Intro
Preface
Contents
1 Mathematical Preliminaries: Vector and Tensor Analysis
1.1 Conventions and Symbols
1.1.1 Summation Convention
1.1.2 Kronecker's Delta and Permutation Symbol
1.1.3 Matrix and Determinant
1.2 Vector
1.2.1 Definition of Vector
1.2.2 Operations of Vectors
1.2.3 Coordinate Transformation of Vector
1.3 Tensor
1.3.1 Definition of Tensor
1.3.2 Quotient Law
1.3.3 Notations of Tensors
1.3.4 Orthogonal Tensor
1.4 Operations of Tensors
1.4.1 Notations in Tensor Operations
1.4.2 Trace
1.4.3 Various Tensors

1.5 Eigenvalues and Eigenvectors
1.6 Calculations of Eigenvalues and Eigenvectors
1.6.1 Eigenvalues
1.6.2 Eigenvectors
1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor
1.8 Cayley-Hamilton Theorem
1.9 Scalar Triple Products with Invariants
1.10 Positive Definite Tensor
1.11 Polar Decomposition
1.12 Isotropic Tensor-Valued Tensor Function
1.13 Representation of Tensor in Principal Space
1.14 Two-Dimensional State
1.15 Tensor Functions
1.16 Partial Differential Calculi
1.17 Differentiation and Integration in Tensor Field

1.18 Representation in General Coordinate System
1.18.1 Primary and Reciprocal Base Vectors
1.18.2 Metric Tensor and Base Vector Algebra
1.18.3 Tensor Representations
2 Description of Motion
2.1 Motion of Material Point
2.2 Time-Derivatives
2.3 Variations and Rates of Geometrical Elements
2.3.1 Deformation Gradient and Variations of Line, Surface and Volume Elements
2.3.2 Velocity Gradient and Rates of Line, Surface and Volume Elements
2.4 Material-Time Derivative of Volume Integration
3 Description of Tensor (Rate) in Convected Coordinate System

3.1 Reference and Current Primary and Reciprocal Base Vectors
3.2 Description of Deformation Gradient Tensor by Embedded Base Vectors
3.3 Pull-Back and Push-Forward Operations
3.4 Convected Time-Derivative
3.4.1 General Convected Derivative
3.4.2 Corotational Rate
3.4.3 On Adoption of Convected Rate Tensor in Hypoelastic Constitutive Equation
3.4.4 Time-Integration of Convected Rate Tensor
4 Deformation/Rotation Tensors
4.1 Deformation Tensors
4.2 Strain Tensors
4.3 Volumetric and Isochoric Parts of Deformation Gradient Tensor
4.4 Strain Rate and Spin Tensors

4.5 Logarithmic (True) and Infinitesimal (Nominal) Strains
5 Stress Tensors and Conservation Laws
5.1 Stress Tensor
5.2 Conservation Law of Mass
5.3 Conservation Law of Linear Momentum
5.4 Conservation Law of Angular Momentum
5.5 Equilibrium Equation
5.6 Equilibrium Equation of Angular Moment
5.7 Virtual Work Principle
5.8 Conservation Law of Energy
5.9 Work Conjugacy
5.10 Various Simple Deformations
5.10.1 Uniaxial Loading
5.10.2 Simple Shear
5.10.3 Combination of Tension and Distortion
6 Objectivity and Objective (Rate) Tensors
6.1 Objectivity