Fundamentals of Continuum Mechanics of Soils

1 Introduction.- 1.1 Scope.- 1.2 Historical Notes.- 1.3 Classical Soil Mechanics versus Mechanics of Soils.- 1.4 Theory versus Experiment.- 1.5 Levels of Investigation.- 1.6 The Continuum.- 1.7 Homogeneity and Isotropy.- 1.8 Soils as Multi-phase Mixtures.- 1.9 The Methodology of Continuum Mechanics.- 2 Deformation and Strain.- 2.1 Deformation and Displacement.- 2.2 Strain.- 2.3 Strain Measures.- 2.4 Invariants of the Deformation Tensor.- 2.5 Small Deformations and Infinitesimal Strains.- 2.6 The Strain Invariants.- 2.7 The Hencky Measure of Strain.- 2.8 The Properties of the Hencky Measure.- 2.9 Compatibility Equations.- 3 Kinematics.- 3.1 Material Derivatives.- 3.2 Velocity and Speed.- 3.3 Acceleration.- 3.4 Material Derivatives of Displacement Gradients.- 3.5 Strain Rates.- 3.6 The Fundamental Theorem of Deformations.- 3.7 Rigid Deformation and Motion.- 3.8 Homogeneous Strain.- 3.9 Pure Strain.- 3.10 Isochoric Deformation and Motion.- 3.11 Irrotational Motion.- 3.12 Laminar Motion.- 3.13 Spherical Deformation.- 3.14 Simple Straining.- 3.15 Uniaxial Straining.- 3.16 Plane Strain.- 3.17 Simple Shear.- 3.18 Simple Torsion of a Circular Cylinder.- 3.19 Telescoping Deformation.- 3.20 Rotational Deformation.- 3.21 Steady Motion.- 4 Balance Equations for Homogeneous Media.- 4.1 Mass.- 4.2 The General Balance Equation.- 4.3 Density Balance.- 4.4 Forces Acting on Deformable Bodies.- 4.5 Tractions and Body Forces.- 4.6 Balance of Linear Momentum.- 4.7 Balance of Moment of Momentum.- 4.8 The Pressure Tensor.- 4.9 The Stress Tensor.- 4.10 The Stress Invariants.- 5 Energetics.- 5.1 Energy Considerations.- 5.2 Kinetic Energy.- 5.3 Potential Energy.- 5.4 Internal Energy.- 5.5 The Total Energy Balance.- 5.6 Historical Notes on Irreversible Processes of the Continuum.- 5.7 The Thermodynamic State.- 5.8 Thermodynamic Tensions.- 5.9 Entropy and Temperature.- 5.10 The Thermodynamic Functions.- 5.11 The Production of Entropy.- 5.12 Particular Cases of the Thermodynamic State.- 6 Multi-phase Mixtures.- 6.1 Extensive and Intensive Variables.- 6.2 Density, Volume, Mass and Weight of Constituents.- 6.3 Diffusion Velocity and Barycentric Velocity.- 6.4 The General Balance of Multi-Phase Mixtures.- 6.5 Multi-phase Density Balance.- 6.6 Multi-phase Balance of Linear Momentum.- 6.7 Multi-phase Balance of Moment of Momentum.- 6.8 Multi-phase Balance of Internal Energy.- 6.9 The Caloric Equations.- 6.10 The Production of Entropy.- 7 Constitutive Equations.- 7.1 Scope.- 7.2 Principles of Formulating Constitutive Equations.- 7.3 The Rheological Equation.- 7.4 Linearity and Non-linearity of Constitutive Equations.- 7.5 The Dual Rheological Equation.- 7.6 Viscoelastic Models.- 7.7 Dual Volumetrie Stress-Strain versus Shear Stress-Strain Relationship.- 7.8 Energy Considerations in View of the Dual Equations.- 7.9 The Isotropie Stress-Strain Relationship.- 7.10 Dilatancy.- 7.11 Isotropie Non-linear versus Linear Viscoelasticity.- 8 The Soil.- 8.1 Single-phase versus Multi-phase Considerations.- 8.2 Soil Constituents.- 8.3 The Water.- 8.4 Water Solutions.- 8.5 Vapor Pressure.- 8.6 The Air.- 8.7 Compressibility of Gases.- 8.8 Air-containing Pores.- 8.9 The Solid Particles.- 8.10 Specific Surface.- 8.11 The Mineralogical Structure of Clays.- 8.12 Electric Charges and Exchange Capacity.- 8.13 The Diffuse Double Layer.