Geometry, Mechanics, and Dynamics

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Geometry, Mechanics, and Dynamics

Paul Newton Phil Holmes Alan Weinstein, Editors

Springer

Geometry, Mechanics, and Dynamics

This page intentionally left blank

Geometry, Mechanics, and Dynamics Editors:

Paul Newton, Phil Holmes, and Alan Weinstein

Paul Newton Department of Aerospace and Mechanical Engineering and Department of Mathematics University of Southern California, Los Angeles, CA 90089-1191 USA [email protected]

Philip Holmes Department of Applied and Computational Mathematics Engineering Quadrangle Princeton University Princeton, NJ 08544-1000 USA [email protected]

Alan Weinstein Department of Mathematics University of California, Berkeley Berkeley, CA 94720 USA [email protected] Cover Illustration: Permission has been granted for use of the thunderstorm photograph on the cover 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 northwest Kansas in August, 1996. The view is to the east from Norton, Kansas. It was taken while Kyle was at Saint Louis Univeristy (SLU) and was featured as a cover photo for the SLU Department 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, Public Relations, California Institute of Technology, Pasadena, CA 91109, USA.

Mathematics Subject Classification (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 without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression 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 technical editors, Wendy McKay and Ross Moore, using modifications of a Springer LATEX macro package, and other packages for the integration of graphics and consistent stylistic features within articles from diverse sources. www.springer-ny.com Springer-Verlag

New York, Berlin, Heidelberg

A member of BertelsmannSpringer Science+Business Media GmbH

To Jerry Marsden on the occasion of his 60th birthday, with admiration, affection, and best wishes for many more years of creativity.

Photo by Robert J. Paz


Contents Preface

I

ix

Elasticity and Analysis

1 Some Open Problems in Elasticity by John Ball

1 3

2 Finite Elastoplasticity Lie Groups and Geodesics on SL(d) by Alexander Mielke 61 3 Asynchronous Variational Integrators by Adrian Lew and Michael Ortiz

II

Fluid Mechanics

91 111

4 Euler–Poincar´ e Dynamics of Perfect Complex Fluids by Darryl D. Holm

113

5 The Lagrangian Averaged Euler (LAE-α) Equations with Free-Slip or Mixed Boundary Conditions by Steve Shkoller

169

6 Nearly Inviscid Faraday Waves by Edgar Knobloch and José M. Vega

181

7 The Variational Multiscale Formulation of LES with Application to Turbulent Channel Flows by Thomas J. R. Hughes and Assad A. Oberai

223

III

241

Dynamical Systems

8 Patterns of Oscillation in Coupled Cell Systems by Martin Golubitsky and Ian Stewart

243

9 Simple Choreographic Motions of N Bodies: A Preliminary Study by Alain Chenciner, Joseph Gerver, Richard Montgomery and Carles Sim´ o

287

10 On Normal Form Computations by J¨ urgen Scheurle and Sebastian Walcher

309

vii

viii

IV

Geometric Mechanics

327

11 The Optimal Momentum Map by Juan-Pablo Ortega and Tudor S. Ratiu

329

12 Combinatorial Formulas for Products of Thom Classes by Victor Guillemin and Catalin Zara

363

13 Gauge Theory of Small Vibrations in Polyatomic Molecules by Robert G. Littlejohn and Kevin A. Mitchell

407

V

429

Geometric Control

14 Symmetries, Conservation Laws, and Control by Anthony M. Bloch and Naomi E. Leonard

431

VI

461

Relativity and Quantum Mechanics

15 Conformal Volume Collapse of 3-Manifolds and the Reduced Einstein Flow by Arthur E. Fischer and Vincent Moncrief

463

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

523

VII

537

Jerrold Marsden, 1942–

Curriculum Vitae

539

Some Research Highlights

541

Graduate Students and Post Doctoral Scholars

545

Publications

549

Contributors

569

Preface Jerry Marsden, one of the world’s pre-eminent mechanicians and applied mathematicians, celebrated his 60th birthday in August 2002. The event was marked by a workshop on “Geometry, Mechanics, and Dynamics” at the Fields Institute for Research in the Mathematical Sciences, of which he was the founding Director. Rather than merely produce a conventional proceedings, with relatively brief accounts of research and technical advances presented at the meeting, we wished to acknowledge Jerry’s influence as a teacher, a propagator of new ideas, and a mentor of young talent. Consequently, starting in 1999, we sought to collect articles that might be used as entry points by students interested in fields that have been shaped by Jerry’s work. At the same time we hoped to give experts engrossed in their own technical niches an indication of the wonderful breadth and depth of their subjects as a whole. This book is an outcome of the efforts of those who accepted our invitations to contribute. It presents both survey and research articles in the several fields that represent the main themes of Jerry’s work, including elasticity and analysis, fluid mechanics, dynamical systems theory, geometric mechanics, geometric control theory, and relativity and quantum mechanics. The common thread running through this broad tapestry is the use of geometric methods that serve to unify diverse disciplines and bring a wide variety of scientists and mathematicians together, speaking a language which enhances dialogue and encourages cross-fertilization. We hope that this 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 contributing to one of these fields and to the seasoned practitioner who seeks a broader view. Jerry is a unique figure in mathematical circles because his work has significantly influenced four often (alas!) separate research communities: pure mathematicians, applied mathematicians, physicists, and engineers. Foundations of Mechanics (with Ralph Abraham [294]), first published in 1967 while Jerry was a graduate student at Princeton, has for the past 35 years been a landmark and inspiration in the field of mechanics; during that time, Jerry and his collaborators have done extraordinary work in a huge variety of sub-fields of mechanics, geometry, and dynamics. Ralph Abraham recalls: “The first edition of Foundations of Mechanics included, in my Preface, a few words on the genesis of the book as Jerry’s notes of my lectures in early 1966. I well recall the first meeting of that graduate course. At the outset I announced a desire for volunteers to make notes ix

x

Preface

which might be duplicated for the use of students, as there was at that time no text we could follow. And at the end of that first meeting, only one volunteer: Jerry. He was a new face for me, and seemed rather young and quiet, and I told him I hoped that others would volunteer for a team effort on the notes. Well, there were no other volunteers, which was just as well. For shortly after each lecture Jerry would deliver a thick sheaf of handwritten notes, usually without a single error. Many details omitted in my talks were filled in with proofs, references, and so on, in the now-famous Marsden style. And the rest, as they say, is history. By now many people know that Jerry is an ideal coworker and coauthor, and I was lucky to be an early benefactor of his wonderful talents and personality.” A talented and prolific expositor, Jerry has written numerous other books, from elementary to advanced level, in addition to his many research articles. Mathematical Foundations of Elasticity (with Tom Hughes [300]) introduced a generation of engineers with appetites for abstraction to a unified and global approach to the subject, and his recent book Introduction to Mechanics and Symmetry, (with Tudor Ratiu [303]) has been remarkably useful to a wide range of scientists and engineers. When Jerry won the 1990 Norbert Wiener Prize (jointly with Michael Aizenman), he noted in his response to the citation that Wiener was “classifiable neither as a pure nor an applied mathematician. He had breadth and depth that worked together in a mutually supportive way.” The same is true of Jerry: it is no accident that he began his career in mathematical physics, moved to a mathematics department, and is now working in the Division of Engineering and Applied Science at Caltech. Jerry’s influence on mathematical education has also been significant. His books on calculus and complex variables are widely used and, with their skillful blend of concreteness and abstraction, have influenced generations of undergraduates. Thorough and wide-ranging in their coverage, they leave the conscientious student with a solid grounding in both theoretical techniques and physical intuition. Jerry’s Ph.D. and postdoctoral students, some of them now leaders in their fields, have made significant contributions in many areas themselves. In addition, Jerry has worked tirelessly for the mathematical community, serving on editorial boards and arranging conferences and workshops, all the while teaching a stellar array of undergraduate and graduate students and post-docs, first at UC Berkeley, and now at Caltech. His extraordinarily influential paper with David Ebin [13], on the analysis of ideal fluid flows remains a classic in the field. It followed upon Arnold’s 1966 paper1 on ideal fluid flows, which showed how the Euler dynamics for 1 Arnold, V. I. [1966], Sur la g´ eometrie differentielle des groupes de Lie de dimension infinie et ses applications ` a l’hydrodynamique des fluids parfaits, Ann. Inst. Fourier, Grenoble, 16, 319–361.

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rigid bodies and fluids could be viewed as geodesic flow on SO(3) with a left-invariant metric, and on Diff vol (Ω) — the volume preserving diffeomorphism group of a region Ω in R3 — with the right invariant metric defined by the fluid kinetic energy. Ebin and Marsden [13] put this work in the context of Sobolev (H s ) manifolds and showed that Arnold’s geodesic flow on H s − Diff vol (Ω), the volume preserving diffeomorphisms of Ω to itself of Sobolev class H s , comes from a smooth geodesic spray. This allowed them to show that the initial value problem for the Euler equations could be solved using Picard iteration and techniques from ordinary differential equation theory. Stephen Smale has remarked: “Jerry Marsden has many fine achievements to his credit. But I am particularly fond of his early work with David Ebin on the equations of fluid mechanics. There are many sides to this study. It gave formal ideas of Arnold great substance and provided an elegant way of presenting old and new fundamental work on the existence of solutions of Navier–Stokes and Euler equations. The rigorous group setting and one of the first important uses of infinite dimensional manifolds are there as well. Quite a milestone in mathematics!” His body of work on “reduction theory,” begun with Alan Weinstein, was an outgrowth of ideas developed by Smale (following Jacobi and others), who introduced the use of symmetry ideas in the context of tangent and cotangent bundles of configuration spaces with Hamiltonians in the form of kinetic plus potential energy. The Marsden and Weinstein paper [30] unified approaches of both Smale2 and Arnold by putting this “reduction theory” in the context of symplectic manifolds. For instance, in the related Poisson context (developed by his student Richard Montgomery) if one starts with a cotangent bundle T ∗ Q and a Lie group G acting on Q, then the quotient (T ∗ Q)/G is a bundle over T ∗ (Q/G) with fiber g∗ , the dual of the Lie algebra of G. As described by Marsden [170], “Thus, one can say — perhaps with only a slight danger of oversimplification — that reduction theory synthesizes the work of Smale, Arnold (and their predecessors of course) into a bundle, with Smale as the base and Arnold as the fiber.” Reduction theory has now been used successfully in a wide variety of fields, and we refer the reader to the overview articles by Marsden [170; 227] as well as many of the articles in this volume for current applications. From these works emerge more than specific theorems and techniques, deep and elegant as they may be. Viewing Jerry Marsden’s contributions as a whole, one finds a clear, pedagogical, and fundamental approach to the subject of mechanics that blends geometry, analysis, and dynamics in powerful, yet practical ways. Thus, while developing abstract techniques in dynamical systems theory, Jerry also helped understand specific orbit trajectories (with a group at the Jet Propulsion Lab) that were used in the Genesis Discovery Mission, launched on August 8, 2001, [244]. In 2 Smale,

S. [1970], Topology and mechanics, Invent. Math., 10, 305–331; 11, 45–64.

xii

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the course of developing the averaged fluid equations (with Holm, Ratiu, and Shkoller), he also contributed to their use in turbulent flow computations [267]. While working out infinite dimensional versions of the Melnikov method and Smale–Birkhoff theory to prove the existence of “Smale horseshoes” in the context of partial differential equations, the chaotic oscillations of a forced beam were being analyzed [71]; and while developing symplectic-energy-momentum preserving variational integrators based on discrete variational principles, Jerry contributed to specific projects (with Michael Ortiz) to simulate the crushing of aluminum cans and analyze fracture mechanics and collision problems [253; 284]. In each of the general areas noted above, we have solicited survey and research articles that illustrate more specifically how some of the methods pioneered by Marsden are currently being used. For Elasticity and Analysis, the paper by J. M. Ball entitled “Some open problems in elasticity” is a self-contained overview which highlights some general open problems in elasticity theory, including some new results showing that local minimizers of the total elastic energy satisfy a weak form of the equilibrium equations. This is followed by the article of A. Mielke, “Finite elastoplasticity, Lie groups and geodesics on SL(d)” which interprets notions of nonlinear plasticity theory in terms of Lie groups, among other things. The contribution of A. Lew and M. Ortiz, entitled “Asynchronous variational integrators” describes a new class of algorithms for nonlinear elastodynamics which is based upon a discrete version of Hamilton’s principle. D. D. Holm’s article in Fluid Mechanics, “Euler–Poincaré dynamics of perfect complex fluids,” describes the use of Lagrangian reduction by stages to derive the Euler–Poincaré equations for non-dissipative motion of exotic fluids such as liquid crystals, superfluids, Yang-Mills magnetofluids and spin-glass systems. Inclusion of defects, such as vortices, in the order parameters is also treated. S. Shkoller’s contribution, “The Lagrangian averaged Euler (LAE −α) equations with free-slip or mixed boundary conditions”, presents a simple proof of well-posedness of the Euler-α equations with novel boundary conditions. E. Knobloch’s and J. Vega’s article, “Nearly inviscid Faraday waves”, explores some of the consequences of introducing small viscosity in the study of surface-gravity-capillary waves excited by vertical vibration of a fluid layer. The contribution of T. J. R. Hughes and A. A. Oberai, “The variational multiscale formulation of LES with applications to turbulent channel flows”, studies turbulent two-dimensional equilibrium and three-dimensional non-equilibrium channel flows using a variational multi-scale formulation of Large Eddy Simulation (LES). In Dynamical Systems Theory, M. Golubitsky and I. Stewart address “Patterns of oscillation in coupled cell systems”. The dynamics of coupled cell 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 assumptions 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. Sim´ o, “Simple choreographic motions of N bodies: A preliminary study”. They describe the existence of new periodic solutions to the N body problem in which all N masses trace the same curve without colliding. J. Scheurle and S. Walcher’s “On normal form computations” closes this section by reviewing computational procedures involved in transforming a vector field into a suitable normal form about a stationary point. For Geometric Mechanics, the paper by J. P. Ortega and T. Ratiu entitled “The optimal momentum map” discusses the (dare we say) classical Marsden–Weinstein reduction procedure and the use of a new optimal momentum map which more efficiently encodes symmetry information of the underlying Hamiltonian system. V. Guillemin’s and C. Zara’s “Combinatorial formulas for products of Thom classes” obtains combinatorial descriptions of the equivariant Thom class dual to the Morse–Whitney stratification of compact Hamiltonian G-manifolds. The paper of R. Littlejohn and K. Mitchell, “Gauge theory of small vibrations in polyatomic molecules,” considers molecular vibrations in the context of gauge theory and fiber bundle theory. In Geometric Control Theory, the paper by A. Bloch and N. Leonard, entitled “Symmetries, conservation laws, and control” traces the role of Marsden’s ideas on reduction and symmetries in the setting of nonlinear control theory. Specific applications to the dynamics of rigid spacecraft with a rotor and the dynamics of underwater vehicles are considered in detail. Finally, for Relativity and Quantum Mechanics, A. E. Fischer and V. Moncrief’s article entitled “Conformal volume collapse of 3-manifolds and the reduced Einstein flow ” describes the Hamiltonian reduction of Einstein’s equations of general relativity and the process of volume collapse. They prove that collapse occurs either along circular fibers, embedded tori, or completely to a point, but surprisingly, always with bounded curvature. This is followed by M. Gotay’s contribution “On quantizing semisimple basic algebras” which examines whether there exists a consistent quantization of the coordinate ring of a basic coadjoint orbit of a semisimple Lie group. We hope that this collection of articles gives the reader some appreciation of both the unity and diversity of the topics influenced by Jerry Marsden’s approach to mechanics. But here we wish to do more than survey his mathematical and scientific contributions; we also want to celebrate Jerry as a colleague and friend. It therefore seems appropriate to conclude with some personal reminiscences.

xiv

Preface

Phil Holmes: I first met Jerry in the summer of 1976, at a conference on dynamical systems at Southampton University, organised by David Rand and Brian Griffiths. He joined Nancy Kopell, John Guckenheimer and Ken Cooke as one of four mathematicians from the USA invited to that meeting. I had completed my Ph.D. in Engineering (experimental studies of dispersive wave propagation in structures) at the Institute of Sound and Vibration a couple of years earlier, and had begun working on nonlinear vibration problems with David Rand. We had done some single and finite degree of freedom problems, and I wanted to begin looking at PDEs in structural mechanics from a dynamical systems perspective. I believe it was in late 1975 that someone told me Jerry was working on a book about bifurcations and dimension reduction for such problems. I wrote to ask for more information and back came a huge package, re-taped and tied with string by UK customs, containing a 500+ page photocopy of the typescript of “The Hopf Bifurcation and its Applications” by Marsden and McCracken [297]. In financially-constrained Britain I had never seen more than fifty pages of xerox copies (all copies were xerox copies in those days) at one time, without special permission. I started reading, and I’m still reading Jerry’s papers and trying to catch up. Jerry and I began corresponding. We met at the Southampton conference and I subsequently visited him in Berkeley during a hectic job-seeking tour of the USA in the Fall of 1976, and again during his visit to Heriot–Watt University in Edinburgh as a Carnegie Fellow in the spring of 1977. This resulted in our first joint paper [54], and was the beginning of a twentyfive year collaboration and friendship which I hope will last at least another twenty five. For me, one of the high points of this was our paper [71], in which we gave one of the first examples of a PDE with chaotic solutions (Smale horseshoes), via an infinite-dimensional extension of the Smale–Birkhoff theorem and Melnikov’s method. (John Guckenheimer gave another at about the same time via center-manifold reduction of a reactiondiffusion equation at a codimension-two bifurcation point.) After I had settled in the USA at Cornell University, Jerry invited me to Berkeley for the Spring semester of 1981, during which we wrote a series of papers [73; 77; 82] extending Melnikov type analyses to multi-degree-of-freedom Hamiltonian systems (although not without leaving a few gaps in our proofs to be filled by others, in the time-honored tradition of Poincaré). While we have not written joint papers in the last ten years, his work at the interface of mechanics and mathematics has remained an inspiration for my own, and we have met once or twice every year and had countless scientific, editorial, organizational, and mathematico-political discussions and collaborations. Jerry is a mainstay of the Journal of Nonlinear Science, which I now edit, and I’m proud to serve as an advisor to the Springer Applied Mathematical Sciences Series which Jerry edits with Larry Sirovich and Stu Antman. I was even prouder to nominate him for the AMS–SIAM

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xv

Norbert Wiener Prize in 1990, and to support his successful nominations to the Royal Society of Canada and the American Academy of Arts and Sciences. But rather than these well-deserved honors, I especially wish to celebrate Jerry’s 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 bright but sometimes erratically-educated graduate students the necessary background and methods, teaching colleagues and collaborators about new advances, and teaching oneself all the things that no one else did. Jerry is a master teacher: in his many textbooks at all levels, and in his conference presentations and lecture courses, always delivered with elegance, polish, and a little humor. (He is almost the only person I know who can put content into powerpoint — although he’s also careful to explain that it’s not actually powerpoint.) At Berkeley and Caltech Jerry has had, and continues to have, a succession of wonderful Ph.D. students and postdocs, many of whom have gone on to propagate his grand project of geometrizing mechanics (their names appear elsewhere in this volume). He has been equally generous with his time with young visitors (for many of whom, including myself, he raised the funds to invite), with the students of others, and simply with people who write or approach him to ask questions at conferences and workshops. I’m happy that we’ve been able to include articles contributed by several such colleagues in this Festschrift. Certainly, Jerry’s interest and involvement in the early struggles of a mechanic poorly trained in mathematics was enormously encouraging to me. In those far-off days, from misty England, he seemed to me a senior scientist: a Professor from distant and fabled Berkeley, David Lodge’s Euphoric State U. Now that we are both almost seniors, he no longer seems that much older, but he still knows a lot more geometry and analysis, and I’m still taking notes in the second row and having trouble with the homework. Paul Newton: Like many of us, I first met Jerry in print. As a Freshman at Harvard in 1977, I learned Stokes’ theorem, Green’s theorem and the divergence theorem from his (and Tromba’s) beautiful “Vector Calculus.” Those who are familiar with the first edition and who are aware of Jerry’s fascination with weather patterns will suspect that his favorite aspect of the book must have been the cloud formations on its cover. Mine was the elegant formulation of these theorems in terms of differential forms, something I had never seen in high school! I remember using this work to such an extent (so much for my social life) that today it is held together only by being wedged between two volumes on my shelf. Fast forward eight years to 1985. While a postdoc at Stanford, I participated (inconspicuously) in a seminar series on dynamics which Jerry,

xvi

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together with Lieberman, ran at Berkeley, and it was there that I first took note of what I now know to be his uncommon blend of open-mindedness and depth of thought, coupled with a generosity of spirit, demonstrated so vividly in his mentoring of young mathematicians. But it was when we both ended up in Los Angeles at roughly the same time (1993 in my case) that we became friends. I was busy working out the details of how the geometric phase manifested itself in the context of vortex dynamics problems, and Jerry’s encouragement and insights were invaluable in helping me move from an early interest in nonlinear dispersive wave models to more general issues in applied dynamical systems theory. Our interests overlapped again when I showed him some problems involving the motion of vortices on a sphere (with applications to weather patterns!). This led him (with Sergey Pekarsky) to begin applying nonlinear stability techniques to relative equilibrium configurations of vortices on a sphere. As I read and re-read many of his books and papers, I seem to fall farther and farther behind. But occasionally I look up to where it all began, the punished copy of “Vector Calculus” on my bookshelf, and I wonder if I would have become a mathematician had my professor chosen Edwards and Penney instead of Marsden and Tromba. Alan Weinstein: My work with Jerry has two facets: (1) symplectic reduction, Poisson geometry, and applications to stability of continuum mechanical systems; (2) calculus books. Our original work on reduction, which is probably one of the two or three most influential papers I have written, was stimulated by our listening to lectures of Smale around 1970; Smale had developed the theory in the special case of lifted actions on cotangent bundles. I think that my interest in abstract symplectic geometry, combined with some interest in physics, meshed perfectly with Jerry’s interest in relativity and applied Hamiltonian dynamics. (One should always mention in this context that symplectic reduction was discovered independently at about the same time by Ken Meyer, though I think it might be fair to say that he conceived of this construction in narrower terms than we did.) About 10 years later, we were attending the “dynamics seminar” in the Berkeley physics department, where Allan Kaufman and Robert Littlejohn were studying recent work of John Greene and Phil Morrison on the Hamiltonian structure of equations in plasma physics. These authors had a Poisson structure for the Maxwell–Vlasov equations which they found by trial and error and for which they checked the Jacobi identity by hand; according to Morrison, this took them 4 months of work, mostly calculations. Jerry and I spent about 6 months developing the right application of reduction to this problem, after which we could derive the correct bracket in 4 minutes, with the Jacobi identity coming for free. Another payoff was that we discovered an error in the Morrison–Greene formula.

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xvii

It was Jerry’s interest in applications which kept us going through much of the 80’s, applying these Poisson brackets to applications of Arnol’d’s general method for analyzing the stability of continuum-mechanical motions. Much of this work was also done in collaboration with Darryl Holm and Tudor Ratiu, and I eventually dropped out of the “group”, but the subject has continued to evolve in the hands of Jerry and his collaborators (now including notably Steve Shkoller), the latest developments being the study of “α-Euler equations” and wide applications of Lagrangian (as opposed to Hamiltonian) methods for the study of stability. On the calculus side, I recall that our collaboration started with a discussion at the end of a tennis game. Jerry was in contact with a new publisher who wanted to do a calculus book, and we had some new ideas for calculus teaching, based on the use of bifurcation ideas to replace the early introduction of limits (which eventually appeared only in a spin-off book called Calculus Unlimited ). We went through several publishers and many versions of the book, and I never had time to play tennis again. I think that Jerry kept it (and squash) up, though.

