, p)} = R5 (u, s, p) R6 (u, s, p)

(2.258) (2.259)

Substituting Eq. (2.250) into Eq. (2.256), we have |

| X 5 { A3 } =

{

b

{ P 3 } f (u, p)du

(2.260)

| |−1 [X 5 ] = [X 3 ] − [X 4 ] X 1 [X 2 ]

(2.261)

{ P 3 } = { P 2 } + [X 4 ][X 1 ]−1 { P 1 }

(2.262)

a

where

{ A2 } = [X 1 ]−1

{

b

) ( { P 1 } + [X 2 ][X 5 ]−1 { P 3 } f (u, p)du

(2.263)

a

By substituting Eqs. (2.246), (2.248) and (2.263) into (2.244), and using the boundary condition (2.240), the following integral equation can be obtained

2.4 Transient Crack Problem of a Coating-Substrate Structure

{

b

[h 1 (u, x, p) + h 2 (u, x, p)] f (u, p)du = −eδx σ0 (x)/ p

67

(2.264)

a

where { +∞ 1 h 1 (u, x, p) = lim H1 (y, s, p)eis(x−u) ds y→0 2π −∞ { 2 +∞ H2 (u, x, s, p) cos(sy)ds h 2 (u, x, p) = lim y→0 π 0 H1 (y, s, p) =

2 E

C1 j q1 j eλ1 j y

(2.265) (2.266)

(2.267)

j=1

H2 (u, x, s, p) = {X 6 }[X 1 ]−1 ({ P 1 } + [X 2 ][X 5 ]−1 { P 3 }) { } {X 6 } = C21 eλ21 x C22 eλ22 x C23 eλ23 x C24 eλ24 x

(2.268) (2.269)

To derive the singular integral equation, an asymptotic analysis is conducted. When y = 0 and s → ∞, the asymptotic form of H1 (y, s, p) can be obtained as H1∞ (0, s, p) =

−2μ0 δ 1 4μ0 i + 1+κ 1+κ s

(2.270)

It can be found that the first term in H1∞ is equal to H1∞ for the static case given by Erdogan and Wu (1997). It should be noted that Erdogan and Wu (1997) only considered the first constant term in (2.270). However, the second-order asymptotic term is retained in (2.270) to promote the convergence of numerical calculations. By adding and subtracting H1∞ from the integrand, h 1 (u, x) can be obtained as | | 1 1 4μ0 + k1 (u, x, p) h 1 (u, x, p) = π 1+κ x −u { +∞ k1 (u, x, p) = [H1 (0, s, p) − H1∞ (0, s, p)]eis(u−x) ds

(2.271) (2.272)

−∞

Finally, by substituting Eqs. (2.266) and (2.271) into Eq. (2.264) we have 1 π

{ a

b

|

| 4μ0 1 + k1 (u, x, p) + πh 2 (u, x, p) f (u, p)du = −eδx σ0 (x)/ p 1+κ x −u (2.273)

68

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

2.4.3 Crack-Tip SIFs The singular integral Eq. (2.273) can be solved by the method given by Kadioglu et al. (1998). To simplify Eq. (2.273) into a standard form, the intervals are normalized using Eq. (2.114). The solution to the Eq. (2.273) can be written as g(r, p) f (r, p) = √ 1 − r2

(2.274)

The SIFs of the crack tips can be defined as / 4μ0 δa ∗ e (x, 0, p) = p) = lim 2(a − x)σ yy x→a 1+κ

/

b−a g(−1, p) 2 / / 4μ0 δb b − a ∗ ∗ e g(1, p) K I (b, p) = lim 2(x − b)σ yy (x, 0, p) = − y→b 1+κ 2

K I∗ (a,

(2.275)

(2.276)

The Laplace inversion of Eqs. (2.275) and (2.276) is conducted numerically by the method provided by Miller and Guy (1966).

2.5 Representative Examples 2.5.1 Example 1: Arbitrarily Oriented Crack in an Infinite Nonhomogeneous Medium Figure 2.1 shows an infinite nonhomogeneous plane with arbitrarily oriented declined crack with the tips located at x = a and x = b. When a normal pressure σ0 is applied √ to the the crack faces, the mixed-mode SIFs normalized by K 0 = σ0 a0 are shown in Figs. 2.8, 2.9 and 2.10, where a0 = (b − a)/2 is the half length of crack. The Poisson’s ratio is 0.3. The mode I and mode II SIFs are shown in Fig. 2.8. It can be observed that the mode I SIFs K I (a) and K I (b) usually are greater than the mode II SIFs K II (a) and K II (b). Obviously, the mode I SIFs dominate. Furthermore, it. can be found that K I (b) in the stiffer crack tip is always larger than K I (a) in the softer crack tip except that they are equal at θ = π/2 for the symmetry. The influences of nonhomogeneity parameter a0 γ and oriented crack angle θ on the SIFs are shown in Figs. 2.8, 2.9 and 2.10. It can be seen from Fig. 2.9, K I (a) increases as the angle θ increases. But K I (a) doesn’t vary monotonously with nonhomogeneity parameter a0 γ . It first decreases with the increase of parameter a0 γ , when θ is less than some angle (about π/3) and then increases with the increase of a0 γ , when θ is greater than the angle. This phenomenon becomes more obvious with the increase of a0 γ . From Fig. 2.10, an interesting characteristic can be found:

2.5 Representative Examples

69

Fig. 2.8 Normalized SIFs versus crack orientation angle θ

K I (b) first increases before θ reaches an angle (about π/3) and then decreases with the increase of angle θ . This phenomenon is consistent with that found by Erdogan 1994). Note that K I (b) always increases with the increase of (Konda and Erdogan / a0 γ for any E 20 E 10 . A simple comparison of mode I and mode II SERRs is shown in Fig. 2.11. Similar to the phenomena of the corresponding SIFs shown in Fig. 2.8, the normalized mode I SERRs.

Fig. 2.9 Mode I SIFs at crack tips a (dash lines) and b (solid lines)

70

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

Fig. 2.10 Mode II SIFs at crack tips a (dash lines) and b (solid lines)

Fig. 2.11 Mode I and mode II SERRs at crack tips

/ / G I (a) G 0 and G I (b) G 0 are usually greater than/the normalized mode II SERRs G II (a)/G 0 and G II (b)/G 0 , where G 0 = π(κ+1)K 0 (8μ0 ).

2.5 Representative Examples

71

2.5.2 Example 2: Interface Crack Between the Coating and the Substrate As shown in Fig. 2.2, the substrate-coating structure under concentrated load is considered. A crack of length 2a is located along the interface between the coating and the substrate. The Poisson’s ratio is assumed to be 0.3. The influences of the distance r0 from the concentrated force p0 to crack center, the nonhomogeneity parameter √ β, the thickness ratio h 1 / h 2 and h 2 /a on the SIFs normalized by K 0 = p0 a are displayed in Figs. 2.12 and 2.13. It can be found that: (1) The influence of r0 /a on the mode I SIFs of the two crack tips have different characteristics: for K I (−a), it decreases with the increasing r0 /a, while for Fig. 2.12 Influences of r0 /a and βa on the mode I SIFs for h 1 / h 2 = 10

Fig. 2.13 Influences of r0 /a and βa on the mode I SIFs for h 1 / h 2 = 2

72

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

K I (a) it has a peak at r0 /a = 0.5. When r0 /a > 0.5, K I (a) decreases rapidly until about r0 /a = 1.5 and then it slowly becomes zero. (2) Both K I (−a) and K I (a) decrease with the increase of βa when r0 /a < 1.0. Namely, it helps to decrease the mode I SIFs when the properties of the coating change softly from its upper surface to the interface. When r0 /a > 1.0 for K I (−a) and r0 /a > 1.5 for K I (a), the mode I SIFs are not sensitive to the parameter βa. (3) Through further observation, it can be found that K I (a) is always greater than K I (−a), except that they are equal at r0 /a = 0 for the symmetry. That means the crack tips close to concentrated loads are easier to open.

2.5.3 Example 3: Cross-Interface Crack Perpendicular to the Interface in a Double-Layered Structure As shown in Fig. 2.3, a nonhomogeneous structure with a crack crossing the interface is considered. The Poisson’s ratio is 0.3. The three sets of nonhomogeneity parameters represent different distribution characteristics of the properties, as shown in Fig. 2.14. Figure 2.14a, b show the curves of E 30 /E 20 = 0.1–10 for E 20 /E 10 = 0.1 and E 20 /E 10 = 10, respectively. Figure 2.14c shows the curve of E 20 /E 10 = 0.1–10 for E 30 /E 20 = 1. It should be mentioned that h 1 may not be equal h 2 . Figure 2.15 shows the variation in SIFs with the normalized half-length of the crack a0 = (b − a)/(2h 1 ) when E 20 /E 10 = 0.1 and E 30 /E 20 increases from 0.1 to 10.0 [see the group of materials in Fig. 2.14a]. The normalized crack center is c0 = (b + a)/(2h 1 ). It can be found that the SIFs increase with the increase of a0 , and for the same a0 , K I (a)/K 0 also decreases with the increase of E 30 /E 20 . Moreover, for the same E 30 /E 20 and a0 , K I (a)/K 0 is usually greater than K I (b)/K 0 except when E 20 /E 10 = 0.1 and E 30 /E 20 = 10. It should be noted that the curve of K I (a)/K 0 coincide with that of K I (b)/K 0 for E 20 /E 10 = 0.1 and E 30 /E 20 = 10. The reason is that when E 20 /E 10 = 0.1 and E 30 /E 20 = 10 [see Fig. 2.14a], the nonhomogeneity material properties of two functionally graded layers are symmetric about the interface. Therefore, in this case, the curves of K I (a)/K 0 and K I (b)/K 0 coincide with each other. It can also be seen from Fig. 2.15 that the maximum difference between K I (a)/K 0 and K I (b)/K 0 takes place when E 20 /E 10 = 0.1 and E 30 /E 20 = 0.1. This is related to two factors: ➀ The crack tip a enters the stiffer region with the increment of crack length, which may result in increasing of K I (a)/K 0 and decreasing of K I (b)/K 0 ; ➁ The ratio of the moduli of the upper surface of Layer 1 to that of the lower surface of Layer 2 gets to the maximum, when E 20 /E 10 = 0.1 and E 30 /E 20 = 0.1. Figure 2.16 depicts the variations of SIFs with a0 when E 30 /E 20 = 10 and E 30 /E 20 changes from 0.1 to 10.0 [see the group of materials in Fig. 2.14b]. First of all, we found some characteristics similar to Fig. 2.15: the SIFs increases with the increase of a0 and K I (a)/K 0 decreases with the increase of E 30 /E 20 for the same a0 . However, for the same E 30 /E 20 and a0 , K I (a)/K 0 is usually less than K I (b)/K 0

2.5 Representative Examples

Fig. 2.14 Variations of elastic modulus with x Fig. 2.15 Variations of the SIFs with a0 for E 20 /E 10 = 0.1

73

74

2 Exponential Models for Crack Problems in Nonhomogeneous Materials

Fig. 2.16 Variations of the SIFs with a0 for E 20 /E 10 = 10

except when E 30 /E 20 = 10 and E 30 /E 20 = 0.1. For the symmetry of material properties [see Fig. 2.14b], K I (a)/K 0 is equal to K I (b)/K 0 when E 20 /E 10 = 10 and E 30 /E 20 = 0.1. Figure 2.17 depicts the variations in SIFs with a0 when E 20 /E 10 = 10 and E 30 /E 20 increases from 0.1 to 10.0 [see the group of material in Fig. 2.14b]. First of all, we found some characteristics similar to Fig. 2.16: SIFs increasing with the increase of a0 and K I (a)/K 0 decreasing with the increase of E 30 /E 20 for the same a0 . However, for the same E 30 /E 20 and a0 , K I (a)/K 0 is usually less than K I (b)/K 0 except when E 20 /E 10 = 10 and E 30 /E 20 = 0.1. For the symmetry of material properties, K I (a)/K 0 is equal to K I (b)/K 0 when E 20 /E 10 = 10 and E 30 /E 20 = 0.1. Figure 2.17a, b provide the SIFs for a crack intersecting the interface in a functionally graded coating-substrate structure by assuming E 30 /E 20 = 1.0 and E 20 /E 10 = 0.1–10.0 [see the group of materials in Fig. 2.14c]. The result shows that: ➀ For the same crack length, K I (a)/K 0 decreases but K I (b)/K 0 increases with the increase of E 20 /E 10 ; ➁ According to the numerical results, K I (a)/K 0 > K I (b)/K 0 when E 20 /E 10 < 1.0, K I (a)/K 0 > K I (b)/K 0 when E 20 /E 10 > 1.0 and naturally K I (a)/K 0 = K I (b)/K 0 when E 20 /E 10 = 1.0; ➂ Both K I (a)/K 0 and K I (b)/K 0 increase with the increase of the crack length. Therefore, this indicates that the normalized SIFs are larger when the crack tip lies in the relatively stiffer material.

2.5.4 Example 4: Inclined Crack Crossing the Interface The geometry of the arbitrarily oriented crack problem is shown in Fig. 2.4. Poisson’s ratio is assumed to be 0.3. When the crack enters the second layer from the first layer, the influence of the crack center position on the mixed-mode SIFs will be studied. The corresponding results are shown in Figs. 2.18 and 2.19. For convenience, the normalized crack length (b0 −a0 )/ h is assumed to be 0.2, where a0 and b0 are the local

2.5 Representative Examples

75

Fig. 2.17 Variations of K I (a)/K 0 with a0 for a E 30 /E 20 = 0.1 and b E 30 /E 20 = 1

coordinates of the two crack tips. When the mechanical properties are symmetrical with respect to the interface, E 20 /E 10 = 0.1 and E 30 /E 20 = 10, Figs. 2.18 and 2.19 show the normalized SIFs changes with the crack center coordinate (b0 + a0 )/(2h) changing from −0.3 to 0.3 when θ = π/4 and θ = 2π/5, respectively. It can be seen from Figs. 2.18 and 2.19 that the curves of K I (a0 )/K 0 and K I (b0 )/K 0 are symmetric about (b0 + a0 )/(2h) = 0, and the same phenomenon can be found for K II (a0 )/K 0

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2 Exponential Models for Crack Problems in Nonhomogeneous Materials

and K II (b0 )/K 0 . It should be pointed out that the crack center reaches at the interface when (b0 + a0 )/(2h) = 0. Therefore, the crack-tip (b0 , 0). gets to the interface when (b0 + a0 )/(2h) = −0.1, and the crack-tip (a0 , 0) getsto the interface when (b0 + a0 )/(2h) = 0.1. Particularly, it should be noted that an interesting phenomenon can be observed: the curve of the mixed-mode SIFs usually attain its extreme value when the crack tip reaches at the interface. Fig. 2.18 Variations of the SIFs with (a0 + b0 )/(2h) for θ = π/4

Fig. 2.19 Variations of the SIFs with (a0 + b0 )/(2h) for θ = 2π/5

2.5 Representative Examples

77

2.5.5 Example 5: Vertical Crack in a Nonhomogeneous Coating-Substrate Structure Subjected to Impact Loading Under the impact of load σ0 H (t), the numerical calculations of dynamic SIFs are carried out for the internal crack in Fig. 2.7. The plane strain state is considered. √ For convenience, the dynamic SIFs are normalized by K 0 = σ0 a0 , where a0 = (b − a)/(2h √ 1 ) is normalized half-length of the crack. Let’s define the wave velocity as c = μ0 /ρ0 and assume Poisson’s ratio as ν = 0.3. Under the impact load, the dynamic SIFs increase to a peak and then drops. Shown in Figs. 2.20 and 2.21 are the dimensionless SIFs with different nonhomogeneity constants δh 1 . To illustrate the effect of δh 1 , we consider the central crack in the FGMs coating, namely (a + b)/(2h 1 ) = 0.5. Through calculations by taking a0 = 0.25, the calculated results are shown in Figs. 2.20 and 2.21. It can be seen from Fig. 2.20, that both the peak and steady value of K I (a, t)/K 0 increase with the increase of δh 1 . However, for K I (b, t)/K 0 , different situations are found in Fig. 2.21. Here, δh 1 = 0 corresponds to the homogeneous coating, δh 1 < 0 means that the medium gets less stiff from the FGMs coating to the substrate, while δh 1 > 0 just means the opposite case. Therefore, when the crack tip lies in the stiffer side, the peak and steady values of normalized dynamic SIFs are relatively greater. In addition, Fig. 2.20 provides the comparison between K I (a, t)/K 0 and K I (b, t)/K 0 with the change of δh 1 . Depicting the maximum difference the peak SIFs at the two crack tips corresponds to δh 1 = ln(0.1), and the minimum difference corresponds to δh 1 = ln(1.0) = 0, which makes it possible to optimize the gradient according to actual needs. Through further observation, it can be noticed that when δh 1 = ln(1.0) = 0, the peak and steadystate values are (1.30, 1.09) for K I (a, t)/K 0 and (1.25, 1.05) for K I (b, t)/K 0 ; for a homogeneous strip of finite thickness with a central crack, the corresponding peak and steady-state values are about (1.34, 1.20) for both K I (a, t)/K 0 and K I (b, t)/K 0 given by Itou(1980).

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2 Exponential Models for Crack Problems in Nonhomogeneous Materials

Fig. 2.20 Variations of the normalized dynamic SIFs at the tip a with ct/a0

Fig. 2.21 Variations of the normalized dynamic SIFs at both tips with ct/a0

Appendix 2A The left side of Eq. (2.56) is related to the loads, and their expression are given by {

p1 (x)

}

1 = 2π p2 (x)

{

∞

−∞

{ [X]{ Q}e ds + isx

q1 (x)

}

q2 (x)

where q1 (x) and q2 (x) are crack face loading [see Eq. (2.53)]. The matrices [X] and { Q} are given by

References

79

| [X] = [X 1 ]

X 2 X −1 3 X4 X5

| , [X 1 ] =

|

C11 C12 C13 C14 D11 D12 D13 D14

|

⎤ C21 ξ21 (h 2 ) C22 ξ22 (h 2 ) C23 ξ23 (h 2 ) C24 ξ24 (h 2 ) ⎢ D21 ξ21 (h 2 ) D22 ξ22 (h 2 ) D23 ξ23 (h 2 ) D24 ξ24 (h 2 ) ⎥ ⎥ X2 = ⎢ ⎣ E 21 ξ21 (h 2 ) E 22 ξ22 (h 2 ) E 23 ξ23 (h 2 ) E 24 ξ24 (h 2 ) ⎦ G 21 ξ21 (h 2 ) G 22 ξ22 (h 2 ) G 23 ξ23 (h 2 ) G 24 ξ24 (h 2 ) ⎤ ⎤ ⎡ ⎡ C21 C22 C23 C24 C11 C12 C13 C14 ⎢ D21 D22 D23 D24 ⎥ ⎢ D11 D12 D13 D14 ⎥ ⎥ ⎥ ⎢ [X 3 ] = ⎢ ⎣ E 21 E 22 E 23 E 24 ⎦, [X 4 ] = ⎣ E 11 E 12 E 13 E 14 ⎦ G 21 G 22 G 23 G 24 G 11 G 12 G 13 G 14 ⎤ ⎡ C11 ξ11 (−h 1 ) C12 ξ12 (−h 1 ) C13 ξ13 (−h 1 ) C14 ξ14 (−h 1 ) ⎢ D11 ξ11 (−h 1 ) D12 ξ12 (−h 1 ) D13 ξ13 (−h 1 ) D14 ξ14 (−h 1 ) ⎥ ⎥ [X 5 ] = ⎢ ⎣ E 11 ξ11 (−h 1 ) E 12 ξ12 (−h 1 ) E 13 ξ13 (−h 1 ) E 14 ξ14 (−h 1 ) ⎦ G 11 ξ11 (−h 1 ) G 12 ξ12 (−h 1 ) G 13 ξ13 (−h 1 ) G 14 ξ14 (−h 1 ) ⎡

ξ1 j (y) = eλ1 j y , ξ2 j (y) = e(λ2 j +β)y ( j = 1, 2, 3, 4) { Q} =

κ −1 p0 e−isr0 { 1 0 1 0 }T μ1

References Bleeck, O., D. Munz, W. Schaller, et al. 1998. Effect of a graded interlayer on the stress intensity factor of cracks in a joint under thermal loading. Engineering Fracture Mechanics 60: 615–623. Chen, J., Z.X. Liu, and Z.Z. Zou. 2002. Transient internal crack problem for a nonhomogeneous orthotropic strip (Mode I). International Journal of Engineering Science 40: 1761–1774. Chi, S.H., and Y.L. Chung. 2003. Cracking in coating-substrate composites with multi-layered and FGM coatings. Engineering Fracture Mechanics 70: 1227–1243. Delale, F., and F. Erdogan. 1983. The crack problem for a nonhomogeneous plane. Journal of Applied Mechanics 50: 609–614. Erdogan, F., and G.D. Gupta. 1972. On the numerical solution of singular integral equations. Quarterly of Applied Mathematics 30: 525–534. Erdogan, F., and B.H. Wu. 1996. Crack problem in functionally graded material layers under thermal stresses. Journal of Thermal Stresses 19: 237–265. Erdogan, F., and B.H. Wu. 1997. The surface crack problem for a plate with functionally graded properties. Journal of Applied Mechanics 64: 449–456. Guo, L.C., Z.H. Wang, and L. Zhang. 2014. A fracture mechanics problem of a functionally graded layered structure with an arbitrarily oriented crack crossing the interface. Mechanics of Materials 46: 69–82. Guo, L.C., and N. Noda. 2008. Fracture mechanics analysis of functionally graded layered structures with a crack crossing the interface. Mechanics of Materials 40 (3): 81–99.

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2 Exponential Models for Crack Problems in Nonhomogeneous Materials

Guo, L.C., L.Z. Wu, L. Ma, et al. 2004a. Fracture analysis of a functionally graded coating-substrate structure with a crack perpendicular to the interface—Part I: Static problem. International Journal of Fracture 127: 21–38. Guo, L.C., L.Z. Wu, and L. Ma. 2004b. The interface crack problem under a concentrated load for a functionally graded coating-substrate composite system. Composite Structures 63 (3–4): 397–406. Guo, L.C., L.Z. Wu, T. Zeng, et al. 2004c. Mode I crack problem for a functionally graded orthotropic strip. European Journal of Mechanics A/Solids 23 (2): 219–234. Guo, L.C., L.Z. Wu, T. Zeng, et al. 2004d. The Dynamic fracture behavior of a functionally graded coating-substrate system. Composite Structures 64 (3–4): 433–441. Gupta, G.D., and F. Erdogan. 1974. The problem of edge crack in an infinite strip. Journal of Applied Mechanics 41: 1001–1006. Itou, S. 1980. Transient response of a finite crack in a strip with stress-free edges. Journal of Applied Mechanics 47: 801–805. Kadioglu, S., S. Dag, and S. Yahsi. 1998. Crack problem for a functionally graded layer on an elastic foundation. International Journal of Fracture 94: 63–77. Konda, N., and F. Erdogan. 1994. The mixed-mode crack problem in a nonhomogeneous elastic medium. Engineering Fracture Mechanics 47: 533–545. Miller, M.K., and W.T. Guy. 1966. Numerical inversion of the laplace transform by use of jacobi polynomials. SIAM Journal on Numerical Analysis 3: 624–635. Shbeeb, N.I., W.K. Binienda, and K. Kreider. 1999a. Analysis of the Driving Force for multiple cracks in an infinite nonhomogeneous plate, part I: Theoretical analysis. Journal of Applied Mechanics 66: 492–500. Shbeeb, N.I., W.K. Binienda, and K. Kreider. 1999b. Analysis of the Driving Force for multiple cracks in an infinite nonhomogeneous plate, part II: Numerical Solutions. Journal of Applied Mechanics 66: 501–506. Theocaris, P.S., and N.I. Ioakimidis. 1977. Numerical integration methods for the solution of singular integral equation. Quarterly of Applied Mathematics 35 (1): 173–183.

