Geometry, Mechanics, and Dynamics

Geometry, Mechanics,and Dynamics

Editors:Paul Newton, Phil Holmes, and Alan Weinstein

Paul NewtonDepartment of Aerospace andMechanical Engineering andDepartment of MathematicsUniversity of Southern California,Los Angeles, CA 90089-1191USAnewton@spock.usc.edu

Philip HolmesDepartment of Applied andComputational MathematicsEngineering QuadranglePrinceton UniversityPrinceton, NJ 08544-1000USApholmes@rimbaud.princeton.edu

Alan WeinsteinDepartment of MathematicsUniversity of California, BerkeleyBerkeley, CA 94720 USAalanw@math.berkeley.edu

Cover Illustration: Permission has been granted for use of the thunderstorm photograph on thecover by Kyle Poage, General Forecaster, National Weather Service, Dodge City, KS 67801,USA. The photo is of a spectacular thunderstorm that occurred at sunset over northwestKansas in August, 1996. The view is to the east from Norton, Kansas. It was taken whileKyle was at Saint Louis Univeristy (SLU) and was featured as a cover photo for the SLUDepartment of Earth and Atmospheric Sciences homepage (8 September 1998).http://www.eas.slu.edu/Photos/photos.html

The photograph of Jerry Marsden on page v was taken by photographer Robert J. Paz, PublicRelations, California Institute of Technology, Pasadena, CA 91109, USA.

Mathematics Subject Classication (2000), 00B1D, 37-02, 53-02, 58-02, 70-02, 73

IP data to come

ISBN 0-387-91185-6 Printed on acid-free paper.

c 2002 by Springer-Verlag New York, Inc.All rights reserved. This work may not be translated or copied in whole or in part with-out the written permission of the publisher (Springer-Verlag NewYork, Inc., 175FifthAvenue,New York, NY 10010, USA), except for brief excerpts in connection with reviews or schol-arly analysis. Use in connection with any form of information storage and retrieval, electronicadaptation, computer software, or by similar or dissimilar methodology now known or here-after developed is forbidden. The use in this publication of trade names, trademarks, servicemarks, and similar terms, even if they are not identied as such, is not to be taken as anexpression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America.

9 8 7 6 5 4 3 2 1 SPIN 10882145

Typesetting: Pages were created from author-prepared LATEX manuscripts by the technicaleditors, Wendy McKay and Ross Moore, using modications of a Springer LATEX macro pack-age, and other packages for the integration of graphics and consistent stylistic features withinarticles from diverse sources.

www.springer-ny.com

Springer-Verlag New York, Berlin, Heidelberg

A member of BertelsmannSpringer Science+Business Media GmbH

To Jerry Marsdenon the occasion of his 60th birthday,

with admiration, aection, and best wishesfor many more years of creativity.

Photo by Robert J. Paz

Contents

Preface ix

I Elasticity and Analysis 1

1 Some Open Problems in Elasticityby John Ball 3

2 Finite Elastoplasticity Lie Groups and Geodesics on SL(d)by Alexander Mielke 61

3 Asynchronous Variational Integratorsby Adrian Lew and Michael Ortiz 91

II Fluid Mechanics 111

4 EulerPoincare Dynamics of Perfect Complex Fluidsby Darryl D. Holm 113

5 The Lagrangian Averaged Euler (LAE-) Equationswith Free-Slip or Mixed Boundary Conditionsby Steve Shkoller 169

6 Nearly Inviscid Faraday Wavesby Edgar Knobloch and Jose M. Vega 181

7 The Variational Multiscale Formulation of LESwith Application to Turbulent Channel Flowsby Thomas J. R. Hughes and Assad A. Oberai 223

III Dynamical Systems 241

8 Patterns of Oscillation in Coupled Cell Systemsby Martin Golubitsky and Ian Stewart 243

9 Simple Choreographic Motions of N Bodies:A Preliminary Studyby Alain Chenciner, Joseph Gerver, Richard Montgomeryand Carles Simo 287

10 On Normal Form Computationsby Jurgen Scheurle and Sebastian Walcher 309

vii

viii

IV Geometric Mechanics 327

11 The Optimal Momentum Mapby Juan-Pablo Ortega and Tudor S. Ratiu 329

12 Combinatorial Formulas for Products of Thom Classesby Victor Guillemin and Catalin Zara 363

13 Gauge Theory of Small Vibrations in PolyatomicMoleculesby Robert G. Littlejohn and Kevin A. Mitchell 407

V Geometric Control 429

14 Symmetries, Conservation Laws, and Controlby Anthony M. Bloch and Naomi E. Leonard 431

VI Relativity and Quantum Mechanics 461

15 Conformal Volume Collapse of 3-Manifoldsand the Reduced Einstein Flowby Arthur E. Fischer and Vincent Moncrief 463

16 On Quantizing Semisimple Basic Algebras, I: sl(2, R)by Mark J. Gotay 523

VII Jerrold Marsden, 1942 537

Curriculum Vitae 539

Some Research Highlights 541

Graduate Students and Post Doctoral Scholars 545

Publications 549

Contributors 569

Preface

Jerry Marsden, one of the worlds pre-eminent mechanicians and appliedmathematicians, celebrated his 60th birthday in August 2002. The eventwas marked by a workshop on Geometry, Mechanics, and Dynamics atthe Fields Institute for Research in the Mathematical Sciences, of which hewas the founding Director. Rather than merely produce a conventional pro-ceedings, with relatively brief accounts of research and technical advancespresented at the meeting, we wished to acknowledge Jerrys inuence as ateacher, a propagator of new ideas, and a mentor of young talent. Conse-quently, starting in 1999, we sought to collect articles that might be usedas entry points by students interested in elds that have been shaped byJerrys work. At the same time we hoped to give experts engrossed in theirown technical niches an indication of the wonderful breadth and depth oftheir subjects as a whole.

This book is an outcome of the eorts of those who accepted our invi-tations to contribute. It presents both survey and research articles in theseveral elds that represent the main themes of Jerrys work, includingelasticity and analysis, uid mechanics, dynamical systems theory, geo-metric mechanics, geometric control theory, and relativity and quantummechanics. The common thread running through this broad tapestry is theuse of geometric methods that serve to unify diverse disciplines and bring awide variety of scientists and mathematicians together, speaking a languagewhich enhances dialogue and encourages cross-fertilization. We hope thatthis book will serve as a guide to these exciting and rapidly evolving areas,and that it will be a resource both for the student intent on contributingto one of these elds and to the seasoned practitioner who seeks a broaderview.

Jerry is a unique gure in mathematical circles because his work hassignicantly inuenced four often (alas!) separate research communities:pure mathematicians, applied mathematicians, physicists, and engineers.Foundations of Mechanics (with Ralph Abraham [294]), rst published in1967 while Jerry was a graduate student at Princeton, has for the past 35years been a landmark and inspiration in the eld of mechanics; duringthat time, Jerry and his collaborators have done extraordinary work ina huge variety of sub-elds of mechanics, geometry, and dynamics. RalphAbraham recalls: The rst edition of Foundations of Mechanics included,in my Preface, a few words on the genesis of the book as Jerrys notes ofmy lectures in early 1966. I well recall the rst meeting of that graduatecourse. At the outset I announced a desire for volunteers to make notes

ix

which might be duplicated for the use of students, as there was at thattime no text we could follow. And at the end of that rst meeting, onlyone volunteer: Jerry. He was a new face for me, and seemed rather youngand quiet, and I told him I hoped that others would volunteer for a teameort on the notes. Well, there were no other volunteers, which was justas well. For shortly after each lecture Jerry would deliver a thick sheaf ofhandwritten notes, usually without a single error. Many details omitted inmy talks were lled in with proofs, references, and so on, in the now-famousMarsden style. And the rest, as they say, is history. By now many peopleknow that Jerry is an ideal coworker and coauthor, and I was lucky to bean early benefactor of his wonderful talents and personality.

A talented and prolic expositor, Jerry has written numerous otherbooks, from elementary to advanced level, in addition to his many researcharticles. Mathematical Foundations of Elasticity (with Tom Hughes [300])introduced a generation of engineers with appetites for abstraction to a uni-ed and global approach to the subject, and his recent book Introduction toMechanics and Symmetry, (with Tudor Ratiu [303]) has been remarkablyuseful to a wide range of scientists and engineers. When Jerry won the 1990Norbert Wiener Prize (jointly with Michael Aizenman), he noted in his re-sponse to the citation that Wiener was classiable neither as a pure noran applied mathematician. He had breadth and depth that worked togetherin a mutually supportive way. The same is true of Jerry: it is no accidentthat he began his career in mathematical physics, moved to a mathematicsdepartment, and is now working in the Division of Engineering and AppliedScience at Caltech.

Jerrys inuence on mathematical education has also been signicant.His books on calculus and complex variables are widely used and, withtheir skillful blend of concreteness and abstraction, have inuenced genera-tions of undergraduates. Thorough and wide-ranging in their coverage, theyleave the conscientious student with a solid grounding in both theoreticaltechniques and physical intuition. Jerrys Ph.D. and postdoctoral students,some of them now leaders in their elds, have made signicant contribu-tions in many areas themselves. In addition, Jerry has worked tirelessly forthe mathematical community, serving on editorial boards and arrangingconferences and workshops, all the while teaching a stellar array of under-graduate and graduate students and post-docs, rst at UC Berkeley, andnow at Caltech.

His extraordinarily inuential paper with David Ebin [13], on the analysisof ideal uid ows remains a classic in the eld. It followed upon Arnolds1966 paper1 on ideal uid ows, which showed how the Euler dynamics for

1Arnold, V. I. [1966], Sur la geometrie dierentielle des groupes de Lie de dimensioninnie et ses applications a lhydrodynamique des uids parfaits, Ann. Inst. Fourier,Grenoble, 16 , 319361.

rigid bodies and uids could be viewed as geodesic ow on SO(3) with aleft-invariant metric, and on Divol()the volume preserving dieomor-phism group of a region in R3 with the right invariant metric denedby the uid kinetic energy. Ebin and Marsden [13] put this work in thecontext of Sobolev (Hs) manifolds and showed that Arnolds geodesic owon Hs Divol(), the volume preserving dieomorphisms of to itselfof Sobolev class Hs, comes from a smooth geodesic spray. This allowedthem to show that the initial value problem for the Euler equations couldbe solved using Picard iteration and techniques from ordinary dierentialequation theory. Stephen Smale has remarked: Jerry Marsden has manyne achievements to his credit. But I am particularly fond of his earlywork with David Ebin on the equations of uid mechanics. There are manysides to this study. It gave formal ideas of Arnold great substance and pro-vided an elegant way of presenting old and new fundamental work on theexistence of solutions of NavierStokes and Euler equations. The rigorousgroup setting and one of the rst important uses of innite dimensionalmanifolds are there as well. Quite a milestone in mathematics!

