[Tadmor_E.B,_Miller_R.E.,_Elliott_R.S.]_Continuum_(BookFi.org).pdf

ContinuumMechanics and Thermodynamics

Continuum mechanics and thermodynamics are foundational theories of many fields ofscience and engineering. This book presents a fresh perspective on these important subjects,exploring their fundamentals and connecting them with micro- and nanoscopic theories.

Providing clear, in-depth coverage, the book gives a self-contained treatment of topics di-rectly related to nonlinear materials modeling with an emphasis on the thermo-mechanicalbehavior of solid-state systems. It starts with vectors and tensors, finite deformation kine-matics, the fundamental balance and conservation laws, and classical thermodynamics. Itthen discusses the principles of constitutive theory and examples of constitutive models,presents a foundational treatment of energy principles and stability theory, and concludeswith example closed-form solutions and the essentials of finite elements.

Together with its companion book, Modeling Materials (Cambridge University Press,2011), this work presents the fundamentals of multiscale materials modeling for graduatestudents and researchers in physics, materials science, chemistry, and engineering.

A solutions manual is available at www.cambridge.org/9781107008267, along with alink to the authors website which provides a variety of supplementary material for boththis book and Modeling Materials.

Ellad B. Tadmor is Professor of Aerospace Engineering and Mechanics, University ofMinnesota. His research focuses on multiscale method development and the microscopicfoundations of continuum mechanics.

Ronald E. Miller is Professor of Mechanical and Aerospace Engineering, Carleton University.He has worked in the area of multiscale materials modeling for over 15 years.

Ryan S. Elliott is Associate Professor of Aerospace Engineering and Mechanics,University of Minnesota. An expert in stability of continuum and atomistic systems, hehas received many awards for his work.

ContinuumMechanics andThermodynamics

From Fundamental Concepts toGoverning Equations

ELLAD B. TADMORUniversity of Minnesota, USA

RONALD E. MILLERCarleton University, Canada

RYAN S. ELLIOTTUniversity of Minnesota, USA

CAMBR IDGE UN IVERS I TY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town,Singapore, Sao Paulo, Delhi, Tokyo, Mexico City

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9781107008267

C E. Tadmor, R. Miller and R. Elliott 2012

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2012

Printed in the United Kingdom at the University Press, Cambridge

A catalog record for this publication is available from the British Library

Library of Congress Cataloguing in Publication dataTadmor, Ellad B., 1965

Continuum mechanics and thermodynamics : from fundamental concepts to governing equations /Ellad B. Tadmor, Ronald E. Miller, Ryan S. Elliott.

p. cm.Includes bibliographical references and index.

ISBN 978-1-107-00826-71. Continuum mechanics. 2. Thermodynamics Mathematics. I. Miller, Ronald E. (Ronald Earle)

II. Elliott, Ryan S. III. Title.QA808.2.T33 2012

531 dc23 2011040410

ISBN 978-1-107-00826-7 Hardback

Additional resources for this publication at www.cambridge.org/9781107008267

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.

Contents

Preface page xiAcknowledgments xiiiNotation xvii

1 Introduction 1

Part I Theory 7

2 Scalars, vectors and tensors 92.1 Frames of reference and Newtons laws 92.2 Tensor notation 15

2.2.1 Direct versus indicial notation 162.2.2 Summation and dummy indices 172.2.3 Free indices 182.2.4 Matrix notation 192.2.5 Kronecker delta 192.2.6 Permutation symbol 20

2.3 What is a tensor? 222.3.1 Vector spaces and the inner product and norm 222.3.2 Coordinate systems and their bases 262.3.3 Cross product 292.3.4 Change of basis 312.3.5 Vector component transformation 332.3.6 Generalization to higher-order tensors 342.3.7 Tensor component transformation 36

2.4 Tensor operations 382.4.1 Addition 382.4.2 Magnification 382.4.3 Transpose 392.4.4 Tensor products 392.4.5 Contraction 402.4.6 Tensor basis 44

2.5 Properties of tensors 462.5.1 Orthogonal tensors 462.5.2 Symmetric and antisymmetric tensors 482.5.3 Principal values and directions 482.5.4 CayleyHamilton theorem 51

v

vi Contents

2.5.5 The quadratic form of symmetric second-order tensors 522.5.6 Isotropic tensors 54

2.6 Tensor fields 552.6.1 Partial differentiation of a tensor field 562.6.2 Differential operators in Cartesian coordinates 562.6.3 Differential operators in curvilinear coordinates 602.6.4 Divergence theorem 64

Exercises 66

3 Kinematics of deformation 713.1 The continuum particle 713.2 The deformation mapping 723.3 Material and spatial field descriptions 74

3.3.1 Material and spatial tensor fields 753.3.2 Differentiation with respect to position 76

3.4 Description of local deformation 773.4.1 Deformation gradient 773.4.2 Volume changes 793.4.3 Area changes 803.4.4 Pull-back and push-forward operations 823.4.5 Polar decomposition theorem 833.4.6 Deformation measures and their physical significance 873.4.7 Spatial strain tensor 90

3.5 Linearized kinematics 913.6 Kinematic rates 93

3.6.1 Material time derivative 933.6.2 Rate of change of local deformation measures 963.6.3 Reynolds transport theorem 100

Exercises 101

4 Mechanical conservation and balance laws 1064.1 Conservation of mass 106

4.1.1 Reynolds transport theorem for extensive properties 1094.2 Balance of linear momentum 110

4.2.1 Newtons second law for a system of particles 1104.2.2 Balance of linear momentum for a continuum system 1114.2.3 Cauchys stress principle 1134.2.4 Cauchy stress tensor 1154.2.5 An alternative (tensorial) derivation of the stress tensor 1174.2.6 Stress decomposition 1194.2.7 Local form of the balance of linear momentum 119

4.3 Balance of angular momentum 1204.4 Material form of the momentum balance equations 122

4.4.1 Material form of the balance of linear momentum 1224.4.2 Material form of the balance of angular momentum 1244.4.3 Second PiolaKirchhoff stress 125

Exercises 127

vii Contents

5 Thermodynamics 1295.1 Macroscopic observables, thermodynamic equilibrium

and state variables 1305.1.1 Macroscopically observable quantities 1315.1.2 Thermodynamic equilibrium 1335.1.3 State variables 1335.1.4 Independent state variables and equations of state 136

5.2 Thermal equilibrium and the zeroth law of thermodynamics 1375.2.1 Thermal equilibrium 1375.2.2 Empirical temperature scales 138

5.3 Energy and the first law of thermodynamics 1395.3.1 First law of thermodynamics 1395.3.2 Internal energy of an ideal gas 143

5.4 Thermodynamic processes 1475.4.1 General thermodynamic processes 1475.4.2 Quasistatic processes 147

5.5 The second law of thermodynamics and the direction of time 1485.5.1 Entropy 1495.5.2 The second law of thermodynamics 1505.5.3 Stability conditions associated with the second law 1525.5.4 Thermal equilibrium from an entropy perspective 1535.5.5 Internal energy and entropy as fundamental thermodynamic

relations 1565.5.6 Entropy form of the first law 1595.5.7 Reversible and irreversible processes 161

5.6 Continuum thermodynamics 1685.6.1 Local form of the first law (energy equation) 1705.6.2 Local form of the second law (ClausiusDuhem inequality) 175

Exercises 177

6 Constitutive relations 1806.1 Constraints on constitutive relations 1816.2 Local action and the second law of thermodynamics 184

6.2.1 Specific internal energy constitutive relation 1846.2.2 ColemanNoll procedure 1866.2.3 Onsager reciprocal relations 1906.2.4 Constitutive relations for alternative stress variables 1916.2.5 Thermodynamic potentials and connection with experiments 192

6.3 Material frame-indifference 1956.3.1 Transformation between frames of reference 1966.3.2 Objective tensors 2006.3.3 Principle of material frame-indifference 2026.3.4 Constraints on constitutive relations due to material frame-indifference 2036.3.5 Reduced constitutive relations 2076.3.6 Continuum field equations and material frame-indifference 2136.3.7 Controversy regarding the principle of material frame-indifference 213

6.4 Material symmetry 2156.4.1 Simple fluids 2186.4.2 Isotropic solids 221

viii Contents

6.5 Linearized constitutive relations for anisotropic hyperelastic solids 2256.5.1 Generalized Hookes law and the elastic constants 229

6.6 Limitations of continuum constitutive relations 236Exercises 237

7 Boundary-value problems, energy principles and stability 2427.1 Initial boundary-value problems 242

7.1.1 Problems in the spatial description 2437.1.2 Problems in the material description 245

7.2 Equilibrium and the principle of stationary potential energy (PSPE) 2477.3 Stability of equilibrium configurations 249