Chapter 3

General Model for Nonhomogeneous Materials with General Elastic Properties

For FGMs with arbitrarily distributed properties spatially, the crack problems are very difficult to solve analytically. Up to now, there are mainly three different types of multi-layered models (Guo and Noda, 2007; Huang et al., 2003; Jin and Paulino, 2001; Wang et al., 2002; Cheng et al., 2012; Zhong and Cheng, 2008). Firstly, a homogenous multi-layered model was adopted by Itou (2001), Jin and Paulino (2001) and Wang et al. (2002), as shown in Fig. 3.1. The main idea of the homogenous multi-layered model is to divide the FGMs strip into many sub-layers with piecewise constant material properties. In this model, the material properties are discontinuous at the interfaces of sub-layers. The second model is referred to as linear multi-layered model (Huang et al., 2003; Cheng et al., 2012), which divides the FGMs into several sub-layers, in each part the mechanical properties are assumed to be a linear function, as shown in Fig. 3.2. The third model is the piecewise-exponential model (PE model) as proposed by Guo and Noda (2007). From the viewpoint of fracture mechanics, the governing partially differential equations for nonhomogeneous materials are usually very difficult to solve analytically, when the material properties vary spatially. In fact, the analytical solution of the fracture parameters can be solved only when the material properties are assumed to have very limited functions. The exponential function is just an ideal function, which can convert the governing partially differential equations into a general partially differential equation with constant coefficients, and it is easy to obtain an analytical solution. That is why the properties of nonhomogeneous materials are assumed to be an exponential function in most theoretical analysis of crack problems. This chapter focuses on the PE model and demonstrates its superiority in analyzing the crack problem of the FGMs with general elastic properties.

© Science Press 2023 L. C. Guo et al., Fracture Mechanics of Nonhomogeneous Materials, https://doi.org/10.1007/978-981-19-4063-7_3

81

82

3 General Model for Nonhomogeneous Materials with General Elastic …

Fig. 3.1 Homogenous multi-layered model

Fig. 3.2 Linear multi-layered model

3.1 Piecewise-Exponential Model for the Mode I Crack Problem 3.1.1 Piecewise-Exponential Model (PE Model) Figure 3.3 shows a functionally graded (nonhomogeneous) strip of thickness h with arbitrarily distributed properties along the thickness direction. An internal crack of length 2c is located perpendicularly to the free surface. Poisson’s ratio ν is assumed constant, and the shear modulus μ = μ(x) is an arbitrary function varying in xdirection. Correspondingly, the Kolosov constant is κ = 3 − 4ν for plane strain and κ = (3 − ν)/(1 + ν) for plane stress. Since the actual material properties of FGMs may be very complex and arbitrarily distributed in the x-direction, the piecewise-exponential model (PE model) is established, as shown in Fig. 3.4. The gradual medium is divided into M nonhomogeneous layers along the x-direction, and each layer has properties varying exponentially. By this way, the real material properties can be approximated by a series of exponential functions. It should be emphasized that in this new model the real material properties

3.1 Piecewise-Exponential Model for the Mode I Crack Problem

83

Fig. 3.3 Geometry of functionally graded strip with arbitrary properties

are used on both surfaces of each nonhomogeneous layer. Therefore, the nature of continuously varying properties of FGMs can be approximated accurately. As shown in Fig. 3.4, the functionally graded region is divided into M layers. Each layer is marked with the subscript n (n = 1, 2, . . . , M). The layers containing the crack tips are denoted by n 1 and n 2 , respectively. Starting from the top surface of the structure, the n-th layer is located between the region x = h n−1 and x = h n . It should be noted that h 0 = 0. The real shear modulus of the functionally graded strip with arbitrary properties is μ = μ(x)

(3.1)

If we assume the shear modulus of each layers in Fig. 3.4 as

Fig. 3.4 Piecewise-exponential model (PE Model) for functionally graded strip with arbitrarily distributed properties

84

3 General Model for Nonhomogeneous Materials with General Elastic …

μn (x) = μn0 eδn x (n = 1, 2, . . . , M)

(3.2)

and use the real material properties on both surfaces of each layer, then we have {

μn (h n−1 ) = μ(h n−1 ) = μn0 eδn h n−1 μn (h n ) = μ(h n ) = μn0 eδn h n

(n = 1, 2, . . . , M)

(3.3)

From the above two equations, μn0 and δn can be solved as | | 1 μ(h n ) δn = ln h n − h n−1 μ(h n−1 )

(3.4)

μn0 = μ(h n )e−δn h n

(3.5)

Correspondingly, the constitutive relations for each layer can be written as | | ⎧ ∂u n μn (x) ∂vn ⎪ ⎪ (1 + κ) = σ + (3 − κ) ⎪ nx x ⎪ κ −1 ∂x ∂y ⎪ ⎪ ⎪ | | ⎨ ∂vn ∂u n μn (x) (1 + κ) + (3 − κ) σnyy = (n = 1, 2, . . . , M) ⎪ κ −1 ∂y ∂x ⎪ ⎪ ) ( ⎪ ⎪ ⎪ ∂v ∂u n ⎪ ⎩ τnx y = μn (x) + n ∂y ∂x

(3.6)

And the boundary conditions can be expressed as σ1x x (0, y) = 0, σ1x y (0, y) = 0 (−∞ < y < ∞) σ M x x (h M = h, y) = 0, σ M x y (h M = h, y) = 0 (−∞ < y < ∞)

(3.7) (3.8)

σnx y (x, 0) = 0 (n = 1, . . . , M)

(3.9)

σnyy (x, 0) = −σ0 (x) (a < x < b; n = n 1 , . . . , n 2 )

(3.10)

vn (x, 0) = 0 (0 < x < a or b < x < h)

(3.11)

The displacement and stress continuity conditions between two neighboring layers are as follows {

σnx x (h n , y) = σ(n+1)x x (h n , y) σnx y (h n , y) = σ(n+1)x y (h n , y)

{ ,

u n (h n , y) = u n+1 (h n , y) vn (h n , y) = vn+1 (h n , y)

(n = 1, 2, . . . , M − 1)

(3.12)

3.1 Piecewise-Exponential Model for the Mode I Crack Problem

85

3.1.2 Solutions to Stress and Displacement Fields Substituting the constitutive Eq. (3.6) into the equilibrium equation and using Fourier transform method, we can obtain the displacement components as ⎧ 2 {∞ E ⎪ 1 ⎪ ⎪ u n (x, y) = 2π E n1 j (s)An1 j eλn1 j y−isx ds ⎪ −∞ ⎪ ⎪ j=1 ⎪ ⎪ 4 ⎪ {∞ E ⎪ ⎪ ⎪ + π2 0 E n2 j (α)An2 j eλn2 j (α)y cos(αy)dα ⎨ j=1

2 {∞ E ⎪ 1 ⎪ ⎪ vn (x, y) = 2π Fn1 j (s)An1 j eλn1 j y−isx ds ⎪ −∞ ⎪ ⎪ j=1 ⎪ ⎪ 4 ⎪ { ⎪ 2 ∞ E ⎪ ⎪ + Fn2 j (α) An2 j eλn2 j (α)y sin(αy)dα ⎩ π 0

(n = 1, . . . , M) (3.13)

j=1

Please note that only the displacement components for y < 0 are provided in Eq. (3.13) due to the symmetry of the FGM plate in Fig. 3.3. Here, s and α are the Fourier variables. The parameters λn1 j (n = 1, . . . , M; j = 1, 2) and λn2 j (n = 1, . . . , M; j = 1, 2, 3, 4) are the roots of the characteristic equation. The parameters λn1 j , E n1 j , Fn1 j (n = 1, . . . , M; j = 1, 2) and λn2 j , E n2 j , Fn2 j (n = 1, . . . , M; j = 1, . . . , 4) are given in Appendix 3A. And An1 j and An2 j are unknown quantities to be solved. Substituting Eq. (3.13) into the constitutive Eq. (3.6) yields the stress components as ⎧ | 2 ⎪ { E ⎪ δ x ⎪ σnx x (x, y) = e n 1 ∞ Bn1 j (s)An1 j eλn1 j y−isx ds ⎪ −∞ ⎪ 2π ⎪ j=1 ⎪ ⎪ | ⎪ ⎪ 4 ⎪ { E ⎪ ∞ ⎪ + π2 0 Bn2 j (α) An2 j eλn2 j (α)x cos(αy)dα ⎪ ⎪ ⎪ j=1 ⎪ ⎪ | ⎪ ⎪ 2 ⎪ { ⎪ δ x ⎪ σnyy (x, y) = e n 1 ∞ E Cn1 j (s)An1 j eλn1 j y−isx ds ⎪ ⎨ 2π −∞ j=1 | (3.14) 4 ⎪ { E ⎪ ∞ 2 λn2 j (α)x ⎪ +π 0 Cn2 j (α)An2 j e cos(αy)dα ⎪ ⎪ ⎪ j=1 ⎪ ⎪ | ⎪ ⎪ 2 ⎪ {∞ E ⎪ 1 ⎪ σnx y (x, y) = eδn x 2π Dn1 j (s)An1 j eλn1 j y−isx ds ⎪ −∞ ⎪ ⎪ j=1 ⎪ ⎪ | ⎪ ⎪ 4 ⎪ { E ⎪ ∞ ⎪ + π2 0 Dn2 j (α) An2 j eλn2 j (α)x sin(αy)dα ⎪ ⎩ j=1

where the known coefficients Bn1 j , Cn1 j , Dn1 j (n = 1, . . . , M; j = 1, 2) and Bn2 j , Cn2 j , Dn2 j (n = 1, . . . , M; j = 1, . . . , 4) are given in Appendix 3A. Without loss of

86

3 General Model for Nonhomogeneous Materials with General Elastic …

generality, the above characteristic roots are arranged as Re(λn21 ) > 0, Re(λn22 ) > 0, Re(λn23 ) < 0 and Re(λn24 ) < 0 (n = 1, 2, . . . , M). For further derivation, we introduce the following auxiliary function ∂vn (x, 0) (a < x < b; n = n 1 , . . . , n 2 ) ∂x

f (x) =

(3.15)

where f (x) = 0 (x < a or x < b) { b f (x)dx = 0

(3.16)

a

Using Eqs. (3.9) and (3.14), we have 2 E

Dn1 j An1 j = 0 (n = 1, . . . , M)

(3.17)

j=1

Substituting Eq. (3.13) into Eq. (3.15) and then applying Fourier transform leads to the following form −is

2 E

{ Fn1 j (s) An1 j =

j=1

b

f (u)eisu du

(3.18)

a

By solving Eqs. (3.17) and (3.18), we have {

b

An1 j = qn j

f (u)eisu du (n = 1, 2, . . . , M;

j = 1, 2)

(3.19)

s + iλn11 E n11 s + iλn12 E n12 ) , qn2 = ( ) , (n = 1, 2, . . . , M) qn1 = ( λn12 E n12 − λn11 E n11 s λn11 E n11 − λn12 E n12 s

(3.20)

a

where

From the boundary conditions in Eqs. (3.7) and (3.9), we have {

b

[X n (x)]{ An2 } =

{ P 0 (u, α)} f (u)du (n = 1; x = h 0 = 0)

(3.21)

{ P n (u, α)} f (u)du (n = M; x = h M = h)

(3.22)

a

{ [X n (x)]{ An2 } = a

where

b

3.1 Piecewise-Exponential Model for the Mode I Crack Problem

| [X n (x)] =

87

Bn21 eλn21 x Bn22 eλn22 x Bn23 eλn23 x Bn24 eλn24 x Dn21 eλn21 x Dn22 eλn22 x Dn23 eλn23 x Dn24 eλn24 x

| (n = 1, . . . , M) (3.23)

{ An2 } =

{

An21 An22 An23 An24

}T

(3.24)

}T { { P 0 } = R01 (u, α) R02 (u, α)

(3.25)

}T { { P M } = R03 (u, α) R04 (u, α)

(3.26)

From the continuity conditions (3.12), a group of equations can be obtained |

{ } { b{ } | An2 Rn3 (h n , u, α) − R(n+1)3 (h n , u, α) = Z n (h n ) −Z n+1 (h n ) A(n+1)2 Rn4 (h n , u, α) − R(n+1)4 (h n , u, α) a f (u)du (n = 1, . . . , M − 1)

|

{

X n (h n ) −e(δn+1 −δn )h n X n+1 (h n )

|

} An2 A(n+1)2

=

{ b a

{

(3.27)

Rn1 (h n , u, α) − e(δn+1 −δn )h n R(n+1)1 (h n , u, α) Rn2 (h n , u, α) − e(δn+1 −δn )h n R(n+1)2 (h n , u, α)

} f (u)du

(3.28)

where | [Z n (h n )] =

| [X n (h n )] =

E n21 eλn21 x E n22 eλn22 x E n23 eλn23 x E n24 eλn24 x Fn21 eλn21 x Fn22 eλn22 x Fn23 eλn23 x Fn24 eλn24 x

|

Bn21 eλn21 x Bn22 eλn22 x Bn23 eλn23 x Bn24 eλn24 x Dn21 eλn21 x Dn22 eλn22 x Dn23 eλn23 x Dn24 eλn24 x

(3.29) x=h n

| (3.30) x=h n

To make the problem tractable, the next key procedure is to simplify the expressions of R0 j (u, α) and Rn j (h n , u, α) (n = 1, . . . , M; j = 1, 2, 3, 4) in Eqs. (3.21), (3.22), (3.27) and (3.28). Theory of residues will be used and the details are provided in Appendix 3A. For simplicity, Eqs. (3.21), (3.22), (3.27) and (3.28) can be written as ⎧ ⎫ ⎧ ⎫ A12 ⎪ P0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ { b⎨ ⎨ A22 ⎬ P1 ⎬ = f (u)du (3.31) [Φ]4M×4M .. ⎪ .. ⎪ ⎪ a ⎪ ⎪ ⎪ . ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ A M2 4M×1 P M 4M×1 Here, the vector P n and matrix Φ are given by

88

3 General Model for Nonhomogeneous Materials with General Elastic …

⎧ Rn3 (h n , u, α) − ⎪ ⎪ ⎨ Rn4 (h n , u, α) − { Pn} = ⎪ R (h , u, α) − ⎪ ⎩ n1 n Rn2 (h n , u, α) − ⎡

⎫ R(n+1)3 (h n , u, α) ⎪ ⎪ ⎬ R(n+1)4 (h n , u, α) (n = 1, . . . , M − 1) R(n+1)1 (h n , u, α) ⎪ ⎪ ⎭ R(n+1)2 (h n , u, α)

0 0 X 1 (0 ⎢ P (h ) − P (h ) 0 ⎢ 1 1 2 1 ⎢ ⎢ 0 P (h ) − P (h 2 ) 2 2 3 ⎢ ⎢ . . . [Φ]4M×4M = ⎢ . . . ⎢ . . . ⎢ ⎢ 0 0 0 ⎢ ⎢ ⎣ 0 0 0 0 0 0

⎤ ··· 0 0 0 ⎥ ··· 0 0 0 ⎥ ⎥ ⎥ ··· 0 0 0 ⎥ ⎥ . . . . ⎥ . . . . ⎥ . . . . ⎥ ⎥ · · · P M−2 (h M−2 ) − P M−1 (h M−2 ) 0 ⎥ ⎥ ··· 0 P M−1 (h M−1 ) − P M (h M−1 ) ⎦ ··· 0 0 X M (h)

(3.32)

(3.33)

where |

|

P n (h n ) =

|

Z L (h n ) X L (h n )

| ,

|

|

P n+1 (h n ) =

|

Z n+1 (h n )

|

e(δn+1 −δn )h n X n+1 (h n )

The unknowns An2 (n = 1, · · · , M) can be obtained by solving linear Eq. (3.31). Substituting An2 into the stress boundary condition in Eq. (3.10) on the crack face, we have the integral equation {

b

[h n1 (u, x) + h n2 (u, x)] f (u)du = −eδn x σ0 (x) (a < x < b)

(3.34)

a

where { +∞ 1 Hn1 (y, s)eis(x−u) ds h n1 (u, x) = lim y→0 2π −∞ { 2 +∞ Hn2 (u, x, α) cos(αy)dα h n2 (u, x) = lim y→0 π 0 ⎧ 2 E ⎪ ⎪ ⎨ Hn1 (y, s) = Cn1 j qn1 j eλn1 j y ⎪ ⎪ ⎩

j=1

(3.35)

(3.36)

Hn2 (u, x, α) = {W n (x)}T { An2 }

{ }T {W n (x)} = Cn21 eλn21 x Cn22 eλn22 x Cn23 eλn23 x Cn24 eλn24 x

(3.37)

Consider the singularity given in Eq. (3.34), an asymptotic analysis is conducted. As s → ∞, the asymptotic form of Hn1 (y, s) (y = 0) can be obtained as Hn1∞ (0, s) =

4μn0 i (n = 1, . . . , M) 1+κ

(3.38)

3.1 Piecewise-Exponential Model for the Mode I Crack Problem

89

Adding and subtracting Hn1∞ from the integrand, h n1 (u, x) can be manipulated as h n1 (u, x) =

| | 1 4μn0 1 + kn1 (u, x) π 1+κ x −u

(3.39)

[Hn1 (0, s) − Hn1∞ (0, s)]eis(u−x) ds

(3.40)

where { kn1 (u, x) =

+∞

−∞

Finally, the singular integral equation is obtained as 1 π

{ a

b

|

| 4μn0 1 + kn1 (u, x) + π h n2 (u, x) f (u)du 1+κ x −u = −eδn x σ0 (x) (n = n 1 , . . . , n 2 )

(3.41)

3.1.3 Crack-Tip SIFs The method given by Erdogan and Gupta (1972) is used to solve the singular integral Eq. (3.41). To reduce (3.41) to a standard form, the intervals are normalized by setting u=

b+a b−a b+a b−a r+ , x= s+ 2 2 2 2

(3.42)

√ For an internal crack, the auxiliary function is expressed by f (r ) = g(r )/ 1 − r 2 . Then, Eq. (3.41) can be converted into the following form with m-order terms | | m E 1 1 4μn0 f (rk ) + kn1 (rk , s j ) + πh n2 (rk , s j ) m 1 + κ s j − rk k=1 = −eδn s j σ0 (s j ) ( j = 1, . . . , m − 1)

(3.43)

where | (2k − 1)π (k = 1, . . . , m) 2m (rπ) ( j = 1, . . . , m − 1) s j = cos m |

rk = cos

(3.44) (3.45)

90

3 General Model for Nonhomogeneous Materials with General Elastic …

To combine Eq. (3.43) with Eq. (3.16), the unknown f (rk ) can be determined. Please note that the subscript n in Eq. (3.43) is corresponding to the layer where the point with the coordinate x = s j (b − a)/2 + (b + a)/2 is located. The SIFs for an internal crack can be obtained as / / 4μn 1 0 δn a b − a 1 e g(−1) (3.46) K I (a) = lim 2(a − x)σ yy (x, 0) = − x→a 1+κ 2 / / 4μn 2 0 δn b b − a e 2 g(1) (3.47) K I (b) = lim 2(x − b)σ yy (x, 0) = x→b 1+κ 2 For simplicity, we assume Poisson’s ratio is constant in the above derivations of the PE model. Actually, the PE model is also valid for Poisson’s ratio with value varying arbitrarily, i.e., ν = ν(x).

3.2 PE Model for Mixed-Mode Crack Problem 3.2.1 Basic Equations and Boundary Conditions Figure 3.5 shows the problem under consideration. An infinite long FGM strip of thickness L, and L 0 denotes the distance between the upper surface of the strip and the y-axis. There is an arbitrarily oriented crack of length 2c and crack angle θ in the FGMs strip. a0 and b0 denote x0 coordinate of both crack tips, respectively. Without loss of generality, the x0 axis of the coordinate system (x0 O y0 ) can be arranged along the crack line. For actual crack problem of FGMs, since the material properties of FGMs may be arbitrarily distributed along the thickness direction and the crack may be arbitrarily oriented, the FGMs strip is divided into M layers in the x-direction for analysis. Each sub-layer is marked by using a subscript n (n = 1, 2, . . . , M) as shown in Fig. 3.6. The symbols h n−1 and h n denote the x coordinate of the upper and lower surfaces of the n-th layer, respectively. Namely, the n-th layer is located in the region between Fig. 3.5 Geometry of functionally graded strip with arbitrary properties

3.2 PE Model for Mixed-Mode Crack Problem

91

Fig. 3.6 Schematic diagram of a general piecewise-exponential model

x = h n−1 and x = h n , so that the thickness of the n-th layer is h n − h n−1 . It should be noticed that h 0 = −L 0 and h M = L − L 0 . If the modulus of each layer is assumed exponential function, then the continuously varied mechanical properties of actual FGMs can be approached by a series of exponential functions. To approach the actual properties accurately, the real mechanical properties of the FGMs should be applied on both surfaces of each layer. The real shear modulus of the FGMs strip with arbitrary properties can be defined as a general function μ(x). Meanwhile, Poisson’s ratio is assumed constant. According to the above description of the piecewise-exponential model, the shear modulus of each layer can be described by an exponential function as μn = μn0 eδn x = μn0 eβn x0 +γn y0 = μn0 eδn cos θ x0 +δn sin θ y0 (h 0 ≤ x ≤ h M ; n = 1, 2, . . . , M)

(3.48)

As shown in Fig. 2.5, according to the principle of superposition, the stress (1) (1) components in each layer can be divided into two different parts: let σnx x , σnyy (1) and τnx y denote the stress components of the first part that will be solved in the local (2) (2) (2) coordinate system (x 0 Oy0 ); let σnx x , σnyy and τnx y denote stress components of the second part that will be solved in global coordinate system (xOy). The corresponding (1) (2) (2) displacement components are u (1) n , vn and u n , vn , respectively, for first and second part. The constitutive relations for each layer can be expressed as | | ⎧ ∂u (k) ∂vn(k) μn (x) n ⎪ (k) ⎪ σnx x = (1 + κ) + (3 − κ) ⎪ ⎪ κ −1 ∂x ∂y ⎪ ⎪ ⎪ | | ⎨ (k) ∂vn ∂u (k) μn (x) n (k) (k = 1, 2) (1 + κ) + (3 − κ) σnyy = ⎪ κ −1 ∂y ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂vn(k) ∂u (k) ⎪ n (k) ⎩ τnx + ) = μ (x)( n y ∂y ∂x The equilibrium equations for each layer can be written as

(3.49)

92

3 General Model for Nonhomogeneous Materials with General Elastic …

⎧ ∂τ (k) ∂σ (k) ⎪ ⎪ ⎪ nx x + nx y = 0 ⎨ ∂x ∂y (n = 1, 2, . . . , M; k = 1, 2) (k) (k) ⎪ ∂τ ∂σ ⎪ nyy nx y ⎪ ⎩ + =0 ∂y ∂x

(3.50)

The boundary and continuity conditions can be expressed as those in Eqs. (2.110)– (2.114).