His body of work on reduction theory, begun with Alan Weinstein, wasan outgrowth of ideas developed by Smale (following Jacobi and others),who introduced the use of symmetry ideas in the context of tangent andcotangent bundles of conguration spaces with Hamiltonians in the formof kinetic plus potential energy. The Marsden and Weinstein paper [30]unied approaches of both Smale2 and Arnold by putting this reductiontheory in the context of symplectic manifolds. For instance, in the relatedPoisson context (developed by his student Richard Montgomery) if onestarts with a cotangent bundle T Q and a Lie group G acting on Q, thenthe quotient (T Q)/G is a bundle over T (Q/G) with ber g , the dual ofthe Lie algebra of G. As described by Marsden [170], Thus, one can sayperhaps with only a slight danger of oversimplicationthat reductiontheory synthesizes the work of Smale, Arnold (and their predecessors ofcourse) into a bundle, with Smale as the base and Arnold as the ber.Reduction theory has now been used successfully in a wide variety of elds,and we refer the reader to the overview articles by Marsden [170; 227] aswell as many of the articles in this volume for current applications.

From these works emerge more than specic theorems and techniques,deep and elegant as they may be. Viewing Jerry Marsdens contributionsas a whole, one nds a clear, pedagogical, and fundamental approach tothe subject of mechanics that blends geometry, analysis, and dynamics inpowerful, yet practical ways. Thus, while developing abstract techniquesin dynamical systems theory, Jerry also helped understand specic or-bit trajectories (with a group at the Jet Propulsion Lab) that were usedin the Genesis Discovery Mission, launched on August 8, 2001, [244]. In

2Smale, S. [1970], Topology and mechanics, Invent. Math., 10 , 305331; 11 , 4564.

xii Preface

the course of developing the averaged uid equations (with Holm, Ratiu,and Shkoller), he also contributed to their use in turbulent ow compu-tations [267]. While working out innite dimensional versions of the Mel-nikov method and SmaleBirkho theory to prove the existence of Smalehorseshoes in the context of partial dierential equations, the chaotic os-cillations of a forced beam were being analyzed [71]; and while developingsymplectic-energy-momentum preserving variational integrators based ondiscrete variational principles, Jerry contributed to specic projects (withMichael Ortiz) to simulate the crushing of aluminum cans and analyzefracture mechanics and collision problems [253; 284].

In each of the general areas noted above, we have solicited survey andresearch articles that illustrate more specically how some of the methodspioneered by Marsden are currently being used.

For Elasticity and Analysis, the paper by J.M.Ball entitled Someopen problems in elasticity is a self-contained overview which highlightssome general open problems in elasticity theory, including some new resultsshowing that local minimizers of the total elastic energy satisfy a weak formof the equilibrium equations. This is followed by the article of A.Mielke,Finite elastoplasticity, Lie groups and geodesics on SL(d) which inter-prets notions of nonlinear plasticity theory in terms of Lie groups, amongother things. The contribution of A. Lew and M.Ortiz, entitled Asyn-chronous variational integrators describes a new class of algorithms fornonlinear elastodynamics which is based upon a discrete version of Hamil-tons principle.

D.D.Holms article in Fluid Mechanics, EulerPoincare dynamics ofperfect complex uids, describes the use of Lagrangian reduction by stagesto derive the EulerPoincare equations for non-dissipative motion of exoticuids such as liquid crystals, superuids, Yang-Mills magnetouids andspin-glass systems. Inclusion of defects, such as vortices, in the order param-eters is also treated. S. Shkollers contribution, The Lagrangian averagedEuler (LAE) equations with free-slip or mixed boundary conditions,presents a simple proof of well-posedness of the Euler- equations withnovel boundary conditions. E.Knoblochs and J.Vegas article, Nearlyinviscid Faraday waves, explores some of the consequences of introduc-ing small viscosity in the study of surface-gravity-capillary waves excitedby vertical vibration of a uid layer. The contribution of T. J. R.Hughesand A.A.Oberai, The variational multiscale formulation of LES with ap-plications to turbulent channel ows, studies turbulent two-dimensionalequilibrium and three-dimensional non-equilibrium channel ows using avariational multi-scale formulation of Large Eddy Simulation (LES).

In Dynamical Systems Theory, M.Golubitsky and I. Stewart addressPatterns of oscillation in coupled cell systems. The dynamics of coupledcell systems both in biological contexts (animal gaits) and physical contexts(coupled pendula/Josephson junctions) are described, with an emphasis on

Preface xiii

the use of symmetry ideas. In particular, the issue of how the modeling as-sumptions dictate the kinds of equilibria and periodic solutions is explored.This is followed by the paper of A.Chenciner, J.Gerver, R.Montgomery,and C. Simo, Simple choreographic motions of N bodies: A preliminarystudy. They describe the existence of new periodic solutions to the N -body problem in which all N masses trace the same curve without col-liding. J. Scheurle and S.Walchers On normal form computations closesthis section by reviewing computational procedures involved in transform-ing a vector eld into a suitable normal form about a stationary point.

For Geometric Mechanics, the paper by J. P.Ortega and T.Ratiu en-titled The optimal momentum map discusses the (dare we say) classicalMarsdenWeinstein reduction procedure and the use of a new optimal mo-mentum map which more eciently encodes symmetry information of theunderlying Hamiltonian system. V.Guillemins and C. Zaras Combinato-rial formulas for products of Thom classes obtains combinatorial descrip-tions of the equivariant Thom class dual to the MorseWhitney stratica-tion of compact Hamiltonian G-manifolds. The paper of R. Littlejohn andK.Mitchell, Gauge theory of small vibrations in polyatomic molecules,considers molecular vibrations in the context of gauge theory and berbundle theory.

In Geometric Control Theory, the paper by A.Bloch and N. Leonard,entitled Symmetries, conservation laws, and control traces the role ofMarsdens ideas on reduction and symmetries in the setting of nonlinearcontrol theory. Specic applications to the dynamics of rigid spacecraft witha rotor and the dynamics of underwater vehicles are considered in detail.

Finally, for Relativity and Quantum Mechanics, A. E. Fischer andV.Moncriefs article entitled Conformal volume collapse of 3-manifoldsand the reduced Einstein ow describes the Hamiltonian reduction of Ein-steins equations of general relativity and the process of volume collapse.They prove that collapse occurs either along circular bers, embedded tori,or completely to a point, but surprisingly, always with bounded curvature.This is followed by M.Gotays contribution On quantizing semisimple ba-sic algebras which examines whether there exists a consistent quantizationof the coordinate ring of a basic coadjoint orbit of a semisimple Lie group.

We hope that this collection of articles gives the reader some appreciationof both the unity and diversity of the topics inuenced by Jerry Marsdensapproach to mechanics. But here we wish to do more than survey his math-ematical and scientic contributions; we also want to celebrate Jerry as acolleague and friend. It therefore seems appropriate to conclude with somepersonal reminiscences.

xiv Preface

Phil Holmes: I rst met Jerry in the summer of 1976, at a conference ondynamical systems at Southampton University, organised by David Randand Brian Griths. He joined Nancy Kopell, John Guckenheimer and KenCooke as one of four mathematicians from the USA invited to that meet-ing. I had completed my Ph.D. in Engineering (experimental studies ofdispersive wave propagation in structures) at the Institute of Sound andVibration a couple of years earlier, and had begun working on nonlinearvibration problems with David Rand. We had done some single and nitedegree of freedom problems, and I wanted to begin looking at PDEs instructural mechanics from a dynamical systems perspective. I believe itwas in late 1975 that someone told me Jerry was working on a book aboutbifurcations and dimension reduction for such problems. I wrote to ask formore information and back came a huge package, re-taped and tied withstring by UK customs, containing a 500+ page photocopy of the typescriptof The Hopf Bifurcation and its Applications by Marsden and McCracken[297]. In nancially-constrained Britain I had never seen more than ftypages of xerox copies (all copies were xerox copies in those days) at onetime, without special permission. I started reading, and Im still readingJerrys papers and trying to catch up.

Jerry and I began corresponding. We met at the Southampton conferenceand I subsequently visited him in Berkeley during a hectic job-seeking tourof the USA in the Fall of 1976, and again during his visit to HeriotWattUniversity in Edinburgh as a Carnegie Fellow in the spring of 1977. Thisresulted in our rst joint paper [54], and was the beginning of a twenty-ve year collaboration and friendship which I hope will last at least an-other twenty ve. For me, one of the high points of this was our paper[71], in which we gave one of the rst examples of a PDE with chaoticsolutions (Smale horseshoes), via an innite-dimensional extension of theSmaleBirkho theorem and Melnikovs method. (John Guckenheimer gaveanother at about the same time via center-manifold reduction of a reaction-diusion equation at a codimension-two bifurcation point.) After I had set-tled in the USA at Cornell University, Jerry invited me to Berkeley for theSpring semester of 1981, during which we wrote a series of papers [73; 77; 82]extending Melnikov type analyses to multi-degree-of-freedom Hamiltoniansystems (although not without leaving a few gaps in our proofs to be lledby others, in the time-honored tradition of Poincare).

While we have not written joint papers in the last ten years, his work atthe interface of mechanics and mathematics has remained an inspirationfor my own, and we have met once or twice every year and had countlessscientic, editorial, organizational, and mathematico-political discussionsand collaborations. Jerry is a mainstay of the Journal of Nonlinear Science,which I now edit, and Im proud to serve as an advisor to the SpringerApplied Mathematical Sciences Series which Jerry edits with Larry Sirovichand Stu Antman. I was even prouder to nominate him for the AMSSIAM

Preface xv

Norbert Wiener Prize in 1990, and to support his successful nominationsto the Royal Society of Canada and the American Academy of Arts andSciences.