Acknowledgments: The editors owe a debt of gratitude to Wendy McKay and Ross Moore. Without their combined expertise in LATEX and other matters, this volume might not have been presented to Jerry until his 70th birthday. We also wish to thank the editors and staff at Springer-Verlag, particularly Achi Dosanjh and Elizabeth Young, for making this book a reality. Paul Newton Santa Barbara, California

Philip Holmes Princeton, New Jersey

Alan Weinstein Berkeley, California

March 2002


Part I

Elasticity and Analysis

1


1 Some Open Problems in Elasticity John M. Ball To Jerry Marsden on the occasion of his 60th birthday ABSTRACT Some outstanding open problems of nonlinear elasticity are described. The problems range from questions of existence, uniqueness, regularity and stability of solutions in statics and dynamics to issues such as the modelling of fracture and self-contact, the status of elasticity with respect to atomistic models, the understanding of microstructure induced by phase transformations, and the passage from three-dimensional elasticity to models of rods and shells. Refinements are presented of the author’s earlier work Ball [1984a] on showing that local minimizers of the elastic energy satisfy certain weak forms of the equilibrium equations.

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . Elastostatics . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Stored-Energy Function and Equilibrium Solutions 2.2 Existence of Equilibrium Solutions . . . . . . . . . . 2.3 Regularity and the Classification of Singularities . . 2.4 Satisfaction of the Euler–Lagrange Equation and Uniform Positivity of the Jacobian . . . . . . . . . . 2.5 Regularity and Self-Contact . . . . . . . . . . . . . . 2.6 Uniqueness of Solutions . . . . . . . . . . . . . . . . 2.7 Structure of the Solution Set . . . . . . . . . . . . . 2.8 Energy Minimization and Fracture . . . . . . . . . . 3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Continuum Thermomechanics . . . . . . . . . . . . . 3.2 Existence of Solutions . . . . . . . . . . . . . . . . . 3.3 The Relation Between Statics and Dynamics . . . . 4 Multiscale Problems . . . . . . . . . . . . . . . . . . . 4.1 From Atomic to Continuum . . . . . . . . . . . . . . 4.2 From Microscales to Macroscales . . . . . . . . . . . 4.3 From Three-Dimensional Elasticity to Theories of Rods and Shells . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

4 4 4 6 11 15 22 22 23 26 27 27 29 33 36 36 38 42 45

4

John M. Ball

1

Introduction

In this paper I highlight some outstanding open problems in nonlinear (sometimes called finite) elasticity theory. While many of these will be well known to experts on analytic aspects of elasticity, I hope that the compilation will be of use both to those new to the field and to researchers in solid mechanics having different perspectives. Of course the selection of problems is a personal one, and indeed represents a list of those problems that I would most like to be able to solve, but cannot. In particular it concentrates on general open problems, or ones that illustrate general difficulties, rather than those related to very specific experimental situations, which is not to imply that the latter are not important or instructive. I have not included any open problems connected with the numerical computation of solutions, since I recently discussed some of these in Ball [2001]. The only new results of the paper are in connection with the problem of showing that local minimizers of the total elastic energy satisfy the weak form of the equilibrium equations. As I pointed out in Ball [1984a], there are hypotheses under which some forms of the equilibrium equations can be proved to hold, and in Section 2.4 I take the opportunity to present some refinements of this old work. The paper is essentially self-contained, and can be read by those having no knowledge of elasticity theory. For those seeking further background on the subject I have written a short introduction (Ball [1996]) to some of the issues, intended for research students, which I hope is a quick and easy read. For more serious study in the spirit of this paper, the reader is referred to the books of Antman [1995], Ciarlet [1988, 1997, 2000], Marsden ˇ and Hughes [1983] and Silhav´ y [1997]. Other excellent but older books and survey articles are Antman [1983], Ericksen [1977b], Gurtin [1981] and Truesdell and Noll [1965]. Valuable additional perspectives can be found in the 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 friend and in recognition of his important contributions to elasticity, and thus to help celebrate his many talents as a mathematician, thinker and writer.

2 2.1

Elastostatics The Stored-Energy Function and Equilibrium Solutions

Consider an elastic body which in a reference configuration occupies the bounded domain Ω ⊂ R3 . We suppose that Ω has a Lipschitz boundary ∂Ω = ∂Ω1 ∪ ∂Ω2 ∪ N , where ∂Ω1 , ∂Ω2 are disjoint relatively open subsets of ∂Ω and N has two-dimensional Hausdorff 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 : Ω → 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 Sobolev space W 1,1 (Ω; R3 ), so that in particular the deformation gradient Dy(x) is well defined for a.e. x ∈ Ω. For each such x we can identify Dy(x) with the 3 × 3 matrix (∂yi /∂xj ). We require the deformation y to satisfy the boundary condition ¯( · ) , y∂Ω = y (2.1) 1

¯ : ∂Ω1 → R is a given boundary displacement. where y 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 energy corresponding to the deformation y is given by W Dy(x) dx , (2.2) I(y) = 3

Ω

where W = W (A) is the stored-energy function of the material. We suppose 3×3 that W : M+ → R is C 1 and bounded below, so that without loss of generality W ≥ 0. (Here and below, M m×n denotes the space of real m × n n×n denotes the space of those A ∈ M n×n with det A > 0.) matrices, and M+ The Piola–Kirchhoff stress tensor is given by TR (A) = DA W (A) .

(2.3)

By formally computing d I(y + τ ϕ)τ =0 = 0 , dτ we obtain the weak form of the Euler–Lagrange equation for I, that is DA W (Dy) · Dϕ dx = 0 (2.4) Ω

for all smooth ϕ with ϕ|∂Ω1 = 0. This can be shown (cf. Antman and Osborn [1979]) to be equivalent to the balance of forces on arbitary subbodies. If y, ∂Ω1 and ∂Ω2 are sufficiently regular then (2.4) is equivalent to the pointwise form of the equilibrium equations div DA W (Dy) = 0

in Ω ,

(2.5)

together with the natural boundary condition of zero applied traction DA W (Dy)n = 0

on ∂Ω2 ,

(2.6)

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John M. Ball

where n = n(x) denotes the unit outward normal to ∂Ω at x. (More generally, we could have prescribed nonzero tractions of various types on ∂Ω2 , as well as including the potential energy of body forces such as gravity in the expression for the energy (2.2), but for simplicity we have not done this, since the main difficulties we address are already present without these additions.) To avoid interpenetration of matter, it is natural to require that y : Ω → R3 be invertible. To try to ensure that deformations have this property, we suppose that (2.7) W (A) → ∞ as det A → 0+ . So as to also prevent orientation reversal we define W (A) = ∞ if det A ≤ 0. Then W : M 3×3 → [0, ∞] is continuous. Clearly if I(y) < ∞ then det Dy(x) > 0

for a.e. x ∈ Ω .

(2.8)

Since y is not assumed to be C 1 , (2.8) does not imply even local invertibility. For studies of local and global invertibility in the context of elasticity, or relevant to it, see Ball [1981], Bauman and Phillips [1994], Ciarlet and Neˇcas [1985], Fonseca and Gangbo [1995], Giaquinta, Modica and Souˇcek ˇ ak [1988] and Weinstein [1985]. [1994], Meisters and Olech [1963], Sver´ We assume that for any elastic material the stored-energy function W is frame-indifferent, i.e., W (RA) = W (A)

for all R ∈ SO(3) ,

A ∈ M 3×3 .

(2.9)

In addition, if the material has a nontrivial isotropy group S, W satisfies the material symmetry condition W (AQ) = W (A)

for all Q ∈ S ,

A ∈ M 3×3 .

The case S = SO(3) corresponds to an isotropic material. For incompressible materials the deformation y is required to satisfy the constraint det Dy(x) = 1 for a.e. x ∈ Ω . All of the problems and results contained in this article have corresponding incompressible 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 solutions. The first, pioneered by Stoppelli [1954, 1955] and described in the book of Valent [1988], is to use the implicit function theorem in a suitable Banach space X to prove the existence of an equilibrium solution close to a given one, when the data of the problem are slightly perturbed. In order to


7

make this work, it is necessary to use spaces X of sufficiently smooth mappings, for example subspaces of W 2,p (Ω; R3 ) for p > 3 or C 2+α (Ω; R3 ), so as to control the nonlinear dependence on Dy. In addition, the linearized elasticity operator at the given solution should be invertible as a map from X to a suitable target space Y . While this method automatically delivers smooth solutions, it is by its nature restricted to small perturbations (for example, small boundary displacements from a stress-free state), and because of the regularity properties required for the linearized operator it in general only applies to situations when ∂Ω1 and ∂Ω2 do not meet, for example when one of them is empty. In particular, mixed boundary conditions typically encountered in applications, for example (2.1) with ∂Ω1 comprising 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 via the direct method of the calculus of variations. In principle such a minimizer should satisfy the equilibrium equations, at least in weak form, but this turns out to be a subtle matter (see Sections 2.3, 2.4). More generally we could ask for conditions ensuring that there exist some kind of local minimizer. Let ¯ , A = y ∈ W 1,1 (Ω; R3 ) : I(y) < ∞ , y|∂Ω1 = y where the boundary condition is understood in the sense of trace. In the definition of A we could include the requirement that y be invertible; this can be done in various ways, one of which is discussed in Section 2.5. 2.1 Definition. Let 1 ≤ p ≤ ∞. The deformation y ∈ A is a W 1,p local minimizer of I if there exists ε > 0 such that I(z) ≥ I(y) for any z ∈ A with z − yW 1,p ≤ ε. The problem of proving the existence of local, but not global, minimizers is discussed later (see Problem 9). A typical result on global minimization is the following. 2.2 Theorem.

Suppose that W satisfies the hypotheses

(H1) W is polyconvex, i.e., W (A) = g(A, cof A, det A) for all A ∈ M 3×3 for some convex g, 3 (H2) W (A) ≥ c0 |A|2 + |cof A| 2 − c1 for all A ∈ M 3×3 , 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 norm on M 3×3 with TEuclidean corresponding inner product A·B = T A B . Theorem 2.2 is a refinement by M¨ uller, Qi and Yan [1994] of the result in Ball [1977a]. For the problem to be nontrivial we need that H2 (∂Ω1 ) > 0.

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John M. Ball

The hypothesis (H1) is known to be too strong for the following reason. Let f : M m×n → R ∪ {+∞} be Borel measurable and bounded below. We recall f is said to be quasiconvex at A ∈ M m×n if the inequality f A + Dϕ(x) dx ≥ f (A) dx (2.10) Ω

Ω

C0∞ (Ω; Rm ), and n

is quasiconvex if it is quasiconvex at holds for any ϕ ∈ every A ∈ M m×n . Here Ω ⊂ R is any bounded open set whose boundary ∂Ω has zero n-dimensional Lebesgue measure. A standard scaling argument (see, for example, Ball and Murat [1984]) shows that contrary to appearances these definitions do not depend on Ω. Results of Morrey [1952] and Acerbi and Fusco [1984] imply that if f : M m×n → R is quasiconvex and satisfies the growth condition for all A ∈ M m×n , (2.11) C1 |A|p − C0 ≤ f (A) ≤ C2 |A|p + 1 where p > 1 and where C0 and C1 > 0, C2 > 0 are constants, then f (Dy) dx (2.12) F(y) = Ω

attains a global minimum on ¯ }. A = {y ∈ W 1,1 (Ω; Rm ) : y|∂Ω1 = y Here we assume that Ω has Lipschitz boundary ∂Ω, that ∂Ω1 ⊂ ∂Ω is Hn−1 ¯ : ∂Ω1 → Rm is given such that A is nonempty. As measurable and that y shown by Ball and Murat [1984], quasiconvexity of f is necessary for the existence of a global minimizer for all perturbed functionals of the form f Dy(x) + h x, y(x) dx F1 (y) = Ω

with h( · , · ) ≥ 0 continuous. These results strongly suggest that (H1) should be replaced by the requirement that W be quasiconvex, a weaker condition than polyconvexity. For example, it is easily seen that a larger class of W for which Theorem 2.2 holds consists of those of the form W = W1 + f , where W1 satisfies (H1) and (H2), and where f : M 3×3 → R is quasiconvex and satisfies (2.11). That this really is a larger class can be seen by taking f = KF for a large K > 0, where F is quasiconvex but not polyconvex. Such F exist satisfying F (RAQ) = F (A) for all R, Q ∈SO(3), ˇ ak [1991]1 . More A ∈ M 3×3 , and can be constructed by the method of Sver´ example we can take F = H qc to be the quasiconvexification of p p H(A) = min U(A) − 1 , U(A) − λ1 , 1 where λ > 1 and U(A) = AT A 2 . The quasiconvexification H qc of H is defined to be the supremum of all quasiconvex functions ψ ≤ H. 1 For


9

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 nondecreasing sequence of continuous quasiconvex functions f k : M 3×3 → [0, ∞), each satisfying a growth condition 0 ≤ f k (A) ≤ αk |A|p − βk for constants αk > 0, βk . (Kristensen [1994] has shown that a lower semicontinuous function f : M 3×3 → [0, ∞] satisfying (2.13) is the supremum in the sense of such a sequence if and only if f is closed W 1,p quasiconvex

of Pedregal [1994], namely that Jensen’s inequality ν , f ≥ f (¯ ν ) holds for all homogeneous W 1,p gradient Young measures2 ν.) However, as they stand none of the existence theorems for minimizers of integrals of general quasiconvex functions apply to elasticity, since they all assume growth conditions such as (2.11) which are not consistent with the condition (2.7). (The same applies to other results, such as the relaxation theorem of Dacorogna [1982].) In particular, it is not clear whether or not a quasiconvex W satisfying our hypotheses can be written as the supremum of everywhere finite continuous quasiconvex functions. This is not true in general for quasiconvex functions f : M m×n → [0, ∞]; for example we can take 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 configuration (see Tartar [1993]), for example A1 = diag (−2, 1), A3 = diag (2, −1),

A2 = diag (1, 2), A4 = diag (−1, −2).

Then f is quasiconvex, since any y with Dy ∈ {A1 , A2 , A3 , A4 } a.e. has constant gradient (this following, for example, from the general result of Chlebík and Kirchheim [2001]). But the argument of Tartar shows that if f were the supremum of continuous quasiconvex functions f k : M 2×2 → [0, ∞) we would have f (0) = 0, a contradiction. 2 See

Young [1969], Tartar [1979], Ball [1989] for the definition and properties of the Young measure (νx )x∈Ω corresponding to a sequence of mappings z(j) : Ω → Rs satisfying a suitable bound, say z(j) L1 ≤ M < ∞, where Ω ⊂ Rn is open (or measurable). For each x ∈ Ω, νx is a probability measure on Rs giving the limiting distribution of values of z(j) (p) as j → ∞ and p → x. If f : Rs → R is continuous, then the weak limit of f (z (j) ) in L1 (E), where E ⊂ Ω is measurable, is given by the function x → νx , f , whenever the weak limit exists. In particular, if z(j) z in L1 (E), then z(x) = ν¯x for x ∈ E, where µ ¯ denotes the centre of mass of a measure µ. Such a Young measure is homogeneous if ν = νx is independent of x. If 1 < p ≤ ∞, a W 1,p gradient Young measure is a Young measure (νx )x∈Ω corresponding to a sequence z (j) = Dy(j) of gradients bounded in Lp (Ω; M m×n ), where we identify M m×n with Rmn .

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John M. Ball

A further reason for preferring quasiconvexity to polyconvexity is that, unlike quasiconvexity, polyconvexity is not closed with respect to periodic homogenization (Braides [1994]). Problem 1. Prove the existence of energy minimizers for elastostatics for quasiconvex stored-energy functions satisfying (2.7). A principal difficulty here is that there is no known useful characterization 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 ∈ M m×n and a ∈ Rm , n ∈ Rn . For 40 years it seemed possible that in fact rank-one ˇ ak [1992] found his convexity was equivalent to quasiconvexity, until Sver´ well-known counterexample for the dimensions n ≥ 2, m ≥ 3. Then Krisˇ ak’s example to show that for the same dimensions tensen [1999] used Sver´ there is no local characterization of quasiconvexity. In the absence of a characterization leading to a new proof technique, one is forced to make direct use of the definition (2.10), which leads to serious problems of approximation by piecewise affine functions when (2.7) holds. In Ball [1977a] it was shown how the hypotheses (H1), (H2) can be satisfied for a class of isotropic materials including models of natural rubbers, via theorems exploiting the representation W (A) = Φ(v1 , v2 , v3 )

(2.14)

of the stored-energy function W of an isotropic material, where Φ is a symmetric function of the singular values vi = vi (A), that is of the eigenvalues of (AT A)1/2 (for a different proof of such theorems see Le Dret [1990]). However it is not obvious how to verify (H1) when the material is not isotropic, for example when it has cubic symmetry. Problem 2. Are there ways of verifying polyconvexity and quasiconvexity for a useful class of anisotropic stored-energy functions? To illustrate the difficulty in verifying (H1), in the isotropic case it is much more convenient to use the representation (2.14) rather than the equivalent representation W (A) = h(I1 , I2 , I3 ) in terms of the principal invariants Ij = Ij (A). Perhaps it is significant that the function Φ in (2.14) has the same regularity as W , while h is less regular (see Ball [1984], ˇ Sylvester [1985], Silhav´ y [2000]). At any rate the more symmetric form (2.14) lends itself more easily to discussing convexity properties. For nonisotropic materials suitable representations do not seem to be available; for example, in the case of cubic symmetry it does not seem to be convenient to use the usual integrity basis (given, for example, in Green and Adkins [1970]).


2.3

11

Regularity and the Classification of Singularities

The main open question concerning regularity is to decide when global, or local, 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 Ω, and C ∞ up to the boundary (except in the neighbourhood of points x0 ∈ ∂Ω1 ∩ ∂Ω2 where singularities can be expected). Clearly additional hypotheses on W are needed for this to be 3×3 → R is C ∞ , and that true. One might assume, for example, that W : M+ (H1) is strengthened by assuming W to be strictly polyconvex (i.e., that g is strictly convex). Also for regularity up to the boundary we would need to assume both smoothness of the boundary (except perhaps at ∂Ω1 ∩ ∂Ω2 ) ¯ is smooth. The precise nature of these extra hypotheses is to be and that y determined. Problem 3 is unsolved even in the simplest special cases. In fact the only situation in which smoothness of y∗ seems to have been proved is for the pure displacement problem (∂Ω2 empty) with small boundary displacements from a stress-free state. For this case Zhang [1991], following work of Sivaloganathan [1989], gave hypotheses under which the smooth solution to the equilibrium equations delivered by the implicit function theorem 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 possible singularities in minimizers of I given by (2.2) for generic stored-energy functions W . If at the same time one could associate with each such singularity a condition on W that prevented it, one would also, by imposing all such conditions simultaneously, possess a set of hypotheses implying regularity. In fact it is possible to go a little way down this road. Consider first the kind of singularity frequently observed at phase boundaries in elastic crystals, in which the deformation gradient Dy is piecewise constant, with values 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 A0 that is a local minimizer of W ( · ), every such deformation y that is locally a weak solution of the Euler–Lagrange equation is trivial, that is A = B, if and only if W is strictly rank-one convex (i.e., the map t → W (A + ta ⊗ n) is strictly convex for every A and all nonzero a, n). Thus strict rank-one convexity is exactly what is needed to eliminate this particular kind of singularity. Another physically occuring singularity is that of cavitation. For radial cavitation the deformation has the form y : B(0, 1) → R3 , where B(0, 1) is the unit ball in R3 , and x . y(x) = r |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 satisfying y(x) = λx for |x| = 1 (i.e., r(1) = λ) is given by the homogeneous deformation ˜ λ (x) ≡ λx . y However, it was shown in Ball [1982] that for a class of stored-energy functions W satisfying (H1) and the growth condition in (H2) but with 2 ≤ p < 3, q < 32 , I attains a minimum among radial deformations satisfying the boundary condition y(x) = λx for |x| = 1, and that for λ > 0 ¯ satisfies r(0) > 0. Furthermore y ¯ satisfies sufficiently large the minimizer y the weak form of the Euler–Lagrange equation (2.4). As a specific example we can take (2.15) W (A) = |A|2 + h(det A) , for h : (0, ∞) → R smooth with h > 0, limδ→∞ h(δ) = limδ→0+ h(δ) = δ ∞. Cavitation is a common failure mechanism in polymers; for interesting pictures of almost radial cavitation of roughly spherical rubber particles imbedded in a matrix of Nylon-6 see Lazzeri and Bucknall [1995]. M¨ uller, Qi and Yan [1994] show that if (H2) holds then no deformation with finite energy can exhibit cavitation. A somewhat stronger condition, which by the Sobolev embedding theorem obviously prevents not only cavitation but any other discontinuity in y, is that W (A) ≥ c0 |A|p − c1 for all A, where c0 > 0 and p > 3. In fact even if p = 3 any finite-energy deformation is continuous on account of (2.8) and the result of Vodop’yanov, Gol’dshtein and Reshetnyak [1979]. There is an extensive literature on cavitation in elasticity; a sample of the more mathematical developments can be found in the papers of Antman and Negr´ on-Marrero [1987], James and Spector [1991], M¨ uller 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 example of the Lavrentiev phenomenon, whereby the infimum of the energy is different in different function spaces. Here it takes the form yλ ) , inf I < inf I = I(˜ A1

A3

where Ap = y ∈ W 1,p (B(0, 1); R3 ) : y|∂B(0,1) = λx . More generally, there is a theory of minimization for elasticity with W polyconvex in function spaces not allowing cavitation due to Giaquinta, Modica and Souˇcek [1989, 1994, 1998] (see also the less technically demanding approach of M¨ uller [1988]). Thus the same W can have different minimizers in different function spaces; if we enlarge the space to allow not only cavitation but crack formation (see Section 2.8), then we can have different minimizers in at least three different spaces.