3.2.2 Solutions to Stress and Displacement Fields Similarly, as Sect. 3.1, substituting the constitutive equations into the main governing equations and then applying Fourier transform method, the solutions to displacement and stress fields can be drived. Thus, in the local coordinate system x0 O y0 , the displacement and stress components of the first part can be written as ⎧ { ∞E 4 ⎪ 1 ⎪ λ(1) (1) n j (s)y0 −isx 0 ds ⎪ u E n(1)j (s)A(1) (x , y ) = ⎪ 0 0 n n j (s)e ⎪ 2π ⎨ −∞ j=1

(n = 1, 2, . . . , M)

{ ∞E 4 ⎪ ⎪ 1 ⎪ (1) λ(1) (1) ⎪ n j (s)y0 −isx 0 ds Fn(1) (x , y ) = v ⎪ 0 0 j (s) An j (s)e ⎩ n 2π −∞ j=1 ⎧ { ∞ E 4 (1) ⎪ λ (s)y0 −isx0 1 (1) (1) (1) ⎪ ⎪ Bn j (s) An j (s)e n j ds ⎪ σnx0 x0 (x0 , y0 ) = eβn x0 +γn y0 ⎪ ⎪ 2π −∞ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ { ∞ E 4 (1) ⎨ λ (s)y0 −isx0 1 (1) (1) (1) Cn j (s)An j (s)e n j ds σny0 y0 (x0 , y0 ) = eβn x0 +γn y0 2π −∞ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ { ∞ E ⎪ 4 (1) ⎪ ⎪ λ (s)y0 −isx0 (1) (1) (1) ⎪ β x +γ y 1 ⎪ Dn j (s) An j (s)e n j ds ⎪ τnx0 y0 (x0 , y0 ) = e n 0 n 0 ⎩ 2π −∞

(3.51)

(n = 1, 2, . . . , M)

j=1

where the characteristic roots of the ODEs are as follows | |( 3−κ 1| − | γn 1+κ 2 | |( / βn 3 − κ 1| γn + − | γn =− 2 2 1+κ 2 | |( / γn βn 3 − κ 1| =− − + | γn 2 2 1+κ 2 | |( / γn βn 3 − κ 1| =− + + | γn 2 2 1+κ 2

γn βn (1) λn1 = − − 2 2 (1)

λn2

(1)

λn3

(1)

λn4

/

/ + βn / − βn / + βn / − βn

3−κ 1+κ 3−κ 1+κ 3−κ 1+κ 3−κ 1+κ

)2

(

/

+ 4 i s β n − γn )2

(

/

+ 4 i s β n + γn )2

(

/

+ 4 i s β n − γn )2

( + 4 i s β n + γn

/

3−κ 1+κ 3−κ 1+κ 3−κ 1+κ 3−κ 1+κ

) + 4s 2 ) + 4s 2 ) + 4s 2 ) + 4s 2

(3.52)

3.2 PE Model for Mixed-Mode Crack Problem

93

(1) (1) Here, the known expressions Bn(1) j (s), C n j (s), Dn j (s) (n = 1, 2, . . . , M; j = 1, 2, 3, 4) are intermediate quantities (Wang et al., 2014). To satisfy regularity condi(1) 2 2 (1) tions, u (1) n and vn must vanish when x 0 + y0 → ∞. The unknown function An j (s) (n = 1, 2, . . . , M; j = 1, . . . , 4) can be determined by introducing new auxiliary functions and applying the stress continuous conditions at crack faces y0 = 0, then we have (1) (1) lim σny (x0 , y0 ) = lim− σny (x0 , y0 ) (n = 1, 2, . . . , M) 0 y0 0 y0

(3.53)

(1) (1) lim τnx (x0 , y0 ) = lim− τnx (x0 , y0 ) (n = 1, 2, . . . , M) 0 y0 0 y0

(3.54)

y0 →0+

y0 →0

y0 →0+

y0 →0

At the crack faces, the following new auxiliary functions are introduced ⎧ ∂ (1) ⎪ ⎪ u (1) ⎨ f 1 (x0 ) = ∂ x [ y lim n (x 0 , y0 ) − lim− u n (x 0 , y0 )] + y0 →0 0 0 →0 (n = 1, 2, . . . , M) ∂ ⎪ (1) (1) ⎪ ⎩ f 2 (x0 ) = [ lim v (x0 , y0 ) − lim− vn (x0 , y0 )] y0 →0 ∂ x0 y0 →0+ n (3.55) Applying Eq. (3.55) and considering the continuity of the displacements at the x0 axis excluding the crack, we can obtain the following equation: {

b0

f i (x0 )dx0 = 0 (i = 1, 2)

(3.56)

a0

Using Eqs. (3.53)–(3.55) together with the Fourier integral transform method, the unknown functions A(1) n j (s) (n = 1, 2, . . . , M; j = 1, . . . , 4) can be expressed as ⎧ ⎫ ⎪ ⎪ A(1) ⎪ n1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (1) ⎨A ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

n2 (1) ⎪ An3 ⎪ ⎪ ⎪ ⎭ (1) ⎪ An4

⎡ ⎢ ⎢ =⎢ ⎣

Wn11 Wn Wn21 Wn Wn31 Wn Wn41 Wn

Wn12 Wn Wn22 Wn Wn32 Wn Wn42 Wn

⎫ ⎤⎧ { b0 ⎪ ⎪ isx0 ⎪ f 1 (x0 )e dx0 ⎪ ⎪ ⎨ ⎬ ⎥⎪ ⎥ a0 (n = 1, 2, . . . , M) (3.57) ⎥ { b0 ⎪ ⎦⎪ ⎪ ⎪ isx0 ⎪ ⎪ f 2 (x0 )e dx0 ⎭ ⎩ a0

The expressions Wn (n = 1, 2, . . . , M) and Wn jk (n = 1, 2, . . . , M, k = 1, 2, j = 1, . . . , 4) are intermediate quantities (Wang et al., 2014). In global coordinates system (xOy), the displacement and stress components of the second part can be written as

94

3 General Model for Nonhomogeneous Materials with General Elastic …

⎧ { ∞E 4 ⎪ 1 ⎪ λ(2) (2) n j (α)x−iαy dα ⎪ u (x, y) = E n(2)j (α)A(2) ⎪ n n j (α)e ⎪ 2π −∞ j=1 ⎨ { ∞E 4 ⎪ ⎪ 1 ⎪ λ(2) (2) ⎪ n j (α)x−iαy dα (x, y) = F (2) (α) A(2) v ⎪ n n j (α)e ⎩ 2π −∞ j=1 n j

(n = 1, 2, . . . , M)

(3.58) ⎧ { ∞E 4 ⎪ 1 (2) λ(2) (2) δ x ⎪ n n j (α)x−iαy dα ⎪ σ (x, y) = e Bn(2) ⎪ nx x j (α)An j (α)e ⎪ 2π ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ { ∞E 4 ⎨ 1 (2) λ(2) (2) n j (α)x−iαy dα Cn(2) (x, y) = eδn x σnyy j (α) An j (α)e ⎪ 2π −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ { 4 ⎪ ∞ E ⎪ ⎪ λ(2) (2) δn x 1 ⎪ n j (α)x−iαy dα Dn(2)j (α)A(2) (x, y) = e τ ⎪ n j (α)e ⎩ nx y 2π

(3.59)

−∞ j=1

where λ(2) n1 λ(2) n2 λ(2) n3 λ(2) n4

/ / 3−κ δn 1 2 2 4α + (δn ) − 4δn αi =− − κ +1 2 2 / / 1 3−κ δn =− − 4α 2 + (δn )2 + 4δn αi κ +1 2 2 / / 1 3−κ δn 4α 2 + (δn )2 + 4δn αi =− + κ +1 2 2 / / 1 3−κ δn 2 2 =− + 4α + (δn ) − 4δn αi κ +1 2 2

(n = 1, 2, . . . , M)

(n = 1, 2, . . . , M)

(n = 1, 2, . . . , M)

(n = 1, 2, . . . , M)

By using the boundary conditions and continuity conditions, we can obtain {

[Φ(α)]4M×4M A(2) (α)

{

} 4M×1

= a

b

{ [ P(u, α, x0 )]4M×2

} f 1 (u) du f 2 (u)

(3.60)

In Eq. (3.60) the matrices Φ and P can be determined using the boundary conditions and continuity conditions, and the solution process for them can refer to Sect. 3.1.2. Solving the linear equation group in Eq. (3.60), the unknown A(2) nj (n = 1, . . . , M; j = 1, . . . , 4) can be expressed in terms of unknown auxiliary functions f 1 (u) and f 2 (u). Similarly, as shown in Sect. 3.1.3, applying boundary conditions on the crack face, the following singular integral equations can be derived.

3.2 PE Model for Mixed-Mode Crack Problem

95

They can be written as {

b0

{ [kn11 (x0 , u) + Hn11 (x0 , u)] f 1 (u)du+

a0

b0

[kn12 (x0 , u) + Hn12 (x0 , u)

a0

1 4μn0 2π p1 (x0 ) ] f 2 (u)du = 1 + κ u − x0 eβn x0 (n = 1, 2, . . . , M; a0 < x0 < b0 ) (3.61) −

{

b0

a0

[kn21 (x0 , u) + Hn21 (x0 , u)− {

b0

1 4μn0 ] f 1 (u)du+ 1 + κ u − x0

[kn22 (x0 , u)+Hn22 (x0 , u)] f 2 (u)du =

a0

2π p2 (x0 ) eβn x0

(n = 1, 2, . . . , M; a0 < x0 < b0 )

(3.62)

where kni j (x0 , u) and Hni j (x0 , u) (i, j = 1, 2) are known expressions. More details on theses expressions can be found in Wang et al. (2014).

3.2.3 Crack-Tip SIFs The singular integral Eqs. (3.61) and (3.62) can be solved by using numerical method. Here, the method given by Erdogan and Gupta (1972) is applied. The unknown functions f 1 (u) and f 2 (u) can be written as (b0 − a0 )gk (u) (k = 1, 2) f k (u) = √ 2 (u − a0 )(b0 − u)

(3.63)

The mixed-mode SIFs can be expressed as / 2μ0n βn a0 b0 − a0 e g2 (a0 ) K I (a0 ) = lim 2(a0 − =− x→a0 1+κ 2 / / 2μ0n βn b0 b0 − a0 (1+2) e g2 (b0 ) K I (b0 ) = lim 2(x0 − b0 )σ y0 y0 (x0 , 0) = x→b0 1+κ 2 / / 2μ0n βn a0 b0 − a0 (1+2) e g1 (a0 ) K II (a0 ) = lim 2(a0 − x0 )τx0 y0 (x0 , 0) = − x→a0 1+κ 2 / / 2μ0n βn b0 b0 − a0 (1+2) e g1 (b0 ) K II (b0 ) = lim 2(x0 − b0 )τx0 y0 (x0 , 0) = x→b0 1+κ 2 /

x0 )σ y(1+2) (x0 , 0) 0 y0

(3.64)

(3.65)

(3.66)

(3.67)

96

3 General Model for Nonhomogeneous Materials with General Elastic …

3.3 PE Model for Dynamic Crack Problem 3.3.1 Basic Equations and Boundary Conditions Consider an embedded crack of length c along the gradient direction of FGM plate with thickness h subjected to an impact loading, as shown in Fig. 3.7. The impact loading on the crack face is represented by σ0 (x)H (t), where σ0 (x) is a known function and H (t) is the Heaviside function. In this section a dynamic piecewise-exponential model (DPE model) is developed. The variations of the modulus and mass density of plate may follow general forms, respectively. The plate is divided into M non-homogeneous sub-layers along the gradient direction with the modulus and mass density vary exponentially in each layer. μn (x) = μn0 eδn x , ρn (x) = ρn0 eβn x

(3.68)

The real modulus and mass density of the functionally graded plate are represented by μreal (x) and ρreal (x), which are known as an arbitrary continuous function. If the modulus and mass density of each sub-layer are approached by two exponentially functions with δn /= βn , then the above governing partially differential equations will become difficult to solve analytically. Thus, to make the problem solvable and also guarantee the accuracy of the approximation, the following idea is proposed: (1) Let δn = βn then the exponential function eβn x can be deleted due to the reason that eβn x exists simultaneously at both sides of these governing partially differential equations. Thus, the governing partial differential equations can be reduced to simple differential equations with constant parameters and solved analytically. Thus, the approximate modulus and mass density of each sub-layer can be written as

Fig. 3.7 The FGMs plate with arbitrary mechanical properties under an impact loading

3.3 PE Model for Dynamic Crack Problem

97

μn (x) = μn0 eδn x , ρn (x) = ρn0 eδn x

(3.69)

where μn0 , ρn0 and δ can be obtained in the next step. (2) To guarantee the accuracy of the approximation, let us consider that the approximate values of the modulus are exactly equal to the real ones at both upper surface and lower surface of each sub-layer; and the approximate values of mass density are exactly equal to the real ones at the mid-surface of each sub-layer. Namely ⎧ δn h n−1 ⎪ ⎨ μn (h n−1 ) = μreal (h n−1 ) = μn0 e δn h n μn((h n ) = μ )real (h n ) (= μn0 e ) ⎪ ⎩ ρn h n−1 +h n = ρreal h n−1 +h n = ρn0 eδn (h n−1 +h n )/2 2 2

(n = 1, 2, . . . , M)

(3.70)

From the above equation, the parameters μn0 , ρn0 and δ in Eq. (3.70) can be solved as | | ⎧ μreal (h n ) 1 ⎪ δ ln = ⎪ h n −h n−1 μreal (h n−1 ) ⎨ n μn0 = μreal((h n )e−δn)h n ⎪ ⎪ ⎩ ρn0 = ρreal h n−1 +h n e−δn (h n−1 +h n )/2

(3.71)

2

Thus, the real properties can be approached by a series of exponential functions, see Fig. 3.7. In this case, the modulus is continuous, while the mass density is discontinuous on the interface between layers. Since Li et al. (2002) revealed that the modulus has stronger effects than the mass density on the DSIFs, and particularly, the influence of the discontinuity of density will be proved to be slight in this study, and the approaching method of the mass density will be proved to be reasonable. Therefore, arbitrarily varying properties of FGMs can be approached accurately. The constitutive equations for each sub-layer are written in the following form | | ⎧ ∂u n (x, y, t) ∂vn (x, y, t) μn (x) ⎪ ⎪ (1 + κ) + (3 − κ) σnx x (x, y, t) = ⎪ ⎪ κ −1 ∂x ∂y ⎪ ⎪ ⎪ | | ⎨ ∂vn (x, y, t) ∂u n (x, y, t) μn (x) (1 + κ) + (3 − κ) (n = 1, 2, . . . ,M) σnyy (x, y, t) = ⎪ κ −1 ∂y ∂x ⎪ ⎪ | | ⎪ ⎪ ⎪ ∂vn (x, y, t) ∂u n (x, y, t) ⎪ ⎩ + σnx y (x, y, t) = μn (x) ∂y ∂x

The dynamic governing equations of n-th layers are

(3.72)

98

3 General Model for Nonhomogeneous Materials with General Elastic …

⎧ ∂σnx x (x, y, t) ∂σnx y (x, y, t) ∂ 2 u n (x, y, t) ⎪ ⎪ + = ρn (x) ⎨ ∂x ∂y ∂t 2 2 ⎪ ∂σnyy (x, y, t) ∂σnx y (x, y, t) ∂ vn (x, y, t) ⎪ ⎩ + = ρn (x) ∂y ∂x ∂t 2

(n = 1, 2, . . . ,M) (3.73)

The boundary conditions for the transient fracture problem in Fig. 3.7 are given by σ1x x (0, y, t) = 0 (−∞ < y < ∞)

(3.74)

σ1x y (0, y, t) = 0 (−∞ < y < ∞)

(3.75)

σ M x x (h, y, t) = 0 (−∞ < y < ∞)

(3.76)

σ M x y (h, y, t) = 0 (−∞ < y < ∞)

(3.77)

σnx y (x, 0, t) = 0 (n = 1, 2, . . . ,M)

(3.78)

σnyy (x, 0, t) = −σ0 (x)H (t) (a < x < b)

(3.79)

vn (x, 0, t) = 0 (0 < x < a or b < x < h)

(3.80)

The continuity conditions of displacement and stress between sub-layers can be written as { u n (h n , y, t) = u n+1 (h n , y, t) (n = 1, 2, . . . , M − 1) (3.81) vn (h n , y, t) = vn+1 (h n , y, t) { σnx x (h n , y, t) = σ(n+1)x x (h n , y, t) (n = 1, 2, . . . , M − 1) (3.82) σnx y (h n , y, t) = σ(n+1)x y (h n , y, t)

3.3.2 Solutions to Stress and Displacement Fields Substituting the constitutive Eq. (3.72) into the dynamic governing Eq. (3.73) and then using Laplace transform and Fourier transform methods, the displacement components can be obtained in Laplace domain (Guo et al., 2004a, b).

3.3 PE Model for Dynamic Crack Problem

99

⎧ 2 λn1 j y−isx ⎪ 1 { +∞ E E ⎪ u ∗n (x, y, p) = 2π ds ⎪ n1 j (s, p)An1 j e −∞ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎨ 4 { E λ y 2 ∞ E n2 j (α, p) An2 j e n2 j cos(αy)d α +π 0 ⎪ j=1 ⎪ ⎪ ⎪ ⎪ 2 4 ⎪ λn1 j y−isx λn2 j y ⎪ 1 { +∞ E A 2 {∞ E A ∗ ⎪ ds + π sin(αy)d α ⎩ vn (x, y, p) = 2π −∞ n1 j e n2 j e 0 j=1

(n = 1, 2, . . . ,M)

j=1

(3.83) where p is Laplace variable of t, and s and α are the Fourier variables of x and y, respectively. The variables marked by the superscript * denote those obtained from the Laplace transform. The intermediate parameters E n1 j (n = 1, 2, . . . ,M; j = 1, 2), E n2 j (n = 1, 2, . . . ,M; j = 1, . . . , 4) and the characteristic roots λn1 j and λn2 j can be obtained on solving the ODEs (Bai et al., 2013). An1 j and An2 j (n = 1, 2, . . . ,M; j = 1, . . . , 4) are unknown quantities that need to be solved. Since the structure in Fig. 3.3 is symmetric, only the displacement components for the left half part (y 0) γ = κ tip δ − , γ ' = κ tip δ + , κ tip = tip δ δ κ2 (x2 < 0) β=

210

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

} δ=

e(θ −π)ε (x2 > 0) , ϕ = εtip ln r + tip e(θ +π)ε (x2 < 0) tip

θ 2

For nonhomogeneous bimaterials, the auxiliary strains are not compatible with the auxiliary displacements except for the crack-tip location, i.e., εiaux j /=

1 aux (u + u aux j,i ) 2 i, j

(6.9)

Notably, if the interface Γ I is curved, as shown in Fig. 6.1, the auxiliary interface Γ A which is the division of material properties used in Eqs. (6.5) and (6.6), is still the x 1 -axis in the Cartesian coordinate system originating from the crack tip. Namely, whether the interface Γ I is straight or not, the auxiliary interface| Γ A is straight. tip tip | for x 2 > 0 Therefore, the crack-tip stiffness tensor is given by Ci jkl = Ci jkl | m=1 | tip tip | for x 2 < 0. But the material properties in Eq. (6.7) are those and Ci jkl = Ci jkl | m=2 evaluated at point x.

