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 1 st edition 2002 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

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

Tensor of type (g) ......................................... 14

Tensor of type m ......................................... 16

Tensor of type (:) ......................................... 18

Tensor of type (j,) ........................................ 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

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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 of p-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

Chapter 2

Chapter 3

Contents ix

Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 103

EXERCISES .................................................. 106

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

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

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Chapter 4

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 ........ ....................................... 1 99 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

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

Chapter 5

Contents xi

Lagrangian and Eulerian strain tensors ... . . . . . . . . . . . . . . . . . . . .. 279 Rate of rotation .......................................... 280 Decomposition of motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 285

2.3 Rigid body motion ....................................... 286

3. MATERIAL DERIVATIVES OF CmCULATION, FLUX, AND VOLUME .............................................. 287

3.1 About the particle derivative ... . . . . . . . . . . . . . . . . . . . . . . . . . .. 287 Physical quantity ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Vector field ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 290 Tensor field ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 292

3.2 Material derivative of circulation ........................... 293

3.3 Material derivative of flux .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 296

3.4 Material derivative of volume integral ..... . . . . . . . . . . . . . . . .. 299 Lagrangian and Eulerian approaches ......... . . . . . . . . . . . . . . . .. 299 Proper motion case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 303

EXERCISES .................................................. 305

FUNDAMENTAL LAWS; PRINCIPLE OF VIRTUAL WORK 315

1. CONSERVATION OF MASS AND CONTINUITY EQUATION 315

1.1 Axiom of mass conservation .............................. 315

1.2 Continuity equation ...................................... 316 Continuity equation in the Lagrangian description ........ . . . . . .. 316 Continuity equation in the Eulerian description ..... . . . . . . . . . . .. 317 Mass flow rate ........................................... 319

1.3 Material derivative of integral of mass density . . . . . . . . . . . . . .. 320

1.4 Isochoric motion, steady and irrotational flows .............. 322 Isochoric motion ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 322 Steady flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 326 Steady isochoric flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 327 Irrotational flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 328 Isochoric irrotational flow ....... . . . . . . . . . . . . . . . . . . . . . . . . . .. 329

2. FUNDAMENTAL LAWS OF DYNAMICS ................. 330

2.1 Body forces and surface forces .. . . . . . . . . . . . . . . . . . . . . . . . . . . 330

2.2 Principles of linear momentum and moment of momentum. . . .. 333

2.3 Cauchy's stress tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 337

xii Contents

2.4 Cauchy's stress tensor and principles of dynamics ............ 342 Linear momentum principle and equilibrium equations .. . . . . . . . .. 342 Moment of momentum principle ............................ 344 The generalized Cauchy's theorem. . . . . . . . . . . . . . . . . . . . . . . . .. 346 Poisson's theorem ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 347

3. THEOREM OF KINETIC ENERGY ....................... 348

3.1 Theorem of kinetic energy in the Eulerian description ......... 348

3.2 Theorem of kinetic energy in the Lagrangian description 350

4. STUDY OF STRESSES.................................. 356

4.1 Reciprocity of stresses .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 356

4.2 Principal stresses ......................................... 358

4.3 Stress invariants; deviator .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 364

4.4 Stress quadric of Cauchy and Lame stress ellipsoid 370

4.5 Geometrical constructions and Mohr's circles ................ 374 (Mohr's) stress plane ...................................... 374 Stress vector and plane of Mohr . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 375 Description of Mohr's circles ............................... 378 Particular stresses ......................................... 380

5. PRINCIPLE OF VIRTUAL WORK ........................ 384

5.1 Preliminary recalls ....................................... 384

5.2 Rigid body motion ....................................... 386

5.3 Expressions of virtual power (and virtual work) .............. 389

5.4 Principle of virtual work ................................. , 391

6. THERMOMECHANICS AND BALANCE EQUATIONS .... 393

6.1 Balance equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Proper motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 393 Material domain .......................................... 398 Fixed domain ............................................ 401

6.2 First principle of thermodynamics .......................... 402 Principle ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 402 Balance equations and local forms . . . . . . . . . . . . . . . . . . . . . . . . . .. 404 Potential energy of body forces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 406 Internal energy and balance equation ......................... 407

6.3 Second principle of thermodynamics ........... . . . . . . . . . . .. 409 Principle ............................................... 409 Clausius-Duhem inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 411 Dissipation and reversibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 412