13

In cavitation there is a change of topology of the deformed configuration associated with the Lavrentiev phenomenon, but one-dimensional examples in Ball and Mizel [1985] for integrals of the form b f x, y(x), yx (x) dx I(y) = a

show that the phenomenon can occur when the minimizer y is continuous with unbounded gradient. This leads to the question: Problem 4. Can the Lavrentiev phenomenon occur for elastostatics under growth conditions on the stored-energy function ensuring that all finiteenergy deformations are continuous? Of course if y∗ is smooth then the Lavrentiev phenomenon cannot hold under the hypotheses of Theorem 2.2. Some interesting recent progress on Problem 4 is due to Foss [2001], Mizel, Foss and Hrusa [2002], who provide examples of the Lavrentiev phenomenon in two dimensions for a homogeneous isotropic polyconvex stored-energy function W satisfying (2.7) and the corresponding growth condition W (A) ≥ c0 |A|p − c1 for all 2×2 and some p > 2. In these examples the reference configuration A ∈ M+ is given by a sector of a disk described in polar coordinates by Ωα = (r, θ) : 0 < r < 1 , 0 < θ < α , and the boundary conditions are of the ‘container’ type that the origin is fixed, that y(Ωα ) ⊂ Ωβ , and β that y(1, θ) = (1, α θ), where 0 < β < 34 α. Whether such examples can be constructed for mixed boundary conditions of the type (2.1), even in two dimensions, or be associated with singularities in the interior of Ω, remains open. The Lavrentiev phenomenon has important implications for the numerical computation of minimizers (see Ball [2001] and the references therein). For a useful survey of the phenomenon in one and more dimensions see Buttazzo and Belloni [1995]. There are striking examples of variational problems of the form (2.12) for which the global minimizer is not smooth. The first such example was found by Neˇcas [1977], who showed that if m = n2 with n sufficiently large then there exists a strictly convex f = f (Dy) whose corresponding integral F(y) = f (Dy) dx (2.16) B(0,1)

has as global minimizer ∗ (x) = yij

xi xj , |x|

x ∈ B(0, 1)

subject to its own (smooth) boundary-values on ∂B(0, 1). Here y∗ is Lipschitz but not C 1 . Then Hao, Leonardi and Neˇcas [1996] modified the example to work for n ≥ 5 with minimizer xi xj ∗ − n1 |x|δij . = (2.17) yij |x|

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John M. Ball

ˇ ak and Yan [2000] showed that there By a more sophisticated method, Sver´ exists a convex f such that (2.17) gives a minimizer for n = 3. In fact, working in the five-dimensional space of 3 × 3 traceless symmetric matrices ˇ ak and Yan also we thus obtain an example with n = 3, m = 5. Sver´ obtained a similar example for the case n = 4, m = 3. Note that the above maps y∗ are one-homogeneous, i.e., y∗ (sx) = sy∗ (x) for all s ≥ 0. In contrast Phillips [2001] has shown that when n = 2 any one-homogeneous weak solution y to a strongly elliptic system of the form div A(Dy) = 0 is linear. Even if y∗ is not smooth everywhere, we can ask for smoothness outside a closed set of Lebesgue measure zero. That such a result is true is strongly suggested by the classical partial regularity theorem of Evans [1986], which (with the incorporation of a weakening of the growth condition due to Acerbi and Fusco [1988]) states that any global minimizer has this property for a class of integrals of the form (2.16) with f satisfying (2.11) and the strong quasiconvexity condition that for some p ≥ 2 f (A + Dϕ) − f (A) dx ≥ γ |Dϕ|2 + |Dϕ|p dx , Ω

Ω

for all A ∈ M m×n , all ϕ ∈ C0∞ (Ω; Rm ). Recently, Kristensen and Taheri [2001] have proved the same result but for W 1,p local minimizers. However, it is not known how to extend these theorems to the case of elasticity when (2.7) holds (see Ball [1998] for a brief discussion). In proving regularity or partial regularity, it is not sufficient to just use the fact (if it is a fact, see below) that y∗ satisfies the weak form of the Euler–Lagrange equation. This follows from the example of M¨ uller and ˇ ak [2001] of a Lipschitz mapping y : Ω → R2 , with Ω ⊂ R2 a bounded Sver´ domain, that is nowhere C 1 and solves the weak form of the Euler–Lagrange equation for an integral I(y) = F (Dy) dx Ω

with F strictly quasiconvex. As shown by Kristensen and Taheri [2001] y can even be a W 1,∞ local minimizer. There seems to be little indication from experiment that natural rubbers have equilibrium solutions with singularities other than those involving cavitation or other forms of fracture. Thus it seems a reasonable conjecture that minimizers are smooth for models of natural rubber for which the stored-energy function satisfies growth conditions prohibiting discontinuities for deformations of finite energy. In view of the above and other counterexamples for elliptic systems, if minimizers are smooth it must be for special reasons applying to elasticity. Three plausible such reasons are: (a) the fact that the integrand W (Dy) does not depend explicitly on y (though dependence on x is allowed), (b) the fact that the dimensions


15

m = n = 3 are low, (c) the frame-indifference of W (see (2.9)). A fourth possible reason is (d) invertibility of y, which we discuss now. That invertibility could have an effect on regularity was first shown by Bauman, Owen and Phillips [1991], who gave an example of an essentially two-to-one equilibrium solution in a ball for two-dimensional elasticity, with stored-energy function of the form (2.15), that is C 1 but not smooth, and is such that det Dy vanishes at the centre of the ball. An instructive example (resulting from discussions with X. Yan and J. Bevan) is that of minimizing the two-dimensional energy for an incompressible material |Dy|2 dx , I(y) = B

where B = B(0, 1) is the unit disc in R2 , and y : B → R2 , in the set of admissible mappings ¯ , (2.18) A = y ∈ W 1,2 (B; R2 ) : det Dy = 1 a.e. , y∂B = y 1 ¯ : (r, θ) → √2 r, 2θ . Then there exists a global where in polar coordinates y ∗ ¯ ∈ A.) But since minimizer y of I in A. (Note that A is nonempty since y by degree theory there are no C 1 maps y satisfying the boundary condition (2.18), it is immediate that y ∗ is not C 1 . For interesting maximum principles satisfied by smooth equilibrium solutions in two-dimensional elasticity, with stored-energy function of the form (2.15), see Bauman, Owen and Phillips [1991a, 1992].

2.4

Satisfaction of the Euler–Lagrange Equation and Uniform Positivity of the Jacobian

Here we return to the computation formally leading to the weak form (2.4) of the Euler–Lagrange equation, under the assumption (2.7). If y∗ ∈ W 1,∞ (Ω; R3 ) is a W 1,∞ local minimizer of I in ¯ , A = y ∈ W 1,1 (Ω; R3 ) : y∂Ω1 = y and if (2.8) holds in the stronger form that for some ε > 0 det Dy∗ (x) ≥ ε

for a.e. x ∈ Ω ,

(2.19)

then it is easily seen that y∗ satisfies (2.4). In fact we can then pass to the limit τ → 0 using bounded convergence in the difference quotient 1 W (Dy∗ + τ Dϕ) − W (Dy∗ ) dx , (2.20) Ω τ since by (2.19) we have det Dy∗ (x) + τ Dϕ(x) ≥ ε/2 for a.e. x ∈ Ω. However, if only (2.7) is assumed, or if y∗ is not assumed in advance to be in W 1,∞ (Ω; 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 conditions on W , global or suitably defined local minimizers of I satisfy the weak form (2.4) of the Euler–Lagrange equation. Problem 6. Prove or disprove that, under reasonable growth conditions on W , global or suitably defined 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 first sight inconsistent with y∗ being a minimizer, but we know from the one-dimensional examples in Ball and Mizel [1985] and from the example of cavitation that it can pay to have the integrand infinite somewhere so that it is smaller somewhere else. In general, it is not possible to estimate the integrand in the difference quotient (2.20) in terms of W (Dy∗ ), the only relevant quantity that is obviously integrable. This difficulty was pointed out by Antman [1976], who was the first to address the issue of satisfaction of the Euler–Lagrange equation for one-dimensional problems of elasticity when (2.7) holds; in this context a device essentially due to Tonelli [1921] can be used to prove that the Euler–Lagrange equation holds (see also Ball [1981a]) without any supplementary growth conditions on W . It is perhaps worth making the simple observation that a smooth deformation y may satisfy I(y) < ∞ and det Dy(x) > 0 a.e. without (2.19) holding. As an example we may take Ω = B(0, 1) and y(x) = |x|2 x , with W (A) = − log det A + g(A), where g : M 3×3 → R is smooth. For a class of strongly elliptic stored-energy functions having the form W (A) = g(A) + h(det A), where g : M 3×3 → R and h : (0, ∞) → [0, ∞) are smooth with h(δ) → ∞ as δ → 0+ at a polynomial rate, Bauman, Owen and Phillips [1991] show that if y ∈ C 1,β satisfies the energy-momentum weak form of the Euler–Lagrange equation in (2.22) below, then in fact y is a smooth solution of the Euler–Lagrange equation (2.5) and the strict positivity condition (2.19) holds. As was pointed out in Ball [1984a], it is possible to derive different first-order necessary conditions for a minimizer when (2.7) holds. (Later Giaquinta, Modica and Souˇcek [1989] derived the same first-order conditions in their framework of Cartesian currents, under somewhat stronger hypotheses.) We give here improved versions of these results. We consider the following conditions that may be satisfied by W : 3×3 , where K > 0 is a (C1) DA W (A)AT ≤ K W (A) + 1 for all A ∈ M+ constant; and 3×3 (C2) AT DA W (A) ≤ K W (A) + 1 for all A ∈ M+ , where K > 0 is a constant.


17

As usual, | · | denotes the Euclidean norm on M 3×3 , for which the inequalities |A · B| ≤ |A| · |B| and |AB| ≤ |A| · |B| hold. But of course the conditions are independent of the norm used up to a possible change in the constant K. 2.3 Proposition.

Let W satisfy (C2). Then W satisfies (C1).

Proof. Since W is frame-indifferent the matrix DA W (A)AT is symmetric (this is equivalent to the symmetry of the Cauchy stress tensor — see (2.28) below). Hence DA W (A)AT 2 = DA W (A)AT · A DA W (A) T = AT DA W (A) · AT DA W (A ]T 2 ≤ AT DA W (A) ,

from which the result follows. Example.

Let

1 . det A Then W satisfies (C1) and (2.9), but not (C2). W (A) = (AT A)11 +

2.4 Theorem. For some 1 ≤ p < ∞ let y ∈ A ∩ W 1,p (Ω; R3 ) be a W 1,p local minimizer of I in A. (i) Let W satisfy (C1). Then [DA W (Dy)DyT ] · Dϕ(y) dx = 0

(2.21)

Ω

for all ϕ ∈ C 1 (R3 ; R3 ) such that ϕ and Dϕ are uniformly bounded and satisfy ϕ(y)|∂Ω1 = 0 in the sense of trace. (ii) Let W satisfy (C2). Then W (Dy)1 − DyT DA W (Dy) · Dϕ dx = 0

(2.22)

Ω

for all ϕ ∈ C01 (Ω; R3 ). We use the following simple lemma. 2.5 Lemma.

(a) If W satisfies (C1) then there exists γ > 0 such that if 3×3 C ∈ M+

then

and

|C − 1| < γ

DA W (CA)AT ≤ 3K W (A) + 1

3×3 for all A ∈ M+ .

18

John M. Ball

3×3 (b) If W satisfies (C2) then there exists γ > 0 such that if C ∈ M+ and |C − 1| < γ then T 3×3 A DA W (AC) ≤ 3K W (A) + 1 for all A ∈ M+ . (2.23)

Proof. We prove (a); the proof of (b) is similar. We first show that there exists γ > 0 such that if |C − 1| < γ then W (CA) + 1 ≤ 32 (W (A) + 1)

3×3 for all A ∈ M+ .

(2.24)

1 ) sufficiently small For t ∈ [0, 1] let C(t) = tC + (1 − t)1. Choose γ ∈ (0, 6K so that |C − 1| < γ√ implies that |C(t)−1 | ≤ 2 for all t ∈ [0, 1]. This is possible since |1| = 3 < 2. For |C − 1| < γ we have that

1

d W (C(t)A) dt dt 0 1 = DA W C(t)A · (C − 1)A dt

W (CA) − W (A) =

0

=

0

1

T DA W C(t)A C(t)A · (C − 1)C(t)−1 dt

≤K 0

W C(t)A + 1 · C − 1 · C(t)−1 dt

1

≤ 2Kγ

1

W C(t)A + 1 dt .

0

Let θ(A) = sup|C−1| 0 for a.e. x ∈ Ω and limτ →0 yτ − yW 1,p = 0. Hence I(yτ ) ≥ I(y) for |τ | sufficiently small. But 1 I(yτ ) − I(y) τ 1 d 1 W 1 + sτ Dϕ y(x) Dy(x) ds dx = τ Ω 0 ds 1 = DW 1 + sτ Dϕ y(x) Dy(x) · Dϕ y(x) Dy(x) ds dx . Ω

0

Since by Lemma 2.5(a) the integrand is bounded by the integrable function 3K W Dy(x) + 1 sup |Dϕ(z)| , z∈R3

we may pass to the limit τ → 0 using dominated convergence to obtain (2.21). (ii) This follows in a similar way to (i) from Lemma 2.5 (b). Since most of the details have already been written down in Bauman, Owen and Phillips [1991a] we just sketch the idea. Let ϕ ∈ C01 (Ω; R3 ). For sufficiently small τ > 0 the mapping θτ defined by θτ (z) := z + τ ϕ(z) belongs to C 1 (Ω; R3 ), satisfies det Dθτ (z) > 0, and coincides with the identity on ∂Ω. Standard arguments from degree theory imply that θτ is a diffeomorphism of Ω to itself. Thus the ‘inner variation’ yτ (x) := y(zτ ) ,

x = zτ + τ ϕ(zτ )

defines a mapping yτ ∈ A, and −1 Dyτ (x) = Dy(zτ ) 1 + τ Dϕ(zτ )

a.e. x ∈ Ω.

Since y ∈ W 1,p it follows easily that yτ − yW 1,p → 0 as τ → 0. Changing variables we obtain −1 det 1 + τ Dϕ(z) dz , W Dy(z) 1 + τ Dϕ(z) I(yτ ) = Ω

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 the following Invertibility Hypothesis. y is a homeomorphism of Ω onto Ω := y(Ω), Ω is a bounded domain, and the change of variables formula f y(x) det Dy(x) dx = f (z) dz (2.26) Ω

Ω

20

John M. Ball

holds whenever f : R3 → R is measurable, provided that one of the integrals in (2.26) exists. Sufficient conditions for this hypothesis to hold are given in Ball [1981] ˇ ak [1988]. and Sver´ 2.6 Theorem. Assume that the hypotheses of Theorem 2.4 (i) and the Invertibility Hypothesis hold. Then σ(z) · Dϕ(z) dz = 0 (2.27) Ω

for all ϕ ∈ C 1 (R3 ; R3 ) such that ϕ|y(∂Ω1 ) = 0, where the Cauchy stress tensor σ is defined by σ(z) := T y−1 (z) , and

z ∈ Ω

−1 T(x) = det Dy(x) DA W (Dy x) Dy(x)T .

(2.28)

¯ is bounded, we can assume that ϕ and Proof. Since by assumption y(Ω) Dϕ are uniformly bounded. Thus (2.27) follows immediately from (2.21), (2.26) and (2.28). Thus Theorem 2.4(i) asserts that y satisfies the spatial (Eulerian) form of the equilibrium equations. Theorem 2.4 (ii), on the other hand, involves the so-called energy-momentum tensor W (A)1 − AT DA W (A), and is a multi-dimensional version of the Du Bois Reymond or Erdmann equation of 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 satisfies (C1) or (C2). Then for some s > 0, s s 3×3 W (A) ≤ M A + A−1 for all A ∈ M+ . Proof. Let V ∈ M 3×3 be symmetric. For t ≥ 0 d W (etV ) = DA W (etV )etV · V dt = etV DA W (etV ) · V ≤ K W (etV ) + 1 |V| .

(2.29)

From this it follows that W eV + 1 ≤ W (1) + 1 eK|V| .

(2.30)


21

Now set V = ln U, where U = UT > 0, and denote by vi the eigenvalues of U. Since 3

12 ln U = (ln vi )2 i=1

≤

3

ln vi ,

i=1

it follows that eK| ln U| ≤ v1K + v1−K v2K + v2−K v3K + v3−K 3 3

13 viK + vi−K ≤ 3−3 i=1

≤C

3 i=1

vi3K +

i=1 3

vi−3K

3

i=1

≤ C1 |U|3K + |U−1 |3K , where C > 0, C1 > 0 are constants. From (2.30) we thus obtain W (U) ≤ M |U|3K + |U−1 |3K , where M = C1 W (1) + 1 . The result now follows from the polar de3×3 composition A = RU of an arbitrary A ∈ M+ , where R ∈ SO(3), T U = U > 0. It is easily seen that if W is isotropic then both (C1) and (C2) are equivalent to the condition that (v1 Φ,1 , v2 Φ,2 , v3 Φ,3 ) ≤ K Φ(v1 , v2 , v3 ) + 1 for all vi > 0 and some K > 0, where Φ is given by (2.14) and Φ,i = ∂Φ/∂vi . In particular, both (C1) and (C2) hold for the class of Ogden materials (Ogden [1972a,b]), for which Φ has the form Φ(v1 , v2 , v3 ) =

M i=1

ai ϕ(αi ) +

N

bi ψ(βi ) + h(v1 v2 v3 )

i=1

where ϕ(α) = v1α + v2α + v3α ,

ψ(β) = (v2 v3 )β + (v3 v1 )β + (v1 v2 )β ,

ai > 0, bi > 0, αi = 0, βi = 0, and where h : (0, ∞) →[0, ∞) is convex, with h(δ) → ∞ as δ → 0, provided that δh (δ) ≤ K1 h(δ) + 1 for all δ > 0.

22

2.5

John M. Ball

Regularity and Self-Contact

An interesting approach to the problem of invertibility in mixed boundaryvalue problems (i.e., to the non-interpenetration of matter) is due to Ciarlet and Neˇcas [1985]. They proposed minimizing W (Dy) dx I(y) = Ω

subject to the boundary condition (2.1) and the global constraint det Dy(x) dx ≤ volume y(Ω) , Ω

and they gave hypotheses under which the minimum was attained, these hypotheses being weakened by Qi [1988]. They further showed that any minimizer is one-to-one almost everywhere, and that assuming sufficient regularity of the free boundary y(∂Ω2 ) the tangential components of the normal stress vector vanish there. Consequently they identified the above constrained boundary-value problem as corresponding to the case of smooth (i.e., frictionless) self-contact. A related but somwhat different formulation has recently been proposed by Pantz [2001a]; see also Giaquinta, Modica and Souˇcek [1994]. Problem 7. Justify the Ciarlet-Neˇcas minimization problem, or an appropriate modification of it, as a model of smooth self-contact. The problem here is to construct suitable variations in the neighbourhood of a region of self-contact of a minimizer to establish that in some sense the tangential stress components vanish there. This is non-trivial because in principle the two parts of the boundary in contact with one another could be wildly deformed and interlocked in a very complex configuration. If such a result could be obtained, a more ambitious target would be to establish the regularity properties of the free boundary in both the self-contacting and non self-contacting regions.

2.6

Uniqueness of Solutions

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


23

minimum energy configurations with a continuum of different sets of phase boundaries, the extent of non-uniqueness is of course much greater. For pure displacement boundary conditions, with a strictly polyconvex (or strictly quasiconvex) stored-energy function satisfying favourable growth conditions, the situation as regards uniqueness is less clear. John [1972b] proved uniqueness for smooth deformations with uniformly small strains (but possibly large rotations). In the same paper he gave a heuristic counter-example to uniqueness for the case of an annular two-dimensional body, and this has been made rigorous by Post and Sivaloganathan [1997] (see Section 2.7), who also proved nonuniquenesss for an analogous threedimensional problem with Ω a torus. But what if Ω is homeomorphic to a ball? In this case we have already seen that cavitation provides one counterexample to uniqueness, though the cavitating solution is discontinuous. Problem 8. Prove or disprove the uniqueness of sufficiently smooth equilibrium solutions to pure displacement boundary-value problems for homogeneous bodies when the stored-energy function W is strictly polyconvex and Ω is homeomorphic to a ball. The answer to this problem probably depends on both the geometry of Ω and the boundary conditions. For example, suppose that Ω is a ball, and that the boundary conditions correspond to severely squeezing the ball until it has a dumb-bell shape consisting of two roughly ball-shaped regions connected by a narrow passage. In this case one might expect, though it is not obvious how to prove it, that there might be equilibrium solutions in which material from one half of Ω is pulled through into the other half, but prevented from returning by the constriction. On the other hand, an elegant result of Knops and Stuart [1984] implies uniqueness for the case when the boundary displacements are linear and Ω is star-shaped (see also Taheri [2001b]).

2.7

Structure of the Solution Set

Problem 9. Devise general methods for proving the existence of local minimizers of I that are not global minimizers, and of other weak solutions of 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 variations in a suitable subset of A. For example, under the hypotheses of Theorem 2.2 suppose that we want to prove the existence of a local minimizer with respect to some metric d on A. Assume that d is such that if z (j) ∈ A with z (j) z in W 1,1 (Ω; R3 ) and sup I(z (j) ) < ∞ then d(z (j) , z) → 0. ¯ and boundary ∂U . Let U ⊂ A be open with respect to d, with closure U ¯ By the direct method, I attains a minimum yˆ on U . Suppose now that we

24

John M. Ball

can prove that inf I > inf I > inf I. ∂U

U

A

Then yˆ ∈ U and is a local, but not global, minimizer with respect to d. I believe that it should be possible to implement this method in some realistic examples, but have not seen it done. The only results on local minimizers in nonlinear elasticity using the direct method that I am aware of are due to Post and Sivaloganathan [1997], who prove the existence of local but not global minimizers for certain two-dimensional problems (see Section 2.6) for which the domain Ω has nontrivial topology by global minimization in a weakly closed homotopy class, and to Taheri [2001a], who generalizes the results in Post and Sivaloganathan [1997] to a wider class of domains. The second approach is to find by some method a sufficiently smooth solution yˆ to the equilibrium equations and attempt to show directly that it is a local minimizer. For local minimizers in W 1,∞ (Ω; R3 ) (weak local minimizers) this can be done in principle by checking positivity of the second variation. However for local minimizers in W 1,p (Ω; R3 ) with 1 ≤ p < ∞, or in Lq (Ω; R3 ), 1 ≤ q ≤ ∞, the task is made much more difficult by the absence of a known generalization to higher dimensions of the Weierstrass fundamental sufficiency theorem of the one-dimensional calculus of variations (for a discussion see Ball [1998]). Sometimes it is possible to circumvent the lack of such a theory. For example, in a dead-load traction 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 cerˆ with Dˆ tain y y = A = constant are local (but not global) minimizers in L1 (Ω; R3 ), by an argument exploiting the geometric incompatibility of A with deformation gradients having lower energy. How can one prove the existence of equilibrium solutions that are not local minimizers? In exceptional cases one may know an equilibrium solution explicitly (for example a trivial solution) and be able to show that it does not satisfy some necessary condition for a local minimizer. If we can also prove the existence of a global minimizer then we have at least two equilibrium solutions. This can be done, for example, for the case of some mixed boundary-value problems when the stored-energy function is polyconvex but not quasiconvex at the boundary (see Ball and Marsden [1984]). Another approach would be to try to use Morse theory or mountain-pass methods, but it is not clear how to do this so that, for example, appropriate conditions of Palais-Smale type can be verified; for results in an interesting model problem see Zhang [2001]. More generally, one can ask for a description of how the set of equilibrium solutions varies as a function of relevant parameters such as boundary displacements or loads. For the pure traction problem near a stress-free state an interesting study of this type is that of Chillingworth, Marsden and Wan [1982, 1983] and Wan and Marsden [1983].


25

Problem 10. Develop local and global bifurcation theories for nonlinear elastostatics that apply to mixed displacement-traction boundary conditions. As an illustration, the most well-known bifurcation problem in elasticity is that of buckling of a thin rod. Although this problem has been treated from the perspective of rod theory in hundreds of papers since the time of Euler [1744], there is no rigorous three-dimensional theory that justifies the usual picture of buckling, for example the existence of critical buckling loads or displacements, with corresponding branches of bifurcating buckled solutions. There are at least two difficulties in providing such a theory. The first is that unless the problem is formulated in a somewhat unrealistic way, there is no sufficiently explicit trivial compressed solution about which to linearize the equilibrium equations. For example, suppose that in a stressfree reference configuration a homogeneous isotropic elastic rod occupies the region Ω = (0, L) × D, where D ⊂ R2 is the cross-section. A natural boundary-value problem to consider, corresponding to clamped ends, consists of the equilibrium equations (2.4) and the boundary conditions (2.1), (2.6), with the choices ∂Ω1 = {0, L} × D, ∂Ω2 = (0, L) × ∂D, and ¯ (0, x ) = (0, x ) , y

¯ (L, x ) = (λL, x ) , y

x ∈ D ,

(2.31)

where λ > 0. For λ = 1 the homogeneous deformation y(x1 , x ) = (λx1 , x ) does not in general satisfy the zero traction condition (2.6). For example, for the compressive case λ < 1 the rod will typically want to bulge, leading to boundary layers near x1 = 0 and x1 = L. In order to have a homogeneously deformed trivial solution y(x) = (λx1 , µx2 , µx3 ) , one can replace (2.31) by the conditions y1 (0, x ) = 0 ,

y1 (L, x ) = λL ,

x ∈ D ,

corresponding to the less realistic case of frictionless end-faces constrained to lie in the planes {0} × R2 and {λL} × R2 . To prevent sliding of the end-faces one could add the further constraint that y2 dx = y3 dx = 0 at x1 = 0, L . D

D

In this case the natural boundary conditions at x1 = 0, L for the variational problem are that the stress vector t across the end-faces has constant transverse components t2 , t3 which are equal at x1 = 0, L. If we try to prescribe compressive loads at x1 = 0, L rather than displacements we encounter other difficulties (see Ball [1996a] for a discussion of one of these). The second more serious difficulty has already been mentioned, namely the lack of regularity of solutions to the linearized equilibrium equations as

26

John M. Ball

one approaches points of ∂Ω1 ∩∂Ω2 , or points of discontinuity of the applied traction in a pure traction formulation of the problem, which prohibits use of the implicit function theorem in natural spaces. Perhaps it might be possible to work in spaces with suitable weights in the neighbourhood of ∂Ω1 ∩ ∂Ω2 . But it seems odd that fine details of what goes on near ∂Ω1 ∩ ∂Ω2 should have a significant bearing on the buckling phenomenon, so perhaps there is a different approach to be discovered that circumvents this difficulty. Once a local bifurcation picture has been established, the next thing to understand is what happens to bifurcating solutions for large parameter values. For the case when ∂Ω1 ∩ ∂Ω2 is empty global results have recently been 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 variational models of fracture. Since typical fracture problems are described by deformations that have jump discontinuities across two-dimensional crack surfaces, fracture cannot in general be modelled in the context of Sobolev spaces. A generalization of the energy functional (2.2) to deformations allowing for fracture is W (Dy) dx + g y+ − y− , νy dH2 , (2.32) I(y) = Ω

Sy

where y belongs to the class SBV(Ω) of mappings of special bounded variation, i.e., those whose gradient is a bounded measure having no Cantor part. In (2.32) Sy denotes the set of jump points of y, νy the measure theoretic normal to Sy , and y± the traces of y on either side of Sy . The second integral represents the surface energy of cracks, as postulated in the Griffith theory of fracture (see, for example, Cherepanov [1998]), the simplest case g = constant corresponding to a contribution to the energy proportional to the total crack surface area H(Sy ). Despite much deep work on such models (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]), and their apparent potential for making an impact on understanding fracture, there have been only isolated attempts to discover their implications for practical problems of fracture mechanics (see, for example, Francfort and Marigo [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 elastostatics.


