AN INTRODUCTION TO CONTINUUM MECHANICS

SOLID MECHANICS AND ITS APPLICATIONS Volume 22

Series Editor: G.M.L. GLADWELL Solid Mechanics Division, Faculty of Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3G 1

Aims and Scope of the Series

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative research­ers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids.

The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design.

The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

An Introduction to Continuum Mechanics - after Truesdell and Noll by

DONALD R. SMITH University of California - San Diego, Department of Mathematics, La Jolla, U.S.A.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library ofCongress Cataloging-in-Publication Data

Smlth. Dona ld R. An introductlon to continuum mechanlcs ! by Donald R. Smlth.

p. cm. -- <S011d mechanlCS and ltS appl1catlons ; v. 22) "Adapted prlnclpal ly from the wrltlngs of C. Truesdel 1." Inc 1 udes index.

1. Contlnuum mechanlcs. 1. Truesdell. C. <Cllfford). 1919-II. Title. III. Series. QC155.7.S58 1993 531--dc20

ISBN 978-90-481-4314-6

Printed on acid-free paper

AII Rights Reserved © 1993 by Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1993 Softcover reprint of the hardcover 1 st edition 1993

93-26846

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

ISBN 978-90-481-4314-6 ISBN 978-94-017-0713-8 (eBook) DOI 10.1007/978-94-017-0713-8

Dedicated to

C. TRUESDELL and W. NOLL

in gratitude for

The Non-Linear Field Theories of Mechanics (1965, 1992).

CONTENTS

Page

Preface

Acknowledgments ................................................... .

Chapter 0: Preliminary Results

0.1 Tensors on Euclidean Spaces .................................. . 0.2 Polar Decomposition ......................................... . 0.3 Isotropic FUnctions ........................................... . 0.4 Gradients of Principal Invariants ............................. . 0.5 Chain Rules of Differentiation ................................ . 0.6 Gradient of a Field .......................................... . 0.7 Divergence Theorem 0.8 Laplacian of a Field 0.9 Coordinate Systems

Chapter 1: Framings

1.1 Framings, or Observers ....................................... . 1.2 Changes of Frame ............................................. . 1.3 Frame Indifference ........................................... . 1.4 Practical Considerations

Chapter 2: Bodies and Motions

2.1 Motion of a Body ............................................ . 2.2 Velocity and Acceleration .................................... . 2.3 Mass and Momentum ........................................ . 2.4 Forces and Moments 2.5 Euler's Laws of Motion

Chapter 9: Kinematics

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3 17 24 32 38 41 45 53 55

69 72 78 81

87 88 91 95 98

3.1 Local Deformation Tensor ..................................... 107 3.2 Referential and Spatial Fields . . .. .. . . . .. . .. . . .. .. . . .. .. . . . .. . 117 3.3 Continuity Equation .. . . . .. .. .. . . .. .. .. .. . .. .. . . .. .. . .. . . . .. . 125 3.4 Change of Surface Integration Variable . .............. ....... 130 3.5 Rigid Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

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Chapter 4: The Cauchy Stress Tensor

4.1 Cauchy's Postulate ......................................... . 4.2 Existence of the Stress Tensor ............................... . 4.3 Cauchy's Equations of Motion ............................... . 4.4 Symmetry of the Stress Tensor .............................. . 4.5 Special Stress Fields ........................................ . 4.6 Stress Power ................................................ .

Chapter 5: Examples on Stress Constitutive Relations 5.1 Linearly Viscous Fluid ...................................... . 5.2 Elastic Fluid ............................................... . 5.3 Navier/Stokes Fluid ......................................... . 5.4 Ideal Fluid ................................................. . 5.5 Elastic Material 5.6 Viscous Material 5.7 Incompressible Linear Viscoelastic Material

Chapter 6: Noll's Simple Material

145 150 153 156 158 161

165 166 167 168 170 174 175

6.1 Noll's Constitutive Axioms and Simple Material ............. 179 6.2 Reduction by Material Indifference . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.3 Reduction by Polar Decomposition .. ..... ..... ..... .. ... .... . 194 6.4 Universal Homogeneous Deformations ....................... 198 6.5 Boundary Value Problems and Variational Principles . . . . . . . . 203

Chapter 7: Internally Constrained Materials 7.1 Determinism for Constrained Materials ..................... . 7.2 Simple Internal Constraints ................................. . 7.3 Simply Constrained Simple Materials ........................ .

Chapter 8: Material Classification from Symmetry 8.1 The Symmetry Group ...................................... . 8.2 Isotropic Materials .......................................... . 8.3 Solids ...................................................... . 8.4 Fluids 8.5 Exact Motions

Chapter 9: Canonical Stress Functions for Isotropic Materials 9.1 Isotropic Elastic Material ....................... , ........... . 9.2 Incompressible Isotropic Elastic Material .................... . 9.3 Reiner/Rivlin Fluid ......................................... .

