Analytic Methods In Geomechanics

Description: 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.

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 I-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 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.