Very Slow Flows of Solids

1 Numerical simulation of very slow flows: an overview.- 1.1 Rationale of modeling.- 1.2 Thermo-mechanical models.- 1.3 Elasticity and viscosity.- 1.4 Cohesive forces versus stresses.- 1.5 Relation between t and n: the stress tensor.- 1.6 Regular, linearized problems.- 1.7 Analytic methods.- 1.8 The finite-difference method and the finite-element method.- 1.9 Algorithm for computation.- 1.10 Programming: some precautions.- References.- 2 Diffusion and advection of heat with a single space variable.- 2.1 The temperature equation.- 2.2 Closed form solutions without singular point.- 2.3 Closed form solutions with a singular point at the origin.- 2.4 Source functions.- 2.5 Use of Laplace transforms.- 2.6 Numerical computation.- 2.7 Moving medium, steady regime.- 2.8 Ice-sheet without bottom melting, sine oscillations of surface temperature.- 2.9 Response to a Dirac impulse in the surface temperature.- References.- 3 Rotation and strain. Invariants of stress and of strain rates.- 3.1 The finite rotation matrix.- 3.2 Angular velocity vector.- 3.3 Lagrangian and Eulerian descriptions.- 3.4 Finite strain.- 3.5 Strain rates.- 3.6 Compatibility conditions.- 3.7 Transformation of a tensor when the coordinate system is changed.- 3.8 Dilatation. The case of an incompressible fluid.- 3.9 Stress equations.- 3.10 Inertia forces and Coriolis forces. Scale models.- 3.11 Principal stresses and principal directions.- 3.12 The stress deviator.- 3.13 Invariants of stress and strain rate.- 3.14 Shear stress on any plane.- References.- 4 Microscopic processes of creep.- 4.1 The macroscopic point of view: work-hardening and creep.- 4.2 The different space scales.- 4.3 Dislocations.- 4.4 Displacement and multiplication of dislocations.- 4.5 Dislocation creep.- 4.6 Vacancies and self-interstitials.- 4.7 Diffusional climb of edge dislocations.- 4.8 Stacking faults and cross-slip.- 4.9 Secondary creep of a polycrystal.- 4.10 Crystal orientation and fabrics.- 4.11 Kinking and twinning.- 4.12 Diffusional creep.- 4.13 Chalmers’ microcreep and Harper-Dorn creep.- 4.14 Pressure solution deformation.- References.- 5 Viscosity as a model for rocks creeping at high temperature.- 5.1 Principles of continuum mechanics.- 5.2 Most general viscous behavior, either isotropic or anisotropic.- 5.3 Isotropic viscosity.- 5.4 Field structure of rocks.- 5.5 Pore pressure in rocks, tectonic stresses, and earthquakes.- 5.6 Data for rock salt.- 5.7 Data for Yule marble.- 5.8 Data for quartzites.- 5.9 Hydrolytic weakening of quartz and silicates.- 5.10 Data for granite.- 5.11 Data for peridotites.- 5.12 Rheology of the Earth’s upper mantle.- 5.13 Different kinds of polar ices.- 5.14 Data for mineral ice Ih, and for isotropic rock ice.- 5.15 Textures in glaciers and recrystallization creep of multi-maxima ice.- References.- 6 Stokes’ problems solved with Fourier transforms: isostatic rebound, glacier sliding.- 6.1 Overview on viscous flows.- 6.2 General equations for the Stokes’ problem.- 6.3 Plane flow.- 6.4 Biharmonic functions.- 6.5 Fourier transforms.- 6.6 Isostatic rebound with an isoviscous asthenosphere.- 6.7 Application to the glacio-isostatic uplift of Fennoscandia.- 6.8 Sliding with melting-refreezing on a sine profile.- 6.9 Sliding with melting refreezing on any microrelief.- 6.10 Discussion of Nye’s sliding theory.- 6.11 Sliding without cavitation of power-law viscous ice.- 6.12 Temperatures at the microscopic scale, and permeability of temperate ice.- 6.13 A sliding theory which takes wetness and permeability into account.- References.