Notes on Numerical Modeling in Geomechanics
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This book is an introduction to numerical analysis in geomechanics and is intended for advanced undergraduate and beginning graduate study of the mechanics of porous, jointed rocks and soils. Although familiarity with the concepts of stress, strain and so on is assumed, a review of the fundamentals of solid mechanics including concepts of physical laws, kinematics and material laws is presented in an appendix. Emphasis is on the popular finite element method but brief explanations of the boundary element method, the distinct element method (also known as the discrete element method) and discontinuous deformation analysis are included. Familiarity with a computer programming language such as Fortran, C++ or Python is not required, although programming excerpts in Fortran are presented at the end of some chapters.
This work begins with an intuitive approach to interpolation over a triangular element and thus avoids making the simple complex by not doing energy minimization via a calculus of variations approach so often found in reference books on the finite element method. The presentation then proceeds to a principal of virtual work via the well-known divergence theorem to obtain element equilibrium and then global equilibrium, both expressed as stiffness equations relating force to displacement. Solution methods for the finite element approach including elimination and iteration methods are discussed. Hydro-mechanical coupling is described and extension of the finite element method to accommodate fluid flow in porous geological media is made. Example problems illustrate important concepts throughout the text. Additional problems for a 15-week course of study are presented in an appendix; solutions are given in another appendix.
Table of Contents
1 Introduction 2 Interpolation Over a Triangle 2.1 Linear Theory 2.2 Explicit Formulas 2.3 Linear Strain Triangle 2.4 Programming Comments 3 Derivatives of Interpolation Functions 3.1 Strains 3.2 Hydraulic Gradient 3.3 Axial Symmetry 3.4 Programming Comment 4 Linear Interpolation Over a Quadrilateral 4.1 The Generic 4-Node Quadrilateral 4.2 The Isoparametric 4-Node Quadrilateral 4.3 Programming Comment 5 Derivatives for a Linear Displacement Quadrilateral 5.1 Chain Rule Application 5.2 Strain Displacement Matrix 5.3 Programming Comment 6 Element Equilibrium and Stiffness 6.1 Equations from Elasticity 6.2 Principle of Virtual Work 6.3 Element Equilibrium 7 Global Equilibrium and Stiffness 7.1 Global Equilibrium 7.2 Global Assembly 7.3 Programming Comment 8 Static Condensation and a 4CST Element 8.1 Static Condensation 8.2 Programming Comment 9 Equation Solving 9.1 Gauss Elimination 9.2 Elimination Boundary Conditions 9.3 Gauss-Seidel Iteration 9.4 Iteration Boundary Conditions 9.5 Programming Comments for Elimination 9.6 Programming Comments for Iteration 10 Material Non-linearity 10.1 Incremental (Tangent Stiffness) Approach 10.2 Iterative (Modified Newton-Raphson) Approach 10.3 Programming Comment 11 Time Integration 12 Finite Element Seepage Formulation 12.1 Incompressible flow through a Rigid, Porous Solid 12.2 Compressible Flow through a Deformable, Porous Solid 13 Hydro-mechanical Coupling 13.1 Effective Stress Concept 13.2 Finite Element Formulation 14 Boundary Element Formulations 14.1 Indirect Formulation 14.2 Direct Formulation 15 Distinct Element Formulations 15.1 DEM Formulation 15.2 DDA Formulation 16 Conclusion Appendix A: REVIEW OF FUNDAMENTAL CONCEPTS 1 Physical Laws 1.1 Conservation of Mass 1.2 Balance of Linear Momentum 1.3 Balance of Angular Momentum 1.4 Balance of Energy 2 Analysis of Stress 2.1 Surface Tractions and Stresses 2.2 Stress Vector and State of Stress 2.3 Cauchy Stress Formulas and Sign Convention 2.4 Equality of Shear Stresses 2.5 Rotation of Reference Axes 2.6 Principal Stresses 2.7 Maximum and Minimum Shear Stresses 2.8 Stress Invariants, Hydrostatic Stress and Deviatoric Stress 2.9 Mohr’s Circle in Two Dimensions 2.10 Stress Equations of Motion 2.11 Initial Stress and Stress Change 3 Analysis of Strain 3.1 Normal Strain 3.2 Shear strain 3.3 Small Strain – Displacement Relations 3.4 Geometric Interpretation of Small Strains 3.5 Change of Reference Axes 3.6 Principal Strains, Maximum Shear Strain and Mohr’s Circle 4 Stress Strain Laws – Elasticity 4.1 Hooke’s Law in One Dimension – Young’s Modulus and Shear Modulus 4.2 Hooke’s Law in Three Dimensions – Other Elastic Moduli 4.3 More on Elastic Anisotropy 5 Plane Stress, Plane Strain and Axial Symmetry 5.1 Plane Stress and Plane Strain 5.2 Rock Weight and Gravity Stress 5.3 Axial Symmetry 6 Limits to Elasticity – Strength 6.1 Mohr-Coulomb Failure 6.2 Hoek-Brown Failure 6.3 Drucker-Prager Failure 6.4 Compressive Strength Under Confining Pressure 6.5 Energy and Stability 6.6 A Statistical Strength Model 6.7 Strain Beyond the Elastic Limit Appendix B: Study Questions Appendix C: Question Replies Subject Index
William Pariseau obtained his B.S. degree in Mining Engineering at the University of Washington (Seattle) following the geological option and subsequently earned a Ph.D. in Mining Engineering at the University of Minnesota with emphasis on rock mechanics and with a minor in applied mathematics. Prior to his Ph.D., he obtained practical experience working for the City of Anchorage, the Alaska Department of Highways, the Mineral Resources Division of the U.S. Bureau of Mines (Spokane), the Anaconda Copper Co. in Butte, Montana, the New York-Alaska Gold Dredging Corp. in Nyac, Alaska. He served in the United States Marine Corps (1953-1956). He maintained a strong association with the former U.S. Bureau of Mines, first with the Pittsburgh Mining Research Center and later with the Spokane Mining Research Center. He is a registered professional engineer and has consulted for a number of commercial and government entities. Currently, he is a professor emeritus and former holder of the Malcolm McKinnon endowed chair in mining engineering at the University of Utah. He joined the Department in 1971 following academic appointments at the Montana College of Science and Technology and the Pennsylvania State University. He has been a visiting academic at Brown University, Imperial College, London, and at the Commonwealth Science and Industrial Research Organization (CSIRO), Australia. He and colleagues have received a number of rock mechanics awards; he was recognized as a distinguished university research professor at the University of Utah in 1991. In 2010, he was recognized for teaching in the College of Mines and Earth Sciences with the Outstanding Faculty Teaching Award. The same year, he was honored by the Old Timers Club with their prestigious Educator Award. He was honored as a Fellow of the American Rock Mechanics Association in 2015.