Deformation of Earth Materials An Introduction to the Rheology of Solid Earth 1st Edition by Shun Ichiro Karato 1107406056 9781107406056

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ISBN 10: 1107406056

ISBN 13: 9781107406056

Author: Shun Ichiro Karato

This graduate textbook presents a comprehensive, unified treatment of the materials science of deformation as applied to solid Earth geophysics and geology. The deformation of Earth materials is presented in a systematic way covering elastic, anelastic and viscous deformation. Advanced discussions on relevant debates are also included to bring readers a full picture of science in this interdisciplinary area. This textbook is ideal for graduate courses on the rheology and dynamics of solid Earth, and includes review questions with solutions so readers can monitor their understanding of the material presented. It is also a much-needed reference for geoscientists in many fields including geology, geophysics, geochemistry, materials science, mineralogy and ceramics.

Deformation of Earth Materials An Introduction to the Rheology of Solid Earth 1st Table of contents:

Part I General background
1 Stress and strain
1.1. Stress
1.1.1. Definition of stress
1.1.2. Principal stress, stress invariants
1.1.3. Normal stress, shear stress, Mohr’s circle
1.2. Deformation, strain
1.2.1. Definition of strain
1.2.2. Meaning of strain tensor
1.2.3. Principal strain, strain ellipsoid
1.2.4. The Flinn diagram
1.2.5. Foliation, lineation (Fig. 1.7)
1.2.6. Various deformation geometries
1.2.7. Macroscopic, and microscopic stress and strain
2 Thermodynamics
2.1. Thermodynamics of reversible processes
2.1.1. The first and the second principles of thermodynamics
2.1.2. Activity, fugacity
Activity
Fugacity
2.1.3. Chemical equilibrium: the law of mass action
2.1.4. Phase transformations: the Clapeyron slope, the Ehrenfest slope
2.1.5. Phase diagrams
Solid-solution, eutectic melting
Solvus
Effects of non-stoichiometry: a phase diagram for an open system
2.2. Some comments on the thermodynamics of a stressed system
2.3. Thermodynamics of irreversible processes
2.3.1. Flux and the generalized forces
2.3.2. Some notes on the driving forces for plastic deformation
2.3.3. Stationary (steady-state) state: the principle of minimum entropy production rate
2.4. Thermally activated processes
2.4.1. Absolute rate theory
Basic theory
Effects of stress on thermal activation
3 Phenomenological theory of deformation
3.1. Classification of deformation
3.2. Some general features of plastic deformation
3.3. Constitutive relationships for non-linear rheology
3.4. Constitutive relation for transient creep
3.5. Linear time-dependent deformation
3.5.1. General theory
3.5.2. Some simple models
3.5.3. Effects of distributed relaxation times
Part II Materials science of deformation
4 Elasticity
4.1. Introduction
4.2. Elastic constants
4.2.1. Hooke’s law and elastic constants
4.2.2. The Cauchy relation
4.3. Isothermal versus adiabatic elastic constants
4.4. Experimental techniques
4.4.1. Static measurements of elastic constants
4.4.2. Ultrasonic techniques
4.4.3. Opto-elastic techniques
4.4.4 . Velocity es timates from the phonon density of stat e
4.5. Som e genera l trends in ela sticity: Birch’ s law
4.6. Effects of chemical composition
4.7. Elasti c constants in several cryst al struct ures
4.7.1. A polyhed ral model of ionic cryst als and elastic anisotropy
4.7.2. Elast ic anisotropy ca used by int rinsic anisot ropy of bonding
Elasticit y of hcp m etals
Elasticity of materials with B1 and B2 structures
4.8. Effects of phase transformations
5 Crystalline defects
5.1. Defects and plastic deformation: general introduction
5.2. Point defects
5.2.1. Generalities
Definition
Equilibrium concentration of point defects
5.2.2. Point defects in ionic solids
A Schottky pair
A Frenk el pair
Defect format ion in an open system (the Kro¨ ger–Vink diagram)
5.3. Dislocations
5.3.1. Generalities: definition of crystal dislocations, slip systems
von Mises condition and the independent slip systems
Stress–strain field, energy of a dislocation, force on a dislocation
Partial dislocations, dissociation of a dislocation
Dislocation density versus stress relationship
Geometrically necessary dislocations
Kinks and jogs
Cha rges on disloc ation s, dislocation -point def ect intera ction
Observation of dislocations
Dislocation density measurement
5.4. Grain boundaries
5.4.1. Generalities: geometry and boundary energy
5.4.2. Thickness of grain boundaries
5.4.3. Grain boundary ledges (steps)
5.4.4. Grain boundary charges
