Nonlinear homogenization and its applications to composites, polycrystals and smart materials

Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials NATO Science Series A S...

Author: P. Ponte Castañeda | Józef Joachim Telega | Barbara Gambin | North Atlantic Treaty Organization. Scientific Affairs Division

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Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials

NATO Science Series A Series presenting the results of scientific meetings supported under the NATO Science Programme. The Series is published by IOS Press, Amsterdam, and Kluwer Academic Publishers in conjunction with the NATO Scientific Affairs Division Sub-Series I. II. III. IV. V.

Life and Behavioural Sciences Mathematics, Physics and Chemistry Computer and Systems Science Earth and Environmental Sciences Science and Technology Policy

IOS Press Kluwer Academic Publishers IOS Press Kluwer Academic Publishers IOS Press

The NATO Science Series continues the series of books published formerly as the NATO ASI Series. The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, although other types of meeting are supported from time to time. The NATO Science Series collects together the results of these meetings. The meetings are co-organized bij scientists from NATO countries and scientists from NATO’s Partner countries – countries of the CIS and Central and Eastern Europe. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series has been re-organised and there are currently Five Sub-series as noted above. Please consult the following web sites for information on previous volumes published in the Series, as well as details of earlier Sub-series. http://www.nato.int/science http://www.wkap.nl http://www.iospress.nl http://www.wtv-books.de/nato-pco.htm

Series II: Mathematics, Physics and Chemistry – Vol. 170

Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials edited by

P. Ponte Castañeda Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, U.S.A.

J.J. Telega Polish Academy of Sciences, Institute of Fundamental Technological Research, Warsaw, Poland and

B. Gambin Polish Academy of Sciences, Institute of Fundamental Technological Research, Warsaw, Poland

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

1-4020-2623-4 1-4020-2621-8

©2005 Springer Science + Business Media, Inc. Print ©2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at:

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Table of Contents

Foreword

xi

List of Participants

xiii

Titles of Contributed Papers

xix

Invited Lectures TOPOLOGY OPTIMIZATION WITH THE HOMOGENIZATION AND THE LEVEL-SET METHODS G. Allaire 1 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Setting of the problem. . . . . . . . . . . . . . . . . . . . . . 2 3 Homogenized formulation . . . . . . . . . . . . . . . . . . . 3 4 Numerical algorithm for the homogenization method. . . . . 5 5 Shape derivative . . . . . . . . . . . . . . . . . . . . . . . . 7 6 Numerical algorithm for the level-set method . . . . . . . . 7 7 Comparisons and conclusions . . . . . . . . . . . . . . . . . 12 THIN FILMS OF ACTIVE MATERIALS K. Bhattacharya 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Bulk materials . . . . . . . . . . . . . . . . . . . . . . . . . 3 Homogeneous thin ﬁlms . . . . . . . . . . . . . . . . . . . . 3.1 A theory of thin ﬁlms . . . . . . . . . . . . . . . . . 3.2 Microstructure in martensitic thin ﬁlms . . . . . . . 3.3 Tents and tunnels . . . . . . . . . . . . . . . . . . . 4 Heterogeneous thin ﬁlms . . . . . . . . . . . . . . . . . . . . 4.1 The general problem . . . . . . . . . . . . . . . . . . 4.2 Conductivity in a strip . . . . . . . . . . . . . . . . . 4.3 A model problem for shape-memory ﬁlms . . . . . . 4.4 Recoverable strains in polycrystalline shape-memory ﬁlms . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 SMA/Elastic Multilayers . . . . . . . . . . . . . . . v

15 15 18 20 20 24 26 28 28 29 32 33 35

vi

TABLE OF CONTENTS

5

4.6 Heterogeneous thin ﬁlms . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

THE PASSAGE FROM DISCRETE TO CONTINUOUS VARIATIONAL PROBLEMS: A NONLINEAR HOMOGENIZATION PROCESS A. Braides and M.S. Gelli 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Some convergence results. Continuum energies on Sobolev spaces . . . . . . . . . . . . . . . . . . . . . 1.2 Continuous energies on functions of bounded variation. Some new results contained in this paper . . . . . . 2 Notation and preliminaries . . . . . . . . . . . . . . . . . . 2.1 Functions of bounded variation . . . . . . . . . . . . 2.2 Γ-convergence . . . . . . . . . . . . . . . . . . . . . . 3 Statement of the main result . . . . . . . . . . . . . . . . . APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS OF MECHANICS A. Cherkaev 1 Variational problems with locally unstable solutions . . . . 1.1 Nonconvex variational problems . . . . . . . . . . . 1.2 Unstable solutions . . . . . . . . . . . . . . . . . . . 2 Constrained minimizing sequences and control problems . . 2.1 The LEGO of laminates . . . . . . . . . . . . . . . . 2.2 Lamination closure . . . . . . . . . . . . . . . . . . . 2.3 Diﬀerential schemes and control problem . . . . . . 3 Variations and analysis of ﬁelds . . . . . . . . . . . . . . . . 3.1 Structural variation . . . . . . . . . . . . . . . . . . 3.2 Analysis of optimality conditions . . . . . . . . . . . 3.3 Minimal extension . . . . . . . . . . . . . . . . . . . 4 Bounds and duality . . . . . . . . . . . . . . . . . . . . . . . 4.1 Variational problems and bounds for eﬀective properties 4.2 Translation method and developments . . . . . . . . 4.3 Duality and bounds for expansion coeﬃcients . . . . 4.4 Duality and bounds for viscoelastic materials . . . . 4.5 Duality and structural optimization . . . . . . . . .

37 41

45 45 46 49 51 52 53 53

65 65 65 70 75 75 79 79 81 81 83 87 89 89 91 94 97 99

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS T. Iwaniec, F. Giannetti, G. Moscariello, C. Sbordone and L. Kovalev 107 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2 Second order nondivergence equations . . . . . . . . . . . . 111

vii

TABLE OF CONTENTS

3 4 5 6 7 8 9 10 11 12 13

Second order divergence equations . . . . The general Beltrami equations . . . . . . Quasiconformal mappings . . . . . . . . . G-convergence of the operators ∂z − µj ∂z Divergence equations revisited . . . . . . . G-limits versus weak-star topology . . . . A jump from ∂z − ν∂z to ∂z − µ∂z . . . . A pair of primary solutions . . . . . . . . Independence of Φz (z) and Ψz (z) . . . . Proof of Theorem 2 . . . . . . . . . . . . . Nondivergence equations revisited . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

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. . . . . . . . . . .

HOMOGENIZATION AND OPTIMAL DESIGN IN STRUCTURAL MECHANICS T. Lewi´ nski 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The layout problem in 2D elasticity. Minimum compliance design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The layout problem in the thin plate theory. Minimum compliance design . . . . . . . . . . . . . . . . . . . . . . . 4 Plates subjected to simultaneous in-plane and transverse loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Thin shells in bending . . . . . . . . . . . . . . . . . . . . . 6 Membrane shells . . . . . . . . . . . . . . . . . . . . . . . . 7 Shape design of membrane shells . . . . . . . . . . . . . . . 8 Michell surface structures . . . . . . . . . . . . . . . . . . . 9 3D bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 115 117 118 119 122 125 126 130 132 136

139 139 141 149 154 159 161 162 163 164

HOMOGENIZATION AND DESIGN OF FUNCTIONALLY GRADED COMPOSITES FOR STIFFNESS AND STRENGTH R. Lipton 169 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 169 2 Homogenization and design of structural properties . . . . 169 2.1 Relaxed optimal structural design . . . . . . . . . . 170 2.2 Topology optimization . . . . . . . . . . . . . . . . . 172 2.3 Design of functionally graded materials . . . . . . . 173 3 Homogenization and design of properties related to strength 176 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . 177 3.2 Design of functionally graded materials for stiﬀness and strength . . . . . . . . . . . . . . . . . . . . . . 178

viii

TABLE OF CONTENTS

4

5

6

Design of functionally graded composites subject to integral constraints on the equivalent stress . . . . . . . . . . . . . . 4.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . 4.2 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . Design of stiﬀ composite structures subject to point wise stress constraints . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Proofs of Theorems 5.1 and 5.2 . . . . . . . . . . . . Numerical example . . . . . . . . . . . . . . . . . . . . . . .

180 182 183 184 185 187

HOMOGENIZATION FOR NONLINEAR COMPOSITES IN THE LIGHT OF NUMERICAL SIMULATIONS H. Moulinec and P. Suquet 193 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 1.1 Individual constituents and local problem . . . . . . 193 1.2 Eﬀective potentials . . . . . . . . . . . . . . . . . . . 194 1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . 195 1.4 Secant moduli . . . . . . . . . . . . . . . . . . . . . 196 2 Interpretation of the variational bounding technique of Ponte Casta˜ neda as a secant method . . . . . . . . . . . . . . . . . 197 2.1 The variational bound . . . . . . . . . . . . . . . . . 197 2.2 Proof of the variational bound . . . . . . . . . . . . 198 2.3 Interpretation of the variational problem associated with w ˜ + . . . . . . . . . . . . . . . . . . . . . . . . . 199 2.4 The two approximations involved in the modiﬁed secant method . . . . . . . . . . . . . . . . . . . . . 199 3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . 200 3.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . 200 3.2 Conﬁgurations and ensemble averages . . . . . . . . 201 3.3 Eﬀective ﬂow stress . . . . . . . . . . . . . . . . . . 204 3.4 Strain and stress averages over individual phases. . . 208 3.5 Fluctuations of the strain and stress ﬁelds . . . . . . 212 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 EXISTENCE AND HOMOGENIZATION FOR THE PROBLEM - div a(x, Du) = f WHEN a(x, ξ) IS A MAXIMAL MONOTONE GRAPH IN ξ FOR EVERY x F. Murat 225 OPTIMAL DESIGN IN 2-D CONDUCTIVITY FOR QUADRATIC FUNCTIONALS IN THE FIELD P. Pedregal 229 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 2 Variational reformulation . . . . . . . . . . . . . . . . . . . 233

ix

TABLE OF CONTENTS

3 4

constrained quasiconvexiﬁcation . . . . . . . . . . . . . . . . 235 Analysis of the relaxed problem . . . . . . . . . . . . . . . . 240

LINEAR COMPARISON METHODS FOR NONLINEAR COMPOSITES P. Ponte Casta˜ neda 247 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 2 Eﬀective behavior . . . . . . . . . . . . . . . . . . . . . . . . 249 3 Eﬀective estimates via linear comparison composites . . . . 251 3.1 Secant bounds . . . . . . . . . . . . . . . . . . . . . 253 3.2 Tangent second-order estimates . . . . . . . . . . . . 257 3.3 Generalized secant second-order estimates . . . . . . 259 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 264 MODELS OF MICROSTRUCTURE EVOLUTION IN SHAPE MEMORY ALLOYS T. Roub´ıˇ cek 1 Introduction, crystallic alloys, shape-memory eﬀect. . 2 Atomistic-level modelling. . . . . . . . . . . . . . . . . 3 Continuum-mechanical level: microscopic models . . . 3.1 Free energy . . . . . . . . . . . . . . . . . . . . 3.2 Dissipation energy . . . . . . . . . . . . . . . . 3.3 Dynamics of isothermal phase transformation 3.4 Thermodynamical evolution . . . . . . . . . . . 4 Mesoscopic-level models . . . . . . . . . . . . . . . . . 5 Polycrystalline models . . . . . . . . . . . . . . . . . . 6 Macroscopic models . . . . . . . . . . . . . . . . . . .

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STOCHASTIC HOMOGENIZATION: CONVEXITY AND NONCONVEXITY J.J. Telega 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical preliminaries . . . . . . . . . . . . . . . . . . 2.1 Description of random media . . . . . . . . . . . . . 2.2 Elements of stochastic calculus . . . . . . . . . . . . 3 Stochastic homogenization based on G- and H- convergence 4 Γ- convergence and stochastic homogenization . . . . . . . 4.1 Convex case . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nonconvex case . . . . . . . . . . . . . . . . . . . . . 5 Two-scale and multi-scale stochastic convergence in the mean . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Basic deﬁnition and properties . . . . . . . . . . . .

269 269 275 276 276 279 281 286 288 292 293

305 305 306 306 311 314 315 315 327 329 329

x

TABLE OF CONTENTS

5.2

6

7

Stationary diﬀusion in random porous medium made of nonhomogeneous material . . . . . . . . . . . . . 5.3 Homogenization of nonlinear elliptic problem . . . . On homogenization of stochastic diﬀerential equations . . . 6.1 Some mathematical preliminaries . . . . . . . . . . . 6.2 A nonlinear random parabolic equation and its homogenization . . . . . . . . . . . . . . . . . . . . . Final remarks and some open problems . . . . . . . . . . .

Index

331 336 338 338 339 344 349

FOREWORD After more than 30 years, the term ”homogenization” is now commonly used in several ﬁelds of science and engineering. From the mathematical point of view, homogenization deals with sequences of functionals or operators, not necessarily linear, depending on a small parameter. The meaning of this parameter depends upon the problem considered. For instance, it may characterize the microstructure, as in the size of a typical ﬁber in a metal-matrix composite, or of a grain in an ice polycrystal. To perform a homogenization process means ﬁnding the limit problem when the small parameter tends to zero (in a proper sense). However, the general theory of functionals and operators with dependence on a small parameter tending to zero has a much wider range of applicability. Consider, for example the development of two-dimensional models for thin plates and shells from their three-dimensional counterparts. A combination of several small parameters, as in reiterated homogenization, and for thin structures with microstructures is also important. Although several books and conference proceedings have already appeared dealing with either the mathematical aspects, or applications of homogenization theory, there seems to be no comprehensive volume dealing with both aspects. The present volume is meant to ﬁll this gap, at least partially, and deals with recent developments in nonlinear homogenization emphasizing applications of current interest. It contains thirteen key lectures presented at the NATO Advanced Workshop on Nonlinear Homogenization and Its Applications to Composites, Polycrystals and Smart Materials. The list of thirty one contributed papers is also appended. The key lectures cover both fundamental, mathematical aspects of homogenization, including nonconvex and stochastic problems, as well as several applications in micromechanics, thin ﬁlms, smart materials, and structural and topology optimization. One lecture deals with a topic important for nanomaterials: the passage from discrete to continuum problems by using nonlinear homogenization methods. Some papers reveal the role of parameterized or Young measures in description of microstructures and in optimal design. Other papers deal with recently developed methodsboth analytical and computational-for estimating the eﬀective behavior and ﬁeld ﬂuctuations in composites and polycrystals with nonlinear constitutive behavior. All in all, the volume oﬀers a cross-section of current activity in nonlinear homogenization including a broad range of physical and engineering applications. The careful reader will be able to identify challenging open problems in this still evolving ﬁeld. For instance, there is the need to improve bounding techniques for nonconvex problems, as well as for solving xi

xii geometrically nonlinear optimum shape-design problems, using relaxation and homogenization methods. First of all, we would like to thank the key speakers and other participants for making the workshop possible. Second, we would like to thank the NATO Science Programme (Dr. F. Pedrazzini, program manager) for providing the essential funding and organizational support for the meeting. Third, we are grateful to the Institute of Fundamental Technological Research of the Polish Academy of Sciences (Prof. W. Nowacki) for serving as kind host for the workshop. Finally, we would like to acknowledge the support of the Applied Mathematics Program (Dr. H. G. Kaper, program manager) of the U.S. National Science Foundation for a Group Travel Award DMS03-05443, which funded the participation of 8 young researchers from several institutions in the US. Our special thanks go to Dr A. Galka (Institute of Fundamental Technological Research) for his organizational eﬀorts and word processing of the Proceedings volume.

Pedro Ponte Casta˜ neda Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, USA

March 2004

J´ozef Joachim Telega Barbara Gambin Polish Academy of Sciences Institute of Fundamental Technological Research Warsaw, Poland

List of Participants 1

Allaire Gregoire

2

Andrianov Igor

3

Bhattacharya Kaushik

4

Bielski Wlodzimierz

5

Bojarski Jaroslaw

6

Braides Andrea

7

Cherkaev Andrej

8

Chenchiah Issac Vikram

9

Czarnecki Slawomir

10

Deger Deniz

11

Dolzmann Georg

Centre de Math´ematiques Appliqu´ees, Ecole Polytechnique, 91128 PALAISEAU Cedex , FRANCE e-mail: [email protected] Willi-Suth-Allee 28, D-50769, K¨ oln, GERMANY e-mail: igor [email protected] California Institute of Technology, Pasadena CA 91125, USA e-mail: [email protected] Institute of Geophysics, Polish Academy of Sciences, Warsaw, POLAND e-mail: [email protected] Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, POLAND e-mail: [email protected] Dipartimento di Matematica, Universit`a di Roma ‘Tor Vergata’, via della ricerca scientiﬁca, 100133 ROMA, ITALY e-mail: [email protected] Department of Mathematics, University of Utah, S 1400 E, Salt Lake City, UT, USA e-mail: [email protected] 104 44 CALTECH, Pasadena, CA 91125, USA e-mail: [email protected] Warsaw Universitry of Technology. Faculty of Civil Engineering. Institute of Structural Mechanics, Warsaw, POLAND e-mail: [email protected] University of Istanbul, Faculty of Science, Physics Department 34459 Vezneciler, Istanbul, TURKEY e-mail: [email protected] Mathematics Department, University of Maryland, College Park, MD 20910, USA e-mail: [email protected] xiii

xiv 12

Donoso Alberto

13

Galka Antoni

14

Gambin Barbara

15

Goldsztein Guillermo

16

Gwiazda Piotr

17

HE Qi-Chang

18

Idiart Martin

19

Ivanova Jordanka

20

Iwaniec Tadeusz

´ Bell´on Becario de Investigaci´on Area de Matem´atica Aplicada, E.T.S. Ingenieros Industriales Universidad de Castilla-La Mancha, Ediﬁcio Polit´ecnico s/n, 13071 Ciudad Real, SPAIN e-mail: [email protected] Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, POLAND e-mail: [email protected] Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, POLAND e-mail: [email protected] School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30062, USA e-mail: [email protected] Warsaw University, Faculty of Mathematics & Informatic & Mechanics, Warsaw, POLAND e-mail: [email protected] Laboratoire de Mecanique, Universite de Marne la Vallee, 19 rue A. Nobel, F-77420 Champs sur Marne, FRANCE e-mail: [email protected] University of Pennsylvania, 297 Towne Blg., 220 S. 33rd St., Philadelphia PA 19104, USA. e-mail: [email protected] Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl.4, 1113 Soﬁa, BULGARIA e-mail: [email protected] Department of Mathematics, Syracuse University, 215 Carnegie Bldg, Syracuse, NY 13244, USA e-mail: [email protected]

xv 21

Kalamajska Agnieszka

22

Kalkan Nevin

23

Kruglenko Eleonora

24

Kuczma Mieczyslaw

25

Kutylowski Ryszard

26

Lopez-Pamies Oscar

27

Lewi´ nski Tomasz

28

Lipton Robert

29

Mercier Sebastien

30

Mig´ orski Stanislaw

Warsaw University, Faculty of Mathematics & Informatic & Mechanics, Warsaw, POLAND e-mail: [email protected] Uniersity of Istanbul, Faculty of Science, Physics Department, 34459 Vezneciler, Istanbul, TURKEY e-mail: [email protected] Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, POLAND e-mail: [email protected] Institute of Structural Engineering University of Zielona G´ ora, Podg´ orna str. 50, 65-246 Zielona G´ ora , POLAND e-mail: [email protected] Instytut In˙zynierii L¸adowej Politechniki Wroclawskiej, Wybrze˙ze Wyspia´ nskiego 27, 50-370 Wroclaw, POLAND e-mail: [email protected] 518 Woodland Terrace, Philadelphia, PA, 19104, USA e-mail: [email protected] Warsaw University of Technology, Faculty of Civil Eng., Warsaw, POLAND e-mail: [email protected] Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA e-mail: [email protected] LPMM/Universite de Metz, Ile du Saulcy 57045 Metz, FRANCE e-mail: [email protected] Jagiellonian University, Faculty of Mathematics, Physics and Computer Science, Institute of Computer Science, ul. Nawojki 11, 30 072 Krak´ ow, POLAND e-mail: [email protected]

xvi 31

Murat Fran¸cois

32

Nazarenko Lidia.V.

33

Novikov Alexei

34

Nowak Zdzislaw

35

Oleszkiewicz Ewa

36

Pedregal Pablo

37

Ponte Casta˜ neda Pedro

38

Raitums Uldis

39

Rychago Mikhail

40

Roubiˇcek Tomas

Laboratoire Jacques-Louis Lions, Universite Pierre et Marie Curie (Paris VI), Boite courrier 18775252 Paris cedex 05, FRANCE e-mail: [email protected] Institute of Hydromechanics of NAS of Ukraine, Kiev, UKRAINE e-mail: [email protected] Applied & Computational Mathematics , 1200 E. California Boulevard, MC 217-50 Pasadena, CA 91125, USA e-mail: [email protected] Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, POLAND e-mail: [email protected] Poznan University of Technology, Pozna´ n, POLAND e-mail: [email protected] Departamento de Matemticas, E.T.S. Ingenieros Industriales Universidad de Castilla-La Mancha, Ediﬁcio Polit´ecnico s/n, 13071 Ciudad Real, SPAIN e-mail: [email protected] Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, 220 S. 33rd. St., Philadelphia, PA 19104-6315 Oﬃce: Towne Building 235, USA email: [email protected] Institute of Mathematics and Computer Science, 29 Rainis bouleward, LV-1459 Riga, LATVIA e-mail: [email protected] Vladimir State Pedagogical University, Vladimir, RUSSIA e-mail: [email protected] Mathematical Institute, Charles University, Praque, CZECH REPUBLIC e-mail: [email protected]ﬀ.cuni.cz

xvii 41

Shulga Svetlana

42

Slastikov Valeriy

43

Stefaniuk Robert

44

Suquet Pierre

45

Telega J´ozef Joachim

46

Tokarzewski Stanislaw

48

Wojnar Ryszard

48

Zatorska Anna

49

Zimmer Johannes

Vladimir State Pedagogical University, Vladimir, RUSSIA e-mail: [email protected] Courant Institute of Mathematical Sciences, 251 Mercer street, New York, NY 10012, USA e-mail: [email protected] Warsaw University, Faculty of Mathematics & Informatic & Mechanics, Warsaw, POLAND e-mail: [email protected] Laboratoire de Mecanique et d’Acoustique, CNRS, 31 chemin Joseph Aiguier, 13402 Marseille cedex 20, FRANCE e-mail: [email protected] Institute of Fundamental Technological Research, PAS, ul. Swietokrzyska 21, 00 049 Warsaw, POLAND e-mail: [email protected] Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, POLAND e-mail: [email protected] Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, POLAND e-mail: [email protected] Warsaw University, Faculty of Mathematics & Informatic & Mechanics, Warsaw, POLAND e-mail: [email protected] California Institute of Technology, Division of Engineering and Applied Science, Mail Stop 104-44, Pasadena CA 91125, USA e-mail: [email protected]

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Titles of Contributed Papers 1. Andrianov I.V., Danishevs’kyy, HOMOGENIZATION AND DISCRETECONTINUUM APPROACH IN THE MECHANICS OF SOLIDS 2. Bielski W., NONLINEAR MAGNETO-ELASTIC COMPOSITE MEDIA: EFFECTIVE PROPERTIES, LINEARIZATION AND DUALITY 3. Bilger N., Ausender F., Bornert M., Masson R., Michel J.C., Moulinec H., Suquet P., Zaoui A., MICROMECHANICAL INVESTIGATIONS OF THE EFFECT OF DEFECT CLUSTERING ON THE BRITTLE-DUCTILE TRANSITION OF A NUCLEAR VESSEL STEEL 4. Bojarski J., OPTIMIZATION OF A CONVEX PROBLEM WITH NONCONVEX CONSTRAINTS : THE MAGNETOSTATIC FIELD 5. Cherkaev E., SYMMETRIES AND BIFURCATIONS IN OPTIMAL DESIGN 6. Czarnecki S., Lewi´ nski T., MINIMUM COMPLIANCE PROBLEMS OF THREE-DIMENSIONAL BODIES. THEORETICAL AND NUMERICAL ASPECTS 7. Deger D., Ulutas K., THE THICKNESS DEPENDENCE OF OPTICAL PROPERTIES OF Bi THIN FILMS 8. Dolzmann G., MACROSCOPIC RESPONSE OF NEMATIC ELASTOMERS: A VARIATIONAL APPROACH VIA RELAXATION OF A FREE ENERGY 9. Donoso A., Pedregal P., OPTIMAL DESIGN OF CONDUCTING GRADED MATERIALS 10. Gambin B., Galka A., AN OPTIMAL DESIGN OF FUNCTIONALLY GRADED MATERIAL: ONE-DIMENSIONAL EXAMPLE 11. Goldsztein G.H. ,PERFECTLY PLASTIC HETEROGENEOUS MATERIALS 12. Gwiazda P., Zatorska- Goldstein A., AN EXISTENCE RESULT FOR THE LERAY-LIONS TYPE OPERATORS WITH DISCONTINOUS COEFFICIENTS 13. HE Q.-C. , Le Quang H., UNIFORM STRAIN FIELDS AND EXACT RELATIONS IN ELASTOPLASTIC FIBER-REINFORCED COMPOSITES 14. Ivanova J., Bontcheva L., Parashkevova L., OPTIMAL DESIGN OF FUNCTIONALLY GRADED PLATES WITH THERMO-ELASTIC PLASTIC BEHAVIOUR xix

xx 15. Kalkan N., INFLUENCE OF DIFFERENT PROCESS PARAME-TERS ON PHYSICAL PROPERTIES OF INDIUM OXIDE THIN FILMS 16. Kalamajska A., Stefaniuk R., NEW GEOMETRIC CONDITIONS FOR QUASICONVEXITY AND THEIR NUMERICAL VERIFICATION 17. Kuczma M.S., NUMERICAL SIMULATION OF SHAPE MEMORY MATERIALS 18. Molinari A., Mercier S., VISCOPLASTIC AND ELASTIC VISCOPLASTIC TANGENT MODELS: THEORY AND APPLICATIONS 19. Denkowski Z., Mig´ orski S., ON THE CONVERGENCE OF SOLUTIONS TO MULTIVALUED PARABOLIC AND HYPERBOLIC EQUATIONS WITH APPLICATIONS 20. Nazarenko L.V. , PREDICTION OF BEHAVIOUR OF POROUS ANISOTROPIC COMPOSITES UNDER MICROFAILURE 21. Nowak Z., CRITICAL LENGTH SCALE OF THE REPRESENTATIVE VOLUME ELEMENT BASED ON THE MARKED CORRELATION FUNCTION 22. Novikov A., NETWORK APPROXIMATION OF THE EFFECTIVE CONDUCTIVITY OF HIGH CONTRAST TWO-PHASE COMPOSITES 23. Raitums U., ON THE RANGE OF SOME G-CLOSED SETS OF NEMITSKII OPERATORS 24. Rychago M., HOMOGENIZATION OF MONOTONE OPERATORS 25. Shulga S. B., HOMOGENIZATION FOR NONLINEAR DOUBLEPOROSITY MODELS 26. Slastikov V., GEOMETRICALLY CONSTRAINED MAGNETIC DOMAIN WALLS 27. Telega J.J., Wojnar R., ELECTROKINETICS AND HOMOGENIZATION IN RANDOM POROUS MEDIA 28. Telega J.J., Gambin B., Galka A., NONLINEAR PIEZOELECTRIC COMPOSITES: DETERMINISTIC AND STOCHASTIC HOMOGENIZATION 29. Telega J.J., Tokarzewski S., Gambin B., Galka A., NONLINEAR COMPOSITES WITH TEMPERATURE- DEPENDENT CONDUCTIVITY COEFFICIENTS OF CONSTITUENTS 30. Ulutas K., Deger D., THE THICKNESS DEPENDENCE OF ELECTRICAL PROPERTIES OF Bi THIN FILMS 31. Zimmer J. , ON THE COMPUTATION OF NONCONVEX HULLS

INVITED LECTURES

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TOPOLOGY OPTIMIZATION WITH THE HOMOGENIZATION AND THE LEVEL-SET METHODS

G. ALLAIRE

CMAP, Ecole Polytechnique 91128 Palaiseau, France

Abstract. After a brief review of the homogenization and level-set methods for structural optimization we make some comparisons of their numerical results. The typical problem is to ﬁnd the optimal shape of an elastic body which is both of minimum weight and maximal stiﬀness under speciﬁed loadings. This problem is known to be ”ill-posed”, namely there is generically no optimal shape and the solutions computed by classical numerical algorithms are highly sensitive to the initial guess and mesh-dependent. The homogenization method makes this problem well-posed by allowing microperforated composites as admissible designs. It induces new numerical algorithms which capture an optimal shape on a ﬁxed mesh. The homogenization method is able to perform topology optimization since it places no explicit or implicit restriction on the topology of the optimal shape. The level-set method instead does not change the ill-posed nature of the problem. It is a combination of the level-set algorithm of Osher and Sethian with the classical shape gradient (or boundary sensitivity). Although this last method is not speciﬁcally designed for topology optimization, it can easily handle topology changes. Its cost is also moderate since the shape is captured on a ﬁxed Eulerian mesh. We discuss their respective advantages and drawbacks.

1. Introduction Shape optimization of elastic structures is a very important and popular ﬁeld. The classical method of shape sensitivity (or boundary variation) has been much studied (see e.g. [10], [13], [16]). It is a very general method which can handle any type of objective functions and structural models, but it has two main drawbacks: its computational cost (because of remesh-

1 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 1–13. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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ing) and its tendency to fall into local minima far away from global ones. The homogenization method (see e.g. [1], [2], [6], [7], [8]) is an adequate remedy to these drawbacks but it is mainly restricted to linear elasticity and particular objective functions (compliance, eigenfrequency, or compliant mechanism). Recently yet another method appeared based on the ﬁrst approach of shape sensitivity but using the versatile level-set method for computational eﬃciency (see [4], [11], [15], [17]). The level-set method has been devised by Osher and Sethian [12] for numerically tracking fronts and free boundaries. In this paper we review and compare the homogenization method and this new level-set method for structural optimization. 2. Setting of the problem. We consider the following structural optimization problem : ﬁnd the optimal shape that minimizes a weighted sum of its elastic compliance and weight. As usual the compliance (i.e. the work done by the load) is a global measure of the design’s rigidity. We introduce a working domain Q ⊂ IRd in which all admissible shapes Ω are included, i.e. Ω ⊂ Q. The boundary of a shape Ω is made of three disjoint parts ∂Ω = Γ ∪ ΓN ∪ ΓD, with Dirichlet boundary conditions on ΓD, and Neumann boundary conditions on Γ ∪ ΓN . We assume that ΓD and ΓN are parts of ∂Q and are supposed to be ﬁxed. Only Γ is allowed to vary in the optimization process. The displacement ﬁeld u in Ω is the unique solution of the linearized elasticity system  −div (A e(u)) = 0 in Ω    u=0 on ΓD (1)  (A e(u))n = f on ΓN   (A e(u))n = 0 on Γ, where A is the elasticity tensor of a linearly isotropic elastic material (with bulk and shear moduli κ and µ). Recall that the deformation tensor is e(u) = 12 (∇u + (∇u)t) and the stress tensor is σ = Ae(u). The compliance of the structure Ω is

c(Ω) = ΓN

f ·u=

Ω

Ae(u) · e(u) =

Ω

A−1 σ · σ.

(2)

Introducing a positive Lagrange multiplier , our structural optimization problem is to minimize, over all subsets Ω ⊂ Q, the objective function J(Ω) equal to the weighted sum of the compliance and weight of Ω. In other words we want to compute minimizers of

inf

Ω⊂Q

J(Ω) = c(Ω) + |Ω| .

(3)

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3

The Lagrange multiplier has the eﬀect of balancing the two contradictory objectives of rigidity and lightness of the shape (increasing its value decreases the weight). As is well known, in absence of any supplementary (topological) constraints on the admissible designs Ω, the objective function J(Ω) may have no minimizer, i.e. there is no optimal shape (see e.g. [1], [9]). The physical reason for this non-existence is that it is often advantageous to cut inﬁnitely many small holes (rather than just a few big ones) in a given design in order to decrease the objective function. Thus, achieving the minimum may require a limiting procedure leading to a ”generalized” design consisting of composite materials made by microperforation of the original material. 3. Homogenized formulation To cope with this physical behavior of nearly optimal shapes, we enlarge the space of admissible designs by permitting perforated composites from the start : this process is called relaxation. Such composite structures are determined by two functions θ(x) and A∗ (x) : θ is the local volume fraction of the original material, taking values between 0 and 1, and A∗ is the eﬀective Hooke’s law determined by the microstructure of perforations . In this section we brieﬂy recall the main results on this so-called homogenization approach (see [1], [7], [9] and references therein). A minimizing sequence of the objective function (3) can be regarded as a composite material obtained by microperforation of the original material A. The eﬀective behavior of such a composite material is characterized by a material density θ(x) ∈ [0, 1] and a Hooke’s law A∗ (x) such that the average or macroscopic behavior of solutions of (1) are determined by the homogenized problem   σ = A∗ (x)e(u) e(u) = 12 (∇u + ∇tu)     in Q  divσ = 0

u=0

on ΓD on ΓN on ∂Q \ (ΓD ∪ ΓN ) .

   σn = f    σn = 0

(4)

The homogenized compliances is ∗

c˜(θ, A ) = ΓN

f ·u=

Q

A∗ (x)−1 σ · σ,

where the stress σ is solution of the homogenized equation (4). Remark that, for a given value θ of the density, there are many diﬀerent possible eﬀective Hooke’s law A∗ in a set Gθ , the so-called G-closure set at volume

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fraction θ, which is the set of all possible homogenized Hooke’s law with density θ. We thus obtain the relaxed or homogenized functional

˜ A∗ ) = c˜(θ, A∗ ) + J(θ,

min

0≤θ≤1, A∗ ∈Gθ

θ(x) dx . Q

˜ A∗ ) has to be minimized over all admissible The relaxed functional J(θ, composite designs, i.e. over all density θ and eﬀective Hooke’s law A∗ ∈ Gθ . Although Gθ is not known explicitly, the minimization of the compliance c˜(θ, A∗ ) can be done analytically since optimal composites are shown to be the so-called sequential laminates (which have explicit elasticity tensors). Indeed, we rewrite the compliance as ∗

c˜(θ, A ) =

min

divσ=0 in Q σn=f on ΓN σn=0 on Γ

Q

A∗ (x)−1 σ · σ.

(5)

Then, the two minimizations, in (θ, A∗ ) and in σ, can be switched. Since the microstructure can be optimized pointwise in the domain, the relaxed formulation becomes

min

divσ=0 in Q σn=f on ΓN σn=0 on Γ

min∗

Q 0≤θ≤1, A ∈Gθ

A∗ −1 σ · σ + θ dx.

(6)

For a ﬁxed stress σ, the minimization of A∗ −1 σ · σ on Gθ is a classical problem in the theory of homogenization and composite materials [1], [9]. It amounts to ﬁnd the most rigid composite of given density θ under the stress σ. In two dimensions, the result is min A∗ −1 σ · σ = A−1 σ · σ +

A∗ ∈Gθ

(κ + µ)(1 − θ) (|σ1 | + |σ2 |)2 4κµθ

(7)

where σ1 and σ2 are the eigenvalues of the 2 by 2 symmetric matrix σ. Furthermore, optimality in (7) is achieved for a so-called rank-2 sequential laminate aligned with the eigendirections of σ. In three dimensions, the result is more complicated, and we give it in the special case of Poisson’s ratio equal to zero, i.e. 3κ = 2µ (the general case is not much diﬀerent in essence) (1 − θ) ∗ min g (σ) A∗ −1 σ · σ = A−1 σ · σ + ∗ A ∈Gθ 4µθ with

∗

g (σ) =

2 (|σ 3 |)

1 | + |σ2 | + |σ if |σ3 | ≤ |σ1 | + |σ2 | 2 (|σ1 | + |σ2 |)2 + |σ3 |2 if |σ3 | ≥ |σ1 | + |σ2 |

(8)

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where the eigenvalues of σ are labeled in such a way that |σ1 | ≤ |σ2 | ≤ |σ3 |. Furthermore, optimality in the ﬁrst regime of (8) is achieved by a rank3 sequential laminate aligned with the eigendirections of σ, while in the second regime it is achieved by a rank-2 sequential laminate aligned with the two ﬁrst eigendirections of σ. After this crucial step, the minimization in θ can easily be done by hand, which completes the explicit calculation of the relaxed formulation. From a mathematical point of view, one can prove that the relaxed formulation (6) admits a minimizer, that any minimizing sequence of the original problem (3) converges (in the sense of homogenization) to a minimizer of (6), and that the two inﬁmum values of (6) and (3) are equal. However, in general there is no uniqueness of the minimizer. 4. Numerical algorithm for the homogenization method.

Figure 1.

Boundary conditions of a 2-d cantilever.

The ﬁrst main advantage of the homogenization method is to change a diﬃcult ”free-boundary” problem into a much easier ”sizing” optimization problem in a ﬁxed domain. The computational cost is thus very low compared to traditional algorithms since the mesh is ﬁxed (shapes are captured rather than tracked). The second main advantage is that the homogenized formulation is well-posed. In practice, the resulting optimal shape is independent of the initial guess and the homogenization method thus performs topology optimization (the ﬁnal optimal shape may have a topology completely diﬀerent from that of the initial guess). Let us describe brieﬂy our favorite algorithm. It is an alternate direction algorithm: we start with an initial design (usually full material everywhere), then, at each iteration, we compute the stress σ solution of a linear elasticity problem with a Hooke’s law corresponding to the previous design, and we update the design variables θ and A∗ in terms of σ by using the explicit formula for the optimal laminated composite material in (7) or (8). We iterate this process until convergence which is detected when the density

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Figure 2. Homogenization method: optimal shape of the cantilever: composite (left) and penalized (right).

variation becomes smaller than some threshold. This algorithm converges smoothly in a relatively small number of iterations (between 10 and 100, depending on the desired accuracy). Furthermore, in practice it is insensitive to the choice of initial guess and convergent under mesh reﬁnement, suggesting that the numerical algorithm always picks up the same global minimum. However, as expected, it usually produces homogenized optimal designs that include large region of composite materials with intermediate density. From a practical point of view, this is an undesirable feature since the primal goal is to ﬁnd an optimal shape, i.e. a density taking only the values 0 or 1 ! The remedy is to introduce a penalization technique that will get rid of composite materials. The strategy is the following : after convergence has been reached on a homogenized optimal design, we run a few more iterations (around 10) of our algorithm during which we force the density to take values close to 0 or 1. More speciﬁcally, denoting by θopt the true optimal density, the penalization procedure amounts to upopt ) . There is no speciﬁc reason date the density at the value θpen = 1−cos(πθ 2 to choose a cosine-shape function for the penalized density, except that it works ﬁne and yields surprisingly nice shapes featuring ﬁne patterns instead of composite regions. The success of this method is due to the fact that the relaxed design is characterized not only by a density θ but also by a microstructure A∗ which is hidden at the sub-mesh level. The penalization has the eﬀect of reproducing this microstructure at the mesh level. Of course it is strongly mesh-dependent in the sense that the ﬁner the mesh the more complicated the resulting ”almost optimal” structure. The homogenization method can be generalized to several other types of objective functions, including sum of compliances for multiple loads, eigenfrequencies, least square criteria for a target displacement or stress (see e.g. [1], [3]). However, in its rigorous setting it is restricted to a linear elasticity model.

HOMOGENIZATION AND LEVEL-SET METHODS

7

5. Shape derivative In this section we come back to the classical setting of shape sensitivity for structural optimization. We brieﬂy recall how to compute a shape derivative for the objective function J(Ω) deﬁned by (3). In order to deﬁne a shape derivative we follow the approach of Murat-Simon [10] (see also [13], [16]). Starting from a reference domain Ω0 , we consider domains of the type Ω = ( Id + τ )(Ω0 ),

(9)

where τ ∈ W 1,∞ (IRd, IRd) (for suﬃciently small τ , ( Id + τ ) is a diﬀeomorphism). We further restrict the class of domains by asking that they all share the same parts of the boundary ΓN and ΓD: speciﬁcally, the map τ must vanish on ΓN ∪ΓD. The shape derivative of J(Ω) at Ω0 is then deﬁned as the Fr´echet derivative in W 1,∞ (IRd, IRd) at 0 of the map τ → J(( Id + τ )(Ω0 )). This notion is well deﬁned and a standard computation shows that the shape derivative of (3) is

∂J (Ω0 ), τ = ∂Ω

Γ

− Ae(u) · e(u) τ · n ds,

(10)

where u is the solution of (1) in Ω0 , n is the unit exterior normal and H the curvature of Γ. Remark that there is no adjoint state involved in (10) (indeed the minimization of (3) is a self-adjoint problem ). Of course, the shape derivative can be computed for other objective functions and other model problems including non-linear elasticity [4], [5]. 6. Numerical algorithm for the level-set method We review the numerical implementation of a gradient method for the minimization of problem (3) as proposed in [4], [5]. The idea is to combine the shape derivative of Section 5 and the level-set method of Osher and Sethian [12]. In order to describe the boundary of Ω we introduce a level-set function ψ deﬁned on the working domain Q such that   ψ(x) = 0 ⇔ x ∈ ∂Ω ∩ Q 

ψ(x) < 0 ⇔ x ∈ Ω ψ(x) > 0 ⇔ x ∈ (Q \ Ω) .

The normal n to the shape Ω is recovered as ∇ψ/|∇ψ| and the curvature H is given by the divergence of n (these quantities are evaluated by ﬁnite diﬀerences since our mesh is uniformly rectangular). Remark that, although n and H are deﬁned on Γ, the level set method allows to deﬁne easily their extension in the whole domain Q. We ﬁll the void part Q \ Ω with a very

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G. ALLAIRE

Figure 3. ations.

Short cantilever with the level-set method: initialization and successive iter-

weak material with Hooke’s law B = 10−3 A and we perform the elasticity analysis on a ﬁxed rectangular mesh in Q (using Q1 ﬁnite elements). Since n and H, as well as the displacement u, are computed everywhere in Q, formula (10) delivers a vector ﬁeld V throughout the domain and not only on the free boundary Γ, namely V = v n with

v = − Ae(u) · e(u) .

After evaluating the gradient of J(Ω), or equivalently this vector ﬁeld V , we transport the level set function ψ along this gradient ﬂow −V = −v n. Since n = ∇ψ/|∇ψ|, we end up with the following Hamilton-Jacobi equation ∂ψ − v|∇ψ| = 0, ∂t

(11)

where the time variable t plays the role of the descent step in the gradient algorithm. Transporting ψ by (11) is equivalent to move the boundary of ∂J Ω (the zero level set of ψ) along the descent gradient direction − ∂Ω . We solve (11) using a standard explicit upwind ﬁnite diﬀerence scheme (see e.g. [14]). Finally, our algorithm is an iterative method, structured as follows: 1. Initialization of the level-set function ψ0 as the signed distance function to the boundary of an initial guess Ω0 . 2. Iteration until convergence, for k ≥ 0: (a) Computation of uk by solving a linear elasticity problem in Q with Hooke’s law Ak(x) = A where ψk(x) < 0 Ak(x) = B where ψk(x) > 0.

HOMOGENIZATION AND LEVEL-SET METHODS

9

Figure 4. Medium cantilever with the level-set method: initialization with many holes and successive iterations.

(b) Deformation of the shape through the transport of the level set function: ψk+1 (x) = ψ(∆tk , x) where ψ(t, x) is the solution of (11) with velocity vk = −Ae(uk)·e(uk) and initial condition ψ(0, x) = ψk(x). The time step ∆tk is chosen such that J(Ωk+1 ) ≤ J(Ωk). This algorithm never creates new holes or boundaries if the time step ∆tk satisﬁes a CFL condition for (11) (there is no nucleation mechanism for new holes). However the level set method is well known to handle easily topology changes, i.e. merging or cancellation of holes. In numerical practice, the number of holes always decrease in dimension d = 2, so the initialization must contain enough holes in order to obtain a good optimal shape (compare Figures 4 and 5). However, in dimension d = 3 new holes can appear by pinching thin plate-like objects, so the initial design is less critical (also still important). In any case, our algorithm is able to perform topology optimization. The algorithm converges smoothly to a (local) minimum which depends, of course, on the initial topology. The numerical results are very similar to those obtained by the homogenization method. In order to speed up the convergence we perform several (of the order of 20 in numerical practice) time steps of the transport equation for each elasticity analysis. The exact number of time steps is controlled by the decrease of the objective function. From time to time, for stability reasons, we also reinitialize the level set function ψ in order that it be the signed distance function to the boundary

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Figure 5. Medium cantilever with the level-set method: initialization with few holes and successive iterations.

of the current shape Ω (see [14]). We give some numerical results for the compliance objective function (3) (more examples, including load dependent designs can be found in [5]). The boundary conditions for cantilever problems are displayed on Figure 1. The results are shown on Figures 4 and 5 for an increasing number of iterations. Figure 6 shows the history of the objective function for a medium cantilever optimized with the two methods discussed here: homogenization and levelset. A three-dimensional example is given on Figure 7. 21

Level-set Homogenization

20 19 18 17 16 15 14 13 12 0

Figure 6.

10

20

30

40

50

60

70

Convergence of the objective function for the two iterative methods.

HOMOGENIZATION AND LEVEL-SET METHODS

Figure 7.

Figure 8.

Three-dimensional cantilever by the level-set method.

Three-dimensional gripping mechanism by the level-set method.

11

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G. ALLAIRE

7. Comparisons and conclusions We implemented the homogenization method and the level-set method in two and three space dimensions for shape and topology optimization. They both share the following advantages: 1. they allow for drastic topology changes during the optimization process, 2. their cost is moderate in terms of CPU time since they are Eulerian shape capturing methods. However, they are very diﬀerent with respect to the following points. 1. Inﬂuence of the initial design: since the homogenization method is a relaxation method, it is independent of the initial guess and it numerically converges to a global maximum ; on the contrary, the level-set method is very sensitive to the initial guess and easily get caught in local minima. 2. Generality of application: the homogenization method is mostly restricted to some speciﬁc objective functions and to the linear elasticity setting ; on the other hand, the level-set method can handle very general objective functions and mechanical models, including nonlinear elasticity (see Figure 9). f=3

Figure 9. Optimal cantilever in non-linear elasticity (the bars under compression are thicker than those under traction).

The two methods discussed above are not in competition ; rather they are complementary. One should ﬁnd a correct topology for a simple objective function in linear elasticity by applying the homogenization method, and then use it as an initial guess for the level-set method in the context of a more involved objective function and non-linear elasticity model.

HOMOGENIZATION AND LEVEL-SET METHODS

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Allaire, G. (2001) Shape optimization by the homogenization method, Springer Verlag, New York. Allaire, G., Bonnetier, E., Francfort, G., Jouve, F. (1997) Shape optimization by the homogenization method. N¨ umerische Mathematik 76, 27-68. Allaire, G., Jouve, F., Maillot, H. (2003) Topology optimization for minimum stress design with the homogenization method, preprint. Allaire, G., Jouve, F., Toader, A.-M. (2002) A level-set method for shape optimization, C. R. Acad. Sci. Paris, S´ erie I, 334, pp.1125-1130. Allaire, G., Jouve, F., Toader, A.-M. (2003) Structural optimization using sensitivity analysis and a level-set method, preprint. Allaire, G., Kohn, R.V. (1993) Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Europ. J. Mech. A/Solids 12, 6, 839-878. Bendsoe, M. (1995) Methods for optimization of structural topology, shape and material, Springer Verlag, New York. Bendsoe, M., Kikuchi, N. (1988) Generating Optimal Topologies in Structural Design Using a Homogenization Method. Comp. Meth. Appl. Mech. Eng. 71, 197-224. Cherkaev, A. (2000) Variational Methods for Structural Optimization, Springer Verlag, New York. Murat, F., Simon, S. (1976) Etudes de probl`emes d’optimal design. Lecture Notes in Computer Science 41, 54-62, Springer Verlag, Berlin. Osher, S., Santosa, F. (2001) Level set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J. Comp. Phys., 171, 272-288. Osher, S., Sethian, J.A. (1988) Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comp. Phys. 78, 12-49. Pironneau, O. (1984) Optimal shape design for elliptic systems, Springer-Verlag, New York. Sethian, J.A. (1999) Level Set Methods and fast marching methods: evolving interfaces in computational geometry, ﬂuid mechanics, computer vision and materials science, Cambridge University Press. Sethian, J., Wiegmann, A. (2000) Structural boundary design via level set and immersed interface methods. J. Comp. Phys., 163, 489-528. Sokolowski, J., Zolesio, J.P. (1992) Introduction to shape optimization: shape sensitivity analysis, Springer Series in Computational Mathematics, Vol. 10, Springer, Berlin. Wang, M. Y., Wang, X., Guo, D. (2003) A level-set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., 192, 227-246.

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THIN FILMS OF ACTIVE MATERIALS

K. BHATTACHARYA

Division of Engineering and Applied Science Mail Stop 104-44, California Institute of Technology Pasadena, CA 91125

Abstract. This paper summarizes some recent developments in the study of the mechanics of thin ﬁlms motivated by the use of active materials in making microactuators.

1. Introduction The development of suitable microactuators is a key challenge in the area of micromachines or microelectromechanical systems (MEMS). A successful microactuator must develop a large work output from a smaller and smaller volume, and therefore work output per unit volume is a key ﬁgure of merit for any actuator material. The limiting frequency of operation is the second important ﬁgure of merit. Shape-memory alloys exhibit the largest work output per cycle per unit volume amongst a variety of actuator systems [33]. Further, the frequency of operation is reasonable in thin ﬁlms of these alloys since heat transfer is accelerated at small sizes. Similar good performance is anticipated in other recently developed active materials like ferromagnetic shape-memory alloys and high-strain ferroelectrics. Preliminary attempts to use active materials as microactuators, though encouraging, have not lived up to this exceptional promise. This paper summarizes some recent theoretical investigations that were motivated by this concern. The remarks here are conﬁned to shape-memory alloys though similar ideas hold for other active materials. The heart of the shape-memory eﬀect lies in a martensitic phase transformation, which is a solid to solid phase transformation between a high temperature austenite phase and low temperature martensite phase. In shape-memory alloys, the austenite has greater crystallographic symmetry than the martensite and consequently there are multiple variants of

15 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 15–44. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

16

Figure 1.

K. BHATTACHARYA

The behavior of a single crystal shape-memory rod can be quite unusual.

martensite: crystallographically equivalent lattices that are oriented diﬀerently with respect to the austenite. The variants of martensite form ﬁnephase microstructure, and speciﬁc features of this microstructure give rise to the shape-memory eﬀect. Most recent attempts at making microactuators using shape-memory alloys have used polycrystalline thin ﬁlms of Nickel-Titanium deposited by sputtering on various substrates (see for example the various papers in [2]). The properties of these ﬁlms are poorer than those of bulk Ni-Ti, and there are various reasons for this. The martensitic microstructure can be quite diﬀerent in thin ﬁlms compared to that in bulk materials due to surface relaxation and the constraint imposed by the substrate. The behavior of polycrystalline materials can also be diﬀerent since crystallographic texture which depends on the processing techniques can be very diﬀerent in thinﬁlms (prepared by sputtering ) compared to bulk (prepared by rolling and drawing). It is thus not clear that Ni-Ti is the best material in thin ﬁlm applications and that sputtering is the best way of making the ﬁlm. Further, most of these actuators use a cantilever-type design which in its simplest implementation places the material in bending. Unfortunately, bending a thin structure requires considerably less energy than stretching or shearing. One can readily see this from this very page 16 bending or rolling it up requires much less energy than stretching or shearing it. The bending stiﬀness (proportional to the rotational moment of inertia) scales as the third power of thickness, while the stretching or shearing stiﬀness (proportional to the cross-sectional area) scales as the thickness. Thus bending a slender structure produces impressive displacements, but very little energy. One thus needs more careful designs which use stretch and shear. These concerns motivate the theoretical examination of martensitic thin ﬁlms . The mechanics literature is rich with various membrane, plane and shell (or string and rod) theories describing the behavior of slender structures. It is not immediately clear which, if any, of these theories describe the behavior of martensitic thin ﬁlms. These materials are anisotropic and can undergo large and discontinuous shear deformation due to the phase transformation. Figure 1 shows schematically the response of a single crystal rod of a typical shape-memory alloy when subjected to a moment. If the moment is applied about one normal direction, then the deformed shape is

THIN FILMS OF ACTIVE MATERIALS

17

a circular arc; but if the moment is applied about the other the deformed shape is a ‘rounded V’ with two straight arm connected by a small arc. In the former the deformation is like that of a classical Euler-Bernoulli beam dominated by bending, while in the latter the deformation is dominated by a large shear induced by the phase transformation. It is therefore instructive to revisit the derivation of theories of thin ﬁlms in the context of martensitic materials. Classically, these theories are derived by a formal asymptotic method: one starts with an ansatz about the deformation of a thin structure (for example, in Euler-Bernoulli beams one assumes that the center-line is inextensible and that plane cross-sections remain plane and perpendicular to the axis), substitutes this into the three dimensional energy and expands the energy in powers of the thickness. This gives a reduced theory; however this theory is only as good as the starting ansatz. This raises the question: what is the right ansatz for an anisotropic, highly nonlinear material like a single crystal of martensitic material? The theory of Γ-convergence provides an alternative. This is a notion of convergence for functionals introduced by Di Giorgi [19] (see for example [20]) and provides a method for rigorously deriving asymptotic theories when one has a variational principle. It has recently been used by several authors (e.g. [1, 3, 35, 36, 26, 13]) to derive theories of slender structures. Bhattacharya and James [10] used a similar procedure to derive a theory of martensitic thin ﬁlms. Their starting point is a three-dimensional theory of martensitic materials which has been studied quite extensively in the last few years [21, 4, 16, 6, 5, 7, 32] (also see [8, 29, 37] for expository reviews). This theory is variational and described brieﬂy in Section 2. The Helmholtz free energy density of these materials is not convex due to the transformation and the presence of multiple variants, and thus energy minimization leads to ﬁne-scale microstructure . The theory has been very successful in predicting various aspects of the microstructure and the shape-memory eﬀect . Bhattacharya and James [10] start with such a theory for a single crystal ﬁlm of thickness h and lateral extent S, and write the total energy per unit thickness eh of the ﬁlm. This energy consists of the Helmholtz free energy and interfacial energy . They study the minimizers uh of the energy subject to some boundary condition and extract from uh the various quantities that are necessary and suﬃcient for determining the limiting energy e0 = limh→0 eh. They ﬁnd that the e0 is determined by two vector ﬁelds: y which describes the deformation of the mid-plane and b which describes the traverse shear and normal compression. The energy e0 inherits an energy well structure from eh and consists of interfacial energy and energy associated with stretching and shearing the ﬁlm. They continue this asymptotic procedure to show that bending energy is two orders (powers of h) smaller.

18

K. BHATTACHARYA

They use the energy e0 to study the microstructure in ﬁlms, and show that there are interfaces that are possible in ﬁlms which are not possible in bulk. They propose a strategy to take advantage of such interfaces in designing micropumps. Building on this strategy, Bhattacharya et al [9] proposed speciﬁc designs and a strategy for fabrication. Section 3 describes this theory. Section 3.1 provides a derivation of the theory, Section 3.2 describes the energy minimizing microstructure and Section 3.3 a strategy for making tent and tunnel based micropumps. Shu [39] has extended this theory to heterogeneous ﬁlms. He considers a ﬁlm made up of multiple layers, each of which can be made of many grains. Further one or more of these layers could be martensitic. Thus, there are three length-scales: the thickness of the ﬁlm (or layers), the thickness of the grains in each layer and the thickness of the martensitic domain. Each of these length-scale is small compared to the lateral extent of the ﬁlm and hence one can describe the eﬀective properties of such ﬁlms. However, the eﬀective energy depends critically on the ratios between the three length-scale. Section 4 examines this theory. Section 4.1 describes the general situation,Section 4.2 gives a simple example to show that the eﬀective conductivity of a heterogeneous strip depends on the aspect ratio of the grains, Section 4.3 considers recoverable strains in a model problem that mimics shape-memory ﬁlms, Section 4.4 estimates recoverable strains in shape-memory ﬁlms and ﬁnds that sputtering Ni-Ti produces a texture that is poor from the point of view of the shape-memory eﬀect, Section 4.5 shows that multilayers of shape-memory alloys and elastic materials can lead to a two-way shape-memory eﬀect and ﬁnally, Section 4.6 describes the general results. We conclude in Section 5 by mentioning other recent developments in this area. 2. Bulk materials Consider a single crystal of a martensitic material occupying a region Ω ⊂ IR3 and undergoing a deformation u : Ω → IR3 . The Cauchy-Born hypothesis states that the deformation gradient ∇u is a measure of the distorsion of the underlying crystal lattice [22]. Therefore it is natural to assume that the Helmholtz free energy density W depends on the deformation gradient and temperature θ so that the total free energy of the crystal is given by

W (∇u, θ)dx. Ω

We shall henceforth conﬁne ourselves to isothermal situations and suppress the temperature from our notation.

THIN FILMS OF ACTIVE MATERIALS

19

The energy density W is required to be frame-indiﬀerent, i.e., superposed rigid rotations do not change the energy or W (RF) = W (F) for all rotation matrices R. Further, it has a multi-well structure because of the martensitic transformation. In order to describe this, it is convenient to choose the high temperature austenite phase as the reference conﬁguration, and so it is described by the deformation gradient I (identity). The low temperature martensite phase has k symmetry-related variants which are described by the matrices U1 , U2 , . . . , Uk. The ﬁrst variant is described by the Bain or distortion matrix U1 which takes the lattice of the austenite to that of the martensite. It can readily be obtained from the lattice parameters of the austenite and martensite lattices. The rest of the matrices are obtained from U1 by symmetry: Ui = QU1 QT , where Q is a rotation that maps the austenite lattice to itself. The energy density W has local minima at each of these matrices I, U1 , . . . , Uk with an absolute minimum at I above the transformation temperature and at Ui below it. Frame-indiﬀerence then implies that the minimizers of W consist of matrices of the form R above the transformation temperature, and RU1 , RU2 , . . . , RUk below the transformation temperature where R is any rotation matrix. Consider two examples. 1. A material undergoing cubic to tetragonal transformation like NiAl or Ni2 MnGa has three variants (k = 3) with U1 = Diag[β, α, α], U2 = Diag[α, β, α], U3 = Diag[α, α, β]. The measured values for Ni64 Al36 are β = 1.1302, α = 0.9392 and those for Ni2 MnGa are β = 0.9555, α = 1.0163. 2. A material undergoing cubic to monoclinic (DO3 to 18R) transformation like CuAlNi or CuZnAl has twelve variants (k = 12) with 

α U1 =  δ 0

δ γ 0



0 0. γ

and U2 . . . U12 obtained by symmetry. The measured values for Cu14wt.%Al-4w5%Ni are are α = 1.073, β = 1.032, γ = 0.9133, and δ = 0.051. and Cu68 Zn17 Al15 are α = 1.073, β = 1.023, γ = 0.9093, and δ = 0.038. In short, the energy density of a material undergoing a martensitic phase transformation has a multi-well structure with wells or minima at the austenite and all the diﬀerent variants of martensite. Energy minimization with such an energy density is a very rich problem. Classical solutions may not exist; instead minimizing sequences may develop oscillations at ﬁner and ﬁner scale [4]. Physically this manifests itself as the ﬁne scale microstructure that one sees in martensitic materials. There has been much

20

K. BHATTACHARYA

S

h

Ωh x3 z(x) x2 x1

S Ω1

1

z3 z2 z1

Figure 2.

The ﬁlm in the reference conﬁguration (top left) and the change of variables.

success in trying to understand the microstructure of martensite from this point of view (see for example [8] and the references there). In particular, one can have interfaces between phases or variants across which the deformation gradient can jump from one well to another. One may wish to assign an interfacial energy to such interfaces. A simple way of doing so is adding a Van der Waals type interfacial energy that penalizes changes in the deformation gradient. Thus we say that the total energy of the single crystal is written as Ω

κ2 |∇2 u|2 + W (∇u) dx.

(2.1)

Above, ∇2 u denotes the full 3 × 3 × 3 matrix of second derivatives of u. 3. Homogeneous thin ﬁlms 3.1. A THEORY OF THIN FILMS

Consider a ﬁlm occupying a region Ωh = S × (− h2 , h2 ) in the reference conﬁguration shown in the upper left corner of Figure 2. Following the discussion in Section 2, we say that the total energy of the ﬁlm per unit thickness subjected to a deformation u : Ωh → IR3 is eh(u) =

1 h

Ωh

κ2 |∇2 u|2 + W (∇u) dx.

(3.1)

21

THIN FILMS OF ACTIVE MATERIALS Unreleased Released region S

Substrate

Figure 3.

Towards micropumps: deposit the ﬁlm and then release it in some region S.

The state of the ﬁlm is described by the deformation uh which minimizes the energy eh amongst all possible deformations u subject to appropriate boundary conditions. Subsequent subsections will describe a strategy for making micropumps where the ﬁlms are released from the substrate in chosen regions, but attached to it outside (see Figure 3). In this situation, it is suﬃcient to study the released part of the ﬁlm subject to the following boundary conditions: the top and bottom surfaces are stress-free while the lateral surface ∂S × (− h2 , h2 ) is subjected to a ﬁxed displacement boundary condition. Therefore, the discussion here is conﬁned to these boundary conditions. The behavior of ﬁlms where the thickness is much smaller than the lateral extent is described by asymptotic limit of uh as h → 0. Bhattacharya and James [10] have shown that ¯ 1 , x2 ) ¯ (x1 , x2 ) + x3 b(x uh(x1 , x2 , x3 ) ≈ y

(3.2)

as h → 0. In other words, the deformation is described by two vector ¯ : S → IR3 which describes the deformation of the mid-plane and ﬁelds y ¯ : S → IR3 which describes the deformation of the thickness (see Figure b ¯ by minimizing the ¯ and b 4). Further, one can obtain these functions y energy e0 (y, b) =

S

κ2 (|∇2py|2 + 2|∇pb|2 ) + W (y,1 |y,2 |b) dx1 dx2

(3.3)

amongst all functions y, b : S → IR3 subject to appropriate boundary conditions. Here ∇p denotes gradient with respect to the planar variables (x1 , x2 ), (a1 |a2 |a3 ) denotes the 3×3 matrix with columns a1 , a2 and a3 , and ∂y . Thus the ﬁrst column in the argument of W is the derivative of y y,i = ∂x i with respect to x1 , the second column is the derivative of y with respect to x2 and the third column is b. All these results are summarized in Theorem 3.1 below. The functions y and b are independent of the thickness direction, as is the energy e0 . They thus yield a simpler two-dimensional eﬀective energy whose minimizers approximate those of the fully three dimensional

22

K. BHATTACHARYA hb

y

he^3

Figure 4. The deformation of a released thin ﬁlm is deﬁned by two vector ﬁelds y which describes the mid-plane deformation and b which deﬁnes the deformation of the thickness.

theory based on eh. This is therefore the appropriate theory to study the microstructure and behavior of thin ﬁlms. The eﬀective energy e0 consists of interfacial energy (term with co-eﬃcient κ2 ) and the energy of stretching and shearing the ﬁlm. The latter is described by the same energy density W as in the fully three-dimensional theory. It is important to note e0 does not contain any terms that penalize bending – these are of much smaller order. Before applying this theory to martensitic thin ﬁlms, it is necessary to examine the interfacial energy a little more carefully. The above treatment kept the interfacial energy coeﬃcient κ constant and ﬁnite, but let the thickness of the ﬁlm h to go to zero. This is appropriate when κ >> h. But in martensitic materials, κ is also small (that is the reason microstructure is so ﬁne!), much smaller than the lateral extent of the ﬁlm (the characteristic length of S). In fact κ can even be comparable to the thickness of the ﬁlm. It is necessary therefore to revisit the asymptotics and let both κ and h go to zero. Shu [39] has found a most remarkable result here: as both κ, h → 0, the eﬀective theory is independent of the ratio of κ/h: it can be zero, ﬁnite or inﬁnite! He has shown that energy minimizing deformations still have ¯ minimize the energy ¯, b the form (3.2) where y

e˜0 (y, b) = S

W (y,1 |y,2 |b)dx1 dx2

(3.4)

amongst all functions y, b : S → IR3 subject to appropriate boundary conditions (see Theorem 3.2 below). This eﬀective energy (3.4) will be used in Sections 3.2 and 3.3 to study the microstructure and behavior of thin ﬁlms. Notice that e˜0 does not contain any derivatives of b. Therefore, one can minimize it out of the problem and replace W with W0 as the integrand of e˜0 where W0 (f |g) = min W (f |g|b). b

23

THIN FILMS OF ACTIVE MATERIALS

It turns out that if W is frame-indiﬀerent, then W0 : IR3×2 → IR is not quasiconvex [35]. This reﬂects the physical fact that very thin ﬁlms can wrinkle or fold on a very ﬁne scale. Therefore one must relax the resulting functional by replacing W0 with QW0 , the quasiconvex envelope of W0 . In summary, min e˜0 = min e¯ y

y,b

e¯(y) =

S

where

QW0 (y,1 |y,2 )dx1 dx2 .

(3.5)

This subsection ends with a precise statement of the results above. It is convenient to make the change of variables shown in Figure 2: x3 z1 = x1 , z2 = x2 , z3 = (3.6) h ˜ : Ω1 → IR3 as and deﬁne the deformation of the scaled domain u ˜ (z1 , z2 , z3 ) = u(z1 , z2 , hz3 ). u

(3.7)

In the new variables, eh is written as:

h

e (˜ u) =

Ω1

κ

2

˜ |2 |∇2pu

2 1 ˜ ,3 |2 + 4 |˜ + 2 |∇pu u,33 |2 h h 1 ˜ ,1 |˜ ˜ ,3 | +W u u,2 | u dz h

(3.8)

where the commas now denote partial diﬀerentiation with respect to zi. The following theorems hold. Theorem 3.1 (Bhattacharya and James [10]). Suppose W satisﬁes the growth conditions c1 (|F|2 − 1) ≤ W (F) ≤ c2 (|F|p + 1)

(3.9)

for some c1 , c2 > 0 and for some 3 < p < 6. Then eh deﬁned in (3.8) has ˜ h in W 2,2 (Ω1 ; IR3 ) subject to the boundary condition a minimizer u ˜ = y0 (z1 , z2 ) + u

N hiz3i i=1

h h . bi(z1 , z2 ) on ∂S × − , i! 2 2

Further {˜ uh} has subsequence (not relabeled) such that ¯ ˜h → y u

in W 2,2 (Ω1 ; IR3 ),

1 h ¯ ˜ → b u h ,3

in W 1,2 (Ω1 ; IR3 ),

1 h ˜ → 0 u h2 ,33

in L2 (Ω1 ; IR3 ),

(3.10)

24

K. BHATTACHARYA

¯ are independent of z3 and minimize the limiting as h → 0 where (¯ y, b) 0 energy e deﬁned in (3.3) in W 2,2 (Ω1 ; IR3 ) × W 1,2 (Ω1 ; IR3 ) subject to the boundary conditions y = y0 b = b1

on ∂S, on ∂S.

(3.11)

Theorem 3.2 (Shu [39]). Suppose W satisﬁes the growth condition (3.9) and let κ = κ(h), limh→0 κ = 0 and limh→0 κ/h is well-deﬁned (though possibly zero or inﬁnite). Also let eh, e˜0 , e¯ be as deﬁned in (3.8), (3.4) and (3.5) respectively. Then, u) = min e˜0 (y, b) = min e¯(y). lim min eh(˜

h→0

˜ u

(y,b)

y

˜ ∈ W 2,2 (Ω1 ; IR3 ) subject to (3.10), y ∈ W 1,q (S; IR3 ) subject to Above, u (3.11)i and b ∈ Lq (S; IR3 ). 3.2. MICROSTRUCTURE IN MARTENSITIC THIN FILMS

The derivation above shows that the behavior of a thin ﬁlm is described by vector ﬁelds y and b (see Figure 4) that minimize the eﬀective energy e˜0 deﬁned in (3.4). The energy density W that appears in e˜0 is the same Helmholtz free energy density that appears in the three-dimensional theory of martensite. It has a multiwell structure as described earlier, and is minimized on matrices of the form RUi where R is a rotation matrix and Ui are given for a material. Therefore, energy minimizing deformations are those that satisfy (y,1 |y2 |b) = RUi for some R and i which may vary with position (x1 , x2 ) in the ﬁlm. 3.2.1. Deformations with a single phase In a single phase deformation, (y,1 |y,2 |b) = RUi for some ﬁxed Ui. Suppose for a moment that Ui = I. Then at each point of the ﬁlm (y,1 |y,2 ) must be the ﬁrst two columns of a rotation matrix. There is an exact characterization of functions y that satisfy this condition: they are the so-called ‘isometric mappings of the plane’ and are all the deformations that one can illustrate by deforming a ﬂat sheet of paper (simple examples are deformations that take a ﬂat sheet into a cone or cylinder). The function b is chosen as y,1 × y,2 . If Ui = I, then deform the ﬁlm uniformly through Ui and then perform the deformation described above as shown in Figure 5. Thus this theory allows a very rich class of deformations with a single variant and captures naturally the ﬂoppiness of thin ﬁlms.

THIN FILMS OF ACTIVE MATERIALS

25

or

U R(z1,z 2) Figure 5. The thin ﬁlm theory allows a very rich class of energy minimizing deformations with a single variant: distort the ﬁlm uniformly by the relevant matrix I, U1 , . . . , or Uk and then perform any deformation that one can illustrate by deforming a ﬂat sheet of paper (simple examples are deformations that take a ﬂat sheet into a cone or cylinder). ^ e

3

^ n

(y,1 | y,2 |b) = RF

^ e

(y,1 | y,2 |b) = G

Figure 6. A two-phase deformation where the distortion jumps across an interface with ˆ is compatible if and only if a line e ˆ in the plane of the ﬁlm is deformed equally normal n by both sides.

3.2.2. Deformations with two variants or two phases: interface conditions More interesting are deformations that involve both phases, since they have the possibility of exhibiting a thermally–induced change of shape. Consider an interface that separates two variants or two phases in a ﬁlm with normal ˆ3 shown schematically in Figure 6. Here, F, G are matrices in the set e {I, U1 , . . . , Uk }. A two–phase deformation y of this type is continuous if and only if the two sides deform the interface equally. In other words, the following invariant line condition must hold: ˆ·e ˆ3 = 0 (RF − G)ˆ e=0 e

(3.12)

ˆ. The energetic argument in Section 2.1 implies that a thin ﬁlm for some e can overcome any incoherence in the thickness direction at the cost of a small elastic energy (of order h2 , in contrast to the membrane energy which is of order h). Bhattacharya and James [10] have shown that given F, G and the norˆ 3 , one can ﬁnd a rotation R and a unit vector e ˆ that satisfy mal to the ﬁlm e

26

K. BHATTACHARYA

(3.12) if and only if ˆ3 · (cof A)ˆ e3 ≤ 0 e

(3.13)

where A = FT F − GT G and cof A denotes the matrix of cofactors of A. ˆ and (ii) If (3.13) holds, then one can ﬁnd (i) two independent directions e ˆ that satisfy (3.12). a one-parameter family of Q for each e To understand (3.13) consider a unit square and stretch it by an amount λ1 > 0 along one side and an amount λ2 > 0 on another. It is intuitively clear that it is possible to ﬁnd an invariant (unstretched) line if and only if one of the two stretches is greater than one while the other is less than one, i.e., if and only if (λ1 − 1)(λ2 − 1) ≤ 0. The equation (3.13) expresses exactly this condition in terms of the given distortion matrices F and G ˆ3 . and the ﬁlm normal e Consider an austenite-martensite interface as an illustration of this interface condition. Unlike in bulk material, it is typically possible to form an exact austenite-martensite interface in a thin ﬁlm [10, 27, 9]. For example, variant 1 of martensite can form an exact interface with the austenite on a − (001)c ﬁlm of a material undergoing cubic to tetragonal transformation if (α2 − 1)(β 2 − 1) ≤ 0; Ni64 Al36 and Ni2 MnGa satisfy this condition and the interface direction (ˆ e) is (0.5462, ±0.8377, 0)

and

(0.5236, ±0.852, 0)

respectively. − (001)c ﬁlm of Cu-Al-Ni and Cu-Zn-Al alloy with the interface direction (ˆ e) equal to (−0.225, 0.9744, 0)

and

(−0.2696, 0.9630, 0)

respectively. In fact, at special compositions these alloy satisfy condition (3) with an equality and can form an exact austenite-martensite interface on any ﬁlm. Such interfaces can be utilized to induce some unusually large changes of shape in a ﬁlm that would be compromised by the ﬁne twinning that almost always occurs during bulk transformation. 3.3. TENTS AND TUNNELS

The variety of interfaces that a martensitic thin ﬁlm can form enable a novel strategy for making micropumps. The idea is to deposit a ﬁlm on a

THIN FILMS OF ACTIVE MATERIALS

Figure 7.

27

Tunnel and tent.

substrate, then release it in some region and look for the following behavior: the ﬁlm is ﬂat in one phase or variant (top of Figure 7), while it bulges up to a tunnel or a tent (bottom of Figure 7 ) as it transforms to another, perhaps under some back pressure (see James and Rozzini [30] for an analysis of pressurized ﬁlms). The crystallography and the region to be released are to be chosen carefully so that each ridge in the ﬁgure is a material interface and the deformations are energy minimizing. If this is possible, then it can be exploited to make micropumps, microvalves and other micromachine actuators. Such a tunnel is possible if and only if the two phases are compatible (in the thin ﬁlm sense) across an interface and the relative deformation is a pure stretch normal to the interface [10]; this in turn is equivalent to the conditions, ˆ3 · (cofA)ˆ e e3 = 0

and

ˆ3 · (A)ˆ trace(A) − e e3 > 0.

(3.14)

where A = FT F−GT G for F, G relevantly chosen in the set {I, U1 , . . . , Uk}. Bhattacharya et al [9] have used this idea to propose a magnetic ﬁeld induced tunnel using the ferromagnetic shape-memory alloy Ni2 MnGa and electric ﬁeld induced tunnel using the ferroelectric materials BaTi03 and PbTiO3 . Similarly, it is possible to form a n-sided pyramidal tent [10] with 1. faces consisting of martensite variants, surrounded by ﬂat austenite, if the ﬁlm is a plane of n-fold symmetry of the austenite and (3.14) is satisﬁed with G = I and F = U1 or,

28

K. BHATTACHARYA

Figure 8.

A heterogeneous ﬁlm with three distinct length-scales.

2. faces consisting of martensite variants i, j, . . ., surrounded by ﬂat martensite variant m, if the ﬁlm is a plane of n-fold symmetry of the martensite variant m and (3.14) is satisﬁed with G = Um and F = Ui. Hane [27] (also see [9]) has shown that it is possible to form a four-sided tent on a (001)c ﬁlm of Cu-Zn-Al by releasing a square region which is oriented at 15.6◦ to the (100)c direction. Cui and James [17] have shown that it is also possible to form a four-sided tent on a (001)c ﬁlm of a specially heattreated Cu-Al-Ni ﬁlm. Their preliminary experimental test using Cu-Al-Ni foils is very encouraging. 4. Heterogeneous thin ﬁlms 4.1. THE GENERAL PROBLEM

Consider the heterogeneous thin ﬁlm shown in Figure 8. It occupies a reference region Ωh with lateral extent S and thickness h as before, but now the ﬁlm consists of multiple layers and each layer consists of multiple grains. Assume that the the in-plane texture is periodic with period d (in other words, the representative area element in the plane has size d). Therefore the properties of the ﬁlm, in particular the free energy density, depends on position:

W = W F,

x1 x2 x3 , , d d h

where it is convenient to normalize the in-plane variables (x1 , x2 ) by d and out-of-plane variable x3 by h. The total energy per unit thickness of the

THIN FILMS OF ACTIVE MATERIALS

29

heterogeneous thin ﬁlm under the deformation u : Ωh → IR3 is eh(u) =

1 h

Ωh

κ2 |∇2 u|2 + W (∇u,

x1 x2 x3 , , ) dx. d d h

(4.1)

As before the ﬁrst term is a Van der Waals type interfacial energy which is taken to be homogeneous for convenience. There are three important length-scales in the problem: the thickness of the ﬁlm h which is also comparable to the layer thickness, the typical grain size d, and if any layer is martensitic the size of the domain which is governed by κ. These three length-scales might have diﬀerent ratios between them, but they are all much smaller than the lateral extent of the ﬁlm. Therefore one can ﬁnd an eﬀective theory for the ﬁlm in the limit when each of κ, d and h tend to zero, but with possibly diﬀerent limiting ratios. In order to do so take κ = κ(h) > 0, d = d(h) > 0,

with

lim κ(h) = lim d(h) = 0,

h→0

h→0

and study the limiting behavior of eh as h → 0. This behavior depends on the limiting ratios α = lim

h→0

κ , d

β = lim

h→0

h κ , α = lim h→0 d h

(4.2)

each of which can be zero, ﬁnite or inﬁnite. Shu [39] has used the framework of Γ-convergence to show that the limiting behavior of the ﬁlm is determined by an eﬀective two-dimensional theory. The deformation of the ﬁlm is determined by one vector ﬁeld y : S → IR3 which one can regard as the mid-plane deformation as shown in Figure 4, and the eﬀective energy of ﬁlm is given by

0

e (u) =

S

¯ W

∂u ∂u dx1 dx2 . dx1 dx2

(4.3)

¯ is the eﬀective energy density which only depends on the in-plane graW ¯ describes the dient of deformation u and not explicitly on the position. W overall behavior of the heterogeneous thin ﬁlm after taking into account the martensitic microstructure, grains and multilayers. It is obtained from W , and depends critically on the limiting rations α, β and α . It is instructive to consider some simple examples before considering the general problem. 4.2. CONDUCTIVITY IN A STRIP

Consider a (two-dimensional) strip shown in Figure 9. It has thickness h and length L, and is made up of two types of grains – grey grains with

30

K. BHATTACHARYA

Q

h

P λd Figure 9.

θ

θ

(1−λ)d A strip made of two types of grains.

volume fraction λ and white grains with volume fraction (1 − λ) – arranged periodically with period d. Suppose the grains are anisotropic with conductivity matrix Pij and Qij in the grey and white grains respectively. If this strip is very thin (h/L small) and there are many grains (d/L small) then the eﬀective behavior of this strip is one-dimensional and one can describe ¯ conduction in this strip with one eﬀective scalar conductivity constant C. Shu [39] has shown that this eﬀective conductivity can critically on the aspect ratio of the grains β = h/d, so that C¯ = C¯ β . If the grains are tall and narrow (h >> d or β = ∞), then a typical point in the interior of the grain does not see the strip-like geometry but the other grains. So it behaves as in bulk. Therefore one has to ﬁrst homogenize over the grains in two dimensions, and then consider a strip of this homogenized materials. In contrast if the grains are ﬂat and wide (h << d or β = 0), then a typical point in the interior of the grain sees the strip-like geometry but not the other grains. Therefore one has to ﬁnd the behavior of thin strips of each grain and then homogenize in one dimension. The basic idea can be understood by considering the strip to be a polycrystal with grey and white grains as also shown in Figure 9. Each grain has a high conductivity and low conductivity direction indicated respectively by the long and short lines in the ﬁgure. The high conductivity direction is oriented at some angle θ in the grey grains, and at an angle −θ in the white grains. If the grains are tall and narrow (h >> d or β = ∞), the current through this strip can zig-zag always taking advantage of the high conductivity direction in the bulk of the strip with two narrow boundary layers at the top and the bottom (see Figure 10 for an especially simple situation when λ = 1/2). Thus the eﬀective conductivity is high in these strips. If on the other hand the grains are ﬂat and wide ((h < d or β = 0), the boundary layer engulfs the entire strip forcing the current to ﬂow in low conductivity directions, and thus the eﬀective conductivity is small. For the thin strip shown in Figure 9, the eﬀective conductivity C¯ β satisﬁes [39] C¯ 0 ≤ C¯ β ≤ C¯ ∞

for 0 < β < ∞

31

THIN FILMS OF ACTIVE MATERIALS

Figure 10.

A schematic of the current in a ﬁlm with tall and narrow grains.

where C¯ 0 is the eﬀective conductivity when the grains are very ﬂat ( in the limit β = h/d → 0) and C¯ ∞ is the eﬀective conductivity when the grains are very long (in the limit β = h/d → ∞). These are given by the formulas C¯ 0 =

¯ P¯ Q ¯ + (1 − λ)P¯ λQ

C ∞C ∞ − C ∞ C¯ ∞ = 11 22 ∞ 12 , C22 2

where C∞ = (λP + (1 − λ)Q · S) · (λI + (1 − λ)S)−1 ,

S=

P11 Q11

0

P12 −Q12 Q11 ,

P11 P22 − P¯ = P22

1

2 P12

=

∆P P22

2 ¯ = Q11 Q22 − Q12 = ∆Q . and Q Q22 Q22

Further, if Q12 ∆P − P12 ∆Q = 0, then C¯ 0 = C¯ β = C¯ ∞ . Finally, if the volume fraction λ = 12 , then 

|P12 | ¯ 0 P22

|Q12 | Q22

−1

C¯ 0 ≤ C¯ β ≤ C¯ 0 1 − β C ( ¯ + ¯ ) P Q

C¯ ∞ ≥ C¯ β ≥ C¯ ∞

| Q12 ∆P − P12 ∆Q | 1− 2 β (P11 ∆Q + Q11 ∆P )

for β small,

for β large.

This example clearly illustrates the competition between the aspect ratio of the grains in a ﬁlm with columnar texture. There is clearly an analogous calculation that one can do for elasticity. The elastic modulus of ﬁlms

32

K. BHATTACHARYA ^

W

W

-1 f2

f1

1

-1 f2

(a) _ W

β*

ε*

1

f1

(b)

ε

β ε

-ε

(c)

f

β*

β

(d)

Figure 11. The behavior of a strip depends on the aspect ratio β = h/d. (a) The (microscopic) energy of a single crystal before it forms microstructure. (b) The (mesoscopic) energy of a single crystal after it forms microstructure. (c) The eﬀective energy of the strip for various aspect ratios: strips with critical aspect ratio less than β ∗ have a recoverable strain , but strips with aspect ratio greater than β ∗ have no recoverable strain. (c) The recoverable strain decreases with increasing aspect ratio and becomes zero beyond the critical value β ∗ = 12 cot θ.

with long columnar grains would be much larger than the elastic modulus of ﬁlms with ﬂat pan-cake shaped grains. There is some experimental evidence consistent with this observation. 4.3. A MODEL PROBLEM FOR SHAPE-MEMORY FILMS

Consider once again the strip shown in Figure 9. Each of the grains in the strip is made up of a model phase transforming material. The microscopic energy density of a single crystal of the model material is shown in Figure 11a: the deformation u in this model material is scalar so that the deformation gradient is a two-vector f = ∇u = (u,1 , u,2 ) = (f1 , f2 ), and the material has two variants with transformation matrix (1, 0) and (−1, 0). Assume that the interfacial energy is negligible (κ = 0) so that the material can freely form microstructure in each grain. The behavior of ˆ which the grain is then determined by the mesoscopic energy density W is the energy of the grain after taking account the microstructure and is shown in Figure 11b. Notice that this energy is zero on the line joining the

THIN FILMS OF ACTIVE MATERIALS

33

two variants. These are the average deformation gradients one can make by mixing the variants, and are exactly the strains that are recoverable by the shape-memory eﬀect [12]. Shu [39] has calculated the eﬀective behavior of a strip made up of two types of grains, the grey grains where the f1 -axis is oriented at an angle θ and the white grains where it is oriented at an angle −θ, and has found that it depends qualitatively on the aspect ratio β = d/h of the grains. The eﬀective behavior of the strip is one-dimensional and the energy governing it is shown in Figure 11c for the case λ = 12 . If the aspect ratio β is smaller than some critical value β ∗ , then the eﬀective energy is zero on some segment (−, ) and grows away from it. is the maximum recoverable strain of the strip, and depends on β as shown in Figure 11d. If on the other hand, β is greater than β ∗ , the recoverable strain is zero and in fact the eﬀective energy is quadratic. This means that the strip behaves like an elastic material – with no shape-memory – even though each of the grains is made up of a shape-memory material. β ∗ and ∗ (the maximum recoverable strain when β = 0) depend on the orientation θ: ∗ = cos θ

β∗ =

1 cot θ. 2

This example again shows the eﬀect of the aspect ratio of the grains in columnar ﬁlms. For any given texture, ﬁlms with ﬂat pan-cake shaped grains have better shape-memory property than those with long rod shaped grains. 4.4. RECOVERABLE STRAINS IN POLYCRYSTALLINE SHAPE-MEMORY FILMS

The observations of Section 4.3 carry over to actual models of shapememory ﬁlms. It is very diﬃcult to calculate the exact eﬀective properties, however it is possible characterize the recoverable strain using bounds and estimates following Shu and Bhattacharya [42] in a geometrically linear theory. Shape-memory thin ﬁlms are often made by sputtering [25, 28, 44, 34, 15, 2], and the grains are typically columnar (see for example Figure 3 of [28]). Further, the microstructure is usually smaller than the grains (see for example Figure 5 in [34]). Table I contrasts the behavior of ﬁlms with long or rod-like (h >> d) grains and ﬁlms with ﬂat or pan-cake shaped (h << d) grains. It lists the estimated recoverable strains for ﬁlms with diﬀerent textures in Ti-Ni and Cu-Zn-Al. Note that they are larger for ﬂat grains compared to long grains. Also note that neither the random nor {110} texture which is commonly obtained by sputtering Ti-Ni and Cu-Zn-Al [25, 44] are ideal textures for large recoverable strain. The ideal

34

K. BHATTACHARYA TABLE 1. The estimated uniaxial recoverable extension for various textures in thin ﬁlms. Recoverable Strains (%)

Texture

Ti-Ni

Cu-Zn-Al

long

ﬂat

long

ﬂat

random

2.3

2.3

1.7

1.7

{111} ﬁlm

5.3

8.1

1.9

5.9

{100} ﬁlm

2.3

2.3

7.1

7.1

{110} ﬁlm

2.3

2.3

1.7

1.7

i

ε (polycrystal)

S2

(k)

R

e

e Pi

loading direction

(1)

S1

e

(2)

ε (grain 1) R

Figure 12.

Recoverable strains in a polycrystal made of two grains.

textures appear to be {100} for Cu-Zn-Al (this texture can be produced by melt-spinning) and {111} for Ti-Ni. These predictions above are based on a model of estimating the recoverable strain in shape-memory polycrystals [11, 12, 39]. Consider a single crystal. The martensite has k variants with transformation strains e(1) , ..., e(k) as shown schematically in Figure 12. The set of recoverable strains in a single crystal, S, is the set of strains that one can make by coherently mixing variants of martensite. This set is known exactly for most materials [11], and shown schematically as the rectangle S1 in Figure 12. The maximum recoverable strain R in some loading direction is given by the maximum projection of the set S in that direction. Now consider a bulk polycrystal made of N grains with orientations R1 , R2 , · · · , RN and volume fraction λ1 , λ2 , · · · , λN . Each grain has its own set of recoverable strains Si = Ri S RiT which is obtained by the rotation of the basic set S through Ri (Figure 12). The set of recoverable strains in

THIN FILMS OF ACTIVE MATERIALS

35

this polycrystal are diﬃcult to calculate, but one can obtain a surprisingly good estimate by considering the inner bound Pi, the set that is obtained from the intersection of all diﬀerent sets Si (the shaded region in Figure 12). The inner bound of the maximum recoverable strain iR in the given loading direction is the maximum projection of the set Pi in that direction. Further, iR can be determined by solving a linear programming problem which is easy to compute. For example, for uniaxial tension in the direction ξ, iR =

max

RjT eRj ∈S ∀ j=1,···,N

(ξ · eξ).

(4.4)

Let us now return to the thin ﬁlm with columnar grains. Consider ﬁrst the case when the grains are ﬂat and pan-cake shaped (h << d or β = 0). The results of Section 4.2 and 4.3 suggest that one to ﬁrst obtain the behavior of thin ﬁlms of each grain and then homogenize in the plane. Section 2 showed that in thin ﬁlms, the martensitic microstructure is governed by an ‘invariant line condition’ (in contrast to the bulk which is governed by the invariant plane condition). Therefore the set of recoverable strains of a single crystal thin ﬁlm is given not by S but by S f . In the geometrically linear theory one the invariant line condition implies that variants i and j are compatible if and only if the in-plane projections of the transformation strain are compatible, or equivalently if f · cof(e(i) − e(j) )f ≤ 0 where f is the ﬁlm normal relative to the cubic basis and “cof e” denotes the cofactor of the matrix e. S f coincides with S for materials undergoing cubic-tetragonal, cubic-trigonal and cubic-orthorhombic transformation. For cubic-monoclinic martensites, S f depends on the ﬁlm normal f , is usually larger than the bulk S, and may be explicitly determined for {100}, {110} and {111} ﬁlms (see [38]). Having obtained the set of recoverable strains for a single crystal ﬁlm, the inner bound on the recoverable strain is again obtained by (4.4); but by replacing the set S with a set S f . Now consider the other extreme case of long and rod shape grains (h >> d or β = ∞). The eﬀective behavior is obtained by homogenizing in three dimensions and then taking the thin-ﬁlm limit. The inner bound on the recoverable strain is obtained from (4.4). Thus, the recoverable strains for such ﬁlms is essentially the same as bulk materials. 4.5. SMA/ELASTIC MULTILAYERS

Consider a multilayer ﬁlm made of alternating layers of an elastic material and a shape-memory alloy. Suppose there is a mismatch strain of M between elastic material and the high temperature austenite phase. Then, Shu

36

K. BHATTACHARYA _ ϕ

_ ϕ

(a) T>T c κ >> h

εa _ ϕ

ε

T>T c κ << h

ε M

(b) T> h

εm

(c)

εa _ ϕ

ε

ε M

(d) T
ε

εm

ε

Figure 13. The behavior of a shape-memory alloy/elastic multilayers. The bold line is the eﬀective energy density of the multilayer, the light continuous line that of the shape-memory material and the light broken line that of the elastic material. (a) and (b) show the behavior when the layer thickness is much smaller than domain width: notice the strong two-way shape-memory eﬀect with strain a at the high temperature and m at the low temperature. (c) and (d) show the behavior when the layer thickness is much larger than the domain width: notice the weak two-way eﬀect.

