In general, all solid materials can be considered as non-homogeneous because their properties can vary with locations in a three-dimensional space. One special type of solid material is characterized by the variations of its physical and mechanical components, structures and properties along only one given coordinate; the material properties have very small or no variations in any other direction perpendicular to the given coordinate. These types of solid materials are called functionally graded materials (FGMs). For example, plant and tree stems, animal bones and other biological hard tissues have gradient variations in their microstructures and functions. Bamboo is a self-optimizing graded structure constructed with a cell-based system fbr sensing external mechanical stimuli. Learning from nature, material scientists have increasingly aimed to design and fabricate graded materials that are more damage-resistant than their conventional homogeneous counterparts. As a design concept, FGMs were originally proposed as an alternative to conventional ceramic thermal barrier coatings to overcome their well-documented shortcomings and to meet the demands of new technologies.
The mechanical responses of FGMs have an important significance in many engineering fields and are of great interest to material scientists, and design and manufacturing engineers. The problems of fracture and crack propagation in FGMs are particularly important and have been studied in depth. The boundary element method (BEM), also known as the boundary integral equation method, is now firmly established in many engineering disciplines and is increasingly used as an effective and accurate numerical tool. Fracture mechanics has been the most active, specialized area of research in BEM and is probably the one most exploited by industry. The traditional BEM is based on the Kelvin's fundamental solution and meets the difficulties encountered when analyzing the fracture mechanics of FGMs.
Since 1983, the second author of this book has devoted much of his research to understanding the elasticity of a multilayered medium and has achieved important results. One of these results is the analytical and closed-form formulation of fundamental solutions fbr a multilayered elastic medium and a transversely isotropic bi-material. These solutions can be applied to investigate and analyze many problems in multi layered media encountered in the science and engineering disciplines using the BEM. Since 2000, the authors have dedicated their efforts to the development of the new BEMs based on these fundamental solutions under the funding of The University Grants Committee of Hong Kong, The University of Hong Kong and the National Natural Science Foundation of China.

Chapter 1 Introduction 1

1.1Functionally graded materials 1

1.2 Methods for fracture mechanics 3

      1.2.1 General 3

      1.2.2 Analytical methods 4

      1.2.3 Finite element method 5

      1.2.4 Boundary element method 6

      1.2.5 Meshless methods 7

1.3 Overview of the book 7

References 9

Chapter 2 Fundamentals of Elasticity and Fracture Mechanics 11

2.1Introduction 11

2.2 Basic equations of elasticity 12

2.3 Fracture mechanics 14

2.3.1General 14

2.3.2 Deformation modes of cracked bodies 15

2.3.3 Three-dimensional stress and displacement fields 16

2.3.4 Stress fields of cracks in graded materials and on the interface of bi-materials 18

2.4 Analysis of crack growth 19

      2.4.1 General 19

      2.4.2 Energy release rate 20

      2.4.3 Maximum principal stress criterion 21

      2.4.4 Minimum strain energy density criterion 23

      2.4.5 The fracture toughness of graded materials 24

2.5Summary 25

References 26

Chapter 3 Yue's Solution of a 3D Multilayered Elastic Medium 27

3.1 Introduction 27

3.2 Basic equations 29

3.3 Solution in the transform domain 31

      3.3.1 Solution formulation 31

      3.3.2 Solution expressed in terms of g 36

      3.3.3 Asymptotic representation of the solution matrices Φ (p, z) and Ψ (p,z) 36

3.4 Solution in the physical domain 37

      3.4.1 Solutions in the Cartesian coordinate system 37

      3.4.2 Closed-form results for singular terms of the solution 39

3.5 Computational methods and numerical evaluation 41

      3.5.1General 41

      3.5.2 Singularities of the fundamental solution 42

      3.5.3 Numerical integration 42

      3.5.4 Numerical evaluation and results 43

3.6 Summary 47

Appendix 1 The matrices of elastic coefficients 47

Appendix 2 The matrices in the asymptotic expressions of Φ (p, z) and Ψ (p, z) 48