5.4.5. Impurities on grain boundaries
6 Experimental techniques for study of plastic deformation
6.1. Introduction
6.2. Sample preparation and characterization
6.2.1. Sample preparation
6.2.2. Microstructural characterization
6.3. Control of thermochemical environment and its characterization
6.3.1. Pressure generation and its measurements
6.3.2. Generation of high temperature and its measurements
6.3.3. Control and characterization of the chemical environment
6.4. Generation and measurements of stress and strain
6.4.1. Generation of deviatoric stress and strain
6.4.2. Measurements of stress and strain
6.5. Methods of mechanical tests
6.5.1. Constant stress, constant strain-rate tests
6.5.2. Stress-relaxation tests
6.5.3. Stress-dip tests
6.5.4. Indentation hardness tests
6.6. Various deformation geometries
6.6.1. Uni-axial (tri-axial) compression/tension
6.6.2. Simple shear deformation
Saw-cut sample assembly (a sandwich method)
Torsion tests
7 Brittle deformation, brittle–plastic and brittle–ductile transition
7.1. Brittle fracture and plastic flow: a general introduction
7.1. Brittle fracture and plastic flow: a general introduction
7.2. Brittle fracture
7.2.1. Micro-cracks and faulting
7.2.2. Coulomb–Navier’s law of fracture strength
Coulomb–Navier’s law of friction
Byerlee’s law, the role of pore fluids and a simple model of brittle fracture strength
7.3. Transitions between different regimes of deformation
7.3.1. Generalities
7.3.2. Semi-brittle regime
7.3.3. Geological observations on the brittle–plastic transition
8 Diffusion and diffusional creep
8.1. Fick’s law
8.2. Diffusion and point defects
8.3. High-diffusivity paths
8.4. Self-diffusion, chemical diffusion
8.5. Grain-size sensitive creep (diffusional creep, superplasticity)
8.5.1. Experimental observations and historical notes
8.5.2. Basic theory of diffusional creep
Nabarro–Herring creep, Coble creep
Diffusional creep in a compound
Interface reaction-controlled creep
Pressure-solution creep
8.5.3. Small-strain and large-strain phenomena
Grain-boundary sliding and ‘‘superplasticity’’
Models of superplasticity
Ashby–Verrall model (ASHBY and VERRALL, 1973)
Mukherjee model (MUKHERJEE, 1971)
Transient phenomena in diffusional creep (Lifshitz–Shikin theory)
8.5.4. Several issues on diffusional creep
How to identify diffusional creep or superplasticity
Some issues on the extrapolation of experimental data on diffusional creep
9 Dislocation creep
9.1. General experimental observations on dislocation creep
9.2. The Orowan equation
9.3. Dynamics of dislocation motion
9.3.1. Intrinsic resistance
The Peierls stress
Thermal and athermal motion of defects
Effects of dissociation
9.3.2. Extrinsic resistance
Interaction with impurity atoms
Suzuki effect
9.3.3. Resistance due to mutual interaction
9.3.4. Dislocation climb, cross-slip
Climb of an edge dislocation
Cross-slip of a screw dislocation
9.4. Dislocation multiplication, annihilation
9.4.1. Dislocation multiplication
9.4.2. Recovery, annealing of dislocations
9.4.3. Evolution of dislocation density and transient creep
9.5. Models for steady-state dislocation creep
9.5.1. Generalities
9.5.2. Some models
Climb control model
Subgrain boundary recovery model
Weertman model
Nabarro model
Influence of pipe diffusion
Glide control model
Harper–Dorn creep
9.6. Low-temperature plasticity (power-law breakdown)
9.7. Deformation of a polycrystalline aggregate by dislocation creep
9.8. How to identify the microscopic mechanisms of creep
9.8.1. From steady-state flow law
9.8.2. From dislocation microstructures
9.8.3. From a stress-dip test
9.9. Summary of dislocation creep models and a deformation mechanism map
Summary of high-temperature creep models
Deformation mechanism maps
10 Effects of pressure and water
10.1. Introduction
10.2. Intrinsic effects of pressure
10.2.1. Experimental observations
Introduction
Some representative results
10.2.2. Models for pressure dependence of plastic deformation
General considerations
Models for activation volume
Pressure dependence of activation volume
Pressure-induced change in the mechanism of defect motion
10.3. Effects of water
10.3.1. General introduction
10.3.2. Mechanisms of dissolution of water (hydrogen)
10.3.3. Experimental observations on the role of water on plastic deformation
Quartz
Olivine
10.3.4. Models for water (hydrogen) weakening
Diffusional creep
Dislocation creep
Other volatiles
10.3.5. Interplay between pressure and water (hydrogen) effects
10.3.6. Some issues in investigating water (hydrogen) in minerals
10.3.7. Principles governing the distribution of water in minerals in Earth