27

Two key issues are fracture initiation and stability, which are both related to the study of local minimizers for the functional (2.32). A technical obstacle in such a study is the lack of a general method of calculating a general variation of I about a given y in the direction of nearby deformations having possibly very different sets of jump points. An understanding of local minimizers would also clarify the status of the nonlinear elastostatics model based on (2.2) with respect to that based on (2.32), and thereby demystify the apparent sensitivity of the elastostatics model to growth behaviour for very large strains.

3 3.1

Dynamics Continuum Thermomechanics

We recall briefly the elements of continuum thermomechanics. The basic balance laws are the balance of linear momentum d ρ yt dx = tR dS + b dx , (3.1) dt E R ∂E E the balance of angular momentum d ρR x ∧ yt dx = x ∧ tR dS + x ∧ b dx , dt E ∂E E and the balance of energy 1 d 2 ρ |y | + U dx = b · y dx + tR · yt dS t t dt E 2 R E ∂E r dx − qR · n dS . + E

(3.2)

(3.3)

∂E

Here y = y(x, t) denotes the deformation, tR the Piola–Kirchhoff stress vector, ρR > 0 the (constant) density in the reference configuration, b the body force, U the internal energy, qR the heat flux vector and r the heat supply. The balance laws are assumed to hold for all Lipschitz domains E ⊂ Ω, and the unit outward normal to ∂E is denoted by n. In addition to the balance laws, thermomechanical processes are required to obey the Second Law of Thermodynamics, which we assume to hold in the form of the Clausius–Duhem inequality qR · n r d dS + dx (3.4) η dx ≥ − dt E θ θ ∂E E for all E, where η is the entropy and θ the temperature. Standard arguments now show that tR = TR n, where TR is the Piola–Kirchhoff stress tensor,

28

John M. Ball

and that for sufficiently smooth processes (3.1), (3.3), (3.4) reduce to the pointwise forms ρR ytt − div TR − b = 0 , d 1 |yt |2 + U − b · yt − div (yt TR ) + div qR − r = 0 dt 2 qR r ηt + div − ≥ 0, θ θ

(3.5) (3.6) (3.7)

and that (3.2) is equivalent to the symmetry of the Cauchy stress tensor T = (det Dy)−1 TR (Dy)T . Eliminating r from (3.6), (3.7), using (3.5) and denoting by ψ = U − θη

(3.8)

the Helmholtz free energy, we obtain that for sufficiently smooth processes −ψt − θt η + TR · Dyt −

qR · grad θ ≥ 0. θ

(3.9)

Adopting the prescription of Coleman and Noll [1963], we assume that given an arbitrary deformation y = y(x, t) and temperature field θ = θ(x, t) we can choose a body force b = b(x, t) and heat supply r = r(x, t) to balance (3.5), (3.6), so that (3.9) becomes an identity to be satisfied by the constitutive equations. For the case of a thermoelastic material, for which TR , η, ψ, qR are assumed to be functions of Dy, θ, grad θ, this leads to the relations (3.10) ψ = ψ(Dy, θ), TR = DA ψ, η = −Dθ ψ , and then (3.9) reduces to the inequality −

qR · grad θ ≥ 0. θ

(3.11)

(Note that, although this inequality must be satisfied by the constitutive equation 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 no longer valid.) For thermoelastic materials the balance of angular momentum is satisfied identically as a consequence of the requirement that TR is frame-indifferent, i.e., for all R ∈ SO(3) , TR RA, θ = RTR A, θ , which is equivalent to the condition that ψ(RA, θ) = ψ(A, θ) ,

for all R ∈ SO(3) .

(3.12)


29

The condition of material symmetry becomes ψ(AQ, θ) = ψ(A, θ) ,

for all Q ∈ S ,

(3.13)

where S is the isotropy group. The equations of isothermal thermoelasticity are obtained from (3.5), (3.10) by assuming that θ(x, t) = θ0 = constant. Thus the balance of linear momentum becomes ρR ytt − div DA W (Dy) − b = 0 ,

(3.14)

where W (A) = ψ(A, θ0 ). As regards the entropy inequality, we again adopt the Coleman and Noll point of view, choosing r to balance (3.6). (Here we follow Dafermos [2000], who gives a similar reduction for isentropic thermoelasticity.) Since, from (3.11), qR = 0 when grad θ = 0, (3.7) becomes 1

2 2 ρR |yt |

+ψ

t

− b · yt − div (yt TR ) ≤ 0 .

(3.15)

For the more general case of a thermoviscoelastic material (of strain-rate type), TR , η, ψ, qR are assumed to be functions of Dy, Dyt , θ, grad θ. By the same method we find that ψ = ψ(Dy, θ), and that S · Dyt − where

η = −Dθ ψ ,

qR · grad θ ≥ 0, θ

TR = DA ψ + S Dy, Dyt , θ, grad θ .

In the isothermal case we obtain the equation of motion ρR ytt − div DA W (Dy) − div S(Dy, Dyt ) − b = 0 ,

(3.16)

where W = ψ(Dy, θ0 ), S(Dy, Dyt ) = S(Dy, Dyt , θ0 , 0), together with the energy inequality (3.15). The frame-indifference of S takes the form S(Dy, Dyt ) = Dy Σ(U, Ut ) ,

(3.17) 1

for some matrix-valued function Σ, where U = (DyT Dy) 2 .

3.2

Existence of Solutions

Problem 12. Prove the global existence and uniqueness of solutions to initial boundary-value problems for properly formulated dynamic theories of 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 supplemented by boundary conditions such as (2.1), (2.6) and by the initial conditions (3.18) y(x, 0) = y0 (x), yt (x, 0) = y1 (x). If the body force is conservative, so that b = −grad y h(x, y),

(3.19)

then (3.14) formally comprises a Hamiltonian system, and could be alternatively obtained by applying Hamilton’s principle to the functional T 1 2 2 ρR |yt | − W (Dy) − h(x, y) dx dt . 0

Ω

In particular, solutions formally satisfy the balance of energy E(y, yt ) = E(y0 , y1 ) ,

where

1

E(y, v) = Ω

2 2 ρR |v|

t ≥ 0,

(3.20)

+ W (Dy) + h(x, y) dx .

However, weak solutions of the quasilinear wave equation (3.14) do not in general satisfy (3.20), since singularities such as shock waves can dissipate energy. Correspondingly, although equality holds in the dissipation inequality (3.15) for smooth solutions, in general it does not do so for weak solutions. Interpreted in the sense of distributions or measures, (3.15) acts as an admissibility criterion for weak solutions. In one dimension (3.14) takes the form ρR ytt − σ(yx )x − b = 0 ,

(3.21)

where σ(yx ) = W (yx ), which setting u1 = ρR yt , u2 = yx is equivalent to the system of two conservation laws ut − f (u)x = g ,

where f (u) =

σ(u2 ) ρ−1 R u1

,

g=

(3.22) b . 0

This system is strictly hyperbolic if σ = W > 0, so that W is strictly convex. Two approaches have been employed to study (3.22), the Glimm scheme, Glimm [1965], and variants of it such as front-tracking (introduced by Dafermos [1972]), and the method of compensated compactness as pioneered by Tartar [1979, 1982] and DiPerna [1983, 1985].


31

The Glimm scheme and variants apply to strictly hyperbolic systems of the form (3.22) with u ∈ Rn , f : Rn → Rn , g ∈ Rn . They involve a semi-explicit construction of the solutions in terms of approximation of the initial data by piecewise constant functions, together with an analysis of wave interactions. They are restricted to initial data having small total variation, and thus, via total variation estimates on the solution, to solutions of small total variation. Glimm’s original work assumed that the system was ‘genuinely nonlinear’, but this restriction was removed by Liu [1981]. Thanks to work of Bressan [1988, 1995], Bressan and Colombo [1995], Bressan [Crasta and Piccoli], Bressan and Goatin [1999], Bressan and Le Floch [1997], Bressan and Lewicka [2000], Bressan, Liu and Yang [1999] and Liu and Yang [1999b,a,c], the solutions obtained in these ways are now known to be unique in appropriate function classes. For genuinely nonlinear systems of two conservation laws, such as (3.21) with W > 0, W = 0, more is known (see [Dafermos, 2000, Chapter XI]). Most of this work is for solutions on the whole real line; for a treatment of (3.21) with displacement boundary conditions see Liu [1977]. The method of compensated compactness, on the other hand, has up to now been restricted to systems of at most two conservation laws, such as (3.21). Starting from a sequence of approximate solutions obtained from the method of vanishing viscosity (or by a variational time-discretization scheme, see Demoulini, Stuart and Tzavaras [2000]), it uses information coming from the existence of a suitable family of ‘entropies’ (quantities for which there is a corresponding conservation law satisfied by smooth solutions) to pass to the limit using weak convergence. However, there is no corresponding uniqueness theorem. These results are described in the books of Bressan [2000], Dafermos [2000] and Serre [2000]. In a recent development, Bianchini and Bressan [2001] have made a breakthrough by obtaining for the first time total variation estimates directly from the vanishing viscosity method. For the three-dimensional equations (3.14) very little is known. Hughes, Kato and Marsden [1977] proved that if W satisfies the strong ellipticity condition D2 W (A)(a ⊗ n, a ⊗ n) ≥ µ|a|2 |n|2

(3.23)

3×3 for all A ∈ M+ , a, n ∈ R3 , where µ > 0, then for smooth initial data (3.18) defined on the whole of R3 with det Dy0 > 0, there exists a unique smooth solution on a small time interval [0, T ), T > 0. This result was extended to pure displacement boundary conditions by Kato [1985]. For related results see Dafermos and Hrusa [1985] and [Dafermos, 2000, Chapter V]. There seem to be no short-time existence results known for mixed displacement-traction boundary conditions. Interesting results concerning large time existence for sufficiently smooth and small initial data on the whole of R3 have been obtained by John [1988]. For corresponding results for 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 = ρR yt , (3.14) becomes the system Dij = ρ−1 R vi,j ,

(3.24)

pt = div DA W (Dy) + b ,

(3.25)

At = D ,

which is hyperbolic if D2 W (A)(a ⊗ n, a ⊗ n) > 0

3×3 for all A ∈ M+ 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 existence 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 = 0 which is satisfied by all weak solutions. Exploiting this in the context of a general system having involutions, Dafermos [1996, 2000] proves a theorem implying that if W is quasiconvex and satisfies (3.23) then any Lipschitz solution A, p of (3.24), (3.25) on R3 × [0, T ], T > 0, is unique within the class of weak solutions admissible with respect to the entropy W , of uniformly small local oscillation, and satisfying the same initial data as A, p. An unpublished idea of LeFloch, found independently by Qin [1998], leads to the observation that for polyconvex W the hypothesis of uniformly small oscillation can be removed. These results are interesting because they so far represent the only use of quasiconvexity and polyconvexity in the ˇ ak [1995] for an idea of how quasiconvexity context of dynamics. See Sver´ (in an augmented space) might be used to prove existence by passage to the limit using weak convergence, in the spirit of compensated compactness. For the full system of three-dimensional nonlinear thermoelasticity (3.5)–(3.7), which has the additional conservation law (3.6), the state of knowledge (or rather lack of it) is similar. For these systems an additional difficulty is that of ensuring invertibility of solutions, and in particular the condition det Dy(x, t) > 0. For a thermoviscoelastic material, one can hope that a sufficiently wellbehaved viscous part S of the stress will guarantee existence without any convexity conditions on W . Indeed, in the one-dimensional isothermal case, for which the equation of motion takes the form ρR ytt − σ(yx , yxt )x − b = 0 ,

(3.26)

for which Dafermos [1969] and Antman and Seidman [1996] have proved existence and uniqueness for a general class of σ. The special case of the equation (3.27) ρR ytt − σ(yx )x − yxxt = 0 .


33

has been treated in numerous papers (see, for example, Greenberg, MacCamy and Mizel [1967], Andrews [1980], Pego [1987]). For corresponding results for thermoviscoelasticity see Racke and Zheng [1997]. For the isothermal case in three dimensions it is natural to consider in place of (3.27) the equation ρR ytt − div DA W (Dy) − ∆yt = 0 , for which a theory of existence is available (see Rybka [1992], Friesecke and Dolzmann [1997]). However the corresponding viscous stress S = Dyt is not of the form (3.17), and so is not frame-indifferent. The only existence theory for weak or strong solutions of (3.16) with S frame-indifferent appears to be that of Potier-Ferry [1981, 1982], who, for pure displacement boundary conditions, established global existence and uniqueness of solutions for initial data close to a smooth equilibrium having strictly positive second variation. Potier-Ferry assumed that the linearized elasticity operator at the equilibrium is strongly elliptic, and that a corresponding positivity condition holds for the linearized viscous stress. He uses methods of Sobolevskii [1966] (for an alternative treatment also based on Sobolevskii’s work see Xu and Marsden [1996] and Xu [2000]). A recent monograph covering various aspects of the analysis of thermoelasticity is that of Jiang and Racke [2000]. A different approach to the existence of solutions in elasticity is to weaken the concept of solution to that of a measure-valued solution, in which the unknown is a Young measure νx,t in appropriate variables satisfying an integral identity obtained by formally passing to the weak limit in a sequence of approximate solutions. Using a variational time-discretization method, the global existence of such solutions has been proved by Demoulini [2000] for the viscoelastic equation (3.16) with S frame-indifferent, and by Demoulini, Stuart and Tzavaras [2001] for the equations (3.14) of elastodynamics with W polyconvex (exploiting the idea of Le Floch and Qin [1998]). However they are unable to handle the constraint det Dy > 0. Of course the significance of such results would be greatly enhanced if there were examples known of cases in which there was no corresponding weak solution.

3.3

The Relation Between Statics and Dynamics

For suitable boundary conditions, the Second Law of Thermodynamics endows the equations of motion of continuum thermodynamics with a Lyapunov function, that is a function of the state variables that is nonincreasing along solutions. For example, suppose that the mechanical boundary conditions are that y = y(x, t) satisfies ¯( · ) y( · , t)∂Ω = y 1

34

John M. Ball

and the condition that the applied traction on ∂Ω2 is zero, and that the thermal boundary condition is θ( · , t)∂Ω = θ0 , where θ0 > 0 is a constant. Assume that the heat supply r is zero, and that the body force is given by (3.19). Then from (3.1), (3.3) and (3.4) with E = Ω we find that the ballistic free energy 1 2 (3.28) E= 2 ρR |yt | + U − θ0 η + h dx Ω

is nonincreasing along solutions. (This is a result of Duhem [1911] for the case of thermoelasticity. See Coleman and Dill [1973], Ericksen [1966] and Ball [1986, 1992] for further discussion and references.) For a thermoviscoelastic material, if v( · , t) → 0, y( · , t) → y( · ) and θ( · , t) → θ0 as t → ∞ the integrand in (3.28) formally tends to W (Dy) + h(x, y), where W (Dy) = ψ(Dy, θ0 ) and ψ is the Helmholtz free energy ψ(Dy, θ0 ) = U (Dy, θ0 ) − θ0 η(Dy, θ0 ) . This motivates minimization of W (Dy) + h(x, y) dx . I(y) = Ω

(For pure zero traction boundary conditions, when uniformly rotating equilibria are to be expected, we do not expect that yt → 0; the corresponding entropy maximization problem is studied by Lin [1990]). As applied to such problems, the calculus of variations can be viewed as representing a crude version of dynamics in which true dynamic orbits are replaced by all paths in a phase space of mappings {y, yt , θ} along which E is nonincreasing. Problem 13. elasticity.

Develop a qualitative dynamics for dynamic theories of

Of course a prerequisite for such a qualitative dynamics is a global existence theory for solutions. Given such a theory, the points at issue are the usual ones for dissipative dynamical systems, namely whether solutions converge to equilibrium states as t → ∞, the structure of regions of attraction, the existence of stable and unstable manifolds of equilibria, the existence of a global attractor, and so on. In particular one can ask whether dynamic orbits generically realize suitably defined local minimizing sequences for the ballistic free energy. This is especially interesting in the case when the ballistic free energy does not attain a minimum (as in models of elastic crystals — see Section 4.2). In fact for the one-dimensional viscoelastic model of this type studied by Ball, Holmes, James, Pego, and Swart [1991] and Friesecke and McLeod [1996], it is known that no dynamic solutions realize global minimizing sequences; it is unclear whether or not this is a one-dimensional phenomenon.


Problem 14. of equilibria.

35

Develop criteria for the dynamic stability and instability

Koiter [1976] is among those who have drawn attention to the problem of justifying the energy criterion for stability, that an equilibrium solution is stable if it is a local minimizer of the corresponding elastic energy (for example, of the ballistic free energy for a thermoviscoelastic material). To keep the discussion simple, consider the case of isothermal motion of a thermoviscoelastic material, for which the equation of motion is given by (3.16), and assume that the body force is zero. The corresponding Lyapunov function is 1 2 E(y, v) = 2 ρR |v| + W (Dy) dx , Ω

for which y = y∗ , v = 0 is a local minimizer provided y∗ is a local minimizer of W (Dy) dx . E(y) = Ω

Since there are different types of local minimizer corresponding to different metrics d on different spaces X of deformations y, in particular W 1,∞ local minimizers and W 1,p local minimizers for 1 ≤ p < ∞, it is not clear which kinds of local minimizers are needed to ensure dynamical stability. As emphasised, for example, by Knops and Wilkes [1973], the standard argument for establishing Lyapunov stability with respect to a metric d requires more than just that y∗ is a strict local minimizer with respect to d (that is I(y) > I(y∗ ) whenever y = y∗ and d(y, y∗ ) is sufficiently small). What is needed, in addition to the continuity of I with respect to d and the continuity in time of dynamic orbits with respect to d, is that y∗ lies in a potential well, namely that for some ε > 0 inf

d(y,y∗ )=ε

I(y) > I(y∗ ) .

For a way of verifying that a strict local minimizer in a space based on W 1,p for 1 < p < ∞ satisfies this requirement when W is strictly polyconvex see Ball and Marsden [1984], and for the case when W is strictly quasiconvex see Evans and Gariepy [1987] and Sychev [1999]. However, we do not know in general how to prove that a given y∗ is a strict W 1,p local minimizer. Further, for W that are not quasiconvex almost nothing is known. These questions are related to the open problem, already mentioned in Section 2.7, of generalizing to higher dimensions the Weierstrass fundamental sufficiency theorem. The only rigorous result justifying the energy criterion in any generality (in particular, in three dimensions) seems to be that of Potier-Ferry [1982], who for pure displacement boundary conditions establishes asymptotic stability, with respect to the norm of W 2,p × W 2,p , p > 3, of smooth equilibria having strictly positive second variation, under the hypotheses described in Section 3.2.

36

John M. Ball

Finally, little is known about criteria justifying instability of an equilibrium solution. An instructive example is that of Friesecke and McLeod [1997], who for a problem of one-dimensional viscoelasticity exhibit an equilibrium solution u ¯ which, with respect to a topology in which the dynamics is well-posed, (a) is dynamically stable, but (b) is such that there is a continuous path in the phase space (not a dynamical solution) leaving u ¯ along which the energy strictly decreases, so that in particular u ¯ is not a local minimizer.

4 4.1

Multiscale Problems From Atomic to Continuum

Problem 15. Establish the status of elasticity theory with respect to atomistic models. Is it possible to derive elasticity theory from atomistic models? Such models range from full quantum many-body theory to approximations such as density-functional theory, Thomas–Fermi theory, and models in which electronic effects are not explicitly considered but incorporated into interatomic potentials. There is an extensive physics and materials literature on such models and on methods for bridging the atomistic and continuum lengthscales (see Phillips [2001] for an introduction). Here I will concentrate on what little is known rigorously for the case of elastic crystals. However, another important class of materials is that of cross-linked polymers, which involves some different issues that are considered from the point of view of statistical mechanics in Deam and Edwards [1976], Edwards and Vilgis [1988]. For crystals, the first question to answer is why they occur in the first place, that is why at low temperature the minimum energy configuration of a very large number of atoms is spatially periodic. This is the famous unsolved ‘crystal problem’, nicely reviewed by Radin [1987]. Likewise, there is no fundamental understanding of the statistical mechanics of crystals, which would explain their stability and instabilities. Given this impasse, in attempts to pass from atomistic to continuum models of crystals some initial atomic order is always assumed. In the context of free-energy minimization, one can draw a distinction between two kinds of approach. In the first, an appropriate limit of a discrete energy functional is sought along sequences of explicit atomic configurations. For example the atoms may be assumed to occupy a periodic lattice in a reference configuration, and to be displaced according to a given sufficiently smooth continuum deformation y (the Cauchy–Born hypothesis), the number of atoms being sent to infinity with a suitable scaling for the energy. In this approach there is no attempt to explain why the atoms adopt the assumed configurations. A


37

recent example is the work of Blanc, Le Bris and Lions [2001], who obtain in a suitable scaling a limiting energy of the form I(y) = W (Dy) dx Ω

in the cases of (a) a two-body interaction, (b) Thomas–Fermi theory. As is well-known the case of two-body interactions leads to a W satisfying the Cauchy relations, namely that the linearized elasticity coefficients at a natural state (say Dy = 1) cijkl =

∂ 2 W (1) ∂Aij ∂Akl

possess the symmetries cijkl = cjikl = cklij = cikjl . These symmetries are known not to hold in general (see Love [1927], Weiner [1983]). Blanc, LeBris, and Lions also obtain second-order terms in the expansion of the energy with respect to the scaling parameter, identifying these with surface energies. For fundamental results on linear deformations with the Cauchy–Born hypothesis see Lieb and Simon [1977] and Fefferman [1985]; in these papers the dependence on the deformation gradient enters implicitly through the given Bravais lattice. For more recent extensions of the results of Lieb and Simon [1977] see Catto, Le Bris and Lions [1998]. The second approach is to consider the true minimizers of the discrete problems, and to try to understand what functional their limit minimizes. One example of this approach is the interesting study by Friesecke and Theil [2001] of a model two-dimensional problem of a lattice of particles linked by harmonic springs between their nearest and next nearest neighbours. They determine open regions of atomic parameters in which the Cauchy–Born hypothesis holds in the appropriate limit, and open regions in which it does not. Another interesting recent study in this spirit is that of Penrose [2001], who considers a two-dimensional model problem of a lattice of rotatable disks with one-body wall interactions and angle-dependent two-body interactions. By suitably restricting the statistical ensemble, so that, for example, certain angles between atoms in the deformed configuration are constrained to lie in certain intervals (these constraints being designed to deter dislocations), he proves the existence of an elastic free energy W corresponding to taking the thermodynamic limit with prescribed linear boundary data. He also deduces a convexity property of W weaker than rank-one convexity, and suggests that W might in fact be quasiconvex. Other work in this spirit is that of Braides, Dal Maso and Garroni [1999], Braides and Gelli [2001a,b] and Foccardi and Gelli [2001], who calculate the Γ-limits of certain discrete functionals with nearest-neighbour

38

John M. Ball

or pairwise interactions, obtaining a limiting functional allowing fracture of the general form (2.32) together with a corresponding function space on which to minimize it (namely SBV(Ω) or GSBV(Ω)). This acts as a reminder that, since the predictions of minimization problems can depend on the function space, as we have seen in Section 2.3 in connection with the Lavrentiev phenomenon, a proper atomistic to continuum derivation should deliver not only the limiting governing equations or energy, but the appropriate function space as well. For a proposed scheme for the passage from atomistic to continuum models for thin films, rods or tubes see Friesecke and James [2000]. There seems to be no work on atomistic derivations of dynamic theories of elasticity, or even of elastostatics in the context of deformations that are not global energy minimizers. In some situations it is desirable to simultaneously use a continuum and a discrete model. For example, one may wish to study the interaction of a defect or other localized region (such as the vicinity of a crack tip), in which atomistic effects may be important, with the surrounding bulk material, where a continuum theory is appropriate. One way of handling the resulting matching problem is the quasicontinuum method (see Tadmor, Ortiz and Phillips [1996]). A rigorous understanding of such methods is lacking.

