Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics / Edition 5

by Mikhail Itskov

Overview

This is the fifth and revised edition of a well-received textbook that aims at bridging the gap between the engineering course of tensor algebra on the one hand and the mathematical course of classical linear algebra on the other hand. In accordance with the contemporary way of scientific publication, a modern absolute tensor notation is preferred throughout. The book provides a comprehensible exposition of the fundamental mathematical concepts of tensor calculus and enriches the presented material with many illustrative examples. As such, this new edition also discusses such modern topics of solid mechanics as electro- and magnetoelasticity. In addition, the book also includes advanced chapters dealing with recent developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics. Hence, this textbook addresses graduate students as well as scientists working in this field and in particular dealing with multi-physical problems. In each chapter numerous exercises are included, allowing for self-study and intense practice. Solutions to the exercises are also provided.

Product Details

ISBN-13:	9783030075361
Publisher:	Springer International Publishing
Publication date:	12/27/2018
Series:	Mathematical Engineering
Edition description:	Softcover reprint of the original 5th ed. 2019
Pages:	300
Sales rank:	652,746
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Prof. Itskov studied Automobile Engineering at the Moscow State Automobile and Road Technical University, Russia. In 1990 he received his doctoral degree in mechanics, and in 2002 he obtained his habilitation degree in mechanics from the University of Bayreuth, Germany. Since 2004 he has been full professor for continuum mechanics at the RWTH Aachen University, Germany. His research interests comprise tensor analysis, non-linear continuum mechanics, in particular the application to anisotropic materials, as well as the mechanics of elastomers and soft tissues in a broad sense.

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

Editorial Reviews

From the reviews:

"Most books on continuum mechanics begin with a more or less lengthy introduction into this mere mathematical topic. … With the present introduction to tensors, Itskov has filled this gap. … It is written in a clear and modern style … . contains a lot of interesting representations, which will be appreciated by researchers in the field. … All in all, the book can be clearly recommended both for students and researchers in the field of continuum mechanics." (Technische Mechanik, Vol. 28 (3), 2008)

From the Publisher