Fracture Mechanics in Layered and Graded Solids (eBook)

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Zhongqi Yue, University of Hong Kong; Hongtian Xiao, Shandong University of Sciences and Technology, China.

Contents 7
Chapter 1 Introduction 13
1.1 Functionally graded materials 13
1.2 Methods for fracture mechanics 15
1.2.1 General 15
1.2.2 Analytical methods 16
1.2.3 Finite element method 17
1.2.4 Boundary element method 18
1.2.5 Meshless methods 19
1.3 Overview of the book 19
References 21
Chapter 2 Fundamentals of Elasticity and Fracture Mechanics 23
2.1 Introduction 23
2.2 Basic equations of elasticity 24
2.3 Fracture mechanics 26
2.3.1 General 26
2.3.2 Deformation modes of cracked bodies 27
2.3.3 Three-dimensional stress and displacement fields 28
2.3.4 Stress fields of cracks in graded materials and on the interface of bi-materials 30
2.4 Analysis of crack growth 31
2.4.1 General 31
2.4.2 Energy release rate 32
2.4.3 Maximum principal stress criterion 33
2.4.4 Minimum strain energy density criterion 35
2.4.5 The fracture toughness of graded materials 36
2.5 Summary 37
References 38
Chapter 3 Yue’s Solution of a 3D Multilayered Elastic Medium 39
3.1 Introduction 39
3.2 Basic equations 41
3.3 Solution in the transform domain 43
3.3.1 Solution formulation 43
3.3.2 Solution expressed in terms of g 48
3.3.3 Asymptotic representation of the solution matrices 
48 
3.4 Solution in the physical domain 49
3.4.1 Solutions in the Cartesian coordinate system 49
3.4.2 Closed-form results for singular terms of the solution 51
3.5 Computational methods and numerical evaluation 53
3.5.1 General 53
3.5.2 Singularities of the fundamental solution 54
3.5.3 Numerical integration 54
3.5.4 Numerical evaluation and results 55
3.6 Summary 59
Appendix 1 The matrices of elastic coefficients 59
Appendix 2 The matrices in the asymptotic expressions of F(., z) and .(., z) 
60 
Appendix 3 The matrices Gs[m, z,F] and Gt [m, z,F] 
62 
References 63
Chapter 4 Yue’s Solution-based Boundary Element Method 65
4.1 Introduction 65
4.2 Betti’s reciprocal work theorem 66
4.3 Yue’s solution-based integral equations 68
4.4 Yue’s solution-based boundary integral equations 70
4.5 Discretized boundary integral equations 71
4.6 Assembly of the equation system 76
4.7 Numerical integration of non-singular integrals 79
4.7.1 Gaussian quadrature formulas 79
4.7.2 Adaptive integration 80
4.7.3 Nearly singular integrals 81
4.8 Numerical integration of singular integrals 82
4.8.1 General 82
4.8.2 Weakly singular integrals 82
4.8.3 Strongly singular integrals 86
4.9 Evaluation of displacements and stresses at an internal point 88
4.10 Evaluation of boundary stresses 89
4.11 Multi-region method 89
4.12 Symmetry 91
4.13 Numerical evaluation and results 93
4.13.1 A homogeneous rectangular plate 94
4.13.2 A layered rectangular plate 95
4.14 Summary 97
References 97
Chapter 5 Application of the Yue’s Solution-based BEM toCrack Problems 99
5.1 Introduction 99
5.2 Traction-singular element and its numerical method 100
5.2.1 General 100
5.2.2 Traction-singular element 101
5.2.3 The numerical method of traction-singular elements 103
5.3 Computation of stress intensity factors 108
5.4 Numerical examples and results 109
5.5 Summary 115
References 115
Chapter 6 Analysis of Penny-shaped Cracks in Functionally Graded Materials 117
6.1 Introduction 117
6.2 Analysis methods for crack problems in a FGM system of infinite extent 118
6.2.1 The crack problem in a FGM 118
6.2.2 The multi-region method for crack problems of infinite extent 119
6.2.3 The layered discretization technique for FGMs 120
6.2.4 Numerical verifications 121
6.3 The SIFs for a crack parallel to the FGM interlayer 122
6.3.1 General 122
6.3.2 A crack subjected to uniform compressive stresses 123
6.3.3 A crack subjected to uniform shear stresses 126
6.4 The growth of the crack parallel to the FGM interlayer 129
6.4.1 The strain energy density factor of an elliptical crack 129
6.4.2 Crack growth under a remotely inclined tensile loading 129
6.5 The SIFs for a crack perpendicular to the FGM interlayer 133
6.5.1 General 133
6.5.2 Numerical verifications 134
