1 Introduction page ............................................ 1
1.1 Continuum properties .................................... 2
1.1.1 Deformation and strain ........................... 3
1.1.2 The stress-field ................................. 3
1.1.3 Constitutive relations ........................... 3
1.2 Earth processes ......................................... 6
1.3 Elements of Earth structure ............................. 8
1.3.1 Mantle .......................................... 12
1.3.2 Core ............................................ 13
1.4 State of the Earth ..................................... 14
Part I: Continuum Mechanics in Geophysics .............. 19
2 Description of Deformation .................................. 21
2.1 Geometry of deformation ................................ 21
2.1.1 Deformation of a vector element ................. 23
2.1.2 Successive deformations ......................... 24
2.1.3 Deformation of an element of volume ............. 24
2.1.4 Deformation of an element of area ............... 25
2.1.5 Homogeneous deformation ......................... 25
2.2 Strain ................................................. 27
2.2.1 Stretch ......................................... 27
2.2.2 Principal fibres and principal stretches ........ 28
2.2.3 The decomposition theorem ....................... 29
2.2.4 Pure rotation ................................... 30
2.2.5 Tensor measures of strain ....................... 32
2.3 Plane deformation ...................................... 34
2.4 Motion ................................................. 36
2.5 The continuity equation ................................ 38
2.4 Appendix: Properties of the deformation gradient
determinant ............................................ 39
3 The Stress-Field Concept .................................... 41
3.1 Traction and stress .................................... 41
3.2 Local equations of linear motion ....................... 44
3.2.1 Symmetry of the stress tensor ................... 44
3.2.2 Stress jumps (continuity conditions) ............ 46
3.3 Principal basis for stress ............................. 48
3.4 Virtual work rate principle ............................ 51
3.5 Stress from a Lagrangian viewpoint ..................... 53
4 Constitutive Relations ...................................... 54
4.1 Constitutive relation requirements ..................... 54
4.1.1 Simple materials ................................ 55
4.1.2 Material symmetry ............................... 56
4.1.3 Functional dependence ........................... 57
4.2 Energy balance ......................................... 57
4.3 Elastic materials ...................................... 60
4.4 Isotropic elastic material ............................. 61
4.4.1 Effect of rotation .............................. 61
4.4.2 Coaxiality of the Cauchy stress tensor and
the Eulerian triad .............................. 62
4.4.3 Principal stresses .............................. 62
4.4.4 Some isotropic work functions ................... 63
4.5 Fluids ................................................. 64
4.6 Viscoelasticity ........................................ 67
4.7 Plasticity and flow .................................... 69
5 Linearised Elasticity and Viscoelasticity ................... 71
5.1 Linearisation of deformation ........................... 71
5.2 The elastic constitutive relation ...................... 72
5.2.1 Isotropic response .............................. 73
5.2.2 Nature of moduli ................................ 73
5.2.3 Interrelations between moduli ................... 74
5.2.4 An example of linearisation ..................... 74
5.2.5 Elastic constants ............................... 75
5.2.6 The uniqueness Theorem .......................... 76
5.3 Integral representations ............................... 79
5.3.1 The reciprocal Theorem .......................... 80
5.3.2 The representation Theorem ...................... 81
5.4 Elastic Waves .......................................... 83
5.4.1 Isotropic media ................................. 83
5.4.2 Green's tensor for isotropic media .............. 85
5.4.3 Interfaces ...................................... 86
5.5 Linear viscoelasticity ................................. 88
5.6 Viscoelastic behaviour ................................. 91
5.7 Damping of harmonic oscillations ....................... 92
6 Continua under Pressure ..................................... 95
6.1 Effect of radial stratification ........................ 95
6.1.1 Hydrostatic pressure ............................ 96
6.1.2 Thermodynamic relations ......................... 97
6.2 Finite strain deformation ............................. 100
6.3 Expansion of Helmholtz free energy and equations
of state .............................................. 102
6.4 Incremental stress and strain ......................... 105
6.4.1 Perturbations in stress ........................ 106
