Continuum and Computational Mechanics for Geomechanical Engineers

Description

The field of rock mechanics and rock engineering utilizes the basic laws of continuum mechanics and the techniques developed in computational mechanics. This book describes the basic concepts behind these fundamental laws and their utilization in practice irrespective of whether rock/rock mass contains discontinuities. This book consists of nine chapters and six appendices. The first four chapters are concerned with continuum mechanics aspects, which include the basic operations, definition of stress and strain tensors, and derivation of four fundamental conservation laws in the simplest yet precise manner. The next two chapters are the preparation for computational mechanics, which require constitutive laws of geomaterials relevant to each conservation law and the procedures for how to determine required parameters of the constitutive laws. Computational mechanics solves the resulting ordinary and partial differential equations. In Chapter 7, the methods of exact (closed-form) solutions are explained and they are applied to ordinary/partial differential equations with solvable boundary and initial conditions. In Chapter 8, the fundamentals of approximate solution methods are explained for one dimension first and then how to extend them to multi-dimensional problems. The readers are expected to learn and clearly understand how they are derived and applied to various problems in geomechanics. The final chapter involves the applications of the approximate methods to the actual problems in practice for geomechanical engineers, which cover the continuum to discontinuum, including the stress state of the earth as well as the ground motions induced by earthquakes. Six appendices are provided to have a clear understanding of continuum mechanics operations and procedures for how to deal with discontinuities/interfaces often encountered in rock mechanics and rock engineering. Cover Half Title Series Page Title Page Copyright Page Table of Contents About the author Acknowledgments Preface 1 Fundamental operations 1.1 Scalar 1.2 Vector 1.3 Vector operations 1.3.1 Addition and subtraction 1.3.2 Dot product 1.3.3 Cross product 1.3.4 Unit vector 1.3.5 Coordinate systems and base vectors 1.3.6 Vector operations on a Cartesian coordinate system 1.4 Tensors of rank n 1.4.1 Definition of tensors of rank n 1.4.2 Tensor operations 1.5 Matrix representation of tensors 1.5.1 Matrix representation of vectors 1.5.2 Matrix representation of tensors 1.6 Coordinate transformation 1.7 Derivation of tensorial quantities 1.7.1 Derivative of a scalar function 1.7.2 Divergence 1.7.3 Rotation 1.7.4 Gradient of a vector: second-order tensor 1.7.5 Divergence of a tensor (second-order tensor) References 2 Stress analysis 2.1 Definition of a stress vector 2.2 Stress tensor 2.3 Relationship between a stress vector and a stress tensor: Cauchy’s law 2.4 Stress transformation 2.5 Principal stresses and stress invariants 2.6 Geometrical representation of stress tensor on the Mohr circle for the 2D condition References 3 Deformation and strain 3.1 Preliminaries 3.2 Derivation of a strain tensor using the Lagrangian description 3.3 Derivation of a train tensor using the Eulerian description 3.4 Relationship between the small strain theory and the finite strain theory 3.5 Geometrical interpretations of a strain tensor 3.5.1 Uniaxial deformation 3.5.2 Simple shear deformation References 4 Fundamental conservation laws 4.1 Fundamental conservation laws for one-dimensional cases 4.1.1 Mass conservation law 4.1.2 Momentum conservation law 4.1.3 Energy conservation laws 4.1.4 Fundamental governing equations for coupled hydro-mechanical phenomena 4.2 Multi-dimensional conservation laws 4.2.1 Mass conservation laws for seepage and diffusion phenomena 4.2.2 Momentum conservation law 4.2.3 Angular momentum conservation law 4.2.4 Energy conservation law 4.3 Derivation of governing equations in the integral form 4.3.1 Mass conservation law 4.3.2 Momentum conservation law 4.3.3 Angular momentum conservation law 4.3.4 Energy conservation law References 5 Constitutive laws 5.1 One dimensional (1D) constitutive laws 5.1.1 1D linear constitutive laws 5.1.2 1D non-linear constitutive laws for solids 5.2 Multi-dimensional constitutive laws 5.2.1 Fourier’s law 5.2.2 Fick’s law 5.2.3 Darcy’s law 5.2.4 Hooke’s law 5.2.5 Newton’s law 5.2.6 Kelvin–Voigt’s law 5.2.7 Navier-Stokes law 5.3 Non-linear behavior (elasto-plasticity and elasto-visco-plasticity) for solids 5.3.1 Elasto-plastic law 5.3.2 Elasto-visco-plasticity 5.3.3 Yield/failure criteria 5.4 Equivalent models for discontinua 5.4.1 Equivalent elastic compliance model (Singh’s model) 5.4.2 Crack tensor model (CTM) 5.4.3 Damage model 5.4.4 Microstructure models 5.4.5 Homogenization technique References 6 Laboratory tests 6.1 Laboratory tests on mechanical properties 6.1.1 Uniaxial compression tests 6.1.2 Direct and indirect tensile strength tests (Brazilian tests) 6.1.3 Triaxial compression tests 6.1.4 Post-failure behavior in uniaxial and triaxial compression tests 6.1.5 Direct shear tests 6.1.6 Tilting tests 6.1.7 Experimental techniques for creep tests 6.2 Thermal properties of rocks and their measurements 6.3 Tests for seepage parameters 6.3.1 Falling head tests 6.3.2 Transient pulse test method 6.4 Tests for diffusion parameters References 7 Methods for exact (closed-form) solutions 7.1 Basic approaches 7.1.1 Intuitive function methods 7.1.2 Solution by separating variables 