[39] has shown that the eﬀective behavior is very interesting and depends very strongly on the ratio between the microstructure scale κ and the ﬁlm (layer) thickness h. The results are shown schematically in Figure 13. If the microstructure is very large compared to ﬁlm (layer) thickness κ >> h, then this multilayer shows a strong two-way shape-memory eﬀect. The basic idea is that large interfacial energy prevents the shape-memory material from forming any microstructure, and thus there is no relaxation of mismatch strains. This is shown in Figs. 13a and 13b. At high temperature (Figure 13a), the energy density of the shape-memory material is shown as the light continuous line, and the elastic layer as the light dashed line. Notice that the shape-memory material has multiple wells with a deep well at the austenite characterized by a stress-free strain 0 while the elastic material has a quadratic energy with stress-free strain of M (due to the mismatch). The eﬀective energy of the ﬁlm at this high temperature is given by the average (weighted by the volume fraction) and is shown as the solid continuous line. Notice that it has a strong minimum at a. Thus at high temperature, the ﬁlm has overall strain a. At low temperature (Figure 13a), the energy density of the shape-memory material is again shown as the light continuous line, and the elastic layer as the light dashed line. Notice

THIN FILMS OF ACTIVE MATERIALS

37

that the shape-memory material has multiple wells with equally deep wells at the two variants of martensite while the elastic material has a quadratic energy with stress-free strain of M . 1 The eﬀective energy of the ﬁlm at this low temperature is shown as the solid continuous line, and this has a strong minimum at m since the mismatch strain biases one martensitic variant over the other. Thus at low temperature, the ﬁlm has overall strain m. Therefore, cycling the temperature causes the strain to cycle between a and m, thus showing a two-way eﬀect. Further the moduli at a and m – given by the curvature of the eﬀective energies at these points – is large, and therefore the two-way eﬀect is very strong. Contrast this with the situation when the microstructure is very small compared to ﬁlm (layer) thickness κ << h. This is shown schematically in Figs. 13c and 13d. The shape-memory material can readily form microstructure, and thus can relax mismatch strains. Thus the energy of the martensitic material both in the high and low temperature states is given by the (quasi) convexiﬁcation of the multiwell energy, and the eﬀective energy of the ﬁlm is (quasi) convex with small modulus (curvature). The diﬀerence between overall strain at high temperature a and low temperature m is small and the moduli is also very small. Therefore, the two-way eﬀect is weak. In summary multilayers of shape-memory alloy and elastic materials can be used to obtain a two-way shape-memory eﬀect. Further the layer thickness must be smaller than the typical size of the martensitic twins, for this eﬀect to be strong. Various groups have used a similar idea to obtain a two-way eﬀect. They typically use two layers – one elastic and one shape-memory – and the ﬁlm rolls up quite dramatically due the asymmetry thereby relaxing some of the mismatch. Unfortunately the ﬁlm has negligible bending stiﬀness and this large displacement can not be used to provide actuation. However, the calculations here shows that these multilayers can be used in a longitudinal mode for produce actuation with large energy. 4.6. HETEROGENEOUS THIN FILMS

The examples in Sections 4.2 and 4.3 showed the interplay between the ﬁlm thickness h and grain size d, while the example in Section 4.5 showed the interplay between ﬁlm thickness and microstructure scale κ. This subsection summarizes the results of Shu [39] for the general situation described in Section 4.1. Recall that the energy per unit thickness of the ﬁlm is given by eh in (4.1), and depends on three small scales h, and κ and d. In the 1 The mismatch strain M depends of temperature but this is small compared to the transformation strain and can be neglected.

38

K. BHATTACHARYA TABLE 2. Summary of the eﬀective behavior of a heterogeneous thin ﬁlm. κ >> d κ∼d κ << d

AT T

A H

TH h << d

AT A H T T H h∼d

AT A T H HT h >> d

limit that all these scales go to zero, the behavior of the ﬁlm is described ¯ depends by the eﬀective energy e0 in (4.3). The eﬀective energy density W critically on the ratios α, β, α of the length-scales deﬁned in (4.2). ¯ for the many diﬀerent cases. A Table II explains how one can obtain W denotes averaging, H homogenizing and T means taking the thin-ﬁlm limit. ¯ is obtained by ﬁrst taking TH denotes that the eﬀective energy density W thin-ﬁlm limit, and then homogenizing in the plane of the ﬁlm. On the other hand, HT denotes homogenization ﬁrst followed by thin-ﬁlm limit. A Finally, a stacked symbol like H denotes performing these two operations simultaneously. First consider the case when the interfacial energy is very large or equivalently when the microstructure scale κ is larger than the other two scales (κ >> h, d or α, α = ∞). This is the ﬁrst row of Table II. In this situation the interfacial energy prevents any strain oscillations, and the eﬀect of heterogeneities is completely smeared out. Therefore, one should average W over all the heterogeneities to obtain a homogeneous material, and then ﬁnd the behavior of a thin ﬁlm of this homogeneous material. In this situation, the ratio of h and d is simply irrelevant. Second consider the case when the interfacial energy is very weak compared to the grain size (κ << d or α = 0). This is the third row of Table II. In this situation, the ratio between h and d is very important. If the grains are ﬂat (h << d or β = 0), then one has to ﬁrst ﬁnd the thin-ﬁlm behavior of each single crystal and then homogenize in the plane. If on the other hand the grains are tall (h >> d or β = ∞), then one has to homogenize in three dimensions and then ﬁnd the behavior of a ﬁlm of this homogeneous materials. All the remaining cases are when some set of length-scales become comparable: then there is no easy way of describing the eﬀective energy. The remainder of this section makes these results precise using the framework of Γ-convergence. In order to do so, recall the change of variables

39

THIN FILMS OF ACTIVE MATERIALS

(3.6) and (3.7) (Figure 2) and rewrite

h

e (y) =

κ

Ω1

2 1 + 2 |∇py,3 |2 + 4 |y,33 |2 h h 1 z1 z2 ˜ ,3 |, , , h +W y,1 |y,2 | u dz h d d

|∇2py|2

2

for any y ∈ W 2,2 (Ω1 ; IR3 ). It is necessary to extend this deﬁnition to y ∈ W 1,2 (Ω1 ; IR3 ) by

eh(y) =

eh(y) +∞

if y ∈ W 2,2 (Ω1 ; IR3 ), otherwise.

It is also necessary to extend the deﬁnition of e0 to W 1,2 (Ω1 ; IR3 ) as

0

e (y) =

¯ +∞

S W (∇py)dzp

if y ∈ V, otherwise

where zp = (z1 , z2 ) and V = {y : y ∈ W 1,2 (Ω1 ; IR3 ) and y,3 = 0 for a.e. z in Ω1 }. Assume, 1. W (F, z) is Carath´eodory and non-negative. 2. There exist c1 , c2 > 0 and 3 < p < 6 such that for all F ∈ IM 3×3 and a.e. z ∈ IR2 × (0, 1), c1 (|F|p − 1) ≤ W (F, z) ≤ c2 (|F|p − 1) 3. There exists c > 0 such that for all F, G ∈ IM 3×3 and a.e.z ∈ IR2 × (0, 1), |W (F, z) − W (F, z)| ≤ c(1 + |F|p−1 + |G|p−1 )|F − G|. ¯ ∈ IR3 and Z = (0, 1)2 . The following Introduce the following notation: F theorems are due to Shu [39]. z

Theorem 4.1 (Strong interfacial energy). Let W = W (F, dp , z3 ), and κh → ∞ as h → 0. Then, eh Γ-converges to e0 as h → 0 with

κ d

→∞

¯ = QW ¯ (F) ˜ 0 (F), ¯ W ¯ ˜ (F|b), ¯ = inf W ˜ 0 (F) W b∈IR3

˜ (F) = W

W (F, z)dz; Z×(0,1)

¯ ∈ IM 3×2 and ˜ 0 is the lower quasi-convex envelope of W ˜ 0, F where QW zp 2 Z = (0, 1) . Further, the result also holds if W = W (F, d ) and κd → ∞ or W = W (F, z3 ) and κh → ∞.

40

K. BHATTACHARYA

Theorem 4.2 (Columnar grains and weak interfacial energy). Suppose z W = W (F, dp ), κd → 0 and hd → β as h → 0. Then, eh Γ-converges to e0 as h → 0 with: (i) Flat grains. β = 0: ¯ ¯ = QW H (F), ¯ (F) W 0 ¯ = inf W H (F|b), ¯ W0H (F) b∈IR3

H

W (F) =

inf

− W (F + ∇ω, zp)dz,

inf

ˆ k∈IN ω∈W 1,p (kZ) ˆ kZ 0

¯ is the lower quasi-convex envelope of W H , Zˆ = (0, 1)3 where QW0H (F) 0 ¯ and F = (F|b) ∈ IM 3×3 . (ii) Tall grains. β = ∞: ¯ = ¯ (F) W ¯ 0 , z0 ) = W0 (F p

¯ + ∇pω, zp)dzp, − W0 (F

inf

inf

k∈IN ω∈W 1,p (kZ) kZ 0

¯ 0 |b, z0 ). inf W (F p

b∈IR3

(iii) Comparable grains. 0 < β < ∞: ¯ = inf ¯ (F) W

¯ + ∇p ω|ω,3 , zp)dz inf − W (F

˜1 Ω1 k∈IN ω∈A k k

where Ω1k = kZ × (0, 1) and A˜1k = {ω ∈ W 1,p(Ω1k ; IR3 ) : ω|∂kZ×(0,1) = 0}.

Above −Ω · · · =

1 |Ω| Ω · · ·.

Theorem 4.3 (Homogeneous layers ). Suppose W = W (F, z3 ), h → 0. Then, eh Γ-converges to e0 as h → 0 with:

κ h

→ α as

(i) Weak interfacial energy. α = 0: ¯ = inf ¯ (F) W

¯ + ∇p ω|ω,3 , z3 )dz inf − W (F

˜1 Ω1 k∈IN ω∈A k k

where Ω1k = kZ × (0, 1) and A˜1k = {ω ∈ W 1,p(Ω1k ; IR3 ) : ω|∂kZ×(0,1) = 0}. (ii) Strong interfacial energy α = ∞: ¯ ¯ = QW ¯ (F) ˜ 0 (F), W ¯ ¯ = inf W ˜ 0 (F) ˜ (F|b), W ˜ (F) = W

b∈IR3 1 0

W (F, z3 )dz3 ,

˜ 0 is the lower quasi-convex envelope of W ˜ 0. where QW

41

THIN FILMS OF ACTIVE MATERIALS

(iii) Moderate interfacial energy ∞ > α > 0:

¯ = inf ¯ (F) W

inf −

˜1 Ω1 k∈IN ω∈A k k

¯ + ∇pω|ω,3 , z3 ) dz α2 |∇2 ω|2 + W (F

with Ω1k, A˜1k as above. z

Theorem 4.4 ([39] All scales comparable). Suppose W = W (F, dp , z3 ), κ κ h d → α and d → β as h → 0 with 0 < α, β < ∞. Then, e Γ-converges to 0 e as h → 0 with ¯ = inf ¯ (F) W

inf −

k∈IN ω∈Aβ Ωβ k k

¯ + ∇pω|ω,3 , dzp, z3 α |∇ ω| + W F β 2

2

2

where Ωβk = kZ × (0, β), Aβk = {ω ∈ ∩W 2,2 (Ωβk ; IR3 ) : ω|Σβ = and Σβ = ∂kZ × (0, β).

dz

∂ω ∂n Σ

= 0}

β

5. Conclusion The study of thin ﬁlms of active materials remains a very active area of research. We conclude by mentioning other interesting results. Our discussion above was motivated by microactuators including micropumps. Yet, we limited ourselves to boundary conditions where the top and bottom surfaces are stress-free. James and Rizzoni [30] considered the problem where the ﬁlm is subjected to pressure on one surface and clamped on the edges. They started from a three dimensional theory with an energy given by that in (3.1) plus the potential energy associated with the applied pressure. They derive a limiting two dimensional theory similar to the result in Section 3.1 with an energy (3.3) plus the potential energy associated with the applied pressure. They further study minimizers of this energy and ﬁnd interesting behavior of the ﬁlm as a function of applied pressure. Shu [41] has extended this theory to heterogeneous materials, and studied the implications for the fabrication of micropumps using sputtered NiTi ﬁlms. Braides, Francfort and Fonseca [13] as well as Shu [41] have used similar ideas to derive eﬀective theories for rough ﬁlms. Further, Shu [40] has used these asymptotic theories to study the morphological and chemical instabilities in thin ﬁlms. We have conﬁned ourselves in this paper to martensitic materials and shape-memory materials. These materials are temperature actuated. This is awkward in practice though feasible. Alternately one can think about ferromagnetic shape memory alloys like Ni2 MnGa or ferroelectric materials like BaTiO3 . These materials possess multiple wells and are capable of large strain actuation [31, 14] just like martensites, but are also magnetically

42

K. BHATTACHARYA

or electrically polarized so that they activated through the application of magnetic or electric ﬁeld. The rigorous derivation of an eﬀective theory for such materials is technically diﬃcult due to the presence of both mechanical deformations and electromagnetic ﬁeld, and remains open. Gioia and James [24] have studied the limiting behavior of a ferromagnetic ﬁlm (with no deformation). Shu and Bhattacharya [43] have studied the limiting behavior of ferroelectric materials under special conditions. Based on these results, there is heuristic insight into the limiting behavior of ferromagnetic shapememory alloys and ferroelectrics. Further, it can be shown that tent-like microstructures exist under special situations in such materials [9]. We have already noted that Cui and James [17] have observed the tentlike microstructure in CuAlNi foils. Encouraged by these results and the calculations in ferromagnetic shape-memory alloys and in ferroelectric materials, there are ongoing eﬀorts at the University of Minnesota and the California Institute of Technology to build actuators based on these ideas. DeSimone, Kohn, M¨ uller and Otto [18] have studied the behavior of soft ferromagnetic ﬁlms where the thickness and the modulus of the anisotropy energy (the analog of our W in ferromagnetism) are small. Finally, recall from Section 3.2.1 that our ﬁlm was free to deform into any developable surface. This was because our limiting theory is a membrane theory that does not penalize the bending. Though Kirchhoﬀ introduced a plate theory over a century and a half ago, a rigorous derivation of such a theory had remained elusive till recently. Friesecke, James and Mu¨ller [23] have recently done so. The key idea is a rigidity theory which provides a sharp estimate on the strain that his necessary for the deformation to deviate from a rigid rotation in addition to ideas of Γ-convergence. Acknowledgments. It is a pleasure to recall the enjoyable collaborations with R.D. James and Y.C. Shu. I am happy to acknowledge the partial support of the National Science Foundation (CMS-9457573), the Air Force Oﬃce of Scientiﬁc Research (MURI: F49620-98-1-0433) and the Army Research Oﬃce (MURI: DAAD19-01-1-0517)

THIN FILMS OF ACTIVE MATERIALS

43

References 1.

Acerbi, E., Buttazzo, G. and Percivale, D. (1991) A variational deﬁnition for the strain energy of an elastic string. J. Elasticity, 25, 137–148. 2. Ahlers, M., Kostorz, G. and Sade, M. (1999) Proceedings of the International Conference on Martensitic Transformations 1998, Mat. Sci. Engng. A 273-275. 3. Anzellotti, G., Baldo, S. and Percivale, D. (1994) Dimension reduction in variational provlems, symptotic development of Γ-convergence and thin structures in elasticity, Asymp. Anal., 9, 61–100. 4. Ball, J.M. and James, R.D. (1987) Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal., 100, 13–52. 5. Ball, J.M. and James, R.D. (1992) Proposed experimental tests of a theory of ﬁne microstructure, Phil. Trans. Royal Soc. London, 338A, 389–450. 6. Bhattacharya, K. (1991) Wedge-like microstructure in martensites, Acta Metall. Mater., 39, 2431-2444. 7. Bhattacharya, K. (1992) Self-accommodation in martensite, Arch. Rat. Mech. Anal., 120, 201-244. 8. Bhattacharya, K. (2003) Microstructure of martensite. Why it forms and how it gives rise to the shape-memory eﬀect. Oxford University Press, in press. 9. Bhattacharya, K., Desimone, A., Hane, K., James, R.D. and Palmstrøm, C.P. (1999) Tents and tunnels on martensitic ﬁlms, Mat. Sci. Engng. A, 273-275, 685–689. 10. Bhattacharya, K. and James, R.D. (1999) A theory of thin ﬁlms of martensitic materials with applications to microactuators, J. Mech. Phys. Solids, 47, 531-576. 11. Bhattacharya, K. and Kohn, R.V. (1996) Symmetry, texture and the recoverable strain of shape-memory polycrystals, Acta Mater., 44, 529-542. 12. Bhattacharya, K. and Kohn, R.V. (1997) Energy minimization and the recoverable strains of polycrystalline shape-memory alloys, Arch. Rat. Mech. Anal., 139, 99-180. 13. Braides, A., Fonseca, I. and Francfort, G. (2000) 3D-2D asymptotic analysis for inhomogeneous thin ﬁlms, Indiana U Math. J., 49, 1367-1404. 14. Burcsu, E., Ravichandran, G. and Bhattacharya, K. (2000) Large strain electrostrictive actuation in barium titanate, Appl. Phys. Lett., 77, 1698-1700. 15. Chang, L.W. and Grummon, D.S. (1997) Phase evolution in sputtered thin-ﬁlms of Ti-X(Ni,Cu)(1-X) 1. Diﬀusive transformations and 2. Displacive transformation, Phil. Mag. A. 76, 163–189 and 191–219. 16. Chipot, M. and Kinderlehrer, D. (1988) Equilibrium conﬁgurations of crystals, Arch. Rat. Mech. Anal., 103, 237–277. 17. Cui, C.J. and James, R.D. (2000) Private communication. 18. DeSimone, A., Kohn, R.V., M¨ uller, S. and Otto, F. (2002) A reduced theory for thin-ﬁlm micromagnetics, Comm. Pure Appl. Math., 55, 1408-1460. 19. De Giorgi, E. (1975) Sulla convergenza di alcune successioni di integrali del tipo dell’area, Rendi Condi di Mat., 8, 277–294. 20. Dal Malo, G. (1992) Introduction to Gamma Convergence, Springer Verlag. 21. Ericksen, J.L. (1980) Some phase transitions in crystals, Arch. Rat. Mech. Anal., 73, 99–124. 22. Ericksen, J.L. (1984) The Cauchy and Born hypothesis for crystals, in Phase transformations and material instabilities in solids (ed. M.E. Gurtin), Academic Press, 61-78. 23. Friesecke, G., James, R.D. and M¨ uller, S. (2002) A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55, 1461-1505. 24. Gioia, G. and James, R.D. (1997) Micromagnetics of very thin ﬁlms, Proc. Royal Soc. Lond A, 453, 213-223. 25. Gisser, K.R.C., Busc, J.D.h, Johnson, A.D. and Ellis, A.B. (1992) Oriented NickelTitanium shape memory alloy ﬁlms prepared by annealing during deposition, Appl. Phys. Lett., 61, 1632–1634.

44 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

38. 39. 40. 41. 42. 43. 44.

K. BHATTACHARYA Fonseca, I. and Francfort, G. (1998) 3D-2D asymptotic analysis of an optimal design problem for thin ﬁlms, J. Reine Angew. Math., 505, 173–202. Hane, K.F. (1999) Bulk and thin ﬁlm microstructures in untwinned martensites, J. Mech. Phys. Solids, 47, 1917-1939. Ishida, A., Takei, A. and Miyazaki, S. (1993) Shape memory thin ﬁlm of Ti-Ni formed by sputtering, Thin Solid Films, 228, 210–214. James, R.D. and Hane, K.F. (2000) Martensitic transformations and shape-memory materials, Acta Mater., 48, The Millenium Special Issue, 197-222. James, R.D. and Rozzoni, R. (2000) Pressurized shape memory thin ﬁlms, J. Elasticity, 50, 399-436. James, R.D. and Wuttig, M. (1998) Magnetostriction of martensite, Phil. Mag A, 77, 1273-1299. Kohn, R.V. and M´ uller, S. (1992) Branching of twins near an austenite-martensite interface, Phil. Mag. A, 66, 697–715. Krulevitch, P., Lee, A.P., Ramsey, P.B., Trevino, J.C., Hamilton, J. and Northrup, M.A. (1996) Thin ﬁlm shape memory alloy microactuators, J. MEMS, 5, 270–282. Krulevitch, P., Ramsey, P.B., Makowiecki, D.M., Lee A.P., Northrup, M.A. and Johnson, G.C. (1996) Mixed sputter deposition of Ni-Ti-cu shape memory ﬁlms, Thin Solid Films, 274, 101–105. Le Dret, H. and Raoult, A. (1995) The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., 74, 549–578. Le Dret, H. and Raoult, A. (1996) The membrane shell model in nonlinear elasticity: a variational asymptotic derivation, J. Nonlinear Sc., 6, 59-84. M¨ uller, S. (1999) Variational models for microstructure and phase transitions. In Calculus of variations and geometric evolution problems: lectures given at the 2nd session of the Centro Internazionale Estivo (C.I.M.E.) held in Cetraro, Italy, June 15-22, 1996 (F. Bethuel, G. Huisken, A. Polden, S. M¨ uller and K. Steﬀen, Authors) (S. Hildebrandt and M. Struwe, Editors). Lecture notes in mathematics 1713, Springer-Verlag , 85-210. Shu, Y.C. (1999) Ph.D. Thesis, California Institute of Technology . Shu, Y.C. (2000) Heterogeneous thin ﬁlms of martensitic materials, Arch. Rat. Mech. Anal., 153, 39-90. Shu, Y.C. (2002) Strain relaxation in an alloy with a rough free surface, J. Elasticity, 66, 63-92. Shu, Y.C. (2002) Shape-Memory Micropumps, Mat. Trans. Jap. Inst. Metals, 43, 1037-1044. Shu, Y.C. and Bhattacharya, K. (1998) The inﬂuence of texture on the shapememory eﬀect in polycrystals, Acta Mater., 46, 5457-5473. Shu, Y.C. and Bhattacharya, K. (2001) Domain patterns and macroscopic behavior of ferroelectric materials, Phil. Mag. B, 81, 2021-2054. Su, Q., Hua, S.Z. and Wuttig, M. (1994) Martensitic transformation in Ni50 Ti50 ﬁlms, J. Alloys Compounds, 211/212, 460–463.

THE PASSAGE FROM DISCRETE TO CONTINUOUS VARIATIONAL PROBLEMS: A NONLINEAR HOMOGENIZATION PROCESS Continuum limits with bulk and surface energies

A. BRAIDES

Dipartimento di Matematica Universit` a di Roma ‘Tor Vergata’ via della Ricerca Scientiﬁca 1, 00133 Roma, Italy AND M. S. GELLI

Dipartimento di Matematica Universit` a di Pisa via Buonarroti 2, 56100 Pisa, Italy

1. Introduction In the past years a number of researches have been devoted to the study of discrete systems with a large number of interactions viewed as a variational limit of energies indexed by the number of nodes of the system. In this framework the setting in which we have a fairly complete set of results is that of central interactions for lattice systems; i.e., systems where the reference positions of the interacting points lie on a prescribed lattice, whose parameters change as the number of points increases. In more precise terms, we consider an open set Ω ⊂ Rn and take as reference lattice Zε = Ω ∩ εZn. The general form of a pair-potential energy is then Eε(u) =

ε fij (u(i), u(j)),

(1.1)

i,j∈Zε

where u : Zε → Rm. The analysis of energies of the form (1.1) has been performed under various hypotheses on fij . The ﬁrst simplifying assumption is that F is invariant under translations (in the target space); that is, ε ε (u, v) = gij (u − v). fij

45 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 45–63. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

(1.2)

46

A. BRAIDES AND M. S. GELLI

Furthermore, an important class is that of homogeneous interactions (i.e., invariant under translations in the reference space); this condition translates into ε ε (u, v) = g(i−j)/ε (u, v). (1.3) fij If both conditions are satisﬁed, we may rewrite the energies Eε above as Eε(u) =

k∈Zn i,j∈Zε ,i−j=εk

εnψkε

u(i) − u(j)

ε

,

(1.4)

where ψkε (ξ) = ε−ngkε (εξ). In this new form the interactions appear through the (discrete) diﬀerence quotients of the function u. Upon identifying each function u with its piecewise-constant interpolation (extending the deﬁnition of u arbitrarily outside Ω), we can consider Eε as deﬁned on (a subset of) L1 (Ω; Rm), and hence consider the Γ-limit with respect to the L1loc -topology. Under some coerciveness conditions the computation of the Γ-limit will give a continuous approximate description of the behaviour of minimum problems involving the energies Eε for ε small (see further for deﬁnitions, and [11] for a quick introduction to the subject). 1.1. SOME CONVERGENCE RESULTS. CONTINUUM ENERGIES ON SOBOLEV SPACES

Growth conditions on energy densities ψkε imply correspondingly boundedness conditions on gradient norms of piecewise-aﬃne interpolations of functions with equi-bounded energy. The simplest type of growth condition that we encounter is on nearest neighbours; i.e., for |k| = 1. If p > 1 exists such that c1 |z|p − c2 ≤ ψkε (z) ≤ c2 (1 + |z|p) (1.5) (c1 , c2 > 0 for |k| = 1), and if ψkε ≥ 0 for all k then the energies are equicoercive: if (uε) is a bounded sequence in L1 (Ω; Rm) and supε Eε(uε) < +∞, then from every sequence (uεj ) we can extract a subsequence converging to a function u ∈ W 1,p(Ω; Rm). In this section we will consider energies satisfying this assumption. Hence, their Γ-limits are deﬁned in the Sobolev space W 1,p(Ω; Rm). First, we remark that the energies Eε can also be seen as an integration with respect to measures concentrated on Dirac deltas at the points of Zε × Zε. If each ψkε satisﬁes a growth condition ψkε (z) ≤ cεk(1 + |z|p), then we have Eε(u) ≤

Ω×Ω

(1 + |u(x) − u(y)|p)dµε,

47

THE PASSAGE FROM DISCRETE TO CONTINUOUS . . .

where µε =

∞

k=1

cεk

i−j=εk, i,j∈Zε

1 δ . εp (i,j)

A natural condition for the ﬁniteness of the limit of Eε is the equi-boundedness of these measures (as ε → 0), regardless to the set Ω; namely, lim sup ε→0

(K − εk)n−1 ε ck < +∞ p+n−1 ε |k|≤K/ε

for each ﬁxed K > 0 (for ﬁxed Ω this condition is applied with K = diam Ω). However, under such assumption, example can be shown exhibiting a nonlocal Γ-limit, of the form

F (u) =

f (∇u(x)) dx + Ω

Ω×Ω

ψ(u(x) − u(y))dµ(x, y),

where µ is the weak∗ -limit of the measures µε outside the ‘diagonal’ of Rn × Rn (see [10] for a detailed example). Under some decay conditions, such long-range behaviour may be ruled out. The following compactness result proved by Alicandro and Cicalese [1] shows that a wide class of discrete systems possesses a ‘local’ continuous limit. We state it in a general ‘spacedependent’ case. Theorem 1.1 (compactness) Let ψkε satisfy c1 |z|p − c2 ≤ ψeεi (x, z)

(1.6)

for all (x, z) ∈ Ω × Rm and i ∈ {1, . . . , n}, ψεk(x, z) ≤ cεk(1 + |z|p)

(1.7)

for all (x, z) ∈ Ω × Rm, and k ∈ Zn, where c1 > 0 and cεk satisfy (H1): lim sup

ε→0+ ξ∈Zn

cεξ < +∞;

(H2): for all δ > 0 Mδ > 0 exists such that lim sup ε→0+

Let Eε be deﬁned by Eε(u) =

k∈Zn i∈Rεk

εnψεk i,

u(i + εk) − u(ı) , ε|k|

|ξ|>Mδ

cεk < δ.

48

A. BRAIDES AND M. S. GELLI

where Rεk := {i ∈ Zε : i + εk ∈ Zε}. Then for every sequence (εj ) of positive real numbers converging to 0, there exists a subsequence (εjk ) and a f : Ω × Rd×N satisfying c(M p − 1) ≤ f (x, M ) ≤ C(M p + 1), with 0 < c < C, such that (Eεjk (·)) Γ-converges with respect to the Lp(Ω)topology to the functional F : Lp(Ω) → [0, +∞] deﬁned as F (u) =

   f (x, ∇u) dx  

if u ∈ W 1,p(Ω; Rm)

Ω

+∞

(1.8)

otherwise.

Moreover, for any u ∈ W 1,p(Ω) and A ∈ A(Ω),

Γ- lim Fεjk (u, A) = k

A

f (x, ∇u) dx.

In the case of energies deﬁned by a scaling process; i.e., when ψkε (x, z) = ψk

z

ε

,

then the limit ϕ is independent of the subsequence, and is characterized by the asymptotic homogenization formula 1 min FT (u), u|∂QT = M i N T→+∞ T

ϕ(M ) = lim

(1.9)

where FT (u) =

ψk

k∈Zn i∈Rk (QT )

u(i + k) − u(i) |k|

1

and u|∂QT = M i means that “near” the boundary of QT the function u is the discrete interpolation of the aﬃne function M x (see [1] for further details). This formula can be compared with that giving the ‘pointwise’ limit treated by Blanc, Le Bris and Lions [8], and is analogous to that used by Friesecke and Theil to check the validity of the Cauchy-Born assumption [26]. It is worth examining this formula in some special cases. First, if only nearest-neighbour interactions are present then it reduces to ϕ(M ) =

n i=1

ψi∗∗ (M ei),

THE PASSAGE FROM DISCRETE TO CONTINUOUS . . .

49

where ψi = ψei and f ∗∗ denotes the lower semicontinuous and convex envelope of f . Note that convexity is not a necessary condition for lower semicontinuity at the discrete level: this convexiﬁcation operation should be interpreted as an eﬀect due to oscillations at a ‘mesoscopic scale’ (i.e., much larger than the ‘microscopic scale’ ε but still vanishing as ε → 0). If not only nearest neighbours are taken into account then the mesoscopic oscillations must be coupled with microscopic ones (see [17, 30]). A simple example is given by the one-dimensional case with next-to-nearest neighbours; i.e. when only f1 and f2 are non zero. In this case ϕ = ψ ∗∗ , where 1 min{f1 (ξ1 ) + f1 (ξ2 ) : ξ1 + ξ2 = 2ξ}. 2

ψ(ξ) = f2 (2ξ) +

The second term obtained by minimization is due to oscillations at the microscopic level: nearest neighbours rearrange so as to minimize their interaction coupled with that between second neighbours (see [11] for a simple treatment of these one-dimensional problems). A similar optimization argument is used in [26] to show the non-validity of the Cauchy-Born rule for some types of very simple lattice interactions in dimension two. Finally, it must be noted that formula (1.9) does not simplify even in the simplest case of three levels of interactions in dimension one, thus showing that this eﬀect, typical of nonlinear homogenization, is really due to the lattice interactions and not restricted to vector-valued functions as in the case of homogenization on the continuum (see [14]). 1.2. CONTINUOUS ENERGIES ON FUNCTIONS OF BOUNDED VARIATION. SOME NEW RESULTS CONTAINED IN THIS PAPER

In this paper we extend the homogenization formula as above for systems whose continuous counterpart is naturally deﬁned on some set of functions of bounded variation (see below for deﬁnitions). In particular we will consider energies deﬁned on the Ambrosio-De Giorgi SBV spaces (see [6, 23, 4, 9]. In the one-dimensional case we can picture a function u ∈ SBV (a, b) as a piecewise-Sobolev functions. If denoting by S(u) its set of discontinuity points, then (local, homogeneous and translation-invariant) energies on SBV (a, b) are of the form E(u) =

b

a

f (u (x)) dx +

g([u](t)),

(1.10)

t∈S(u)

where u± (t) are the right and left-hand side limits of u at t. The pioneering example for this case is due to Chambolle [18], who treated the limit of some ﬁnite-diﬀerence schemes in Computer Vision (see

50

A. BRAIDES AND M. S. GELLI

[7]), producing as the continuum counterpart the one-dimensional version of the Mumford-Shah functional ([29]). That is however a ‘limit’ case when the potential is a truncated quadratic function and the decoupling process between bulk and surface parts is obtained by considering the eﬀect of the quadratic and constant parts separately. A diﬀerent approach is initiated in a subsequent paper by Braides, Dal Maso and Garroni [13], who consider potentials of convex-concave type and express the limit in terms of diﬀerent scalings of the two parts (the same type of interactions are analyzed by Truskinovsky [33]). The general case of nearest-neighbour interactions has been treated by Braides and Gelli in [16], where the following result is proved (for missing deﬁnitions see the sections below). Theorem 1.2 (Nearest-neighbour interactions) Let Ω = (a, b) and let ψε : R → [0, +∞] satisfy ψε(z) ≥ c1 |z|p − c2

for all z < 0

(1.11)

for some p > 1 and c1 > 0, and let Eε be given by

Eε(u) =

εψε

u(i + 1) − u(u ) i

ε

i,i+1∈Zε

.

Let Tε ∈ R be an arbitrary sequence satisfying lim Tε = +∞,

lim εTε = 0,

ε→0+

ε→0+

(1.12)

and let Fε, Gε : R → [0, +∞] be deﬁned by   ψε(z)

Fε(z) =

Gε(z) =

z ≤ Tε

 +∞ z > Tε    εψε z if z > εTε  

ε

+∞

(1.13)

(1.14)

otherwise.

Assume that there exist F, G : R → [0, +∞] such that (note that this assumption is always satisﬁed, upon extracting a subsequence) Γ- lim Fε∗∗ = F on R,

(1.15)

Γ- lim sub− Gε = G on R \ {0},

(1.16)

ε→0+

ε→0+

THE PASSAGE FROM DISCRETE TO CONTINUOUS . . .

51

where sub− g is the lower semicontinuous and subadditive envelope of g. Then, (Eε)ε Γ-converges to E with respect to the convergence in L1loc (0, L) and the convergence in measure, where

E(u) =

 b   F ( u) ˙ dx + G([u]) + σDu+  c (0, L)    a S(u)      

if u ∈ BVloc (0, L) Dcu− = 0 and [u] > 0 on S(u)

+∞

otherwise in L1 (0, L),

where F and G are deﬁned by (for notation convenience we set G(0) = 0) F (z) := inf{F (z1 ) + G0 (z2 ) : z1 + z2 = z}, G(z) := inf{F ∞ (z1 ) + G(z2 ) : z1 + z2 = z}, ∞

and σ := F (1), where F ∞ (z) = lim

z→+∞

F(z) z

and G0 (z) = lim

z→0+

G(z) z .

The fundamental issue here is the separation of scale eﬀect highlighted by equations (1.13)–(1.14), that allows to derive the bulk and surface energy densities of the limit from the discrete interactions. In this paper, always remaining in the one-dimensional framework, we show how in the case of long-range interactions this scale separation can be coupled with the nonlinear homogenization process described in formula (1.9) (see Theorem 3.2 further on). The general n-dimensional vector-valued case can be dealt with by using the localization and integral representation methods of Γ-convergence ([2]). Particular cases when a simpler description of the limit energy densities is possible are treated in [19, 20, 15]. Some special cases when also non-central interactions are considered are dealt with in [3, 25]. Higher-order eﬀects are considered in [21, 12]. 2. Notation and preliminaries For a set A of R we denote int A the interior of A. We write sgn t and [t] to denote the sign of t and the integer part of t, respectively. We write L1 (A) or |A| to denote the Lebesgue measure of A ⊂ R, # for the counting measure and δt for the Dirac mass at t. We use standard notation for Sobolev and Lebesgue spaces. If φ is a measurable function then −B φ dx is its mean value on the set B. If µ is a (signed) Borel measure then µ+ and µ− denote its positive and negative parts, respectively, and |µ| its total variation; if B is a Borel set, then the measure µ B is deﬁned as µ B(A) := µ(A ∩ B). The letter c will denote a strictly positive constant whose value may vary from line to line.

52

A. BRAIDES AND M. S. GELLI

2.1. FUNCTIONS OF BOUNDED VARIATION

We recall that the space BV (a, b) of functions of bounded variation on (a, b) is deﬁned as the space of functions u ∈ L1 (a, b) whose distributional derivative Du is a signed Borel measure. For each such u there exists f ∈ countable) set S(u) ⊂ (a, b), a sequence of real numbers L1 (a, b), a (at most (at)t∈S(u) with t |at| < +∞ and a non-atomic measure Dcu singular with respectto the Lebesgue measure such that the equality of measures Du = f L1 + t∈S(u) atδt +Dcu holds. It can be easily seen that for such functions the left hand-side and right hand-side approximate limits u− (t), u+ (t) exist at every point, and that S(u) = {t ∈ R : u− (t) = u+ (t)} and at = u+ (t) − u− (t) =: [u](t). We will write u = f , which is an approximate gradient of u. Dcu is called the Cantor part of Du. A sequence uj converges weakly to u in BV (a, b) if uj → u in L1 (a, b) and supj |Duj |(a, b) < +∞. The space SBV (a, b) of special functions of bounded variation is deﬁned 0; i.e., whose as the space of functions u ∈ BV (a, b) such that Dcu = distributional derivative Du can be written as Du = u L1 + t∈S(u) (u+ (t)− u− (t))δt. This notation describes a particular case of a SBV -functions space as introduced by De Giorgi and Ambrosio [23]. We will mainly deal with functionals whose natural domain is that of piecewise-W 1,p functions, which is a particular sub-class of SBV (a, b) corresponding to the conditions u˙ ∈ Lp(a, b) and #(S(u)) < +∞, but we nevertheless use the more general SBV notation for future reference and for further generalization to higher dimensions (see [5]). For an introduction to BV and SBV functions we refer to the book by Ambrosio, Fusco and Pallara [6], while approximation methods for free-discontinuity problems are discussed by Braides [9]. A class of energies on SBV (a, b) are those of the form (1.10) with f, g : R → [0, +∞]. Lower semicontinuity conditions on E are equivalent to requiring that f is lower semicontinuous and convex and g is lower semicontinuous and subadditive; i.e., g(x + y) ≤ g(x) + g(y). The latter can be interpreted as a condition penalizing fracture fragmentation, whereas convexity penalizes oscillations. If ϕ is not lower semicontinuous and convex (respectively, subadditive) then we may consider its lower semicontinuous and convex (respectively, subadditive) envelope; i.e., the greatest lower semicontinuous and convex (respectively, subadditive) function not greater than ϕ, that we denote by ϕ∗∗ (respectively, sub− ϕ). For a discussion on the role of these conditions for the lower semicontinuity of E we refer to [9] Section 2.2 or [11]. Energies in BV must satisfy further compatibility conditions between f and g.

53

THE PASSAGE FROM DISCRETE TO CONTINUOUS . . . 2.2. Γ-CONVERGENCE

We recall the deﬁnition of De Giorgi’s Γ-convergence in a metric space space (X, d): given a family of functionals Fn : X → [0, +∞], n ∈ N, for u ∈ X we deﬁne the Γ-lower limit and Γ-upper limit of (Fn) as

F (u) = Γ(d)- lim inf Fn(u) := inf lim inf Fn(un) : lim d(un, u) = 0 , n n n (2.1) and

F (u) = Γ(d)- lim sup Fn(u) := inf lim sup Fn(un) : lim d(un, u) = 0 . n

n

n

(2.2) and are lower semicontinuous. If these two Note that the functions quantities coincide then their common value is called the Γ-limit of the sequence (Fn) at u, and is denoted by Γ- limn Fn(u) or Γ(d)- limn Fn(u). Equivalently, F (u) = Γ-limn Fn(u) if and only if the two following conditions are satisﬁed: (i) (lower semicontinuity inequality) for all sequences (un) converging to u in X we have F (u) ≤ lim inf n Fn(un); (ii) (existence of a recovery sequence) there exists a sequence (un) converging to u in X such that F (u) ≥ lim supn Fn(un). We will use as d the L1 -metric or a metric giving convergence in measure. For a comprehensive study of Γ-convergence we refer to the books of Dal Maso [22] and Braides [11] (see also [14] Part II). The reason for the introduction of this notion is explained by the following fundamental theorem. F

F

Theorem 2.1 Let F = Γ-limn Fn, and let a compact set K ⊂ X exist such that inf X Fn = inf K Fn for all n. Then ∃ min F = lim inf Fn. X

n

X

Moreover, if (un) is a converging sequence with limn Fn(un) = limn inf X Fn then its limit is a minimum point for F . 3. Statement of the main result With ﬁxed L > 0, consider an open interval (0, L) of R and for n ∈ N let λn = L/n. This parameter will play the role played by ε in the Introduction. We use the notation xin = iλn. We deﬁne An(0, L) as the set of discrete functions u : λnZ ∩ [0, L] → R. This set will be identiﬁed as the subset of L1 (0, L) of functions constant almost everywhere on each interval (xin − λn λn i , xn + ), i ∈ {1, . . . , n}. 2 2

54

A. BRAIDES AND M. S. GELLI

Let K ∈ N be ﬁxed and for n ∈ N and j ∈ {1, . . . , K} let ψnj : R → (−∞, +∞] be given Borel functions bounded below. Deﬁne En : L1 (0, L) → [0, +∞] as

En(u) =

 K n−j i+j i  u(x ) − u(x )  n n  λnψ j n

jλn

 j=1 i=0  

x ∈ An(0, L)

otherwise in L1 (0, L). (3.1) We will describe the asymptotic behaviour of En as n → +∞ when the energy densities are potentials of Lennard-Jones type. More precisely, we will make the following assumption: +∞

(growth conditions: superlinearity at −∞, mixed type at +∞) there exists a convex function Ψ : R → [0, +∞] such that lim

z→−∞

Ψ(z) = +∞ |z|

and there exist constants c1j , c2j > 0 such that c1j (Ψ(z) − 1) ≤ ψnj (z) ≤ c2j max{Ψ(z), |z|} for all z ∈ R.

(3.2)

Under this hypothesis it will be possible to describe explicitly the behaviour of the energies En by means of their Γ-limit. The exact statement of the result will be given at the end of this section. Remark 3.1 Hypothesis (3.2) is designed to cover the case of LennardJones potential (and potential of the same shape), where ψnj = ψ is equal for all j and n, and k1 k2 ψ(z) = 12 − 6 z z for some k1 , k2 > 0. In this case, we can take    1

Ψ(z) =

 

z 12

+∞

if z > 0 otherwise.

Another case included in hypothesis (3.2) is when all functions satisfy a uniform growth condition of order p > 1; i.e., cj (|z|p − 1) ≤ ψnj (z) ≤ C cj (|z|p + 1) for all j and n.

55

THE PASSAGE FROM DISCRETE TO CONTINUOUS . . .

Note that in the case K = 1 (nearest-neighbour interaction) the right hand-side of (3.2) can be dropped (see Theorem 1.2). Before stating our main result, we have to introduce the counterpart of the energy densities Fn and Gn in Theorem 1.2 for the case K > 1. Following the idea already performed in [17], we consider clusters of N subsequent points (N large) and deﬁne an average discrete energy for each of those clusters, so that the energy En may be approximately regarded as a ’nearest neighbour interaction energy’ acting between such clusters, to which the above description applies. Actually, we ﬁx a sequence (Nn) of natural numbers with the property lim Nn = +∞,

lim

n

n

Nn = 0, n

(3.3)

and we deﬁne K N n −j 1

ψn(z) = min

Nn j=1

ψnj

u(i + j) − u(i)

i=0

j

: u : {0, . . . , Nn} → R,

u(x) = zx if x = 0, . . . , K, Nn − K, . . . , Nn .

(3.4)

By using the energies ψn we will regard a system of Nn neighbouring points as a single interaction between the two extremal ones, up to a little error which is negligible as Nn → +∞. We can now state our convergence result, whose thesis is exactly the same as that of Theorem 1.2 with εn := Nnλn. Theorem 3.2 Let ψnj satisfy (3.2) and let (En) be given by (3.1). Let ψn be given by (3.4) and let εn = Nnλn. For all n ∈ N let Tn ∈ R be deﬁned as in (1.12), and let Fn, Gn : R → [0, +∞] be deﬁned by   ψn(z)

Fn(z) =

Gn(z) =

z ≤ Tn

 +∞ z>T  n z   εnψn if z > εnTn  

εn

+∞

(3.5)

(3.6)

otherwise.

Assume that there exist F, G : R → [0, +∞] such that Γ- lim Fn∗∗ = F on R,

(3.7)

Γ- lim sub− Gn = G on R \ {0}.

(3.8)

n

n

56

A. BRAIDES AND M. S. GELLI

Note that this assumption is always satisﬁed, upon extracting a subsequence. Then, (En)n Γ-converges to E with respect to the convergence in L1loc (0, L) and the convergence in measure, where

E(u) =

 L   F ( u) ˙ dx + G([u]) + σDu+  c (0, L)    0 S(u)      

if u ∈ BVloc (0, L) Dcu− = 0 , and [u] > 0 on S(u) otherwise in L1 (0, L),

+∞

where F and G are deﬁned by (for notation convenience we set G(0) = 0) F (z) := inf{F (z1 ) + G0 (z2 ) : z1 + z2 = z}, ∞

G(z) := inf{F ∞ (z1 ) + G(z2 ) : z1 + z2 = z},

and σ := F (1). Proof. Upon adding a ﬁxed positive constant we may assume that all ψnj are non negative. We begin by proving the liminf inequality. We thus ﬁx un, u ∈ L1loc (0, L) such that un → u in measure and supn En(un) < +∞. Upon extracting a subsequence we may suppose that un → u a.e. and that the limit limn En(un) exists. By using (3.2) and reasoning as in the proof of Theorem 1.2 we get that u ∈ BVloc (0, L) and un → u weakly in BVloc (0, L) (see [16] for details). It will be enough to show that for all 0 < a < b < L ﬁxed we have

b

F (u(x)) ˙ dx + a

G([u](t)) + σDu+ c (a, b) ≤ lim En(un).

(3.9)

n

t∈S(u)∩(a,b)

As already mentioned, for all n ∈ N we will estimate the energy En(un) with a ‘nearest neighbour interaction’ one, with energy density ψn and discretization step εn = λnNn. Thus, with ﬁxed Nn, we will choose w ∈ {1, . . . , Nn} in such a way that we can ﬁnd a suitable piecewise aﬃne interpolation of un on the lattice (λnw + εnZ) ∩ (a, b), call it vn, so that vn still converges to u in measure and En(un) = En1 (vn) + o(1). For all w ∈ {1, . . . , Nn} let Zn(w) = { ∈ w + NnZ : λn ∈ (a, b)},

Φn(w) =

K +2K−1

λnψnj

u (λ (i + j)) − u (λ i) n n n n

∈Zn (w) j=1 i= −2K

+2K−1 s= −2K

|un(λn(s + 1)) − un(λns)|.

jλn (3.10)

57

THE PASSAGE FROM DISCRETE TO CONTINUOUS . . .

+2K−1 |un(λn(s + 1)) − un(λns)| = |Dun|([( − 2K)λn, ( + Note that s= −2K 2K)λn]), where we consider the discrete function un identiﬁed with the L1 function piecewise constant. Since each interval [λns, λn(s + 1)] belongs at most to 4K interval of the type [λn( − 2K), λn( + 2K)] we have that Nn

+2K−1

|un(λn(s + 1)) − un(λns)| ≤ 4K|Dun|(a, b).

w=1 ∈Zn (w) s= −2K

Moreover, we also have Nn

K +2K−1

u (λ (i + j)) − u (λ i) n n n n

λnψnj

jλn

w=1 ∈Zn (w) j=1 i= −2K

=

Nn 4K

u (λ ( − 2K + j + i )) − u (λ ( − 2K + i ) n n n n

λnψnj

jλn

i =0 w=1 ∈Zn (w)

≤ 5KEn(un).

(3.11)

Hence, Nn

Φn(w) ≤ 5KEn(un) + 4K|Dun|(a, b) ≤ c

w=1

and we can ﬁnd wn ∈ {1, . . . , Nn} such that Φn(wn) ≤

c . Nn

For all ∈ Zn(wn) we deﬁne zn =

un(λn( + Nn − K)) − un(λn( + K)) λn(Nn − 2K)

and u ˜ n on {0, . . . , Nn} by

u ˜ n(i) =

 1   (un(λn( + i)) − un(λn( + K))) + zn K    λn     

if i = K, . . . , Nn − K

zn i

otherwise.

Finally, we deﬁne vn as the continuous piecewise-aﬃne interpolate function such that vn(a) = u(a) and

vn = zn on (λn, λn( + Nn)),

∈ Zn(wn).

58

A. BRAIDES AND M. S. GELLI

Note that vn → u in measure and u ˜ n is a test function for the minimum

problem deﬁning ψn(zn). We then have K +N n −K−j

En(un) ≥

∈Zn (wn ) j=1

≥

jλn

i= +K

K N n −j

u (λ (i + j)) − u (λ i) n n n n

λnψnj

λnψnj

u ˜ (i + j) − u ˜ (i) n

n

j

∈Zn (wn ) j=1 i=0

−

K K−1

u ˜ (i + j) − u ˜ (i) n

λnψnj

n

j

∈Zn (wn ) j=1 i=0

−

N n −j

K

λnψnj

u ˜ (i + j) − u ˜ (i) n

n

. (3.12)

j

∈Zn (wn ) j=1 i=Nn −K−j+1

As for the ﬁrst term in (3.12), for any ∈ Zn(wn), we have K N n −j

u ˜ (i + j) − u ˜ (i) n

ψnj

n

j

j=1 i=0

≥ Nnψn(zn ),

so that, denoting Zn = {i ∈ Z : λnwn + εni ∈ Zn(wn)}, K N n −j

∈Zn (wn ) j=1 i=0

u ˜ (i + j) − u ˜ (i) n

λnψnj

εnψn

n

j

≥

v (ε (i + 1)) − v (ε i) n n n n

.

εn

i∈Zn

(3.13)

By plugging estimate (3.13) in (3.12) we get En(un)

≥

v (ε (i + 1)) − v (ε i) n n n n

εnψn

εn

i∈Zn

−

K K−1

λnψnj

u ˜ (i + j) − u ˜ (i) n

n

j

∈Zn (wn ) j=1 i=0

−

K

N n −j

∈Zn (wn ) j=1 i=Nn −K−j+1

=:

i∈Zn

u ˜ (i + j) − u ˜ (i)

λnψnj

n

v (ε (i + 1)) − v (ε i) n n n n

εnψn

εn

n

j − In1 − In2 .

(3.14)

59

THE PASSAGE FROM DISCRETE TO CONTINUOUS . . .

It remains to give an estimate of the terms In1 , In2 . We will conﬁne ourselves to prove that In1 = o(1), the proof of the same estimate for In2 being analogous. Indeed, by using hypothesis (3.2), we have K K−1

In1 ≤ c2j

λnΨ

u (λ ( + i + j)) − u (λ ( + i)) n n n n

jλn

∈Zn (wn ) j=1 i=0

u (λ ( + i + j)) − u (λ ( + i)) n n n n

+

j

(3.15) and, by convexity, we have also In1

≤

K

c2j

∈Zn (wn ) j=1

K−1 i+j−1 i=0

s=i

u (λ ( + s + 1)) − u (λ ( + s)) 1

n n n n λnΨ j jλn

+|un(λn( + s + 1)) − un(λn( + s))|

≤ c

K 2K−j i+j−1 1

∈Zn (wn ) j=1 i>K−j s=i

+

j

u (λ ( + s + 1)) − u (λ ( + s)) n n n n

λnψn1

λn

c + Φn(wn). Nn

(3.16)

By taking into account our choice of wn, it follows In1 ≤

v (ε (i + 1)) − v (ε i) c

n n n n 1+ εnψn . Nn ε n i∈Z

(3.17)

n

Eventually we get En(un) ≥ (1 −

v (ε (i + 1)) − v (ε i) c c n n n n ) εnψn − Nn i∈Z εn Nn n

and it suﬃces to pass to the liminf and use Theorem 1.2 on the interval (a, b) to have the desired inequality. To prove the limsup inequality we will ﬁrst prove that Γ- lim sup En(u) ≤ n

L

F (u(x)) ˙ dx + 0

G([u](t))

t∈S(u)

on the functions u ∈ SBV (0, L) with #S(u) < +∞ and then we will use a relaxation argument. For the sake of simplicity as a ﬁrst step we provide

60

A. BRAIDES AND M. S. GELLI

a recovery sequence for functions of the form u(x) = ξx + zχ(x0 ,L) . Thus, let us consider the case of u(x) = ξx. By proceeding as in the proof of 1 2 Theorem 1.2 (for details see [16]) we can ﬁnd ξn, ξn ∈ R and aset ofindices Jn ⊂ {0, . . . , εLn } such that, set βn := #(Jn) and Mn := L/εn , there holds εnβnξn1 + εn(Mn − βn)ξn2 = Lξ + o(1) εnβnψn(ξn1 ) + εn(Mn − βn)ψn(ξn2 ) = LF (ξ) + o(1).

(3.18)

For s = 1, 2, let vns be a minimum point for the problem deﬁning ψn(ξni ), and deﬁne un as un(iλn) := λnvns (i − ) + ξn2 λn +

(ξn1 − ξn2 )χJn (j)Nnλn

j=1

if i ∈ [, + Nn). It can be proved that un → u in L1 (0, L) and En(un) ≤ εnβnψn(ξn1 ) + εn(Mn − βn)ψn(ξn2 ) +

K K−1

λnψnj

u ((i + j + )λ ) − u ((i + )λ ) n n n n

jλn

∈Zn (0) j=1 i=0 N n −j

+

λnψnj

u ((i + j + )λ ) − u ((i + )λ ) n n n n

jλn

i=Nn −K−j+1

.

Since by (3.18) limn εnβnψn(ξn1 ) + εn(Mn − βn)ψn(ξn2 ) = LF (ξ), it remains to prove that the second terms in (3.19) tend to 0. To do this we will proceed exactly as in the estimate of the terms In1 , In2 in the liminf inequality, by taking into account hypothesis (3.2) and using the convexity of Ψ(z) + |z| to consider only interactions of order one. Thus, it holds En(un) ≤ LF (ξ) + o(1) +c

K K−1 j−1 1

∈Zn (0) j=1 i=0 s=0

+c

K

un (λn (+i+s+1))−un (λn (+i+s)) λn

1 n (λn (+i+s)) λnψn1 un (λn (+i+s+1))−u λn j −K−j s=0

N n −K

∈Zn (0) j=1 i=Nn

j

λnψn1 j−1

c + (1 + εn(βn|ξn1 | + (Mn − βn)|ξn2 |)) Nn c ≤ (F (ξ) + 1 + εn(βn|ξn1 | + (Mn − βn)|ξn2 |)). Nn

(3.19)

61

THE PASSAGE FROM DISCRETE TO CONTINUOUS . . .

By the growth condition on Ψ (and then on ψn, F ) the terms εn(βn|ξn1 | + (Mn − βn)|ξn2 |) are equibounded. Then it suﬃces to take the limsup as n goes to +∞. Let us come now to the case u(x) = ξx + zχ(xo ,L) (x) with z = 0 and G(z) < +∞. Analogously to the proof of Theorem 1.2 we can ﬁnd natural numbers Ln ∈ N and real numbers zns ∈ [0, +∞) for s = {1, . . . , Ln} with the following properties lim Lnεn = 0,

lim

n

n

Ln

zns = z,

G(z) = lim n

s=1

Ln

Gn(zns ).

(3.20)

s=1

s Let o ∈ Zn(0) be such that xo ∈ [ oλn, ( o + Nn)λn) . If vn is a minimum point for the problem deﬁning ψn z/εn for s = 1, . . . , Ln and un is the recovery sequence for ξx previously deﬁned, we set for i ∈ [λn, ( + Nn)λn) with ∈ Zn(0)

 un(iλn)     

wn(iλn) =

    

λnvns (i − ) + un(0 λn) +

if < o s

un(iλn + (o + LnNn)λn) +

s s =1 zn

Ln

s s=1 zn

if ∈ [o, o + LnNn) if ≥ o + LnNn.

By using (3.20) it can be easily checked that wn converges to u in L1 (0, L) and, by proceeding as above, En(un) = LF (ξ) + G(z) + o(1). This procedure can be applied, up to slight modiﬁcations, to any function in SBV (0, L), piecewise aﬃne and with a ﬁnite number of jumps. So the thesis follows by usual density and relaxation arguments (see e.g. [11]) . Remark 3.3 (next-to-nearest neighbour interactions) In the case that ψnj = 0 for all j > 2 then we can equivalently take in place of ψn the function ψ˜n deﬁned as in the Sobolev case, by 1 ψ˜n(z) = ψn2 (z) + min{ψn1 (z1 ) + ψn1 (z2 ) : z1 + z2 = 2z}. 2 With the previous remark in mind, we can examine the behaviour of next-to-nearest neighbour systems for Lennard-Jones potentials. Example 3.4 (next-to-nearest neighbour Lennard Jones interactions) Let ψ be as in Remark 3.1, for any n let ψn1 (z) = ψ(z), ψn2 (z) = ψ(2z)

62

A. BRAIDES AND M. S. GELLI

and ψnj = 0 for j ≥ 3. By using the previous remark, we obtain  +∞         1 + 2−12

F (z) =

if z ≤ 0 k1 −

1 + 2−6 k2 z6

z 12      k2    − 2 (1 + 2−12 )(1 + 2−6 )2 =: mF 4k1

where

if 0 < z ≤ zmin if z > zmin ,

(1 + 2−12 )k 1/6 1

zmin = 2

(1 + 2−6 )k2

is the minimum point for ψ˜ = ψ˜n given by the previous remark. The two values zmin and mF can be compared to the corresponding minimum point ˜ that are (2k1 /k2 )1/6 and −k2 /4k1 , respectively. and minimum value for ψ, 2 It is also interesting to note that k2 1 ˜ = − 2 = min ψ. lim ψ(w) w→+∞ 8k1 2 References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12.

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APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS OF MECHANICS Laminates, diﬀerential schemes, variations, extensions, bounds, and duality

A. CHERKAEV

Department of mathematics, University of Utah Salt Lake City UT 84112, U.S.A. Abstract. This paper reviews recent developments of mathematical methods for nonconvex variational problems of mechanics, particularly, problems of optimal layouts of material in a heterogeneous medium. These problems are characterized by locally unstable solutions which are interpreted as optimal microstructured media. We discuss variational formulations of these problems, properties of their solutions and several approaches to address them: minimizing sequences and the technique of laminates, laminate closures, and the diﬀerential scheme ; necessary conditions by structural variations and minimal extension technique; the lower bounds and bounds for the variety of eﬀective tensors. Several examples are presented. Particularly, the bound for the tensor of thermal expansion coeﬃcients is found. Special attention is paid to the use of duality for reformulation of minimax problems as minimal ones.

1. Variational problems with locally unstable solutions 1.1. NONCONVEX VARIATIONAL PROBLEMS

Introduction Nonconvex variational problems in mechanics describe optimal layouts of several materials in a structure. A typical problem is a minimization of the energy of a heterogeneous structure by a layout of phases. This problem is met in many applications. Structural optimization asks for an optimal “mixture” of a solid material and void or for the best structure of a composite. The martensite alloys, polycrystals and similar materials can exist in several forms (phases) and Gibbs principle states that the phase with minimal energy is realized. Biostructures adapt themselves to the environment in a best way. Optimal layouts in man-made structures response to an engineering requirements, minimization of the

65 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 65–105. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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energy of natural materials corresponds to the realization of the thermodynamical Gibbs principle, and optimality of biological morphologies reﬂects evolutionary perfection. A minimization problem of this type helps to establish bounds on eﬀective properties of a composite. If the mixed materials are linear, a composite is equivalent to a linear material in the sense that if loaded, it stores the same energy as a homogeneous material with stiﬀness C∗ . The problem of G-closure asks about the set of the eﬀective tensors of all microstructures and bounds on that set. In order to ﬁnd the bound for C∗ , we minimize the energy stored in a composite medium, or the sum of the energies corresponding to several linearly independent loadings. The bound for the range of C∗ follows from the lower bound of the energy. Variational problems in elasticity The state of a classical elastic material is deﬁned by the equations of equilibrium of the stress tensor σ: ∇ · σ = 0 σ = σT .

(1)

The stress is related to the tensor of deformation and further to the vector of displacement u by the constitutive equation σ = F (),

= (∇u).

(2)

These relations are the Euler-Lagrange equations for a variational problem

J = min (u)

Ω

W() dx,

= (∇u)

(3)

where W() is the energy of deformation, if the constitutive relation (2) can be written in the form F =

∂ W() ∂

(4)

Boundary conditions are imposed on the displacement u. Alternatively, the equilibrium can be described by the dual variational problem

Jσ = min σ(φ)

Ω

Wσ(σ) dx,

σ = ∇ × (∇ × φ)T

(5)

where Wσ (σ) is a dual form of energy called the stress energy or the complementary energy, and the potential representation in (5)2 accounts for the equilibrium constraints (1).

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

67

A linear elastic material corresponds to the constitutive relations: σ = C : or σij =

Cijknnk ,

k,n

1 = (∇u + ∇uT ) 2

(6)

and the stored energy function 1 W = T : C : 2

(7)

where is the strain, C is the fourth-rank stiﬀness tensor , and (:) is the convolution by two indices. The density of the stress energy corresponds to: 1 Wσ = σ T : S : σ, σ = ∇ × (∇ × φ)T 2 where S is the compliance tensor that is inverse to C, S : C = 1. Both problems (3) and (5) describe the same elastic equilibrium and deal with ﬁelds that are derivable from some vector potentials and therefore satisfy the integrability conditions. The sum of the functionals in (3) and (5) is equal to the work of external forces on the displacements of the points of medium, or W + Wσ = : σ Minimization of the energy J with prescribed nonzero displacements on the boundary corresponds to minimization of the integral stiﬀness of the loaded elastic domain. Similarly, minimization of the energy Jσ with prescribed nonzero forces on the boundary corresponds to minimization of the integral compliance of the domain or maximization of its stiﬀness. Notations Below, in Section 1.2, we discuss general properties of nonconvex variational problems; the analysis is applicable to both forms of elastic energy. We will write the variational problem in the form

J = min u

W (∇u) dx

(8)

Ω

stressing the dependence of the Lagrangian W on the gradient of a vector potential. For quadratic energies, we will often use the form W = 12 v T Dv where v is a ﬁeld; for example v = ∇u or v = σ, or v = , or v is a combination of these ﬁelds. Accordingly, D is a tensor characterizing the properties, that can be either stiﬀness or compliance tensor.

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Stability to perturbations The energy of a classical material is stable in the following sense: If an unbounded domain ﬁlled with the material is subject to an aﬃne external elongation at inﬁnity (that corresponds to the constant strain), the strain is constant everywhere. The minimum of the energy (8) is achieved at an aﬃne function u(x) = Ax + B satisfying the boundary conditions. The energy of such materials is called quasiconvex (see the Section 1.2 for the deﬁnition) and the constitutive relations are elliptic. The ellipticity implies that the solution u(x) in a ﬁnite domain Ω is smooth if both the domain Ω and boundary conditions are smooth. Problems of optimal design, composites, natural polymorphic materials (martensites), polycrystals, smart materials, biological materials, etc. lead to variational problems with locally unstable solutions. In such problems, the minimizer is not aﬃne even if the external loading is homogeneous. These variational problems are called nonquasiconvex; they were studied in recent books by Dagorogna [26], Cherkaev [20], Milton [63], Allaire [1], Bendsøe and Sigmund [11] from diﬀerent viewpoints; extensive references can be found there. The problems of nonlinear elasticity are also generally nonquasiconvex, see [27]. The unstable solutions may correspond to the minimization of an objective diﬀerent from the energy, see for example [20]. Multiwell Lagrangians A transparent example of a nonquasiconvex problem is given by the following problem of structural design: Find a layout of N elastic materials that minimizes the total energy of a domain Ω. It is assumed that Ω is ﬁlled with several materials with the energy functions Wi(∇u), i = 1, . . . N where N is the number of phases. The energy W of the body is equal to W (∇u) =

N

χi(x)Wi(∇u)

(9)

i=1

where χi is the characteristic function of the subdomain Ωi occupied with ith material, 1 if x ∈ Ωi Ω = Ωi χi(x) = 0 if x ∈ Ωi i

It is assumed that the boundary displacement u(s) (s is the coordinate at the boundary ∂Ω) is given and the volume fractures mi of materials are prescribed as follows mi = χi ,

χ =

where is the symbol of averaging.

1 |Ω|

χ(x) dx Ω

(10)

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Using the deﬁnition (9) of the energy, we formulate the problem as I0 = min min χi (x) u(x)

N Ω

χi(x)Wi(∇u) dx +

i=1

N

γi

i=1

Ω

χi(x) dx − mi (11)

or I0 = min min χi (x) u(x)

N Ω

χi[Wi(∇u) + γi] dx −

i=1

N

γimi

(12)

i=1

where γi are the Lagrange multipliers of the constraints (10). An optimal layout χi of materials minimizes the sum of the energy Wi and the “cost” γi of the materials, adapting itself to the applied load. Following Kohn and Strang [44], this problem is transformed to a nonconvex variational problem for minimizer u if the sequence of minimal operations is interchanged and the minimization over χi is performed ﬁrst with “frozen” values of ∇u. The problem becomes

I0 = min u(x)

Ω

F (∇u, γi)dx −

N

γimi

(13)

i=1

where F (∇u, γi) = min {Wi(∇u) + γi} . i=1,...,N

is a nonconvex function of ∇u. The second term in (13) is independent of u and deﬁnes the amounts of materials in the mixture linking them to the costs of materials. We can assume that the costs γi are somehow speciﬁed and analyze the problem

I = min u(x)

Ω

F (∇u, γi)dx

(14)

and then deﬁne the costs to arrive at the correct volume fractions mi. The Lagrangian F is equal to the minimum of several functions Wi + γi. It is called multiwell Lagrangian and the components Wi are called wells. The costs γi must be chosen so that no well dominates: Minimum corresponds to diﬀerent wells Wi(∇u) + γi for diﬀerent values of ∇u. Formally, the range of γi is restricted by the requirements that optimal volume fractions are nonnegative, mi ≥ 0. The nonconvexity (more exactly, nonquasiconvexity , see below) of F poses several speciﬁc problems. The Euler equation for this problem is not elliptic in certain domains Vfrb of the range of ∇u. These domains must be avoided; the optimal solution ∇u jumps over the forbidden region Vfrb .

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1.2. UNSTABLE SOLUTIONS

Nonconvex energy leads to nonmonotone constitutive relations and therefore to nonunique constitutive relations: If W is nonconvex, equations (2), (4) for ∇u have more than one solution. At equilibrium, one stress σ corresponds to several strains. The nonuniqueness is the source of instability of a solution. The variational principle (8) selects the solution with the least energy from the stationary solutions of (4). This optimal solution ∇u typically oscillates between several values that correspond to diﬀerent wells Wi of the multiwell energy W , and the spatial scale of oscillation can be inﬁnitesimal. Questions about unstable solutions In dealing with nonconvex variational problems, we cannot deﬁne the solution ∇u in every point; instead, we are trying to answer several indirect questions about the solution, which we formulate here repeating them twice in mathematical and mechanical terms. (1) What are the regions of v = ∇u that correspond to oscillatory and smooth solutions, respectively? (1a) For what stresses and strains does the composite correspond to smaller energy than any pure phase? (2) What are the optimal values of v = ∇u in each well that alternate in an optimal solution? (2a) What are the strains and stresses inside the materials that form an optimal composite? (k)

(3) What are minimizing sequences χi of partitions for an optimal solution? (3a) What is the microstructure of an optimal composite? Oscillatory solutions can be described in terms of some averages, by passing to a ”relaxed” variational problem with a ”relaxed” Lagrangian. The relaxed Lagrangian Wrelax (∇u) is equal to the average over a small volume Lagrangian W (∇w) of an optimal rapidly oscillating solution w(x) with a ﬁxed mean value, w(x) = u, Wrelax (∇u) =

inf

w(x): w =u

W (∇w) .

It is assumed that w(x) is either quasiperiodic or stochastically homogeneous. This Lagrangian is called the quasiconvex envelope of the original multiwell Lagrangian, and it corresponds to a unique solution of the stationarity equation. Mechanically, the relaxed relation corresponds to the relations that link together locally averaged stresses and strains in a heterogeneous material with optimal microstructures. This averaged description poses several problems as well:

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

71

(4) How to compute or bound the quasiconvex envelope that describes the relaxed problem? (4a) What is the constitutive relation (or its estimate) between the averaged stresses and strains in an optimal composite? (5) How to deﬁne and obtain suboptimal solutions? (5a) What ﬁnitescale composites approximate the optimal inﬁnitesimal microstructure? The problem of suboptimality is not easy because of complicated microgeometry. We need to simplify (coarse) this geometry by sacriﬁcing not more than a certain portion of the objective. Answers: One-dimensional problem tional problem has the form

b

min u(x)

The unstable one-dimensional varia-

F (x, u, u )dx

(15)

a

where x ∈ [a, b] is an independent variable. We assume that F (x, u, v) is a nonconvex function of v(x) = u (x) at least for some values of x and u. Here u and v are n-dimensional vector functions of a real argument x. The one-dimensional nonconvex problem (15) is relaxed by replacing the Lagrangian F by its convex envelope Cv F (x, u, v). The convex envelope Cv F (v) of a scalar function F of a n-dimensional vector v ∈ Rn is solution to the following problem (see [75]) Cv F (v) =

min

m1 ,...mn+1 ,ξ1 ,...,ξn+1

n

miF (v + ξi)

(16)

i=1

where mk are nonnegative parameters, mk ≥ 0 such that m1 +. . .+mn+1 = 1, and ξi are n-dimensional vectors such that n+1

miξi = 0.

i=1

The convex envelope Cv F (x, u, v) of the Lagrangian F (x, u, v) is computed with respect to the variable v while u and x are treated as parameters. Consideration of the relaxed problem helps to answer the above questions as follows: (1) The minimizer u(x) is oscillatory and its derivative v = u alternates its values inﬁnitely fast if the value of the convex envelope is smaller than the value of the function, Cv F (x, u, v) < F (x, u, v). If these two coincide, Cv F (x, u, v) = F (x, u, v), the Lagrangian is convex, and the minimizer is smooth. The derivative v of an optimal solution never takes the values in the forbidden region,v ∈ Vfrb where Vfrb = {v : Cv L(x, u, v) < L(x, u, v)} .

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A. CHERKAEV

(1a) An oscillatory solution indicates that a composite is optimal, a smooth solution means that a pure phase is optimal. (2) An oscillatory optimal solution takes at most n + 1 values v + ξi in a proximity of each point (Caratheodory theorem , see [75]); these values correspond to diﬀerent convex wells and are called supporting points of the envelope. Each well supports not more than one point of v + ξi. (2a) The values v +ξi can be interpreted as strains (stresses) inside the pure material of the optimal composite. Each material is characterized by a pair of stress and strain. (3) The details of the partition of the interval are of no importance, only the measure mi of the subintervals where v = u takes speciﬁc values v + ξi is important. The fractions mi vary according to the values of u and x, adapting the composite to the varying conditions. (4) The computation of the convex envelope is an algebraic problem (16). The constitutive relation between the average stresses and strains is monotone in the sense that the Weierstrass E-function is nonnegative. ECv F (v, vˆ) = (v − vˆ)T

d v) ≥ 0 Cv F (v) + Cv F (v) − Cv F (ˆ dv

∀v, vˆ.

(4a) Since the convex envelope is linear at least in one direction, the dual ∂ Cv F stays constant when v varies. This convariable (stress or strain) ∂v stancy is interpreted as the optimality condition for the layout. (5) Suboptimal solutions may correspond to a ﬁnite size of partition of the interval [a, b] or to continuous solutions that oscillate with a ﬁnite frequency. Quasiconvex envelope The multivariable case is more complex because the third argument v of the Lagrangian F (x, u, v) – the matrix v = ∇u – is subject to linear diﬀerential constraints. In contrast with the one-dimensional case where v = u is an arbitrary integrable function, the partial derivatives v = ∇u are subject to integrability conditions ∇ × v = ∇ × ∇u = 0. These conditions restrict the neighboring values of v = ∇u of a continuous vector potential u. Generally, multivariable variational problems deal with divergencefree, curlfree, or otherwise linearly constrained ﬁelds that are subject to corresponding integrability conditions. Following Murat [71] and Dacorogna [26], it is convenient to consider the general form of such constraints A∇u =

n d j=1 k=1

aijk

∂vj = 0, ∂xk

i = 1, . . . , r

(17)

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

73

where A = {aijk} is a constant r × n × d third-rank tensor of constraints. We will assume the form L = L(v) of the Lagrangian, where v is subject to (17). Remark 1.1 The diﬀerential constraints on strain ∇ × (∇ × ) = 0, called compatibility conditions, deal with a linear form of second derivatives. Here, for the sake of simplicity, we will deal mostly with the constraints in the form (17); most results can be adjusted to a more general case of constraints that involve the second derivatives. The integrability conditions (17) introduce the dependence on a partition since they depend on the normal n and tangent t to the dividers of Ωi as well as on the properties of the neighbors. The tangent (t) components t · · t of strain and the normal (n) components of vector σ · n are continuous. These continuity conditions bond the neighboring ﬁelds in a structure. Particularly, the supporting points v + ξi of convex envelope (16) are interconnected because some of them must neighbor in the microstructure. Therefore the construction of the envelope must be modiﬁed to the quasiconvex envelope, see [7, 20, 29, 44]. Consider an inﬁnitely small cubic neighborhood ω of an inner point x ∈ Ω. Assume that the mean ﬁeld v is given and that the pointwise ﬁelds are (almost) ω-periodic and subject to (17). Quasiconvex envelope QL(v) is the minimum over all admissible perturbations with zero mean of the integral over ω of the Lagrangian L(v), 1 ξ(x)∈Ξ ω

QL(v) = min

L(v + ξ)dx

(18)

ω

where ω is an inﬁnitesimal cube, and the set Ξ is deﬁned as

Ξ= ξ:

ξ(x)dx = 0, ω

A∇ξ = 0,

ξ ∈ L∞ (Ω) .

(19)

A Lagrangian L(v) is quasiconvex, if L(v) = QL(v). The quasiconvexity of a Lagrangian means stability of the aﬃne solution to all localized zeromean ﬁnite perturbations ξ that are consistent with the diﬀerential constraints . The solution is stable to the local perturbations if QL(v) = L(v) and is unstable otherwise, when QL(v) < L(v). In the construction of the quasiconvex envelope, one treats x as a constant and assumes the periodicity of the perturbations ξ, which corresponds to the assumption that ω is a inﬁnitesimal neighborhood in Ω. In the one-dimensional case, the integrability conditions disappear and the quasiconvex envelope becomes the convex envelope.

74

A. CHERKAEV

In contrast with the convex envelope, the quasiconvex envelope is a solution to a variational not an algebraic problem. Correspondingly, the solution depends on the geometry of the partition of a cube into subdomains occupied with diﬀerent materials (the microgeometry). The above questions cannot be answered as simply as in the one-dimensional case; in the rest of the paper we discuss the progress in understanding of them. Methods of investigation of nonconvex variational problems The diversity of the above questions corresponds to a number of methods. Below, we outline recent developments of several explicit approaches to optimal mixtures, methods that specify the problem reducing it to computable algorithms. We assume that a physical problem is formulated as a minimal variational problem (8). Independently we discuss methods to reformulate a minimax problem as a minimal one, see Section 4 (i). The lamination technique (Section 2) deals with an a priori constrained class of microstructures (laminates) and uses various optimization schemes to search for optimal structures. The diﬀerential scheme (Section 2.3) allows for treatment of the problem of the best microstructure as a regular control problem. (ii). The classical variational conditions (Section 3) are based on classical Weierstrass-type structural variations. They are used to analyze ﬁelds in optimal structures and describe or approximate regions of stable and oscillatory solutions. One obtains the range of stresses and strains in each of the mixed material and evaluates suboptimal solutions. The minimal extension based on these conditions provides a Lagrangian that is stable against a special class of perturbations, an upper bound for the quasiconvex envelope. (iii). The technique of bounds (Section 4) replaces the variational problem with a rough ﬁnite-dimensional optimization problem that constrains the quasiconvex envelope from below. The bound takes into account differential constraints replacing them with special integral inequalities on admissible ﬁelds. In order to obtain the bound, duality is often used. These techniques attack the problem from diﬀerent directions but none of them gives the complete solution. The lamination technique is based on assumptions about the type of optimal geometry and the found structures are generally not unique. The variational technique is a more direct approach but it assumes a special type of local perturbations. Generally, the bounds are not expected to be exact, either. None of the above questions is fully answered so far: There is a lot of uncharted territory ahead. However, several nonquasiconvex problems are fully understood, particularly we know what are optimal composites for optimal two-phase conducting and elastic composites, see examples in [20, 63].

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

75

2. Constrained minimizing sequences and control problems 2.1. THE LEGO OF LAMINATES

Generally, the ﬁelds in a microstructure are given by solutions of elasticity problem with periodic boundary conditions and a layout χi of the materials. The eﬀective properties of a composite are computed through the integrals of this solution. For a general type of geometry, the solution can be found only numerically. However, there is an exceptional class of laminate microgeometries for which the elastic ﬁelds can be explicitly computed at each point. Correspondingly, the eﬀective properties can be explicitly computed as well. Laminates are easily generalized to the structures called laminates of a rank – a multi-scale structure of laminates from laminates, from laminates. The ﬂexibility and richness of this class and relative simplicity of calculation of the ﬁelds and eﬀective properties made it subject to detailed investigation by many authors such as Bruggeman, Hashin, Milton, Lurie, Gibiansky, Norris, Avellaneda, Murat, Tartar, Francfort, Bendsøe, Lipton, Kikuchi, Sigmund, and others. Problems in which laminates are optimal The main feature of laminate structures is constancy of the ﬁelds in layers. The ﬁelds also satisfy the compatibility conditions (17) that link together the ﬁeld in the neighboring layers and the normal to layers. For piece-wise constant ﬁelds, these conditions take the form B·n=0 (20) where n is the normal to the layers in the laminate, B is the r × d tensor of discontinuities: B = {Bik},

Bik =

n

aijk[vj ]+ −,

(21)

j=1

A = {aijk} is the tensor of diﬀerential constraints (17), and [Z]+ − is the jump of the value of Z at the boundary of the layers. The compatibility conditions depend on the normal n. We show now that if the number r of linearly independent constraints is less than the dimension d, the compatibility conditions can be satisﬁed for any ﬁelds in the layers, if the structure is properly chosen. In this case, the quasiconvex envelope coincides with convex envelope, QF = CF which is supported by the ﬁelds v + ξi that are constant within each well (material). Let us show the compatibility of the N -well problem in the case when r
76

A. CHERKAEV

We show that any N arbitrary ﬁelds are compatible in spite of constraints (20). Consider a hierarchical structure of a laminate of (N − 1)-rank that is the (N − 1) times repeated sequential laminate. First we observe that any two ﬁelds v1 and v2 are compatible in a simple laminate with specially chosen normal n1 . Indeed, since the number r of linearly independent constraints is less than the dimension d, rank of the matrix B is equal to r. Therefore, the linear homogeneous system (20) always has a solution – a d-dimensional normal n1 – because the rank of B is smaller than d. Next, we replace the two ﬁelds with an average ﬁeld v12 treating them as a homogeneous ﬁeld in a new composite material. Then we make the ﬁeld v12 in this laminate compatible with the third ﬁeld v3 using the second-rank laminate structure and choosing its normal n2 . This can be done asymptotically if the width of the new layers is much larger than the width of the layers in the ﬁrst two materials. Continuing the process, we build a laminate of (N − 1)-rank that contains N arbitrary ﬁelds and the conditions (21) are satisﬁed. Finally, the convex envelope of N -well potential is supported by at most N ﬁelds. Since these ﬁelds are compatible in the constructed laminate structure, they also represent a minimizer of the constrained problem – the quasiconvex envelope. The structure of the described laminate is nonnunique if more than two phases are mixed together. Indeed, one can combine the ﬁelds in a different sequence; correspondingly, the normals will be diﬀerent. Similarly to a simple laminate which is the simplest structure for two-well potentials, the nonunique (N − 1)-rank laminate is a basic element for problems with the multi-well potentials. Particularly, the relaxation of any nonconvex Lagrangian L(∇u) that depends on a gradient of a scalar u (curl-free minimizer) or on one or two divergence-free ﬁelds L(j1 , j2 ), ∇ · j1 = ∇ · j2 = 0 is done by constructing a convex envelope of the Lagrangian which coincides with the quasiconvex envelope because (22) holds. Laminates and minimizing sequences If the number of constraints is larger than dimension, r ≥ d (as it is in elastic ﬁelds), arbitrary constant ﬁelds v + ξi cannot be compatible. In this case, the quasiconvex envelope is not smaller than the convex envelope, QF (v) ≥ CF (v) but is still not larger than the function itself, QF (v) ≤ F (v). In order to compute an upper bound of the quasiconvex envelope, we still can minimize the energy over parameters of k-rank-laminate structures. The minimizers are the explicit geometrical parameters of the laminate. Fields inside the layers are constant and the explicit calculations of them is possible. This alone makes the laminates an attractive tool. The minimum of the Lagrangian over all laminates is called the laminate closure (see below, Section 2.2). It is not known yet for what Lagrangians the laminate closure coincides with the quasiconvex

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APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

envelope. There are numerous examples of such coincidence starting from [52, 78]; however, there are counterexamples, see Sverak [80] and Milton [63] (where the elasticity problem is addressed). The laminate technique is even used for complicated models of materials with time-dependent properties, as in a recent paper by Lurie [50]. Laminates from linear materials The laminate technique is well developed for piece-wise quadratic energies that correspond to piece-wise linear constitutive equations. In this case, eﬀective properties of laminates are explicit tensor-functions of constituent materials, their fractions mi, and the normal n to the layers; they are independent of the ﬁelds in the structures. Consider a laminate from two phases with the energies 1 1 W1 (v) = v T D1 v + γ1 and W2 (v) = v T D2 v + γ2 2 2 where D1 and D2 are the matrices of properties, γ1 and γ2 are the costs of materials, and v is a vector of the ﬁelds. Introduce the subspace q for the discontinuous components of v in a laminate with the normal n = (n1 , . . . , nd). The subspace q of discontinuous components of v is spanned by all vectors qi orthogonal to the continuous components of v T

q p = 0,

where p = {pij } :

pij =

d

aijknk .

(23)

k=1

The vector qv is discontinuous on the boundary with the normal n while vector pv stay continuous. One can show [20] that the laminate corresponds to the Lagrangian !

W0 (v) = min

m1 ∈[0,1]

1 T v Dlam (n, m1 , D1 , D2 )v + m1 γ1 + (1 − m1 )γ2 2

"

where Dlam is the eﬀective tensor of the laminate that depends on the mass fractions m1 and m2 = 1 − m1 of materials in it, normal n to laminates, and properties D1 and D2 of the mixed materials as in the following way Dlam (n, m1 , D1 , D2 ) = m1 D1 + m2 D2 −m1 m2 (D1 − D2 ) H (D1 − D2 )

H(m, q, D1 , D2 ) = q q T (m2 D1 + m1 D2 )q

−1

qT ,

(24) (25)

The dependence on normal enters the formula through the dependence (23) of q = q(n).

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A. CHERKAEV

k In order to obtain the tensor Dlm of eﬀective properties of laminates of the kth rank, the iterative operation is used:

(k)

(k−1)

Dlm (nk , m1 ) = Dlam nk, mk , Dlm (k−1)

(k−1) (1) , Dlm (2)

,

(26)

(k−1)

where Dlm (1) and Dlm (2) are two laminates obtained at the previous (k − 1) iteration and function Dlam is deﬁned in (24). The two laminates (k−1) (k−1) Dlm (1) and Dlm (2) correspond to two diﬀerent sets of structural parameters: normals and volume fractions of phases. The resulting structures with explicitly known properties are the laminates of kth rank. Optimization of the properties of these structures over the volume fractions and normals on each iteration leads to an upper bound of the quasiconvex envelope, or to the laminate closure. Special structures: Matrix laminates Formula (26) is especially simple when the composite obtained on each step is laminated k times with the pure phase D1 . In this case, a matrix of layers of D1 is enveloping the kernel D2 ; the structure is called matrix laminate of k-rank [20], its eﬀective properties tensor Dml is

Dml = D1 + m2 (D2 − D1 )−1 + m1 Pk where Pk =

k

αiqi qiT D1 qi

−1

qiT ,

−1

,

(27)

(28)

i=1

and αi are nonnegative parameters such that αi ≥ 0,

k

αi = 1.

(29)

i=1

The energy of these or otherwise specialized laminates provides a computable upper bound of the quasiconvex energy. Moreover, it was shown that this bound is exact for several problems, such as G-closure of two conducting phases, [51–53], the optimal elastic composite of maximal and minimal stiﬀness, suggested in [35, 36], and the optimal composite developed in [1, 2, 9, 10] that minimizes a sum of energies of several loadings [5, 28, 31]. The iteration of the scheme leads to nesting sequence of multiply coated structures that are optimal in a problem of coupled bounds, [21, 25]. In these problems, the upper bound coincides with an independently obtained lower bound (see below, Section 4)

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

79

2.2. LAMINATION CLOSURE

The lamination closure consists of all tensors that can be obtained by sequential laminates of an arbitrary order. The boundary of the closure can be described as a minimal set of tensors that includes the stiﬀness tensors of the original materials and stays convex under a class of transformations. It was studied in many papers starting from Francfort and Milton [30], Milton [61], see also a recent development in [20, 63]. The constitutive equations for laminates with a ﬁxed normal can be solved (see [6]) for the discontinuous components of stresses and strains, 







t·σ·t n·σ·n  n · σ · t  = Tn(C)  t · σ · n  . n··n t··t

(30)

After this transform, the eﬀective coeﬃcients of laminates Tn(Clam ) with a ﬁxed normal n can be described as a convex envelope stretched on the matrices Tn(Ci) of the coeﬃcients of the original materials Ci, Tn(Clam ) =

miTn(Ci),

i

because the vector in the right-hand side of (30) is constant in the laminate structure. The lamination closure LC of the set of stiﬀness tensors Ci of the original materials is the minimal set of tensors C that: (i) results in convex set of transforms Tn(C) for any normal n, and (ii) includes the tensors Ci, Tn(C)|C∈Lc is convex ∀n,

Ci ∈ LC.

The procedure of explicit calculation of lamination closure is complicated unless the additional restrictions on the geometry are imposed, see for example [30], [60], [61], [46] or the asymptotics are considered, as in [65]. 2.3. DIFFERENTIAL SCHEMES AND CONTROL PROBLEM

Diﬀerential scheme The rank of laminates can be inﬁnite. In this case, eﬀective properties are found using the so-called diﬀerential scheme. Diﬀerential scheme further restricts the class of laminates but allows for formulation of a regular control problem. It was used starting from Bruggeman [14, 15], developed by Norris [74], Lurie and Cherkaev [55], Avellaneda [3], Hashin [40], and other authors. Assume that an inﬁnitesimal amount of a material is added to a composite in a laminate and consider the variation of eﬀective properties. One can show [20] that the evolution of the eﬀective tensor is described by the

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A. CHERKAEV

tensor-valued diﬀerential equation µ

d ∆(µ) = Ψ(∆(µ), D, n). dµ

(31)

where µ ∈ (0, 1] is the current amount of the materials in the mixture, ∆(µ) is the current value of the eﬀective tensor of the material under construction, and D = D(µ) is the tensor properties of the material added at the “instance” µ. The formula for Ψ can be easily derived from (24) in the form Ψ(∆(µ), D, n) = [(∆(µ) − D) − (∆(µ) − D)H(∆(µ) − D)], where

H = q[qT Dq]−1 q T

(32)

and q = q(n(µ)).

Functions n(µ) and D = D(µ) can be treated as controls. The optimization objective is to minimize the energy σ : D(1) : σ of the ﬁnal mixture. Generalizations The scheme can be easily generalized if we allow to add to the mixture not only pure materials D(µ) but also a known structures of them, such as the laminates in a smaller scale. In this case, the added material D(µ) becomes an eﬀective tensor of these structures which is assumed to be a known function of the properties Di of initially given materials, their volume fractions ci = ci(µ) and the orientation na = na(µ). If twomaterials laminates are added, the tensor D(µ) is given by (24), (23) where m1 = c1 (µ), m2 = 1 − c1 (µ), and n = n(µ). The formula is naturally generalized if N -phase-laminate is added: D(µ) = Dlam (c1 (µ), . . . , cN−1 (µ), na(µ)).