215 217 220

227 234 242 245 249

255 262 263

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Chapter 10: Classical Infinitesimal Theory of Elasticity 10.1 Infinitesimal Deformations ................... .............. . 271 10.2 Linearized Constitutive Functions ...... .............. ...... 275 10.3 Classical Linearly Elastic Isotropic Material . . . . . . . . . . . . . . . . 281

Chapter 11: Shear of an Isotropic Elastic Rectangular Block 11.1 Kinematics .............. .............. ..................... 287 11.2 Infinitesimal Classical Theory .............................. 290 11.3 Elastic Simple Material . . .. . . . .. . . . .. . . . . . . . .. . . . .. .. .. . . .. . 292 11.4 Incompressible Elastic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Chapter 12: Torsion of an Isotropic Elastic Circular Cylinder 12.1 Kinematics . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 12.2 Elastic Isotropic Simple Material ............................ 312 12.3 Elastic Isotropic Incompressible Material ... ................ 316 12.4 Other Static Deformations . .. . . . . .. . . .. . . . . . . .. .. .. . .. . . . . . 320

References

Quotation References

Name Index

Subject Index

331

337

339

341

Donald R. Smith

PREFACE

In 1965 appeared the monumental work The Non-Linear Field Theories of Mechanics in which C. TRUESDELL and W. NOLL set forth the remarkable reorganization and further development and expansion of classical continuum mechanics by NOLL and his associates in the flowstream of NEWTON, JAMES BERNOULLI, EULER, CAUCHY and others. This mighty book by TRUESDELL & NOLL, reissued in a second edition in 1992, was my introduction to the rational mechanics of continuous media and I am grateful to the authors for it.

The present text has evolved from a short course of lectures I have given about every other year since 1975 at the University of California at San Diego and at the Technical University of Munich. The duration of the course is typically only a quarter or a semester, and the goal is to provide a brief introduction to rational continuum mechanics in a form that is suitably expressed and appropriate for students of engineering and science. Good introductory texts already exist such as the texts by TRUESDELL (1977,1991), WANG (1979), GURTIN (1981), and BOWEN (1989), but the field has assumed such a vast proportion that no single brief introductory text can provide a comprehensive survey of it all. The selection of material in such a text must be conditioned strongly by the interests and tastes of the author.

In his initial work NOLL focused on the simplest case of the (pure) me­chanics of nonpolar bodies for which all torques are moments of forces, without internal structure. NOLL's ideas have since been generalized by NOLL and oth­ers to the case of mixtures, to structured and polar materials which exhibit body couples and couple stresses in addition to moments of forces, and to thermomechanics including the theory of energy and heat. The present book is an introductory text on continuum mechanics for senior undergraduate and beginning graduate students of mathematics, science and engineering. The

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presentation is tightly focused on the simplest case of the classical mechanics of nonpolar materials, leaving aside the effects of internal structure, tempera­ture and electromagnetism and excluding other mathematical models such as statistical mechanics, relativistic mechanics and quantum mechanics.

Within the limitations of the simplest mechanical theory I have tried to provide a development that is largely self-contained. An introduction to tensor algebra and tensor calculus on a Euclidean space is included in Chapter 0 along with a discussion of general curvilinear coordinate systems. The change of integration variable formula for volume integrals is described in Section 3.3, and a discussion is given in Section 3.4 of the change of integration variable for surface integrals which is later used in Section 4.3 in introducing the PIOLA/KIRCHHOFF stress tensor. The potency of the theory along with the usefulness of general coordinates are illustrated by the material in Chapter 12 on the torsion of an isotropic elastic cylinder, culminating in an explanation of the fact that an incompressible isotropic elastic cylinder will lengthen in proportion to the square of the twist when twisted severely. This is the Poynting effect in its classic form and is an example of various results that cannot be explained by the corresponding classical infinitesimal theories.

Some topics are omitted altogether and others are given only brief coverage. For example there is only a brief mention of NOLL's important theory of forces. There is a discussion of NOLL's classification of materials based on material symmetry and leading to definitions of solid and fluid, but the subsequent extensive theories of fluids and solids are largely left aside here. Yet the book contains points of contact with these and other omitted topics along with references to the literature, so an instructor can easily include such omitted topics in a course based on this book. Though the book is primarily an introduction to continuum mechanics, I believe that the lure and attraction inherent in the subject may also recommend the book as a vehicle by which the student can obtain a broader appreciation of certain important methods and results from classical and modern analysis.

Del Mar, California Donald R. Smith

Acknowledgments

This book is my presentation of the work of many people, and it is a pleasure for me to acknowledge their work here. Many authors have been instrumental in the development of continuum mechanics, including ISAAC NEWTON (1642-1727), JAMES BERNOULLI (1654-1705), LEONHARD EULER (1707-1783), AUGUSTIN CAUCHY (1789-1857) and WALTER NOLL (1925- ). Throughout the book I have acknowledged the authors of the work discussed here, as far as I know of them. It is almost certain that I have, out of ignorance, slighted some authors who should have been mentioned or mentioned more prominently. I ask their forgiveness.

My general dependence on other authors, even to the limited extent to which I am aware of it, is too vast for complete citation here. I mention here only WALTER NOLL and CLIFFORD TRUESDELL who, combined, are responsible for more than a third of the references listed at the end of this book. TRUESDELL & NOLL's The Non-Linear Field Theories of Mechanics (1965) was my introduction to the rational mechanics of continuous media. The present book could not have been written except for the work and writings of both NOLL and TRUESDELL.

Parts of this book were written while I was visiting the Technical Uni­versity of Munich; I thank ROLAND BULIRSCH for making that visit possible and for encouraging me to give a course of lectures in Munich on the sub­ject of this book. I thank CAROLYN GEIMAN SMITH for her encouragement, and thank the Department of Mathematics, University of California at San Diego, for providing an environment in which this work could be pursued and completed. I thank SIA NEMET-NASSER, GILBERT HEGEMIER, JOSEPH GODDARD and ANNE HOGER for encouraging their graduate students of en­gineering over a period of years to take various of my courses dealing with the material of this book and also for allowing me in 1992 to teach this material in a basic graduate course in the Department of Applied Mechanics and Engi­neering Sciences at the School of Engineering, University of California at San Diego.

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