PREFACE THE AUTHOR
CHAPTER 1: ELEMENTARY TENSOR ANALYSIS ........................... 1
1.1 Introduction ............................................... 1
1.2 General Tensors, Cartesian Tensors, and Tensor Rank ........ 2
1.3 A Brief Review of Vector Analysis .......................... 2
1.4 Dyadic Form of Second-Order Tensors ........................ 5
1.5 Derivatives of Tensors ..................................... 7
1.6 Divergence and Stokes Theorems ............................. 8
1.6.1 Divergence Theorem or Gauss Theorem ................. 8
1.6.2 Stokes Theorem ...................................... 9
1.7 Some Formulas in Cylindrical Coordinates .................. 10
1.8 Some Formulas in Spherical Coordinates .................... 12
1.9 Summary and Further Reading ............................... 13
1.10 Problems .................................................. 13
CHAPTER 2: ELASTICITY AND ITS APPLICATIONS ..................... 17
2.1 Introduction .............................................. 17
2.2 Basic Concepts for Stress Tensor .......................... 18
2.3 Piola-Kirchhoff Stresses .................................. 19
2.4 Coordinate Transformation of Stress ....................... 21
2.5 Basic Concepts for Strain Tensor .......................... 23
2.6 Rate of Deformation ....................................... 25
2.7 Compatibility Equations ................................... 26
2.8 Hill's Work-Conjugate Stress Measures ..................... 26
2.9 Constitutive Relation ..................................... 27
2.10 Isotropic Solids .......................................... 27
2.11 Transversely Isotropic Solids ............................. 29
2.12 Equations of Motion and Equilibrium ....................... 30
2.13 Compatibility Equation in Terms of Stress Tensor .......... 32
2.14 Strain Energy Density ..................................... 33
2.15 Complementary Energy ...................................... 34
2.16 Hyperelasticity and Hypoelasticity ........................ 34
2.17 Plane Stress, Plane Strain, and the Airy Stress Function .. 36
2.18 Stress Concentration at a Circular Hole ................... 40
2.19 Force Acting at the Apex of a Wedge ....................... 43
2.20 Uniform Vertical Loading on Part of the Surface ........... 45
2.21 Solution for Indirect Tensile Test (Brazilian Test) ....... 46
2.22 Jaeger's Modified Brazilian Test .......................... 48
2.23 Edge Dislocation .......................................... 49
2.24 Dislocation Pile-up and Crack ............................. 51
2.25 Screw Dislocation and Faulting ............................ 53
2.26 Mura Formula for Curved Dislocation ....................... 58
2.27 Summary and Further Reading ............................... 60
2.28 Problems .................................................. 60
CHAPTER 3: COMPLEX VARIABLE METHODS FOR 2-D ELASTICITY ......... 63
3.1 Introduction .............................................. 63
3.2 Coordinate Transformation in Complex Variable Theory ...... 66
3.3 Homogeneous Stresses in Terms Analytic Functions .......... 67
3.4 A Borehole Subject to Internal Pressure ................... 67
3.5 Kirsch Solution by Complex Variable Method ................ 68
3.6 Definiteness and Uniqueness of the Analytic Function ...... 69
3.7 Boundary Conditions for the Analytic Functions ............ 70
3.8 Single-valued Condition for Multi-connected Bodies ........ 72
3.9 Multi-connected Body of Infinite Extend ................... 75
3.10 General Transformation of Quantities ...................... 76
3.11 Elastic Body with Holes ................................... 78
3.12 Stress Concentration at a Square Hole ..................... 82
3.13 Mapping Functions for Other Holes ......................... 87
3.14 Summary and Further Reading ............................... 88
3.15 Problems .................................................. 89
CHAPTER 4: THREE-DIMENSIONAL SOLUTIONS IN ELASTICITY ........... 93
4.1 Introduction .............................................. 93
4.2 Displacement Formulation .................................. 94
4.2.1 Helmholtz Decomposition ............................ 94
4.2.2 Lame's Strain Potential for Incompressible Solids .. 96
4.2.3 Galerkin Vector .................................... 97
4.2.4 Love's Displacement Potential for Cylindrical
Solids ............................................. 98
4.2.5 Papkovitch-Neuber Displacement Potential ........... 99
4.2.6 2-D Papkovitch-Neuber vs. Kolosov-Muskhelisvili
Methods ........................................... 101