- 8.14 The Gouy-Chapman Double-layer Theory of Planar Surfaces.- 8.15 Limitations of the Gouy-Chapman Theory.- 8.16 Two Interacting Surfaces in Electrolyte Solution.- 8.17 The Work of Interacting Surfaces.- 8.18 Osmotic Pressure and Consolidation.- 8.19 Pore Water Pressure.- 8.20 Swelling Pressure.- 8.21 Factors Affecting the Behavior of Clays in Consolidation.- 8.22 Properties of Clays as Predicted by the Diffuse Double Layer.- 8.23 The Structure of Clays.- 8.24 Interfacial Forces.- 8.25 Air-Water Interface.- 8.26 Capillarity.- 8.27 Suction and Shrinkage.- 9 Soil as a Multi-phase Mixture.- 9.1 Introduction.- 9.2 Volume and Weight Relations in Soils.- 9.3 Density Balance.- 9.4 Balance of Linear Momentum.- 9.5 Balance of the Internal Energy.- 10 Flow in Soils.- 10.1 Introduction.- 10.2 Force Fields.- 10.3 Flow Potentials.- 10.4 Review of Linear and Non-linear, Saturated and Unsaturated Flow.- 10.5 Darcy’s Law.- 10.6 Flow in Saturated Soils.- 10.7 Modes of Saturated Flow.- 10.8 The Coefficient of Permeability.- 10.9 Seepage in Saturated Soils.- 10.10 Unsaturated Flow in Multi-phase Fluids.- 10.11 Flow in Unsaturated Soils.- 10.12 Flow in Unsaturated Non-swelling Soils.- 10.13 The Boltzmann Transformation Solution.- 10.14 Flow in Unsaturated Swelling Soils.- 11 Volumetrie Stress-Strain Phenomena.- 11.1 The Volumetrie Stress-Strain Relationship.- 11.2 Volume Changes in Soils.- 11.3 Consolidation of Saturated Soils.- 11.4 Terzaghi’s Theory of Consolidation.- 11.5 Discussion of Terzaghi’s Theory of Consolidation.- 11.6 The Consolidometer.- 11.7 The Consolidation Test.- 11.8 The Void Ratio-Pressure Dependence.- 11.9 The Pressure and Strain Tensors in Uniaxial Consolidation.- 11.10 The Triaxial Testing Device.- 11.11 The Spherical Consolidation.- 11.12 The Pressure and Strain Tensors in the Triaxial Test.- 11.13 The Void Ratio-Pressure Curve.- 11.14 Normally Consolidated and Overconsolidated Soils.- 11.15 Consolidation of Unsaturated Soils.- 11.16 Hysteresis.- 11.17 Phenomenological Linear Volumetric Stress-Strain Relationship.- 11.18 Modeling the Linear Volumetric Stress-Strain Relationship.- 11.19 Constant Spherical Pressure.- 11.20 General Spherical Pressure.- 11.21 The Volumetric Plastic Restraint.- 11.22 Effective Pressure.- 11.23 Total Pressure.- 11.24 The Internal Energy and Energy Rate of Spherical Phenomena.- 11.25 The Excess Stored Specific Free Energy.- 11.26 The Excess Stored Specific Free Energy of Solids in Linear Viscoelastic Media.- 11.27 Isotropic Functional Relationship Applied to the Excess Stored Specific Free Energy.- 11.28 The Excess Stored Specific Free Energy (Particular Cases).- 12 Shear Stress-Strain Phenomena.- 12.1 Introduction.- 12.2 Density Effects on Shear Stresses.- 12.3 The Shear Stress-Strain Relationship.- 12.4 Shear versus Volumetric Stress-Strain Relationship.- 12.5 Deviatoric Tests.- 12.6 The Conventional Triaxial Shear Test.- 12.7 The Unconfined Compression Test.- 12.8 The Simple Shear Test.- 12.9 The Direct Shear Test.- 12.10 The Torsion Test and Testing Device.- 12.11 Torsion of Solid Cylindrical Samples.- 12.12 Torsion of Hollow Cylindrical Samples.- 12.13 Introduction to the Pure Deviatoric Test.- 12.14 The Pure Deviatoric Test and its Equipment.- 12.15 Stresses and Strains in the Pure Deviatoric Test.- 12.16 Linear Deviatoric Stress-Strain Relationship.- 12.17 The Linear Deviatoric Constitutive Equation for Soils.