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This page intentionally left blank Geophysical Continua presents a systematic treatment of deformation in the Earth from seismic to geologic time scales and demonstrates the linkages between different aspects of the Earth’s interior that are often treated separately. A unified treatment of solids and fluids is developed to include thermodynamics and electrodynamics, in order to cover the full range of tools needed to understand the interior of the globe. A close link is made between microscopic and macroscopic properties manifested through elastic, viscoelastic and fluid rheologies and their influence on deformation. Following a treatment of geological deformation, a global perspective is taken on lithospheric and mantle properties, seismology, mantle convection, the core and Earth’s dynamo. The emphasis throughout the book is on relating geophysical observations to interpretations of earth processes. Physical principles and mathematical descriptions are developed that can be applied to a broad spectrum of geodynamic problems. Incorporating illustrative examples and an introduction to modern computational techniques, this textbook is designed for graduate-level courses in geophysics and geodynamics. It is also a useful reference for practicing Earth scientists. Supporting resources for this book, including exercises and full-colour versions of figures, are available at www.cambridge.org 9780521865531. Brian Kennett is Director and Distinguished Professor of Seismology at the Research School of Earth Sciences in The Australian National University. Professor Kennett’s research interests are directed towards understanding the structure of the Earth through seismological observations. He is the recipient of the 2006 Murchison Medal of the Geological Society of London, and the 2007 Gutenberg Medal of the European Geosciences Union, and he is a Fellow of the Royal Society of London. Professor Kennett is the author of three other books for Cambridge University Press: Seismic Wave Propagation in Stratified Media (1983), The Seismic Wavefield: Introduction and Theoretical Development (2001), and The Seismic Wavefield: Interpretation of Seismograms on Regional and Global Scales (2002). Hans-Peter Bunge is Professor and Chair of Geophysics at the Department of Earth and Environmental Sciences, University of Munich, and is Head of the Munich Geo-Center. Prior to his Munich appointment, he spent 5 years on the faculty at Princeton University. Professor Bunge’s research interests lie in the application of high performance computing to problems of Earth and planetary evolution, including core, mantle and lithospheric dynamics. A member of the Bavarian Academy of Sciences, Bunge is also President of the Geodynamics Division of the European Geosciences Union (EGU). Geophysical Continua Deformation in the Earth’s Interior B.L.N. Kennett Research School of Earth Sciences The Australian National University H.-P. Bunge Department of Geosciences Ludwig Maximilians University, Munich Cambridge University Press Cambridge New York Melbourne Madrid Cape Town Singapore São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org 9780521865531 © B.L.N. Kennett and H.-P. Bunge 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 978-0-511-40890-8 ISBN-13 978-0-521-86553-1 eBook (EBL) hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents 1 Introduction 1.1 Continuum properties 1.1.1 Deformation and strain 1.1.2 The stress-field 1.1.3 Constitutive relations 1.2 Earth processes 1.3 Elements of Earth structure 1.3.1 Mantle 1.3.2 Core 1.4 State of the Earth page 1 2 3 3 3 6 8 12 13 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 Appendix: Properties of the deformation gradient determinant 39 vi Contents 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 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 Elastic Waves 83 Isotropic media 83 Green’s tensor for isotropic media 85 Interfaces 86 Linear viscoelasticity 88 Viscoelastic behaviour 91 Damping of harmonic oscillations 92 Contents vi 6 Continua under Pressure 95 Effect of radial stratification 95 Hydrostatic pressure 95 Index ab initio calculations, 104, 172, 175, 252 absolute plate motion, 299 acceleration, 44, 76, 96, 110, 118, 127, 132 