There is a large gap between the engineering course in tensor algebra on the one hand and the treatment of linear transformations within classical linear algebra on the other hand. The aim of the book is to bridge this gap by means of the consequent and fundamental exposition. The book is addressed primarily to engineering students with some initial knowledge of matrix algebra. Thereby the mathematical formalism is applied as far as it is absolutely necessary. Numerous exercises provided in the book are accompanied by solutions enabling an autonomous study. The last chapters of the book deal with modern developments in the theory of isotropic an anisotropic tensor functions and their applications to continuum mechanics and might therefore be of high interest for PhD-students and scientists working in this area.In the last decades, the absolute notation for tensors has become widely accepted and is now a current state of the art for publications in solid and structural mechanics. This is opposed to a majority of books on tensor calculus referring to index notation. The latter one complicates the understanding of the matter especially for readers with initial knowledge. Thus, this book aims at being a modern textbook on tensor calculus for engineers in line with the contemporary way of scientific publications.
Author(s): Mikhail Itskov
Publisher: Springer
Year: 2007
Language: English
Pages: 244
1.1 Notion of the Vector Space ......Page 11
1.2 Basis and Dimension of the Vector Space ......Page 13
1.3 Components of a Vector; Summation Convention ......Page 15
1.4 Scalar Product; Euclidean Space; Orthonormal Basis ......Page 16
1.5 Dual Bases ......Page 18
1.6 Second-Order Tensor as a Linear Mapping ......Page 22
1.7 Tensor Product; Representation of a Tensor with Respect to a Basis ......Page 26
1.8 Change of the Basis; Transformation Rules ......Page 28
1.9 Special Operations with Second-Order Tensors ......Page 29
1.10 Scalar Product of Second-Order Tensors ......Page 35
1.11 Decompositions of Second-Order Tensors ......Page 37
Exercises ......Page 38
2.1 Vector- and Tensor-Valued Functions; Differential Calculus ......Page 42
2.2 Coordinates in Euclidean Space; Tangent Vectors ......Page 44
2.3 Coordinate Transformation. Co-; Contra- and Mixed Variant Components ......Page 47
2.4 Gradient; Covariant and Contravariant Derivatives ......Page 49
2.5 Christoffel Symbols; Representation of the Covariant Derivative ......Page 53
2.6 Applications in Three-Dimensional Space: Divergence and Curl ......Page 56
Exercises ......Page 64
3.1 Curves in Three-Dimensional Euclidean Space ......Page 65
3.2 Surfaces in Three-Dimensional Euclidean Space ......Page 72
3.3 Application to Shell Theory ......Page 79
Exercises ......Page 85
4.1 Complexification ......Page 86
4.2 Eigenvalue Problem; Eigenvalues and Eigenvectors ......Page 87
4.3 Characteristic Polynomial ......Page 90
4.4 Spectral Decomposition and Eigenprojections ......Page 92
4.5 Spectral Decomposition of Symmetric Second-Order Tensors ......Page 97
4.6 Spectral Decomposition of Orthogonal and Skew-Symmetric Second-Order Tensors ......Page 99
4.7 Cayley-Hamilton Theorem ......Page 103
Exercises ......Page 104
5.1 Fourth-Order Tensors as a Linear Mapping ......Page 106
5.2 Tensor Products; Representation of Fourth-Order Tensors with Respect to a Basis ......Page 107
5.3 Special Operations with Fourth-Order Tensors ......Page 109
5.4 Super-Symmetric Fourth-Order Tensors ......Page 112
5.5 Special Fourth-Order Tensors ......Page 114
Exercises ......Page 116
6.1 Scalar-Valued Isotropic Tensor Functions ......Page 118
6.2 Scalar-Valued Anisotropic Tensor Functions ......Page 122
6.3 Derivatives of Scalar-Valued Tensor Functions ......Page 125
6.4 Tensor-Valued Isotropic and Anisotropic Tensor Functions ......Page 131
6.5 Derivatives of Tensor-Valued Tensor Functions ......Page 138
6.6 Generalized Rivlin’s Identities ......Page 142
Exercises ......Page 144
7.1 Introduction ......Page 147
7.2 Closed-Form Representation for Analytic Tensor Functions and Their Derivatives ......Page 151
7.3 Special Case: Diagonalizable Tensor Functions ......Page 154
7.4 Special case: Three-Dimensional Space ......Page 156
7.5 Recurrent Calculation of Tensor Power Series and Their Derivatives ......Page 163
Exercises ......Page 165
8.1 Polar Decomposition of the Deformation Gradient ......Page 167
8.2 Basis-Free Representations for the Stretch and Rotation Tensor162 ......Page 168
8.3 The Derivative of the Stretch and Rotation Tensor with Respect to the Deformation Gradient ......Page 171
8.4 Time Rate of Generalized Strains ......Page 175
8.5 Stress Conjugate to a Generalized Strain ......Page 177
8.6 Finite Plasticity Based on the Additive Decomposition of Generalized Strains ......Page 179
Exercises ......Page 184
Solutions ......Page 185
References ......Page 237
C ......Page 240
J ......Page 241
R ......Page 242
T ......Page 243
Z ......Page 244