But rather than these well-deserved honors, I especially wish to celebrateJerrys continuing emphasis on mentoring and encouraging young people.Few people outside academia, and few Deans and Presidents within it,realise that a large part of research is actually teaching: teaching brightbut sometimes erratically-educated graduate students the necessary back-ground and methods, teaching colleagues and collaborators about new ad-vances, and teaching oneself all the things that no one else did. Jerry is amaster teacher: in his many textbooks at all levels, and in his conferencepresentations and lecture courses, always delivered with elegance, polish,and a little humor. (He is almost the only person I know who can put con-tent into powerpointalthough hes also careful to explain that its notactually powerpoint.)

At Berkeley and Caltech Jerry has had, and continues to have, a succes-sion of wonderful Ph.D. students and postdocs, many of whom have goneon to propagate his grand project of geometrizing mechanics (their namesappear elsewhere in this volume). He has been equally generous with histime with young visitors (for many of whom, including myself, he raised thefunds to invite), with the students of others, and simply with people whowrite or approach him to ask questions at conferences and workshops. Imhappy that weve been able to include articles contributed by several suchcolleagues in this Festschrift. Certainly, Jerrys interest and involvementin the early struggles of a mechanic poorly trained in mathematics wasenormously encouraging to me. In those far-o days, from misty England,he seemed to me a senior scientist: a Professor from distant and fabledBerkeley, David Lodges Euphoric State U. Now that we are both almostseniors, he no longer seems that much older, but he still knows a lot moregeometry and analysis, and Im still taking notes in the second row andhaving trouble with the homework.

Paul Newton: Like many of us, I rst met Jerry in print. As a Freshmanat Harvard in 1977, I learned Stokes theorem, Greens theorem and thedivergence theorem from his (and Trombas) beautiful Vector Calculus.Those who are familiar with the rst edition and who are aware of Jerrysfascination with weather patterns will suspect that his favorite aspect ofthe book must have been the cloud formations on its cover. Mine wasthe elegant formulation of these theorems in terms of dierential forms,something I had never seen in high school! I remember using this work tosuch an extent (so much for my social life) that today it is held togetheronly by being wedged between two volumes on my shelf.

Fast forward eight years to 1985. While a postdoc at Stanford, I par-ticipated (inconspicuously) in a seminar series on dynamics which Jerry,

xvi Preface

together with Lieberman, ran at Berkeley, and it was there that I rst tooknote of what I now know to be his uncommon blend of open-mindednessand depth of thought, coupled with a generosity of spirit, demonstrated sovividly in his mentoring of young mathematicians.

But it was when we both ended up in Los Angeles at roughly the sametime (1993 in my case) that we became friends. I was busy working outthe details of how the geometric phase manifested itself in the contextof vortex dynamics problems, and Jerrys encouragement and insights wereinvaluable in helping me move from an early interest in nonlinear dispersivewave models to more general issues in applied dynamical systems theory.Our interests overlapped again when I showed him some problems involvingthe motion of vortices on a sphere (with applications to weather patterns!).This led him (with Sergey Pekarsky) to begin applying nonlinear stabilitytechniques to relative equilibrium congurations of vortices on a sphere.

As I read and re-read many of his books and papers, I seem to fall fartherand farther behind. But occasionally I look up to where it all began, thepunished copy of Vector Calculus on my bookshelf, and I wonder if Iwould have become a mathematician had my professor chosen Edwardsand Penney instead of Marsden and Tromba.

Alan Weinstein: My work with Jerry has two facets: (1) symplectic re-duction, Poisson geometry, and applications to stability of continuum me-chanical systems; (2) calculus books.

Our original work on reduction, which is probably one of the two orthree most inuential papers I have written, was stimulated by our listen-ing to lectures of Smale around 1970; Smale had developed the theory inthe special case of lifted actions on cotangent bundles. I think that myinterest in abstract symplectic geometry, combined with some interest inphysics, meshed perfectly with Jerrys interest in relativity and appliedHamiltonian dynamics. (One should always mention in this context thatsymplectic reduction was discovered independently at about the same timeby Ken Meyer, though I think it might be fair to say that he conceived ofthis construction in narrower terms than we did.)

About 10 years later, we were attending the dynamics seminar in theBerkeley physics department, where Allan Kaufman and Robert Little-john were studying recent work of John Greene and Phil Morrison on theHamiltonian structure of equations in plasma physics. These authors hada Poisson structure for the MaxwellVlasov equations which they found bytrial and error and for which they checked the Jacobi identity by hand;according to Morrison, this took them 4 months of work, mostly calcula-tions. Jerry and I spent about 6 months developing the right application ofreduction to this problem, after which we could derive the correct bracketin 4 minutes, with the Jacobi identity coming for free. Another payo wasthat we discovered an error in the MorrisonGreene formula.

Preface xvii

It was Jerrys interest in applications which kept us going through muchof the 80s, applying these Poisson brackets to applications of Arnolds gen-eral method for analyzing the stability of continuum-mechanical motions.Much of this work was also done in collaboration with Darryl Holm andTudor Ratiu, and I eventually dropped out of the group, but the subjecthas continued to evolve in the hands of Jerry and his collaborators (nowincluding notably Steve Shkoller), the latest developments being the studyof -Euler equations and wide applications of Lagrangian (as opposed toHamiltonian) methods for the study of stability.

On the calculus side, I recall that our collaboration started with a discus-sion at the end of a tennis game. Jerry was in contact with a new publisherwho wanted to do a calculus book, and we had some new ideas for cal-culus teaching, based on the use of bifurcation ideas to replace the earlyintroduction of limits (which eventually appeared only in a spin-o bookcalled Calculus Unlimited). We went through several publishers and manyversions of the book, and I never had time to play tennis again. I thinkthat Jerry kept it (and squash) up, though.

Acknowledgments: The editors owe a debt of gratitude to Wendy McKayand Ross Moore. Without their combined expertise in LATEX and othermatters, this volume might not have been presented to Jerry until his 70thbirthday. We also wish to thank the editors and sta at Springer-Verlag,particularly Achi Dosanjh and Elizabeth Young, for making this book areality.

Paul NewtonSanta Barbara, California

Philip HolmesPrinceton, New Jersey

Alan WeinsteinBerkeley, California

March 2002

Part I

Elasticity and Analysis

1

1Some Open Problems inElasticity

John M. Ball

To Jerry Marsden on the occasion of his 60th birthday

ABSTRACT Some outstanding open problems of nonlinear elasticity aredescribed. The problems range from questions of existence, uniqueness,regularity and stability of solutions in statics and dynamics to issues suchas the modelling of fracture and self-contact, the status of elasticity withrespect to atomistic models, the understanding of microstructure inducedby phase transformations, and the passage from three-dimensional elasticityto models of rods and shells. Renements are presented of the authorsearlier work Ball [1984a] on showing that local minimizers of the elasticenergy satisfy certain weak forms of the equilibrium equations.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 42 Elastostatics . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 The Stored-Energy Function and Equilibrium Solutions 42.2 Existence of Equilibrium Solutions . . . . . . . . . . 62.3 Regularity and the Classication of Singularities . . 112.4 Satisfaction of the EulerLagrange Equation and

Uniform Positivity of the Jacobian . . . . . . . . . . 152.5 Regularity and Self-Contact . . . . . . . . . . . . . . 222.6 Uniqueness of Solutions . . . . . . . . . . . . . . . . 222.7 Structure of the Solution Set . . . . . . . . . . . . . 232.8 Energy Minimization and Fracture . . . . . . . . . . 26

3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Continuum Thermomechanics . . . . . . . . . . . . . 273.2 Existence of Solutions . . . . . . . . . . . . . . . . . 293.3 The Relation Between Statics and Dynamics . . . . 33

4 Multiscale Problems . . . . . . . . . . . . . . . . . . . 364.1 From Atomic to Continuum . . . . . . . . . . . . . . 364.2 From Microscales to Macroscales . . . . . . . . . . . 384.3 From Three-Dimensional Elasticity to Theories of Rods

and Shells . . . . . . . . . . . . . . . . . . . . . . . . 42References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3

4 John M. Ball

1 Introduction

In this paper I highlight some outstanding open problems in nonlinear(sometimes called nite) elasticity theory. While many of these will be wellknown to experts on analytic aspects of elasticity, I hope that the compila-tion will be of use both to those new to the eld and to researchers in solidmechanics having dierent perspectives. Of course the selection of prob-lems is a personal one, and indeed represents a list of those problems thatI would most like to be able to solve, but cannot. In particular it concen-trates on general open problems, or ones that illustrate general diculties,rather than those related to very specic experimental situations, which isnot to imply that the latter are not important or instructive. I have notincluded any open problems connected with the numerical computation ofsolutions, since I recently discussed some of these in Ball [2001].

The only new results of the paper are in connection with the problem ofshowing that local minimizers of the total elastic energy satisfy the weakform of the equilibrium equations. As I pointed out in Ball [1984a], thereare hypotheses under which some forms of the equilibrium equations canbe proved to hold, and in Section 2.4 I take the opportunity to presentsome renements of this old work.

The paper is essentially self-contained, and can be read by those havingno knowledge of elasticity theory. For those seeking further backgroundon the subject I have written a short introduction (Ball [1996]) to someof the issues, intended for research students, which I hope is a quick andeasy read. For more serious study in the spirit of this paper, the reader isreferred to the books of Antman [1995], Ciarlet [1988, 1997, 2000], Marsdenand Hughes [1983] and Silhavy [1997]. Other excellent but older booksand survey articles are Antman [1983], Ericksen [1977b], Gurtin [1981] andTruesdell and Noll [1965]. Valuable additional perspectives can be found inthe books of Green and Zerna [1968], Green and Adkins [1970], and Ogden[1984].

It is an honour to dedicate this article to Jerry Marsden, both as a friendand in recognition of his important contributions to elasticity, and thus tohelp celebrate his many talents as a mathematician, thinker and writer.

2 Elastostatics

2.1 The Stored-Energy Function and EquilibriumSolutions

Consider an elastic body which in a reference conguration occupies thebounded domain R3. We suppose that has a Lipschitz boundary = 1 2 N , where 1, 2 are disjoint relatively open subsetsof and N has two-dimensional Hausdor measure H2(N) = 0 (i.e., N

1. Some Open Problems in Elasticity 5

has zero area). Deformations of the body are described by mappings

y : R3 ,where y(x) =

(y1(x), y2(x), y3(x)

)denotes the deformed position of the

material point x = (x1, x2, x3). We assume that y belongs to the Sobolevspace W 1,1(;R3), so that in particular the deformation gradient Dy(x)is well dened for a.e. x . For each such x we can identify Dy(x) withthe 3 3 matrix (yi/xj).

