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#### Fracture Mechanics of Nonhomogeneous Materials

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Preface
Chapter 1　Fundamental theory of fracture mechanics of nonhomogeneous
materials　1
1.1　Internal crack　2
1.1.1　Basic equations for nonhomogeneous materials　2
1.1.2　Crack-tip fields for homogeneous materials　3
1.1.3　Crack-tip fields for nonhomogeneous materials　6
1.1.4　Crack-tip fields for nonhomogeneous orthotropic materials　12
1.2　Interface crack　14
1.2.1　Crack-tip fields of an interface crack　14
1.2.2　Crack-tip fields of an interface crack between two nonhomogeneous media　19
1.3　Three-dimensional curved crack　23
1.3.1　Internal crack　23
1.3.2　Interface crack　25
References　26
Chapter 2　Exponential models for crack problems in nonhomogeneous materials　28
2.1　Crack model for nonhomogeneous materials with an arbitrarily oriented crack　29
2.1.1　Basic equations and boundary conditions　29
2.1.2　Full field solution for a crack in the nonhomogeneous medium　31
2.1.3　Stress intensity factors (SIFs) and strain energy release rate (SERR)　36
2.2　Crack problems in nonhomogeneous coating-substrate or double-layered structures　38
2.2.1　Interface crack in nonhomogeneous coating-substrate structures　38
2.2.2　Cross -interface crack parallel to the gradient of material properties　45
2.2.3　Arbitrarily oriented crack in a double-layered structure　54
2.3　Crack problems in orthotropic nonhomogeneous materials　69
2.3.1　Basic equations and boundary conditions　69
2.3.2　Solutions to stress and displacement fields　71
2.3.3　Crack-tip SIFs　77
2.4　Transient crack problem of a coating-substrate structure　78
2.4.1　Basic equations and boundary conditions　78
2.4.2　Solutions to stress and displacement fields　79
2.4.3　Crack-tip SIFs　84
2.5　Representative examples　85
2.5.1　Example 1: Arbitrarily oriented crack in an infinite nonhomogeneous medium　85
2.5.2　Example 2: Interface crack between the coating and the substrate　88
2.5.3　Example 3: Crossing-interface crack perpendicular to the interface in a double-layered structure　89
2.5.4　Example 4: Inclined crack crossing the interface　94
2.5.5　Example 5: Vertical crack in a nonhomogeneous coating-substrate structure subjected to impact loading　96
Appendix 2A　98
References　99
Chapter 3　General model for nonhomogeneous materials with general elastic properties　101
3.1　Piecewise-exponential model for the mode I crack problem　102
3.1.1　Piecewise-exponential model (PE model)　102
3.1.2　Solutions to stress and displacement fields　105
3.1.3　Crack-tip SIFs　111
3.2　PE model for mixed-mode crack problem　112
3.2.1　Basic equations and boundary conditions　112
3.2.2　Solutions to stress and displacement fields　114
3.2.3　Crack-tip SIFs　119
3.3　PE model for dynamic crack problem　119
3.3.1　Basic equations and boundary conditions　119
3.3.2　Solutions to stress and displacement fields　123
3.3.3　Crack-tip SIFs　126
3.4　Representative examples　127
3.4.1　Example 1: Mode I crack problem for nonhomogeneous materials with general elastic properties　127
3.4.2　Example 2: Mixed-mode crack problem for nonhomogeneous materials with general elastic properties and an arbitrarily oriented crack　134
3.4.3　Example 3: Dynamic Mode I crack problem for nonhomogeneous materials with general elastic properties　139
Appendix 3A　145
References　151
Chapter 4　Fracture mechanics of nonhomogeneous materials based on piecewise-exponential model　153
4.1　Thermomechanical crack models of nonhomogeneous materials　154
4.1.2　Crack model for nonhomogeneous materials under thermal shock load　157
4.2　Viscoelastic crack model of nonhomogeneous materials　170
4.2.1　The correspondence principle for viscoelastic FGMs　170
4.2.2　Viscoelastic models for nonhomogeneous materials　173
4.2.3　PE model for the viscoelastic nonhomogeneous materials　174
4.3　Crack model for nonhomogeneous materials with stochastic properties　177
4.3.1　Stochastic micromechanics-based model for effective properties　177
4.3.2　Probabilistic characteristics of effective properties at transition region　182
4.3.3　Crack in nonhomogeneous materials with stochastic mechanical properties　183
4.4　Examples　188
4.4.1　Example 1: Steady thermomechanical crack problem　188
4.4.2　Example 2: Viscoelastic crack problem　195
4.4.3　Example 3: Crack problem in FGMs with stochastic mechanical properties　198
References　202
Chapter 5　Fracture of nonhomogeneous materials with complex interfaces　205
5.1　Interaction integral (I-integral)　207
5.1.1　J-integral　207
5.1.2　I-integral　208
5.1.3　Auxiliary field　208
5.1.4　Extraction of the SIFs　210
5.2　Domain-independent I-integral (DII-integral)　211
5.2.1　Domain form of the I-integral　211
5.2.2　DII-integral　214
5.3　DII-integral for orthotropic materials　220
5.4　Consideration of dynamic process　223
5.5　Calculation of the T-st
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