6.5.3 The SIFs for a crack subjected to uniform compressive stresses 136
6.5.4 The SIFs for a crack subjected to uniform shear stresses 141
6.6 The growth of the crack perpendicular to the FGM interlayer 151
6.6.1 The crack growth under a remotely inclined tensile loading 151
6.6.2 The crack growth under a remotely inclined compressive loading 155
6.7 Summary 157
References 158
Chapter 7 Analysis of Elliptical Cracks in Functionally Graded Materials 160
7.1 Introduction 160
7.2 The SIFs for an elliptical crack parallel to the FGM interlayer 161
7.2.1 General 161
7.2.2 Elliptical crack under a uniform compressive stress 163
7.2.3 Elliptical crack under a uniform shear stress 173
7.3 The growth of an elliptical crack parallel to the FGM interlayer 181
7.4 The SIFs for an elliptical crack perpendicular to the FGM interlayer 187
7.4.1 General 187
7.4.2 Elliptical crack under a uniform compressive stress 188
7.4.3 Elliptical crack under a uniform shear loading 193
7.5 The growth of an elliptical crack perpendicular to the FGM interlayer 205
7.5.1 Crack growth under a remotely inclined tensile loading 205
7.5.2 Crack growth under a remotely inclined compressive loading 209
7.6 Summary 213
References 214
Chapter 8 Yue’s Solution-based Dual Boundary Element Method 216
8.1 Introduction 216
8.2 Yue’s solution-based dual boundary integral equations 217
8.2.1 The displacement boundary integral equation 217
8.2.2 The traction boundary integral equation 219
8.2.3 The dual boundary integral equations for crack problems 220
8.3 Numerical implementation 221
8.3.1 Boundary discretization 221
8.3.2 The discretized boundary integral equation 224
8.4 Numerical integrations 225
8.4.1 Numerical integrations for the displacement BIE 225
8.4.2 Numerical integrations for the traction BIE 226
8.5 Linear equation systems for the discretized dual BIEs 230
8.6 Numerical verifications 235
8.6.1 Calculation of stress intensity factors 235
8.6.2 The effect of different meshes and the coefficientD on the SIF values 236
8.7 Summary 238
Appendix 4 Finite-part integrals and Kutt’s numerical quadrature 238
A4.1 Introduction 238
A4.2 Kutt’s numerical quadrature 239
References 240
Chapter 9 Analysis of Rectangular Cracks in the FGMs 243
9.1 Introduction 243
9.2 A square crack in FGMs of infinite extent 243
9.2.1 General 243
9.2.2 A square crack parallel to the FGM interlayer 245
9.2.3 A square crack having a 45. 
248 
9.2.4 A square crack perpendicular to the FGM interlayer 250
9.3 A square crack in the FGM interlayer 251
9.4 A rectangular crack in FGMs of infinite extent 253
9.4.1 General 253
9.4.2 A rectangular crack parallel to the FGM interlayer 254
9.4.3 A rectangular crack with long sides perpendicular to the FGM interlayer 257
9.4.4 A rectangular crack with short sides perpendicular to the FGM interlayer 258
9.5 A square crack in a FGM of finite extent 260
9.6 Square cracks in layered rocks 264
9.6.1 General 264
9.6.2 The crack dimensions and the parameters of layered rocks 265
9.6.3 A square crack subjected to a uniform compressive load 265
9.6.4 A square crack subjected to a non-uniform compressive load 269
9.7 Rectangular cracks in layered rocks 273
9.7.1 General 273
9.7.2 A rectangular crack subjected to a linear compressive load 273
9.7.3 A rectangular crack subjected to a nonlinear compressive load 276
9.8 Summary 279
References 279
Chapter 10 Boundary element analysis of fracturemechanics in transversely isotropic bi-materials 281
10.1 Introduction 281
10.2 Multi-region BEM analysis of cracks in transversely isotropic bi-materials 282
10.2.1 General 282
10.2.2 Calculation of the stress intensity factors 282
10.2.3 A penny-shaped crack perpendicular to the interface of transversely isotropic bi-materials 284
10.2.4 An elliptical crack perpendicular to the interface of transversely isotropic bi-materials 290
10.3 Dual boundary element analysis of a square crack in transversely isotropic bi-materials 295
10.3.1 General 295
10.3.2 Numerical verification 296
10.3.3 Numerical results and discussions 296
10.4 Summary 311
Appendix 5 The fundamental solution of transversely isotropic bi-materials 311
References 316