IRREDUCIBLE TENSORIAL SETS
1 Vectors and Tensors in a Finite-Dimensional Space 1 11 Notion of the Vector Space 1 12 Basis and Dimension of the Vector Space 2 13 Components of a Vector, Summation Convention 5 14 Scalar Product, Euclidean Space, Orthonormal Basis 6 15 Dual Bases 7 16 Second-Order Tensor as a Linear Mapping 12 17 Tensor Product, Representation of a Tensor with Respect to a Basis 17 18 Change of the Basis, Transformation Rules 19 19 Special Operations with Second-Order Tensors 20 110 Scalar Product of Second-Order Tensors 26 111 Decompositions of Second-Order Tensors 28 112 Tensors of Higher Orders 30 Exercises 31 2 Vector and Tensor Analysis in Euclidean Space 35 21 Vector- and Tensor-Valued Functions, Differential Calculus 35 22 Coordinates in Euclidean Space, Tangent Vectors 37 23 Coordinate Transformation Co-, Contra- and Mixed Variant Components 41 24 Gradient, Covariant and Contravariant Derivatives 43 25 Christoffel Symbols, Representation of the Covariant Derivative 48 26 Applications in Three-Dimensional Space: Divergence and Curl 52 Exercises 60 3 Curves and Surfaces in Three-Dimensional Euclidean Space 63 31 Curves in Three-Dimensional Euclidean Space 63 32 Surfaces in Three-Dimensional Euclidean Space 69 33 Application to Shell Theory 76 Exercises 82 4 Eigenvalue Problem and Spectral Decomposition of Second-Order Tensors 85 41 Complexification 85 42 Eigenvalue Problem, Eigenvalues and Eigenvectors 87 43 Characteristic Polynomial 90 44 Spectral Decomposition and Eigenprojections 92 45 Spectral Decomposition of Symmetric Second-Order Tensors 97 46 Spectral Decomposition of Orthogonal and Skew-Symmetric Second-Order Tensors 99 47 Cayley-Hamilton Theorem 103 Exercises 105 5 Fourth-Order Tensors 107 51 Fourth-Order Tensors as a Linear Mapping 107 52 Tensor Products, Representation of Fourth-Order Tensors with Respect to a Basis 108 53 Special Operations with Fourth-Order Tensors 111 54 Super-Symmetric Fourth-Order Tensors 114 55 Special Fourth-Order Tensors 116 Exercises 118 6 Analysis of Tensor Functions 121 61 Scalar-Valued Isotropic Tensor Functions 121 62 Scalar-Valued Anisotropic Tensor Functions 125 63 Derivatives of Scalar-Valued Tensor Functions 128 64 Tensor-Valued Isotropic and Anisotropic Tensor Functions 138 65 Derivatives of Tensor-Valued Tensor Functions 144 66 Generalized Rivlin’s Identities 149 Exercises 152 7 Analytic Tensor Functions 155 71 Introduction 155 72 Closed-Form Representation for Analytic Tensor Functions and Their Derivatives 159 73 Special Case: Diagonalizable Tensor Functions 162 74 Special Case: Three-Dimensional Space 165 75 Recurrent Calculation of Tensor Power Series and Their Derivatives 171 Exercises 174 8 Applications to Continuum Mechanics 177 81 Polar Decomposition of the Deformation Gradient 177 82 Basis-Free Representations for the Stretch and Rotation Tensor 178 83 The Derivative of the Stretch and Rotation Tensor with Respect to the Deformation Gradient 181 84 Time Rate of Generalized Strains 185 85 Stress Conjugate to a Generalized Strain 187 86 Finite Plasticity Based on the Additive Decomposition of Generalized Strains 190 Exercises 195 9 Solutions 197 91 Exercises of Chap 1 197 92 Exercises of Chap 2 210 93 Exercises of Chap 3 222 94 Exercises of Chap 4 226 95 Exercises of Chap 5 237 96 Exercises of Chap 6 242 97 Exercises of Chap 7 253 98 Exercises of Chap 8 259 References 261