4.3 Stress Formulations ...................................... 101
4.3.1 Beltrami and Beltrami-Schaefer Stress Functions ... 101
4.3.2 Maxwell Stress Functions .......................... 103
4.3.3 Morera Sress Function ............................. 104
4.3.4 Other Beltrami Stress Functions ................... 104
4.4 Some 3-D Solutions in Geomechanics ....................... 106
4.4.1 Hollow Sphere Subject to Internal and External
Pressures ......................................... 106
4.4.2 Kelvin's Fundamental Solution ..................... 109
4.4.2.1 Papkovitch-Neuber Potential Method ....... 109
4.4.2.2 Love's Displacement Potential Method ..... 111
4.4.3 Boussinesq's Fundamental Solution ................. 113
4.4.3.1 Love's and Lame's Strain Potential
Methods .................................. 114
4.4.3.2 Papkovitch-Neuber Potential Method ....... 115
4.4.4 Cerruti's Fundamental Solution .................... 118
4.4.5 Mindlin's Fundamental Solution in Half-space ...... 122
4.4.6 Lorentz's Fundamental Solution .................... 128
4.4.7 Melan's Fundamental Solution ...................... 130
4.5 Harmonic Functions and Indirect Method ................... 132
4.6 Harmonic Functions in Spherical Coordinates .............. 136
4.7 Harmonic Functions in Cylindrical Coordinates ............ 137
4.8 Biharmonic Functions ..................................... 138
4.9 Muki's Formulation in Cylindrical Coordinates ............ 139
4.9.1 Muki's Vector Potentials .......................... 140
4.9.2 Method of Solution by Hankel Transform ............ 141
4.9.3 Boussinesq Solution by Hankel Transform ........... 144
4.10 Summary and Further Reading .............................. 147
4.10.1 Summary ........................................... 147
4.10.2 Further Reading ................................... 148
4.10.2.1 General Method of Solutions for 3-D
Elasticity ............................... 148
4.10.2.2 Integral Transform in Solving 3-D
Problems ................................. 148
4.10.2.3 General Method of Solutions for
Circular Cylinders ....................... 148
4.10.2.4 General Method of Solutions for Spheres .. 149
4.11 Problems ................................................. 149
CHAPTER 5: PLASTICITY AND ITS APPLICATIONS .................... 159
5.1 Introduction ............................................. 159
5.2 Flow Theory and Deformation Theory ....................... 160
5.3 Yield Function and Plastic Potential ..................... 161
5.4 Elasto-plastic Constitutive Model ........................ I62
5.5 Rudnicki-Rice (1975) Model ............................... 163
5.6 Drucker's Postulate, PMPR, and Il'iushin's Postulate ..... 163
5.7 Yield Vertex ............................................. 165
5.8 Mohr-Coulomb Model ....................................... 168
5.9 Lode Angle or Parameter .................................. 169
5.10 Yield Criteria on the π-Plane ............................ 171
5 11 Other Soil Yield Models .................................. 174
5.12 Cap Models ...............................................
5.13 Physical Meaning of Cam-Clay Model ....................... 177
5.14 Modified Cam-Clay ........................................ 178
5.15 A Cam-clay Model for Finite Strain ....................... 181
5.16 Plasticity by Internal Variables ......................... i83
5.17 Viscoplasticity .......................................... 184
5.17.1 One-dimensional Modeling .......................... 184
5.17.2 Three-dimensional Models .......................... 186
5.17.1 Consistency Condition for Perzyna Model ........... 187
5.17.4 Consistency Model of Wang et al. (1997) ........... 188
5.17.5 Adachi-Oka (1984) Model ........................... 189
5.18 Summary and Further Reading .............................. 191
5.19 Problems ................................................. 192
CHAPTER 6: FRACTURE MECHANICS AND ITS APPLICATIONS ............ 197
6.1 Introduction ............................................. 197
6.2 Stress Concentration at a Elliptical Hole ................ 198
6.3 Stress Concentration at a Tensile Crack .................. 202
6.4 Stress Field near a Shear Crack .......................... 205
6.5 The General Stress and Displacement Field for Mode I
Cracks ................................................... 207
6.6 The General Stress and Displacement Field for Mode II
Cracks ................................................... 211
6.7 The General Stress and Displacement Field for Mode III
Cracks ................................................... 212
6.8 The Energy Release Rate at Crack Tips .................... 214