We require the deformation y to satisfy the boundary condition

y1

= y( ) , (2.1)

where y : 1 R3 is a given boundary displacement.We suppose for simplicity that the body is homogeneous, i.e., the material

response is the same at each point. In this case the total elastic energycorresponding to the deformation y is given by

I(y) =

W(Dy(x)

)dx , (2.2)

where W = W (A) is the stored-energy function of the material. We supposethat W : M33+ R is C1 and bounded below, so that without loss ofgenerality W 0. (Here and below, Mmn denotes the space of real mnmatrices, and Mnn+ denotes the space of thoseA Mnn with detA > 0.)The PiolaKirchho stress tensor is given by

TR(A) = DAW (A) . (2.3)

By formally computing

dd

I(y + )=0

= 0 ,

we obtain the weak form of the EulerLagrange equation for I, that is

DAW (Dy) Ddx = 0 (2.4)

for all smooth with |1 = 0. This can be shown (cf. Antman and Os-born [1979]) to be equivalent to the balance of forces on arbitary subbodies.If y, 1 and 2 are suciently regular then (2.4) is equivalent to thepointwise form of the equilibrium equations

divDAW (Dy) = 0 in , (2.5)

together with the natural boundary condition of zero applied traction

DAW (Dy)n = 0 on 2 , (2.6)

6 John M. Ball

where n = n(x) denotes the unit outward normal to at x. (More gener-ally, we could have prescribed nonzero tractions of various types on 2, aswell as including the potential energy of body forces such as gravity in theexpression for the energy (2.2), but for simplicity we have not done this,since the main diculties we address are already present without theseadditions.)

To avoid interpenetration of matter, it is natural to require that y : R3 be invertible. To try to ensure that deformations have this property, wesuppose that

W (A) as detA 0+ . (2.7)So as to also prevent orientation reversal we dene W (A) = if detA 0.Then W : M33 [0,] is continuous. Clearly if I(y) 0 for a.e. x . (2.8)

Since y is not assumed to be C1, (2.8) does not imply even local invertibility.For studies of local and global invertibility in the context of elasticity, orrelevant to it, see Ball [1981], Bauman and Phillips [1994], Ciarlet andNecas [1985], Fonseca and Gangbo [1995], Giaquinta, Modica and Soucek[1994], Meisters and Olech [1963], Sverak [1988] and Weinstein [1985].

We assume that for any elastic material the stored-energy function W isframe-indierent, i.e.,

W (RA) = W (A) for all R SO(3) , A M33 . (2.9)

In addition, if the material has a nontrivial isotropy group S, W satisesthe material symmetry condition

W (AQ) = W (A) for all Q S , A M33 .

The case S = SO(3) corresponds to an isotropic material.For incompressible materials the deformation y is required to satisfy the

constraintdetDy(x) = 1 for a.e. x .

All of the problems and results contained in this article have correspondingincompressible versions, some of which we cite in the references. However,in general we do not state these explicitly.

2.2 Existence of Equilibrium Solutions

There are two traditional routes to proving the existence of equilibrium so-lutions. The rst, pioneered by Stoppelli [1954, 1955] and described in thebook of Valent [1988], is to use the implicit function theorem in a suitableBanach space X to prove the existence of an equilibrium solution close to agiven one, when the data of the problem are slightly perturbed. In order to

1. Some Open Problems in Elasticity 7

make this work, it is necessary to use spaces X of suciently smooth map-pings, for example subspaces of W 2,p(;R3) for p > 3 or C2+(;R3), soas to control the nonlinear dependence on Dy. In addition, the linearizedelasticity operator at the given solution should be invertible as a map fromX to a suitable target space Y . While this method automatically deliv-ers smooth solutions, it is by its nature restricted to small perturbations(for example, small boundary displacements from a stress-free state), andbecause of the regularity properties required for the linearized operator itin general only applies to situations when 1 and 2 do not meet, forexample when one of them is empty. In particular, mixed boundary con-ditions typically encountered in applications, for example (2.1) with 1comprising the two end-faces of a cylindrical rod, are in general not allowed,at least with the techniques as currently developed (see Section 2.7).

The second route is to prove the existence of a global minimizer of I viathe direct method of the calculus of variations. In principle such a minimizershould satisfy the equilibrium equations, at least in weak form, but thisturns out to be a subtle matter (see Sections 2.3, 2.4). More generallywe could ask for conditions ensuring that there exist some kind of localminimizer.

LetA = {y W 1,1(;R3) : I(y) 0 such that I(z) I(y) for any z Awith z yW 1,p .

The problem of proving the existence of local, but not global, minimizersis discussed later (see Problem 9). A typical result on global minimizationis the following.

2.2 Theorem. Suppose that W satises the hypotheses

(H1) W is polyconvex, i.e., W (A) = g(A, cof A,detA) for all A M33for some convex g,

(H2) W (A) c0(|A|2 + |cof A| 32 ) c1 for all A M33, where c0 > 0.

Then, if A is nonempty, there exists a global minimizer y of I in A.Here and below we take | | to be the Euclidean norm on M33 withcorresponding inner product AB = T(ATB). Theorem 2.2 is a renementby Muller, Qi and Yan [1994] of the result in Ball [1977a]. For the problemto be nontrivial we need that H2(1) > 0.

The hypothesis (H1) is known to be too strong for the following reason.Let f : Mmn R {+} be Borel measurable and bounded below. Werecall f is said to be quasiconvex at A Mmn if the inequality

f

A+D(x)

dx

f(A) dx (2.10)

holds for any C0 (;Rm), and is quasiconvexif it is quasiconvex atevery A Mmn. Here Rn is any bounded open set whose boundary has zero n-dimensional Lebesgue measure. A standard scaling argument(see, for example, Ball and Murat [1984]) shows that contrary to appear-ances these denitions do not depend on . Results of Morrey [1952] andAcerbi and Fusco [1984] imply that if f : Mmn R is quasiconvex andsatises the growth condition

C1|A|p C0 f(A) C2 |A|p + 1 for all A Mmn, (2.11)

where p > 1 and where C0 and C1 > 0, C2 > 0 are constants, then

F(y) =

f(Dy) dx (2.12)

attains a global minimum on

A = {y W 1,1(;Rm) : y|1 = y}.Here we assume that has Lipschitz boundary , that 1 is Hn1measurable and that y : 1 Rm is given such that A is nonempty. Asshown by Ball and Murat [1984], quasiconvexity of f is necessary for theexistence of a global minimizer for all perturbed functionals of the form

F1(y) =

f

Dy(x)

+ h

x,y(x)

dx

with h( , ) 0 continuous. These results strongly suggest that (H1)should be replaced by the requirement that W be quasiconvex, a weakercondition than polyconvexity. For example, it is easily seen that a largerclass of W for which Theorem 2.2 holds consists of those of the formW = W1 + f , where W1 satises (H1) and (H2), and where f : M33 Ris quasiconvex and satises (2.11). That this really is a larger class can beseen by taking f = KF for a large K > 0, where F is quasiconvex but notpolyconvex. Such F exist satisfying F (RAQ) = F (A) for all R,Q SO(3),A M33, and can be constructed by the method of Sverak [1991]1. More

1For example we can take F = H qc to be the quasiconvexication ofH (A) = min U (A) 1 p , U (A) 1 p ,

where > 1 and U (A) = A T A 12 . The quasiconvexication H qc of H is dened to bethe supremum of all quasiconvex functions H .

generally we could take f to satisfy

f(A) C1|A|p C0 (2.13)

for some p > 1, C1 > 0, C0 and to be the supremum of a nondecreasingsequence of continuous quasiconvex functions fk : M33 [0,), eachsatisfying a growth condition

0 fk(A) k|A|p kfor constants k > 0, k. (Kristensen [1994] has shown that a lower semi-continuous function f : M33 [0,] satisfying (2.13) is the supremumof such a sequence if and only if f is closedW 1,p quasiconvexin the senseof Pedregal [1994], namely that Jensens inequality

, f

f() holds forall homogeneous W 1,p gradient Young measures2 .)

However, as they stand none of the existence theorems for minimizers ofintegrals of general quasiconvex functions apply to elasticity, since they allassume growth conditions such as (2.11) which are not consistent with thecondition (2.7). (The same applies to other results, such as the relaxationtheorem of Dacorogna [1982].) In particular, it is not clear whether or not aquasiconvex W satisfying our hypotheses can be written as the supremumof everywhere nite continuous quasiconvex functions. This is not true ingeneral for quasiconvex functions f : Mmn [0,]; for example we cantake m = n = 2, f(A) = 0 if A {A1,A2,A3,A4} and f(A) = otherwise, where the Ai are diagonal matrices in a Tartar conguration(see Tartar [1993]), for example

A1 = diag (2, 1), A2 = diag (1, 2),A3 = diag (2,1), A4 = diag (1,2).

Then f is quasiconvex, since any y with Dy {A1,A2,A3,A4} a.e. hasconstant gradient (this following, for example, from the general result ofChlebik and Kirchheim [2001]). But the argument of Tartar shows that iff were the supremum of continuous quasiconvex functions fk : M22 [0,) we would have f(0) = 0, a contradiction.

2SeeYoung [1969], Tartar [1979], Ball [1989] for the denition and properties of theYoung measure ( x )x corresponding to a sequence of mappings z( j ) : R s satis-fying a suitable bound, say z ( j ) L 1 M < , where R

n is open (or measurable).For each x , x is a probability measure on R s giving the limiting distribution ofvalues of z( j ) (p) as j and p x. If f : R s R is continuous, then the weak limitof f (z ( j ) ) in L 1 (E ), where E is measurable, is given by the function x x , f ,whenever the weak limit exists. In particular, if z( j ) z in L 1 (E ), then z(x) = fl x forx E , where fl denotes the centre of mass of a measure . Such a Young measure ishomogeneous if = x is independent of x. If 1 < p , a W 1,p gradient Young mea-sure is a Young measure ( x )x corresponding to a sequence z( j ) = D y ( j ) of gradientsbounded in L p (; M m n ), where we identify M m n with R mn .

