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

by Mikhail Itskov

Overview

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 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 and 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.

This second edition is completed by a number of additional examples and exercises. The text and formulae are thoroughly revised and improved where necessary.

Product Details

ISBN-13:	9783319988061
Publisher:	Springer-Verlag New York, LLC
Publication date:	09/15/2018
Series:	Mathematical Engineering
Sold by:	Barnes & Noble
Format:	eBook
File size:	24 MB
Note:	This product may take a few minutes to download.

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