Notes on Numerical Modeling in Geomechanics

✍ Scribed by William G. Pariseau

• Publisher: CRC Press
• Year: 2022
• Tongue: English
• Leaves: 293
• Edition: 1
• Category: Library

✦ Synopsis

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

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Acknowledgments
About the author
1 Introduction
2 Interpolation over a triangle
2.1 Linear theory
2.2 Explicit formulas
2.3 Linear strain triangles
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 for a quadrilateral
4.1 The generic 4-node quadrilateral
4.2 The isoparametric 4-node quadrilateral
5 Derivatives for a linear displacement quadrilateral
5.1 Chain rule application
5.2 Strain–displacement matrix
5.3 Programming comments
6 Element equilibrium and stiffness
6.1 Equations from elasticity
6.2 Principle of virtual work
6.3 Element equilibrium
7 Global equilibrium and global 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 nonlinearity
10.1 Incremental (tangent stiffness) approach
10.2 Iterative (modified Newton–Raphson) approach
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
Appendix B: Study questions
Appendix C: Question replies
Index