Contents xiii

6.4 Conclusion and constitutive equations ....................... 414

EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 417

Chapter 6 LINEAR ELASTICITY 455

1. ELASTICITY AND TESTS ............................... 455

2. GENERALIZED HOOKE'S LAW IN LINEAR ELASTICITY . 458

2.1 Generalized Hooke's law ............ . . . . . . . . . . . . . . . . . . . .. 458

2.2 Quadratic forms and strain energy function .................. 459

2.3 Isotropic material and Lame coefficients ..................... 463 Constitutive equations ...................................... 463 Young's modulus and Poisson's ratio ......................... 466 Bulk modulus ........................................... 469 Shear modulus ........................................... 471 Hooke's law's expression in a general coordinate system .......... 472 Navier's equations of motion ................................ 475

3. EQUATIONS AND PRINCIPLES IN ELASTOSTATICS ..... 477

3.1 Navier's equation; the Beltrami equations of compatibility. . . . 478

3.2 Principle ofsuperposition .............. . . . . . . . . . . . . . . . . .. 481

3.3 Saint-Venant's principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 484

4. CLASSICAL PROBLEMS .......... .. . . .. . .. . .. .. . .. . . ... 485

4.1 Plane problems .......................................... 485 Plane stress problems ................ . . . . . . . . . . . . . . . . . . . . .. 485 Plane strain problems ...................................... 488

4.2 Classical problems in elastostatics .......................... 491 Uniaxial stresses .......................................... 491 Torsion of a circular cylinder body ............................ 493 Torsion of cylindrical shafts ................................. 496

EXERCISES ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 501

SUMMARY OF FORMULAE ............................................. 541

xiv Contents

BIBLIOGRAPHY ......................................................... 575

GLOSSARY OF SYMBOLS ............................................... 577

INDEX .................................................................... 585

PREFACE

This book is designed for students in engineering, physics and mathematics. The material can be taught from the beginning of the third academic year. It could also be used for self­study, given its pedagogical structure and the numerous solved problems which prepare for modem physics and technology.

One of the original aspects of this work is the development together of the basic theory of tensors and the foundations of continuum mechanics.

Why two books in one?

Firstly, Tensor Analysis provides a thorough introduction of intrinsic mathematical entities, called tensors, which is essential for continuum mechanics. This way of proceeding greatly unifies the various subjects. Only some basic knowledge of linear algebra is necessary to start out on the topic of tensors. The essence of the mathematical foundations is introduced in a practical way.

Tensor developments are often too abstract, since they are either aimed at algebraists only, or too quickly applied to physicists and engineers. Here a good balance has been found which allows these extremes to be brought closer together. Though the exposition of tensor theory forms a subject in itself, it is viewed not only as an autonomous mathematical discipline, but as a preparation for theories of physics and engineering. More specifically, because this part of the work deals with tensors in general coordinates and not solely in Cartesian coordinates, it will greatly help with many different disciplines such as differential geometry, analytical mechanics, continuum mechanics, special relativity, general relativity, cosmology, electromagnetism, quantum mechanics, etc ..

Secondly, the foundations of Continuum Mechanics constitute the most important part of this work. It involves chapters on the Lagrangian and Eulerian descriptions, deformations, kinematics of continua, fundamental laws, the principle of virtual work, and linear elasticity. These chapters lay the groundwork for other or more technical subjects as fluid mechanics, strength of materials, plasticity and viscoelasticity, thermoelasticity and thermodynamics, nonlinear continuum mechanics, finite element methods in continuum mechanics, etc ..

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xvi Preface

The reader will quickly discover the importance of these chapters and, given their controlled and logical progression as well as their role of introducing the above mentioned disciplines, will eagerly take up the challenge.

Unlike other authors who denote tensors by (several) lines above and below bold letters, I simply represent both general tensors and vectors by bold letters. This simplification is possible since the reader gradually gains experience in dealing with tensors and because tensors of orders higher than 2 are not frequently used in continuum mechanics. In addition, the difference in level of operation dot '.' or '.' has no mathematical meaning, it has only a pedagogical value, namely: bringing down dots specifies that the corresponding operation results are real numbers. This stipulated simplification leads the reader to think carefully about operations between tensors and about the types of different tensors. Of course, scalars are not designated by bold letters. According to usage the vertical brackets completely enclose the elements of matrices, whereas they partly enclose the normal mathematical expressions.

Terms of the continuum mechanics terminology are sometimes translated into French for French speaking readers.

The important propositions and the formulae to be framed are shown by W and~.

The summary of formulae and glossary of symbols should make the assimilation of notions easier.

All the proofs and the 95 solved exercises are described in detail.

Acknowledgements. Many thanks are due to my former students who let me expound a part of the material that resulted in this book.

I wish to express my gratitude to Kluwer Academic Publishers for their cooperation.

I would particularly appreciate it if readers would let me know of any errors or further suggestions.

Any faculty of science or engineering school, interested in my analysis of continuum mechanics and in the way differential geometry and theoretical mechanics have been developed in my previous books, can contact me for a possible collaboration.

Yves R. Talpaert

Y talpaert@aol.com