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Tensor Analysis With Applications In Mechanics - Victor A. Eremeyev

Tensor Analysis With Applications In Mechanics

Hardcover

Published: 18th May 2010
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The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies. In this case general curvilinear coordinates become necessary. The principal basis of a curvilinear system is constructed as a set of vectors tangent to the coordinate lines. Another basis, called the dual basis, is also constructed in a special manner. The existence of these two bases is responsible for the mysterious covariant and contravariant terminology encountered in tensor discussions.A tensor field is a tensor-valued function of position in space. The use of tensor fields allows us to present physical laws in a clear, compact form. A byproduct is a set of simple and clear rules for the representation of vector differential operators such as gradient, divergence, and Laplacian in curvilinear coordinate systems.This book is a clear, concise, and self-contained treatment of tensors, tensor fields, and their applications. The book contains practically all the material on tensors needed for applications. It shows how this material is applied in mechanics, covering the foundations of the linear theories of elasticity and elastic shells.The main results are all presented in the first four chapters. The remainder of the book shows how one can apply these results to differential geometry and the study of various types of objects in continuum mechanics such as elastic bodies, plates, and shells. Each chapter of this new edition is supplied with exercises and problems - most with solutions, hints, or answers to help the reader progress. An extended appendix serves as a handbook-style summary of all important formulas contained in the book.

Forewordp. v
Prefacep. vii
Tensor Analysisp. 1
Preliminariesp. 3
The Vector Concept Revisitedp. 3
A First Look at Tensorsp. 4
Assumed Backgroundp. 5
More on the Notion of a Vectorp. 7
Problemsp. 9
Transformations and Vectorsp. 11
Change of Basisp. 11
Dual Basesp. 12
Transformation to the Reciprocal Framep. 17
Transformation Between General Framesp. 18
Covariant and Contravariant Componentsp. 21
The Cross Product in Index Notationp. 22
Norms on the Space of Vectorsp. 24
Closing Remarksp. 27
Problemsp. 27
Tensorsp. 29
Dyadic Quantities and Tensorsp. 29
Tensors From an Operator Viewpointp. 30
Dyadic Components Under Transformationp. 34
More Dyadic Operationsp. 36
Properties of Second-Order Tensorsp. 40
Eigenvalues and Eigenvectors of a Second-Order Symmetric Tensorp. 44
The Cayley-Hamilton Theoremp. 48
Other Properties of Second-Order Tensorsp. 49
Extending the Dyad Ideap. 56
Tensors of the Fourth and Higher Ordersp. 58
Functions of Tensorial Argumentsp. 60
Norms for Tensors, and Some Spacesp. 66
Differentiation of Tensorial Functionsp. 70
Problemsp. 77
Tensor Fieldsp. 85
Vector Fieldsp. 85
Differentials and the Nabla Operatorp. 94
Differentiation of a Vector Functionp. 98
Derivatives of the Frame Vectorsp. 99
Christoffel Coefficients and their Propertiesp. 100
Covariant Differentiationp. 105
Covariant Derivative of a Second-Order Tensorp. 106
Differential Operationsp. 108
Orthogonal Coordinate Systemsp. 113
Some Formulas of Integrationp. 117
Problemsp. 119
Elements of Differential Geometryp. 125
Elementary Facts from the Theory of Curvesp. 126
The Torsion of a Curvep. 132
Frenet-Serret Equationsp. 135
Elements of the Theory of Surfacesp. 137
The Second Fundamental Form of a Surfacep. 148
Derivation Formulasp. 153
Implicit Representation of a Curve; Contact of Curvesp. 156
Osculating Paraboloidp. 162
The Principal Curvatures of a Surfacep. 164
Surfaces of Revolutionp. 168
Natural Equations of a Curvep. 170
A Word About Rigorp. 173
Conclusionp. 175
Problemsp. 175
Applications in Mechanicsp. 179
Linear Elasticityp. 181
Stress Tensorp. 181
Strain Tensorp. 190
Equation of Motionp. 193
Hooke's Lawp. 194
Equilibrium Equations in Displacementsp. 200
Boundary Conditions and Boundary Value Problemsp. 202
Equilibrium Equations in Stressesp. 203
Uniqueness of Solution for the Boundary Value Problems of Elasticityp. 205
Betti's Reciprocity Theoremp. 206
Minimum Total Energy Principlep. 208
Rita's Methodp. 216
Rayleigh's Variational Principlep. 221
Plane Wavesp. 227
Plane Problems of Elasticityp. 230
Problemsp. 232
Linear Elastic Shellsp. 237
Some Useful Formulas of Surface Theoryp. 239
Kinematics in a Neighborhood of ¿p. 242
Shell Equilibrium Equationsp. 244
Shell Deformation and Strains; Kirchhoff's Hypothesesp. 249
Shell Energyp. 256
Boundary Conditionsp. 259
A Few Remarks on the Kirchhoff-Love Theoryp. 261
Plate Theoryp. 263
On Non-Classical Theories of Plates and Shellsp. 277
Formularyp. 287
Hints and Answersp. 315
Bibliographyp. 355
Indexp. 359
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789814313124
ISBN-10: 9814313122
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 380
Published: 18th May 2010
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 22.86 x 15.49  x 2.54
Weight (kg): 0.68