Table of Contents
1 Vectors and Tensors in a Finite-Dimensional Space 1
1.1 Notions of the Vector Space 1
1.2 Basis and Dimension of the Vector Space 3
1.3 Components of a Vector, Summation Convention 5
1.4 Scalar Product, Euclidean Space, Orthonormal Basis 6
1.5 Dual Bases 8
1.6 Second-order Tensor as a Linear Mapping 12
1.7 Tensor Product, Representation of a Tensor with Respect to a Basis 16
1.8 Change of the Basis, Transformation Rules 19
1.9 Special Operations with Second-Order Tensors 20
1.10 Scalar Product of Second-Order Tensors 26
1.11 Decompositions of Second-Order Tensors 27
1.12 Tensors of Higher Orders 29
Exercises 30
2 Vector and Tensor Analysis in Euclidean Space 35
2.1 Vector-and Tensor-Valued Functions, Differential Calculus 35
2.2 Coordinates in Euclidean Space, Tangent Vectors 37
2.3 Coordinate Transformation. Co-, Contra-and Mixed Variant Components 40
2.4 Gradient, Covariant and Contravariant Derivatives 42
2.5 Christoffel Symbols, Representation of the Covariant Derivative 46
2.6 Applications in Three-Dimensional Space: Divergence and Curl 49
Exercises 57
3 Curves and Surfaces in Three-Dimensional Euclidean Space 59
3.1 Curves in Three-Dimensional Euclidean Space 59
3.2 Surfaces in Three-Dimensional Euclidean Space 66
3.3 Application to Shell Theory 73
Exercises 79
4 Eigenvalue Problem and Spectral Decomposition of Second-Order Tensors 81
4.1 Complexification 81
4.2 Eigenvalue Problem, Eigenvalues and Eigenvectors 82
4.3 Characteristic Polynomial 85
4.4 Spectral Decomposition and Eigenprojections 87
4.5 Spectral Decomposition of Symmetric Second-Order Tensors 92
4.6 Spectral Decomposition of Orthogonal andSkew-Symmetric Second-Order Tensors 94
4.7 Cayley-Hamilton Theorem 98
Exercises 100
5 Fourth-Order Tensors 103
5.1 Fourth-Order Tensors as a Linear Mapping 103
5.2 Tensor Products, Representation of Fourth-Order Tensors with Respect to a Basis 104
5.3 Special Operations with Fourth-Order Tensors 106
5.4 Super-Symmetric Fourth-Order Tensors 109
5.5 Special Fourth-Order Tensors 111
Exercises 114
6 Analysis of Tensor Functions 115
6.1 Scalar-Valued Isotropic Tensor Functions 115
6.2 Scalar-Valued Anisotropic Tensor Functions 119
6.3 Derivatives of Scalar-Valued Tensor Functions 122
6.4 Tensor-Valued Isotropic and Anisotropic Tensor Functions 129
6.5 Derivatives of Tensor-Valued Tensor Functions 135
6.6 Generalized Rivlin's Identities 140
Exercises 142
7 Analytic Tensor Functions 145
7.1 Introduction 145
7.2 Closed-Form Representation for Analytic Tensor Functions and Their Derivatives 149
7.3 Special Case: Diagonalizable Tensor Functions 152
7.4 Special case: Three-Dimensional Space 154
7.5 Recurrent Calculation of Tensor Power Series and Their Derivatives 161
Exercises 163
8 Applications to Continuum Mechanics 165
8.1 Polar Decomposition of the Deformation Gradient 165
8.2 Basis-Free Representations for the Stretch and Rotation Tensor 166
8.3 The Derivative of the Stretch and Rotation Tensor with Respect to the Deformation Gradient 169
8.4 Time Rate of Generalized Strains 173
8.5 Stress Conjugate to a Generalized Strain 175
8.6 Finite Plasticity Based on the Additive Decomposition of Generalized Strains 178
Exercises 182
Solutions 185
References 239
Index 243