7.3.1 Definition of a stable equilibrium configuration 2507.3.2 Lyapunovs indirect method and the linearized equations

of motion 2517.3.3 Lyapunovs direct method and the principle of minimum

potential energy (PMPE) 255Exercises 259

Part II Solutions 263

8 Universal equilibrium solutions 2658.1 Universal equilibrium solutions for homogeneous simple

elastic bodies 2658.2 Universal solutions for isotropic and incompressible hyperelastic

materials 2688.2.1 Family 0: homogeneous deformations 2698.2.2 Family 1: bending, stretching and shearing of a rectangular block 2708.2.3 Family 2: straightening, stretching and shearing of a sector

of a hollow cylinder 2708.2.4 Family 3: inflation, bending, torsion, extension and shearing

of an annular wedge 2708.2.5 Family 4: inflation or eversion of a sector of a spherical shell 2748.2.6 Family 5: inflation, bending, extension and azimuthal shearing

of an annular wedge 2758.3 Summary and the need for numerical solutions 275Exercises 275

9 Numerical solutions: the finite element method 2779.1 Discretization and interpolation 2779.2 Energy minimization 281

9.2.1 Solving nonlinear problems: initial guesses 2829.2.2 The generic nonlinear minimization algorithm 2839.2.3 The steepest descent method 2849.2.4 Line minimization 2859.2.5 The NewtonRaphson (NR) method 2879.2.6 Quasi-Newton methods 2889.2.7 The finite element tangent stiffness matrix 289

ix Contents

9.3 Elements and shape functions 2899.3.1 Element mapping and the isoparametric formulation 2939.3.2 Gauss quadrature 2989.3.3 Practical issues of implementation 3019.3.4 Stiffness matrix assembly 3079.3.5 Boundary conditions 3099.3.6 The patch test 3119.3.7 The linear elastic limit with small and finite strains 313

Exercises 315

10 Approximate solutions: reduction to the engineering theories 31710.1 Mass transfer theory 31910.2 Heat transfer theory 32010.3 Fluid mechanics theory 32110.4 Elasticity theory 322Afterword 323

11 Further reading 32411.1 Books related to Part I on theory 32411.2 Books related to Part II on solutions 326

Appendix A Heuristic microscopic derivation of the total energy 327

Appendix B Summary of key continuummechanics equations 329

References 334Index 343

Preface

This book on Continuum Mechanics and Thermodynamics (CMT) (together with the com-panion book, by Tadmor and Miller, on Modeling Materials (MM) [TM11]) is a compre-hensive framework for understanding modern attempts at modeling materials phenomenafrom first principles. This is a challenging problem because material behavior is dictated bymany different processes, occurring on vastly different length and time scales, that interactin complex ways to give the overall material response. Further, these processes have tradi-tionally been studied by different researchers, from different fields, using different theoriesand tools. For example, the bonding between individual atoms making up a material isstudied by physicists using quantum mechanics, while the macroscopic deformation ofmaterials falls within the domain of engineers who use continuum mechanics. In the enda multiscale modeling approach capable of predicting the behavior of materials at themacroscopic scale but built on the quantum foundations of atomic bonding requires adeep understanding of topics from a broad range of disciplines and the connections betweenthem. These include quantum mechanics, statistical mechanics and materials science, aswell as continuum mechanics and thermodynamics, which are the focus of this book.

Together, continuummechanics and thermodynamics form the fundamental theory lyingat the heart ofmany disciplines in science and engineering. This is a nonlinear theory dealingwith the macroscopic response of material bodies to mechanical and thermal loading.There are many books on continuum mechanics, but we believe that several factors set ourbook apart. First, is our emphasis on fundamental concepts. Rather than just presentingequations, we attempt to explainwhere the equations come from andwhat are the underlyingassumptions. This is important for those seeking to integrate continuum mechanics withina multiscale paradigm, but is also of great value for those who seek to master continuummechanics on its own, and even for experts who wish to reflect further upon the basisof their field and its limitations. To this end, we have adopted a careful expository style,developing the subject in a step-by-step fashion, building up from fundamental ideas andconcepts to more complex principles. We have taken pains to carefully and clearly discussmany of the subtle points of the subject which are often glossed over in other books.

A second difference setting our CMT apart from other books on the subject is the inte-gration of thermodynamics into the discussion of continuum mechanics. Thermodynamicsis a difficult subject which is normally taught using the language of heat engines andCarnot cycles. It is very difficult for most students to see how these concepts are relatedto continuum mechanics. Yet thermodynamics plays a vital role at the foundation of con-tinuum mechanics. In fact, we think of continuum mechanics and thermodynamics as asingle unified subject. It is simply impossible to discuss thermomechanical processes in

xi

xii Preface

materials without including thermodynamics. In addition, thermodynamics introduces keyconstraints on allowable forms of constitutive relations, the fundamental equations describ-ing material response, that form the gateway to the underlying microscopic structure of thematerial.

The third difference is that we have written CMT with an eye to making it accessible toa broad readership. Without oversimplifying any of the concepts, we endeavor to explaineverything in clear termswith as little jargon as possible.We do not assume prior knowledgeof the subject matter. Thus, a reader from any field with an undergraduate education inengineering or science should be able to follow the presentation. We feel that this isparticularly important as it makes this vital subject accessible to researchers and studentsfrom physics, chemistry and materials science who traditionally have less exposure tocontinuum mechanics.

The philosophy underlying CMT and its form provide it with a dual role. On its own,it is suitable as a first introduction to continuum mechanics and thermodynamics forgraduate students or researchers in science and engineering. Together with MM, it providesa comprehensive and integrated framework for modern predictive materials modeling. Withthis latter goal in mind, CMT is written using a similar style, notation and terminology tothat of MM, making it easy to use the two books together.

Acknowledgments

Aswe explained in the preface, this book is really one part of a two-volume project coveringmany topics in materials modeling beyond continuum mechanics and thermodynamics(CMT). In the following few pages, we choose to express our thanks to everyone involvedin the entire project, whether their contribution directly affected the words on these pagesor only the words in the companion volume (Modeling Materials or MM for short). Wemention this by way of explanation, in case a careful reader is wondering why we thankpeople for helping us with topics that clearly do not appear in the table of contents. Thepeople thanked belowmost certainly helped shape our understanding of materials modelingin general, even if not with respect to CMT specifically.

Our greatest debt goes to our wives, Jennifer, Granda and Sheila, and to our children:Maya, Lea, Persephone and Max. They have suffered more than anyone during the longcourse of this project, as their preoccupied husbands and fathers stole too much time fromso many other things. They need to be thanked for such a long list of reasons that we wouldlikely have to split these two books into three if we were thorough with the details. Thanks,all of you, for your patience and support. We must also thank our own parents Zehev andCiporah, Don and Linda, and Robert and Mary for giving us the impression perhapsmistaken that everybody will appreciate what we have to say as much as they do.

The writing of a book is always a collaborative effort with so many people whose namesdo not appear on the cover. These include students in courses, colleagues in the corridors andoffices of our universities and unlucky friends cornered at conferences. The list of peoplethat offered a little piece of advice here, a correction there or a word of encouragementsomewhere else is indeed too long to include, but there are a few people in particular thatdeserve special mention.

Some colleagues generously did calculations for us, verified results or provided othercontributions from their own work. We thank Quiying Chen at the NRC Institute forAerospace Research in Ottawa for his time in calculating UBER curves with densityfunctional theory. Tsveta Sendova, a postdoctoral fellow at the University of Minnesota(UMN), coded and ran the simulations for the two-dimensional NEB example we present.Another postdoctoral fellow at UMN, Woo Kyun Kim, performed the indentation andthermal expansion simulations used to illustrate the hot-QC method. We thank Yuri Mishin(GeorgeMasonUniversity) for providing figures, and ChristophOrtner (Oxford University)for providing many insights into the problem of full versus sequential minimization ofmultivariate functions, including the example we provide in the MM book. The hot-QCproject has greatly benefited from the work of Laurent Dupuy (SEA Saclay) and FredericLegoll (Ecole Nationale des Ponts et Chaussees). Their help in preparing a journal paper on

xiii

xiv Acknowledgments

the subject has also proven extremely useful in preparing the chapter on dynamic multiscalemethods. Furio Ercolessi must be thanked in general for his fantastic web-based notes onso many important subjects discussed herein, and specifically for providing us with hismolecular dynamics code as a teaching tool to provide with MM.

Other colleagues patiently taught us the many subjects in these books about which weare decidedly not experts. Dong Qian at the University of Cincinnati and Michael Parks atSandia National Laboratories very patiently and repeatedly explained the nuances of variousmultiscale methods to us. Similarly, we would like to thank Catalin Picu at the RensselaerPolytechnic Institute for explaining CACM, and Leo Shilkrot for his frank conversationsabout CADD and the BSM. Noam Bernstein at the Navy Research Laboratories (NRL) wasinvaluable in explainingDFT in away that an engineer could understand, and PeterWatson atCarleton University was instrumental in our eventual understanding of quantummechanics.Roger Fosdick (UMN) discussed, at length, many topics related to continuum mechanicsincluding tensor notation, material frame-indifference, Reynolds transport theorem and theprinciple of action and reaction. He also took the time to read and comment on our take onmaterial frame-indifference.