(33)

Remark 2.1 The diﬀerential scheme assumes inﬁnitely many scales of averaging and leads to “physically unrealistic” structures. However, this and similar schemes are used to show the attainability of a homogenized constitutive relations or to ﬁnd the limiting constitutive relations. From this viewpoint, we should not worry about the realistic character of the microstructures more than about manufacturing Sierpinski gasket. The structures obtained by the diﬀerential scheme form a subset of lamination closure. The herring-bone-type structures, random laminates, etc. are not included in the scheme, unless it is modiﬁed. An obvious modiﬁcation would allow us to consider more complex structures than simple laminates. On the other hand, the described structures are obtained by a regular diﬀerential equation and their variety is easy to describe and optimize.

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

81

Optimal control problem Using the diﬀerential scheme, one can formulate a regular control problem as in [55]. The problem is: Minimize φ(D(1)) subject to diﬀerential constraint (32), the representation (33), and the integral constraints 1

mi =

0

cidµ,

i = 1, . . . k − 1

that express the constraints on available amounts of materials. The controls are: the normal n(µ) of the laminate at the instance µ, and the characteristics of the added laminate: its relative volume fractions cik(µ) and the normal υ(µ). Evidently, the added laminate can be replaced by some other known microstructure, including those obtained by the differential scheme itself. Remark 2.2 A distinguished feature of this problem is the irreversibility of the mixing. One can show that the constructed optimal structure belongs to the boundary of the G-closure all the time; if it does not, the trajectory can be improved. What is changing is the amounts of the already used materials. The solution to this control problem exists and it can be found from either Pontryagin’s maximum principle or Bellman’s dynamic programming. The set of extremal properties obtained by diﬀerential scheme is yet another narrowing of the laminate closure that can be called the sequentiallamination envelope. 3. Variations and analysis of ﬁelds 3.1. STRUCTURAL VARIATION

Structural variations method focuses on evaluation of the ﬁelds in optimal structures. This method investigates stability of solutions in a special class of variations. An optimal layout may form either a ﬁnite or inﬁnitesimally ﬁne structure, but it still consists of the patterns from initially given materials. Variational method characterizes the stress in the materials (phases) within an optimal structure. Basic technique of calculus of variations is used. No assumptions are made about the geometry of microstructures in optimal layouts. On the contrary, the very appearance of microstructures in an optimal design is deduced from analysis of the necessary conditions. The earlier development of structural variations was done by Lurie [48] (see also related approach by Murat [69]); the technique was developed in [59] and in the recent works [16, 17, 20, 43, 45]. Structural variations – Weierstrass-type test Consider again the problem (11) of optimal layout of several materials in the domain Ω that minimizes the energy of the elastic equilibrium. The ﬁnite number of given materials

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A. CHERKAEV

requires special variational technique because the variation of the properties caused by interchanging of the materials are ﬁnite, not inﬁnitesimal. The only small parameter is the measure of the support of the variation. Consider the following local variation. Place an inﬁnitesimal inclusion of one of the admissible materials Cincl into an arbitrary interior point of the domain ΩH of the host material Chost . Alternatively, place a dilute matrix-laminate composite (27) made of the material Chost (the matrix) and an inﬁnitesimal fraction of the material Cincl (the inclusions). Then, perform the following calculations: (i) Compute the perturbation of the ﬁelds and the increment of the objective functional – energy. To compute the increment we may either use modiﬁed Eshelby-type formulas [68] or simply compute eﬀective properties of a dilute matrix-laminate structure (27) when the volume fraction of the inclusions becomes inﬁnitesimal. The corresponding formula for the variation δD of average properties follows from (23) when m2 = δm 1

−1

δD = (Dhost − Dincl )−1 + Pk(Dincl , n, α)

δm.

Here δm is the inﬁnitesimal volume of the inclusion, and the term Pk(Dincl , n, α) is deﬁned in (28) where one puts D1 = Dincl . Applied to the stress energy Wσ, the increment δWσ of the energy becomes 1 δWσ = σ : (δS) : σ δm + o(δm). (34) 2 The increment δJ of the objective is equal to δJ = δWσ + (γincl − γhost )δm + o(δm) where the second term accounts for the diﬀerence in costs of the materials. (ii) Next, the increment δWσ in (34) is maximized by choosing the “most dangerous” variations, that is by choosing structural parameters α1 , . . . , αk and n1 , . . . , nk. (Here we follow the method suggested by Lurie in [48]). These parameters enter the problem through term Pk(Dincl , n, α) (see (28)) that represents the shape and orientation of the inclusions. The resulting most dangerous variation ∆(σ, Dhost ) = max δJ, n,α

α = {α1 , . . . , αk},

n = {n1 , . . . , nk}

(35)

depends only on the ﬁeld σ at the point of the subdomain Ωhost where the inclusion is inserted. Remark 3.1 If the number of available materials is greater than two, we generalize the variation scheme by allowing more complex inclusion such as

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

83

a laminate composite of several materials and maximize the increment by the composition of Dincl ; in this case, the cost γincl of the inclusion becomes γincl =

βmγm

m

where γm is the cost of mth material in the composition, and βm is its volume fraction in the inclusion. If the composite inclusions are used, ∆(σ, Dhost ) is the result of maximization of the increment over the structural parameters and the composition of the inclusion. Remark 3.2 One can argue that an optimal structure could be a neverending sequence of embedded laminates, or similar fractal structure, see [4], [13] and there is no solid neighborhood of a material to put the inclusion in. However, this sequence is a result of an asymptotic process and the inclusion can be smaller that the domain of a pure phase; its size should go to zero faster than the size of the domain. The ﬁelds in materials are deﬁned almost everywhere except the points of accumulation; correspondingly, the conditions could be applied almost everywhere. 3.2. ANALYSIS OF OPTIMALITY CONDITIONS

Increment ∆(σ, Dhost ) of an optimal conﬁguration is nonnegative for all trial inclusions, therefore the uniform in x inequality holds: ∆(σ, Dhost ) ≥ 0

∀x ∈ Ωhost .

Indeed, if the increment can be made negative by inserting an inclusion to the design, the layout is not optimal and the variation improves it. Solving the optimality conditions ∆(σ, Dhost ) ≥ 0 for σ, we obtain inequalities for the region Vhost of optimality of the tested material in the form σhost ∈ Vhost if D = Dhost . The procedure is repeated for all given materials. In this way, we construct the sets V1 , . . . , Vn where the ﬁeld σ in the corresponding materials satisﬁes the optimality conditions. If the materials are isotropic, these sets depend only on invariants of the ﬁelds σ. Notice that the optimality conditions assume the form of inequalities. This feature is expected because the set D1 , . . . , DN of values of the controls consists of several isolated points. The detailed analysis of the optimal ﬁelds in optimal conducting [20] and elastic [16, 17] designs reveals the following properties of two-material mixtures (two-well Lagrangian) from linear elastic materials: strong and expensive material C1 is never understressed. A norm Ns(σ) of the stress

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A. CHERKAEV

in the strong and expensive material in an optimal structure is bounded from below: (36) Ns(σ) ≥ ηs(C2 , C1 , γ2 − γ1 ), if σ ∈ V1 . Similarly, weak and cheap material C1 is never overstressed: in an optimal structure, a norm Nw(σ) of the stress tensor in that material is bounded from above if σ ∈ V2 . (37) Nw(σ) ≤ ηw(C1 , C2 , γ1 − γ2 ), Fulﬁllment of these necessary conditions is equivalent to the requirement that a norm of the ﬁeld in each material is uniformly bounded everywhere in the optimal design. We show the expressions for Nw and Ns assuming for simplicity in formulas that the Poisson ratios in the materials equal zero. These conditions are as follows: the eigenvalues σ1 , σ2 of the stress ﬁeld in the weak material belong to the intersection of two elliptic neighborhoods of zero: Nw(σ) = min{ασ12 + βσ22 , ασ22 + βσ12 }, where α1 > 0, α2 > 0, and ηw > 0 are the constants that depend only on the material’s properties of the inserted and the host materials, see [16]. The eigenvalues of the stress tensor in the strong material lie outside of the convex envelope of the ellipses:

Ns(σ) = C

σ12 σ22 σ22 σ12 + , + α β α β

,

where ηs depends on the material’s properties. The sets V1 and V2 are dual. Forbidden region There is a nonempty supplement Vfrb to the sets V1 and V2 where none of the materials is optimal. This is the region where the quasiconvex envelope of the Lagrangian is strictly smaller than the Lagrangian itself. If the applied average ﬁeld σ belongs to this region, the pointwise ﬁeld splits into several pieces σi, each in an allowed region Vi, σ =

ciσi,

σi ∈ Vi,

σ ∈ Vfrb .

ci ≥ 0,

c1 + . . . cN = 1,

i

and the optimality conditions are satisﬁed in every point. Because of this split, the structure appears that sends the pointwise ﬁelds in the materials away from the forbidden region Vfrb . This phenomenon explains appearance of composites in optimal structures. In the optimal composite zone, the stresses inside the materials belong to the boundaries of the Vi sets everywhere, while the mean stress belongs to the forbidden region. An optimal structure adjusts itself to the stress conditions by varying volume fractions of the phases and the normals to the boundaries.

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

85

Necessary conditions and optimal microstructures The optimality conditions (36) and (37) also explain the inﬁnitesimal scale of alternations. Indeed, we expect that the stress in each material remains on the boundary of + its permitted regions in some subdomains Ω+ w and Ωs of the design domain Ω. In these subdomains, the stress ﬁeld satisﬁes the elasticity equations and, in addition, the conditions: Ns(σ) = ηs Nw(σ) = ηw

if x ∈ Ω+ s, if x ∈ Ω+ w.

(38)

These conditions overdetermine the system for the stress and cannot be sat+ isﬁed in the domains Ω+ w and Ωs with nonzero interiors. Indeed, the stress of any ﬁxed layout is uniquely determined from the elasticity equations; varying the division line between phases, one can enforce the equalities (38) along some lines but not everywhere in a domain with nonzero interi+ ors. To solve this contradiction we suggest that the domains Ω+ w and Ωs of ﬁnite measures are divided by a dense (fractal-type) boundary that passes inﬁnitely close to each point in these domains. This leads to the appearance of a microstructure in an optimal design. The ﬁelds in diﬀerent materials within the structure belong to the disconnected sets Vi that surround the forbidden region; at the same time, they are competitive with each other, which means that the equations: n · [σa − σb] · n = t · [σa − σb] · n = t · [Saσa − Saσb] · t = 0,

σa ∈ Va, σb ∈ Vb

hold on the dividing line. Here the indices a and b denote the neighboring materials. The jump over forbidden region Vfrb is only possible if the normal n to the dividing surface is specially chosen or composite has a special microstructure. Particularly, one can check that the norms of the ﬁelds in the ﬁrst and the second materials in laminates and in second-rank orthogonal laminates belong to the boundaries of their sets V1 and V2 if structural parameters are optimally adjusted to the applied ﬁeld σ. The structural parameters are: the orientation of the layers and their fraction(s). When the applied ﬁeld varies, the norms N1 and N2 stay constant. The same is true (see [20]) for the Hashin-Shtrikman coated spheres structures [41] that are optimal if the external stress is isotropic. Three-dimensional optimal structures The analysis can be extended to a three-dimensional case, see [17]. The permitted regions are similar: The eigenvalues of the optimal stress in the weak material correspond to the intersection of three oblate spheroids, Nw ≤ ηw where

Nw(σ) = min α(σ12 + σ22 ) + βσ32 , α(σ22 + σ32 ) + βσ12 , α(σ32 + σ12 ) + βσ22 .

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A. CHERKAEV

In each point, stress belongs either to surface of a spheroid, or to the line of intersection of two spheroids, or to a point of intersection of all three of them. The permitted region for the eigenvalues of the stress in the strong material corresponds to the convex envelope stretched on the three larger prolate spheroids dual to the ﬁrst ones, Ns ≥ ηs,

Ns(σ) = C

σ12 + σ22 σ32 σ22 + σ32 σ12 σ32 + σ12 σ22 + , + , + α β α β α β

.

This envelope consists of the parts of original spheroids, the cylindrical surfaces between pairs of them, and a plane triangle supported by three symmetric points of the three ellipsoids. The two norms are dual. The constraints on the optimal three-dimensional stress ﬁeld matches the variety of optimal structures independently found in [33], [1] in which the necessary conditions are satisﬁed as equality pointwise. It is shown that the optimal structures are the matrix laminates of third-rank, that can degenerate into second-rank laminates and further into simple laminates. Optimal simple laminates correspond to the ﬁelds in both phases that belong to the boundaries of spheroids of the permitted ﬁelds. Optimal second-rank cylindrical matrix laminates correspond to a ﬁeld in the weak material that belongs to the intersection of two spheroids, and to a ﬁeld in the strong material that belongs to the cylindrical part of complex envelope stretched on two spheroids. Optimal third-rank matrix laminates correspond to an isotropic constant ﬁeld in the weak material that belongs to the intersection of all three spheroids, and a ﬁeld in the strong material that belongs to the ﬂat part of the complex envelope stretched on three spheroids. This analysis again shows the duality of the structures and ﬁelds in optimal micro-geometries. Types of optimal micro-geometry in three-material composites Three-material mixtures can be optimal only if the cost of the intermediate material is accurately chosen, see [20]. The too expensive intermediate material never enters the optimal composition, and the too cheap material will be used together with the worst and the best materials, but not with these two together. The region of permitted ﬁelds in the intermediate material lies between the permitted regions of outside materials; therefore the norm of the intermediate material in an optimal mixture is at a distance from both zero and inﬁnity. This implies that the three materials in an optimal structure cannot meet in an isolated point because then the norm of ﬁelds in all material would go either to zero or to inﬁnity in the proximity of this point.

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87

We conclude that either the three materials never meet in an optimal microstructure because two of them are inclusions in the third one, or they meet in a dense set of points as in laminate of the second-rank. Suboptimal projects The necessary conditions technique allows to evaluate suboptimal designs, see [16]. The optimality is naturally expressed through the ﬁelds in materials, not through the microstructure which can be nonunique and which parameters are hard to quantify. Checking the ﬁelds in a design, we can ﬁnd out how close these ﬁelds are to the boundaries of the permitted regions Vi and conclude about suboptimality of the design. In a suboptimal structure, the ﬁelds in phases do not always belong to the regions Vi but one can measure the norm of the distance between the actual ﬁeld and its region of optimality Vi and judge about the closeness of a design to the optimal one. The ability to quantify suboptimal projects is speciﬁc for this method and cannot be extended to the laminate technique. 3.3. MINIMAL EXTENSION

The structural variation methods allows us to construct an upper bound of the quasiconvex envelope of the Lagrangian obtained by the minimal extension procedure [20]. The minimal extension provides a Lagrangian that is stable to a speciﬁed class of variations. As other variational methods, the extension is based on an a priori assumptions about the class of used variations, therefore it does not result in a “ﬁnal” or universal extension. Minimal extension SF (σ) is the maximal function that is smaller than the original Lagrangian F (σ), SF (σ) ≤ F (σ)

∀σ

and cannot be improved by any local variations,

min δlocal

variation

F (σ)dx = 0. Ω

In other words, the extended Lagrangian SW () has the following properties: (i) It preserves the cost of the variational problem (3); . (ii) It leads to a stationary solution deﬁned for all ﬁelds (including those in the forbidden region), which cannot be improved by the class of considered variations. Remark 3.3 The last property distinguishes the minimal extension from quasiconvex envelope. The quasiconvex envelope is the maximal Lagrangian that is smaller then the original Lagrangian and corresponds to a solution

88

A. CHERKAEV

that cannot be improved by any local variations. The deﬁnition of minimal extension weakens the last requirement by specifying the class of trial local variations, thus making the extension computable by a regular procedure. In other words, the quasiconvex envelope is a limit of the minimal extension when the class of variations includes “everything”. Let us illustrate the approach on the same problem of optimal mixture of two linearly elastic materials. It is convenient to represent extended Lagrangian SW (σ) in the form 1 SW (σ) = σ : Sextd : σ + γextd , 2

Sextd = Sextd (σ).

(39)

Here Sextd (σ) is a tensor of properties that depends on σ. The tensor Sextd can be interpreted as an anisotropic compliance tensor of composite, made of initially given materials. The structure of the optimal composite and its eﬀective tensor Sextd varies together with the external ﬁeld σ. The compliance Sextd and the cost γextd must be chosen so that no structural variation can improve the cost of the variational problem and that the most dangerous variation leaves the cost unchanged. The cost term of the extension accounts for composition of the mixture γextd =

N

mi γ i .

(40)

i=1

When the mean ﬁeld σ belongs to one of the permitted regions Vi, the extended Lagrangian SW (σ) coincides with the original Lagrangian: SW (σ) = W (σ)

∀σ ∈ Vi, i = 1, . . . , P,

or Sextd = Si,

γextd = γi

∀σ ∈ Vi, i = 1, . . . , P.

When the mean ﬁeld σ belongs to the forbidden region Vfrb we deﬁne the extended Lagrangian (the tensor Sextd ) from the requirement that no structural variation improve the objective and the most “dangerous” variation keeps the objective unchanged. The scheme is as follows: a trial inclusion from the given materials or their composition is inserted into the unknown optimal material Sextd (σ) that corresponds to the ﬁeld σ ∈ Vfrb . We call the extension neutral with respect to the variation if ∆extd (σ, Sextd ) = 0 ∀σ ∈ Vfrb

(41)

where ∆ is the maximal increment computed as in (35). The condition of neutrality (41) implicitly determines the optimal tensor Sextd (σ) and the extended Lagrangian.

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

89

Thus, the minimal extension SW of the Lagrangian W is deﬁned by a variational inequality: SW (σ, Sextd ) = Wi(σ), SW (σ, Sextd ) ≤ Wi(σ),

∆extd (σ, Sextd ) ≥ 0, ∆extd (σ, Sextd ) = 0,

∀σ ∈ Vi, ∀σ ∈ ∪Vi

Remark 3.4 Applied to one-dimensional variational problems, a similar scheme of minimal extension results in an extension equal to the convex envelope of the Lagrangian. Assuming that the extension is based on Weierstrass variation instead of the structural variation, it is easy to check that the extension is equal to the convex envelope Cv L(x, u, v) of the Lagrangian L(x, u, u ). In the multivariable case, the described extension gives an upper boundary of the “ﬁnal” extension (the quasiconvex envelope of the Lagrangian) which may or not coincide with it. An example of exact extension is given in [20]. On the other hand, one could think of a wider class of variations that could lead to another extension with larger ∆(σ). 4. Bounds and duality 4.1. VARIATIONAL PROBLEMS AND BOUNDS FOR EFFECTIVE PROPERTIES

The sets of the eﬀective properties of all possible structures from given materials is called the G-closure of the set of these materials. To obtain the bounds for eﬀective properties, we consider variational problems of energy minimization by a periodic layout. Bound related to the energy minimization The ﬁrst problem is minimization of the energy of an aﬃne external ﬁeld is applied to the structure. Assume for deﬁniteness that the strain energy W(C(χ), ) is minimized. The energy W(C(χ), ) of a periodic layout equals to the energy of the equivalent homogeneous material (composite): W(C(χ), ) = W(C∗ , )

(42)

and deﬁnes the eﬀective properties tensor C∗ as the coeﬃcients in the righthand side of (42). In order to constraint the set of tensors C∗ , we ﬁnd a lower bound for the energy of the type W(C(χ), ) ≥ B(CB , χi , ),

∀

∀χ

where B is an explicit function of the mean ﬁeld and volume fractions mi = χi . One can show that B is a second-degree homogeneous function

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A. CHERKAEV

of ,

i χi , i

B = : CB

: .

(43)

In this procedure, the energy of an optimal composite is deﬁned by the quasiconvex envelope of the multiwell Lagrangian and the lower bound should restrict this envelope from below. Then, we pass from the bounds for an optimal energy to constraints on the range of optimal eﬀective properties tensor and conclude that C∗ ≥ CB . The tensor CB depends on invariants of the applied ﬁeld, see (43). We eliminate this dependence and obtain the G-closure. Duality and bounds The energy and its estimate are deﬁned up to additive constants. To deal with this uncertainty, we take into account the dual form of the energy – the Legendre transform of it. The dual energy has the form Wσ (σ) = max { : σ − W()} ,

(44)

and it is an involution W() = max { : σ − Wσ(σ)} . σ

(45)

The diﬀerential constraints (1) on σ and the constraint in (3) on are also mutually dual. The sum of the energy Wσ and its dual form W is equal to the work of applied forces W() + Wσ(σ) = : σ, and is completely deﬁned. On the other hand, the sum of the quadratic energy and its dual is still a quadratic form of the vector (σ, ) of double dimension and can be estimated by the same procedure as a single energy. Physically, the estimation of two forms of the energy correspond to estimation of the reaction of a structure to two ways of loading. A structure can be loaded by either external tension forces or external elongations, or both: forces in one direction and elongation in the other. Hence, either the average stress σ or the average strain in a structure are prescribed. The estimate of the strain energy W () is expressed through the prescribed average strain , and the estimate of the dual stress energy W ∗ (σ) – through the prescribed average stress σ . Several loadings To tighten the bounds, we can minimize the sum of energies caused by several mutually orthogonal homogeneous external loadings

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

91

applied to the periodic structure, which is expressed by the Lagrangian of the type: Π(χ, 1 , . . . , n) =

n

W(χ, i)).

i=1

The layout χ remains the same for all loadings; in particular, the jumps of the ﬁelds caused by diﬀerent independent external ﬁelds occur at the same dividing surfaces; therefore the pointwise ﬁelds in the structures are interrelated. This relation is taken into account by the translation method (described in the next section) that tightens the lower bound for the sum of energies. (k) A more general minimized quantity Πσ is the sum of the energy of the periodicity cell and its dual form; it has the form Π(k) σ = Π(χ, 1 , . . . , k ) + Πσ (χ, σk+1 , . . . , σn). One must consider several problems of this type for diﬀerent k = 0, . . . , n to completely characterize the set of eﬀective coeﬃcients. When the loading varies, the optimal structure varies too; the procedure must be applied for all possible combination of the loadings. The resulting set of coeﬃcients describes the set of eﬀective tensors of the structure that optimally respond to any given loading combination; they form the boundary of the G-closure. 4.2. TRANSLATION METHOD AND DEVELOPMENTS

We show the technique for derivation of lower bounds working on the example of bounding the quadratic strain energy W. Convex envelope and harmonic-mean (Wiener) bound The simplest lower bound for the nonconvex Lagrangian can be obtained by neglecting the diﬀerential constraints on the strain ﬁeld . Deleting the constraints, we enlarge the set of minimizers and achieve a lower minimum. If these constraints are neglected, the ﬁeld becomes constant within each material and the calculation of the minimum of a multiwell Lagrangian becomes an elementary algebraic problem; its solution is given by the convex envelope CL of the Lagrangian L. Because the wells are convex, the convex envelope is supported by at most one point in a well. Therefore the convex envelope CW() at the point has the form CW () = min

i ,mi

where

n i

mi = 1,

n

miW(i)

(46)

i

mi ≥ 0,

=

n i

mii

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A. CHERKAEV

and the bound is given by the inequality W(C∗ , ) ≥ CW ()

∀C∗ .

(47)

As we mentioned earlier, this bound is achievable at a laminate structure , if the rank of aijkuj is smaller than d. For quadratic energies of the type W () = 12 T Di + c where c is an undeﬁned constant, the bound to be computed is 1 CW () = T CH + cB , 2

CH =

miCi−1

−1

= Ci−1 −1

(48)

i

where CH is a harmonic mean and cB is an additive constant. Because of arbitrariness of the ﬁeld , the above bound implies the inequality: C∗ ≥ CH known in elasticity as one of Hill’s bounds. Remark 4.1 The presence of the additive constant c in this energy does not poses a problem because the strain ﬁelds can be made arbitrary large and the constant c can be neglected. However, in the next problem (Section 4.3) cB should be eliminated by estimating the sum of energy and its dual form. The complementary bound for the eﬀective properties is obtained by the same procedure, estimating the dual energy Wσ = 12 σ : S : σ + cσ where S = C −1 is the compliance and cσ is a constant. It has the form S∗ ≥ SH

or C∗ ≤ C .

Alternatively, one can estimate the sum of these two energies obtaining the above bounds at once and not worrying about the additive constant, because c + cσ = 0 due to the duality relations (44), (45). Improved bounds The bound obtained by a convex envelope can be improved if some relations which follow from the diﬀerential constraints (17) are taken into account. Indeed, the vector Θ = {1 , . . . , k, σk+1 , σn} combined from components of all ﬁelds is not a free vector but is related to the solution of an elasticity problem (2). As such, it is constrained by inequalities of the type φ (Θ 1 , . . . , Θ n , m1 , . . . , mn, D1 , . . . , Dn) ≤ 0.

(49)

Here, i is the average ﬁeld within ith phase, m1 , . . . , mn are the volume fractions, and D1 , . . . , Dn are material properties of the phases.

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

93

To obtain the bounds we need to ﬁnd (prove) inequality that holds for all admissible layouts . This inequality should be nontrivial: it should not hold for arbitrary vectors Θ but for the solutions of the elasticity equations. The inequality (49) must be added to the procedure of estimation of the lower bound (47) with the Lagrange multiplier t ≥ 0. The bound becomes W ≥ max C(W − + tφ) − tφ.

(50)

t≥0

The Hashin-Shtrikman bound [42], the translation bounds [20], and the bound by Nesi [73] are all the examples of such bounds. They all relax pointwise diﬀerential constraints by replacing them with integral inequalities. This technique was implemented to obtain bounds for the eﬀective compliance tensor S∗ well-studied starting from the classical bounds by Reuss, Voigt, and Hill. The bounds were tighten for isotropic materials by Hashin and Shtrikman [41] and Walpole [83]; then these bounds were coupled and further tighten by Berryman and Milton [12] and (for two-dimensional case) by Cherkaev and Gibiansky in [22]. The coupled bound were obtained exploring the diﬀerential constraints on the stress and strain tensors using the translation method [20]. Similar bounds for conducting materials were obtained by by Hashin and Shtrikman, then these bounds were coupled and further tighten by Lurie and Cherkaev [52], [53] and Tartar [81]. The key component of the technique is the inequality (49). The quadratic in Θ inequalities of the type ΘT T Θ ≤ Θ T T Θ

(51)

where T is not nonnegatively deﬁned matrix, can be found either immediately from the divergence theorem, [52] or by using the theory of compensated compactness [26, 62, 70, 71, 81]; numerous examples can be found in [20] and [63]. For example, the quadratic inequalities (51) imposed on the stress and strain tensors in two-dimensional elasticity are: det σ = detσ ,

det ≤ det ,

(52)

(the inequality sign in the second relation is due to second-oder diﬀerential constraints on ). Accounting for quadratic inequalities, we get the translated bounds of the type 1 W(C∗ ) ≥ T CP + cB , 2

CP =

N i=1

−1

mi(Ci + T )

−1

−T

94

A. CHERKAEV

where the matrix T satisfy (51) and the inequalities Ci + T ≥ 0 for all Ci. One can see that the property tensors are translated by the matrix T , thereafter comes the name of the method [62]. The known quadratic inequalities (translators) provide the exact bounds that match the lamination closure for a number of two-phase composites. They are too rough to provide exact bounds for multimaterial mixtures but they are sometimes exact even for these problems [66] and they always improve the harmonic mean bounds. There is no known technique to regularly derive nonquadratic inequalities for the average ﬁelds. The hunt for new translators is a nonregular problem of ﬁnding inequalities for the solutions of partial diﬀerential equations with periodic piece-wise constant coeﬃcients that are valid independently of the geometry of the structure. An inequality of such a type recently found in [73] states that in two-dimensional conductivity problem the determinant of the matrix of gradients of the two solutions does not change its sign anywhere in the periodicity cell. Adding this inequality to the translation bound, Nesi [73] obtained new more restrictive bounds for multimaterial mixtures. Meanwhile, the technique of the translation bounds is developed in another direction: the bounds for eﬀective properties are applied to various problems. Among these problems are: minimization of the sum of elastic energies in two [5] and three dimensions [31], see also [46], minimization of a functional diﬀerent from the energy, [19], [67], and compliance minimization in the worst possible scenario of loading [24]. 4.3. DUALITY AND BOUNDS FOR EXPANSION COEFFICIENTS

Here, we apply the method to ﬁnd bounds for the anisotropic eﬀective stiﬀness and extension tensors of a multiphase composite made of expandable materials, following [18]. These bounds link an anisotropic eﬀective compliance S∗ and anisotropic extension tensor α∗ of a composite. One meets such problems when dealing with composites made of materials that experience phase transition or thermal expansion. The bounds for the expansion coeﬃcients are less developed than bounds for the stiﬀness. The existing bounds [77], [76], [39] deal with the isotropic case, and the bounds by Gibiansky and Torquato [39] are extremely close to the results of numerical optimization by Sigmund and Torquato [79]. The complicated algebraic structure of the isotropic bounds makes their generalization for general anisotropic case not too attractive. In next section, we derive general bounds for the anisotropic thermal expansion tensor which are given by rather elegant tensorial expressions of a clear algebraic structure. Study of these anisotropic multiphase thermal expansion is important

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95

for applications because most composites (for instance, laminates) are anisotropic. The bound for anisotropic expansion coeﬃcients estimates the maximum and minimum of the eﬀective expansion in any direction; they can be used in structural optimization. Composite from expanding phases An expandable material is subject to a transformation impact and an elastic load. The constitutive relation for such a material is described as = S : σ + α,

∇ · σ = 0,

∇ × (∇ × )T = 0.

(53)

The expansion tensor α is a symmetric second-rank tensor of deformation due to the temperature change or the phase transition. In thermal elasticity, α is proportional to the temperature change; equation (53) is normalized with this respect (the temperature change is equal to one). For isotropic thermal-elastic materials, α is a spherical tensor; for materials under austenite-martensite transformation, α is close to a deviator (tracefree) tensor. The form of α∗ in a composite is unknown. We want to bound the range of eﬀective tensors, knowing only properties of the phases and their volume fractions in the mixture. The energy of an expandable material can be presented in two mutually dual forms 1 1 W(C, Γ, ) = : C : + : Γ + cv , Wσ (S, α, σ) = σ : S : σ + σ : α − cp 2 2 where Γ = −C : α is the expansion stress tensor, the constant tensor fully determined by the eigenstrain α and the stiﬀness tensor C. The diﬀerence between the parameters cv and cp: 1 cv − cp = α : C : α 2 can be derived from the duality relations (44). Note that α enters a lowerorder term in the energy which makes the estimation more delicate than the one for the compliance S that determines the main term. A composite with perfect bonds between phases is characterized by the eﬀective relation between volume averaged stress σ and strain that is similar to (53) with the tensors S and α being replaced by the tensors of eﬀective moduli S∗ and α∗ , respectively. The expression for the energy changes accordingly. The eﬀective tensors depend on the moduli and expansion coeﬃcients of the mixed materials and on microstructure, but are independent of the acting ﬁelds. The bounds for the eﬀective moduli are independent of the microstructure; they are represented by the inequalities of the type G(S∗ , α∗ , Sph, αph, mph) ≥ 0

96

A. CHERKAEV

where mph = {m1 , . . . mN } are the volume fractions of the phases in the composite, Sph = {S1 , . . . SN } and αph = {α1 , . . . αN } are the moduli of the phases. In order to obtain the bound, we deal with the following questions: (i) What functional should be estimated? (ii) What expression bound the functional from below? (iii) How to pass from the bound for the functional to the bounds for the eﬀective coeﬃcients? (iv) What are the bounds when void is present in the mixture? The method We estimate the sum Wσ + W of the energy and its dual form from below by using the translation method. Namely, we neglect the diﬀerential constraints in (53) replacing them with inequalities of the type σ : Tσ : σ ≥ σ : Tσ : σ which are considered as algebraic constraints. Here, Tσ is the matrix translator (for explicit form of T , see (52)). The matrix Tσ is nonpositive deﬁnite and it provides the above inequality due to diﬀerential constraints on the ﬁeld σ. The minimization problem becomes algebraic, and the standard minimization procedure yields to the inequality 1 T Wσ (S∗ , α∗ , σ )+W(C∗ , α∗ , ) ≥ ΘT PB Θ+qB Θ+rB , 2

∀Θ = (σ , )T

(54) where the tensors PB = PB (mph, Cph) and qB = qB (mph, Cph, αph) of the fourth- and second-rank, respectively, and the constant rB = rB (mph, Cph, αph) are explicitly calculated. The left-hand side of the (54) is also a quadratic function of the averaged ﬁelds Θ = [σ , ] with the coeﬃcients being the eﬀective properties C∗ , α∗ of the composite. Eliminating the dependence of Θ, we obtain the bounds for the eﬀective properties as described below. New bounds The inequality (54) yields the following inequalities for the eﬀective coeﬃcients. A matrix inequality is given by:

S∗ + Tσ Tσ

Tσ C∗ + T

− PB ≥ 0

∀T :

Si + T σ Tσ

Tσ Ci + T

≥ 0,

(55)

where i = 1, . . . , N , #

PB =

S + Tσ Tσ

Tσ C + T

−1 $−1

,

and Tσ and T are the translators similar to Tσ. This inequlity is obtained from (54) when Θ → ∞. Inequality (55) estimates the leading term in the energy; it does not depend on αph and coincides with the translation bound for the eﬀective elastic tensor. It contains, as particular cases, the

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

97

Hill bounds and the Hashin-Shtrikman-Walpole bounds for isotropic S∗ . Notice that tensorial inequality (55) for the sum of energy and its dual naturally includes lower bounds for S∗ and C∗ and coupling between them. The range of α∗ is determined by the scalar inequality (α∗ − αE (T )) : PE (S∗ , T ) : (α∗ − αE (T )) ≤ rE (T )

∀T as in (55)

(56)

where explicitly calculated coeﬃcients: fourth-rank tensor PE , the secondorder tensor αE , and the scalar rE depend on the properties of the phases, volume fractions, eﬀective tensor S∗ , and translator T σ. It is obtained from the requirement that the minimum of the diﬀerence d(Θ) between the leftand right-hand sides of (54) over Θ is nonnegative for all Θ. We compute the minimum of the quadratic d(Θ) over Θ and exclude Θ. The bounds are independent of the structure of a composite and depend only on the moduli of the phases and their volume fractions. The bounds for S∗ are independent of the extension tensors of the phases, but the bounds for α∗ depend on the compliance and expansion coeﬃcients of the phases and on the eﬀective compliance tensor S∗ of a composite. For each admissible tensor T , the coeﬃcients of the eﬀective tensor α∗ are bounded by an ellipsoid centered at αE (T ), and the bound (56) states that they belong to the intersection of all such ellipsoids. Special cases The results for the mixtures with voids are easily obtained. This case poses diﬃculties for previously suggested bounds, see [79]. In this case, the coeﬃcients in (56) are simpliﬁed to

PE = S˜∗ − S˜−1 −1

−1

,

αE = −S˜−1 −1 : Γ ,

rE = ΓT : S : Γ − Γ : S˜−1 −1 : Γ where S˜ = S + Tσ and Γ = −S −1 α. The previously obtained bounds by Schapery [77], Rozen and Hashin [76], and Gibiansky and Torquato [39] follow from our bounds. Particularly, for the two-phase mixtures, the constant rE vanishes which leads to the explicit relation α∗ = αE , which agrees with the result by Rozen and Hashin [76]. If the eﬀective tensor S∗ approaches its bound, some eigenvalues of tensor PE go to inﬁnity and the eﬀective expansion coeﬃcients tends to the coeﬃcients of αE , which agrees with the result by Gibiansky and Torquato [39]. 4.4. DUALITY AND BOUNDS FOR VISCOELASTIC MATERIALS

Reformulation of a saddle problem The duality and the Legendre transform allows to reformulate several minimax variational problems as minimal

98

A. CHERKAEV

problems and to establish new minimal variational principles. In turn, these principles permit applying the translation method technique. For example, the translation bounds for viscoelastic material tensors were established in [23]. When a viscous-elastic material is subject to a harmonic excitation, its state is described by equations of complex elasticity which look exactly as the usual elasticity equations but the ﬁelds and properties are complexvalued tensors. The real part and imaginary parts C and C of this tensor represent the stiﬀness and viscosity of a material. The approach is based on an observation that the real and imaginary part of complex elasticity equation can be viewed as the Euler equation for a minimax variational problem with a quadratic Lagrangian L( , ) =

1 2

T

C C

C −C

(57)

where the symbols and denote the real and imaginary parts. The variational problem for the real and imaginary parts of the ﬁelds is

max min

L( , )dx

(58)

ω

where and satisfy inhomogeneous boundary conditions. Problem (58) is of the min-max type which prevents the immediate use of the technique of bounds. Performing Legendre transform with respect to the real or imaginary part of the complex ﬁeld, or with respect to both, one transforms the Lagrangian to one of four forms; two of these forms correspond to minimax problems, and two other correspond to minimal problems for the transformed Lagrangian. The dual with respect to form of Lagrangian (57) is Lσ (σ , ) =

1 2

σ

T

C −1 C −1 C

C −1 C C + C (C −1 )C

σ

(59)

and the variational problem becomes a minimization problem

min min σ

ω

Lσ (σ , )dx

Euler equations of this transformed Lagrangian still give the real and imaginary part of complex conductivity equation. The functional is a positive deﬁned quadratic function of the ﬁelds and the obtained variational principle expresses minimum of the energy release rate (entropy production) per period of oscillation. The technique of bounds is applicable to the Lagrangian (59), it allows to obtain the coupled bounds for the real and imaginary part of the eﬀective tensor of a viscoelastic material, see [37, 38, 64].

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS

99

4.5. DUALITY AND STRUCTURAL OPTIMIZATION

Optimal design problems often lead to minimax variational problems, see for example [82] or are formulated as non-self-adjoint extremal problem for a self-adjoint operator, see [49] and [20]. Duality is used to relax a multilinear minimax problem of optimal design that cannot be immediately regularized by the Legendre transform. To illustrate the approach, consider the simplest problem of minimization of a functional related to a solution of conductivity problem. One can choose the layout of several isotropic conductors in a domain Ω to achieve the minimum. In this way, a structural problem of minimization of a weakly lower semicontinuous functional of the solution of the boundary value problem is formulated as a control problem: Minimize Φ(u)dx + φ(u)ds (60) I= Ω

∂Ω

where Φ and φ are continuous functions, and u solves the boundary value problem: ∂ ∇· F (χ, ∇u) = 0 (61) ∂∇u that links together the control χ and solution u. Adding this diﬀerential constraint with the Lagrange function υ to the functional (60) and integrating by parts, we obtain the following min-max problem min min max I(χ, u, υ) χ

where

I(χ, u, υ) = Ω

u

υ

(Φ(u) + ∇υ · F (χ, ∇u))dx +

∂Ω

(φ(u) + υF (∇u) · n)ds.

If the materials are linear, F (∇u) = C(χ)∇u. Local problem To ﬁnd the best structure, we pass to the local problem that describes an optimal microstructure in a neighborhood ω a point x0 ∈ Ω. We obtain the min-max problem L = min min max I; χ

u

υ

I= ω

∇υ T C(χ)∇u dx

where the mean ﬁelds ∇u and ∇υ and amounts mi = χi of the materials must be prescribed (they are determined later from the solution of the problem in large). The formulated local problem describes the basic element of an optimal structure, while the global problem describes the distribution of these elements and variation of their properties on the large scale.

100

A. CHERKAEV

In order to transform the local minimax problem to the minimal one, a three-step procedure is needed because the Legendre transform with respect to a linear term ∇υ is degenerative. (i) Observe that the objective of the local problem linearly depends on both magnitudes |∇u | and |∇υ | which implies that the magnitudes of the ﬁelds in the local problem do not aﬀect the distribution of the properties. Therefore, we normalize the ﬁelds assuming that |∇u | = 1 and |∇υ | = 1. (ii) Introduce new potentials 1 a = √ (u + υ) 2

1 and b = √ (u − υ) 2

and rewrite the local problem as the diﬀerence !

I= ω

"

1 T 1 ∇a C(χ)∇a − ∇bT C(χ)∇b dx. 2 2

(62)

One can check that the gradients of a and b are orthogonal, ∇a ·∇b = 0. (iii) Finally, perform the Legendre transform of the quadratic Lagrangian (62) with respect to ∇b, introducing the dual to ∇b divergence-free variable j (∇ · j = 0), we arrive at the minimization problem !

J = min

a,χ,j ω

1 T 1 ∇a C(χ)∇a + j T C −1 (χ)j − ∇bT j 2 2

"

dx

(63)

that requires the minimization of the energy (the ﬁrst term of the Lagrangian) of the ﬁeld ∇a and the complementary energy (second term) caused by an orthogonal current ﬁeld j. Optimal composite minimizes the sum of the energy of the ﬁeld ∇a and the complementary energy of the orthogonal ﬁeld j = ∇ × θ; the mean values of both ﬁelds are given. This requirement implies that an optimal composite must have the minimal resistance in a direction and the minimal conductivity (or the maximal resistance) in an orthogonal direction. The result is evident: the best structure is a laminate oriented so that the normal to the layer is oriented along b. In terms of the original ﬁelds, the normal bisects the directions of gradients ∇u and ∇υ of the primary and dual potentials. The technique remains the same for the elasticity operator. An optimal structure minimizes a weighted sum of diﬀerence of the stress and strain energy caused by two transversal ﬁelds. The structures are not completely described yet but it can be shown that laminates of a rank are optimal in asymptotic cases, see [20, 63, 65].

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Conclusion The outlined techniques provide partial answers to the questions about solutions of nonquasiconvex variational problems. Each method is being actively developed in recent years, and still none of them is complete today. Acknowledgement The author thanks Graeme Milton for comments and references and acknowledges support from National Science Foundation, Army Research Oﬃce, and support from NATO Research Oﬃce for the traveling and participation. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15.

Allaire, G. (2002) Shape optimization by the homogenization method, Vol. 146 of Applied Mathematical Sciences, New York: Springer-Verlag. Allaire, G. and Kohn, R. V. (1993) Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions, Quarterly of Applied Mathematics LI(4), 675–699. Avellaneda, M. (1987) Iterated homogenization, diﬀerential eﬀective medium theory, and applications, Communications on Pure and Applied Mathematics (New York) 40, 803–847. Avellaneda, M., Cherkaev, A. V., Gibiansky, L. V., Milton, G. W. and Rudelson, M. (1996) A complete characterization of the possible bulk and shear moduli of planar polycrystals, Journal of the Mechanics and Physics of Solids 44(7), 1179–1218. Avellaneda, M. and Milton, G. W. (1989) Bounds on the Eﬀective Elastic Tensor of Composites based on Two-Point Correlations, In: D. Hui and T. J. Kozik (eds.): Composite Material Technology, 89–93. Backus, G. E. (1970) A geometrical picture of anisotropic elastic tensors. Reviews of Geophysics and Space Physics 8, 633–671. Ball, J. M. and Murat, F. (1984) W 1,p -quasiconvexity and variational problems for multiple integrals, Journal of Functional Analysis 58(3), 225–253. See errata [8]. Ball, J. M. and Murat, F. (1986) Erratum: W 1,p -quasiconvexity and variational problems for multiple integrals, Journal of Functional Analysis 66(3), 439. Bendsøe, M. P., D´ıaz, A. R. and Kikuchi, N. (1993) Topology Optimization and Generalized Layout Optimization of Elastic Structures. In: M. P. Bendsøe and C. A. Mota Soares (eds.): Topology Design of Structures, Sesimbra, Portugal. Dordrecht. Boston. London, 159–205. Bendsøe, M. P. and Kikuchi, N. (1988) Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering 71(2), 197–224. Bendsøe, M. P. and Sigmund, O. (2003) Topology optimization. Theory, methods and applications. Springer-Verlag, Berlin. Berryman, J. G. and Milton G. W. (1988) Microgeometry of random composites and porous media. Journal of Physics D: Applied Physics 21, 87–94. Bhattacharya, K., Firoozye, N. B., James, R. D. and Kohn, R. V. (1994) Restrictions on microstructure, Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences 124(5), 843–878. Bruggemann, D. A. G. (1935) Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielectrizit´ atkonstanten und Leitf¨ ahigkeiten der Mischk¨ orper aus isotropen Substanzen, Annalen der Physik (1900) 22, 636–679. Bruggemann, D. A. G. (1937) Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. III. Die elastische Konstanten der Quasiisotropen

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22. 23. 24. 25.

26. 27. 28. 29. 30. 31. 32. 33.

34.

A. CHERKAEV Mischk¨ orper aus isotropen Substanzen, Annalen der Physik (1900) 29, 160–178. Cherkaev, A. and K¨ uc¸u ¨k, I. (2004) Detecting stress ﬁelds in an optimal structure Part I: Two-dimensional case and analyzer. Int. J. Structural Multidisciplinary Optimization 26(1-2), 1–15. Cherkaev, A. and K¨ uc¸u ¨k, I. (2004) Detecting stress ﬁelds in an optimal structure Part II: Three-dimensional case, Int. J. Structural Multidisciplinary Optimization 26(1-2), 16–27. Cherkaev, A. and Vinogradov, V. (2004) Bounds for expansion coeﬃciens of composites, Proceedings of ICTAM-2004. submitted. Cherkaev, A. V. (1998) Variational Approach to structural optimization. In: C. T. Leondes (ed.): Structural Dynamical Systems: Computational Technologies and Optimization, Gordon and Breach Science Publ., 199–237. Cherkaev, A. V. (2000) Variational methods for structural optimization, Berlin; Heidelberg; London; etc., Springer-Verlag. Cherkaev, A. V. and Gibiansky, L. V. (1992) The exact coupled bounds for eﬀective tensors of electrical and magnetic properties of two-component two-dimensional composites, Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences 122(1-2), 93–125. First version [34] (in Russian). Cherkaev, A. V. and Gibiansky, L. V. (1993) Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite, Journal of the Mechanics and Physics of Solids 41(5), 937–980. Cherkaev, A. V. and Gibiansky, L. V. (1994) Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli, Journal of Mathematical Physics 35(1), 127–145. Cherkaev, E. and Cherkaev, A. (2004) Principle compliance and robust optimal design, International Journal of Elasticity, to appear. Clark, K. E. and Milton G. W. (1994) Modelling the eﬀective conductivity function of an arbitrary two-dimensional polycrystal using sequential laminates. Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences 124(4), 757–783. Dacorogna, B. (1989) Direct methods in the calculus of variations, Berlin; Heidelberg; London; etc., Springer-Verlag. DeSimone, A. and Dolzmann, G. (2000) Material instabilities in nematic elastomers, Phys. D, 136(1-2), 175–191. D´ıaz, A. R. and Lipton, R. (1997) Optimal Material Layout for 3D Elastic Structures, Structural Optimization 13, 60–64. Preliminary version in [47]. Fonseca, I. (1988) The lower quasiconvex envelope of the stored energy function for an elastic crystal, Journal de Math´ ematiques Pures et Appliqu´ees, 67(2), 175–195. Francfort, G. A. and Milton, G. W. (1994) Sets of conductivity and elasticity tensors stable under lamination, Communications on Pure and Applied Mathematics (New York), 47(3), 257–279. Francfort, G. A., Murat, F. and Tartar, L. (1995) Fourth-order moments of nonnegative measures on S 2 and applications, Archive for Rational Mechanics and Analysis, 131(4), 305–333. Gibiansky, L. V. and Cherkaev, A. V. (1984) Design of composite plates of extremal rigidity, Report 914, Ioﬀe Physico-technical Institute, Acad of Sc, USSR, Leningrad, USSR. English translation in [35]. Gibiansky, L. V. and Cherkaev, A. V. (1987) Microstructures of composites of extremal rigidity and exact estimates of the associated energy density, Report 1115, Ioﬀe Physico-Technical Institute, Acad of Sc, USSR, Leningrad, USSR. English translation in [36]. Gibiansky, L. V. and Cherkaev, A. V. (1988) The set of tensor pairs of dielectric and magnetic permeabilities of two-phase composites, Technical Report 1286, Ioﬀe Physico-technical Institute, Acad of Sc, USSR, Leningrad, USSR. English translation in [21].

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36.

37.

38.

39. 40. 41. 42. 43. 44. 45.

46. 47. 48. 49. 50. 51.

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Gibiansky, L. V. and Cherkaev, A. V. (1997) Design of Composite Plates of Extremal Rigidity. In: A. V. Cherkaev and R. V. Kohn (eds.): Topics in the mathematical modelling of composite materials, Vol. 31 of Progress in nonlinear diﬀerential equations and their applications, Boston, MA: Birkh¨ auser Boston, pp. 95–137. Translation. The original publication in [32]. Gibiansky, L. V. and Cherkaev, A. V. (1997), Microstructures of composites of extremal rigidity and exact bounds on the associated energy density. In: A. V. Cherkaev and R. V. Kohn (eds.): Topics in the mathematical modelling of composite materials, Vol. 31 of Progress in nonlinear diﬀerential equations and their applications. Boston, MA: Birkh¨ auser Boston, 273–317. Translation. The original publication is in [33]. Gibiansky, L. V. and Milton, G. W. (1993) On the eﬀective viscoelastic moduli of two-phase media: I. Rigorous bounds on the complex bulk modulus. Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences 440(1908), 163–188. Gibiansky, L. V., Milton G. W., and Berryman J. G. (1999) On the eﬀective viscoelastic moduli of two-phase media. III. Rigorous bounds on the complex shear modulus in two dimensions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455(1986), 2117–2149. Gibiansky, L. V. and Torquato, S. (1997) Thermal expansion of isotropic multiphase composites and polycrystals, Journal of the Mechanics and Physics of Solids 45(7), 1223–1252. Hashin, Z. (1988) The diﬀerential scheme and its application to cracked materials, Journal of the Mechanics and Physics of Solids 36(6), 719–734. Hashin, Z. and Shtrikman, S. (1962) A variational approach to the theory of the elastic behavior of polycrystals, Journal of the Mechanics and Physics of Solids, 10, 343–352. Hashin, Z. and Shtrikman, S. (1963) A variational approach to the theory of the elastic behavior of multiphase materials, Journal of the Mechanics and Physics of Solids, 11, 127–140. Khludnev, A. M., Ohtsuka, K. and Sokolowski, J. (2002) On derivative of energy functional for elastic bodies with cracks and unilateral conditions, Quart. Appl. Math., 60(1), 99–109. Kohn, R. V. and Strang, G. (1986) Optimal design and relaxation of variational problems, Communications on Pure and Applied Mathematics (New York) 39, 1– 25 (part I), 139–182 (part II) and 353–357 (part III). Lewi´ nski, T. and Sokolowski, J. (2000) Topological derivative for nucleation of noncircular voids. The Neumann problem. In: Diﬀerential geometric methods in the control of partial diﬀerential equations (Boulder, CO, 1999), Vol. 268 of Contemp. Math. Providence, RI: Amer. Math. Soc., pp. 341–361. Lewi´ nski. T. and Telega. J. J. (2000) Plates, laminates and shells. Asymptotic analysis and homogenization, Vol. 52 of Series on Advances in Mathematics for Applied Sciences. River Edge, NJ: World Scientiﬁc Publishing Co. Inc. Lipton, R. and D´ıaz, A. R. (1995) Moment Formulations for Optimum Layout in 3D Elasticity. In: N. Olhoﬀ and G. I. N. Rozvany (eds.): Structural and Multidisciplinary Optimization, Goslar, Germany. New York, 161–168. Journal version in [28]. Lurie, K. A. (1975) Optimal control in problems of mathematical physics. Moscow, Russia: Nauka. In Russian. Lurie, K. A. (1993) Applied Optimal Control Theory of Distributed Systems, New York; London: Plenum Press. Lurie, K. A. (1997) Eﬀective properties of smart elastic laminates and the screening phenomenon, International Journal of Solids and Structures 34(13), 1633–1643. Lurie, K. A. and Cherkaev, A. V. (1981) G-closure of a set of anisotropic conducting media in the case of two dimensions, Doklady Akademii Nauk SSSR, 259(2), 328– 331. In Russian.

104 52.

53. 54.

55. 56.

57.

58.

59.

60. 61. 62. 63. 64.

65. 66. 67. 68.

A. CHERKAEV Lurie, K. A. and Cherkaev, A. V. (1982) Exact estimates of conductivity of mixtures composed of two isotropical media taken in prescribed proportion, Technical Report Ref. 783, Physical-Technical Institute, Acad. Sci. USSR. In Russian. Published in English as [54]. Lurie, K. A. and Cherkaev, A. V. (1984) Exact estimates of conductivity of a binary mixture of isotropic compounds, Report 894, Ioﬀe Physico-technical Institute, Acad of Sc, USSR. In Russian. Published in English as [56]. Lurie, K. A. and Cherkaev, A. V. (1984) Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences, 99(1-2), 71–87. Published earlier as [52] (in Russian). Lurie, K. A. and Cherkaev, A. V. (1985) Optimization of properties of multicomponent isotropic composites, Journal of Optimization Theory and Applications, 46(4), 571–580. Lurie, K. A. and Cherkaev, A. V. (1986) Exact estimates of the conductivity of a binary mixture of isotropic materials, Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences, 104(1-2), 21–38. Published earlier as [53]. Lurie, K. A., Cherkaev, A. V. , and Fedorov, A. V. (1982) Regularization of optimal design problems for bars and plates. I, II, Journal of Optimization Theory and Applications, 37(4), 499–522, 523–543. The ﬁrst and the second parts were separately published as [59] and [58]. Lurie, K. A., Fedorov, A. V. , and Cherkaev, A. V. (1980) On the existence of solutions of certain optimal design problems for bars and plates, Report 668, Ioﬀe Physico-technical Institute, Acad of Sc, USSR, Leningrad, USSR. In Russian. English translation in [57], part II. Lurie, K. A., Fedorov, A. V. and Cherkaev, A. V. (1980) Relaxation of optimal design problems for bars and plates and eliminating of contradictions in the necessary conditions of optimality, Report 667, Ioﬀe Physico-technical Institute, Acad of Sc, USSR, Leningrad, USSR. In Russian. English translation in [57], part I. Luskin, M. (1996) Approximation of a laminated microstructure for a rotationally invariant, double well energy density, Numerische Mathematik, 75(2), 205–221. Milton, G. W. (1986) Modelling the properties of composites by laminates’. In: J. Ericksen et al. (eds.): Homogenization and Eﬀective Moduli of Materials and Media (Minneapolis, Minn., 1984/1985). Berlin; Heidelberg; London; etc., 150–175. Milton, G. W. (1990) On characterizing the set of possible eﬀective tensors of composites: the variational method and the translation method, Communications on Pure and Applied Mathematics (New York), 43(1), 63–125. Milton, G. W. (2002) The theory of composites, Vol. 6 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge: Cambridge University Press. Milton, G. W. and Berryman, J. G. (1997) On the eﬀective viscoelastic moduli of two-phase media. II. Rigorous bounds on the complex shear modulus in three dimensions. Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences 453(1964), 1849–1880. Milton, G. W. and Cherkaev, A. V. (1995) What elasticity tensors are realizable?, Journal of Engineering Materials and Technology, 117(4), 483–493. Milton, G. W. and Kohn, R. V. (1988) Variational bounds on the eﬀective moduli of anisotropic composites, Journal of the Mechanics and Physics of Solids, 36(6), 597–629. Milton, G. W. and Serkov, S. K. (2000) Bounding the current in nonlinear conducting composites, J. Mech. Phys. Solids 48(6-7), 1295–1324. The J. R. Willis 60th anniversary volume. Movchan, A. B. and Movchan, N. V. (1995) Mathematical modelling of solids with nonregular boundaries, 2000 N.W. Corporate Blvd., Boca Raton, FL 33431-9868: CRC Press.

APPROACHES TO NONCONVEX VARIATIONAL PROBLEMS 69. 70. 71. 72.

73.

74. 75. 76. 77. 78. 79.

80. 81. 82. 83.

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Murat, F. (1977) Contre-examples pour divers probl`emes o` u le contrˆ ole intervient dans les coeﬃcients, Annali di Matematica Pura ed Applicata. Series 4, 49–68. Murat, F. (1981) Compacit´e par compensation: condition n´ecessaire et suﬃsante de continuit´e faible sous une hypoth`ese de rang constant, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 8(1), 69–102. Murat, F. and Tartar, L. (1985) Calcul des variations et homog´en´eisation, 57, 319–369. English translation as [72]. Murat, F. and Tartar, L. (1997) Calculus of variations and homogenization. In: A. V. Cherkaev and R. V. Kohn (eds.): Topics in the mathematical modeling of composite materials, Vol. 31 of Progress in nonlinear diﬀerential equations and their applications. Boston: Birkh¨ auser, 139–173. Translation. The original publication in [71]. Nesi, V. (1995) Bounds on the eﬀective conductivity of two-dimensional composites made of n ≥ 3 isotropic phases in prescribed volume fraction: the weighted translation method, Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences, 125(6), 1219–1239. Norris, A. N. (1985) A diﬀerential scheme for the eﬀective moduli of composites, Mechanics of Materials: An International Journal, 4, 1–16. Rockafellar, R. (1997) Convex analysis, Princeton, NJ: Princeton University Press. Reprint of the 1970 original, Princeton Paperbacks. Rozen, B. W. and Hashin, Z. (1970) Eﬀective Thermal Expansion Coeﬃcients and Speciﬁc Heats of Composite Materials, Int. J. Engng. Sci., 8, 157–173. Schapery, R. A. (1968) Thermal expansion coeﬃcients of composite materials based on energy principles, J. Comp. Mat, 2, 380–404. Schulgasser, K. (1976) Relationship between single-crystal and polycrystal electrical conductivity. Journal of Applied Physics 47, 1880–1886. Sigmund, O. and Torquato, S. (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method, Journal of the Mechanics and Physics of Solids, 45(6), 1037–1067. ˇ ak, V. (1992) Rank-one convexity does not imply quasiconvexity, Proceedings Sver´ of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences, 120(1-2), 185–189. Tartar, L. (1985) Estimation ﬁnes des coeﬃcients homog´en´eis´es. In: P. Kree (ed.): E. De Giorgi colloquium (Paris, 1983). London, 168–187. Telega, J. J. and Lewi´ nski, T. (2000) On a saddle-point theorem in minimum compliance design, J. Optim. Theory Appl., 106(2), 441–450. Walpole, L. (1966) On bounds for the overall elastic moduli of inhomogeneous systems I and II, Journal of the Mechanics and Physics of Solids, 14, 151–162, 289–301.

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ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

T. IWANIEC

Departament of Mathematics Syracuse University Syracuse, NY, 13244 USA F. GIANNETTI, G. MOSCARIELLO, C. SBORDONE

Dipartimento di Matematica ed Applicazioni ”R. Caccioppoli” Universit` a di Napoli Federici II via Cintia, 80126 Napoli, Italy AND L. KOVALEV

Departament of Mathematics Campus Box 1146 Washington University, St. Louis, MO 63130, USA

1. Introduction Quasiconformal mappings continue to play an important role in the modern theory of partial diﬀerential equations (PDEs). Here we shall focus on elliptic equations in two variables for which quasiconformal analysis suites very well. There are a number of central themes about convergence of diﬀerential operators. While quasiconformal mappings have been widely acknowledged in this context there are some ideas still unexplored. Here, using the normal family arguments we shall develop a fairly general method of constructing the G-limits of some diﬀerential operators. The very deﬁnition of a Gconvergence is concerned with the second order elliptic equations [13], [5], [11], [10]. Nowadays this concept evolves much further to include general PDEs. Let us brieﬂy discuss such a general framework. Roughly speaking, G-compactness is based on analysis of the solutions to the limit equation. In many situations, it has very little to do with convergence of the coeﬃcients.

107 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 107–138. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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Typically one seeks the limit of the operators Lj =

Aα j (x)

|α|m

∂ |α| in Ω ⊂ Rn ∂xα

(1.1)

where the Sobolev space W m,p(Ω) is considered the domain of deﬁnition of Lj with range in L p(Ω), Lj : W m,p(Ω) → L p(Ω)

(1.2)

We are ultimately looking for lim Lj to be an operator of the same form, that is ∂ |α| L = Aα(x) α (1.3) ∂x |α|m

It is reasonable to assume that the coeﬃcients are uniformly bounded in Ω |Aα j (x)| C for |α| m and j = 1, 2, . . .

(1.4)

In particular, one could extract a subsequence, still denoted by Aα j , converging to some Aα ∈ L ∞ (Ω), |α| m, in the weak-star topology. This simply means that α η(x)Aj (x) dx → η(x)Aα(x) dx (1.5) Ω

Ω

for every test function η ∈ L 1 (Ω). However, by no means one could conclude that L f = lim Lj f j (1.6) j→∞

whenever f j f weakly in W m,p(Ω), even if a priori we know that the sequence {Lj f j } is converging strongly in L p(Ω), see Section 9 for relevant examples. This situation is quite common in PDEs. We therefore introduce the following general notion [6] Deﬁnition 1.1. Diﬀerential operators Lj are said to G-convergence to L if the following holds. Suppose we are given functions f j converging weakly in W m,p(Ω) to an f ∈ W m,p(Ω) such that {Lj f j } converges strongly in L p(Ω). Then lim Lj f j = L f (1.7) j→∞

We shall make use of the following notation G

Lj −→ L

or

L = G·lim Lj

(1.8)

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

109

G

It is plain that Lj −→ L whenever the coeﬃcients of Lj converge almost everywhere to the corresponding coeﬃcients of L . The G-convergence has already a long history [4], [12], [8]. We shall come to it later on. A route to PDEs in the theory of planar quasiconformal mappings is made via the Beltrami equation ∂f ∂f = µ(z) ∂z ∂z

in Ω ⊂ C

(1.9)

for f = u + ıv ∈ W 1,2 (Ω). More general equations of concern in this theory take the form ∂f ∂f ∂f = µ(z) + ν(z) (1.10) ∂z ∂z ∂z where µ and ν are given complex-valued measurable functions satisfying the ellipticity condition |µ(z)| + |ν(z)| k < 1 a.e. in Ω

(1.11)

for some k, usually written as k=

K −1 , K +1

with 1 ≤ K < ∞

(1.12)

One of the major achievements in geometric function theory was the existence and uniqueness of so-called normalized K-quasiconformal solution of the Beltrami equation (1.9), that is, a homeomorphic solution %→C %, Φ:C

such that Φ(0) = 0 , Φ(1) = 1 and Φ(∞) = ∞

(1.13)

Using these normalized solutions we will be able to identify the G-limits of the complex Beltrami operators and some second order PDEs. In particular, we shall establish the following Theorem 1. The family BK (Ω) of all Beltrami operators ∂ ∂ − µ(z) , ∂z ∂z

with |µ(z)|

K −1 K +1

in Ω

(1.14)

is G-compact. This elegant result can actually be recovered from the seminal work of Spagnolo [13] in a rather straightforward way, though Spagnolo does not rely on quasiconformal mappings. Our approach here tells us how to attack the problem of G-convergence for general Beltrami equations as well.

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With each equation (1.10) we shall associate two K-quasiconformal solutions Φ, Ψ : C → C of so-called adjacent equation, namely  Φz = µ Φz − ν Φz

 Ψz = µ Ψz − ν Ψz

 Φ(0) = 0 and Φ(1) = 1

√  Ψ(0) = 0 and Ψ(1) = ı (ı = −1)

(1.15)

The heart of the matter will be to prove that at almost every point z ∈ C the complex derivatives Φz (z) and Ψz (z) are linearly independent (over the ﬁeld of real numbers). We have succeeded in estabilishing such independence when 1 K < 3, whence we have obtained the following main result. Theorem 2. (G-compactness). The family FK (Ω), 1 K < 3, of the general Beltrami operators ∂ ∂ ∂ − µ(z) − ν(z) ∂z ∂z ∂z with |µ(z)| + |ν(z)| k =

K −1 1 < K +1 2

(1.16)

in Ω

(1.17)

is G-compact. In spite of close analogy with Theorem 1 this more general result is really diﬀerent and the proof is much more involved. It seems plausible that the restriction to 1 K < 3 is redundant but we have not been able to dispense with this restriction. Should the linear independence fail on a set E of positive measure, this set must be of Cantor type. Most of the line segments cannot lie in E . In particular, E has no interior. This observation indicates the complexity of a hypothetical counterexample to the point-wise independence of Φz and Ψz , and to G-compactness when K ≥ 3. Two special cases of Theorem 2 are closely related to the existing results. First, the Beltrami operators in which ν = ν(z) is real-valued come very naturally from the second order divergence type equations [13]. Another special case occurs when ν(z) = µ(z). G-compactness in this case can be deduced from a study of nondivergence elliptic equations [6], [15]. As far as we are aware this is the ﬁrst time the general Beltrami operators without any relation between µ and ν have been successfully treated. While Theorem 1 ensures that the class of the Beltrami operators ∂ ∂ − µ ∂z (linear over C) is closed under G-convergence [14] it can happen ∂z ∂ ∂ that passing to a G-limit of ∂z − νj ∂z will result in an operator of the form ∂ ∂ ∂z − µ ∂z . A number of examples of such a peculiar situation, and of other interests, are presented in Sections 8 and 9.

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

111

2. Second Order Nondivergence Equations Typically we study elliptic boundary value problems in which the nonhomogeneous equation L u = h admits unique solution. Our discussion here is focussed on the elliptic second order equation LA (u) = Tr(A∇2 u) = (2.1) = a11 (x, y) uxx + 2 a12 (x, y) uxy + a22 (x, y) uyy = h with given h ∈ L 2 (Ω) in a domain Ω ⊂ R2 . The measurable symmetric matrix function ! " a11 a12 A= (2.2) a21 a22 satisﬁes the ellipticity bounds √ |ξ|2 √ A(x, y) ξ, ξ K |ξ|2 K

(2.3)

In a smooth domain Ω ⊂ R2 the Dirichlet problem   Tr(A∇2 u) = h ∈ L 2 (Ω) 

(2.4) u = 0 on ∂Ω

admits unique solution u ∈ W 2,2 (Ω). Explicitly in terms of the Green’s function, we have u(z) = GA (z, ξ) h(ξ) dξ (2.5) Ω

Its second order derivatives (Hessian matrix of u) are found by mean of a singular integral def 2 GA (z, ξ) h(ξ) dξ == A h (2.6) ∇ u(z) = Ω

where GA (z, ξ) stands for the Hessian matrix of GA with respect to zvariable. Now a sequence of operators Tr(Aj ∇2 ) is G-converging to Tr(A∇2 ) if for every h ∈ L 2 (Ω) we have Aj h A h

weakly in L 2 (Ω)

(2.7)

Note that in this case the G-limit of Tr(Aj ∇2 ), if exists, is unique. The main result concerning nondivergence elliptic operators is due to [14]. Accordingly, every family {Tr(A∇2 )} of equi-uniformly elliptic operators is

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G-compact. Analogous result in higher dimensions requires Cordes conditions [15]. We point out that in general the G-limit, LA = G·lim LAj , is not obtained from a weak limit of the matrices Aj . Our proof of Corollary 13.1 will shed a new light on how to determine the coeﬃcients of the operator LA . Let us formulate the second order equations in terms of the complex derivatives 1 ∂ ∂ 1 ∂ ∂ ∂ ∂ = +ı = −ı and (2.8) ∂z 2 ∂x ∂y ∂z 2 ∂x ∂y Upon a few elementary algebraic computations we arrive at the formula Tr(A∇2 u) = ( 2 uzz + µ uzz + µ uzz ) TrA where µ = µ(z) =

a11 − a22 + 2ı a12 a11 + a22

(2.9)

(2.10)

The ellipticity bounds at (2.3) imply |µ(z)|

K −1 <1 K +1

(2.11)

or, equivalently K (Tr A)2 ( K + 1 )2 det A

(2.12)

Using the complex gradient f (z) = uz =

ux − ı u y 2

(2.13)

we are reduced to the Beltrami equation 2fz + µ(z) fz + µ(z) fz =

h(z) Tr A

(2.14)

where the Dirichlet problem translates into the boundary condition e [f (z) dz] = 0 ,

on ∂Ω

(2.15)

Later we shall conﬁne ourselves to discussing the case TrA ≡ const. The choice of the constant is immaterial. Let us note, however, that the following normalization √ 1 Tr A ≡ K + √ (2.16) K together with (2.11) brings us back to the ellipticity bounds at (2.3).

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

113

3. Second Order Divergence Equations !

Let A =

α11 α12 α21 α22

" (3.1)

be a measurable symmetric matrix function satisfying the ellipticity bounds at (2.3). For each vector ﬁeld h = (h1 , h2 ) ∈ L 2 (Ω, R2 ) we may consider the equation div A∇u = div h (3.2) for u ∈ W 1,2 (Ω). The concept of G-convergence extends to this type of equations as follows. Deﬁnition 3.1. A sequence div (Aj ∇) is said to G-converge to div (A∇) if the following holds. Suppose we are given functions uj weakly converging to u in W 1, 2 (Ω) and vector ﬁelds hj strongly converging to h in L 2 (Ω, R2 ) such that div Aj ∇uj = div hj . Then div A∇u = div h. Variational principles ensure that for every h ∈ L 2 (Ω, R2 ) the boundary value problem   div A∇u = div h (3.3)  1,2 u ∈ W0 (Ω) admits exactly one solution. Its gradient can be expressed by means of a singular integral of h def KA (z, ξ) h(ξ) dξ (3.4) ∇u(z) == K h = Ω

Now it is a routine matter to see that the G-convergence G

div Aj ∇ −→ div A∇

(3.5)

means Kh K h, weakly in L 2 (Ω, R2 ) for every h ∈ L 2 (Ω, R2 ). A reduction of (3.2) to a ﬁrst order Beltrami type equation goes through the concept of so-called stream function v ∈ W 1,2 (Ω), deﬁned by the following system of two equations   α11 ux + α12 uy = h1 + vy (3.6)  α21 ux + α22 uy = h2 − vx Denote by f = u + ıv and h = h1 + ıh2 . Upon routine computation we arrive at the complex equation fz − µ(z)fz − ν(z)fz = a h + b h

(3.7)

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where µ=

2b =

α22 − α11 − 2ı α12 1 + Tr A + det A

ν =

α22 − α11 − 2ı α12 = µ 1 + Tr A + det A

1 − detA 1 + Tr A + det A

2a =

2 + Tr A 1 + Tr A + det A

(3.8)

(3.9)

For later purposes we note that ν is real-valued and it vanishes if det A ≡ 1. The ellipticity of (3.7) can be examined by using the eigenvalues of A, say √ 1 √ λ 1 λ2 K K

(3.10)

We have | µ(z) | =

λ2 − λ1 (1 + λ1 )(1 + λ2 )

Hence | µ(z) | + | ν(z) | =

| ν(z) | =  λ2 − 1     λ2 + 1

|λ1 λ2 − 1| (1 + λ1 )(1 + λ2 )

(3.11)

if det A 1

  1 − λ1   if det A 1 1 + λ1 These formulas may be summarized by the inequality √ K −1 | µ(z) | + | ν(z) | √ <1 K +1

(3.12)

(3.13)

Two special cases merit mentioning here. The ﬁrst case, when det A ≡ 1, leads us to the Beltrami equation with ν(z) ≡ 0. fz − µ(z)fz =

1 2

h + µh

(3.14)

√ K −1 <1 (3.15) | µ(z) | √ K +1 The second case we want to point out concerns the isotropic matrix A = λI. This yields µ(z) ≡ 0 and the coeﬃcient ν(z) = 1−λ(z) 1+λ(z) . Hence

with

∂f 1 − λ(z) ∂f h(z) − = ∂z 1 + λ(z) ∂z 1 + λ(z)

(3.16)

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

115

4. The General Beltrami Equations It is desirable to begin with the most general ﬁrst order linear operator L =α

∂ ∂ ∂ ∂ +β +γ +δ ∂z ∂z ∂z ∂z

(4.1)

A necessary and suﬃcient condition for L to be elliptic reads as follows |αξ + γξ| = |δξ + βξ|

for all complex numbers ξ = 0

(4.2)

If one looks at L as a point (α, β, γ, δ) in the complex space C4 then the ellipticity condition deﬁnes an open subset of C. This subset turns out to ∂ ∂ ∂ ∂ have exactly four connected components. The operators ∂z , ∂z , ∂z , ∂z lie in 1,p diﬀerent components. We consider the Sobolev spaces W (C), 1 < p < ∞, as natural domains of deﬁnition of these operators ∂ ∂ ∂ ∂ , , , : W 1,p(C) −→ L p(C) ∂z ∂z ∂z ∂z

(4.3)

There is a singular integral operator that makes a transition possible from one component to another, the Beltrami-Ahlfors transform S : L p(C) → L p(C)

(4.4)

deﬁned by 1 (Sω)(z) = − π

C

ω(ξ) dξ , (z − ξ)2

for ω ∈ L p(C)

(4.5)

Here the singular integral exists in the sense of Cauchy principal value for almost every z ∈ C. The transition from one component to another can be achieved via the identity S◦

∂ ∂ = : W 1,p(C) −→ L p(C) ∂z ∂z

Integration by parts gives a formula 2 |fz | = |fz |2 for f ∈ W 1,2 (C) C

C

(4.6)

(4.7)

expressing the fact that S is a unitary operator in L 2 (C), ||Sω ||2 = ||ω ||2 , for all ω ∈ L 2 (C)

(4.8)

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Without loss of generality we shall restrict ourselves to only one elliptic component, say the class of operators containing the Cauchy-Riemann com∂ . plex derivative ∂z In this class the nonhomogeneous equations reduce to the following desirable form (4.9) fz − µ(z)fz − ν(z)fz = h ∈ L p(Ω) in which | µ(z) | + | ν(z) | k < 1 ,

1+k =K 1 1−k

(4.10)

The number K is referred to as the maximal distortion. We shall consider 1,2 the Sobolev space Wloc (Ω) as the natural domain of deﬁnition of the general Beltrami operator B =

∂ ∂ ∂ 1,2 2 (Ω) → Lloc (Ω) − µ(z) − ν(z) : Wloc ∂z ∂z ∂z

(4.11)

Equation (4.9) in the entire complex plane takes an integral form ω − µ(z)Sω − ν(z)Sω = h ∈ L p(C)

(4.12)

for a density function ω = fz ∈ L p(C). It deﬁnes so-called integral Beltrami operator (4.13) B = I − µ S − ν S : L p(C) −→ L p(C) The regularity theory of quasiconformal mappings relies on its inverse. Actually the inverse operator B−1 : L p(C) −→ L p(C)

(4.14)

exists [3] for all p in the range 1+k =

2K 1 2K
(4.15)

As opposed to the case p = 2 this result is rather deep [3]. However, for the sake of simplicity we shall draw conclusions only when p = 2. In this case the existence of the continuous inverse B−1 : L 2 (C) → L 2 (C) is clear from the inequality || µ S + ν S ||2 k < 1. With the aid of B−1 we solve uniquely the Beltrami equation (4.9) in the entire plane for the complex partial derivatives of f , fz = B−1 (h) and fz = S ◦ B−1 (h)

(4.16)

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

117

Having disposed of B−1 we say that the Beltrami operator (4.11) is a G-limit of ∂ ∂ ∂ − µj (z) − νj (z) (4.17) Bj = ∂z ∂z ∂z |µj (z)| + |νj (z)| k < 1 (4.18) if for every h ∈ L 2 (C) we have −1 B−1 j h B h,

weakly in L 2 (C)

(4.19)

Our study of this question heavily depends on quasiconformal mappings. For the convenience of the reader we give a brief account of this topic. 5. Quasiconformal Mappings The recent reference for these topics is [1]. It is clear that the solutions of the homogeneous equation ∂f ∂ ∂f − µ(z) − ν(z) = 0 ∂z ∂z ∂z satisfy so-called distortion inequality Bf =

1,2 (Ω) for f ∈ Wloc

(5.1)

K −1 <1 K +1

(5.2)

| fz (z) | k | fz (z) | , where k =

Such functions, known as K- quasiregular mappings, inherit many topological features from holomorphic functions. Broadly speaking, the celebrated Stoilow factorization theorem captures all of them. It asserts that every K-quasiregular mapping f : Ω → C takes the form f (z) = F (χ(z))

(5.3)

where χ : Ω → C is a K-quasiregular homeomorphism, called K-quasiconformal mapping, and F is a holomorphic function in χ(Ω). It is rather nontrivial fact that the complex derivative fz of a nonconstant quasiregular mapping is nonzero almost everywhere. Many more properties of quasiregular mappings can be derived from Stoilow’s factorization once we appeal to the corresponding properties of holomorphic functions. Lemma 5.1. A c-uniform limit f = lim f j of K-quasiregular mappings f j : Ω → C is again a K-quasiregular mapping. If, in addition, each f j is a homeomorphism then so is f , provided f ≡ const. There are also Caccioppoli type estimates of quasiregular mappings. Making use of them we easily infer that under the terms of Lemma 5.1 the 1,2 (Ω) to fz and fz , respectively. derivatives fzj and fzj converge weakly in Wloc Another remarkable property of quasiconformal mappings is the following Montel type theorem

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Lemma 5.2. A family of all K-quasiconformal mappings Φ : C → C normalized by the conditions Φ(0) = 0, Φ(1) = γ = 0

(5.4)

is a normal family. That is, every sequence {Φj } of such mappings contains a subsequence converging c-uniformly to a mapping Φ in this family. We emphasize explicitly that the limit mapping Φ : C → C is also a K-quasiconformal homeomorphism. In eﬀect such mappings possess an a priori additional normalization at inﬁnity, Φ(∞) = ∞. Given any compact set E ⊂ C, the complex derivatives Φz and Φz form a bounded family in L 2 (E), thus weakly compact as well. One of the fundamental result in the theory of planar quasiconformal mappings is the existence theorem established by C.B. Morrey [10], more recently called the measurable Riemann mapping theorem. Let us state one of its variants Lemma 5.3. ( Normalized solution) The Beltrami equation ∂Φ ∂Φ = µ(z) , with ∂z ∂z

|µ(z)|

K −1 in C K +1

(5.5)

admits exactly one K-quasiconformal solution such that Φ(0) = 0 and Φ(1) = 1. This normalized K-quasiconformal solution is always onto and admits % = C ∪ {∞}, homeomorphic extension to the one-point compactiﬁcation C by setting Φ(∞) = ∞. We postpone the proof until Section 10 where we shall establish a signiﬁcant generalization of this result, see Theorem 10.1. With these prerequisites at hand we are now ready to prove Theorem 1. 6. G-convergence of the Operators ∂z − µj ∂z This section is dedicated not only to a proof of Theorem 1 but, more importantly, to establishing a notation and technique for the sometimes cumbersome details in the forthcoming treatment of more general Beltrami operators. As a ﬁrst step we extend each of the Beltrami coeﬃcients µj (z) to the entire complex plane by setting µj (z) = 0 outside Ω. Let Φj be normalized K-quasiconformal solutions to the Beltrami equations (5.5) with µj in place of µ. Passing to a subsequence if necessary, we may assume that Φj ⇒ Φ uniformly on compact subsets, by Lemma 5.2. Let µ = µ(z) denote the complex dilatation of Φ, that is µ(z) =

Φz (z) , Φz (z)

almost everywhere in C

(6.1)

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

119

where we recall that Φz = 0, almost everywhere. We now claim that the ∂ ∂ ∂ ∂ −µ(z) ∂z is a G-limit of the sequence Bj = ∂z −µj (z) ∂z . operator B = ∂z j 1,2 To this end consider a sequence {f } weakly converging to f in W (Ω) such that the functions hj = Bj f j strongly converge to h in L 2 (Ω). We aim to show that fz − µ(z)fz = h or, equivalently Φz fz − Φz fz = Φz h

(6.2)

The reader familiar with the subject may wish to recognize that the quadratic expression in the left hand side is a null-Lagrangian. In fact, we have the following identity Φz fz − Φz fz = (Φfz )z − (Φfz )z in the sense of distributions, meaning that (Φz fz − Φz fz ) η = (ηz fz − ηz fz ) Φ Ω

(6.3)

(6.4)

Ω

for every test function η ∈ C0∞ (Ω). Equation (6.2) is equivalent to showing that (ηz fz − ηz fz ) Φ = η h Φz (6.5) Ω

Ω

We certainly have such integral formulas for each j = 1, 2, . . . j j j (ηz fz − ηz fz ) Φ = η hj Φjz Ω

(6.6)

Ω

To prove our claim, we need only justify a passage to the limit as j → ∞. Concerning the left hand side of (6.6) we recall that Φj ⇒ Φ uniformly on compact subsets, while fzj fz and fzj fz weakly in L 2 (Ω). Thus Φj fzj Φfz and Φj fzj Φfz , in the sense of distributions (weakly in 2 (Ω) as well). Therefore, the integrals in the left hand side of (6.6) Lloc converge to the corresponding integral at (6.5). As for the integrals in the right hand side of (6.6), we recall that hj → h strongly in L 2 (Ω) and Φjz Φz weakly in L 2 (Ω). This implies hj Φjz hΦz weakly in L 1 (Ω), enough to ensure that these integrals converge to the integral in the right hand side of (6.5). Finally, identity (6.5) shows that h = fz − µ(z)fz with Φz µ= , as desired. Φz 7. Divergence Equations Revisited Essentially the same arguments give a proof of the celebrated theorem of Spagnolo [13] on G-compactness of divergence type operators div (A∇) when normalized by the condition det A(z) ≡ 1.

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Corollary 7.1. The family of all operators div (A∇) with symmetric matrices of measurable coeﬃcients satisfying the ellipticity bounds at (2.3) and normalized by det A(z) ≡ 1 (7.1) is G-compact. Proof. Let  Aj (z) = 

j

j

j

j

α11 , α12

  ,

for

j = 1, 2, . . .

(7.2)

α21 , α22 be a sequence of symmetric matrix-valued functions satisfying ellipticity bounds at (2.3) and normalized by det Aj (z) ≡ 1. To determine the Glimit, we follow very closely the formulas derived in Section 3. Accordingly, we deﬁne  √ j j j  − α − 2ı α K −1 α  11 12  , where |µj (z)| √ µj = 22  j j   K +1 2 + α11 + α22        νj (z) ≡ 0 (7.3)    1   aj (z) ≡   2       2 bj (z) ≡ µj (z) Let Φj : C → C be the normalized quasiconformal solutions to the equation Φjz − µj (z) Φjz = 0 ,

in the entire plane C,

(7.4)

where the Beltrami coeﬃcients µj are understood to be equal to zero outside Ω. This can be achieved by extending each function Aj to be the identity matrix in C \ Ω. We may, and do, assume that Φj ⇒ Φ uniformly on z compact subsets of C. It deﬁnes a Beltrami coeﬃcient µ(z) = Φ Φz , such that √

. Then, as expected, we mimic the formulas at (7.3) to deﬁne |µ(z)| √K−1 K+1 the limit matrix " ! α11 α12 (7.5) A = α21 α22 by requiring that µ(z) =

α22 − α11 − 2ıα12 , 2 + α11 + α22

2 α11 α22 − α12 ≡ 1

(7.6)

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

121

These conditions determine all entries of A uniquely. Explicit formulas are: α11 =

1 − 2 eµ + |µ|2 1 − |µ|2

(7.7)

α22 =

1 + 2 eµ + |µ|2 1 − |µ|2

(7.8)

−2 m µ 1 − |µ|2

(7.9)

α12 = α21 =

We now claim that div (A∇) is a G-limit√ of div (Aj ∇). The reader may implies the same ellipticity wish to verify that the inequality |µ(z)| √K−1 K+1 bounds hold for A as those for Aj , the ones formulated at (2.3). To prove our claim we consider a sequence of functions uj converging weakly to u in W 1,2 (Ω) and the vector ﬁelds hj = (hj1 , hj2 ) converging strongly to h = (h1 , h2 ) in L 2 (Ω, R2 ) such that div (Aj ∇uj ) = div hj

j = 1, 2, . . .

(7.10)

We aim to show that div (A∇u) = div h

(7.11)

To this end we look again at the formulas (3.6) and rewrite (7.10) equivalently as:  j j j j j j  α11 ux + α12 uy = h1 + vy (7.12)  j j j α21 ux + α22 ujy = hj2 − vxj It is clear that the stream functions v j form a bounded sequence in W 1,2 (Ω). We may therefore assume, passing to a subsequence if necessary, that v j converge to v weakly in W 1,2 (Ω). Hence the complex functions f j = uj + ı v j converge to f = u + ı v weakly in W 1,2 (Ω). In view of (3.14) the system (7.12) is equivalent to fzj − µj fzj =

1 j (h + µj hj ) , 2

hj = hj1 + ı hj2

(7.13)

At this point, the reader is urged to notice that in general µj hj fail to converge to µh strongly in L 2 (Ω). Fortunately, this equation when formulated in terms of the normalized solutions takes the form Φjz fzj − Φjz fzj =

1 j j 1 (Φz h + Φjz hj ) (Φz h + Φz h ), 2 2

(7.14)

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weakly in L 1 (Ω). The rest of the reasoning runs as before. We pass to the limit to conclude with the equation 1 1 Φz fz − Φz fz = Φz h + Φz h 2 2

(7.15)

div (A∇u) = div h

(7.16)

which is the same as completing the proof of Corollary 7.1. 8. G -limits Versus Weak-Star Topology We discuss here, by way of examples, possible relations of G-limits to weakstar convergence when the Beltrami coeﬃcients possess some special algebraic structure. Roughly speaking µj will be made (nonlinearly) of certain weakly converging terms. Example 8.1. In a rectangular region Ω = (a, b) × (c, d) we consider realvalued measurable functions µn(z) = where

    

un(x) − vn(y) , un(x) + vn(y)

√1 K

un(x)

√1 K

vn(y)

∗

√ √

z = x + ıy

K

for a < x < b

K

for c < y < d

(8.1)

(8.2)

∗

Suppose that un u and vn v, and deﬁne µ(z) =

u(x) − v(y) u(x) + v(y)

Then the G-limit of the Beltrami operators ∂ ∂ ∂z − µ(z) ∂z .

∂ ∂z

(8.3) ∂ − µn(z) ∂z is equal to

Proof. First notice the ellipticity bounds |µn(z)|

K −1 , K +1

n = 1, 2, . . .

(8.4)

We extend µ and each µn to the entire complex plane by letting them be equal to zero in C \ Ω. Let Φn : C → C be the normalized solutions of n the equations Φn z = µn(z)Φz and let Φ be their c-uniform limit; pass to

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

123

a subsequence if necessary. We need only show that Φ is the normalized solution of the equation Φz = µ(z)Φz (8.5) Thus we begin with the normalized solutions Φn : C → C of the equations n Φn z = µn(z)Φz ,

n = 1, 2, . . .

(8.6)

To justify passage to the limit we examine the complex antiderivatives x y F n(z) = un(t) dt + ı vn(s) ds (8.7) a ∗

c

∗

As un u and vn v we see that F n(z) converge point-wise everywhere in Ω to x y F (z) = u(t) dt + ı v(s) ds (8.8) a

c

These mappings are actually Lipschitz continuous, so uniformly bounded as well. Precisely, we have |F n(z1 ) − F n(z2 )| K |z1 − z2 |,

for z1 , z2 ∈ Ω

(8.9)

Their complex derivatives take the form 2Fzn = Fxn + ıFyn = un(x) − vn(y)

(8.10)

2 (8.11) K Hence, each µn(z) is none other than the Beltrami coeﬃcient of F n(z) 2Fzn = Fxn − ıFyn = un(x) + vn(y)

µn(z) =

Fzn(z) , Fzn(z)

n = 1, 2, . . .

(8.12)

Now equations (8.6) become n n Fzn Φn z − Fz Φz = 0

Let us take a look at the integral form of these equations: n n (ηz Φn = 0 z − ηz Φz ) F

(8.13)

(8.14)

Ω

for every test function η ∈ C0∞ (Ω). This makes it legitimate to pass to the limit as n → ∞, (ηz Φz − ηz Φz ) F = 0 (8.15) Ω

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which is the same as Φz =

Fz Φz = µ(z)Φz , Fz

almost everywhere in Ω

(8.16)

Finally, the equation Φz = µ(z)Φz holds in C \ Ω by trivial means since µ(z) ≡ 0 and Φz ≡ 0 outside Ω. This is the desired conclusion. One particular case merits additional considerations Example 8.2. Take vn(y) ≡ 1 for c < y < d and all n = 1, 2, . . .. Deﬁne each un(x) in the closed interval [0, 3] by the rule:    3,        un(x) = 3,          1,

if

3k − 2 3k − 3 <x< n n

if

3k − 2 3k − 1 <x< n n

if

3k − 1 3k <x< n n

(8.17)

for k = 1, 2, . . . , n. This sequence converges weakly to a constant function. Precisely, we have ∗

un(x) u(x) =

3+3+1 7 = 3 3

(8.18)

By virtue of Example 8.1 the G-limit of µn is also a constant function in Ω = (0, 3) × (c, d) G−lim µn = µ(z) =

u(x) − 1 4 = u(x) + 1 10

On the other hand, the Beltrami coeﬃcients  3k − 3 1   , if <x<   2 n     un(x) − 1  1 3k − 2 µn(z) = = , if <x<  un(x) + 1  2 n      3k − 1   0, <x< if n

(8.19)

3k − 2 n 3k − 1 n

(8.20)

3k n

converge in the weak-star topology of L ∞ (Ω) to a constant function 4 µ∞ (z) = 13 ( 12 + 12 + 0) = 13 , which is diﬀerent from µ(z) ≡ 10 .