11 Physical mechanisms of seismic wave attenuation
11.1. Introduction
11.2. Experimental techniques of anelasticity measurements
11.2.1. General introduction
11.2.2. Wave-propagation method
11.2.3. Quasi-static measurements
11.2.4. Low-frequency oscillation methods
11.3. Solid-state mechanisms of anelasticity
11.3.1. Point defect mechanisms
Anelastic relaxation due to local motion of point defects
Transient diffusional creep
11.3.2. Dislocation mechanisms
11.3.3. Grain-boundary mechanisms
Grain-boundary sliding
Anelasticity due to motion of twin boundaries, subboundaries
Grain-scale thermoelasticity
11.3.4. Experimental studies on solid-state mechanisms of anelasticity in Earth materials
11.4. Anelasticity in a partially molten material
11.4.1. General introduction
11.4.2. Attenuation due to viscous motion of melt
11.4.3. Thermoelasticity in a partially molten material
11.4.4. Experimental studies on anelasticity in partial melts
11.4.5. Seismological observations
12 Deformation of multi-phase materials
12.1. Introduction
12.2. Some simple examples
12.3. More general considerations
12.3.1. Variational principle
12.3.2. A more general stress–strain distribution
General introduction
Direct calculation
Self-consistent approach
Jordan–Handy model
Numerical modeling
12.3.3. Effective stress exponent and activation enthalpy
12.4. Percolation
12.4.1. General introduction
12.4.2. The Bethe lattice
12.5. Chemical effects
12.6. Deformation of a single-phase polycrystalline material
12.7. Experimental observations
12.8. Structure and plastic deformation of a partially molten material
12.8.1. Geometry of partial melt
12.8.2. Deformation of a partially molten material
13 Grain size
13.1. Introduction
13.2. Grain-boundary migration
13.2.1. Driving forces
13.2.2. Mobility of grain boundaries
Intrinsic mobility
Impurity drag
Effects of secondary phase particles
Effects of a fluid phase
13.3. Grain growth
13.3.1. Normal grain growth
Basic theory
13.3.2. Effects of secondary phase particles
Zener pinning
Ostwald ripening
13.3.3. Abnormal grain growth
13.3.4. Effect of stress: the limiting grain size
13.3.5. Some examples
13.4. Dynamic recrystallization
13.4.1. Experimental observations
Microstructural observations
Mechanical aspects
13.4.2. Models for dynamic recrystallization
Strain-induced grain-boundary migration and impingement
Subgrain rotation and subgrain growth
13.4.3. Application of paleopiezometers
13.5. Effects of phase transformations
13.5.1. Kinetics of a phase transformation
Nucleation–growth
13.6. Grain size in Earth’s interior
14 Lattice-preferred orientation
14.1. Introduction
14.2. Lattice-preferred orientation: definition, measurement and representation
14.2.1. Measurement of lattice-preferred orientation
14.2.2. Representation of lattice-preferred orientation
Pole figure
Inverse pole figure
Orient ation distr ibution fu nction
14.2.3. Statistics and fabric strength
14.2.4. Symmetry of fabrics
14.3. Mechanisms of lattice-preferred orientation
14.3.1. Fabric developments by deformation
Rotation of a crystal by constrained deformation
14.3.2. Fabric development due to oriented crystallization
14.3.3. Effects of dynamic recrystallization on crystallographic fabric
14.3.4. Fabric development in diffusional creep or superplasticity
14.3.5. Controlling framework for LPO
14.4. A fabric diagram
14.5. Summary
15 Effects of phase transformations
15.1. Introduction
15.2. Effects of crystal structure and chemical bonding: isomechanical groups
15.2.1. Direct mechanical tests on plastic properties of deep Earth materials
Some experimental observations
15.2.2. Isomechanical groups (classification of mechanical properties, scaling law, systematics)
Qualitative aspects (microstructures)
Quantitative aspects (scaling laws for creep)
Scaling laws and normalization of parameters (ASHBY and BROWN, 1982)
Systematics in high-temperature creep in oxides and silicates
15.3. Effects of transformation-induced stress–strain: transformation plasticity
15.3.1. Experimental observations
15.3.2. Models for transformation plasticity
Greenwood–Johnson model: an unrelaxed model
Poirier model: a relaxed model
15.3.3. Experimental observations
15.4. Effects of grain-size reduction
15.5. Anomalous rheology associated with a second-order phase transformation
15.6. Other effects
16 Stability and localization of deformation
16.1. Introduction
16.2. General principles of instability and localization
16.2.1. Criteria for instability and localization: infinitesimal amplitude analyses
16.2.2. Development of strain localization: finite amplitude instability