We are especially indebted to those colleagues that were willing to take the time tocarefully read and comment on drafts of various sections of the books a thankless anddelicate task. James Sethna (Cornell University) and Dionisios Margetis (University ofMaryland) read and commented on the statistical mechanics chapter. Noam Bernstein(NRL) must be thanked more than once, for reading and commenting on both the quantummechanics chapter and the sections on cluster expansions. Nikhil Admal, a graduate studentworking with Ellad at UMN, contributed significantly to our understanding of stress andread and commented on various continuum mechanics topics, Marcel Arndt helped bytranslating an important paper on stress by Walter Noll from German to English andworked with Ellad on developing algorithms for lattice calculations, while Gang Lu atthe California State University (Northridge) set us straight on several points about densityfunctional theory. Other patient readers to whom we say thank you include Mitch LuskinfromUMN (numerical analysis of multiscale methods and quantummechanics), Bill Curtinfrom Brown University (static multiscale methods), Dick James from UMN (restrictedensembles and the definition of stress) and Leonid Berlyand from Pennsylvania StateUniversity (thermodynamics).

There are a great many colleagues who were willing to talk to us at length about varioussubjects in these books. We hope that we did not overstay our welcome in their officestoo often, and that they do not sigh too deeply anymore when they see a message from usin their inbox. Most importantly, we thank them very much for their time. In addition tothose already mentioned above, we thank David Rodney (Institut National Polytechnique deGrenoble), Perry Leo and Tom Shield (UMN), Miles Rubin and Eli Altus (Technion), JoelLebowitz, Sheldon Goldstein and Michael Kiessling (Rutgers)1 and Andy Ruina (Cornell).Wewould also be remiss if we did not take the time to thankArt Voter (Los Alamos National

1 Ellad would particularly like to thank the Rutgers trio for letting him join them on one of their lunches to discussthe foundations of statistical mechanics a topic which is apparently standard lunch fare for them along withthe foundations of quantum mechanics.

xv Acknowledgments

Laboratory), John Moriarty (Lawrence Livermore National Laboratory) and Mike Baskes(Sandia National Laboratories) for many insightful discussions and suggestions of valuablereferences.

There are some things in these books that are so far outside our area of expertise that wehave even had to look beyond the offices of professors and researchers. Elissa Gutterman,an expert in linguistics, provided phonetic pronunciation of French and German names. Asnone of us are experimentalists, our brief foray into pocket watch testing would not havebeen very successful without the help of Steve Truttman and Stan Conley in the structureslaboratories at Carleton University. The story of our cover images involves so many people,it deserves its own paragraph.

As the reader will see in the introduction to both books, we are fond of the symbolicconnection between pocket watches and the topics we discuss herein. There are manybeautiful images of pocket watches out there, but obtaining one of sufficient resolution,and getting permission to use it, is surprisingly difficult. As such, we owe a great debtto Mr. Hans Holzach, a watchmaker and amateur photographer at Beyer ChronometrieAG in Zurich. Not only did he generously agree to let us use his images, he took overthe entire enterprise of retaking the photos when we found out that his images did nothave sufficient resolution! This required Hans to coordinate with many people that wealso thank for helping make the strikingly beautiful cover images possible. These includethe photographer, Dany Schulthess (www.fotos.ch), Mr. Rene Beyer, the owner of BeyerChronometrie AG in Zurich, who compensated the photographer and permitted photos tobe taken at his shop, and also to Dr. Randall E. Morris, the owner of the pocket watch,who escorted it from California to Switzerland (!) in time for the photo shoot. The fact thattotal strangers would go to such lengths in response to an unsolicited e-mail contact is atestament to their kind spirits and, no doubt, to their proud love of the beauty of pocketwatches.

We cannot forget our students.Many continue to teach us things every day just by bringingus their questions and ideas. Others were directly used as guinea pigs with early drafts ofparts of these books.2 Ellad would like to thank his graduate students and postdoctoralfellows over the last five years who have been fighting with this project for attention,specifically Nikhil Admal, Yera Hakobian, Hezi Hizkiahu, Dan Karls, Woo Kyun Kim,Leonid Kucherov, Amit Singh, Tsvetanka Sendova, Valeriu Smiricinschi, Slava Sorkin andSteve Whalen. Ron would likewise like to thank Ishraq Shabib, Behrouz Shiari and DenisSaraev, whose work helped shape his ideas about atomistic modeling. Ryan would liketo thank Kaushik Dayal, Dan Gerbig, Dipta Ghosh, Venkata Suresh Guthikonda, VincentJusuf, Dan Karls, Tsvetanka Sendova, Valeriu Smirichinski and Viacheslav (Slava) Sorkin.Harley Johnson and his 20082009 and 20102011 graduate classes at the University ofIllinois (Urbana-Champaign) who used the books extensively provided great feedback toimprove the manuscripts, as did Bill Curtins class at Brown in 20092010. The 2009and 2010 classes of Rons Microstructure and Properties of Engineering Materials classcaught many initial errors in the chapters on crystal structures and molecular statics and

2 Test subjects were always treated humanely and no students were irreparably harmed during the preparation ofthese books.

xvi Acknowledgments

dynamics. Some students of Ellads Continuum Mechanics course are especially notedfor their significant contributions: Yilmaz Bayazit (2008), Pietro Ferrero (2009), ZhuangHoulong (2008), Jenny Hwang (2009), Karl Johnson (2008), Dan Karls (2008), Minsu Kim(2009), Nathan Nasgovitz (2008), Yintao Song (2008) and Chonglin Zhang (2008).

Of course, we should also thank our own teachers. Course notes from Michael Ortiz,Janet Blume, Jerry Weiner, Nicolas Triantafyllidis and Tom Shield were invaluable tous in preparing our own notes and this book. Thanks also to Ellad and Rons formeradvisors at Brown University, Michael Ortiz and Rob Phillips (both currently at Caltech)and Ryans former advisors Nicolas Triantafyllidis and John A. Shaw at the Universityof Michigan (Nick is currently at the Ecole Polytechnique, France), whose irresistibleenthusiasm, curiosity and encouragement pulled us down this most rewarding of scientificpaths.

Ryan would like to thank the University of Minnesota and the McKnight Foundationwhose McKnight Land-Grant Professorship helped support his effort in writing this book.Further, he would like to sincerely thank Patrick Le Tallec, Nicolas Triantafyllidis, RenataZwiers, Kostas Danas, Charis Iordanou and everyone at the Laboratoire de Mecanique desSolides (LMS), the Ecole Polytechnique, France for their generous support, hosting andfriendship during Ryan and Sheilas Paris adventure of 2010. Finally, Ryan would like toacknowledge the support of the National Science Foundation.

We note that many figures in these books were prepared with the drawing packageAsymptote (see http://asymptote.sourceforge.net/), an open-source effort that we thinkdeserves to be promoted here. Finally, we thank our editor Simon Capelin and the entireteam at Cambridge, for their advice, assistance and truly astounding patience.

Notation

This book is devoted to the subject of continuummechanics and thermodynamics. However,together with the companion book by Tadmor and Miller, Modeling Materials (MM)[TM11], it is part of a greater effort to create a unified theoretical foundation for multiscalemodeling of material behavior. Such a theory includes contributions from a large numberof fields including those covered in this book, but also quantum mechanics, statisticalmechanics and materials science. We have attempted as much as possible to use the mostcommon and familiar notation from within each field as long as this does not lead toconfusion. To keep the amount of notation to a minimum, we generally prefer to appendqualifiers to symbols rather than introducing new symbols. For example, f is force, whichif relevant can be divided into internal, f int , and external, f ext , parts.

We use the following general conventions:

Descriptive qualifiers generally appear as superscripts and are typeset using a Roman (asopposed to Greek) nonitalic font.

The weight and style of the font used to render a variable indicates its type. Scalarvariables are denoted using an italic font. For example, T is temperature. Array variablesare denoted using a sans serif font, such as A for the matrix A. Vectors and tensors (inthe mathematical sense of the word) are rendered in a boldface font. For example, isthe stress tensor.

Variables often have subscript and superscript indices. Indices referring to the compo-nents of a matrix, vector or tensor appear as subscripts in italic Roman font. For example,vi is the ith component of the velocity vector. Superscripts will be used as counters ofvariables. For example, F e is the deformation gradient in element e. Iteration countersappear in parentheses, for example f (i) is the force in iteration i.