7.1.3 Complex variable method 7.2 Closed-form solutions for solids 7.2.1 Visco-elastic rock sample subjected to uniaxial loading 7.2.2 Visco-elastic layer on an incline 7.2.3 One-dimensional bar embedded in rock 7.2.4 Circular cavity in the elastic rock under a far-field hydrostatic stress 7.2.5 Unified analytical solutions for circular/spherical cavity in an elasto-plastic rock 7.2.6 Foundations-bearing capacity 7.2.7 Two-dimensional closed-form solution methods 7.2.8 Three-dimensional closed-form solutions 7.3 Closed-form solutions for fluid flow through porous rocks 7.3.1 Some considerations on the Darcy law for rocks and discontinuities 7.3.2 Permeability tests based on a steady-state flow 7.3.3 Permeability tests based on a non-steady-state flow (transient flow tests) 7.4 Temperature distribution in the vicinity of geological active faults 7.5 Closed-form solutions for diffusion problems 7.5.1 Drying testing procedure 7.5.2 Saturation testing technique 7.6 Evaluation of creep-like deformation of semi-infinite soft rock layer References 8 Methods for approximate solutions 8.1 Comparison of exact and approximate solutions 8.1.1 Exact (closed-form) solution 8.1.2 Finite difference method 8.1.3 Finite element method 8.1.4 Comparisons 8.2 1D hyperbolic problem: equation of motion 8.2.1 Weak form formulation 8.2.2 Discretization 8.2.3 Specific example 8.2.4 1D parabolic problem: creep problem 8.2.5 1D elliptic problem: static problem 8.2.6 Computational examples 8.3 Parabolic problems: heat flow, seepage and diffusion 8.3.1 Introduction 8.3.2 Governing equation 8.3.3 Weak form formulation 8.3.4 Discretization 8.3.5 Steady-state problem 8.3.6 Specific example 8.3.7 Example 1: simulation of a solid body with heat generation 8.3.8 Example 2: simulation of a diffusion problem 8.4 FEM for 1D pseudo-coupled parabolic problems: heat flow and thermal stress; swelling and swelling pressure 8.4.1 Introduction 8.4.2 Governing equations 8.4.3 Coupling of heat and stress fields 8.4.4 Weak form formulation 8.4.5 Discretization 8.4.6 Specific example 8.4.7 Example: simulation of heat generation and associated thermal stress 8.5 Hydro-mechanical coupling: seepage and effective stress problem 8.5.1 Introduction 8.5.2 Governing equations 8.5.3 Weak form formulation 8.5.4 Discretization 8.5.5 Specific example 8.5.6 Example: simulation of settlement under sudden loading 8.6 Biot problem: coupled dynamic response of porous media 8.6.1 Introduction 8.6.2 Governing equations 8.6.3 Weak form formulation 8.6.4 Discretization 8.6.5 Specific example 8.6.6 Example: simulation of dynamic response of saturated porous media 8.7 Introduction of boundary conditions in a simultaneous equation system 8.7.1 Formulation 8.7.2 Actual implementation and solution of Eq. (8.216b) 8.8 Rayleigh damping and its implementation 8.9 Non-linear problems 8.10 Multi-dimensional situations 8.10.1 Shape functions 8.10.2 Numerical integration 8.11 Special numerical methods for media having discontinuities 8.11.1 No-tension finite element method 8.11.2 Pseudo discontinuum finite element method 8.11.3 Smeared crack element 8.11.4 Finite element method with joint or interface element (FEM-J) 8.11.5 Discrete finite element method (DFEM) 8.11.6 Displacement discontinuity method (DDM) 8.11.7 Discrete element method (DEM) 8.11.8 Discontinuous deformation analysis method (DDA) References 9 Applications of approximate methods in geo-engineering problems 9.1 Applications in continuum 9.1.1 The stress state of earth and earth’s crust 9.1.2 Evaluation of the tunnel face effect 9.1.3 Three-dimensional simulation of the excavation of a railway tunnel supported with forepoles, rockbolts, shotcrete and steel ribs 9.1.4 Effect of bolting pattern in underground excavations 9.1.5 Numerical studies on the indentation (impression) experiment 9.1.6 T he evaluation of the long-term response of an underground cavern 9.1.7 Long-term stability of the Derinkuyu underground city, Cappadocia, Turkey 9.1.8 Stability analyses of Tomb of Pharaoh Amenophis III, Luxor, Egypt 9.1.9 Dynamic response of a large underground cavern 9.1.10 Response and stability of abandoned room and pillar mine under static and earthquake loading 9.1.11 Modal analyses of shafts at the Horonobe Underground Laboratory 9.1.12 Temperature and stress distributions around an underground opening 9.1.13 Water-head variations in rock mass around an underground cavern 9.1.14 Breakout formation in boreholes in sedimentary rocks due to moisture loss 9.2 Applications in discontinuum 9.2.1 Earthquake fault rupture simulation 9.2.2 Pseudo-dynamic analyses on the interaction of structures and earthquake faults 9.2.3 Dynamic stability conditions of a single rock block 9.2.4 Stability of a slope against planar sliding 9.2.5 Stability of rock slope against columnar toppling 9.2.6 Stability of rock slope against flexural toppling and its stabilization 9.2.7 Retrofitting of unlined tunnels 9.2.8 Analysis of backfilling of abandoned mines 9.2.9 Simulation of creep-like deformation of the Babadağ landslide by DFEM 9.2.10 Simulation of creep-like deformation of a rock block at the Nakagusuku Castle References Appendix 1: Gauss divergence theorem Appendix 2: Geometrical interpretation of the Taylor expansion Appendix 3: Reynolds transport theorem Appendix 4: The Gauss elimination method and its implementation Appendix 5: Constitutive modeling of discontinuities and interfaces Appendix 6: Thin band element for modeling discontinuities and interfaces in numerical analyses Index