Continuum Micromechanics: Theory and Application to Multiscale Tectonics

Preface
Acknowledgments
List of Computer Programs and Movies
Mathcad Worksheets
MATLAB Programs and Packages
Movies
Contents
Chapter 1: Background, Mathematic Preliminaries and Notations
1.1 The Necessity of a Multiscale Approach
1.2 A Micromechanical Approach
1.3 Mathematic Preliminaries, Notation, and Convention
1.3.1 Vectors
1.3.2 Matrices
1.3.3 Tensors
1.3.4 Coordinate System Transformation of Cartesian Tensors
1.3.5 Matrix Exponentiation
1.3.6 Differential Operators, Convention, and Related Theorems
1.4 Fourth-Order Tensors
1.5 Notes and Key References
References
Chapter 2: Orientation of Fabric Elements
2.1 Orientation of a Line
2.2 Orientation of a 3D Object
2.2.1 Spherical Angles (θ,훟,ϑ)
2.2.2 Euler Angles (α,β,γ)
2.2.3 Direction of a3 and a Rotation Angle (ξ,φ,γ)
2.3 The Invariant Area Element on a Unit Sphere
2.4 Rotation around a Coordinate Axis
2.5 Rotation around a General Axis
2.6 Euler Angles as a Set of Rotations
2.7 Generation of a Population of Lines Following a Specified Distribution
2.7.1 A Set of Lines with Uniform Distribution
2.7.2 A Set of Lines Having a Gaussian Point Maximum
2.7.3 Line Set Forming a Small or Great Circle Girdle
2.8 Generation of a Population of 3D Fabric Elements Following a Specified Distribution
2.8.1 A Set of 3D Orientations Forming a Uniform Distribution
2.8.2 3D Objects Having Preferred Orientations
2.9 Misorientation Between Two Objects
2.10 Notes and Key References
References
Chapter 3: Stress, Strain, and Elasticity
3.1 Stress
3.2 Equilibrium Equations
3.3 Sign Conventions
3.4 Strain of a Line
3.5 Displacement Field and Strain Tensor in a Continuous Body
3.6 The Eulerian Displacement Gradients, Infinitesimal Strain, and Infinitesimal Rotation
3.7 Hooke´s Law
3.7.1 In Isotropic Materials
3.7.2 In Anisotropic Materials
3.8 Matrix Expression of the Elastic Stiffness Components
3.9 Boundary Value Problems of Linear Elasticity
3.10 Multiscale Stress and Strain in Real Materials
3.11 The Effective Rheology on the Macroscale
3.12 Notes and Key References
Appendix: Green Function for an Infinite Anisotropic Elastic Body
References
Chapter 4: Deformation: Strain and Rotation
4.1 The Position Gradient Tensor
4.2 Polar Decomposition
4.3 Finite Strain Tensors, Stretch, and Shear of Lines
4.3.1 Stretch of Material Lines
4.3.2 Shear Between a Pair of Initially Orthogonal Lines
4.3.3 Shear Strain and Shear Direction of a Material Plane
4.4 Other Useful Decompositions of Finite Deformation
4.5 Infinitesimal Deformations
4.6 Notes and Key References
References
Chapter 5: Flow: Strain Rate and Vorticity
5.1 Material and Spatial Coordinates
5.2 Displacement Field, Velocity Field, Spatial and Material Derivative
5.3 Strain Rate Tensor
5.4 Velocity Gradient Tensor, Strain Rate, and Vorticity
5.5 Some Simple Flow Fields
5.6 Flow Described in Different Reference Frames
5.7 Decomposition of Vorticity
5.8 Some Examples of Vorticity Decomposition
5.9 Flow Field Arising from Multiple Slip Systems
5.10 Notes and Key References
References
Chapter 6: Flow and Finite Deformation in Tabular Zones
6.1 Flow Apophyses
6.2 The Relationship Between Flow and Finite Deformation
6.3 Kinematic Models for Homogeneous Tabular Zones
6.3.1 Simple Shearing
6.3.2 Monoclinic General Shearing
Plane-Strain General Shearing
Monoclinic Transpression
Triclinic Transpression
6.4 Kinematics of Combining Simple and Pure Shearing Components in Tabular Zones
6.4.1 Superposition of Simple Shearing Motions
6.4.2 Kinematically Permissible Superposition of Simple and Pure Shearing Components in Constructing the Macroscale Flow in Ta...