6.1.3 Relation Between the I-integral and the SIFs The relation between the energy release rate and the SIFs is J=

|K |2 1 , E ∗ cosh2 (πεtip )

|K |2 = K 12 + K 22

{ ) ( tip E m (plane stress) 1 1 1 1 ' tip = + ' , Em = Em (plane strain) E∗ 2 E 1' E2 tip 1−(ν )2

(6.10)

(6.11)

m

The I-integral is derived from the J-integral for the superimposed load of the actual field and the auxiliary field. Similarly, the I-integral can be expressed as I =

2 K 1 K 1aux + K 2 K 2aux E∗ cosh2 (πεtip )

(6.12)

By taking the auxiliary SIF vector {K 1aux , K 2aux } to be {1, 0} and {0, 1} successively, one can calculate the I-integrals I (1) and I (2) . Then, the SIFs can be calculated through the following relations: K1 =

E ∗ cosh2 (πεtip ) (1) I , 2

K2 =

E ∗ cosh2 (πεtip ) (2) I 2

(6.13)

6.2 Domain-Independent I-integral (DII-Integral)

211

6.2 Domain-Independent I-integral (DII-Integral) 6.2.1 DII-Integral for Materials with Continuous Properties Generally, the contour integral needs to be converted into an equivalent domain aux integral in numerical calculations. According to the relation σi j εiaux j = σkl εkl , the I-integral can thus be simplified as { I = lim

Γ→0

aux aux (σ jk ε jk δ1i − σiaux j u j,1 − σi j u j,1 )n i dΓ

(6.14)

Γ

As shown in Fig. 6.1, the contours Γ and Γ B are divided by the interface Γ I into two parts, i.e., Γ = Γ 1 + Γ 2 and Γ B = Γ B1 + Γ B2 . The domain A0 is enclosed by the contour Γ 0 . Taking the limit Γ → 0 leads to A0 → A. The domains A0 and A are divided into two parts by the material interface Γ I , i.e., A0 = A01 + A02 and A0 = A1 + A2 . For a traction-free crack, I-integral can be rewritten by using a smooth weight function q with values varying from 1 on Γ to 0 on Γ B as { I = − lim

Γ1 →0 Γ01

{

{ P1i n i qdΓ − lim

Γ2 →0 Γ02

P1i n i qdΓ +

1 2 ◯ ◯ (P1i − P1i )n i qdΓ

(6.15)

ΓI

aux aux where P1i = σ jk ε jk δ1i − σiaux j u j,1 − σi j u j,1 and the closed integral paths are Γ01 = + − − ΓB1 + ΓC + Γ1 + ΓI and Γ02 = ΓB2 + ΓI + Γ2− + ΓC− . The variables or expressions on the interface marked by the superscripts ➀ and ➁ mean that they belong to the domains A1 and A2 , respectively. By applying the divergence theorem to Eq. (6.15) in the domains A01 and A02 , the equivalent domain integral is defined by

{ I =−

{ (P1i q),i dA +

A

1 2 ◯ ◯ (P1i − P1i )n i qdΓ

(6.16)

ΓI

The first integral is of the same form as that for the interface crack between two homogenous materials. By applying the kinematic equation εi j = (u i, j + u j,i )/2 and the equilibrium equations σi j,i = 0 and σiaux j,i = 0, the second term can be simplified as { { aux Inonh = − P1i,i qdA = (σi j u aux (6.17) j,i1 − σi j,1 εi j )qdA A

A

As shown in Fig. 6.1, the components of the unit normal vector are n1 = 0 and n2 = 1 on the interface Γ I . According to the definition of the auxiliary fields, the traction, displacement and displacement gradient with respect to x 1 across the interface Γ I

212

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

are continuous, i.e., 1 2 1 2 1 2 ◯ ◯ ◯ ◯ ◯ ◯ σ2 j = σ2 j = σ2 j , u j = u j = u j, u j,1 = u j,1 = u j,1 1 2 1 2 1 2 aux◯ aux◯ aux◯ aux◯ aux◯ aux◯ σ2 j = σ2 j = σ2aux = uj = u aux = u j,1 = u aux j , uj j , u j,1 j,1 (6.18) Applying the continuity conditions to simplify the third integral in Eq. (6.16) yields

IΓI

⎫ ⎧ { ⎨ 1 + n (σ aux u )◯ 1 − n (σ aux ε )◯ 1 ⎬ ◯ ) n i (σi j u aux i j,1 1 i j j,1 ij ij qdΓ = 0 (6.19) = 2 − n (σ aux u )◯ 2 + n (σ aux ε )◯ 2⎭ ⎩ −n (σ u aux )◯ ΓI

i

ij

i

j,1

j,1

ij

1

ij

ij

Then, the I-integral I can be simplified as {

{

I =−

P1i q,i dA + A

aux (σi j u aux j,i1 − σi j,1 εi j )qdA

(6.20)

A

According to the symmetry of the stress tensor, the first integrand in Eq. (6.17) can be written as tip

tip

aux aux σi j u aux j,i1 = σi j (Si jkl σkl ),1 = σi j Si jkl σkl,1

(6.21)

Substituting the constitutive equation εi j = Si jkl (x)σkl and Eq. (6.21) into the I-integral yields { I =

aux aux (σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i )q,i dA A

{ tip

+

aux σi j [Si jkl − Si jkl (x)]σkl,1 qdA

(6.22)

A

It is noteworthy that the I-integral does not include any derivatives of material properties, which facilitates the practical implementation of the I-integral because it is usually difficult to obtain the derivatives of properties for many practical materials.

6.2.2 DII-Integral for Materials with Complex Interfaces In the above section, it can be found that the I-integral does not require the component material properties to be differentiable. However, in many practical materials such as particle reinforced materials, the interface crack may meet some other interfaces

6.2 Domain-Independent I-integral (DII-Integral)

213

Fig. 6.2 Integral domain cut by two interfaces ΓI12 and ΓJ

created by particles. In this section, the I-integral formulation is derived, when there is another curved interface in the integral domain. 1. Interface Integrals As shown in Fig. 6.2, there is another curved interface Γ interface near the crack tip. Thus, the domain A (the domain in Γ 0 when Γ → 0) is divided into four parts by Γ I = Γ I12 + Γ I34 and Γ J = Γ J13 + Γ J24 , i.e., A1 , A2 , A3 and A4 enclosed by the contours Γ 01 , Γ 02 , Γ 03 and Γ 04 , respectively. Here, the contours are as Γ01 = − + Γ2− + ΓC− , Γ03 = ΓB3 + ΓB1 + ΓC+ + Γ1− + ΓI12 + ΓJ13 , Γ02 = ΓB2 + ΓJ24 + ΓI12 − − − ΓI13 + ΓJ34 and Γ04 = ΓB4 + ΓI34 + ΓJ24 . The I-integral can be written as { I = − lim

Γ1 →0 Γ01

{ −

{

{ P1i n i qdΓ − lim

Γ2 →0 Γ02

P1i n i qdΓ −

P1i n i qdΓ

Γ03

∗ P1i n i qdΓ + Iinterface

(6.23)

Γ04 ∗ Here, Iinterface is a line integral along the interfaces expressed by ∗ ∗ ∗ ∗ ∗ Iinterface = I12 + I34 + I24 + I13

where ∗ I12 = ∗ I24

=

{ ΓI12

{

ΓJ24

{ 1 2 3 4 ◯ ◯ ◯ ◯ ∗ (P1i − P1i )n i qdΓ, I34 = (P1i − P1i )n i qdΓ 2 ◯

4 ◯

(P1i − P1i )n i qdΓ,

ΓI34

∗ I13

=

{

ΓJ13

1 3 ◯ ◯ (P1i − P1i )n i qdΓ

(6.24)

214

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

Here, the variables or expressions on the interface marked by the superscripts ➀, ➁, ➂ and ➃ belong to the domains A1 , A2 , A3 and A4 , respectively. Applying divergence theorem to Eq. (6.23) yields { I =−

{

A

aux ∗ σi j [Si jkl − Si jkl (x)]σkl,1 qd A + Iinterface tip

P1i q,i dA +

(6.25)

A

2. Interface Integrals According to Sect. 6.2.1, the integrals on the interfaces Γ I12 and Γ I34 can be proved ∗ ∗ = I34 = 0. The auxiliary stresses and displacements are defined to be zero, i.e., I12 by only using two homogeneous material components on both sides of the interface Γ I , so that the auxiliary stresses and displacement gradients are continuous across 1 3 aux◯ 1 = σ aux◯ 3 = σ aux and uaux◯ =u = uaux . Then, the interface Γ , i.e., σ aux◯ J13

∗ can be rewritten in tensor form as the integral I13 ∗ = I13

{

}

,i

,i

} 3 1 ◯ ◯ 3 − σ◯ 1 ) · uaux − σ aux : (ε◯ 3 − ε◯ 1 )n q dΓ n · σ aux · (u,1 − u,1 ) + n · (σ ◯ 1 ,1

,i

(6.26)

ΓJ13

According to the equilibrium condition on the interface Γ J13 , the tractions on both sides of the interface should be equal. That is 3 1 = n · σ◯ n · σ◯

(6.27)

Since the interface is perfectly bonded, the derivatives of the actual displacements with respect to the curvilinear coordinate ξ2 are equal on both sides of the interface, i.e., (

)1 ( )3 ∂u ◯ ∂u ◯ = ∂ξ2 ∂ξ2

(6.28)

∗ According to these continuity conditions, it can be proved that I13 = 0 and = 0. The proof process can be found in Sect. 5.2.2. Consequently, the interface integral is finally simplified as ∗ I24

∗ Iinterface =0

(6.29)

As the interface crosses the crack, it can be proved that the interface integral is zero. Due to no contribution of the interface integral, the I-integral can also be simplified as the form in Eq. (6.22). The interface does not affect the value of the I-integral so that the present I-integral is domain-independent for interfaces.

6.2 Domain-Independent I-integral (DII-Integral)

215

6.2.3 DII-Integral for a Curved Interface Crack If the crack face in the integral domain is curved, as shown in Fig. 6.3, the integral domain A should be divided into seven domains A1 ~ A7 in order to apply the divergence theorem (Wu et al. 2011). According to the analysis in Sect. 6.2.2, the interface integrals on the interfaces Γ I , Γ J and the auxiliary interface Γ A are zero so that the DII-integral can be simplified finally as { I =

aux aux (σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i )q,i dA A

{ tip

+

aux σi j [Si jkl − Si jkl (x)]σkl,1 qd A + Icrackface

(6.30)

A

where Icrackface is a line integral given by { Icrackface =

aux aux (σ jk ε jk δ1i − σiaux j u j,1 − σi j u j,1 )n i qdΓ

(6.31)

+ − ΓC+ +ΓC− +ΓCA +ΓCA

+ − Here, ΓCA and ΓCA are the top and bottom surface of a fictitious crack which is tangent to the crack tip. For a traction-free crack, the crack-face integral can be simplified as

Fig. 6.3 A curved interface crack

216

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

{ Icrackface =

{ aux (σiaux j εi j n 1 − n i σi j u j,1 )qdΓ −

ΓC+ +ΓC−

n i σi j u aux j,1 qdΓ

(6.32)

+ − ΓCA +ΓCA

Since several practical materials and structures consist of two or more constituent materials, there exist lots of complicated material interfaces between the constituents. When a crack propagates along the material interface, some complicated interfaces may appear around the crack tip. Therefore, it is very cumbersome to extract the fracture parameters by using an integral domain to exclude all the interfaces for such materials. The DII-integral method can avoid these difficulties.

6.2.4 Consideration of Dynamic Fracture Process The dynamic J-integral is given by { J = lim

Γ→0

| | (W + L)δ1i − σi j u j,1 n i dΓ

(6.33)

Γ

where W = σ jk ε jk /2 and L = ρ u˙ 2j /2 are the strain energy density and kinetic energy density, respectively. Additionally, ρ and u˙ j are the mass density and velocity, respectively. Because the near-tip asymptotic fields for a stationary crack are exactly the same as those for a static crack, the auxiliary field is defined by using the analytical solutions to a static crack in an infinite plate. Definitely, the auxiliary field is defined in Sect. 6.1.2. By superimposing the auxiliary field upon the actual field and extracting the mutual terms of the J-integral, one obtains the I-integral as { | I = lim

Γ→0

Γ

| 1 aux aux aux aux (σ jk ε jk + σ jk εaux )δ + ρ u ˙ u ˙ δ − σ u − σ u 1i j j 1i jk j,1 j,1 n i dΓ jk jk 2 (6.34)

For the perfectly-bonded interfaces, it can be proved that the interface integral is zero. By ignoring the body forces, the I-integral can be simplified as (Huang et al. 2016, 2018) { | | aux aux σi j u aux I = j,1 + σi j u j,1 − σ jk ε jk δ1i q,i dA A

+

{ { A

} tip aux σi j [Si jkl − Si jkl (x)]σkl,1 + ρ u¨ j u aux j,1 qdA

(6.35)

6.3 T-stress Evaluation

217

Finally, the DSIFs can be extracted from the DII-integral according to the equations given in Sect. 6.1.3.

6.3 T-stress Evaluation The DII-integral in Eq. (6.30) can be used to extract the T-stress of an interface crack just by adopting a suitable auxiliary field.

6.3.1 Auxiliary Field As shown in Fig. 6.4, the tangent line to the interface crack at crack-tip location as the fictitious (or auxiliary) interface ΓA is used. The auxiliary displacements, stresses and strains are defined by 1 ' E 1 + E 2' 1 =− ' E 1 + E 2'

u aux 1 =− u aux 2

) r F( ' 2 ln + (1 + νtip ) sin2 θ π d ) F( ' ' (1 − νtip )θ − (1 + νtip ) sin θ cos θ π

aux σ11 =− aux σ22 =− aux σ12 =−

' E tip

E 1'

E 2'

2F cos3 θ πr

E 1'

E 2'

2F cos θ sin2 θ πr

+ ' E tip + ' E tip

2F cos2 θ sin θ E 1' + E 2' πr

aux εiaux j = Si jkl (x)σkl (i, j, k, l = 1, 2)

(6.36)

(6.37) (6.38)

where F is a point force applied for the auxiliary field, and d is a reference length. ' ' and Poisson’s ratio νtip are given by The crack-tip elastic modulus E tip { ' E tip

=

E m |r|=0 (plane stress) ' ,v = Em | (plane strain) tip 1−νm2 | r =0

{

νm |r|=0 (plane stress) νm | (plane strain) 1−νm | r =0

(6.39)

218

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

Fig. 6.4 An interface crack

6.3.2 Extraction of the T-stress √ In the asymptotic expansion of the stress σ ij in Eq. (1.33), the term O( r ) and higher order terms do not contribute to the I-integral by taking the limit Γ → 0. Only the non-vanishing contribution is given by the T-stress. The relation between the I-integral and the T-stress is I =

T2 T1 F ' F = E1 E 2'

(6.40)

As a result, the T-stress Tm can be obtained by calculating the I-integral.

6.4 DII-Integral for 3D Interface Cracks 6.4.1 Definition of the I-integral on the Crack Front In 3D conditions, the crack front is usually a curve. As shown in Fig. 6.5, the pointwise energy release rate at point s takes the form { ( J (s) = lim cl (s) Γ→0

Γ(s)

) 1 σik εik δl j − u i,l σi j n j dΓ 2

(6.41)

where, the subscripts i, j, k, l = 1, 2, 3. By superimposing an auxiliary field onto the actual field and extracting the mutual part from the J-integral, one can obtain the I-integral as

6.4 DII-Integral for 3D Interface Cracks

219

Fig. 6.5 A curved crack front and related curvilinear coordinate system

{ I (s) = lim cl (s) Γ→0

Pl j n j dΓ

(6.42)

Γ(s)

where Pl j =

1 aux aux (σik εik + σikaux εik )δl j − u i,l σiaux j − u i,l σi j 2

(6.43)

6.4.2 Auxiliary Fields for 3D Interface Crack In the local curvilinear coordinate system as shown in Fig. 6.5, an arbitrary point p with the Cartesian coordinates (x 1 , x 2 , x 3 ) and its three curvilinear coordinates ξ 1 , ξ 2 and ξ 3 are defined by the following relations: ξ1 = r · e 1 , ξ2 = r · e 2 , ξ3 =

{s

dl

(6.44)

0

where the point s is on the crack front with minimum distance to point p, r is a position vector from the point s to p. The components of the auxiliary fields (σ aux = σiaux j ei e j , aux aux ε aux = εiaux e e and u = u e ) in (ξ , ξ , ξ ) coordinate system can be defined i j i 1 2 3 j i as ⎧ I tip (r,θ,εtip ,κm ) √ r ⎨ fi tip (K 1aux = 1, K 2aux = 0) tip 2π m cosh(πε ) (6.45) u iaux = 4μ II tip tip √ ⎩ fi tip(r,θ,ε ,κm ) r (K aux = 0, K aux = 1) 1 2 tip 2π 4μm cosh(πε )

u aux 3

aux 2K III = μ(s)

/

r θ sin 2π 2

220

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

where the angular functions f iI and f iII (i = 1, 2) are given in Eq. (6.8). Then the auxiliary stresses and strains are defined by tip

aux aux σiaux j = C i jkl (u k,l + u l,k )/2 (i, j, k, l = 1, 2, 3)

(6.46)

aux εiaux j = Si jkl (x) · σkl (i, j, k, l = 1, 2, 3)

(6.47)

/ Here, r = ξ12 + ξ22 and θ = tan−1 (ξ2 /ξ1 ) are the polar coordinates. Notably, the material properties used in the auxiliary stresses and displacements for any point are evaluated at point s. The above auxiliary fields are chosen to be the in-plane and anti-plane asymptotic fields that are defined in the curvilinear coordinate system. Specifically, the abovementioned auxiliary fields depend on the coordinate ξ1 and ξ2 only, and all partial derivatives of the auxiliary fields with respect to ξ3 are vanished. Therefore, the auxiliary stresses satisfy the relation ∂σi1aux ∂σ aux + i2 = 0, (i = 1, 2, 3) ∂ξ1 ∂ξ2

(6.48)

Consequently, for a curved crack front, the auxiliary fields do not satisfy equilibrium equations and gradient equations, i.e., ∇ • σ aux /= 0, ε aux /=

1 (∇uaux + uaux ∇) 2

(6.49)

6.4.3 Extraction of the SIFs The relation between the energy release rate and the SIFs is J (s) =

KK E ∗ (s) cosh2 (πεs )

+

2 K III ∗ 2μ (s)

(6.50)

1 1 2 1 1 2 = ' + ' , ∗ = + E ∗ (s) E 1 (s) E 2 (s) μ (s) μ1 (s) μ2 (s) Similarly, the I-integral can be expressed by the SIFs as I (s) = 2

aux K III K III K 1 K 1aux + K 2 K 2aux + μ∗ (s) E ∗ (s) cosh2 (πεs )

(6.51)

6.4 DII-Integral for 3D Interface Cracks

221

aux Through setting the auxiliary SIF vector {K 1aux , K 2aux , K III } to be {1, 0, 0}, {0, 1, 0} and {0, 0, 1} successively, the I-integrals I(1) (s), I(2) (s) and I(3) (s) can be calculated. Then, the SIFs K I , K II and K III can be obtained by the following relations:

E ∗ (s) cosh2 (πεs ) I(1) (s) 2 E ∗ (s) cosh2 (πεs ) I(2) (s) K2 = 2 K III = μ∗ (s)I(3) (s) K1 =

(6.52)

6.4.4 Domain Form of the I-integral The domain expression is naturally compatible with the finite element formulation of the field equations. By considering a small segment ls on a curved crack front as shown in Fig. 6.6 and at a point p on it, the virtual crack advance is assumed to be δξ 1 = cl (p)Δa(p), where Δa(p) is the magnitude of the advance crack. The total energy caused by the I-integral I (p) due to the crack advance δξ 1 on the segment l s is defined as { I = I ( p)Δa( p)dl (6.53) ls

If the segment l s is sufficient small, the I-integral I (p) varies slightly along ls . Due to this consideration, I (p) in Eq. (5.94) can be replaced by I (s). Therefore, I (s) can be obtained by the relation

Fig. 6.6 Integral volume around the crack front

222

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

I (s) =

I ΔA

(6.54)

Pl j n j cl ( p)qdΓ

(6.55)

{ where ΔA = ls Δa( p)dl is the total crack advance area. Since the segment ls is sufficiently small, the material properties at point s are used in the definitions of the auxiliary stresses and displacements for any point in the integral volume V. As shown in Fig. 6.6, it can be easily proved for a traction-free crack that { I ( p) = − lim

Γ→0

Γ0

where Γ0 = ΓB + ΓC+ + Γ − + ΓC− = ∂ A0 is a closed contour in ξ 1 − ξ 2 plane, and q is an arbitrary function with values varying smoothly from 1 on Γ to 0 on Γ B . By sweeping the area A0 (area enclosed by Γ 0 ) on l s and keeping them in ξ 1 − ξ 2 plane, we can get a tubular of volume V 0 enclosed by the closed surface S 0 . The closed surface S 0 consists of four curved surfaces S 0 , SC+ , SC− and S B that are generated by sweeping Γ − , ΓC+ , ΓC− and ΓB along l s , respectively, and two planar surfaces (sections) A0 and A'0 . As a result, I can be rewritten as I = − lim

∲

S→0 S

Pl j n j ql dS, ql = cl qΔa

(6.56)

0

where ql is a smooth test function with values varying from cl Δa on S to 0 on S B . Taking the limit S → 0 leads to V 0 → V. Due to the existence of the interface S I , the integral volume V is divided into two sub-volumes V 1 and V 2 . Similarly, the curved surfaces S and S B are divided into two sub-interfaces as well, i.e., S = S 1 + S 2 and S B = S B1 + S B2 . In sub-volumes V 1 and V 2 , the material properties are assumed to be continuous. Applying the divergence theorem to Eq. (5.97) in the sub-volumes V 1 and V 2 yields { I =− V1 +V2

{ (Pl j ql, j + Pl j, j ql )dV +

1 2 ◯ ◯ (Pl j − Pl j )ql n j dV

(6.57)

SI

According to the continuity conditions of the actual and auxiliary stresses and 1 2 1 2 ◯ ◯ ◯ ◯ 1 = (σ aux )◯ 2 and ◯ displacement gradients σ2 j = σ2 j , u i,1 = u i,1 , (σ2aux j ) 2j 1 = (u aux )◯ 2 , the interface integral (the second term) in Eq. (6.57) vanishes. (u aux )◯ i,1

i,1

According to the equilibrium of the actual stresses without body forces, i.e., σi j, j = 0,

6.4 DII-Integral for 3D Interface Cracks

223

I can be simplified as { I =

aux aux (u i,l σiaux j ql, j + u i,l σi j ql, j − σi j εi j ql,l )dV V

{ aux aux (u i,l σiaux j, j + u i,l j σi j − σi j,l εi j )ql dV

+

(6.58)

V

Whether the material properties are homogeneous or nonhomogeneous, the second term in Eq. (5.99) is not zero for a curved crack front, because the auxiliary fields are defined by using the in-plane and anti-plane crack-tip asymptotic fields.

6.4.5 DII-Integral As shown in Fig. 6.7, the integral volume V and its outer boundary S 0 are divided by two interfaces S I = S I12 + S I34 and S J = S J13 + S J24 into four parts, i.e., V = V 1 + V 2 + V 3 + V 4 and S 0 = S 01 + S 02 + S 03 + S 04 . Then, the integral in Eq. (5.99) can be expressed as { I = − lim

S→0 S01 +S02 +S03 +S04

Pl j n j ql dS

{

= − lim

S→0 S01 +SI12 +SJ13

Fig. 6.7 Integral volume cut by two interfaces S I and S J

{

Pl j n j ql dS − lim

S→0 − S02 +SJ24 +SI12

Pl j n j ql dS

224

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

{

{

−

∗ Pl j n j ql dS + Iinterface

Pl j n j ql dS − − S03 +SJ13 +SI34

(6.59)

− − S04 +SI34 +SJ24

where the interface integral is given by {

∗ Iinterface =

{

1 ◯ Pl j n j ql dS +

SI12 +SJ13

2 ◯ Pl j n j ql dS

− SJ24 +SI12

{

3 ◯ Pl j n j ql dS +

+ − SJ13 +SI34

{

4 ◯ Pl j n j ql dS

(6.60)

− − SI34 +SJ24

According to the position relations between the interfaces, the interface integral can be expressed as ∗ Iinterface =

{

1 2 ◯ ◯ (Pl j − Pl j )n j ql dS +

SI12

{

+

{

3 4 ◯ ◯ (Pl j − Pl j )n j ql dS

SI34

2 ◯

4 ◯

{

(Pl j − Pl j )n j ql dS + SJ24

1 3 ◯ ◯ (Pl j − Pl j )n j ql dS

(6.61)

SJ13

Referring to Sect. 5.6.5 and the continuity conditions on the interfaces, it can be ∗ = 0. Then, the I-integral can be expressed finally as proved that Iinterface 1 I (s) = ΔA

{ | V

aux aux u i,l σiaux j ql, j + u i,l σi j ql, j − σi j εi j ql,l aux aux aux + (u i,l σiaux j, j + u i,l j σi j − f i u i,l − σi j,l εi j )ql

| dV

(6.62)

The DII-integral for an interface crack in Eq. (5.111) is of the same form as that for an internal crack (see Sect. 5.6.5). A region with arbitrary interfaces can be chosen as the integral domain so that the present DII-integral can facilitate the numerical implementation for interface crack problems in the materials with complex interfaces.