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

125

9. A Jump From ∂z − ν∂z To ∂z − µ∂z It is somewhat surprising that the G-limit of the operators ∂z − νn ∂z with |νn(z)| ≡ const = 0 may turn into the class of operators ∂z − µ ∂z with |µ(z)| ≡ const = 0. This jump cannot be reversed, by Theorem 1. In light of such examples more caution is required in guessing whether a given class of linear PDEs is closed under G-convergence. Example 9.1. We have

√2 z n ∂ ∂ − ∂z 2 z n ∂z whereas the coeﬃcients νn(z) =

G−converge

−−−−−−−→ √ n 2z 2 zn

z ∂ ∂ + ∂z 2z ∂z

(9.1)

converge weakly to zero as n → ∞.

Proof. For computational convenience we modify slightly the νn-coeﬃcients so that the problem reduces to showing that ∂ − Bn == ∂z def

*

n + 1 z n−1 ∂ 2n z n−1 ∂z

G−converge

−−−−−−−→

z ∂ def ∂ + == B (9.2) ∂z 2z ∂z

For the equivalence, we simply observe that the diﬀerence between νn−1 and the modiﬁed coeﬃcients is uniformly converging to zero. Precisely, we have * * * 1 n + 1 z n−1 n + 1 − (9.3) →0 νn−1 (z) − = 2 2n z n−1 2n Now suppose that we are given functions f n converging to f weakly in def 1,2 2 (C) and such that hn == Bnf n → h strongly in Lloc (Ω). Wloc We want to infer that Bf = h. Let us begin with some formulas, lengthy though elementary computation being omitted, * n + 1 z n−1 n n n f (9.4) h (z) = fz − 2n z n−1 z * n + 1 z n−1 n hn(z) = (f n)z − f (9.5) 2n z n−1 z The following identity is the key to make a passage to the limit as n → ∞; * z n+1 2n + 2 z n (9.6) h − n+1 hn = n z z +* , , +* ! ! n+1 " " n + 1 zn n z 2n + 2 z n n + 1 z2 n n + f − f − n+1 f f n z 2n z 2 n zn z z z z

z

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Each term converges in the sense of distributions, we only outline the ar2 (C), hence zn+1 hn 0 guments. By assumption hn → h strongly in Lloc zn+1 2 weakly in Lloc (C). By Sobolev imbedding theorem f n → f strongly in n 2 2 Lloc (C). In particular, zzn f n 0 weakly in Lloc (C). The limit equation takes the form +* , √ z √ z 1 z2 2 h − 0= 2 f + f − [ 0 ]z − [ 0 ]z (9.7) z z 2 z2 z z

which is the same as

z (9.8) fz 2z The interested reader may wish to mimic the above computation to generalize the above example as follows. h = fz +

Example 9.2. For k = 12 , 23 , 34 , 45 , . . . we have √ zn ∂ ∂ − k n ∂z z ∂z

G−converge

−−−−−−−→

∂ z ∂ + k ∂z z ∂z

(9.9)

as n → ∞. The divergence type equations discussed in Section 3 have one interesting thing to tell us about these examples. If we assume that √ the coeﬃcients n (z) √1 λn(z) with K, then in fact νn are real-valued, say νn = 1−λ 1−λn (z) K we are dealing with the isotropic equations div (λn ∇), see formula (3.16). The G-closure of all such isotropic operators consists, rather unexpectedly, of all divergence operators div (A∇) with A satisfying ellipticity bounds at (2.3). This result can be found in [9]. In other words, the class of general Beltrami operators with the coeﬃcients ν = ν(z) real-valued is G-closed. Before turning to the general Beltrami operators we ﬁrst establish some new facts about quasiconformal mappings. We do this in the next two sections. 10. A Pair of Primary Solutions A somewhat more sophisticated approach is needed to handle general Beltrami operators. As these operators are linear only over the ﬁeld of real numbers, two independent K-quasiconformal solutions will have a part to play ∂ ∂ ∂ in G-convergence. Associated with each operator B = ∂z − µ(z) ∂z − ν(z) ∂z

∂ ∂ ∂ is its adjacent operator B ∗ = ∂z −µ(z) ∂z +ν(z) ∂z . The game is to establish a linear class of quasiconformal mappings F : C → C which solve the same homogeneous equation B ∗ F = 0. Precisely, we consider the 2-dimensional space F = { F = αΦ + βΨ ; α, β ∈ R} (10.1)

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

where Φ and Ψ, called primary solutions, satisfy Φ(0) = 0, Φ(1) = 1 Φz − µ Φz + ν Φz = 0, Ψ(0) = 0, Ψ(1) = ı Ψz − µ Ψz + ν Ψz = 0,

127

(10.2)

We point out that each F = αΦ+βΨ, except for the trivial case α = β = 0, will be a K-quasiconformal normalized solution to the equation B∗F = 0 ,

F (0) = 0 and F (1) = α + ıβ = 0

(10.3)

Theorem 10.1. The adjacent operator B ∗ admits a linear family F of K-quasiconformal solutions. This family is spanned by a pair (Φ, Ψ) of its primary solutions. Our proof below will also give the existence result of Lemma 5.3, while the uniqueness of Φ in case ν ≡ 0 can easily be deduced from Stoilow’s factorization. We will follow very closely the ideas in [7], see also [1]. Let us assume, as a preliminary step, that both µ and ν are supported in an annulus r < |z| < R. µ(z) = ν(z) = 0 ,

whenever |z| r or |z| R

(10.4)

1,2 In particular, every solution F ∈ Wloc (C) to the adjacent Beltrami equation

Fz − µ(z)Fz + ν(z) Fz = 0

(10.5)

is analytic for |z| > R. We then seek solutions normalized by the following three conditions: F (0) = 0,

F (1) = a ∈ C and F (z) = O(|z|) near inﬁnity

(10.6)

Lemma 10.2. The normalization conditions at (10.6) determine unique 1,2 (C) of Equation (10.5). solution F ∈ Wloc Proof. For the uniqueness we consider the case a ≡ 0 to show that F ≡ 0. This quasiregular mapping, being O(|z|) near inﬁnity, must be either constant or a quasiconformal homeomorphism. Indeed we deduce this fact from Stoilow’s factorization theorem and analogous observation for holomorphic functions. However, F (0) = F (1) = 0 whence F ≡ 0, as desired. Our proof of the existence makes appeal to Schauder’s ﬁxed point theorem. We shall ﬁnd a solution in the following form F (z) = a z e Tω, for some ω ∈ L p(C),

p>2

(10.7)

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Here T stands for the Cauchy transform in the complex plane, that is, an integral operator deﬁned by 1 1 1 (Tω)(z) = + ω(τ ) dτ (10.8) π τ C z−τ Let us reveal in advance that the unknown density function ω(τ ) will vanish for |τ | > R and, therefore, Tω will be a continuous function on C. We have the following normalization conditions for granted (Tω)(1) = 0 and ω(τ ) 1 dτ lim (Tω)(z) = z→∞ π |τ|R τ

(10.9) (10.10)

Next we invoke the Beurling-Ahlfors transform at (4.5) to compute complex derivatives of Tω: ∂Tω = ω ∂z

and

∂Tω = Sω ∂z

(10.11)

In this way our equation B ∗ F = 0 reduces to the integral equation for the density function ω, ω − µ Sω +

az ν az

e Tω−Tω Sω

=

(10.12)

µ(z) a ν(z) Tω−Tω e (10.13) − z az To ﬁnd of a solution to this nonlinear equation we ﬁx an exponent p > 2 close enough to 2 so that k ||S ||p < 1. Of course we seek a solution ω ∈ L p(C) which will be supported in the annulus r < |z| < R. To every such ω there corresponds a unique solution Ω ∈ L p(C) of the linear equation az ν e Tω−Tω SΩ = (10.14) Ω − µ SΩ + az =

µ(z) a ν(z) Tω−Tω e (10.15) − z az That we can solve this equation for Ω is a straightforward consequence of the following uniform estimates =

(1 − k ||S ||p) ||Ω ||p -Ω − µSΩ + a z ν e Tω−Tω SΩ- az p

(10.16) (10.17)

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

- |µ(z)| + |ν(z)| - k Cp(r, R) z p Hence ||Ω ||p

k Cp(r, R) def == M 1 − k ||S ||p

129 (10.18)

(10.19)

where the constant M is independent of ω. Of course our solution Ω is supported in the annulus r < |z| < R. These estimates indicate that we should look for ω to be a ﬁxed point of the nonlinear operator deﬁned by the rule Πω = Ω, on a closed convex subset U ⊂ L p(C), U = {ω ∈ L p(C); ||ω ||p M, ω(z) = 0 if |z| r or |z| R}

(10.20)

% is a compact operator, we Since the Cauchy transform T : U → C (C) ﬁnd that Π : U → U is both continuous and compact. Schauder’s ﬁxed point theorem tells us that there exist ω ∈ U such that Πω = ω. This is the solution of (10.12) that we were looking for, completing the proof of Lemma 10.2. With these preliminary result we now return to: Proof of Theorem 10.1. The solution F , constructed in Lemma 10.2, is analytic for |z| > R. If in addition a = 0, then F has non-vanishing derivative at inﬁnity 1 F (z) ω(τ ) def F (∞) == lim = a · exp dτ = 0 (10.21) z→∞ z π |τ|R τ Stoilow’s factorization implies that F is in fact a K-quasiconformal home% onto C. % The uniqueness part of Lemma 10.2 is crucial for omorphism of C establishing linear dependence (over R) of the solutions F = Fa, which are normalized by the condition F (1) = Fa(1) = a ∈ C. Denote by Φ = F1 and Ψ = Fi the solutions normalized by Φ(1) = 1 and Ψ(1) = ı, Φ(0) = Ψ(0) = 0. We want to verify the relation Fα+ıβ (z) = α Φ(z) + β Ψ(z),

for all α, β ∈ R

(10.22)

which is immediate since both sides satisfy the same equation and the same normalization. We are now in a position to remove the restriction about the supports of µ and ν which was imposed in the ﬁrst step of our arguments. Suppose we are given arbitrary measurable coeﬃcients µ and ν satisfying the ellipticity condition |µ(z)| + |ν(z)| k , almost everywhere in C (10.23)

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For each n = 1, 2, . . . we truncate them to the annulus µn(z) = νn(z) =

1 n

|z| n,

µ(z) , 0

if n1 |z| n otherwise

(10.24)

ν(z) , 0

if n1 |z| n otherwise

(10.25)

Let Φn and Ψn be the corresponding normalized solutions. They generate a linear family of solutions Fn = {F n = αΦn + βΨn;

α, β ∈ R}

(10.26)

We may assume that Φn → Φ and Ψn → Ψ c-uniformly to K-quasiconformal homeomorphisms Φ, Ψ : C → C such that Φ(0) = Ψ(0) = 0, Φ(1) = 1, Ψ(1) = ı. For, if not, we pass to a suitable subsequence. Clearly, αΦn + βΨn → αΦ + βΨ. In a nontrivial case when α2 + β 2 = 0, all mappings αΦn + βΨn are K-quasiconformal, so is the limit mapping αΦ + βΨ. The n n n 2 complex derivatives αΦn z +βΨz and αΦz +βΨz converge weakly in Lloc (C) to αΦz +βΨz and αΦz +βΨz , respectively. This ensures that the limit mappings αΦ + βΨ also satisfy the adjacent Beltrami equation. The proof of Theorem 10.1 is complete. The reader may wish to observe that we have lost the uniqueness in passing to the limit as n → ∞. In fact the uniqueness for the general Beltrami equation without compact supports of the coeﬃcients remains unclear. 11. Independence of Φz (z) and Ψz (z) If we ﬁx real parameters α, β ∈ R, α2 + β 2 = 0, then αΦ + βΨ is a quasiconformal mapping. Thus to every nonzero pair of real numbers there corresponds a set E ⊂ C of full measure such that α Φz (z) + β Ψz (z) = 0

for z ∈ E

(11.1)

This suggests that for almost every z ∈ C the complex numbers Φz (z) and Ψz (z) might be linearly independent over the ﬁeld R. In other words we want to ﬁnd a set E of full measure in which (11.1) holds for all α, β ∈ R , except for the trivial case α = β = 0. This is by no means an easy task since the set of parameters α, β is uncountable. It is equivalent to the following condition a.e. in C (11.2) m(Φz Ψz ) = 0,

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

131

Theorem 11.1. Suppose Φ, Ψ : Ω → C generate a linear family of Kquasiconformal mappings, 1 K < 3: αΦ + βΨ : Ω → C,

α2 + β 2 = 0.

(11.3)

a.e. in Ω

(11.4)

Then m(Φz Ψz ) = 0

The proof is preceded by three lemmas. Lemma 11.2. Let F : Ω → C be a K-quasiconformal mapping with 1 K < 3. Then for each compact E ⊂ Ω we have dz |Fw−1 (w)|3 dw < ∞ (11.5) |F (z)| z E F(E) Proof. The ﬁrst inequality is just a matter of the substitution z = F −1 (ω). That the right hand integral is ﬁnite follows from a sharp area distortion theorem of K. Astala [2]. We apply this theorem to the inverse mapping F −1 : F (Ω) → C, which is also K-quasiconformal. Astala’s theorem asserts, 1,p 2K among other things, that F −1 ∈ Wloc (F (Ω)) for every 1 p < K−1 . Thus assuming 1 K < 3 we can take p = 3. The other two lemmas are about measurable functions in any abstract measure space, say (E, σ), where 0 < σ(E) < ∞. Lemma 11.3. Let Γ : E → I be a σ-measurable function valued in a closed interval I ⊂ R. Then there exists γ ∈ I such that for every 0 < |I| we have σ(E) · (11.6) σ{z ∈ E; γ − Γ(z) γ + } 3|I| Proof. Suppose that, on the contrary, for every γ ∈ I there exists 0 < |I| for which σ(E) (11.7) σ{z ∈ E; γ − Γ(z) γ + } < 3|I| Among all such intervals [γ − , γ + ] we can choose a ﬁnite cover of I, say I ⊂

N

[γj − j , γj + j ]

(11.8)

j=1

in such a way that every point of I belongs to at most two intervals, Besicovitch Lemma. It then follows that N j=1

j 3 |I|

(11.9)

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We arrive at the contradiction in the following way σ(E)

N

σ{z ∈ E; γj − j Γ(z) γj + j }

(11.10)

j=1

<

N σ(E) j=1

3|I|

· j σ(E)

(11.11)

Lemma 11.4. Let Γ : E → R be a σ-measurable function. Then there exists γ ∈ R such that dσ(z) =∞ (11.12) E |Γ(z) − γ| Proof. We may assume that the values of Γ(z) lay in a closed interval I ⊂ R, since otherwise we restrict Γ to a suitable subset of E. We take γ ∈ I as in Lemma 11.3. Suppose to the contrary that the integral at (11.12) is ﬁnite. By Chebyshev inequality we see that dσ(z) (11.13) σ{z ∈ E; |Γ(z) − γ| < } E |Γ(z) − γ| where E denotes the set in the left hand side. The integral in the right hand side has order o(), as → 0, contrary to Lemma 11.3. Proof of Theorem 11.1: Assume that m(Φz Ψz ) = 0 on a set E ⊂ Ω of positive measure. We may further assume that Ψz ∈ L ∞ (E) and Ψz = 0 on E, for if not, we replace E by a smaller set of positive measure. Of course, this is legitimate because Ψz = 0 a.e. in Ω. Consider a measurable function Γ =

Φz :E→R Ψz

In view of Lemma 11.4 there exists γ ∈ R such that Ψz dz dz ||Ψz ||L ∞ (E) ∞= E |Φz − γΨz | E |Fz |

(11.14)

(11.15)

where F = Φ − λΨ is a K-quasiconformal mapping, in contradiction with Lemma 11.2. 12. Proof of Theorem 2 Given a sequence of general Beltrami operators in Ω ⊂ C Bj =

∂ ∂ ∂ − µj (z) − νj (z) ∂z ∂z ∂z

(12.1)

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

133

We extend the coeﬃcients µj and νj to the entire complex plane by letting them equal to zero outside Ω. To identify a subsequence and its G-limit we solve the adjacent equations  j j j   Φz − µj Φz + νj Φz = 0, Φj (0) = 0, Φj (1) = 1 (12.2)   j j j j j Ψz − µj Φz + νj Φz = 0, Ψ (0) = 0, Ψ (1) = ı for K-quasiconformal homeomorphisms Φj , Ψj : C → C. This is a pair of primary solutions. Theorem 11.1 ensures that m(Φjz Ψjz ) = 0, almost everywhere in C. We can express uniquely the coeﬃcients µj and νj in terms of the complex derivatives of Φj and Ψj , namely   Ψjz Φjz − Ψjz Φjz   (z) = ı µ  j    2 m(Φjz Ψjz ) (12.3)  j j j j   Φ − Ψ Φ Ψ z z  z z    νj (z) = ı j j 2 m(Φz Ψz ) We may assume, passing to a subsequence if necessary, that the following c-uniform limits exist Φ(z) = lim Φj (z) ,

Ψ(z) = lim Ψj (z)

j→∞

j→∞

(12.4)

By Lemma 5.1 each of these limits is either K-quasiconformal homeomorphism or a constant mapping. The latter case is ruled out because of the normalization conditions Φ(0) = 0 = 1 = Φ(1) and Ψ(0) = 0 = ı = Ψ(1). By the same reason the linear combination F = αΦ + βΨ = lim (αΦj + βΨj ) j→∞

(12.5)

is a K-quasiconformal homeomorphism with F (0) = 0 and F (1) = α+ıβ = 0. Again by virtue of Theorem 11.1, we infer that m(Φz Ψz ) = 0

a.e. in C

(12.6)

Next we mimic the formulas (12.3) to deﬁne the following coeﬃcients  Ψz Φz − Ψz Φz   µ(z) = ı    2 m(Φz Ψz ) (12.7)   Ψ Φ − Ψ Φ  z z z z   ν(z) = ı 2 m(Φz Ψz )

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This simply means that   Φz − µ(z)Φz + ν(z) Φz = 0 

(12.8) Ψz − µ(z)Ψz + ν(z) Ψz = 0

We now need two Lemmas. The ﬁrst one gives us a uniform ellipticity bound for µ(z) and ν(z). The second will complete the proof of Theorem 2. Lemma 12.1. For almost every z ∈ C |µ(z)| + |ν(z)|

K −1 =k<1 K +1

(12.9)

Proof. In fact, the derivation of (12.9) is somewhat tricky. Since the mappings αΦ + βΨ are all K-quasiregular, we have |αΦz + βΨz | k |αΦz + βΨz | ,

a.e. in C

(12.10)

for every α, β ∈ R. In general, the set in which this inequality holds depends on the parameters α and β. As a ﬁrst step we consider only rational numbers α and β, that is, a countable set of parameters. Intersection of a countable family of sets of full measure is again a set of full measure. Thus inequality (12.10) holds on a set E ⊂ C of full measure whenever α and β are rational. It remains valid in this same set E whenever α and β are real numbers, by a density argument. Now, in view of Theorem 11.1, we may further assume that m(Φz Ψz ) = 0 on E. With this choice of the set E we can test the inequality (12.10) by real-valued measurable functions in place of the parameters α and β . Given a real-valued measurable function θ = θ(z) deﬁned in E, we may solve the equation α Φz + β Ψz = eıθ(z)

(12.11)

for the measurable real coeﬃcients  m(eıθ Ψz )   α(z) =    m(Φz Ψz ) (12.12)

  m(eıθ Φz )    β(z) = m(Φz Ψz ) One can certainly select measurable θ = θ(z) in such a way that |µ(z)| + |ν(z)| = |µ eıθ − ν e−ıθ | ,

a.e. in C

(12.13)

In view of the equations (12.8) and (12.11) we then ﬁnd that |µ(z)| + |ν(z)| = |αΦz + βΨz | k |αΦz + βΨz | = k, for z ∈ E as claimed.

(12.14)

135

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

Lemma 12.2. The diﬀerential operators Bj at (12.1) G-converge to B=

∂ ∂ ∂ − µ(z) − ν(z) ∂z ∂z ∂z

(12.15)

Proof. Suppose we are given a sequence {f j } weakly converging to f in the 1,2 2 Sobolev space Wloc (Ω) such that Bj f j converge strongly in Lloc (Ω). It is to be shown that (12.16) lim Bj f j = Bf j→∞

To this end we need a few identities. Let us begin with elementary, though lengthy, diﬀerentiation in the sense of distributions [ (Ψz − ıΦz )f + (Ψz + ıΦz ) f ]z − [ (Ψz − ıΦz )f + (Ψz + ıΦz ) f ]z = [ (Ψz − ıΦz )fz + (Ψz + ıΦz ) fz ] − [ (Ψz − ıΦz )fz + (Ψz + ıΦz ) f z ] = (Ψz − ıΦz )fz + (µΨz − νΨz − ı µΦz + ı νΦz )fz + − (µΨz − νΨz − ı µΦz + ı νΦz )fz − (Ψz + ı Φz ) fz = (Ψz − ı Φz )(fz − µfz − νfz ) − (Ψz + ı Φz ) (fz − µfz − νfz ) We can certainly apply this identity to f j , Φj and Ψj , to obtain [(Ψjz − ıΦjz )f j + (Ψjz + ıΦjz ) f j ]z − [ (Ψjz − ıΦjz )f j + (Ψjz + ıΦjz ) f j ]z = = (Ψjz − ıΦjz ) Bj f j − (Ψjz + ıΦjz ) Bj f j 2 Denote by h the strong Lloc (Ω)-limit of the sequence {Bj f j }. Since

(Φjz , Φjz , Ψjz , Ψjz ) (Φz , Φz , Ψz , Ψz )

(12.17)

2 (C) we are in a position to weakly in L 2 (C) and f j → f strongly in Lloc pass to the limit in this latter equations. The formula in the limit takes the form

[ (Ψz − ı Φz )f + (Ψz + ı Φz ) f ]z − [ (Ψz − ı Φz )f + (Ψz + ı Φz ) f ]z = = (Ψz − ıΦz ) h − (Ψz + ı Φz ) h Upon cancellation of the second order derivatives of Φ and Ψ this formula simpliﬁes as (Ψz − ıΦz )(fz − h) + (Ψz + ı Φz ) fz = = (Ψz − ıΦz ) fz + (Ψz + ı Φz ) (fz − h)

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The rest is a matter of elementary algebraic manipulation in which we take into account the equations at (12.7). Precisely, we ﬁrst express fz − h as a linear combination of fz and fz , the detailed veriﬁcation of this computation being left to the reader. 2 m(Φz Ψz ) (fz − h) =

(12.18)

= ı (Ψz Φz − Ψz Φz ) fz − ı (Ψz Φz − Φz Ψz ) fz

(12.19)

Then in view of (12.7) we conclude with the equation h = fz − µfz − νfz = Bf

(12.20)

completing the proof of Lemma 12.2. This also proves Theorem 2. 13. Nondivergence Equations Revisited Formulas we derived in previous sections also work for the operators LA given at (2.1). One could examine these equations for general matrices but we stick to the case Tr A = a11 (z) + a22 (z) ≡ 2

(13.1)

for simplicity. More importantly, in order to use Theorem 2 we need to restrict ourselves to the case 1 K < 3. These constraints may be summarized as Tr A(z) ≡ 2 and det A(z) d (13.2) where d > 34 , see the inequality (2.12). Corollary 13.1. The family of the operators LA = Tr(A∇2 ) normalized by (13.2) is G-compact. Proof. Suppose we are given a sequence LAj of diﬀerential operators in Ω ⊂ C normalized by conditions (13.2), that is ∂2 ∂2 ∂2 LAj = 4 = + αj (z) + αj (z) ∂z∂z ∂z∂z ∂z∂z 4

∂ ∂ ∂ + αj (z) + αj (z) ∂z ∂z ∂z

◦

∂ ∂z

where 4αj = aj11 − aj22 + 2ı aj12 ,

and

aj11 + aj22 ≡ 2

(13.3)

ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

137

As before, we view Aj as matrix functions deﬁned in the entire plane, letting them be equal to the identity matrix outside Ω. We recall the primary solutions of the adjacent equation.  j j j   Φz + αj Φz − αj Φz = 0, Φj (0) = 0, Φj (1) = 1 (13.4)   j j j Ψz + αj Ψz − αj Ψz = 0, Ψj (0) = 0, Ψj (1) = ı It is important for the subsequent arguments to notice that the complex derivatives Φjz and Ψjz assume only imaginary values. This property is of course preserved under weak convergence in L 2 (C). Thus the limit mappings Φ = lim Φj and Ψ = lim Ψj enjoy the same property, Φjz (z) , Ψjz (z) , Φz (z) , Ψz (z)

∈ ıR

Following formulas (12.7) we ﬁnd that µ(z) = ν(z). It means that the limit mappings solve the same equation of the form: Φz + αΦz − αΦz = 0

and

Ψz + αΨz − αΨz = 0

(13.5)

with some measurable coeﬃcient α = α(z) satisfying the ellipticity bound |α(z)| + |α(z)| k =

K −1 K +1

Theorem 2 tells us that the Beltrami operator

∂ ∂z

(13.6)

∂ ∂ + α ∂z + α ∂z is a G-limit

∂ ∂ ∂ + αj ∂z + αj ∂z . This translates into a G-convergence of the operators ∂z of LAj to the operator LA . Finally, the coeﬃcients of A are determined uniquely from the relations

a11 + a22 = 2 ,

and a11 − a22 + 2ı a12 = 4α

(13.7)

More explicitely, a11 = 1 + 2 e α ,

a22 = 1 − 2 e α

and

a12 = 2 m α (13.8)

A remarkable feature of Corollaries (7.1) and (13.1) is that both normalization conditions: det A = 1 for divergence operators and Tr A ≡ const for a nondivergence equation, are invariant under G-convergence. Acknowledgement Flavia Giannetti, Gioconda Moscariello and Carlo Sbordone want to acknowledge GNAMPA- Gruppo Nazionale per l’Analisi Matematica, la Probabilit e le loro Applicazioni. Tadeusz Iwaniec wants to acknowledge National Science Foundation Grants: DMS-0301582 and DMS-0244297.

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References 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15.

Astala, K., Iwaniec, T. and Martin, G. Elliptic equations and quasiconformal mappings in the plane. to appear. Astala, K. (1994) Area distortion of quasiconformal mappings, Acta Math., 173(1), 37–60. Astala, K., Iwaniec, T. and Saksman, E. (2001) Beltrami operators in the plane, Duke Math. J., 107(1), 27–56. Bensoussan, A., Lions, J.-L. and Papanicolaou, G. (1978) Asymptotic analysis for periodic structures, vol. 5 of Studies in Mathematics and its Applications, NorthHolland Publishing Co., Amsterdam, . De Giorgi, E. and Spagnolo, S. (1973) Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 8, 391–411. D’Onofrio, L. and Greco, L. A counterexample in g-convergence of nondivergence elliptic operators, Proc. Royal Soc. Edinburgh, to appear. Iwaniec, T. (1976) Quasiconformal mapping problem for general nonlinear systems of partial diﬀerential equations. In Symposia Mathematica, vol. XVIII (Convegno sulle Transformazioni Quasiconformi e Questioni Connesse, INDAM, Rome, 1974), Academic Press, London, 501–517. Jikov, V. V., Kozlov, S. M. and Ole˘ınik, O. A. (1994) Homogenization of diﬀerential operators and integral functionals, Springer-Verlag, Berlin, Translated from the Russian by G. A. Yosiﬁan [G. A. Iosif yan]. Marino, A. and Spagnolo, S. (1969) n Un tipo di approssimazione dell’operatore n 1 Di (aij (x)Dj ) con operatori 1 Dj (β(x)Dj ), Ann. Scuola Norm. Sup. Pisa (3), 23, 657–673. Morrey, Ch. B., Jr. (1938) On the solutions of quasi-linear elliptic partial diﬀerential equations, Trans. Amer. Math. Soc., 43(1), 126–166. Murat, F. and Tartar, L. (1997) H-convergence. In Topics in the mathematical modelling of composite materials, Progr. Nonlinear Diﬀerential Equations Appl., Birkh¨ auser Boston, Boston, MA, 31, 21–43. Ole˘ınik, O. A. , Shamaev, A. S. and Yosiﬁan, G. A. (1992) Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, NorthHolland Publishing Co., Amsterdam, 26 . Spagnolo, S. (1968) Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa (3) 22 571-597; errata, ibid. (3), 22, 673. Zhikov, V. V. and Sirazhudinov, M. M. (1981) Averaging of nondivergence secondorder elliptic and parabolic operators and stabilization of the solution of the Cauchy problem, Mat. Sb. (N.S.), 116(158)(2), 166–186, . Zhikov, V. V. and Sirazhudinov, M. M. (1988) Averaging of a system of Beltrami equations, Diﬀerentsial nye Uravneniya, 24(1), 64–73, 181.

HOMOGENIZATION AND OPTIMAL DESIGN IN STRUCTURAL MECHANICS

´ T. LEWINSKI

Warsaw University of Technology, Faculty of Civil Engineering, Institute of Structural Mechanics, al.Armii Ludowej 16, 00-637 Warsaw, Poland E-mail: [email protected]

Abstract. The paper gives a brief review of the available results concerning the minimum compliance problem of two phase plates loaded in-plane, thin plates in bending, plates subjected to arbitrary loads, Mushtari-DonnellVlasov shells, Koiter shells, membrane shells and 3D elastic bodies. An emphasis is put on displacement-based formulations which can be almost directly implemented into the commerical FE codes. Key words: topology optimization, homogenization, plates, shells, compliance

1. Introduction The problem of minimizing the compliance of plates composed of two isotropic materials has been explicitly solved by Gibiansky and Cherkaev (1984, 1987). The origins of this topic should be looked for in earlier papers concerned with a scalar elliptic problem, see Tartar (2000), where details of this development is outlined. A comprehensive mathematical study of the compliance minimization problem within linear elasticity can be found in Allaire (2002). More physical approach, partly based on the methodology of the contemporary mechanics of composites, is presented by Cherkaev (2000). Other aspects of the problem are given by Bendsøe (1995). Minimization of the compliance of thin bending plates and shells is discussed in Lewi´ nski and Telega (2000). The aim of this paper is to discuss selected aspects of the minimum compliance problem of plates, shells and 3D bodies that can be important while forming the numerical algorithms and computer programs.

139 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 139–168. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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The minimum compliance problem of plates loaded in-plane is originally posed in terms of stresses. Consequently, the potential W ∗ given by (34) is expressed in terms of stress resultants. We shall show that the constitutive equations (35) can be inverted to the ﬁnal form (44) and one can ﬁnd the potential W explicitly. Due to this result the minimum compliance problem can be solved with using FE systems, like Abaqus, where the user can write down his own constitutive equations by the user’s procedures, cf. Dzier˙zanowski (2001) and Czarnecki et al. (2003). The important properties of the displacement-based formulation have been cleared up by Lipton (1994). The minimum compliance problem of thin bending plates can be posed and solved similarly, as the in-plane problem. Inversion of the equations (61) is possible, as has been already shown in Sec. 26 of Lewi´ nski and Telega (2000) and Kolanek and Lewi´ nski (2003). The numerical procedures based upon (62), expressed in terms of displacements, are discussed in Czarnecki et al. (2003a,b). In the present paper we recall the diﬀerences between the relaxed formulations of the in-plane and bending problems. The problem of a simultaneous in-plane and transverse loading is discussed in Sec. 4. This problem is new and its solution is linked with ﬁnding new translation matrices. The translation matrix (74) is proposed in a non-diagonal form, capable of taking into account the in-plane-bending coupling. In this manner we introduce an eﬀective potential, as a function generated by the invariants (90). The eﬀective potential can be determined explicitly in the simpler problem of shape optimization, or in the case when the weaker material degenerates to voids. This paves the way to a numerical realization of the method, possibly with using the commercial FE codes. The problem of optimal layout of two materials within the MushtariDonnell-Vlasov shells assumes a surprisingly simple form, similar to the coupled problem of the thin plate theory. The eﬀective potential W ∗ is common for both the problems. The shell characterization concerns the equilibrium problem and the interpretation of the isoperimetric condition. This simplicity is a consequence of the decoupling of the basic cell problems of homogenization of Mushtari-Donnell-Vlasov shells of periodic structure, see Lewi´ nski and Telega (2000). In other thin shell models the above problems are coupled, which makes impossible ﬁnding the potential W ∗ explicitly. One more analogy is considered: between the minimum compliance problems in 2D elasticity and in the membrane shell theory. The relaxed formulation of the minimum compliance problem of the membrane shells involves the eﬀective potential W ∗ of exactly the same form as that of the in-plane problem. This helps in the examination of the membrane shells of minimal compliance. Degeneration of the weaker material leads to the

HOMOGENIZATION AND OPTIMAL DESIGN . . .

141

shape design of membrane shells. The assumption of the amount of the material being small rearranges the problem to the Michell-like form, free of any material characteristics. The 3D minimum compliance problem is considered very brieﬂy. The crucial role is played by the explicit formulae for the eﬀective moduli of nski the stiﬀ 3rd rank orthogonal laminates. They have been found in Lewi´ (2001a,b,c) and used in numerical computations in Czarnecki and Lewi´ nski (2003) and in Czarnecki et al. (2003a,b). An alternative approach is used in D´ıaz and Lipton (1997). The 3D shape design has been developed in Allaire et al. (1997), Olhoﬀ et al. (1998) and Borrvall and Petersson (2001). 2. The layout problem in 2D elasticity. Minimum compliance design The problem of optimal layout of two given materials is formulated here within the plane elasticity problem under which either the plane stress or plane strain state is understood. In both the formulations the mathematical description is identical; the diﬀerences concern interpretation of the elastic moduli. Thus it is suﬃcient to consider further the plane stress problem concerning thin plates loaded in plane. The stress resultants form a tensor ﬁeld N = (N αβ ) referred to the plane domain Ω representing a middle plane of the plate. This domain is parametrized by the Cartesian system (x1 , x2 ) with the orthogonal basis (e1 , e2 ). The components N αβ refer to this basis. We write N ∈ E2s , where E2s represents the set of symmetric second rank tensors. To formulate the optimum design problem one should ﬁx the loading, boundary conditions and physical parameters of both the constituents. We assume that the plate is loaded along a boundary identiﬁed here with a part Γσ of the contour ∂Ω. The density of the loading is p = (p1 , p2 ). The plate is supported along Γu, the remaining part of the contour ∂Ω. The plate is assumed to be of constant thickness h. The plate is formed of two isotropic materials, indexed by γ = 1 or γ = 2. The constitutive equations are assumed in the linear form N = Aε

(1)

A = hC is the stiﬀness tensor and C = (C αβλµ) represents the tensor of reduced elastic moduli of plane stress state. Both the tensors A and C are of 4th rank and possess appropriate symmetry properties, which is written as C ∈ M4s , A ∈ M4s . Moreover, ε = (εαβ ) represents the strain tensor, determined by the displacement ﬁeld u = (u1 , u2 ) by 1 ∂uα ∂uβ + (2) εαβ (u) = 2 ∂xβ ∂xα

´ T. LEWINSKI

142

Two materials are at our disposal. They are located transversely homogeneous below and above the subdomains Ω1 or Ω2 of the middle plane Ω. Let χγ , γ = 1, 2, represent a characteristic function of the domain Ωγ . Thus χγ (x) = 1 if x = (x1 , x2 ) ∈ Ωγ and χγ (x) = 0 otherwise. One can write now (3) A(x) = χ1 (x)A1 + χ2 (x)A2 where Aγ represents the stiﬀness tensor of γ th phase. Let us deﬁne the tensors I 1 , I 2 ∈ M4s of components 1 I1αβλµ = δ αβ δ λµ 2 1 αλ βµ I2αβλµ = δ δ + δ αµδ βλ − δ αβ δ λµ 2

(4)

The tensors above have properties of projectors, see Milton and Kohn (1988). Both the materials are assumed as isotropic. The eﬀective stiﬀness tensors are represented by Aγ = 2kγ I 1 + 2µγ I 2 where kγ =

Eγ h , 2(1 − νγ )

µγ =

Eγ h 2(1 + νγ )

(5)

(6)

Here Eγ are Young moduli and νγ are Poisson ratios. The following ordering assumption is adopted k2 > k1 ,

µ2 > µ1

(7)

Consequently, the tensor A2 − A1 is positive deﬁnite. The amounts of both the materials are ﬁxed. In particular the quantity χ2 (x)dx = C2 (8) Ω

equal to the volume of material 2 is given. A natural measure of the stiﬀness of the structure is its compliance or the value of the linear form (9) f1 (v) = p · vds Γσ

on the solution v = u of the equilibrium problem. One can prove that the compliance f1 (u) increases if a part of a structure is removed or replaced by a weaker material, see Lewi´ nski and Sokolowski (2003). The compliance

HOMOGENIZATION AND OPTIMAL DESIGN . . .

f1 (u) will be treated as is χ2 , characterizing the Let us deﬁne the set ational equation

143

a functional J1 (χ2 ) = f1 (u(χ2 )) whose argument distribution of both the materials in the body. Σ1 (Ω) of ﬁelds N ∈ L2 (Ω, E2s ) such that the vari N αβ εαβ (v)dx = f1 (v) (10) Ω

. /2 is satisﬁed for all v ∈ V1 (Ω). Here V1 (Ω) consists of such v ∈ H 1 (Ω) that traces of v vanish on Γu. The equations inverse to (1) will be written as ε = A−1 N . According to Castigliano’s theorem, see Neˇcas and Hlavaˇcek (1981), the compliance can be expressed by

J(χ2 ) = min{ N : A−1 N dx | N ∈ Σ1 (Ω) } (11) Ω

The formula above involves the equilibrium problem. The tensor A−1 is represented by −1 (12) A−1 (x) = χ1 (x)A−1 1 + χ2 (x)A2 Due to isotropy, see (5) 1 1 A−1 γ = Kγ I 1 + Lγ I 2 2 2

(13)

where Kγ = (kγ )−1 , Lγ = (µγ )−1 . Now we are ready to formulate the problem of optimum design of a two-component plate. A search for the stiﬀest plate (loaded in plane) is realized by min{J(χ2 ) | χ2 ∈ L∞ (Ω; {0, 1}), χ2 satisﬁes (8) }

(14)

The problem above needs relaxation, as explained in the articles by Allaire and Lipton (2003) in these Proceedings, see also Allaire (2002) and Cherkaev (2000). Relaxation of (14) consists in i) replacing the design variable χ2 by its weak-∗ limits, called m2 belonging to L∞ (Ω, [0, 1]), ii) replacing the tensor functions x → A(x) of class L∞ (Ω, M4s ) by their H-limits A∗ belonging to the same class, see Sec. 2.1.2 in Allaire (2002). Various sequences {(χ2 )n} determining the sequences of tensor valued functions An can tend to the same weak-∗ limit m2 ∈ L∞ (Ω; [0, 1]) and simultaneously determine various H-limits A∗ ∈ L∞ (Ω; M4s ). The set of these H-limits is denoted by Gm2 . According to the conjecture of Dal Maso and Kohn, proved by Raitums (2001), the set Gm2 is characterized pointwise as follows. Let us consider an εY -periodic composite; ε > 0 and Y is a rectangular periodicity cell parametrized by the Cartesian coordinate system (y1 , y2 ).

144

´ T. LEWINSKI

Let χYγ be a characteristic function of the subdomain Yγ ⊂ Y occupied by material γ; γ = 1, 2. Distribution of the moduli within Y are given by A(y) = χY1 (y)A1 + χY2 (y)A2

(15)

Y −1 A−1 (y) = χY1 (y)A−1 1 + χ2 (y)A2

Let us deﬁne the deformation operator 1 ∂uα ∂uβ y + εαβ (u) = 2 ∂yβ ∂yα

(16)

We say that ε ∈ L2 (Y, Es2 ) is kinematically admissible if there exists u = (u1 , u2 ) deﬁned on Y such that (16) holds. If additionally u assumes equal values at opposite sides of Y then we write εy (u) ∈ K1per (Y ), where K1per (Y ) is the set of kinematically admissible deformations of the Y -periodic problem. According to the theory of homogenization the tensor of elastic moduli of an εY -periodic composite is given by the formula ε : (Ahε) = min { εy : (A(y)εy) |

(17)

A is given by (15), ε ∈ y

K1per(Y

), ε = ε} y

for all ε ∈ E2s . The brackets · represent averaging over Y . Considering all ﬁelds χY2 ∈ L∞ (Y, {0, 1}) satisfying the condition χY2 = 0 1 we deﬁne the set P of tensors Ah of such periodic structure. This set is not closed. The closure: P = G means admission of periodic composites of hierarchical microstructure. It turns out that the set Gm2 has the following characterization. If m2 ∈ L∞ (Ω, [0, 1]) then (18) Gm2 = {A∗ ∈ L∞ (Ω; M4s ) A∗ (x) ∈ Gm2 (x) a.e. in Ω } see (2.10) in Allaire (2002). For a ﬁxed x ∈ Ω the quantity A∗ (x) represents a certain tensor Ah from the set G with = m2 (x). To construct a formula dual to (17) we deﬁne the set Σper (Y ) = n = (nαβ ) ∈ L2 (Y ; E2s ) n : εy (v) = 0 ∀ v ∈ D(Y, E2s ), 1 nαβ νβ take opposite values on opposite sides of Y

(19)

where ν is a unit vector outward normal to Y ; D(Y, E2s ) represents the space of C ∞ functions with compact support in Y and with values in E2s . The tensor A−1 h has the following representation 0

−1 1 N : (A−1 | n ∈ Σper (20) 1 (Y ), n = N } h N ) = inf{ n : A n

HOMOGENIZATION AND OPTIMAL DESIGN . . .

145

Now we are ready to formulate the relaxed form of the problem (14) ∞ 2 min{J(m2 ) | m2 ∈ L (Ω; [0, 1]), m2 (x)dx = C2 } (21) Ω

where 2 2 ) = min{ J(m

2W ∗ (N , m2 )dx | N ∈ Σ1 (Ω) }

(22)

Ω

with

0 1 2W ∗ (N , ) = inf { n : (A−1 (y)n) | n ∈ Σper 1 (Y ),

(23)

n = N , χY2 ∈ L∞ (Y ; {0, 1}), χY2 = } Here A−1 (y) is given by (15) while is a number from [0, 1]. The formula (23) refers to the case of division of Y into Y1 and Y2 . The inﬁmum in (23) is not achieved on such composites. It is achieved by hierarchical composites of 2nd rank. The task of ﬁnding the function W ∗ (N , ) explicitly seems to be extremely diﬃcult. None the less this function was explicitly found in the report by Gibiansky and Cherkaev (1987). The method used there is called now the translation method, according to the suggestion of Milton (2002). We omit here the derivation and recall the ﬁnal result. First, let us deﬁne the invariants of N 1 I(N ) = √ trN , 2 (24) /1 1 . II(N ) = √ (trN )2 − 4 det N 2 2 and set ζN

II(N ) = , |I(N )|

ζN

NI − NII = NI + NII

(25)

for I(N ) = 0. Here NI , NII are principal values of N ; NI NII by the convention. Let us introduce a useful notation for the quantity f which assumes two values: f1 and f2 f = (1 − )f1 + f2 ,

∆f = |f2 − f1 |,

[f ] = (1 − )f2 + f1

(26)

Here is a number from [0, 1]. Let us introduce the auxiliary functions of the parameter K1 K2 + L2 K ˘ , K() = L2 + [K]

L1 L2 + K2 L ˘ L() = K2 + [L]

(27)

´ T. LEWINSKI

146 ζ1 () = and further

K2 + [L] , ∆L

ζ2 () =

∆K [K] + L2

˘ , aL() = K()

aR() = K2 ,

cL() = L2 ,

˘ cR() = L() ,

AL() =

(1 − )∆L(L2 + [K]) [K + L]

2 (1 − )(∆L)2 ∆K AR() = (K2 + [L])[K + L] Next we deﬁne the function   HL(ζ, ) if ζ ∈ [0, ζ2 ()] H(ζ, ) = Hi(ζ, ) if ζ ∈ (ζ2 (), ζ1 ())   HR(ζ, ) if ζ ζ1 () where

HL(ζ, ) = aL() + cL()ζ 2 HR(ζ, ) = aR() + cR()ζ 2

(28)

(29)

(30)

(31)

(32)

and the following two formulae for Hi(ζ, ) are equivalent Hi(ζ, ) = HL(ζ, ) + AL()(ζ − ζ2 ())2 Hi(ζ, ) = HR(ζ, ) + AR()(ζ − ζ1 ())2 No we are ready to give the ﬁnal formula for the potential (23)  1 2    I (N )H(ζN , ) if I(N ) = 0 2 2W ∗ (N , ) =   2 ˘  1 L()II (N ) if I(N ) = 0 2

(33)

(34)

The intervals [0, ζ2 ], [ζ2 , ζ1 ], [ζ1 , +∞] are called the third, second and ﬁrst regimes respectively. We note that W ∗ (·, ) is smooth and convex. The problem (22) can be interpreted as an equilibrium problem of an eﬀective body of non-linear and hyperelastic physical properties, with the constitutive equation of the form ε=

∂W ∗ (N , m2 ) ∂N

(35)

HOMOGENIZATION AND OPTIMAL DESIGN . . .

147

It is worth noting that the equation above can be explicitly inverted. To show this inverted relation we introduce the invariant of the strain tensor εI − εII II(ε) , ζε = (36) ζε = |I(ε)| εI + εII for the case of I(ε) = 0. Let us introduce now the auxiliary functions of k2 + [µ] ζ˘1 () = , ∆µ

ζ˘2 () =

∆k µ2 + [k]

(37)

the notation [·], ∆, being deﬁned by (26). New functions are k1 k2 + µ2 k , µ2 + [k] a ˘R() = k2 , a ˘L() =

c˘L() = µ2 , c˘R() = and

(38)

µ1 µ2 + k2 µ k2 + [µ]

(1 − )∆µ(µ2 + [k]) A˘L() = , [k + µ] 2 (1 − )∆k(∆µ)2 A˘R() = [k + µ](k2 + [µ])

(39)

Moreover, ˘L() + c˘L()ζ 2 , FL(ζ, ) = a ˘R() + c˘R()ζ 2 FR(ζ, ) = a and

(40)

2 Fi(ζ, ) = FL(ζ, ) − A˘L() ζ − ζ˘2 ()

(41a)

2 Fi(ζ, ) = FR(ζ, ) − A˘R() ζ − ζ˘1 ()

(41b)

or, alternatively

The function F (ζ, ) is formed by stitching the three parabolas  ˘   FL(ζ, ) if ζ ∈ [0, ζ2 ()] F (ζ, ) = Fi(ζ, ) if ζ ∈ (ζ˘2 (), ζ˘1 ())   F (ζ, ) if ζ ζ˘ () R

1

(42)

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148

We deﬁne now the potential W (ε, m2 ) I 2 (ε)F (ζε, m2 ) if I(ε) = 0 W (ε, m2 ) = c˘R(m2 )II 2 (ε) if I(ε) = 0

(43)

This potential occurs in the following constitutive equation N=

∂W (ε, m2 ) ∂ε

The potentials W and W ∗ are dual to each other, or W (ε, m2 ) = sup ε : N − W ∗ (N , m2 ) N ∈ E2s W ∗ (N , m2 ) = sup N : ε − W (ε, m2 ) ε ∈ E2s

(44)

(45)

Both the functions are smooth and convex in the ﬁrst argument. Moreover, the following identity min{ W ∗ (N , m2 )dx | N ∈ Σ1 (Ω) } = Ω

= max{f1 (v) −

(46) W (ε(v), m2 )dx | v ∈ V1 (Ω) }

Ω

conﬁrms that the equilibrium problem can either be expressed in terms of stress resultants or in terms of displacements. The optimization problem (21)–(22) can be numerically solved provided that an equilibrated version of FEM is at our disposal. Then the values of (N αβ ) at nodes are the unknowns. Such an approach is neither easy nor typical. That is why having the explicit expression for W (ε, m2 ) is so important. The result (43) is crucial here. It makes it possible to pose the whole problem in terms of displacements. Let us express the equilibrium problem (22) in terms of displacements Find u ∈ V1 (Ω) such that N = ∂W (ε(u), m2 ) 2 ∂ε (P1 ) and N = (N αβ ) satisﬁes the equilibrium equation (10) The problem (21) can be reformulated as follows min{f1 (u) | m2 ∈ L∞ (Ω; [0, 1]), 1 (PR) u is a solution of (P2 ) and m2 (x)dx = C2 } 1 Ω

(47)

HOMOGENIZATION AND OPTIMAL DESIGN . . .

149

and just this problem can be attacked by the displacement-based algorithms of the minimum compliance problem of hyperelasticity, cf. Bendsøe (1995), Bendsøe and Sigmund (1999), Dzier˙zanowski (2001). The unknown m2 is the only design variable of problem (47). Thus the ﬁnal design is determined by distribution of m2 within Ω. This solution, however, should be augmented with an information on the underlying microstructure. To this end one should ﬁnd the subdomains in which 0 ζε ζ˘2 (regime III) ζ˘2 ζε ζ˘1 ζε ζ˘1

(regime II or sliding regime) (regime I)

as well as the domains where m2 = 0 (there material 1 is present) or m2 = 1 (where material 2 is used). Distribution of m2 determines the subdomains corresponding to the subsequent regimes uniquely. However, a recovery of the microstructure is not unique. The simplest realizations are provided by 2nd rank laminates which in the regime II simplify to the ﬁrst rank laminates. The boundaries ζε = ζ˘1 and ζε = ζ˘2 are lines of smooth change of 1st rank laminate into 2nd rank laminates, see Gibiansky and Cherkaev (1987), Allaire and Kohn (1993). Allaire and Aubry (1999) proved that the 2nd rank microstructure cannot be replaced by a usual composite structure. Thus the inﬁmum in (23) cannot be replaced by minimum and the relaxation cannot be achieved without hierarchical microstructures. Note that the laminate-based interpretation is a little artiﬁcial, since the potential W depends only on trε and det ε, hence it refers to an isotropic body and not to orthotropic one, as suggested by the laminate interpretation. The laminate interpretation is possible due to a certain paradox: the potential of a physically nonlinear isotropic body can be equally well viewed as a potential of a physically linear orthotropic nonhomogeneous body. 3. The layout problem in the thin plate theory. Minimum compliance design The non-homogeneous plate considered in Sec. 2 can be subjected to a transverse loading of intensity q thus transmitting the in-plane problem to an anti-plane problem, as will be discussed here. Let us assume that the plate thickness h is constant. The plate is clamped along Γu and free of loading along the remaining part Γσ of the contour ∂Ω. According to the Kirchhoﬀ model adopted here the plate deformations are determined by deﬂection w(x), x ∈ Ω. Let n represent a unit vector outward normal to ∂Ω. A function v deﬁned in Ω is said to represent a kinematically admissible

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150

deﬂection, which is written as v ∈ V2 (Ω), provided that v ∈ H 2 (Ω) and ∂v = 0 (in the sense of trace) along Γu. The transverse loading v = 0, ∂n implies bending and torsion represented by the tensor ﬁeld M = (M αβ ) on Ω. The ﬁeld M is said to be statically admissible, and we write M ∈ Σ2 (Ω) if M ∈ L2 (Ω; E2s ), div div M ∈ L2 (Ω) and M αβ καβ (v)dx = f2 (v) ∀ v ∈ V2 (Ω) (48) Ω

where

∂2v ∂xα∂xβ

(49)

q(x)v(x)dx

(50)

καβ (v) = − f2 (v) = Ω

As in Sec. 2 we assume that the plate is formed of two isotropic materials 1 and 2 distributed transversely homogeneous. Thus the middle plane Ω is divided into Ω1 and Ω2 , as in Sec. 2. The bending stiﬀness tensor involved in the constitutive equation M = Dκ κ (w)

(51)

D(x) = χ1 (x)D 1 + χ2 (x)D 2

(52)

is decomposed by the formula

similar to (3), where, due to isotropy kγ I 1 + 22 µγ I 2 (53) D γ = 22 2 2 h h 2 Here kγ = 2γ = kγ , µ µγ with kγ , µγ deﬁned by (6); the ordering 12 12 k1 and µ 22 > µ 21 . assumption (7) implies 2 k2 > 2 Note that the case of designing a plate of alternate thickness h2 and kγ and µ 2γ depending on hγ h1 is comprised by (52). Then (53) holds with 2 with E1 = E2 , ν1 = ν2 being given. The equation (48) refers to the case of Ω being a plane of symmetry of the plate body. Otherwise the in-plane and bending problems cannot be decoupled. The isoperimetric condition (8) is still adopted. The compliance of the plate in bending is expressed by the Castigliano formula

(54) J(χ2 ) = min{ M : D −1 M dx | M ∈ Σ2 (Ω) } Ω

HOMOGENIZATION AND OPTIMAL DESIGN . . .

151

the minimum compliance problem has still the form (14) with J given by (54). Prior to performing relaxation of the above problem we recall the homogenization result for thin plates of periodic structure. The rescaled cell of periodicity Y is assumed to be formed of two materials resulting in bending stiﬀnesses Dγ , γ = 1, 2; the distribution of the stiﬀnesses D(y) being given by (15) where A should be replaced by D. The deformation measures within Y are deﬁned by y καβ (w) = −

∂2w ∂yα∂yβ

(55)

for w ∈ H 2 (Y ). The ﬁeld κ y on Y is said to be kinematically admissible if there exists w ∈ H 2 (Y ) such that (55) holds. Moreover, the traces of ∂w w and assume equal values at opposite sides of Y . Then we write ∂yα κ y ∈ K2per (Y ). The distribution of two materials within Y determines the eﬀective bending stiﬀness tensor D h by the formula κ y : (D(y)κ κ y ) | D is given by (52), κ : (D hκ ) = min {κ κ ∈ y

K2per (Y

(56)

), κ κ = κ} y

to hold for every κ ∈ E2s . Let ν = (ν1 , ν2 ) be a unit vector outward normal to ∂Y and τ = (τ1 , τ2 ) be a unit vector tangent to ∂Y . For m = (mαβ ) deﬁned on Y we denote mn = mαβ νανβ , qˆ = να

mτ = mαβ νατβ ,

∂mαβ ∂mτ + ∂yβ ∂s

(57)

Let us deﬁne an aﬃne set Σper 2 (Y

) = {m ∈ L

2

(Y, E2s )

2 αβ ∂ m ∂yα∂yβ = 0 ,

mn assumes equal −, and qˆ assumes opposite values at opposite faces of Y } Now we are ready to give the formula reciprocal to (56) 0

−1 1 (y)m | m ∈ Σper M : (D −1 2 (Y ), m = M } h M ) = inf{ m : D (58)

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152

Tensor D −1 (y) is given by (15) putting D instead of A. The problem (21) in which 2 (59) J(m2 ) = min{ 2W ∗ (M , m2 )dx | M ∈ Σ2 (Ω) } Ω

with 0

1 2W ∗ (M , ) = inf { m : D −1 (y)m | m ∈ Σper 2 (Y ), m = M ,

χY2

∞

∈ L (Y ; {0, 1}),

χY2

(60) = }

for 0 1 is a relaxed problem with respect to the initial problem (54). The inﬁmum in (60) is not achieved within the class of ﬁrst rank (or usual) periodic composites: one cannot indicate for each M ∈0 E2s a function χY2 and a corresponding ﬁeld m = (mαβ ) for which 2W ∗ = m : D −1 (y) m) . This equality can only be reached for specially chosen M . In general the equality (60) is achieved by introduction of 2nd rank laminated (or rather ribbed) composite plates. The explicit form of W ∗ (M , ) has been for the ﬁrst time found by Gibiansky and Cherkaev (1984). This function can be put in a form similar to (34), see (26.3.44) in Lewi´ nski and Telega (2000). The problem (59) can be viewed as an equilibrium problem of an effective physically nonlinear plate of hyperelastic properties given by the inverted constitutive equation κ=

∂W ∗ (M , m2 ) ∂M

(61)

∂W (κ κ , m2 ) ∂κ κ

(62)

or by their primal form M=

The explicit form of (62) has been found in Lewi´ nski and Telega (2000, Sec. 26.6, Eq. (26.6.35)) and that is why will not be repeated here. Theoretically, the function W (κ κ , m2 ) can be found by the Legendre transformation, as in (45), but the derivation leading to the mentioned result should rather be based upon (61) as shown in Lewi´ nski and Telega (2000). Both ∗ the functions W (M , ) and W (κ κ , ) are smooth and convex with respect to the ﬁrst argument. The formulae (61) and (62) resemble the previous results (34) and (43) for the in-plane problem, but there is no full similarity. The diﬀerences arise because the diﬀerential constraints imposed on the ﬁelds n in (23) are diﬀerent than the diﬀerential constraints concerning the ﬁelds m in (60). κ ) = 0 None the less the potential W (κ κ , m2 ) is deﬁned regime-wise; for I(κ

153

HOMOGENIZATION AND OPTIMAL DESIGN . . .

the three regimes appear ζκ ζ˘1 (m2 ) II : ζ˘2 (m2 ) ζκ ζ˘1 (m2 ) ˘ 2) III : 0 ζκ ζ(m I:

The case of I(κ κ ) = 0 can be included into the regime I. The quantities ˘ kα, µ 2α, α = 1, 2 ζγ (m2 ) depend on m2 and on the data 2 ∆2 k ζ˘1 () = ∆2 µ

ζ˘2 () =

∆2 k µ 22 + [2 k]

(63)

We note that ζ˘1 () does not depend on and the formula for ζ˘1 is not similar to (37). The equilibrium problem of an eﬀective hyperelastic plate included in (59), can be reformulated to the displacement-based form κ (w), m2 ) Find w ∈ V2 (Ω) such that M = ∂W (κ 2 (P2 ) ∂κ κ and M satisﬁes the variational equilibrium equation (48) The relaxed problem considered, i.e. (21) and (59), is equivalent to min{f2 (w) | m2 ∈ L∞ (Ω; [0, 1]), w is a solution 2 (PR) of (P2 ) and m2 (x)dx = C2 } 2 Ω The ﬁeld m2 , the area fraction of the stronger phase, is the only design variable of our optimization problem. Upon ﬁnding m2 one should detect the subdomains corresponding to the regimes I, II, III as well as the subdomains where the plate is isotropic: m2 = 0 or m2 = 1. The main result of Gibiansky and Cherkaev says that the subdomain of regime II is characterized by a 1st rank ribbed microstructure in which principal directions of tensors M and κ coincide. In the regime I one observes deviation of the principal directions of M with respect to the principal directions of κ . The 3rd regime is characterized a 2nd rank microstructure. The regions I-III corresponding to the regimes I-III are uniquely determined by the distribution of m2 (whether the latter distribution is unique is an open problem). However, the underlying microstructure is not unique. In particular, the region III can be viewed as built of a composite plate in which one material is a matrix and the second one is located in the oval inclusions, see Vigdergauz (1994) and Grabovsky and Kohn (1995).

´ T. LEWINSKI

154

Knowing that the 2nd rank ribbed plates suﬃce to saturate the inﬁmum in (60) one can rearrange the relaxed problem (PR2 ) to a setting in which the constitutive equations are linear, but are parametrized by the three design variables: θ2 , ω2 and φ. Here θ2 , ω2 represent area fractions of the stronger material for the subsequent laminations and φ measures inclinations of the ribs. This formulation is treated now as standard, see Sec. 1.2.2 in Bendsøe (1995) concerning the plane stress problem and see Czarnecki and Lewi´ nski (2001), where the Kirchhoﬀ plate problem has been dealt with. It is important to note that the numerical results found by the method described above (the method of controlling the microstructural variables) and by the method based on (PR2 ) practically coincide, cf. Czarnecki et al. (2003b). This suggests that the problem is uniquely solvable, a conjecture still not proved. The formulation (PR2 ) can be directly implemented into commercial FE codes, see Czarnecki et al. (2003a,b), a full version of these papers being in print. 4. Plates subjected to simultaneous in-plane and transverse loading Assume that the plate considered in previous sections is subjected to the boundary loading p = (p1 , p2 ) along Γσ and to the transverse loading q. ∂w vanish. The The plate is clamped on Γu, where the ﬁelds u1 , u2 , w and ∂n virtual ﬁelds v = (v1 , v2 ) and v are said to be kinematically admissible if (v, w) ∈ V with V = V1 × V2 . The equilibrium conditions are not coupled: they assume the previous variational form (10) and (48). Also the constitutive equations (1) and (51) are decoupled. The compliance of the plate equals f (u, w) = f1 (u) + f2 (w)

(64)

The plate is built of two isotropic materials, distributed transversely homogeneous. The given moduli (6) determinethein-plane stiﬀness tensors h2 Aγ . The distribution of Aγ and the bending stiﬀness tensors D γ = 12 all stiﬀnesses is determined by one function χ2 , see (3) and (52). The compliance (64) can be expressed by the Castigliano formula .

/ (65) J(χ2 ) = min { N : A−1 N + M : D −1 M dx Ω

(N , M ) ∈ Σ1 (Ω) × Σ2 (Ω)}

here J(χ2 ) = f (u, w) and (u, w) is a solution to the equilibrium problem.

HOMOGENIZATION AND OPTIMAL DESIGN . . .

155

The minimum compliance problem has still the form (14) with J(χ2 ) given by (65) and with the isoperimetric condition in the unchanged form (8). The problem thus formulated necessitates relaxation by homogenization. To construct the relaxed formulation the plate is endowed with a microstructure of eﬀective properties corresponding pointwise to a certain periodic plate with the rescaled cell of periodicity Y . The function χYγ ∈ L∞ (Y ; {0, 1}) determines distribution of the γ-th material within Y . Let us introduce the compound quantities , + A−1 (y) 0 T S = (N , M )T σ = (n, m) , a= (66) −1 0 D (y) The eﬀective potential is expressed in terms of the eﬀective volume fraction and of the stress and couple resultants N and M by per 2W ∗ (S, ) = inf {σ : (aσ) | σ ∈ Σper 1 (Y ) × Σ2 (Y ),

(67)

σ = S, χY2 ∈ L∞ (Y ; {0, 1}), χY2 = } where 0 1. The result (67) corresponds to the formulae in Sec. 3.6 of Lewi´ nski and Telega (2000), for eﬀective compliances of transversely asymmetric (or unbalanced) plates. Relaxation by homogenization rearranges the problem (65) to the form (21) in which 2 (68) J(m2 ) = min{ 2W ∗ (S, m2 )dx | S ∈ Σ1 (Ω) × Σ2 (Ω) } Ω

This result can be inferred from the relaxed formulation of the minimum compliance problem of shells, see Sec. 28 in Lewi´ nski and Telega (2000). To make the above formulation useful one should express W ∗ (S, ) explicitly in terms of N , M and . Due to isotropy we expect that W ∗ should be expressible in terms of the invariants of N , invariants of M and the quantity tr(N M ). The ﬁrst steps towards this aim were taken in Dzier˙zanowski and Lewi´ nski (2003). Before recalling the main results of this work let us remind here that the explicit formulae for W ∗ (N , ) and W ∗ (M , ) were found in Gibiansky and Cherkaev (1984, 1987) in two steps. First, these potentials are estimated from below. Then this lower bound is proved to be achievable by certain microstructures. Both the lower bounds were found by the translation method, see Gibiansky (1999). The translation method for the bending problem is based on the estimate m : T m m : (T m )

∀ m ∈ Σper 2 (Y )

(69)

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´ T. LEWINSKI

where T = I2 − I1

(70)

This tensor is associated with det m by m : (T m) = −2 det m

(71)

The proof of (69) involves the diﬀerential constraints imposed on the ﬁelds m being statically admissible, cf. Gibiansky and Cherkaev (1984). The same translation T is helpful in bounding the in-plane potential. We make use of the identity n : (T n) = n : (T n )

∀ n ∈ Σper 1 (Y )

(72)

which is valid due to the diﬀerential constraints imposed on the stress resultants n being statically admissible, see Gibiansky and Cherkaev (1987). For ﬁnding a rational lower bound of (67) the following identity is crucial n : (T m) = n : (T m )

(73)

per Here n ∈ Σper 1 (Y ) and m ∈ Σ2 (Y ). This identity links two, seemingly independent, tensor ﬁelds n and m, cf. Dzier˙zanowski and Lewi´ nski (2003). Let us form the translation matrix + , αT γT T (α, β, γ) = (74) γT βT

where α, β, γ are certain real numbers. By applying the translation method with the translation matrix (74) one arrives at the bound W ∗ (S, ) W (S, )

(75)

2W (S, ) = max{S : (aT (α, β, γ)S) | (α, β, γ) ∈ Z }

(76)

3 4−1 + T (α, β, γ) aT (α, β, γ) = (a − T (α, β, γ))−1

(77)

with and

Z = {(α, β, γ) ∈ R3 | (a − T (α, β, γ)) is positive semideﬁnite } To make the dimensions of all components of S unique we deﬁne √ √ 12 5 = 12 M , 5= m, M m h h 2 = (n, m 5) , σ T

2 = (N , M 5) S

T

(78)

(79)

HOMOGENIZATION AND OPTIMAL DESIGN . . .

157

and modify the tensors a and T appropriately. Instead of (76) we have 2γ 2γ 2 ) = max{S 2 : (2 2 (α, β, 2W (S, aT (α, β, 2)S) 2) ∈ Z2 } (80) where 2γ 2T (α, β, 2) = a

6

−1 7−1 2 2γ 2 2 − T (α, β, γ a 2) + T2 (α, β, 2)

(81)

2 h h 2 2 is a counterpart of Z. √ and γ 2= γ, β = β. The set Z 12 12 2 and T2 we introduce the For a convenient representation of tensors a tensorial basis 1 E 1 = √ (e1 ⊗ e1 + e2 ⊗ e2 ) 2 1 E 2 = √ (e1 ⊗ e1 − e2 ⊗ e2 ) (82) 2 1 E 3 = √ (e1 ⊗ e2 + e2 ⊗ e1 ) 2 2 and T2 : Now we decompose tensors a 2= a

3

2 aij E i ⊗ E j

T2 =

i,j=1

where

and

3

T2ij E i ⊗ E j

(83)

i,j=1

1 [2 aij ] = diag [K(y), L(y), L(y), K(y), L(y), L(y)] 2 K(y) = K1 χY1 (y) + K2 χY2 (y) , L(y) = L1 χY1 (y) + L2 χY2 (y)

(84)

(85)

The matrix [Tij ] is of the form       2 [Tij ] =      

−α 0 0 −2 γ 0 0



 γ 2 0    0 0 α 0 0 γ 2   2 −2 γ 0 0 −β 0 0    0 γ 2 0 0 β2 0  0 0 γ 2 0 0 β2 0

α 0

0

(86)

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158

A speciﬁc, simple form of the matrices (84), (86) makes it possible to ﬁnd 2 the explicit representation of the set Z: 1 1 1 2γ 2 = {(α, β, (87) Z 2) − Kσ α Lσ, 0 β2 Lσ, 2 2 2 2 42 γ 2 (Kσ + 2α)(Kσ + 2β) 2 σ − 2α), 42 γ 2 (Lσ − 2β)(L

2 2: a 2T S and the representation of S

σ = 1, 2 }

. / 2 :a 2 = a1 (N 1 )2 + a2 (N 2 )2 + (N 3 )2 2T S S . / +d1 (M 1 )2 + d2 (M 2 )2 + (M 3 )2

+b1 (M 1 N 1 ) + b2 N 2 M 2 + N 3 M 3

(88)

2 γ where aσ , dσ , bσ are parameters depending on , α, β, 2, Kσ, Lσ; σ = 1, 2. i i Moreover, the components N , M refer to the basis (82) N=

3

N iE i ,

M=

i=1

3

M iE i

(89)

i=1

Maximization in (80) determines the parameters aσ, dσ, bσ as functions of the invariants trN , Here

devN ,

trM ,

devM ,

trN trM ,

/1 . devN = (N 2 )2 + (N 3 )2 2

tr(N M )

(90)

(91)

is a norm of the deviator of N . Eventually, the potential W assumes a nonlinear hyperelastic form, depending on the invariants (90). Maximization in (80) cannot be performed analytically, except for such special cases like shape design (K1 → ∞, L1 → ∞). Moreover, achievability of the bound (75) is still an open problem. By replacing W ∗ by its approximant W we rearrange the problem (65) to the form 2 (92) J(m2 ) = min{ 2W (S, m2 )dx | S ∈ Σ1 (Ω) × Σ2 (Ω) } Ω

The problem (92) represents an equilibrium problem of an eﬀective plate of hyperelastic properties. Its primal formulation reads:

HOMOGENIZATION AND OPTIMAL DESIGN . . .

159

Find (u, w) ∈ V such that the variational equilibrium equation is fulﬁlled ∀ (v, v) ∈ V (93) N αβ εαβ (v) + M αβ καβ (v) dx = f (v, v) Ω

the constitutive equations being given by ε=

∂W ∂N

κ=

∂W ∂M

(94)

The potential W is deﬁned regime-wise with the regimes depending on the invariants (90). Therefore, an explicit inversion of (94) seems to be impossible. The minimum compliance problem is set as follows min{f (u, w) | m2 ∈ L∞ (Ω; [0, 1]),

(95)

m2 (x)dx = C2 }

(u, w) is a solution to (93) – (94) and Ω

Numerical procedures for (95) are not developed up till now. 5. Thin shells in bending Let us consider the problem of minimal compliance of shells obeying the deformational constraints of the Mushtari-Donnell-Vlasov shell model. This approach leads to essential simpliﬁcations of the minimum compliance problem of two-phase shells. Consider a shell of constant thickness h. The middle surface S is treated as an image of a plane domain Ω. The latter domain is parametrized by Cartesian coordinates ξ 1 , ξ 2 which are mapped on S thus forming a curvilinear parametrization with the metric tensor (gαβ ) and curvature tensor (bαβ ), deﬁned as usual. Deformation of the shell is determined by deformation of its middle surface, given by the displacement ﬁeld (u1 , u2 , w) of components referred to the basis on S; uα represents a displacement tangent to the ξ α coordinate and w stands for the displacement measured along the direction normal to S. The strain measures are deﬁned by 1 αβ (u, w) = uαβ + uβα − bαβ w 2 (96) καβ (w) = −wαβ where ( )α represents a covariant derivative, see Naghdi (1963). We note that the expression for καβ is approximate, since the tangent displacements do not enter this formula. Consequently, the conditions αβ (u, w) = 0,

καβ (w) = 0

´ T. LEWINSKI

160

do not lead to the rigid body motions (we expect translations and inﬁnitesimal rigid body rotations as the only integrals of the above system). The formulation will be reasonable for the boundary conditions capable of canceling additional, spurious zero-energy modes. The model of MushtariDonnell-Vlasov is said to be applicable for shells of slowly varying curvatures. It is well-posed at least for clamped shells, see Bernadou and Oden (1988). The shell is subjected to a surface and boundary loading. Its work on the trial ﬁelds (v1 , v2 , v) is represented by f (v, v). The stress and couple resultants are statically admissible, then we write S ∈ Σ(Ω), S = (N , M )T if √ (97) N αβ αβ (v, v) + M αβ καβ (v) gdξ = f (v, v) ∀ (v, v) ∈ V Ω

where g = det(gαβ ), ξ = (ξ 1 , ξ 2 ). Here V represents the space of kinematically admissible trial displacements (v1 , v2 , v). The regularity conditions concerning N , M and v, v can be assumed as usual, see Lewi´ nski and Telega (2000). Let us assume that the shell is formed of two isotropic materials, distributed homogeneously in the transverse direction. Within the model considered the constitutive equations are decoupled, as follows N = A , where

M = Dκ

A(ξ) = χ1 (ξ)A1 + χ2 (ξ)A2 D(ξ) = χ1 (ξ)D1 + χ2 (ξ)D 2

(98)

(99)

Aγ , D γ are given by the formulae (5), (6), (53), where

 1 αβ λµ   g g  2  1 αλ βµ  I2αβλµ = g g + g αµg βλ − g αβ g λµ  2 I1αβλµ =

The volume to be occupied by the material no 2 is ﬁxed: = χ2 (ξ) g(ξ)dξ = C2

(100)

(101)

Ω

The following layout problem is dealt with /√ . inf{ N : (A−1 N ) + M : (D −1 M ) gdξ | (N , M ) ∈ Σ, Ω

χ2 ∈ L∞ (Ω; {0, 1}), Eq. (101) }

(102)

HOMOGENIZATION AND OPTIMAL DESIGN . . .

161

It turns out that the relaxed form of this problem is similar to the relaxed form of the problem of a plate subjected to arbitrary loading, see (68). The √ formulation is similar to (21), where dx should be replaced with gdξ. The 2 2 ) has the form (68), where S ∈ Σ(Ω); here Σ(Ω) refers to the functional J(m problem of the Mushtari-Donnell-Vlasov shell. Moreover, the potential W ∗ has the form compatible with (67). This latter fact follows from the theorem of homogenization of Mushtari-Donnell-Vlasov shells of periodic structure, see Lewi´ nski and Telega (2000, Sec. 17.4). The basic cell problems: in-plane 1 ), (P 2 ) in Sec. 17.4 of Lewi´ nski and and bending ones decouple, see (Ploc loc Telega (2000), since the derivatives of the function w do not intervene into αβ (u, w) and the derivatives of uα are omitted in the formula for καβ . The mathematical structure of the local problems of homogenization is conditioned by the highest derivatives entering the formulae for deformation measures. Thus the relaxed formulation of the optimum layout problem is similar to the optimum layout problem in the plate problem, see Sec. 4. The form of the potential W ∗ does not depend on tensor (bαβ ). The curvature of the middle plane aﬀects only the equilibrium problem. The formulae (96) do not satisfy, as has been stressed above, the criterion of the rigid body motions. One can indicate a family of thin shell models which satisﬁes this criterion, see Budiansky and Sanders (1963). In particular, this criterion is satisﬁed within the framework of the Sanders and Leonard theory, called also the theory of Koiter, see Bernadou (1996). In this model the formula for καβ has the form καβ (u, w) = −wαβ − bγαβ uγ − bγαuγβ − bγβ uγα + bγαbγβ w

(103)

The appearance of the ﬁrst derivatives of the ﬁelds uγ implies coupling of the basic cell problems of the theory of homogenization of shells of periodic structure, see Sec. 17.1 in Lewi´ nski and Telega (2000) and Telega and Lewi´ nski (1998). This coupling takes also place in the dual formula, see (17.2.19) of Lewi´ nski and Telega (2000), where σ = (n, mT ) ∈ Σper (Y ) and this set is not a Cartesian product of two sets independently deﬁning the conditions of n and m, respectively, as in (67). Here n and m are coupled by the diﬀerential equations, cf. (17.2.18) of Lewi´ nski and Telega (2000). Just that is why we do not know any translation matrix that could reveal the form of W ∗ (N , M , ), deﬁned in Sec. 28.2 in Lewi´ nski and Telega (2000). 6. Membrane shells The theory of membrane shells is presented in the book by Sanchez-Hubert and Sanchez-Palencia (1997). This theory is relatively simple and hence is attractive in the shape and topology optimization. The starting point is

´ T. LEWINSKI

162

the relaxed form of the minimum compliance problem of Mushtari-DonnellVlasov shells. Neglecting there the bending eﬀects leads to the membrane formulation. One arrives at the problem of the form (95) in which one √ should put dx = gdξ and (u, w) represents a solution of the equilibrium problem of an eﬀective membrane: √ N αβ αβ (v, v) gdξ = f (v, v) ∀ (v, v) ∈ V (104) Ω

where N is interrelated with (u, w) by =

∂W ∗ (N , m2 ) ∂N

(105)

The potential W ∗ (N , m2 ) is given by the formula (23). We know that the condition (105) can be inverted to the form (44), where W is given by (43). Thus the main diﬃculty of the minimum compliance problem of a membrane shell – disclosing the formula for W ∗ (and for W ) – has been overcome while solving a similar problem in 2D elasticity. Curvature of the shell aﬀects the equilibrium problem (104) only, but does not aﬀect the formula (105). 7. Shape design of membrane shells The passage from two-material design to shape design has been shown by Allaire and Kohn (1993) for the in-plane elasticity problem. Since this passage is local and since the eﬀective constitutive equation (105) for membrane shell problem coincides with the eﬀective constitutive equation for the in-plane elasticity problem, the result of Allaire and Kohn can be easily applied to the membrane shell case. Assumption of k1 = 0, µ1 = 0 in the expression (34) deﬁning the potential W ∗ (N , ) leads to 1− G(N ) , 2

(106)

1 1 W0∗ (N ) = K2 I 2 (N ) + L2 II 2 (N ) , 4 4 1 G(N ) = (K2 + L2 ) (|NI | + |NII |)2 4

(107)

W ∗ (N , ) = W0∗ (N ) + where

The potential W0∗ (N ) refers to the homogeneous shells made of the material of moduli (k2 , µ2 ).

HOMOGENIZATION AND OPTIMAL DESIGN . . .

163

Let Σm(Ω) represents the set of stress resultants N satisfying the variational equation of equilibrium (104) of a membrane shell. The problem of minimization of compliance of the membrane shell assumes the form √ (108) min{ 2W ∗ (N , m2 (ξ)) gdξ | m2 ∈ L∞ (Ω; [0, 1]) ,

Ω

N ∈ Σm(Ω) ,

√ m2 gdξ = C2 }

Ω

where W ∗ is given by the formula (106). It is known that the plane problem (108) possesses at least one solution, see Allaire and Kohn (1993). This theorem holds also for membrane shells. The engineering aspects of the problem of optimum design of membrane structures are discussed in Barnes (1988). 8. Michell surface structures The passage to the limit with C2 0 degenerates the problem (108) to the form √ J1 = min{ (|NI | + |NII |) gdξ | N ∈ Σm(Ω) } (109) Ω

This fact has been proved by Allaire and Kohn (1993) in the context of 2D elasticity and similar arguments apply here giving (109). Let us note that the functional in (109) does not depend of the elastic moduli. Strang and Kohn (1983) proved that the problem (109) is equivalent to the dual one J1 = max{f (v, v) | (v, v) ∈ B } where B = { ∈ E2s

|I | 1 ,

|II | 1 }

(110) (111)

is a locking locus, see Telega and Jemiolo (1998). Among the equilibrium problems of membrane shells one can single out a subclass of statically determinate problems. In the statically determinate problems three components of the force ﬁeld N can be found by three equilibrium equations and natural boundary conditions. Then the set Σm is one element and the value J1 can be found by computing the integral in (109). Such problems are also geometrically determinate, since from three equations: 11 = 1, 12 = 0, 22 = −1 one can ﬁnd three ﬁelds v1 , v2 , v and then compute the virtual work f (v1 , v2 , v). Let us note that the problem of torsion of a shell of revolution realized by the tractions tangent to the contours of its openings is just statically

164

´ T. LEWINSKI

determinate. The integral J1 can be then easy found by the formula (109) and then conﬁrmed by (110), see Lewi´ nski (2003). Due to the simplicity of this problem one make one step further and examine which shape of the meridian results in the smallest compliance J1 . It turns out that the optimal meridian must be of circular shape and consequently the stiﬀest shell subjected to torsion is just a spherical shell considered by Michell (1904), cf. Lewi´ nski (2003). 9. 3D bodies The problem (14) can be generalized to the 3D elasticity setting. The relaxation of this problem is studied in Allaire (2002). According to the Theorem 4.1.12 of this book the 3D optimal body is endowed with a 3rd rank stiﬀ orthogonally laminated microstructure. The feature of the optimal solution is colinearity of the trajectories of stresses and lamination directions. The numerical algorithm based upon this property requires having in hand explicit formulae for the eﬀective moduli of the composite considered, for arbitrary volume fractions in the three subsequent laminations. These formulae were derived in Lewi´ nski (2001a,b,c), see Czarnecki et al. (2003a,b), by using the formula of Francfort and Murat (1986). It has been proved that the popular updating scheme (see Bendsøe, 1995, Sec. 1.2.2) assures convergence of the iterative process. The results presented in Czarnecki et al. (2003a,b) will be published soon. Other method of ﬁnding the numerical solutions has been proposed in D´ıaz and Lipton (1997), where the method of moments has been applied, see Francfort et al. (1995). The numerical algorithms of 3D shape optimization have been developed in Allaire et al. (1997), Olhoﬀ et al. (1998), Bendsøe and Sigmund (1999), Borrvall and Petersson (2001). Final remarks The problem of optimal design of the stiﬀest linearly elastic structures in the form of thin plates loaded in plane, thin plates loaded transversely, thin shells and 3D prisms is formulated by (21)-(22), where N should be understood as a generalized stress tensor. This formulation involves the dual potential W ∗ of a hypothetical hyperelastic structure. The numerical methods for solving this problem are eﬀective provided that the potential W ∗ is known or is characterized by a ﬁnite-dimensional minimization problem. In the latter case one comes across diﬃculties in ﬁnding the derivatives of W ∗ with respect to m2 or the components of the stress tensor (or moment tensor). These diﬃculties can usually be overcome by using the known techniques of the control theory.

HOMOGENIZATION AND OPTIMAL DESIGN . . .

165

Now we are ready to a numerical treatment of the above minimum compliance problem for thin plates and Mushtari-Donnell-Vlasov shells. Other thin shell models seem to be less important, due to known ambiguity in the deﬁnition of the tensor of changes of curvature. The relaxed formulation of the minimum compliance problem for the plates with transverse shear deformation, assuming two values of the plate thickness, was given in D´ıaz et al. (1995) and solved by means of the moment formulation, see Francfort et al. (1995). This topic is not closed and should be developed by considering optimal layout problems of sandwich plates and shells of soft core and of transverse laminated structure. Acknowledgement The fruitful and invaluable discussions with J.J. Telega are gratefully acknowledged. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

Allaire, G. (2002) Shape Optimisation by the Homogenisation Method, Springer, New York Allaire, G. and Aubry, S. (1999) On optimal microstructures for a plane shape optimization problem, Struct. Optimiz., 17, pp. 86–94 Allaire, G., Bonnetier, E., Francfort, G. and Jouve, F. (1997) Shape optimisation by the homogenisation method, Numer. Math., 76, pp. 27–68 Allaire, G. and Kohn, R.V. (1993) Optimal design for minimum weight and compliance in plane stress using extremal microstructures, Eur. J. Mech., A/Solids, 12, pp. 839–878 Barnes, M.R. (1988) Form-ﬁnding and analysis of prestressed nets and membranes, Computers and Structures, 30, pp. 685–695 Bendsøe, M.P. (1995) Optimization of Structural Topology, Shape and Material, Springer, Berlin Bendsøe, M.P. and Sigmund O. (1999) Material interpolation schemes in topology optimization, Arch. Appl. Mech., 69, pp. 635–654 Bernadou, M. (1996) Finite Element Methods for Thin Shell Problems, Wiley, Chichester, Masson, Paris Bernadou M. and Oden J.T. (1988) An existence theorem for a class of nonlinear shallow shell problems, J. Math. Pures Appl., 60, pp. 285–308 Borrvall, T. and Petersson, J. (2001) Large-scale topology optimization in 3D using parallel computing. Comp. Meth. Appl. Mech. Eng., 190, pp. 6201–6229 Budiansky, B. and Sanders, J.L. (1963) On the best ﬁrst order linear shell theory, In: Progress in Applied Mechanics. The Prager Anniv. Volume, The MacMillan, pp, 129–140 Cherkaev, A. (2000) Variational methods for structural optimization, Springer, New York Czarnecki, S., Dzier˙zanowski, G. and Lewi´ nski, T. (2003a) Two-phase topology optimization of plates and three-dimensional bodies, Proc. 5th World Congress of Structural and Multidisciplinary Optimization. CD ROM, 19-23 May 2003, Lido di Jesolo, Venice, Italy Czarnecki, S., Dzier˙zanowski, G., Lewi´ nski, T. and Telega, J.J. (2003b) Topology optimization of shells and surface Michell structures, Proc. 5th World Congress of

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16.

17. 18. 19. 20.

21. 22. 23. 24. 25. 26.

27.

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´ T. LEWINSKI Structural and Multidisciplinary Optimization. CD ROM, 19-23 May 2003, Lido di Jesolo, Venice, Italy Czarnecki, S. and Lewi´ nski, T. (2001) Optimal layouts of a two-phase isotropic material in thin elastic plates. In: Waszczyszyn, Z. and Pamin, J. Eds., Proc. 2 nd European Conference on Computational Mechanics, ECCM-2001, Krak´ow, 26-29 VI 2001, CD ROM Czarnecki, S. and Lewi´ nski, T. (2002) On forming of the stiﬀest structures (general lecture), In: Obr¸ebski, J. Ed., Lightweight Structures in Civil Engineering. Proc. of the Int. IASS Symp., Warsaw, 24-28 VI, pp. 455–467, The book and CD ROM Obr¸ebski, J.B. Micropublisher, Warszawa Czarnecki, S. and Lewi´ nski, T. (2003) Computational shaping of the least compliant two-phase three-dimensional bodies, 15th Int. Conference on Computer Methods in Mechanics, 3-6 June 2003, Gliwice-Wisla, CD ROM., Ed. Burczy´ nski. T. D´ıaz, A. and Lipton, R. (1997) Optimal material layout for 3D elastic structures, Struct. Optimiz., 13, pp. 60–64 D´ıaz, A., Lipton, R. and Soto, C.A. (1995) A new formulation of the problem of optimum reinforcement of Reissner-Mindlin plates. Comp. Meth. Appl. Mech. Engrg., 123, pp. 121–139 Dzier˙zanowski, G. (2001) Application of MatLab in the problems of optimization of elastic plates loaded in plane. Minimization of the compliance of a two-phase plate, pp. 201-212, In: Polska Mechanika u Progu XXI Wieku, Ed.: Szcze´sniak, W., Oﬁcyna Wyd. PW, Warszawa (in Polish) Dzier˙zanowski, G. and Lewi´ nski, T. (2003) Layout optimization of two isotropic materials in elastic shells, J. Theor. Appl. Mech., 41, pp. 459–472 Eschenauer, H.A., Kobelev, V.V. and Schumacher, A. (1994) Bubble method for topology and shape optimization of structures, Struct. Optimiz., 8, pp. 42–51 Francfort, G.A. and Murat, F. (1986) Homogenization and optimal bounds in linear elasticity, Arch. Rat. Mech. Anal., 94, pp. 307–334 Francfort, G.A., Murat, F. and Tartar, L. (1995) Fourth-order moments of nonnegative measures on S 2 and applications, Arch. Rat. Mech. Anal., 131, pp. 305–333 Gibiansky, L.V. (1999) Eﬀective properties of a plane two-phase elastic composites: coupled bulk-shear moduli bounds, In: V.Berdichevsky, V.Jikov, G.Papanicolaou, Eds. Homogenization, World Scientiﬁc, Singapore, pp. 214–258 Gibiansky, L.V. and Cherkaev, A.V. (1984) Designing composite plates of extremal rigidity, In: Fiziko-Tekhnichesk, Inst. Im. A.F.Ioﬀe. AN SSSR, preprint No 914, Leningrad (in Russian), English translation in: Cherkaev, A.V. and Kohn, R.V. eds., Topics in the Mathematical Modelling of Composite Materials, Birkh¨ auser, Boston 1997 Gibiansky, L.V. and Cherkaev, A.V. (1987) Microstructures of elastic composites of extremal stiﬀness and exact estimates of the energy stored in them, In: FizikoTekhnichesk. Inst. Im. A.F.Ioﬀe. AN SSSR, preprint No 1115, Leningrad (in Russian), pp. 52, English translation in: Cherkaev, A.V. and Kohn, R.V. eds., Topics in the Mathematical Modelling of Composite Materials, Birkh¨ auser, Boston 1997 Grabovsky, Y. and Kohn, R.V. (1995) Microstructures minimizing the energy of a two-phase elastic composite in two space dimensions. I. The confocal ellipse construction. II. The Vigdergauz microstructure, J. Mech. Phys. Solids., 43, pp. 933–947, 949–972 Kohn, R.V. and Strang, G. (1986) Optimal design and relaxation of variational problems, Comm. Pure Appl. Math., 39, pp. 113–137, 139–183, 353–379 Kolanek, K. and Lewi´ nski, T. (2003) Circular and annular two-phase plates of minimal compliance, Computer Assisted Mechanics and Engineering Sciences, 10, pp. 177–199 Lewi´ nski, T. (2001a) The iterative formula for the eﬀective moduli tensor of twophase laminates, In: Theoretical Foundations of Civil Engineering-VIII. Ed. By Szcze´sniak, W., pp. 347–352, Oﬁcyna Wydawnicza PW, Warszawa 2001

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HOMOGENIZATION AND DESIGN OF FUNCTIONALLY GRADED COMPOSITES FOR STIFFNESS AND STRENGTH

R. LIPTON

Department of Mathematics Louisiana State University, Baton Rouge, LA 70803

1. Introduction Several technologically important applications have beneﬁted from the use of composite materials. Novel manufacturing processes have recently produced composites with gradations in their local properties, see [21] and [28]. The ability to manufacture composites with tailored micro-structure provides the opportunity for the design and construction of structures with signiﬁcantly enhanced properties. In many applications the discrete structure of the composite is on a length scale that is small relative to the characteristic length scale of the loading. For this case we present a rigorous framework for the design of graded composites for optimal stiﬀness subject to stress constraints, see Theorems 4.1, 4.2, 5.1 and 5.2. This problem is naturally related to the well known problems of relaxed optimal structural design and topology optimization. The fundamental mathematical feature common to these problems is the use of homogenization theory in their formulation. The connections and distinctions between each of the problems is described in the following section. 2. Homogenization and design of structural properties The fundamental problem of design of heterogeneous structures is the determination of the optimal spatial dependence for the composition. This design problem has received signiﬁcant attention in the engineering community and is the topic of a rapidly developing literature on functionally graded materials, see [41] and [50]. Related problems have also received signiﬁcant attention from both the applied mathematics and structural op-

169 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 169–192. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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timization communities in the 1980s and 1990s under the headings of the homogenization method for topology optimization and relaxation methods for structural optimization, see [2], [3], [7], [12], [13], [27], [40], [47], [49] and [55]. Homogenization methods applied to the design of composite components for optimal structural performance can be found in the works of [9], [16], [18], [20], [48], [56]. This list is by no means complete and further references to the literature can be found in the books and articles [4], [6], [8], [14], [15], [30], [39], and [58]. In each of these treatments the problem of determining the optimal spatial dependence for the composition is obtained in the context of linear elasticity and through the use of eﬀective constitutive relations. The natural mathematical setting for the deﬁnition and interpretation of eﬀective constitutive properties are given by the theories of H convergence, see [46] and G-convergence, see [57]. In the following subsections we introduce the notions of relaxed structural optimization, topology optimization and graded material optimization in the context of optimal compliance design. 2.1. RELAXED OPTIMAL STRUCTURAL DESIGN

We start by describing the problem of relaxed optimal structural design for minimum compliance. Here one considers a structural element occupying a prescribed domain Ω made from N linearly elastic materials with elasticity tensors A1 , A2 , A3 , . . . , AN . A generic point in Ω is denoted by x. The elastic phases occupy the N subsets Ω1 , Ω2 , . . . , ΩN where Ω1 ∪Ω2 ∪· · ·∪ΩN = Ω. A particular choice of component elasticity tensors is speciﬁed by the array A = (A1 , A2 , A3 , . . . , AN ). The local elasticity tensor C(A, x) is piecewise constant and takes the value C(A, x) = Ai in the ith material. Denoting the indicator function of the ith material by χi the local elasticity tensor th material and is written C(A, x) = N i=1 χi(x)Ai. Here χi = 1 in the i zero outside. The design space consists of all partitions of Ω into Lebesgue materials measurable subsets Ωi, i = 1, 2, . . . , N occupied by the diﬀerent subject to the resource constraints meas(Ωi) ≤ γi. Here i γi ≥ meas(Ω). and the vector of resource constraints is written γ = (γ1 , γ2 , . . . , γN ). For a body load f in W −1,2 (Ω, R3 ) the elastic displacement u is the 1,2 W0 (Ω, R3 ) solution of −div (C(A, x)(u)) = f .

