Tensor algebra and tensor analysis for engineers : with applications to continuum mechanics

✍ Scribed by Mikhail Itskov

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 279
• Series: Mathematical engineering
• Edition: 3ed.
• Category: Library

✦ Synopsis

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 this modern textbook is to bridge this gap by means of the consequent and fundamental exposition. The book primarily addresses engineering students with some initial knowledge of matrix algebra. Thereby the mathematical formalism is applied as far as it is absolutely necessary. Numerous exercises are provided in the book and are accompanied by solutions, enabling self-study. The last chapters of the book deal with modern developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics and are therefore of high interest for PhD-students and scientists working in this area.This third edition is completed by a number of additional figures, examples and exercises. The text and formulae have been revised and improved where necessary. Read more... Vectors and Tensors in a Finite-Dimensional Space -- Vector and Tensor Analysis in Euclidean Space -- Curves and Surfaces in Three-Dimensional Euclidean Space -- Eigenvalue Problem and Spectral Decomposition of Second-Order Tensors -- Fourth-Order Tensors -- Analysis of Tensor Functions -- Analytic Tensor Functions -- Applications to Continuum Mechanics -- Solutions

✦ Table of Contents

Cover......Page 1
Tensor Algebra and Tensor Analysis for Engineers......Page 4
Preface to the Third Edition......Page 8
Preface to the Second Edition......Page 10
Preface to the First Edition......Page 12
Contents......Page 14
1.1 Notion of the Vector Space......Page 17
1.2 Basis and Dimension of the Vector Space......Page 18
1.3 Components of a Vector, Summation Convention......Page 21
1.4 Scalar Product, Euclidean Space, Orthonormal Basis......Page 22
1.5 Dual Bases......Page 23
1.6 Second-Order Tensor as a Linear Mapping......Page 28
1.7 Tensor Product, Representation of a Tensor with Respect to a Basis......Page 33
1.8 Change of the Basis, Transformation Rules......Page 35
1.9 Special Operations with Second-Order Tensors......Page 36
1.10 Scalar Product of Second-Order Tensors......Page 42
1.11 Decompositions of Second-Order Tensors......Page 44
1.12 Tensors of Higher Orders......Page 46
Exercises......Page 47
2.1 Vector- and Tensor-Valued Functions, Differential Calculus......Page 51
2.2 Coordinates in Euclidean Space, Tangent Vectors......Page 53
2.3 Coordinate Transformation. Co-, Contra- and Mixed Variant Components......Page 57
2.4 Gradient, Covariant and Contravariant Derivatives......Page 59
2.5 Christoffel Symbols, Representation of the Covariant Derivative......Page 64
2.6 Applications in Three-Dimensional Space: Divergence and Curl......Page 68
Exercises......Page 76
3.1 Curves in Three-Dimensional Euclidean Space......Page 79
3.2 Surfaces in Three-Dimensional Euclidean Space......Page 85
3.3 Application to Shell Theory......Page 92
Exercises......Page 98
4.1 Complexification......Page 100
4.2 Eigenvalue Problem, Eigenvalues and Eigenvectors......Page 102
4.3 Characteristic Polynomial......Page 105
4.4 Spectral Decomposition and Eigenprojections......Page 107
4.5 Spectral Decomposition of Symmetric Second-Order Tensors......Page 112
4.6 Spectral Decomposition of Orthogonal and Skew-Symmetric Second-Order Tensors......Page 114
4.7 Cayley-Hamilton Theorem......Page 118
Exercises......Page 120
5.1 Fourth-Order Tensors as a Linear Mapping......Page 122
5.2 Tensor Products, Representation of Fourth-Order Tensors with Respect to a Basis......Page 123
5.3 Special Operations with Fourth-Order Tensors......Page 126
5.4 Super-Symmetric Fourth-Order Tensors......Page 129
5.5 Special Fourth-Order Tensors......Page 131
Exercises......Page 133
6.1 Scalar-Valued Isotropic Tensor Functions......Page 135
6.2 Scalar-Valued Anisotropic Tensor Functions......Page 139
6.3 Derivatives of Scalar-Valued Tensor Functions......Page 142
6.4 Tensor-Valued Isotropic and Anisotropic Tensor Functions......Page 152
6.5 Derivatives of Tensor-Valued Tensor Functions......Page 158
6.6 Generalized Rivlin's Identities......Page 163
Exercises......Page 166
7.1 Introduction......Page 168
7.2 Closed-Form Representation for Analytic Tensor Functions and Their Derivatives......Page 172
7.3 Special Case: Diagonalizable Tensor Functions......Page 175
7.4 Special Case: Three-Dimensional Space......Page 178
7.5 Recurrent Calculation of Tensor Power Series and Their Derivatives......Page 184
Exercises......Page 187
8.1 Polar Decomposition of the Deformation Gradient......Page 189
8.2 Basis-Free Representations for the Stretch and Rotation Tensor......Page 190
8.3 The Derivative of the Stretch and Rotation Tensor with Respect to the Deformation Gradient......Page 193
8.4 Time Rate of Generalized Strains......Page 197
8.5 Stress Conjugate to a Generalized Strain......Page 199
8.6 Finite Plasticity Based on the Additive Decomposition of Generalized Strains......Page 202
Exercises......Page 207
9.1 Exercises of Chap.1......Page 208
9.2 Exercises of Chap.2......Page 221
9.3 Exercises of Chap.3......Page 233
9.4 Exercises of Chap.4......Page 237
9.5 Exercises of Chap.5......Page 248
9.6 Exercises of Chap.6......Page 253
9.7 Exercises of Chap.7......Page 264
9.8 Exercises of Chap.8......Page 270
References......Page 272
Index......Page 275