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

Couverture
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
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