(2.1)

The stress at each point in the composite σ is given by the constitutive relation σ(x) = C(A, x)(u(x)). (2.2) Here (u) is the strain tensor given by (u)ij = (ui,j + uj,i)/2,

(2.3)

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where ui is the ith component of displacement u and ui,j is its partial derivative in the j th coordinate direction. The equilibrium equation (2.1) holds in the weak sense, i.e., for every v in W01,2 (Ω, R3 )

C(A, x)(u) : (v) dx = Ω

Ω

f · v dx.

(2.4)

The contractions C(A, x)(u) : (v) and f · v are given by Cijkl(A, x)(u)ij (v)kl and fivi respectively, where repeated subscripts indicate summation. The design criteria considered in this example is the compliance of the structure given by F (u) = Ω f ·u dx. The objective of the structural design problem is to optimize F (u) over the class of admissible conﬁgurations subject to the prescribed resource constraints, i.e., inf

{F (u)}.

Admissable designs

(2.5)

The design problem given above is not readily amenable to numerical solution. The fundamental reason for this is that most problems of this type do not possess conﬁgurations for which the inﬁmum in (2.5) is attained, see [38], [45], and [55]. Thus any approach that seeks to identify optimal conﬁgurations is likely to fail. Instead one seeks to identify minimizing sequences of conﬁgurations that approach the inﬁmum in (2.5). Here the objective is to identify nearly optimal conﬁgurations. Methods developed for this purpose are known as relaxation methods, see [11] and [27]. Here the original design problem is replaced by a “relaxed version” that is used to identify minimizing sequences of conﬁgurations for the original design problem. The generic feature of minimizing sequences is the appearance of zones in which inﬁnitesimally ﬁne oscillations of material properties occur. This motivates the extension of the design space to the set of all G limits of sequences of local elasticity tensors associated with admissible conﬁgurations of the component materials see [40] and [47]. Roughly speaking, a G limit can be thought of as a local elasticity tensor that takes values in the set of eﬀective elasticity tensors associated with mixtures of the N component materials, see [53]. This extension of the design space suﬃces to produce the desired relaxed problem for optimizing structural compliance. The extended design space is referred to as the G-closure of the set of local elasticity tensors C(A, x), see [15]. In order describe the extended design space one needs to know all the eﬀective elastic tensors or at least the extremal ones that are relevant for compliance optimization. For N=2 the extremal set required for the relaxed compliance optimization problem is known provided that the two elastic materials are well ordered, see [2], [5],

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[22], [26], [43] and [44]. For all other cases the description of the parts of the G-closure set that are required to relax the original problem remains an interesting open question. Numerical methods have been developed for the relaxed optimal design of structures made from two well ordered isotropic elastic materials. The methods make exclusive use of the eﬀective elastic tensors that participate in the relaxation process, see [2], [3], [16], [18], [24], and [29]. The eﬀective elastic tensors required in the relaxed problem are those associated with the well known ﬁnite rank laminar micro-structures. Eﬀective elastic tensors for ﬁnite rank laminates have explicit formulas given in in terms of the local volume fraction of each phase and geometric parameters describing the anisotropy of the microstructure, see [19]. The volume fractions and geometric parameters that describe the eﬀective elastic tensor are free to change across the structural domain. The local volume fractions of each material satisfy integral constraints consistent with the resource constraints of the original unrelaxed optimal design problem. Existence of an optimal design within the class of these eﬀective properties is guaranteed since they are precisely the part of the G closure set participating in the relaxation, see the references [4], [15], [39], [47] and [58]. One then uses the local volume fractions and geometric parameters describing the eﬀective elasticity tensor ﬁeld to recover the minimizing sequence of discrete micro-structures. 2.2. TOPOLOGY OPTIMIZATION

Topology optimization addresses the problem of ﬁnding the best structural shape and topology required to minimize the compliance of a structure when it is subjected to prescribed boundary loads. The simplest example is given by a cantilever supported at one end and loaded at the other end. Here one is required to remove a speciﬁed amount of material from the cantilever in order to reduce the weight of the structure. The removal of material is equivalent to the addition of a hole or void. The goal is to add holes or voids in such a way that the resulting structure is the stiﬀest among all possible ways that one can remove the speciﬁed amount of material. The ultimate objective is to ﬁnd the best design using a ﬁnite number of holes or voids where the number is not known a-priori, hence the name topology optimization. As in the optimal structural design case one may consider partitions of the design domain into sub-domains consisting either of material, hole or void. This is a problem of shape optimization and one may design using the characteristic functions of the sets describing the holes or voids as design variables. However it is diﬃcult to add or remove holes or voids using the classical shape optimization methods, [7]. In the homogenization method

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of topology design the microstructure is introduced by hand and provides a suitable parametrization over which one can optimize over topology. This strategy is proposed and developed in the work of [7]. Here one replaces the characteristic function of the holes or voids with a relaxed design variable given by a density of material. Points where the density is zero correspond to points in holes or voids, points where the density is one corresponds to solid material and points having intermediate values of the density are interpreted as points inside a porous composite material. In this approach the type of porous composite microstructure is speciﬁed at the outset and is given by a locally periodic array of square holes. The square hole can occupy any fraction of the square period cell between zero and one. An eﬀective elastic tensor is associated with the porous material. For density values between zero and one the eﬀective elastic tensor is a known function of the density. It provides an interpolation between the elastic properties of solid material and void. Indeed, when the density is zero the eﬀective tensor assumes the value zero and when the density is one it recovers the elastic tensor of the solid material. An integral constraint is placed on the density that corresponds to the amount of material that can be removed from the structure. The new design problem is given in terms of densities and the optimal density distribution can be solved numerically using gradient or optimality criteria methods [8]. Since the new design parametrization includes all of the original characteristic function designs one recovers a better design. This method has been very successful as it typically produces designs for which the density takes the values zero or one over most of the domain. In this way a viable candidate for the optimal topology can be identiﬁed. An elegant alternative approach using optimal layered microgeometries is given in [2] and [3]. An accounting of the developments and extensions to the homogenization method of topology optimization are given in [4], [8], [29] and [30]. 2.3. DESIGN OF FUNCTIONALLY GRADED MATERIALS

The goal of functionally graded materials design is to identify graded material properties that deliver a desired level of structural performance. In many applications there is a separation of scales and the discrete entities forming up the microstructure exist on scales signiﬁcantly smaller than the characteristic length scale of the loading. Under this hypothesis functionally graded materials are modelled using eﬀective thermophysical properties that depend upon features of the underlying micro-geometry. The eﬀective thermophysical properties are given by eﬀective constitutive laws relating average ﬂux to average gradient, see [41]. It is shown in this article how one can rigorously apply homogenization

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methods to the design of functionally graded materials when the scale of the discrete microstructure is suﬃciently small. The ﬁrst step is to propose a homogenized design problem for which an optimal design exists. The homogenized design problem should be chosen to accurately reﬂect the continuous grading of material properties seen in the applications. The second step is to produce a sequence of discrete micro-structures with elasticity tensors that G-converge to the optimal design for the homogenized design problem. Since the compliance function is continuous with respect to G-convergence, one is then able to identify discrete graded micro-structures that deliver structural compliance close to that of the optimal homogenized design. This procedure is similar in spirit to topology optimization in that a convenient parametrization of the design space is found over which one can do optimization. Here the parametrization is accomplished using eﬀective elastic tensors. However the goal of this problem is distinct from that of topology optimization in that an explicit graded microstructure is sought. The similarity between functionally graded materials design and relaxed structural optimization is also due to the use of eﬀective elastic tensors as design variables. However the design space for the functionally graded design problem is constructed diﬀerently than the one used for relaxed structural optimization. One recalls that the design space for the relaxed optimal design problem is constructed through the extension of all simple function designs to the set of all G limits. This can be thought of as a bottom up construction of the design space. On the other hand functionally graded design is better suited to a top down construction of the design space. Here one designs with a ﬁxed set of eﬀective elastic tensors corresponding to a prescribed class of micro-structures. Since the application requires a continuous gradation of microstructure, the idea is to enforce a continuous variation of eﬀective elastic tensors across the design domain. This restriction guarantees that the design space is G closed, see sections 4 and 5. We start by specifying an admissible set of eﬀective elastic tensors that correspond to a set of locally periodic micro-structures. These microstructures are prescribed at the outset and are assumed to be made by a manufacturing process such as the lattice block materials made by JAMCORP. It will be assumed that the set of micro-structures can be speciﬁed uniquely through a vector β = (β1 , . . . , βn) of geometric parameters. An example is a periodic array of spheroids speciﬁed by the orientation of their principle axis and aspect ratio. The periodic microstructure is speciﬁed in a unit period cell Q centered at the origin. Points in the cell are denoted by y. The characteristic function of the ith phase in the unit cell is denoted by χi(β, y), i = 1, . . . , N . The piecewise constant elastic tensor taking the

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values Ai in the ith phase is given by C0 (A, β, y). The geometric parameter that describes the micro-geometry can change across the structural domain Ω. This variation corresponds to the gradation of eﬀective material properties through a gradation in microstructure. Denoting points in Ω by x the eﬀective elastic tensor is given by CH (A, β(x)) :

= Q

C0 (A, β(x), y)((w(x, y)) + ) : ((w(x, y)) + ) dy (2.6)

where for x ﬁxed, w(x, y) is Q periodic in the y variable and is the W 1,2 (Q, R3 ) solution of

−div C0 (A, β(x), y)((w(x, y) + ) = 0

(2.7)

for any constant strain . Here (w(x, y)) = (1/2)(∂iwj (x, y) + ∂j wi(x, y)) and all derivatives are taken with respect to the y variable. It is assumed that the volume occupied by each phase in the microstructure changes continuously with respect to the geometric vector β, i.e.,

lim

δβ→0 Q

|χi(β, y) − χi(β + δβ, y)| dy = 0.

(2.8)

The admissible set Ad of geometric vectors is assumed to be closed and bounded. In the applications the gradation of composition is assumed to be continuous. Here the admissible gradation of microstructure is taken to be uniformly H¨ older continuous in the closure of Ω, i.e., for ﬁxed constants C and α, (2.9) |β(x + h) − β(x)| ≤ C|h|α. Standard arguments easily show that CH (A, β(x)) changes continuously across the domain. The local volume fraction of the ith phase in the composite is given by θi(x) = Q χi(β(x), y) dy. From (2.8) and (2.9) it is evident that θi(x) changes continuously across the design domain. For a body load f in W −1,2 (Ω, R3 ) the elastic displacement uH in the homogenized composite is the W01,2 (Ω, R3 ) solution of

−div CH (A, β(x))(uH ) = f .

(2.10)

The stress at each point in the composite σ H is given by the constitutive relation (2.11) σ H (x) = CH (A, β(x))(uH (x)).

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A resource constraint is placed on the amount of each phase appearing the design. It is given by Ω

θi(x) dx ≤ γi, i = 1, . . . , N.

(2.12)

The vector of constraints (γ1 , . . . , γN ) is denoted by γ. The set of controls β(x) ∈ Ad that satisfy (2.9) and the resource constraints (2.12) is denoted by Adγ . The compliance of the homogenized design is given by F (uH ) = Ω f · uH dx and the homogenized functionally graded optimal design problem is given by P =

inf

β(x)∈Adγ

{F (uH )}.

(2.13)

An optimal design for (2.13) exists, this is shown in Section 4 Theorem 4.1. Most importantly it is shown how to use the optimal design for the homogenized problem to construct a composite with a discrete graded microstructure with compliance arbitrarily close to P , see Theorem 4.2 and section 4.2. We close by noting that the discussion given in this section holds for any design problem with objective function that is continuous with respect to G-convergence. This includes problems of vibration suppression and control of the elastic deformation in the mean square norm. The method presented here is suitable for design problems involving a very large number of small heterogeneities or other ﬁne micro-structures. In the context of ﬁber or particle reinforced composites, when it is known that the scale of the microstructure is comparable to the design domain and the number of ﬁbers is not too large the reader is referred to the treatment given in [1]. In the next section we discuss approaches that incorporate the use of homogenization theory for the design of properties that can be related to the strength of a structural component. 3. Homogenization and design of properties related to strength Equally important to the design process are considerations related to the strength and durability of the resulting structural component. We restrict attention to the linearly elastic regime and focus on the design of material composition to minimize the possibility of failure initiation. In this context the objective is to develop numerical and theoretical methods that allow one to simultaneously design for optimal structural response while constraining a relevant measure of the stress to lie below some preset level.

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3.1. BACKGROUND

We begin with a summary of work in areas related to this topic. In the context of topology optimization the work given in [17] treats the problem of stress constrained design of structures. In this work a homogenized ﬁrst failure criteria is proposed. The criteria incorporates an elegant ﬁrst approximation to the microscopic stress ﬂuctuation due to the interaction between the homogenized stress and the microstructure. The homogenized criteria is used together with eﬀective elastic tensors to provide a convenient parametrization for topology optimization. The microstructure participating in the description of the eﬀective elasticity tensor and the homogenized failure criteria is a laminar microstructure of the second rank. It turns out that the formulation proposed in [17] does not apply to pointwise stress constraints, but does apply to problems with mean square stress constraints when the pore phase is ﬁlled with suﬃciently compliant elastic material. Other related work addresses the relaxation of structural optimization problems involving objective functions depending on gradients and stresses. The goal is to ﬁnd minimizing sequences of discrete micro-structures that minimize the mean square deviation of the stress or gradient from a prescribed target ﬁeld. This type of problem is ﬁrst proposed in [59] in the context of thermal conductivity. For a dense Gδ set of targets the suitable homogenized problem is found. The precise relationship between the mean square deviation of the temperature gradients in the homogenized problem and the temperature gradients in the corresponding sequence of progressively ﬁner discrete micro-structures is given. This is accomplished through the introduction of the notion of strong L2 closure, the reader is also referred to the subsequent developments reported in [42], [54]. It is shown in [59] that minimizing sequences are exclusively associated with rank one laminar microstructures. Here the local layer orientation is parallel to the gradient ﬁeld. As of this writing this class of targets resists an explicit representation. However numerical experiments using layered materials, see [37] and [61], suggests a conjecture that 0 lies in the Gδ set. For composites made of two isotropic phases the work of [37] and [61] shows that that minimizing sequences of discrete micro-structures can be found within the class of rank one micro-structures for any choice of target ﬁeld. The more recent work [23] provides an explicit formula for the relaxation of the mean square deviation. This is used to rule out the appearance of minimizing sequences of layered conﬁgurations with more than one scale of oscillation and establishes that minimizing sequences of layered materials are exclusively given by rank one laminates. Another recent development is given in [52]. Here for any choice of target the notion of constrained quasiconvexity [51] is applied and is used to explicitly compute the constrained

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quasiconvex envelope of the mean square deviation [52]. When the boundary conditions for the temperature are linear, optimal upper bounds are given in [35] for the L2 norm of the gradient of the temperature. The bounds are given in terms of the volume fractions of each material in the composite. The microstructure attaining the upper bound depends upon the contrast of the two materials. For contrast above a certain threshold (depending on volume fractions), rank two laminates are optimal otherwise rank one laminates attain the upper bound, see [35]. Very recently the problem of minimization of the Lp norm of the gradient of the temperature has been addressed for multi-phase isotropic nonlinear conductors. Here the temperature is prescribed on the boundary and there are no heat sources inside the composite. For p = 2 and for linear heat conductors it is shown that no relaxation through homogenization is required and that the optimal design is given by a conﬁguration, see [33]. Moreover a simple algorithm can used to construct optimal designs. Analogous statements can be made for p > 2 for conductors with the appropriate nonlinear behavior [33]. The relaxation of optimal design problems for N anisotropic dielectric materials subject to a ﬁnite number of weighted L2 constraints on the electric ﬁeld is taken up in [32]. For objective functions that are continuous with respect to G-convergence it is shown that the relaxed problem is given in terms of eﬀective dielectric tensors and their derivatives (when viewed as functions of their component dielectric properties). This is in sharp contrast to the unconstrained case when only eﬀective conductivity tensors are required for relaxation. A local characterization of all G-limits and their derivatives participating in the relaxation is given in [34]. The relaxation of optimal compliance design problems in the presence of point wise constraints on the stress is addressed in [34]. The notion of stress constrained G-closure is introduced and the relaxed design problem is described. It is shown that the set of G-limits with derivatives satisfying the appropriate point wise constraints contain the stress constrained G-closure [34]. 3.2. DESIGN OF FUNCTIONALLY GRADED MATERIALS FOR STIFFNESS AND STRENGTH

In this paper a rigorous methodology is given for the design of microgeometry to hedge against structural failure. Here the idea is to constrain the stress to lie below some nominal level while at the same time optimizing for maximum structural stiﬀness. For a given stress tensor σ we consider the equivalent stress given by > √ σ eq = Π(σ) = Πσ : σ, (3.1)

HOMOGENIZATION AND DESIGN

179

where Π is a positive deﬁnite fourth rank tensor, see [60]. Examples of (σ eq )2 include the square of the Von Mises equivalent stress given by Π(σ) = [(σ11 −σ22 )2 +(σ22 −σ33 )2 +(σ11 −σ33 )2 ]/2+3((σ12 )2 +(σ13 )2 +(σ23 )2 ). (3.2) We ﬁrst develop a rigorous methodology for the design of graded microstructure for optimal compliance subject to integral constraints on the square of the equivalent stress. As in the previous section the geometric parameters describing the microstructure are H¨older continuous across the design domain. The problem formulation and solution is described in section 4, see Theorems 4.1 and 4.2. A numerical example is given in section 6. Previous work in this area treated the optimal design of piecewise constant distributions of microstructure. In that context, the problem of design using layered micro-structures for minimizing the mean square norm of the electric ﬁeld is addressed in [37] and [61]. The problem of design of long ﬁber reinforced shafts for maximum torsional rigidity subject to mean square stress constraints is taken up in [36]. The second design problem focuses on the maximization of structural stiﬀness subject to pointwise constraints on the equivalent stress. Our approach depends on the new homogenization constraints recently obtained in [31]. The constraints are given in terms of quantities dubbed macro-stress modulation functions. These quantities provide a bridge between macroscopic and microscopic scales and capture the excursion of the local stress ﬂuctuation about the homogenized or macroscopic stress. The macro-stress modulation functions are expressed in terms of the solutions w(x, y) of (2.7). We introduce the square of the “local” equivalent stress in the ith phase deﬁned for every constant symmetric 3 × 3 tensor given by (3.3) F i(x, y, ) = χi(β(x), y)Π(Ai((w(x, y)) + ). The macro-stress modulation function is given by f i(x, SH (A, β(x))σH (x)) = F i(x, · , SH (A, β(x))σH (x))L∞ (Q) .

(3.4)

Here σ H is the homogenized stress (2.11) and SH (A, β(x)) is the eﬀective compliance tensor ﬁeld given by SH (A, β(x)) = [CH (A, β(x))]−1 . Roughly speaking, point wise bounds on macro-stress modulation functions give point wise bounds on the equivalent stress inside the composite when the microstructure is suﬃciently ﬁne. To make this precise we denote the elasticity tensor for a composite made from N elastic materials with microstructure of length scale ε by Cε(A, x). Here Cε(A, x) = Ai in the ith elastic material. The characteristic function of the ith phase is

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R. LIPTON

denoted by χεi(x) and the stress ﬁeld in the composite is denoted by σ ε(x). An immediate consequence of Theorem 4.1 and Corollary 4.3 of [31] is the following Theorem 3.1 Given a measurable subset ω of Ω such that max {f i(x, SH (A, β(x))σH (x))} < t, i

(3.5)

for x in ω. Then one can construct a sequence of discrete graded microstructures such that the sequence {Cε(A, x)}ε>0 G converges to CH (A, β(x)) given by (2.6) and lim sup Π(σ ε(x)) ≤ t,

(3.6)

ε→0

for almost every x in ω. From Theorem 3.1 it is evident that point wise control on the macro-stress modulation functions provide control over the point wise values of the equivalent stress in the composite. In section we 5 use Theorem 3.1 to formulate a rigorous design strategy for maximum stiﬀness design subject to point wise constraints on the equivalent stress, see Theorems 5.1 and 5.2. 4. Design of functionally graded composites subject to integral constraints on the equivalent stress In this section we present a rigorous strategy for the design of graded composites for minimum compliance subject to an integral constraint on the square of the equivalent stress. The strategy uses a formula given in terms of the derivative of eﬀective properties to provide a suitable homogenized integral constraint on the equivalent stress. The ith phase gradient of CH (A, β(x)) is given by [34] iCH (A, β(x)) : ∇

= Q

χi(β(x), y)((w) + ) ⊗ ((w) + ) dy, i = 1 . . . , N, (4.1)

where is any constant 3 × 3 constant strain and w = w(x, y) is the Q periodic solution of (2.7). The homogenized square of the equivalent stress is given in terms of the ﬂuctuation tensor Q(A, β(x)) deﬁned by

Q(A, β(x)) =

N

iCH (A, β(x))S(A, β(x)) − I, (S(A, β(x))AiΠAi∇

i=1

(4.2)

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181

where I is the fourth order identity. The homogenized design problem is given by H=

inf

β(x)∈Adγ

{F (uH );

Q

{Q(A, β(x)) + I}σ H : σ H dx ≤ K 2 },

(4.3)

here σ H is the stress in the homogenized composite given by (2.11) and uH is the W01,2 (Ω, R3 ) solution of (2.10). It is pointed out that one recovers the unconstrained compliance optimization problem (2.13) by choosing K = ∞ in (4.3). Thus one regards (2.13) as a special case of (4.3). The ﬁrst property of the homogenized design problem is given by Theorem 4.1 There exists an optimal design for the homogenized design ˆ problem, i.e., there is an optimal control β(x) ∈ Adγ for which the inﬁmum in (4.3) is attained. The optimal design can be used to construct a sequence of local microstructures with stresses that satisfy integral constraints on the square of the equivalent stress and have compliance controllably close to H given by (4.3). In order to state this precisely we introduce as before the elasticity tensor for a composite made from N elastic materials with microstructure of length scale ε by Cε(A, x). Here Cε(A, x) takes the value Ai in the ith phase. The characteristic function for the ith phase is denoted by χεi(x). The displacement uε in the composite is the W01,2 (Ω, R3 ) solution of −div (Cε(A, x)(uε)) = f .

(4.4)

The stress ﬁeld in the composite is denoted by σ ε(x) = Cε(A, x)(uε) and the compliance is given by F(uε) = Ω f · uε dx. The amount of the ith material in the graded composite is given by Ω χεi(x) dx. ˆ ∈ Adγ for (4.3) can be used to Theorem 4.2 The optimal control β(x) construct a sequence of functionally graded composites with elasticity tenˆ such that sors {Cε(A, x)}ε>0 that G converge to CH (A, β(x)) lim F(uε) = H,

ε→0

lim

lim

ε→0 Ω

ε→0 Ω

χεi(x) dx =

Ω

Π(σ ε) dx ≤ K 2 and θi(x) dx ≤ γi, i = 1, . . . , N.

(4.5)

The explicit procedure for construction of the sequence {Cε(A, x)}ε>0 described in Theorem 4.1 is presented in subsection 4.2.

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R. LIPTON

Theorems 4.1 and 4.2 provide a new rigorous strategy for designing suﬃciently stiﬀ composites subject to mean square stress constraints. The method is illustrated in section 6 where a numerical example is given. 4.1. PROOF OF THEOREM 4.1

One starts by considering a minimizing sequence {β n(x)}∞ n=1 for the homogenized design problem H given by (4.3). Since {β n(x)}∞ n=1 is an equacontinuous family one readily deduces that there is a subsequence (also denoted ˆ by {β n(x)}∞ n=1 ) converging uniformly on Ω to a control β(x) ∈ Adγ . Next, ˆ y), Q periodic for any 3 × 3 constant strain we consider the function w(x, in the y variable that is the W 1,2 (Q, R3 ) solution of

ˆ ˆ y)((w(x, y)) + ) = 0. −div C0 (A, β(x),

(4.6)

Here all derivatives are with respect to the y variable and x appears as a parameter. Recalling (2.8) it is evident that

lim

n→∞ Q

ˆ |χi(β n(x), y) − χi(β(x), y)| dy = 0

(4.7)

for every x in Ω. We apply (4.7) together with standard arguments to ﬁnd that ˆ lim |(w(x, y)) − (wn(x, y))|2 dy = 0, (4.8) n→∞ Q

where wn(x, y) are Q periodic in the y variable and are the W 1,2 (Q, R3 ) solutions of

−div C0 (A, β n(x), y)((wn(x, y)) + ) = 0.

(4.9)

It is now evident from (4.8) and the formulas (2.6) and (4.1) that ˆ lim Q(A, β n(x)) = Q(A, β(x))

n→∞

ˆ lim CH (A, β n(x)) = CH (A, β(x))

n→∞

(4.10)

for every x in Ω. From (4.10) one has that {CH (A, β n(x))}∞ n=1 G converges 1,2 H 3 ˆ to C (A, β(x)), see [57]. We consider the W0 (Ω, R ) solutions uH n and H ˆ of u

) =f (4.11) −div CH (A, β n(x))(uH n and

ˆ uH ) = f −div CH (A, β(x))(ˆ

(4.12)

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183

respectively. From the continuity of the compliance with respect to G convergence we deduce that uH ). lim F(uH n ) = F(ˆ

(4.13)

n→∞

∞ H ∞ In view of (4.10), standard arguments show that {(uH n )}n=1 and {σn }n=1 ˆ converge strongly in L2 (Ω)3×3 to (ˆ uH ) and σ ˆ H = CH (A, β(x))(ˆ uH ) respectively. An application of the Lebesgue convergence theorem gives

lim

n→∞ Ω

{Q(A, β n(x)) + I}σnH : σnH dx =

Ω

ˆ {Q(A, β(x)) + I}ˆ σH : σ ˆ H dx (4.14)

and Theorem 4.1 follows. 4.2. PROOF OF THEOREM 4.2

We describe the class of discrete graded micro-structures used in the design process. For a control β(x) in Adγ we consider the associated tensor C0 (A, β(x), y) used in the description of the homogenized problem. Conκ sider a family of partitions Pκj of Ω into subregions O j , = 1, . . . , Nκj Nκ

κ

of diameter κj and Ω = ∪ =1j O j with κj → 0 as j → ∞. For ﬁxed κj deﬁne the step function approximation β κj (x) = β(x ) for points x in the κ κ sub domain O j . Here x is a sample point chosen from O j . Now denote the scale of the microstructure by 1/νj where 1/νj goes to zero as j goes to inﬁnity and (1/νj )/κj → 0. The elasticity tensor C0 (A, β κj (x), xνj ) describes a graded microstructure with length scale of the grading given by κj and microstructure on the length scale 1/νj . The characteristic function of the ith phase is given by χi(β κj (x), xνj ). Direct application of Theorem 4.1 of [31] gives the following Theorem 4.3 The sequence {C0 (A, β κj (x), xνj )}∞ j=1 G-converges to H C (A, β(x)). We introduce the W01,2 (Ω, R3 ) solutions uj of

−div C0 (A, β κj (x), xνj )(uj ) = f .

(4.15)

The stress ﬁeld in the composite is denoted by σ j (x) = C0 (A, β κj (x), xνj )(uj ). From the deﬁnition of G convergence uj uH weakly in W01,2 (Ω, R3 ) and σ j σ H weakly in L2 (Ω)3×3 where uH is the solution of (2.10) and σH is the stress given by (2.11). Since the compliance is continuous under G convergence we have lim F(uj ) = F(uH ).

j→∞

(4.16)

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R. LIPTON

Appealing to Theorem 2.3 of [31] it follows immediately that

lim

j

j→∞ Ω

Π(σ ) dx = Ω

{Q(A, β(x)) + I}σ H : σ H dx.

(4.17)

Last, elementary arguments show that

lim

κj

j→∞ Ω

χi(β (x), xνj ) dx =

Ω

θi(x) dx

(4.18)

Theorem 4.2 now follows from (4.16), (4.17) and (4.18 ). 5. Design of stiﬀ composite structures subject to point wise stress constraints In this section we provide a rigorous strategy for construction of graded composites for optimal compliance while at the same time controlling point wise values of the equivalent stress. The methodology developed here makes use of the macro-stress modulation functions f i(x, SH (A, β(x))σH (x)) in the formulation of the homogenized design problem. Set M (x, SH (A, β(x))σ H (x)) = max{f i(x, SH (A, β(x))σ H (x))} i

(5.1)

The homogenized design problem for compliance optimization with point wise stress control is given by H∞ =

inf

β(x)∈Adγ

{F (uH ); M (x, SH (A, β(x))σH (x)) ≤ K 2 },

(5.2)

here σ H is the stress in the homogenized composite given by (2.11) and uH is the W01,2 (Ω, R3 ) solution of (2.10). The ﬁrst property of the homogenized design problem is given by Theorem 5.1 There exists an optimal design for the homogenized design ˆ problem, i.e., there is an optimal control β(x) ∈ Adγ for which the inﬁmum in (5.2) is attained. The optimal design for H∞ can be used to construct a sequence of local micro-structures with stresses that satisfy point wise constraints on the equivalent stress and have compliance controllably close to H∞ given by (5.2). In order to state this precisely, we introduce as before, the elasticity tensor for a composite made from N elastic materials with microstructure of length scale ε by Cε(A, x). Here Cε(A, x) takes the value Ai in the ith phase. The characteristic function for the ith phase is denoted by χεi(x). The displacement uε in the composite is the W01,2 (Ω, R3 ) solution of (4.4) and the stress ﬁeld in the composite is denoted by σ ε(x) = Cε(A, x)(uε(x)).

HOMOGENIZATION AND DESIGN

185

ˆ Theorem 5.2 The optimal control β(x) ∈ Adγ for (5.2) can be used to construct a sequence of functionally graded composites with elasticity tenˆ such that sors {Cε(A, x)}ε>0 that G converge to CH (A, β(x)) lim F(uε) = H∞ ,

ε→0

lim sup Π(σ ε(x)) ≤ K 2 , a.e. in Ω and ε→0

lim

ε→0 Ω

χεi(x) dx

= Ω

θi(x) dx ≤ γi, i = 1, . . . , N.

(5.3)

The explicit procedure for construction of the sequences {Cε(A, x)}ε>0 described in Theorem 5.2 follows the one given in section 4.2 and is presented in subsection 5.1. Theorems 5.1 and 5.2 provide a method for identifying graded microgeometries that deliver desired structural stiﬀness subject to point wise stress constraints . The method is summarized as follows. First, solve the homogenized design problem H∞ numerically using steepest decent methods to ˆ recover the optimal control β(x). Second, for a given tolerance δ > 0 use Theorem 5.2 to construct a graded composite microstructure with characteristic length scale ε suﬃciently ﬁne so that outside of a subset A of volume δ one has that Π(σ ε(x)) < K 2 + δ and H∞ − δ < F(uε) < H∞ + δ

Ω

χεi dx < γi + δ, i = 1, . . . , N.

(5.4)

5.1. PROOFS OF THEOREMS 5.1 AND 5.2

To prove Theorem 5.1 consider a minimizing sequence of controls for H∞ . Passing to a subsequence if necessary we obtain a sequence {β n(x)}∞ n=1 ˆ converging uniformly to an admissible control β(x). Following the arguments in subsection 4.1 the associated sequence {CH (A, β n(x))}∞ n=1 conH ˆ verges point wise to C (A, β(x)) and therefore G converges to it as well. ˆ H of (4.11) and (4.12) respecConsider the W01,2 (Ω, R3 ) solutions uH n and u tively. From the continuity of the compliance with respect to G convergence it follows as noted in subsection 4.1 that limn→∞ F(uH uH ). n ) = F(ˆ H ∞ H ∞ Also as in subsection 4.1, the sequences {(un )}n=1 and {σn }n=1 conˆ verge strongly in L2 (Ω)3×3 to (ˆ uH ) and σ ˆ H = CH (A, β(x))(ˆ uH ). Passing ∞ to a subsequence if necessary we deduce that the sequences {(uH n )}n=1 H ∞ H H and {σn }n=1 converge point wise to the limits (ˆ u (x)) and σ ˆ (x) =

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R. LIPTON

ˆ CH (A, β(x))(ˆ uH (x)) respectively. Set gn(x, y) = (wn(x, y)) + (uH n (x)) ˆ are solutions of (4.9) ˆ and gˆ(x, y) = (w(x, y)) + (ˆ uH (x)). Here wn and w uH (x)). Arguments analogous to with = (uH n (x)) and (4.6) with = (ˆ those given in subsection 4.1 show that {gn(x, y)}∞ n=1 converges strongly in L2 (Q)3×3 to gˆ for almost every x in Ω. From this one ﬁnds that

lim

n→∞ Q

ˆ |χi(β n(x), y)Π(gn(x, y)) − χi(β(x), y)Π(ˆ g (x, y))|dy = 0.

(5.5)

Passing to subsequences if necessary note that (5.5) implies the point wise ˆ y)Π(ˆ g (x, y)) a.e. and one convergence χi(β n(x), y)Π(gn(x, y)) → χi(β(x), easily deduces that ˆ f i(x, SH (A, β(x))ˆ σH (x)) ≤ K 2 .

(5.6)

Theorem 5.1 now follows. The proof of Theorem 5.2 uses the same method of construction given in section 4.2. For a control β(x) in Adγ we consider the associated tensor C0 (A, β(x), y) used in the description of the homogenized problem. Here the eﬀective elastic tensor CH (A, β(x)) and the associated homogenized stress σ H satisfy the constraints given in H∞ . As in section 4.2 consider κ a family of partitions Pκj of Ω into subregions O j , = 1, . . . , Nκj of diNκ

κ

ameter κj and Ω = ∪ =1j O j with κj → 0 as j → ∞. For ﬁxed κj deﬁne the step function approximation β κj (x) = β(x ) for points x in the subκ κ domain O j . Here x is a sample point chosen from O j . Now denote the scale of the microstructure by 1/νj where 1/νj goes to zero as j goes to inﬁnity and (1/νj )/κj → 0. The elasticity tensor C0 (A, β κj (x), xνj ) describes a graded microstructure with length scale of the grading given by κj and microstructure on the length scale 1/νj . The characteristic function of the ith phase is given by χi(β κj (x), xνj ). From Theorem 4.3 the H sequence {C0 (A, β κj (x), xνj )}∞ j=1 G converges to C (A, β(x)). As in section 4.2 consider the W01,2 (Ω, R3 ) solutions uj of (4.15) The stress ﬁeld in the composite is denoted by σ j (x) = C0 (A, β κj (x), xνj )(uj ). From the deﬁnition of G convergence uj uH weakly in W01,2 (Ω, R3 ) and σ j σ H weakly in L2 (Ω)3×3 where uH is the solution of (2.10) and σH is the stress given by (2.11). Since the compliance is continuous under G convergence we have lim F(uj ) = F(uH ).

j→∞

(5.7)

Appealing to Theorem 4.1 and Corollary 4.3 of [31] it follows immediately that lim sup Π(σ j ) ≤ t, for t > K 2 , a.e. in Ω. (5.8) j→∞

HOMOGENIZATION AND DESIGN

As before one has lim

κj

j→∞ Ω

187

χi(β (x), xνj ) dx =

Ω

θi(x) dx.

(5.9)

Theorem 5.2 now follows from (5.7), (5.8) and (5.9 ). 6. Numerical example We provide an example of compliance design subject to integral constraints on the stress. The design domain is taken to be the cross section of a long cylindrical shaft. The cross section Ω is given by the square (−1, 1)2 . The shaft is reinforced with a microstructure made up of long cylindrical ﬁbers of circular cross section. The generators of the ﬁbers are parallel to those of the shaft. The common shear stiﬀness of the ﬁbers is denoted by Gf and the matrix shear stiﬀness is denoted by Gm. The shaft is subjected to torsion loading and the problem is to design the microstructure so that the composite shaft has maximum torsional rigidity subject to a prescribed stress constraint. The period cell for the microstructure is given by the unit square centered at the origin. For this problem the design vector β is just the radius of the circular cross section of the ﬁber denoted by a. Here the admissible set of radii is Ad = {0 ≤ a ≤ 0.4}. In this context (2.9) becomes |a(x) − a(x + h)| ≤ C|h|α.

(6.1)

To proceed with the numerical implementation we replace (6.1) with a constraint of the form a(x)W 1,p (Ω) ≤ C, (6.2) for p chosen such that α = 1 − 2/p. This choice is motivated by Morrey’s inequality which implies a(x)C0,α (Ω) ≤ Kβ(x)W 1,p (Ω) .

(6.3)

The constraint given by (6.2) provides a convenient context for the discretization of a(x) using ﬁnite elements while at the same time insuring that the constraint (6.1) holds. We set p = 3 and the admissible set of controls is speciﬁed by Adγ = {a(x) ∈ Ad; a(x)W 1,3 (Ω) < C, Ω πa2 ≤ γ}. The local ﬁber area fraction in the homogenized composite is given by θf (x) = π(a(x))2 . The local area fraction of matrix is given by θm(x) = 1 − θf (x). For torsion loading the elastic behavior is described by the stress potential ϕH (x) deﬁned over the cross section of the homogenized shaft. The eﬀective shear compliance of the shaft is a function of the design variable a(x) and is denoted by S(a(x)). The eﬀective shear stiﬀness is described in terms of

188

R. LIPTON

the solution of the unit cell problem similar to (2.6), see [36]. The stress potential is the W01,2 (Ω) solution of −div(S(a(x))∇ϕH ) = 1.

(6.4)

In the context of the torsion problem the homogenized integral stress constraint is given in terms of a ﬂuctuation tensor analogous to (4.2) that is denoted by Q(a(x)). The ﬂuctuation is deﬁned in terms of the solution of the unit cell problem similar to (4.2), see [36]. The torsional rigidity is given by T (ϕH ) = 2 Ω ϕH dx The homogenized design problem is given by HT =

sup {T (ϕ );

H

a(x)∈Adγ

Ω

(Q(a(x)) + I)∇ϕH · ∇ϕH dx ≤ K 2 }.

(6.5)

The maximization of the torsional rigidity is equivalent to compliance minimization and the optimization problem is well posed by virtue of Theorem 4.1. The optimal design is denoted by a ˆ(x). We denote the stress potential for the composite shaft made from a ﬁber composite with microstructure on a length scale ε by ϕε. For a prescribed tolerance δ > 0, the proof of Theorem 4.2 shows how to construct a graded composite shaft made from discrete ﬁbers with ε suﬃciently small such that the torsional rigidity is greater than HT − δ, the stress ∇ϕε in the composite satisﬁes ε2 2 Ω |∇ϕ | dx ≤ K + δ and the total cross-sectional area of ﬁbers is less than γ + δ. In the following numerical optimization the shear compliance of the ﬁbers and matrix are given in dimensionless units by 2 and 4 respectively. The homogenized stress constraint is speciﬁed by K 2 = 0.22 and the total area occupied by the ﬁbers is constrained to lie below 25% of the total volume of the shaft cross-section. The optimization over admissible distributions of radii a(x) is done numerically using the method of steepest decent. The torsional rigidity of the optimized design is 0.7. The density distribution of the ﬁbers in the numerically optimized design is denoted by opt(x) = 1 − θ opt(x) is presented θfopt(x) = π(aopt(x))2 . A grey scale plot of θm f in Figure 1. Here the density of ﬁbers is free to range between 0 and 0.5. The lightest regions correspond to local ﬁber fraction θfopt = 0.5 and the darkest regions correspond to θfopt < 0.05. The level curves of θfopt(x) are also plotted in Figure 1. It is pointed out that our treatment of the H¨ older constraint (6.1) is ad hoc. One of the goals of future research is to provide a systematic numerical approach that implements the rigorous design strategies presented in this paper.

189

HOMOGENIZATION AND DESIGN

The Final Density Distribution

1

0.95

0.8

0.9

0.6

0.85

0.4

0.8 0.2

0.75 0

0.7 −0.2

0.65 −0.4

0.6 −0.6

0.55

−0.8

−1

0.5 −1

Figure 1.

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Stress constrained design for maximum torsional rigidity.

7. Acknowledgements The author thanks Yuri Grabovsky, Martin Bendsœ, Robert Kohn, John Taylor and Ani Velo for stimulating discussions about stress constrained design. This work is supported by NSF grant DMS-0296064 and by the AFOSR through grant F49620-99-1-0009. The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions herein are those of the authors and should not be interpreted as necessarily representing the oﬃcial policies or endorsements, either expressed or implied of the Air Force Oﬃce of Scientiﬁc Research or the US Government. References 1.

Aboudi, J., Pindera, M.J., and Arnold, S.M. (1997) Microstructural optimization of functionally graded composites subjected to thermal gradient via the coupled higher-order theory, Composites part B Vol. 28B, 93-108.

190 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16.

17. 18. 19. 20. 21. 22.

23.

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HOMOGENIZATION FOR NONLINEAR COMPOSITES IN THE LIGHT OF NUMERICAL SIMULATIONS

H. MOULINEC AND P. SUQUET

L.M.A./ C.N.R.S. 31 Chemin Joseph Aiguier. 13402. Marseille. Cedex 20. France. Abstract. The aim of this study is to provide insight, through numerical simulation, into the relations between the heterogeneity of the strain ﬁeld at the microstructural level and the eﬀective properties of nonlinear composites. Keywords: Composites, plasticity variational principles, computational methods, fast Fourier transforms, ﬁeld ﬂuctuations. 1. Introduction 1.1. INDIVIDUAL CONSTITUENTS AND LOCAL PROBLEM

The composites under consideration in this study are made of N diﬀerent homogeneous constituents or phases r = 1, ..N . In a representative volume element V of the composite the individual phases occupy diﬀerent domains Vr with characteristic functions χ(r) (x) and volume fraction c(r) . The spatial average over the r.v.e. V of a function f and the partial average over the domains Vr are deﬁned as N 1 1 f = f (x) dx, f r = f (x) dx, f = c(r) f r . |V | V |Vr| Vr r=1

The constitutive relations of the individual phases derive from a strainenergy potential w (r) : σ=

∂w(r) (ε) in phase r. ∂ε

(1)

In all cases considered in this study, the energy-functions w (r) are convex and most often, even strictly convex.

193 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 193–223. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

194

H. MOULINEC AND P. SUQUET

The local problem, by which the local stress and strain ﬁelds can be determined at the microscopic level, reads as: Find a displacement ﬁeld u and a stress ﬁeld σ solving :   ∂w(r)  σ(x) = (ε(x)) in Vr, div(σ) = 0 in V,  ∂ε (2) 

  1 T ε= ∇u + ∇ u , ε = ε. 2

In order to close the problem (2) where boundary conditions arre missing, periodic boundary conditions will be imposed : u − ε.x periodic,

σ.n anti-periodic on ∂V.

(3)

For brevity, the abridged notation v# (respectively τ .n − #) will be used for a periodic ﬁeld v (respectively an anti-periodic, equilibrated, ﬁeld τ ). The homogenized or eﬀective constitutive relations for the composite are, by deﬁnition, the relations between the average stress σ = σ and the average strain ε = ε ., where σ and ε are the solutions of (2) and (3). 1.2. EFFECTIVE POTENTIALS

When the energy functions w (r) are convex, when the scale are well-separated (and under additional technical assumptions which will not be discussed here), there exists a homogenized local behavior deriving from a homogenized potential (see Marcellini [10] and Muller [14] for a rigorous derivation): σ=

∂w 2 (ε), ∂ε

with

where

w(ε) 2 = inf

v∈K(ε)

< w(x, ε(v)) >,

K(ε) = {v = ε.x + v ∗ , v ∗ # }.

(4)

(5)

Alternatively : ε=

where w

∗

=

N

∂w 2∗ (σ), ∂σ

w 2∗ (σ) =

inf

τ∈S(σ)

< w∗ (x, σ) >,

(6)

(w (r) )∗ (x, .)χ(r) (x), (w(r) )∗ being the convex Legendre

r=1

transform of w (r) and S(σ) = {τ ,

div(τ ) = 0,

τ = σ,

τ .n − # }.

It can be shown that w 2 and w 2∗ are two dual convex functions.

HOMOGENIZATION FOR NONLINEAR COMPOSITES

195

Elementary bounds Elementary bounds are obtained by specializing the trial ﬁelds in (4) and (6). Taking u = ε.x leads to the Voigt bound, whereas taking τ = σ gives the Reuss bound : ≤< w > (ε) (Voigt), w(ε) 2

w 2∗ (σ) ≤< w∗ > (σ)

(Reuss).

(7)

The Voigt bound has been improved by diﬀerent authors (Willis [28], Ponte Casta˜ neda [16], Suquet [22], Olson [15]). One of these upper bounds will be discussed in the next section. In contrast, almost no reﬁnement of the Reuss bound is known (to the author) with the noticeable exception of Talbot and Willis [26]. 1.3. EXAMPLES

An important case where the constitutive relations of the phases can be written in the form (1) is that of elasto-plastic materials when their constitutive relations are described in the framework of the deformation theory of plasticity. As is well known the deformation theory of plasticity is not rigorously equivalent to the incremental theory of plasticity except for proportional loadings without unloading. Admittedly, this condition is rarely met at the microscopic scale in composite materials, but the understanding of this simpliﬁed case gives insight into the more complicated incremental case. Within the context of the deformation theory of plasticity, the energy takes the form: 1 w(r) (ε) = k (r) trε2 + f (r) (ε2eq ), (8) 2 where εeq is the von Mises strain ε2eq = 23 eij eij , e being the strain deviator. The stress-strain relation deriving from the energy (8) read : trσ = 3k (r) trε,

(r)

s = 2µsct (εeq )e,

(9) (r)

where s is the stress deviator and where the secant shear modulus µsct is deﬁned as : 2 (r) (10) µsct (εeq ) = f (r) (ε2eq ). 3 Power-law materials Incompressible power-law materials correspond to a particular form of the strain energy (8) with : (r)

w

(r)

σ ε0 (ε) = 0 m+1

εeq ε0

m+1 when

trε = 0,

+∞

otherwise.

(11)

This form is particularly useful in the context of bounds for composite materials, as will be seen later. The stress strain relations for power-law

196

H. MOULINEC AND P. SUQUET

materials read as : 2 e s = , σeq 3 εeq or equivalently

εeq = ε0

σeq σ0

σeq = σ0

εeq ε0

m

n ,

with

n=

,

1 . m

In the present study the rate sensitivity exponent m varies from 0 to 1. m = 1 corresponds to a linear newtonian ﬂuid, m = 0 to a rigid plastic material. When all constituents of a nonlinear composite have the same rate-sensitivity exponent m and the same reference strain-rate ε0 but diﬀer(r) ent ﬂow-stress σ0 , the composite is an incompressible power-law material and its eﬀective energy can be written as σ 20 ε0 εeq m+1 w(ε) 2 = when trε = 0, +∞ otherwise. (12) m + 1 ε0 where the eﬀective ﬂow-stress σ 20 is, in three-dimension, a function of ε which is positively homogeneous of degree 0. The Voigt and Reuss bounds read, for power-law composites : 1/n

< σ0

>n ≤ σ 2 0 ≤ < σ0 > .

1.4. SECANT MODULI

An equivalent but more general writing of (10) is (r)

σ(x) = Lsct (ε(x)) : ε(x) in phase r,

(13)

(r)

where Lsct is the tensor of secant moduli in phase r deﬁned as: (r)

Lsct (ε) = 3k (r) J + 2µ(r) K,

(r)

µ(r) = µsct (εeq ),

(14)

J and K being the fourth-order tensors associated with the projection of a symmetric second-order tensor on its hydrostatic part and its purely deviatoric part respectively. Two observations can be made about secant moduli: (r) − Because of the dependence of Lsct on ε which itself varies spatially, a nonlinear composite can be regarded as a linear composite with in(r) ﬁnitely many phases (with moduli Lsct (ε(x)). (r) − Lsct depends on ε only through the scalar invariant εeq .

HOMOGENIZATION FOR NONLINEAR COMPOSITES

197

2. Interpretation of the variational bounding technique of Ponte Casta˜ neda as a secant method When the f (r) ’s are concave (and the w(r) ’s are convex) a bound which is more accurate than Voigt and Reuss bounds can be obtained. This bound was ﬁrst derived in general form by [17]. Then an alternative and simple proof of the same result was given by [23] in a form which made clear the connection with the secant moduli in the linear comparison composite (see also [7] for a similar and independent observation). 2.1. THE VARIATIONAL BOUND

The bound can be formulated in two steps as follows: Assume that the energy functions w(r) ’s are convex and can be written in the form (8) where the f (r) ’s are concave. Then : 1. The following inequality holds N

≤w 2 (ε) = inf w(ε) 2 +

v∈K(ε)

! (r)

c

r=1

" 0 2 1 1 (r) 0 2 1 (r) k trε r + f ( εeq r) . (15) 2

2. The Euler’s equations associated with the right-hand side of (15) are found to be exactly the elasticity equations in a a ﬁctitious linear composite, called the linear comparison composite (LCC) where the phases have the same geometry as in the actual nonlinear composite but exhibit a linear behavior characterized by a stiﬀness tensor L(r) L(r) = 3k (r) J + 2µ(r) K,

(16)

where µ(r) is the solution of the following nonlinear problem: (r)

(r)

µ(r) = µsct (εeq

),

(r)

εeq

3 4 2 1/2 = εeq , r

(17)

and where the strain ﬁeld ε entering (17) is the strain ﬁeld in the linear comparison composite. Before proceeding to the detailed derivation of these two results a few comments on the relation between this bound and the secant method should be made. − The moduli of phase r in the linear comparison composite are the (r) secant moduli of phase r evaluated at some “eﬀective strain” ε for phase r. For this reason the method which consists of replacing the actual eﬀective constitutive relations for the nonlinear composite by 2 : ε is called a secant method. the relation σ = L

198

H. MOULINEC AND P. SUQUET

− The strain ﬁeld entering (17) is the strain ﬁeld in the linear comparison composite and not in the actual nonlinear composite. − The classical version of the secant method (Berveiller and Zaoui [2]) evaluates the secant moduli of phase r at the average strain of phase r (in the LCC). The “eﬀective strain” used in (17) is diﬀerent and makes use 3 of 4the second moment in each phase of the strain ﬁeld in the LCC ( εeq 2 ). The resulting secant method has been called the “modiﬁed” r

secant method by Suquet [23]. − Problem (17) is a nonlinear problem since the moduli µ(r) depend on the strain ﬁeld ε which itself depends on the moduli of the individual phases. The solution (µ(r) ), r = 1, · · · , N ) of problem (17) depends on 2 also depends on the overall strain ε. Therefore the eﬀective stiﬀness L the overall strain. − Although this does not jump to the eyes, the bound (15) is exactly the bound of Ponte Casta˜ neda [17] derived by completely diﬀerent means. The present derivation ([23]) makes only use of the concavity inequality. 2.2. PROOF OF THE VARIATIONAL BOUND

Recall the variational deﬁnition of w: 2 = inf w(ε) 2

v∈K(ε)

= inf

v∈K(ε)

N

6 (r)

c

r=1

< w(x, ε(v)) >

7 1 (r) 2 (r) 2 k trε + f (εeq ) . 2 r

(18)

The proof of Suquet [23], which we will follow here, proceeds in two steps. Step 1: proof of (15) The ﬁrst term in (15) (dilatational term) is exactly the ﬁrst term in (18). As for the second term, the concavity of any realvalued function f deﬁned on V reads as: tf (x) + (1 − t)f (y) ≤ f (tx + (1 − t)y)

∀x, y ∈ V, ∀t ∈ [0, 1].

(19)

Since the average over phase r is a linear combination of values of f (r) at diﬀerent points in Vr , an immediate consequence of the concavity of f (r) and of property (19) is: 4

0 1 3 (20) f (r) (ε2eq ) ≤ f (r) ε2eq r . r

This completes the proof of (15).

HOMOGENIZATION FOR NONLINEAR COMPOSITES

199

2.3. INTERPRETATION OF THE VARIATIONAL PROBLEM ASSOCIATED 5+ WITH W

To derive the Euler’s equations associated 0with1 the variational problem (15) we have to take the variation of j (r) (ε) = ε2eq r when ε varies in K(ε). Let ε∗ be such that ε∗ = 0. Then j (r) (ε + hε∗ ) − j (r) (ε) lim = h→0 h

#

∂j (r) ∗ ,ε ∂ε

$ r

4 = e, e∗ r , 3

where e and e∗ are the deviators of ε and ε∗ respectively. Let ε denote the strain ﬁeld achieving the inﬁmum in (15). Then the Euler’s equations for w 2+ read as : N

! (r)

k

c

(r)

r=1

" 0 ∗1 4 (r) 3 2 4 ∗ εeq e, e r = 0, trε trε r + f 3 r

for all ε∗ in K(0). In other words there exists a stress ﬁeld σ and a displacement ﬁeld u such that :  σ (x) = k(x)trε i + 2µ(x)e (x), div(σ ) = 0 in V,  ε =

1 2

∇u + ∇T u ,

with k(x) =

N

k

(r) (r)

χ

u − ε.x #,

(x),

µ(x) =

N

r=1

2 µ(r) = f (r) 3

σ .n − #.



µ(r) χ(r) (x),

r=1

3

εeq

2

4 r

(r)

= µsct

3

εeq

2

4 r

.

2.4. THE TWO APPROXIMATIONS INVOLVED IN THE MODIFIED SECANT METHOD

The modiﬁed secant method is not more diﬃcult to implement than the classical secant method, but it is more accurate and has a clear status (upper bound). In addition the predictions of the classical secant method for porous materials is unrealistic (see Suquet [24] for more details). For these reasons the modiﬁed secant method is becoming popular ([1], [20], [6] for instance). However, it should be made very clear that using the modiﬁed secant method (or variational method) to estimate the eﬀective properties of nonlinear composites, requires two diﬀerent levels of approximation:

200

H. MOULINEC AND P. SUQUET

1) The ﬁrst level of approximation consists of replacing the nonlinear composite, which behaves as a linear composite with inﬁnitely many phases, by a two-phase linear composite with elastic moduli which are nonlinear functions of the local strain ﬁeld in the phases. This approximation is central in the modiﬁed secant method. 2) The second level approximation, which is in fact optional, consists of estimating the eﬀective properties of the linear comparison composite (r) and the eﬀective equivalent strain εeq in the phases. Bounds and estimates, such as the Hashin-Shtrikman bounds or the self-consistent scheme, are commonly used for this purpose, but they add another approximation to the method. 3. Numerical simulations 3.1. OBJECTIVE

This section has two diﬀerent objectives at the global (macroscopic) and local (microscopic) scales. The ﬁrst objective is to assess the accuracy of the approximations made by the modiﬁed secant method by comparing its predictions for the eﬀective properties of nonlinear composites with “exact” computational results. The second objective is to compare local statistical informations, e.g. ﬁrst moments and ﬂuctuations of the local strain and stress ﬁelds in the LCC and in the actual nonlinear composite. For the purpose of these comparisons, we consider two-phase ﬁbrous composites composed of incompressible power-law phases, with a microstructure close to the ideal composite cylinder assemblage. To assess separately the accuracy of the two approximations involved in the secant method, as summarized in section 2.4, three diﬀerent computations have been performed. − The stress and strain ﬁelds in the actual nonlinear composite have been been computed exactly (up to numerical errors) using a computational method based on the fast Fourier transform (see Moulinec and Suquet [12] for a general introduction to the method and [13] for a more detailed presentation of the method applied to power-law materials). The corresponding results are referred to as “exact (FFT)” in the subsequent ﬁgures and labelled with crosses in the graphs. − The stress and strain ﬁelds in the linear comparison composite have also been computed exactly (up to numerical errors) using the same computational method based on the fast Fourier transform. The elastic properties of the LCC are determined by imposing the closure condition (17) to the exact ﬁelds in the LCC. In practice, this is done

HOMOGENIZATION FOR NONLINEAR COMPOSITES

201

(r)

iteratively. The elastic moduli µn at step n of the procedure are computed from the strain ﬁeld εn−1 at step (n − 1) using the relation (17). Then a linear elastic problem for the LCC with isotropic incompress(r) ible phases with shear moduli µn is solved and a new strain ﬁeld εn is determined. The procedure is repeated until convergence is reached. Incompressibility is taken into account according to the computational scheme of Suquet and Moulinec [25]. The corresponding results are referred to as “LCC (FFT)” in the subsequent ﬁgures and labelled with circles in the graphs. − Finally, the homogenized ﬂow stress and the ﬁrst and second moments of the stress and strain ﬁelds can be evaluated analytically using both the approximation of a LCC and a linear analytical scheme to estimate the eﬀective properties of the LCC (see examples in Suquet [24] and Ponte Casta˜ neda and Suquet [19]). The Hashin-Shtrikman formalism is known to yield reasonably accurate predictions of the linear eﬀective properties of composite cylinder assemblages, at least when the contrast is moderate. The Hashin-Shtrikman bound used in the sequel reads as : µ(1) − µ(2) µ 2HS = µ(2) + c(1) . (21) (1) − µ(2) (2) µ 1+c 2µ(2) It is a lower bound for the eﬀective shear modulus µ 2 of the composite when µ(1) ≥ µ(2) and an upper bound in the opposite case. The results obtained by using this bound to estimate the eﬀective properties of the LCC are referred to as “LCC (HS+ or -)” in the subsequent ﬁgures and correspond to solid lines in the graphs. 3.2. CONFIGURATIONS AND ENSEMBLE AVERAGES

Two-phase ﬁbrous composites are considered. As shown in ﬁgure 1, their microstructure is of matrix-inclusion type and obtained by placing randomly non-overlapping and self-similar composite cylinders with three different sizes in a square unit cell. These microstructures approach an ideal composite cylinder assemblage (the diﬀerence being that, ideally, the cylinders’radii form an inﬁnite sequence tending to 0, whereas only three ﬁnite sizes are considered here). 3.2.1. Statistical homogeneity and isotropy Achieving numerically statistical homogeneity and isotropy is a diﬃcult issue: how can one be sure that the computational results are representative of the exact results for a given class of microstructures? Both properties (homogeneity and isotropy) cannot be obtained exactly but asymptotically

202

H. MOULINEC AND P. SUQUET

as some parameter goes to +∞ (either the size of the rve or the number of conﬁgurations). As brieﬂy analyzed in Moulinec and Suquet [12], Michel et al [11] and in more details in Kanit et al [9], there are (at least) two practical ways of checking statistical homogeneity. The ﬁrst possibility is to consider unitcells of increasing size and to investigate the stationarity of the computational results with respect to the unit-cell size. The question is therefore: how large the unit-cell should be? The second option is to consider a large number of diﬀerent realizations and to average the results among all conﬁgurations (ensemble average). None of the conﬁgurations is, taken separately, representative of the class of microstructures but, assuming that the ergodic property holds and is not biased by the choice of the conﬁgurations, the ensemble average of the results is representative. The question is: how many conﬁgurations of a given size should be considered in the ensemble average? The critical size (or number of conﬁgurations) above which the results are stationary (up to a given oﬀset) depends on the criterion which is used to check statistical homogeneity. This criterion can be related to the geometry of the composite (e.g. N-point correlation functions) or related to the mechanical response of the composite (eﬀective properties, average per phase of a ﬁeld or higher-order moments per phase of this ﬁeld). The threshold also depends very strongly on the type of material nonlinearity which is considered. In particular, the unit-cell sizes found in the analysis of linear properties are not suﬃcient to ensure stationarity of nonlinear properties. The question of the correct size for rigid-plastic materials even remains unclear, although homogenization for rigid-plastic materials can be rigorously proved to be legitimate (Bouchitte and Suquet [3]). We followed the second option by considering diﬀerent conﬁgurations of the microstructures, all having the same size. At a given volume fraction it was observed that twenty diﬀerent conﬁgurations containing approximately 490 ﬁbers of diﬀerent size (see below) were suﬃcient to ensure stationarity of the ensemble averages of the eﬀective properties for the diﬀerent conﬁgurations. The same type of question arises regarding isotropy of the conﬁgurations. Isotropy is even harder to check than statistical homogeneity. Again two options are possible, a purely geometric approach or a mechanical approach. We followed a mechanical approach by checking that the computational results were “consistent” with isotropy in the following sense (which is not a complete proof of isotropy). In two dimensions of space, the response of an isotropic material to a pure shear stress should be a pure shear strain. When the conﬁgurations are considered individually this rule is not exactly met. The maximal deviation measured by ε11 /ε12 (see be-

HOMOGENIZATION FOR NONLINEAR COMPOSITES

203

low for more details about the loading conditions) in the most severe case (highest nonlinearity) is about 4.5 %. However, when ensemble averages are taken the deviation from the rule is of the order of 0.5 %. The ensemble averages of the computational results were considered as isotropic in this weak sense. 3.2.2. Conﬁgurations Fourty diﬀerent realizations similar to those shown in ﬁgure 1 have been generated using an algorithm described in more details in [13]. In twenty of them the ﬁber volume fraction c(1) varies from 0.2 to 0.24, with an average c(1) = 0.20626. In the other twenty conﬁgurations the ﬁber volume fraction varies from 0.4 to 0.45 with average c(1) = 0.4124. These slight variations in volume fraction are an artefact of the algorithm used to place randomly the composite cylinders, the number of cylinders of a given size being diﬀerent from one conﬁguration to another. Ensemble averages of the computational results are performed in the following way. The conﬁgurations are labelled with an index α. In a given

c(1) = 0.2105 Figure 1.

c(1) = 0.4209

Typical microstructures. Phase 1 (ﬁbers) in white.

conﬁguration α the volume fraction of phase r is : 3 4 (r) c(r) α = χα , (r)

where χα denotes the characteristic function of phase r in the speciﬁc (r) conﬁguration α. The ensemble average of the cα gives the average volume

204

H. MOULINEC AND P. SUQUET

fraction of phase r over all conﬁgurations : 1 (r) cα , A A

c(r) =

A = number of conﬁgurations.

α=1

Similarly the conﬁgurational and ensemble averages of a quantity F over phase r are deﬁned as: (r)

Fα =

1 3 (r) cα

4 F χ(r) α ,

F

(r)

=

A 1

c(r)

(r)

c(r) α Fα .

α=1

3.3. EFFECTIVE FLOW STRESS

The individual constituents are incompressible power-law materials with the same exponent m (or n). The composite is incompressible and its effective energy w 2 is a positively homogeneous function of degree m + 1 with respect to the overall strain ε. In the present case (two-dimensional problem, overall isotropy), w 2 can be characterized in terms of a single eﬀective coeﬃcient σ 20 , referred to as the eﬀective ﬂow stress of the composite. The applied loading is a pure in-plane shear stress : σ = σ 12 (e1 ⊗ e2 + e2 ⊗ e1 ) . (1)

(2)

Two diﬀerent constrasts were used in the computations, σ0 /σ0 = 5 (1) (2) (ﬁbers stronger than the matrix), and σ0 /σ0 = 0.2 (ﬁbers weaker than the matrix). Five diﬀerent values of the rate-sensitivity exponent m were investigated, m = 1/n with n = 1, 2, 3, 5, 10. All together, 800 hundred computations were performed (2 diﬀerent volume fractions, 20 conﬁgurations per volume fraction, 5 values of m, two diﬀerent contrasts and two schemes, exact and LCC). 3.3.1. Dependence on the volume fraction The dependence of the eﬀective ﬂow stress on the ﬁber volume fraction is shown in ﬁgures 2 and 3. The results of the exact computation for the 20 diﬀerent conﬁgurations (crosses) are compared with the predictions of modiﬁed secant method (LCC). The eﬀective properties of the LCC are either computed exactly (circles) or approximated by the Hashin-Shtrikman bound (solid line). A few points are worth noticing: − The dispersion in the computational results is relatively small and the diﬀerence between the various methods is much larger than the variations due to the diﬀerences in conﬁgurations (the 20 diﬀerent crosses, or circles, can be hardly distinguished). Legitimate conclusions about the methods themselves can be drawn from these graphs.

205

HOMOGENIZATION FOR NONLINEAR COMPOSITES 2.4 2.2

0

/

2.0

(2)

(1)

0

0

/

(2) 0

=5

1.9

(2)

/

0

0

0

=5

n=3

1.7 1.6

1.8

Exact (FFT) HS-

1.6

Exact (FFT) LCC (FFT) LCC (HS-)

1.5 1.4 1.3

1.4

1.2

1.2

(1)

1.1

c

1.0 0.0

0.1

0.2

0.3

0.4

0.5

c

1.0 0.0

0.1

0.2

(a) 2.0 0

/

0.3

(1)

0.4

0.5

(b) 2.0

(2)

(1)

0

0

1.8

/

(2) 0

=5

1.9

0

/

(2)

(1)

0

0

1.8

n=5

/

(2) 0

=5

n=10

1.7

1.7 1.6

1.6

Exact (FFT) LCC (FFT) LCC (HS-)

1.5 1.4

Exact (FFT) LCC (FFT) LCC (HS-)

1.5 1.4 1.3

1.3

1.2

1.2 (1)

1.1 1.0 0.0

(2)

/

1.8

n=1 2.0

1.9

(1)

0

0.1

0.2

(c)

0.3

0.4

(1)

1.1

c

0.5

1.0 0.0

c 0.1

0.2

0.3

0.4

0.5

(d)

Figure 2. Fibers stronger than the matrix. Eﬀective ﬂow stress of the composite as a function of the ﬁber volume fraction. Comparison between the exact nonlinear results (cross), the exact results for the LCC (circles) and the estimate (solid line) obtained by using the Hashin-Shtrikman lower bound in the evaluation of the eﬀective properties of the LCC

− When n = 1, the phases are linear and there is no diﬀerence between the exact and the LCC computations performed with the FFT method. The diﬀerence between the exact computation and the prediction of the Hashin-Shtrikman lower bound measures the accuracy of the approx-

206

H. MOULINEC AND P. SUQUET

imation provided by the HS bound. This approximation is reasonably good. However, it should be kept in mind that the contrast considered here is moderate (contrast = 5), whereas in the LCC the contrast between the secant moduli can be much larger.

1.0

0

/

(2)

n=1

0

1.0

0.8

0.8

0.6

0.6

0.4

0.4 Exact (FFT) LCC (HS+)

0.2 0.0 0.0

(1) 0

(2) 0

/

(1)

0.1

0.2

0.3

0.0 0.5 0.0

0.4

/

(2)

Exact (FFT) LCC (FFT) LCC (HS+) (1) 0

(2) 0

/

0

/

0.1

0.2

(2)

n=5

1.0

0.8

0.8

0.6

0.6

0.0 0.0

(1) 0

/ 0.1

(2) 0

(1)

0.2

(c)

0.3

/

(2)

0.4

0.5

0.4

0.0 0.5 0.0

n=10

0

Exact (FFT) LCC (FFT) LCC (HS+)

0.2

c

= 0.2

0

0.4

Exact (FFT) LCC (FFT) LCC (HS+)

0.2

0.3

(b)

0

0.4

(1)

c

= 0.2

(a) 1.0

n=3

0

0.2

c

= 0.2

0

(1) 0

/ 0.1

(2) 0

(1)

c

= 0.2 0.2

0.3

0.4

0.5

(d)

Figure 3. Fibers weaker than the matrix. Eﬀective ﬂow stress of the composite as a function of the ﬁber volume fraction. Comparison between the exact nonlinear results (cross), the exact results for the LCC (circles) and the upper bound (solid line) obtained by using the Hashin-Shtrikman upper bound in the evaluation of the eﬀective properties of the LCC.

− The results of the modiﬁed secant method without any additional approximation (LCC(FFT)) lie always above the exact nonlinear results.

HOMOGENIZATION FOR NONLINEAR COMPOSITES

207

This is expected from the general theory since, in the case of powerlaw phases, a bound on the energy w 2 provides a bound on the eﬀective coeﬃcient σ 20 . − There are signiﬁcant quantitative diﬀerences between the case of strong and weak ﬁbers. When the ﬁbers are stronger than the matrix the error introduced by the secant method is large: for instance it can be seen (2) in ﬁgure 2 that the reinforcement eﬀect due to the ﬁbers (2 σ0 /σ0 − 1) predicted by the secant method is, for n = 10, twice the actual reinforcement in the nonlinear composite. When the ﬁbers are weaker than the matrix, the error introduced by the secant method is, in comparison, smaller. It is tempting to conclude from this observation that the secant method is accurate for weak, but incompressible, inclusions. Incompressibility plays a role, since in the compressible case and in the limiting case of pores, it is well-known that the predictions of the modiﬁed secant method can be very far from the actual nonlinear properties (especially under hydrostatic loadings). − The use of the Hashin-Shtrikman bound has a diﬀerent eﬀect in the case of weaker and stronger ﬁbers. When the ﬁbers are weaker than the matrix (ﬁgure 3), the expression (21) is a rigorous upper bound for the eﬀective shear modulus of the LCC and the solid line LCC (HS+) lies above the computational results LCC(FFT). The combination of this upper bound with the secant method, which is itself an upper bound, yields a rigorous upper bound for the actual eﬀective properties of the nonlinear composite. − When the ﬁbers are stronger than the matrix (ﬁgure 2), the expression (21) is a lower bound for the eﬀective shear modulus of the LCC and the solid line LCC (HS-) lies below the computational results LCC(FFT). The error made by using a lower bound partly compensates the error due to the upper bound character of the modiﬁed secant method. This “compensation of errors” could lead to erroneous conclusions regarding the accuracy of the modiﬁed secant method: although the prediction of the secant method with the Hashin-Shtrikman lower bound appears to be in good agreement with the computational results, this adequacy is more a result of the “compensation of errors” than a deﬁnite proof of the method’s accuracy. 3.3.2. Dependence on the rate-sensitivity exponent The dependence of the eﬀective ﬂow stress on the rate-sensitivity parameter m is analyzed in ﬁgures 4 and 5. Since the ﬁber volume fraction varies slightly from one conﬁguration to another, a comparison with the analytical scheme at a given volume fraction is only possible by taking the ensemble average of the eﬀective ﬂow stress for each group of 20 conﬁgurations. These

208

H. MOULINEC AND P. SUQUET

ensemble averages are shown in ﬁgures 4 and 5. A few points are worth of notice. 1.5 0

1.4

/

2.0

(2)

(1) 0

0

/

(2) 0

=5

0

1.8

(1)

(2)

/

(1) 0

0

/

(2) 0

(1)

c =0.20626

c =0.4124

1.3

1.6

1.2

Exact (FFT) LCC (FFT) LCC (HS-)

1.4 Exact (FFT) LCC (FFT) LCC (HS-)

1.1

1.2

m 1.0 0.0

=5

0.2

0.4

0.6

0.8

m 1.0 1.0 0.0 (1)

0.2 (2)

Figure 4. Fibers stronger than the matrix: σ0 /σ0 function of the rate sensitivity exponent m.

0.4

0.6

0.8

1.0

= 5. Eﬀective ﬂow stress as a

− The plots corresponding to the two versions of the LCC, computational (FFT) or analytical (HS) are almost parallel: the initial shift in the linear case (m=1) remains approximately the same throughout the range of values of m which have been investigated. This observation holds independently of the contrast. − When the ﬁbers are stronger than the matrix, the actual eﬀective ﬂow stress is seen to be an increasing function of m. This qualitative trend is correctly rendered by the LCC models (computational and analytical) with a quantitative shift. − When the ﬁbers are weaker than the matrix, the actual eﬀective ﬂow stress is again seen to be an increasing function of m. The two LCC models predict a decrease of the ﬂow stress with m and therefore fail to capture correctly the actual dependence of σ 20 on m. 3.4. STRAIN AND STRESS AVERAGES OVER INDIVIDUAL PHASES. (r)

(r)

The second invariant of the average strain εeq and of the average stress σ eq over the phases can be compared for the diﬀerent models. The ensemble averages σ (r) and ε(r) of the stress and strain ﬁelds obtained numerically for the diﬀerent conﬁgurations are evaluated ﬁrst. Then the von-Mises strain or stress for these ensemble averages are taken.

209

HOMOGENIZATION FOR NONLINEAR COMPOSITES 0.9 0

/

0.7

(2)

(1) 0

0

/

(2) 0

= 0.2

0

(1)

/

(2)

(1) 0

0

/

(2) 0

= 0.2

(1)

c =0.20626

c =0.4124

0.8

0.6

Exact (FFT) LCC (FFT) LCC (HS+)

0.7

Exact (FFT) LCC (FFT) LCC (HS+)

0.5

m 0.6 0.0

0.2

0.4

0.6

0.8

m 0.4 1.0 0.0 (1)

0.2 (2)

Figure 5. Fibers weaker than the matrix: σ0 /σ0 function of the rate sensitivity exponent m.

0.4

0.6

0.8

1.0

= 0.2. Eﬀective ﬂow stress as a

3.4.1. Fibers stronger than the matrix When the ﬁbers are stronger than the matrix, all predictions for the average strain in the individual phases give very similar results, as can be seen in ﬁgure 6. This can be explained by the fact that the ﬁbers being stiﬀ, their average strain is small and goes to 0 as m goes to 0. This corresponds to the observation (Suquet [22]) that, in the rigid-plastic limit, the volume of matrix located around the inclusions ﬂows and does not transmit enough stress to the inclusions to cause their yielding. The inclusions behave, in the limit as m goes to 0, as rigid inclusions. Consequently the average strain in the matrix is governed by the average condition: 1 ε(2) (2) ε. c The agreement for the ﬁrst moment of the stress is not as good as for the strain. The analytical predictions LCC(HS-) for the secant method show the same trends as the computational results LCC(FFT) and the two curves are almost parallel. The results for the actual nonlinear composite show that the stress carried by the matrix increases slowly as m goes to 0, whereas the stress carried by the ﬁbers decreases rapidly as m goes to 0, resulting in the overall drop of the eﬀective ﬂow stress seen in ﬁgure 4. The modiﬁed secant method, either in its exact computational version LCC(FFT) or in its approximate analytical version LCC(HS-), fails to predict correctly these variations of the stress with m.

210 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0

H. MOULINEC AND P. SUQUET

(r)

/

eq

eq

0

0

phase 2

Exact (FFT) LCC (FFT) LCC (HS-) phase 1

m 0.2

0.4

0.6

Figure 7.

1.0

(r)

/

(1)

0

phase 1

phase 2 Exact (FFT) LCC (FFT) LCC (HS-)

0.2

0.4

0.6

0

phase 2 Exact (FFT) LCC (FFT) LCC (HS-) phase 1

m 0.2 (2)

2.2 2.0 1.8 0 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 m 0.0 0.8 1.0 0.0 (1)

(1)

c =0.4124 (1) (2) / =5

eq

0

0.4

0.6

0.8

1.0

= 5. Average strain in the phases.

(r)

c =0.20626 (1) (2) / =5

eq

/

eq

Fibers stronger than the matrix: σ0 /σ0

(r) eq

0.8

2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 (1)

Figure 6.

2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0

(1)

c =0.20626 (1) (2) / =5

eq

/

(1)

c =0.4124 (1) (2) / =5

eq

0

0

phase 1 phase 2

Exact (FFT) LCC (FFT) LCC (HS-)

0.2 (2)

Fibers stronger than the matrix: σ0 /σ0

0.4

0.6

m 0.8

1.0

= 5. Average stress in the phases.

3.4.2. Fibers weaker than the matrix When the ﬁbers are weaker than the matrix, the above observations apply, provided that the strain and the stress are interchanged: when m is close to 0, the average stress in the matrix is well aproximated by σ (2) and all models reproduce this result.

1 c(2)

σ,

211

HOMOGENIZATION FOR NONLINEAR COMPOSITES 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0

2.2 2.0 eq eq 1.8 0 0 1.6 1.4 phase 2 1.2 Exact (FFT) 1.0 LCC (FFT) 0.8 LCC (HS+) 0.6 phase 1 0.4 0.2 m 0.0 0.2 0.4 0.6 0.8 1.0 0.0 (r)

(1)

/

c =0.20626 (1) (2) / = 0.2

(1)

Figure 8.

(2)

Fibers weaker than the matrix: σ0 /σ0

(r)

(1)

/

eq

c =0.4124 (1) (2) / = 0.2

eq

0

0

phase 2 Exact (FFT) LCC (FFT) LCC (HS+) phase 1

m 0.2

0.4

0.6

0.8

1.0

= 0.2. Average stress in the phases.

The predictions of the secant methods LCC (computational and analytical) for the average strains do not show the same trends as the computations for the actual nonlinear composites. In particular the secant method fails to predict the increase of the strain in the ﬁbers when m goes to 0, which corresponds physically to the ﬁbers ﬂowing more easily.

2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0

(r) eq

Figure 9.

2.0

/

(1)

c =0.20626 (1) (2) / = 0.2

eq

0

0

(r)

1.8

eq

/

(1)

c =0.4124 (1) (2) / = 0.2

eq

1.6

0

0

1.4 1.2 phase 1

phase 1

1.0 0.8 0.6

phase 2 Exact (FFT) LCC (FFT) LCC (HS+)

0.2

0.4

0.6

phase 2

0.4

m 0.8

Exact (FFT) LCC (FFT) LCC (HS+)

0.2 0.0 1.0 0.0 (1)

(2)

Fibers weaker than the matrix: σ0 /σ0

0.2

0.4

0.6

m 0.8

1.0

= 0.2. Average strain in the phases

212

H. MOULINEC AND P. SUQUET

3.5. FLUCTUATIONS OF THE STRAIN AND STRESS FIELDS

3.5.1. About strain heterogeneity As is well known, the assumption on which the Voigt bound is built, namely the uniformity of the strain throughout the r.v.e., is a very crude approximation. Even in linear elasticity it is readily seen that the actual strain ﬁeld in a composite is heterogeneous (i.e. nonuniform). This heterogeneity can be schematically classiﬁed in two categories. The interphase heterogeneity corresponds to the fact that diﬀerent phases undergo diﬀerent average strains. The intraphase heterogeneity refers to the fact that, even within a single phase, the strain ﬁeld is far from being uniform. The importance of the ﬁrst type of strain heterogeneity has long been recognized, even in linear elasticity. Eshelby [4] has shown for instance that the exact strain ﬁeld in an ellipsoidal inclusion embedded in an inﬁnite linear elastic matrix subject to a remote average strain is uniform, but diﬀerent from the remote average strain. This result has been used in several diﬀerent ways to estimate the average strains in individual phases, under approximations on the shape of the phases, on their distribution and on the nature of their interactions. These approximations (which in certain cases yield rigorous bounds) have generated a number of schemes for linear elastic composites. The second type of heterogeneity (intraphase) has received less attention than the ﬁrst one and this is easily explained by considering the case of linear elasticity. The eﬀective stiﬀness of a linear elastic composite is given as : N 2 = L(x) : A(x) = c(r) L(r) : A(r) , (22) L r=1

where A(x) denotes the strain localization tensor which gives the local strain ﬁeld ε(x) = A(x) : ε in terms of the average strain. A(r) denotes the average of A(x) over phase r and the last equality in (22) shows that, in linear elasticity, it is suﬃcient to determine the average localization strain tensor A(r) , or in other words the average strain in each phase, to determine 2 of the composite. the eﬀective stiﬀness L The details of the strain ﬁeld (or stress ﬁeld), which do not matter in linear problems, become important when one is interested in nonlinear problems triggered by the local value of the strain (or stress) ﬁelds and not only by their average value (extreme examples of strong nonlinearities are fracture or damage). For this reason, nonlinearity introduces a deﬁnite complication in the analysis of composite materials. It becomes necessary to understand more precisely the structure of the nonuniformity of the ﬁelds within each in-

HOMOGENIZATION FOR NONLINEAR COMPOSITES

213

dividual phase. A measure of the deviation from uniformity is provided by the quadratic ﬂuctuations of the ﬁelds. Following Ponte Casta˜ neda [18], these quadratic ﬂuctuations can be measured by means of the fourth-order (r) (r) tensors C ε and C σ : 3

4 (r) (r) = ε − ε ε , ε(r) = ε r , ⊗ ε − C (r) ε r

C (r) σ =

3

σ − σ (r) ⊗ σ − σ(r)

4 r

.

There is a straightforward relation between the second moments, the ﬁrst moments and the ﬂuctuations of a ﬁeld. For instance : (r) C (r) ⊗ ε(r) . ε = ε ⊗ ε r − ε

3.5.2. Isotropic measure of the ﬂuctuations An isotropic measure of the amplitude of the ﬂuctuations can be obtained by considering their quadratic norms: 6 δε(r)

=

and

6 δσ(r)

(r)

(r)

=

2 (r) (r) (εij − εij )(εij − εij 3

71/2

3 (r) (r) (σij − σ ij )(σij − σ ij 2 (r)

, 71/2 .

(r)

δε and δσ are invariants of C ε and C σ , as can be seen by noting that:

2 2 (r) 2 (r) 2 δε(r) = K :: C (r) ε = εeq − εeq , 3

2 3 (r) 2 (r) 2 δσ(r) = K :: C (r) σ = σ eq − σ eq . 2 In all cases (ﬁbers stronger or weaker than the matrix) the ﬂuctuation of the strain ﬁeld in the matrix blows up as m goes to 0. This corresponds physically to the concentration of the strain in bands, the width of which becomes smaller when m goes to 0. These strain concentration phenomena (or even strain localization) is the rule, rather than the exception, in composite systems and has been reported in previous work ([12], [19], [24]). The situation in phase 1 is diﬀerent when the ﬁbers are stronger or weaker than the matrix. When the ﬁbers are stronger, the bands where the strain tends to concentrate avoid the ﬁbers. The strain in the ﬁbers is therefore rather uniform (and small). This can be seen in ﬁgure 10. The secant method LCC(FFT) predicts well the strain ﬂuctuation in this case.

214

H. MOULINEC AND P. SUQUET

In contrast, when the ﬁbers are weaker than the matrix, the localization bands are “attracted” by the ﬁbers and follow a path which runs through several ﬁbers. This explains why the strain ﬂuctuations in phase 1 blow-up as m tends to 0, as can be seen in ﬁgure 12. The secant method LCC(FFT) is unable to predict these large ﬂuctuations. The blowing-up of the strain ﬂuctuations can be viewed from a more mathematical perspective by considering the functional spaces in which the variational problem (4) is well-posed for power-law phases. When 0 < m < +∞ the space of vector ﬁelds with ﬁnite energy reads : {u ∈ L1 (V )N ,

εij (u) ∈ Lm+1 (V ),

div(u) = 0},

which, by Korn’s inequality in Lm+1 (V ) (Geymonat and Suquet [5]) coincides with the space : {u ∈ W 1,m+1 (V )N ,

div(u) = 0}.

In particular, any vector ﬁeld with ﬁnite energy has well-deﬁned traces on every smooth surface inside V and is therefore continuous (in the sense of traces) across such a surface. When m = 0, the space of vector ﬁelds with ﬁnite energy is (see Suquet [21] or Temam [27] for more details) : BDinc (V ) = {u ∈ L1 (V )N ,

εij (u) ∈ M 1 (V ),

div(u) = 0 },

where M 1 (V ) stands for the space of bounded measures on V . The possibility for ε(u) to be a measure corresponds to the fact that on every smooth surface, u has 2 traces (inner and outer) which may be diﬀerent. In other words, in the limit where m goes to 0, the L1 norm of the strain ﬁeld remains bounded but its L2 norm may blow-up. This L2 norm corresponds to the second moment of the strain ﬁeld. On the other hand, in the LCC (either computational or analytical) the energy in each phase is 0 given 1 by the right-hand side of (15) which, in order to be ﬁnite requires ε2eq r to be ﬁnite. Therefore the well-posedness of the variational problem for the LCC requires the trial vector ﬁelds to have a square integrable strain ﬁeld (and therefore a ﬁnite second moment or ﬂuctuation tensor). It is therefore not surprising to observe that the strain ﬂuctuations in the LCC shown in ﬁgures 10 and 12 have a ﬁnite limit as m goes to 0. The LCC models (computational or analytical) are therefore unable to capture the blowing-up of the strain ﬁeld in the actual nonlinear composite. As for the stress ﬁeld, its ﬂuctuations in the actual nonlinear composite remain ﬁnite as m goes to 0. This can be explained by the fact that plasticity imposes a strong constraint on the stress ﬁeld (the von Mises stress has

215

HOMOGENIZATION FOR NONLINEAR COMPOSITES

to remain bounded). Surprisingly, when the ﬁbers are stronger than the matrix, the stress ﬂuctuations in the LCC(FFT) appear to blow-up in the inclusion phase. We have no physical explanation for this phenomenon, which could be an artefact of the secant method. 1.0

2.5

(r)

/

0.8

(r)

(1)

c =0.20626 (1) (2) / =5

eq

0

0

0.6

/

2.0

(1)

c =0.4124 (1) (2) / =5

eq

0

0

1.5 phase 2

0.4

phase 2

1.0

Exact (FFT) LCC (FFT) LCC (HS-)

0.2

Exact (FFT) LCC (FFT) LCC (HS-)

0.5 phase 1

0.0

phase 1

0.0

m -0.2 0.0

0.2

0.4

0.6

0.8

m -0.5 1.0 0.0 (1)

0.2 (2)

Figure 10. Fibers stronger than the matrix: σ0 /σ0 strain ﬂuctuations.

2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 0.0

(r)

/

2.2 2.0 1.8 0 0 1.6 Exact (FFT) 1.4 LCC (FFT) 1.2 LCC (HS-) 1.0 0.8 0.6 0.4 0.2 m 0.0 -0.2 0.6 0.8 1.0 0.0 (1)

c =0.20626 (1) (2) / =5

eq

phase 1

phase 2

0.2

0.4

(1)

0.4

0.8

(1)

c =0.4124 (1) (2) / =5

eq

0

0

Exact (FFT) LCC (FFT) LCC (HS-)

phase 1

m

phase 2

0.2 (2)

Figure 11. Fibers stronger than the matrix: σ0 /σ0 stress ﬂuctuations.

1.0

= 5. Isotropic measure of the

(r)

/

0.6

0.4

0.6

0.8

1.0

= 5. Isotropic measure of the

216

H. MOULINEC AND P. SUQUET

1.6

1.6

(r)

/

1.4

(r)

(1)

c =0.20626 (1) (2) / = 0.2

eq

0

1.2

0

0.8 0.6

(1)

c =0.4124 (1) (2) / = 0.2

eq

0

1.2

Exact (FFT) LCC (FFT) LCC (HS+)

1.0

/

1.4

Exact (FFT) LCC (FFT) LCC (HS+)

1.0 0.8 0.6

phase 2

0

phase 2

phase 1

phase 1

0.4

0.4

0.2

0.2

0.0 -0.2 0.0

m 0.2

0.4

0.6

0.0

m

-0.2 1.0 0.0

0.8

(1)

0.2 (2)

Figure 12. Fibers weaker than the matrix: σ0 /σ0 strain ﬂuctuations.