The Einstein summation convention will be followed on repeated indices (e.g. vivi =v21 + v

22 + v

23 ), unless otherwise clear from the context. (See Section 2.2.2 for more

details.) A subscript is used to refer to multiple equations on a single line, for example,

Eqn. (3.32)2 refers to the second equation in Eqn. (3.32) (ai(x, t) . . . ). Important equations are emphasized by placing them in a shaded box.Below, we describe the main notation and symbols used in the book, and indicate the pageon which each is first defined.

xvii

xviii Notation

Mathematical notation

Notation Description Page

equal to by definition 22:= variable on the left is assigned the value on the right 283 for all 22 contained in 22 a subset of 107iff if and only if 22O(n) orthogonal group of degree n 32SL(n) proper unimodular (special linear) group of degree n 217SO(n) proper orthogonal (special orthogonal) group of degree n 32R set of all real numbers 22R

n real coordinate space (n-tuples of real numbers) 25| | absolute value of a real number 25 norm of a vector 25, inner product of two vectors 25Dx;u nonnormalized directional derivative with respect to x in

the direction u57

f [] square brackets indicate f is a linear function of its arguments 24AT transpose of a second-order tensor or matrix: [AT ]ij = Aji 19AT transpose of the inverse of A: AT (A1)T 43a b dot product (vectors): a b = aibi 25a b cross product (vectors): [a b]k = ijkaibj 29a b tensor product (vectors): [a b]ij = aibj 39A : B contraction (second-order tensors): A : B = AijBij 44A B transposed contraction (second-order tensors): A B = AijBji 44A(ij ) symmetric part of a second-order tensor: A(ij ) = 12 (Aij + Aji) 48A[ij ] antisymmetric part: A[ij ] = 12 (Aij Aji) 48A ,

A th eigenvalue and eigenvector of the second-order tensor A 49

Ik (A) kth principal invariant of the second-order tensor A 49d inexact differential 159detA determinant of a matrix or a second-order tensor 21trA trace of a matrix or a second-order tensor: trA = Aii 19, grad gradient of a tensor (deformed configuration) 570, Grad gradient of a tensor (reference configuration) 77curl curl of a tensor (deformed configuration) 58Curl curl of a tensor (reference configuration) 77div divergence of a tensor (deformed configuration) 59Div divergence of a tensor (reference configuration) 772 Laplacian of a tensor (deformed configuration) 60e local node number on element e for global node number 294

xix Notation

General symbols Greek

Symbol Description Page

stretch parameter 78, i set of extensive state variables 135i, ii set of intensive state variables obtained from 173, i set of intensive state variables work conjugate with 157ij Kronecker delta 19ijk permutation symbol 20, ij small strain tensor 93 polar coordinate in polar cylindrical system 61 zenith angle in spherical system 62i curvilinear coordinates in a general coordinate system 60 bulk viscosity (fluid) 220 Lame constant 235 shear viscosity (fluid) 220 shear modulus (solid) 235 Poissons ratio 235, I parent space for a finite element 294 total potential energy of a system and the applied loads 247 mass density (deformed configuration) 1060 mass density (reference configuration) 107, ij Cauchy stress tensor 116 , ij Kirchhoff stress tensor 123 azimuthal angle in spherical system 63, i deformation mapping 72 specific Helmholtz free energy 193, i spin axial vector 98

General symbols Roman

Symbol Description Page

a, ai acceleration vector (material description) 94a, ai acceleration vector (spatial description) 94B bulk modulus 241B, Bij left CauchyGreen deformation tensor 85B matrix of finite element shape function derivatives 302b, bi body force (material description) 122b, bi body force (spatial description) 112Cv molar heat capacity at constant volume 144C, CIJ right CauchyGreen deformation tensor 79

xx Notation

C, CIJKL referential elasticity tensor 226cv specific heat capacity at constant volume 320c, cijkl spatial (or small strain) elasticity tensor 228c, cmn elasticity matrix (in Voigt notation) 230D, DiJkL mixed elasticity tensor 227D matrix representation of the mixed elasticity tensor 303d, dij rate of deformation tensor 96E total energy of a thermodynamic system 141E Youngs modulus 235E, EIJ Lagrangian strain tensor 87E finite element strain operator matrix 302ei orthonormal basis vectors 23e, eij EulerAlmansi strain tensor 90F frame of reference 196F ext , F exti total external force acting on a system 10F , FiJ deformation gradient 78F matrix representation of the deformation gradient 301f column matrix of finite element nodal forces 281G material symmetry group 216g specific Gibbs free energy 195gi , gi contravariant and covariant basis vectors, respectively 28H0 , H0i angular momentum about the origin 120h outward heat flux across a body surface 173h specific enthalpy 194I identity tensor 41I identity matrix 20J Jacobian of the deformation gradient 79J Jacobian of the finite element parent space mapping 295J affine mapping from the parent element space to physical space 295K macroscopic (continuum) kinetic energy 140K finite element stiffness matrix 287k thermal conductivity 210L, Li linear momentum 110l, lij spatial gradient of the velocity field 95M ext0 ,M

ext0i total external moment about the origin acting on a system 120

N number of particles or atoms 110nd dimensionality of space 16n number of moles of a gas 144Pdef deformation power 172Pext external power 170P , PiJ first PiolaKirchhoff stress tensor 122P matrix representation of the first PiolaKirchhoff stress 301p pressure (or hydrostatic stress) 119

xxi Notation

Q heat transferred to a system during a process 140Qt orthogonal transformation between frames of reference 197Q,Qi orthogonal transformation matrix 31q, qi spatial heat flux vector 174q0 , q0I reference heat flux vector 175R rate of heat transfer 170R, RiJ finite rotation (polar decomposition) 83r radial coordinate in polar cylindrical (and spherical)

system61

r spatial strength of a distributed heat source 173r0 reference strength of a distributed heat source 175S entropy 150Sext external entropy input rate 176S int internal entropy production rate 176S shape function for finite element node (physical space) 280S, SIJ second PiolaKirchhoff stress tensor 125S matrix of finite element shape functions 279s specific entropy 175sext specific external entropy input rate 176sint specific internal entropy production rate 176s, sijkl spatial (or small strain) compliance tensor 230s, smn compliance matrix (in Voigt notation) 231s shape function for finite element node (parent space) 294T temperature 137T , Ti nominal traction (stress vector) 123t, ti true traction (stress vector) 113t, ti true external traction (stress vector) 112U internal energy 140U , UIJ right stretch tensor 83u spatial specific internal energy 170u0 reference specific internal energy 175u, ui displacement vector 91u, ui finite element approximation to the displacement field 279u column matrix of finite element nodal displacements 278V0 volume (reference configuration) 79V volume (deformed configuration) 79V , Vij left stretch tensor 83v specific volume 132v, vi velocity vector (material description) 94v, vi velocity vector (spatial description) 94W work performed on a system during a process 140W strain energy density function 194w, wij spin tensor 97

xxii Notation

X , XI position of a continuum particle (reference configuration) 72x, xi position of a continuum particle (deformed configuration) 72X column matrix of finite element nodal coordinates 278z axial coordinate in polar cylindrical system 61

1 Introduction

A solid material subjected to mechanical and thermal loading will change its shape anddevelop internal stress and temperature variations. What is the best way to describe thisbehavior? In principle, the response of a material (neglecting relativistic effects) is dic-tated by that of its atoms, which are governed by quantum mechanics. Therefore, if wecould solve Schrodingers equation for all of the atoms in the material (there are about1022=10 000 000 000 000 000 000 000 atoms in a gram of copper) and evolve the dynamicsof the electrons and nuclei over macroscopic times (i.e. seconds, hours and days), wewould be able to predict the material behavior. Of course, when we say material, we arealready referring to a very complex system. In order to predict the response of the mate-rial we would first have to construct the material structure in the computer, which wouldrequire us to use Schrodingers equation to simulate the process by which the material wasmanufactured. Conceptually, it may be useful to think of materials in this way, but we canquickly see the futility of the approach: the state of the art of quantum calculations involvesjust hundreds of atoms over a time of nanoseconds.

Fortunately, in many cases it is not necessary to keep track of all the atoms in a materialto describe its behavior. Rather, the overall response of such a collection of atoms is oftenmuchmore readily amenable to an elegant, mathematical description. Like the pocket watchon the cover of this book, the complex and intricate inner workings of a material are oftennot of interest. It is the outer expression of these inner workings the regular motion of thewatch hands or macroscopic material response that is of primary concern. To this end,lying at the opposite extreme to quantum mechanics, we find continuum mechanics andthermodynamics (CMT). The CMT disciplines completely ignore the discreteness of theworld, treating it in terms of macroscopic observables time and space averages over theunderlying swirling hosts of electrons and atomic nuclei. This leads to a theory couched interms of continuously varying fields. Using clear thinking inspired by our understandingof the basic laws of nature (which have been validated by experiments) it is possible toconstruct a remarkably coherent and predictive framework for material behavior. In fact,CMThave been so successful that with the exception of electromagnetic phenomena, almostall of the courses in an engineering curriculum from aerodynamics to solid mechanics aresimply an application of simplified versions of the general CMT theory to situations ofspecial interest. Clearly there is something to this macroscopically averaged view of theworld. Of course, the continuum picture becomes fuzzy and eventually breaks down whenwe attempt to apply it to phenomena governed by small length and time scales.1 Those are

1 Having said that, it is important to note that continuum mechanics works remarkably well down to extremelysmall scales. Micro electro mechanical systems (MEMS) devices, which are fully functioning microscopic

1

DineshKumarHighlight



2 Introduction

exactly the multiscale situations that we explore in depth in the companion book to thisone titled Modeling Materials: Continuum, Atomistic and Multiscale Techniques (MM)[TM11]. Here, we focus on CMT.

Continuum mechanics involves the application of the principles of classical mechanicsto material bodies approximated as continuous media. Classical mechanics itself has a longand distinguished history. As Clifford Truesdell, one of the fathers of modern continuummechanics, states in the introduction to his lectures on the subject [Tru66a]:

The classical nature of mechanics reflects its greatness: Ever old and ever new, it continuesto pour out for us understanding and application, linking a changing world to unchangedlaw.