- 7 Open flow in a cylindrical channel of a power-law viscous fluid, and application to temperate valley glaciers.- 7.1 General equations for steady flow, when stresses and strain rates are x-independent.- 7.2 Is secondary flow possible?.- 7.3 Power-law viscosity: governing equation for the stress function, and analytical solutions.- 7.4 Governing equation for the velocity, and singularities at the edges.- 7.5 Numerical computation.- 7.6 The inverse problem. Yon Neumann’s stability criterion.- 7.7 Kinematic waves on glaciers.- 7.8 Mathematical developments of the theory, and real facts.- 7.9 Empirical sliding laws.- 7.10 Subglacial hydraulics.- 7.11 Sliding law with cavitation.- 7.12 Stability of a temperate glacier.- References.- 8 Coupled velocity and temperature fields: the ice-sheet problem.- 8.1 Thermal runaway.- 8.2 A pseudo-unidimensional model for the asthenosphere.- 8.3 The inverse problem for an ice-sheet: I — Balance velocities.- 8.4 The inverse problem for an ice-sheet: II — Balance temperatures in the pseudo-sliding approximation.- 8.5 Steady temperatures, abandoning the pseudo-sliding approximation.- 8.6 The forward problem: the bottom boundary layer model.- 8.7 Steady states, reversible evolution, and surges of an ice-sheet.- 8.8 Previous assessments of stability, and thermal stability of the BBL.- 8.9 The global forward problem for an ice-sheet: governing equations.- 8.10 The global forward problem: computation of stable steady states.- References.- 9 Thermal convection in an isoviscous layer and in the Earth’s mantle.- 9.1 Buoyancy forces: general equations.- 9.2 Stability of a viscous layer uniformly heated from below.- 9.3 Marginal convective flow in an isoviscous layer.- 9.4 Convection at high Rayleigh numbers: experimental evidence.- 9.5 The boundary layer theory.- 9.6 Mathematical validity of the boundary layer theory, of the mean field theory, and of the Boussinesq, isoviscous approximation.- 9.7 Mantle viscosity.- 9.8 Geothermal heat, and the location of heat sources.- 9.9 Whole mantle convection or two-layer convection?.- 9.10 Nourishment of mid-ocean ridges, small scale convection, and local flows.- References.- 10 Computation of very slow flows by the finite-difference method.- 10.1 Choice of master functions.- 10.2 Difference schemes.- 10.3 Computational algorithms.- 10.4 Boundary conditions at artificial boundaries.- 10.5 Curved boundaries.- 10.6 Coupled velocity and temperature fields: flow in a single direction and upwind differences.- 10.7 Convective flow: staggered grids and symmetric difference schemes.- 10.8 Evolution of convection with time.- 10.9 Non-linear instabilities.- References.- 11 Elasto-statics.- 11.1 Isotropic linear elasticity.- 11.2 Isothermal and adiabatic elasticity.- 11.3 General equations.- 11.4 Principle of correspondence.- 11.5 Plane strain and plane stress.- 11.6 Use of Fourier transforms for plane strain problems.- 11.7 Source fields in elasticity.- 11.8 Saint-Venant’s principle; application to the screw dislocation problem.- 11.9 Edge dislocations.- 11.10 Singularities at the tips of cracks and faults.- 11.11 Generalization and limitations of linear, perfect elasticity.- References.- 12 Plates and layered media.- 12.1 Equilibrium of a thin plate floating on a fluid.- 12.2 Elastic thin plate.- 12.3 Lithosphere modeled as an elastic plate.- 12.4 Lithosphere modeled as an elastic-plastic plate.- 12.5 Unbending of an elastic-perfectly plastic plate.- 12.6 Buckling of a thin elastic plate embedded in a viscous medium.- 12.7 Incipient folding of a thin layer with larger viscosity than the surrounding medium.- 12.8 Layered medium.