10 John M. Ball

A further reason for preferring quasiconvexity to polyconvexity is that,unlike quasiconvexity, polyconvexity is not closed with respect to periodichomogenization (Braides [1994]).

Problem 1. Prove the existence of energy minimizers for elastostaticsfor quasiconvex stored-energy functions satisfying (2.7).

A principal diculty here is that there is no known useful characteriza-tion of quasiconvexity. If W is quasiconvex then W is rank-one convex,that is the map t W (A + ta n) is convex for each A Mmn anda Rm,n Rn. For 40 years it seemed possible that in fact rank-oneconvexity was equivalent to quasiconvexity, until Sverak [1992] found hiswell-known counterexample for the dimensions n 2, m 3. Then Kris-tensen [1999] used Sveraks example to show that for the same dimensionsthere is no local characterization of quasiconvexity. In the absence of acharacterization leading to a new proof technique, one is forced to makedirect use of the denition (2.10), which leads to serious problems of ap-proximation by piecewise ane functions when (2.7) holds.

In Ball [1977a] it was shown how the hypotheses (H1), (H2) can be sat-ised for a class of isotropic materials including models of natural rubbers,via theorems exploiting the representation

W (A) = (v1, v2, v3) (2.14)

of the stored-energy function W of an isotropic material, where is a sym-metric function of the singular values vi = vi(A), that is of the eigenvaluesof (ATA)1/2 (for a dierent proof of such theorems see Le Dret [1990]).However it is not obvious how to verify (H1) when the material is notisotropic, for example when it has cubic symmetry.

Problem 2. Are there ways of verifying polyconvexity and quasiconvexityfor a useful class of anisotropic stored-energy functions?

To illustrate the diculty in verifying (H1), in the isotropic case it ismuch more convenient to use the representation (2.14) rather than theequivalent representation W (A) = h(I1, I2, I3) in terms of the principalinvariants Ij = Ij(A). Perhaps it is signicant that the function in(2.14) has the same regularity as W , while h is less regular (see Ball [1984],Sylvester [1985], Silhavy [2000]). At any rate the more symmetric form(2.14) lends itself more easily to discussing convexity properties. For non-isotropic materials suitable representations do not seem to be available; forexample, in the case of cubic symmetry it does not seem to be convenientto use the usual integrity basis (given, for example, in Green and Adkins[1970]).

1. Some Open Problems in Elasticity 11

2.3 Regularity and the Classication of Singularities

The main open question concerning regularity is to decide when global, orlocal, minimizers of I are smooth. A special case is

Problem 3. When is the minimizer y in Theorem 2.2 smooth?

Here smooth means C in , and C up to the boundary (except inthe neighbourhood of points x0 1 2 where singularities can beexpected). Clearly additional hypotheses on W are needed for this to betrue. One might assume, for example, that W : M33+ R is C, and that(H1) is strengthened by assuming W to be strictly polyconvex (i.e., that gis strictly convex). Also for regularity up to the boundary we would needto assume both smoothness of the boundary (except perhaps at 12)and that y is smooth. The precise nature of these extra hypotheses is to bedetermined. Problem 3 is unsolved even in the simplest special cases. In factthe only situation in which smoothness of y seems to have been provedis for the pure displacement problem (2 empty) with small boundarydisplacements from a stress-free state. For this case Zhang [1991], followingwork of Sivaloganathan [1989], gave hypotheses under which the smoothsolution to the equilibrium equations delivered by the implicit function the-orem was in fact the unique global minimizer y of I given by Theorem 2.2.

An even more ambitious target would be to somehow classify possiblesingularities in minimizers of I given by (2.2) for generic stored-energyfunctions W . If at the same time one could associate with each such singu-larity a condition on W that prevented it, one would also, by imposing allsuch conditions simultaneously, possess a set of hypotheses implying regu-larity. In fact it is possible to go a little way down this road. Consider rstthe kind of singularity frequently observed at phase boundaries in elasticcrystals, in which the deformation gradient Dy is piecewise constant, withvalues A,B on either side of a plane {x n = k}. It was shown in Ball[1980] that, under the natural assumption that there is some matrix A0that is a local minimizer of W ( ), every such deformation y that is locallya weak solution of the EulerLagrange equation is trivial, that is A = B, ifand only if W is strictly rank-one convex (i.e., the map t W (A+ tan)is strictly convex for every A and all nonzero a,n). Thus strict rank-oneconvexity is exactly what is needed to eliminate this particular kind ofsingularity.

Another physically occuring singularity is that of cavitation. For radialcavitation the deformation has the form y : B(0, 1) R3, where B(0, 1)is the unit ball in R3, and

y(x) = r(|x|) x|x| .

Thus if r(0) > 0, y is discontinuous at x = 0, where a hole of radius r(0)is formed. If (H1) holds, then since polyconvexity implies quasiconvexity,

12 John M. Ball

the minimizer of I among smooth (W 1,3 is enough, see below) y satisfy-ing y(x) = x for |x| = 1 (i.e., r(1) = ) is given by the homogeneousdeformation

y(x) x .However, it was shown in Ball [1982] that for a class of stored-energyfunctions W satisfying (H1) and the growth condition in (H2) but with2 p < 3, q < 32 , I attains a minimum among radial deformations satis-fying the boundary condition y(x) = x for |x| = 1, and that for > 0suciently large the minimizer y satises r(0) > 0. Furthermore y satisesthe weak form of the EulerLagrange equation (2.4). As a specic examplewe can take

W (A) = |A|2 + h(detA) , (2.15)for h : (0,) R smooth with h > 0, lim h() = lim0+ h() =. Cavitation is a common failure mechanism in polymers; for interestingpictures of almost radial cavitation of roughly spherical rubber particlesimbedded in a matrix of Nylon-6 see Lazzeri and Bucknall [1995]. Muller,Qi and Yan [1994] show that if (H2) holds then no deformation with niteenergy can exhibit cavitation. A somewhat stronger condition, which bythe Sobolev embedding theorem obviously prevents not only cavitation butany other discontinuity in y, is that W (A) c0|A|p c1 for all A, wherec0 > 0 and p > 3. In fact even if p = 3 any nite-energy deformation iscontinuous on account of (2.8) and the result of Vodopyanov, Goldshteinand Reshetnyak [1979].

There is an extensive literature on cavitation in elasticity; a sample of themore mathematical developments can be found in the papers of Antmanand Negron-Marrero [1987], James and Spector [1991], Muller and Spector[1995], Polignone and Horgan [1993a,b], Sivaloganathan [1986, 1995, 1999],Sivaloganathan and Spector [2000a,b, 2001], Pericak-Spector and Spector[1997], Stringfellow and Abeyaratne [1989] and Stuart [1985, 1993].

An interesting feature of cavitation is that it provides a realistic exam-ple of the Lavrentiev phenomenon, whereby the inmum of the energy isdierent in dierent function spaces. Here it takes the form

infA1

I < infA3

I = I(y) ,

where Ap ={y W 1,p(B(0, 1);R3) : y|B(0,1) = x

}. More generally,

there is a theory of minimization for elasticity with W polyconvex in func-tion spaces not allowing cavitation due to Giaquinta, Modica and Soucek[1989, 1994, 1998] (see also the less technically demanding approach ofMuller [1988]). Thus the same W can have dierent minimizers in dierentfunction spaces; if we enlarge the space to allow not only cavitation butcrack formation (see Section 2.8), then we can have dierent minimizers inat least three dierent spaces.

1. Some Open Problems in Elasticity 13

In cavitation there is a change of topology of the deformed congurationassociated with the Lavrentiev phenomenon, but one-dimensional examplesin Ball and Mizel [1985] for integrals of the form

I(y) = ba

f(x, y(x), yx(x)

)dx

show that the phenomenon can occur when the minimizer y is continuouswith unbounded gradient. This leads to the question:

Problem 4. Can the Lavrentiev phenomenon occur for elastostatics un-der growth conditions on the stored-energy function ensuring that all nite-energy deformations are continuous?

Of course if y is smooth then the Lavrentiev phenomenon cannot holdunder the hypotheses of Theorem 2.2. Some interesting recent progresson Problem 4 is due to Foss [2001], Mizel, Foss and Hrusa [2002], whoprovide examples of the Lavrentiev phenomenon in two dimensions fora homogeneous isotropic polyconvex stored-energy function W satisfying(2.7) and the corresponding growth condition W (A) c0|A|p c1 for allA M22+ and some p > 2. In these examples the reference congurationis given by a sector of a disk described in polar coordinates by ={(r, ) : 0 < r < 1 , 0 < <

}, and the boundary conditions are

of the container type that the origin is xed, that y() , andthat y(1, ) = (1, ), where 0 < 0

detDy(x) for a.e. x , (2.19)then it is easily seen that y satises (2.4). In fact we can then pass to thelimit 0 using bounded convergence in the dierence quotient

1

[W (Dy + D)W (Dy)]dx , (2.20)

since by (2.19) we have det(Dy(x) + D(x)

) /2 for a.e. x .However, if only (2.7) is assumed, or if y is not assumed in advance to bein W 1,(;R3), then it is not obvious how to pass to the limit.

16 John M. Ball

Problem 5. Prove or disprove that, under reasonable growth conditionson W , global or suitably dened local minimizers of I satisfy the weak form(2.4) of the EulerLagrange equation.

Problem 6. Prove or disprove that, under reasonable growth conditionson W , global or suitably dened local minimizers of I satisfy (2.19).

If W (A) as |A| and if y W 1,, or if (2.19) does not hold,then W (Dy) is essentially unbounded. This is at rst sight inconsistentwith y being a minimizer, but we know from the one-dimensional examplesin Ball and Mizel [1985] and from the example of cavitation that it can payto have the integrand innite somewhere so that it is smaller somewhereelse. In general, it is not possible to estimate the integrand in the dierencequotient (2.20) in terms of W (Dy), the only relevant quantity that isobviously integrable. This diculty was pointed out by Antman [1976],who was the rst to address the issue of satisfaction of the EulerLagrangeequation for one-dimensional problems of elasticity when (2.7) holds; inthis context a device essentially due to Tonelli [1921] can be used to provethat the EulerLagrange equation holds (see also Ball [1981a]) without anysupplementary growth conditions on W .

