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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: 25-Mar-2015
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319163420
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: 25-Mar-2015
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319163420
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This is the fourth and revised edition of a well-received book that aims at bridging the gap between the engineering course of tensor algebra on the one side and the mathematical course of classical linear algebra on the other side. In accordance with the contemporary way of scientific publications, 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. 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 monograph addresses graduate students as well as scientists working in this field. 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(5)
1.6 Second-Order Tensor as a Linear Mapping
13(5)
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(5)
2 Vector and Tensor Analysis in Euclidean Space
37(32)
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(6)
2.5 Christoffel Symbols, Representation of the Covariant Derivative
51(3)
2.6 Applications in Three-Dimensional Space: Divergence and Curl
54(15)
3 Curves and Surfaces in Three-Dimensional Euclidean Space
69(28)
3.1 Curves in Three-Dimensional Euclidean Space
69(7)
3.2 Surfaces in Three-Dimensional Euclidean Space
76(8)
3.3 Application to Shell Theory
84(13)
4 Eigenvalue Problem and Spectral Decomposition of Second-Order Tensors
97(24)
4.1 Complexification
97(2)
4.2 Eigenvalue Problem, Eigenvalues and Eigenvectors
99(3)
4.3 Characteristic Polynomial
102(2)
4.4 Spectral Decomposition and Eigenprojections
104(5)
4.5 Spectral Decomposition of Symmetric Second-Order Tensors
109(3)
4.6 Spectral Decomposition of Orthogonal and Skew-Symmetric Second-Order Tensors
112(4)
4.7 Cayley-Hamilton Theorem
116(5)
5 Fourth-Order Tensors
121(14)
5.1 Fourth-Order Tensors as a Linear Mapping
121(1)
5.2 Tensor Products, Representation of Fourth-Order Tensors with Respect to a Basis
122(3)
5.3 Special Operations with Fourth-Order Tensors
125(3)
5.4 Super-Symmetric Fourth-Order Tensors
128(2)
5.5 Special Fourth-Order Tensors
130(5)
6 Analysis of Tensor Functions
135(34)
6.1 Scalar-Valued Isotropic Tensor Functions
135(4)
6.2 Scalar-Valued Anisotropic Tensor Functions
139(3)
6.3 Derivatives of Scalar-Valued Tensor Functions
142(10)
6.4 Tensor-Valued Isotropic and Anisotropic Tensor Functions
152(7)
6.5 Derivatives of Tensor-Valued Tensor Functions
159(5)
6.6 Generalized Rivlin's Identities
164(5)
7 Analytic Tensor Functions
169(22)
7.1 Introduction
169(4)
7.2 Closed-Form Representation for Analytic Tensor Functions and Their Derivatives
173(3)
7.3 Special Case: Diagonalizable Tensor Functions
176(3)
7.4 Special Case: Three-Dimensional Space
179(6)
7.5 Recurrent Calculation of Tensor Power Series and Their Derivatives
185(6)
8 Applications to Continuum Mechanics
191(22)
8.1 Deformation of a Line, Area and Volume Element
191(2)
8.2 Polar Decomposition of the Deformation Gradient
193(1)
8.3 Basis-Free Representations for the Stretch and Rotation Tensor
194(3)
8.4 The Derivative of the Stretch and Rotation Tensor with Respect to the Deformation Gradient
197(4)
8.5 Time Rate of Generalized Strains
201(3)
8.6 Stress Conjugate to a Generalized Strain
204(3)
8.7 Finite Plasticity Based on the Additive Decomposition of Generalized Strains
207(6)
9 Solutions
213(68)
9.1 Exercises of Chap. 1
213(13)
9.2 Exercises of Chap. 2
226(12)
9.3 Exercises of Chap. 3
238(8)
9.4 Exercises of Chap. 4
246(10)
9.5 Exercises of Chap. 5
256(6)
9.6 Exercises of Chap. 6
262(12)
9.7 Exercises of Chap. 7
274(5)
9.8 Exercises of Chap. 8
279(2)
References 281(4)
Index 285
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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