The unchanged laws that Truesdell refers to are the balance principles of mechanics:conservation of mass and the balance of linear and angular momentum. Together withthe first law of thermodynamics (conservation of energy), these principles lead to a set ofcoupled differential equations governing the evolution of material systems.2 The resultinggeneral theory of continuum mechanics and thermodynamics is applicable to arbitrarymaterials undergoing arbitrarily large deformations. We develop this theory and exploreits applications in two main parts. Part I on theory focuses on the basic theory underlyingCMT, going from abstract mathematical ideas to the response of real materials. Part II onsolutions focuses on the application of the theory to solve actual problems.

Part I begins with Chapter 2 on scalars, vectors and tensors and the associated notationused throughout the book. This chapter deals with basic physical andmathematical conceptsthat must be understood before we can discuss themechanics of continuum bodies. First andforemost we must provide basic definitions for space and time. Without such definitionsit is meaningless to speak of the positions of physical objects and their time evolution.Newton was well aware of this and begins his Principia [New62] with a preface called theScholium devoted to definitions. In many ways Newtons greatness lies not in his famouslaws (which are based on earlier work) but in his ability to create a unified framework outof the confusion that preceded him by defining his terms.3 Once space and time are agreedupon, the next step is to identify suitable mathematical objects for describing physicalvariables. We seek to define such things as the positions of particles, their velocities andmore complex quantities like the stress state at a point in a solid. A key property of allsuch variables is that they should exist independently of the particular coordinate systemin which they are represented. Variables that have this property are called tensors or tensorfields. Anyone with a mathematical or scientific background will have come across theterm tensor, but few really understand what a tensor is. This is because tensors are often

machines smaller than the diameter of a human hair (100 microns), are for the most part described quiteadequately by continuum mechanics. Even on the nanoscale where the discrete nature of materials is apparent,continuum mechanics is remarkably accurate to within a few atomic spacings of localized defects in the atomicarrangement.

2 The second law of thermodynamics also plays an essential role. However, in the (standard) presentation of thetheory developed here it does not explicitly enter as a governing equation of the material. Rather, it serves torestrict the possible response to external stimuli of a material (see Chapter 6).

3 Amazingly, more than 300 years after Newton published Principia, the appropriate definitions for space andtime in classical mechanics remain controversial. We discuss this in Section 2.1.



3 Introduction

defined with a purely rules-based approach, i.e. a recipe is given for checking whethera given quantity is or is not a tensor. This is fine as far is it goes, but it does not leadto greater insight. The problem is that the idea of a tensor field is complex and to gaina true and full understanding one must immerse oneself in the rarefied atmosphere ofdifferential geometry. We have placed ourselves squarely between these two extremesand have attempted to provide a more nuanced fundamental description of tensors whilekeeping the discussion as accessible as possible. For this reason we mostly adopt theCartesian coordinate system in our discussions, introducing the more general covariant andcontravariant notation of curvilinear coordinates only where necessary.

Our next step takes us away from the abstract world of tensor algebra and calculus to thedescription of physical bodies. As noted above, we know that in reality bodies are made ofmaterial and material is made of atoms which themselves are made of more fundamentalparticles and who knows perhaps those are made of strings or membranes existingin a higher-dimensional universe. Continuum mechanics ignores this underlying discretestructure and provides a model for the world in which a material is infinitely divisible. Cuta piece of copper in two and you get two pieces of copper, and so on ad infinitum. Thedownside of this simplification is that it actually becomes more complicated to describethe shape and evolution of bodies. For a discrete set of particles all we need to know isthe positions of the particles and their velocities. In contrast, how can we describe theposition that an evolving blob of material occupies in space? This broadly falls underthe topic of kinematics of deformation covered in Chapter 3. The study of kinematics isconcerned exclusively with the abstract motion of bodies, taking no consideration of theforces that may be required to impart such a motion. As a result, kinematics is purely thegeometric, descriptive aspect of mechanics, phrased in the language of configurations thata blob of material can adopt. In a sense one can think of a configuration being the sheetmusic of mechanics. The external mechanical and thermal loading are what ultimatelyrealize this configuration, just as the musicians and their instruments ultimately bring asymphony to life.

A continuum body can take on an infinity of possible configurations. It is convenient toidentify one of these as a reference configuration and to refer all other configurations tothis one. Once a reference configuration is selected, it is possible to define the concept ofstrain (or more generally local deformation). This is the change in shape experienced bythe infinitesimal environment of a point in a continuum body relative to its shape in thereference configuration. Since it is shape change (as opposed to rigid motion) that materialbodies resist, strain becomes a key variable in a continuum theory. An important aspect ofcontinuummechanics is that shape change can be of arbitrary magnitude. This is referred tosomewhat confusingly as finite strain as if contrasting the theory with another one dealingwith infinite strain. Really the distinction is with theories of infinitesimal strain (likethe theories of strength of materials and linear elasticity taught as part of an engineeringcurriculum). This makes continuummechanics a nonlinear theory very general in the sortof problems it can handle, but also more difficult to solve.

Having laid out the geometry of deformation, we must next turn to the laws of natureto determine how a body will respond to applied loading. This topic naturally divides intotwo parts. Chapter 4 focuses on this question from a purely mechanical perspective. This




4 Introduction

means that we ignore temperature and think only of masses and the mechanical forcesacting on them. At the heart of this description are three laws taken to be fundamentalprinciples in classical mechanics: conservation of mass and the balance of linear and ofangular momentum. Easily stated for a system of particles, the extension of these lawsto continuous media leads to some interesting results. The big name here is Cauchy, whothrough some clever thought experiments was able to infer the existence of the stress tensorand its properties. Cauchy was concerned with what we today would call the true stressor for obvious reasons the Cauchy stress. This is the force per unit area experienced bya point in a continuum when cut along some plane passing through that point. The notionof configurations introduced above means that the stress tensor can be recast in a varietyof forms that, although lacking the clear physical interpretation of Cauchys stress, havecertain mathematical advantages. In particular, the first and second PiolaKirchhoff stresstensors represent the stress relative to the reference configuration mentioned above.

The second set of the laws of nature that must be considered to fully characterize acontinuum mechanics problem are those having to do with temperature, i.e. the laws ofthermodynamics discussed in Chapter 5. In reality, a material is not just subjected tomechanical loading which leads to stresses and strains in the body; it also experiencesthermal loading which can lead to an internally varying temperature field. Furthermore, themechanical and thermal effects are intimately coupled into what can only be described asthermomechanical behavior. Thermodynamics is for most people a more difficult subjectto understand than pure mechanics. This is another consequence of the simplificationafforded by the continuum approximation. Concepts like temperature and entropy thathave a clear physical meaning when studied at the level of discrete particles become farmore abstract at the macroscopic level where their existence must be cleverly inferredfrom experiments.4 The three laws of thermodynamics (numbered in a way to make Cprogrammers happy) are the zeroth law, which deals with thermal equilibrium and leads tothe concept of temperature, the first law, which expresses the conservation of energy anddefines energy, and the second law, which deals with the concept of entropy and the directionof time (i.e. why we have a past and a future). Unlike a traditional book on thermodynamics,we develop these concepts with an eye to continuummechanics. We do not talk about steamengines, but rather show how thermodynamics contributes a conservation law to the fieldequations of continuum mechanics, and how restrictions related to the second law impactthe possible models for material behavior the so-called constitutive relations describednext.

The theory we have summarized so far appears wonderfully economical. Using a handfulof conservation laws inferred from experiments, a very general theoretical formulation isestablished which (within a classical framework) fully describes the behavior of materialssubjected to arbitrary mechanical and thermal loading. Unfortunately, this theory is notclosed. By this we mean that the theoretical formulation of continuum mechanics andthermodynamics possesses more unknowns than equations to solve for them. If one thinksabout this for a minute, it is not surprising we have not yet introduced the particular nature

4 A student wishing to truly understand thermodynamics is strongly encouraged to also explore this subject fromthe perspective of statistical mechanics as is done in Chapter 7 of [TM11].




5 Introduction

of the material into the discussion. Clearly the response of a block of butter will be differentthan that of steel when subjected to mechanical and thermal loading. The equations relatingthe response of a material to the loading applied to it are called constitutive relations andare discussed in Chapter 6. Since we are dealing with a general framework which allows forarbitrary finite deformation, the constitutive relations are generally nonlinear. Continuummechanics cannot predict the particular form of the constitutive relations for a givenmaterial these are obtained either empirically through experimentation or more recently using mul-tiscale modeling approaches as described in MM [TM11]. However, continuum mechanicscan place constraints on the allowable forms for these relations. This is very important,since it dramatically reduces the set of possible functions that can be used for interpretingexperiments or multiscale simulations. One constraint already mentioned above is the re-strictions due to the second law of thermodynamics. For example, it is not possible to have amaterial in which heat flows from cold to hot.5 Another fundamental restriction is related tothe principle of material frame-indifference (or objectivity). Material frame-indifferenceis a difficult and controversial subject with different, apparently irreconcilable, schoolsof thought. Most students of continuum mechanics even very advanced students find this subject quite difficult to grasp. We provide a new presentation of material frame-indifference that we feel clarifies much of the confusion and demonstrates how the differentapproaches mentioned above are related and are in fact consistent with each other. A thirdrestriction on the form of constitutive relations is tied to the symmetry properties of thematerial. This leads to vastly simplified forms for special cases such as isotropic materialswhose response is independent of direction. Even simpler forms are obtained when theequations are linearized, which in the end leads to the venerable (generalized) Hookeslaw a linear relation between the Cauchy stress and the infinitesimal strain tensor.

