ContinuumMechanicsandThermodynamicsassets.cambridge.org/97811070/08267/frontmatter/9781107008267... · ContinuumMechanicsandThermodynamics ... and classical thermodynamics. It

Continuum Mechanics 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.

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Cambridge University Press978-1-107-00826-7 - Continuum Mechanics and Thermodynamics: From Fundamental Concepts to Governing EquationsEllad B. Tadmor, Ronald E. Miller and Ryan S. ElliottFrontmatterMore information

http://www.cambridge.org/9781107008267

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Continuum Mechanics andThermodynamics

From Fundamental Concepts toGoverning Equations

ELLAD B. TADMORUniversity of Minnesota, USA

RONALD E. MILLERCarleton University, Canada

RYAN S. ELLIOTTUniversity of Minnesota, USA






C A M B R I D G E U N I V E R S I T Y P R E S S

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

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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

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ISBN 978-1-107-00826-7 Hardback

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Contents

Preface page xiAcknowledgments xiiiNotation xvii

1 Introduction 1

Part I Theory 7

2 Scalars, vectors and tensors 92.1 Frames of reference and Newton’s 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 Cayley–Hamilton 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 Newton’s second law for a system of particles 1104.2.2 Balance of linear momentum for a continuum system 1114.2.3 Cauchy’s 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 Piola–Kirchhoff 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 (Clausius–Duhem 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 Coleman–Noll 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 Hooke’s 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 Lyapunov’s indirect method and the linearized equations

of motion 2517.3.3 Lyapunov’s 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 Newton–Raphson (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 continuum mechanics 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, continuum mechanics and thermodynamics form the fundamental theory lyingat the heart of many 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 explain where the equations come from and what 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 terms with 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

As we 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 below most 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(George Mason University) for providing figures, and Christoph Ortner (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 explaining DFT in a way that an engineer could understand, and Peter Watson atCarleton University was instrumental in our eventual understanding of quantum mechanics.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 Luskinfrom UMN (numerical analysis of multiscale methods and quantum mechanics), 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).We would also be remiss if we did not take the time to thank Art 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 2008–2009 and 2010–2011 graduate classes at the University ofIllinois (Urbana-Champaign) who used the books extensively provided great feedback toimprove the manuscripts, as did Bill Curtin’s class at Brown in 2009–2010. The 2009and 2010 classes of Ron’s “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 Ellad’s 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 Ron’s formeradvisors at Brown University, Michael Ortiz and Rob Phillips (both currently at Caltech)and Ryan’s 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 Sheila’s “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 continuum mechanics 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, σ 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 =v2

1 + v22 + v2

3 ), unless otherwise clear from the context. (See Section 2.2.2 for moredetails.)

• 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 25〈Dx•;u〉 nonnormalized directional derivative with respect to x in

the direction u

57

f [•] square brackets indicate f is a linear function of its arguments 24AT transpose of a second-order tensor or matrix: [AT ]ij = Aji 19A−T transpose of the inverse of A: A−T ≡ (A−1)T 43a · b dot product (vectors): a · b = aibi 25a × b cross product (vectors): [a × b]k = εijkaibj 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 ) = 1

2 (Aij + Aji) 48A[ij ] antisymmetric part: A[ij ] = 1

2 (Aij − Aji) 48λA

α , ΛAα αth eigenvalue and eigenvector of the second-order tensor A 49

Ik (A) kth principal invariant of the second-order tensor A 49d inexact differential 159det A 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) 57∇0•, 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) 77∇2• Laplacian of a tensor (deformed configuration) 60⇀αe local node number on element e for global node number α 294






xix Notation�

General symbols – Greek

Symbol Description Page

α stretch parameter 78Γ, Γi set of extensive state variables 135Γi, Γi

i set of intensive state variables obtained from Γ 173γ, γi set of intensive state variables work conjugate with Γ 157δij Kronecker delta 19εijk permutation symbol 20ε, εij small strain tensor 93θ polar coordinate in polar cylindrical system 61θ zenith angle in spherical system 62θi curvilinear coordinates in a general coordinate system 60κ bulk viscosity (fluid) 220λ Lame constant 235μ shear viscosity (fluid) 220μ shear modulus (solid) 235ν Poisson’s ratio 235ξ, ξI parent space for a finite element 294Π total potential energy of a system and the applied loads 247ρ mass density (deformed configuration) 106ρ0 mass density (reference configuration) 107σ, σij Cauchy stress tensor 116τ , τij Kirchhoff stress tensor 123φ azimuthal angle in spherical system 63ϕ, ϕi deformation mapping 72ψ specific Helmholtz free energy 193ψ, ψi 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 Cauchy–Green 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 Cauchy–Green deformation tensor 79






xx Notation�

C, CIJ K L 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, DiJ kL 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 Young’s modulus 235E, EIJ Lagrangian strain tensor 87E finite element strain operator matrix 302ei orthonormal basis vectors 23e, eij Euler–Almansi strain tensor 90F frame of reference 196F ext , F ext

i 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 ext

0 ,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 Piola–Kirchhoff stress tensor 122P matrix representation of the first Piola–Kirchhoff stress 301p pressure (or hydrostatic stress) 119






xxi Notation�

ΔQ heat transferred to a system during a process 140Qt orthogonal transformation between frames of reference 197Q,Qαi 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 176Sα shape function for finite element node α (physical space) 280S, SIJ second Piola–Kirchhoff 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) 231sα shape function for finite element node α (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) 94ΔW 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






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ContinuumMechanicsandThermodynamicsassets.cambridge.org/97811070/08267/frontmatter/9781107008267... · ContinuumMechanicsandThermodynamics ... and classical thermodynamics. It