Appendix 3 The matrices G_(s) [m, z, Φ] and G_(t) [m, z, Φ] 50

References 51

Chapter 4 Yue's Solution-based Boundary Element Method 53

4.1 Introduction 53

4.2 Betti's reciprocal work theorem 54

4.3 Yue's solution-based integral equations 56

4.4Yue's solution-based boundary integral equations 58

4.5 Discretized boundary integral equations 59

4.6 Assembly of the equation system 64

4.7 Numerical integration of non-singular integrals 67

      4.7.1 Gaussian quadrature formulas 67

      4.7.2 Adaptive integration 68

      4.7.3 Nearly singular integrals 69

4.8 Numerical integration of singular integrals 70

      4.8.1 General 70

      4.8.2 Weakly singular integrals 70

      4.8.3 Strongly singular integrals 74

4.9 Evaluation of displacements and stresses at an internal point 76

4.10 Evaluation of boundary stresses 77

4.11 Multi-region method 77

4.12 Symmetry 79

4.13 Numerical evaluation and results 81

      4.13.1 A homogeneous rectangular plate 82

      4.13.2 A layered rectangular plate 83

4.14 Summary 85

References 85

Chapter 5 Application of the Yue*s Solution-based BEM to Crack Problems 87

5.1 Introduction 87

5.2 Traction-singular element and its numerical method 88

      5.2.1 General 88

      5.2.2 Traction-singular element 89

      5.2.3 The numerical method of traction-singular elements 91

5.3 Computation of stress intensity factors 96

5.4 Numerical examples and results 97

5.5 Summary 103

References 103

Chapter 6 Analysis of Penny-shaped Cracks in Functionally Graded Materials 105

6.1 Introduction 105

6.2 Analysis methods for crack problems in a FGM system of infinite extent 106

      6.2.1 The crack problem in a FGM 106

      6.2.2 The multi-region method for crack problems of infinite extent 107

      6.2.3 The layered discretization technique for FGMs 108

      6.2.4 Numerical verifications 109

6.3 The SIFs for a crack parallel to the FGM interlayer 110

      6.3.1 General 110

      6.3.2 A crack subjected to uniform compressive stresses 111

      6.3.3 A crack subjected to uniform shear stresses 114

6.4 The growth of the crack parallel to the FGM interlayer 117

      6.4.1 The strain energy density factor of an elliptical crack 117

      6.4.2 Crack growth under a remotely inclined tensile loading 117

6.5 The SIFs for a crack perpendicular to the FGM interlayer 121

      6.5.1 General 121

      6.5.2 Numerical verifications 122

      6.5.3 The SIFs for a crack subjected to uniform compressive stresses 124

      6.5.4 The SIFs for a crack subjected to uniform shear stresses 129

6.6 The growth of the crack perpendicular to the FGM interlayer 139

6.6.1 The crack growth under a remotely inclined tensile loading 139

6.6.2 The crack growth under a remotely inclined compressive loading 143

6.7 Summary 145

References 146

Chapter 7 Analysis of Elliptical Cracks in Functionally Graded Materials 148

7.1Introduction 148

7.2 The SIFs for an elliptical crack parallel to the FGM interlayer 149

      7.2.1 General 149

      7.2.2 Elliptical crack under a uniform compressive stress 151

      7.2.3 Elliptical crack under a uniform shear stress 161

7.3The growth of an elliptical crack parallel to the FGM interlayer 169

7.4The SIFs for an elliptical crack perpendicular to the FGM interlayer 175

      7.4.1 General 175

      7.4.2 Elliptical crack under a uniform compressive stress 176

      7.4.3 Elliptical crack under a uniform shear loading 181

7.5 The growth of an elliptical crack perpendicular to the FGM interlayer 193

      7.5.1 Crack growth under a remotely inclined tensile loading 193

      7.5.2 Crack growth under a remotely inclined compressive loading 197

7.6 Summary 201

References 202

Chapter 8 Yue's Solution-based Dual Boundary Element Method 204

8.1 Introduction 204

8.2 Yue's solution-based dual boundary integral equations 205

      8.2.1 The displacement boundary integral equation 205

      8.2.2 The traction boundary integral equation 207

      8.2.3 The dual boundary integral equations for crack problems 208

8.3 Numerical implementation 209

      8.3.1 Boundary discretization 209

      8.3.2 The discretized boundary integral equation 212

8.4 Numerical integrations 213

      8.4.1 Numerical integrations for the displacement BIE 213

      8.4.2 Numerical integrations for the traction BIE 214

8.5 Linear equation systems for the discretized dual BIEs 218

8.6 Numerical verifications 223

      8.6.1 Calculation of stress intensity factors 223

      8.6.2 The effect of different meshes and the coefficient D on the SIF values 224

8.7 Summary 226

Appendix 4 Finite-part integrals and Kutt's numerical quadrature 226

      A4.1 Introduction 226

      A4.2 Kutt's numerical quadrature 227

References 228

Chapter 9 Analysis of Rectangular Cracks in the FGMs 231

9.1Introduction 231

9.2 A square crack in FGMs of infinite extent 231

      9.2.1 General 231

      9.2.2 A square crack parallel to the FGM interlayer 233

      9.2.3 A square crack having a 45° angle with the FGM interlayer 236

      9.2.4 A square crack perpendicular to the FGM interlayer 238

9.3 A square crack in the FGM interlayer 239

9.4 A rectangular crack in FGMs of infinite extent 241

      9.4.1General 241

      9.4.2 A rectangular crack parallel to the FGM interlayer 242

      9.4.3 A rectangular crack with long sides perpendicular to the FGM interlayer 245

      9.4.4 A rectangular crack with short sides perpendicular to the FGM interlayer 246

9.5 A square crack in a FGM of finite extent 248

9.6Square cracks in layered rocks 252

      9.6.1General 252

      9.6.2 The crack dimensions and the parameters of layered rocks 253

      9.6.3 A square crack subjected to a uniform compressive load 253

      9.6.4 A square crack subjected to a non-unifbrm compressive load 257

9.7 Rectangular cracks in layered rocks 261

      9.7.1 General 261

      9.7.2 A rectangular crack subjected to a linear compressive load 261

      9.7.3 A rectangular crack subjected to a nonlinear compressive load 264

9.8 Summary 267

References 267

Chapter 10 Boundary element analysis of fracture mechanics in transversely isotropic bi-materials 269

10.1Introduction 269

10.2 Multi-region BEM analysis of cracks in transversely isotropic bi-materials 270

      10.2.1 General 270

      10.2.2 Calculation of the stress intensity factors 270

      10.2.3 A penny-shaped crack perpendicular to the interface of transversely isotropic bi-materials 272

      10.2.4 An elliptical crack perpendicular to the interface of transversely isotropic bi-materials 278

10.3 Dual boundary element analysis of a square crack in transversely isotropic bi-materials 283

      10.3.1 General 283

      10.3.2 Numerical verification 284

      10.3.3 Numerical results and discussions 284

10.4 Summary 299

Appendix 5 The fundamental solution of transversely isotropic bi-materials 299

References 304

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