The addition of constitutive relations to the conservation and balance laws derived beforecloses the theory. It is now possible to write down a system of coupled, nonlinear partialdifferential equations that fully characterize a thermomechanical system. Together withappropriate boundary conditions (and initial conditions for a dynamical problem) a well-defined (initial) boundary-value problem can be constructed. This is described in Chapter 7.Special emphasis is placed in this chapter on purely mechanical static problems. In thiscase, the boundary-value problem can be conveniently recast as a variational problem, i.e. aproblem where instead of solving a complicated system of nonlinear differential equations,a single scalar energy functional has to be minimized. This variational principle, referred toas the principle of minimum potential energy (PMPE), is of great importance in continuummechanics as well as more general multiscale theories such as those discussed in MM[TM11]. A key component of the derivation of the PMPE is the theory of stability, whichis concerned with the conditions under which a mechanical system is in stable equilibriumas opposed to unstable equilibrium. (Think of a pencil lying on a table as opposed to onebalanced on its end.) We only give a flavor of this rich and complex theory, sufficient forour purposes of elucidating the derivation of PMPE.

5 This is true for thermomechanical systems. However, if electromagnetic effects are considered, the applicationof an appropriate electric potential to certain materials can lead to heat flow in the wrong direction withoutviolating the second law.



6 Introduction

The discussion of stability and PMPE concludes the first part of the book. At this stage,we are able to write down a complete description of any problem in continuum mechanicsand we have a clear understanding of the origins of all of the equations that appear in theproblem formulation. Unfortunately, the complete generality of the continuum mechanicsframework, with its attendant geometric and material nonlinearity, means that it is almostalways impossible to obtain closed-form analytical solutions for a given problem. So howdo we proceed? There are, in fact, three possible courses of action, which are described inPart II on Solutions. First, in certain cases it is possible to obtain closed-form solutions.Even more remarkably, some of these solutions are universal in that they apply to allmaterials (in a given class) regardless of the form of the constitutive relations. In addition totheir academic interest, these solutions have important practical implications for the designof experiments that measure the nonlinear constitutive relations for materials. The knownuniversal solutions are described in Chapter 8.

The second option for solving a continuum problem (assuming the analytical solution isunknown or, more likely, unobtainable) is to adopt a numerical approach. In this case, thecontinuum equations are solved approximately on a computer. The most popular numericalapproach is the finite element method (FEM) described in Chapter 9. In FEM the continuumbody is discretized into a finite set of domains, referred to as elements, bounded bynodes whose positions and temperatures constitute the unknowns of the problem.6 Whensubstituting this representation into the continuum field equations, the result is a set ofcoupled nonlinear algebraic equations for the unknowns. Entire books are written on FEMand our intention is not to compete with those. We do, however, offer a derivation of thekey equations that is different frommost texts. We focus on static boundary-value problemsand approach the problem from the perspective of the PMPE. In this setting, the FEMsolution to a general nonlinear continuum problem corresponds to the minimization of theenergy of the system with respect to the nodal degrees of freedom. This is a convenientapproach which naturally extends to multiscale methods (like those described in Chapter 12of [TM11]) where continuum domains and atomistic domains coexist.

The third and final option for solving continuum problems is to simplify the equationsby linearizing the kinematics and/or the constitutive relations. This approach is discussedin Chapter 10. As noted at the start of this introduction, this procedure leads to almost allof the theories studied as independent subjects in an engineering curriculum. For example,few students understand the connection between heat transfer and elasticity theory. Theability of continuum mechanics to provide a unified framework for all of these subjects isone of the reasons that this is such an important theory. Most students who take a continuummechanics course leave with a much deeper understanding of engineering science (oncethey have recovered from the shell shock). We conclude in Chapter 11 with some suggestedfurther reading for readers wishing to expand their understanding of the topics covered inthis book.

6 It is amusing that the continuum model is introduced as an approximation for the real discrete material, but thatto solve the continuum problem one must revert back to a discrete (albeit far coarser) representation.


PART I

THEORY

2 Scalars, vectors and tensors

Continuum mechanics seeks to provide a fundamental model for material response. It issensible to require that the predictions of such a theory should not depend on the irrelevantdetails of a particular coordinate system. The key is to write the theory in terms of variablesthat are unaffected by such changes; tensors1 (or tensor fields) are the measures thathave this property. Tensors come in different flavors depending on the number of spatialdirections that they couple. The simplest tensor has no directional dependence and is calleda scalar invariant to distinguish it from a simple scalar. A vector has one direction. For twodirections and higher the general term tensor is used.

Tensors are tricky things to define. Many books define tensors in a technical mannerin terms of the rules that tensor components must satisfy under coordinate system trans-formations.2 While certainly correct, we find such definitions unilluminating when tryingto answer the basic question of what is a tensor?. In this chapter, we provide an intro-duction to tensors from the perspective of linear algebra. This approach may appear rathermathematical at first, but in the end it provides a far deeper insight into the nature oftensors.

Before we can begin the discussion of the definition of tensors, we must start by definingspace and time and the related concept of a frame of reference, which underlie thedescription of all physical objects. The notions of space and time were first tackled byNewton in the formulation of his laws of mechanics.

2.1 Frames of reference and Newtons laws

In 1687, Isaac Newton published his Philosophiae Naturalis Principia Mathematica orsimply Principia, in which a unified theory of mechanics was presented for the first time.According to this theory, themotion ofmaterial objects is governed by three laws. Translatedfrom the Latin, these laws state [Mar90]:

1 The term tensor was coined by William Hamilton in 1854 to describe the norm of a polynome in his theoryof quaternions. It was first used in its modern sense by Woldemar Voigt in 1898.

2 More correctly, tensors are defined in terms of the rules that their components must satisfy under a change ofbasis. A rectilinear coordinate system consists of an origin and a basis. The distinction between a basis and acoordinate system is discussed further below. However, we will often use the terms interchangeably.

9




I Every body remains in a state, resting or moving uniformly in a straight line, exceptinsofar as forces on it compel it to change its state.

II The [rate of] change of momentum is proportional to the motive force impressed, andis made in the direction of the straight line in which the force is impressed.

III To every action there is always opposed an equal reaction.

Mathematically, Newtons second law (also called the balance of linear momentum) is

F ext =d

dt(mv), (2.1)

where F ext is the total external force acting on a system,m is its mass and v is the velocityof the center of mass. For a body with constant mass, Eqn. (2.1) reduces to the famousequation, F ext = ma, where a is acceleration. (The case of variable mass systems isdiscussed further on page 13.)

Less well known than Newtons laws of motion is the set of definitions that Newtonprovided for the fundamental variables appearing in his theory (force, mass, space, time,motion and so on). These appear in the Scholium to the Principia (a chapter with explana-tory comments and clarifications). Newtons definitions of space and time are particularlyeloquent [New62]:

Space Absolute space, in its own nature, without reference to anything external, remainsalways similar and unmovable.

Time Time exists in and of itself and flows equably without reference to anythingexternal.

These definitions were controversial in Newtons time and continue to be a source of activedebate even today. They were necessary to Newton, since otherwise his three laws weremeaningless. The first law refers to the velocity of objects and the second law to therate of change of velocity (acceleration). But velocity and acceleration relative to what?Newton was convinced that the answer was absolute space and absolute time. This viewwas strongly contested by the relationists led by Gottfried Leibniz, who as a point ofphilosophy believed that only relative quantities were important and that space was simplyan abstraction resulting from the geometric relations between bodies [DiS02].

Newtons bucket The argument was settled (at least temporarily) by a simple thought ex-periment that Newton described in the Principia.3 Take a bucket half filled with waterand suspend it from the ceiling with rope. Twist the rope by rotating the bucket as far aspossible. Wait until the water settles and then let go. The unwinding rope will cause thebucket to begin spinning. Initially, the water will remain still even though the bucket isspinning, but then slowly due to the friction between the walls of the bucket and the water,the water will begin to spin as well until it is rotating in unison with the bucket. When the

3 The story of this experiment and how it inspired later thinkers such as Ernst Mach and Albert Einstein iseloquently told in Brian Greenes popular science book on modern physics [Gre04].

11 2.1 Frames of reference and Newtons laws

water is spinning its surface will assume a concave profile, higher near the bucket wallsthan in the center. The rotation of the bucket and water will continue as the rope unwindsand begins to wind itself up in the opposite direction. Eventually, the bucket will slow to astop, but the water will continue spinning for a while, before the entire process is repeatedin the opposite direction. Not an experiment for the cover of Nature, but quite illuminatingas we shall see.