It is perhaps worth making the simple observation that a smooth defor-mation y may satisfy I(y) < and detDy(x) > 0 a.e. without (2.19)holding. As an example we may take = B(0, 1) and

y(x) = |x|2x ,

with W (A) = log detA+ g(A), where g : M33 R is smooth.For a class of strongly elliptic stored-energy functions having the form

W (A) = g(A) + h(detA), where g : M33 R and h : (0,) [0,)are smooth with h() as 0+ at a polynomial rate, Bauman, Owenand Phillips [1991] show that if y C1, satises the energy-momentumweak form of the EulerLagrange equation in (2.22) below, then in fact yis a smooth solution of the EulerLagrange equation (2.5) and the strictpositivity condition (2.19) holds.

As was pointed out in Ball [1984a], it is possible to derive dierentrst-order necessary conditions for a minimizer when (2.7) holds. (LaterGiaquinta, Modica and Soucek [1989] derived the same rst-order condi-tions in their framework of Cartesian currents, under somewhat strongerhypotheses.) We give here improved versions of these results.

We consider the following conditions that may be satised by W :

(C1)DAW (A)AT K(W (A) + 1) for all A M33+ , where K > 0 is aconstant; and

(C2)ATDAW (A) K(W (A) + 1) for all A M33+ , where K > 0 is aconstant.

1. Some Open Problems in Elasticity 17

As usual, | | denotes the Euclidean norm on M33, for which the in-equalities |A B| |A| |B| and |AB| |A| |B| hold. But of course theconditions are independent of the norm used up to a possible change in theconstant K.

2.3 Proposition. Let W satisfy (C2). Then W satises (C1).

Proof. Since W is frame-indierent the matrix DAW (A)AT is symmetric(this is equivalent to the symmetry of the Cauchy stress tensor see (2.28)below). HenceDAW (A)AT2 = [DAW (A)AT] [A(DAW (A))T]

=[ATDAW (A)

] [ATDAW (A)]T ATDAW (A)2 ,

from which the result follows.

Example. Let

W (A) = (ATA)11 +1

detA.

Then W satises (C1) and (2.9), but not (C2).

2.4 Theorem. For some 1 p 0 such that if

C M33+ and |C 1| < then DAW (CA)AT 3K(W (A) + 1) for all A M33+ .

18 John M. Ball

(b) If W satises (C2) then there exists > 0 such that if C M33+ and|C 1| < thenATDAW (AC) 3K(W (A) + 1) for all A M33+ . (2.23)Proof. We prove (a); the proof of (b) is similar. We rst show that thereexists > 0 such that if |C 1| < then

W (CA) + 1 32 (W (A) + 1) for all A M33+ . (2.24)

For t [0, 1] let C(t) = tC+(1 t)1. Choose (0, 16K ) suciently smallso that |C 1| < implies that |C(t)1| 2 for all t [0, 1]. This ispossible since |1| = 3 < 2. For |C 1| < we have that

W (CA)W (A) = 10

ddt

W (C(t)A) dt

= 10

DAW(C(t)A

) [(C 1)A] dt= 10

DAW(C(t)A

)(C(t)A

)T ((C 1)C(t)1) dt K

10

[W(C(t)A

)+ 1

] C 1 C(t)1 dt 2K

10

(W(C(t)A

)+ 1

)dt .

Let (A) = sup|C1|

1. Some Open Problems in Elasticity 19

In particular detDy (x) > 0 for a.e. x and lim0 y yW 1,p = 0.Hence I(y ) I(y) for | | suciently small. But1

(I(y ) I(y)

)=

1

10

dds

W([1+ sD

(y(x)

)]Dy(x)

)dsdx

=

10

DW([1+ sD

(y(x)

)]Dy(x)

) [D(y(x))Dy(x)] dsdx .Since by Lemma 2.5(a) the integrand is bounded by the integrable function

3K(W(Dy(x)

)+ 1

)supzR3

|D(z)| ,

we may pass to the limit 0 using dominated convergence to obtain(2.21).

(ii) This follows in a similar way to (i) from Lemma 2.5 (b). Since most ofthe details have already been written down in Bauman, Owen and Phillips[1991a] we just sketch the idea. Let C10 (;R3). For suciently small > 0 the mapping dened by

(z) := z+ (z)

belongs to C1(;R3), satises detD (z) > 0, and coincides with theidentity on . Standard arguments from degree theory imply that is adieomorphism of to itself. Thus the inner variation

y (x) := y(z ) , x = z + (z )

denes a mapping y A, andDy (x) = Dy(z )

[1+ D(z )

]1 a.e. x .Since y W 1,p it follows easily that y yW 1,p 0 as 0. Changingvariables we obtain

I(y ) =

W(Dy(z)

[1+ D(z)

]1) det(1+ D(z)) dz ,from which (2.22) follows using (2.23) and dominated convergence.

In order to give an interpretation of Theorem 2.4 (i), let us make thefollowing

Invertibility Hypothesis. y is a homeomorphism of onto := y(), is a bounded domain, and the change of variables formula

f(y(x)

)detDy(x) dx =

f(z) dz (2.26)

20 John M. Ball

holds whenever f : R3 R is measurable, provided that one of the integralsin (2.26) exists.

Sucient conditions for this hypothesis to hold are given in Ball [1981]and Sverak [1988].

2.6 Theorem. Assume that the hypotheses of Theorem 2.4 (i) and theInvertibility Hypothesis hold. Then

(z) D(z) dz = 0 (2.27)

for all C1(R3;R3) such that |y(1) = 0, where the Cauchy stresstensor is dened by

(z) := T(y1(z)

), z

andT(x) =

(detDy(x)

)1DAW (Dy

(x))Dy(x)T . (2.28)

Proof. Since by assumption y() is bounded, we can assume that andD are uniformly bounded. Thus (2.27) follows immediately from (2.21),(2.26) and (2.28).

Thus Theorem 2.4(i) asserts that y satises the spatial (Eulerian) formof the equilibrium equations. Theorem 2.4 (ii), on the other hand, involvesthe so-called energy-momentum tensor W (A)1 ATDAW (A), and is amulti-dimensional version of the Du Bois Reymond or Erdmann equationof the one-dimensional calculus of variations.

The hypotheses (C1) and (C2) imply that W has polynomial growth.More precisely, we have

2.7 Proposition. Suppose W satises (C1) or (C2). Then for somes > 0,

W (A) M(As + A1s) for all A M33+ .Proof. Let V M33 be symmetric. For t 0 d

dtW (etV)

= (DAW (etV)etV) V=(etVDAW (etV)) V

K(W (etV) + 1) |V| . (2.29)From this it follows that

W(eV)+ 1 (W (1) + 1)eK|V| . (2.30)

1. Some Open Problems in Elasticity 21

Now set V = lnU, where U = UT > 0, and denote by vi the eigenvaluesof U. Since

lnU = ( 3i=1

(ln vi)2) 1

2

3

i=1

ln vi ,it follows that

eK| lnU| (vK1 + vK1 )(vK2 + vK2 )(vK3 + vK3 ) 33

( 3i=1

vKi +3

i=1

vKi) 1

3

C( 3

i=1

v3Ki +3

i=1

v3Ki)3

C1[|U|3K + |U1|3K] ,

where C > 0, C1 > 0 are constants. From (2.30) we thus obtain

W (U) M(|U|3K + |U1|3K) ,where M = C1

(W (1) + 1

). The result now follows from the polar de-

composition A = RU of an arbitrary A M33+ , where R SO(3),U = UT > 0.

It is easily seen that if W is isotropic then both (C1) and (C2) areequivalent to the condition that(v1,1 , v2,2 , v3,3) K((v1, v2, v3) + 1)for all vi > 0 and some K > 0, where is given by (2.14) and ,i = /vi.In particular, both (C1) and (C2) hold for the class of Ogden materials(Ogden [1972a,b]), for which has the form

(v1, v2, v3) =Mi=1

ai(i) +Ni=1

bi(i) + h(v1v2v3)

where

() = v1 + v2 + v

3 , () = (v2v3)

+ (v3v1) + (v1v2) ,

ai > 0, bi > 0, i = 0, i = 0, and where h : (0,) [0,) is convex,with h() as 0, provided that h() K1(h() + 1) for all > 0.

22 John M. Ball

2.5 Regularity and Self-Contact

An interesting approach to the problem of invertibility in mixed boundary-value problems (i.e., to the non-interpenetration of matter) is due to Ciarletand Necas [1985]. They proposed minimizing

I(y) =

W (Dy) dx

subject to the boundary condition (2.1) and the global constraint

detDy(x) dx volume (y()) ,and they gave hypotheses under which the minimum was attained, thesehypotheses being weakened by Qi [1988]. They further showed that anyminimizer is one-to-one almost everywhere, and that assuming sucientregularity of the free boundary y(2) the tangential components of thenormal stress vector vanish there. Consequently they identied the aboveconstrained boundary-value problem as corresponding to the case of smooth(i.e., frictionless) self-contact. A related but somwhat dierent formulationhas recently been proposed by Pantz [2001a]; see also Giaquinta, Modicaand Soucek [1994].

Problem 7. Justify the Ciarlet-Necas minimization problem, or an ap-propriate modication of it, as a model of smooth self-contact.

The problem here is to construct suitable variations in the neighbourhoodof a region of self-contact of a minimizer to establish that in some sense thetangential stress components vanish there. This is non-trivial because inprinciple the two parts of the boundary in contact with one another couldbe wildly deformed and interlocked in a very complex conguration. If sucha result could be obtained, a more ambitious target would be to establishthe regularity properties of the free boundary in both the self-contactingand non self-contacting regions.

2.6 Uniqueness of Solutions

For mixed boundary-value problems of elasticity nonuniqueness of equilib-rium solutions is common-place, the most familiar examples being thoseassociated with buckling of rods, plates and shells. Buckling can occureven for pure zero-traction boundary conditions, such as in the eversionof part of a spherical shell. For the pure zero-traction problem one caneven have nonuniqueness among homogeneous dilatations (see Ball [1982]).Nonuniqueness of these types is expected to hold, and to some extent canbe proved rigorously, when the stored-energy function satises favourablehypotheses such as strict polyconvexity (though see Section 2.7). For stored-energy functions corresponding to elastic crystals, for which there are many

1. Some Open Problems in Elasticity 23

minimum energy congurations with a continuum of dierent sets of phaseboundaries, the extent of non-uniqueness is of course much greater.

