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

  • Formatas: PDF+DRM
  • Serija: Mathematical Engineering
  • Išleidimo metai: 15-Sep-2018
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319988061
Kitos knygos pagal šią temą:
Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum MechanicsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Mathematical Engineering
  • Išleidimo metai: 15-Sep-2018
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319988061
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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.
1 Vectors and Tensors in a Finite-Dimensional Space
1(36)
1.1 Notion of the Vector Space
1(2)
1.2 Basis and Dimension of the Vector Space
3(2)
1.3 Components of a Vector, Summation Convention
5(1)
1.4 Scalar Product, Euclidean Space, Orthonormal Basis
6(2)
1.5 Dual Bases
8(4)
1.6 Second-Order Tensor as a Linear Mapping
12(6)
1.7 Tensor Product, Representation of a Tensor with Respect to a Basis
18(3)
1.8 Change of the Basis, Transformation Rules
21(1)
1.9 Special Operations with Second-Order Tensors
22(6)
1.10 Scalar Product of Second-Order Tensors
28(2)
1.11 Decompositions of Second-Order Tensors
30(2)
1.12 Tensors of Higher Orders
32(1)
Exercises
32(5)
2 Vector and Tensor Analysis in Euclidean Space
37(36)
2.1 Vector- and Tensor-Valued Functions, Differential Calculus
37(2)
2.2 Coordinates in Euclidean Space, Tangent Vectors
39(4)
2.3 Coordinate Transformation. Co-, Contra- and Mixed Variant Components
43(2)
2.4 Gradient, Covariant and Contravariant Derivatives
45(7)
2.5 Christoffel Symbols, Representation of the Covariant Derivative
52(3)
2.6 Applications in Three-Dimensional Space: Divergence and Curl
55(15)
Exercises
70(3)
3 Curves and Surfaces in Three-Dimensional Euclidean Space
73(26)
3.1 Curves in Three-Dimensional Euclidean Space
73(7)
3.2 Surfaces in Three-Dimensional Euclidean Space
80(7)
3.3 Application to Shell Theory
87(9)
Exercises
96(3)
4 Eigenvalue Problem and Spectral Decomposition of Second-Order Tensors
99(24)
4.1 Complexification
99(2)
4.2 Eigenvalue Problem, Eigenvalues and Eigenvectors
101(3)
4.3 Characteristic Polynomial
104(2)
4.4 Spectral Decomposition and Eigenprojections
106(5)
4.5 Spectral Decomposition of Symmetric Second-Order Tensors
111(3)
4.6 Spectral Decomposition of Orthogonal and Skew-Symmetric Second-Order Tensors
114(4)
4.7 Cayley-Hamilton Theorem
118(1)
Exercises
119(4)
5 Fourth-Order Tensors
123(14)
5.1 Fourth-Order Tensors as a Linear Mapping
123(1)
5.2 Tensor Products, Representation of Fourth-Order Tensors with Respect to a Basis
124(2)
5.3 Special Operations with Fourth-Order Tensors
126(4)
5.4 Super-Symmetric Fourth-Order Tensors
130(2)
5.5 Special Fourth-Order Tensors
132(2)
Exercises
134(3)
6 Analysis of Tensor Functions
137(36)
6.1 Scalar-Valued Isotropic Tensor Functions
137(5)
6.2 Scalar-Valued Anisotropic Tensor Functions
142(3)
6.3 Derivatives of Scalar-Valued Tensor Functions
145(10)
6.4 Tensor-Valued Isotropic and Anisotropic Tensor Functions
155(6)
6.5 Derivatives of Tensor-Valued Tensor Functions
161(6)
6.6 Generalized Rivlin's Identities
167(2)
Exercises
169(4)
7 Analytic Tensor Functions
173(22)
7.1 Introduction
173(4)
7.2 Closed-Form Representation for Analytic Tensor Functions and Their Derivatives
177(3)
7.3 Special Case: Diagonalizable Tensor Functions
180(3)
7.4 Special Case: Three-Dimensional Space
183(6)
7.5 Recurrent Calculation of Tensor Power Series and Their Derivatives
189(3)
Exercises
192(3)
8 Applications to Continuum Mechanics
195(22)
8.1 Deformation of a Line, Area and Volume Element
195(2)
8.2 Polar Decomposition of the Deformation Gradient
197(1)
8.3 Basis-Free Representations for the Stretch and Rotation Tensor
198(3)
8.4 The Derivative of the Stretch and Rotation Tensor with Respect to the Deformation Gradient
201(3)
8.5 Time Rate of Generalized Strains
204(3)
8.6 Stress Conjugate to a Generalized Strain
207(2)
8.7 Finite Plasticity Based on the Additive Decomposition of Generalized Strains
209(5)
Exercises
214(3)
9 Solutions
217(74)
9.1 Exercises of Chap. 1
217(14)
9.2 Exercises of Chap. 2
231(16)
9.3 Exercises of Chap. 3
247(7)
9.4 Exercises of Chap. 4
254(12)
9.5 Exercises of Chap. 5
266(6)
9.6 Exercises of Chap. 6
272(11)
9.7 Exercises of Chap. 7
283(6)
9.8 Exercises of Chap. 8
289(2)
References 291(4)
Index 295
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.
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