Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics (Mathematical Engineering)

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. Tensor Algebra and Tensor Analysis for Engineers......Page 3 Preface to the Third Edition......Page 7 Preface to the Second Edition......Page 9 Preface to the First Edition......Page 11 Contents......Page 13 1.1 Notion of the Vector Space......Page 16 1.2 Basis and Dimension of the Vector Space......Page 17 1.3 Components of a Vector, Summation Convention......Page 20 1.4 Scalar Product, Euclidean Space, Orthonormal Basis......Page 21 1.5 Dual Bases......Page 22 1.6 Second-Order Tensor as a Linear Mapping......Page 27 1.7 Tensor Product, Representation of a Tensor with Respect to a Basis......Page 32 1.8 Change of the Basis, Transformation Rules......Page 34 1.9 Special Operations with Second-Order Tensors......Page 35 1.10 Scalar Product of Second-Order Tensors......Page 41 1.11 Decompositions of Second-Order Tensors......Page 43 1.12 Tensors of Higher Orders......Page 45 Exercises......Page 46 2.1 Vector- and Tensor-Valued Functions, Differential Calculus......Page 50 2.2 Coordinates in Euclidean Space, Tangent Vectors......Page 52 2.3 Coordinate Transformation. Co-, Contra- and Mixed Variant Components......Page 56 2.4 Gradient, Covariant and Contravariant Derivatives......Page 58 2.5 Christoffel Symbols, Representation of the Covariant Derivative......Page 63 2.6 Applications in Three-Dimensional Space: Divergence and Curl......Page 67 Exercises......Page 75 3.1 Curves in Three-Dimensional Euclidean Space......Page 78 3.2 Surfaces in Three-Dimensional Euclidean Space......Page 84 3.3 Application to Shell Theory......Page 91 Exercises......Page 97 4.1 Complexification......Page 99 4.2 Eigenvalue Problem, Eigenvalues and Eigenvectors......Page 101 4.3 Characteristic Polynomial......Page 104 4.4 Spectral Decomposition and Eigenprojections......Page 106 4.5 Spectral Decomposition of Symmetric Second-Order Tensors......Page 111 4.6 Spectral Decomposition of Orthogonal and Skew-Symmetric Second-Order Tensors......Page 113 4.7 Cayley-Hamilton Theorem......Page 117 Exercises......Page 119 5.1 Fourth-Order Tensors as a Linear Mapping......Page 121 5.2 Tensor Products, Representation of Fourth-Order Tensors with Respect to a Basis......Page 122 5.3 Special Operations with Fourth-Order Tensors......Page 125 5.4 Super-Symmetric Fourth-Order Tensors......Page 128 5.5 Special Fourth-Order Tensors......Page 130 Exercises......Page 132 6.1 Scalar-Valued Isotropic Tensor Functions......Page 134 6.2 Scalar-Valued Anisotropic Tensor Functions......Page 138 6.3 Derivatives of Scalar-Valued Tensor Functions......Page 141 6.4 Tensor-Valued Isotropic and Anisotropic Tensor Functions......Page 151 6.5 Derivatives of Tensor-Valued Tensor Functions......Page 157 6.6 Generalized Rivlin's Identities......Page 162 Exercises......Page 165 7.1 Introduction......Page 167 7.2 Closed-Form Representation for Analytic Tensor Functions and Their Derivatives......Page 171 7.3 Special Case: Diagonalizable Tensor Functions......Page 174 7.4 Special Case: Three-Dimensional Space......Page 177 7.5 Recurrent Calculation of Tensor Power Series and Their Derivatives......Page 183 Exercises......Page 186 8.1 Polar Decomposition of the Deformation Gradient......Page 188 8.2 Basis-Free Representations for the Stretch and Rotation Tensor......Page 189 8.3 The Derivative of the Stretch and Rotation Tensor with Respect to the Deformation Gradient......Page 192 8.4 Time Rate of Generalized Strains......Page 196 8.5 Stress Conjugate to a Generalized Strain......Page 198 8.6 Finite Plasticity Based on the Additive Decomposition of Generalized Strains......Page 201 Exercises......Page 206 9.1 Exercises of Chap.1......Page 207 9.2 Exercises of Chap.2......Page 220 9.3 Exercises of Chap.3......Page 232 9.4 Exercises of Chap.4......Page 236 9.5 Exercises of Chap.5......Page 247 9.6 Exercises of Chap.6......Page 252 9.7 Exercises of Chap.7......Page 263 9.8 Exercises of Chap.8......Page 269 References......Page 271 Index......Page 274 Annotation 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, enablingself-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 formulaehave beenrevised and improved where necessary This book provides a comprehensive exposition of tensor analysis. It presents modern developments in the theory of isotropic and anisotropic tensor functions and includes numerous exercises with solutions.