For pure displacement boundary conditions, with a strictly polycon-vex (or strictly quasiconvex) stored-energy function satisfying favourablegrowth conditions, the situation as regards uniqueness is less clear. John[1972b] proved uniqueness for smooth deformations with uniformly smallstrains (but possibly large rotations). In the same paper he gave a heuristiccounter-example to uniqueness for the case of an annular two-dimensionalbody, and this has been made rigorous by Post and Sivaloganathan [1997](see Section 2.7), who also proved nonuniquenesss for an analogous three-dimensional problem with a torus. But what if is homeomorphic to aball? In this case we have already seen that cavitation provides one coun-terexample to uniqueness, though the cavitating solution is discontinuous.

Problem 8. Prove or disprove the uniqueness of suciently smooth equi-librium solutions to pure displacement boundary-value problems for homo-geneous bodies when the stored-energy function W is strictly polyconvex and is homeomorphic to a ball.

The answer to this problem probably depends on both the geometryof and the boundary conditions. For example, suppose that is a ball,and that the boundary conditions correspond to severely squeezing the balluntil it has a dumb-bell shape consisting of two roughly ball-shaped regionsconnected by a narrow passage. In this case one might expect, though itis not obvious how to prove it, that there might be equilibrium solutionsin which material from one half of is pulled through into the other half,but prevented from returning by the constriction. On the other hand, anelegant result of Knops and Stuart [1984] implies uniqueness for the casewhen the boundary displacements are linear and is star-shaped (see alsoTaheri [2001b]).

2.7 Structure of the Solution Set

Problem 9. Devise general methods for proving the existence of localminimizers of I that are not global minimizers, and of other weak solutionsof the equilibrium equations.

For the existence of local minimizers there are two natural approaches.First we could try to use the direct method of the calculus of variationsin a suitable subset of A. For example, under the hypotheses of Theorem2.2 suppose that we want to prove the existence of a local minimizer withrespect to some metric d on A. Assume that d is such that if z(j) A withz(j) z in W 1,1(;R3) and sup I(z(j))

24 John M. Ball

can prove thatinfU

I > infU

I > infA

I.

Then y U and is a local, but not global, minimizer with respect to d. Ibelieve that it should be possible to implement this method in some realisticexamples, but have not seen it done. The only results on local minimizers innonlinear elasticity using the direct method that I am aware of are due toPost and Sivaloganathan [1997], who prove the existence of local but notglobal minimizers for certain two-dimensional problems (see Section 2.6)for which the domain has nontrivial topology by global minimization ina weakly closed homotopy class, and to Taheri [2001a], who generalizes theresults in Post and Sivaloganathan [1997] to a wider class of domains.

The second approach is to nd by some method a suciently smooth so-lution y to the equilibrium equations and attempt to show directly that itis a local minimizer. For local minimizers in W 1,(;R3) (weak local min-imizers) this can be done in principle by checking positivity of the secondvariation. However for local minimizers in W 1,p(;R3) with 1 p < ,or in Lq(;R3), 1 q , the task is made much more dicult bythe absence of a known generalization to higher dimensions of the Weier-strass fundamental suciency theorem of the one-dimensional calculus ofvariations (for a discussion see Ball [1998]). Sometimes it is possible tocircumvent the lack of such a theory. For example, in a dead-load trac-tion problem arising from the bi-axial load experiments of Chu and James[1993, 1995] on CuAlNi single crystals, it is proved in Ball and James [2002],Ball, Chu and James [2002] (see also Ball, Chu and James [1995]) that cer-tain y with Dy = A = constant are local (but not global) minimizers inL1(;R3), by an argument exploiting the geometric incompatibility of Awith deformation gradients having lower energy.

How can one prove the existence of equilibrium solutions that are notlocal minimizers? In exceptional cases one may know an equilibrium solu-tion explicitly (for example a trivial solution) and be able to show that itdoes not satisfy some necessary condition for a local minimizer. If we canalso prove the existence of a global minimizer then we have at least twoequilibrium solutions. This can be done, for example, for the case of somemixed boundary-value problems when the stored-energy function is poly-convex but not quasiconvex at the boundary (see Ball and Marsden [1984]).Another approach would be to try to use Morse theory or mountain-passmethods, but it is not clear how to do this so that, for example, appropriateconditions of Palais-Smale type can be veried; for results in an interestingmodel problem see Zhang [2001].

More generally, one can ask for a description of how the set of equilibriumsolutions varies as a function of relevant parameters such as boundary dis-placements or loads. For the pure traction problem near a stress-free statean interesting study of this type is that of Chillingworth, Marsden and Wan[1982, 1983] and Wan and Marsden [1983].

1. Some Open Problems in Elasticity 25

Problem 10. Develop local and global bifurcation theories for nonlinearelastostatics that apply to mixed displacement-traction boundary conditions.

As an illustration, the most well-known bifurcation problem in elasticityis that of buckling of a thin rod. Although this problem has been treatedfrom the perspective of rod theory in hundreds of papers since the timeof Euler [1744], there is no rigorous three-dimensional theory that justiesthe usual picture of buckling, for example the existence of critical bucklingloads or displacements, with corresponding branches of bifurcating buckledsolutions. There are at least two diculties in providing such a theory. Therst is that unless the problem is formulated in a somewhat unrealistic way,there is no suciently explicit trivial compressed solution about which tolinearize the equilibrium equations. For example, suppose that in a stress-free reference conguration a homogeneous isotropic elastic rod occupiesthe region = (0, L) D, where D R2 is the cross-section. A naturalboundary-value problem to consider, corresponding to clamped ends, con-sists of the equilibrium equations (2.4) and the boundary conditions (2.1),(2.6), with the choices 1 = {0, L} D, 2 = (0, L) D, and

y(0,x) = (0,x) , y(L,x) = (L,x) , x D , (2.31)

where > 0. For = 1 the homogeneous deformation y(x1,x) = (x1,x)does not in general satisfy the zero traction condition (2.6). For example, forthe compressive case < 1 the rod will typically want to bulge, leading toboundary layers near x1 = 0 and x1 = L. In order to have a homogeneouslydeformed trivial solution

y(x) = (x1, x2, x3) ,

one can replace (2.31) by the conditions

y1(0,x) = 0 , y1(L,x) = L , x D ,

corresponding to the less realistic case of frictionless end-faces constrainedto lie in the planes {0} R2 and {L} R2. To prevent sliding of theend-faces one could add the further constraint that

D

y2 dx =D

y3 dx = 0 at x1 = 0, L .

In this case the natural boundary conditions at x1 = 0, L for the variationalproblem are that the stress vector t across the end-faces has constant trans-verse components t2, t3 which are equal at x1 = 0, L. If we try to prescribecompressive loads at x1 = 0, L rather than displacements we encounterother diculties (see Ball [1996a] for a discussion of one of these).

The second more serious diculty has already been mentioned, namelythe lack of regularity of solutions to the linearized equilibrium equations as

26 John M. Ball

one approaches points of 12, or points of discontinuity of the appliedtraction in a pure traction formulation of the problem, which prohibitsuse of the implicit function theorem in natural spaces. Perhaps it mightbe possible to work in spaces with suitable weights in the neighbourhoodof 1 2. But it seems odd that ne details of what goes on near1 2 should have a signicant bearing on the buckling phenomenon,so perhaps there is a dierent approach to be discovered that circumventsthis diculty.

Once a local bifurcation picture has been established, the next thing tounderstand is what happens to bifurcating solutions for large parametervalues. For the case when 1 2 is empty global results have recentlybeen obtained by Healey and Rosakis [1997], Healey and Simpson [1998]and Healey [2000].

2.8 Energy Minimization and Fracture

Many of the problems described above have generalizations to variationalmodels of fracture. Since typical fracture problems are described by de-formations that have jump discontinuities across two-dimensional cracksurfaces, fracture cannot in general be modelled in the context of Sobolevspaces. A generalization of the energy functional (2.2) to deformations al-lowing for fracture is

I(y) =

W (Dy) dx+Sy

g(y+ y, y

)dH2 , (2.32)

where y belongs to the class SBV() of mappings of special bounded vari-ation, i.e., those whose gradient is a bounded measure having no Cantorpart. In (2.32) Sy denotes the set of jump points of y, y the measure theo-retic normal to Sy, and y the traces of y on either side of Sy. The secondintegral represents the surface energy of cracks, as postulated in the Griththeory of fracture (see, for example, Cherepanov [1998]), the simplest caseg = constant corresponding to a contribution to the energy proportional tothe total crack surface area H(Sy). Despite much deep work on such mod-els (see, for example, Acerbi, Fonseca and Fusco [1997], Ambrosio [1989,1990], Ambrosio and Braides [1995], Ambrosio, Fusco and Pallara [1997,2000], Ambrosio and Pallara [1997], Braides [1998], Braides and Coscia[1993, 1994], Braides, Dal Maso and Garroni [1999], Buttazzo [1995]), andtheir apparent potential for making an impact on understanding fracture,there have been only isolated attempts to discover their implications forpractical problems of fracture mechanics (see, for example, Francfort andMarigo [1998], Bourdin, Francfort and Marigo [2000]).

Problem 11. Clarify the status of models based on the energy functional(2.32) with respect to classical fracture mechanics and to nonlinear elasto-statics.

1. Some Open Problems in Elasticity 27

Two key issues are fracture initiation and stability, which are both re-lated to the study of local minimizers for the functional (2.32). A technicalobstacle in such a study is the lack of a general method of calculating ageneral variation of I about a given y in the direction of nearby deforma-tions having possibly very dierent sets of jump points. An understandingof local minimizers would also clarify the status of the nonlinear elastostat-ics model based on (2.2) with respect to that based on (2.32), and therebydemystify the apparent sensitivity of the elastostatics model to growth be-haviour for very large strains.