6.5 Representative Interfacial Fracture Problems 6.5.1 Straight Interface Crack Example 1: A square plate consisting of two homogenous materials The SIFs are calculated for an interface crack between two homogeneous elastic semi-infinite planes, as shown in Fig. 6.8. The exact solution for K 1 and K 2 at the right crack tip for an infinite plate is given by

6.5 Representative Interfacial Fracture Problems

225

Fig. 6.8 A square plate with an interface crack under remote tension

√ K = K 1 + iK 2 = σ0 (1 + 2iε) πa(2a)−iε

(6.63)

The data used in the numerical analysis are given by: W = 30; a = 1; E 1 /E 2 = 2–1000; ν 1 = ν 2 = 0.3; σ 0 = 1; u 1 (x1 = ±W ) = 0. The generalized plane strain is considered. The mesh consists of 847 Q8 and 24 T6qp elements with a total of 871 elements and 2634 nodes. The four-layer elements around the crack tip are adopted as the integral domain for the √ calculation of the I-integral. Table 6.1 lists the SIFs K(a) normalized by K 0 = σ0 πa. It can be found that all the relative errors are within 0.15% for K 1 /K 0 and 1.5% for K 2 /K 0 compared with the exact solution. Excellent agreement indicates that the present method is reliable for interfacial fracture problems. Table 6.1 Normalized SIFs of an interface crack under remote tension (Example 1) E1 E2

α

β

Present W/a = 30

Sukumar et al. (2004) W/a = 30

Analytical W/a = ∞

K 1 (a)/K 0 K 2 (a)/K 0 K 1 (a)/K 0 K 2 (a)/K 0 K 1 (a)/K 0 K 2 (a)/K 0 2 0.333 0.095 1.0026

−0.0403

1.002

−0.0411

1.0011

−0.0397

4 0.600 0.171 1.0048

−0.0729

1.004

−0.0743

1.0035

−0.0720

8 0.778 0.222 1.0070

−0.0951

1.007

−0.0967

1.0059

−0.0939

20 0.905 0.259 1.0090

−0.1112

1.009

−0.1127

1.0081

−0.1098

40 0.951 0.272 1.0098

−0.1172

1.010

−0.1185

1.0090

−0.1157

100 0.980 0.280 1.0103

−0.1209

1.010

−0.1220

1.0096

−0.1194

1000 0.998 0.285 1.0106

−0.1230

1.010

−0.1239

1.0100

−0.1217

226

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

Fig. 6.9 A center crack located between two nonhomogeneous material plates under the remote tension (plane strain)

Example 2: A plate consisting of two nonhomogeneous materials In this example, an interface crack between two nonhomogeneous materials is selected to check the domain-independence of the I-integral for material nonhomogeneity. As shown in Fig. 6.9, the specimen of length 2L and width 2 W contains a center interface crack of length 2a. The Young’s moduli of material 1 and material 2 are defined as E 1 = E 0 eδx1 , E 2 = E 0 e−δx1 , δ =

1 2W

ln

E 1 (W ) E 1 (−W )

(6.64)

The other data are as follows: L = 200; W = 50; a/W = (0.4, √0.6, 0.8); E 0 = 1000; E(W )/E(−W ) = (10,100); ν 1 = ν 2 = 0.3; σ 0 = 1; K 0 = σ0 πa; and generalized plane strain. The mesh consists of 1470 elements and 4529 nodes. The seven different integral domains, which consist of the elements entirely and partially in a reference contour of radius RI , are employed to verify the domain-independence of the I-integral where he is the radial edge-length of the crack-tip element. Table 6.2 lists normalized SIFs computed using different integral domains. It can be observed that all the relative errors are within 0.15% between the maximal and minimal values of K 1 /K 0 and 1% between those of K 2 /K 0 . It indicates that the I-integral exhibits domain-independence for the materials with continuously varying properties. Example 3: A plate consisting of four materials Figure 6.10 shows a multi-layered plate with an interface crack of length 2a. The other data are as follows: L = 200; W = 50; a/W = (0.4, 0.6); E 1 = 10,000; √ E2 = E 1 /10; E 3 = E 1 /2; E 4 = E 1 /5; ν 1 = ν 2 = ν 3 = ν 4 = 0.3; σ 0 = 1; K 0 = σ0 πa;

6.5 Representative Interfacial Fracture Problems

227

Table 6.2 Normalized SIFs of an interface crack between two nonhomogeneous plates under far field tension (Example 2) RI he

a/W = 0.4 E(W )/E(−W ) = 10

a/W = 0.4 E(W )/E(−W ) = 100

a/W = 0.6 E(W )/E(−W ) = 100

a/W = 0.8 E(W )/E(−W ) = 100

K 1 (a) K0

K 1 (a) K0

K 1 (a) K0

K 1 (a) K0

K 2 (a) K0

K 2 (a) K0

K 2 (a) K0

K 2 (a) K0

4

1.3227

−0.1203

1.8251

−0.1922

1.8695

−0.1938

2.1666

−0.2604

8

1.3218

−0.1200

1.8244

−0.1915

1.8683

−0.1930

2.1649

−0.2595

16

1.3232

−0.1201

1.8261

−0.1917

1.8701

−0.1932

2.1660

−0.2595

32

1.3235

−.1200

1.8266

−0.1914

1.8693

−0.1930

2.1662

−0.2589

64

1.3219

−0.1195

1.8243

−0.1904

1.8697

−0.1922

2.1683

−0.2586

128

1.3228

−0.1196

1.8258

−0.1907

1.8701

−0.1919

2.1673

−0.2582

256

1.3228

−0.1196

1.8257

−0.1906

1.8697

−0.1919

2.1667

−0.2579

and generalized plane strain. The normalized SIFs at x 1 = a are listed in Table 6.3. It can be observed that for different integral domains, all the relative errors are within 0.1% among the values of K 1 /K 0 and 0.8% among those of K 2 /K 0 , regardless of the existence of vertical interfaces. Fig. 6.10 A multi-layered plate with an interface crack

228

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

Table 6.3 Normalized SIFs of an interface crack when the crack tip is close to another vertical interface (Example 3) RI / h e

a/ W = 0.4

a/ W = 0.6

K 1 (a)/K 0

K 2 (a)/K 0

K 1 (a)/K 0

K 2 (a)/K 0

5

0.7792

0.1007

1.2068

0.1428

20

0.7789

0.1007

1.2066

0.1430

50

0.7786

0.1012

1.2058

0.1433

100

0.7785

0.1015

1.2060

0.1434

150

0.7787

0.1014

1.2061

0.1434

200

0.7787

0.1015

1.2061

0.1434

6.5.2 A Circular-Arc Shaped Interface Crack In order to show the validation of the DII-integral for curved interface crack problems, as shown in Fig. 6.11, a square specimen with a circular-arc shaped interface crack along the boundary of inclusion is considered. The length 2 W remains fixed at 50 times of the inclusion radius R to simulate an infinite plate. The data used in numerical analysis are as follows: W = 25; R = 1; E 1 = 72.4 × 103 ; E 2 = 3.45 × 103 ; ν 1 = 0.22; ν 2 = 0.35; θ 0 = 10°–90°; σ 0 = 1; and generalized plane strain. The analytical solution of the complex SIF for the problem of an infinite plate with such a configuration is given by (Perlman and Sih 1967; Choi and Earmme 1992) √ (1 − 2iε)β K 1 − i K 2 = σ0 πR sin θ0 1+α

|

M12 − M2 − 1 − γ N1 e2εθ0 − α − 2αγ

| + e−(2ε+i)θ0

eε(π+θ0 )−i[θ0 /2−ε ln(2R sin θ0 )]

α=

μ1 +μ2 κ1 , μ2 +μ1 κ2

β=

μ1 +μ1 κ2 , μ2 +μ1 κ2

γ =

(6.65) μ1 1+κ2 , μ2 1+κ1

ε=

ln α 2π

M1 = cos θ0 + 2ε sin θ0 , N1 = cos θ0 − 2ε sin θ0 ) ( 3 1 − ε2 cos(2θ0 ) + 2ε sin(2θ0 ) M2 = + ε 2 + 4 4 The influence of the surface integral along crack faces and auxiliary crack faces on the I-integral is approximately zero when the integral domain is not too large for a curved crack. Therefore, in the above numerical example, the line integral Icrackface that appears in Eq. (6.30) is ignored. The numerical and analytical results are listed in Fig. 6.12. The results show that the maximum relative errors of K 1 /K 0 and √K 2 /K 0 between the numerical and analytical values are about 2%, where K 0 = σ0 πRθ0 . Then, six different integral domains of RI /he = 3–96 are selected to verify the convergence of the I-integral for a curved interface crack. Here, only θ 0 = 30° and θ 0 = 60° are considered, and the results are listed in Table 6.4. The relative errors of K 1 /K 0 and K 2 /K 0 are no more than 0.5 and 1%, respectively, for RI /R < 4.5%. It

6.5 Representative Interfacial Fracture Problems

229

Fig. 6.11 A rectangular plate with a circular-arc crack along the interface of a circular inclusion

Fig. 6.12 Normalized SIFs at the left crack tip varying with angle θ0

implies that the I-integral ignoring the crack-face integral is reliable for a small-size integral domain. Then, as shown in Fig. 6.11, a square plate of unit length (i.e., W = 1/2) is considered by replacing the stress load σ 0 with a small displacement u 2 = 0.01 at the√top surface. The volume fraction of the inclusion is taken to√be 20%, i.e., R = 0.2/π. Figure 6.13 shows the SIFs normalized by K 0 = σave πRθ0 where σave is the average normal stress on the top edge. The present results obtained by using the I-integral method agree well with those given by Liu and Xu (2000) and in addition, the present SIFs vary more smoothly as the angle θ0 increases.

230

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

Table 6.4 Normalized SIFs for different integral domains RI / h e

RI /R (%)

θ0 = 30◦ K 1 /K 0

θ0 = 60◦ K 2 /K 0

K 1 /K 0

K 2 /K 0

3

0.142

1.2322

0.5580

0.5482

0.9819

6

0.284

1.2326

0.5603

0.5498

0.9855

12

0.568

1.2325

0.5619

0.5503

0.9871

24

1.136

1.2319

0.5631

0.5501

0.9873

48

2.272

1.2309

0.5636

0.5493

0.9863

96

4.544

1.2294

0.5633

0.5477

0.9823

0.25

1.0

0.47

0.55

Relative errors Err (%)

Fig. 6.13 Normalized SIFs at the crack tip A

6.5.3 T-stress Evaluation of Biomaterial Strips As shown in Fig. 6.14a, a bi-material strip specimen is analyzed as a benchmark example. The strip contains two layers of width h1 and h2 . The length of the strip L is fixed at ten times of the width of the lower layer h2 , i.e., L = 10h2 , to simulate an infinite length strip. A horizontal interface crack is of length L/ 2. The centralized forces P0 = 1 are applied at x2 = h 1 /2 and x2 = −h 2 /2 on the left edge of the strip. The T-stress in the upper strip is given by (Kim and Vlassak 2006) ( )) 1 6M − Ph 1 4Pη + 3Pη2 3Mη2 M3 Δ− + T1 = Σ + + 2 2 2h 1 η h1 h 1 I0 2h 21 ( ) √ P cos(ω − ω f ) M sin(γ + ω − ω f ) + − 2ξ (6.66) √ √ h1 A h 21 I (

where M3 is the moment caused by the constraint on the right edge of the strip

6.5 Representative Interfacial Fracture Problems

231

Fig. 6.14 Bi-material strip specimens under edge loads 2 , P = P0 − M3 = P0 h 1 +h 2

η=

h1 , h2

Σ=

1+α , 1−α

C 2 M3 , h1

Δ=

M = −C3 M3

1+2Ση+Ση2 2η(1+Ση)

1 1 , I = , 2 3 1 + Σ(4η + 6η + 3η ) 12(1 + Ση3 ) (( ) ) ) ( ) ( 1 2 1 1 1 1 Δ I0 = Σ Δ − + Δ− + 3 − Δ− + η η 3 η η 3η ( ) ( √ ) Σ γ = sin −1 6Ση2 (1 + η) AI , C2 = ΣI0 η1 − Δ + 21 , C3 = 12I 0

(6.67) (6.68)

A=

(6.69)

(6.70)

The bi-material parameters α and β are defined respectively by α=

μ1 (κ2 − 1) − μ2 (κ1 − 1) μ1 (κ2 + 1) − μ2 (κ1 + 1) ,β = μ1 (κ2 + 1) + μ2 (κ1 + 1) μ1 (κ2 + 1) + μ2 (κ1 + 1)

(6.71)

The real non-dimensional constants ω, ω f and ξ used in numerical computations are given in Table 6.5. The T-stress in the lower strip T2 = T1 /Σ

(6.72)

In order to test the stability of numerical results, 13 integral domains of different size are selected, i.e., RI / h e is taken to be 4, 8, 15, 30, 60, 100, 150, 200, 300, 400, 600, 800 and 1000, sequentially. Let T1Anal denote the T-stress in the upper layer obtained by the formulation given by Kim and Vlassak (2006). The T-stress results using different integral domains normalized by T1Anal , are shown in Fig. 6.15. It can

232

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

Table 6.5 Material constants and normalized T-stresses computed using Eq. (6.66) (T Anal ) and by 1 the I-integral method (T M 1 ) ω(◦ ) ω f

h1 h2

E1 E2

v1

1

7/3

1/3 1/3 0.4 0.1 47.8

v2

α

β

ξ

1.0957 0.5441

T1Anal h 2 /P0 (analytical) T1M h 2 /P0 (I-integral) 0.0709

0.0702

20/9 1/4 1/8 0.4 0.2 43.0

1.0055 0.5399

0.0784

0.0773

4

2/5 2/5 0.6 0.1 49.8

1.1860 0.9684

0.1310

0.1317

4

1/4 1/4 0.6 0.2 45.1

1.0991 0.9643

0.1424

0.1410

1/3 1/3 0.4 0.1 49.4

1.1227 0.3271 −0.5201

−0.5147

20/9 1/4 1/8 0.4 0.2 45.1

1.0431 0.3224 −0.5097

−0.5043

4

2/5 2/5 0.6 0.1 51.8

1.2583 0.6438 −0.5470

−0.5413

4

1/4 1/4 0.6 0.2 47.8

1.1843 0.6389 −0.5340

−0.5291

0.5 7/3

be noted that T 1 converges towards a stable value for each geometry and material parameters when RI / h e ≥ 150. And when RI / h e ≥ 150, all the relative errors of T1 /T1Anal are within 2.0% for the different integral domains. Table 6.5 shows the average values of the T-stress (T1M ) for RI /he = 150–1000 and the analytical values (T1Anal ). All the T-stress results are normalized by σ0 = P0 / h 2 . It can be found that the maximum relative error between T1M and T1Anal is about 1.5% for all cases. Good agreement indicates that accurate results can be obtained by using the present I-integral method. As shown in Fig. 6.14b, a bi-material strip of length L containing two interfaces perpendicular to each other is investigated to verify the domain-independence of the I-integral. The tip of the edge interface crack is offset by a distance of δ horizontally from the vertical interface. The data used in the numerical analysis are as follows: L = 10; h1 = h2 = 1; δ = 0.05; P0 = 1; and generalized plane strain. The T-stress is determined by using 13 integral domains of RI /he = 4–1000, in which integral domains of RI /he > 200 contain the vertical interface. Figure 6.16 shows that the T-stresses are normalized by T1K varying with RI /he , where T1K denotes the T-stress obtained for RI /he = 1000. It can be noted that when RI / h e ≥ 150, the variations of T1 /T1K are no more than 0.2% as the integral domain size increases. The vertical interface has no influence on the I-integral. From the above results, it can be noted that the T-stress converges towards a stable value only when the integral domain size reaches a threshold. In numerical computations, the threshold size of the integral domain for obtaining a stable Tstress value should be examined at first. The integral domain should be much larger for the extraction of the T-stress than for the extraction of the SIFs. In order to achieve satisfactory precision, the integral domain should have enough large size. In general, it is difficult to select an enough large integral domain without interfaces for the materials with complex interfaces. Therefore, the domain-independence of the I-integral for material interfaces is of practical significance.

6.5 Representative Interfacial Fracture Problems Fig. 6.15 Influences of the integral domain size on the T-stress for a h 1 / h 2 = 1 and b h 1 / h 2 = 0.5

Fig. 6.16 Variation of T1 /T1K with RI /he

233

234

6 Interfacial Fracture of Nonhomogeneous Materials with Complex …

References Choi, N.Y., and Y.Y. Earmme. 1992. Evaluation of stress intensity factors in circular arc-shaped interfacial crack using L integral. Mechanics of Materials 14: 141–153. Cisilino, A.P., and J.E. Ortiz. 2005. Three-dimensional boundary element assessment of a fibre/matrix interface crack under transverse loading. Computers and Structures 83: 856–869. Gosz, M., J. Dolbow, and B. Moran. 1998. Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracks. International Journal of Solids and Structures 35 (15): 1763–1783. Huang, K., L.C. Guo, H.J. Yu, et al. 2016. A domain-independent interaction integral method for evaluating the dynamic stress intensity factors of an interface crack in nonhomogeneous materials. International Journal of Solids and Structures 100: 547–557. Huang, K., L.C. Guo, and H.J. Yu. 2018. Investigation on mixed-mode dynamic stress intensity factors of an interface crack in bi-materials with an inclusion. Composite Structures 202: 491–499. Kim, J.H., and J.J. Vlassak. 2006. T-stress of a bi-material strip under generalized edge loads. International Journal of Fracture 142: 315–322. Liu, Y.J., and N. Xu. 2000. Modeling of interface cracks in fiber-reinforced composites with the presence of interphases using the boundary element method. Mechanics of Materials 32: 769–783. Matos, P.P.L., R.M. McMeeking, P.G. Charalambides, et al. 1989. A method for calculating stress intensities in bimaterial fracture. International Journal of Fracture 40: 235–254. Merzbacher, M.J., and P. Horst. 2009. A model for interface cracks in layered orthotropic solids: Convergence of modal decomposition using the interaction integral method. International Journal for Numerical Methods in Engineering 77: 1052–1071. Nahta, R., and B. Moran. 1993. Domain integrals for axisymmetric interface crack problems. International Journal of Solids and Structures 30 (15): 2027–2040. Nakamura, T. 1991. Three-dimensional stress fields of elastic interface cracks. Journal of Applied Mechanics 58: 939–946. Perlman, A.B., and G.C. Sih. 1967. Elastostatic problems of curvilinear cracks in bonded dissimilar materials. International Journal of Engineering Science 5: 845–867. Smelser, R.E., and M.E. Gurtin. 1977. On the J-integral for bi-material bodies. International Journal of Fracture 13: 382–384. Sukumar, N., Z.Y. Huang, J.H. Prévost, et al. 2004. Partition of unity enrichment for bimaterial interface cracks. International Journal for Numerical Methods in Engineering 59: 1075–1102. Williams, M.L. 1959. The stresses around a fault or crack in dissimilar media. Bulletin of the Seismological Society of America 49 (2): 199–204. Wu, L.Z., H.J. Yu, L.C. Guo, et al. 2011. Investigation of stress intensity factors for an interface crack in multi-interface materials using an interaction integral method. Journal of Applied MechanicsTransactions of the ASME 78: 061007. Yu, H.J., L.Z. Wu, L.C. Guo, et al. 2010. Interaction integral method for the interfacial fracture problems of two nonhomogeneous materials. Mechanics of Materials 42: 435–450. Yu, H.J., L.Z. Wu, and H. Li. 2012. T-stress evaluations of an interface crack in the materials with complex interfaces. International Journal of Fracture 177: 25–37.

Chapter 7

Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

When the components possess disparate thermal expansion coefficients, the composite subjected to temperature changes will develop residual thermal stresses which will act as the driving force for crack growth. The investigations on the thermal fracture of FGMs (Noda and Jin 1993, 1995; Jin and Noda 1994) showed that the singularity of the crack-tip stress field is inverse square root singularity, i.e., σi j ∼ r −1/2 , for nonhomogeneous materials subjected to thermal loading. The Iintegral method was successfully developed to extract the SIFs and T-stress in the thermal fracture analysis of nonhomogeneous materials with differentiable properties (Song and Paulino 2006; Johnson and Qu 2007; Amit and Kim 2008). However, the I-integral losses its domain-independence feature across the interface, where the thermal expansion coefficient is discontinuous (Guo et al. 2012; Zhang et al. 2018, 2019). By designing a new auxiliary field which is distinct from the analytical solution of the asymptotic fields of a crack, a new domain-independent I-integral was established for the thermal fracture of composite materials (Yu and Kitamura 2015; Yu et al. 2015; Guo et al. 2020). This chapter introduces the I-integral for nonhomogeneous thermoelastic materials with continuous or discontinuous properties.

7.1 Internal Crack Under Thermal Loading 7.1.1 Basic Equations of Thermoelasticity For thermoelasticity, the change of the temperature produces the deformation which can be formulated as εithj = α th (ϑ − ϑ0 )δi j = α th ∆ϑδi j

(7.1)

where ϑ and ϑ 0 are current and initial temperatures, respectively. In the twodimensional condition, α th = α for plane stress and α th = (1 + ν)α for plane © Science Press 2023 L. C. Guo et al., Fracture Mechanics of Nonhomogeneous Materials, https://doi.org/10.1007/978-981-19-4063-7_7

235

236

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

strain, where α is the thermal expansion coefficient. The total strain εi j includes the mechanical strain εimj generated by the mechanical loading and the thermal strain εithj caused by thermal loading, i.e., εi j = εimj + εithj

(7.2)

The total strain can be expressed by the gradients of the displacements as εi j =

1 (u i, j + u j,i ) 2

(7.3)

The elastic stress only causes the mechanical strain and thus, the constitutive equations become εimj =

1 3 − κ(x) σi j − σkk δi j 2μ(x) 1 + κ(x)

(7.4)

Ignoring body forces, the stresses σi j need to satisfy the equilibrium equations σi j, j = 0

(7.5)

7.1.2 I-integral for Thermoelasticity For a crack in two-dimensional linear thermoelastic solid, as shown in Fig. 7.1, the J-integral is defined as { ( J = lim

Γ→0

Γ

) 1 m σ jk ε jk δ1i − σi j u j,1 n i dΓ 2

(7.6)

The superimposing of the actual state (u i ) and an auxiliary state (u iaux ) leads to a new equilibrium state for which the J-integral is J (u i + u iaux ). By extracting the cross terms from the J-integral, one can obtain the I-integral as { | I = lim

Γ→0

Γ

| 1 aux m aux aux aux (σ ε + σ jk ε jk )δ1i − σi j u j,1 − σi j u j,1 n i dΓ 2 jk jk

(7.7)

Therefore, the mutual energy momentum tensor for a thermoelastic solid is P1i =

1 aux m aux aux (σ ε + σ jk εaux jk )δ1i − σi j u j,1 − σi j u j,1 2 jk jk

(7.8)

7.1 Internal Crack Under Thermal Loading

237

Fig. 7.1 A contour integral around the crack tip

7.1.3 Auxiliary Field Since the thermal loading does not alter the singularity of the near-tip stress field, this paper also adopts the incompatibility formulation to define the auxiliary displacement, stress and strain, i.e., / u iaux

=

| r | aux I K I f i (θ ) + K IIaux f iII (θ ) 2π

σiaux j =

K Iaux giIj (θ ) + K IIaux giIIj (θ ) √ 2π r

aux εiaux j = Si jkl (x)σkl

(7.9)

(7.10) (7.11)

The angular functions f iI , f iII , giIj , and giIIj depend only on the crack-tip material constants which are identical to those in Eqs. (5.10) and (5.11). Here, K Iaux and K IIaux are the auxiliary mode-I and mode-II SIFs, respectively, and C ijkl and S ijkl are the elastic stiffness and compliance tensors which satisfy the relations Ci j pq S pqkl = δik δ jl . For isotropic media Si jkl =

1 κ −3 κ −3 δik δ jl + δi j δkl , Ci jkl = 2μδik δ jl − μ δi j δkl 2μ 8μ κ −1

where κ and μ are the Kolosov constant and shear modulus, respectively.

