Continuum Mechanics of Anisotropic Materials

<p>Chapter 1. Introduction </p><p><p>Chapter 2. Mechanical modeling of materials </p><p>2.1 Introduction</p><p>2.2 Models and the real physical world</p><p>2.3 Guidelines for modeling objects and solving mechanics problems</p><p>2.4 The types of models used in mechanics</p><p>2.5 The particle model</p><p>2.6 The rigid object model</p><p>2.7 The deformable continuum model </p><p>2.8 Lumped parameter models </p><p>2.9 Statistical models</p><p>2.10 Cellular automata</p><p>2.11 The limits of reductionism</p><p>2.12 References</p><p>Appendix 2A Laplace transform refresher</p><p>Appendix 2B First order differential equations</p><p>Appendix 2C Electrical analogs of the spring and dashpot models</p><p><p>Chapter 3. Basic continuum kinematics</p><p>3.1 The deformable material model, the continuum</p><p>3.2 Rates of change and the spatial representation of motion</p><p>3.3 Infinitesimal motions</p><p>3.4 The strain conditions of compatibility</p><p><p>Chapter 4. Continuum formulations of conservation laws </p><p>4.1 The conservation principles</p><p>4.2 The conservation of mass</p><p>4.3 The state of stress at a point</p><p>4.4 The stress equations of motion</p><p>4.5 The conservation of energy</p><p><p>Chapter 5. Formulation of constitutive equations </p><p>5.1 Guidelines for the formulation of constitutive equations</p><p>5.2 Constitutive ideas</p><p>5.3 Localization</p><p>5.4 Invariance under rigid object motions</p><p>5.5 Determinism</p><p>5.6 Linearization</p><p>5.7 Coordinate invariance</p><p>5.8 Homogeneous versus inhomogeneous constitutive models</p><p>5.9 Restrictions due to material symmetry</p><p>5.10 The symmetry of the material coefficient tensors</p><p>5.11 Restrictions on the coefficients representing material properties</p><p>5.12 Summary of results</p><p>5.13 Relevant literature </p><p><p>Chapter 6 Modeling material symmetry </p><p>6.1 Introduction </p><p>6.2 The representative volume element (RVE)</p><p>6.3 Crystalline materials and textured materials</p><p>6.4 Planes of mirror symmetry</p><p>6.5 Characterization of material symmetries by planes of symmetry</p><p>6.6 The forms of the 3D symmetric linear transformation A </p><p>6.7 The forms of the 6D symmetric linear transformation </p><p>6.8 Curvilinear anisotropy</p><p>6.9 Symmetries that permit chirality</p><p>6.10 Relevant literature </p><p><p>Chapter 7. Four linear continuum theories </p><p>7.1 Formation of continuum theories</p><p>7.2 The theory of fluid flow through rigid porous media</p><p>7.3 The theory of elastic solids</p><p>7.4 The theory of viscous fluids</p><p>7.5 The theory of viscoelastic materials</p><p>7.6 Relevant literature </p><p><p>Chapter 8 Modeling material microstructure </p><p>8.1 Introduction </p><p>8.2 The representative volume element (RVE)</p><p>8.3 Effective material parameters</p><p>8.4 Effective elastic constants</p><p>8.5 Effective permeability</p><p>8.6 Structural gradients</p><p>8.7 Tensorial representations of microstructure </p><p>8.8 Relevant literature </p><p><p>Chapter 9. Poroelasticity </p><p>9.1 Poroelastic materials</p><p>9.2 The stress-strain-pore pressure constitutive relation</p><p>9.3 The fluid content-stress-pore pressure constitutive relation</p><p>9.4 Darcy’s Law</p><p>9.5 Matrix material and pore fluid incompressibility constraints</p><p>9.6 The undrained elastic coefficients</p><p>9.7 Expressions of mass and momentum conservation</p><p>9.8 The basic equations of poroelasticity</p><p>9.9 The basic equations of incompressible poroelasticity</p><p>9.10 Some example isotropic poroelastic problems</p><p>9.11 An example: the unconfined compression of an anisotropic disc</p><p>9.12 Relevant literature </p><p><p>Chapter 10 Mixture </p><p>10.1 Introduction</p><p>10.2 Kinematics of mixtures</p><p>10.3 The conservation laws for mixtures</p><p>10.4 A statement of irreversibility in mixture processes</p><p>10.5 Donnan equilibrium and osmotic pressure</p><p>10.6 Continuum model for a charged porous medium; the governing equations</p><p>10.7 Linear irreversible thermodynamics and the four constituent mixture</p><p>10.8 Modeling swelling and compression experiments on the intervertebral disc</p><p>10.9 Relevant literature</p><p><p>Chapter 11. Kinematics and mechanics of large deformations </p><p>11.1 Large deformations</p><p>11.2 Large homogeneous deformations</p><p>11.3 Polar decomposition of the deformation gradients</p><p>11.4 The strain measures for large deformations</p><p>11.5 Measures of volume and surface change in large deformations</p><p>11.6 Stress measures</p><p>11.7 Finite deformation elasticity</p><p>11.8 The isotropic finite deformation stress-strain relation </p><p>11.9 Finite deformation hyperelasticity</p><p>11.10 Incompressible elasticity </p><p>11.11 Relevant literature</p><p><p> </p><p>Chapter 12. Plasticity Theory</p><p>12.1 Extension of von Mises criterion to anisotropic materials</p><p>12.2 Yield criteria for pressure sensitive anisotropic materials</p><p>12.3 Some particular deformation characteristics exhibited by granular materials (dilatancy/contractancy, anisotropy, hardening/softening, and shear localization).</p><p>12.4 Dilatant double shearing kinematics</p><p>12.5 Evolution equations for the material parameters</p><p>12.6 Numerical biaxial compression test of anisotropic granular materials</p><p>12.6 Numerical triaxial compression test of anisotropic granular materials</p><p>12.7 Plasticity theories for crystalline materials</p><p><p>Appendix A. Matrices and tensors </p><p>A.1 Introduction and rationale</p><p>A.2 Definition of square, column and row matrices</p><p>A.3 The types and algebra of square matrices</p><p>A.4 The algebra of n-tuples</p><p>A.5 Linear transformations</p><p>A.6 Vector spaces</p><p>A.7 Second rank tensors</p><p>A.8 The moment of inertia tensor</p><p>A.9 The alternator and vector cross products </p><p>A.10 Connection to Mohr’s circles</p><p>A.11 Special vectors and tensors in six dimensions</p><p>A.12 The gradient operator and the divergence theorem</p><p>A.13 Tensor components in cylindrical coordinates</p>