Nonlinear mesoscopic elasticity : the complex behaviour of granular media including rocks and soil

Nonlinear Mesoscopic Elasticity......Page 5
Contents......Page 7
Preface......Page 13
Acknowledgements......Page 15
1.1 Systems......Page 17
1.2 Examples of Phenomena......Page 20
1.3 The Domain of Exploration......Page 22
1.4 Outline......Page 23
References......Page 24
2 Microscopic/Macroscopic Formulation of the Traditional Theory of Linear and Nonlinear Elasticity......Page 27
2.1 Prefatory Remarks......Page 28
2.2.1 A Microscopic Description......Page 29
2.2.2 Microscopic Description and Thermodynamics......Page 36
2.2.3 From Microscopic Model to Continuum Elasticity......Page 40
2.3.1 Displacement, Strain, and Stress......Page 45
2.3.2 Dynamics of the Displacement Field......Page 47
2.3.4 Inhomogeneous Elastic Systems......Page 48
2.4.2 Series Expansion for ES......Page 49
2.4.3 Series Expansion for EZ......Page 50
2.4.4 Series Expansion for FT......Page 51
2.4.5 Assemble the Pieces......Page 52
2.5 Energy Scales......Page 53
References......Page 54
3 Traditional Theory of Nonlinear Elasticity, Results......Page 55
3.1.1 Quasistatic Response; Linear......Page 57
3.1.2 Quasistatic Response; Nonlinear......Page 58
3.2 Dynamic Response; Linear......Page 60
3.3 Quasistatic/Dynamic Response; Nonlinear......Page 61
3.4.1 Basic Equations......Page 63
3.4.2 Wave Propagation......Page 64
3.4.3 Resonant Bar......Page 71
3.5 Exotic Response; Nonlinear......Page 75
3.6.1 Green Function, Free Space......Page 80
3.6.2 Green Function, Resonant Bar......Page 81
References......Page 83
4.1 Background......Page 85
4.2.1 Hertz–Mindlin Contacts......Page 86
4.2.3 Hertzian Asperities......Page 88
4.2.4 Van der Waals Surfaces......Page 89
4.2.5 Other......Page 91
4.3 Effective Medium Theory......Page 92
4.4.1 Hertzian Contacts......Page 99
4.4.2 Van der Waals Surfaces......Page 103
4.4.3 Generalization and Caveats......Page 110
References......Page 111
5.1 Temperature......Page 113
5.2.1 Saturation/Strain Coupling......Page 118
5.2.2 Saturation/Strain Response......Page 124
References......Page 128
6.1.1 Finite Displacement Elastic Elements: The Model......Page 129
6.1.2 Finite Displacement Elastic Element: Implementing the Model......Page 132
6.3 Finite Displacement Elastic Elements: Dynamic Response......Page 138
6.3.1 Finite Displacement Elastic Element: Resonant Bar......Page 139
6.3.2 Finite Displacement Elastic Element: Wave Mixing......Page 146
6.5 Summary......Page 152
6.6.1 Hertzian Contacts......Page 153
6.6.2 The Masing Rules......Page 155
6.6.3 The Endochronic Formalism......Page 157
References......Page 160
7.1 Fast/Slow Linear Dynamics......Page 161
7.1.1 Quasistatic Response......Page 162
7.1.2 AC Response......Page 165
7.2 Fast Nonlinear Dynamics......Page 169
7.3 Auxiliary Fields and Slow Dynamics......Page 174
7.3.1 X = The Conditioning Field......Page 175
7.3.2 X = Temperature......Page 178
7.4 Summary......Page 179
References......Page 181
8 Q and Issues of Data Modeling/Analysis......Page 183
8.1.2 Extracting Elastic Energy......Page 184
8.1.3 Other......Page 185
8.2.1 Nonlinear Dampling: Traditional Theory......Page 186
8.3 Why Measure Q?......Page 187
8.4.1 Measurement of Q in a Linear System......Page 188
8.4.2 Measurement of Q in a Nonlinear System......Page 190
8.5.1 Modeling a Resonant Bar......Page 192
8.5.2 Data Processing......Page 194
References......Page 213
9.1 Spectroscopies......Page 215
9.1.2 Nonlinear, Homogeneous......Page 216
9.2.1 Time of Flight......Page 218
9.2.2 Normal Mode......Page 219
9.2.3 Normal Mode, a Numerical Example......Page 222
9.2.4 Time Reversal......Page 225
9.2.5 Time Reversal, a Numerical Example......Page 231
9.3.1 Time of Flight......Page 234
9.3.2 Nonlinear Normal-Mode Tomography......Page 236
9.3.3 Nonlinear Time-Reversal Tomography......Page 239
References......Page 241
10.1 Some Basic Observations......Page 243
10.2 Quasistatic Stress-Strain Data; Hysteresis......Page 247
10.3 Coupling to Auxiliary Fields......Page 252
10.3.1 Saturation......Page 253
10.3.2 Temperature......Page 258
10.4 Inversion......Page 263
10.4.1 Simple σ – ε Protocol and Minimalist Model......Page 264
10.4.2 Elaborate σ – ε Protocol and Minimalist Model......Page 267
10.4.3 The Relationship of σ – ε Data to Dynamics......Page 270
References......Page 274
11.1.1 Pressure-Dynamic......Page 277
11.1.2 Temperature-Dynamic......Page 280
11.1.3 Saturation-Dynamic......Page 281
11.2 Dynamic–Dynamic......Page 283
11.2.1 Dynamic–Dynamic: Wave Mixing......Page 284
11.2.2 Dynamic–Dynamic, Resonant Bar, Preliminaries: Fast Dynamics and Slow Dynamics......Page 290
11.2.3 Dynamic–Dynamic: Resonant Bar, Data......Page 296
11.3.1 Anomalous Fast Dynamics......Page 324
11.3.2 Slow Dynamics......Page 325
References......Page 327
12.1.1 Wave Mixing in the Earth......Page 329
12.1.2 The Earth as Resonant Bar......Page 331
12.2 Passive Probes......Page 334
References......Page 337
13.1 Overview......Page 339
13.2 Historical Context......Page 340
13.3 Simple Conceptual Model of a Crack in an Otherwise Elastically Linear Solid......Page 343
13.4.1 Nonlinear Wave Modulation Spectroscopy (NWMS)......Page 346
13.4.3 Robust NWMS......Page 352
13.4.4 NWMS Summation......Page 354
13.4.6 Nonlinear Ringdown Spectroscopy (NRS)......Page 357
13.4.7 Slow Dynamics Diagnostics (SDD)......Page 361
13.5 Progressive Mechanical Damage Probed by NEWS Techniques......Page 363
13.6.1 Harmonic Imaging......Page 369
13.6.2 Modulation Imaging......Page 370
13.6.3 Imaging Applying Time Reversal Nonlinear Elastic Wave Spectroscopy (TR NEWS)......Page 373
13.7.1 Time Reversal + Phase Inversion......Page 377
13.7.2 Scaling Subtraction/Variable Amplitude Method......Page 380
References......Page 381
Color Plates......Page 385
Index......Page 407