7.1.4 Extraction of the SIFs for Thermoelastic Media For thermoelastic materials, the J-integral can also be expressed with the SIFs as

238

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

J=

K I2 + K II2 ' E tip

(7.12)

The J-integral for the superimposed state J (u i + u iaux ) can be expressed by J=

(K I + K Iaux )2 + (K II + K IIaux )2 ' E tip

(7.13)

Expanding the J-integral, one can obtain the relation between the I-integral and the SIFs I =

2 aux + K II K IIaux ) ' (K I K I E tip

(7.14)

Taking the vector [K Iaux , K IIaux ] to be [1, 0] and [0, 1], sequentially, one can solve the mode I and modeII SIFs separately by computing I-integral twice.

7.1.5 Domain Form of the I-integral 1. Conversion of the I-integral into domain form As shown in Fig. 7.2, it is considered that an integral domain A is cut by an interface Γ I into two sub-domains A1 and A2 . In each sub-domain, the thermomechanical properties are assumed to be continuously differentiable. By introducing a weight function q with the value varying from 1 on Γ ε to 0 on ΓB = ΓB1 + ΓB2 + ΓB3 , the I-integral can be expressed as { I = − lim

Γε →0 Γ O 1

|

| 1 aux m aux aux (σ jk ε jk + σ jk εaux )δ − σ u − σ u 1i j,1 i j j,1 n i qdΓ jk ij 2

Fig. 7.2 Integral domain cut by an interface ΓI

7.1 Internal Crack Under Thermal Loading

|

{ −

Γ2O

239

| 1 aux m aux aux (σ jk ε jk + σ jk εaux )δ − σ u − σ u 1i j,1 i j j,1 n i qdΓ + IΓC + Iinterface jk ij 2 (7.15)

The detailed derivations of Eq. (7.15) can refer to Sect. 5.2.1. Here, the closed integral paths are Γ1O = Γ − + ΓC− + ΓB1 + ΓI + ΓC+ + ΓB3 and Γ2O = ΓB2 + ΓI− . As Γ → 0, the region enclosed by the contour Γ1O becomes the domain A1 . The line integral along the crack faces is |

{ IΓC = ΓC− +ΓC+

| 1 aux m aux aux aux (σ ε + σ jk ε jk )δ1i − σi j u j,1 − σi j u j,1 n i qdΓ 2 jk jk

(7.16)

The interface integral Iinterface is a line integral with respect to the jump of P1i along ΓI and its expression is given by { | Iinterface = − ΓI

|◯ 2 1 aux m aux aux aux n i qdΓ (σ jk ε jk + σ jk ε jk )δ1i − σi j u j,1 − σi j u j,1 2

{ |

|◯ 1 1 aux m aux aux aux n i qdΓ + (σ jk ε jk + σ jk ε jk )δ1i − σi j u j,1 − σi j u j,1 2 ΓI || { || 1 aux m aux aux aux (σ ε + σ jk ε jk )δ1i − σi j u j,1 − σi j u j,1 n i qdΓ (7.17) =− 2 jk jk ΓI

2 − (∗)◯ 1 denotes the jump of a Here, the double bracket symbol [[∗]] = (∗)◯ quantity across the interface. The superscripts ➀ and ➁ means that they belong to the domains A1 and A2 , respectively. By applying the divergence theorem, the I-integral is converted into an equivalent domain integral as I =−

| } 2 { {| ∑ 1 aux m aux aux (σ jk ε jk + σ jk εaux )δ − σ u − σ u dA 1i j,1 i j jk ij j,1 q 2 ,i S=1 AS

+ IΓC + Iinterface

(7.18)

2. Interface integral According to the definition of the auxiliary fields, it can be observed that aux m aux σ jk ε jk = σ jk S jkrs σr s = σr s εraux s

Then, the mutual energy momentum tensor can be simplified as

(7.19)

240

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces aux m aux P1i = σ jk ε jk δ1i − σiaux j u j,1 − σi j u j,1

(7.20)

Correspondingly, one can simplify the interface integral as { Iinterface =

aux aux m [[σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i ]]n i qdΓ

(7.21)

ΓI

According to the continuity conditions on the interface [Eqs. (5.31) and (5.32)] and derivation process in Sect. 5.2, it can be proved easily that { Iinterface =

{ th [[σiaux j εi j ]]n 1 qdΓ

=

ΓI

[[σiiaux α th (ϑ − ϑ0 )]]n 1 qdΓ

(7.22)

ΓI

According to the definitions of the auxiliary fields, the auxiliary stresses are continuous across the interface. For a non-adiabatic interface across which the temperature is continuous, the interface integral reduces to { Iinterface =

σiiaux [[α th ]](ϑ − ϑ0 )n 1 qdΓ

(7.23)

ΓI

If the thermal expansion coefficient is continuous across the interface ([[α th ]] = 0) or the temperature on the interface does not change ( ϑ|ΓI = ϑ0 |ΓI ), the interface integral becomes zero. Otherwise, the interface integral must be considered. 3. DII-integral for mechanical interface For a thermoelastic solid without body force, we have aux aux σi j,i = 0, σiaux j,i = 0, σi j εi j,1 = σi j u j,i1 tip

tip

aux aux σi j u aux j,i1 = σi j (Si jkl σkl ),1 = σi j Si jkl σkl,1

(7.24) (7.25)

Then, the divergence of P1i given by m aux th aux aux aux aux P1i,i = σiaux j,1 εi j + σi j (εi j,1 − εi j,1 ) − σi j,i u j,1 − σi j u j,i1 − σi j,i u j,1 − σi j u j,i1 (7.26)

can be simplified as tip

m aux th aux aux aux th P1i,i = σiaux j,1 εi j − σi j εi j,1 − σi j u j,i1 = σi j,1 [Si jkl (x) − Si jkl ]σkl − σi j εi j,1 (7.27)

The I-integral can be rewritten as

7.1 Internal Crack Under Thermal Loading

I =

2 { ∑ S=1 A

+

241

aux aux m (σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i )q,i dA

S

2 { ∑ S=1 A

tip

σiaux j,1 [Si jkl − Si jkl (x)]σkl qd A +

2 { ∑ S=1 A

S

th σiaux j εi j,1 qd A + IΓC + Iinterface

S

(7.28) The first two terms are the mechanical parts, and the third term is the thermal part which is given by I th =

2 { ∑ S=1 A

S

th σiaux j εi j,1 qd A =

2 { ∑ S=1 A

σiiaux [α th (ϑ − ϑ0 )],1 qdA

(7.29)

S

For a traction-free crack, the crack-face integral can be easily proved to be zero, i.e., IΓC = 0. Then, the I-integral can be expressed finally as I = I m + I th + Iinterface { aux aux m aux tip I m = {(σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i )q,i + σi j,1 [Si jkl − Si jkl (x)]σkl q}dA A

I th =

2 { ∑ S=1 A

{ Iinterface =

th σiiaux [α,1 (ϑ − ϑ0 ) + α th (ϑ − ϑ0 ),1 ]qdA

S

σiiaux [[α th ]](ϑ − ϑ0 )n 1 qdΓ

(7.30)

ΓI

Notation that the mechanical part I m is not related to the derivatives of material properties, and thus it can be expressed as the integrals on the entire integral domain. The domain integrals in the I-integral do not contain the derivatives of the elastic properties, but contain the derivative of the thermal expansion coefficient. Next, the influences of the discontinuity of the mechanical and thermal properties on the I-integral in Eq. (7.30) through defining two types of interfaces, i.e., • Strong mechanical interface: across which elastic constants are discontinuous. • Weak mechanical interface: across which elastic constants are continuous but their gradients are discontinuous. • Strong thermal interface: across which thermal expansion coefficient is discontinuous. • Weak thermal interface: across which thermal expansion coefficient is continuous but its gradient is discontinuous.

242

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

According to the definitions of the interfaces, let us discuss whether or not the I-integral can still be applicable when the integral domain contains an interface. (1) There are no derivatives of the mechanical properties in Eq. (7.30) so that the I-integral is effective for weak mechanical interfaces. (2) The interface integral does not contain the jumps of elastic constants or even the elastic constants. If the thermal parameters are continuous across the interfaces, the interface integral vanishes. Therefore, the I-integral is domain-independent for strong mechanical interfaces. (3) The I-integral contains the gradient of the thermal expansion coefficient (see the term I th ). Therefore, the integral domain should be divided into several subdomains by the weak thermal interfaces. After such treatment, the I-integral is effective for the weak thermal interfaces. Moreover, the thermal expansion coefficient is continuous for a weak thermal interface, i.e., α 1 = α 2 . Therefore, the I-integral is domain-independent for weak thermal interfaces. (4) The integral domain must be divided by the strong thermal interfaces into subdomains and the interface integral must be computed. Therefore, the present I-integral is domain-dependent for the strong thermal interfaces. In all, the I-integral is effective for all of the above-mentioned four interfaces. The interface integral is zero for strong mechanical interfaces, weak mechanical interfaces and weak thermal interfaces while the interface integral must be calculated for strong thermal interfaces.

7.2 Interface Crack Under Thermal Loading 7.2.1 Definition of the I-integral For an interface crack in the two-dimensional linear thermoelastic solid, as shown in Fig. 7.3, the J-integral and I-integral are identical with Eqs. (7.6) and (5.9), respectively, for an internal crack. Namely, the I-integral is given by { | I = lim

Γε →0

Γε

| 1 aux m aux aux aux (σ ε + σ jk ε jk )δ1i − σi j u j,1 − σi j u j,1 n i dΓ 2 jk jk

(7.31)

The domain form of the I-integral ensures high accuracy in numerical calculations. By using the divergence theorem, an equivalent domain integral can be deduced from Eq. (7.31) as I =

2 { ∑ S=1 A

S

aux aux m (σiaux j u i,1 + σi j u i,1 − σik εik δ1 j )q, j dA

7.2 Interface Crack Under Thermal Loading

243

Fig. 7.3 An interface crack between the two nonhomogeneous materials

+

2 { ∑ S=1 A

aux aux m aux m (σiaux j u i, j1 + σi j u i, j1 − σi j,1 εi j − σi j εi j,1 )qd A + IΓI

(7.32)

S

where { IΓI = − { =

aux m aux [[σ jk ε jk δ1i − σiaux j u j,1 − σi j u j,1 ]]n i qdΓ ΓI

1 aux m aux ◯ [(σ jk ε jk δ1i − σiaux j u j,1 − σi j u j,1 )

ΓI

2 ]n qdΓ aux m aux ◯ − (σ jk ε jk δ1i − σiaux i j u j,1 − σi j u j,1 )

(7.33)

The continuity conditions of the stresses and displacements on the interface Γ I can be written as { 1 2 1 2 1 2 ◯ ◯ ◯ ◯ ◯ ◯ σi2 = σi2 = σi2 , ui = ui = ui , u i,1 = u i,1 = u i,1 1 2 1 2 1 2 aux ◯ aux ◯ aux ◯ aux ◯ aux ◯ aux ◯ aux σi2 = σi2 = σi2aux , u i = ui = u iaux , u i,1 = u i,1 = u i,1 (7.34) The auxiliary fields have the same continuity conditions on the interface because the auxiliary fields are defined by using the analytical solutions to the near-tip fields of an interface crack. According to the above continuity conditions and n1 = 0 on

244

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

the interface Γ I , it is easily proved that IΓI = 0. According to the derivation in the above section, the I-integral for an interface crack between two nonhomogeneous materials can be simplified as { I =

aux aux m (σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i )q,i dA A

{

{ tip σi j [Si jkl

+

−

aux Si jkl (x)]σkl,1 qd A

+

A

σiiaux [α th (ϑ − ϑ0 )],1 qd A

(7.35)

A

As the domain of the integral is chosen arbitrarily in the above derivation process, the I-integral is domain-independent for an interface crack between two nonhomogeneous materials.

7.2.2 Auxiliary Field The analytical solutions to the crack-tip fields of an interface crack in a pure elastic solid are adopted as the auxiliary fields for thermo-elasticity. Namely, the auxiliary fields are given by

u iaux

=

⎧ ⎨ ⎩

f iI (r,θ,εtip ,κm ) / r tip 2π 4μm cosh(πεtip ) tip f iII (r,θ,εtip ,κm ) / r tip 2π 4μm cosh(πεtip ) tip

(K 1aux = 1, K 2aux = 0) (K 1aux = 0, K 2aux = 1)

(i = 1, 2; m = 1, 2) (7.36)

tip

1 aux aux σiaux j = 2 C i jkl (u k,l + u l,k ) (i, j, k, l = 1, 2)

(7.37)

aux εiaux j = Si jkl (x) · σkl (i, j, k, l = 1, 2)

(7.38)

The angular functions f iI are given in Sect. 6.1.2.

7.2.3 Extraction of Thermal SIFs The relation between the thermal SIFs and the I-integral for thermoelasticity is I =

2 K 1 K 1aux + K 2 K 2aux E∗ cosh2 (πεtip )

(7.39)

By taking K 1aux = 1, K 2aux = 0 and K 1aux = 0, K 2aux = 1, respectively, the SIFs K 1 and K 2 can be obtained according to the relations

7.2 Interface Crack Under Thermal Loading

K1 =

245

E ∗ cosh2 (πεtip ) I1 , 2

K2 =

E ∗ cosh2 (πεtip ) I2 2

(7.40)

7.2.4 I-integral for a Thermoelastic Solid with Complex Interfaces As shown in Fig. 7.4, a crack is located on ΓI = ΓI12 + ΓI34 and a curved interface ΓJ = ΓJ13 + ΓJ24 crosses the interface Γ I . The integral domain A enclosed by the contour Γ O , is divided by Γ I and Γ J into four sub-domains, A1 , A2 , A3 , and A4 which are enclosed by Γ1O = ΓI12 + Γ J13 + ΓB4 + ΓC+ − Γ2 , Γ2O = ΓB1 + ΓJ24 − ΓI12 − Γ1 +ΓC− , Γ3O = ΓB3 −ΓJ13 +ΓI34 and Γ4O = ΓB2 −ΓI34 −ΓJ24 , respectively. Accordingly, the I-integral can be expressed as I = − lim

Γ→0

4 { ∑ S=1

Γ SO

∗ P1i n i qdΓ + Iinterface

(7.41)

∗ Here, Iinterface is a line integral along the interfaces Γ I and Γ J which is given by ∗ ∗ ∗ ∗ ∗ Iinterface = I12 + I34 + I24 + I13

where

Fig. 7.4 Integral domain with two interfaces

(7.42)

246

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces ∗ I12 = ∗ I13

=

{ ΓI12

{

ΓJ13

{ 1 2 3 4 ◯ ◯ ◯ ◯ ∗ (P1i − P1i )n i qdΓ, I34 = (P1i − P1i )n i qdΓ 1 ◯

ΓI34

3 ◯

(P1i − P1i )n i qdΓ,

∗ I24

=

{

ΓJ24

2 4 ◯ ◯ (P1i − P1i )n i qdΓ

(7.43)

Here, the expressions marked by the superscripts ➀, ➁, ➂, and ➃ means that they belong to the domains A1 , A2 , A3 , and A4 , respectively. By applying the continuity conditions on the horizontal interface, it is easily to prove that ∗ ∗ I12 = I34 =0

(7.44)

According to the continuity conditions of a non-adiabatic interface, one obtains ∗ I13

{

3 − (εth )◯ 1 ]n qdΓ th ◯ σiaux 1 j [(εi j ) ij

= ΓJ13

{

=

3 − (α th )◯ 1 ](ϑ − ϑ )n qdΓ σiiaux [(α th )◯ 0 1

(7.45)

4 − (α th )◯ 2 ](ϑ − ϑ )n qdΓ σiiaux [(α th )◯ 0 1

(7.46)

ΓJ13

and ∗ I24 =

{ ΓJ24

By applying the divergence theorem, the I-integral can be expressed as { I =

aux aux m (σiaux j u i,1 + σi j u i,1 − σik εik δ1 j )q, j dA A

{ tip

+

aux σi j [Si jkl − Si jkl (x)]σkl,1 qdA A

+

4 { ∑ S=1 A

σiiaux [α th (ϑ − ϑ0 )],1 qdA + IΓJ

S

∗ ∗ where IΓJ = I13 + I24 is the interface integral along the interface Γ J .

(7.47)

7.3 T-stress Evaluation for Nonhomogeneous Thermoelasticity

247

7.3 T-stress Evaluation for Nonhomogeneous Thermoelasticity How to evaluate the T-stress from the I-integral is derived in the above section. The contour integral around the crack tip can be written as { I = lim

Γ→0

aux aux (σ jk εaux jk δ1i − σi j u j,1 − σi j u j,1 )n i dΓ

(7.48)

Γ

Similarly, the I-integral can be converted into the domain form { I =

aux aux m (σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i )q,i dA A

{ tip

+

σiaux j,1 [Si jkl − Si jkl (x)]σkl qd A A

+

∑{ {

σiiaux [α th (ϑ − ϑ0 )],1 qdA

S A S

σiiaux [[α th ]](ϑ − ϑ0 )n 1 qdΓ

+

(7.49)

ΓI

The interface integral cannot be eliminated due to the mismatch of thermal expansion coefficient on the interface.

7.3.1 Auxiliary Field Similarly to the auxiliary for elasticity, the auxiliary displacements and stresses for thermoelasticity to extract the T-stress are given by F(κtip + 1) r F ln − sin2 θ 8πμtip d 4πμtip F(κtip − 1) F =− θ+ sin θ cos θ 8πμtip 4πμtip

u aux 1 =− u aux 2

F F F aux aux aux cos3 θ, σ22 = − πr cos θ sin2 θ, σ12 = − πr cos2 θ sin θ σ11 = − πr

(7.50) (7.51)

where d is the coordinate of a fixed point on the x 1 -axis, as shown in Fig. 7.5. The aux auxiliary strains are given by εiaux j = Si jkl (x)σkl , which is incompatible with the auxiliary displacement fields.

248

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

Fig. 7.5 A crack in a 2D nonhomogeneous solid

7.3.2 Extraction of T-stress The actual stress fields are given by σi j = K I giIj (θ ) + K II giIIj (θ ) + T δ1i δ1 j + O(r 1/2 ) + · · ·

(7.52)

The stresses, strains and displacement gradients can be written in the series σi j , εi j or u i, j = O(r −1/2 ) + O(r 0 ) + O(r 1/2 ) + o(r 1/2 )

(7.53)

In the expansions, O(r 1/2 ) represents the term involving r 1/2 , and o(r 1/2 ) represents higher order terms. According to the definitions of the auxiliary fields, the auxiliary stresses, auxiliary strains and auxiliary displacement gradients are of the aux aux −1 singularity r −1 , i.e., σiaux j , εi j or u i, j ∼ O(r ). Similarly, the I-integral can be written as the following series 1

1

I = I (− 2 ) + I (0) + I ( 2 ) + I (h) 1

1

(7.54)

where I (− 2 ) , I (0) , I ( 2 ) , and I (h) denote the parts of the I-integral contributed by O(r −1/2 ), O(r 0 ), O(r 1/2 ) and higher-order terms, respectively in the expressions of variables σi j , εi j and u j,1 . As the integral path shrinks to the crack tip, the integral path can be taken as dΓ = r dθ . Substituting the terms O(r 1/2 ) and o(r 1/2 ) in Eq. (7.53) into the I-integral

7.3 T-stress Evaluation for Nonhomogeneous Thermoelasticity

249

in Eq. (7.48) yields 1 I(2)

{π = lim

r →0 −π

[O(r 1/2 )O(r −1 ) − O(r 1/2 )O(r −1 ) − O(r −1 )O(r 1/2 )]n i r dθ

= lim O(r 1/2 ) = 0

(7.55)

r →0

I

(h)

{π = lim

r →0 −π

[o(r 1/2 )O(r −1 ) − o(r 1/2 )O(r −1 ) − O(r −1 )o(r 1/2 )]n i r dθ

= lim o(r 1/2 ) = 0

(7.56)

r →0

As for the singular term O(r −1/2 ), the integrations from θ = −π to +π of the angular functions of the three terms in Eq. (7.48) are cancelled out, resulting in 1

I (− 2 ) = 0. According to the above analysis, the only term that contributes to the I-integral is O(1). Substituting the stresses σi j = T δ1i δ1 j into Eq. (7.48) yields

I

(0)

{π = lim

r →0 −π

aux aux (σ11 ε11 n 1 − σ11 u aux 1,1 n 1 − σi j u i,1 n j )r dθ

(7.57)

According to the definition of the auxiliary fields, one obtains aux ε11 =−

( ) F cos θ κ(x) + 1 − 4 sin2 θ 8πr μ(x)

(7.58)

( ) F cos θ κtip + 1 − 4 sin2 θ 8πr μtip

(7.59)

u aux 1,1 = −

Substituting Eqs. (7.58) and (7.59) into Eq. (7.57), I (0) can be written as

I

(0)

{π = − lim

r →0 −π

n i σiaux j u j,1 r dθ

(7.60)

th th The thermal strains ε11 = ε22 = α th (ϑ − ϑ0 ) are generally not singular so that they do not affect the singular term of the displacement gradients. The thermal strains affect the constant term of the displacement gradient u1,1 but do not affect u2,1 . According to the asymptotic expressions of the displacements given in Sect. 5.5 and the strain–displacement relations, we have

250

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces m th u 1,1 = ε11 = ε11 + ε11 =

T +α th (ϑ − ϑ0 ), u 2,1 = 0 E'

(7.61)

' ' 2 Here, E tip = E tip for plane stress and E tip = E tip /(1 − νtip ) for plane strain. Using Eq. (7.61) and the auxiliary stress fields, Eq. (7.60) can be simplified as

I

(0)

{π = lim

r →0 −π

|

=

F T cos2 θ [ ' +α th (ϑ − ϑ0 )]r dθ πr E |

T th ' +αtip (ϑ − ϑ0 )tip F E tip

(7.62)