6.9 Fracture Toughness for Rocks ............................. 218
6.10 J-integral and the Energy Release Rate ................... 219
6.11 Westergaard Stress Function and Superposition ............ 223
6.12 Growth of Slip Surface in Slopes ......................... 227
6.13 Energy Release Rate for Earthquake ....................... 233
6.14 Wing Crack Model under Compressions ...................... 235
6.15 Bazant's Size Effect Law via J-integral .................. 237
6.16 Continuum Damage Mechanics ............................... 240
6.17 Solids Containing Microcracks ............................ 242
6.17.1 Compliance Change due to a Single Crack ........... 242
6.17.2 Effective Compliance for Cracked Bodies ........... 243
6.17.3 Non-interacting Result for Planar Transverse
Isotropy .......................................... 243
6.17.4 Planar Transverse Isotropy by Self-consistent
Method ............................................ 244
6.17.5 Planar Transverse Isotropy by Differential
Scheme ............................................ 245
6.17.6 Non-interacting Result for Cylindrical
Transverse Isotropy ............................... 245
6.17.7 Non-interacting Result for Isotropically Cracked
Solids ............................................ 246
6.18 Rudnicki-Chau (1996) Multiaxial Microcrack Model ......... 247
6.19 Summary and Further Reading .............................. 249
6.20 Problems ................................................. 250
CHAPTER 7: VISCOELASTICTY AND ITS APPLICATIONS ................ 257
7.1 Introduction ............................................. 257
7.2 Boltzmann's Integral Form of Stress and Strain ........... 258
7.3 Stieltjes Convolution Notation ........................... 260
7.4 Stress-Strain Relation in Differential Equation Form ..... 261
7.4.1 Maxwell Model ..................................... 262
7.4.2 Kelvin-Voigt Model ................................ 263
7.4.3 Three-Parameter Models ............................ 263
7.4.4 Generalized Maxwell and Kelvin Models ............. 265
7.5 Stress-strain Relation in Laplace Transform Space ........ 265
7.5.1 Viscoelastic Solids with Elastic Bulk Modulus ..... 267
7.5.2 Maxwell Solids .................................... 267
7.5.3 Kelvin-Voigt Solids ............................... 268
7.5.4 Standard Linear Solid and Three-Parameter Models .. 268
7.6 Correspondence Principle ................................. 270
7.6.1 Boussinesq Problem for Maxwell Half-space ......... 271
7.6.2 Boussinesq Problem for Kelvin-Voigt Half-space .... 273
7.6.3 Boussinesq Problem for Three-Parameter Model A .... 274
7.7 Creeping and Relaxation Tests ............................ 275
7.7.1 Maxwell Material .................................. 275
7.7.1.1 Creeping Test ............................ 275
7.7.1.2 Relaxation Test .......................... 276
7.7.2 Kelvin-Voigt Material ............................. 277
7.7.2.1 Creeping Test ............................ 277
7.7.2.2 Relaxation Test .......................... 277
7.7.3 Three-parameter Model A or Standard Linear Solid .. 278
7.7.3.1 Creeping Test ............................ 278
7.7.3.2 Relaxation Test .......................... 280
7.7.3.3 Relaxation Test in Compression ........... 281
7.8 Calibration of the Viscoelastic Model .................... 281
7.9 Viscoelastic Crack Models for Steam Injection ............ 284
7.9.1 Superposition of Auxiliary Problems I and II ...... 284
7.9.2 Center of Dilatation in Two-dimensional
Bimaterial ........................................ 284
7.9.3 Stress Intensity Factor of Auxiliary Problem II ... 287
7.9.4 Inverse Laplace Transform ......................... 287
7.9.5 Numerical Results ................................. 288
7.10 Summary and Further Reading .............................. 289
7.11 Problems ................................................. 290
CHAPTER 8: LINEAR ELASTIC FLUID-INFILTRATED SOLIDS AND
POROELASTICITY ................................................ 295
8.1 Introduction ............................................. 295
8.2 Biot's Theory of Poroelasticity .......................... 298
8.2.1 McNamee and Gibson Cylindrical Form ............... 298
8.2.2 Rice-Cleary (1976) Linearized Constitutive
Relation .......................................... 298
8.2.3 Rudnicki (1986) Constitutive Relation ............. 301
8.2.4 Rudnicki's (1985) Anisotropic Diffusive Solids .... 302
8.3 Biot-Verruijt Displacement Function ...................... 304
8.4 McNamee-Gibson-Verruijt Displacement Function ............ 306
8.5 Schiffman-Fungaroli-Verruijt Displacement Function ....... 307
