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Nonsmooth Mechanics and Analysis NONSMOOTH MECHANICS AND ANALYSIS Theoretical and Numerical Advances Advances in Mechanics and Mathematics VOLUME 12 Series Editors: David Y. Gao Virginia Polytechnic Institute and State University, U.S.A. Ray W. Ogden University of Glasgow, U.K. Advisory Editors: I. Ekeland University of British Columbia, Canada S. Liao Shanghai Jiao Tung University, P.R. China K.R. Rajagopal Texas A&M University, U.S.A. T. Ratiu Ecole Polytechnique, Switzerland W. Yang Tsinghua University, P.R. China NONSMOOTH MECHANICS AND ANALYSIS Theoretical and Numerical Advances Edited by P. ALART Université Montpellier II, Montpellier, France O. MAISONNEUVE Université Montpellier II, Montpellier, France R.T. ROCKAFELLAR University of Washington, Seattle, Washington, U.S.A. 1 3 Library of Congress Control Number: 2005932761 ISBN-10: 0-387-29196-2 e-ISBN 0-387-29195-4 ISBN-13: 978-0387-29196-3 Printed on acid-free paper. ¤ 2006 Springer Science+Business Media, 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 Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, 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 springeronline.com Contents Preface ix A short biography of Jean Jacques Moreau xiii Part I Convex and nonsmooth analysis 1 Moreau’s proximal mappings and convexity in Hamilton-Jacobi theory 3 R. T. Rockafellar 2 Three optimization problems in mass transportation theory 13 G. Buttazzo 3 Geometrical and algebraic properties of various convex hulls 25 B. Dacorogna 4 Legendre-Fenchel transform of convex composite functions 35 J.-B. Hiriart-Urruty 5 What is to be a mean? 47 M. Valadier Part II Nonsmooth mechanics 6 Thermoelastic contact with frictional heating 61 L.-E. Andersson, A. Klarbring, J.R. Barber and M. Ciavarella 7 A condition for statical admissibility in unilateral structural analysis 71 G. Del Piero 8 Min-max Duality and Shakedown Theorems in Plasticity 81 vi NONSMOOTH MECHANICS AND ANALYSIS Q. S. Nguyen 9 Friction and adhesion 93 M. Raous 10 The Clausius-Duhem inequality, an interesting and productive inequality 107 M. Fr´emond 11 Unilateral crack identification 119 G. E. Stavroulakis, M. Engelhardt and H. Antes 12 Penalty approximation of Painlev´eproblem 129 M. Schatzman 13 Discrete contact problems with friction 145 F. Maceri and P. Bisegna Part III Fluid mechanics 14 A brief history of drop formation 163 J. Eggers 15 Semiclassical approach of the “tetrad model” of turbulence 173 A. Naso and A. Pumir Part IV Multibody dynamics: numerical aspects 16 The geometry of Newton’s cradle 185 C. Glocker and U. Aeberhard 17 Nonsmooth analysis for contact mechanics simulations 195 P. Alart, D. Dureisseix and M. Renouf 18 Numerical simulation of a multibody gas 209 M. Jean 19 Granular media and ballasted railway tracks 221 Contents vii C.Cholet,G.Saussine,P.-E.GautierandL.-M.Cl´eon 20 Scaling behaviour of velocity fluctuations in slow granular flows 233 F. Radja¨ı and S. Roux Part V Topics in nonsmooth science 21 Morphological equations and sweeping processes 249 J.-P. Aubin and J. A. Murillo Hern´andez 22 Higher order Moreau’s sweeping process 261 V. Acary and B. Brogliato 23 An existence result in non-smooth dynamics 279 M.D.P. Monteiro Marques and L. Paoli 24 Finite time stabilization of nonlinear oscillators subject to dry friction 289 S. Adly, H. Attouch and A. Cabot 25 Canonical duality in nonsmooth, constrained concave minimization 305 D. Y. Gao Appendix: Theoretical and Numerical Nonsmooth Mechanics International Colloquium 315 Index 319 Preface In the course of the last fifty years, developments in nonsmooth analy- sis and nonsmooth mechanics have often been closely linked. The present book acts as an illustration of this. Its objective is two-fold. It is of course intended to help to diffuse the recent results obtained by various renowned specialists. But there is an equal desire to pay homage to Jean Jacques Moreau, who is undoubtedly the most emblematic figure in the correlated, not to say dual, advances in these two fields. Jean Jacques Moreau appears as a rightful heir to the founders of differential calculus and mechanics through the depth of his thinking in the field of nonsmooth mechanics and the size of his contribution to the development of nonsmooth analysis. His interest in mechanics has focused on a wide variety of subjects: singularities in fluid flows, the initiation of cavitation, plasticity, and the statics and dynamics of gran- ular media. The ‘Ariadne’s thread’ running throughout is the notion of unilateral constraint. Allied to this is his investment in mathematics in the fields of convex analysis, calculus of variations and differential mea- sures. When considering these contributions, regardless