1.0

(r)

/

0.8

(1)

c =0.20626 (1) (2) / = 0.2

eq

0

0

Exact (FFT) LCC (FFT) LCC(HS+)

0.6 0.4

0.8

1.0

/

(1)

c =0.4124 (1) (2) / = 0.2

eq

1.2

0

1.0

0

Exact (FFT) LCC (FFT) LCC(HS+)

0.8 phase 2

0.4

0.2

0.2

phase 1

0.0

m -0.2 0.0

(r)

1.4

0.6

= 0.2. Isotropic measure of the

0.6

phase 2

0.4

0.2

0.4

0.6

0.8

phase 1

0.0

m

-0.2 1.0 0.0 (1)

0.2 (2)

Figure 13. Fibers weaker than the matrix: σ0 /σ0 stress ﬂuctuations.

0.4

0.6

0.8

1.0

= 0.2. Isotropic measure of the

3.5.3. Anisotropic measure of the ﬂuctuations Ponte Casta˜ neda [18], has recently proposed an homogenization scheme which is exact to second-order in the contrast of the phases and where the anisotropy of the ﬂuctuations of the ﬁelds in each phase plays an important role. More speciﬁcally, following this author, directional fourth-order

217

HOMOGENIZATION FOR NONLINEAR COMPOSITES

tensors are formed from the average strains in each phase: E (r) ε =

2 ε(r) ε(r) ⊗ (r) , 3 ε(r) εeq eq

(r) F (r) ε = K − Eε .

The projections of the ﬂuctuations on these tensors deﬁne the “parallel” and “orthogonal” components of the ﬂuctuations * * 2 (r) 2 (r) (r) (r) (r) (r) E ε :: C ε , δε⊥ = F ε :: C ε , δε|| = 3 3 (r)

(r)

The strain ﬂuctuations in phase r will be said to be anisotropic if δε⊥ = δε|| . Note that: * δε(r) =

(r)

(r)

(δε|| )2 + (δε⊥ )2 .

Similarly, anisotropic measures of the stress ﬂuctuations can be deﬁned as: * * 3 (r) 3 (r) (r) (r) (r) (r) E σ :: C σ , δσ⊥ = F σ :: C σ , δσ|| = 2 2 where E (r) σ =

3 σ(r) σ (r) ⊗ (r) , 2 σ (r) σ eq eq

(r) F (r) σ = K − Eσ .

The anisotropic ﬂuctuations of the strain and stress ﬁelds are shown in ﬁgures 14 to 17. Only computational results are given since the evaluation of the anisotropic ﬂuctuations with the Hashin-Shtrikman formalism, although feasible, is rather complicated in practice (anisotropic moduli would have to be taken for the phases) and has not been implemented in this study. A general observation for both cases (ﬁbers stronger or weaker than the matrix) is that there is no signiﬁcant anisotropy of the ﬂuctuations in the LCC, whereas a signiﬁcant anisotropy is observed in the actual nonlinear composite. A possible interpretation is that the anisotropy of the ﬁeld ﬂuctations is related to the anisotropy of the LCC. In the secant method the LCC is taken to be composed of isotropic phases and this is certainly a limitation of the method. Other methods, such as the tangent, aﬃne or SOE method, where the elastic moduli of the individual phases are anisotropic, could be more promissing in that respect. Fibers stronger than the matrix When the ﬁbers are stronger than the matrix, the ﬂuctuations of the strain ﬁeld in the ﬁbers are small. Their anisotropy, which exists, should not play a crucial role.

218 0.08

H. MOULINEC AND P. SUQUET

(1)

/

c1=0.20626 (1) (2) / =5 0 0 Fibers

eq

0.06

Exact (FFT) LCC (FFT)

0.15

(1)

/

c1=0.4194 (1) (2) / =5 0 0 Fibers

eq Exact (FFT) LCC (FFT)

0.1

0.04 (1)

(1)

||

0.02

0.05

(1)

||

(1)

m 0.0 0.0

0.2

0.4

0.6

0.8

m 1.0 0.00.0

0.2

0.4

(a) 1.0

(2)

/

eq

0.8 (2) ||

0.4 0.2 Exact (FFT) LCC (FFT)

0.0 0.0

0.2

2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 (2) 0.4 m 0.2 0.0 0.8 1.0 0.0

c1=0.20626 (1) (2) / =5 0 0 Matrix

0.6

0.4

0.6

0.8

1.0

(b)

0.6

(2)

/

c1=0.4194 (1) (2) / =5 0 0 Matrix

eq

(2) ||

(2)

Exact (FFT) LCC (FFT)

0.2

0.4

(c)

0.6

m 0.8

1.0

(d) (1)

(2)

Figure 14. Fibers stronger than the matrix: σ0 /σ0 = 5. (a) and (b) Strain ﬂuctuations in the ﬁbers. (c) and (d) Strain ﬂuctuations in the matrix.

The anisotropy of the strain ﬂuctuations in the matrix is certainly more (r) signiﬁcant. The strain ﬂuctuation δε⊥ in the directions orthogonal to the (r)

overall strain is limited whereas the ﬂuctuation δε|| in the direction of the overall strain is large and responsible for the blowing-up of the norm of the ﬂuctuations analyzed in the previous paragraph. These observations are consistent with the predictions of Idiart and Ponte Casta˜ neda [8]. The stress ﬂuctuations remain limited in the actual nonlinear composite

219

HOMOGENIZATION FOR NONLINEAR COMPOSITES 1.8

1.8

(1)

/

1.6

c1=0.20626 (1) (2) / =5 0 0 Fibers

eq

1.4

Exact (FFT) LCC (FFT)

1.2 1.0

(1)

/

1.6

c1=0.4124 (1) (2) / =5 0 0 Fibers

eq

1.4

Exact (FFT) LCC (FFT)

1.2 1.0

(1)

0.8

(1)

0.8

0.6

0.6

(1) ||

0.4

m 0.2

0.4

||

0.4

0.2 0.0 0.0

(1)

0.6

0.8

0.2

m

0.0 1.0 0.0

0.2

(a) 0.4

0.6

0.8

1.0

(b) 0.6

(2)

/

0.4

(2)

/

c1=0.20626 (1) (2) / =5 0 0 Matrix

eq

0.3

c1=0.4124 (1) (2) / =5 0 0 Matrix

eq

0.4

(2) (2)

(2)

||

0.2

||

(2)

0.2 0.1 Exact (FFT) LCC (FFT)

0.0 0.0

0.2

0.4

0.6

Exact (FFT) LCC (FFT)

m 0.8

1.0

0.0 0.0

0.2

(c)

0.4

0.6

m 0.8

1.0

(d) (1)

(2)

Figure 15. Fibers stronger than the matrix: σ0 /σ0 = 5. (a) and (b) Stress ﬂuctuations in the ﬁbers. (c) and (d) Stress ﬂuctuations in the matrix.

and they are smaller than the strain ﬂuctuations. We have no interpretation of the large stress ﬂuctuations observed in the LCC (which could be just an artefact of the method). Fibers weaker than the matrix When the ﬁbers are weaker than the matrix, the ﬂuctuations of the strain in both phases are signiﬁcant. Again this is due to the fact that the bands where the strain concentrates run through the weaker phase. The orthogonal ﬂuctations of the stress ﬁeld are higher

220 1.4 1.2

H. MOULINEC AND P. SUQUET 1.4

(1)

/

c1=0.20626 (1) (2) / = 0.2 0 0 Fibers

eq

1.0

c1=0.4124 (1) (2) / = 0.2 0 0 Fibers

eq

1.0

Exact (FFT) LCC (FFT)

0.8

(1)

0.6

/

1.2

Exact (FFT) LCC (FFT)

0.8

(1)

(1)

0.6

(1)

(1)

||

||

0.4

0.4

0.2 0.0 0.0

0.2

m 0.2

0.4

0.6

m

0.0 1.0 0.0

0.8

0.2

0.4

(a) 1.0

1.0

(2)

/

eq

(2)

/

c1=0.20626 (1) (2) / = 0.2 0 0 Matrix

0.8

0.8

1.0

c1=0.4124 (1) (2) / = 0.2 0 0 Matrix

eq

0.8

0.6

0.6 (2)

0.4

(2)

(2)

(2)

0.4

||

0.2

||

0.2 Exact (FFT) LCC (FFT)

0.0 0.0

0.6

(b)

0.2

0.4

0.6

Exact (FFT) LCC (FFT)

m 0.8

1.0

0.0 0.0

(c)

0.2

0.4

0.6

m 0.8

1.0

(d) (1)

(2)

Figure 16. Fibers weaker than the matrix: σ0 /σ0 = 0.2. (a) and (b) Strain ﬂuctuations in the ﬁbers. (c) and (d) Strain ﬂuctuations in the matrix.

than the parallel ﬂuctuations. Both types of ﬂuctuations seem to increase rapidly as m approaches 0, but not as rapidly as the strain ﬂuctuations.

221

HOMOGENIZATION FOR NONLINEAR COMPOSITES 0.1

0.2 c1=0.20626 (1) (2) / = 0.2 0 0 Fibers 0.15

(1)

/

eq

0.08 0.06

(1)

/

c1=0.4124 (1) (2) / = 0.2 0 0 Fibers

eq

(1) (1)

(1)

||

||

0.1 0.04 0.02 Exact (FFT) LCC (FFT)

0.0 0.0

(1)

0.05

0.2

0.4

Exact (FFT) LCC (FFT)

m

0.6

0.0 1.0 0.0

0.8

0.2

(a) 0.6

0.4

0.6

1.0 eq

(2)

(2)

/

c1=0.20626 (1) (2) / = 0.2 0 0 Matrix

0.8

(2)

(2)

0.6

||

0.4

(2)

0.2 0.2

Exact (FFT) LCC (FFT)

0.2

0.4

1.0

c1=0.4124 (1) (2) / = 0.2 0 0 Matrix

eq

||

0.0 0.0

0.8

(b)

(2)

/

0.4

m

0.6

Exact (FFT) LCC (FFT)

m 0.8

1.0

0.0 0.0

(c)

0.2

0.4

0.6

m 0.8

1.0

(d) (1)

(2)

Figure 17. Fibers weaker than the matrix: σ0 /σ0 = 0.2. (a) and (b) Stress ﬂuctuations in the ﬁbers. (c) and (d) Stress ﬂuctuations in the matrix.

4. Conclusions 1. The modiﬁed secant method (or variational bounding technique), based on a linear comparison composite, has the rigorous status of an upper bound when no further assumption is introduced. It can lead to a signiﬁcant overestimation of the eﬀective properties for strong nonlinearities. This overestimation can be hidden in certain circumstances (composites reinforced by strong ﬁbers) when a lower bound (or estimate) is used for the eﬀective properties of the linear comparison

222

H. MOULINEC AND P. SUQUET

composite. 2. The intraphase strain ﬂuctuations and, to a lesser extent, the stress ﬂuctuations, are signiﬁcant and certainly essential in understanding the response of nonlinear composites. They may even become inﬁnite in the rigid-plastic limit. They are strongly anisotropic with larger ﬂuctuations in the direction of the overall strain, as anticipated by [18]. The modiﬁed secant method cannot reproduce the amplitude and the anisotropy of these ﬂuctuations. Other methods making use of an anisotropic linear comparison composite have to be explored in more details. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Bardella, L. (2003). An extension of the secant method for the homogenization of the nonlinear behavior of composite materials. Int. J. Engng Sc., 41:741–768. Berveiller, M. and Zaoui, A. (1979). An extension of the self-consistent scheme to plastically-ﬂowing polycrystals. J. Mech. Phys. Solids, 26:325–344. Bouchitte, G. and Suquet, P. (1991). Homogenization, Plasticity and Yield design. In Maso, G. D. and Dell’Antonio, G., editors, Composite Media and Homogenization Theory, pages 107–133. Birkha¨ user, Boston. Eshelby, J. (1957). The determination of the elastic ﬁeld of an ellipsoidal inclusion and related problems. Proc. R. Soc. London A, 241:376–396. Geymonat, G. and Suquet, P. (1986). Functional spaces for Norton-Hoﬀ materials. Math. Meth. Appl. Sc., 8 :185–211. Gilormini, P., Nebozhyn, M. and Ponte Casta˜ neda, P. (2001). Accurate estimates for the Creep Behavior of Hexagonal Polycrystals. Acta Materialia, 49:329–337. Hu, G. (1996). A method of plasticity for general aligned spheroidal void or ﬁberreinforced composites. Int. J. Plasticity, 12:439–449. Idiart, M. and Ponte Casta˜ neda, P. (2004). Field ﬂuctuations and macroscopic properties for nonlinear composites. Submitted. Kanit, T., Forest, S., Galliet, I., Mounoury, V. and Jeulin, D. (2003). Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Structures, 40:3647–3679. Marcellini, P. (1978). Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl., 117:139–152. Michel, J., Moulinec, H. and Suquet, P. (1999). Eﬀective properties of composite materials with periodic microstructure: a computational approach. Comp. Meth. Appl. Mech. Engng., 172:109–143. Moulinec, H. and Suquet, P. (1998). A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comp. Meth. Appl. Mech. Engng., 157:69–94. Moulinec, H. and Suquet, P. (2003). Intraphase strain heterogeneity in nonlinear composites: a computational approach. Eur. J. Mech.: A/ Solids, 22:751–770. Muller, S. (1987). Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rat. Mech. Anal., 99:189–212. Olson, T. (1994). Improvements on Taylor’s upper bound for rigid-plastic bodies. Mater. Sc. Engng A, 175:15–20. Ponte Casta˜ neda, P. (1991). The eﬀective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids, 39:45–71. Ponte Casta˜ neda, P. (1992). New variational principles in plasticity and their application to composite materials. J. Mech. Phys. Solids, 40:1757–1788.

HOMOGENIZATION FOR NONLINEAR COMPOSITES 18. 19. 20. 21. 22. 23. 24. 25.

26. 27. 28.

223

Ponte Casta˜ neda, P. (2002). Second-order homogenization estimates for nonlinear composites incorporating ﬁeld ﬂuctuations. I - Theory. J. Mech. Phys. Solids, 50:737–757. Ponte Casta˜ neda, P. and Suquet, P. (1998). Nonlinear composites. Advances in Applied Mechanics, volume 34, pages 171–302. Academic Press, New York. Segurado, J., LLorca, J. and Gon´ alez, C. (2002). On the accuracy of mean-ﬁeld approaches to simulate the plastic deformation of composites. Scripta Materialia, 46:525–529. Suquet, P. (1978). Sur un nouveau cadre fonctionnel pour les ´equations de la plasticit´e. C.R. Acad. Sc. Paris, A, 286:1129–1132. Suquet, P. (1993). Overall potentials and extremal surfaces of power law or ideally plastic materials. J. Mech. Phys. Solids, 41:981–1002. Suquet, P. (1995). Overall properties of nonlinear composites : a modiﬁed secant moduli theory and its link with Ponte Casta˜ neda’s nonlinear variational procedure. C.R. Acad. Sc. Paris, 320, S´erie IIb:563–571. Suquet, P. (1997). Eﬀective properties of nonlinear composites. In Suquet, P., editor, Continuum Micromechanics, volume 377 of CISM Lecture Notes, pages 197–264. Springer Verlag, New York. Suquet, P. and Moulinec, H. (1997). Numerical simulation of the eﬀective properties of a class of cell materials. In Golden, K., Grimmett, G., James, R., Milton, G., and Sen, P., editors, Mathematics of multiscale materials, volume 99 of IMA Lecture Notes, pages 277–287. Springer-Verlag, New-York. Talbot, D. and Willis, J. (1994). Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry. Proc. R. Soc. Lond. A, 447:365–384. Temam, R. (1987). Mathematical problems in Plasticity. Gauthier-Villars, Paris. Willis, J. (1991). On methods for bounding the overall properties of nonlinear composites. J. Mech. Phys. Solids, 39:73–86.

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EXISTENCE AND HOMOGENIZATION FOR THE PROBLEM −div a(x, Du) = f WHEN a(x, ξ) IS A MAXIMAL MONOTONE GRAPH IN ξ FOR EVERY x

F. MURAT

Laboratoire Jacques-Louis Lions Universit´e Pierre et Marie Curie (Paris VI) Boˆıte courrier 187 75252 Paris cedex 05 France In this lecture I will report on recent joint work [2], [3] with Gilles Francfort and Luc Tartar on the existence of a solution for some nonlinear problem and on its homogenization. We consider the problem of ﬁnding u such that

u ∈ W01,p(Ω), −div a(x, Du) = f

in D (Ω),

(1)

for a given f ∈ W −1,p (Ω). Here Ω is a bounded open set of RN , p is a real number with 1 < p < +∞, and p = p/(p − 1). The classical setting is the case where a : (x, ξ) ∈ Ω × RN → a(x, ξ) ∈ RN is a Carath´eodory function (i.e. a single-valued function, continuous in ξ for almost every x ∈ Ω, and measurable in x for every ξ ∈ RN ), which is monotone, i.e. satisﬁes, for almost every x ∈ Ω and for every ξ1 ∈ RN and ξ2 ∈ RN (a(x, ξ1 ) − a(x, ξ2 ))(ξ1 − ξ2 ) ≥ 0, (2) and which further satisﬁes for almost every x ∈ Ω and for every ξ ∈ RN  p   a(x, ξ)ξ ≥ α|ξ| − |a(x)|,

a(x, ξ)ξ ≥ β|a(x, ξ)|p − |b(x)|,

  a(x, 0) = 0,

225 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 225–228. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

(3)

226

F. MURAT

for some α > 0, β > 0, a ∈ L1 (Ω) and b ∈ L1 (Ω) (the second assertion of (3) is a diﬀerent way of stating the classical growth condition |a(x, ξ)| ≤ γ|ξ|p−1 + |h(x)| for some γ > 0 and some h ∈ Lp (Ω)). In such a classical setting it is well known that there exists a solution to (1). In our work we investigate the case where, for a given x ∈ Ω, ξ → a(x, ξ) is no more a single-valued monotone continuous function from RN into RN , but actually a multi-valued maximal monotone graph of RN × RN . In such a case, the crucial issues are, ﬁrstly, the deﬁnition of an adequate setting, especially as far as the proper measurability assumptions on the graph that replaces a(x, ξ) are concerned, secondly the choice of an appropriate approximation process, so as to prove an existence result, and ﬁnally the statement and proof of results on maximal monotone graphs that allow us to perform the homogenization of the problem. The only paper on these topics that we are aware of is that of V. Chiad` o Piat, G. Dal Maso & A. Defranceschi [1], in which delicate measurability assumptions are made in the deﬁnition of the graph and delicate measurability selection theorems are used in the proofs. In our paper [2] we provide a deﬁnition which can be proved to be equivalent to theirs, but which, in our opinion, provides a simpler framework. Furthermore we only use in the proofs classical theorems of single-valued analysis. As far as existence is concerned, we prove the existence of a function u and of a vector ﬁeld d such that  1,p   u ∈ W0 (Ω), d ∈ Lp (Ω)N ,

in D (Ω),

e = Du, −div d = f

  (e, d) ∈ A,

(4)

when the graph A ⊂ Lp(Ω)N × Lp (Ω)N is such that for every (e, d) ∈ A one has for almost every x ∈ Ω  p   d(x)e(x) ≥ α|e(x)| − |a(x)|,

d(x)e(x) ≥ β|d(x)|p − |b(x)|,

  (0, 0) ∈ A,

(5)

for some α > 0, β > 0, a ∈ L1 (Ω) and b ∈ L1 (Ω), and when the graph A further satisﬁes one of the the following two equivalent conditions. – First condition. The graph A is deﬁned by (e, d) ∈ A

⇐⇒

d(x) − e(x) = φ(x, d(x) + e(x)), a.e. x ∈ Ω,

(6)

227

EXISTENCE AND HOMOGENIZATION . . .

where φ : (x, λ) ∈ Ω × RN → φ(x, λ) ∈ RN is a given (single-valued) Carath´eodory function which is deﬁned on all of Ω × RN and satisﬁes

x → φ(x, λ) is measurable on Ω for every λ ∈ RN , |φ(x, λ1 ) − φ( x, λ2 )| ≤ |λ1 − λ2 |, a.e. x ∈ Ω, ∀λ1 , λ2 ∈ RN .

(7)

– Second condition. The graph A ⊂ Lp(Ω)N × Lp (Ω)N is pointwise monotone, i.e. satisﬁes for every (e1 , d1 ) ∈ A and (e2 , d2 ) ∈ A (d1 (x) − d2 (x))(e1 (x) − e2 (x)) ≥ 0,

a.e. x ∈ Ω,

(8)

and is such that for every g ∈ Lp(Ω)N and for every t > 0, there exists a (unique) (e, d) such that

(e, d) ∈ A, e(x) + tj(d(x)) = g(x),

(9)

a.e. x ∈ Ω,

where j : RN → RN is the function deﬁned by j(λ) = |λ|p −2 λ. Assuming that hypothesis (5) holds, we prove that the two conditions above are equivalent in the following sense. If the graph A is deﬁned by (6) for some function φ which is deﬁned on all of Ω × RN and which satisﬁes (7), then A satisﬁes (8) and (9). Conversely, if the graph A satisﬁes (8) and (9), then there exists a fonction φ which is deﬁned on all of Ω × RN and which satisﬁes (7), such that A is deﬁned by (6). These two equivalent conditions provide a convenient deﬁnition of a “multi-valued maximal monotone graph of RN × RN depending on x”. The justiﬁcation of this assertion is twofold. On the one hand, if A satisﬁes (8) and (9), it is easy to prove that A is a maximal monotone graph of Lp(Ω)N × Lp (Ω)N . The converse is also true, but this is a more involved result in the theory of maximal monotone graphs. On the other hand, if φ is independent of x, deﬁned on all of RN and satisﬁes (7), it is easy to prove that A deﬁned by (6) is a maximal monotone graph of RN × RN . The converse is also true, but the proof uses Kirzbraun’s theorem. If the graph A satisﬁes (5) and if one of the two equivalent conditions given above holds true, we prove in [2] the existence of a solution to (4). We give two diﬀerent proofs of this existence result. Each proof uses a diﬀerent approximation procedure, which relies on one of the two conditions above. As regards homogenization, we will prove in a forthcoming paper [3] the following compactness result with respect to the H−convergence (or in other terms a homogenization result) concerning this class of graphs. From

228

F. MURAT

every sequence ε of graphs Aε which satisfy (5) uniformly (i.e. for the same α > 0, β > 0, a ∈ L1 (Ω) and b ∈ L1 (Ω)), and which are such that one of the two equivalent conditions given above is satisﬁed, one can extract a subsequence, still denoted by ε, and there exists a graph A0 in the same class, such that for every f ∈ W −1,p (Ω), any accumulation point (u, d) (in the weak topology of W01,p(Ω) × Lp (Ω)N ) of the solutions (uε, dε) to  1,p   uε ∈ W0 (Ω), dε ∈ Lp (Ω)N ,

e = Du , −div dε = f

ε ε   (e , d ) ∈ A , ε ε ε

in D (Ω),

(10ε)

is a solution (u0 , d0 ) to (100 ) (observe that uε is bounded in W01,p(Ω) and that dε is bounded in Lp (Ω)N ). This provides a new (and in our opinion simpler) proof of the result of V. Chiad` o Piat, G. Dal Maso & A. Defranceschi [1].

References 1. 2. 3.

Chiad` o Piat, V., Dal Maso, G. and Defranceschi, A. (1990) G-convergence of monotone operators, Ann. Inst. H. Poincar´e Anal. Non lin´eaire, 7, 123–160. Francfort, G., Murat, F. and Tartar, L. (2003) Monotone operators in divergence form with x-dependent multivalued graphs, Boll. Un. Mat. Ital., to appear. Francfort, G., Murat, F. and Tartar, L. (2004) Homogenization of monotone operators in divergence form with x-dependent multivalued graphs, in preparation.

OPTIMAL DESIGN IN 2-D CONDUCTIVITY FOR QUADRATIC FUNCTIONALS IN THE FIELD

P. PEDREGAL

ETSI Industriales Universidad de Castilla-La Mancha 13071 Ciudad Real, SPAIN

1. Introduction We would like to consider a typical optimal design problem in conductivity for the optimal layout of the distribution of two given and diﬀerent conducting materials with conductivities α and β (0 < α < β) on a design domain Ω, so as to minimize a certain cost functionaldepending on the underlying electric ﬁeld. In precise terms, we seek to minimize the functional

I(χ) = Ω

a(x, χ(x)) |∇u(x) − G(x)|2 dx

(1.1)

under the constraints − div ([αχ + β(1 − χ)] ∇u) = u=

f in Ω, u0 on ∂Ω,

where the coeﬃcient a(x, χ), and the data G, f and u0 are given and known. Usually the characteristic function χ, indicating where the α-material is to be placed, is subjected to a volume constraint Ω

χ(x) dx ≤ t0 |Ω|

which enforces a restriction on the amount of material α at our disposal. t0 ∈ (0, 1) is given. It is important to stress that we do not make any hypothesis about the sign of a in the cost functional, so that by playing with the minus signs we can also consider the problem

Maximize in χ :

I(χ) = Ω

a(x, χ(x)) |∇u(x) − G(x)|2 dx

229 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 229–246. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

230

P. PEDREGAL

under the constraints − div ([αχ + β(1 − χ)] ∇u) = u=

f in Ω, u0 on ∂Ω.

It is well-known that these optimization problems do not have optimal solutions within the class of characteristic functions ([10]) so that a relaxation must be performed. Homogenization theory has been the main tool to deﬁne and expand the range of admissible designs to incorporate composites as structural elements. It plays a relevant role when functionals depend on u but there is no explicit dependence on ∇u (except for a few important cases). Several recent books treat this perspective in an exhaustive way ([1], [16]). When the integrand in the cost functional ϕ explicitly depends on ∇u, then homogenization theory has been pushed to somehow treat this more general case. See [7], [9]. A paradigmatic problem suggested and studied in [15] focuses on the functional 1 I(χ) = 2

Ω

|∇u(x) − G(x)|2 dx

(1.2)

where G is a given target ﬁeld. This is a particular situation of (1.1) by taking a(x, χ) ≡ 1. One of the main results whose proof we would like to sketch here is a full relaxation of the initial problem with cost functional given in (1.2). Namely, consider the two functions

2 2 1 β 2 F (1) + F (2) − ((2 − t)β + tα) det F , ϕ(t, F ) = tβ(β − α)

2

2

ψ(t, F ) = αβ (αt + β(1 − t)) F (1) + (α(1 − t) + βt) F (2)

− t(1 − t)(β − α)2 + 2αβ det F, and the variational problem

Minimize in (t, U ) : Ω

ϕ(t(x), ∇U (x)) − 2G(x) · ∇U (1) (x) + |G(x)|2 dx

subject to U ∈ H 1 (Ω),

U (1) = u0 on ∂Ω,

0 ≤ t(x) ≤ 1,

Ω

ψ(t(x), ∇U (x)) ≤ 0,

t(x) dx ≤ t0 |Ω| .

Theorem 1.1 This variational problem is equivalent to (a relaxation for) the original optimal design problem (with cost functional given in (1.2)) in the sense that:

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OPTIMAL DESIGN IN 2-D CONDUCTIVITY ...

1. the inﬁma to both problems coincide; 2. there are optimal solutions for the latter. In addition, there is a precise way of building minimizing sequence for the original optimal design problem from optimal solutions of its relaxation. A similar result is valid for a general cost as in (1.1) with completely explicit integrands for the relaxation. All of these formulae, as well as many important properties related to relaxation, were obtained in a series of papers ([2], [13], [14]). Namely, it we take h(x) = βaα(x) − αaβ (x),

2

2

ψ(t, F ) = αβ (αt + β(1 − t)) F (1) + (α(1 − t) + βt) F (2)

− t(1 − t)(β − α)2 + 2αβ det F,

 2 h(x) (2) 2 2 (1)   β F + F − 2β det F   tβ(β − α)2     a (x)   + ββ det F,       if h(x) ≥ 0, ψ(t, F ) ≤ 0, 2 2 ϕ(x, t, F ) = −h(x)  α2 F (1) + F (2) − 2α det F    (1 − t)α(β − α)2    a (x)   + αα det F,      if h(x) ≤ 0, ψ(t, F ) ≤ 0,  

+∞,

otherwise.

Theorem 1.2 The variational problem

Minimize in (t, U ) : Ω

ϕ(x, t(x), ∇U (x)) dx

subject to U ∈ H 1 (Ω),

U (1) = u0 on ∂Ω,

0 ≤ t(x) ≤ 1,

Ω

ψ(t(x), ∇U (x)) ≤ 0,

t(x) dx ≤ t0 |Ω| .

is equivalent to (a relaxation for) our original optimal design problem in the sense explained in the statement of Theorem 1.1. It is interesting to analyze diﬀerent possibilities for the coeﬃcients aα and aβ depending on whether they have the same sign and are positive or negative, or if one is positive and the other negative, or even if either one vanishes. All these diﬀerent possibilities will be exhaustively explored in [5]. The interesting example in [8] is also included in our analysis. It has been shown ([15]) that in fact there is a simpler relaxation for the u purely quadratic case a ≡ 1 valid for a dense Gδ set of gradients G = ∇˜

232

P. PEDREGAL

in H01 (Ω). Indeed when G = ∇˜ u belongs to that unknown Gδ set, the optimization problem Minimize in t(x) :

I(t) =

1 2

Ω

|∇u(x) − ∇˜ u(x)|2 dx

subject to − div ([αt + β(1 − t)] ∇u) = u= t(x) dx ≤ Ω

f in Ω, u0 on ∂Ω, t0 |Ω| ,

where now t is allowed to vary in the full interval [0, 1], admits optimal solutions and hence it is equivalent to the original optimal design problem. Being equivalent to the initial problem, the value of both inﬁma is the same. In addition, optimal microstructures for the original problem come in the form of ﬁrst-order laminates where layers align themselves parallel to the optimal electric ﬁeld ∇u(x), and the optimal t(x) furnishes the local volume fraction. Moreover, minimizing sequences of electric ﬁelds for the original problem always converge strongly to the optimal solution of the second problem. In this way, the optimal solution for this last problem encodes the necessary information to reconstruct optimal microstructures for the original optimal design problem. For the particular case G ≡ 0 (also treated in [9]) we provide some analytical and numerical evidence (alghough not a rigorous proof yet) based on our relaxation (Theorem 1.1) that the zero ﬁeld belongs to that Gδ set. In fact this same evidence also holds for a general quadratic functional (always taking G ≡ 0) with a general coeﬃcient a, and even for a general target ﬁeld G. Conjecture 1.3 Suppose the coeﬃcients aα ≡ a(x, 1), aβ ≡ a(x, 0) are both positive. The optimization problem

Minimize in t(x) : Ω

[t(x)aα(x) + (1 − t(x))aβ (x)] |∇u(x) − G(x)|2 dx

subject to − div ([αt(x) + β(1 − t(x))]∇u(x)) = u= 0 ≤ t(x) ≤ Ω t(x) dx ≤

f in Ω, u0 on ∂Ω, 1 in Ω, t0 |Ω| ,

is equivalent to (a full relaxation for) the previous optimal design problem in the following sense:

OPTIMAL DESIGN IN 2-D CONDUCTIVITY ...

233

1. the inﬁma for both problems are the same; 2. the second optimization problem always has an optimal solution (t˜, u ˜). ˜ In addition, ﬁrst-order laminates with local volume fraction t(x) and layers parallel to ∇˜ u(x) are optimal microstructures for the original optimal design problem. Finally, minimizing ﬁelds for the original problem always converge strongly to optimal ﬁelds of this relaxation. As indicated, we will take advantage of Theorem 1.2 to explore the relaxed problem and simplify it appropriately to performe some numerical simulations. This simpliﬁcation leads us to conjecture the simpler form of the relaxation as described in the formal statement of Conjecture 1.3. The structure of the paper is as follows. In Section 2, we brieﬂy explain the vector variational reformulation of such problems and its relaxation. Next, we describe the constrained quasiconvexiﬁcations involved (see [2], [4], [13], [14]). We then study, as a model situation, the relaxed problem (Section 4) corresponding to a ≡ 1, and explain speciﬁcally the reasons leading to our conjecture above. We close our contribution with several simulations based on these results. 2. Variational reformulation The variational reformulation of such problems has been explained in some recent references, in particular in the ones cited in the Introduction. Notice, to begin with, that by writing a |∇u − G|2 = a |∇u|2 − 2aG∇u + a |G|2 and noting that the second term is linear in ∇u, we realize that it suﬃces to treat the case G ≡ 0. We therefore restrict attention to this case from now on. If we let Λγ be the subspace determined by

Λγ = A ∈ M2×2 : γA(1) + T A(2) = 0

where A(i) is the i-th row of A and T is the π/2-counterclockwise rotation in the plane, and w is the solution of the problem ∆w = f

in Ω,

w = 0 on ∂Ω,

then we deﬁne the integrand  (1) 2    aα(x) A , (1) 2 ϕ(x, A) = a (x) A ,  β  

+∞,

if A ∈ Λα,x if A ∈ Λβ,x otherwise,

234

P. PEDREGAL

where we further put

Λγ,x = Λγ +

0 T ∇w(x)

aα(x) ≡ a(x, 1), aβ (x) ≡ a(x, 0).

,

Likewise we set   1/ |Ω| ,

V (x, A) =



0, +∞,

if A ∈ Λα,x, if A ∈ Λβ,x \ Λα,x, otherwise.

The non-convex, vector, variational problem

Minimize in U :

J(U ) = Ω

ϕ(x, ∇U (x)) dx

subject to Ω

V (x, ∇U (x)) dx ≤ t0 ,

U (1) = u0

on ∂Ω,

is equivalent to our initial optimal design problem. It has been shown ([2]) that a full relaxation for this reformulation can be expressed in terms of gradient Young measures as follows

Minimize in ν :

J(ν) =

Ω Λα,x ∪Λβ,x

ϕ(x, A) dνx(A) dx

subject to ν = {νx}x∈Ω is a (H 1 (Ω))-gradient Young measure supported in Λα,x∪Λβ,x,

V (x, A) dνx(A) dx ≤ t0 ,

Ω Λα,x ∪Λβ,x

∇U (x) =

Λα,x ∪Λβ,x

A dνx(A),

U (1) = u0

on ∂Ω.

Our aim is to examine this generalized variational problem. Let t(x) = V (x, A) dνx(A) Λα,x ∪Λβ,x

be the local volume fraction associated with the α-material. For such a ﬁxed t(x) complying with Ω

t(x) dx ≤ t0 ,

OPTIMAL DESIGN IN 2-D CONDUCTIVITY ...

and U such that ∇U (x) =

235

Λα,x ∪Λβ,x

U (1) = u0

A dνx(A),

on ∂Ω,

we consider the restricted minimum problem consisting of minimizing J(ν) among all (H 1 (Ω)-gradient Young measures having local volume fraction t(x) and ﬁrst moment given by the gradient of U . The relaxation can also be stated in terms of pairs (t, ∇U ) ([11])

Minimize in (t, U ) : Ω

ϕ(x, t(x), ∇U (x)) dx

subject to

U (1) = u0

on ∂Ω, Ω

t(x) dx ≤ t0 |Ω|

where the integrand ϕ is deﬁned, for (x, t, F ) ﬁxed, through the minimum

min µ

Λα,x ∪Λβ,x

ϕ(x, A

(1)

) dµ(A) : F =

Λα,x ∪Λβ,x

A dµ(A), t = µ(Λα,x)

taken over the class of homogeneous (H 1 -) gradient Young measures. Due to the local nature of gradient Young measures and their translation invariance, the x-variable is like a parameter here, so that we will concentrate on examining the optimization problem

min µ

(1)

Λα ∪Λβ

ψ(A

) dµ(A) : F =

Λα ∪Λβ

A dµ(A), t = µ(Λα) .

(2.1)

Here ψ = ϕ(x, ·), µ = νx has to be a homogeneous gradient Young measure supported in Λα ∪ Λβ , t = t(x), F = ∇U (x). Our ﬁrst step is the explicit computation of the integrand ϕ(x, t, F ). 3. Constrained quasiconvexiﬁcation For ﬁxed (x, t, F ) we would like to compute explicitly the minimum (2.1). Even though we can hardly specify when a probability measure is a gradient Young measure, it is remarkable that we can indeed compute this minimum in closed form for the quadratic case. Our ﬁrst observation is that we can restrict, without loss of generality, to the case f ≡ 0. Note that the x-dependence occur in the deﬁnition of the manifolds Λγ,x and in the coeﬃcient of the quadratic cost functional. But the translation to deﬁne Λγ,x in terms of Λγ takes place in the second row while the cost functional depends only on the ﬁrst row and hence there is no interference between the two. We will therefore assume that f ≡ 0.

236

P. PEDREGAL

Our strategy to compute the minimum in (2.1) is as follows. We will investigate a broader minimum problem in the sense that we will allow a larger class of probability measures compete in the minimization process. This larger class is nothing but the class of polyconvex measures. Said diﬀerently, we will retain the important properties we know gradient Young measures should verify. Once this minimum is computed, we will check a posteriori that the optimal measures, which commute with the determinant, are indeed laminates ([12]). This will entitle us to assert that the minimum computed with this larger class is the same than the one for gradient Young measures. Hence, we are concerned with the optimization problem

Minimize in ν :

Λα ∪Λβ

ϕ(A(1) ) dν(A)

subject to ν = tνα + (1 − t)νβ commutes with determinant, supp (νγ ) ⊂ Λγ ,

γ = α, β, F = t Λα

A dνα(A) + (1 − t)

Λβ

A dνβ (A).

Let us ﬁrst examine the constraints. We introduce the following variables

Gγ =

R2

|λ|2 dνγ(1) (λ),

γ = α, β,

(3.1)

(1)

where νγ is the probability measure resulting from the projection of νγ onto the ﬁrst row variable. On the other hand if we let

Fγ =

Λγ

A dνγ (A)

we have that Fγ ∈ Λγ (because Λγ is a 2-dimensional subspace) and F = tFα + (1 − t)Fβ . From these two conditions, it is immediate to obtain

1 1 (1) βF (1) + T F (2) , Fβ = αF (1) + T F (2) . t(β − α) (1 − t)(α − β) (3.2) The important constraint on the commutation with det leads to

Fα(1) =

det F = t Λα

det A dνα(A) + (1 − t)

Λβ

det A dνβ (A).

2

But notice that det A = γ A(1) if A ∈ Λγ , so that by using (3.1), det F = tαGα + (1 − t)βGβ .

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OPTIMAL DESIGN IN 2-D CONDUCTIVITY ...

On the other hand, by Jensen’s inequality , Gγ ≥

R2

2 (1) λ dνγ (λ)

2

= Fγ(1) ,

and by using (3.2), we can write

2

t2 (β −α)2 Gα ≥ βF (1) + T F (2) ,

2

(1−t)2 (β −α)2 Gβ ≥ αF (1) + T F (2) .

After some algebra, we can rewrite the three restrictions as − det F + tαGα + (1 − t)βGβ =

2 β 2 F (1)

2 −2β det F + + F (2) − t2 (β − α)2 Gα ≤ 2 2 −2α det F + α2 F (1) + F (2) − (1 − t)2 (β − α)2 Gβ ≤

0, 0, 0.

Regarding the cost functional, we can rewrite it in terms of the G variables as taαGα + (1 − t)aβ Gβ . Altogether we face the mathematical programming problem Minimize in G = (Gα, Gβ ) :

taαGα + (1 − t)aβ Gβ

subject to − det F + tαG + (1 − t)βG =

β α (1) 2 (2) 2 2 2 2 −2β det F + β F + F − t (β − α) Gα ≤ 2 2 −2α det F + α2 F (1) + F (2) − (1 − t)2 (β − α)2 Gβ ≤

0, 0, 0.

where α, β, aα, aβ are data of the initial problem, and t and F (and x) are ﬁxed, but arbitrary as the minimum value of this problem deﬁnes the relaxed integrand ϕ(x, t, F ). A key issue here is whether the optimal vector G can be recovered from an admissible laminate ν in (3.1), and, in particular, we need to provide a precise way of going from G to such a laminate. Notice that this optimization problem is a linear, mathematical programming problem in the variables G. Three issues are important: 1 when the admissible set is non-empty; 2 the point(s) where the minimum value is attained depending on the coeﬃcients aα and aβ ; 3 laminates for which the optimal G are recovered. The ﬁrst two items are elementary. Indeed, the admissible set will be non-empty provided that, for the point where the two inequality constraints

238

P. PEDREGAL

become equalities, the constraint in the form of equality becomes nonpositive. This is very easy to see geometrically (Figure 3.1). The condition reads 2 2 αβ (αt + β(1 − t)) F (1) + (α(1 − t) + βt) F (2) −

t(1 − t)(β − α)2 + 2αβ det F ≤ 0.

(3.3)

The second issue is also elementary. The admissible set (when it is nonempty) represents a segment with two extreme points (Figure 3.1). The linear cost functional will attain its minimum on one of them, or become constant, depending on the particular values of the coeﬃcients aα, aβ . Computations are again elementary in two diﬀerent situations depending on the sign of the expression h(x) ≡ βaα(x) − αaβ (x) : 1.

h(x) ≥ 0:

ϕ(x, t, F ) =

2 h(x) (2) 2 2 (1) β F + F − 2β det F tβ(β − α)2

+

aβ (x) det F, β

if the pair (t, F ) veriﬁes (3.3). The value is +∞ if the pair (t, F ) does not verify (3.3). 2. h(x) ≤ 0:

2 −h(x) aα(x) (2) 2 2 (1) ϕ(x, t, F ) = α F + F det F − 2α det F + 2 α (1 − t)α(β − α)

whenever (3.3) holds.

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OPTIMAL DESIGN IN 2-D CONDUCTIVITY ...

The third issue about whether these extreme points can be obtained as the second moments for appropriate measures να and νβ as in (3.1) so that the convex combination tνα + (1 − t)νβ is a laminate, is a little bit more involved. See [2], [14] for details. We simply state the result for the vertex when 2 2 − det F + β 2 F (1) + F (2) − t2 (β − α)2 Gα = 0. Put 4 4 2 2 2 2 2 −2αβ A(1) A(2) − 2αβ(α + β) A(1) det A − 2(α + β) A(2) det A,

g(A) = α2 β 2 A(1) + A(2) + (α2 + 6αβ + β 2 ) det A

and !

"

2 2 > 1 1 r1 (A) = + αβ A(1) − A(2) − g(A) , 2 2(β − α) det A ! " 2 2 > 1 1 r2 (A) = + αβ A(1) − A(2) + g(A) . 2 2(β − α) det A

Take the matrices

Aα,t =

Aβ,i =

Aβ,j,t =

zt αT zt

wi βT wi

wj,t βT wj,t

,

,

1 1 (1) βA + T A(2) , t β−α 1 (−1) (1) wi = αA + T A(2) , 1 − ri(A) β − α 1 rj (A) (−1) (1) wj,t = αA + T A(2) , t 1 − rj (A) β − α

,

zt =

where i = j and A(k) stands for the k-th row of A. Finally, put si,j =

(1 − ri(A)) [t (1 − rj (A)) − (1 − t)rj (A)] , t (1 − rj (A)) − (1 − ri(A)) rj (A)

i = j.

Lemma 3.1 ([14]) There are two second-order laminates supported in three matrices (except when t = ri(A) for i = 1 or i = 2 that the laminate collapses to a ﬁrst-order laminate) which are optimal microstructures. Namely, bearing in mind the notation before the statement, the two laminates

νi,j = si,j δAβ,i for i = j, where

1 − si,j − t t + (1 − si,j ) δAα,t + δAβ,j,t , 1 − si,j 1 − si,j det (Aα,t − Aβ,j,t) = 0,

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P. PEDREGAL

t 1 − si,j − t det Aβ,i − Aα,t − Aβ,j,t 1 − si,j 1 − si,j

= 0,

are optimal, and so are any convex combination of these two. A similar result holds for the other extreme point. Altogether we have essentially proved our main relaxation theorem. Consider the three functions h(x) = βaα(x) − αaβ (x),

2

2

ψ(t, F ) = αβ (αt + β(1 − t)) F (1) + (α(1 − t) + βt) F (2)

2 + 2αβ det F, − t(1 − t)(β − α)  2 h(x) (2) 2 2 (1)   β F + F − 2β det F   tβ(β − α)2     a (x)   + ββ det F,       if h(x) ≥ 0, ψ(t, F ) ≤ 0, 2 2 ϕ(x, t, F ) = −h(x) 2 (1) (2)  α F + F − 2α det F   (1 − t)α(β − α)2     a (x)   + αα det F,     if h(x) ≤ 0, ψ(t, F ) ≤ 0,    +∞, otherwise. Theorem 1.2 The variational problem

Minimize in (t, U ) : Ω

ϕ(x, t(x), ∇U (x)) dx

subject to U ∈ H 1 (Ω),

U (1) = u0 on ∂Ω,

0 ≤ t(x) ≤ 1,

Ω

ψ(t(x), ∇U (x)) ≤ 0,

t(x) dx ≤ t0 |Ω| .

is equivalent to (a relaxation for) our original optimal design problem in the sense explained in the statement of Theorem 1.1. 4. Analysis of the relaxed problem The ﬁnal step is to analyze the relaxed problem we have obtained at the end of the preceding section. For simplicity and to avoid to get lost among notation, we will restrict attention to the particular case when we take a(x, χ) ≡ 1 since the general case for Conjecture 1.3 is similar and all of our observations here are also valid for the case we have envisioned from the beginning.

241

OPTIMAL DESIGN IN 2-D CONDUCTIVITY ...

In this situation the function h(x) becomes β − α which is always positive, ψ does not change

2

2

ψ(t, F ) = αβ (αt + β(1 − t)) F (1) + (α(1 − t) + βt) F (2)

− t(1 − t)(β − α)2 + 2αβ det F and ϕ does not depend on x. We will put

2 2 1 β 2 F (1) + F (2) − ((2 − t)β + tα) det F ϕ(t, F ) = tβ(β − α)

when ψ(t, F ) ≤ 0. We have used the notation ϕ(t, F ) instead of ϕ(t, F ) for notational simplicity. The problem we would like to examine is

Minimize in (t, U ) : Ω

ϕ(t(x), ∇U (x)) dx

subject to U ∈ H 1 (Ω),

ψ(t(x), ∇U (x)) ≤ 0,

U (1) = u0 on ∂Ω,

0 ≤ t(x) ≤ 1,

Ω

t(x) dx ≤ t0 |Ω| .

This vector variational problem is convex (in the vectorial sense and taking into account the additional variable t ([6])) and regular in the sense that it admits optimal solutions since it is a relaxation. Moreover all functions involve are quadratic in the vector gradient variable and smooth. It is however a rather complex problem to analyze. One possibility is to look at optimality conditions introducing several multipliers to keep record of restrictions. This makes the problem more complicated precisely because of all the restrictions we have to enforce. Instead of this approach, we make the following simple but relevant observation. It is an elementary fact which tells a lot about the relationship

2

between ϕ, ψ and the quadratic cost F (1) .

Lemma 4.1 For each i = 1, 2 and for ﬁxed t, the optimal solution of the quadratic, mathematical programming problem Minimize in F (i) :

ϕ(t, F )

subject to ψ(t, F ) ≤ 0 occurs when (αt + β(1 − t)) F (1) + T F (2) = 0. In addition, the associated optimal structures (gradient Young measures) are ﬁrst-order laminates with volume fraction t for the α-material and ori

2

entation of layers parallel to F (1) . The optimal value in both cases is F (1) .

242

P. PEDREGAL

The idea is then to replace the complicated constraint ψ(t(x), ∇U (x)) ≤ 0 by the much simpler one (αt(x) + β(1 − t(x))) ∇U (1) (x) + T ∇U (2) (x) = 0, hoping to keep track of the minimum we are seeking. But this last condition amounts, after all, to

div [αt(x) + β(1 − t(x))] ∇U (1) (x) = 0 in

Ω,

and as pointed out in the lemma the cost simpliﬁes to 2 ∇U (1) (x) dx. Ω

If we remember that U (1) is our original ﬁeld u, we are led to consider

Minimize in t : Ω

|∇u(x)|2 dx

subject to div ([αt(x) + β(1 − t(x))] ∇u(x)) = u= 0 ≤ t(x) ≤ t(x) dx ≤ Ω

0 in Ω, u0 on ∂Ω, 1 in Ω, t0 |Ω| .

We have not been able to show that this last optimization problem has optimal solutions. This is essentially Tartar’s result in [15] for such Gδ set as explained in the Introduction. Yet we believe that this is true as stated in Conjecture 1.3. As a matter of fact, if we believe it so, then we can exploit optimality conditions for this last problem which are more easily handled and in a standard way, to see the behavior of those simulations. In all experiments we have conducted the convergence was ﬁne and the simulations stable and robust. We show here several such simulations corresponding to a non-vanishing right-hand side f in the equilibrium equation and to a divergence free vector ﬁeld G. All of these simulations have been performed by A. Donoso as part of his PhD thesis at Universidad de Castilla-La Mancha. Some of them were presented in a poster session in this same conference. In all examples, the design domain is the unit square Ω = (0, 1)2 and we take a vanishing boundary data u0 ≡ 0. The values for the parameters α and β are 1 and 2,

OPTIMAL DESIGN IN 2-D CONDUCTIVITY ...

243

respectively. Diﬀerent simulations correspond to diﬀerent right-hand sides f o diﬀerent values for the global resource constraint t0 . In addition, the color level in these pictures indicates the optimal volume fraction t(x) while the curves are level curves of the optimal, underlying electric ﬁeld so that optimal microstructures are ﬁrst order laminates with volume fraction t(x) for the α-material and layers align themselves orthogonal to those level curves (parallel to the ﬁeld). We have tried to indicate this in the two points A and B of Figure 4.1. White color is always associated with the α phase. Finally, the strong convergence of minimizing ﬁelds to optimal ﬁelds is due to the fact that the optimal underlying gradient Young measures (ﬁrst-order laminates) reduces to a Dirac delta measure in the ﬁrst row (see [12]). The ﬁrst set of simulations are those in [9]. In the ﬁrst example (Figure 4.1), the right-hand side f is taken to be identically 1 and t0 = 0.6. The next two cases correspond to

f (x) =

1, if x ∈ (1/4, 3/4)2 , 0, otherwise.

optimal structures are given in Figure 4.2 for the case t0 = 0.85 and in Figure 4.3 for t0 = 0.6.

244

P. PEDREGAL

OPTIMAL DESIGN IN 2-D CONDUCTIVITY ...

245

Finally, for Figures 4.4 and 4.5 we have

1 f (x) = x1 − 2

2

1 − x2 − 2

2

,

f (x) = e−100(x1 −0.5) + e−100(x2 −0.5)

respectively. In both cases t0 = 0.6.

More simulations can be found in [5].

2

2

246

P. PEDREGAL

Acknowledgements This work is supported, in part, by BFM20010738 of MCyT (Spain) and by GC-02-001 of JCCM (Castilla-La Mancha). It is also a pleasure to acknowledge the support of the Organizing Committee of this Workshop. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16.

Allaire, G. (2002) Shape optimization by the homogenization method, Springer. Aranda, E. and Pedregal, P. (2003) Constrained envelope for a general class of design problems, Disc. Cont. Dyn. Syst. A, Supplement Volume, 30–41. Ball, J.M.. (2002) Some open problems in elasticity, in: Geometry, Mechanics and Dynamics, eds. P. Newton, P. Holmes, A. Weinstein, Springer, to appear. Bellido, J. C., and Pedregal, P. (2002) Explicit quasiconvexiﬁcation of some cost functionals depending on derivatives of the state in optimal design , Disc. Cont. Dyn. Syst. A, 8, 967-982. Donoso, A., and Pedregal, P. (2003) (in preparation). Fonseca, I., Kinderlehrer, D., Pedregal, P. 1994 Energy functonals depending on elastic strain and chemical composition, Calc. Var., 2, 283-313. Grabovsky, Y., (2001) Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals, to appear in Advan. Appl. Math. Kohn, R. V. and Strang, G. (1986) Optimal design and relaxation of variational problems, I, II and III, CPAM, 39, 113-137, 139-182 and 353-377. Lipton, R. and Velo, A., 2000 Optimal design of gradient ﬁelds with applications to electrostatics, in Nonlinear Partial Diﬀerential Equations and Their Applications, College de France Seminar, D. Cioranescu, F. Murat, and J.L Lions eds, Chapman and Hall/CRC Research Notes in Mathematics. Murat, F. (1977) Contre-exemples pur divers problmes ou le contrle intervient dans les coeﬃcients, Ann. Mat. Pura ed Appl., Serie 4, 112, 49-68. Pedregal, P. (2000) Optimal design and constrained quasiconvexity, SIAM J. Math. Anal., 32, 854-869. Pedregal, P. (2000) Variational methods in nonlinear elasticity, SIAM, Philadelphia. Pedregal, P. (2001) Fully explicit quasiconvexiﬁcation of the mean-square deviation of the gradient of the state in optimal design, ERA-AMS, 7, 72-78. Pedregal, P. (2003) Constrained quasiconvexiﬁcation of the square of the gradient of the state in optimal design, QAM, in press. Tartar, L. (1994) Remarks on optimal design problems, in Calculus of Variations, Homogenization and Continuum Mechanics, G. Buttazzo, G. Bouchitte and P. Suquet, eds., World Scientiﬁc, Singapore, 279-296. Tartar, L. (2000) An introduction to the homogenization method in optimal design, Springer Lecture Notes in Math., 1740, 47-156.

LINEAR COMPARISON METHODS FOR NONLINEAR COMPOSITES

˜ P. PONTE CASTANEDA

Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania, Philadelphia, PA 19104-6315 U.S.A.

Abstract. Recently developed methods for estimating the eﬀective behavior of nonlinear composites are reviewed. The methods follow from variational principles expressing the eﬀective behavior of the given nonlinear composites in terms of the behavior of suitably chosen “linear comparison” composites. This allows the use of classical bounds and estimates (e.g., Hashin-Shtrikman, self consistent approximations) for linear materials to generate corresponding bounds and estimates for nonlinear ones. The ﬁrst method (Ponte Casta˜ neda 1991) makes use of the “secant” moduli of the phases, evaluated at the second moments of the strain ﬁeld over the phases, and delivers bounds, but these bounds are exact only to ﬁrst order in the heterogeneity contrast. The second method (Ponte Casta˜ neda 1996) makes use of the “tangent” moduli, evaluated at the phase averages (or ﬁrst moments) of the strain ﬁeld, and yields estimates that are exact to second-order in the contrast, but that can violate the bounds in some special cases. These special cases turn out to correspond to situations, such as percolation phenomena, where ﬁeld ﬂuctuations become important. The third method (Ponte Casta˜ neda 2002) delivers estimates that are exact to second order in the contrast, making use of “generalized secant” moduli incorporating both ﬁrst- and second-moment information, in such a way that the bounds are never violated.

1. Introduction Well-established methods are available to estimate the eﬀective or overall behavior of linear elastic composite materials. These so-called homogenization methods include the variational principles of Hashin and Shtrikman

247 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 247–268. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

248

˜ P. PONTE CASTANEDA

(1962), which are particularly well suited to estimate the eﬀective behavior of composites with particulate random microstructures. There is also the self-consistent approximation, developed in several diﬀerent physical contexts by various authors (e.g., Hershey 1954, Kron¨er 1958, Willis 1977), which is known to be fairly accurate for polycrystals and other materials with granular microstructures. For nonlinear (e.g., plastic, viscoplastic, etc.) composites, rigorous methods have not been available until fairly recently, even though eﬀorts along these lines have been going on for some time, particularly in the context of ductile polycrystals (e.g., Hill 1965, Hutchinson 1976). Making use of a nonlinear extension of the HashinShtrikman (HS) variational principles, due to Willis (1983), the ﬁrst bounds of the HS type for nonlinear composites were derived by Talbot and Willis (1985). Ponte Casta˜ neda (1991) proposed a more general variational approach making use of optimally chosen “linear comparison composites.” This approach is not only capable of delivering bounds of the HS type for nonlinear composites, but, in addition, can be used to generate bounds and estimates of other types, such as self-consistent estimates and three-point bounds (Ponte Casta˜ neda 1992). A diﬀerent, but equivalent method for the special class of power-law materials has been proposed by Suquet (1993). Talbot and Willis (1992) provided a simultaneous generalization of the variational principles of Talbot and Willis (1985) and the linear comparison composite method of Ponte Casta˜ neda (1991), which has the potential to give improved estimates for certain special, non-standard situations. Suquet (1995) has given an alternative interpretation (“modiﬁed secant”) of the variational estimates of Ponte Casta˜ neda (1991) in terms of the secant moduli of the phases, evaluated at the second-moments of the ﬁelds in the phases. Ponte Casta˜ neda (1996) proposed a second approach that makes use of more general linear comparison composites, incorporating the tangent moduli of the phases, evaluated self-consistently at the phase averages of the ﬁelds in the linear comparison composite. While this method does not yield bounds, it appears to give more accurate results. In particular, this method was the ﬁrst to yield general homogenization estimates capable of reproducing exactly to second-order in the contrast the asymptotic expansions of Suquet and Ponte Casta˜ neda (1993)—and for this reason it is known as the “second-order” method. Moreover, the method has been given a more rigorous variational interpretation by Ponte Casta˜ neda and Willis (1999). Masson et al. (2000) proposed an “aﬃne” method, which corresponds to certain further approximations in the context of the second-order method. For the most part, it has been found by comparison to numerical simulations (Moulinec and Suquet 1998, Michel et al. 1999, 2000) that the

LINEAR COMPARISON METHODS . . .

249

second-order estimates can be quite accurate even at high values of the heterogeneity contrast. But Leroy and Ponte Casta˜ neda (2001) found that the second-order estimates can violate rigorous bounds near the percolation limit for certain classes of composites. Motivated by this ﬁnding and the observation that the earlier second-order method included only information on the phase averages, or ﬁrst moments of the ﬁelds, an improved “second-order” homogenization method was developed recently by Ponte Casta˜ neda (2002). This method incorporates the dependence of the linear comparison composite on the ﬁeld ﬂuctuations, or more speciﬁcally, on the second-moments of the ﬁeld ﬂuctuations in the phases, in such a way that the resulting estimates satisfy known bounds and are exact to second-order in the contrast. Other recent review articles on nonlinear composites include Ponte Casta˜ neda and Suquet (1998) and Willis (2000).

2. Eﬀective behavior Consider a material composed of N diﬀerent phases, which are distributed randomly in a specimen occupying a volume Ω at a length scale that is much smaller than the size of the specimen and the scale of variation of the loading conditions. The constitutive behavior of the nonlinear phases will be characterized by convex energy functions w(r) (r = 1, ..., N ), with subquadratic growth, such that the local stress-strain relation is determined by ∂w (x, ε), σ= ∂ε

w(x, ε) =

N

χ(r) (x) w (r) (ε),

(1)

r=1

where the functions χ(r) are equal to 1 if the position vector x is inside phase r (i.e., x ∈ Ω(r) ) and zero otherwise. The relations (1) can be used to describe several constitutive models including deformation theory of plasticity, in which case ε and σ are identiﬁed with the inﬁnitesimal strain and stress, respectively. The relation applies equally well to viscoplastic materials, in which case the associated deformations are ﬁnite and ε and σ are associated with the Eulerian strain rate and Cauchy stress, respectively. For these cases, the behavior of the constituent phases is often assumed to be isotropic and incompressible, such that the phase potentials take the form w (r) (ε) = φ(r) (εe), where εe denotes the equivalent von-Mises strain (rate). Making use of the symbols . and . (r) to denote volume averages over the composite (Ω) and over phase r (Ω(r) ), respectively, the eﬀective

250

˜ P. PONTE CASTANEDA

behavior of the composite is determined by the eﬀective energy function 5 (ε) = min w(x, ε) = min W ε∈K ε∈K

N

c(r) w(r) (ε) (r) ,

(2)

r=1

where the scalars c(r) = χ(r) denote the volume fractions of the given phases and K the set of kinematically admissible strain (rate) tensors ε, such

that there is a displacement (velocity) ﬁeld v satisfying ε = 12 ∇v + ∇vT 5 physically corresponds to the energy in Ω and v = εx on ∂Ω. Thus, W stored (dissipated) in the composite when subjected to an aﬃne displacement (velocity) on the boundary with prescribed average strain (rate) ε = ε . It can be shown (Hill 1963) that the average stress σ = σ is then related to the average strain (rate) ε via the relation σ=

5 ∂W . ∂ε

(3)

There is an exactly dual formulation (see, for example, Ponte and Suquet 1998) which makes use of the local complementary energy potential u, such that ε = ∂u/∂σ. The potential u is deﬁned in terms of w via the Legendre transformation: . (4) u(x, σ) = stat {σ · ε − w(x, ε)} = w∗ (x, σ), ε where the stat(ionary) operation means taking the derivative of the terms inside the curly brackets with respect to ε, solving for ε as a function of σ, and substituting the result back inside the brackets to obtain a function of σ. Note that the requisite smoothness hypotheses have been made about the function w, and that, because of the convexity hypothesis on the w(r) , there is no ambiguity in the above deﬁnition—and the function u is also convex. In terms of the stress potential u, the eﬀective constitutive relation for the nonlinear composite may then be alternatively written as ε=

2 ∂U , ∂σ

(5)

2 is the eﬀective stress potential for the composite, deﬁned by where U 2 (σ) = min u(x, σ) = min U σ ∈S σ ∈S

N

c(r) u(r) (σ) (r) .

(6)

r=1

In this relation, S = {σ, div σ = 0 in Ω, σ = σ} denotes the set of statically admissible stresses. Again, under the above-mentioned hypotheses

251

LINEAR COMPARISON METHODS . . .

on the w(r) , the two formulations are exactly equivalent in the sense of 2 =W 5∗ . Legendre duality: U For later reference, some additional notation is introduced next. The per-phase averages of the stress and strain are deﬁned via: ε(r) = ε (r) and σ (r) = σ (r) , (7) N (r) (r) (r) (r) and σ = and are such that ε = N r=1 c ε r=1 c σ . In addition, the second moments of the strain and stress over phase r are given by ε ⊗ ε (r) and σ ⊗ σ (r) , in terms of which expressions may be obtained for the corresponding phase ﬂuctuation covariance tensors: (r) . Cε = (ε − ε(r) ) ⊗ (ε − ε(r) ) (r) = ε ⊗ ε (r) − ε(r) ⊗ ε(r) ,

(8)

(r)

and similarly for Cσ . Analogous expressions may also be given for the overall ﬂuctuation covariance tensors Cε and Cσ .

3. Eﬀective estimates via linear comparison composites 5 and U 2 are diﬃcult to compute, beIn general, the eﬀective potentials W cause they amount to solving sets of nonlinear partial diﬀerential equations with randomly oscillating coeﬃcients. In this work, approximations will be developed for these potentials by making use of variational principles for suitably deﬁned “linear comparison composites.” Following Ponte Casta˜ neda (1996, 2002), a linear comparison compositeis introduced with energy function wT given by: wT (x, ε) =

N

(r)

χ(r) (x) wT (ε),

(9)

r=1 (r)

where the phase potential wT is given by the second-order Taylor approximation to the nonlinear phase potential w (r) : 1 (r) (r) wT (ε) = w(r) (ε(r) ) + ρ(r) (ε(r) ) · (ε − ε(r) ) + (ε − ε(r) ) · L0 (ε − ε(r) ). (10) 2 (r)

In this relation, ε(r) is a uniform reference strain and L0 is a constant modulus tensor—both of which are taken to be arbitrary at this stage. Also, the symbol ρ(r) is used to denote the ﬁrst derivative of the phase potential w(r) , such that: ρ(r) (ε(r) ) =

∂w(r) (r) (ε ). ∂ε

(11)

˜ P. PONTE CASTANEDA

252

It is useful to note here that the phase potential (10) corresponds to a ﬁctitious linear “thermoelastic” material with stress-strain relation given by: (r) (r) (12) σ = ρ(r) (ε(r) ) + L0 (ε − ε(r) ) = τ (r) (ε(r) ) + L0 ε, where τ (r) is the polarization, deﬁned by: (r)

τ (r) (ε) = ρ(r) (ε) − L0 ε.

(13)

Next an “error” or “corrector” function is deﬁned via the expression: (r) (r) (r) ε(r) ) − wT (ˆ ε ) , (14) V (r) (ε(r) , L0 ) = stat w (r) (ˆ εˆ (r) where the stationary operation means again optimizing with respect to the relevant variable. It then follows that the potential w(r) of phase r may be approximated as: (r)

(r)

w (r) (ε) ≈ wT (ε) + V (r) (ε(r) , L0 ),

(15)

(r)

which is valid for any ﬁxed values of ε(r) and L0 . This approximation can be averaged over the composite to obtain, via (2), a corresponding expression for the eﬀective potential of the nonlinear composite: 5 (ε) ≈ W 5T (ε; ε(s) , L(s) ) + W 0

N (r) c(r) V (r) (ε(r) , L0 ),

(16)

r=1

where

5T (ε; ε(s) , L(s) ) = minwT (x, ε) . W 0 ε∈K

(17)

is the eﬀective potential associated with the linear comparison composite (r) deﬁned by relations (9) and (10) above. Note that the L0 are assumed to be such that the minimum in (17) is attained—a suﬃcient condition for (r) this is that the L0 be positive deﬁnite. 5 from the eﬀecThe expression (16) allows the approximation of W 5 tive potential WT of a linear “thermoelastic” composite with the same microstructure as the original nonlinear composite, and is valid for any (r) choice of the variables ε(r) and L0 . This suggests, of course, optimizing 5. with respect to these variables to generate the best possible estimate for W However, as we will see below, the best choice for these variables depends on the choice of the corrector function V (r) .

253

LINEAR COMPARISON METHODS . . .

σ generalized secant

ρˆ (r ) tangent

ρ (r )

secant

ε (r )

εˆ (r )

ε

Figure 1. The “generalized secant” linear approximation (18) to the nonlinear stress-strain relation of phase r versus the corresponding “secant” and “tangent” approximations.

To see this, it is necessary to consider the optimality conditions for the ˆ(r) in the deﬁnition of the functions V (r) , which are given by the variables ε relations: (r) (r) ε(r) ) − ρ(r) (ε(r) ) = L0 (ˆ ε − ε(r) ). (18) ρ(r) (ˆ In fact, depending on the functional form of the nonlinear potential (r) w(r) , and the values of ε(r) and L0 , there may be several stationary points associated with (14), and therefore several possible solutions to equations ˆ(r) . Although a complete study of all the possibilities (18) for the variables ε has not yet been carried out, some choices have been tested, leading to three essentially diﬀerent types of estimates, roughly corresponding to the three diﬀerent types of interpolations shown in Figure 1 for the nonlinear constitutive relations of the phases: secant, tangent and generalized secant. 3.1. SECANT BOUNDS

The ﬁrst choice (Ponte Casta˜ neda 1991) is the one that makes the stationary point in (14) an extremum, typically a maximum in the context of

˜ P. PONTE CASTANEDA

254

plasticity, where w(r) has subquadratic growth at inﬁnity. Then, the function V (r) can be written in the form: (r) (r) (r) ε(r) ) − wT (ˆ ε ) , (19) V (r) (ε(r) , L0 ) = max w (r) (ˆ εˆ (r) and the estimation (16) leads to an upper boundthat can be optimized to give: N (s) (r) (s) 5 5 c(r) V (r) (ε(r) , L0 ) . (20) W (ε) ≤ min WT (ε; ε , L0 ) + (s) L0 , ε(s) r=1 It is known (Willis 1992, Talbot and Willis 1992) that the best choice of the variables ε(s) in the context of the bound (20) is usually ε(s) = 0, which leads to the bounds originally proposed by Ponte Casta˜ neda (1991) for isotropic composites: N (s) (r) 5 5 c(r) V (r) (0, L ) , (21) W (ε) ≤ min W0 (ε; L ) + 0

(s)

L0

0

r=1

where

50 (ε; L(s) ) = statw0 (x, ε) . W (22) 0 ε∈K is the eﬀective energy associated with a linear elastic comparison composite with local potentials given by w0 (x, ε) =

N r=1

(r)

χ(r) (x) w0 (ε),

1 (r) (r) and w0 (ε) = ε · L0 ε. 2

(23)

When the classical bounds of Hashin and Shtrikman (1963) are used for the linear comparison composite (22), the expression (21) can be used to generate (and in some cases improve) the bounds of the Hashin-Shtrikmantype ﬁrst given by Talbot and Willis (1985) for nonlinear media. However, more general bounds and estimates, including three-point bounds and selfconsistent estimates, may be easily generated (Ponte Casta˜ neda 1992) via (21) by means of the corresponding bounds and estimates for the linear comparison composite. The bounds (21) also include the bounds given by Suquet (1993) for the special class of power-law materials. One advantageous feature of the bound (21), ﬁrst noted by deBotton and Ponte Casta˜ neda (1992,1993), is that the eﬀective constitutive relation (3) for the nonlinear composite is the same as that of the linear comparison composite:

˜0 L ˆ (s) ε, σ=L (24) 0

LINEAR COMPARISON METHODS . . .

255

ˆ (s) of the linear comparison composite satisfy where the moduli tensors L 0 the optimality condition in expression (21). This means that the eﬀective constitutive relation for the nonlinear composite is precisely the same as that of the linear comparison composite, where the local properties of the linear comparison composite are determined as the solution of the procedure ˆ (r) . Clearly, since these variables depend (21) for the optimized variables L 0 on the applied strain ε, the above relation is nonlinear on this variable, as expected. Furthermore, it follows from the derivation of the bound (21) that the strain ﬁeld ε(x) in the linear comparison composite is an estimate for the corresponding ﬁeld in the actual nonlinear composite. Although such an estimate may not be very accurate in a local sense, the statistics of this ﬁeld in the linear composite may provide good estimates for the corresponding statistics in the actual nonlinear composite. Thus, the phase average and second moments in the nonlinear composite may be estimated via standard relations for the linear comparison composite:

ˆ (s) ε, ε(r) = A(r) L 0

and ε ⊗ ε (r) =

50 (s) 2 ∂W ˆ , L 0 c(r) ∂L(r) 0

(25)

ˆ (s) of the phase modulus which, again, are evaluated at the optimal values L 0 tensors in the phases of the linear comparison composite. The second relation in (25) has been derived at diﬀerent levels of generality by diﬀerent authors, including Bobeth and Diener (1987), Kreher (1990) and Parton and Buryachenko (1990). (r) It is also noted that the optimal choice of the variables L0 can be given an interpretation in terms of the second moment of the strain ﬁeld in the linear comparison composite (Suquet 1995, Hu 1996, Ponte Casta˜ neda and Suquet 1998). First note that the condition for the optimal choice of the ˆ(r) in the maximum problem in (19) is given by variables ε ∂w(r) (r) (r) (r) ˆ , (ˆ ε ) = L0 ε ∂ε

(26)

(r)

which physically means that the L0 should be chosen to be the secant (r) (r) modulus tensor Ls of nonlinear phase r. (Note that the choice of Ls in (26) is not unique, and that (26) should be more properly regarded as a (r) ˆ(r) .) Assuming smoothness with respect to L0 , the prescription for the ε condition for the optimal choice of these variables is generated by substi˜ 0 in (21) to obtain: tuting the expression (22) for W ˆ(r) = ε ⊗ ε (r) . ˆ(r) ⊗ ε ε

(27)

˜ P. PONTE CASTANEDA

256

(r)

In other words, the optimal values of the modulus tensors L0 in phase r of the linear comparison composite is related to the secant modulus (26) of the ˆ(r) corresponding nonlinear material in phase r, evaluated at some strain ε deﬁned through the relation (27), where the right-hand side is the secondorder moment of the strain ﬁeld over phase r in the linear comparison composite. It is emphasized that the expression (27) cannot be satisﬁed in general. This is because the left-hand side of this equation is of rank 1, while the right-hand side is generally of full rank. For incompressible, isotropic (r) phases, the optimal choice of the modulus tensors L0 is also incompressible and isotropic, and it suﬃces to optimize with respect to the magnitude (r) of this tensor, which is the (scalar) shear modulus µ0 . It then follows (Su(r) ˆ is its magnitude, which quet 1995) that all that matters in the tensor ε (r) optimizing with respect to µ0 , is determined by the relation: ˆ(r) e =

>

2e (r) .

(28) (r)

More generally, for anisotropic phases, the tensors L0 should be chosen anisotropic, but it is not clear at this time what is the most general type of anisotropy that should be selected, given that complete arbitrariness in (r) the selection of the coeﬃcients of L0 leads to the inconsistency mentioned in the context of relation (27). For incompressible behavior, one simple (r) possibility is to select the L0 such that: (r)

(r)

(r)

L0 = 2λ0 E(r) + 2µ0 F(r) ,

(29)

where 2 (r) (r) ˇ ⊗ε ˇd , E(r) = ε 3 d

(r)

ˇd = with ε

(r)

εd

(r) εe

,

and F(r) = K − E(r)

(30)

are projection operators (Ponte Casta˜ neda 1996), such that E(r) E(r) = E(r) , (r) (r) (r) (r) (r) (r) (r) F F = F , E F = F E = 0, J + E(r) + F(r) = I. In the above relations, K and J are the standard fourth-order, isotropic, shear and hydrostatic projection tensors, respectively. With the choice (29) for the (r) 5 , optimization with respect modulus tensors L0 in expression (21) for W (r) (r) to the variables λ0 and µ0 leads to the relations * (r) εˆ

=

2 (r) E · ε ⊗ ε (r) , 3

* (r) εˆ⊥

=

2 (r) F · ε ⊗ ε (r) , 3

(31)

LINEAR COMPARISON METHODS . . .

257

ˆ(r) have been where the “parallel” and “perpendicular” components of ε

1 1 (r) (r) (r) 2 (r) 2 2 (r) 2 (r) (r) (r) ˆ ·E ε ˆ ˆ ·F ε ˆ and εˆ⊥ = 3 ε , respecdeﬁned by εˆ = 3 ε (r)

(r)

(r)

ε⊥ )2 = (ˆ εe )2 . Note that the roots tively, in such a way that (ˆ ε )2 + (ˆ leading to positive values of εˆ and εˆ⊥ have been selected in these relations. In any case, the bound (21) can then be shown to reduce to: ˜ (ε) ≤ W

N

ˆ(r) . c(r) w(r) ε

(32)

r=1

For the special class of polycrystalline materials, deBotton and Ponte Casta˜ neda (1995) have given an alternative, more specialized version of the above bounds. For more general anisotropic composites, Suquet (Ponte Casta˜ neda and Suquet 1998) has given an alternative generalization of the formula (32), which makes use of all the components of the second-moment tensor, at the expense of having to introduce a generalization of the local potentials w(r) , which are deﬁned on the fourth-order second moment tensors. Finally, it should be noted that there are exactly dual versions of all the results in this section, including an expression analogous to (21) for the ˜ 0 , as 2 (σ) in terms of the eﬀective compliance tensor M eﬀective potential U well as expressions analogous to (25) for the phase average stresses σ (r) , and the corresponding second moments σ ⊗ σ (r) . 3.2. TANGENT SECOND-ORDER ESTIMATES

A second class of estimates (Ponte Casta˜ neda 1996) is generated by making ˆ(r) = ε(r) in relations (18), which, in turn, makes the functions the choice ε V (r) vanish identically. (s) Then, keeping the variables L0 ﬁxed for the time being, and optimizing with respect to the variables ε(r) in the general estimate (16) leads to the result: 5 (ε) ≈ stat W 5T (ε; ε(s) , L(s) ) , (33) W 0 ε(s) 5T is still given by relation (17). Stationarity with respect to the where W variables ε(s) in this expression then gives the conditions: (s) (s) c(s) Lt (ε(s) ) − L0 (ε (s) − ε(s) ) = 0, (34) which can be satisﬁed by setting: ε(s) = ε(s) ,

(35)

˜ P. PONTE CASTANEDA

258

where the symbol ε(s) has been used to denote the phase averages of the strain ﬁeld ε (s) . Using the condition (35), the estimate (33) can be shown (Ponte Casta˜ neda and Suquet 1998) to reduce to: 5 (ε) ≈ W

! " N 1 (r) (r) (r) (r) (r) (r) w (ε ) + ρ (ε ) · (ε − ε ) . c 2 r=1

(36)

ˆ(r) has It is interesting to note that the above choice for the variables ε the disadvantage (Ponte Casta˜ neda and Willis 1999) that the stationarity (s) condition with respect to the variables L0 , given by (ε − ε(r) ) ⊗ (ε − ε(r) ) (r) = 0,

(37)

cannot be satisﬁed in general (i.e., unless the strain ﬁeld is constant in each phase). Because of this, the alternative, physically motivated prescription (r)

(r)

L0 = Lt (ε(r) )

(38)

was proposed by Ponte Casta˜ neda (1996) to close the system of equations deﬁning the eﬀective behavior of the nonlinear composite. Recalling the deﬁnition of the “tangent” modulus: . ∂ 2 w(r) (r) (ε), Lt (ε) = ∂ε∂ε

(39)

it is seen that prescription (38) is fully consistent with expression (18) in ˆ(r) → ε(r) in this the sense that it corresponds to taking the limit as ε expression. Once again, the phase averages and second moments of the ﬁelds in the nonlinear composite may be estimated using standard relations for a linear thermoelastic comparison composite : ε(r) = A(r) ε + a(r) ,

5T 2 ∂W (r) and Cε = (r) , c ∂L(r)

(40)

0

(r)

where the choices (35) and (38) must be made for the variables ε(r) and L0 deﬁning the properties of the phases in the linear comparison composite. It is remarked that there is a corresponding result for the eﬀective poten2 (σ), but, unfortunately, the result is not equivalent to the result (36). tial U In other words, there is a duality gap, which is known (Ponte Casta˜ neda 1996) to increase with increasing nonlinearity. The details are omitted here for brevity. However, it is remarked that the corresponding problems for the

LINEAR COMPARISON METHODS . . .

259

2T (σ), are dual to each other. 5T (σ) and U linear comparison composite, W Because of this, the expressions for the phase averages of the stress ﬁelds (r) σ (r) , and their corresponding ﬂuctuations Cσ are consistent with relations (40), provided that the quantities are normalized with respect to the average stress and strain quantities, σ and ε, in the linear comparison composite. In connection with the duality gap, it is important to note that it follows from the stationarity of the prescription (33) with respect to the variables (r) ε(s) (but not with respect to the L0 ) that the overall stress-strain relation (3) for the nonlinear composite may be written in the form:

1 3

4(r) ∂ ε(r) pq (r) (r) (r) ρ(r) ε(r) + , εij − εij Nijklpq εkl − εkl 2 ∂ε r=1 (41) where the ε(r) are determined from relations (40)1 , and where

σ=

N

c(r)

(r)

Nijklpq =

∂ 3 w(r) ε(r) . ∂ εij ∂ εkl∂ εpq

Notice that the “correction” term in the average stress-strain relation (r) depends on the phase ﬂuctuation tensors Cε . This correction term is actually a measure of the duality gap involved in the procedure. It is further noted that the so-called “aﬃne” method corresponds to dropping the correction terms in the constitutive relation (41), as pointed out by Masson et al. (2000). 3.3. GENERALIZED SECANT SECOND-ORDER ESTIMATES

The two prior sections have shown two types of estimates that can be derived from the general estimates (16). The ﬁrst type (20) has the advantage that it leads to bounds, but unfortunately, it has the disadvantage that it is only exact to second-order in the heterogeneity contrast, failing to agree to second-order with the small-contrast results of Suquet and Ponte Casta˜ neda (1993). On the other hand, the second class of estimates (36) has the advantage that it does recover the small-contrast expansion results exactly to second order, but it also suﬀers from some limitations, including the fact that it has been found to violate the bounds in some special situations (Leroy and Ponte Casta˜ neda 2001), and that it exhibits a large duality gap for strongly nonlinear systems, leading to physically unrealistic predictions for some extreme examples, such as porous media with an ideally plastic matrix phases (Ponte Casta˜ neda 1996). Another, perhaps more constructive, way to look at these estimates is to observe that the ﬁrst class of estimates (20) makes use of the second

˜ P. PONTE CASTANEDA

260

moments of the ﬁelds over the phases, while the second class of estimates (36) makes use of the ﬁrst moments, or phase averages. This suggests that perhaps better estimates may be obtained by trying to make use of both the ﬁrst and second moments of the ﬁelds over the phases. Noting that the ﬁrst and second moments are clearly involved in the general estimates (16) naturally suggest optimizing with respect to both the variables ε(r) (r) and L0 . However, as already mentioned, unfortunately, there is no general improvement in the bounds by optimizing with respect to both types of (r) variables, relative to just optimizing with respect to the L0 . This means that the corrector deﬁnition (14) must be enlarged to include other stationarity conditions than the extremal ones. On the other hand, it has also ˆ(r) = ε(r) in the corrector functions used for the been seen that the choice ε tangent second-order estimates (36) also leads to problems because the re(r) sulting estimates are not stationary with respect to the variables L0 . This suggests the need to consider the most general stationary conditions (18), leading to the the notion of a “generalized secant” modulus tensor, which is seen (refer to Figure 1) to be somewhat intermediate between the “secant” modulus, deﬁned by (26) and used in the context of the bound (21), and the “tangent” modulus deﬁned by (39) and used in the second-order estimate (33). (s) Thus, optimizing over all the variables ε(s) and L0 in expression (16), the following stationary variational estimate is obtained: N (r) 5T (ε; ε(s) , L(s) ) + 5 (ε) ≈ stat c(r) V (r) (ε(r) , L0 ) . (42) W W 0 (s) (s) L0 , ε r=1 Unfortunately, to date, no estimates fully satisfying the optimality conditions on both types of variables have been found. For this reason, the focus here will be on estimates that are stationary with respect to the variables (s) L0 , with the variables ε(s) held ﬁxed: N (r) 5T (ε; ε(s) , L(s) ) + 5 (ε) ≈ stat W c(r) V (r) (ε(r) , L ) . (43) W (s)

L0

0

0

r=1

ˆ(r) = ε(r) , and formally interchanging Then, making the assumption that ε (r) ˆ(r) and L0 the order of the stationary points with respect to the variables ε 5T and (14) in expression (43), where use is made of expressions (17) for W (r) for the V , leads to the following optimality conditions for the variables (r) L0 : ε(r) − ε(r) ) = (ˆ ε(r) − ε(r) ) ⊗ (ˆ

5T 2 ∂W c(r) ∂L(r) 0

= (ε − ε(r) ) ⊗ (ε − ε(r) ) (r) , (44)

LINEAR COMPARISON METHODS . . .

261

which can be seen to be a set of conditions on the second moments of the strain ﬁeld in the phases relative to the reference strains ε(r) . It is noted that these conditions are much less restrictive than conditions (37)—and therein lies the advantage of this choice of stationary points in the functions V (r) . However, as already discussed in the context of relations (27), the conditions (44) are too restrictive because they require the fourth-order tensors (ε − ε(r) ) ⊗ (ε − ε(r) ) (r) to be of rank 1. This means that the (r) choice of the tensors L0 cannot be completely general, and that therefore, (r) special classes of anisotropic tensors L0 must be considered. Then, as with the special case of relation (31), only certain traces—depending on the (r) choice of the tensors L0 —of relations (44) should be enforced. It should be emphasized, however, that the optimal choice for composites with isotropic phases is not isotropic for these estimates. Thus, Ponte Casta˜ neda (2002) (r) made use of the choice (29) for the L0 to estimate the eﬀective response of power-law composites with isotropic phases. In any case, the conditions ˆ(r) should be determined from certain, (44) suggest that the variables ε appropriately chosen traces of the second moments of the ﬂuctuations of the strain ﬁeld in the phases, which is a little diﬀerent from condition (27) identifying them with appropriate traces of the second moments of the relevant ﬁelds. Using appropriate traces of the equation (44), together with the expres5 can then be shown sion (14) for the functions V (r) , the estimate (43) for W to reduce to: 5 (ε) ≈ W

N

c(r) w(r) (ˆ ε(r) ) − ρ(r) (ε(r) ) · (ˆ ε(r) − ε(r) ) ,

(45)

r=1

where the variables ε(r) still remain to be speciﬁed. As already mentioned, it has not yet been possible to select the variables ε(r) optimally in the context of expression (42), which would lead to the requirements that: (s) (s) ˆ(s) ) = 0, (46) c(s) Lt (ε(s) ) − L0 (ε (s) − ε (s)

where the symbols Lt have been used once again to denote the tangent moduli of the phases. For this reason, thus far, resort has been made only of ad hoc choices for the variables ε(r) . The ﬁrst and perhaps most sensible choice is to enforce stationarity of only part of the estimate (45), namely, the part corresponding to the linear 5T . As we have seen in the context of the “secondcomparison problem W order” estimates (33), the result of this calculation is the condition (35),

˜ P. PONTE CASTANEDA

262 or

ε(r) = ε(r) .

(47)

It is important to emphasize that this choice does not satisfy the more general stationarity condition (46), but, on the other hand, is physically appealing, and considerably simpler from a computational point of view. Combining conditions (35) for the ε(r) with the conditions (44) for the ˆ(r) leads to the requirement that: ε . (r) (ˆ ε(r) − ε(r) ) ⊗ (ˆ ε(r) − ε(r) ) = (ε − ε(r) ) ⊗ (ε − ε(r) ) (r) = Cε ,

(48)

which must be interpreted in terms of appropriate traces. Similarly, the secant-type condition (18) now specializes to: (r)

ρ(r) (ˆ ε(r) ) − ρ(r) (ε(r) ) = L0 (ˆ ε(r) − ε(r) ),

(49)

(r)

where the tensor L0 has a certain number (less than or equal to 6) of independent components, and the expression (45) for the eﬀective potential of the nonlinear composite reduces to: 5 (ε) ≈ W

N

c(r) w(r) (ˆ ε(r) ) − ρ(r) (ε(r) ) · (ˆ ε(r) − ε(r) ) ,

(50)

r=1

which was ﬁrst given by Ponte Casta˜ neda (2002). Again, estimates for the phase averages and covariance of the ﬂuctuations of the strain ﬁelds in the nonlinear composite may be estimated using relations (40) for the linear thermoelastic comparison composite, where the (r) ˆ (r) and ε(r) , respectively . variables L0 and ε(r) must be set equal to L 0 Starting from a linear comparison composite with local complementary energy potential: 1 (r) (r) uT (σ) = u(r) (σ (r) ) + γ (r) (σ (r) ) · (σ − σ (r) ) + (σ − σ (r) ) · M0 (σ − σ (r) ), 2 (51) (r) (r) where the σ are uniform reference stresses, the M0 are uniform compliance tensors, and γ (r) = ∂u(r) /∂σ, it is possible to proceed in a completely analogous fashion, and derive a corresponding estimate for the eﬀective complementary energy potential of the nonlinear composite in terms of the 2T of the linear comparison composite deﬁned by reeﬀective potential U lations (51). Then, it can be shown that the nonlinear estimate reduces to: N 2 ˆ (r) ) − γ (r) (σ (r) ) · (σ ˆ (r) − σ (r) ) , U (σ) ≈ c(r) u(r) (σ (52) r=1

263

LINEAR COMPARISON METHODS . . .

where the variables σ (r) in (51) have been identiﬁed with the averages σ (r) = σ (r) of the stress in the various phases of the linear thermoelastic comparison composite deﬁned by (51). On the other hand, the variables ˆ (r) are obtained from appropriate traces of the relation: σ . (r) ˆ (r) − σ(r) ) = (σ − σ (r) ) ⊗ (σ − σ (r) ) (r) = Cσ , (53) ˆ (r) − σ (r) ) ⊗ (σ (σ ˆ (r) and σ(r) are related through the generalized secant condiwhere the σ tions (r) ˆ (r) ) − γ (r) (σ (r) ) = M0 (σ ˆ (r) − σ (r) ) (54) γ (r) (σ (s)

(r)

for the comparison compliance tensors M0 , and the Cσ denote the covariance matrices of the stress ﬂuctuations in phase r. 2 is not equivalent to the estimate Unfortunately, the estimate (52) for U 5 . In fact, it is known (Ponte Casta˜ (50) given earlier for W neda 2002) that there is also a duality gap in this theory, which although much smaller than the corresponding gap for the tangent second-order estimates, it is nevertheless nonzero. However, once again, the corresponding linear comparison 5T are exactly dual to each other. This means that 2T and W problems for U the estimates for the the phase averages and covariance of the ﬂuctuations of the stress and strain ﬁelds are consistent with each other provided that they are normalized with respect to the appropriate average stress and strain quantities. It is emphasized, however, that the constitutive relation of the linear comparison composite, given by: σ=

N

c(r) ρ(r) ε(r)

(55)

r=1

is not consistent with the constitutive relation that would be obtained from expressions (50) and (52), via relations (3) and (5). In spite of its simplicity, the estimate (55)—unlike estimates (50) and (52)— is not exact to secondorder in the contrast, and is therefore expected to be less accurate than the corresponding energy estimates. For completeness, it is remarked that another, simpler choice for the variables ε(r) is given by the prescription: ε(r) = ε,

(56)

which also fails to satisfy the more general stationarity condition (46). ˆ(r) Combining conditions (56) for the ε(r) with the conditions (44) for the ε leads to the result that: ε(r) − ε) = (ε − ε) ⊗ (ε − ε) (r) . (ˆ ε(r) − ε) ⊗ (ˆ

(57)

˜ P. PONTE CASTANEDA

264

On the other hand, the secant-type condition (18) now specializes to (r)

ε(r) ) − ρ(r) (ε) = L0 (ˆ ε(r) − ε), ρ(r) (ˆ

(58)

and the expression (45) for the eﬀective potential of the nonlinear composite reduces to: 5 (ε) ≈ W

N

c(r) w (r) (ˆ ε(r) ) − ρ(r) (ε) · (ˆ ε(r) − ε(r) ) .