8.6 Schiffman-Fungaroli Displacement Function ................ 308
8.7 Laplace-Hankel Transform Technique ....................... 309
8.8 Point Forces and Point Fluid Source in Half-space ........ 310
8.8.1 Vertical Point Force Solution ..................... 310
8.8.2 Horizontal Point Force Solution ................... 312
8.8.3 Fluid Point Source Solution ....................... 314
8.9 Cleary's Fundamental Solution of Point Forces in Full
Space .................................................... 315
8.9.1 Canonical Representation of Point Force Solution .. 315
8.9.2 Determination of Evolution Functions .............. 316
8.9.3 Determination of Unknown Constant F∞ .............. 321
8.9.4 Final Solutions ................................... 322
8.10 Rudnicki's Fundamental Solutions in Full Space ........... 323
8.10.1 Impulsive Fluid Source ............................ 323
8.10.2 Canonical Form of Displacement Solution ........... 323
8.10.3 Error Function Representation ..................... 324
8.10.4 Suddenly Applied Fluid Mass Source ................ 325
8.10.5 Equivalence of Fluid Mass Dipole and Body Force ... 327
8.10.6 Fluid Mass Dipoles ................................ 328
8.10.7 Point Force Solution by Rudnicki (1986) ........... 328
8.11 Thermoelasticity vs. Poroelasticity ...................... 330
8.12 Summary and Further Reading .............................. 330
8.12.1 Summary ........................................... 330
8.12.2 Further Reading ................................... 331
8.13 Problems ................................................. 331
CHAPTER 9: DYNAMICS AND WAVES IN GEOMATERIALS ................. 337
9.1 Introduction ............................................. 337
9.2 Seismic Waves ............................................ 338
9.3 Waves in Infinite Elastic Isotropic Solids ............... 338
9.4 Helmholtz Theorem and Wave Speeds ........................ 340
9.5 Rayleigh Waves ........................................... 341
9.5.1 Characteristics Equation for Rayleigh Wave Speed .. 341
9.5.2 Rayleigh Wave in Solids Satisfying Poisson
Condition ......................................... 342
9.5.3 Segel (1977) Method for Arbitrary Poisson's
Ratio ............................................. 345
9.6 Love Waves ............................................... 346
9.6.1 Non-existence of SH-wave in Homogeneous Half-
space ............................................. 347
9.6.2 Love Waves in an Elastic Layer on a Half-space .... 347
9.6.3 Dispersion Characteristics of Love Waves .......... 350
9.7 Stoneley Waves ........................................... 351
9.8 Elastic-plastic Waves .................................... 353
9.8.1 Acceleration Waves in Solids ...................... 353
9.8.2 Shear Banding as Stationary Acceleration Wave ..... 354
9.8.3 Acoustic Tensor for Geomaterials .................. 355
9.8.4 Wave Speed Analysis ............................... 357
9.9 Waves in Viscoelastic Solids ............................. 357
9.9.1 Complex Moduli ....................................
9.9.2 Longitudinal and Transverse Waves Speeds .......... 358
9 10 Dynamic Fracture Mechanics ............................... 359
9.10.1 Dynamic Solutions for a Stationary Crack .......... 360
9.10.2 Asymptotic Fields near a Moving Crack-tip ......... 361
9.10.3 Dynamic Energy Release Rate ....................... 362
9.10.4 Dynamic Fracture Toughness ........................ 363
9.11 Vibrations and Soil Dynamics ............................. 364
9.12 Summary and Further Reading .............................. 365
9.12.1 Summary
9.12.2 Further Reading ...................................
9.12.2.1 Waves in Solids and Elastodynamics ....... 366
9.12.2.2 Seismic Waves on Earth ................... 360
9.12.2.3 Waves in Porous Media .................... 361
9.12.2.4 Dynamic Fracture Mechanics ............... 361
9.12.2.5 Dynamic Fragmentation .................... 361
9.13 Problems ................................................. 371
Appendices .................................................... 371
Appendix A: Nanson Formula ............................... 371
Appendix B: Laplace Transform ............................ 373
Appendix C: Legendre Transform and Work Increments ....... 382
Selected Biographies .......................................... 385
References .................................................... 403
Author Index .................................................. 425
Subject Index ................................................. 433
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