6.4.2 Perturbations in boundary conditions ........... 107
6.5 Elasticity under pressure ............................. 107
7 Fluid Flow ................................................. 110
7.1 The Navier-Stokes equation ............................ 110
7.1.1 Heat flow ...................................... 111
7.1.2 The Prandtl number ............................. 112
7.2 Non-dimensional quantities ............................ 113
7.2.1 The Reynolds number ............................ 115
7.2.2 Stokes Flow .................................... 115
7.2.3 Compressibility ................................ 115
7.2.4 The Peclet number .............................. 117
7.3 Rectilinear shear flow ................................ 117
7.4 Plane two-dimensional flow ............................ 118
7.5 Thermal convection .................................... 121
7.5.1 The Rayleigh and Nusselt numbers ............... 121
7.5.2 The Boussinesq approximation ................... 121
7.5.3 Onset of convection ............................ 122
7.5.4 Styles of convection ........................... 125
7.6 The effects of rotation ............................... 126
7.6.1 Rapid rotation ................................. 127
7.6.2 The Rossby and Ekman numbers ................... 128
7.6.3 Geostrophic flow ............................... 128
7.6.4 The Taylor-Proudman theorem .................... 129
7.6.5 Ekman layers ................................... 129
8 Continuum Equations and Boundary Conditions ................ 131
8.1 Conservation equations ................................ 131
8.1.1 Conservation of mass ........................... 132
8.1.2 Conservation of momentum ....................... 132
8.1.3 Conservation of energy ......................... 133
8.2 Interface conditions .................................. 134
8.3 Continuum electrodynamics ............................. 135
8.3.1 Maxwell's equations ............................ 135
8.3.2 Electromagnetic constitutive equations ......... 136
8.3.3 Electromagnetic continuity conditions .......... 137
8.3.4 Energy equation for the electromagnetic
field .......................................... 138
8.3.5 Electromagnetic disturbances ................... 139
8.3.6 Magnetic fluid dynamics ........................ 141
8.4 Diffusion and heat flow ............................... 145
8.4.1 Equilibrium heat flow .......................... 146
8.4.2 Time-varying problems .......................... 148
Part II: Earth Deformation .................................... 151
9 From the Atomic Scale to the Continuum ..................... 153
9.1 Transport properties and material defects ............. 153
9.1.1 Grains and crystal defects ..................... 153
9.1.2 General transport properties ................... 156
9.1.3 Atomic diffusion ............................... 157
9.2 Lattice vibrations .................................... 158
9.3 Creep and rheology .................................... 162
9.3.1 Crystal elasticity ............................. 162
9.3.2 Deformation behaviour .......................... 163
9.4 Material properties at high temperatures and
pressures ............................................. 166
9.4.1 Shock-wave techniques .......................... 166
9.4.2 Pressure concentration by reduction of area .... 168
9.5 Computational methods ................................. 171
9.5.1 Electronic structure calculations .............. 171
9.5.2 Atomistic simulations .......................... 174
9.5.3 Simulation of crystal structures ............... 175
9.5.4 Finite temperature ............................. 176
9.5.5 Influence of defects ........................... 178
10 Geological Deformation ..................................... 180
10.1 Microfabrics .......................................... 181
10.1.1 Crystal defects ................................ 181
10.1.2 Development of microstructure .................. 182
10.1.3 Formation of crystallographically preferred
orientations ................................... 184
10.2 Macroscopic structures ................................ 186
10.2.1 Multiple phases of deformation ................. 187
10.2.2 Folding and boudinage .......................... 189
10.2.3 Fractures and faulting ......................... 193
10.2.4 Development of thrust complexes ................ 204
11 Seismology and Earth Structure ............................. 207
11.1 Seismic Waves ......................................... 207
11.1.1 Reflection and refraction ...................... 208
11.1.2 Attenuation effects ............................ 209
11.2 Seismic sources ....................................... 212
11.3 Building the response of the Earth to a source ........ 216
11.3.1 Displacements as a normal mode sum ............. 218
11.3.2 Free oscillations of the Earth ................. 220
11.4 Probing the Earth ..................................... 231
11.4.1 Seismic phases ................................. 231