(59)

r=1

2 , which again is not equivalent An analogous estimate may be obtained for U 5 . Worse than that, the corresponding linear to the estimate (59) for W 5T are also not equivalent (if the average 2 comparison potentials UT and W 2T ), so that this stress σ is used to deﬁne the reference stresses σ (r) in U version of the method, although a bit simpler and still second-order in the contrast, is somewhat less clean. Finally, it should be mentioned that the new estimates (50) and (52) 5 and U 2 of nonlinear composites require corfor the eﬀective potentials W 2T of the lin5T and U responding estimates for the eﬀective potentials W ear thermoelastic composites, from which the required estimates for the phase averages and covariance tensors of the ﬁelds can be computed. In this connection, it is relevant to recall that estimates of the self-consistent and Hashin-Shtrikman type for linear thermoelastic composites have been generated by Budiansky (1970) and Laws (1973), and Willis (1981), respectively. 4. Concluding Remarks In this paper, we have reviewed several closely related nonlinear homogenization methods that have been proposed by the author in recent years. The ﬁrst method, originally known as the “variational” method (Ponte Casta˜ neda 1991), and now also known as the “modiﬁed secant” method (Suquet 1995), has now been applied to a number of problems. A fairly exhaustive review of applications prior to 1998, mostly to porous and metalmatrix composites, are contained in the review article by Ponte Casta˜ neda and Suquet (1998). In recent years, applications have been given for polycrystals (Nebozhyn et al. 1999, 2001), including texture evolution in these materials (Gilormini et al. 2003). The second method, known as “tangent second-order” method (Ponte Casta˜ neda1996) was also reviewed in Ponte Casta˜ neda and Suquet (1998), and has been applied recently to polycrystals (Bornert and Ponte Casta˜ neda 1998, Bornert et al. 2001), as well as to reinforced elastomers (Ponte Casta˜ neda and Tiberio 2000). However, this

LINEAR COMPARISON METHODS . . .

265

method has been largely superseded by the third method, known as the “second-order” method (Ponte Casta˜ neda 2002). As already mentioned, this method is also exact to second-order in the heterogeneity contrast, but has the added advantage of being more accurate for systems with large contrasts and strong nonlinearities, where the ﬁeld ﬂuctuations in the phases can be large, and has the property that it leads to estimates satisfying the variational bounds of the ﬁrst method. The new second-order method has been applied to estimate not only the eﬀective, or macroscopic behavior, but also to assess the ﬁeld ﬂuctuations in two-phase nonlinear composites (Idiart and Ponte Casta˜ neda 2003), viscoplastic polycrystals (Liu and Ponte Casta˜ neda 2004), including comparisons with FFT numerical simulations (Lebensohn et al. 2004), as well as in polymeric composites and porous elastomers (Lopez-Pamies and Ponte Casta˜ neda 2003, 2004). Finally, at the theoretical level, it should be mentioned that open questions still remain, particularly regarding the optimal choice of the reference strains ε(r) , which deﬁnes the linear comparison composite in the context of the general estimates (43), as well as related issues concerning the existence of a duality gap. Acknowledgment This work was supported by NSF grants CMS-02-01454 and DMS-0204617. References 1. Bobeth, M. and Diener, G. (1987) Static elastic and thermoelastic ﬁeld ﬂuctuations in multiphase composites. J. Mech. Phys. Solids 35, 37–149. 2. Bornert, M. and Ponte Casta˜ neda, P. (1998) Second-order estimates of the self-consistent type for viscoplastic polycrystals. Proc. R. Soc. Lond. A 356, 3035–3045. 3. Bornert, M., Ponte Casta˜ neda, P. and Zaoui, A. (2001) Second-order estimates for the eﬀective behavior of viscoplastic polycrystalline materials. J. Mech. Phys. Solids 49, 2737–2764. 4. Budiansky, B. (1970) Thermal and thermoelastic properties of composites. J. Comp. Mater. 4, 286–295. 5. deBotton, G. and Ponte Casta˜ neda, P. (1992) On the ductility of laminated materials. Int. J. Solids Structures, 29, 2329–2353. 6. deBotton, G. and Ponte Casta˜ neda, P. (1993) Elastoplastic constitutive relations for ﬁber-reinforced solids. Int. J. Solids Structures 30, 1865-1890. 7. deBotton, G. and Ponte Casta˜ neda, P. (1995) Variational estimates for the creep behavior of polycrystals. Proc. R. Soc. London A 448, 421–442. 8. Gilormini, P., Liu, Y. and Ponte Casta˜ neda, P. (2003) Variational selfconsistent estimates for texture evolution in viscoplastic polycrystals— Application to titanium. Acta Mater. 51, 5245–5437. 9. Hashin, Z. and Shtrikman, S. (1962) On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids 10, 335–

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342. 10. Hashin, Z. and Shtrikman, S. (1963) A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids 11, 127–140. 11. Hershey, A. V. (1954) The elasticity of an isotropic aggregate of anisotropic cubic crystals. ASME J. Appl. Mech. 21, 236–240. 12. Hill, R. (1963) Elastic properties of reinforced solids : Some theoretical principles. J. Mech. Phys. Solids 11, 357–372. 13. Hill, R. (1965) Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13, 89–101. 14. Hu, G. (1996) A method of plasticity for general aligned spheroidal void of ﬁber-reinforced composites. Int. J. Plasticity 12, 439–449. 15. Hutchinson, J. W. (1976) Bounds and self-consistent estimates for creep of polycrystalline materials. Proc. R. Soc. Lond. A 348, 101–127. 16. Idiart, M. and Ponte Casta˜ neda, P. (2003) Field ﬂuctuations and macroscopic properties for nonlinear composites. Int. J. Solids Structures 40, 7015–7033. 17. Kreher, W. (1990) Residual stresses and stored elastic energy of composites and polycrystals. J. Mech. Phys. Solids 38, 115–128. 18. Kr¨ oner, E. (1958) Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einkristalls. Z. Phys. 151, 504–518. 19. Laws, N. (1973) On the thermostatics of composite materials. J. Mech. Phys. Solids 21, 9–17. 20. Lebensohn, R., Liu, Y. and Ponte Casta˜ neda, P. (2004) Macroscopic properties and ﬁeld ﬂuctuations in model power-law polycrystals: full-ﬁeld solutions versus second-order estimates. Proc. R. Soc. Lond. A 460, 1381–1405. 21. Leroy, Y. and Ponte Casta˜ neda, P. (2001) Bounds on the self-consistent approximation, for nonlinear media and implications for the second-order method. C.R. Acad. Sc. Paris IIb 329, 571–577. 22. Liu, Y. and Ponte Casta˜ neda, P. (2004) Second-order theory for the eﬀective behavior and ﬁeld ﬂuctuations in viscoplastic polycrystals. J. Mech. Phys. Solids 52, 467–495. 23. Liu, Y. and Ponte Casta˜ neda, P. (2004) Homogenization estimates for the average behavior and ﬁeld ﬂuctuations in cubic and hexagonal viscoplastic polycrystals. J. Mech. Phys. Solids 52, 1175–1211. 24. Lopez-Pamies, O. and Ponte Casta˜ neda, P. (2003) Second-order estimates for the large deformation response of particle-reinforced rubbers. C.R. Mecanique 331, 1–8. 25. Lopez-Pamies, O. and Ponte Casta˜ neda, P. (2004) Second-order estimates for the eﬀective response and loss of ellipticity of porous rubbers at large deformations. J. Elasticity, submitted. 26. Masson, R., Bornert, M., Suquet, P. and Zaoui, A. (2000) An aﬃne formulation for the prediction of the eﬀective properties of nonlinear composites and polycrystals. J. Mech. Phys. Solids 48, 1203–1227. 27. Michel, J., Moulinec, H. and Suquet, P. (1999) Eﬀective properties of composite materials with periodic microstructure: a computational approach. Comp. Meth. Appl. Mech. Engng. 172, 109–143. 28. Michel, J., Moulinec, H. and Suquet, P. (2000) A computational method based on augmented Lagrangians and Fast Fourier Transforms for composites with high contrast. Comput. Modelling Engng. Sc. 1, 79–88. 29. Moulinec, H. and Suquet, P. (1998) A numerical method for computing the

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49. Willis, J. R. (1992) On method for bounding the overall properties of nonlinear composites: correction and addition. J. Mech. Phys. Solids 40, 441–445. 50. Willis, J. R (2000) The overall response of nonlinear composite media. Eur. J. Mech. A/Solids 19, S165–S184.

MODELS OF MICROSTRUCTURE EVOLUTION IN SHAPE MEMORY ALLOYS

ˇ T. ROUB´ICEK

Mathematical Institute, Charles University, Sokolovsk´ a 83, CZ-186 75 Praha 8, Czech Republic, and Institute of Information Theory and Automation, Acad. of Sci., Pod vod´ arenskou vˇeˇz´ı 4, CZ-182 08 Praha 8, Czech Republic.

Abstract: This contribution surveys menagerie of models for twinning-like microstructure and its evolution in crystallic alloys exhibiting shape-memory eﬀects. Various levels of description of microstructure are distinguished, as well as various evolutionar mechanisms (iso- or aniso-thermal, viscous or/and plastic). Keywords. Martensitic phase transformation, twinning, mathematical and computational modelling, nonconvex stored energy, rate-independent dissipation, activation, inelastic response, thermomechanical evolution.

1. Introduction, crystallic alloys, shape-memory eﬀect. Shape-memory alloys (=SMAs) belong to so-called smart materials which enjoy important applications especially in aerospace or mechanical engineering and human medicine, and have therefore been subjected to intensive theoretical and experimental research in past decades. Sometimes, regularly orchestrated atoms in stoichiometric ratio (as, e.g., Ni2 MnGa or NiTi) are linked by chemical bonds, and then one speaks about intermetalics (or intermetalic alloys); as usual, we will use the abbreviation “SMA” for intermetalics, too. SMAs exhibit speciﬁc hysteretic (=rate-independent memory) stress/stain/temperature response, which is called a shape-memory eﬀect (=SME); cf. Figure 4 and Remark 3.9 below. The mechanism behind SME is quite simple: atoms tend to be arranged in various crystalographical conﬁguration (in particular, having diﬀerent symmetry groups) under diﬀerent temperatures. Mostly, at higher temperatures atoms tend to form a grid with higher symmetry (typically cubic) which is referred to as the austen-

269 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 269–304. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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one variant of martensite

parent austenite (cubic)

another variant of martensite

tw pl in an ni e ng

ite phase (cf. Figure 1 left) while at lower temperatures they tend to form a lower-symmetrical grid called martensite phase. Due to symmetry, the lower-symmetrical grid may occur in several variants (cf. Figure 1 middle) which can be even combined (we speak about a coherent co-existence) with each other, forming thus so-called twins of two variants (cf. Figure 1 right).

twinned martensite composed from two variants

Figure 1. A schematic 2-dimensional situation: Left: a “cubic” grid. Middle: its martensitic transformation. Right: twins created by matching two slightly rotated triangles of both martensitic variants.

Depending on a particular alloy, the decay of symmetry of martensite comparing to the cubic grid of the parent austenite can vary. If one side of this cube is elongated while the others are equally shrunk (or vice versa), it results to tetragonal martensite which can live in 3 variants. An example is InTl, or Nickel- of Iron-based alloys (as NiMnGa, NiAl, FePt, etc.). If all sides are diﬀerent from each other but angles are still kept 90◦ , we get orthorhombic martensite which can live in 6 variants; e.g. AuCd and some Copper-based alloys (as CuNiAl, e.g.). If even one angle diﬀers from 90◦ , we get the less symmetrical martensite, monoclinic, which forms 12 variants. An example is some copper-based alloys (as CuZn or CuAlZn, e.g.), or the commercially most successful alloy, namely NiTi; under the name NiTiNOL (NOL=Naval Ordnance Laboratory), it has been invented in [43]. Besides, a trigonal martensite has all sides mutually equal but angles diﬀerent from 90◦ , and it forms 4 variants. Depending on temperature, the parent austenite may co-exist with martensite, or sometimes even two or more types of martensite can occur simultaneously. The parent austenite in a stress-free conﬁguration represents a natural state of the material. From a viewpoint of continuum mechanics, we can thus speak about a reference conﬁguration of a specimen occupying a domain Ω ⊂ IR3 and, as usual, y : Ω → IR3 denotes the deformation and u : Ω → IR3 the displacement, related to each other by y(x) = x + u(x), x ∈ Ω. Hence deformation gradient equals F = ∇y = I + ∇u, where

MODELS OF SHAPE MEMORY ALLOYS

271

I ∈ IR3×3 denotes the identity matrix and ∇ is the Lagrangean gradient operator. Referring to Figure 1(right), the two triangles arise from the parent phase by an aﬃne displacement. The natural condition of continuity of the overall piece-wise aﬃne displacement u which create the right-handside trapezoid from the left-hand-side square leads to Hadamard’s rankone condition on the piece-wise constant deformation gradient (taking two values on Figure 1(right), say F1 and F2 ), i.e. Rank(F1 − F2 ) = 1,

or equivalently

F1 − F2 = a ⊗ n

(1)

for some (so-called transformation strain) vector a and the unit vector n referring to the twinning plane normal, cf. Figure 1(right). The layers with the deformation gradients F1 and F2 can alternately be matched, forming thus a laminate, cf. the right/up triangle on Figure 2(left). Depending on width of the particular layer, both phase(variant)s may have diﬀerent volume fractions, say λ1 , λ2 ≥ 0, λ1 + λ2 = 1. Viewing Figure 2(left) from a distance, the average deformation gradient is then λ1 F1 +λ2 F2 . This can be further matched, with only a small deformation (hence small stored energy) in the transient region, with another deformation gradient, say F0 , which should now satisfy further rank-one connection condition:

Rank F0 − (λ1 F1 + λ2 F2 ) = 1,

λ1 + λ2 = 1, λ1 , λ2 ≥ 0.

(2)

The reader can think, e.g., about F0 corresponding to a parent martensite; canonically, in a stress-free state, F0 would then be an identity matrix (up to an orientation-preserving rotation) and the boundary between the twined martensite and the austenite is then called the habit plane, cf. Figure 2(left). Yet, laminates can be combined in layers-within-layers to second-order (or even higher-order) laminates, or some other self-organization as wedges or branching can be observed and explained by mere crystallographic arguments, cf. [22, 23, 26, 28, 30, 58, 97, 106, 115, 116, 129, 151, 162, 165, 216]. In reality, the stored energy even need not be locally completely minimized so that, as a result, in real specimens, we can often observe very complicated conﬁgurations that may seem rather chaotic, cf. Figure 2(right).

ha

F1 F 2 F1 martensite

t pl win a n ni e ng

ˇ T. ROUB´ICEK

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tp

bi la ne

cf. Fig.1 (left)

F0 austenite

cf. Fig.1 (right) 100µ m

Figure 2. Left: matching austenite with a laminate from two martensitic variants. Right: a rather chaotic arrangement of martensite in a CuAlNi single ˇ crystal; courtesy of V´ aclav Nov´ ak and Petr Sittner, Institute of Physics, Academy of Sciences of the Czech Republic.

Mathematical and computational modelling of SMAs represents a certain tool of theoretical understanding of transformation processes and may both complete experimental results and predict response of new materials or applications in engineering workpieces even before casted or built. However, modelling of such “chaos” as on Fig. 2(right) and even its evolution may seem (and, to a large extent, really is) hopeless. Anyhow, there is a great theoretical/applicational challenge for such modelling, which lead to enormous eﬀort producing wide menagerie of models during past decades. As seen also from Figure 2, the real situation is essentially a multi-scale one. Eﬀective description of the microstructure in SMA can be done, depending on a purpose and on available data as well as on computational abilities, at various levels compromising rigor with phenomenology. The classiﬁcation of the levels is not understood in a uniﬁed manner in the literature but let us agree here to use the following convention, also cf. Figure 3: 1. Atomic level:

2. Microscopic level: 3. Mesoscopic level: 4. Meso/macrocontinuum-mechanics microstructure -scopic level: description of polycrystalic described by material microstructure volume fractions

5. Macroscopic level and a final product:

Figure 3. Various levels of modelling of single-crystal and polycrystalinne SMAs up to a ﬁnal product, here a knitted NiTi peripheral vascular stent, ELLA-CS ﬁrm, Hradec Kr´ alov´e, Czech Republic (URL: http://www.ellacs.cz).

1. Atomic level: the description counts barycenter of particular atoms and an inter-atomic potential. Scales about 10 nm can be thus modeled. 2. Microscopic level: continuum mechanics is used to describe deformation, strain, and stress at particular “points”. Scales about 1-100 µm

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can thus be modeled unless one may rely that the modelled small volume is enough representative for the whole homogeneously-deformed single-crystal. 3. Mesoscopic level: an “averaged” deformation is treated by tools of continuum mechanics while the microstructure is described essentially by volume fractions which mix deformation gradients of particular phases (or phase variants). This may describe scales of large single crystals as indeed often used in labs. 4. Meso/macro-scopic level: macroscopic deformation with volume fractions are still main ingredients to describe conﬁguration at a given “macroscopic point” but no speciﬁc orientation or anisotropy is recorded. Such models addresses essentially polycrystals on the scale of engineering interest. 5. Macroscopic level: all detailed information about the microstructure is suppressed. This compromises a detailed description on previous levels with eﬃcient computational realizability needed usually in advanced engineering applications. Models on this level may suppress even any spatial dependence and then, contrary to previous distributedparameter models, they work with lumped parameters. So far, we spoke rather about natural, stress-free conﬁguration of particular phase(variant)s. However, loading by outer force or by a displacement of the boundary (a so-called hard device) may impose some deformation of the atomic grid so that atoms are not in their energetically optimal positions. Hence, some energy is stored in the interatomic links. The elastic response bears always a great phenomenology, usually recorded by a tensor of elastic constants assigned to each phase. When loaded even more, the martensitic phase(variant)s can usually transform to each other (or possibly also to a higher-symmetrical austenite). Depending whether only particular variants of martensite(s) or also a parent austenite is involved (i.e. depending on temperature), the resulted response is called quasi-plasticity and pseudo-elasticity (sometimes called also super- or ferro-elasticity), respectively; cf. Fig. 4. In both cases, we will speak about phase transformation (=PT); strictly speaking, only the latter case is a (diﬀusionless ﬁrst-order) PT, the former case being referred to rather as a reorientation of martensite. Disregarding some exceptional materials, PT requires certain activation stress (typically tens of MPa) and thus a certain energy which is then dissipated to the chaotic atomic vibrations, i.e. to heat. Chaotic vibrations of the whole grid are viewed macroscopically as temperature and, to much lesser extend, as acoustic emission in the frequency range of MHz, cf. [124]. PT processes are, except very fast time scales, rate independent (at least if inﬂuence of possible temperature variation to PTs is neglected) and represent thus a certain plastic response of the material, beside the usual plas-

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ticity by slip which is however activated by much higher stress (typically hundreds of MPa). In contrast to plasticity, PT in SMAs is reversible in the sense that it can be repeated in many (sometimes millions) cycles without observable changes of structure of SMA.

stress

M

stress

M

A strain

M

strain

M

Figure 4. Schematic stress/strain response of SMA: quasiplasticity at lower temperature (left) vs. pseudoelasticity at higher temperature (right). “A” refers to austenite, “M” to martensite.

The dynamics of the martensitic PT is a complicated, and still not a fully understood process. It is now quite established that the “internal friction is also controlled by dislocations and their interactions with other lattice defects”, cf. [91, p.48]. These 1- or 0-dimensional singularities in the atomic grid make the dissipative mechanism to a large extend independent from the stored mechanism in the 3-dimensional bulk, similarly like amount of Carbon (as an impurity) in Iron dramatically inﬂuences plastic response of the resulting steel (by making movement of dislocations harder) without changing substantially elastic constants. Particular models diﬀer not only by description of the microstructure (as classiﬁed on Figure 1) but also by the ways how they treat the dissipative mechanism. Some of them ignore it at all, some of them try to derive it from the stored energy, and some of them allows to prescribed it rather independently, admitting thus a certain phenomenology. Also, some models ignore changes of temperature, assuming implicitly that processes are slow enough so that all produced heat can be transferred out much faster, while the other models counts with fully thermodynamical evolution. The purpose of this paper is primarily to give an overview of models of various sorts to provide an orientation for non-experts and possibly to widen knowledge of experts dealing often with only a rather speciﬁc class of models. Besides, the presentation focuses to the models [16, 143, 167, 169, 181, 182, 183]. Due to a limited scope, the reference list certainly cannot be considered as complete, however, and is partly inﬂuenced by author’s own activities. Numbering of the following Sections is in accord with the classiﬁcation on Figure 3.

MODELS OF SHAPE MEMORY ALLOYS

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2. Atomistic-level modelling. On this level, modelling plays with suﬃciently large ensembles of individual atoms. The conﬁguration of the atoms at a speciﬁc time t is thus described by their positions and impulses; say ri ∈ IRd will be the position of i-th atom, mi its mass, and midri/dt its impulse. The only essential ingredience of the model is the potential energy of the ensemble V (r) depending on the position vector r = (ri). The dynamic of r, i.e. r = r(t), is then governed by the Newton system mi

d2 r i = −Vri (r) dt2

(3)

where Vri is the partial gradient of V with respect to ri. Of course, (3) is to be completed by the initial conditions for r and for the impulse dr/dt at time t = 0. In this setting, called “standard ensemble”, (3) is conservative, d i.e. the sum of the potential and the kinetic energy V (r) + 12 i mi( dt ri)2 is d conserved. Anyhow, by augmenting (3) suitably (e.g. by a term like ±ε dt ri), some outer cooling or heating (i.e. decrease or increase of kinetic energy) can computationally be realized to activate thermally PT. To hit general martensitic grids as well as their elastic responses, an embedded-atom method (EAM) has been proposed by Daw and Baskes [56], based on the philosophy that the energy of an atom is counted as the work needed to embed this atom into the local electron density induced by the other atoms summed with a core-core (pairwise) repulsion potential. Threedimensional simulations of up to millions-of-atoms ensemble are possible, see [14, 59, 79, 80, 101, 102, 136, 137, 168, 184].

Figure 5. FeNi nanoparticle of approx. 104 atoms: Left: cubic austenite, Middle: tetragonal martensite resulting by PT after cooling, Right: a (part of a) cross-section with a twinning plane like on Fig. 1(right); courtesy of Marcel Arndt, Universit¨ at Bonn, Germany.

Possibilities of atomistic modelling are indeed interesting. In particular, link between computational statistical mechanics and continuum mechanics can be seen very explicitly, some parameters for the continuum models can be observed from the atomistic simulations, under-surface view can easily be got, and, e.g., by varying a composition of an alloy in computer, one

ˇ T. ROUB´ICEK

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can investigate new alloys without casting them. Time-scales possible to simulate, however, ranges rather nanoseconds only. Remark 2.1 (Lennard-Jones potential) The simplest option, inspired from gas physics, takes into account only nearest atoms and the potential of inter-atomic attractive/repulsive forces is in the Lennard-Jones form: V (r) =

Vij (ri, rj ) with Vij (ri, rj ) = εij

i<j

σij 12 σij 6 − ri−rj ri−rj

(4)

where εij and σij are physical constant. In fact, V in (4) counts rather nearest neighbours only. Interestingly, even in a two-dimensional case (d = 2) and very small ensembles (few tens of atoms), (4) can create cubic/“hexagonal” PT by changing the over-all energy, i.e. by outer “heating/cooling” the ensemble, see [105]. Anyhow, the pair potential of type (4) cannot model shear anisotropy and thus cannot compete with the EAM. On the other hand, such a potential is suitable for limit passage to a continuous model at least in 1D case, see [35]. 3. Continuum-mechanical level: microscopic models As outlined in Introduction, mechanisms through which the material stores and dissipates energy are determinative for inelastic response of SMAs. Thus, corresponding response functions are ultimately the main ingredients for any model. 3.1. FREE ENERGY

The speciﬁc energy stored in the inter-atomic links in the continuum ˆ θ) is phenomenologically described as a function of the deψˆ = ψ(F, formation gradient F and, in anizothermal case, a temperature θ. The ˆ θ) = ψ(RF, ˆ frame-indiﬀerence, i.e. ψ(F, θ) for any R ∈SO(d), the group of orientation-preserving rotations, requires that ψˆ in fact depends only on the (right) Cauchy-Green stretch tensor C := F T F , cf. Examples 3.2 and 3.3. By d, we denote the dimension of the specimen domain Ω ⊂ IRd, though physically relevant case is d = 3 only. As F =I+∇u, we can express the speciﬁc stored energy in terms of the displacement gradient as

ψ = ψ(x, ∇u, θ) = ψˆ (I + ∇u)Q(x), θ

(5)

where Q(x) ∈ SO(d) is the rotation of the reference (austenite) grid at a given point x; in case of a singlecrystal specimen Q is independent of x while in polycrystals Q is piecewise constant referring to particular grains. The Piola-Kirchhoﬀ stress σ : IRd×d → IRd×d is given by

277

MODELS OF SHAPE MEMORY ALLOYS

∂ σ = ∂(∇u) ψ(x, ∇u, θ). Sometimes, a higher-gradient contribution (often called capillarity in analogy with ﬂuids or, perhaps better, an interfacial energy) of the type µ2 |∇mu|2 , µ ≥ 0, m ≥ 2, augments the stored energy to take into account a longer-range interaction in the atomic grid; sometimes a general positive-deﬁnite quadratic form instead of 12 | · |2 are considered, too.

Example 3.1 (One-dimensional model). Based on Landau-Devonshire’s theory, ﬁrst SMA model seems to have been proposed by Falk [62] for d = 1 by putting

ˆ θ) = αF 6 − βF 4 + (γ + δθ)F 2 − cθln θ , ψ(F, θ0

(6)

with some parameters α, β, δ ∈ IR+ , γ ∈ IR, c heat capacity, and θ0 a reference temperature, and independently also in [150]. Later, such higher-order ˆ θ) has been formulated in d = 3 situation for copper-based expansion of ψ(·, alloys (as CuAlNi) in terms of small-strain tensor has been proposed in [64]. A lot of authors studied this model both theoretically and numerically, cf. [5, 40, 41, 196, 199] or also [36, Chap.4]. Example 3.2 (Ericksen-James potential). Cubic/tetragonal PT of In-20.7 at% Tl alloy was proposed in [60] (see also, e.g., [53, 134]) to employ a frame-indiﬀerent potential ψˆ in the form ˆ θ) = (α0 + α1 θ) ψ(F,

+ β

3C11 trC

!

+ γ

!

−1

3C11 trC

−

3C11 trC

2

−1

+

3C22 trC − 1 2

22 1 + 3C trC

3C22 trC

3C33 trC 2

−1

2

−1

−1

+

3C33 trC

+

−1

3C33 trC

−1

2 "

2 "2

2 2 2 2 2 2 + δ C12 +C13 +C23 +C21 +C31 +C32 + ε(trC − 3)2 − cθln

(7)

θ θ0

,

where again C = F F and α0 , α1 , β, ...ε are phenomenological coeﬃcients, c is the heat capacity. This potential has 4 wells of the form SO(3)Uα with U1 = I := diag(1, 1, 1) and U2,...,4 = diag(η1 , η1 , η2 ) up to permutations with η’s numbers close to 1, related with lattice parameters a0 , a, and b by η1 = a/a0 and η2 = c/a0 ; here, a0 is the size of the cubic cell of the austenite while (a × a × b) is related to prism of the tetragonal martensitic cell. Free energy for InTl (and also for another indium-based alloys as, e.g., InCd, InPb, or InSn) was also proposed in [117]. Example 3.3 (St.Venant-Kirchhoﬀ-like form). Another type of potentials with more explicit reference to measured data and more universal use can be constructed as follows. We consider that the material can occur in L

ˇ T. ROUB´ICEK

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stress-free conﬁgurations that are determined by distortion matrices U , = 1, ..., L, which are independent of θ, i.e. thermal expansion is neglected. These are related with lattice parameters that are usually known with 3or 4-digit accuracy. One can imagine U1 = I corresponding to the cubic parent austenite in the stress-free conﬁguration taken as the reference one, while the others U are related with particular martensitic variants. E.g., for a tetragonal martensite, U , = 2, ..., L = 4 are as in Example 3.2, while in case of orthorhombic martensite, all sides of this prism diﬀer from each other, which gives rise to 6 diagonal matrices, then L = 7. A monoclinic martensite creates even 12 (i.e. L = 13) nondiagonal matrices. The frame-indiﬀerent free energy of particular phase(variant)s is considered as a function of Green strain tensor related to the distortion of this phase(variant), namely ε = 12 ((U )−1 F F U −1 − I). In the simplest case (cf. [165, Sect.6.6], e.g.), one can consider a function quadratic in terms of ε of the form ψˆ (F, θ) =

d i,j,k,l=1

θ

ε ij Cijkl ε kl − c θln

θ0

+ d ,

(8)

} is the 4th-order tensor of elastic moduli satisfying where C = {Cijkl the usual symmetry relations depending also on symmetry of the speciﬁc phase(variant) , and θ is a temperature (sometimes considered as a parameter only), while c the heat capacity of this phase(variant) and d is some oﬀset. The overall stored energy is assembled as ψˆ (F, θ) L − ˆ θ) := −kB θ ln kB θ ˆ θ) := min ψˆ (F, θ) or ψ(F, e (9) ψ(F,

=1,...,L

=1

where kB is the Boltzmann constant (per unit volume). Both options exhibit the same multi-well character, the latter option being backed up by statistical physics while the former one being computationally simpler and keeping the wells precisely at the orbits SO(3)U . For θ as a ﬁxed parameter, the former option in (9) has been used in [17] for CuAlNi undergoing cubic/orthorhombic PT and in [15] for NiMnGa with cubic/tetragonal PT. However, the data required for this potential are available in many other alloys but the measurements of the elastic tensor C

which are standardly done only for the austenite (with few exceptions, as CuAlNi in [219] or CuZnAl in [76]) and thus one must consider C 1 = ... = C L as a certain approximation. As to c and d in (8), see Remark 3.11. An often used “geometrically-linear” modiﬁcation [106] replaces ε by the small strain tensor 12 (F + F − U − U ), which oﬀers several simpliﬁcations but has limited applications, cf. [27, 113] for a comparison.

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3.2. DISSIPATION ENERGY

PT in SMAs is characterized by a speciﬁc dissipation which results to a hysteretic response in stress/strain/temperature diagrammes. This is, to a large extent, rate-independent, activated process similar like standard slip plasticity in metals and proper modelling of the dissipation is equally important as the stored energy, in particular to model evolution. Unfortunately, for dissipation mechanisms our understanding is much worse than for the stored energy and “much remains unknown concerning the nucleation and evolution of microstructure, and the resultant hysteresis” [33]. As the dissipation seems essentially inﬂuenced by various impurities and dislocations, there is even more phenomenology needed than for the stored energy. A conventional approach to obtain a certain hysteretic-like response thanks to the multi-well character of the stored energy in SMA is based on a mere viscosity-like dissipation, i.e. the dissipation rate is considered in a general form (usually for n = 1) is

ν ∇n

∂u 2 , ∂t

ν ≥ 0.

(10)

To achieve greater generality and allowing a discussion of the “viscosity” approach, we will admit also n ≥ 1, which also may correspond with the presence of the higher-order capillarity-type term µ2 |∇mu| in the stored energy. Beside the frame-indiﬀerence objection [10], cf. also citedemoulini and Remark 3.8, there are still some doubts about the dissipation potential (10) which, as we will discuss below, cannot model rate-independent dissipation that could record a phenomenology independent of stored energy. Hencefore, an attempt to build a phenomenology into the continuumlevel model based on partial-diﬀerential equations (or inequalities) was proposed in [181, Formula (33)] and was further developed in [16, 167, 169]. This (to some extent simpliﬁed) standpoint is that the amount of dissipated energy within the particular PT (here it is meant also M/Mtransformation, i.e. of one variant of the martensite transforms to another, sometimes referred rather as a re-orientation of martensite) can be described by a single, phenomenologically given number (of the dimension J/m3 =Pa). This philosophy is to design at least the energetics in accord with experiments, if the activated PT dynamics cannot be understood in detail by more rigorous arguments, and has independently been adopted in physics, see [93, 206, 214]. For this, we need to identify the particular phase(variant)s and thus deﬁne a continuous mapping L : Ω × IRd×d → L L L where L := { ζ ∈ IR ; ζ ≥ 0, = 1, ..., L, i=1 ζ = 1 } is a simplex with L vertices. Like (5), we assume ˆ + A)Q(x)), with Lˆ : IRd×d → L, L(x, A) = L((I

(11)

ˇ T. ROUB´ICEK

280

with Q referring to (5). Again, Lˆ is related with the material itself and thus is expected to be frame indiﬀerent. We have in mind that the components {Lˆ1 , ..., LˆL} of Lˆ = (Lˆ1 , . . . , LˆL) form a partition of unity on IRd×d such that L (F ) is equal 1 if F is in the -th phase, i.e. F is in a neighbourhood of -th well SO(d)U of ψˆ (which can be identiﬁed according to the stretch ˆ ) in tensor F F closed to U U like in [78, 144, 146]). Of course, L(F the (relative) interior of L indicates F in the spinodal region where no deﬁnite phase is speciﬁed. Hence λ plays the role of what is often called a vector of order parameters or a vector-valued internal variable, cf. [141]. The concrete form of Lˆ does not seem to be important as long as Lˆ enjoys the above properties. The phenomenology itself is considered through the choice of a “norm” on IRL (not necessarily Euclidean and even not symmetric), let us denote it by | · |L; its physical dimension will be Jm−3 =Pa. The desired meaning is to set up the speciﬁc energy E k needed for PT of a phase(variant) to k as |e − ek|L, where e = (0, .., 0, 1, 0, ...0) ∈ IRL is the unit vector with 1 at the position . The (pseudo)potential of dissipative forces that corresponds this phenomenology is ∂L(∇u) dx. Ω

∂t

(12)

L

This means, considering a process over the time interval [t1 , t2 ], the overall dissipated energy by all underwent PTs in the whole specimen Ω will be

t2 t1

∂ L(∇u) dxdt = Ω

∂t

L

Var

Ω t∈[t1 ,t2 ]

L ∇u(t, x) dx

(13)

where the total variation “Var” with respect to the (possibly nonsymmetric) norm | · |L counts which PTs (and how many times) has been undergone in the point x. The phenomenological dissipated energies E k are then to be got from experiments, which by far need not be simple, however. Essentially, one needs to perform a single-crystal experiment where particular PTs can be clearly distinguished. Then the energies E k can be read from hysteresis on the stress/strain diagram. A less ambitious idea is to say a bit speculatively, e.g., that particular E k are proportional to stretch change, i.e. to a distance of Ui Ui from Uj Uj and then only one E k is to be determined from experimental data, or to combine model-ﬁtting technique with experiments. The last option was used in [206, p.331] for Ti-Ni single crystal, leading to E k = 4.7MPa if either or k refers to austenite (i.e. energy for A/M PT) while E k = 0 otherwise (i.e. M/M-reorientation is considered as nondissipative, which however is not completely true, of course).

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Let us remark that a lightly diﬀerent construction of Lˆ but of a similar spirit, relying on particular PTs rather than particular phase(variant)s, has been used in [13, 16]. 3.3. DYNAMICS OF ISOTHERMAL PHASE TRANSFORMATION

Fixing temperature θ as a constant, the stored energy ϕ(·) = ψ(·, θ) in the general form (5) augmented possibly by the capillarity-like term 12 µ|∇mu|2 together with the conventional viscosity-type dissipation (10) and possibly ∂ u|2 , ≥ 0 a mass density, leads to the with the speciﬁc kinetic energy 12 | ∂t system of d-equations

∂u ∂2u − div σ(∇u) + ν(−1)n∆n + µ(−1)m∆mu = f, ∂t2 ∂t

(14)

with the “elastic” part of Piola-Kirchhoﬀ stress response σ(x, ∇u) := ∂ ∂(∇u) ϕ(x, ∇u, θ) and f is the volume loading force. Of course, one is to complete (14) by initial and boundary conditions, say u(t, ·) = u0 and ∂ ∂t u(t, ·) = v0 for the initial time t = 0 and, for simplicity, homogeneous Dirichlet boundary conditions on the boundary ∂Ω of Ω; of course, usual loading regimes rather uses f = 0 but control u (the “hard device”) or the normal stress on ∂Ω (the “soft device”). By varying m, n, µ, ν, and , we can cover a large variety of models. Actually, the model (14) has been extensively studied in the literature for various particular cases. Classical viscosity/capillarity-like model (i.e. µ, ν > 0, n = 1, m = 2) has been studied, e.g., in [2], and [107, 166, 187] for = 0, [85, 209, 213] for d = 1. The model with no capillarity (i.e. µ = 0) has been studied, e.g., in [9, 44, 55, 57, 72, 95, 163, 185, 203, 207, 213]. The model with the zero-derivative term (i.e. µ > 0 and m = 0) has been used in [21, 73, 203, 211] to imitate roughly (without physical basis) multidimensional-like eﬀects even if d = 1, showing a paradoxical behavior exhibited by the model without capillarity, namely that sharp interfaces cannot move if once developed. Zero-capillarity model but with viscosity realized through a Volterra integral operator was proposed in [65]. For µ = ν = 0, the system is fully hyperbolic with non-monotone highest-order term and with complex troubles related with such systems especially if d > 1. Anyhow, for d = 1, this situation has been addressed in [208] by augmenting (14) by a “microkinetic” term −β∆∂ 2 u/∂t2 . Simulations was performed in [111, 112] in d = 3 case. In the above references, the existence of the weak solution to (14) is guaranteed in the cases m ≥ 2, n ≥ 0, µ > 0, ν ≥ 0, m ≥ 0, n ≥ 1, µ ≥ 0, ν > 0.

or

(15) (16)

ˇ T. ROUB´ICEK

282

∂ The energy balance can be obtained by testing (14) by v = ∂t u and using Green’s formula over Ω, which is allowable if (16) holds and n ≥ m. Assuming, for simplicity, homogeneous boundary conditions, this gives:

∂u n ∂u 2 f · dxdt , (17) ∇ dxdt = E(u0 , v0 ) + ∂t ∂t Q Q ? @A B ? @A B ? @A B

∂u (T ) + ν E u(T ), ∂t @A

?

B

total energy at time level T

energy dissipated

total energy at time level 0

work made by external forces

where Q = Ω × (0, T ) and, for brevity, we denoted the total energy by

E(u, v) := Ω

2 µ |v| + ϕ(∇u, θ) + |∇mu|2 dx. 2 2

If (16) does not hold or if n < m, then max(n,m) L2 (0, T ; H0 (Ω; IRd))

∂ ∂t u

(18)

need not belong to the func-

∂ tion space and testing (14) by v = ∂t u is not permissible in general. (The notation concerning Lebesgue spaces Lp possibly valued in Banach spaces, and Sobolev spaces H k ≡ W k,2 is standard, H0k ≡ W0k,2 indicates zero traces on the boundary ∂Ω.) In this case, unless the data are suﬃciently regular, one can expect (17) to hold as an inequality “≤” only, which can be proved in a standard manner by a limit passage of some approximate solutions and the weak lower semicontinuity of the energy E(·, ·). The vanishing-viscosity analysis performed in [169] seems interesting:

Proposition 3.4 (See [169].) Denote by uν the solution to (14). Let m ≥ 2, n ≥ 0, ν 0. Then one can select a subsequence, denoted again by {uν }, such that uν → u weakly in L∞ (0, T ; H m(Ω; IRd)) ∩ W 1,∞ (0, T ; L2 (Ω; IRd)), and any u thus obtained is a weak solution to the “non-viscous”, nonlinear hyperbolic problem

∂2u − div σ(∇u) + µ(−1)m∆mu = f, ∂t2

and lim sup ν ν0

u(0) = u0 ,

∂u (0) = u1 , (19) ∂t

∂u ∂u n ∂uν 2 (T ) + f · dxdt. ∇ dxdt ≤ E(u0 , v0 ) − E u(T ), Q

∂t

∂t

Q

∂t

Moreover, if m ≥ 2 (in case d = 1) or m ≥ 3 (in case d = 2, 3), and if the data are smooth enough in the sense σ ∈ C 2 (IRd×d), u0 ∈ H 2m(Ω; IRd), v0 ∈ H m(Ω; IRd) and f ∈ H 1 (0, T ; L2 (Ω; IRd)), then even ∂uν 2 lim ν ∇n dxdt = 0.

ν0

Q

∂t

(20)

If, in addition, f ∈ L2 (0, T ; H 1 (Ω; IRd)), then (20) holds for m=2=d, too.

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283

Asymptotical behavior for µ = ν 2 → 0 was studied in [84, 194] (n = 1, m = 2 and d = 1) and for µ → 0 in [107] ( = 0, n = 1, m = 2). The above Proposition says that, for small “viscosity” coeﬃcient ν > 0, the solution uν imitates behavior of a solution u to (19). It clearly suggests, at least if the stored energy involves a capillarity-type term (possibly of a suﬃciently high order if d = 3) and the corresponding viscosity-like dissipative mechanism occurs, this mere “viscous” mechanism is not suitable to control the dissipation expected in SMA. Indeed, if ν is small, the model (14) asymptotically stops to dissipate, which says that, in particular, under a cyclical loading the area of hysteresis loops in stress/strain diagrams degenerates to zero as ν 0, which would be in disagreement with general experimental evidence. The model (14) can produce some loops in the stress/strain diagram only if ν has a deﬁnite, positive value. Yet, if this speciﬁc value of ν really determines the area of the loop within a given cycling loading f = f (t), this cannot be kept ﬁxed under a time scaling, i.e. one cannot assume rate-independent hysteresis loops produced by the model above, cf. also the numerical experiments [42]. This is again in disagreement with experiments. When no capillarity is counted, i.e. µ = 0, the response seems to be much diﬀerent as complex shock waves may develop and dissipate energy by its own mechanism, which indicates that (20) cannot be expected in such cases, as analyzed from a physical viewpoint in [2, 3, 210], mathematically in [21, 73], as well as computationally for d = 1 in [211, 212, 213]; for d = 3 see also [111] where, however, no energy balance is incorporated. The works [211, 212, 213] indicate that the energy dissipated within the phase transformation is for suﬃciently slow loading regimes essentially determined by the landscape of the stored energy, which does not seem in agreement with experimental evidence as mentioned in Sect. 3.2 and thus neglecting capillarity does not seem to solve the task, either. In view of this, one is challenged to build a rate-independent (and storedenergy-independent) phenomenology yielding energy balance (17) but with the dissipated energy replaced by (13). This leads to the system of d equations

∂u ∂2u − div σ + σ(∇u) + ν(−1)n∆n + µ(−1)m∆mu = f, p ∂t2 ∂t

(21)

∂ u(0) = u1 , and suitable together with initial conditions u(0) = u0 and ∂t boundary conditions, and involving a “plastic” stress σp subjected to a diﬀerential inclusion

σ p ∈ SL

∂

∂t

L(∇u) L (∇u),

(22)

284

ˇ T. ROUB´ICEK

where SL(λ) := {λ∗ ∈ IRL; ∀v ∈ IRL : |v|L ≥ |λ|L + λ∗ · (v − λ)} is the subdiﬀerential of the convex function | · |L and L denotes the derivative of L(x, ·) (assumed to exist). Proposition 3.5 (See [16, 167] (resp. [169]) for more details.) Let n = m = 2, µ ≥ 0, ν > 0 (resp. µ > 0, m = 3, ν = 0). Then the system (22) completed with the initial and boundary conditions has a weak solution (resp. a “very weak” solution that can even be attained by “viscous” solutions for n ≥ 1 when ν → 0).

Figure 6. Three snapshots from 3D simulations of isothermal (θ = 40◦ C) cubic/tetragonal PT in a NiMnGa single crystal governed by (21)– (22) with potential (8)–(9), L = 4, under up-down face compression/tension; courtesy of Marcel Arndt, Universit¨ at Bonn, Germany.

Remark 3.6 (Maximum dissipation principle). In view of (22), σp = ∂ L(∇u)). As SL is a maximal monotone setωL (∇u) with some ω ∈ SL( ∂t L ∂ valued mapping on IR , this inclusion is equivalent to (ω − z) · ( ∂t L(∇u) − v) ≥ 0 for all pairs (z, v), z ∈ SL(v). In particular, for v = 0 we obtain ∂L(∇u) ∂L(∇u) ω = max ξ. ∂t ∂t ξ∈SL (0)

(23)

This states that the dissipation of all PTs is maximal provided that the ∂ L(∇u) ∈ IRL×d×d is kept ﬁxed while the vector volume-fraction rate ∂t of “driving pressures” ω ∈ IRL (of dimension [Jm−3 =Pa]) varies freely over all admissible driving pressures from SL(0). This just resembles Hill’s maximum-dissipation principle [87]. In plasticity theory, this principle can alternatively be expressed as normality in the sense that the plasticdeformation rate belongs to the cone of outward normals to the elasticity domain, see also [131, 132, 188] or [83, Sect.3.2], [135, Sect.2.4.4] or [189, ∂ Sect.2.6]. Here, this would result in the observation that the rate ∂t L(∇u) belongs to the normal cone of SL(0) at the point ω. The boundary of SL(0) determines the activation threshold which triggers PTs. This is intimately related with the fact that SL is a maximal responsive set-valued map, cf. [61] for details. The maximum-dissipation principle can be detected even for the hyperbolic model (19) if µ = 0, as shown in [3] in case d = 1.

285

MODELS OF SHAPE MEMORY ALLOYS

Remark 3.7 (Maximum-dissipation principle once again). Alternatively, if = ν = 0, we can come to the maximum-dissipation principle by a degree-α homogeneous regularization of the dissipative potential (12) in the ∂ sense Rα,u(v) = Ω |L (∇u)∇v|α Ldx with α > 1; hence u → R1,u( ∂t u) is the original rate-independent potential (12). This would lead to a regularization of (22) by replacing it with ∂L(∇u) α−2 ∂L(∇u) σp = α L (∇u) ∂t ∂t

(24)

L

if the norm | · |L is smooth. Then, denoting a solution to (21) with (24) by uα and deﬁning the dissipation rate as the diﬀerence between storedd E(uα), cf. (18), and the power of the external force Ω f (t, ·) · energy rate dt ∂ ∂ ∂t uα dx, cf. [172], the rate ∂t uα maximizes Rα,u under the condition that ∂ ∂ Rα,uα ( ∂t uα) is the dissipation rate, i.e. ∂t uα = v where v solves the problem Maximize Rα,uα (v) subject to Rα,uα (v) =Dt,uα (v) Dt,u(v) :=

Ω

   

where

  f (t, ·) · v−ϕ (∇u):∇v−µ∇mu : ∇mv dx. 

(25)

Indeed, at least formally, the 1st-order optimality condition for (25) gives (1+χ)Rα,u (v) = χDt,uα with Ft,uα being the linear functional written α formally as Dt,u = f (t, ·)+div(ϕ (∇u)) − µ(−1)m∆mu and χ∈IR being a ∂ ( ∂ u), ∂ u = uα and using Rα,u Lagrange multiplier. Testing it by v= ∂t ∂t ∂t α−2 ∂ ∂ ∂ ∂ Ω α|L (∇u)∇ ∂t u|L L (∇u)∇ ∂t uL (∇u) : ∇ ∂t u dx = αRα,u( ∂t u), we get ∂ ∂ (1 + χ)αRα,uα ( ∂t uα) + χDt,uα ( ∂t uα) = 0. Comparing it with the constraint in (25), we get χ = α/(1 − α) and thus Rα,u ( ∂ u ) = αDt,uα ; see [202] for α ∂t α the “viscous” case α = 2 or [66, 172] for more general cases. In fact, (25) has a structure of ﬁnding extremum of a linear functional Dt,uα over the boundary of a convex set {v; Rα,uα (v) ≤ Dt,uα (v)} and 2nd-order analysis results to the Hessian (1 + χ)αRα,u (v) which is negatively deﬁnite if it α ∂ exists at v; note that (1 + χ)α = χ < 0. This shows that this ∂t uα realizes a global maximum of (25). Passing formally α 1 to get a rate-independent limit, we arrive to (21)–(22) with = ν = 0 and eventually also (23). For a general viewpoint on the maximum-dissipation principle see [172]. Remark 3.8 (Nonlocal, frame-indiﬀerent higher-order terms.) Instead of the energies Ω |∇mu|2 dx and Ω |∇nu|2 dx in (18) and (10), one can also consider a nonlocal energy in the form a(∇u) :=

1 4

Ω Ω

2

K(x, ξ)∇u(x) − ∇u(ξ) dxdξ

(26)

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286

with a symmetric, non-negative kernel K : Ω×Ω → IR. Models of this type have been proposed for d=1 in [177, 179] and use for d=3 in [13, 15, 16]. Depending on the choice of K(x, ξ), (26) can still eﬀect have a compactifying on the deformation gradient like the terms Ω |∇mu|2 dx and Ω |∇nu|2 dx but, unlike them, might allow for sharp interfaces. This is in good agreement with the fact that interfaces observed in SMAs are often atomically sharp, cf. Fig. 3. Besides, (26) is frame indiﬀerent if | · | in (26) is the Frobenius norm on IRd×d. 3.4. THERMODYNAMICAL EVOLUTION

A conventional departure point for derivation of the thermodynamically consistent system is the speciﬁc free energy ψ(x, ∇u, ∇mu, θ). Like in (5), ˆ + ∇u)Q(x), θ) + µ |∇mu|2 one can consider here ψ(x, ∇u, ∇mu, θ) = ψ((I 2 with ψˆ from (6), (7) or (9). For simplicity, one can consider a 1st-order expansion of ψ around a reference temperature θ0 in the form ψ(x, ∇u, ∇mu, θ) = φ0 (x, ∇u) + (θ−θ0 )φ1 (x, ∇u) +

θ µ m 2 ; |∇ u| − cθln 2 θ0

note that (6) or (7) is already in this form. Speciﬁc entropy is deﬁned by s=−

∂ψ θ = −φ1 (∇u) + c 1 + ln( ) . ∂θ θ0

(27)

Through Gibbs’ relation, the speciﬁc internal energy is: e = ψ + θs = φ0 (∇u) +

µ m 2 |∇ u| + cθ . 2

(28)

The conventional energy balance says that d dt

(x) ∂u 2 e(x) + dx = 2 ∂t Ω

Ω

f·

∂u of the dx =: power external forces. ∂t

(29)

∂ ∂ Testing (3.5) by ∂t u gives, after using (28) and s = − ∂θ ψ and also (29), and considering isotropical heat conduction through Fourier’s law, i.e. the heat ﬂux is −κ∇θ with κ denoting heat conductivity coeﬃcient of the material, the energy balance (3.10) then yields the following entropy equation:

θ

∂u 2 ∂s + div(κ∇θ) = ν ∇n =: the dissipation rate. ∂t ∂t

(30)

Substituting s from (27) gives the heat equation for temperature c

∂u ∂u 2 ∂θ adiabatic & − div(κ∇θ) = −θφ1 (∇u) : ∇ + ν ∇n =: the dissipation heat. (31) ∂t ∂t ∂t

MODELS OF SHAPE MEMORY ALLOYS

287

When completed by initial and boundary conditions, (31) with (14) with ∂ ψ represent a closed system. The thermodynamical consistency σ = ∂(∇u) relies on the Clausius-Duhem inequality for the total entropy Ω s dx can then be obtained by multiplying (30) by 1/θ, integrating it over Ω, and applying Green’s formula. Such model for µ, ν > 0, n = 1, m = 2 was studied in [90, 159, 160] and numerically in [42, 147] for d = 1. The model with no capillarity (i.e. µ = 0) was studied in [46, 153, 221]. Models with no viscosity-like term, i.e. ν = 0, in the anisothermal extension have been studied in [5, 75, 196, 197], cf. also [36, Chap.5], and numerically in [40, 41] for d = 1 and ϕ from Example 3.1. For µ = 0 = ν and d = 1, see [178]. Remark 3.9 The mentioned shape memory eﬀect is the phenomenon that the cooled multi-variant martensite can be easily “plastically” deformed and, when heated up, it “remembers” the original “parent” form (=a singlevariant austenite) in which the body has been casted and takes this shape. This eﬀect can be created by the free energy of the form (8)–(9) simply if c1 in (8), i.e. the heat capacity of the austenite, is greater than c2 = ... = cL, i.e. the heat capacity of the martensite, so that the austenite become energetically preferred (comparing to the martensite) if θ is large enough and vice versa. Sometimes, these constants mutually diﬀer even by tens of percents, e.g. NiTi has c1 = 5.92 MJ/(Km3 ) and c2 = ... = c13 = 4.50 MJ/(Km3 ); see the thesis [8] cited in [199]. Remark 3.10 In case of (9-right), the entropy s = −∂ψ/∂θ can be evaluated as θ s = c(u, θ) 1 + ln( ) , θ0

L

where

e−ψ (u,θ) c

. −ψ (u,θ)

=1 e

c(u, θ) = =1 L

(32)

Eﬀective heat capacity c(u, θ) is thus a convex combination of the original capacities c ; in statistical physics this is usually addressed as thermal average. Remark 3.11 (Clausius-Clapeyron relation). Disregarding hysteresis, the slope SCC := dσ/dθ of the stress/temperature response is often available from experiments, and it is related with the free energy by the so-called Clausius-Clapeyron relation: SCC is the speciﬁc entropy variation per the transformation strain in the selected direction, cf. [130]. Also, the temperature, under which austenite is equilibrated with martensite, is usually experimentally available. This enables us to identify c and d in (8). Remark 3.12 (Plastic-like modiﬁcation). In view of the discussion in Sect. 3.3, the modiﬁcation by augmenting the viscous-like dissipation by the activated, rate-independent term (22), and thus the heat equation (31)

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288

∂ augments by the | ∂t L(∇u)|L, cf. (12), and the dissipated energy in the energy balance then would involve (13). The analysis of this extension has not been done, however. For d = 1, such sort of model has been proposed in [6, 7, 38] and (with a thermal activation) in [108]. For a thorough thermodynamical framework of martensitic PT see [127].

4. Mesoscopic-level models On a mesoscopic level, we want to see an “averaged character” of fast oscillations of the gradient ∇u of minimizers u to the stored energy u → ψ(x, ∇u, θ) + 12 µ|∇mu|2 dx in “the limit” if µ → 0. This can be described Ω by a probability measure νx on IRd×d possibly depending on (i.e. being parameterized by) x ∈ Ω; cf. e.g. [22, 23, 180]. We then call ν = {νx}x∈Ω a Young measure [220] if, in addition, x → νx is weakly measurable. To illustrate it, let us come back to Figure 2(left) zoomed in such a way that the twined martensite in right-up halfspace is “homogenized” to a “mixture” of two deformation gradients F1 and F2 with volume fractions λ1 and λ2 (cf. (2)). This would result to piece-wise homogeneous Young measure

νx =

δF0 λ1 δF1 + λ2 δF2

if x is left-down from the habit plane, if x is right-up from the habit plane,

(33)

see Figure 2(left), where δF denotes the Dirac measure supported at the matrix F ∈ IRd×d. In our context, relevant Young measures are only those that are attainable by gradients, i.e. ν =w*-limk→∞ {δ∇uk (x) }x∈Ω for some sequence in the Sobolev space W 1,p(Ω; IRd) with p referring to the p-power growth/coercivity of F → ψ(F, θ); e.g. p = 4 in Examples 3.2 and 3.3. Let us denote by Gp(Ω; IRd×d) the set of such parameterized measures. For example, ν = {νx}x∈Ω given by (33) belongs to Gp(Ω; IRd×d) due to the rank-one connections (1)–(2). This is an example of a so-called 1st-order laminate. Thisprocess can be re-iterated: an exam ple for a 2nd-order laminate is νx = 2i=0 ai(x)δFi (x) with 2i=0 ai(x) = 1, ai(x) ≥ 0, and F0 (x), F1 (x), F 2 (x) satisfying (1)–(2) is again a gradi2 ent Young measure provided i=0 ai(x)Fi(x) = ∇u(x) for some u ∈ d 1,p W (Ω; IR ); cf. e.g. [22, 54, 151, 161, 162] for details. Unfortunately, not every ν ∈ Gp(Ω; IRd×d) is of the form of a laminate, which can be interpreted that microstructures might be much more chaotic; this is connected ˇ ak’s counterexample [201] that rank-one convexity does with famous Sver´ not imply quasiconvexity. Even worse, their mesoscopic description in terms of parameterized probability measures is not possible in an eﬃcient way, which is related with lack of eﬃcient characterization of quasiconvex functions; cf. [109, 110, 162]. Anyhow, starting from [152], there are numerical

289

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studies involving gradient Young measures but they eventually have to deal with laminates; cf. [11, 12, 17, 118, 119, 154, 180, 183]. In special cases, sufﬁciency of 1st-order laminates can rigorously be established [32]. Nevertheless, for analysis, we will work with all possible microstructures, i.e. with the whole set G p(Ω; IRd×d). Referring to (11), mesoscopic volume fractions at acurrent “macroscopic” point λ = λ(x) will then be naturally calculated as IRd×d L(x, A)νx(dA). Therefore, the mesoscopic conﬁguration will be a triple q := (u, ν, λ) of macroscopic displacement u, the microstructure ν, and the volume fraction λ, and the set Q of admissible conﬁgurations is

Q := (u, ν, λ) ∈ W 1,p(Ω; IRd)×G p(Ω; IRd×d)×L∞ (Ω; IRL); ∀(a.a.)x∈Ω : ∇u(x) =

Aνx(dA), λ(x) =

IRd×d

(34)

L(x, A)νx(dA) .

IRd×d

Neglecting any rate-dependent (i.e. here kinetic and viscous) eﬀects, in [144], Mielke and Theil introduced a suitable and eﬃcient deﬁnition of a solution to rate-independent processes generally applicable e.g. to plasticity, ferromagnetics, delamination, damage, and to SMAs, too, cf. [142, 143, 145, 146] and for numerical study [45]. Mielke-Theil’s deﬁnition plays merely with energetics of the process q : [0, T ] → Q, requiring, beside the initial condition q(0) = q0 , its stability (35) and the energy inequality (36) in the sense: ∀ t ∀˜ q∈Q: ∀t ≥ s :

G(t, q(t)) ≤ G(t, q˜) + R(q(t) − q˜),

G(t, q(t)) + VarR(q; s, t) ≤ G(s, q(s)) −

t3 ∂F s

∂τ

4

, q(τ ) dτ.

(35) (36)

Here G(t, q) = V (q) + δQ(q) − F (t), q is the Gibbs stored energy with V the stored energy (cf. (38)), δQ the indicator function of Q from (34), and F the loading, and VarR(q; s, t) is the total variation of the process q over the time interval [s, t] with respect to the dissipation potential R(q) = R(u, ν, λ) := Ω |λ|Ldx. It was shown in [144, 145, 146] that, in qualiﬁed cases (in particular, Q convex, covering our model unfortunately only for d = 1), the two inequalities (35)–(36) can be written in the form of a doubly nonlinear problem ∂R

dq

dt

+ ∂[V + δQ](q) = ∂R

dq

dt

+ V (q) + NQ(q)

F

(37)

where ∂ denotes the subdiﬀerential and NQ the normal cone. In [182], (37) was applied to a scalar model for SMAs. The deﬁnite advantages of the formulation (35)–(36) is applicability to the case d > 1 and easier analysis d as dt q is not explicitly involved.

ˇ T. ROUB´ICEK

290

Proposition 4.1 (See [143].) Assuming the relaxed stored energy is regularized by a higher-order term in λ

V (q) = V (u, ν, λ) := Ω

IRd×d

ϕ(x, A)νx(dA) + ρ|∇λ|2 dx

(38)

with ρ > 0, the initial state q0 is stable, and the loading F (·) is a W 1,1 function. Then there is a solution process q : [0, T ] → Q (i.e. (35) and (36), even as an equality, hold and q(0) = q0 ) such that, moreover, u(·), λ(·), and G(·, q(·)) are measurable in time. To be more precise, [143] assumes still a certain “nonbuckling” condition, namely that u, forming a stable conﬁguration with some λ ﬁxed, is determined uniquely; however, this additional condition seems redundant if a ﬁner proof would be made. Let us remark that the regularization (38) can be interpreted [143, 182] as a limit from the Ericksen-Timoshenko model scrutinized in [177, 179]. Gradients of mesoscopical volume fractions has already been used in Fr´emond’s model [69, p.364] or [71, Formula (7.20)], cf. Remark 4.4. The evolution of microstructure described by Young measures has been used in [17] (quasi-static 3D model), and [183] (a 2D model). A 3-atomic 2nd-order laminate like (33) in terms of small deformations has been considered in [164, 200] (a lumped-parameter model). The dissipative mechanism R is used in [121] (a 3D model) and in [154] (applied to dislocation structure in plasticity). An example of a 1st-order laminate evolution with controlled hysteresis in λ/ε-diagram is on Fig. 7. simple-laminate microstructure reconstruted from calculations:

volume fraction

0%

deformation deformed specimen with volume fraction (displayed in gray scale):

100%

Figure 7. Isothermal 3D simulations of re-orientation of monoclinic martensite induced by shear of a single crystal governed by (35)–(36); courtesy of Martin Kruˇz´ık, Academy of Sciences of the Czech Republic.

Remark 4.2 (Maximum-dissipation principle). For d = 1, like in Remark 3.7, we can formally work with (37) and come to the maximumdissipation principle by a degree-α homogeneous regularization of the dis sipative potential Rα(q) = Ω |λ|α dx with α > 1; hence R ≡ R1 is the L

MODELS OF SHAPE MEMORY ALLOYS

291

original rate-independent potential. Then, deﬁning again the dissipation d V (q) and the power of rate as the diﬀerence between stored-energy rate dt d d external force F, dt q , the rate dt q maximizes Rα under the condition that d d Rα( dt q) is the dissipation rate, i.e. dt q = q˙ where q˙ solves the problem Maximize subject to

Rα(q) ˙ ˙ F − ω, q ˙ = Rα(q)

(39)

for ω ∈ V (q)+NQ(q) a driving force. Indeed, at least formally, the 1st-order optimality condition for (39) gives (1 + χ)Rα (q) ˙ = χ(ω − F ) with χ ∈ IR d ( d q), d q = a Lagrange multiplier. Testing it by q˙ = dt q and using Rα dt dt d d d αRα( dt q), we get (1 + χ)αRα( dt q) = χω − F, dt q . Comparing it with the ( d q) = α(F − ω). constraint in (39), we get χ = α/(1 − χ) and thus Rα dt Like in Remark 3.7, the 2nd-order analysis shows that it is a maximum and passing formally α 1 to get a rate-independent limit, we arrive to d ∂R( dt q) + ω F , which gives (37) when taking ω ∈ V (q) + NQ(q) into account, and, like (23), also 3 dq

dt

4

, ω − F = max

ξ∈∂R(0)

3 dq

dt

4

,ξ .

(40)

Remark 4.3 (Volume-fraction models). As the Young measure ν does not occur explicitly in the dissipation potential R, some models eliminates ν by minimization of the stored energy, hence the “shortened” conﬁguration q = (u, λ) is used. Minimization of the energy under ﬁxed volume fraction has been used in [114]. Later, this idea repeats in [67, 78, 122, 146, 195] and, according to [125], called cross-quasiconvexiﬁcation, mostly in terms of small deformations only. In [1] a partial minimization uses by 1st- and 2nd-order laminates only. Rather paradoxically, some models apply just opposite philosophy: minimize volume fraction under ﬁxed macroscopical strain ∇u, see [155]. Remark 4.4 (Mixture-like models). There are a lot of models which treat martensitic/austenitic structures as mixtures of particular phases of martensite or/and austenite, suppressing the issue about rank-one connections eithercompletely or incorporating it in some sense by adding an enL ergy L

=1 k=1 H k λ λk with a positive matrix [H k] preferring wells with rank-one connections [74, 156, 157], like it was used in [93, 94] to express an inﬂuence of an interfacial energy). Instead of the form (9), those models consider rather the “mixture” of ψˆ : ˆ θ, λ) = L λ · ψˆ (F, θ), ψ(F,

=1

λ = (λ1 , ..., λL) ∈ ∆L,

(41)

i.e. ψˆ is a convex combination of energy of pure phases ψˆ through volume fractions λ = (λ1 , ..., λL). This has been proposed by Fr´emond [68], see also

ˇ T. ROUB´ICEK

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[69, 70] and for its analysis and numerical implementation [48, 49, 50, 51, 52, 71, 88, 218]. Fr´emond’s model [68, 69] proposes a general dissipation potential R(u, ∇u) with higher-order term like in (38), possibly nonsmooth at λ = 0 to describe activation phenomena. In literature, the analysis, however, deals with viscous-like dissipation only. Further simpliﬁcation (equal elastic moduli and, mostly, small deformations) leads to modiﬁcation of (41) to the free energy of the type ˆ θ, λ) = ψ(F,

d

(e − et )ij Cijkl(e − et )kl + ψ0 (θ, λ),

i,j,k,l=1

et =

L

λ e ,

(42)

=1

where e = 12 (F + F ) is the small strain tensor, and e = 12 (U + U ) with the distortion matrices U of particular pure phase(variant)s as in Example 3.3. For such sort of models see [20, 34, 77, 81, 82, 86, 99, 100, 120, 128, 133, 138, 156, 157, 190, 199, 214]. The dissipation usually involves volume fractions λ’s, sometimes in a rate-independent manner, see [34, 66, 77, 86, 99, 100, 214]. Obviously, the mutual shift of the bottom of austenitic vs. martensitic wells, cf. Remark 3.9, originated by diﬀerences in c ’s in (8) would result to ψ0 (θ, λ) = (λ1 c1 + (1 − λ1 )c2 )θln(θ/θ0 ) + λ1 d1 + (1 − λ1 )d2 in (42) when assuming c2 = ... = cL and d2 = ... = dL the constants of martensite variants and hence λ1 is the volume fraction of austenite Instead, a term ψ0 (θ, λ) in the form like Cl (θ/θ0 − 1)λ1 with θ0 a transformation temperature and Cl latent heat of PT is often considered, see [77, 81, 82, 100, 138, 190, 199, 206]. Various other mixture-based models are often used among physicists and engineers, see e.g. [1, 96] (λ activated by a stress through “wiggly” stored energies) or [92, 93, 94, 148, 149] (thermally activated PT based on statistical-physics). In this context a bit “exotic” concept of mixture, proposed in [103], treats binary SMAs, namely NiTi, as a mixture of its constituents, here Nickel and Titanium. 5. Polycrystalline models In contracts to single crystals that are interesting rather for lab experiments only, the real industrial interest is in polycrystallic SMAs. Behaviour of polycrystals may however substantially diﬀer from single-crystal they are composed from. E.g., the recoverable strain may or need not substantially decrease in polycrystals. Extensive heritage of recoverable strain in polycrystals was theoretically explained in [31] as a result of remarkable change of symmetry during PT, i.e. cubic/monoclinic PT is the best for preserving

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markable single-crystal SME also in polycrystals, while cubic/tetragonal PT may suppress these eﬀects. Polycrystalline modelling is certainly even greater art than the previously presented single-crystal models. Any attempt for rigorous averaging of nonlinear inelastic response of particular grains is extremely diﬃcult (and done only in special static situations assumed to be fully governed by minimum-energy principle [39, 195]) and is often substituted by rather artiﬁcial hypotheses. Widely used is Taylor’s [204] deformation-gradient homogeneity assumption, see e.g. [205, 206]. On the other hand, models assuming homogeneous stress (i.e. no interactions between grains) are developed, too; see [192]. In between, there are models distributed both stress and strain in a more sophisticated/speculative way to particular grains whose response is modelled as a lumped parameter with a greater or lesser simpliﬁcation. E.g., in [193] the load partition is done through a multilevel scheme joining particular sub-level both serially and parallelly, and thus the distribution can be controlled by various geometrical factors related with the particular SMA and the polycrystalline texture to get obtaining an agreement with neutron diﬀraction experiments. Other model [190] averages volume fractions, taking into account intergranular interactions, see also [138]. Models based on (42) after an averaging were proposed in [34, 37, 39, 157]. Inﬂuence of thermal coupling has been addressed in [206]. A “multiple natural conﬁguration” based model employing volumefraction parameters evolving in a rate independent manner has been developed in [170, 171, 173, 217]. Simple and, in some sense, rigorous modelling approach to nonlinear averaging is to use randomly rotated grains of a certain (possibly also randomized) shape. The particular grains are assumed to have a homogeneous stress, strain, and possibly a (mesoscopically-described) microstructure but otherwise the models deal with distributed parameters and the geometric interaction between grains is counted rigorously. This can be covered by the model presented in Section 4 if the rotation through Q in (5) and (11) is taken piecewise constant, in accord with the texture of the polycrystal, the analysis having been covered by [143] in the framework of the model (35)–(36). Three-dimensional calculations with randomized grain orientation and shape are in [98] or, with non-uniform probability distribution related with experimental texture data, in [74, 99, 100]. For d = 2, see [120] (based on (42)) or [166] (based on (7) and a regular texture of grains). 6. Macroscopic models This level is characterized by neglecting any information about the microstructure, hence the conﬁguration on this level is fully characterized by

ˇ T. ROUB´ICEK

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displacement u only. Steady state of single crystals based on minimum energy principle leads to the eﬀective free energy Ω ϕˆqc (∇u, θ)dx with ψˆqc (F, θ) =

1 v∈W01,p (Ω) |Ω|

inf

Ω

ˆ + ∇v, θ)dx. ψ(F

(43)

ˆ θ), and there The function ψˆqc (·, θ) is called a quasiconvexiﬁcation ψ(·, has been an intensive research in this area, starting since Morrey [139]; cf. e.g. [4, 22, 47, 54, 109, 110, 151, 162, 180, 201]. A sophisticated construction of ψˆqc for tetragonal/orthorhombic (as in YBa2 Cu3 O7 ), tetragonal/monoclinic (as in ZnO2 ) or cubic/monoclinic (as in CuZnAl) transformation was proposed in [29]. Some macroscopical models of this sort address polycrystals and has been already mentioned in Sect. 5. A realistic evolution model based on the macroscopical displacement u without any microstructural information cannot be expected in general cases. Anyhow, in a special case that the particular phase(variant)s can be well distinguished in accord with certain speciﬁc features of the deformation gradient, a rate-independent model like (35)–(36) is scrutinized in [121]. A certain extreme is represented by lumped-parameter models (d = 0) which are completely phenomenological and the rate-independent hysteretic response within special loading regime (usually a uni-axial loading of an SMA-wire) is achieved through Duham’s operator, based on ordinary differential equations, in [191] or a Preisach operator [92]. Identiﬁcation of a Preisach or rather Krasnoselski˘ı-Pokrovski˘ı operator in SMA-context was addressed in [104] or [24, 25, 123, 215], respectively. The top of the hierarchy of SMA models is certainly application to engineering workpieces. Here, however, the greatest compromises must be done between a rigorous SMA modelling and usually complicated geometry and limited project time/ﬁnancing and computational facilities available. The application of SMAs are very broad, ranging aerospace technologies, various actuators even in our households, and, perhaps the most important, medicine (orthodontics, stents, artiﬁcial bone implants, etc.). In view of Figure 3, let us mention only models of vascular stents which are, contrary to Figure 3, cut by laser, see in [18, 19, 99, 100, 176]. Conclusion. Without ambition for completeness, a state-of-art in modelling PT in SMAs has been presented, and we saw that it is indeed a certain art, and, as such, all models are to greater or lesser extent artiﬁcial. We saw a menagerie of various models, addressing diﬀerent levels (cf. Figure 3) and emphasizing diﬀerent principles (minimization of stored energy vs. maximization of dissipation, phenomenology vs. rational crystallography and

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295

thermodynamics), sometimes contradicting each others, having (ambitions for) diﬀerent range of applicability (varying from nearly none, through lab single-crystal experiments, to commercially exploitable modelling of ﬁnal products), and sometimes having ability to ﬁt with particular experiments (to explain microstructure, or stress/strain/temperature response, or even to hit energetics correctly). The mutual relations between these models are mostly missing, except the steady-state models based on pure minimization of stored energy where relations between macro/meso/micro-scopic models has been intensively scrutinized, see e.g. [4, 22, 54, 151, 162, 180]. This rather chaotic and constantly developing state of art reﬂects difﬁculty of the addressed topic, its multilevel character, and even a certain (although quite deterministic) “chaos” one can quite typically observe in microstructure evolution when loading SMA specimens by outer stress or displacement or by varying temperature. A (quite missing) rigorous interconnections of various levels in evolution case is perhaps the greatest challenge both theoretically and computationally. Acknowledgments. The author is thankful for stimulating discussions during long term collaboration to M. Arndt, M. Kruˇz´ık, A. Mielke, V. Nov´ ak, K.R. Rajagopal, ˇ P. Sittner, and A.R. Srinivasa. Besides, useful comments of J. Zimmer have been incorporated, too. This research has been partly covered by the grants A 107 5005 ˇ ˇ ˇ (GA AV CR) and MSM 11320007 (MSMT CR).

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STOCHASTIC HOMOGENIZATION: CONVEXITY AND NONCONVEXITY

J. J. TELEGA

Polish Academy of Sciences, Institute of Fundamental Technological Research, ´ etokrzyska 21, 00-049 Warsaw, Poland Swi¸

1. Introduction In the determination of eﬀective properties of heterogeneous media with random distribution of microinhomogeneities we distinguish, grosso modo, two approaches (similarly to the deterministic case). The ﬁrst approach is typical for physical and engineering papers where one usually does not introduce a small parameter ε > 0 characterizing mircoinhomogeneities and the notion of eﬀective properties is rather intuitive. The second approach is mathematically rigorous and eﬀective properties, for instance elastic moduli, are derived by performing the limit passage ε → 0 (in an appropriate sense). Just this case will be considered in the present paper. More precisely, we will consider application of G, H - and Γ - convergence to stochastic homogenization problems. We will also present the stochastic two - and multi-scale convergence in the mean as well as comments on application of stochastic partial diﬀerential equations. The book published recently by Torquato [58] summarizes results achieved by using the ﬁrst approach. Pellegrini [42] devised a simple procedure of ﬁnding eﬀective properties of weakly nonlinear dielectric composite. According to this author, under the assumption of an additional hypothesis of ergodicity per phase, a volume phase average can be reinterpreted as an average over some probability distribution for the electric ﬁeld in the phase under cosideration. Identifying volume averages to statistical averages, the problem of ﬁnding the eﬀective potential reduces to computing the probability distribution of the ﬁeld in each phase, by means of which the phase averages can be carried out. An extension to strongly nonlinear dielectrics has been elaborated by the same author in [43]. In essence, the approach used relies on simultaneous closure conditions on the aver-

305 P. Ponte Casta˜neda et al. (eds.), Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 305–347. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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age and the second moment of the ﬁeld, due to a Gaussian approximation for the probability distribution of the electric ﬁeld in each phase. Earlier mathematically rigorous results, pertaining mainly to linear problems, are presented in the papers [24,25,26,39] and in the book by Jikov et al. [27]. It seems that these results can be extended to quasi-linear problems, similarly to the periodic case, cf. Galka et al. [21], Telega et al. [57]. In this paper we do not consider ﬂows through random porous media. For a synthesis, the reader is refered to [51], cf. also [49,50,55,59]. In this contribution, except Sec. 5.2, we focus on nonlinear problems. The plan of the paper is as follows. In Sec. 2 we introduce indispensable concepts pertaining to description of random media and elements of stochastic calculus. In Sec. 3 we brieﬂy mention stochastic homogenization based on G - and H- convergence. Stochastic Γ- convergence in convex and nonconvex cases is considered in Sec.4. A class of random integral functionals is introduced and application to stationary thermoelasticity is given. Two-scale stochastic convergence in the mean is discussed in Sec. 5. This notion is applied to linear diﬀusion in random porous medium and a nonlinear elliptic problem. The notion of multi-scale stochastic convergence (or reiterated stochastic convergence) is also mentioned. In Sec. 6 we comment on an alternative approach based on homogenization of stochastic partial diﬀerential equations including linear ones. Some open problems are suggested in Sec. 7. 2. Mathematical preliminaries An essential problem in the study of random media constitutes the description of microstructure. Therefore in Section 2.1 we gathered some useful relevant data. Section 2.2 introduces indispensable concepts of stochastic calculus. 2.1. DESCRIPTION OF RANDOM MEDIA

Natural and man-made microinhomogeneous, particularly porous materials usually possess formidably complex architecture. Obviously, in modelling, one has to assume reasonable idealizations, cf. Telega and Bielski [49,50,51] and the relevant references cited therein. Random media constitute an important class of such idealizations. We pass to a brief description of reconstruction of random media, cf. [3,44,58]. We observe that Adler and Thovert [4] described random objects such as fractures and fracture networks using the methods of the geometry of random ﬁelds, cf. [2]. Let (Ω, Σ, µ) denote a probability space, where Σ is a complete σ-algebra and µ is the probability measure. Assume that Ω is acted on by an ndimensional dynamical system T (x) : Ω → Ω, such that for each x ∈

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Rn, both T (x) and T (x)−1 are measurable, and such that the following

conditions are satisﬁed [14,27]: (a) T (0) is the identity map on Ω and for x1 , x2 ∈ Rn, T (x1 + x2 ) = T (x1 )T (x2 ); (b) for each x ∈ Rn and measurable set F ∈ Σ, µ(T (x)−1 F ) = µ(F ), i.e., µ is an invariant measure for T ; (c) for each F ∈ Σ, the set {(x, ω) ∈ Rn × Ω | T (x)ω ∈ F } is a dx × dµ measurable subset of Rn × Ω, where dx denotes the Lebesgue measure on Rn. We observe that T (x)−1 = T (−x). The dynamical system satisfying (a)–(c) is also called a measure preserving ﬂow, cf. Sab [46]. Consequently, we can introduce random homogeneous ﬁelds, starting from the random variable f ∈ L1 (Ω): f˜(x, ω) ≡ f (T (x)ω).

(2.1)

We observe that f˜ is also called the statistically homogeneous (i.e. stationary) random process. Statistical homogeneity means that two geometric points of the space are statistically undistinguishable. In other words, the statistical properties of the medium are invariant under the action of translations. In this way we have a group {U (x) : x ∈ Rn} of isometries on L2 (Ω) = L2 (Ω, Σ, µ) deﬁned by (U (x)f )(ω) = f (T (x)ω), x ∈ Rn, ω ∈ Ω, f ∈ L2 (Ω). The function x → U (x) is continuous in the strong topology, i.e., for each f ∈ L2 (Ω), U (x)f → f strongly in L2 (Ω) as x → 0. The strong convergence holds provided that the probability space is separable, what we tacitly assume throughout this paper. The function f (T (x)ω) of x ∈ Rn is called a realization of the ﬁeld f . A dynamical system is said to be ergodic, if every invariant function, i.e. satisfying f (T (x)ω) = f (ω), is constant almost everywhere in Ω. Example 2.1. (i) Spherical inclusions or voids in a matrix (Sab [45,46]). Consider an inﬁnite composite medium consisting of a matrix with identical spherical particles (or voids) randomly embedded in the matrix. Then a realization ω ∈ Ω is identiﬁed with the set ω = {am : m ∈ N} of the centers am of the particles (or voids). Here N denotes the set of natural numbers. Let N b(ω, Q) denote the number of centers that fall in the open domain Q ⊂ Rn. Σ is deﬁned as the smallest σ-algebra containing the subsets of Ω of the form: {ω ∈ Ω : N b(ω, Q1 ) = k1 , . . . , N b(ω, Qi) = ki}, where Q1 , . . . , Qi are disjoint domains and k1 , . . . , ki are positive integers. The probability measure µ is uniquely deﬁned on Σ by its values on these

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subsets; the translations act on Ω as follows: ∀x, ∀ω = {am : m ∈ N}, T (x)ω = {am + x | m ∈ N}. To guarantee the statistical homogeneity of the composite, µ must be invariant under the action of T (x), for all x. This will be the case if, for instance, µ obeys the Poisson distribution: µ[N b(ω, Q1 ) = k1 , . . . , N b(ω, Qi) = ki] = µ[N b(ω, Q1 ) = k1 ] × . . . × µ[N b(ω, Qi) = ki] with

(a|Q|)k exp(−a|Q|). k! Here a > 0 is a constant and |Q| denotes the measure of Q (its volume). It is known that the Poisson distribution is ergodic, cf. Papanicolaou [38]. Unfortunately, it is not always a good model for high concentration of particles (voids) because of possible overlapping. (ii) Statistically periodic media are modelled by a probability space (Ω , Σ , µ ) on which acts a measure preserving Zn - group T (z), z ∈ Zn, which is a set of bijective maps from Ω into itself satisfying the group property in Zn and the following invariance property (Sab [46]): µ(N b(ω, Q) = k) =

∀z ∈ Zn, ∀F ∈ Σ , T (z)F = {ω|T (−z)ω ∈ F } ∈ Σ and µ (T (z)F ) = µ (F ). We recall that Z denotes the set of integers. A statistically periodic medium is ergodic if constants are the only real random variables such that: ∀z ∈ Zn, f ◦ T (z) = f a. s. (almost surely). For other examples the reader is referred to Dal Maso and Modica [19,20], and Sab [45,46]. We observe that statistically periodic media are a special case of statistically homogeneous media. Indeed, it suﬃces to take Ω = Ω × Y with Y = [0, 1)n, Σ to be Σ ⊗ L(Y), the completion with respect to µ = µ ⊗ dy of the σ-algebra product of Σ and Borel σ-algebra on Y . Now T (x) deﬁned by: ∀x ∈ Rn, ∀ ω ∈ Ω , ∀y ∈ Y, T (x)(ω , y) = (T [x + y]ω , x + y − [x + y]) is a measure preserving ﬂow on (Ω, Σ, µ); moreover, the medium is ergodic if and only if T is ergodic. (iii) Periodic media are also a special case of statistically homogeneous

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ergodic media, cf. Sab [46]. Now Y is the set Y = [0, 1)n (the unit basic cell), whilst Σ and µ are the Borel σ-algebra and Lebesgue measure on Y , respectively. Then T (x) acts on Y as follows ∀x ∈ Rn, y ∈ Y, T (x)y = x + y − [x + y]. We recall that [x] denotes the integer part of x. In this particular case, a random variable is identiﬁed with a measurable map on Y , f˜ is identiﬁed with the periodic continuation of f to the whole space whilst the expectation is identiﬁed to the volume average on Y and dµ = dy. (iv) Quasiperiodic media. Let Ω = Tm, m > n, be an m-dimensional torus endowed with the Lebesgue measure. To deﬁne the dynamical system T (x) we ﬁx an (m × n)-matrix Λ = (Λij ) and set, cf. Pankov [36], T (x)ω = ω + Λx mod Zm. The map T (x) preserves the measure µ. For T (x) to be ergodic, it is necessary and suﬃcient that Λk = 0 for any k ∈ Zn, k = 0. Any measurable function f on Ω may be identiﬁed with a unique measurable 1−periodic function on Rm. However, in this case we have a lot of essentially diﬀerent realizations f (ω + Λx). Realizations of this type are called quasiperiodic functions, if f (ω) is continuous. 2 Let us ﬁx a set G ∈ Σ. The random domain G(ω) is the set G(ω) = {x ∈ Rn | T (x)ω ∈ G}. It is usually assumed that G(ω) is almost surely (a. s.) an open domain in Rn. We also introduce the characteristic function χ(ω) of the domain G and the realization χ(T (x)ω) of this random variable. Then x → χ(T (x)ω), for ﬁxed ω ∈ Ω, is the characteristic function of the domain G(ω). To formulate the ergodic theorem we introduce the notion of the mean value for functions deﬁned in Rn. Let g ∈ L1loc (Rn), i.e. the function f is integrable on every measurable bounded set K ⊂ Rn. A number M {g} is called the mean value of g if

lim

ε→0 K

g(ε−1 x) dx = |K|M {g}.

(2.2)

Here |K| denotes the Lebesgue measure of K. Of crucial importance is the following famous result, see Jikov et al. [27]. Birkhoﬀ ergodic theorem. Let f ∈ Lα(Ω), α ≥ 1. Then for almost all

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ω ∈ Ω the realization f (T (x)ω) possesses a mean value in the following sense: x (2.3) f (T ( )ω) M {f (T (x)ω)} weakly in Lα loc . ε Moreover, the mean value M {f (T (x)ω)}, considered as a function of ω ∈ Ω, is invariant, and df

f =

f (ω) dµ = Ω

Ω

M {f (T (x)ω)} dµ.

In particular, if the system T (x) is ergodic, then M {f (T (x)ω)} = f

for almost all ω ∈ Ω.

2

Let Q be a given deterministic, bounded domain in Rn and G(ω) the statistically homogeneous random domain introduced previously. Media with a microstructure, including porous media, are characterized by a small parameter ε = l/L, where l is the characteristic microscopic dimension whilst L denotes the characteristic macroscopic dimension. Obviously, for a real material ε = ε0 > 0. Homogenization is nothing else as smearing out heterogeneities, for instance pores, inclusions or ﬁssures. Mathematically it means that we have to perform the limit passage ε → 0. Wright [59] assumes the following deﬁnition Qε(ω) = Q \ Gε(ω), where Gε(ω) = {x ∈ Rn | ε−1 x ∈ G(ω)}. Such deﬁnitions of the random domain Qε(ω) are suitable for theoretical considerations. In practice, we have to describe G(ω) or Gε(ω) more precisely. Many papers have been written on this subject. The reader is advised to consult an excellent review paper by Adler and Thovert [3], cf. also Roberts [44] and Telega and Bielski [50,51]. In essence, the methods of the geometry of random ﬁelds are exploited, see Adler [2]. More precisely, the approach reviewed in Adler and Thovert [3] consists in an approximation of the characteristic function χ(T (x)ω), denoted simply by Z(x),   1

Z(x) =

if x belongs to the pore space,

 0 otherwise.

(2.4)

The function Z(x) is called a phase function ( it depends on ω). The determination of the pore space is equivalent to the determination of the function Z(x) and of some of its properties. Adler and Thovert [3] reviewed computational and experimental methods enabling to determine this function. Among various available techniques let us mention high resolution microtomography, very suitable for systematic studies of porous architectures of geomaterials and trabecular bones, cf. the relevant references in [50,51].

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To reconstruct homogeneous random ﬁelds one may use Gaussian or non-Gaussian (like Markov) random ﬁelds. Let us brieﬂy discuss the most frequently used case of GRFs (Gaussian random ﬁelds), cf. also Roberts [44]. Once Z(x) is determined, the porosity is calculated according to Φ = Z(x) .

(2.5)

Another important quantity is the second moment of the phase function which is also called the correlation function Rz (u) = [Z(x) − Φ][Z(x + u) − Φ] /(Φ − Φ2 ).

(2.6)

Here u = |u| denotes the modulus of the translation vector u. For the sake of simplicity macroscopic isotropy has been assumed, hence the modulus u of |u|. For more details the reader is referred to Adler and Thovert [3] and Roberts [44]. Unfortunately, a porous medium (or a medium with inclusions) is incompletely, i.e., in a nonunique manner, deﬁned by only its two ﬁrst moments: Φ and Rz . Rigorously, it is precisely deﬁned by an inﬁnite series of centered moments (or correlation functions Rk ): Rk(u1 , u2 , . . . , uk−1 ) = (Φ − Φ2 )−k/2 [Z(x) − Φ]

k−1 C

[Z(x + ui) − Φ] (2.7)

i=1

Porous media are often not statistically homogeneous and their local properties may depend on the location. To classify various types of heterogeneities one may use the classiﬁcation proposed for fractals. For a review, the reader is referred to Adler and Thovert [3]. Here it suﬃces to recall that homogeneous random ﬁelds are invariant under the action of an arbitrary translation. Heterogeneous random ﬁelds lack this property. 2.2. ELEMENTS OF STOCHASTIC CALCULUS

In the case of periodic homogenization a fundamental role in the derivation of macroscopic constitutive equations is played by the so-called local or cell problem posed on a basic cell, usually denoted by Y , cf. Bensoussan et al. [9]. The periodic homogenization obviously involves the diﬀerentiation with respect to y ∈ Y . Now, in the case of stochastic homogenization the role of Y is played by Ω. We assume that Ω is a separable as a measure space. To carry out stochastic homogenization we need to introduce elements of local stochastic calculus. For more details the reader is referred to Andrews and Wright [8] and Bourgeat et al. [14].