6.4.3 Spatial Variation of Simple and Pure Shearing Components
6.5 Limitations of Kinematic Modeling
6.6 Notes and Key References
References
Chapter 7: Constitutive Equations
7.1 Newtonian Viscosity
7.2 Power Law Viscous Behavior
7.3 Flow Laws for Rocks
7.4 Wet Quartzite Flow Laws from Experiments
7.5 Tangent Viscous Stiffness and Linearization
7.6 Plasticity as a Limit Behavior of Power Law Viscosity
7.7 Anisotropic Secant Compliance Tensor for a Polycrystal Material
7.8 Boundary and Initial-Value Problems of Viscosity
7.9 Notes and Key References
References
Chapter 8: Rotation of Rigid Objects in Homogeneous Flows
8.1 Rotation and Coordinate System Transformation Revisited
8.2 From Angular Velocity to Finite Rotation
8.2.1 Angular Velocity
8.2.2 The Relationship Between Angular Velocity and Finite Rotation
8.3 Angular Velocity of a Rigid Object in Slow Viscous Flows the Jeffery Equation
8.4 Analytical Solutions for Spheroids in Monoclinic Flows
8.4.1 Equations Governing the Motion of Spheroidal Objects
8.4.2 Solutions of Spheroidal Objects in Monoclinic Flows
8.5 The Behavior of Rigid Spheroids in Monoclinic Flows
8.6 Numerical Approach
8.6.1 The Runge-Kutta Method
8.6.2 Rodrigues Rotation Approximation
8.6.3 Runge-Kutta-Rodrigues Approximation
8.6.4 Implementation
8.7 Notes and Key References
References
Chapter 9: Further Analysis of Spheroids in Simple Shearing Flows
9.1 Jeffery Orbits
9.2 Revolution Around the Distinct Axis
9.3 Rotation of a Population of Rigid Spheroids
9.4 Forces Acting on a Prolate Object in Simple Shearing
9.5 Deformation of a Prolate Object in Simple Shearing Flows
9.6 Concluding Remarks
9.7 Notes and Key References
References
Chapter 10: Eshelby´s Inclusion and Inhomogeneity Problem
10.1 Eshelby´s Elastic Inclusion/Inhomogeneity Problem
10.2 Eshelby Tensors and the Auxiliary Interaction Tensor
10.3 Extension to Newtonian Viscous Materials
10.4 Expressions of Eshelby Tensors for Linear Isotropic Materials
10.4.1 Eshelby Tensor Expressions for Penny-Shaped, Rod-Like, and Spherical Objects in Isotropic Elastic Materials
10.4.2 Eshelby Tensor Expressions for Penny-Shaped, Rod-Like, and Spherical Bodies in Isotropic Newtonian Materials
10.5 Strain and Rotation of a Deformable Ellipsoid
10.6 Notes and Key References
References
Chapter 11: Viscous Inclusions in Anisotropic Materials
11.1 Limitation of the Penalty Approach
11.2 Green Functions for Viscous Materials
11.3 Viscous Eshelby Tensors and Auxiliary Quantities
11.4 Formal Solutions for the Interior and Exterior Fields
11.5 Isotropic Systems
11.5.1 Viscous Tensor Identities
11.5.2 An Explicit Approach to Evaluate Exterior Solutions for Isotropic Systems
11.6 Some Analytic Solutions for Isotropic Systems
11.6.1 Kinematics of an Ellipsoid in 3D Flows
11.6.2 Deviatoric Stresses and Pressure
11.6.3 An Ellipse in 2D Flows
11.7 Equations for Ellipsoid Rotation in Anisotropic Viscous Materials
11.7.1 Angular Velocity Equations
11.7.2 Shear Spin When the Inclusion Is Instantaneously a Spheroid or Sphere
11.8 Summary
11.9 Notes and Key References
Appendices
Integral Expressions of Gij,Gij, l, and Hi
Relations Among Tensor Quantities for Isotropic Incompressible Materials
Derivation of Equations for Incompressible Isotropic Materials
References
Chapter 12: Two-Dimensional Inclusion Problems
12.1 Previous Approach
12.2 Generalized Plane Flows in Anisotropic Viscous Materials
12.3 Formulation
12.4 Application to Materials with a Planar Anisotropy
12.5 Pressure Field Around a 3D Inclusion in a Viscous Matrix with Planar Anisotropy
12.6 Concluding Remarks
12.7 Notes and Key References
References
Chapter 13: Effective Stiffnesses of Heterogeneous Materials
13.1 Scales, Micro- and Macroscale Fields
13.2 Macroscale Averages and Hill´s Lemma
13.3 Determination of Effective Stiffnesses for Linear Materials: Principles
13.4 Determination of Effective Stiffnesses for Linear Materials: Methods
13.5 Examples for Computing Effective Elastic Stiffness Tensor of Polycrystal Aggregates from Single Crystal Stiffness Coeffic...