Influence of time dependence of materials parameters
Influence of elasticity (machine stiffness)
16.2.3. Orientation of shear zones
16.3. Mechanisms of shear instability and localization
16.3.1. General considerations
16.3.2. Specific mechanisms of instability and localization
Geometrical (necking) instability
Adiabatic instability (thermal runaway instability)
Localization due to grain-size reduction
Localization due to intrinsic instability of dislocation motion
Localization due to grain-boundary migration associated with dynamic recrystallization
Localization due to anisotropic microstructure (LPO) development
Instability and localization in deformation of a two-phase material
16.4. Long-term behavior of a shear zone
16.5. Localization of deformation in Earth
Part III Geological and geophysical applications
17 Composition and structure of Earth’s interior
17.1. Gross structure of Earth and other terrestrial planets
17.2. Physical conditions of Earth’s interior
17.2.1. Pressure distribution
17.2.2. Temperature distribution
17.3. Composition of Earth and other terrestrial planets
17.3.1. Planetary atmospheres and the volatiles in Earth and terrestrial planets
17.3.2. The composition and structure of Earth
Models of Earth’s composition
Crust
Mantle
Core
Several points are noteworthy about the inner core:
17.4. Summary: Earth structure related to rheological properties
18 Inference of rheological structure of Earth from time-dependent deformation
18.1. Time-dependent deformation and rheology of Earth’s interior
18.2. Seismic wave attenuation
Basics
Seismic wave attenuation and long-term rheology
18.3. Time-dependent deformation caused by a surface load: postglacial isostatic crustal rebound
18.3.1. Basic theory
18.3.3. Importance of non-linear rheology
18.3.4. Importance of transient creep
18.3.5. Observations and some results
18.4. Time-dependent deformation caused by an internal load and its gravitational signature
18.4.1. Basic theory
Elastic (or rigid body) support
Static equilibrium (isostasy)
Dynamic topography
18.4.2. Observations and some results
18.5. Summary
19 Inference of rheological structure of Earth from mineral physics
19.1. Introduction
19.2. General notes on inferring the rheological properties in Earth’s interior from mineral physi
19.2.1. Constitutive relations
19.2.2. Spatial variation of stress and strain rate
19.2.3. Spatial variation in thermochemical variables and grain size
19.3. Strength profile of the crust and the upper mantle
19.3.1. The brittle–plastic transition (see also Chapter 7)
19.3.2. Crust
19.3.3. Upper mantle
19.3.3.1 The strength of the lithosphere
19.3.3.2 The lithosphere–asthenosphere boundary
19.3.3.3 Rheological contrast between the continental and oceanic upper mantle
19.3.3.4 The Lehmann discontinuity
19.4. Rheological properties of the deep mantle
19.4.1. Transition zone
19.4.2. Lower mantle
19.4.3. Some topics on deep mantle, related to rheological properties
19.5. Rheological properties of the core
19.5.1. Outer core
19.5.2. Inner core
20 Heterogeneity of Earth structure and its geodynamic implications
20.1. Introduction
20.2. High-resolution seismology
20.2.1. Fundamentals of high-resolution seismology
Seismic tomography (velocity tomography)
Topography and the nature of discontinuities
Attenuation tomography
20.2.2. Some basic results
20.3. Geodynamical interpretation of velocity (and attenuation) tomography
20.3.1. Origin of lateral heterogeneity: some fundamentals
Generalities: variation of seismic wave velocities with physical/chemical factors
Thermal origin
Effects of phase transformations
Chemical origin
Direct effects of partial melting
Grain size
20.3.2. Topography on discontinuities and its geodynamic significance
20.3.3. Summary of inversion scheme
20.3.4. Origin and geodynamic significance of lateral heterogeneity: Some examples
Upper mantle
Transition zone
Lower mantle
21 Seismic anisotropy and its geodynamic implications
21.1. Introduction
21.2. Some fundamentals of elastic wave propagation in anisotropic media
21.3. Seismological methods for detecting anisotropic structures
21.3.1. Azimuthal anisotropy
21.3.2. Polarization anisotropy
21.4. Major seismological observations
21.5. Mineral physics bases of geodynamic interpretation of seismic anisotropy
21.5.1. Anisotropy due to a layered structure
21.5.2. Lattice-preferred orientation (see Chapter 14)
21.6. Geodynamic interpretation of seismic anisotropy
21.6.1. Generalities
21.6.2. Seismic anisotropy and its geodynamic implications

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