### Table of Contents

**Introduction**

Concepts and Mathematical Preliminaries

Introduction

Summation Convention

Dummy Index and Dummy Variables

Free Indices

Vector and Matrix Notation

Index Notation and Kronecker Delta

Permutation Tensor

Operations Using Vector, Matrix, and Einstein's Notation

Change of Reference Frame, Transformations, Tensors

Some Useful Relations

Summary

**
**Kinematics of Motion, Deformation and Their Measures

Description of Motion

Lagrangian and Eulerian Descriptions

Material Particle Displacements

Continuous Deformation and Restrictions on the Motion

Material Derivative

Acceleration of a Material Particles

Coordinate Systems and Bases

Covariant Basis

Contravariant Basis

Alternate Way to Visualize Co- and Contra-Variant Bases

Jacobian of Deformation

Change of Description, Co- and Contra-Variant Measures

Notations For Covariant and Contravariant Measures

Deformation, Measures of Length and Change in Length

Covariant and Contravariant Measures of Strain

Changes in Strain Measures Due To Rigid Rotation of Frames

Invariants of Strain Tensors

Expanded Form of Strain Tensors

Physical Meaning of Strains

Polar Decomposition: Rotation and Stretch Tensors

Deformation of Areas and Volumes

Summary

Definitions and Measures of Stresses

Cauchy Stress Tensor

Contravariant and Covariant Stress Tensors

General Remarks

Summary of Stresses and Considerations in Their Derivations

General Considerations

Summary of Stress Measures

Conjugate Strain Measures

Relations between Stress Measures and Useful Relations

Summary

**
**Rate of Deformation, Strain Rate, and Spin

Tensors

Rate of Deformation

Decomposition of [□ L], the Spatial Velocity Gradient Tensor

Interpretation of the Components of [□D]

Rate of Change or Material Derivative of Strain Tensors

Physical Meaning of Spin Tensor [ □W ]

Vorticity Vector and Vorticity

Material Derivative of Determinant of J

Material Derivative of Volume

Rate of Change of Area: Material Derivative of Area

Stress And Strain Measures for Convected Time Derivatives

Convected Time Derivatives

Conjugate Convected Time Derivatives of Stress And Strain Tensors

Summary

**
**Conservation and Balance Laws in Eulerian Description

Introduction

Mass Density

Conservation Of Mass: Continuity Equation

Transport Theorem

Conservation Of Mass: Continuity Equation

Balance of Linear Momenta

Kinetics of Continuous Media: Balance of Angular Momenta

First Law of Thermodynamics

Second Law of Thermodynamics

A Summary of Mathematical Models

Summary

**
**Conservation and Balance Laws In Lagrangian Description

Introduction

Mathematical Model for Deforming Matter in Lagrangian Description

Conservation Of Mass: Continuity Equation

Balance of Linear Momenta

Balance of Angular Momenta

First Law of Thermodynamics

Second law of thermodynamics in terms of Φ

Second law of thermodynamics in terms of Ψ

Summary of Mathematical Models

First and Second Laws for Thermoelastic Solids

Summary

General Considerations in the Constitutive Theories

Introduction

Axioms of Constitutive Theory

Objective

Solid Matter

Fluids

Preliminary Considerations in the Constitutive Theories

General Approach of Deriving Constitutive Theories

Summary

Ordered Rate Constitutive Theories for Thermoelastic Solids

Introduction

Entropy inequality in Φ: Lagrangian description

Constitutive Theories for Thermoelastic Solids

Constitutive Theories Using Generators and Invariants

Strain energy density π: Lagrangian description

Stress in terms of Green strain based on π: Lagrangian

Stress in terms of Cauchy strain based on π: Lagrangian

Constitutive Theories for the Heat Vector: Lagrangian

Alternate Derivations: Strain In Terms Of Stress

Alternate Derivations: Heat Vector In Terms Of Stress

Summary

**
**Thermoviscoelastic Solids without Memory

Introduction

Constitutive Theories Using Helmholtz Free Energy Density

Constitutive Theories Using Gibbs Potential

Comparisons of constitutive theories using Φ and Ψ** **

**
**Thermoviscoelastic Solids with Memory

Introduction

Constitutive Theories Using Helmholtz Free Energy Density

Constitutive Theories Using Gibbs Potential

Comparisons of constitutive theories using Φ and Ψ

**
**Ordered Rate Constitutive Theories for Thermofluids

Introduction

Second Law of Thermodynamics: Entropy Inequality

Dependent Variables and Their Arguments

Development of Constitutive Theory for Thermo Fluids

Rate Constitutive Theory of Order *N*

Rate Constitutive Theory of Order Two

Rate Constitutive Theory of Order One

Generalized Newtonian and Newtonian Fluids

Incompressible Ordered Thermo Fluids of Orders N, 2 And 1

Incompressible Generalized Newtonian, Newtonian Fluids

Conjugate Measures, Validity of Rate Constitutive Theories

Summary

**
**Ordered Rate Constitutive Theories for Polymers

Introduction

Second Law of Thermodynamics: Entropy Inequality

Dependent Variables and Their Arguments

Development of Constitutive Theory for Polymers

Rate Constitutive Theory of Orders `*M*' and `*N'*

Rate Constitutive Theory of Orders *M*=1 and* N*=1

Rate Constitutive Theory of Orders *M*=1 and *N*=2

Constitutive Theories for Incompressible Polymers

Numerical Studies Using Giesekus Constitutive Model

**
**Ordered Rate Constitutive Theories for Hypoelastic Solids

Introduction

Second Law of Thermodynamics: Entropy Inequality

Dependent Variables and Their Arguments

Development of Constitutive Theory for Hypo-Elastic Solids

Rate Constitutive Theory of Order `*N*'

Rate Constitutive Theory of Order Two

Rate Constitutive Theory of Order One

Compressible Generalized Hypo-Elastic Solids of Order One

Incompressible Ordered Hypo-Elastic Solids

Incompressible Generalized Hypo-Elastic Solids: Order One

Summary

**
**Mathematical Models with Thermodynamic Relations

Introduction

Thermodynamic Pressure: Compressible Matter

Mechanical Pressure: Incompressible Matter

Specific Internal Energy

Variable Transport Properties or Material Coefficients

Final Form of the Mathematical Models

Summary

**
**Principle of Virtual Work

Introduction

Hamilton's Principle in Continuum Mechanics

Euler-Lagrange Equation: Lagrangian Description

Euler-Lagrange Equation: Eulerian Description

Summary and Remarks

Appendices