Analytic Methods in Geomechanics

Elementary Tensor Analysis
Introduction 
General Tensors, Cartesian Tensors, and Tensor Rank
A Brief Review of Vector Analysis
Dyadic Form of Second Order Tensors
Derivatives of Tensors
Divergence and Stokes Theorems 
Some Formulae in Cylindrical Coordinates
Some Formulae in Spherical Coordinates 
Summary and Further Reading
Problems

Elasticity and Its Applications 
Introduction 
Basic Concepts for Stress Tensor 
Piola–Kirchhoff Stresses 
Coordinate Transformation of Stress
Basic Concepts for Strain Tensor 
Rate of Deformation 
Compatibility Equations 
Hill’s Work-conjugate Stress Measures 
Constitutive Relation 
Isotropic Solids
Transversely Isotropic Solids 
Equations of Motion and Equilibrium 
Compatibility Equation in Terms of Stress Tensor 
Strain Energy Density
Complementary Energy 
Hyperelasticity and Hypoelasticity
Plane Stress, Plane Strain and the Airy Stress Function
Stress Concentration at a Circular Hole
Force Acting at the Apex of a Wedge 
Uniform Vertical Loading on Part of the Surface 
Solution for Indirect Tensile Test (Brazilian Test)
Jaeger’s Modified Brazilian Test
Edge Dislocation 
Dislocation Pile-up and Crack 
Screw Dislocation and Faulting
Mura Formula for Curved Dislocation 
Summary and Further Reading
Problems

Complex Variable Methods for 2-D Elasticity
Introduction 
Coordinate Transformation in Complex Variable Theory
Homogeneous Stresses in Terms Analytic Functions
A Borehole Subject to Internal Pressure
Kirsch Solution by Complex Variable Method 
Definiteness and Uniqueness of the Analytic Function
Boundary Conditions for the Analytic Functions
Single-valued Condition for Multi-connected Bodies
Multi-connected Body of Infinite Extend
General Transformation of Quantities
Elastic Body with Holes 
Stress Concentration at a Square Hole 
Mapping Functions for Other Holes
Summary and Further Reading
Problems


Three-Dimensional Solutions in Elasticity
Introduction 
Displacement Formulation 
Stress Formulations 
Some 3-D Solutions in Geomechanics 
Harmonic Functions and Indirect Method
Harmonic Functions in Spherical Coordinates 
Harmonic Functions in Cylindrical Coordinates 
Biharmonic Functions 
Muki’s Formulation in Cylindrical Coordinates
Summary and Further Reading
Problems


Plasticity and Its Applications
Introduction 
Flow Theory and Deformation Theory
Yield Function and Plastic Potential 
Elasto-plastic Constitutive Model
Rudnicki–Rice (1975) Model
Drucker’s Postulate, PMPR, and Il’iushin’s Postulate 
Yield Vertex 
Mohr–Coulomb Model
Lode Angle or Parameter
Yield Criteria on the π-Plane
Other Soil Yield Models
Cap Models
Physical Meaning of Cam-Clay Model 
Modified Cam-Clay
A Cam-Clay Model for Finite Strain
Plasticity by Internal Variables
Viscoplasticity 
Summary and Further Reading
Problems

Fracture Mechanics and Its Applications
Introduction 
Stress Concentration at a Elliptical Hole 
Stress Concentration at a Tensile Crack 
Stress Field near a Shear Crack 
The General Stress and Displacement Field for Mode I Cracks
The General Stress and Displacement Field for Mode II Cracks 
The General Stress and Displacement Field for Mode III Cracks
The Energy Release Rate at Crack Tips 
Fracture Toughness for Rocks
J-integral and the Energy Release Rate 
Westergaard Stress Function and Superposition 
Growth of Slip Surface in Slopes 
Energy Release Rate for Earthquake 
Wing Crack Model under Compressions 
Bazant’s Size Effect Law via J-integral 
Continuum Damage Mechanics 
Solids Containing Microcracks 
Rudnicki–Chau (1996) Multiaxial Microcrack Model
Summary and Further Reading
Problems

Viscoelasticty and Its Applications 
Introduction 
Boltzmann’s Integral Form of Stress and Strain 
Stieltjes Convolution Notation 
Stress-Strain Relation in Differential Equation Form 
Stress-strain Relation in Laplace Transform Space 
Correspondence Principle 
Creeping and Relaxation Tests 
Calibration of the Viscoelastic Model 
Viscoelastic Crack Models for Steam Injection 
Summary and Further Reading
Problems

Linear Elastic Fluid-Infiltrated Solids and Poroelasticity 
Introduction 
Biot’s Theory of Poroelasticity
Biot–Verruijt Displacement Function 
McNamee–Gibson–Verruijt Displacement Function 
Schiffman–Fungaroli–Verruijt Displacement Function 
Schiffman–Fungaroli Displacement Function
Laplace–Hankel Transform Technique 
Point Forces and Point Fluid Source in Half-space 
Cleary’s Fundamental Solution of Point Forces in Full Space 
Rudnicki’s Fundamental Solutions in Full Space
Thermoelasticity vs. Poroelasticity 
Summary and Further Reading
Problems

Dynamics and Waves In Geomaterials 
Introduction 
Seismic Waves
Waves in Infinite Elastic Isotropic Solids 
Helmholtz Theorem and Wave Speeds 
Rayleigh Waves 
Love Waves 
Stoneley Waves 
Elastic-plastic Waves 
Waves in Viscoelastic Solids 
Dynamic Fracture Mechanics 
Vibrations and Soil Dynamics 
Summary and Further Reading 
Problems

Appendices 
Appendix A: Nanson Formula 
Appendix B: Laplace Transform 
Appendix C: Legendre Transform and Work Increments

Selected Biographies

References

Author Index

Subject Index