4.2

From Microscales to Macroscales

Materials undergoing phase transformations involving a change of shape at a critical temperature typically develop characteristic patterns of microstructure, in which the deformation gradient has large variations on a fine length-scale that varies from material to material but can be as small as a few atomic spacings. Such microstructures often contain twinned regions consisting of many parallel layers separated by sharp interfaces, the deformation gradient alternating between two distinct values in adjacent layers. Why does fine microstructure form? To what extent can we predict its morphology? What are the properties of the material at a macroscale much larger than the microscale of the microstructure? Whereas it would be desirable to answer such questions in the context of a suitably formulated dynamical theory, it is neither clear what such a theory should be (especially as regards the kinetics of interfaces), nor do we currently have the techniques to give answers corresponding to any such theory. Hence we will discuss some more specific open problems that arise in static models of such phase transformations. Consider a single crystal of thermoelastic material that undergoes a diffusionless phase transformation involving a change of shape at the temperature θc . For definiteness, suppose that there is an interval E of temperatures, containing θc , such that for θ ∈ E, θ > θc , the minimum energy


39

configuration (called austenite) of the crystal is cubic. Taking the reference configuration to be undistorted austenite at θ = θc , we suppose that for θ ∈ E, θ ≤ θc , a minimum energy configuration (called martensite) is given by the transformation strain U (θ). We suppose that the Helmholtz free3×3 energy function ψ(A, θ) attains a finite minimum with respect to A ∈ M+ for each θ ∈ E, so that by adding to it a suitable function of θ we may assume for the purposes of free-energy minimization that min ψ(A, θ) = 0 .

3×3 A∈M+

Let

3×3 K(θ) = A ∈ M+ : ψ(A, θ) = 0

be the set of energy-minimizing deformation gradients. Note that by (3.12) and (3.13) SO(3)K(θ)S = K(θ) , where we take S to be the subgroup P 24 of SO(3) consisting of the 24 rotations mapping a cube into itself. It is thus reasonable to suppose that for θ ∈ E,   , θ > θc , α(θ)SO(3) N (4.1) K(θ) = SO(3) ∪ i=1 SO(3)Ui (θc ) , θ = θc ,   N SO(3)U (θ) , θ < θ , i c i=1 where α( · ) describes the thermal expansion, with α(θc ) = 1, and where the Ui (θ), 1 ≤ i ≤ N , are the distinct positive definite symmetric matrices of the form QT U (θ)Q, Q ∈ P 24 . We assume that N is independent of θ ∈ E. If Ui (θc ) = 1 the transformation is first-order, while if Ui (θc ) = 1 it is secondorder. We say that each Ui describes a different variant of martensite. For example, in the case of a cubic-to-tetragonal transformation we may take U (θ) = U1 (θ) = diag (η3 , η1 , η1 ) , where η1 (θ) > 0, η3 (θ) > 0, and then we find that N = 3, and that U2 (θ) = diag (η1 , η3 , η1 ) ,

U3 (θ) = diag (η1 , η1 , η3 ) .

For other transformations we get different numbers of variants; for example, for cubic-to-orthorhombic transformations N = 6, and for cubic to monoclinic transformations N = 12. Note that in adopting (4.1) we exclude large shears leaving the crystal lattice invariant (see Ericksen [1977b]), the inclusion of which would lead to K(θ) consisting of an infinite number of energy wells for each θ, and to an energy-minimization problem of a different character to that based on (4.1) (see Fonseca [1988]).

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John M. Ball

The total free energy corresponding to the deformation y : Ω → R3 is given by ψ(Dy, θ) dx . Iθ (y) = Ω

Zero-energy microstructures (at the temperature θ) correspond to sequences of deformations y(j) such that (4.2) lim Iθ y(j) = 0 . j→∞

If we assume a mild growth condition on ψ, such as ψ(A, θ) ≥ c0 |A|p − c1

3×3 for all A ∈ M+ ,

where c0 > 0, c1 and p > 1 are constants, then (4.2) is equivalent to the statement that Dy(j) → K(θ) in measure, or that the Young measure (νx )x∈Ω corresponding to (a subsequence of) Dy(j) satisfies supp νx ⊂ K(θ)

a.e. x ∈ Ω .

The set of macroscopic deformation gradients corresponding to zero-energy 3×3 such that Dy(j) microstructures is the set of gradients Dy : Ω → M+ p (j) Dy in L for some sequence y satisfying (4.2). Equivalently, following the results of Kinderlehrer and Pedregal [1991, 1994], this is the set of gradients Dy such that Dy(x) ∈ K(θ)qc a.e. x ∈ Ω , where for a compact set K ⊂ M 3×3 , K qc denotes the quasiconvexification of K. Equivalent definitions of K qc are K qc = {¯ ν : ν is a homogeneous W 1,∞ Young measure with supp ν ⊂ K} = {A ∈ M 3×3 : ϕ(A) ≤ max ϕ(B) for all quasiconvex ϕ} B∈K = {E ⊃ K : E quasiconvex}. Here AA = y ∈ W 1,1 (Ω; R3 ) : y|∂Ω = Ax , and a set E is quasiconvex if it is the zero set ϕ−1 (0) of a nonnegative quasiconvex function ϕ. We also have that 3×3 : inf Iθ (y) = 0 . K qc = A ∈ M+ AA

We can similarly define the polyconvexification K pc and the rank-one 3×3 by convexification K rc of a compact set K ⊂ M+ K pc = A ∈ M 3×3 : ϕ(A) ≤ max ϕ(B) for all polyconvex ϕ , 3×3 B∈M K rc = A ∈ M 3×3 : ϕ(A) ≤ max ϕ(B) for all rank-one convex ϕ . 3×3 B∈M

Clearly K rc ⊂ K qc ⊂ K pc .

(4.3)


Problem 16.

41

Determine K(θ)qc for θ ≤ θc .

For θ > θc we have that K(θ)qc = K(θ) (cf. Ball and James [1992]). For θ ≤ θc the problem is open. In particular, K(θ)qc is not known for the cubic-to-tetragonal case either when θ < θc or θ = θc . Ball and James [1992] computed K qc for the case of two wells K = SO(3)U ∪ SO(3)V , with U = UT > 0, V = VT > 0, det U = det V and with SO(3)U rankone connected to SO(3)V, which occurs for orthorhombic-to-monoclinic transformations. In this case by linear changes of variables we can assume that (4.4) U = diag (η1 , η2 , η3 ) , V = diag (η2 , η1 , η3 ) , where η1 > 0, η2 > 0, η3 > 0 and η1 = η2 . (This includes the case U = U1 (θ), V = U2 (θ) of two tetragonal wells.) The answer in this case is (see 3×3 such that Ball and James [2003]) that K qc consists of those A ∈ M+   a c 0 AT A =  c b 0  , 0 0 η32 where ab − c2 = η12 η22 , a + b + 2|c| ≤ η12 + η22 . The proof is by calculating K pc , showing that K pc ⊂ K rc , and using (4.3). Friesecke [2000] has announced that he can calculate K(θ)pc , θ < θc , in the cubic-to-tetragonal case. However, whether in general K(θ)pc = K(θ)qc is unknown. (Despite this, in their study of nonclassical austenite-martensite interfaces Ball and Carstensen [1999] were able to show that for θ < θc the identity matrix is rank-one connected to K(θ)qc if and only if it is rank-one connected to K(θ)pc .) Problem 17. For free-energy functions ψ(A, θ) of elastic crystals, determine for which ∂Ω1 ⊂ ∂Ω and g : ∂Ω1 → R3 the minimum of Iθ (y) = ψ(Dy, θ) dx Ω

in A = y ∈ W 1,1 (Ω; R3 ) : y∂Ω = g is attained, and for which it is not.

1

A solution to this problem (also to the corresponding problem including applied loads on ∂Ω2 ) would help clarify the validity of the hypothesis of Ball and James [1987] that the formation of microstructure is associated with non-attainment of minimum energy. For example, is non-attainment generic or exceptional? It is probably overly optimistic to expect a general answer to Problem 17. A simpler special case for which the answer is in general unknown is when

42

John M. Ball

∂Ω1 = ∂Ω, g(x) = Ax, and A ∈ K(θ)qc \K(θ). In this case the problem is equivalent to asking whether for such A there exists a deformation y satisfying y|∂Ω = Ax and Dy(x) ∈ K(θ)

almost everywhere .

For the corresponding two-well problem in which K(θ) is replaced by K = SO(3)U ∪ SO(3)V, with U, V given by (4.4), there is no y with y|∂Ω = Ax and Dy(x) ∈ K almost everywhere. This non-attainment result was proved by Ball and Carstensen [1999] using the result of Ball and James [1991] that any y with Dy(x) ∈ K qc a.e. is a plane strain, the point being that a plane strain y cannot coincide with a linear mapping Ax on the boundary of a three-dimensional region Ω unless Dy(x) = A a.e. . In the corresponding two-dimensional problem the answer is different. In fact, if Ω ⊂ R2 and K = SO(2)U ∪ SO(2)V, where U = diag (η1 , η2 ) ,

V = diag (η2 , η1 ) ,

ˇ ak [1996] modified the theory uller and Sver´ and η1 > 0, η2 > 0 then M¨ of convex integration due to Gromov [1986] to show that there exists y with y|∂Ω = Ax, Dy(x) ∈ K a.e. for any A ∈ K qc . (For variations on the method see Dacorogna and Marcellini [1999], M¨ uller and Sychev [2001], Sychev [2001] and Kirchheim [2001].) Whether these exotic minimizers exist in three dimensions, and if so whether they are physically relevant, is unclear. If, as seems likely, they do exist, then it is natural to ask whether they are admissible, in the sense that they can be obtained as limits of minimizers for a corresponding functional incorporating interfacial energy, for example 2 ε ψ Dy(x), θ + ε2 D2 y(x) dx Iθ (y) = Ω

in the limit ε → 0. For general information on the models and techniques described in this section see Ball and James [2003], Bhattacharya [2001], Hane [1997], M¨ uller [1999], Luskin [1996] and Pedregal [1991, 2000].

4.3

From Three-Dimensional Elasticity to Theories of Rods and Shells

A rod is a three-dimensional body whose form is close to that of a curve in R3 . We can describe a reasonably wide class of such bodies as those which occupy in a reference configuration the bounded domain Ωh = r(s) + Q(s)(0, x ) : s ∈ (0, L), x ∈ hD , where r : (0, L) → R3 is a smooth embedded curve parametrized by arclength, so that |r (s)| = 1, where the cross-section D ⊂ R2 is a bounded


43

domain with 0 ∈ D, and where Q : (0, L) → SO(3) is a smooth mapping with Q(s)e1 = r (s) for each s ∈ (0, L), which describes how the crosssection is rotated. The parameter h > 0 measures the thickness of the rod. The simple case of an initially straight rod of circular cross-section corresponds to the choice r(s) = se1 , D = B(0, 1) ⊂ R2 , Q(s) = 1, so that Ωh = (0, L)e1 × hD. A shell is a three-dimensional body whose form is close to that of a twodimensional surface. A class of such bodies consists of those occupying in a reference configuration the bounded domain Ωh = r(s1 , s2 ) + τ n(s1 , s2 ) : (s1 , s2 ) ∈ S ,

|τ | ≤ h ,

where S ⊂ R2 is a bounded domain, and r : S → R3 is a smooth embedded oriented surface with unit normal n(s1 , s2 ). Here h > 0 is the thickness of the shell. A plate is a flat shell, corresponding to Ωh = S × (−h, h)

(4.5)

When h is small, such thin rods and shells are traditionally described respectively by one-dimensional rod and two-dimensional shell models, in which the independent variables are respectively (s, t) and (s1 , s2 , t), where t is time. There is an immense literature on the many such theories, well surveyed in the books of Antman [1995], and Ciarlet [1997, 2000]. However there are only the beginnings of a rigorous theory justifying such models with respect to three-dimensional elasticity. Problem 18. Give a rigorous derivation of models of rods, plates and shells, showing that their solutions well approximate appropriate solutions to three-dimensional elasticity for small values of the thickness parameter h. There seem to be no results of this type for dynamical theories of elasticity, so we concentrate on what is known for elastostatics. Here one would ideally like results showing that the solution sets for boundary-value value problems of three-dimensional elasticity converge as h → 0 to corresponding sets for an appropriate rod or shell theory, together with appropriate error estimates. In passing to the limit h → 0 other parameters such as loads may need to be scaled with h. Taking into account such scaling, one would like the convergence and error estimates to be uniform with respect to parameters such as loads entering the boundary conditions, so that in particular the description of buckling according to the three-dimensional theory could be correlated with that for the rod or shell theory identified. One of the many difficulties to be overcome in order to achieve such results is to understand how boundary-layers behave in the limit h → 0. Such boundary layers will occur, for example, at the ends of a rod, where, according to Saint-Venant’s principle one expects the limiting rod theory to see

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only resultant loads and moments applied to the ends. Higher-order corrections in h can be expected to yield more sophisticated theories, for example involving numbers of directors (vectors depending on the independent variables giving a better description of the three-dimensional deformation). An isolated theory that addresses some of these difficulties is Mielke’s treatment Mielke [1988], Mielke [1990] of Saint-Venant’s problem for an initially straight rod of uniform cross-section and prescribed resultant loads and moments at its two ends, in which via a six-dimensional centre manifold he identifies a Cosserat theory of rods whose solutions attract for long rods those of three-dimensional elasticity having uniformly small strains and the same resultant loads and moments. In this connection, for threedimensional nonlinear elasticity Ericksen [1977b,a, 1983] has derived equations describing beautiful semi-inverse solutions for helical deformations of a rod. For developments see Muncaster [1979, 1983], and for an existence theory for the corresponding problem defined on cross-sections see Ball [1977]. For plates with a St. Venant-Kirchhoff stored-energy function Monneau [2001] has devised a scheme which shows that for periodic boundary conditions and sufficiently small external forces, there is a solution of the threedimensional equilibrium equations which converges as h → 0 to the solution of the corresponding Kirchhoff-Love plate theory, together with error estimates. However, the principal method that has so far produced rigorous results of the desired type is Γ-convergence (see De Giorgi and Franzoni [1979], Dal Maso [1993]). The first application of Γ-convergence to nonlinear elasticity was that of Acerbi, Buttazzo and Percivale [1991], who used it to derive a one-dimensional model for an elastic string. Then Le Dret and Raoult [1995a,b, 1996, 1998] and Ben Belgacem [1997] used it to derive a corresponding two-dimensional membrane theory (see also Braides, Fonseca and Francfort [2000]). Le Dret and Raoult [2000] have also investigated which Cosserat theories of plates Γ-converge to the membrane theory limit as the thickness goes to zero. Bhattacharya and James [1999] used Γ-convergence to derive equations for thin films of martensitic material, an interesting conclusion being that in the two-dimensional theory there can be exact austenite–martensite interfaces (for developments see Shu [2000]). In interesting recent work Friesecke, James, and M¨ uller [2001] have derived a theory of nonlinear bending of plates starting from the nonlinear elastic energy W Dy(x) dx , I h (y) = Ωh

where Ωh is given by (4.5). This is a more delicate problem than for the membrane theory since for the boundary conditions for which the bending theory is expected to be valid, I h (yh ) is expected to be of order h3 for minimizers yh , whereas for boundary conditions leading to the membrane


45

theory we expect that I h (yh ) is of order h. Hence the limit h → 0 corresponds to a singular perturbation. The corresponding bending theory has energy functional 1 24

Q2 (II) ds1 ds2 , S

where II denotes the second fundamental form IIij = n,i · y,j ,

n = y,1 ∧ y,2 ,

and where Q2 (A) = min3 Q3 (A + a ⊗ e3 + e3 ⊗ a) , a∈R Q3 (A) = D2 W 1 (A, A) . The proof is via a refinement of a rigidity result for SO(3) of John [1961, 1972a] (see also Kohn [1982]). John [1965, 1971] also rigorously obtains equations for shells of isotropic material, assuming that the radius of curvature of the shell is large and the maximum strain is uniformly small, providing interior estimates for the validity of the approximation. For other related work see Pantz [2000, 2001b]. For plates satisfying general boundary conditions one expects some theory incorporating both the membrane and bending cases, but the form this should take is unclear. For work in this direction see Ciarlet [2000] and Ciarlet and Roquefort [2000].

Acknowledgments: I am indebted to S. S. Antman, J. J. Bevan, K. Bhattacharya, C. M. Dafermos, G. Friesecke, R. D. James, M. Jungen, J. Kristensen, V. J. Mizel, O. Penrose, A. Raoult, J. Sivaloganathan, and A. Taheri for valuable suggestions and comments. The article was completed while I was visiting the Tata Institute for Fundamental Research in Bangalore, to whose members and staff I am grateful for their support and warm hospitality.

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2 Finite Elastoplasticity Lie Groups and Geodesics on SL(d) Alexander Mielke To Jerry Marsden on the occasion of his 60th birthday ABSTRACT The notions of nonlinear plasticity with finite deformations is interpreted in the sense of Lie groups. In particular, the plastic tensor P = F p−1 is considered as element of the Lie group SL(d). Moreover, the plastic dissipation defines a left–invariant Finsler metric on the tangent bundle of this Lie group. In the case of single crystal plasticity this metric is given in terms of the different slip systems and is piecewise affine on each tangent space. For von Mises plasticity the metric is a left–invariant Riemannian metric. A main goal is to study the associated distance metric and the geodesic curves.

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . Formulations of Elastoplasticity . . . . . . . . . Mathematical Formulation Using Lie Groups Dissipation Functionals . . . . . . . . . . . . . . 4.1 von Mises Plasticity . . . . . . . . . . . . . . . 4.2 Single-Crystal Plasticity . . . . . . . . . . . . . 5 The Metric and Geodesics on G . . . . . . . . . 6 Geodesics for von Mises Dissipation . . . . . . A Set-Valued Calculus of Variations . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

1

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

61 65 72 74 74 75 76 84 86 88

Introduction

We give a formulation of perfect plasticity in the context of finite strain. This formulation is based on two functionals, namely the stored-energy density ψ (also called the Helmholtz free energy) and the dissipation func The deformation of a body Ω ⊂ Rd is a mapping ϕ : Ω → Rd ; tional ∆. 61

62

A. Mielke

and the deformation gradient Dϕ ∈ Rd×d is denoted by F . In perfect finite elastoplasticity the function ψ depends on the deformation gradient F and the plastic variable P = F −1 p solely via the elastic part F e of the deforma tion gradient, i.e. F e = F P = F F −1 p . We write ψ = ψ(F , P ) = Ψ(F P ). This multiplicative split was introduced in Lee [1969] and its use is nowadays well-established, see e.g. Simó and Ortiz [1985]; Simó [1988]; Miehe and Stein [1992]; Hackl [1997]; Gurtin [2000]. The main point in this paper is to work out the Lie group structure which is central to finite plasticity. For the usage of Lie group theory in mechanics we refer to Abraham and Marsden [1978]; Arnol’d [1989]; Marsden and Hughes [1983]. In particular, we are led to the study of left-invariant mechanical systems on Lie groups, which play a fundamental rˆ ole in many areas in mechanics, such as in rigid-body motion Krishnaprasad and Marsden [1987], in the deformations of thin rods Mielke and Holmes [1988]; Mielke [1991] and in flow problems for inviscid fluids or plasmas Ebin and Marsden [1970]; Marsden and Weinstein [1983]; Holm, Marsden, Ratiu, and Weinstein [1985]. Since minimization of the dissipation is a fundamental concept in plasticity we encounter geodesic curves with respect to the left-invariant dissipation metric. In the present work all appearing Lie groups are matrix groups, so the paper can be read without previous knowledge in Lie group theory. The deformation gradient F is considered as an element of the general linear group GL(d) = GL+ (d, R) = F ∈ Rd×d : det F > 0 , and the plastic variable P is considered as an element of a Lie subgroup G ⊂ GL(d); in many cases G is the special linear group SL(d) = P ∈ GL(d) : det P = 1 . is a mapping from TG into [0, ∞) such that The dissipation functional ∆ ∆(P , P˙ ) = ∆(P −1 P˙ ) with

∆(αξ) = α∆(ξ)

for all (P , P˙ ) ∈ TG and α ≥ 0. The first condition expresses the fact that previous plastic deformations do not influence the dissipation (i.e. for each G ∈ G the dissipation of the process t → GP (t) is the same). The second condition gives rate independency of the model. The function ∆ is defined onthe Lie algebra g = TI G and is assumed to be convex and coercive, i.e., r1 ξ ≤ ∆(ξ) ≤ r2 ξ . (or equivalently Ψ and ∆) together with The two functionals ψ and ∆ the principle of maximal dissipation (cf. Simó and Ortiz [1985]; Ziegler and Wehrli [1987]; Simó [1988]) define the full equations of elastoplasticity. We

2. Finite Elastoplasticity

63

give here an alternate form using the subdifferential ∂∆(ξ): ∂ ψ(F , P ) = DΨ(F P )P T , (1.1) ∂F ∂ ψ(F , P ) = ∂∆(P −1 P˙ ) + (F P )T DΨ(F P ) . 0 ∈ ∂∆(P −1 P˙ ) + P T ∂P Σ=

The second equation is posed on g. Together with the relation F = Dϕ and the elastic equilibrium condition − div Σ = f ext and suitable boundary conditions this defines the classical plastic flow rule. The equivalence between these equations an the classical mechanical flow rules involving the yield surface is given in Section 2. As in Carstensen, Hackl, and Mielke [2001] we propose an integrated version of (1.1) which implies (1.1) but not vice versa. To this end we need to define a non-symmetric distance metric on G as follows: 0 , P 1 ) = inf I∆ P ( · ) P ∈ C1 [0, 1] , G , P (0) = P 0 , P (1) = P 1 D(P 1 with I∆ P ( · ) = ∆ P (t)−1 P˙ (t) dt . (1.2) 0

is a so-called Finsler–Minkowski metric generated The distance metric D ˙ . The ˙ → ∆ G−1 G : TG → [0, ∞); G , G by the infinitesimal metric ∆ metric is invariant with respect to the left translation. In the special situ

1 ation where ∆(ξ) = B ξ , ξ 2 with B = B∗ > 0 we obtain a Riemannian metric. over the body Ω ⊂ Rd we define the nonlinear By integrating ψ and ∆ functionals Ψ Dϕ P − f ext (t , · ) · ϕ dx − E t, ϕ, P = text (t, · ) · ϕ da , Ω Γtract Ω P 0 , P 1 = P 0 (x) , P 1 (x) dx . D D Ω

Here, E is the elastic and potential energy stored in the body under given Ω gives the minimal ϕ : Ω → Rd and P : Ω → G. The functional D dissipation arising from a change of the plastic variable from P 0 : Ω → G to the new state P 1 : Ω → G. The global non-differentiated form of the plastic flow problem now takes the form Ω P (t) , P ∗ global stability : E t, ϕ(t), P (t) ≤ E(t, ϕ∗ , P ∗ ) + D for all (ϕ∗ , P ∗ ) with ϕ∗ |Γdispl = ϕ0 (t, · ) ; t energy balance : E t, ϕ(t), P (t) + s Ω ∆ P (τ, x)−1 P˙ (τ, x) dx dτ t ∂ E τ, ϕ(τ ), P (τ ) dτ . = E s, ϕ(s), P (s) + s ∂τ (1.3)

64

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A similar energetic formulation for phase transformation problems was given in Mielke and Theil [1999]; Mielke, Theil, and Levitas [2002]; Govindjee, Mielke, and Hall [2001]. Such a formulation has the major advantage, that it immediately suggests an incremental formulation which has a variational form. If 0 = t0 < t1 < . . . tN = T is a time discretization and P 0 : Ω → G the initial value, then the time stepping algorithm reads Incremental Problem. Given (ϕk , P k ) at time tk , find (ϕk+1 , P k+1 ) such that it minimizes the functional Ω (P k , P ). Tk+1 : (ϕ, P ) → E(tk+1 , ϕ, P ) + D In Lemma 2.1 we show that all solutions of this incremental problem satisfy global stability as well as a discretized version of the energy balance in (1.3). Incremental formulations for rate-independent processes which solely depend on minimization are well-known in abstract sweeping processes (cf. Monteiro Marques [1993]) and in linearized elastoplasticity, cf. e.g. Han and Reddy [1995]. The usage of variational formulations in finite elastoplasticity is only recent Ortiz and Repetto [1999]; Carstensen, Hackl, and Mielke [2001]; Miehe and Schotte [2000]; Miehe [2000]. The usage of leftinvariant dissipation metrics in this context is new. In the present work we only deal with the case of ideal plasticity without hardening, in subsequent work Mielke [2002] the general case with hardening will treated, where also more general associative plastic flow rules are considered. The above theory generates a need of a characterization of the Finsler– : G × G → [0, ∞) as defined in (1.2). By left Minkowski distance metric D ˙ invariance (i.e. ∆(P , P ) = ∆(P −1 P˙ )) we have 0 , P 1 ) = D(P −1 P 1 ) D(P 0

,P). with D(P ) := D(I

Recalling ∆(ξ) ≤ r2 ξ we immediately obtain an upper bound D(P ) ≤ r2 log P whenever a logarithm ξ ∈ g with P = eξ exists. This already indicates that the Lie algebraic exponential map does not provide the geodesic curves, i.e., the curves t → P (t) = etξ are not minimizers in (1.2) in general. In Section 6 we provide a geodesic exponential map which is surjective, see also M in (1.5) below. and its Sections 4 to 6 are concerned only with the study of the metric D shortest paths. In Theorem 5.1 we show that the infimum in (1.2) is always attained, i.e., there is always a curve P ∈ CLip [0, 1] , G with P (0) = P 0 0 , P 1 ) = I P ( · ) . As usual, a geodesic and P (1) = P 1 such that D(P ∆ curve P : (t0 , t1 ) → G is a curve which is locally a shortest path. The difficulty for Finsler metrics arises from two facts; first of all, ∆ : g\{0} → [0, ∞) may be non differentiable and, secondly, the subdifferential Q = ∂∆(0) ⊂ g ∗ at 0 may be not strictly convex, where Q = ∂∆(0) = η ∈ g ∗ η , ξ ≤ ∆(ξ) for all ξ .