After calculating the I-integral, the T-stress can be extracted using ' T = E tip

I ' th − E tip αtip (ϑ − ϑ0 )tip F

(7.63)

7.4 Generalized DII-Integral for Elasticity 7.4.1 Requirements for the Establishment of the DII-Integral Due to the mismatch of the thermal expansion coefficient on the interface, the interface integral cannot be eliminated from the domain of the I-integral in previous sections. In order to avoid solving the interface integrals for materials with complex interfaces, a new DII-integral is discussed in this section. In order to design the DII-integral for an elastic solid, the interface integral must be eliminated, i.e., { IΓI = − [[P1i n i ]]qdΓ (7.64) ΓI

where [[*]] denotes the jump of the quantity (*). In order to establish a generalized DII-integral for pure elastic solids, the conditions that the auxiliary field must satisfy on the curve Γ I are discussed first. After that, according to these requirements, a generalized auxiliary field is designed. Then, the relation between the generalized I-integral and the SIFs is derived. It is difficult to set the smooth function q = 0 on all the interfaces for materials with complex interfaces, and thus the condition of q /= 0 on Γ I is considered in the following discussion. If q /= 0, IΓI = 0 requires P1i n i to be continuous across the curve Γ I , i.e., [[P1i n i ]] = 0. Expanding [[P1i n i ]] yields

7.4 Generalized DII-Integral for Elasticity

[[P1i n i ]] =

251

n1 n1 aux aux [[σiaux [[σi j εiaux j εi j ]] + j ]] − [[n i σi j u j,1 ]] − [[n i σi j u j,1 ]] (7.65) 2 2

The actual stresses and displacements across the interface satisfy two continuity conditions. The first continuity condition is that the interface is in equilibrium, i.e., [[n i σi j ]] = 0

(7.66)

The second continuity condition is that the perfect interface requires the displacement gradients with respect to the local coordinate ξ 2 to be ||

∂u j ∂ξ2

|| =0

(7.67)

where ξ 1 and ξ 2 are the orthogonal curvilinear coordinates, as shown in Fig. 7.6. For simplicity, the auxiliary stress tensor is assumed to be symmetric, i.e., σiaux j = aux σ ji . By applying the strain–displacement relations εi j = (u i, j + u j,i )/2 and the chain rule, one obtains aux σiaux j εi j = σi j

∂u j ∂ξk ∂ξk ∂ xi

(7.68)

Substituting Eqs. (7.67)–(7.68) into the first term in Eq. (7.65) yields n1 n1ni [[σiaux j εi j ]] = 2 2 Fig. 7.6 The curvilinear coordinates based on a curved interface

|| σiaux j

∂u j ∂ξ1

|| +

) n 1 ∂u j ( aux −n 2 [[σ1aux (7.69) j ]] + n 1 [[σ2 j ]] 2 ∂ξ2

252

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

Analogously, the second term in Eq. (7.65) can be expressed through defining the stresses σˆ klaux = Ci jkl (x)εiaux j as n1 n1ni [[σi j εiaux j ]] = 2 2

|| || ) ∂u j n 1 ∂u j ( + σˆ iaux −n 2 [[σˆ 1aux ˆ 2aux (7.70) j j ]] + n 1 [[σ j ]] ∂ξ1 2 ∂ξ2

Notation that σˆ iaux j is not the auxiliary stress but a variable related to the auxiliary strain. The third term in Eq. (7.65) is simplified by the chain rule [[n i σiaux j u j,1 ]]

|| || ) ∂u j ( aux aux ∂u j − n2 n 1 [[σ1aux = n 1 n i σi j j ]] + n 2 [[σ2 j ]] ∂ξ1 ∂ξ2

(7.71)

Substituting Eqs. (7.66) and (7.69)–(7.71) into Eq. (7.65) yields ( ||) || n1 aux ∂u j aux ∂u j [[n i σi j [[P1i n i ]] = − ]] − n i σˆ i j 2 ∂ξ1 ∂ξ1 ( ) n 1 n 2 ∂u j + [[σ1aux ˆ 1aux j ]] − [[σ j ]] 2 ∂ξ2 ( || aux || ) ∂u j ∂u j 1 + n 22 n 21 aux [[σ2aux [[ σ ˆ ]] + ]] − n σ + i ij j 2j ∂ξ2 2 2 ∂ x1

(7.72)

Since n 1 /= 0 for an arbitrary material interface Γ I , the equation [[P1i n i ]] = 0 requires || aux aux n i σiaux ˆ iaux ˆ 1aux ˆ 2aux j = ni σ j , [[σ1 j ]] = [[σ j ]], [[σ2 j ]] = [[σ j ]] = 0,

∂u aux j ∂ x1

|| =0 (7.73)

To reduce the number of continuity conditions across the interface and simplify the auxiliary stress–strain relations, the relation σiaux ˆ iaux j = σ j is defined in the entire integral domain. Namely, the stiffness tensor in the auxiliary stress–strain relation is identical to that in the actual stress–strain relation. Then, Eq. (7.73) reduces to || aux aux σiaux j = C i jkl (x)εkl , [[σ2 j ]] = 0,

∂u aux j ∂ x1

|| =0

(7.74)

According to the above analysis, if the auxiliary fields satisfy Eq. (7.74), then the I-integral is domain-independent for material interfaces. Equation (7.74)1 gives the auxiliary stress–strain relation, and Eqs. (7.74)2 and (7.74)3 provide the continuity conditions of the auxiliary stress and displacement. For a weak interface Γ I across which material properties are continuous, the continuity conditions are as follows (Yu et al. 2015)

7.4 Generalized DII-Integral for Elasticity

253

|| [[σi j ]] = [[εi j ]] =

∂u j ∂ xi

|| =0

(7.75)

Substituting these relations into Eq. (7.65) yields ) ∂u j ( n 1 n1 aux [[σˆ 1aux [[σ1aux j ]] − j ]] − n 2 [[σ2 j ]] ∂ x1 2 2 || aux || ∂u j ) n 1 ∂u j ( aux aux [[σˆ 2 j ]] + [[σ2 j ]] − n i σi j + 2 ∂ x2 ∂ x1

[[P1i n i ]] =

(7.76)

Equation (7.76) reveals that if the stress–strain relations in Eq. (7.74)1 are used in the entire integral domain, the continuity conditions in Eqs. (7.74)2 and (7.74)3 are also necessary for a weak interface. According to the conditions in Eq. (7.74), the auxiliary strain is defined by using Eq. (7.74)1 after defining the appropriate auxiliary stresses and displacements. The auxiliary fields defined in Chaps. 5 and 6 just satisfy the conditions given in Eq. (7.74), which results in domain-independent I-integrals for material interfaces. To extract the SIFs in previous studies on the I-integral, the auxiliary stress and displacement are generally chosen to be the corresponding asymptotic near-tip expressions of an interior crack in an infinite homogeneous body or those of an interface crack along the interface between two semi-infinite homogeneous media. Yet, it is still not clear whether other applicable definitions of the auxiliary field can be found. To clarify this point, the exact expressions of the applicable auxiliary fields are discussed.

7.4.2 Design of the Generalized Auxiliary Fields The continuity conditions derived in Sect. 7.4.1 provide a framework for designing applicable auxiliary fields to establish the DII-integral. Based on these conditions, a generalized expression of the auxiliary field is derived for an isotropic elastic solid. 1. Auxiliary field for an isotropic elastic solid If an elastic solid only contains the interfaces parallel to the crack, the generated elastic fields naturally satisfy the continuity conditions in Eqs. (7.74)2 and (7.74)3 . The generalized auxiliary field is derived here for a homogeneous solid for which the material constants are prescribed to be the crack-tip ones. In order to match with the singular terms in the actual fields, the auxiliary stress must have an inverse square root singularity. In the crack-tip polar coordinate system, the auxiliary stresses are assumed to be 1 σαβ = √ σ˜ αβ (θ ), (α, β = r, θ ) 2πr

(7.77)

254

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

The auxiliary stresses are assumed to satisfy the equilibrium equations without body forces 1 ∂σr θ σrr − σθθ ∂σrr + + =0 ∂r r ∂θ r ∂σr θ 1 ∂σθθ 2σr θ + + =0 ∂r r ∂θ r

(7.78)

and the compatibility conditions (

) ∂2 1 ∂2 1 ∂ + 2 2 (σrr + σθθ ) = 0 + ∂r 2 r ∂r r ∂θ

(7.79)

Simplifying these governing equations yields d4 σ˜ θθ 5 d2 σ˜ θθ 9 + + σ˜ θθ = 0 4 2 dθ 2 dθ 16 σ˜ r θ = −

4 d2 σ˜ θθ 2 dσ˜ θθ , σ˜ rr = + 2σ˜ θθ 3 dθ 3 dθ 2

(7.80)

(7.81)

The general solutions to stresses are given by θ 3θ 3θ θ + c2 sin + c3 cos + c4 sin 2 2 2 2

(7.82)

θ c2 θ c1 3θ 3θ sin − cos + c3 sin − c4 cos 3 2 3 2 2 2

(7.83)

θ 5c2 θ 3θ 3θ 5c1 cos + sin − c3 cos − c4 sin 3 2 3 2 2 2

(7.84)

σ˜ θθ = c1 cos σ˜ r θ = σ˜ rr =

where c1 , c2 , c3 , and c4 are arbitrary constants. By applying the coordinate conversion formula, the generalized forms of the auxiliary stresses in the Cartesian coordinate system are obtains as

aux σ12

| c1

| (4 cos θ2 + cos 5θ2 ) − c3 cos θ2 =√ c θ 5θ θ 2πr + 32 (4 sin 2 + sin 2 ) + c4 sin 2 | c1 | 1 (4 cos θ2 − cos 5θ2 ) + c3 cos θ2 aux 3 σ22 =√ c θ 5θ θ 2πr + 32 (4 sin 2 − sin 2 ) − c4 sin 2 | | c1 5θ c2 5θ θ θ 1 sin − cos − c3 sin − c4 cos =√ 2 3 2 2 2 2πr 3 aux σ11

1

3

The auxiliary displacement is computed according to the relation:

(7.85)

(7.86) (7.87)

7.4 Generalized DII-Integral for Elasticity

255

κtip − 3 aux 1 aux 1 tip aux σiaux σ δi j (u i, j + u aux j,i ) = Si jkl σkl = j + 2 2μtip 8μtip kk

(7.88)

where μtip is the shear modulus, respectively, and the crack tip Kolosov constant equals κtip = 3 − 4νtip for plane strain and κtip = (3 − νtip )/(1 + νtip ) for plane stress, and νtip is the Poisson’s ratio. The auxiliary displacements are obtained as /

| | r c31 cos θ2 (1 + 2κtip − 2 cos θ ) − c3 cos θ2 2π − c32 sin θ2 (1 + 2κtip + 2 cos θ ) − c4 sin θ2 / | c | r − 31 sin θ2 (1 − 2κtip + 2 cos θ ) + c3 sin θ2 1 = μtip 2π − c32 cos θ2 (1 − 2κtip − 2 cos θ ) − c4 cos θ2

u aux 1 = u aux 2

1 μtip

(7.89)

(7.90)

2. Relations between the SIFs and the generalized DII-integral The establishment of the DII-integral aims to extract the SIFs. This section derives the relation between the generalized DII-integral and the SIFs. Since Γ in the I-integral is an arbitrary path shrinking to the crack tip, Γ can be taken to be a circular path, i.e., Γ = r dθ . According to Eq. (7.74)1 , the I-integral in Eq. (5.9) can be expressed as {π I = lim

r →0 −π

aux aux (σiaux j εi j n 1 − n i σi j u j,1 − n i σi j u j,1 )r dθ

(7.91)

Let’s consider a sufficiently small region Ωε around the crack tip, in which the material properties are continuously differentiable. The actual crack-tip asymptotic displacements and stresses in an isotropic elastic body are expressed as /

| r | ˆI K I f i (θ ) + K II fˆiII (θ ) 2π ) ( κ +1 κ −3 cos θ + δi2 sin θ + O(r 3/2 ) + · · · + T r δi1 8μ 8μ

ui =

σi j =

K I gˆ iIj (θ ) + K II gˆ iIIj (θ ) + T δi1 δ j1 + O(r 1/2 ) + · · · √ 2πr

(7.92)

(7.93)

where the angular functions fˆiI , fˆiII , gˆ iIj , and gˆ iIIj are defined as (Gdoutos 2005) fˆ1I = fˆ2I =

κtip −cos θ 2μtip κtip −cos θ 2μtip

cos θ2 , fˆ1II = sin θ2 , fˆ2II =

2+κtip +cos θ 2μtip 2−κtip −cos θ 2μtip

θ 2 cos θ2

sin

(7.94)

256

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

( ) II ( ) I gˆ 11 = 41 (3 cos θ2 + cos 5θ2 ), gˆ 11 = 14 (−7 sin θ2 − sin 5θ2) II I = 41 (5 cos θ2 − cos 5θ2 ), gˆ 22 = 14 (− sin θ2 + sin 5θ2 ) gˆ 22 I II = 41 − sin θ2 + sin 5θ2 , gˆ 12 = 14 3 cos θ2 + cos 5θ2 gˆ 12

(7.95)

In region Ωε , the compliance tensor S ijkl (x) is expressed in a Maclaurin series expansion as Si jkl (x) = Si jkl + r Si(1) jkl (θ ) + tip

r 2 (2) S (θ ) + · · · 2 i jkl

(7.96)

The strain is computed by applying the constitutive equations. By substituting the actual field [Eqs. (7.92) and (7.93)] and the auxiliary field [Eqs. (7.85)–(7.87), (7.89), and (7.90)] into the I-integral in Eq. (7.91), we obtain I =

) | (c 1 + κtip | 2 + c4 K II (c1 + c3 )K I − 4μtip 3

(7.97)

The auxiliary mode-I and mode-IISIFs are taken to be K Iaux = c1 + c3 and = −c2 /3 − c4 , respectively, so that the I-integral reduces to

K IIaux

I =

1 + κtip (K I K Iaux + K II K IIaux ) 4μtip

(7.98)

which is identical to the traditional I-integral. 3. Generalized auxiliary field By introducing the two constants cI and cII , the constants c1 , c2 , c3 , and c4 can be expressed in the form of the auxiliary SIFs K Iaux and K IIaux as ) ( c1 = 34 cI K Iaux , c3 = 1(− 43 cI K)Iaux c2 = − 43 cII K IIaux , c4 = − 1 − 41 cII K IIaux

(7.99)

By applying Eq. (7.99), the generalized auxiliary fields are expressed as / u iaux

=

| r | aux I K I f i (θ ) + K IIaux f iII (θ ) 2π

σiaux j =

K Iaux giIj (θ ) + K IIaux giIIj (θ ) √ 2πr

aux εiaux j = Si jkl (x)σkl

(7.100)

(7.101) (7.102)

where the angular functions f iI , f iII , giIj , and giIIj depend only on the crack-tip material constants and are expressed as follows

7.4 Generalized DII-Integral for Elasticity

257

I II f iI = cI fˆiI + (1 − cI ) f i , f iII = cII fˆiII + (1 − cII ) f i giIj = cI gˆ iIj + (1 − cI )g iI j , giIIj = cII gˆ iIIj + (1 − cII )g iIIj

(7.103)

Here, the angular functions fˆiI , fˆiII , gˆ iIj and gˆ iIIj are identical to those in Eqs. (5.13) I

II

and (5.14), and the angular functions f i , f i , g iI j and g iIIj are given by I

I

f 1 = − μ1tip cos θ2 , f 2 = II

f1 =

1 μtip

sin θ2 ,

II

f2 =

1 μtip 1 μtip

θ 2 cos θ2

sin

g I11 = −g I22 = − cos θ2 , g I12 = − sin g II11 = −g II22 = − sin θ2 , g II12 = cos θ2

(7.104) θ 2

(7.105)

These two constants cI and cII in the generalized auxiliary field can be assigned freely to obtain the different applicable auxiliary fields, and the generalized auxiliary field is the linear combination of two special auxiliary fields, as follows. (1) Crack face traction-free auxiliary field (cI = cII = 1) In the first auxiliary field, the constants are prescribed to be cI = cII = 1, so that the auxiliary traction on the crack face is zero, i.e., σ2aux j (r, ±π) = 0. Thus, this auxiliary field is referred as the crack face traction-free auxiliary field. In this condition, the auxiliary stresses and displacements are identical to the well-known crack-tip asymptotic stress and displacement fields of a crack in an infinite elastic body. (2) Zero mean stress auxiliary field (cI = cII = 0) In the second auxiliary field, the constants are prescribed to be cI = cII = 0, so that the mean stress is zero, i.e., σiiaux = 0. Thus, this auxiliary field is considered as the zero mean stress auxiliary field. Notably, the crack face traction is non-zero for this auxiliary field, σ2aux j (r, ±π) / = 0. | I The angular functions are given in Fig. 7.7. It can be observed that g22 (0)|cI =0 = | | | | | I II II I I (0)|cI =1 = 1, g22 (0)|cII =0 = g22 (0)|cII =1 = 0, g12 (0)|cI =0 = g12 (0)|cI =1 = 0, g22 | | II II (0)|cII =0 = g12 (0)|cII =1 = 1. Therefore, the stress angular function gij (θ ) and g12 generally varies with cI and cII while gij (θ = 0) is independent of the constants cI and cII . The generalized auxiliary field contains two free constants cI and cII , and the generalized auxiliary field can be expressed as the linear combination of two special auxiliary fields. One is the crack face traction-free auxiliary field that has been widely used in the I-integral studies, and the other is the zero-mean-stress auxiliary field which has mean auxiliary stress of zero.

258

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

Fig. 7.7 Two representative stress angular functions

7.4.3 Generalized DII-Integral Let’s consider an integral domain A containing a strong interface Γ I . For the generalized auxiliary field, the interface integral is zero (IΓI = 0). By applying the relations aux σiaux j εi j = σi j εi j , IΓI = 0 and A = A1 + A2 , the I-integral { | I = lim

Γε →0

Γε

| 1 aux m aux aux aux (σ ε + σ jk ε jk )δ1i − σi j u j,1 − σi j u j,1 n i dΓ 2 jk jk

(7.106)

can be converted into an equivalent form { I =

aux aux (σiaux j u j,1 q,i + σi j u j,1 q,i − σi j εi j q,1 )dA A

{ (

+ A

aux aux σiaux j,i u j,1 + σi j u j,i1 + σi j,i u j,1 aux aux +σi j u aux j,i1 − σi j,1 εi j − σi j εi j,1

) qd A + IΓC

(7.107)

In the case of no body forces (σi j,i = 0), the generalized DII-integral is simplified as

7.5 A New DII-Integral for Thermoelasticity

259

{ I =

aux aux (σiaux j u j,1 q,i + σi j u j,1 q,i − σi j εi j q,1 )dA A

{ tip

+

aux σi j [Si jkl − Si jkl (x)]σkl,1 qd A + IΓC A

= ID + IΓC

(7.108)

By substituting the traction-free condition on the crack face (σ2 j (r, ±π) = 0) and the values ni on the crack face (n 1 = 0 on ΓC+ and ΓC− , n 2 = −1 on ΓC+ , and n 2 = 1 on ΓC− ) into Eq. (5.20), the crack face integral is simplified as { IΓC =

{ σ2aux j u j,1 qdΓ −

ΓC+

σ2aux j u j,1 qdΓ

(7.109)

ΓC−

According to the definition of the generalized auxiliary field, the crack face integral can be simplified as { IΓC = ΓC

{ } − q (1 − cII )K IIaux (u + 2,1 + u 2,1 ) dΓ √ − aux + 2πr −(1 − cI )K I (u 1,1 + u 1,1 )

(7.110)

where u i+ = u i (r, π) and u i− = u i (r, −π). The crack face integral is IΓC = 0 for the crack-face traction-free auxiliary field, whereas IΓC is non-zero √ for the zero-meanstress auxiliary field. According to Eq. (7.92), the term O( r ) in the displacement expression has no contribution to u i+ + u i− , whereas the term O(r ) has a non-zero contribution to u i+ +u i− . The main contribution to IΓC comes from the term O(r ) of the displacement so that IΓC has an r 1/2 singularity, which is validated by a homogeneous plate with a central crack under tension as shown in Fig. 7.8. The zero-mean-stress auxiliary field is successfully used in the establishment of the DII-integral for the interface with isotropic mismatch strain. The generalized DII-integral is the linear combination of K I and K II , and the relation does not involve the free constants cI and cII . Namely, the values of the DII-integral are theoretically independent of cI and cII .

7.5 A New DII-Integral for Thermoelasticity 7.5.1 DII-Integral for an Internal Crack The thermal strain is given by εithj = α th (ϑ − ϑ0 )δi j for isotropic materials, where ϑ 0 and ϑ are the initial and current temperatures, respectively. The I-integral using the crack-face traction-free auxiliary field (see Sect. 7.4.2) can be simplified finally as

260

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

Fig. 7.8 Normalized I-integrals versus the integral domain auxiliary / size for the zero-mean-stress √ field (The analytical solutions to the SIFs are K I = σ0 πacos2 β and K II = σ0 πacosβ sin β. (I) (II) The analytical solutions to the I-integral are given by IAna = K 12 /E ' and IAna = K II2 /E ' , where RI is radius of integration domain, and h 0 is the edge length of the crack-tip element.)

{ I =

aux aux m (σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i )q,i dA A

{ tip

+

aux σi j [Si jkl − Si jkl (x)]σkl,1 qd A A

+

∑{

th σiiaux [α,1 (ϑ − ϑ0 ) + α th (ϑ − ϑ0 ),1 ]qdA + Iinterface

(7.111)

S A S

The material constituents generally possess different thermal expansion coefficients, resulting in Iinterface /= 0 due to σiiaux /= 0. According to the definition of the auxiliary field, the interface integral { Iinterface = −

aux m aux [[σ jk ε jk δ1i − σiaux j u j,1 − σi j u j,1 ]]n i qdΓ

(7.112)

ΓI

can be simplified as { Iinterface =

σiiaux [[α th ]](ϑ − ϑ0 )n 1 qdΓ

(7.113)

ΓI

If an auxiliary field satisfies the relation σiiaux = 0, the interface integral gets vanished. Such auxiliary field is just the zero-mean-stress auxiliary field which are given by

7.5 A New DII-Integral for Thermoelasticity

/ | | r 1 θ θ aux aux K cos − K II sin =− μtip 2π I 2 2 / | | r 1 θ θ aux aux K = sin cos u aux + K 2 II μtip 2π I 2 2 | | 1 θ θ aux aux K Iaux cos + K IIaux sin σ22 = −σ11 =√ 2 2 2πr | | 1 θ θ aux aux K Iaux sin − K IIaux cos σ12 = σ21 = −√ 2 2 2πr

261

u aux 1

(7.114)

(7.115)

It is noteworthy that the above auxiliary stress in this auxiliary field satisfies the equilibrium equations without body forces, i.e., σiaux j,i = 0 and the compatibility aux aux equations σii, j j = 0, but the crack face traction is non-zero, i.e., σ2 j (r, ±π) / = 0. By using the zero-mean stress auxiliary field, the interface integral vanishes and the I-integral in Eq. (7.18) is simplified as { I =

aux aux (σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i )q,i dA A

{ tip

+

aux σi j [Si jkl − Si jkl (x)]σkl,1 qd A + IΓC

(7.116)

A

The non-zero crack-face integral IΓC is introduced in details in Sect. 7.4.3. The thermal parameters are involved in the interface integral and thus Eq. (7.111) is a DII-integral for mechanical interfaces, abbreviated as DII-MI. Since the interface integral is zero for the zero-mean-stress auxiliary field, the I-integral in Eq. (7.116) is a DII-integral for thermomechanical interfaces, abbreviated as DII-TMI. Compared with the DII-MI, the DII-TMI eliminates the integral with respect to the thermal property gradient, including the gradient of the thermal expansion coefficient and the gradient of temperature. Moreover, it can be observed that the formula itself in Eq. (7.116) does not contain any terms related to the thermal properties. These features facilitate the practical implementation of the DII-TMI in dealing with materials with complex thermomechanical interfaces. The thermal loading does not affect the relations between the I-integral and the SIFs. According to the analysis in Sect. 7.4.2, the I-integral can be expressed as I =

1 + κtip (K I K Iaux + K II K IIaux ) 4μtip

(7.117)

Taking the vector {K Iaux , K IIaux } to be {1, 0} and {0, 1}, sequentially, one can solve the mode I and modeII SIFs separately by twice computing the DII-TMI.