- 12.9 Self-gravitating layered Earth, with lateral density contrasts.- 12.10 Poloidal and toroidal plate velocity fields, and absolute velocities.- 12.11 Driving forces acting on plates.- References.- 13 Variational theorems, and the Finite Element Method.- 13.1 Variational formulations.- 13.2 Variational formulation for a viscous body.- 13.3 Boundary conditions in variational formulation.- 13.4 Sliding of a power-law viscous medium on a smooth sine profile.- 13.5 Drag on a sphere moving in a power-law viscous fluid.- 13.6 Piecewise polynomials as trial functions: the Finite Element Method.- 13.7 Choice of the master functions, and of the finite element.- 13.8 System matrix equation, in case of non-Newtonian viscosity.- 13.9 Some hints on the techniques of the F.E.M..- 13.10 The Galerkin method, and its application to convective heat transfer.- 13.11 Incremental procedures, with a Laplacian point of view.- References.- 14 The rigid plastic model.- 14.1 Yield criteria.- 14.2 The elastic-plastic model for large strains.- 14.3 Perfect plasticity.- 14.4 Plane strain: stress and velocity fields in deforming regions.- 14.5 Discontinuities and plastic waves.- 14.6 Punching of a semi-infinite rigid-plastic medium by a flat indenter.- 14.7 Could the solution above model punching of Asia by India?.- 14.8 Nye’s flow.- 14.9 Rigid-plastic layer pressed between rough plates.- 14.10 The “perfect-plastic model” for ice-sheets.- References.- 15 Viscoelasticity and transient creep.- 15.1 Objective time derivatives.- 15.2 Overview on bodies with memory.- 15.3 The Maxwell body.- 15.4 Correspondence principle for simple viscoelastic bodies.- 15.5 Peltier’s theory of glacio-isostasy.- 15.6 Boltzmannian bodies.- 15.7 Different kinds of transient creep.- 15.8 Recoverable creep and anelasticity.- 15.9 Transient creep in rock salt.- 15.10 Transient creep in ice.- 15.11 Attempts to set up a rheological model.- 15.12 A new model, with a buffer strain and no yield strength.- References.- 16 Homogenization, and the transversely isotropic power-law viscous body.- 16.1 Anisotropic linear rheology.- 16.2 Invariants for transverse isotropy.- 16.3 Constitutive law at large scale of temperate glacier ice.- 16.4 Microscopic models for transient creep.- 16.5 Steady creep law of a polycrystal by homogenization.- 16.6 The self-consistent method for isotropic polycrystals.- 16.7 Third-power law transversely isotropic viscosity.- References.- Appendix I Some important numerical methods.- I.1 Numerical quadrature.- I.2 Runge-Kutta and predictor-corrector algorithms.- I.3 Solution of tridiagonal systems.- I.4 Large sets of linear equations.- Appendix II Vector analysis.- II.1 Divergence, gradient, and curl.- II.2 Laplacian and vector Laplacian.- II.3 Gradient of a vector.- Appendix III Cylindrical and spherical coordinates.- III.1 Vectorial operators, strain rates and stress equations in cylindrical coordinates.- III.2 Vectorial operators, strain rates and stress equations in spherical coordinates.- III.3 Axisymmetric flow in spherical coordinates.- Appendix IV Fourier and Fourier-Bessel transforms.- IV.1 Fourier transforms.- IV.2 Parseval theorem, convolutions, and filters.- IV.3 Fourier-Bessel transforms.- Appendix V Spherical harmonics and the gravity field.- V.1 Surface spherical harmonics.- V.2 Expansion of a vector field into spherical harmonics.- V.3 Solution of Laplace equation. Geoid height anomalies and free-air gravity anomalies.- V.4 Gravity anomalies due to density anomalies.- Appendix VI Laplace transforms.- VI.1 Definition and main properties.- VI.2 Inversion of a Laplace transform.- VI.3 Table of Laplace transforms.- Name index.