11.4.2 Normal mode frequencies ........................ 239
11.4.3 Comparison with observations ................... 243
11.4.4 Imaging three-dimensional structure ............ 247
11.5 Earthquakes and faulting .............................. 253
12 Lithospheric Deformation ................................... 257
12.1 Definitions of the lithosphere ........................ 257
12.2 Thermal and mechanical structure ...................... 258
12.2.1 Thermal conduction in the oceanic
lithosphere .................................... 258
12.2.2 Mechanical deformation ......................... 261
12.2.3 Estimates of the elastic thickness of the
lithosphere .................................... 265
12.2.4 Strength envelopes and failure criteria ........ 266
12.3 Plate boundaries and force systems .................... 271
12.3.1 Nature of plate boundaries ..................... 271
12.3.2 Plate boundary forces .......................... 272
12.4 Measures of stress and strain ......................... 274
12.4.1 Stress measurements ............................ 274
12.4.2 Strain measurements ............................ 276
12.5 Glacial rebound ....................................... 280
12.6 Extension and convergence ............................. 283
12.6.1 Extension ...................................... 283
12.6.2 Convergence .................................... 288
13 The Influence of Rheology: Asthenosphere to the Deep
Mantle ..................................................... 294
13.1 Lithosphere and asthenosphere ......................... 294
13.1.1 Seismic imaging ................................ 295
13.1.2 Seismic attenuation ............................ 297
13.1.3 Seismic anisotropy ............................. 299
13.1.4 Asthenospheric flow ............................ 302
13.1.5 The influence of a low-viscosity zone .......... 302
13.2 Subduction zones and their surroundings ............... 308
13.2.1 Configuration of subduction zones .............. 309
13.2.2 Flow around the slab ........................... 311
13.2.3 Temperatures in and around the subducting
slab ........................................... 314
13.2.4 Subduction and orogeny ......................... 316
13.3 The influence of phase transitions .................... 319
13.4 The deeper mantle ..................................... 323
13.4.1 Viscosity variations in the mantle and the
geoid .......................................... 323
13.4.2 The lower boundary layer ....................... 328
14 Mantle Convection .......................................... 330
14.1 Convective forces ..................................... 330
14.1.1 Boundary layer theory .......................... 330
14.1.2 Basic equations ................................ 332
14.1.3 Boundary conditions ............................ 333
14.1.4 Non-dimensional treatment ...................... 334
14.1.5 Computational convection ....................... 335
14.2 Convective planform ................................... 339
14.3 Thermal structure and heat budget ..................... 342
14.3.1 Thermal boundary layers and the geotherm ....... 342
14.3.2 Plates ......................................... 346
14.3.3 Hot spots and plumes ........................... 348
14.4 Circulation of the mantle ........................ 353
14.4.1 Present-day and past plate motion models ....... 354
14.4.2 Implications of plate motion models for
mantle circulation ............................. 357
14.4.3 Mantle circulation models ...................... 361
14.5 Mantle rheology ....................................... 371
14.5.1 Temperature dependence ......................... 372
14.5.2 Strain dependence .............................. 373
14.6 Coupled lithosphere-mantle convection models .......... 375
14.7 Thermochemical convection ............................. 377
15 The Core and the Earth's Dynamo ............................ 379
15.1 The magnetic field at the surface and at the top of
the core .............................................. 380
15.2 Convection and dynamo action .......................... 384
15.2.1 Basic equations ................................ 384
15.2.2 Boundary conditions ............................ 387
15.2.3 Interaction of the flow with the magnetic
field .......................................... 388
15.2.4 Deviations from the reference state ............ 389
15.2.5 Non-dimensional treatment ...................... 390
15.3 Numerical dynamos ..................................... 392
15.4 Evolution of the Earth's core ......................... 397
15.4.1 Energy balance ................................. 397
15.4.2 Thermal and compositional effects .............. 399
15.4.3 Inner core growth in a well-mixed core ......... 400
Appendix: Table of Notation ................................... 407
Bibliography .................................................. 413
Index ......................................................... 423
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