Very slow flows of solids : basics of modeling in geodynamics and glaciology


Very slow flows of solids : basics of modeling in geodynamics and glaciology

by Louis A. Lliboutry

(Mechanics of fluids and transport processes, v. 7)

Martinus Nijhoff, 1987

大学図書館所蔵 件 / 19



Includes bibliographical references and indexes



This book is written primarily for Earth scientists faced with problems in thermo mechanics such as the flow and evolution of ice-sheets, convection currents in the mantle, isostatic rebound, folding of strata or collapse of cavities in salt domes. Failure, faults, seismic waves and all processes involving inertial terms will not be dealt with. In general such scientists (graduate students beginning a Ph. D. for instance) have too small a background'in continuum mechanics and in numerical computation to model conveniently these problems, which are not elementary at all. Most of them are not linear, and therefore seldom dealt with in treatises. If the study of reality were clearly cut into two successive steps: first to make a physical model, setting up a well-posed problem in thermo-mechanics, and second to solve it, the obvious solution would be to find a specialist in computational mechanics who could spend enough time on a problem which, although maybe crucial for on-going fundamental research, has little practical interest in general, and cannot be considered properly as a noteworthy progress in Mechanics. But this is not the way Science develops. There is a continuous dialectic between the building up of a model and its mathematical treatment. The model should be simple enough to be tractable, but not oversimplified. Its sensitivity to the different components it is made of should be investigated, and more thought is needed when the results contradict hard facts.


  • 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.

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