- 12.18 Isotropic Strain Functions of Shear Constitutive Equations.- 12.19 Isotropic Stress Functions of Shear Constitutive Equations.- 12.20 Spherical Components in the Pure Deviatoric Test.- 12.21 Pore Pressures in the Pure Deviatoric Test.- 12.22 The Effect of the Rate of Loading in the Pure Deviatoric Test.- 12.23 Critical Void Ratio and Pure Deviatoric Loading.- 12.24 The Free Energy and Energy Rate of Deviatoric Phenomena.- 12.25 The Disbursed Specific Free Energy.- 12.26 The Disbursed Free Energy Applied to Linear Stress-Strain Relations.- 12.27 The Disbursed Free Energy in the Pure Deviatoric Test of Linear Viscoelastic Soils.- 12.28 The Disbursed Free Energy in Pure Deviatoric Tests for Isotropic Stress-Strain Relations of Soils.- 12.29 Note on General Non-linear Stress-Strain Relations.- 12.30 Closing Remarks on Shear Stresses.- 13 Failure.- 13.1 Brief Review of Failure Theories.- 13.2 Failure Criteria.- 13.3 The Dual Specific Internal Energy.- 13.4 Stored and Disbursed Specific Free Energy.- 13.5 Specific Free Energy Balance of the Elastic Medium (A Particular Case).- 13.6 Specific Free Energy Balance of Linear Viscoelastic Media.- 13.7 The Pure Deviatoric Test of a Linear Viscoelastic Medium.- 13.8 Free Energy Balance with Non-linear Stress-Strain Relations.- 13.9 The Pure Deviatoric Test with Isotropic Relations.- 13.10 Appraisal of the Presented Failure Criteria.- 13.11 The Study of Failure through the Pure Deviatoric Test.- 13.12 Drained Pure Deviatoric Shear Tests.- 13.13 Consolidated Undrained Pure Deviatoric Shear Tests.- 13.14 Undrained Pure Deviatoric Shear Tests.- 13.15 Slip Surfaces.- 13.16 Lateral Earth Pressure.- 13.17 The Coefficient of Lateral Earth Pressure.- Appendix A Tensor Mathematics.- A.1 Introduction.- A.2 The Indicial Notation.- A.3 Transformation of Coordinates.- A.4 The Summation Convention.- A.5 The Kronecker Delta.- A.6 Contravariant and Covariant Tensors.- A.7 Symmetric and Skew-symmetric Tensors.- A.8 Addition, Subtraction and Multiplication.- A.9 Contraction.- A.10 The Line Element.- A.11 The Angle between Vectors.- A.12 Lowering and Raising Indices.- A.13 The Christoffel Symbols.- A.14 Covariant Differentiation of Tensors.- A.15 Principal Directions of Second-order Tensors.- A.16 Differential Operators.- A.17 Orthogonal and Cartesian Coordinates.- A.18 Invariants.- A.19 Integrals of Tensor Fields.- A.20 Geometrical Representation of Second-order Tensors.- A.21 Axially Symmetric Second-order Tensors.- Appendix B Cylindrical Coordinates.- B.1 Introduction.- B.2 Definition of the Cylindrical Coordinate System.- B.3 The Fundamental Tensor.- B.4 The Christoffel Symbols.- B.5 Covariant Derivatives.- B.6 Basic Operations of First-order Tensors in Cylindrical Coordinates.- B.7 Elements of Differential Geometry.- B.8 Equations of Kinematics.- B.9 The Strain Tensor.- B.10 The Balance Equations.- Appendix C Rheological Modeling.- C.l Introduction.- C.2 The Hookean Elastic Element.- C.3 The Newtonian Viscous Element.- C.4 Coupling of Rheological Elements.- C.5 St Venant’s Element of Plastic Restraint.- C.6 The Prandtl Body.- C.7 The Maxwell Body.- C.8 The Kelvin Body.- C.9 The Burgers Body.- C.10 The Relations Between Excitation and Response.- C.11 The Relaxation and Creep Functions.- C.12 The General Rheological Models.- C.13 Elastic and Dissipative Excitations.- C.14 The Plastic Restraint.- References.