Analytic Methods in Geomechanics

Analytic Methods in Geomechanics

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Features

  • Provides an indispensible guide to the fundamentals of geomechanics
  • Requires only minimal training in engineering mathematics, strength of materials, and engineering mechanics
  • Describes the use of complex variable technique for solving 2-D elasticity problems
  • Details solutions for 3-D elasticity, including Boussinesq, Kelvin, and Midlin
  • Introduces the use of Helmholtz free energy in macroscopic deformation
  • Includes discussions of continuum damage mechanics and microcrack models in fracture mechanics

Summary

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.

Table of Contents

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

Author Bio(s)

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.

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