Numerical Modelling in Geomechanics

Numerical Modelling in Geomechanics, 2002, Scan, OCR......Page 1
10. A Finite Element Model for Water Saturated and Partially......Page 6
17. Modelling of Debris Flows and Flow Slides .................................................. 321......Page 7
1 Steady State Problems of Elliptic Type......Page 10
1. Introduction......Page 11
2.1. Introduction to strong, weak and variational formulations for 1D elliptic......Page 12
2.2. l . Flow in Porous Media......Page 15
2.2.2. The Linear Elasric Problem......Page 16
3.1. Infroduction: elements, nodes and shape functions......Page 17
3.2. Finite elements in one dimension......Page 18
3.2.1 . Mapping: isoparametric elements......Page 21
3.3. Finite elements in two and three dimensions......Page 22
3.4. A note on numerical integration techniques on finite element spaces......Page 25
4.1. Gaierkin and Weighted Residual Approximtions......Page 26
4.2. Example I: ID elastic bar in tension......Page 28
4.3. Example 2: Flow in Porous Media......Page 29
4.4. Example 3: Linear Elasticity......Page 30
5. References......Page 31
2 Transient Problems of Parabolic Type......Page 32
l. Introduction: Strong and weak formulations......Page 33
2. Finite Difference Methods: An introduction to the study of stability......Page 35
3.1. Introduction......Page 39
3.2, Single step ak\'gotithms: A Newmark scheme......Page 42
3.3. Matrix Stability Analysis......Page 43
4. Linear multistep algorithms: Runge Kutta schemes......Page 45
4.1. Fourth-order Runge-Kutta scheme......Page 46
4.2. Emr estimation and adaptiw time-stepping......Page 47
5. References......Page 48
3 Constitutive Modelling for Rate - independent Soils: A Review......Page 50
1. Introduction......Page 51
2. Notation......Page 52
3. Constitutive equations in rate form: general principles......Page 53
4. Non-linearity and incremental non-linearity......Page 55
5. Incrementally linear models......Page 57
6.1. Variable moduli models with loading-unloading conditions......Page 59
6.2.1. Basic principles and constitutive equations......Page 60
6.2.2. Perfect plasticity......Page 62
6.2.3. Isotropic hardening plasticity......Page 63
6.2.4. Anisotropic hardening plasticity......Page 66
6.3. Bounding Surface models and generalized plasticity......Page 71
6.4. Plasticity with extended hrdening rules......Page 73
7. Incrementally multi-linear models......Page 75
8.1. General concepts......Page 77
8.2. The theory of hypoplasticity......Page 78
9. References......Page 82
4 Coupling Equations for Water Saturated and Partially Saturated Geomaterials......Page 92
l. Introduction......Page 93
2.1. Microscopic balance equations......Page 94
2.2. Kinematic equations......Page 96
23. Mass balrutce equations......Page 99
2.4. Linear momentum balance equations......Page 100
2.6. Energy Mnce equation ad eniropy inequadity......Page 101
2.7. Constitutive equahgons......Page 102
2.8. Initial and boundary conditions......Page 104
3. References......Page 105
5 Continuum Damage Modelling and Some Computational Issues......Page 108
2.1. Incremental Damage Model......Page 109
2.2. Damage Model with Crack Closure......Page 111
2.3. Integrated Damage Model......Page 113
3. Damage Induced Anisotropy......Page 116
3.1. Principle......Page 117
3.2. Coupling with plasticity......Page 118
3.3. Crack closure eflects......Page 119
3.4. Damage and smeared crack models......Page 120
4. Non Local Damage......Page 121
4.2. Gradient damage model......Page 122
5.1. Secant Stiffness Algorithm......Page 123
5.2. Consistent Tangent Stiffness and Convergence......Page 126
5.3. Solution Control......Page 128
6. Conclusion Example of Computation......Page 131
7. References......Page 133
6 Non Linear Problems: An introduction......Page 136
1. Introduction......Page 137
2. The Newton-Raphson method......Page 138
2.1. The Newton-Rephson method in ofinite element context......Page 140
2.2. Modified Newton-Raphson......Page 145
2.3. Quasi Newton methods......Page 146
2.4. Displacement control......Page 149
2.5. Implementation of the Newton-Raphson method in a finite element code......Page 150
3. References......Page 152
7 Small Strain vs Large Strain Formulation in Computational Mechanics......Page 154
I. Introduction......Page 155
2.1. Virtual power equation......Page 156
2.3. Comments......Page 157
3.2. The first Pwla-Kirchhoflstress tensor......Page 158
3.3. The second Piola -Kirch hoflstress tensor......Page 159
3.4. New form of the external virtual power......Page 160
4. Summary......Page 161
5.2. Comparison between large and small strain for the internal virtuulpower......Page 162
5.3. Comparison between large strain and small strain for the external virtual......Page 163
5.4. Some remarks about constitutive equathns......Page 164
7. References......Page 165