262

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

7.5.2 DII-Integral for an Interface Crack 1. I-integral for an interface crack As shown in Fig. 7.9, the I-integral for an interface crack is given by { I = lim

Γ→0

P1i n i dΓ, P1i = Γ

1 aux m aux aux (σ ε + σ jk εaux jk )δ1i − σi j u j,1 − σi j u j,1 2 jk jk (7.118)

The I-integral is converted into domain form by applying the divergence theorem when the integrand P1i is continuously differentiable in the integral domain. As shown in Fig. 7.9, consider an integral domain containing material interfaces ΓI and ΓJ = ΓJ1 + ΓJ2 . The integral domain A is divided into four sub-domains A1 , A2 , A3 and A4 by ΓC , ΓI , ΓJ and ΓA , where ΓA denotes the tangential extension of the crack face. In each sub-domain, the material properties in each sub-domains are continuously differentiable. Meanwhile, ΓB is divided into ΓB1 ΓB2 and ΓB3 (ΓB3 = 1 2 + ΓB3 ), and Γ is divided into Γ1 and Γ2 . The sub-domains A1 , A2 , A3 and A4 are ΓB3 enclosed by Γ1O , Γ2O , Γ3O and Γ4O , respectively. Here, Γ1O = ΓB1 +ΓC+ +Γ1− +ΓI+ +ΓJ1 , − − 1 2 Γ2O = ΓB2 + ΓJ2 + ΓI− + Γ2− + ΓC− , Γ3O = ΓB3 + ΓJ1 + ΓA and Γ4O = ΓB3 + ΓA− + ΓJ2 . By introducing an arbitrary weight function q with the value varying smoothly from 1 on Γε to 0 on ΓB , the I-integral can be converted into an equivalent integral

Fig. 7.9 The integral domain cut by the interfaces ΓI and ΓJ

7.5 A New DII-Integral for Thermoelasticity

I = − lim

Γ→0

4 { ∑ S=1

ΓS

P1i n i qdΓ + IΓC + IΓI + IΓJ + IΓA

263

(7.119)

The appearance of the line integral IΓA is due to the discontinuity of the auxiliary fields u iaux and σiaux j across ΓA . Applying the divergence theorem to Eq. (7.119) yields I =−

4 { ∑ S=1 A

(P1i q),i dA + IΓC + IΓI + IΓJ + IΓA

(7.120)

S

The line integrals are given by { IL =

[(P1i ) L + − (P1i ) L − ]n i qdΓ (L = ΓI , ΓJ , ΓA , ΓC )

(7.121)

L

For materials containing complex material interfaces, ΓJ can be an irregular and complex path so that the avoidance of calculating the interface integral IΓJ can dramatically reduce the difficulty of the numerical calculations. Subsequently, a new auxiliary field is designed to eliminate the interface integral. 2. New auxiliary field for an interface crack Generally, the line integral IΓJ is non-zero if adopting the traditional auxiliary field. As a result, it is troublesome to calculate the interface integrals for the materials with complex interfaces. To overcome this difficulty, a new auxiliary field is designed to eliminate the integral IΓJ . The continuity conditions that IΓJ = 0 for thermoelasticity are as follows || aux || ∂u j aux aux σi j = Ci jkl (x)εi j , =0 ∂ x1 ΓJ aux th [[σ2aux j ]]ΓJ = 0, [[σii α (ϑ − ϑ0 )]]ΓJ = 0

(7.122)

If materials contain complex interfaces with mismatched coefficient of thermal expansion ([[α th ]]ΓJ /= 0) or the temperature is discontinuous across the interface ([[ϑ − ϑ0 ]]ΓJ /= 0), the additional condition for thermo-elasticity [Eq. (7.122)4 ] is taken to be σiiaux = 0. Because the I-integral must be a non-zero finite value, the auxiliary stresses should have the oscillatory singularity. Thus, one can express the auxiliary stresses in the crack-tip polar coordinate system as follows | 1 | iε σrraux (r, θ ) = √ r σ˜ r 1 (θ ) + r −iε σ˜ r 2 (θ ) 2πr | 1 | iε aux σθθ (r, θ ) = √ r σ˜ θ 1 (θ ) + r −iε σ˜ θ 2 (θ ) 2πr

264

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

| 1 | iε σraux r σ˜ r θ 1 (θ ) + r −iε σ˜ r θ2 (θ ) θ (r, θ ) = √ 2πr where i =

(7.123)

√ −1 is imaginary unit, and ε is a bi-material constant defined as ) ( tip tip tip κ1 μ2 + μ1 1 ε= ln tip tip tip 2π κ2 μ1 + μ2

(7.124)

It is assumed that the auxiliary stresses satisfy the equilibrium equations without body forces {

∂σrraux ∂r ∂σraux θ ∂r

+ +

1 r 1 r

∂σraux θ ∂θ ∂σθaux θ ∂θ

+ +

σrraux −σθaux θ = r 2σraux θ = 0 r

0

(7.125)

Substituting Eq. (7.123) into Eq. (7.125) yields ) | ⎧ iε |( 1 + i|( ε σ˜ r 1 (θ ))+ σ˜ r'θ 1 (θ ) − σ˜ θ 1 (θ ) ⎨r 2 | (7.126) σ˜ θ2 (θ ) ) = 0 +r −iε )21 − i ε σ˜ r 2 (θ ) + σ|˜ r'θ 2 (θ ) − |( | ⎩ iε |( 3 ' ' −i ε 3 + i ε σ ˜ (θ ) + σ ˜ (θ ) + r − iε σ ˜ (θ ) + σ ˜ (θ ) = 0 r r θ 1 r θ 2 θ1 θ2 2 2 The terms in Eq. (7.126) multiplied by r iε and r −iε must be zero due to the arbitrary of the material mismatch. Thus the angular functions σ˜ r 1 (θ ), σ˜ r 2 (θ ), σ˜ r θ 1 (θ ), and σ˜ r θ 2 (θ ) can be expressed by using σ˜ θ1 (θ ) and σ˜ θ2 (θ ) as σ˜ r θ 1 (θ ) =

σ˜ ' (θ ) − 3θ 1 , +i ε 2

σ˜ r 1 =

σ˜ r θ 2 (θ ) =

σ˜ ' (θ ) − 3θ 2 , −iε 2

σ˜ r 2 =

) ( 3 σ˜ θ''1 (θ )+ +iε σ˜ θ 1 (θ ) 2 ( )( ) 1 3 +iε +iε 2 ( 2 ) 3 σ˜ θ''2 (θ )+ −i ε σ˜ θ 2 (θ ) 2)( ( ) 1 3 −iε −iε 2 2

(7.127)

Substituting Eq. (7.127) into σiiaux = 0, the angular functions σ˜ θ 1 (θ ) and σ˜ θ2 (θ ) can be solved as ) ) ( ( 3 3 m m + i ε θ + C2 sin + iε θ σ˜ θ 1 (θ ) = C1 cos 2 2 ) ) ( ( 3 3 m m − i ε θ + C4 sin − iε θ (7.128) σ˜ θ2 (θ ) = C3 cos 2 2 The superscript m is used to distinguish the auxiliary fields of two sides of the interface crack, where m = 1 for x2 > 0 and m = 2 for x2 < 0 Cim (i = 1, 2, 3, 4) are arbitrary constants. After solving the angular functions in Eq. (7.127), the auxiliary stresses can be expressed in the polar coordinate system as

7.5 A New DII-Integral for Thermoelasticity aux σθθ

σrraux σraux θ

265

| ( ) ( ) | } iε θ + C2m sin 23 + iε θ ) | r iε C1m| cos 23 + ( ) ( =√ −i ε C3m cos 23 − i ε θ + C4m sin 23 − iε θ 2πr +r aux = −σθθ ) (3 ) | } { iε | m ( 3 m 1 i ε θ − C cos + i ε r C1| sin 2 + 2 2 (3 ) (3 θ ) | =√ m m −i ε C3 sin 2 − i ε θ − C4 cos 2 − i ε θ 2πr +r {

1

(7.129)

In polar coordinate system, the auxiliary displacement (u iaux ) and auxiliary stress satisfy the gradient relations as follows

(σiaux j )

tip

aux σ aux − νm σθθ ∂u raux = rr , tip ∂r Em

tip

1 ∂u aux σ aux − νm σrraux u raux θ + = θθ tip r r ∂θ Em

(7.130)

Then the Cartesian components of the auxiliary field are given by | | } 1 1 m m r iε −C 1 cos( 2 − i ε)θ + C 2 sin( 2 − i ε)θ | | −i ε −C3m cos( 21 + iε)θ + C4m sin( 21 + i ε)θ 2πr +r aux (r, θ ) = −σxaux σ yy x (r, θ ) | } { iε| m 1 r C1| sin( 21 − iε)θ + C2m cos( 21 − i ε)θ | aux σx y (r, θ ) = − √ (7.131) −iε C3m sin( 21 + i ε)θ + C4m cos( 21 + iε)θ 2πr +r / { ir iε | m | | r 1 C1 cos( 21 + i ε)θ + C2m sin( 21 + i ε)θ aux 2ε−i | m | u x (r, θ ) = tip ir −iε 1 1 m μm 2π − 2ε+i C3 cos( 2 − i ε)θ + C4 sin( 2 − i ε)θ / { ir iε | | | 1 1 m m r 1 −C sin( + i ε)θ + C cos( + i ε)θ aux 1 2 2ε−i | 2 2 | u y (r, θ ) = tip 1 1 ir −iε m m μm 2π − 2ε+i −C3 sin( 2 − i ε)θ + C4 cos( 2 − iε)θ (7.132)

σxaux x (r, θ ) = √

1

{

aux + − The auxiliary variables can be designed to satisfy σiaux j (r, 0 ) = σi j (r, 0 ) or 1 2 1 1, 2, 3, 4) as Ci = Ci or Ci = Ci2 = tip tip = u 1 /u 2 in the following calculation, u iaux (r, 0+ ) = u iaux (r, 0− ). Taking Γ to be a circular path, the I-integral can be rewritten as

u iaux (r, 0+ ) = u iaux (r, 0− ) by taking Cim (i = tip tip u 1 /u 2 , respectively. Here, we take Ci1 = Ci2 aux + − which leads to σiaux j (r, 0 ) / = σi j (r, 0 ) but {π I = lim

r →0 −π

m aux aux (σiaux j εi j n 1 − σi j u j,1 n i − σi j u j,1 n i )r dθ

(7.133)

The actual crack-tip filed in an isotropic elastic body is given in Sect. 1.2.1. Substituting the actual and auxiliary fields [Eqs. (7.131) and (7.132)] into the Iintegral yields | | I = P1 (E mtip , νmtip ) r −2iε (K I − i K II )(C31 − iC41 ) + r 2iε (K I + i K II )(C11 − iC21 )

266

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

| | + P2 (E mtip , νmtip ) (K I − i K II )(C11 − iC21 ) + (K I + i K II )(C31 + iC41 ) (7.134) where the non-zero constants P1 and P2 only depend on the material properties near the crack tip. It is expected to eliminate the first item which contains r −2iε and r 2iε to guarantee the I-integral to be domain-independent of r . Through setting C11 = −iC21 and C31 = iC41 and defining the auxiliary SIFs K 1aux = i(C41 − C21 ),

K 2aux = −C41 − C21

(7.135)

The I-integral can be simplified as I =2

tip tip tip tip μ2 E 1 e2πε + μ1 E 2 ( tip tip tip E 1 E 2 μ1 (1

+ e2πε )

K 1aux K 1 + K 2aux K 2

)

(7.136)

Correspondingly, the auxiliary fields can be expressed finally as σxaux x (r, θ )

| ) )| ( ( tip θ μm e−εθ θ aux aux K 1 cos − ε ln r + K 2 sin − ε ln r = − tip √ 2 2 2πr μ1

aux (r, θ ) = −σxaux σ yy x (r, θ ) | ( ( ) )| tip μm e−εθ θ θ aux aux −K σxaux (r, θ ) = cos sin − ε ln r + K − ε ln r √ 2 1 y tip 2 2 2πr μ1 (7.137) / r 1 e−εθ u aux (r, θ ) = x tip 1 + 4ε 2 2π μ | 1 ) )| ( ( θ θ aux aux aux aux (K 2 − 2εK 1 ) sin + ε ln r − (K 1 + 2εK 2 ) cos + ε ln r 2 2 / r 1 e−εθ u aux (r, θ ) = y tip 1 + 4ε 2 2π μ | 1 ) )| ( ( θ θ (K 1aux + 2εK 2aux ) sin + ε ln r + (K 2aux − 2εK 1aux ) cos + ε ln r 2 2 (7.138) aux The auxiliary strain is defined as εiaux j = Si jkl (x)σkl . If the materials on the two sides of the crack are identical with each other, the bimaterial constant ε becomes zero. As a result, the auxiliary fields in Eqs. (7.137) and (7.138) are identical with the zero-mean-stress auxiliary field for an internal crack [Eqs. (7.114) and (7.115)], respectively. Correspondingly, the relation between the I-integral and the SIFs in Eq. (7.136) reduces to Eq. (7.117). It implies that one can deem the DII-integral for an internal crack in a thermoelastic solid as a special case of the DII-integral of an interface crack.

7.6 Typical Thermal Fracture Problems

267

3. DII-integral for an interface crack for thermoelasticity For the new auxiliary field (the zero-mean-stress auxiliary field), Eq. (7.122)2 is satisfied on ΓJ , but not satisfied on ΓI , ΓA and ΓC . As a result, the line integrals along ΓI , ΓA and ΓC are non-zero, i.e., IΓJ = 0, IΓI /= 0, IΓA /= 0, and IΓC /= 0. By applying the equilibrium equations σiaux j,i = 0 and σi j,i = 0, the I-integral is simplified as { I =

aux aux m (σiaux j u j,1 + σi j u j,1 − σ jk ε jk δ1i )q,i dA A

{ tip

+

aux σi j [Si jkl − Si jkl (x)]σkl,1 qdA + IΓC + IΓI + IΓA

(7.139)

A

The I-integral in Eq. (7.139) is referred to as the DII-integral for thermal–mechanical interfaces (DII-TMI) as well. Correspondingly, the I-integral using the traditional auxiliary field [Eq. (7.47)] is referred to as the DII-integral for mechanical interfaces (DII-MI) as well. The actual stresses σ2 j and the auxiliary displacement gradients u aux j,1 are continuous across ΓI and ΓA , resulting in [[σ2 j u aux j,1 ]] = 0 across ΓI and ΓA . For a tractionfree crack (σ2 j (r, ±π ) = 0 on ΓC ), the line integral I L (L = ΓI , ΓA , ΓC ) can be simplified as { IL =

{ (σ2aux j )m=1 u j,1 qdΓ

L+

−

(σ2aux j )m=2 u j,1 qdΓ

(7.140)

L−

Notably, if the interface ΓI where the crack lies is curved, the interface integral IΓI is not along the interface ΓI but along a line tangential to the crack front. Similarly, ΓA is the extension of the crack front as well.

7.6 Typical Thermal Fracture Problems 7.6.1 Internal Crack Under Thermal Loading As shown in Fig. 7.10, a functionally graded coating plate with the coating of 0.5 W in width and the substrate of 0.5 W in width contains an edge crack of length a. The Young’s modulus, thermal conductivity and thermal expansion coefficient of the coating are given by E(x) = E 1 eωx/ W , λ(x) = λ1 eβ x/ W and α(x) = α1 eδx/ W (x ≤ 0.5W ). The parameters of the substrate are given by E 2 = E 1 eω/2 , λ2 = λ1 eβ/2 and α2 = α1 eδ/2 . Poisson’s ratio is ν = 0.33 for both coating and substrate. The initial temperature for the entire plate is ϑ 0 , and the stable temperatures are prescribed to be ϑ 1 at x = 0 and ϑ 2 at x = W.

268

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

Fig. 7.10 A cracked functionally graded coating plate

Figure 7.11 shows the mode-I TSIFs normalized by the factor K 0 = E 1 α1 (ϑ2 − √ ϑ0 ) πa/(1 − ν) for different thermal expansion coefficients under the uniform temperature increment. The TSIFs obtained by the I-integral method is agreed well with the analytical ones that was given by Guo et al. (2008). Then, the material constants of the top and bottom half-plate are prescribed to be (E 1 , α 1 , λ1 , ν 1 ) and (E 2 , α 2 , λ2 , ν 2 ), respectively. The plane strain is considered and the other parameters are as follows: W = 30, a = 12, ϑ 1 = ϑ 2 = 10ϑ 0 , E 2 = 5E 1 , α 2 = 5α 1 , λ2 = 5λ1 , ν 1 = ν 2 = 0.3. Table 7.1 shows the TSIFs and T-stresses computed by using the different integral domains of radius RI = 0.4–4.5 (i.e., RI /he = 4–45, where he = 0.1 denotes the edge length of the element at the crack tip). The results show that the maximum relative errors are less than 1% whether the integral domain is cut by an interface (RI /he > 30) or not. Fig. 7.11 Normalized TSIFs varying with crack length

7.6 Typical Thermal Fracture Problems

269

Table 7.1 Mode-I TSIFs and T-stresses obtained by different integral domains RI 0.4

0.8

1.2

1.6

2.4

3.2

4.5

K 1 /K 0

2.0724

2.0721

2.0721

2.0721

2.0721

2.0721

2.0721

T-stress

−1.8086

−1.7927

−1.7921

−1.7919

−1.7918

−1.7918

−1.7918

7.6.2 Particulate Plate with an Internal Crack In order to show the superiority of the DII-integral using the zero-mean- stress auxiliary field (DII-TMI) over the DII-integral via crack-face traction-free auxiliary field (DII-MI), as shown in Fig. 7.12, we investigate a square plate of length 2 W composed of 25 uniformly distributed square particles of length 2 l. A center crack of length 2a and angle β measured counterclockwise lies in the center of the plate. The temperatures along the left and right edges of the plate are prescribed to be ϑ 1 and ϑ 2 , respectively, and ϑ 0 denotes the initial temperature. The plate is in the plane strain condition and the other data used in numerical simulations are given by: W = 1, l = 1/9, a/W = 0.4, β = (0°, 36°), ϑ 1 = ϑ 0 = 0, ϑ 2 = −20, u 2 (x2 = ±W ) = 0. The elastic moduli, Poisson’s ratios, the coefficients of the thermal expansion are as follows: (1) Matrix: E 0 = 105 , ν 0 = 0.2 and α 0 = 10−5 . (2) Particle: E 1 = 4 × 105 , ν 1 = 0.2 and α 1 = 4 × 10−5 . Next, the DII-MI without Iinterface and the DII-TMI are used to compute the SIFs in order to verify the influence of the interfaces on these two DII-integrals. As shown in Fig. 7.13, the twelve integral domains of RI /he = 3–384 are adopted to solve √ the SIFs. Figure 7.14 shows the SIF values normalized by K 0 = E 0 α0 (ϑ1 − ϑ2 ) W . The results reveal that the normalized SIFs computed by the DII-TMI (the I-integral using the zero-mean-stress auxiliary field) remain constant, whereas those computed by the DII-MI (the I-integral using the crack-face traction-free auxiliary field) with ignoring the interface integral vary dramatically when the integral domains contain the interfaces with mismatched thermal properties.

7.6.3 Multi-interface Plate with an Interface Crack As shown in Fig. 7.15, a square multi-interface plate of length 2L = 8 is divided into five parts by material interfaces ΓI , ΓJ1 , ΓJ2 and ΓJ3 . A center interface crack of length 2a = 4 lies on the interface ΓI . The displacements u 2 on the top and bottom edges are constrained to be zero. A uniform temperature change ∆ϑ = −100 is applied to the whole plate. The material constants are as follows: E 1 = 1; E 2 = 1.5; E 3 = 10; a1 = 1; a2 = 1.5; a3 = 10; v1 = v2 = v3 = 0.3.

270

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

Fig. 7.12 A square plate reinforced by square particles with an internal crack

Fig. 7.13 Integral domains of RI / h e = 3, 6, 12, 24, 48, 96, 144, 192, 240, 288, 336 and 384 around the right crack tip

7.6 Typical Thermal Fracture Problems

271

Fig. 7.14 Normalized SIFs computed using the DII-TMI and the DII-MI without the interface integral

272

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

Fig. 7.15 A rectangular multi-interface plate with a central interface crack

The mesh consists of 1825 elements and 5594 nodes. The edge length of the element at the crack tip is√h e = 0.011. Figure 7.16 shows the SIFs normalized by the factor K 0 = E 1 a1 ∆ϑ πa/(1 − v1 ) obtained by the DII-TMI and the DII-MI without the interface integral Iinterface along ΓJ1 , ΓJ2 and ΓJ3 . As shown in Fig. 7.16, the SIFs obtained by the DII-MI are identical with those obtained by the DII-TMI when the integral domains do not contain the interfaces ΓJ1 , ΓJ2 and ΓJ3 . As the integral domain size increases, the SIFs obtained by the DII-MI without Iinterface change dramatically, while the values of the SIFs obtained by the DII-TMI almost do not change. The results show that the interface integral Iinterface does not affect the DII-TMI but affects the DII-MI significantly. Due to its domain-independence for thermal–mechanical interfaces, the new DII-integral using the zero-mean-stress auxiliary field is quite effective for an interface crack in the material with complex interfaces.

7.6 Typical Thermal Fracture Problems

Fig. 7.16 Normalized SIFs at a right crack tip and b left crack tip

273

274

7 Thermal Fracture of Nonhomogeneous Materials with Complex Interfaces

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