1st Edition

Analytic Methods in Geomechanics

By Kam Tim Chau Copyright 2013
    458 Pages 177 B/W Illustrations
    by CRC Press

    A multidisciplinary field, encompassing both geophysics and civil engineering, geomechanics deals with the deformation and failure process in geomaterials such as soil and rock. Although powerful numerical tools have been developed, analytical solutions still play an important role in solving practical problems in this area. Analytic Methods in Geomechanics provides a much-needed text on mathematical theory in geomechanics, beneficial for readers of varied backgrounds entering this field.

    Written for scientists and engineers who have had some exposure to engineering mathematics and strength of materials, the text covers major topics in tensor analysis, 2-D elasticity, and 3-D elasticity, plasticity, fracture mechanics, and viscoelasticity. It also discusses the use of displacement functions in poroelasticity, the basics of wave propagations, and dynamics that are relevant to the modeling of geomaterials. The book presents both the fundamentals and more advanced content for understanding the latest research results and applying them to practical problems in geomechanics.

    The author gives concise explanations of each subject area, using a step-by-step process with many worked examples. He strikes a balance between breadth of material and depth of details, and includes recommended reading in each chapter for readers who would like additional technical information. This text is suitable for students at both undergraduate and graduate levels, as well as for professionals and researchers.

    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

    Biography

    Professor K.T. Chau, Ph.D., is the chair professor of geotechnical engineering in the Department of Civil and Environmental Engineering at the Hong Kong Polytechnic University. He obtained his Ph.D. in theoretical and applied mechanics from Northwestern University in Chicago and an executive certificate from the Graduate School of Business of Stanford University. Dr. Chau is a fellow of the Hong Kong Institution of Engineers (HKIE), the chairman of the Elasticity Committee (2009–2012) of the Engineering Mechanics Institute (EMI) of ASCE, and chairman of the TC103 of the ISSMGE. His research interests have included geomechanics and geohazards, including bifurcation and stability theories in geomaterials, rock mechanics, fracture and damage mechanics in brittle rocks3-D elasticity, earthquake engineering and mechanics, landslides and debris flows, tsunami and storm surges, and rockfalls and dynamic impacts, seismic pounding, vulnerability of tall buildings with transfer systems, and shaking table tests. He is the author of more than 100 journal papers and 200 conference publications.