8 Implicit Integration of Constitutive Equations in Computational Plasticity......Page 168
1. Introduction......Page 169
2. Evolution equations......Page 170
3. Numerical integration: basic principles......Page 171
4.1. Operdor split and product formula algorithm......Page 172
4.2. Problem I: elastic predictor......Page 173
4.3. Problem 2: phtic corrector......Page 174
5. Return mapping algorithm in principal elastic strain space......Page 175
6. Consistent tangent matrix......Page 176
7. Examples of application......Page 178
8. References......Page 182
9 Non Linear Problems: Advanced Techniques......Page 186
2. The Arc-length method......Page 187
2.1. Analytical description of the method......Page 190
2.2.1. Spherical versus cylindrical arc-length......Page 194
2.2.4. Bathe-Dvorkin version......Page 195
3. Line-search methods......Page 196
4. References......Page 198
10 A Finite Element Model for Water Saturated and Partially Saturated Geomaterials......Page 200
2. Balance equations for an isothermal saturatedfpartially saturated medium......Page 201
2.1. Muss balance equation......Page 202
2.2......Page 204
3. Constitutive equations......Page 205
4. Weak form: Variational appmacb......Page 207
5. Time discretization......Page 209
6. Consistent linewization......Page 211
7. Finite element discretization in space......Page 213
8. Conclusions......Page 214
9. References......Page 215
11 Alternative Formulations in Soil Dynamics......Page 216
1. Introduction......Page 217
2.1. Introduction......Page 218
2.2, First Order Equations. Riemann Invuriants and Radiation BCs......Page 219
2.3. Elastodynamics: Second order equation......Page 221
3.2. Discretization......Page 222
4. 2-step Taylor-Galerkin algorithm......Page 224
5.2. Vibmtion isolation......Page 227
7. References......Page 231
12 Objective Modelling of Strain Localization......Page 236
2. Problems with objective description of strain locdization......Page 237
3. Classification of models......Page 239
3.1. Kinematic description......Page 240
3.2. Constitutive models......Page 241
4. Regutarized softening continua......Page 242
4.1. Nonlocal models......Page 243
4.2. Explicit gradient models......Page 246
5. Concluding remarks......Page 248
6. References......Page 249
13 Numerical Modelling of Strong Discontinuities......Page 250
2. Triangular element with embedded displacement discontinuity......Page 251
3. Traction-separation law in damage format......Page 254
4. Evaluation of internal forces and tangent stiffness......Page 255
5. Tracing of the discontinuity path......Page 257
6. Resolution of discontinuities by extended finite elements......Page 259
8. References......Page 262
14 Practical Aspects of the Finite Element Method......Page 264
2. The Mysteries of Bending......Page 265
3. Risks of Reduced Integration......Page 266
4. Volumetric locking and failure loads......Page 268
5. Why we cannot choose all elements for geotechnical analysis: the Zienkiewicz-Taylor patch test......Page 271
6. Our favourite solver (do we have any?)......Page 279
6.1. Conjugate Gradient Method with Preconditioning......Page 282
6.2. Skyline storage scheme......Page 283
8. References......Page 284
15 Mechanical Effects of Chemical Degradation of Bonded Geomaterials in Boundary Value Problems......Page 286
2. Extension of plasticity theory to cope with weathering effects......Page 287
3.1. Weathering induced subsi&?nce in a circular foundation......Page 292
3.2. Weathering eflectb on a slope......Page 297
33. Weathering eflects in abandoned mines......Page 302
4. Conclusions......Page 307
5. References......Page 308
16 Finite Element Modelling of Landslides......Page 310
l. Introduction......Page 311
2. The mathematical model......Page 312
2.1. The eflective stress tensor......Page 313
2.2. Kinematics......Page 314
2.3. Balance of mass......Page 315
2.4. Balance of linear momentum......Page 316
2.6. The U-p, formulation......Page 317
2.8. A note on the incompressible undrained limit......Page 318
3.1. Discretization of the U-p, model......Page 319
4.1. Failure of a cut slope under rain actiorz......Page 321
4.2. Liquefaction failure of a dyke under earthquake action......Page 322
5. Conclusions......Page 326
6. References......Page 327
17 Modelling of Debris Flows and Flow Slides......Page 330
1. Introduction......Page 331
2. Mathematical Model: Depth Integrated Equations......Page 336
3. Depth integrated rheological models......Page 337
3.1. Bingham fluids......Page 340
3.2. Frictional Fluids: the role of pore pressures......Page 341
4. Initial conditions......Page 342
5. Discretization......Page 343
6.1. Tip No. 7flowslide at Abe$an (1 966)......Page 344
6.2. Air Liquefaction Flow Slides......Page 345
7, Acknowledgements......Page 347
8. References......Page 348
Index......Page 350