Continuum Mechanics of Anisotropic Materials

Stephen C. Cowin is Distinguished Professor of Mechanical Engineering at The City College of The City University of New York

Chapter 1. Introduction



Chapter 2. Mechanical modeling of materials

2.1 Introduction

2.2 Models and the real physical world

2.3 Guidelines for modeling objects and solving mechanics problems

2.4 The types of models used in mechanics

2.5 The particle model

2.6 The rigid object model

2.7 The deformable continuum model

2.8 Lumped parameter models

2.9 Statistical models

2.10 Cellular automata

2.11 The limits of reductionism

2.12 References

Appendix 2A Laplace transform refresher

Appendix 2B First order differential equations

Appendix 2C Electrical analogs of the spring and dashpot models



Chapter 3. Basic continuum kinematics

3.1 The deformable material model, the continuum

3.2 Rates of change and the spatial representation of motion

3.3 Infinitesimal motions

3.4 The strain conditions of compatibility



Chapter 4. Continuum formulations of conservation laws

4.1 The conservation principles

4.2 The conservation of mass

4.3 The state of stress at a point

4.4 The stress equations of motion

4.5 The conservation of energy



Chapter 5. Formulation of constitutive equations

5.1 Guidelines for the formulation of constitutive equations

5.2 Constitutive ideas

5.3 Localization

5.4 Invariance under rigid object motions

5.5 Determinism

5.6 Linearization

5.7 Coordinate invariance

5.8 Homogeneous versus inhomogeneous constitutive models

5.9 Restrictions due to material symmetry

5.10 The symmetry of the material coefficient tensors

5.11 Restrictions on the coefficients representing material properties

5.12 Summary of results

5.13 Relevant literature



Chapter 6 Modeling material symmetry

6.1 Introduction

6.2 The representative volume element (RVE)

6.3 Crystalline materials and textured materials

6.4 Planes of mirror symmetry

6.5 Characterization of material symmetries by planes of symmetry

6.6 The forms of the 3D symmetric linear transformation A

6.7 The forms of the 6D symmetric linear transformation

6.8 Curvilinear anisotropy

6.9 Symmetries that permit chirality

6.10 Relevant literature



Chapter 7. Four linear continuum theories

7.1 Formation of continuum theories

7.2 The theory of fluid flow through rigid porous media

7.3 The theory of elastic solids

7.4 The theory of viscous fluids

7.5 The theory of viscoelastic materials

7.6 Relevant literature



Chapter 8 Modeling material microstructure

8.1 Introduction

8.2 The representative volume element (RVE)

8.3 Effective material parameters

8.4 Effective elastic constants

8.5 Effective permeability

8.6 Structural gradients

8.7 Tensorial representations of microstructure

8.8 Relevant literature



Chapter 9. Poroelasticity

9.1 Poroelastic materials

9.2 The stress-strain-pore pressure constitutive relation

9.3 The fluid content-stress-pore pressure constitutive relation

9.4 Darcy’s Law

9.5 Matrix material and pore fluid incompressibility constraints

9.6 The undrained elastic coefficients

9.7 Expressions of mass and momentum conservation

9.8 The basic equations of poroelasticity

9.9 The basic equations of incompressible poroelasticity

9.10 Some example isotropic poroelastic problems

9.11 An example: the unconfined compression of an anisotropic disc

9.12 Relevant literature



Chapter 10 Mixture

10.1 Introduction

10.2 Kinematics of mixtures

10.3 The conservation laws for mixtures

10.4 A statement of irreversibility in mixture processes

10.5 Donnan equilibrium and osmotic pressure

10.6 Continuum model for a charged porous medium; the governing equations

10.7 Linear irreversible thermodynamics and the four constituent mixture

10.8 Modeling swelling and compression experiments on the intervertebral disc

10.9 Relevant literature



Chapter 11. Kinematics and mechanics of large deformations

11.1 Large deformations

11.2 Large homogeneous deformations

11.3 Polar decomposition of the deformation gradients

11.4 The strain measures for large deformations

11.5 Measures of volume and surface change in large deformations

11.6 Stress measures

11.7 Finite deformation elasticity

11.8 The isotropic finite deformation stress-strain relation

11.9 Finite deformation hyperelasticity

11.10 Incompressible elasticity

11.11 Relevant literature



 

Chapter 12. Plasticity Theory

12.1 Extension of von Mises criterion to anisotropic materials

12.2 Yield criteria for pressure sensitive anisotropic materials

12.3 Some particular deformation characteristics exhibited by granular materials (dilatancy/contractancy, anisotropy, hardening/softening, and shear localization).

12.4 Dilatant double shearing kinematics

12.5 Evolution equations for the material parameters

12.6 Numerical biaxial compression test of anisotropic granular materials

12.6 Numerical triaxial compression test of anisotropic granular materials

12.7 Plasticity theories for crystalline materials



Appendix A. Matrices and tensors

A.1 Introduction and rationale

A.2 Definition of square, column and row matrices

A.3 The types and algebra of square matrices

A.4 The algebra of n-tuples

A.5 Linear transformations

A.6 Vector spaces

A.7 Second rank tensors

A.8 The moment of inertia tensor

A.9 The alternator and vector cross products

A.10 Connection to Mohr’s circles

A.11 Special vectors and tensors in six dimensions

A.12 The gradient operator and the divergence theorem

A.13 Tensor components in cylindrical coordinates