3 Dynamics

3.1 Continuum Thermomechanics

We recall briey the elements of continuum thermomechanics. The basicbalance laws are the balance of linear momentum

ddt

E

Ryt dx =E

tR dS +E

bdx , (3.1)

the balance of angular momentum

ddt

E

Rx yt dx =E

x tR dS +E

x bdx , (3.2)

and the balance of energy

ddt

E

(12R|yt|2 + U

)dx =

E

b yt dx+E

tR yt dS

+E

r dxE

qR ndS . (3.3)

Here y = y(x, t) denotes the deformation, tR the PiolaKirchho stressvector, R > 0 the (constant) density in the reference conguration, b thebody force, U the internal energy, qR the heat ux vector and r the heatsupply. The balance laws are assumed to hold for all Lipschitz domainsE , and the unit outward normal to E is denoted by n. In additionto the balance laws, thermomechanical processes are required to obey theSecond Law of Thermodynamics, which we assume to hold in the form ofthe ClausiusDuhem inequality

ddt

E

dx E

qR n

dS +E

r

dx (3.4)

for all E, where is the entropy and the temperature. Standard argumentsnow show that tR = TRn, where TR is the PiolaKirchho stress tensor,

28 John M. Ball

and that for suciently smooth processes (3.1), (3.3), (3.4) reduce to thepointwise forms

Rytt divTR b = 0 , (3.5)ddt(12 |yt|2 + U

) b yt div (ytTR) + divqR r = 0 (3.6)t + div

(qR

) r 0, (3.7)

and that (3.2) is equivalent to the symmetry of the Cauchy stress tensor

T = (detDy)1 TR(Dy)T .

Eliminating r from (3.6), (3.7), using (3.5) and denoting by

= U (3.8)

the Helmholtz free energy, we obtain that for suciently smooth processes

t t +TR Dyt qR grad

0 . (3.9)

Adopting the prescription of Coleman and Noll [1963], we assume that givenan arbitrary deformation y = y(x, t) and temperature eld = (x, t)we can choose a body force b = b(x, t) and heat supply r = r(x, t) tobalance (3.5), (3.6), so that (3.9) becomes an identity to be satised by theconstitutive equations. For the case of a thermoelastic material, for whichTR, , ,qR are assumed to be functions of Dy, , grad , this leads to therelations

= (Dy, ), TR = DA, = D , (3.10)and then (3.9) reduces to the inequality

qR grad

0 . (3.11)

(Note that, although this inequality must be satised by the constitutiveequation for qR, for processes involving shocks (3.11) is not equivalent to(3.7), since the cancellations in the argument used to obtain (3.9) are nolonger valid.) For thermoelastic materials the balance of angular momen-tum is satised identically as a consequence of the requirement that TR isframe-indierent, i.e.,

TR(RA,

)= RTR

(A,

), for all R SO(3) ,

which is equivalent to the condition that

(RA, ) = (A, ) , for all R SO(3) . (3.12)

1. Some Open Problems in Elasticity 29

The condition of material symmetry becomes

(AQ, ) = (A, ) , for all Q S , (3.13)

where S is the isotropy group. The equations of isothermal thermoelasticityare obtained from (3.5), (3.10) by assuming that (x, t) = 0 = constant.Thus the balance of linear momentum becomes

Rytt divDAW (Dy) b = 0 , (3.14)

where W (A) = (A, 0). As regards the entropy inequality, we again adoptthe Coleman and Noll point of view, choosing r to balance (3.6). (Here wefollow Dafermos [2000], who gives a similar reduction for isentropic ther-moelasticity.) Since, from (3.11), qR = 0 when grad = 0, (3.7) becomes(

12R|yt|2 +

)t b yt div (ytTR) 0 . (3.15)

For the more general case of a thermoviscoelastic material (of strain-ratetype), TR, , ,qR are assumed to be functions of Dy,Dyt, , grad . Bythe same method we nd that

= (Dy, ), = D ,

and that

S Dyt qR grad

0 ,

whereTR = DA + S

(Dy,Dyt, , grad

).

In the isothermal case we obtain the equation of motion

Rytt divDAW (Dy) divS(Dy,Dyt) b = 0 , (3.16)

where W = (Dy, 0), S(Dy, Dyt) = S(Dy,Dyt, 0, 0), together with theenergy inequality (3.15). The frame-indierence of S takes the form

S(Dy,Dyt) = Dy(U,Ut) , (3.17)

for some matrix-valued function , where U = (DyTDy)12 .

3.2 Existence of Solutions

Problem 12. Prove the global existence and uniqueness of solutions toinitial boundary-value problems for properly formulated dynamic theoriesof nonlinear elasticity.

30 John M. Ball

To discuss this problem let us begin with isothermal thermoelasticity.The governing equations are (3.14). These equations need to be supple-mented by boundary conditions such as (2.1), (2.6) and by the initial con-ditions

y(x, 0) = y0(x), yt(x, 0) = y1(x). (3.18)

If the body force is conservative, so that

b = grad yh(x,y), (3.19)then (3.14) formally comprises a Hamiltonian system, and could be alter-natively obtained by applying Hamiltons principle to the functional T

0

(12R|yt|2 W (Dy) h(x,y)

)dxdt .

In particular, solutions formally satisfy the balance of energy

E(y,yt) = E(y0,y1) , t 0 , (3.20)where

E(y,v) =

(12R|v|2 +W (Dy) + h(x,y)

)dx .

However, weak solutions of the quasilinear wave equation (3.14) do notin general satisfy (3.20), since singularities such as shock waves can dis-sipate energy. Correspondingly, although equality holds in the dissipationinequality (3.15) for smooth solutions, in general it does not do so for weaksolutions. Interpreted in the sense of distributions or measures, (3.15) actsas an admissibility criterion for weak solutions.

In one dimension (3.14) takes the form

Rytt (yx)x b = 0 , (3.21)where (yx) = W (yx), which setting u1 = Ryt, u2 = yx is equivalent tothe system of two conservation laws

ut f(u)x = g , (3.22)where

f(u) =((u2)1R u1

), g =

(b0

).

This system is strictly hyperbolic if = W > 0, so that W is strictlyconvex.

Two approaches have been employed to study (3.22), the Glimm scheme,Glimm [1965], and variants of it such as front-tracking (introduced byDafermos [1972]), and the method of compensated compactness as pio-neered by Tartar [1979, 1982] and DiPerna [1983, 1985].

1. Some Open Problems in Elasticity 31

The Glimm scheme and variants apply to strictly hyperbolic systems ofthe form (3.22) with u Rn, f : Rn Rn, g Rn. They involve asemi-explicit construction of the solutions in terms of approximation of theinitial data by piecewise constant functions, together with an analysis ofwave interactions. They are restricted to initial data having small total vari-ation, and thus, via total variation estimates on the solution, to solutionsof small total variation. Glimms original work assumed that the systemwas genuinely nonlinear, but this restriction was removed by Liu [1981].Thanks to work of Bressan [1988, 1995], Bressan and Colombo [1995], Bres-san [Crasta and Piccoli], Bressan and Goatin [1999], Bressan and Le Floch[1997], Bressan and Lewicka [2000], Bressan, Liu and Yang [1999] and Liuand Yang [1999b,a,c], the solutions obtained in these ways are now knownto be unique in appropriate function classes. For genuinely nonlinear sys-tems of two conservation laws, such as (3.21) with W > 0,W = 0, moreis known (see [Dafermos, 2000, Chapter XI]). Most of this work is for so-lutions on the whole real line; for a treatment of (3.21) with displacementboundary conditions see Liu [1977].

The method of compensated compactness, on the other hand, has up tonow been restricted to systems of at most two conservation laws, such as(3.21). Starting from a sequence of approximate solutions obtained fromthe method of vanishing viscosity (or by a variational time-discretizationscheme, see Demoulini, Stuart and Tzavaras [2000]), it uses informationcoming from the existence of a suitable family of entropies (quantities forwhich there is a corresponding conservation law satised by smooth solu-tions) to pass to the limit using weak convergence. However, there is no cor-responding uniqueness theorem. These results are described in the books ofBressan [2000], Dafermos [2000] and Serre [2000]. In a recent development,Bianchini and Bressan [2001] have made a breakthrough by obtaining forthe rst time total variation estimates directly from the vanishing viscositymethod.

For the three-dimensional equations (3.14) very little is known. Hughes,Kato and Marsden [1977] proved that if W satises the strong ellipticitycondition

D2W (A)(a n,a n) |a|2|n|2 (3.23)for all A M33+ , a,n R3, where > 0, then for smooth initial data(3.18) dened on the whole of R3 with detDy0 > 0, there exists a uniquesmooth solution on a small time interval [0, T ), T > 0. This result wasextended to pure displacement boundary conditions by Kato [1985]. Forrelated results see Dafermos and Hrusa [1985] and [Dafermos, 2000, Chap-ter V]. There seem to be no short-time existence results known for mixeddisplacement-traction boundary conditions. Interesting results concerninglarge time existence for suciently smooth and small initial data on thewhole of R3 have been obtained by John [1988]. For corresponding resultsfor incompressible elasticity see Hrusa and Renardy [1988], Ebin and Sax-

32 John M. Ball

ton [1986], Ebin and Simanca [1990, 1992] and Ebin [1993, 1996].In the variables A = Dy, p = Ryt, (3.14) becomes the system

At = D , Dij = 1R vi,j , (3.24)pt = divDAW (Dy) + b , (3.25)

which is hyperbolic if

D2W (A)(a n,a n) > 0 for all A M33+ and nonzero a,n R3.There is no theory of weak solutions for such multi-dimensional systems.In particular, it is unclear what conditions on W are natural for existence,and whether these conditions will be the same as those guaranteeing exis-tence for elastostatics, namely quasiconvexity or polyconvexity. The system(3.24), (3.25) is special in the sense that there is an involution

Aij,k Aik,j = 0which is satised by all weak solutions. Exploiting this in the context of ageneral system having involutions, Dafermos [1996, 2000] proves a theoremimplying that if W is quasiconvex and satises (3.23) then any Lipschitzsolution A,p of (3.24), (3.25) on R3 [0, T ], T > 0, is unique withinthe class of weak solutions admissible with respect to the entropy W , ofuniformly small local oscillation, and satisfying the same initial data asA,p. An unpublished idea of LeFloch, found independently by Qin [1998],leads to the observation that for polyconvex W the hypothesis of uniformlysmall oscillation can be removed. These results are interesting because theyso far represent the only use of quasiconvexity and polyconvexity in thecontext of dynamics. See Sverak [1995] for an idea of how quasiconvexity(in an augmented space) might be used to prove existe