of their nature, one cannot fail to be struck by their clarity, discerning originality and elegance. Precision and rigor of thinking, clarity and elegance of style are the distinctive features of his work. In 2003, Jean Jacques Moreau’s colleagues decided to celebrate his 80th birthday by organizing a symposium in Montpellier, the city in which he has spent the major part of his university life and where he is still very active scientifically in the Laboratoire de M´ecanique et de G´enie Civil. In the appendix, the reader will find a list of those who par- ticipated in the symposium, entitled “Theoretical and Numerical Nons- mooth Mechanics”. This book is one of the outcomes of the symposium. It does not represent the acts, stricto sensu, although it does reflect its structure in five parts. Each of these corresponds to a field of mathemat- ics or mechanics in which Jean Jacques Moreau has made remarkable contributions. The works that follow are those of eminent specialists, x NONSMOOTH MECHANICS AND ANALYSIS united by the fact that they have all, at some time, appreciated and utilized these contributions. The diversity of the topics presented in the following pages may come as a surprise. In a way this reflects the open and mobile mind of the man they pay tribute to; it should allow the reader to appreciate, at least in part, the importance of the scientific wake produced. Specialists in one or a number of the subjects dealt with will enjoy discovering new results in their own fields. But they will also discover subjects outside their usual domain, linked through certain aspects and likely to open up new horizons. Not surprisingly, the first part is devoted to Con- vex and Nonsmooth Analysis, two modern and powerful approaches to the study of theories like optimization, calculus of variations, optimal control, differential inclusions, and Hamilton-Jacobi equations, in par- ticular. Some new aspects are presented regarding the evolution of these theories. When a convex function is propagated by a Hamilton-Jacobi partial differential equation involving a concave-convex hamiltonian, the associated proximal mapping is shown to locally exhibit Lipschitz depen- dence on time. Models of transportation in a city are presented, which in connection with Monge-Kantorovich, embrace optimal transportation networks, pricing policies and the design of the city. The following chap- ter deals with various types of convex hulls involving a differential inclu- sion with given boundary value. Next, we find proof of a formula giving the Legendre-Fenchel transformation of a convex composite function in terms of the transformation of its components. Last is a synthetic survey of balayage involving bistochastic operators, using inequalities that go back to Hardy, Littlewood and P´olya. Some very different applications of nonsmooth analysis in mechanics and thermomechanics are tackled in the second part. The existence and uniqueness of steady state solutions are investigated for thermoelastic contact problems, where a variable contact heat flow resistance is con- sidered as well as frictional heating. A necessary and sufficient condition is obtained, with a direct mechanical interpretation, for statical admissi- bility of loads in unilateral structural analysis. In the case of plasticity, an overall presentation of shakedown theorems in the framework of gen- eralized standard models is derived by min-max duality and some new results are proposed for the kinematic safety coefficient. This is followed by a review of recent studies and applications carried out on the different forms of a model coupling adhesion and friction, based on introducing the adhesion intensity variable. From a general point of view, through various examples in thermomechanics, it is shown how Clausius-Duhem inequality may be productive, by identifying quantities that are related as constitutive laws and by suggesting useful experiments. The reader PREFACE xi will also find a thorough investigation, in two dimensions, of the unilat- eral crack identification problem in elastodynamics. This problem may be quite complicated, nonsmooth and nonconvex. The last two papers in this part concern, firstly, a penalty approximation of the Painlev´eprob- lem with convergence to a solution and, secondly, a block relaxation solutions algorithm, in the case of a stress-based static formulation of a discrete contact problem with friction. Drop formation and turbulence are among the most fascinating topics in fluid mechanics. In the third part a new approach of coarse-grained approximation of velocity during turbulence follows a history of drop formation focusing on the periodic rediscovery of non linear features.
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