65

In this general situation we do not have a full characterization of shortest paths or geodesic curves. However, we derive a nonsmooth counterpart to it follows Noether’s theorem. For instance, from the left invariance of ∆ that for each shortest path P there exists an η ∈ g ∗ such that P T ηP −T ∈ ∂∆(P −1 P˙ )

(1.4)

holds almost everywhere along the curve. We continue to use the notation ( · )T for the transposed matrix, where throughout the Euclidean scalar product in Rd is used. See Theorem 5.2 for the exact statement in Lie group notation. Examples of nonsmooth shortest paths and geodesics are given there as well. In Section 6 we study von Mises plasticity where ∆ generates a leftinvariant Riemann metric on G = SL(d) which is additionally right invariant with respect to SO(d). The latter invariance corresponds to isotropy of the elastoplastic model. Starting from 1 ∆(ξ) = γ2 ξ sym 2 + γ3 ξ anti 2 2 where

ξ sym = 12 (ξ + ξ T ) ,

ξ anti = 12 (ξ−ξ T )

with γ2 , γ3 > 0 we find that all geodesics are smooth and have the form P (t) = M t(α + ω) := P (0) etα etω (1.5) with α = γ12 η T and ω = γ12 + γ13 η anti . Here η is defined such that (1.4) holds. This leads to the formula ,P) D(P ) = D(I 1 = min γ2 ξ sym 2 + γ3 ξ anti 2 2 : P = eξsym −δξanti e(1+δ) ξanti where δ = γ3 /γ2 .

2

Formulations of Elastoplasticity

The aim of this section is to motivate the model for elastoplasticity at finite deformations from the mechanical point of view, for more background we refer to Simó [1988]; Hackl [1997]; Gurtin [2000]. Then we reformulate the mechanical notion by using methods from convex analysis based on subdifferentials. This leads to a formulation involving energy balance and local (infinitesimal) stability. We will then argue that local stability can be replaced by global stability which then justifies the incremental problem in variational form.

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We consider perfect finite plasticity, i.e., we do not take into account plastic hardening effects. The body is given by a domain Ω ⊂ Rd , where the dimension d is usually equal to 3 but d ∈ {1, 2} is also relevant in certain special situations. The deformation is ϕ : Ω → Rd and its gradient F = Dϕ is decomposed multiplicatively into the elastic part F e and the plastic part F p Dϕ = F = F e F p . (2.1) The plastic part F p is not a gradient. It corresponds to the atomic disarrangement in single crystals arising through slips. In polycrystals F p also includes rearrangements of the grains. The first main assumption of perfect finite plasticity is that the storedenergy density (per unit volume) ψ depends on F and F p only through the elastic part of the deformation gradient , P ) = Ψ(F e ) = Ψ(F P ) where P = F −1 . ψ = ψ(F p

(2.2)

For notational simplicity we omit all dependence on the material point x ∈ Ω, but of course our theory can handle general inhomogeneous materials with ψ = Ψ(x, F P ). Using the elastic stress tensor (first Piola–Kirchhoff tensor) Σ=

∂ ψ(F , P ) = DΨ(F P )P T ∂F

(2.3)

the elastic equilibrium equation reads −divΣ = f ext (t, · ) in Ω , ϕ = ϕ0 (t, · ) on Γdispl ,

(2.4)

Σν = text (t, · ) on Γtract = ∂Ω/Γdispl . We additionally need a flow rule for the internal plastic variable P = F −1 p . This is obtained from the principle of maximal dissipation or equivalently from the associated flow rule. The aim of this section is to derive another formulation using a dissipation metric and the calculus of subdifferentials. We start with the conventional formulation. The plastic back stress Q can be defined as Q=−

∂ ψ(F , P ) = −F T DΨ(F P ) . ∂P

(2.5)

Note that the tensor Q = P T Q depends only on the elastic part F e = F P of the deformations. The yield function y in general depends on Q and P and y = 0 defines the yield surface. Again assuming that y is invariant under all plastic deformations we postulate y = y(Q , P ) = Y (Q) = Y (P T Q) .

(2.6)


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The principle of maximal dissipation generates the associated flow rule (cf. Simó and Ortiz [1985]; Ziegler and Wehrli [1987]; Simó [1988]) ∂Y (Q) , P −1 P˙ = λ ∂Q

with λ ≥ 0 , Y (Q) ≤ 0 and λY (Q) = 0 .

(2.7)

Now (2.7) together with the elastic equilibrium condition (2.4) form the full set of equations describing the evolution of elastoplasticity. The flow rule (2.7), which now has the Karush–Kuhn–Tucker form, can be reformulated by introducing 0 for Q ∈ Q d×d : Y (Q) ≤ 0 and XQ (Q) = Q= Q∈R ∞ otherwise . where we always assume that Q is convex and there exist r1 , r2 > 0 such that B(0, r1 ) ⊂ Q ⊂ B(0, r2 ) .

(2.8)

Using the outer normal cone NQ, the flow rule (2.7) now takes the form P −1 P˙ ∈ ∂XQ (Q) = NQ Q.

(2.9)

Note that (2.9) is more general, since Q may have corners while ∂Q = Q : Y (Q) = 0 is C1 if Y ∈ C1 . For general convex functions f : Z → R on a Banach space Z the subdifferential ∂f (z) at the point z ∈ Z is defined via

∂f (z) = z ∗ ∈ Z ∗ : f (z + z) ≥ f (z) + z ∗ , z for all z ∈ Z , (2.10) where Z ∗ is the dual Banach space and z ∗ , z the dual pairing. The outer normal cone Nz C of a closed convex set C in the point z ∈ C is Nz C = z ∗ ∈ Z ∗ : z ∗ (z− z ) ≤ 0 for all z ∈ C (2.11) Using the Legendre transform (Lf )(z ∗ ) = supz∈Z z ∗ , z−f (z) we define the dissipation functional

∆(ξ) = LXQ (ξ) = sup η , ξ : η ∈ Q . (2.12) With (2.8) we conclude r1 ξ ≤ ∆(ξ) ≤ r2 ξ. Moreover ∆ : Rd×d → [0, ∞) is convex (as Q is convex) and homogeneous of degree 1. Under the additional assumption ∆(−ξ) = ∆(ξ), which is not necessary for the present work, the function ∆ defines a metric on Rd×d . Without this condition we call ∆ a nonsymmetric metric.

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We define the dissipation functional ∆(P , P˙ ) = ∆(P −1 P˙ ) . The classical duality theory for convex functions applied to ∆ = LXQ gives ξ ∈ ∂XQ (Q)

⇐⇒

Q ∈ ∂∆(ξ) .

, P˙ ) = P −T ∂∆(P −1 P˙ ) we find two equivaWith ξ = P −1 P˙ and ∂2 ∆(P lent forms of the flow rule Q ∈ ∂∆(P −1 P˙ )

⇐⇒

Q ∈ ∂2 ∆(P , P˙ ) .

(2.13)

We employ a standard relation for subdifferentials of convex functions f which are homogeneous of degree 1: ∂f (z) =

z ∗ ∈ Z ∗ : z ∗ ∈ ∂f (0) and z ∗ , z = f (z) .

(2.14)

Together with the definition of Q in (2.5) the right-hand side of (2.13) gives our final pointwise version of the flow rule: 0∈

∂ , P˙ ) ⊂ Rd×d . ψ(F , P ) + ∂2 ∆(P ∂P

(2.15)

To obtain field equations we consider the integrated functionals E t, ϕ, P =

ψ Dx ϕ(x) , P (x) − f

Ω

Ω P , P˙ = ∆

· ϕ(x) dx

text (t, x) · ϕ(x) da ,

−

ext (t, x)

Γtract

P (x) , P˙ (x) dx . ∆

Ω

Now, the whole problem of elastoplasticity can be written using variational derivatives and the subdifferential as follows: elastic equilibrium : Dϕ E t, ϕ, P = 0 , Ω P , P˙ . plastic flow rule : 0 ∈ DP E t, ϕ, P + ∂2 ∆

(2.16)

Our final formulation, which we derive now, has the major advantage that it does not involve derivatives but relies only on of functionals. minimization ∂ P, · we Applying (2.14) with z ∗ = Q = − ∂P ψ F , P and f = ∆ can calculate the time derivative of the bulk energy e(t) = E t, ϕ(t), P (t)


69

along any solution t → ϕ(t) , P (t) : d e(t) dt ∂ ˙ = E t, ϕ(t), P (t) + Dϕ E t, ϕ(t), P (t) ϕ(t) + DP E t, ϕ, P P˙ (t) ∂t

∂ − Q(t, x) , P˙ (t, x) dx = E t, ϕ(t), P (t) + 0 + ∂t Ω ∂ P (t, x) , P˙ (t, x) dx ∆ = E t, ϕ(t), P (t) − ∂t Ω ∂ Ω P (t) , P˙ (t) . = E t, ϕ(t), P (t) − ∆ ∂t This leads to the energy balance t Ω P (τ ) , P˙ (τ ) dτ ∆ E t, ϕ(t), P (t) − E s, ϕ(s), P (s) + s t ∂ E τ, ϕ(τ ), P (τ ) dτ (2.17) = s ∂τ t ∂ ∂ f ext (τ, x) · ϕ(τ, x) dx + text (τ, x) · ϕ(τ, x) da dτ . =− Ω ∂τ Γtract ∂τ s

Moreover (2.16) implies the local conditions Dϕ E t, ϕ(t), P (t) [ψ] = 0 Ω P (t) , S ≥ 0 DP E t, ϕ(t), P (t) [S] + ∆

(2.18)

for all ψ with ψ|Γdispl = ϕ0 (t, · ) and all S. Note that (2.17) and (2.18) are equivalent to (2.16) and hence to (2.4) and (2.7). Since we are considering rate-independent elastoplasticity we may interpret (2.18) as a local form of a stability condition. In fact, we may assume that ϕ(t) is the (global) minimizer of E t, · , P (t) . Similarly we may assume that P (t) is such that we cannot under the given loading (i.e., t fixed) lower the energy from E t, ϕ(t), P (t) to a value E t, ϕ∗ , P ∗ such that the gain in the energy is larger than the associated dissipation obtained from moving P (t) continuously into P ∗ . To measure this dissipation we introduce a nonsymmetric distance metric on matrices via ! 1 P (τ ) , P˙ (τ ) dτ P ∈ C1 [0, 1] , Rd×d , ∆ D(P 0 , P 1 ) = inf 0

P (0) = P 0 ,

" P (1) = P 1 .

(2.19)

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I , P −1 P 1 but we do not assume sym P0 ,P1 = D Note that we have D 0 P 0 , P 1 = D P 1 , P 0 . The triangle inequality metry, i.e., we may have D 0 , P 2 ) ≤ D(P 0 , P 1 ) + D(P 1 , P 2) D(P

(2.20)

eξ ) = is an immediate consequence of the definition. Moreover we have D(I, 2 ∆(ξ) + O(|ξ| ) for ξ → 0. Defining the integrated version of the metric Ω (P 0 , P 1 ) = D

P 0 (x) , P 1 (x) dx D

Ω

we arrive at our final global formulation of elastoplasticity Ω (P (t) , P ∗ ) E t, ϕ(t), P (t) ≤ E(t, ϕ∗ , P ∗ ) + D for all ϕ∗ with ϕ∗ |Γdispl = ϕ0 (t, · ) and all P ∗ ; t energy balance: E t, ϕ(t), P (t) + s ∆Ω P (τ )−1 P˙ (τ ) dτ t ∂ E τ, ϕ(τ ), P (τ ) dτ . = E s, ϕ(s), P (s) + s ∂τ (2.21) Clearly this formulation implies the previous ones but not vice versa. Equivalence can be obtained, however, under certain convexity assumption like in linear elastoplasticity. A major advantage of the formulation (2.21) is that it immediately gives rise to a suitable incremental formulation when the time interval [0, T ] is discretized via 0 = t0 < t1 < . . . < tN = T and the initial value (ϕ0 , P 0 ) at t0 = 0 is given: global stability:

Incremental Problem: Given (ϕk , P k ) at time tk , find (ϕk+1 , P k+1 ) such that it minimizes the functional Ω (P k , P ) . Tk+1 : (ϕ, P ) → E(tk+1 , ϕ, P ) + D

(2.22)

We show that any solutions of (2.22) satisfy a suitable discretized version of (2.21). 2.1 Lemma. Let (ϕk , P k ), k = 1, . . . , N , be a solution of the incremental problem (2.22). Then, for k = 1, . . . , N we have (i) stability of ϕk , P k at time tk , i.e., Ω P k , P ∗ for all ϕ∗ , P ∗ , E tk , ϕk , P k ≤ E t, ϕ∗ , P ∗ + D


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(ii) the two-sided discretized energy estimate E(tk , ϕk , P k ) − E(tk−1 , ϕk , P k ) tk ∂ E(s, ϕk , P k ) ds = ∂s tk−1 Ω (P k−1 , P k ) ≤ E(tk , ϕk , P k ) − E(tk−1 , ϕk−1 , P k−1 ) + D tk ∂ E(s, ϕk−1 , P k−1 ) ds ≤ ∂s tk−1 = E(tk , ϕk−1 , P k−1 ) − E(tk−1 , ϕk−1 , P k−1 ) . Proof. To shorten the notation we let z k = (ϕk , P k ). The stability (i) is obtained as follows. For all z ∗ we have Ω (P k , P ∗ ) E(tk , z ∗ ) + D Ω (P k−1 , P ∗ ) + D Ω (P k , P ∗ ) − D Ω (P k−1 , P ∗ ) = E(tk , z ∗ ) + D Ω (P k−1 , P k ) + D Ω (P k , P ∗ ) − D Ω (P k−1 , P ∗ ) ≥ E(tk , z k ) + D ≥ E(tk , z k ) . Here, the first inequality uses that z k is a global minimizer (cf. (2.22)) and the second estimate is the triangle inequality (2.20). The lower estimate in (ii) follows from the stability of z k−1 at time tk−1 : Ω P k−1 , P k ≥ E tk , z k − E tk−1 , z k . E tk , z k − E tk−1 , z k−1 + D The upper estimate is a consequence of z k being a minimizer: Ω P k−1 , P k ≤ E tk , z k−1 − E tk−1 , z k−1 . E tk , z k − E tk−1 , z k−1 + D This proves the result.

The minimization problem in the incremental problem (2.22) can be solved in two steps, since the functional Tk+1 has the form P k , P dx − Lext (tk+1 , ϕ) , Ψ ( Dx ϕ)P + D Tk+1 (ϕ , P ) = Ω

where Lext (t, · ) denotes the external loading. We first minimize pointwise in x ∈ Ω with respect to P and obtain the reduced incremental energy density Ψred P k (F ) = min Ψ F P + D P k , P : P ∈ G . To study the existence question for solutions (ϕk+1 , P k+1 ) it is now essend×d → [0, ∞] such tial to prove certain convexity properties of Ψred Pk( · ) : R as rank-one and quasi-convexity, see Ortiz and Repetto [1999]; Carstensen, Hackl, and Mielke [2001] for first results in this direction. For the present this will be done in subsequent work. metric D

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3

A. Mielke

Mathematical Formulation Using Lie Groups

In general situations of plasticity with finite elastic and plastic deformations the gradient F is assumed to lie in GL(3) = GL+ (3, R) = deformation to lie P ∈ R3×3 det P > 0 . The plastic variable P = F −1 p is assumed in a Lie subgroup G of GL(3). For example, G = SL(3) = P ∈ GL(3) : det P = 1 is the case of incompressible plastic deformations. If only one slip system should be taken into account, then we let G = I + αn ⊗ m : α ∈ R , where |n| = |m| = 1 and n · m = 0. Hence, in the following G ⊂ GL(d) will denote a general Lie subgroup which contains the plastic variable: P (t, x) ∈ G. By g = TI G we denote the associated Lie algebra. Our plasticity model is based solely on two constitutive functions: the , P ) and a plastic dissipation potential ∆(P , P˙ ); elastic energy density ψ(F ψ : GL(d) × G → R and

: TG → [0, ∞) . ∆

Here again we neglect any dependence on x ∈ Ω which could be included ∂ ψ(F , P ) is an element easily. As a consequence the elastic stress Σ = ∂F ∂ ∗ of TF GL(d) and the plastic backstress Q = − ∂P ψ(F , P ) is an element of T∗P G. Hence, the flow rule (2.13) has to be considered on T∗ G. To introduce the (symmetry) axiom of our plasticity model we use the left and right translations LG and RG on GL(d): G → G, G → G, LG : RG : P → GP ; P → P G . The derivative DLG maps the tangent space TP G bijectively onto TLG P G. The basic axiom of multiplicative plasticity can now be formulated by introducing an action of G on GL(d) × T G via G × GL(d) × TG → GL(d) × TG , −1 τ: G, (F , P , P˙ ) → RG F , LG P , DLG P˙ . For τ G , (F , P , P˙ ) we use the abbreviation −1 τG (F , P , P˙ ) = RG F , LG P , DLG P˙ = F G−1 , GP , GP˙ . In the whole of this section, we will use the abstract Lie group notation but add for clarity the corresponding matrix notation in the double brackets ... . In addition to the plastic symmetry there are two standard symmetries which are already present in pure elasticity. The first symmetry is called objectivity or frame indifference and is due to the Euclidian symmetry


73

of the space Rd . The rotations R ∈ O(d) act on F e via left translation. The second symmetry is the material symmetry which is incorporated in a subgroup S of O(d) and which acts on F e via right translation. (A1) Objectivity (frame indifference): For all R ∈ SO(d) and all F , P ∈ GL(d) × G we have R F , P ) = ψ(F , P ) = ψ(RF ,P) . ψ(L (A2) Invariance under superimposed plastic deformations: ◦ τG |TG = ∆ , for all G ∈ G; ψ ◦ τG |GL(d)×G = ψ and ∆

G−1 , GP ) = ψ(F ,P) ψ(F

and

GP , GP˙ = ∆ P , P˙ . ∆

(A3) Rate independency and dissipativity: The function P , · : TP G → [0, ∞) is convex and homogeneous of degree 1 ∆ P , LP ξ ≤ r2 ξ . with r1 ξ ≤ ∆ (A4) Material symmetry: For all S ∈ S and all F , P , P˙ ∈ GL(d) × TG we have , P S) , , P ) = ψ(F , R P ) = ψ(F ψ(F S R P , DR P˙ = ∆ P , P˙ = ∆ P S , P˙ S . ∆ S

S

Assumptions (A2) and (A3) imply P , P˙ = ∆ DL−1 P˙ = ∆ P −1 P˙ , ∆ P where ∆ : g = TI G → [0, ∞) is convex and homogeneous of degree 1. Assumptions (A1) and (A2) give , P ) = Ψ(F P ) = Ψ (F P )T F P . ψ(F The material symmetry S ⊂ O(d) in (A4) reads for the reduced functionals ∆ and Ψ (3.1) ∆ AdS ξ = ∆ ξ and Ψ F e S = Ψ F e for S ∈ S, ξ ∈ g and F e ∈ GL(d). Here, −1 ξ = RξR−1 AdR ξ = DLR DRR is the adjoint action of G on g. We will also need the co-adjoint action of G on g ∗ , which is defined by ∗ G → CoAdG = Ad∗G−1 = Ad−∗ G ∈ Lin(g ) .

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The construction of suitable energy densities Ψ is well understood. Typically one assumes that the mapping F → Ψ(F P ) is such that it satisfies the conditions of pure elasticity, that is Ψ(F e ) = +∞ for det F e ≤ 0 and Ψ(F e ) → ∞ for det F e 0 or F e → ∞. Moreover, Ψ is assumed to be quasi- or poly-convex. The construction of plastic dissipation densities is discussed in the next section.

4

Dissipation Functionals

We discuss two different cases, namely von Mises plasticity (cf. Simó and Ortiz [1985]; Simó [1988]; Miehe and Stein [1992]) and single-crystal plasticity (cf. Ortiz and Repetto [1999]; Gurtin [2000]). The former is a model for polycrystals such as metals where the presence of grains averages out the different crystallographic directions; the material is assumed to be isotropic, i.e., S = SO(d).

4.1

von Mises Plasticity

The Lie group G of plastic directors is given by the special linear group G = SL(d) = P ∈ GL(d) : det P = 1 . The associated Lie algebra g = TI G is sl(d) = ξ ∈ Rd×d : trace ξ = 0 . The material symmetry is given by S = SO(d) = R ∈ GL(d) : RT R = I . Denoting by ξ F the matrix Frobenius norm d 2 2 ξ = ξ : ξ = trace ξξ T = ξjk F j,k=1

we obtain AdR ξ F = ξ F for R ∈ SO(d) and ξ ∈ Rd×d . Moreover we have the following result. 4.1 Proposition. Let ξ sym = 12 (ξ + ξ T ) and ξ anti = 12 (ξ − ξ T ) and take any function δ : R2 → [0, ∞) which is convex and homogeneous of degree 1. Then ∆(ξ) = δ ξ sym F , ξ anti F satisfies (3.1) for S = SO(d).


