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

Springer Science & Business Media, 30 avr. 2009 - 247 pages
This second edition is completed by a number of additional examples and exercises. In response of comments and questions of students using this book, solutions of many exercises have been improved for a better understanding. Some changes and enhancements are concerned with the treatment of sk- symmetric and rotation tensors in the ?rst chapter. Besides, the text and formulae have thoroughly been reexamined and improved where necessary. Aachen, January 2009 Mikhail Itskov Preface to the First Edition Like many other textbooks the present one is based on a lecture course given by the author for master students of the RWTH Aachen University. In spite of a somewhat di?cult matter those students were able to endure and, as far as I know, are still ?ne. I wish the same for the reader of the book. Although the present book can be referred to as a textbook one ?nds only little plain text inside. I tried to explain the matter in a brief way, nevert- lessgoinginto detailwherenecessary.Ialsoavoidedtediousintroductions and lengthy remarks about the signi?cance of one topic or another. A reader - terested in tensor algebra and tensor analysis but preferring, however, words instead of equations can close this book immediately after having read the preface.

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### Table des matières

 Vectors and Tensors in a FiniteDimensional Space 1 12 Basis and Dimension of the Vector Space 3 13 Components of a Vector Summation Convention 5 14 Scalar Product Euclidean Space Orthonormal Basis 6 15 Dual Bases 8 16 SecondOrder Tensor as a Linear Mapping 12 17 Tensor Product Representation of a Tensor with Respect to a Basis 16 18 Change of the Basis Transformation Rules 19
 Exercises 100 FourthOrder Tensors 103 52 Tensor Products Representation of FourthOrder Tensors with Respect to a Basis 104 53 Special Operations with FourthOrder Tensors 106 54 SuperSymmetric FourthOrder Tensors 109 55 Special FourthOrder Tensors 111 Exercises 114 Analysis of Tensor Functions 115

 19 Special Operations with SecondOrder Tensors 20 110 Scalar Product of SecondOrder Tensors 26 111 Decompositions of SecondOrder Tensors 27 112 Tensors of Higher Orders 29 Exercises 30 Vector and Tensor Analysis in Euclidean Space 35 22 Coordinates in Euclidean Space Tangent Vectors 37 23 Coordinate Transformation Co Contra and Mixed Variant Components 40 24 Gradient Covariant and Contravariant Derivatives 42 25 Christoffel Symbols Representation of the Covariant Derivative 46 Divergence and Curl 49 Exercises 57 Curves and Surfaces in ThreeDimensional Euclidean Space 59 32 Surfaces in ThreeDimensional Euclidean Space 66 33 Application to Shell Theory 73 Exercises 79 Eigenvalue Problem and Spectral Decomposition of SecondOrder Tensors 80 42 Eigenvalue Problem Eigenvalues and Eigenvectors 82 43 Characteristic Polynomial 85 44 Spectral Decomposition and Eigenprojections 87 45 Spectral Decomposition of Symmetric SecondOrder Tensors 92 46 Spectral Decomposition of Orthogonal and SkewSymmetric SecondOrder Tensors 94 47 CayleyHamilton Theorem 98
 62 ScalarValued Anisotropic Tensor Functions 119 63 Derivatives of ScalarValued Tensor Functions 122 64 TensorValued Isotropic and Anisotropic Tensor Functions 129 65 Derivatives of TensorValued Tensor Functions 135 66 Generalized Rivlins Identities 140 Exercises 142 Analytic Tensor Functions 145 72 ClosedForm Representation for Analytic Tensor Functions and Their Derivatives 149 Diagonalizable Tensor Functions 152 ThreeDimensional Space 154 75 Recurrent Calculation of Tensor Power Series and Their Derivatives 161 Exercises 163 Applications to Continuum Mechanics 165 82 BasisFree Representations for the Stretch and Rotation Tensor 166 83 The Derivative of the Stretch and Rotation Tensor with Respect to the Deformation Gradient 169 84 Time Rate of Generalized Strains 173 85 Stress Conjugate to a Generalized Strain 175 86 Finite Plasticity Based on the Additive Decomposition of Generalized Strains 178 Exercises 182 Solutions 184 References 239 Index 243 Droits d'auteur