Tensor Analysis and Continuum Mechanics

603 Pages·2002·13.191 MB·English

by  Yves R. Talpaert

Tensor Analysis and Continuum Mechanics
Tensor Analysis and
Continuum Mechanics
by
Yves R. Talpaert
Faculties of Science and Schools of Engineering at
Algiers University, Algeria;
Brussels University, Belgium;
Bujumbura University, Burundi;
Libreville University, Gabon;
Lome University, Togo;
Lubumbashi University, Zaire and
Ouagadougou University, Burkina Faso
Springer-Science+Business Media, B.Y.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6190-4 ISBN 978-94-015-9988-7 (eBook)
DOI 10.1007/978-94-015-9988-7
Printed on acid-free paper
All Rights Reserved
© 2002 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2002.
Softcover reprint of the hardcover 1s t edition 2002
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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CONTENTS
PREFACE ........................................................... xv
Chapter 1. TENSORS 1
1. FIRST STEPS WITH TENSORS ......................... .
1.1 Multilinear forms .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Linear mapping .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Multilinear form ........................................... 2
1.2 Dual space, vectors and covectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3
Expression of a covector .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Einstein summation convention. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Change of basis and cobasis '" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Tensors and tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. to
Tensor product of multilinear forms. . . . . . . . . . . . . . . . . . . . . . . . . . to
(?) .........................................
Tensor of type II
Tensor of type (~) ......................................... 12
(g) .........................................
Tensor of type 14
m. ........................................
Tensor of type 16
Tensor of type (:) ......................................... 18
(j,) ........................................
Tensor of type 20
Symmetric and antisymmetric tensors 22
2. OPERATIONS ON TENSORS ........ . . . . . . . . . . . .. .. . . . . . 25
2.1 Tensor algebra ........................................... 25
Addition of tensors ....................................... , 25
Multiplication of a tensor by a scalar ..... . . . . . . . . . . . . . . . . . . . .. 25
Tensor multiplication ...................................... 26
2.2 Contraction and tensor criteria ..... . . . . . . . . . . . . . . . . . . . . . . . .. 27
Contraction ............................................... 27
Tensor criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
vii
viii Contents
3. EUCLIDEAN VECTOR SPACE ............................ 33
3.1 Pre-Euclidean vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33
Scalar multiplication and pre-Euclidean space ................... 33
Fundamental tensor .. , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34
3.2 Canonical isomorphism and conjugate tensor . . . . . . . . . . . . . . . .. 35
Canonical isomorphism ..................................... 35
Conjugate tensor and reciprocal basis .... . . . . . . . . . . . . . . . . . . . . .. 37
Covariant and contravariant representations of vectors ............ 40
Representation of tensors of order 2 and contracted products ....... 42
3.3 Euclidean vector spaces ............................... . . . . . 45
4. EXTERIOR ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
4.1 p-forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
Definition of a p-form ...................................... 49
Exterior product of I-forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50
Expression of a p-form ..................................... 52
Exterior product ofp -forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56
Exterior algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57
4.2 q-vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59
5. POINT SPACES ......................................... 63
5.1 Point space and natural frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63
Point space ............................................... 63
Coordinate system and frame of reference ...................... 64
Natural frame ............................................. 66
5.2 Tensor fields and metric element .. . . . . . . . . . . . . . . . . . . . . . . . . .. 69
Transformations of curvilinear coordinates ..................... 69
Tensor fields .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73
Metric element .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74
5.3 Christoffel symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
Definition of Christoffel symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
Ricci identities and Christoffel formulae ........................ 78
5.4 Absolute differential, Covariant derivative, Geodesic .. . . . . . . . . 80
Absolute differential of a vector, covariant derivatives ............ 80
Absolute differential of a tensor, covariant derivatives . . . . . . . . . . .. 83
Geodesic and Euler's equations ..................... . . . . . . . . 85
Absolute derivative of a vector (along a curve) .................. 86
5.5 Volume form and adjoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Volume form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Adjoint. .. .. . .. .... . .. .. . . ... . . .. .. . .. . ... . .. ..... .. .. . 91
5.6 Differential operators .................................... 92
Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Divergence .............................................. 99
Curl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 101
Contents ix
Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 103
EXERCISES .................................................. 106
Chapter 2 LAGRANGIAN AND EULERIAN DESCRIPTIONS 147
1. LAGRANGIAN DESCRIPTION .......................... 147
1.1 Configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 147
1.2 Deformation and Lagrangian Description ................... 148
1.3 Flow and hypotheses of continuity .... . . . . . . . . . . . . . . . . . . . . . 152
1.4 Trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 153
1.5 Streakline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 154
1.6 Velocity and acceleration of a particle . . . . . . . . . . . . . . . . . . . . . .. 155
1.7 Abstract configuration .................................... 156
2. EULERIAN DESCRIPTION ... . . . . . . . . . . . . . . . . . . . . . . . . . .. 157
2.1 Definition; Comparison between L-and E-descriptions ..... . .. 157
2.2 Trajectory and velocity ................... . . . . . . . . . . . . . . .. 158
2.3 Streamline .............................................. 161
2.4 Steady motion ........................................... 163
EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165
Chapter 3 DEFORMATIONS 171
1. HOMOGENEOUS TRANSFORMATION .......... . . . . . . .. 172
1.1 Definition of homogeneous transformations ...... . . . . . . . . . .. 172
1.2 Convective transport .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Convective transport of a vector ............................ 174
Convective transport of a volume . . . . . . . . . . . . . . . . . . . . . . . . . . .. 174
Simple shear .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 176
1.3 Cauchy-Green deformation tensor and stretch ............... 177
(Right) Cauchy-Green deformation tensor. . . . . . . . . . . . . . . . . . . . 177
Stretch ................................................. 180
Shear angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181
x Contents
Principal stretches ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 182
1.4 Finite strain tensor .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 184
1.5 Polar decomposition ..................................... 186
Pure stretch and rotation .................................. 186
Euler-Almansi strain tensor .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 190
1.6 Rigid body transformation ................................ 191
2. TANGENTIAL HOMOGENEOUS TRANSFORMATION. . . . 193
2.1 Deformation gradient .................................... 193
2.2 Homogeneous transformations of elements .................. 196
Transport of vectors, volume deformation, and area deformation ... 196
Stretches ........ ....................................... 19 9
Strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 201
2.3 Displacement and gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 203
Material displacement gradient .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 204
Spatial displacement gradient .... . . . . . . . . . . . . . . . . . . . . . . . . . .. 206
Curvilinear coordinate system .............................. 208
3. INFINITESIMAL TRANSFORMATION . . . . . . . . . . . . . . . . . .. 2 I 0
3.1 Tensor notions relating to infinitesimal transformations ........ 2 I I
3.2 Compatibility conditions .................................. 216
3.3 Rigid body transformation ....... ......................... 220
EXERCISES ................................................... 222
Chapter 4 KINEMATICS OF CONTINUA 263
1. LAGRANGIAN KINEMATICS 263
1.1 Homogeneous transformation motion ...................... 264
1.2 General motion and gradient 266
2. EULERIAN KINEMATICS ............................... 268
2.1 Homogeneous transformation motion ....................... 268
Velocity field ............................................. 268
Material derivative of a vector .... . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Material derivative of a volume .............................. 269
Eulerian rates ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 271
2.2 General motion and velocity gradient . . . . . . . . . . . . . . . . . . . . . .. 274
Velocity gradient tensor and Eulerian rates ......... . . . . . . . . . .. 274