The key point is the fact that the surface of thewater assumes a concave profile. The reasonfor this appears obvious. When the water is spinning it is accelerating outward (in the sameway that a passenger in a turning vehicle is pushed out to the side) and since there is nowherefor the water to go but up, it climbs up the walls of the bucket. This is certainly correct;however, it depends on the definition of spinning. Spinning relative to what? It cannot bethe bucket itself, because when the experiment starts and the water appears still while thebucket is spinning, one can say that the water is spinning in the opposite direction relative toa stationary bucket and yet the surface of the water is flat. Later when both the bucket andwater are spinning together, so that the relative spin is zero, the water is concave. At the endwhen the bucket has stopped and the water is still spinning relative to it, the surface of thewater is still concave. Clearly, the shape of the water surface cannot be explained in termsof the relative motion of the bucket and water. So what is the water spinning relative to? Youmight say the earth or the fixed stars,4 but Newton countered with a thought experiment.Imagine that the experiment was done in otherwise empty space. Since the experiment withthe bucket requires gravity, imagine instead two globes tied together with a rope. There isnothing in the universe except for the two globes and the rope: an immense vacuum, wherethere was nothing external or sensible with which the globes could be compared [New62].If the rope is made to rotate about an axis passing through its center and perpendicular toit, we expect a tension to be built up in the rope due to the outward acceleration of theglobes exactly as in the bucket experiment. But now there is clearly nothing to relate thespinning of the rope and globes to except absolute space itself. QED as far as Newton wasconcerned.5 Absolute space and time lie at the heart of Newtons theory. It is not surprising,therefore, that Newton considered his discovery of these concepts to be his most importantachievement [Gre04].

Frame of reference In practice, Newton recognized that it is not possible to work directlywith absolute space and time since they cannot be detected, and so he introduced theconcepts of relative space and relative time [New62]:

4 Recall that the word planet comes from the Greek planetai meaning wanderers, because the planets appearto move relative to the fixed backdrop of the stars.

5 Even Leibniz had to accept Newtons argument, although he remained unconvinced about the reality of absolutespace: I find nothing in . . . the Scholium . . . that proves, or can prove, the reality of space in itself. However, Igrant there is a difference between an absolute true motion of a body, and a mere relative change of its situationwith respect to another body [Ale56]. Two hundred years later Ernst Mach challenged Newtons assertion byclaiming that the water in the bucket is spinning relative to all other mass in the universe. Mach argued that ifit were possible to perform Newtons experiment with the globes in an empty universe, then there would be notension in the rope because there would be no other mass relative to which it was spinning. Albert Einstein wasintrigued by Machs thinking, but the conclusion to emerge from the special theory of relativity was that in factthere would be tension in the rope even in an empty universe [Gre04, p. 51].


Relative space is some movable dimension or measure of the absolute spaces, which oursenses determine by its position to bodies and which is commonly taken for immovablespace; such is the dimension of a subterraneous, an aerial, or celestial space, determinedby its position in respect of the earth.

Relative, apparent, and common time is some sensible and external (whether accurate orunequable) measure of duration by the means of motion, which is commonly used insteadof true time, such as an hour, a day, a month, a year.

Today, we refer to this combination of relative space and relative time as a frame ofreference. A modern definition is that a frame of reference is a rigid physical object, suchas the earth, the laboratory or the fixed stars, relative to which positions are measured,and a clock to measure time.

Inertial framesof reference With the definition of absolute space and absolute time, Newtonslaws of motion were made explicit. However, it turns out that Newtons equations also holdrelative to an infinite set of alternative frames of reference that aremoving uniformly relativeto the absolute frame. These are called inertial frames of reference.6

Consider an inertial frame of reference that is moving at a constant velocity v relativeto absolute space. Say that the position of some object is (x1 , x2 , x3) in the absolute frameand (x1 , x

2 , x

3) in the inertial frame.

7 Assume the frames origins coincide at time t = 0.The positions of the object and measured times in both frames are related through

x1 = x1 vt, x2 = x2 , x3 = x3 , t = t,where, without loss of generality, the coordinate systems associated with the two frameshave been aligned so that the relative motion is along the 1-direction. A mapping of thistype is called a Galilean transformation. Note that the velocities of the object along the1-direction measured in the two frames are related through

v1 =dx1dt

= v1 v.

It is straightforward to show that Newtons laws of motion hold in the inertial frame. Thefirst law is clearly still valid since an object moving uniformly relative to absolute spacealso moves uniformly relative to the inertial frame. The third law also holds under theassumption that force is invariant with respect to uniform motion. (This property of force,called objectivity, is revisited in Section 6.3.3.) The fact that the second law holds in allinertial frames requires more careful thought. The law is clearly satisfied for the case wherethe mass of the system is constant. In this case, F ext = ma, which holds in all inertialframes since the acceleration is the same:

a1 =dv1dt

=d(v1 v)

dt=

dv1dt

= a1 ,

6 See also Section 6.3, where the relationship between inertial frames and the transformation between frames ofreference and objectivity is discussed.

7 Locating objects relative to a frame of reference requires the introduction of a coordinate system (see Sec-tion 2.3.2). Here a Cartesian coordinate system is used.


13 2.1 Frames of reference and Newtons laws

where the fact that v is constant was used. What about the case where the mass of thesystem is variable, for example, a rocket which burns its fuel as it is flying or a rollingcart containing sand which is being blown off as the cart moves? In these cases, a directapplication of Newtons second law would appear to show a dependence on the motion ofthe frame, since

d(mv1)dt

=dm

dtv1 + m

dv1dt

=dm

dt(v1 v) + md(v1 v)

dt=

d(mv1)dt

v dmdt

. (2.2)

This result suggests that the rate of change of momentum for variable mass systems isnot the same in all inertial frames since it directly depends on the motion of the frame v.The answer to this apparent contradiction is that there is another principle at work whichis not normally stated but is assumed to be true. This is the principle of conservation ofmass.8 Newtons second law is expressed for a system, a body in Newtons language,and the mass of this body in a classical system is conserved. This appears to suggest thatvariable mass systems are impossible, since m = constant. However, consider the casewhere the system consists of two bodies, A and B, with masses mA and mB . The bodiescan exchange mass between them, so that mA = mA (t) and mB = mB (t), but their sumis conserved, mA + mB = m = constant. In this case, the rate of change of momentumis indeed the same in all inertial frames, since dm/dt in Eqn. (2.2) is zero and therefore,d(mv1)/dt = d(mv1)/dt. If one wants to apply Newtons second law to a subsystem whichis losing or gaining mass, say only body A in the above example, then one must explicitlyaccount for the momentum transferred in and out of the subsystem by mass transfer. Onecan view this additional term as belonging to the force which is applied to the subsystem.This is the principle behind the operation of a rocket (see Exercise 2.1) or the recoil of agun when a bullet is fired.9

We have established that Newtons laws of motion (with the added assumption of con-servation of mass) hold in all inertial frames of reference. This fact was understood byNewton, who stated in Corollary V to his equations of motion [New62]:

When bodies are enclosed in a given space, their motions in relation to one another are thesame whether the space is at rest or whether moving uniformly straight forward withoutcircular motion.

Once one inertial frame is known, an infinite number of other inertial frames can beconstructed through a Galilean transformation. The practical problem with this way ofdefining inertial frames is that it is not possible to know whether a frame of reference ismoving uniformly relative to absolute space, since it is not possible to detect absolute space.For this reason the modern definition of inertial frames does not refer to absolute space,but instead relies on Thomsons law of inertia, which is described shortly.

8 Many books on mechanics take the view that Newtons laws only hold for systems of point particles that bydefinition have constant mass. In this case, conservation of mass is trivially satisfied and need not be mentioned.The view presented here is more general and consistent with the generalization of Newtons laws to continuumsystems which is adopted in the later chapters.

9 Interestingly, the correct treatment of variable mass systems is not uniformly understood even by researchersworking in the field. See, for example, the discussion in [PM92].






Problemswith absolute space Despite the apparent acceptance of absolute space when it wasintroduced, it continued (and continues) to trouble many people. Two main criticisms areraised against it.

1. Metaphysical nature of absolute spaceThe absolute space which Newton introduced is an undetectable, invisible, all filling,fixed scaffolding relative to which positions are measured. A sort of universal globalpositioning system with a capital G. Regardless of ones religious views, one wantsto say Gods frame of reference, and that is in some sense how Newton viewed it. Thealmost spiritual nature of this medium is apparent. Here we have an invisible thingthat cannot be seen or sensed in any way and yet it has a profound effect on ourevery day experiences since it determines the acceleration upon which the physicallaws of motion depend. Newton was strongly criticized for this aspect of his work byphilosophers of science. For example, ErnstMach stated: With respect to the monstrousconceptions of absolute space and absolute time I can retract nothing. Here I have onlyshown more clearly than hitherto that Newton indeed spoke much about these things,but throughout made no serious application of them [Mac60]; or according to HansReichenbach: Newton begins with precisely formulated empirical statements, but addsa mystical philosophical superstructure . . . his theory of mechanics arrested the analysisof the problems of space and time for more than two centuries, despite the fact thatLeibniz, who was his contemporary, had a much deeper understanding of the nature ofspace and time [Rei59]. These claims have more recently been debunked as stemmingfrom a misunderstanding of the role that absolute space plays in Newtons theory,a misunderstanding of Leibnizs theoretical shortcomings and a misunderstanding ofEinsteins theory of relativity in which spacetime plays a similar role to that of Newtonsdefinitions [Ear70, Art95, DiS06].