gravitational, 97, 121, 216 acceleration field, 37, 51, 58 acoustic waves, 110, 142 activation energy, 164, 210 activation enthalpy, 158 adiabat, 330, 352 adiabatic, 12, 59, 97, 98, 144, 168, 328, 342, 343, 345, 380, 400 adiabatic compression, 287 adiabatic gradient, 344 adiabatic moduli, 99, 328 adjoint equations, 367, 371 adjoint problem, 335, 368, 372 advection, 287, 292, 345 advective, 105, 117, 148 advective boundary layer, 315 aftershocks, 200, 203, 204 AK135 model, 10, 11, 229, 232, 244, 251 Alfven waves, 145 -effect, 390 Anderson fracture, 198, 200 anelastic approximation, 286, 332, 385 angular momentum, 2 angular velocity, 354 anisotropic, 95 anisotropy, 15, 17, 186, 200, 207, 270, 294, 299, 301, 302, 380 azimuthal, 299 Arrhenius form, 158, 286, 291 asperities, 255 asthenosphere, 17, 154, 257, 259, 284, 288, 294, 295, 297, 303, 335, 341, 372 asthenospheric flow, 302 atomic level, 153, 162 attenuation, 17, 154, 209 scattering, 209 barriers, 255 bathymetric swells, 349 bending, 186, 190 biharmonic equation, 119, 305 body force, 51, 57, 76, 83, 110, 119, 217 body waves, 231, 247 bottom heating, 125, 126, 341, 345, 363, 364, 366 boudinage, 193 boundary conditions, 107, 123, 131, 146, 217, 220, 222, 225, 259, 280, 287, 290, 312, 313, 333, 354, 371, 388, 395 boundary layer, 303, 359, 396 boundary layer theory, 330 boundary layers, 117 Boussinesq approximation, 122, 286, 287, 303, 391 Brillouin zone, 159, 161 brittle, 16, 193, 261–284, 318, 376 brittle failure, 377 brittle–ductile transition, 16, 193, 261, 267, 268 buckling, 186, 190, 193 bulk modulus, 73, 97, 99, 103, 104, 108, 116, 132, 214, 249, 252, 328 bulk viscosity, 66, 110, 132 bulk-modulus, 13 bulk-sound speed, 98, 116, 249–253, 310, 328 buoyancy, 121, 303, 307, 329, 330, 342 buoyancy flux, 350, 351 Burgers vector, 154, 165 Cauchy strain, 100, 101, 162 Cauchy strain tensor, 32, 33, 67, 72 Cauchy stress, 106 centrifugal force, 127, 323, 386 centrifugal forces, 127 centroid moment tensor, 253, 274 chemical components, 378 chemical heterogeneity, 18, 252, 253, 328 chemical potential, 157, 163, 286, 399 Clapeyron slope, 321, 322, 329 Clausius–Clapeyron equation, 135 coefficient of thermal expansion, 122 collisional resistance, 273 Ключевые слова: chain rule, material, cijpq etpq, gj, statistical mechanic, relative, ri, r dr, good, equation, surface, tr, lithosphere, vice versa, mass, velocity, source, exural rigidity, scale, time, poynting vector, uek, dening scalar, ql, exp, core, u z, rc, behaviour, byerlees law, plate, layer, in?uence, ra qx, magnetic eld, earth, ?eld, int, ha, structure, tage tsub, element, high, dm, fault, p z, dx d, ut, t s, harmonic oscillation, ti j, ?ow, xk, greens tensor, transition, dt vj, order, elastic, core–mantle, dt, ul, jackson, kennett, process, dt dv, dislocation, model, principal, frequency, multi-anvil press, form, magnetic, earthquake, pattern, p v, piolakirchhoff tensor, good correspondence, atomic bond, change, p t, deformation, plane, rotation, ?xi, representation, doramaira unit, convective instability, continuum, t p, gradient, ?uid, it, upper, depth, displacement, relation, constant, presence, relaxation, oblate spheroid, signi?cant, slip, property, spatial frame, rate, small, v t, snells law, lett, region, boundary, fe si, temperature, dr mj, i j, dz, second-rank tensor, post-glacial rebound, volume, solution, spherical harmonic, number, ultrasonic interferometry, heat, v z, modulus, burgers vector, b t, core–mantle boundary, zone, lame modulus, geomagnetic eld, dt ds, function, uz t, solid, wavespeed, result, oceanic island, strain, stress, simple, orogenic belt, net, nn, subduction, visco-plastic formulation, wave, w e, t v, fjk, force, taylorproudman theorem, inge lehmann, seismic, dvx tui, highly heterogeneous, phase, geophys, rock, spheroidal oscillation, reference, magnetic induction, direction, mode, phys rev, ft, case, viscosity, courtesy, set, pressure, ri rc, event, component, s t, p x, ?ux, ii, unit tensor, variation, condition, fk t, internal, dv vc, term, motion, youngs modulus, slab, density, intrinsic anelasticity, energy, constitutive relations, sedimentary basin, thermal, convection, ijd, dz m, normal, vector, carrara marble, mantle, lowercase letter, tensor, gik, dp dr, sm rc, large, grazing incidence, shear