75

Proof. We simply use the fact that AdR leaves invariant the splitting Rd×d = Sym(d) ⊕ Anti(d). We consider the special case that ∆ is given by a scalar product, i.e.,

1 ∆B (ξ) = Bξ , ξ 2 ≥ γ ξ F (4.1) where B : g → g ∗ is symmetric and positive definite (γ > 0). This case is equivalent to assuming that G is equipped with a left-invariant Riemannian metric. From standard invariant theory it follows that every symmetric operator B which is Ad-invariant, i.e. ∆B AdR ξ = ∆B (ξ) for R ∈ SO(d) and ξ ∈ Rd×d , there exist real numbers γ1 , γ2 and γ3 such that B ξ = γ1 trace(ξ)I + γ2 ξ sym + γ3 ξ anti .

(4.2)

Here positive definiteness is equivalent to dγ1 + γ2 > 0,

γ2 > 0 and γ3 > 0 .

Moreover, for ξ ∈ g = sl(d) we have trace ξ = 0 and hence γ1 is irrelevant. On g = sl(d) we have 2 2 1 (4.3) ∆B ξ = γ2 ξ sym F + γ3 ξ anti F 2 . Note that the plastic flow rule gives P˙ = P ξ sym + ξ anti such that ξ sym corresponds to plastic stretching whereas ξ anti corresponds to plastic spin. The controversy about the relevance of plastic spin can be rephrased in our setting in the relative size of γ3 with respect to γ2 . We will discuss this question after the construction of the geodesic curves.

4.2

Single-Crystal Plasticity

In single crystals plasticity occurs through movements of dislocation and the plastic strain F p = P −1 can be thought of as atomic disarrangements. The motion of dislocations is viewed to take place on n crystallographically determined slip systems S α = dα ⊗ nα , α = 1, . . . , n, where nα is the unit normal vector to the αth slip plane and dα is the associated slip direction with |dα | = 1 and dα · nα = 0. The plastic flow then takes the form (see Gurtin [2000]) n

να S α F p F˙ p = α=1

where να ≥ 0 is the slip rate of the slip system S α (negative slip rates are realized here by using the negative slip system −S α ). The crystal symmetry group S ⊂ O(d) associates a permutation πR ∈ Perm(n) to each R ∈ S such that πR◦R and = πR ◦ πR α ⇐⇒ S πR (α) = RS α RT Rd , Rnα = dπR (α) , nπR (α) for α = 1, . . . , n.

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For our plastic director P shows that span S α

˙ = F −1 p we obtain P = −P

#n α=1

να S α , which

α = 1, . . . , n ⊂ g = TI G .

This generates in a natural way the associated Lie group via the smallest Lie algebra g containing all the slip systems. From dα · nα = 0 we know trace(S α ) = 0 and hence g ⊂ sl(d) and G ⊂ SL(d). According 4.2], the dissipation associated to #n #n to [Gurtin, 2000, Section κ ξ = − α=1 να S α is given by α=1 α |να | with positive constants κα satisfying the material symmetry relation κπR (α) = κα for all R ∈ S. Since in general the coefficients να are not uniquely determined by ξ we define ∆(ξ) = min

n !

n " κα |να | ξ = − να S α .

α=1

α=1

To match this construction with our abstract theory we need to show r1 ξ leq∆(ξ) ≤ r2 ξ for all ξ ∈ g. The lower estimate is trivial with r = minα κα , since n n α S = ξ ≤ |ν | |να | ≤ (1/r) ∆(ξ) . α F F α=1

α=1

The upper estimate holds if and only if g equals the span of all S α . The latter condition is nontrivial as is seen for the case d = 3 and n = 2 with S 1 = e1 ⊗ e2 and S 2 = e2 ⊗ e3 . Clearly, span S 1 , S 2 is two-dimensional Lie but the smallest algebra g containing this span is span S 1 , S 2 , S 3 with S 3 = S 1 , S 2 = S 1 S 2 − S 2 S 1 = e1 ⊗ e3 .

5

The Metric and Geodesics on G

To formulate the elastoplastic problem in Section 2 we used the distance on G which is induced by the (nonsymmetric) norm ∆ on g via metric D the left-invariant Finsler–Minkowski metric P , P˙ = ∆ DL−1 P˙ = ∆ P −1 P˙ , ∆ P P 0 , P 1 = inf I P ( · ) P ∈ C1 [0, 1], G , P (0) = P 0 , P (1) = P 1 D ∆ ∆ 1 ˙ with I∆ P ( · ) = ∆ DL−1 (5.1) P (t) P (t) dt . 0

As a short notation we introduce G → [0, ∞) D∆ : I,P P → D ∆

(5.2)


77

∆ P 0 , P 1 = D∆ P −1 P 1 . If ∆ −ξ = such that by left-invariance D 0 ∆ ξ for all ξ ∈ g then D∆ P −1 = D∆ P . In general we cannot expect that there is a C1 function such that the infimum is attained. We also have to take into account curves with corners, namely P ∈ W1,∞ ([0,1] , G) =CLip ([0, 1] , G) where P˙ ∈ L∞ (0, 1) . A curve P ∈ CLip [0, 1] , G is called a shortest path from P (0) to ∆ P (0), P (1) . A curve P ∈ P (1) (with respect to ∆) if I∆ P = D CLip (t1 , t2 ), G is called a geodesic curve (with respect to ∆), if for each t ∈ (t1 , t2 ) there exists ε > 0 such that P |[t−ε,t+ε] is a shortest path. Every part P |[τ1 ,τ2 ] of a shortest path is again a shortest path. Moreover, shortest paths are in general not unique. Example.

Consider the Lie group G = P = diag(a, b) ∈ GL(2) a, b > 0

˙ P , P˙ = |a|/a and the left-invariant metric ∆ ˙ + |b|/b. Then, D∆ diag(a, b) = | log a| + | log b| and any curve t → diag( a(t), b(t) is a shortest path if a : [0, 1] → R and b : [0, 1] → R are monotone. A curve is a geodesic if and only if a and b are locally monotone. In particular, nonmonotone a or b is admissible if different monotonicity regions are separated by intervals of constant values. E.g. a(t), b(t) = max{1, 2−|t|}, 1 is a geodesic over t ∈ R.

The next example, which has been mentioned to the author by D. Mittenhuber Mittenhuber [2000], shows that shortest paths for left-invariant Finsler metrics in general must have corners. Here the notion of a corner means that the function P :[0, 1] → G is not C 1 when the parametrization ˙ is such that ∆ DL−1 P (t) P (t) ≥ c > 0. The existence of nonsmooth shortest paths might indicate that certain experimental fact concerning the switching between different slip systems can be modelled easily using Finsler metrics. This topic will be pursued in future research. " ! a b ∈ GL(2) a > 0, b ∈ R with the Example. Consider G = 0 1 associated Lie algebra g=

" ! α β ∈ gl(2) α, β ∈ R 0 0

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and the left-invariant Finsler metric induced by α β

= |α| + |β|. ∆ 0 0 a b ˙ ˙ the Finsler metric reads ∆ L−1 ˙ + |b|)/a. P P = (|a| 0 1 a1 b1 To find the shortest path between P 0 = I and P 1 = 0 1 we have to minimize 1 a(1) a a(0) 1 ˙ |a(t)| |b(t)| ˙ 1 = and = + dt subject to . b(1) 0 b(0) a(t) a(t) b1 0 Writing P =

The integral over the first term gives the total variation of log a( · ) and is bounded from below by log amax + log(amax /a1 ) where amax = max a(t) t ∈ [0, 1] . The integral over the second term is bounded from below by |b(1) − b(0)|/amax = |b1 |/amax . In fact, these lower bounds are attained by the path   1 + 3t(amax −1), 0 for t ∈ [0, 13 ],  a(t), b(t) = for t ∈ [ 13 , 23 ], amax , (3t − 1)b1   for t ∈ [ 23 , 1]. amax −(a1 − amax )(3t − 2), b1 The shortest path is now obtained by minimizing with respect to amax ≥ max{1, a1 }. This gives D∆

a b

m2 |b| = min log + 0 1 a m  1  log + |b|  a = log a + |b|/a   log(b2 /4a) + 2

m ≥ max{1, a} for a ∈ (0, 1] and |b| ≤ 2, for a ≥ max{1, |b|/2}, for |b| ≥ 2 and a ≤ |b|/2.

In particular, we see that the shortest path is unique (up to reparametrization). Hence, almost all shortest paths have one or two corners. (Note that a˙ b˙ ≡ 0 along all shortest paths.) For the general case we are able to estimate the dissipation metric from above via n # if P = eξ1 . . . eξn . ∆ ξi D∆ P ≤ i=1

(Let P (t) = e . . . e e , for t ∈ k/n, (k + 1)/n , and then use that P (t)−1 P˙ (t) = nξ k+1 .) In fact, we may approximate D∆ I, P as well as we like by increasing n and optimizing ξ 1 , . . . , ξ n . The simple estimate D∆ eξ ≤ ∆ ξ is of restricted use, since the (Lie algebraic) exponential map ξ → eξ from g ξ1

ξk (nt−k)ξk+1


79

into G is not surjective in general. For a surjective geodesic exponential 6.2. Of course we have the local expansion map Mδ : g → G, see Corollary ξ 2 D∆ e = ∆ ξ + O ξ for ξ → 0. Next we show that there always exists a shortest path between any two points on G. After that we use the left-invariance and a nonsmooth version of Noether’s theorem to characterize shortest paths. 5.1 Theorem. For any P 1 ∈ G there exists a shortest path P ( · ) ∈ CLip [0, 1] , G from I to P 1 . Proof. We need to find P ∈ CLip [0, 1] , G with P (0) = I, P (1) = P 1 and I∆ P ( · ) = inf I∆ Q Q ∈ C1 [0, 1] , G , Q(0) = I, Q(1) = P 1 . (5.3) This just means that the infimum is in fact a minimum if the set of candidates is increased from C1 to CLip . The right-hand side in (5.3) is by defini tion D∆ (P 1 ) and we can choose an infimizing sequence Q(k) in C1 [0, 1] , G with Q(k) (0) = I, Q(k) (1) = P 1 and δk := I∆ Q(k) → D∆ P 1 . We now use the rate-independency of the integral and reparametrise the time variable. Let t ˙ (k) (s) ds Q(k) (s) , Q 1+∆ τ (k) (t) = ck 0

with ck = 1/(1 + δk ), then τ (k) ∈ C1 [0, 1] , R satisfies τ (k) (0) = 0,

τ (k) (1) = 1

and τ˙ (k) (t) ≥ ck > 0 .

Hence, the inverse function t = T (k) (τ ) exists and we can set P (k) (τ ) = Q(k) T (k) (τ ) . Clearly, P (k) ∈ C1 [0, 1] , G with P (k) (0) = I,

P (k) (1) = P 1 ,

and

I∆ P (k) = δk

by rate-independency. Thus, (P (k) )k∈N is also an infimizing sequence. Additionally we know P (k) (τ ) , P˙ (k) (τ ) = T˙ (k) (τ ) a(k) (τ ) = ∆

a(k) (τ ) ≤ 1 + δk ck 1 + a(k) (τ )

(k) (k) ˙ Q(k) T (k) (τ ) , Q T (τ ) . where a(k) (τ ) = ∆ This new infimizing sequence has the advantage that ξ (k) := t → ( DL−1 P˙ (t) P (k) (t)

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is bounded in the linear space L∞ (0, 1) , g . From t P (k) (t) = I + 0 P (k) (s)ξ (k) (s) ds

(5.4) we conclude that P (k) is bounded in the linear space C1 [0, 1] , Rd×d . Thus, we may extract a subsequence (kl )l∈N , such that (P (kl ) )l∈N con d×d verges uniformly on [0, 1] to a limit P ∈ CLip [0, 1] , R and (ξ (kl) )l∈N converges weak* to a limit ξ ∈ L∞ (0, 1), g with ξ ∞ ≤ 1 + D∆ P 1 the limits P and ξ satisfy (5.4) as well (or weakly in L2 (0, 1), g ). Clearly, and thus P ∈ W1,∞ [0, 1] , G . From standard theorems in the calculus of variations (cf. Ball [1977], Dacorogna [1989]) we know that convexity in P˙ implies weak lower semicontinuity and we conclude I∆ P ≤ lim inf I∆ P (kl ) = lim inf δkl = D∆ P1 . l→∞

l→∞

This proves the result.

We now want to characterize shortest paths by deriving suitable differential equations like those for geodesic curves in Riemannian geometry. To motivate the analysis we first assume that ∆ : g → [0, ∞) is a C 2 -function (and no longer homogeneous of degree 1). The Euler–Lagrange equation associated to the functional I∆ (P ) is given by d P , P˙ − DP ∆(P , P˙ ) = 0 . (5.5) DP˙ ∆ dt In matrix notation we have P , P˙ = −P −T D∆ P −1 P˙ P˙ T P −T ∈ T∗ G DP ∆ P and

P , P˙ = P −T D∆ P −1 P˙ . DP˙ ∆

Thus, it is possible to write out the equation explicitly. To obtain a proper formulation in Lie group notation it is advantageous to use the Hamilto∗ of the Lagrangian Define the conjunian form on T on TG. G instead P , P˙ and the Hamiltonian gate variables P , Q ∈ T∗ G via Q = DP˙ ∆

P , Q = Q , P˙ − ∆ P , P˙ where P˙ has to be eliminated. In our speH cial case denote by f : g ∗ → g the inverse of the function D∆ : g → g ∗ and obtain −1 ∗ where DL−∗ P˙ = DLP f ( DL−∗ P Q), P = ( DLP ) , , Q) = H( DL−∗ Q) H(P P

with H(η) = sup η , ξ − ∆(ξ) ξ ∈ g . We have f (η) = DH(η) and the Hamiltonian equations which are equivalent to (5.5) are, in matrix notation, ˙ = −Q DH(P T Q) T . P˙ = P DH P T Q , Q (5.6)


81

To obtain a simple Lie algebraic formulation of the left-invariant system we introduce left-invariant coordinates (P , η) ∈ T0 G := G × g ∗ via η = ∗ DL−∗ P Q. Moreover, we need the Lie–Poisson structure JLP on g with JLP (η) ∈ Skew(g, g ∗ ) and

JLP (η)ξ 1 , ξ 2 = η , [ξ 1 , ξ 2 ] for all ξ 1 , ξ 2 ∈ g , where [ξ 1 , ξ 2 ] denotes the Lie bracket on g. According to Abraham and Marsden [1978] (5.5) is equivalent to P˙ = DLP DH(η) ,

η˙ = JLP (η) DH(η) .

(5.7)

is constant along solutions, namely Clearly, the Hamiltonian H d H DL−∗ P Q(t) ≡ 0 . dt Moreover, the Hamiltonian system is invariant under the G-action (τG )G∈G with τG : (P , Q) → (LG P , DL−∗ G Q). By Noether’s theorem (cf. Abraham and Marsden [1978]; Mielke [1991]) it follows that the co-adjoint action generates the associated first integrals. Define the momentum map T∗ G → g ∗ , J : P , Q → Ad∗P −1 ( DL∗P Q) = QP T , then for all solutions we have J P (t), Q(t) ≡ const., see Abraham and Marsden [1978]. Here we used the coadjoint action Ad∗ of G on g ∗ which is defined via ∗

−1 AdG η , ξ = η , AdG ξ = η , DLG DRG ξ for all ξ, η . Note that in matrix notation we have ∗ AdG η = P GT ηG−T where P : Rd×d → g ∗ is the projection defined via trace Pηξ T = trace ηξ T for all ξ ∈ g . Letting J P (t), Q(t) = η ∈ g ∗ and reintroducing P˙ by elimination of Q we find the “half-integrated” form of the Euler–Lagrange equation for left-invariant densities ∆: ∗ ˙ with a constant η ∈ g ∗ . (5.8) D∆ DL−1 P P = AdP η In the Hamiltonian context this can be rewritten as with a constant η ∈ g ∗ . P˙ = DLP DH Ad∗P η

(5.9)

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The generalization of the ODEs for the stationary curves of the functional I∆ (P ) to the case of nonsmooth ∆ involves the subdifferential ∂∆ ξ as defined in (2.10). Recall

Q = ∂∆(0) ⊂ g ∗ and ∂∆ ξ = η ∈ Q ∆(ξ) = η , ξ from (2.14). We derive integrated forms (5.8) and (5.9) for shortest paths by directly adapting the proof of Noether’s theorem to subdifferential techniques. 5.2 Theorem. Let P : [0, 1] → G be a shortest path between P (0) and P (1) with P (0) = P (1). Then there exists a nonzero η ∈ g ∗ such that Ad∗ η ∈ ∂∆ DL−1 P˙ (t) for a.e. t ∈ [0, 1]. (5.10) P (t)

P (t)

∗ ˙ ∗ Equivalently, we have DL−1 P P ∈ ∂XQ AdP η = NAdP η Q, where Nη Q denotes the normal cone, cf. (2.11). Proof. We choose ξ ∈ g and ρ ∈ C∞ c (0, 1), R , and define Qρ (t) = eρ(t)ξ P (t). From ρ(0) = ρ(1) = 0 and the fact that P is a shortest path we conclude I∆ Qρ ≥ I∆ P = D∆ P (0)−1 P (1) . Moreover, we find −1 ˙ ˙ (t) + P˙ (t) = ρκ(t) ˙ + Ξ(t) DL−1 Qρ Q = DLP (t) ρ(t)ξP with κ(t) = Ad−1 P (t) ξ. For fixed ρ consider the functional Kρ : g → [0, ∞) defined by 1 Kρ ξ := ∆ ρ(t) ˙ Ad−1 (5.11) P (t) ξ + Ξ(t) dt . 0

This functional is convex and by our assumption it attains the global min∗ imum in ξ = 0. This ρ (0) ⊂g . implies 0 ∈ ∂K −1 Let f (t, ξ) = ∆ Ξ(t) + ρ(t) ˙ AdP (t) ξ then ∂f (t, ξ) = ρ(t) ˙ Ad−∗ P (t) ∂∆ Ξ(t) and Lemma A.1 below gives " ! 1 ρ(t)η(t) ˙ dt η(t) ∈ Ad∗P (t)−1 ∂∆ Ξ(t) a.e. . ∂Kρ (0) = 0

From 0 ∈ ∂Kρ (0) for all ρ ∈ W01,∞ (0, 1), R and Lemma A.2 below we conclude that ⊂ g∗ E = ess Ad−∗ P (t) ∂∆ Ξ(t) t∈(0,1)

is nonempty. However, for any η ∈ E equation (5.10) is satisfied as Ξ = DLP −1 P˙ . Clearly, η = 0 would imply P˙ (t) = 0 for a.e. t ∈ [0, 1]. Hence, we conclude η = 0.


83

This theorem provides us with a sufficient condition for the differentiability of shortest paths and hence of geodesic curves. If the boundary ∂Q 1 of Q is of class C , then the normal cones Nη Q areone-dimensional rays λN (η) λ ≥ 0 , where N : ∂Q → ξ ∈ g ξ = 1 is continuous. Thus, up to time reparametrization (5.10) is equivalent to P˙ = DLP N Ad∗P η , whose solutions are continuously differentiable in time. 5.3 Let ∂Q be of class C1 (or, equivalently, let ξ ∈ g Corollary. ∆ ξ ≤ 1 be strictly convex ), then all geodesic curves are C1 functions in time. Example. To see the impact to Example 5, where of this result we return α, β > 0 is considered the Abelian Lie group G = diag(α, β) ∈ GL(2) 2 with ∆ ξ = |ξ1| + |ξ2 |, where ξ = (ξ1 , ξ2 ) ∈ g = R . We have ∂∆ ξ = Sign(ξ1 ) , Sign(ξ2 ) , where Sign(ξ1 ) = +1, [−1, 1], and −1 for ξ1 > 0, ξ1 = 0, and ξ1 < 0, respectively. Using Ad∗P (η) = η, as G is Abelian, equation (5.10) takes the form η Signα(t)/α(t) ˙ 1 ∈ η= for fixed η ∈ R2 . ˙ η2 Sign β(t)/β(t) This clearly holds if and only if α˙ and β˙ do not change sign. Then only the cases η = (±1, ±1) are relevant. We might consider solutions of (5.10) as generalized geodesic curves. It is unclear whether they are real geodesic curves since the usual argument P , P˙ in P˙ which is in Riemannian geometry involves strict convexity of D certainly not the case here. The next example shows that we have to expect nonsmooth curves as soon as ∆ : g\{0} → [0, ∞) is not differentiable. Example. Consider G = GL(d) and ∆(ξ) = ξ sym ξ anti such that Q = η sym + η anti : η sym ≤ 1 , η anti ≤ 1 . Using the Sign function 1 ξ : ξ ≤ 1 and Sign(ξ) = ξ ξ for ξ = 0, we find the with Sign(0) = formula ∂∆ ξ = Signsym ξ sym + Signanti ξ anti . Now define the nonsmooth curve P : [−1, 1] → GL(d) via Q(t)Ω(t) = e(σ−ω)t eωt for t ≤ 0, P (t) = for t ≥ 0, Ω(t) = eωt with ω = −ω T , σ = σ T and ω = σ = 1. For t < 0 we find ∗ ˙ ∂∆ DL−1 P P = ∂∆ AdΩ(t) σ = AdΩ(t) σ + Sign(0) anti , and for t > 0 we have ˙ ∂∆ DL−1 P P = ∂∆ ω = Sign(0) sym + ω .

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For the left-hand side in (5.10) we obtain, with η = σ + ω, for t ≤ 0 the relation Ad∗P (t) η = Ad∗Ω(t) Ad∗Q(t) (σ + ω) = Ad∗Ω(t) σ + ω = Ad∗Ω(t) σ + ω and for t ≥ 0 we have Ad∗P (t) η = Ad∗Ω(t) (σ + ω) = Ad∗Ω(t) σ + ω. Hence, we see that (5.10) is satisfied for t ∈ [−1, 1]\{0}.

6

Geodesics for von Mises Dissipation

In the case of von Mises plasticity one assumes isotropy of the dissipation functional on G = SL(d). Moreover, it is assumed to be generated by a scalar product, see (4.1).

1 ∆B ξ = Bξ , ξ 2 . The geodesic curves can be obtained in two ways. On the one hand we can use the theory of Riemannian metrics where it is easy to show that the geodesic curves are obtained also by minimizing the energy 1 2 −1 ˙ 1 dt . 2 ∆ DLP (t) P (t) 0

Then our smooth theory starting with (5.5) applies. For a more general Lie group approach we refer to Mittenhuber [2000]. On the other hand we can specialize the general nonsmooth theory to this situation. We have

∂∆B (0) = η ∈ sl(d) η , B−1 η ≤ 1 1 and ∂∆B ξ = for ξ = 0 . 1 Bξ

Bξ ,ξ 2

Thus, under the assumption P˙ = 0 the half-integrated equation (5.8) takes the form (6.1) P˙ = λ(t) DLP B−1 Ad∗P η , where η ∈ g ∗ is fixed and λ(t) is a positive scalar fixing the time rate. Without loss of generality we take λ ≡ 1 further notation for on. In matrix G = GL(d) or G = SL(d) we have P˙ = P B−1 P T ηP −T . In order to solve (6.1) we need additional information. In von Mises plasticity we can use the isotropy which in the material symmetry manifests group S = SO(d) through ∆B AdR ξ = ∆B ξ . For B this is equivalent to B−1 Ad∗R = AdR−1 B−1 for all R ∈ SO(d). In the case G = SL(d) this leads, according to (4.2), to B−1 η = γ12 η sym + γ13 η anti = 2γ12 η + η T + 2γ13 η−η T with γ2 , γ3 > 0.

(6.2) (6.3)


85

6.1 Theorem. If ∆B is isotropic (see (6.2) or (6.3)) then the unique solution of (6.1) with λ ≡ 1 and P (0) = I is given by (6.4) P (t) = etα etω , with α = γ12 η T and ω = γ12 + γ13 η anti . Proof. We make the ansatz P (t) = etα etω with ω = −ω T ∈ so(d). With Ad∗P 1 P 2 = Ad∗P 2 Ad∗P 1 , we obtain P˙ = etα (α + ω)etω and T T P B−1 Ad∗P η = P Ade−tω B−1 Ad∗etα η = etα B−1 etα ηe−tα etω . Clearly, we have found a solution if αT η = ηαT

and α + ω = B−1 η

holds. The second equation gives αsym =

1 γ2

η sym

and

αanti =

1 γ3

η anti − ω .

