Microstructures in Elastic Media

Phan-Thien, Nhan.; Kim, Sangtae.0-19-509086-1

Table of contents :
Contents……Page 8
1.1 Introduction and Motivation……Page 14
1.2 Stress and Strain……Page 15
1.3 Equations of Equilibrium……Page 17
1.4 Strain Energy……Page 19
1.4.1 Uniqueness……Page 20
1.4.2 Extremum Principles……Page 21
1.5 Betti’s Reciprocal Theorem……Page 23
1.6.1 Classification of Integral Equations……Page 24
1.6.2 Kelvin State……Page 26
1.6.3 Integral Representation……Page 28
1.6.4 Rigid Inclusion……Page 30
1.6.5 Eliminating Single or Double Layer……Page 32
1.7.1 Single Layer……Page 33
1.7.2 Double Layer……Page 36
1.8 Boundary Integral Equations……Page 40
1.8.1 Direct BEM……Page 41
1.8.2 Indirect BEM……Page 43
1.9 Spectral Properties……Page 46
1.9.2 λ = –1……Page 48
1.9.3 λ = +1……Page 49
1.9.4 Type II Problems……Page 50
1.9.5 Spectral Radius of Κ……Page 51
1.10.1 Rigid-Body Displacement……Page 52
1.10.5 Integral Representation……Page 53
1.10.8 Translating Rigid Sphere 1……Page 54
1.10.10 Kelvin’s Solution……Page 55
1.10.12 Papkovich-Neuber Representation……Page 56
1.10.14 Self-Adjoint Property of G……Page 57
1.10.18 Liapunov-Tauber Theorem……Page 58
2.1.1 Papkovich-Neuber Representation……Page 60
2.1.2 Potential Deformation……Page 62
2.1.3 Rotlet Deformation……Page 63
2.1.4 Kelvinlet Deformation……Page 64
2.1.5 Half–Space Solutions……Page 66
2.1.6 Interior Deformation……Page 69
2.2 Multipole Expansion……Page 70
2.2.1 Stresslet……Page 73
2.3.1 Translating a Rigid Sphere……Page 74
2.3.2 Rotating a Rigid Sphere……Page 76
2.3.3 Rigid Sphere in a Linear Deformation……Page 77
2.3.4 Rigid Sphere in a Quadratic Ambient Field……Page 81
2.3.5 Translating an Elastic Spherical Inclusion……Page 82
2.4.3 Navier Solutions……Page 84
2.4.6 Rigid Spherical Inclusion in High-Order Field……Page 85
3.1 Faxén Relations……Page 86
3.2 Rigid Spherical Inclusion……Page 89
3.3 Rigid Ellipsoidal Inclusion……Page 90
3.3.1 Singularity Solution for Translation……Page 92
3.3.2 Singularity Solution for Linear Ambient Field……Page 94
3.3.3 Degenerate Cases……Page 97
3.3.5 Interactions between Two Ellipsoids……Page 99
3.4.2 Faxén Relations for Torque and Stresslet……Page 100
3.4.4 Tractions for the Translating Ellipsoid……Page 101
4 Load Transfer Problem and Boundary Collocation……Page 102
4.1 The Method of Reflection……Page 103
4.2 Load Transfer between Two Spheres……Page 104
4.2.1 Far Field by Reflection……Page 105
4.2.2 Near Touching……Page 110
4.3.1 Spherical Harmonics……Page 113
4.3.2 Kelvin’s General Solutions……Page 115
4.4 Boundary Collocation……Page 119
4.4.1 Twin Multipole Expansions……Page 120
4.4.2 Collocation Equations for Translation Problems……Page 121
4.5 Comparison……Page 125
4.6 Constitutive Relation……Page 130
4.6.1 Constitutive Theory……Page 131
4.6.2 Cubic Lattices……Page 133
4.7 Kelvinlet near a Rigid Sphere……Page 135
4.7.1 The Axisymmetric Kelvinlet……Page 137
4.7.2 The Transverse Kelvinlet……Page 143
4.8.2 Lurié Solution……Page 148
4.8.3 Type I Problems……Page 149
5.1 Introduction……Page 150
5.2 Direct Formulation……Page 152
5.3 Completed Double Layer Boundary Element Method……Page 155
5.3.1 Range Completer……Page 156
5.3.2 Null Functions of (1+Κ)……Page 157
5.3.3 Completion Process……Page 158
5.3.4 Container Surface……Page 160
5.3.5 A Summary……Page 163
5.4.1 Translational Displacement……Page 164
5.4.2 On Picard Iteration……Page 166
5.4.4 Homogeneous Deformation……Page 168
5.5 Stresslet……Page 171
5.6 Spectrum for a Sphere……Page 172
5.6.1 Type I Problems – Ill-posed……Page 177
5.7 Completed Double Layer Traction Problem……Page 178
5.8.4 Gram-Schmidt Orthonormalization……Page 180
5.8.5 Hadamard Ill-posed Problem……Page 181
6 Numerical Implementation……Page 182
6.1 Numerical Quadrature……Page 183
6.2.1 Constant Element……Page 187
6.2.2 Higher Order Element……Page 188
6.3.1 Multivalued Traction……Page 193
6.3.2 Regular Integrals……Page 194
6.3.3 Singular Integrals……Page 196
6.3.4 Rigid-Body Displacement……Page 198
6.3.5 Adaptive Integration Schemes……Page 199
6.3.6 Far-Field Approximation……Page 202
6.4.2 Iterative Methods……Page 204
6.4.3 Domain Decomposition……Page 205
6.5 Distributed Computing under PVM……Page 206
6.5.1 Some Concepts in Distributed Computing……Page 207
6.5.2 Master/Slave Implementation……Page 210
6.6.3 Galerkin Expansion……Page 212
6.6.5 Evaluation of ∫[sub(Δ)] G[sub(ij)]dS and ∫[sub(Δ)] K[sub(ij)]dS……Page 214
7.1.1 Direct Formulation……Page 216
7.1.2 CDL-BIEM……Page 220
7.2 Sphere in Homogeneous Deformation……Page 223
7.3 Two Spheroids……Page 226
7.4 CDL in Half–Space……Page 229
7.5 Container Surface……Page 231
7.6 Deformation of a Cluster……Page 233
7.7.1 Arrays of Spheres……Page 236
7.7.2 Epilogue: Sedimentation through an Array of Spheres……Page 237
References……Page 242
C……Page 251
G……Page 252
K……Page 253
P……Page 254
S……Page 255
Y……Page 256
Z……Page 257