2. Equivalence of inertial framesThe second complaint raised against Newton is that since all inertial frames are equiv-alent from the perspective of Newtonian dynamics and there is no way to tell themapart, it is not sensible to single out one of them, absolute space, as being special.Instead, one must think of all inertial frames as inherently equivalent. The definition ofan inertial frame must therefore change since it can no longer be defined as a frameof reference in uniform motion relative to absolute space. A solution was proposedby James Thomson in 1884, which he called the law of inertia. It is paraphrased asfollows [DiS91]:10

For any system of interacting bodies, it is possible to construct a reference-frameand time scale with respect to which all accelerations are proportional to, and in thedirection of, impressed forces.

This is meant to be added to Newtons laws of motion as a fourth law on equal standingwith the rest. In this way inertial frames are defined as frames in which Newtons secondlaw holds without reference to absolute space. The conclusion from this is that the oftenasked question regarding why the laws of motion hold only relative to inertial frames is

10 Thomsons law is revisited from the perspective of material frame-indifference (objectivity) in Section 6.3.3.


15 2.2 Tensor notation

ill-posed. The laws of motion do not hold relative to inertial frames, they define them[DiS91]. This view on inertial frames is often the one expressed in modern books onmechanics. With this interpretation, an inertial frame is defined as a frame of referencein which Newtons laws of motion are valid.

Relativistic spacetime Thomsons definition of the law of inertia is not the end of the story,of course. Just as the Newtonian picture was falling into place, James Clerk Maxwellwas developing the theory of electromagnetism. One of the uncomfortable conclusions toemerge from Maxwells theory was that electromagnetic waves travel at a constant speed,c = 299 792.458 km/s, relative to all frames of reference, a fact that was confirmedexperimentally for light. This conclusion makes no sense in the Newtonian picture. Howcan something travel at the same speed relative to two frames of reference that are in relativemotion?

Surprisingly, a hint to the answer is already there in Newtons words: time exists inand of itself and flows equably without reference to anything external. Einstein showedthat this was entirely incorrect. Time does not exist in and of itself. It is intimately tiedwith space and is affected by the motion of observers. The result is relativistic spacetime,which is beyond the scope of this book. It is, however, interesting to point out that Einsteinsspacetime, like Newtons absolute space is something. In the absence of gravity, in thespecial theory of relativity, Einstein speaks of an absolute spacetime not much differentphilosophically from Newtons absolute space [DiS06]. In general relativity, spacetimecomes alive [Gre04] and interacts with physical objects. In this way, the criticism thatspace and time are metaphysical is removed.

Within this context, it may be possible to regard Newtons absolute space as a legitimateconcept that can be considered a limiting case of relativistic spacetime. If this is true,then perhaps the original definition of inertial frames in terms of absolute space is tenable,removing the need for Thomsons law of inertia. Philosophers of science are still arguingabout this point.

2.2 Tensor notation

Having introduced the concepts of space, time and frame of reference, we now turn to anuts and bolts discussion regarding the notation of tensor algebra. In the process of doingso we will introduce important operations between tensors. It may seem a bit strange tostart discussing a notation for something that we have not defined yet. Think of it as theintroduction of a syntax for a new language that we are about to learn. It will be useful forus later, when we learn the words of this language, to have a common structure in whichto explain the concepts that emerge. Walter Jaunzemis, in his entertaining introductionto continuum mechanics, put it very nicely: Continuum mechanics may appear as afortress surrounded by the walls of tensor notation [Jau67]. We begin therefore at thewalls.



2.2.1 Direct versus indicial notation

Tensors represent physical properties such as mass, velocity and stress that do not dependon the coordinate system. It should therefore be possible to represent tensors and theoperations on them and between them without reference to a particular coordinate system.Such a notation exists and is called direct notation (or invariant notation). Direct notationprovides a symbolic representation for tensor operations but it does not specify how theseoperations are actually performed. In practice, in order to perform operations on tensors theymust always be projected onto a particular coordinate system where they are representedby a set of components. The explicit representation of tensor operations in terms of theircomponents is called indicial notation. This is the notation that has to be used when tensoroperations involving numerical values are performed.

The number of spatial directions associated with a tensor is called its rank or order.We will use these terminologies interchangeably. A scalar invariant, such as mass, is notassociated with direction at all, i.e. a body does not have a different mass in differentdirections. Therefore, a scalar invariant is a rank 0 tensor or alternatively a zeroth-ordertensor. A vector, such as velocity, is associated with one spatial direction and is therefore arank 1 or first-order tensor. Stress involves two spatial directions, the orientation of a planesectioning a body and a direction in space along which the stress is evaluated. It is thereforea rank 2 or second-order tensor. Tensors of any order are possible. In practice, we will onlybe dealing with tensors up to fourth order.

In both indicial and direct notations, tensors are represented by a symbol, e.g.m for mass,v for velocity and for stress. In indicial notation, the tensors spatial directions are denotedby indices attached to the symbol. Mass has no direction so it has no indices, velocity hasone index, stress two, and so on: m, vi , ij . The number of indices is equal to the rankof the tensor and the range of an index [1, 2, . . . , nd ] is determined by the dimensionalityof space.11 We will be dealing mostly with three-dimensional space (nd = 3); however,the notation we develop applies to any value of nd . The tensor symbol with its numericalindices represents the components of the tensor, e.g. v1 , v2 and v3 are the components of thevelocity vector. A set of simple rules for the interaction of indices provides a mechanism fordescribing all of the tensor operations that we will require. In fact, what makes this notationparticularly useful is that any operation defined by indicial notation has the property that ifits arguments are tensors the result will also be a tensor. We discuss this further at the endof Section 2.3, but for now we state it without proof.

In direct notation, no indices are attached to the tensor symbol. The rank of the tensoris represented by the typeface used to display the symbol. Scalar invariants are displayedin a regular font while first-order tensors and higher are displayed in a bold font (or withan underline when written by hand): m, v, (or m, v, by hand). As noted above, theadvantage of direct notation is that it emphasizes the fact that tensors are independent ofthe choice of a coordinate system (whereas indices are always tied to a particular selection).Direct notation is also more compact and therefore easier to read. However, the lack ofindices means that special notation must be introduced for different operations between

11 See the discussion on finite-dimensional spaces in Section 2.3.

17 2.2 Tensor notation

tensors. Many symbols in this notation are not universally accepted and direct notation isnot available for all operations. We will discuss direct notation in Section 2.4, where tensoroperations are defined.

In some cases, the operations defined by indicial notation can also be written using thematrix notation familiar from linear algebra. Here vectors and second-order tensors arerepresented as column and rectangular matrices of their components, for example

[v] =

v1v2v3

, [] =11 12 1321 22 2331 32 33

.The notation [v] and [] is a shorthand representation for the columnmatrix and rectangularmatrix, respectively, formed by the components of the vector v and the second-order tensor. This notation will sometimes be used when tensor operations can be represented bymatrix multiplication and other matrix operations on tensor components.

Before proceeding to the definition of tensors, we begin by introducing the basic rulesof indicial notation, starting with the most basic rule: the summation convention.

2.2.2 Summation and dummy indices

Consider the following sum:12

S = a1x1 + a2x2 + + and xnd .We can write this expression using the summation symbol :

S =ndi=1

aixi =ndj=1

ajxj =nd

m=1

amxm .

Clearly, the particular choice for the letter we use for the summation, i, j orm, is irrelevantsince the sum is independent of the choice. Indices with this property are called dummyindices. Because summation of products, such as aixi , appears frequently in tensor op-erations, a simplified notation is adopted where the symbol is dropped and any indexappearing twice in a product of variables is taken to be a dummy index, over which a sumis implied. For example,

S = aixi = ajxj = amxm = a1x1 + a2x2 + + and xnd .This convention was introduced by Albert Einstein in the famous 1916 paper in whichhe outlined the principles of general relativity [Ein16]. It is therefore called Einsteinssummation convention or just the summation convention for short.13

12 This section follows the introduction to indicial notation in [LRK78].13 Although the summation convention is an extremely simple idea, it is also extremely useful and is therefore

widely used and quoted. This amused Einstein who is reported to have joked with a friend that apparentlyI havemade a great discovery inmathematics; I have suppressed the summation sign every time that summationmust be made over an index which occurs twice . . . [Wei11].


[Tadmor_E.B,_Miller_R.E.,_Elliott_R.S.]_Continuum_(BookFi.org).pdf

Documents

Transcript of [Tadmor_E.B,_Miller_R.E.,_Elliott_R.S.]_Continuum_(BookFi.org).pdf