13.6 Comparison of the Methods
13.7 Expressions for Effective Stiffness of Multiphase Composites from Noninteracting Approximation
13.8 Notes and Key References
References
Chapter 14: Application Example 1: An Elastic Prolate Object in a Viscous Matrix
14.1 An Elastic Prolate Inclusion in an Elastic Matrix
14.2 Comparison with the Fiber-Loading Theory
14.3 An Elastic Prolate in a Newtonian Viscous Matrix
14.3.1 Equations for Stress and Strain in the Prolate Object
14.3.2 Solution of When Σ Is Constant
Solutions for Shear Stresses
Solutions for Normal Stresses
Solutions for the Mean Stress
14.3.3 Comparison with Earlier Work on Simple Shearing
14.3.4 Solution of for a Rod-Like Prolate Object on the Vorticity Normal Section in a Simple Shearing Flow
14.4 Application to Microboudinage
14.5 Concluding Remarks
14.6 Notes and Key References
Appendix: Solutions of an Elastic Flat Oblate Body in a Newtonian Viscous Matrix
References
Chapter 15: Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix
15.1 The Equations
15.2 Solution for the Strain
15.3 The Deviatoric Stress in the Inclusion
15.4 The Pressure Field in the Inclusion
15.5 Analysis of the Solutions
15.5.1 Stress Relaxation and Creep in the Viscous Inclusion
15.5.2 Dynamic Pressure in the Viscous Inclusion
15.6 Discussion and Geological Implications
15.7 Notes and Key References
References
Chapter 16: Application Example 3: Deformation Around a Heterogeneity-Flanking Structures
16.1 The Motion of Material Particles Around an Ellipsoid from the Exterior Solutions
16.2 Macroscale Flows and Model Geometry
16.3 Modeling Results
16.4 Summary and Concluding Remarks
16.5 Notes and Key References
Appendix: Conversion Between Cartesian and Ellipsoidal Coordinates
References
Chapter 17: Generalization of Eshelby´s Formalism and a Self-Consistent Model for Multiscale Rock Deformation
17.1 Nonlinear Rheology and Partitioning Equations
17.2 Non-ellipsoidal Shape of Rheological Distinct Elements
17.3 Inclusions in a Finite Space
17.4 Interface Properties
17.5 Heterogeneous Matrix and Homogenization
17.6 A Self-Consistent Algorithm
17.7 Multiscale Modelling
17.8 Behaviors of Fabric Elements
17.8.1 Finite Strains in RDEs
17.8.2 Rigid or Deformable Fabric Elements in RDEs
17.8.3 Crystal Lattice Rotation
17.8.4 Empirical Behaviors of Fabric Elements
17.9 A Self-Consistent Multiscale Model for the Deformation of Earth´s Heterogeneous Lithosphere
17.10 A Continuum Micromechanics-Based Multiscale Approach
17.11 Notes and Key References
References