This reduces the first equation to ηω − ωη = γ12 + γ13 η sym η anti − η anti η sym

and we find the desired result.

6.2 Corollary. Under the above assumptions we have

1 and DB P = min B ξ , ξ 2 P = Mδ (ξ) DB RT P R = DB P ! sl(d) → SL(d); (6.5) with δ = γ3 /γ2 and Mδ ξ = ξ → eξsym −δξanti e(1+δ)ξanti . Here Mδ is a surjective map for δ > 0. In particular, the limit γ3 0 (plastic spin is dissipationless) leads to the formula 1 DB0 P = γ2 log P P T 2 F . Proof. For ξ ∈ sl(d) let P (t) = Mδ tξ , then (6.4) holds with α = ξ sym − δ ξ anti

and

ω = (1 + δ)ξ anti .

Moreover, we have P −1 P˙ = e−tω (α + ω)etω . Hence 2 2 1 I∆B P ( · ) = ∆B (α + ω) = γ2 ξ sym F + γ3 ξ anti F 2 . As we know by Theorem 5.1 there exists a shortest path. By the above this path has to have the form P (t) = Mδ tξ and thus (6.5) follows. The special case δ = 0 follows since Mδ ξ depends continuously on δ ∈ R with M0 ξ = eξsym eξanti . Thus, the equation P = eξsym eξanti has a unique solution for ξ sym from P P T = e2ξsym . (We use the unique decomposition P = V R with V = V T > 0 and R ∈ SO(d), also called Cartan decomposition.) Here ξ anti is unique up to solutions of eξ = I.

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In many crystal experiments the plastic spin appears to be very small. Hence, the case γ3 ∞ should be studied since γ3 γ2 makes plastic spin less favorable than plastic stretching. The above distance metric can be rewritten as Dγ2 ,γ3 P = 1 1 γ2 min γ2 σ2 + 2 ω2 2 σ = σ T , ω = −ω T , P = eσ−ω e(1+ δ )ω γ3 and we conjecture that the limit for γ3 → 0 is given by 1 Dno spin (P ) = inf γ22 σ σ = σ T , ω = −ω T , P = eσ−ω eω . In case the matrix P has no representation in the desired form the infimum is taken to be +∞. Finally we return to the Hamiltonian formulation for the geodesic curves. We recall the transformed Hamiltonian system (5.7) P˙ = DLP DH 0 (η) = DLP B−1 η , (6.6) T T η˙ = JLP (η) DH(η) = B−1 η η − η B−1 η

with H(η) = 12 B−1 η , η . Here JLP : g ∗ → Skew(g, g ∗ ), is the Lie–Poisson structure, which in our case reads as JLP η ξ = ξ T η − ηξ T , using matrix notation. Because of the isotropy (6.3) the second equation in (6.6) decouples as follows: η˙ sym = γ (η sym η anti − η anti η sym ) , η˙ anti = 0 ,

where γ =

1 γ2

+

1 γ3

.

(6.7)

This system can be solved explicitly by η(t) = η sym (t) + η anti (t) = e−tγω σetγω + ω , where ω = −ω T and σ = σ T . From this and the first equation in (6.6) we (t)etγω which leads to P ˙ = P 1 (σ − ω). find P (t) via the ansatz P (t) = P γ2 This justifies the ansatz in Theorem 6.1, and shows that there are no other possibilities.

A

Set-Valued Calculus of Variations

In this appendix we provide the two lemmas which were needed to prove Theorem 5.2. In the first lemma we give a formula for the subdifferential of a function which is defined as an integral over convex functions. This result and certain generalizations of it are well-known, so we omit the proof, see Barbu [1994]; Barbu and Precupanu [1986].


87

A.1 Lemma. Let Y be a finite-dimensional Banach space and Y ∗ its dual. Assume that the function f : [0, 1] × Y → [0, ∞) is Lebesgue–Borel measurable, that f (t, · ) is lower semi-continuous and convex, and that 1 f ( · , y) is integrable. Define the functional J(y) = 0 f (t, y) dt, then J : Y → [0, ∞) is convex and !

1

∂J(η) =

w(t) dt ∈ Y ∗ w ∈ L1 (0, 1) , Y ∗ ,

0

" w(t) ∈ ∂f (t, η) for a.e. t ∈ [0, 1] .

The second lemma generalizes the fundamental lemma in the calculus of variations which states that a function a : (0, 1) → R must be constant if 1 it satisfies 0 ρ(t)a(t) ˙ dt = 0 for all ρ ∈ W01,∞ (0, 1) . A.2 Lemma. For each t ∈ [0, 1] let C(t) ⊂ Rn be a convex, compact set such that the mapping (t, a) → XC(t) (a) from [0, 1] × Rn into [0, ∞] is measurable. Moreover, assume that for each ρ ∈ W01,∞ (0, 1) we have 0 ∈ Kρ := ! 1 " ρ(t)a(t) ˙ dt a : [0, 1] → Rn measurable, (t) ∈ C(t) a.e. ⊂ Rn . 0

Then there exists a vector b∗ ∈ Rn such that b∗ ∈ C(t) for a.e. t ∈ [0, 1]. Proof. We first prove the result for piecewise constant maps t → C(t) and ρ with piecewise constant derivative. Assume[0, 1] = ∪ni=1 Ai with Ai ∩ Aj = ∅ for i = j, Ai measurable and |Ai | = Ai dt > 0. Moreover, ˙ = λi /|Ai | for a.e. t ∈ Ai . Then, convexity of assume C(t) = Ci and ρ(t) ˙ dt ∈ λi Ci . Hence, we have to show that ∩ni=1 Ci = ∅ Ci implies Ai ρ(t)a(t) under the assumption 0∈

n

λi Ci =

i=1

n !

" λi ai ai ∈Ci

i=1

#n for all λ1 , . . . , λn ∈ R with i=1 λi = 0. We prove this by induction on n. For n = 2 we have 0 ∈ λ1 C1 +λ2 C2 with λ1 + λ2 = 0 if and only if 0 ∈ C1 − C2 = a1 − a2 aj ∈ Cj . But this is the same as C1 ∩ C2 = ∅. For the induction from n to n + 1 let Bn = ∩ni=1 Ci which is nonempty by the induction hypothesis. We need to show Bn ∩ Cn+1 = ∅. First note the simple identity Bn =

n ! 1

" µi Ci µ1 + . . . + µn = 1 .

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A. Mielke

From 0 ∈

#n+1

0∈

i=1

λi Ci for all λ1 , . . . , λn+1 with

n !

µi Ci − Cn+1

i=1

=

n !

#n+1 i=1

λi = 0 we conclude

n " µi = 1 1

n " µi Ci µi = 1 − Cn+1 = Bn − Cn+1 . 1

i=1

$n+1 But this exactly means Bn ∩Cn+1 = i=1 Ci = ∅.This proves the assertion for piecewise constant functions t → ρ(t), ˙ C(t) . The general case is obtained by discretization. For m ∈ N we define the subintervals for k = 1, . . . , 2m Ikm = (k−1)2−m , k2−m and the sets ! Ckm =

1

" for a.e. t ∈ [0, 1] . a(t) dt a(t) ∈ C (k−1 + t)2−m

0

By taking all ρ ∈ W01,∞ (0, 1) such that ρ˙ is constant on Ikm we have that #2m # 0 ∈ k=1 λk Ckm for k λk = 0. From the above we conclude m

B

m

=

2

Ckm = ∅ .

k=1 m+1 m+1 By construction we have Ckm = 12 C2k−1 + 12 C2k . Since B m+1 ⊂ Cim+1 for m+1 m ⊂ Cj for all j, and finally B m+1 ⊂ B m . Because all all i we find B m m is nonempty and satisfies B are compact, we know that B ∗ = ∩∞ m=1 B ∗ m ∗ m all m ∈ N and k = 1, . . . 2m . B ⊂ B for all m. This implies B ⊂ Ck for By the measurability of the family t → C(t) and the convexity of C(t) we m conclude that for almost all t the sets C[t2 m ] converge for m → ∞ to C(t). ∗ Hence ∅ = B ⊂ C(t) for a.e. t ∈ [0, 1].

References Abraham, R. and J. E. Marsden [1978], Foundations of Mechanics. Addison– Wesley, 2nd edition. Arnol’d, V. I. [1989], Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, 60, 2nd edition, Springer-Verlag, New York. Ball, J. M. [1977], Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Analysis 63, 337–403. Barbu, V. [1994], Mathematical Methods in Optimization of Differential Systems. Mathematics and its Applications, 310. Kluwer Academic Publ., Dordrecht.


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Barbu, V. and Th. Precupanu [1986], Convexity and Optimization in Banach Spaces. D. Reidel Publishing Co., Dordrecht. Carstensen, C., K. Hackl, and A. Mielke [2001], Nonconvex potentials and microstructures in finite-strain plasticity. Proc. Royal Soc. London, September 2001, (in press). Dacorogna, B. [1989], Direct Methods in the Calculus of Variations. SpringerVerlag. Ebin, D. G. and J. E. Marsden [1970], Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. of Math. 92, 102–163. Govindjee, S., A. Mielke, and G. J. Hall [2001], The free-energy of mixing for n-variant martensitic phase transformations using quasi–convex analysis. J. Mech. Physics Solids, (to appear). Gurtin, M. E. [2000], On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Physics Solids 48, 989–1036. Hackl, K. [1997], Generalized Standard Media and Variational Principles in Classical and Finite Strain Elastoplasticity. J. Mech. Phys. Solids 45 667–688. Han, W. and B. D. Reddy [1995], Computational plasticity: the variational basis and numerical analysis. Comp. Mech. Advances 2, 283–400. Holm, D. D., J. E. Marsden, T. Ratiu, and A. Weinstein [1985], Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, pp. 116. Krishnaprasad, P. S. and J. E. Marsden [1987], Hamiltonian structures and stability for rigid bodies with flexible attachments. Arch. Rat. Mech. Anal. 98, 71–93. Lee, E. H. [1969], Elastic-plastic deformation at finite strains J. Appl. Mech, 36, 1–6. Marsden, J. E. and T. J. R. Hughes [1983], Mathematical Foundations of Elasticity. Prentice Hall 1983. Marsden, J. E. and A. Weinstein [1983], Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D 7, 305–323. Miehe, C. [2000], Elements of computational inelasticity at finite strains, (in preparation). Miehe, C. and E. Stein [1992], A Canonical Model of Multiplicative ElastoPlasticity. Formulation and Aspects of the Numerical Implementation European J. Mechanics, A/Solids 11, 25–43. Miehe, C. and J. Schotte [2000], A two-scale micro-macro-aproach to anisotropic finite plasticity of polycrystals. In “Multifield problems, State of the Art; A.–M. S¨ andig, W. Schiehlen, W.L. Wendland (eds); Springer-Verlag 2000”, pp. 104– 111. Mielke, A. [1991], Hamiltonian and Lagrangian Flows on Center Manifolds, Lecture Notes in Mathematics Vol. 1489, Springer-Verlag, Berlin. Mielke, A. [2002], Energetic formulation of multiplicative elastoplasticity using dissipation distances. Preprint Univ. Stuttgart, February 2002. Mielke, A. and P. J. Holmes [1988], Spatially complex equilibria of buckled rods. Arch. Rat. Mech. Anal. 101, 319–348.

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Mielke, A. and F. Theil [1999], A mathematical model for rate-independent phase transformations with hysteresis. In “Models of Continuum Mechanics in Analysis and Engineering, (H.-D. Alber, R. Farwig eds.), Shaker Verlag, 1999”, 117–129. Mielke, A., F. Theil, and V. I. Levitas [2002], A variational formulation of rateindependent phase transformations using an extremum principle. Archive Rat. Mech. Analysis, (in print). Mittenhuber, D. [2000], Pseudo-Riemannian metrics on Lie groups. Preprint TU Darmstadt, Febr. 2000. Mittenhuber, D. [2000], Personal Communication by e-mail, August 2000. Monteiro Marques, M. D. P. [1993], Differential Inclusions in Nonsmooth Mechanics (Shocks and Dry Friction). Birkh¨ auser. Ortiz, M. and E. A. Repetto [1999], Nonconvex energy minimization and dislocation in ductile single crystals. Journal of the Mechanics and Physics of Solids. 47, 397–462. Sim´ o, J. C. [1988], A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part I: Continuum formulation. Comp. Meth. Appl. Mech. Engrg. 66, 199–219. Sim´ o, J. C., and M. Ortiz [1985], A Unified Approach to Finite Deformation Elastoplastic Analysis Based on the Use of Hyperelastic Constitutive Equations, Comput. Meths. Appl. Mech. Engrg. 49, 221–245. Ziegler, H. and C. Wehrli [1987], The Derivation of Constitutive Relations from the Free Energy and the Dissipation Function. In Advances in Applied Mechanics 25, (T. Y. Wu & J. W. Hutchinson, eds.) Academic Press, New York.

3 Asynchronous Variational Integrators A. Lew and M. Ortiz To Jerry Marsden on the occasion of his 60th birthday ABSTRACT We describe a class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are characterized by the following distinguishing attributes: i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; ii) the algorithms derive from a spacetime form of a discrete version of Hamilton’s principle. As a consequence of this variational structure, the algorithms conserve local energy and momenta exactly, subject to solvability of the local time steps. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . The Discrete Problem . . . . . . . . . . . . . . 2.1 Spatial Discretization . . . . . . . . . . . . . 2.2 Time Discretization . . . . . . . . . . . . . . 2.3 Discrete Variational Principle . . . . . . . . . 2.4 Conservation Properties of AVIs . . . . . . . 2.5 Time-Adaption and Space-Time Formulation 3 Numerical Examples . . . . . . . . . . . . . . 3.1 Two-Dimensional Neo-Hookean block . . . . 3.2 Three-Dimensional L-Shaped Beam . . . . . 4 Summary . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

1

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

91 93 95 95 97 98 99 101 102 107 109 109

Introduction

Dynamical systems that exhibit well-separated multiple time scales arise in many areas of application, including molecular dynamics, microstructural 91

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A. Lew and M. Ortiz

evolution in materials, and structural mechanics. In a numerical treatment of these problems, it is natural to consider time stepping algorithms with multiple time steps chosen to resolve the various time scales in the problem. In the engineering finite-element literature, the so-called subcycling algorithms (Neal and Belytschko [1989], Belytschko and Mullen [1976]) fit that description. These algorithms were developed mainly so that stiff elements, or regions of the model, could advance at smaller time steps than more compliant elements. In its original version, the method grouped the nodes of the mesh and assigned a different time step to each group. Adjacent groups of nodes were constrained to have integer time step ratios (see Belytschko and Mullen [1976]), a condition that was relaxed subsequently (Neal and Belytschko [1989]; Belytschko [1981]). Recently, an implicit multi-time step integration method was developed and analyzed by Smolinski and Wu [1998]. Since the original formulation of subcycling algorithms, a sea change has occurred in the understanding of discrete dynamics, most notably since the development of the theory of variational integrators (Kane, Marsden, and Ortiz [1999]; Kane, Marsden, Ortiz, and West [2000]; Marsden, Patrick, and Shkoller [1998] and Marsden and West [2001]). Widely used algorithms, such as some versions of Newmark’s algorithm, can be recast into the discrete mechanics framework as shown by Kane, Marsden, Ortiz, and West [2000]. Even without deliberately adjusting the time step to achieve exact conservation these algorithms possess remarkable energy-conservation properties that probably originate from their symplectic and variational nature. One of the attractive features of variational methods is that if a problem has symmetries and, as a consequence, corresponding conserved quantities exist, then these quantities are automatically conserved by the algorithm. These recent theoretical developments provide a sound basis for formulating powerful multiresolution methods that offer considerable freedom of local time step selection while possessing exact local energy and momenta conservation properties. In this article, we describe one such class of integrators, termed asynchronous variational integrators (AVI), characterized by the following distinguishing attributes: 1. When applied to dynamical systems defined by the finite element method, AVIs permit the selection of different time steps for each element. The local time steps need not bear an integral relation to each other, and the integration of the elements may, therefore, be carried out asynchronously. 2. The time-integration algorithm is given by a spacetime form of a discrete version of Hamilton’s principle. As a consequence of this variational structure, the algorithm conserves local energy and momenta exactly, subject to solvability of the local time steps.

3. Asynchronous Variational Integrators

93

The use of a variable time step sidesteps the restrictions imposed by wellknown theorems (Ge and Marsden [1988]), which rule out the possibility that constant time stepping algorithms be symplectic, and preserve energy and momentum. A symplectic-energy-momentum time integrator for finitedimensional dynamical systems was developed in Kane, Marsden, and Ortiz [1999], where the time step of the complete system was computed in order to preserve the total energy. Conditions for the solvability of the time step were also provided by Kane, Marsden, and Ortiz [1999]. The integrators described in this article are symplectic-energy-momentum preserving. The variational structure of the algorithms furnishes a local energy balance equation which may be satisfied by allowing for different and variable time steps in each element. The resulting algorithms satisfy local, as opposed to merely global, energy and momentum balance. Numerical examples in two and three dimensions demonstrate the remarkable versatility and conservation properties of AVIs. The present article is extracted from a much more extended version (Lew, West, Marsden and Ortiz [2001]), which may be consulted for further details of the theory and algorithmic implementation.

2

The Discrete Problem

In describing the dynamic response of elastic bodies under loading, we select a reference configuration B ⊂ R3 of the body at time t0 . The coordinates of points X ∈ B are used to identify material particles throughout the motion. The motion of the body is described by the deformation mapping X ∈ B.

x = ϕ(X, t) ,

(2.1)

Thus, x is the location of material particle X at time t. The material velocity and acceleration fields follow from (2.1) as ϕ(X, ˙ t) and ϕ(X, ¨ t), X ∈ B, respectively, where a superposed dot denotes partial differentiation with respect to time at fixed material point X. The deformation mapping is subject to essential boundary conditions on the displacement boundary ∂d B ⊂ ∂B. For definiteness, the potential energy of the body is assumed to be of the form W (D1 ϕ, X) dV − ρB · ϕ dV − T · ϕ dS , (2.2) V ϕ( · , t), t = B

B

∂τ B

where W is the strain energy density per unit volume, B(X, t) is the body force per unit mass, T(X, t) is the prescribed traction applied on the traction boundary ∂τ B = ∂B \ ∂d B, ρ is the mass density in B, and Di denotes the partial derivative of a function with respect to its i-th argument. In

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addition, the kinetic energy of the body is assumed to be of the form ρ 2 |ϕ| ˙ dV . (2.3) T ϕ( ˙ · , t) = B 2 The corresponding Lagrangian of the body is L ϕ( · , t), ϕ( ˙ · , t), t = T [ϕ] ˙ − V [ϕ, t] .

(2.4)

Consider now a motion of the body during the time interval [t0 , tf ]. The action attendant to the motion is tf S[ϕ( · , · )] = L(ϕ, ϕ, ˙ t)dt . (2.5) t0

We note that, upon insertion of (2.4) in (2.5), the evaluation of the action functional entails a spacetime integral. Within the framework just outlined, Hamilton’s principle postulates that the motion ϕ(X, t) of the body which joins prescribed initial and final conditions renders the action functional S stationary with respect to all admissible variations, i. e., variations of ϕ(X, t) vanishing at t0 and tf and satisfying the essential boundary conditions on ∂d B. A standard calculation shows that the Euler–Lagrange equations corresponding to Hamilton’s principle are d ˙ t) − D2 L(ϕ, ϕ, ˙ t) = 0 (2.6) D1 L(ϕ, ϕ, dt for all t ∈ [t0 , tf ]. We recall that, in Lagrangian mechanics, energy conservation follows as a consequence of the invariance of the action under time-translation. In order to establish this connection, introduce the shifted action t−α L(ϕ( · , ξ + α), ϕ( ˙ · , ξ + α), ξ) dξ (2.7a) Sα [ϕ] = t0 −α t

L(ϕ( · , ξ), ϕ( ˙ · , ξ), ξ − α) dξ ,

=

(2.7b)

t0

where t ∈ [t0 , tf ]. Thus, Sα is the value of the action for a motion identical in all respects to ϕ but occurring a time α earlier. Differentiating equations (2.7a) and (2.7b) with respect to α and evaluating the result along trajectories gives t t ∂Sα = H =− D3 L(ϕ, ϕ, ˙ ξ)dξ , (2.8) ∂α α=0 t0 t0 where H = D2 L(ϕ, ϕ, ˙ t) · ϕ˙ − L(ϕ, ϕ, ˙ t)

(2.9)

is the total energy of the body. In particular, for an autonomous Lagrangian D3 L = 0, and (2.8) simply states that the total energy of the body is conserved along trajectories.


2.1

95

Spatial Discretization

Next we consider a finite-element triangulation T of B. The corresponding finite-dimensional space of finite-element solutions consists of deformation mappings of the form xa Na (X) , (2.10) ϕh (X) = a∈T

where Na is the shape function corresponding to node a, xa represents the position of the node in the deformed configuration. A key observation underlying the formulation of AVIs is that, owing to the extensive character of the Lagrangian (2.4), the following element-by-element additive decomposition holds: LK , (2.11) L= K∈T

where LK is the contribution of element K ∈ T to the total Lagrangian, which follows by restricting (2.4) to K. Each elemental or local Lagrangian LK can in turn be written as a function of the nodal positions and velocities of the element, i. e., LK ϕh ( · , t), ϕ˙ h ( · , t), t ≡ LK xK (t), x˙ K (t), t , (2.12) where xK is the vector of positions of all the nodes in element K. In particular, for the Lagrangian (2.4) the local Lagrangians have the form LK xK , x˙ K , t) = TK (x˙ K ) − VK (xK , t , (2.13) where VK (xK , t) is the elemental potential energy, and TK (x˙ K ) =

1 2

˙K x˙ T K MK x

(2.14)

is the elemental kinetic energy. Here MK is the element mass matrix, which is constant by conservation of mass and will be assumed to be expressible in diagonal or lumped form.

2.2

Time Discretization

A key feature of the AVIs is that the elements and nodes defining the triangulation of the body are updated asynchronously in time. To this end, we endow each element K ∈ T with a discrete time set K −1 K (2.15) ΘK = t0 = t1K < . . . < tN < tN K K j j j K −1 K < tf ≤ t N with tN K . In addition, we write xK ≡ xK (tK ), tK ∈ ΘK , for K the discrete element coordinates, and % ΘK (2.16) Θ= K∈T

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ϕh t

t

tf

tf

ti

ti X

x

Figure 2.1. Spacetime diagram of the motion of a three-element, one-dimensional mesh. The reference configuration is shown on the left, while the deformed configuration is on the right. The trajectories of the nodes are depicted as dashed lines in both configurations. The horizontal segments above each element K define the set ΘK .

for the entire time set. We shall also need to keep proper time at all nodes in the mesh. To this end, we let % a (2.17) Θa = ΘK = t0 = t1a ≤ . . . ≤ taNa −1 ≤ tN a {K∈T |a∈K}

denote the ordered nodal time set for node a. In these definitions, the sym & bol denotes disjoint union. For simplicity, we assume that tjK = tjK for any pair of elements K and K . The case of time coincidences between elements can be treated simply by taking the appropriate limits. We additionally write xia = xa (tia ), tia ∈ Θa , for the discrete nodal coordinates, and let (2.18) X = xia , a ∈ T , i = 1, . . . Na denote the set of nodal coordinates defining the discrete trajectory. The particular class of AVIs under consideration here is obtained by allowing each node a ∈ T to follow a linear trajectory within each time interval [tia , ti+1 a ]. The corresponding nodal velocities are piecewise constant in time. The nodal trajectories thus constructed are defined in the time a intervals [t0 , tN a ]. An x − t diagram of the motion of a three-element onedimensional mesh is shown in Fig. 2.1 by way of illustration. Higher-order AVI methods can also be devised by considering piecewise polynomial nodal trajectories.


97

We note that the pair of sets (Θ, X) completely defines the trajectories of the discrete system. A class of discrete dynamical systems is obtained by considering discrete action sums of the form (see, e. g., Marsden and West [2001] for a recent review on discrete dynamics and variational integrators) LjK , (2.19) Sd (X, Θ) = K∈T 1≤j

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