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Let f be a measurable function deﬁned on Ω, the function (x, ω) → f (T (x)ω) is dx × dµ measurable on Rn × Ω and consequently we can deﬁne a group {U (x) | x ∈ Rn} of isometries on L2 (Ω) = L2 (Ω, Σ, µ) by (U (x)f )(ω) = f (T (x)ω), x ∈ Rn, ω ∈ Ω, f ∈ L2 (Ω). Then U (x)f → f strongly in L2 (ω) as x → 0, cf. [27]. A ﬁxed n-dimensional dynamical system T on Ω enables us to deﬁne a stochastic diﬀerential calculus on L2 (Ω) which comes from the individual coordinate actions arising from the isometry group {U (x) | x ∈ Rn}. To this end, we observe that when each coordinate of x = (x1 , . . . .xn) varies over R with the other coordinates held equal to zero in U (x), we obtain n one-parameter, strongly continuous, group of isometries Ui on L2 (Ω) which pairwise commute. Thus we may write Ui(xi)f = U (0, 0, . . . , xi, 0, . . . , 0)f = f (T (0, 0, . . . , xi, 0, . . . , 0)). Let D1 , . . . , Dn denote the inﬁnitesimal generators in L2 (Ω) of these oneparameter groups and let D1 , . . . , Dn denote their respective domains. Consequently f ∈ Di if and only if f ∈ L2 (Ω) and Di(f ) = lim

ξ→0

Ui(ξ)f − f ξ

(2.8)

exists strongly as an element of L2 (Ω). We set D(Ω) = D1α1 . . . Dnαn , we set

n D

i=1

Di. For each multi-index α = (α1 . . . αn), let Dα =

D∞ (Ω) = {f ∈ L2 (Ω) | Dαf ∈ D(Ω), for all multi-indices α}. Let us now denote by C ∞ (Ω) the set of all functions f ∈ L∞ (Ω) such that for each multi-index α, Dαf ∈ L∞ (Ω) and ξ −1 (Ui(ξ)Dαf − Dαf ) converges strongly in L∞ (Ω) as ξ → 0, thus deﬁning Diαf , for i = 1, . . . , n. Let f ∈ L1 (Ω) and α be a multi-index. The stochastic weak derivative Dαf is the linear functional on C ∞ (Ω) deﬁned by (D αf )(ϕ) = (−1)|α|

f Dαϕ dµ,

(2.9)

Ω

where |α| = α1 + . . . + αn The T -invariance of the probability measure µ yields Ω

f Dig dµ = −

Ω

gDif dµ

(2.10)

where f ∈ Di, g ∈ Di. If we endow D(Ω) with the natural norm f 2D(Ω) = f 2L2 (Ω) +

i

Dig2L2 (Ω) ,

(2.11)

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then the resulting Banach space can be viewed as a stochastic generalization of the Sobolev space H 1 (Rn). With a slight abuse of notation, we will let Dif ∈ L2 (Ω) denote the stochastic weak derivative Dαi f for f ∈ Di in the case where αi = (δ1i, δ2i . . . δni), δij )- the Kronecker delta. For f ∈ L2 (Ω), resp. v = (v1 , . . . , vn) ∈ [L2 (Ω)]n = L2 (Ω)n), we deﬁne the stochastic gradient ∇ω f , stochastic divergence div ω v, stochastic Laplacian ∆ωf , and curl curlω v by ∇ω f = (D1 f, . . . , Dnf ), div ωv = ∆ω f =

Divi

i

Di2 f, (curl ω v)ij = Divj − Dj vi.

i

Let us pass to the presentation of some properties of stochastically differentiable functions. We recall that a function f ∈ L2 (Ω) is invariant for T if f (T (x)) = f, µ. a. e. on Ω, for all x ∈ Rn (µ. a. e. = almost everywhere with respect to the probability measure µ). The set of all functions in L2 (Ω) invariant for T is a closed subspace of L2 (Ω) and is denoted by I 2 (Ω). We set M 2 (Ω) = [I 2 (Ω)]⊥ , i.e. M 2 (Ω) is the orthogonal complement of I 2 (Ω) in L2 (Ω). Another characterization of the subspace I 2 (Ω) of invariant functions is that it is complemented in L2 (Ω) by a projection E : L2 (Ω) → L2 (Ω) of norm 1 which can be calculated as follows 1 (Ef )(ω) = lim f (T (x)ω) dx, f ∈ L2 (Ω), for µ-a.e. ω ∈ Ω. λ→∞ (2λ)n [−λ,λ]n

(2.12) The last limit exists strongly in L2 (Ω) and also pointwise µ-a.e.; we have M 2 (Ω) = kerE. Let us specify some of the properties mentioned of stochastically diﬀerentiable functions. Lemma 2.1 (i) If f ∈ L2 (Ω) then f ∈ I 2 (Ω) if and only if ∇ωf = 0. (ii) If ϕ ∈ D ∞ (Ω), then for µ- a. e. ω ∈ Ω, the function x → ϕ(T (x)) is in C ∞ (Rn), and for any multi-index α = (α1 , . . . , αn), ∂ αϕ(T (x)ω) = (Dαϕ)(T (x)ω), αn 1 where ∂ α = ∂ |α| /∂xα 1 . . . ∂xn . (iii) C ∞ (Ω) is strongly dense in L2 (Ω). (iv) Let u ∈ L2 (Ω)n, v ∈ L2 (Ω)n, curl ωu = 0, and div ω v = 0. Then

Ω

uivi dµ =

Ω

(Eu) · (Ev) dµ.

(2.13)

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Furthermore, if T is ergodic then (2.13) reads Ω

u · v dµ =

Ω

u dµ ·

Ω

v dµ.

2

The last equation extends the well-known Hill-type relation [27,30,45,58]. If F is a µ-measurable subset of Ω, we denote by C ∞ (F ) the set of all functions in C ∞ (Ω) vanishing on Ω \ F . If f ∈ L1 (Ω), we will say that Dif vanishes on F if (Dif )(ϕ) = 0 for all ϕ ∈ C ∞ (F ). Condition (i) implies that if f ∈ L2 (Ω) agrees on F with an element of I 2 (Ω), then ∇ωf = 0 on F . In order to obtain a converse statement, we say that F is T-open in Ω if C ∞ (F ) is strongly dense in the set L2 (F ) of all elements of L2 (Ω) vanishing on Ω \ F and that F is T-connected in Ω if whenever f ∈ C ∞ (Ω) has a stochastic gradient which vanishes on F , then f agrees on F with an element of L2 (Ω). We have the following result. Lemma 2.2 If F is a µ-measurable subset of Ω that is T-open and Tconnected, then f ∈ L2 (Ω) has a stochastic gradient which vanishes on F if and only if f agrees on F with an element of I 2 (Ω). 2 We observe that stochastic calculus can be extended to the space Lp(Ω), cf.[8]. 3. Stochastic homogenization based on G- and H- convergence Consider the following equation: −div [aε(x, uε)∇x uε] = f, uε = 0 on ∂Q

in Q

(3.1)

Under the assumption of symmetry aij = aji and periodicity of the coeﬃcients aij (y, u) with respect to y ∈ Y , G- convergence (ε → 0) of problem (3.1) was performed by Galka et al. [21], cf. also [57]. In the last papers the integral representation theorem of eﬀective moduli was extended to the quasi-linear case. Application of Pad´e approximants was also studied. Boccardo and Murat [10] investigated the H- convergence of solutions of Eq. (3.1). In this case no assumptions of periodicity and symmetry is required. Under some conditions it was shown that H

aε(·, r) −→ a(·, r), (r ∈ R, r − ﬁxed, ε → 0).

(3.2)

Here H denotes ”H- convergence”. We recall that G- and Γ- convergence coincide provided that aij = aji. For detailed information on H- covergence the reader is referred to the fundamental paper by Murat and Tartar [33]; G-convergence is studied in the books [9,27,30,36], cf. also [17]

STOCHASTIC HOMOGENIZATION

315

Here we shall not discuss the quasi-linear transport equation in the case of random coeﬃcients, i.e. when aε(x, ω, r) = a(x, T (ε−1 x)ω, r)

(3.3)

where x ∈ Q, ω ∈ Ω and r ∈ R. Due to (3.2), only the gradient of uε plays a role in the limit passage, say H- convergence, whilst the function itself, say temperature or electric ﬁeld, is treated as a parameter. For details on stochastic G- convergence in the case (3.3), including multivalued monotone operators the reader is refered to the book by Pankov [36, Chap. 3]. The linear case with examples is systematically presented in the book by Jikov et al. [27, Chapters 7-11]. Another approach consists in using stochastic two-scale convergence in the mean, cf. Sec. 5. Remark 3.1. For results pertaining to stochastic homogenization of linear transport equation the reader is referred to the papers [8,9,24,39] and the books by Jikov et al. [27] and Pankov [36]. Those results can be extended almost immediately to quasi-linear transport equation provided that proper conditions on the material moduli aεij are imposed. 4. Γ- convergence and stochastic homogenization 4.1. CONVEX CASE

In two seminal papers by Dal Maso and Modica [19,20], the authors laid foundations for nonlinear stochastic homogenization of convex functionals, with growth α, α > 1. For α = 2 one recovers previously obtained results for the linear diﬀerential equations with random coeﬃciens. Those ressults were next extended by Sab [46] to convex functionals with linear growth, thus enabling to cope with perfect plasticity, cf. also [47]. We observe that the paper by Sab [46] extends to random perfectly plastic media the results obtained by Bouchitt´e [12] in the periodic case, cf. also [47]. However, Sab [46] did not studied the inﬂuence of boundary loading, which may play an important role, cf. Bouchitt´e and Suquet [13] in the periodic case. Telega and Gambin [53] completed the results obtained by Sab in [46] by admitting boundary loading. It is notewortly that loaded boundary may lower the limit load multiplier of perfectly plastic body made of homogenized material. Let us pass to a presentation of essential results obtained in Gambin et al. [23]. We shall formulate a general stochastic homogenization theorem applicable to performinig homogenization of equations of stationary thermoelasticity. We denote by A0 the family of all bounded open subsets of Rn. Obviously, from the physical point of view n = 1, 2, or 3. For every A ∈ A0

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we denote by W 1,α(A) the Sobolev space of functions of Lα(A) whose ﬁrstorder weak derivatives belong to Lα(A). Let us ﬁx α > 1, β > 1, c1 ≥ c0 > 0. We denote by F = F(c0 , c1 , α, β) the class of all functionals β n n n K : (Lα loc(R ) × Lloc(R ) × A0 → R = R ∪ {−∞} ∪ {+∞}

such that K(u, θ, A) =   

=

 

u|A ∈ W 1,α(A)n, θ|A ∈ W 1,β (A), otherwise.

f [x, e(u(x)), θ(x), ∇θ(x)]dx if

A

+∞,

(4.1)

n Here f : Rn × En s × R × R → R is any function satisfying the following conditions:

(i) f (x, , ξ, q) is Lebesque measurable in x and convex in , ξ and q; (ii) c0 (||α + |ξ|β + |q|β ≤ f (x, , ξ, q) ≤ c1 (||α + |ξ|β + |q|β+1 ) n for each (x, , ξ, q) ∈ Rn × En s ×R×R . We denote by En s the space of symmetric n × n matrices; in the case of linear thermoelasticity α = β = 2. Moreover, e(u) denotes the small strain tensor: 1 eij (u) = u(i,j) = (ui,j + uj,i), (4.2) 2 ∂ui . We recall that Lα(Rn)n = [Lα(Rn)]n. where, as usual, u(i,j) = ∂x j We observe that Dal Maso and Modica [19,20] studied only the integrands of the form f (x, ∇u(x)), cf. also Sab [46]. In order to perform stochastic homogenization of equation of stationary thermoelasticity a more general approach is obviously needed. After Dal Maso and Modica [19,20] we equip F with the metric d so that the set F is a compact metric space. To deﬁne the metric d, we ﬁrst introduce the -Yosida ( > 0) transform of K ∈ F:

TεK(u, θ, A) = inf{K(v, R, A) + + 1ε

A

1 ε

A

|v − u|αdx

β n n n |R − θ|β dx; v ∈ Lα loc(R ) , R ∈ Lloc(R )}.

(4.3)

Now we are in position to deﬁne a distance on F. Let us choose a countable dense subset W = {wj | j ∈ N} × {gj | j ∈ N} of W 1,α(Rn)n × W 1,β (Rn) and a countable subfamily B = {Bk|k ∈ N} of A0 . Here N denotes the set of natural numbers. For instance, B could be chosen as the family

STOCHASTIC HOMOGENIZATION

317

of all bounded open subsets of Rn which are ﬁnite unions of rectangles with rational vertices. Let us deﬁne for K, G ∈ F d(K, G) =

+∞ i,j,k=1

1 2i+j+k

|φ(T1/iK(wj , gj , Bk)) − φ(T1/iG(wj , gj , Bk))|.

(4.4) Here φ : R → R is any increasing, continous bounded function. For instance, we may take φ = arctan [19]. To prove that d is a distance on F it suﬃces to show that if d(K, G) = 0 then F = G. Indeed, Prop. 1.11 and Collorary 1.6 due to Dal Maso and Modica [19], now extended to our more general case, are formulated as follows. Proposition 4.1 β n n n (a) Let K ∈ F, u ∈ Lα loc(R ) , θ ∈ Lloc(R ), A ∈ A0 . Then lim TεK(u, θ, A) = sup TεK(u, θ, A) = K(u, θ, A).

ε→0+

ε>0

(b) Let W be a dense subset of W 1,α(Rn)n × W 1,β (Rn) and B a dense subfamily of A0 . If K, G ∈ F and K(w, g, B) = G(w, g, B) ∀(w, g) ∈ W, ∀B ∈ B then K = G. 2 Now we have to show that the metric space (F, d) is compact, hence complete and separable. To this end we have to introduce the notion of Γconvergence. For more details the reader is referred to [16,18,27,30] and the relevant references cited therein. Let X be a metric space and let {Kδ }δ>0 be a sequence of functions deﬁned on X with values in R. For instance, in our case X = Lα(A)n × Lβ (A). We say that {Kδ } Γ(X)-converges at a point z∞ ∈ X to λ ∈ R if the following two conditions are satisﬁed: (A1 ) λ ≤ lim inf Kδ (zδ ) δ→0+

for any sequence {zδ }δ>0 converging in X to z∞ ; (A2 ) there exists a sequence {zδ }δ>0 converging in X to z∞ such that lim supδ→0+ Kδ (zδ ) ≤ λ. In such a case we write λ = Γ(X)−lim Kδ (z∞ ). More precisely, we should δ→0+

write Γ(X − ) instead of Γ(X), cf. [18-20]. Since only the above notion of Γ-convergence (or epiconvergence) is used in this paper, therefore we prefer to use our simpler notation. If there exists K∞ : X → R such that K∞ (z) = Γ(X)−lim fδ (z), δ→0+

∀z ∈ X

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we say that {Kδ } Γ(X)-converges to K∞ . Then from (A1 ) and (A2 ) we conclude that K∞ (z∞ ) = min{lim inf Kδ (zδ )|zδ converges in X to z∞ } δ→0+

(4.5)

for every z∞ ∈ X. Consequently, the Γ(X)-limit K∞ is determined univocally. Let now {Kδ } be a sequence in F. Then we write Γ(Lα(A)n × Lβ (A)) − lim+ Kδ (u, θ) = K∞ (u, θ) , δ→0

(4.6)

∀(u, θ) ∈ Lα(A)n × Lβ (A), whenever A ∈ A0 . More precisely, in (4.6) we should write (Kδ )A and (K∞ )A instead of Kδ and K∞ , cf. [19]. Indeed, each K ∈ F deﬁnes, for every A ∈ A0 , a functional KA : Lα(A)n × Lβ (A) → R, 2 2 , θ) cf. [18,19]. It suﬃces to extend (u, θ) ∈ Lα(A)n ×Lβ (A) to an element (u β n n n 2 A) does not 2 , θ, of Lα loc(R ) × Lloc(R ). We observe that the value of K(u 2 of (u, θ). 2 , θ) depend on the extension (u As we shall see, the distance d on F has been chosen to be deﬁned by (4.4), since then there is the link between d and Γ-convergence. Primarily, however, we formulate a compactness result. Proposition 4.2 The class F is compact for the Γ(Lα × Lβ )-convergence, i.e. every sequence {Kδ }δ>0 in F contains a subsequence that Γ(Lα × Lβ )converges to a functional K∞ ∈ F. 2 Now we are in position to formulate a theorem which links d, Γ- convergence and ε-Yosida transform. Theorem 4.3 Let {Kδ }δ>0 be a sequence in F and K∞ ∈ F. Then the following conditions are equivalent: (1) (2) (3)

limδ→0+ d(Kδ , K∞ ) = 0; Γ(Lα × Lβ ) − limδ→0+ Kδ = K∞ ; limδ→0+ (TεKδ )(u, θ, A) = (TεK∞ )(u, θ, A) β n for each ε > 0, (u, θ) ∈ Lα loc(A) × Lloc(A), A ∈ A0 .

Proof. The proof parallels that of Prop. 1.21 of Dal Maso and Modica [19], with obvious extensions. Therefore it is omitted here. 2 Random integral functionals Let (Ω, Σ, P ) be a ﬁxed probability space, that is Ω is a set of elementary events, Σ is a σ-ﬁeld of subsets of Ω and P is a probability measure on Σ. A random integral functional is any measurable function K : Ω → F when Ω is equipped with the σ-ﬁeld Σ and F with the Borel σ-ﬁeld ΣF generated by the distance d deﬁned by Eq. (4.4), cf. [19].

STOCHASTIC HOMOGENIZATION

319

If K is a random integral functional, the image measure K# P on F deﬁned by (K# P )(S) = P (K−1 (S)) for every S ∈ ΣF , is called the distribution law of K. We shall write K ∼ G if K and G are random integral functionals having the same distribution law. The additive group Zn and the multiplicative group R+ act on F by the translation operator τz (z ∈ Zn) deﬁned by (τz K)(u, θ, A) =

f (x + z, e(u), θ, ∇θ)dx

(4.7)

A

and by the homothety operator ρε (ε > 0) deﬁned by (ρεK)(u, θ, A) =

f A

x

ε

, e(u), θ, ∇θ dx

(4.8)

1,β α for every (u, θ) ∈ Wloc (Rn)n × Wloc (Rn), and A ∈ A0 . We recall that Z stands for the set of integers. We observe that if the integrand f does not depend on θ, but still depends on ∇θ, then

(τz K)(u, θ, A) = K(τz u, τz θ, τz A), where (τz u)(x) = u(x − z), τz θ(x) = θ(x − z), τz A = {x ∈ Rn|x − z ∈ A}, and (ρεK)(u, A) = εn(ρεu, ρεθ, ρεA), where (ρεu)(x) = 1ε u(εx), (ρεθ)(x) = 1ε θ(εx), ρεA = {x ∈ Rn|εx ∈ A}. In other words, during translation and homothety, the function θ in (4.8) and (4.9) is treated as a parameter similarly to the case of periodic homogenization, cf. [15]. By virtue of Corollary 2.4 due to Dal Maso and Modica [19] we conclude that if K is a random integral functional and z ∈ Rn, ε > 0, then the functions τz K, ρεK : Ω → F deﬁned by (τz K)(ω) = τz (K(ω)),

(ρεK)(ω) = ρε(K(ω)), ∀ω ∈ Ω,

(4.9)

are random integral functionals. Furthemore, if G is another random integral functional such that K ∼ G, then we have τz K ∼ τz G and ρεK ∼ ρεG. We say that {Kε}ε>0 is a stochastic homogenization process modelled on a ﬁxed random integral functional K on Ω if Kε ∼ ρεK for every ε > 0, that is Kε and ρεK have the same distribution law. Let K be a random integral functional. We say that K is stochastically periodic in law if K and τz K have the same law for every z ∈ Zn. Ergodicity is a well-established notion when applied to integrands. Here we need ergodicity in F with respect to Zn. After Dal Maso and Modica [19] we say that a random integral functional indexrandom integral

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J. J. TELEGA

functionalK ∈ F is ergodic if P [K ∈ S] = 0 or 1 for every ΣF -measurable subset S of F such that τz (S) = S for every z ∈ Zn. For K ∈ F, A ∈ A0 , ξ ∈ R and (u0 , θ0 ) ∈ W 1,α(A)n × W 1,β (A) we may consider the following Dirichlet problem:

mξ (K, u0 , θ0 , A) = min (u − u0 , θ − θ0 ) ∈

f (x, e(u(x)), ξ, ∇θ(x))dx ;

A W01,α(A)n

×

W01,β (A)

(4.10)

.

We conclude that mξ (K, u0 , θ0 ) is continuous in K in respect of the metric d. We stress that in (4.10) ξ ∈ R plays the role of a parameter. Let Q1/ε be the cube Q1/ε = {x ∈ Rn : |xi| < 1/ε, i = 1, ..., N } and |Q1/ε| = (2/ε)n its Lebesgue measure. We recall that lq = q · x and l = x, where q ∈ Rn, ∈ En s. After these lengthy, yet necessary preparations, we are in a position to state our main homogenization theorem. Theorem 4.4 Let K be a random integral functional and deﬁne Kε = ρεK. If K is periodic in law, then Kε converges P -almost everywhere as ε → 0+ to a random integral K0 . Moreover, there exist Ω ⊂ Ω of full measure such that the limit lim

ε→0+

mξ (K(ω), l, lq , Q1/ε) = f0 (ω, , ξ, q) |Q1/ε|

(4.11)

exists for every ω ∈ Ω , ξ ∈ R, q ∈ Rn, ∈ En s and K0 (ω)(u, θ, A) =

f0 [ω, e(u(x)), θ(x), ∇θ(x)]dx

(4.12)

A β n 1,α(A)n, for every ω ∈ Ω , A ∈ A0 , (u, θ) ∈ Lα loc(A) ×Lloc(A) with u|A ∈ W 1,β θ|A ∈ W (A). Additionally, if K is ergodic, then K0 or equivalently f0 (ω, , ξ, q) does not depend on ω and

f0 (, ξ, q) = lim

ε→0 Ω

mξ (K(ω), l, lq , Q1/ε) |Q1/ε|

(4.13)

2 for every ξ ∈ R, q ∈ Rn, ∈ En s. Sketch of the proof of the stochastic homogenization theorem Prior to passing to the proof of Th. 4.4 we are going to provide useful comments and additional indispensable tools. First, we observe that similar

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STOCHASTIC HOMOGENIZATION

theorem was formulated by Dal Maso and Modica [20] in a much simpler case where

x (4.14) (ρεK)(ω)(θ, A) = f ω, , ∇θ(x) dx. ε A

The same authors stated a stronger result as Th. 3.2 in their another paper [19]. More precisely, for {Kε}ε>0 a stochastic process modelled on a stochastically periodic random integral functional K, in [19] it was assumed that there exists M > 0 such that the two families of random functions (K(·)(θ, A))θ∈Lβ

loc

(Rn )

,

(K(·)(θ, B))θ∈Lβ

loc

(R n )

are independent wherever A, B ∈ A0 with dist(A, B) > M . Then a counterpart of formula (4.13) holds. In fact, {Kε} converges in probability as ε → 0+ to the single functional K0 ∈ F independent of ω. Now, the functional K0 is easily deduced from (4.13) by deleting and ξ. Let us recall the notions of convergence in probability and convergence in law, cf. [19] and the relevant references cited therein. We say that a sequence of random integral functionals {Kε}ε>0 converges in probability to a random integral functional K∞ if

lim P ω ∈ Ω|d(Kε(ω), K∞ (ω)) > η = 0, ∀η > 0

ε→0+

(4.15)

where d is the distance on F. It is well-known that any sequence converging in probability contains a subsequence which converges pointwise almost everywhere. We say that {Kε}ε>0 converges in law to K∞ if the corresponding laws µε = Kε# P converge weakly-∗ as ε → 0+ to µ∞ = K∞# P , i.e.,

ϕ(K)dµε(K) =

lim

ε→0+

K

ϕ(K)dµ∞ (K)

(4.16)

F

for every continuous function ϕ : F → R. Equivalently we may write µε, ϕ → µ∞ , ϕ

as

ε→0

where ·, · denotes the duality pairing deﬁned on C ∗ (F) × C(F); C(F) denotes the space of continuous functions on the compact space F with the supremum norm and C ∗ (F) is its dual. The interrelationship between these two types of covergence is wellknown, cf. [19, Prop.2.9].

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Proposition 4.5 Let K∞ be a constant random integral functional, that is there exists K0 ∈ F such that K∞ (ω) = K0 for P -almost all ω ∈ Ω. Then convergence in law and convergence in probability toward K∞ are equivalent. 2 Let us comment on the stronger version of the stochastic homogenization theorem due to Dal Maso and Modica [19]. This theorem is unsatisfactory for two important reasons. First, the independence at large distances is not always veriﬁed. Such is the case of chessboard structure with cells of random size sketched in Fig. 3 of [20]. Second, whilst convergence in probability is the best possible if we give as in [19] the hypotheses in terms of law, the problem arises whether there is almost everywhere convergence in the case Kε = ρεK, a result well-known in the case of linear stochastic homogenization, cf. [19,20] and the relevant references therein. Both these diﬃculties are overcome by Th. I of Dal Maso and Modica [19] and our more general Th. 4.4. In order to prove our Th. 4.4 we need a few additional results, cf. [5, 20]. A set function µ : A0 → R is said to be subadditive if µ(A) ≤

µ(Ak)

(4.17)

k∈K

for every A ∈ A0 and for every ﬁnite family {Ak}k∈K in A0 such that Ak ⊂ A ∀k ∈ K,

Aj ∩ Ak = ∅ ∀j, k ∈ K, j = k,

|A −

Ak | = 0.

k∈K

Let M = M(c) be a family of subadditive functions µ : A0 → R such that 0 ≤ µ(A) ≤ c|A| ∀A ∈ A0 where c > 0 is a ﬁxed constant. We denote by ΣM the trace on M of the product Σ-algebra of RA0 . Let (Ω, Σ, P ) be a given probability space. A (Σ, ΣM )-measurable map µ : Ω → M is called a subadditive process. The group Zn acts on M by the formula (τz µ)(A) = µ(τz A) .

(4.18)

If (−µ) is subadditive then µ is called superadditive. We say that a subadditive process is ergodic if P [µ ∈ S] = 0 or 1 for ΣM -measurable subset S of M such that τz S = S for every z ∈ Zn. Essential role in the proof of Th. 4.4 will play the following proposition, which is substantially the subadditive ergodic theorem due to Akcoglu and Krengel [5].

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Proposition 4.6 Let µ : Ω → M be a subadditive process. If µ is periodic in law, that is µ and τz µ have the same law for every z ∈ Zn, then there exists a Σ-measurable function Φ : Ω → R and a subset Ω ⊂ Ω of full measure such that

µ(ω) lim+

ε→0

1 εQ

1 ε Q

µ(ω)(tQ) = Φ(ω) t→+∞ |tQ|

= lim

(4.19)

for every ω ∈ Ω and for every cube Q ∈ Rn. Moreover, if µ is ergodic then Φ is constant. 2 For the proof the reader is referred to Dal Maso and Modica [20]. Proof of Theorem 4.4. We divide it into two steps. Step 1. The random integrand f (ω, x, , ξ, q) does not depend on ξ. Then mξ (K(ω), l, lq , Q1/ε) appearing in Eq. (4.11) does not involve ξ and we simply write m(K(ω), l, lq , Q1/ε). Now our proof is an extension of the proof of Th. I due to Dal Maso and Modica [20]. We recall that lq = q · x, l = x. Let us ﬁx q ∈ Rn, ∈ En s and deﬁne µp (ω)(A) = m(K(ω), l, lq , A),

∀ω ∈ Ω, ∀A ∈ A0

where p = (, q). Then µp (ω) ∈ M(c) with c = c1 (1 + |q|α + ||β ) for every ω ∈ Ω, and µp : Ω → M is (Σ, ΣM )-measurable since m(·, l, lq , A) is continuous on F equipped with the distance d. For every z ∈ Zn, ω ∈ Ω, A ∈ A0 we have (τz µp )(ω)(A) = µp (ω)(τz A)

= min (τz K)(ω)(τ−z u, τ−z θ, A) (u,θ)

| τ−z u − τ−z l ∈ W01,α(A)n, τ−z θ − τ−z lq ∈ W01,β (A)

= min (τz K)(ω)(v + l(z), R + lq (z), A) (v,R)

| v − l ∈ W01,α(A)n, R − lq ∈ W01,β (A) . Since the integrand of K depends only on x, e(u) and ∇θ, therefore (τz K)(v + l(z), R + lq (z), A) = (τz K)(ω)(v, R, A). Hence (τz µp )(ω)(A) = m((τz K)(ω), l, lq , A),

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for every z ∈ Zn, ω ∈ Ω, A ∈ A0 . Thus µp is periodic in law because τz K and K have the same law and m(·, l, lq , A) is continuous on F. In virtue of Prop. 4.6 we conclude that there exist a subset Ωp ⊂ Ω of full measure and a Σ-measurable function Φp : Ω → R such that µp (ω)(tQ) = Φp (ω) t→+∞ |tQ| lim

for every ω ∈ Ωp and for every cube Q ∈ Rn. Let now Q1/ε be the cube n deﬁned earlier and let f0 : Ω × En s × R → R be the function deﬁned by f0 (ω, , q) = lim sup ε→0+

µp (ω)(Q1/ε) |Q1/ε|

n ∀(ω, , q) ∈ Ω × En s ×R .

We observe that the functions p = (, q) →

µp (ω)(A) |A|

(ω ∈ Ω, A ∈ A0 )

are convex and equibounded between 0 and c1 (1 + |q|β + ||α), hence locally equicontinuous. The convexity follows from the convexity in (u, θ) of K(ω)(u, θ, A). Consequently f0 (ω, , q) is convex in (, q). Let us set E

Ω =

p∈Qn

Ωp

where Q is the set of rational numbers. We have P (Ω ) = 1 and

lim

µp (ω)

ε→0+

1 εQ

1 ε Q

= f0 (ω, , q)

n n for every ω ∈ Ω , p = (, q) ∈ En s × R and for every cube Q in R . Furthermore, we get

µp (ω)

1

ε

Q =

1 n

ε

m((ρεK)(ω), l, lq , Q).

Hence, since ρεK = Kε, we obtain lim

ε→0+

m(Kε(ω), l, lq , Q) = f0 (ω, , q) |Q|

n n for every ω ∈ Ω , ∈ En s , q ∈ R and for every cube in R . In virtue of Prop. 4.2, for every ω ∈ Ω there exists an integral functional K0 (ω) ∈ F

STOCHASTIC HOMOGENIZATION

325

such that Kε(ω) Γ(Lα × Lβ )-converges to K0 (ω) as ε → 0+ . More precisely, there exists such a subsequence still denoted by Kε. Let us calculate the integrand g0 (ω, x, , q) of K0 (ω). Fix ω ∈ Ω and set Qρ(x) = {y ∈ Rn : |yi − xi| < ρ, i = 1, ..., N }. Taking into account formula (A) given in Sec. 2 of [23] and the continuity of m(·, l, lq , A) we conclude that there exist a subset N of Rn with |N | = 0 such that m(K0 (ω), l, lq , Qρ(x)) |Qρ(x)| m(Kε(ω), l, lq , Qρ(x)) = lim lim |Qρ(x)| ρ→0+ ε→0+

g0 (ω, x, , q) = lim

ρ→0+

µp (ω)

= lim lim

1 ε Qρ(x)

ρ→0+ ε→0+

1 ε Qρ(x)

= f0 (ω, , q)

n for every x ∈ Rn \ N , ∈ En s , q ∈ R . Thus we get

K0 (ω)(u, θ, A) =

f0 [ω, e(u(x)), ∇θ(x)]dx

A β n n n for every ω ∈ Ω , A ∈ A0 , (u, θ) ∈ Lα loc(R ) × Lloc(R ) such that (u, θ)|A ∈ W 1,α(A)n × W 1,β (A). If K is ergodic, then µp is ergodic and on account of Prop. 4.6, Φp and thus also f0 do not depend on ω. Step 2. Let now f = f (ω, x, , θ, q). On account of Prop. 4.2, for a ﬁxed ω ∈ Ω, there exists a subsequence of {Kε}ε>0 , still denoted by {Kε} such that Kε(ω) Γ(Lα × Lβ )-converges to

K∞ (ω)(u, θ, A) =

f0 [ω, x, e(u(x)), θ(x), ∇θ(x)]dx

Ω β n n n 1,α for each A ∈ A0 , (u, θ) ∈ Lα (A)n × loc(R ) × Lloc(R ) with (u, θ)|A ∈ W 1,β W (A). Let now θ ∈ C 1 (Rn) and consider the function, cf. Braides [15] in the case of periodic homogenization,

fθ (ω, x, , q) = f (ω, x, , θ(x), q). To the function fθ we may apply Step 1 and write lim

ε→0+

m(Kθ (ω), l, lq , Q1/ε) = f0θ (ω, , q) |Q1/ε|

(4.20)

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J. J. TELEGA

n for every ω ∈ Ω , (, q) ∈ En s × R , and

K0θ (u, R, A) =

f0 [ω, e(u(x)), θ(x), ∇R(x)]dx

A

Still by Step 1 and (4.20) we get f∞ (ω, x, , ξ, q) = f0ξ(ω, , q) = f0 (ω, , ξ, q). Ergodicity implies that f0 does not depend on ω. Thus the proof of Th. 4.4 is complete. 2 Remark 4.1. (i) Speciﬁc case of two-phase materials was studied in [19,23]. Consider the scalar case and assume that {Xkε }k∈Zn is a family of independent random variables deﬁned on a probability space (Ω, Σ, P ) with λ > 0 and Λ > 0 such that [19], cf. also [23], P {ω ∈ Ω| Xkε (ω) = λ} = r,

P {ω ∈ Ω| Xkε (ω) = Λ} = 1 − r

for every ε > 0, k ∈ Zn and for r ∈ (0, 1). For every ε > 0, k ∈ Zn, let Qεk be the cube in Rn: {x ∈ Rn | εki ≤ xi < ε(ki + 1), i = 1, . . . , n} and denote by Ikε its characteristic function. We set Xkε (ω)Ikε (x), ω ∈ Ω, x ∈ Rn, aε(ω, x) = k∈Zn

and

Kε(ω)(u, A) =

A

aε(ω, x)|Du(x)|α dx, if u|A ∈ W 1,α(A),

+∞

otherwise.

It can be shown that Th. 4.2 applies. In the one-dimensional case (n=1) the limit functional takes the form du 0 K0 (u, A) = a | |αdx (4.21) dx A

and f0 (p) = [rλ1/(1−α) + (1 − r)Λ1/(1−α) ]1−αpα. We conclude that a0 is the α-harmonic r-weighted mean of λ and Λ. We observe that if α = 2 and r = 1/2, a0 is the harmonic mean of λ and Λ. Thus K0 given by (4.21) coincides with the limit in the deterministic, periodic case. However, in dimension two√for α = 2 and r = 1/2 the corresponding limit is the geometric mean λΛ instead of the harmonic mean, cf. Jikov et al. [27].

327

STOCHASTIC HOMOGENIZATION

(ii) The easiest way of proving the ergodicity of K is to verify a mixing condition (or independence at large distances), cf. [19]. (iii) A random integrand f is ergodic if it satisﬁes the following mixing condition [20]:

lim P {ω ∈ Ω|f (ω, xi, i, ξ, qi) > si ∀i ∈ I,

|z|→+∞ z∈Zn

f (ω, yj + z, ∆j , ξ, rj ) > tj ∀j ∈ J} =

= P {ω ∈ Ω|f (ω, xi, i, ξ, qi) > si ∀i ∈ I} ×

×P {ω ∈ Ω|f (ω, yj , ∆j , ξ, rj ) > tj ∀j ∈ J}

for every pair of ﬁnite families {(xi, i, qi, si}i∈I and {(yj , ∆j , rj , tj }j∈J n in Rn × En s × R × R. Here ξ ∈ R is treated as a parameter. Then one can extend Th. III due to Dal Maso and Modica [20] and prove that, for instance, if f is ergodic, then K is ergodic. Ergodicity of K also follows if a measure preserving ergodic ﬂow on Ω is introduced. Then the integrand is ergodic and the ergodicity of K is satisﬁed, cf. Sab [46]. (iv) It seems that Th. 4.2 can be weakened by assuming the convexity of n integrands only with respect to ∈ En s and q ∈ R , cf. Braides [15]. 4.2. NONCONVEX CASE

Consider now the following class of functionals F, deﬁned as follows: G belongs to F if and only if there exists a function g from Rn × Emn into R, measurable with respect to its ﬁrst variable, and a positive constant L such that, for every a, b in Emn and a.e. x c1 |a|α ≤ g(x, a) ≤ c2 (1 + |a|α),

(4.22)

|g(x, a) − g(x, b) ≤ L(1 + |a|α−1 + |b|α−1 )|a − b|

(4.23)

1,α n where 1 < α < ∞, with, for every A in A0 and u in Wloc (R , Rm)

G(u, A) =

g(x, ∇u(x)) dx.

(4.24)

A

Such a class was investigated by Messaoudi and Michaille [32] provided that A0 is the set of all bounded subsets of Rm with Lipschitz boundary. The operators τz , z∈ Zn, and ρε are deﬁned similarly to Sec. 4.1; Emn denotes the space of m × n matrices.

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J. J. TELEGA

We observe that no convexity assumption on the function g(x, ·) is imposed. In fact, g is quasiconvex. A random integral functional, periodic in law and ergodic is deﬁned similarly to Sec. 4.1. The process {Kε}ε>0 given by Kn(ω)(u, A) =

f (ε−1 n x, ∇u(x)) dx, εn =

A

1 , n

(4.25)

Γ- converges (or epiconverges) almost surely, when εn → 0, towards a functional Kh in F. More precisely, there exists a subset Ω in Σ with µ(Ω ) = 1 such that, for every ω ∈ Ω , every bounded regular domain A Kh(·, A) = Γ(τ ) −lim Kn(ω)(·, A) n→∞

(4.26)

exists in W 1,α(A)m equiped with its weak topology τ or with the strong topology of Lα(A)m. The limit functional Kh is given by Kh(u, A) =

f h(∇u(x)) dx

(4.27)

A

where, for every matrix a in E

f h(a) =

inf

k∈N\{0}

Ω

1 inf{ |kY |

mn

,

f (ω, a + ∇u(x)) dx | u ∈ W01,α(nY )m}.

kY

(4.28) Here Y denotes the unit cube (0, 1)n. This stochastic homogenization result is proved by exploiting methods typical to the periodic case combined with the Akcoglu and Krengel [5] theorem on discrete superadditive (subadditive) processes. Let I be the set of intervals [x, y), where x and y belong to Zn and consider a set function S from I into L1 (Ω, Σ, µ) verifying the following conditions: (i) S is superadditive, that is, for every A ∈ I such that there exists a ﬁnite family {Ai}i∈I of disjoint sets in I whose union A belong to I, then SAi (·) SA(·) ≥ i∈I

(ii) S is covariant, that is, for every A ∈ I, every z ∈ Zn, SA+z = SA ◦ τz . (iii) sup{

1 meas(A)

Ω

SAdP,

A ∈ I,

meas(A) = ∅} < +∞.

STOCHASTIC HOMOGENIZATION

329

S is called a discrete superadditive process. If (−S) is superadditive, S is said to be subadditive. For subadditive processes sup in (iii) is to be replaced by inf and < +∞ by > −∞. Theorem 4.7 (Akcoglu and Krengel [5]). When n tends to +∞, k1n S[0,k)n converges almost surely. Moreover, if τz is ergodic, we have, almost surely 1 1 lim n S[0,k)n (ω) = sup n k→+∞ k k∈N\{0} k

S[0,k)n (ω) dµ .

(4.29)

Ω

Remark 4.2. If S is subadditive then in (4.29) the sup is to be replaced by inf. Remark 4.3. The class F of nonconvex functionals considered in Sec. 4.2 does not cover polyconvex stored energy functions with the condition detF> 0, where F stands for the deformation gradient. Remark 4.4. Licht and Michaille [31] weakened condition (4.22) as follows 0 ≤ g(x, a) ≤ c2 (1 + |a|α), a ∈ Emn. These autors extended also the Akcoglu and Krengel Th. 4.3 to the case of subadditive processes indexed by convex sets. Remark 4.5 Abbdaimi et al. [1] performed stochastic homogenization in the case α = 1, provided that integrand is quasiconvex. Then the integral functional has linear growth and one has to deal with nonreﬂexive spaces W 1,1 , L1 and BV . 5. Two-scale and multi-scale stochastic convergence in the mean The notion of two-scale convergence in the periodic case was introduced by Nguetseng [34] and developed by Allaire [6], cf. also [60]. Next, Bourgeat et al. [14] introduced the concept of stochastic two-scale convergence in the mean. 5.1. BASIC DEFINITION AND PROPERTIES

Deﬁnition 5.1. A sequence {uε}ε>0 in L2 (Q × Ω) is said to stochastically two-scale converge in the mean to u ∈ L2 (Q × Ω) if for all ψ ∈ L2 (Q × Ω)

lim

ε→0 Q×Ω

ε

−1

u (x, ω)ψ(x, T (ε

x)ω) dx dµ =

u(x, ω)ψ(x, ω) dx dµ. Q×Ω

(5.1) Let us pass to providing some basic facts concerning stochastic two-scale convergence in the mean.

330

J. J. TELEGA

Theorem 5.1 Suppose {uε}ε>0 is a bounded sequence in L2 (Q × Ω). Then there exists a subsequence of {uε}ε>0 which stochastically two-scale converges in the mean to u ∈ L2 (Q × Ω). 2 Theorem 5.2 (a) Suppose that {uε} and {ε∇xuε} are bounded sequences in L2 (Q × Ω). Then there exist u ∈ L2 (Q, H 1 (Ω)) and a subsequence, still denoted by {uε}, such that {uε} (resp. {ε∇xuε}) stochastically two-scale converges in the mean to u (resp. ∇ωu). (b) Let X be a norm-closed, convex subset of H 1 (Ω) (for instance a ball). Suppose {uε} is a sequence in L2 (Q × Ω) which satisﬁes the following conditions: (i) uε(·, ω) ∈ X , for µ- a. e. ω ∈ Ω, (ii) there exists an absolute constant C > 0 such that Ω

uε(·, ω)2H1 (Q) dµ ≤ C.

Then there exist u ∈ H 1 (Q, L2 (Ω)), v ∈ L2 (Q × Ω)n, and a subsequence, still denoted by {uε}, which satisfy the following conditions: (iii) for a.e. x ∈ Q, u(x, ·) ∈ I 2 (Ω) and for µ- a. e. ω ∈ Ω, u(·, ω) ∈ X ; (iv) v is contained in the L2 (Q × Ω)n - norm closure of L2 (Q) ⊗ (range of ∇ω), and for a.e. x ∈ Q, v(x, ·) ∈ M 2 (Ω)n, curlω v(x, ·) = 0; (v) {uε} (resp. {∇xuε}) stochastically two-scale converges in the mean to u (resp. ∇x u + v); 2 (vi) ∇x u(x, ·) ∈ I 2 (Ω)n, for a. e. x ∈ Q. Remark 5.1. Let us extend the mapping E deﬁned by Eq. (2.14). To this end, for each y ∈ Rn we deﬁne the mapping T˜(y) : Q × Ω → Q × Ω by T˜(y)(x, ω) = (x, T (y)ω). Then {T˜(y) | y ∈ Rn} is an n-dimensional dynamical system on Q × Ω with an invariant measure dx × dµ. Let I 2 (Q × Ω) denote the set of all functions in L2 (Q × Ω) which are invariant for T˜. We have I 2 (Q × Ω) = L2 (Q, I 2 (Ω)). When (Ω, T ) is replaced by (Q × Ω, T˜) then ˜ Eg(x, ω) = E[g(x, ·)](ω),

(5.2)

where g ∈ L2 (Q × Ω). Formula (2.14) yields 1 λ→0 (2λ)n

˜ Eg(x, ω) = lim

[−λ,λ]n

g(x, T (y)ω) dy.

(5.3)

˜ does not depend on ω ∈ Ω provided that µ is ergodic We observe that Eg for T . ˜ Useful properties of E

331

STOCHASTIC HOMOGENIZATION

˜ deﬁnes a projection of norm 1 of L2 (Q × Ω) onto I 2 (Q × Ω). (a) E (b) If g ∈ L2 (Q × Ω) then

E Q

g(x, ·) dx =

Q

˜ Eg(x, ·) dx.

5.2. STATIONARY DIFFUSION IN RANDOM POROUS MEDIUM MADE OF NONHOMOGENEOUS MATERIAL

The essential novelty of the present section lies in taking into account both the random distribution of micropores and (macroscopic) material inhomogeneity. Thus we extend the results due to Andrews and Wright [8] and Bourgeat et al. [14]. The domains Q and Qε(ω) have been introduced in Sec. 2.1. We set ¯ ε(ω), Qsε(ω) = Q\Q

(5.4)

where the bar over a set denotes its closure. We employ the superscript s for the solid phase. Let Γε(ω) = Q ∩ ∂Qsε(ω), S ε(ω) = ∂Q ∩ ∂Qsε(ω), and F = Ω\G; moreover F (ω) = {x ∈ Rn | T (x) ω ∈ F }. We assume that the set Rn\F (ω) is open and connected for each ω ∈ Ω. Consider the following transport equation −divx [A(x, T (ε−1 x)ω)∇x uε(x, ω)] = f (x) in Qsε(ω), uε(x, ω) = 0

on S ε(ω),

[A(x, T (ε−1 x)ω)∇uε] · n = 0

on Γε(ω).

(5.5)

Here n is the outward unit normal vector to ∂Qsε(ω) and f ∈ L2 (Q). The moduli Aij (i, j = 1, . . . , n) are assumed to satisfy the following conditions: (i) Aij ∈ L∞ (Q × G); (ii) Aij = Aji; (iii) there exist constants β ≥ α > 0 such that for dx × dµ-a. e. (x, ω) ∈ Q × G, α|ξ|2 ≤ Aij (x, ω)ξiξj ≤ β|ξ|2 ∀ξ ∈ Rn. Here |ξ|2 = ξiξi. We observe that uε may be, for instance, a temperature ﬁeld or an electric potential. A function uε satisfying the following variational equation Qsε (ω)

Aij (x, T (ε

−1

x)ω)uε,j (x, ω)v,i(x) dx

=

Qsε (ω)

f v dx,

(5.6)

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J. J. TELEGA

for each v ∈ H 1 (Qsε) with trace 0 on S ε(ω), is called a weak solution of (5.5) provided that uε(·, ω) ∈ H 1 (Qsε). Under the above assumptions uε exists and is unique. We recall that the trace of a function on the boundary (or the interface) practically means its value on this boundary. From (iii) and (5.6) we conclude that there exists C > 0 such that, for all ε > 0 uε(·, ω)2H1 (Qsε ) dµ ≤ C. (5.7) Ω

Similarly to the case of periodic homogenization [26,27,30], the function uε has to be extended to the whole domain Q. After Andrews and Wright [8], we say that a µ−measurable set F ∈ F has the extension property with respect to Q if the following conditions are satisﬁed: there exists ε0 > 0 and C1 > 0 such that for each u(·, ω) ∈ H 1 (Qsε(ω)) with trace 0 on S ε(ω), there exists u˜˜ε(·, ω) ∈ H01 (Q) such that ˜ ˜ε (a) (x, ω) → u (x, ω) is dx × dµ−measurable on Q × Ω; ˜ = u(·, ω), ∀ε ∈ (0, ε0 ], ∀ω ∈ Ω; (b) u˜ε(·, ω) Qsε (ω)

(c) u˜˜ε(·, ω)H1 (Q) ≤ C1 u(·, ω)H1 (Qsε (ω)) , ∀ε ∈ (0, ε0 ], ∀ω ∈ Ω. Remark 5.2. (i) Andrews and Wright [8] consider a more general extension, namely these authors additionally assume that there exists p > 1 and in condition (c) the space H 1 (Q) is to be replaced by W 1,p(Q). Then obviously u˜˜ε(·, ω) ∈ W01,p(Q) and p is called the exponent of smoothness of the extension. (ii) The construction of extension operators is a delicate problem even in the periodic case, cf. [30] and the relevant references cited therein. For a discussion and illustrative examples in the case of randomly perforated media the reader is referred to the papers by Jikov [25,26] and the book by Jikov et al. [27] . 2 s Let fε(·, ω) be a function deﬁned on Qε(ω). Similarly to the periodic case, we want to ﬁnd its limit when ε → 0. We set fε0 (x, ω)

=

  fε(x, ω),

if x ∈ Qsε(ω),

 0,

if x ∈ Q\Qsε(ω).

Hence we conclude that fε0 is the function obtained by extending fε(·, ω) by 0 into the perforations Fε(ω) of Q. The following lemma is a speciﬁc case of Lemma 3.5 formulated in [8]. Lemma 5.3 Let F ∈ Σ be a µ-measurable subset of Ω. Suppose that for each ω ∈ Ω, {vε(·, ω)}ε>0 is a sequence of functions with vε(·, ω) deﬁned on Qsε(ω), and assume that

STOCHASTIC HOMOGENIZATION

333

{v˜˜ε}ε>0 is a sequence of functions deﬁned on Q × Ω for which ˜ v˜ε(·, ω)|Qsε (ω) = vε(·t, ω), ∀ω ∈ Ω; (b) {v˜˜ε} stochastically two-scale converges in the mean to v in L2 (Q × Ω). Then {vε0 } stochastically two-scale converges in the mean to χΩ\F v in L2 (Q× Ω), where χΩ\F v denotes the characteristic function of Ω\F . Proof . From (a) for all (x, ω) ∈ Q × Ω, we have vε0 (x, ω) = χQ\F (ω) (x)v˜˜ε(x, ω) = χΩ\F (T (ε−1 x)ω)v˜˜ε(x, ω). (a)

ε

Thus

vε0

∈ L (Q × Ω), for each ε. Using now (b) we readily get 2

lim

ε→0 Q×Ω

= lim

vε0 (x, ω)ψ(x, T (ε−1 x)ω) dxdµ

ε→0 Q×Ω

= Q×Ω

v˜˜ε(x, ω)(χΩ\F ψ)(x, T (ε−1 x)ω) dxdµ

(χΩ\F vψ)(x, ω) dxdµ

2 for all ψ ∈ L2 (Q × Ω). Now we are in a position to pass with ε → 0 in (5.5) in the sense of stochastic two-scale convergence in the mean. Theorem 5.4 Let F ∈ Σ have the extension property with respect to Q. Then there exists a subsequence of the sequence {uε}ε>0 , still denoted by {uε}, such that there exists u ∈ H 1 (Q, I 2 (Ω)), and ζ ∈ L2 (Q, M 2 (Ω))n, with ζ contained in L2 (Q × Ω)n – norm closure of L2 (Q) ⊗ (range ∇ω), for which {(uε)0 , (∇x uε)0 }ε>0 stochastically two-scale converges in the mean to (χΩ\F u, χΩ\F (ζ + ∇x u)), and such that (u, ζ) is a weak solution of divω {χΩ\F (ω)A(x, ω)[ζ(x, ω) + ∇x u(x, ω)]} = 0,

(5.8)

˜ Ω\F (ω)A(x, ω)(ζ(x, ω) + ∇x u(x, ω))]} = µ(Ω\F )f, −divx {E[χ

(5.9)

u(x, ω) = 0 on ∂Q, (5.10) where (5.8) holds almost everywhere on Q whilst (5.9) and (5.10) hold µ– a.e. on Ω. Proof. Using the extension property of F with respect to Q, we ﬁnd ˜ε deﬁned on Q × Ω such that ε0 > 0, C > 0 and a measurable function u˜ for all 0 < ε ≤ ε0 we have ˜ε(·, ω) ∈ H 1 (Q), u˜ (5.11) 0

˜ε(·, ω)|Qs = uε(·, ω), u˜ ε

(5.12)

˜ε(·, ω)H1 (Q) ≤ Cuε(·, ω)H1 (Qs (ω)) , u˜ ε

(5.13)

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J. J. TELEGA

for all ω ∈ Ω. From (5.7) and (5.13) we conclude that the sequence {u˜˜ε}0<ε≤ε0 is a bounded in L2 (Ω, H 1 (Q)). Hence, by (5.11), (5.12), Th. 5.2 and Lemma 5.1 there exists a subsequence {(uε)0 } and functions (u, ζ) ∈ H01 (Q, I 2 (Ω)) × L2 (Q, M 2 (Ω))n with ζ contained in L2 (Q × Ω)n– norm closure of L2 (Q) ⊗ (range∇ω ) such that {(uε)0 , ∇x (uε)} stochastically twoscale converges in the mean to (χΩ\F u, χΩ\F (ζ + ∇x u)). Let us now verify that (u, ζ) is a weak solution of Eqs. (5.8) and (5.9). First we observe that if v ∈ H01 (Q), then on account of the T –invariance of µ, (5.6) and the fact that χQε (ω) (x) = χF (T (ε−1 x)ω), (x, ω) ∈ Q × Ω, we get

Q×Ω ε

= Ω Qε (ω)

+

=

Ω

ε

˜˜,j (x, ω)v,i(x) dxdµ Aij (x, T (ε−1 x)ω)u

Qsε (ω)

˜˜,j (x, ω)v,i(x) dxdµ Aij (x, T (ε−1 x)ω)u Aij (x, T (ε−1 x)ω)uε,j (x, ω)v,i(x) dxdµ

χF (T (ε−1 x)ω)Aij (x, T (ε−1 x)ω)u˜˜ε,j (x, ω)v,i(x) dxdµ

Q×Ω

f v dx.

+µ(Ω\F )

(5.14)

Q

Hence

χΩ\F (T (ε−1 x)ω)Aij (x, T (ε−1 x)ω)u˜˜ε,j (x, ω)v,i(x) dxdµ

Q×Ω

f v dx.

= µ(Ω\F )

(5.15)

Q

˜ε, ∇x u˜˜ε} stochastically two-scale In virtue of Th. 5.2 we conclude that {u˜ converges in the mean to (u, ζ+∇x u). We will now prove that (u, ζ) satisﬁes Eqs. (5.8) and (5.9). To this end we apply Lemma 2.1(b) of Andrews and Wright [8] which states that there exists a countable subset B of C ∞ (Ω), strongly dense in W 1,p(Ω) for 1 ≤ p < ∞. Let g, h ∈ C0∞ (Q), b ∈ I 2 (Ω) and k ∈ B. For 0 < ε ≤ ε0 the function x → g(x)b(ω) + εh(x)k(T (ε−1 x)ω), x ∈ Q, is an element of C0∞ (Q). We use this as a test function in (5.15) and obtain Q×Ω

˜˜ε,j (x, ω)[b(ω)g,i(x) + χΩ\F (T (ε−1 x)ω)Aij (x, T (ε−1 x)ω)u

335

STOCHASTIC HOMOGENIZATION

+ε(h(x)k(T (ε−1 x)ω)),i]dxdµ

f (x)[g(x)b(ω) + εh(x)k(T (ε−1 x)ω)] dxdµ.

= µ(Ω\F ) Q

(5.16)

We have ε(h(x)k(T (ε−1 x)ω),i = εk(T (ε−1 x)ω)h,i(x) + h(x)Di[k(T (ε−1 x)ω)]. Substituting the last relation into (5.15) and letting ε tend to zero, we conclude from the stochastic two-scale convergence in the mean that

χΩ\F (ω)A(x, ω)[ζ + ∇x u(x, ω)] · [b(ω)∇x g(x) + h(x)∇ωk(ω)] dxdµ

Q×Ω

f (x)g(x)b(ω) dx.

= µ(Ω\F )

(5.17)

Q

We now set g ≡ 0 in (5.17) and conclude from the fact that h ∈ C0∞ (Q) is arbitrary that, for all k ∈ A and for a. e. x ∈ Q, Ω

χΩ\F (ω)[ζ(x, ω) + ∇x u(x, ω)] · ∇ωk(ω) dµ = 0.

Since B is countable, we conclude that (u, ζ) is a weak solution of Eq. (5.8) for a. e. x ∈ Q. In order to verify that (u, ζ) is a weak solution of Eq. (5.9) we apply ˜ given in Sec. 5.1. Indeed, let h ≡ 0 the property (b) of the mapping E, 2 in Eq. (5.17). Since b ∈ I (Ω) is otherwise arbitrary, it follows that for all ϕ ∈ L2 (Ω) we have

Q

χΩ\F (ω)

Q

−

A(x, ω)[ζ(x, ω) + ∇x (x, ω)] · ∇x g(x) dx

f (x)g(x)dx E(ϕ) dµ = 0. Q

Hence

E Q

A(x, ω)[ζ(x, ω)+∇x u(x, ω)]·∇x g(x) dx = µ(Ω\F )

f (x)g(x) dx. Q

By the aforementioned property (b) of E, for µ– a. e. ω ∈ Ω, we get Q

˜ Ω\F (ω)A(x, ω)[ζ(x, ω) + ∇x u(x, ω)]) · ∇x g(x) dx = E(χ

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= µ(Ω\F )

f (x)g(x) dx. Q

Since g ∈ C0∞ (Q) is arbitrary, from the last relation we deduce Eq. (5.9). In order to prove that (u, ζ) is uniquely determined by (5.8)–(5.10) it suﬃces to follow the corresponding part of the proof of Th. 4.1.1. of Bourgeat et al. [14] or the proof of Theorem 3.1 of Andrews and Wright [8]. The proof of Th. 5.3 is thus complete. 2 Remark 5.3. Consider the case when T is ergodic. Then u does not depend on ω and Eq. (5.9) is written as follows −divx

Ω

χΩ\F (ω)A(x, ω)[∇x u(x) + ζ(x, ω)] dµ = f (x),

in Q.

We may set a = ∇x u(x) ∈ Rn and treat x as a parameter. The eﬀective (homogenized) moduli are given by A (x)a =

h

Ω

χΩ\F (ω)A(x, ω)(a + ζ(x, ω)) dµ,

(5.18)

for each a ∈ Rn. On account of linearity we may write ζ = aiψ (i) (x, ω), where ψ (i) (x, ·) ∈ [M 2 (Ω)]3 . Finally we get

Ahij (x) =

Ω

(j)

χΩ\F (ω)Aik(x, ω)(δjk + ψk ) dµ.

(5.19)

(j)

The local functions ψk (i = 1, . . . , n) are solutions to the following problem, easily obtained from Eq. (5.8) (i)

ψj (x, ·) ∈ M 2 (Ω),

(j)

Di[χΩ\F Aik(x, ω)(δjk + ψk )] = 0.

5.3. HOMOGENIZATION OF A NONLINEAR ELLIPTIC PROBLEM

In this section we assume that A = (Aij ), T and Ω are as in Sec. 5.2. Here we follow Bourgeat et al. [14]. Let h1 : Rn → Rn and h2 : R → R be two monotone functions satisfying the conditions: 0 ≤ h1 (d) · d ≤ C1 |d|2 , ∀d ∈ Rn 0 ≤ h2 (a)a ≤ C1 |a|2 ,

∀a ∈ R,

for ﬁxed positive constants C1 and C2 . Without loss of generality we assume that h1 (0) = 0, h2 (0) = 0.

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For f ∈ L2 (Q) and ε > 0 we consider the following nonlinear problem: −div [A(x, T (ε−1 x)ω)∇uε] − div [h1 (∇uε)] + h2 (uε) = f u = 0 on ∂Q.

in Q,

(5.20)

for µ - a.e. ω ∈ Ω. By applying the Minty - Browder theorem [41] we conclude that problem (5.20) is uniquely solvable for µ - a.e. ω ∈ Ω with the solution uε(·, ω) ∈ H01 (Q) µ-a.e. Moreover the H01 - norm of uε(·, ω) is µ - measurable and there is an absolute constant C > 0 such that for all ε > 0, ||uε(·, ω)||H1 (Q) ≤ C, 0

µ − a.e. in Ω .

(5.21)

Passing in (5.20) with ε to zero, in the sense of two-scale stochastic convergence, we get the following result. Theorem 5.5 There exists u ∈ H01 (Q; I 2 (Ω)) and ζ ∈ L2 (Q; M 2 (Ω))n such that {uε}ε>0 , (resp., {∇x uε}) stochastically 2-scale mean converges to u (resp.,{∇x u + ζ }) as ε → 0, and (u,ζ ) is the unique weak solution to the problem −div ω{A(x, ω)[∇x u + ζ ] + h1 (∇x u + ζ )} = 0,

in Q × Ω,

˜ ˜ 2 (u) = f, ω)[∇x u + ζ ] + h1 (∇x u + ζ )} + Eh −div x E{A(x,

(5.22)

in Q. (5.23)

Sketch of the proof. The variational formulation of (5.20) is obtained by exploiting the monotonicity property of h1 and h2 :

A(x, T (ε−1 x)ω)∇uε·∇(ϕθ−uε) dx dµ+

Q×Ω

h1 (∇(ϕθ))·∇(ϕθ−uε)dx dµ

Q×Ω

h2 (ϕθ)(ϕθ − uε)dx dµ ≥

+ Q×Ω

f (ϕθ − uε)dx dµ

Q×Ω

D∞ (Ω).

∈ where ϕ ∈ Next replacing ϕθ by g(x)b(ω) + εh(x)k(T (ε−1 x)ω) and letting ε tends to zero, after some calculations we get the result. Here g, h ∈ C0∞ (Ω), b ∈ I 2 (Ω) and k ∈ B. For the deﬁnition of B see the proof of Th. 5.4. Remark 5.4. Telega and Bielski [50] introduced the notion of multiscale stochastic convergence in the mean (or reiterated stochastic convergence) . Such is the case, for instance, if H01 (Q), θ

−1 Aε(x, ω) = A(x, T1 (ε−1 1 x)ω1 , T2 (ε2 x)ω2 )

where T1 , T2 are two measure preserving ﬂows and ω1 ∈ Ω1 , ω2 ∈ Ω2 . Particularly, one may consider mixed periodic - stochastic homogenization

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if at one level the microstructure is periodic and at another stochastic. Here ε1 , ε2 are positive functions of ε > 0 which converge to 0 as ε does. The following assumption is made, cf. Allaire and Briane [7], The functions ε , ε are assumed to be separated , i.e. they satisfy 1 2 the f ollowing condition : (H) lim ε2 = 0. ε→0 ε 1

6. On homogenization of stochastic diﬀerential equations The methods of stochastic homogenization presented in the previous sections may be viewed as natural extensions of the methods eleborated for periodic problems. An alternative, and not so palpable, approach consists in performing homogenization of suitably formulated stochastic diﬀerential equations. The reader not familiar with the theory of such equations is advised to study the book by Øksendal [35] and Chap.3 of the book [9]. Chapter 3 of the last book is advised an an introduction to homogenization of stochastic partial diﬀerential equations. As far as we know, homogenization of stochastic partial diﬀerential equations has not yet penetrated into micromechanics. 6.1. SOME MATHEMATICAL PRELIMINARIES

Let (Ω, Σ, P ) be a probability space, and Σt be an increasing family of subalgebras of Σ, i.e. Σt1 ⊂ Σt2 if t1 ≤ t2 . A stochastic process wt with values in Rn, which is continuous and satisﬁes: (i) w0 = 0 ∀s ≤ t, (ii) wt|Σs = ws, (iii) (wt − ws)(wt − ws)T |Σs = It−s, s ≤ t, where I denotes the identity matrix, is called a Wiener process with respect to Σt. The property (ii) expresss the fact that wt is a Σt - martingale. Such a process a Gaussian process, i.e., for each t1 , t2 , . . . tk , the random vector wt1 , wt2 , . . . , wtk is Gaussian. Continuity may be understood in the sense of the following deﬁnition. Deﬁnition 6.1. Suppose that {Xt} and {Yt} are stochastic processes on (Ω, Σ, P ) . Then we say that {Xt} is a continuous version of {Yt} if P ({ω|Xt(ω) − Yt(ω) = 0}) = 1 for all t. Kolmogorov’s theorem [35] provides a criterion for the continuity. This theorem answers (positively) the question of continuity of Wiener process. Wiener process play an important role in deﬁning the stochastic integral (Ito integral).

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339

Let us introduce the following spaces: LpΣt (0, T ; Rn) = {Φ(t, ω)|Φ is measurable, ∀t, Φt = Φ(t, ·) is Σt−measurable, T

a.s. |Φt|pdt < ∞}, p ≥ 1 0

MΣpt (0, T ; Rn)

= {Φ ∈

T

LpΣt (0, T ; Rn)|

|Φt|pdt < ∞}.

0

The Ito integral, denoted by I can be deﬁned for functions from MΣ2 t , which is a Hilbert space. The standard notation is T

I(Φ) =

Φt · dwt

(6.1)

0

Remark 6.1 (i) The stochastic integral extends to intergands Φ which are random matrices. (ii) The stochastic integral extends to integrands Φ which belong to L2Σt . (iii) From (6.1.) we conclude that for any t ∈ [0, T ], t

It(Φ) =

Φ(s) · dws.

(6.2)

0

The process It is continuous; moreover It is a Σt-martingale. For conections of the Ito integral with partial diﬀerential equations and martngake formulation the reader is reﬀered to [9,35]. 6.2. A NONLINEAR RANDOM PARABOLIC EQUATION AND ITS HOMOGENIZATION

In this section we consider the following second-order semilinear parabolic partial diﬀerential equation [40]: ∂uε x ∂uε 1 x ∂ aij ( , ξt/ε2 ) (t, x) + g( , ξ t/ε2 , uε(t, x)), (t, x) = ∂t ∂xi ε ∂xj ε ε

(6.3)

where (t, x) ∈ (0, T ) × Rn, uε(0, x) = u0 (x) ∈ L2 (Rn), {ξ t}t>0 is a stationary diﬀusion process with values in Rm. As ussual, ε > 0 is a small parameter intended to tend to zero. We denote by L the inﬁnitesimal generator of ξ: 1 ∂2 ∂ L = qij (y) + bi(y) . 2 ∂yi∂yj ∂yi

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We assume the following conditions: C1 . The functions aij (z, y) and g(z, y, u) are periodic in z of period 1 in all the coordinate directions; the matrix a = (aij (z, y)) is uniformly possitive deﬁnite: ∃c1 > 0, 0 < c−1 1 I ≤ a(z, y) ≤ c1 I. Moreover, the gradient of aij both with respect to y and z exists and is uniformly bounded: ∃c2 > 0, |∇z aij (z, y)| + |∇y aij (z, y)| ≤ c2 . C2 . (i) There exist c3 , c4 , α > 0 such that 0 < c−1 3 I ≤ q(y) ≤ c3 I, |∇qij (y)| ≤ c4 , |b(y)| + |∇b(y)| ≤ c4 (1 + |y|α). (ii) There exist c5 , c6 > 0, β > −1, such that whenever |y| > c5 , (b · y) ≤ c6 , |y| where (b · y) denotes the inner product in Rn. From these assumptions it follows that the process ξ has a unique invariant probability measure µ(dy) = p(y)dy whose density decays at inﬁnity faster than any negative power of |y| (see the relevant reference in [40]). C3 . g(z, y, u) satisﬁes the estimates: ∃c7 > 0, ∃c8 > 0, ∃c9 > 0,

|g(z, y, u)| ≤ c7 |u|, |gu(z, y, u)| ≤ c8 ,

(1 + |u|)|guu(z, y, u)| ≤ c9 ,

and g, gu, guu are jointly continuous. C4 . The relation g(z, y, u)p(y)dzdy = 0 Tn Rn

holds for any u ∈ R. Here T = Rn/Zn stands for the n-dimensional torus (periodic functions are identiﬁed with functions deﬁned on Tn). By virtue of our assumptions the diﬀusion process {ξ t} is a solution to the following stochastic equation: dξ t = b(ξ t)dt + σ(ξ t)dwt,

(6.4)

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STOCHASTIC HOMOGENIZATION

where σ(y) = q1/2 (y) whilst {wt} is a standard m-dimensional Wiener process. One may decompose g(z, y, u) as follows: g(z, y, u) = g(z, y, u) + g(y, u),

(6.5)

where g(y, u) =

g(z, y, u)dz,

(6.6)

Tn

so that

g(z, y, u)dz = 0, Tn

∀y ∈ Rm, u ∈ R,

g(y, u)p(y)dy = 0,

∀u ∈ R.

(6.7)

(6.8)

Rn

The ﬁrst relation implies the existence of a vector function G(z, y, u) such that g(z, y, u) = divz G(z, y, u). Indeed, let G = ∇v , where for each (y, u) ∈ Rm+1 , v(·, y, u) solves the partial diﬀerential equation ∆v = g on Tn . Then the function G(z, y, u) satisﬁes the ﬁrst two estimates in the condition (C3 ). For any u(t, x) we have x x 1 x divx G( , y, u(t, x)) = g( , y, u(t, x)) + Gu( , y, u(t, x))∇x (t, x). ε ε ε ε Under assumptions (C2 ) and (C4 ) relation (6.8) ensures the solvability of the Poisson equation LG(y, u) + g(y, u) = 0, ∀u ∈ R, 2,p m in the space Wloc (R ) . We observe that the solution G(·, u) has polynomial growth in |y| for all u ∈ R. The solution is unique up to an additive constant; for deﬁniteness we assume that it has zero mean with respect to the invariant measure µ(dy) = p(y)dy. Deﬁne now the space

VT := L2 (0, T ; H 1 (Rn)) ∩ C([0, T ]; L2 (Rn)). and let VT denote the space V T equipped with the sup of the weak topology of L2 (0, T ; H 1 (Rn)), and the topology of the space C([0, T ]; L2w(Rn), where L2w(Rn) denotes the weak topology of the space L2 (Rn).

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To formulate the homogenization theorem, we introduce the following two auxiliary parabolic problems: (

∂ (k) ∂ ∂ ∂ aij (z, ξτ ) χ (z, τ ) = − aik(z, τ ), + ∂τ ∂zi ∂zj ∂zi (z, τ ) ∈ Tn × [0, +∞),

and (

∂ ∂ ∂ aij (z, ξτ ) )φ(z, τ, u) = −g(z, ξ τ , u), + ∂τ ∂zi ∂zj (z, τ ) ∈ Tn × [0, +∞),

where u is a real parameter. The functions χ(k) (z, τ ) and φ(z, τ, u) are now deﬁnes as stationary solutions to these problems. They are counterparts of local problems well-known in deterministic periodic homogenization, cf.[9]. The main result established by Pardoux and Piatnitski [40] is summarized as follows. Theorem 6.1 Under the speciﬁed assumptions, the family of laws of the solutions {uε}ε>0 to problem (6.3) converges weakly, as ε → 0, in the space VT , for all T > 0, to the unique solution of the martingale problem with the ˆ drift A(u(s)), where ˆ A(u) = divx (a(I + ∇z χ Tn ∇x u) − divx (gχ Tn (u)) −divx (a∇x φ Tn (u)) + φug Tn (u) + Gug Tn (u),

(6.9)

and the covariance R(u, (s)), where

(R(u)ϕ, ϕ) =

[q(y)(ϕ∇y G(y, u)] · [ϕ∇y G(y, u)]p(y)dy, ϕ ∈ C0∞ (Rn)

Rm

(6.10) Remark 6.2 (i) The notation a(I + ∇z χ Tn stands for

a(z, ξt)(I + ∇z χ(z, t))dz ,

Tn

which does not depend on t, since (ξ, χ) is stationary. Similarly one can ˆ inerpret the remaining terms in the expression for A(u). (ii) Precise meaning of the expression (R(u)ϕ, ϕ) is given by the r.h.s. of (6.10). (iii) For the notion of martingale problem and drift the reader is referred

STOCHASTIC HOMOGENIZATION

343

to [9,35]. Remark 6.3 Jourdan et al. [28] analyzed a nonlinear micro-macro model of polymeric ﬂuids in the case of a shear ﬂow. In the case of nonNewtonian ﬂuids such as polymeric ﬂuids, an appropriate equation links the stress tensor to the velocity ﬁeld through, say, a partial diﬀerential equation or an integral relation. To construct a micro-macro model, one goes down to the microscopic scale and applies kinetic theory to obtain a mathematical model for the evolution of the microstructure of the ﬂuid, here the conﬁgurations of polymer chains. The reader is referred ot the relevant references cited in [28] for detailed information on this type of modeling. In contrast to the purely macroscopic approach where the microscopic models are used to derive macroscopic constitutive equations, most of the time through some simplifuing assumptions (closure assumptions) whose impact on the result is diﬃcult to estimate, the so-called micro-macro approach consists in keeping explicit track of both scales. In the simple case of dumbbell model, where the polymer is modelled by two beads linked by a spring, this micro-macro approach leads to the following system of equations:

ρ

∂u + u · ∇u = −∇q + η∆u + divτ , ∂t

τ =n

divu = 0,

(6.11)

(X ⊗ F(X)) ψ(t, x, X)dX − nkBI,

∂ψ 2 σ2 + u · ∇x ψ = −divX [ ∇x uX − F(X) ψ] + 2 ∆X ψ. ∂t ζ ζ Here u(t, x) is the velocity of the ﬂuid, q(t, x) the pressure, τ (t, x) the stress tensor, and ψ(t, x, X denotes the probability density function of the endto-end vector X of the polymer at position x and time t. The remaining symbols are physical parameters: F(X) is the entropic force a representative polymer chain experiences, ρ and η are the density and viscosity of the ambiant ﬂuid respectively, n denotes the density of polymers, the coeﬃcient σ is deﬁned by σ 2 = 2kB T ζ, with T the temperature and ζ the friction coeﬃcient of the beads with the ﬂuid. We observe that the Fokker-Planck equation (6.11)4 on ψ holds at each macroscopic point x. Jourdain et al.[28] studied a particular case of the so-called FENE model (Finite Extensible Nonlinear Elastic). These anthors assumed that the force within the spring has the following form: F(X) =

HX 1−

|X|2 /(bkB T /H)

where H and b denote constant coeﬃcients.

,

(6.12)

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Under assumption (6.12), the model (6.11) is cast into a system of four equations which comprises two coupled stochastic diﬀerential equations. Existence of a unique solution was proved. 7. Final remarks and some open problems Paroni [37] derived the homogenized linear elasticity tensor of polycrystalline aggregates with random orientation of crystallite. In the proof of the homogenization theorem crucial role is played by div-curl lemma due to Murat and Tartar, cf. [33]. Symmetries of the homogenized material moduli were investigated. Boivin and Depauw [11] proved a homogenization theorem for a Dirichlet eigenvalues of reversible random walks on Zd with stationary and uniformly elliptic conductivities. The authors presented also a review of available results on homogenization of random walks. Pointwise ergodic theorem, being a counterpart of the Birkhoﬀ ergodic theorem for the continuous case (cf. Sec.2 of our paper), was formulated and proved. Also, a central limit theorem for reversible random walks in random environment was proved. For a systematic introduction to central limit theorems in homogenization of random media in the continuous case the reader is referred to the book by Jikov et al. [27]. Komorowski and Olla [29] performed homogenization, by using the fuctional central limit theorem (FCLT), in the case when the particle trajectory is described by Itˆ o’s stochastic diﬀerential equation: √ dxt = b(t, xt)dt + 2κdwt, x0 = 0, where b : R × Rm × Ω → Rm is an m-dimensional stationary Gaussian random ﬁeld (drift) with incompressible (divx (t, xt) = 0) realizations over a probability space T1 = (Ω, Σ, µ. The Brownian motion wt is independent of b and given on another probability space; the parameter κ ≥ 0 is referred to as the molecular diﬀusivity. In [29] the homogenization was performed by ussing the FCLT for the family of continuous trajectory process xεt = εxt/ε2 . FCLT was also applied to a class of time dependent Gaussian, Markovian drifts, the so-called Ornstein-Uhlenbeck drifts. Speciﬁcally, homogenization was carried out for b(·, ·) with the power energy spectrum. In nonlinear homogenization, particularly in the nonconvex case, many open problems arise, even at the level of derivation of macroscopic properties in the periodic case. For instance, consider a hyperelastic composite with periodic microstructure made of, say, Ogden or Saint-Venant Kirchhoﬀ material. As usual, such a microstructure is characterized by a small parameter ε > 0. Essential are growth conditions and the fact that detFε > 0, where Fε stands for the gradient of deformation. How to rigorously derive

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345

the limit (homogenized) elastic potential? Similar question pertains to hyperelastic composites with random microstructure. Still more complex is the case of porous hyperelastic media, since then self-contact at the pore level inevitably occurs (closure, partial or complete, of pores). To avoid interpenetration, the self-injectivity condition has to be taken into account making the problem of homogenization extremely complex. Application of homogenization methods to cracked materials oﬀers new possibilites of ﬁnding their macroscopic properties, cf. [22,52,54]. An exciting class of nonlinear materials with microstructure present soft tissues like myocardium, arterial walls, cartilage, muscles, etc., cf. [56]. These biological materials are characterized by hierarchical ordering of microstructure, ranging from nanoscale to macroscopic scale. As far as we know, nonlinear homogenization methods have not yet been incorporated into the modeling of soft tissues behavior. Acknowledgement. The author is indebted to drs A. Galka and B. Gambin for typesetting of the manuscript. The work was supported by the Ministry of Science and Information Technology (Poland) through the grant No 8 T07A 052 21. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13.

Abddaimi, Y., Michaille, G. and Licht, C. (1997) Stochastic homogenization for an integral functional of a quasiconvex function with linear growth, Asymptotic Anal., 15, 183–202. Adler, J.P. (1981) The geometry of random ﬁelds, Wiley, Chichester. Adler, P.M. and Thovert, J.-F. (1998) Real porous media: Local geometry and macroscopic properties, Appl. Mech. Reviews, 51, 537–585. Adler, P.M.and Thovert, J.-F. 1999 Fracture and fracture networks, Kluwer, Dordrecht. Akcoglu, M.A. and Krengel, U. (1981) Ergodic theorems for superadditive processes, J. reine angew. Math., 323, 53-67. Allaire, G. 1992) Homogenization and two-scale convergence, SIAM J. Math. Anal., 23, 1482–1518. Allaire, G. and Briane, M. (1996) Multiscale convergence and reiterated homogenization, Proc. R. Soc. Edinburgh, 126A, 297–342. Andrews, K.T. and Wright, S. (1998) Stochastic homogenization of elliptic boundary-value problem with Lp –data, Asymptotic Anal., 17, 165–184. Bensoussan, A., Lions J.-L. and Papanicolaou G. (1978) Asymptotic analysis for periodic structures, North-Holland, Amsterdam. Boccardo, L. and Murat, F. (1981) Homog´ en´eisation de probl` emes quasi-lin´eaire, in: Atti del Convegno ”Studia di Problemi - limite della Analisi Funzionale, pp 13-51, Bressanone, 7-9 Settembre , Pitagora Editrice, Bologna 1982. Boivin, D. and Depauw, J. (2003) Spectral homogenization of reversible random walks on Zd in a random enviroment, Stochastic Processes and their Appl., 104, 29–56. Bouchitt´e, G. (1987) Convergence et relaxation de fouctionnalles convexes du calcul des variations ` a croissance lin´eaire, Annales Fac. Sci Toulouse, 8, 7-36. Bouchitt´e, G. and Suquet, P. (1991) Homogenization, plasticity and yield design, in: Composite Media and Homogenization Theory, ed. by G. Dal Maso and G.F.

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INDEX

activated process, 247 activation stress, 241 active materials, 15 adjacent equation, 110 adjacent operator, 127 admissible control, 185 admissible designs, 3, 198 admissible laminate, 205 admissible layouts, 93 almost optimal structure, 6 alternate direction algorithm, 5 Ambrosio-De Giorgi SBV spaces, 49 anisotropic measure of the ﬂuctuations, 296 area distortion, 131 asymptotic homogenization formula, 48 atomic level, 240 atomistic-level modelling, 243 austenite lattice, 19 austenite phase, 15, 238 austenite-martensite interface, 26 average discrete energy, 54 average stress, 218, 225, 227, 231, 232 Bain or distortion matrix, 19 Beltrami equation, 109, 112 Beltrami operator, 107, 109, 117 Birkhoﬀ ergodic theorem, 307 Borel functions, 53 Borel measure, 51 boundary variation, 1 bulk and surface energy densities, 51

349

bulk materials, 18 Carath´eodory function, 47, 195 Carath´eodory theorem, 72 Castigliano formula, 154 Castigliano’s theorem, 143 Cauchy transform, 128 Cauchy-Born hypothesis, 18 Cauchy-Born rule, 49 central interactions for lattice systems, 45 Clausius-Clapeyron relation, 255 Clausius-Duhem inequality, 255 clusters, 54 columnar grains, 35, 40 compactness result, 47 compensated compactness, 93 compliance objective function, 10 compliance of the structure, 2 compliance optimization, 184 composite from expanding phases, 95 condition of neutrality, 88 conductivity in a strip, 29 constrained minimizing sequences, 75 constrained quasiconvexiﬁcation, 203 continuous piecewise-aﬃne interpolate function, 57 continuum energies on Sobolev spaces, 46 control problem, 75, 79 convex energy functions, 217 convex envelope, 71, 91 convex envelope of N -well potential, 76

350 convex Legendre transform , 274 convexiﬁcation operation, 48 Cordes conditions, 112 correlation function, 309 cost functional, 197, 198, 203, 205, 206 cross-quasiconvexiﬁcation, 259 crystallographic symmetry, 15 crystallographic texture, 16 cubic-orthorhombic transformation, 35 cubic-tetragonal transformation, 35 cubic-trigonal transformation, 35 curvature, 7 deformation of the thickness, 21 deformation theory of plasticity, 275 deformations with a single phase, 24 descent gradient direction, 8 description of random media, 304 design of functionally graded materials, 173 design of stiﬀ composite structures, 184 design’s rigidity, 2 diﬀerential constraint, 73, 81, 92 diﬀerential scheme, 65, 79 discrete graded micro-structures, 180, 183 discrete interpolation, 48 dissipative forces, 248 distortion matrices, 246 distribution law, 317 dual variational problem, 66 duality, 65 duality and bounds, 98 duality and structural optimization, 99 duality gap, 226, 231 dynamical system, 304

eﬀective behavior, 215, 217, 226, 233, 234 eﬀective conductivity, 31 eﬀective constitutive relations, 274 eﬀective energy, 22, 217, 222 eﬀective estimates, 219 eﬀective ﬂow stress, 284 eﬀective properties of nonlinear composites, 280 eﬀective stress potential, 218 eﬀective tensor of the laminate, 77 eigendirections, 4, 5 eigenvalue, 5 elastic compliance, 2 elementary bounds, 275 embedded-atom method, 243 energy balance, 250 energy minimization, 19 energy of stretching and shearing the ﬁlm, 22 ensemble averages, 281 entropy equation, 254 epiconvergence, 315 equi-coercive energy, 46 Ericksen-James potential, 245 evolution of microstructure, 258 exact ﬁelds, 280 expandable materials, 94 extended Lagrangian, 87 extension property , 330 FENE model, 341 ferroelectrics, 15 ferromagnetic ﬁlms, 42 ferromagnetic shape-memory alloy, 27 ﬁbers stronger than the matrix, 289 ﬁbers weaker than the matrix, 290 ﬁne twinning, 26 ﬁne-scale microstructure, 17 ﬁnite perturbations, 73

351 ﬁrst moment, 203 ﬁrst-order laminates, 200, 201, 209, 211, 256 ﬂuctuation covariance tensors, 219 ﬂuctuation tensor, 180, 188 ﬂuctuations of the strain and stress ﬁelds, 292 forbidden region, 84 four-sided tent, 28 fourth-rank stiﬀness tensor, 67 Fr´echet derivative, 7 frame-indiﬀerence, 244 free energy, 244 free energy density, 28 functionally graded composites, 169, 180 functionally graded materials, 178 functions of bounded variation, 51 Γ-convergence, 17, 38, 52, 313, 315 Γ-limit, 53 G-closure, 3, 89, 171, 172 G-compactness, 107, 110 G-convergence, 107, 108, 118, 170, 174, 176, 178, 183 G-limits, 178 general Beltrami equations, 115 general Beltrami operators, 110, 132 generalized secant modulus tensor, 228 generalized secant second-order estimates, 227 generalized variational problem, 202 graded material optimization, 170 graded material properties, 173 gradient method, 7 gradient Young measures, 202–204, 209, 211 granular microstructures, 216 gripping mechanism, 11 growth conditions, 23, 46, 54

H-limit, 143 habit plane, 239 Hamilton-Jacobi equation, 8 hard device, 249 harmonic-mean (Wiener) bound, 91 Hashin-Shtrikman lower bound, 285 Hashin-Shtrikman-Walpole bounds, 97 heat equation, 254 Helmholtz free energy, 17, 24 Helmholtz free energy density, 18 heterogeneous ﬁlms, 18 heterogeneous random ﬁelds, 309 heterogeneous structures, 169 heterogeneous thin ﬁlms, 28 hierarchical microstructure, 144 hierarchical structure of a laminate, 76 Hill bounds, 92, 97 Hill-type relation, 312 homogeneous interactions, 45 homogeneous layers, 40 homogenization approach, 3 homogenization of a nonlinear elliptic problem, 334 homogenization theory, 198 homogenized composite, 181 homogenized design problem, 184, 188 homogenized functional, 4 homogenized Hooke’s law, 4 homogenized optimal design, 6 homogenized stress constraint, 188 homothety operator, 317 improved bounds, 92 instability of solution, 70 integral Beltrami operator, 116 integral constraints, 187 interface conditions, 25 interfacial energy, 17, 22, 40

352 intermetalics, 237 internal energy, 254 invariant line condition, 25 isometric mappings of the plane, 24 isothermal phase transformation, 249 isotropic measure of the ﬂuctuations, 293 Jensen’s inequality, 205 Kirchhoﬀ model, 149–151 K-quasiconformal mapping, 117, 118, 131 K-quasiconformal normalized solution, 127 K-quasiregular mapping , 117 Lagrange multiplier, 2, 3 laminate, 205, 207 laminate closure, 78 laminate evolution, 258 laminate structure, 92 laminates, 65, 204 laminates and minimizing sequences, 76 lamination closure, 79 Landau-Devonshire’s theory, 245 large and discontinuous shear deformation, 16 lattice parameters, 19, 245, 246 layout problem in 2D elasticity, 141 layout problem in the thin plate theory, 149 least square criteria, 6 Legendre transform, 90, 218 Lennard-Jones potential, 54, 244 level of approximation, 280 level-set function, 7, 9 level-set method, 2, 7, 8, 12 limiting energy, 24 linear analytical scheme, 281

linear comparison composite, 216, 219, 222, 277, 280 linear comparison methods, 215 linear composite with inﬁnitely many phases, 276 linear elastic comparison composite, 222 linear thermoelastic comparison composite, 226, 230, 231 local problem, 99 low temperature martensite, 19 lower bound, 89 lumped-parameter model, 258, 262 macro-stress modulation functions, 179 macroscopic level, 241 macroscopic models, 261 martensite phase, 15, 238 martensitic microstructure, 16 martensitic phase transformation, 15 martensitic thin ﬁlms, 16, 22, 24 martensitic transformation, 19 martingale, 336 mathematical programming problem, 205, 209 matrix laminates, 78 maximal monotone graph, 193, 195 maximal resistance, 100 maximum dissipation principle, 252 maximum stiﬀness design, 180 maximum-dissipation principle, 253, 258 measure preserving ﬂow, 305 MEMS, 15 meso/macro-scopic level, 241 mesoscopic level, 241 mesoscopic scale, 48 mesoscopic-level models, 256 Michell surface structures, 163 microactuator, 15

353 microgeometry, 178 micromachine actuators, 27 micropumps, 18, 21, 26, 27 microscopic level, 240 microscopic models, 244 microscopic scale, 48 microstructure, 24 microstructure evolution, 237 microstructure of perforations, 3 microvalves, 27 min-max problem, 99 minimal conductivity, 100 minimal extension, 87 minimizers, 17 minimizing sequence, 171, 199 minimum compliance problem, 159 mismatch strain , 35 mixture-like models, 259 modiﬁed secant method, 279 monoclinic martensite, 246 monoclinic transformation, 19 most rigid composite, 4 multi-valued maximal monotone graph in RN × RN , 195 multi-well structure, 19 multiple grains, 28 multiple layers, 18, 28 multipliers, 209 multi-scale stochastic convergence in the mean, 335 multivariable variational problems, 72 multiwell Lagrangians, 68 Mumford-Shah functional, 49 Mushtari-Donnell-Vlasov shell, 140, 159 nearest-neighbour interaction, 48, 54 next-to-nearest neighbour interactions, 61 next-to-nearest neighbour Lennard Jones interactions, 61

nonconvex variational problems, 65 nonconvexity, 69 nondivergence equations, 111, 136 nonlinear composites, 215, 233–236, 273, 280 nonlinear hyperbolic problem, 250 nonmonotonic constitutive relations, 70 nonquasiconvexity, 69 normalization conditions, 127 normalized K-quasiconformal , 109 normalized solution, 118 n-sided pyramidal tent, 27 null-Lagrangian, 119 numerical simulations, 280 objective function, 7 optimal control, 181, 185 optimal design, 84, 176, 181 optimal design problem in conductivity, 197 optimal energy, 90 optimal layout, 141, 197 optimal layout of two materials, 140 optimal microstructures, 85, 200, 201, 207, 211 optimal shape, 2 optimal shape of the cantilever, 6 optimal solutions, 200 optimal structural design, 172 optimal structure, 84, 209, 211 optimality conditions, 83 optimization problem, 200, 210 optimum design problem, 141 orthorhombic martensite, 238, 246 oscillations, 19 pair-potential energy, 45 partition of unity, 248 penalization procedure, 6 perforated composites, 3 periodic composites, 144

354 phase averages, 215–218, 230–232 phase transformation, 241 Piola-Kirchhoﬀ stress, 244 plane elasticity, 141 polyconvex measures, 204 polycrystallic SMA, 260 polycrystalline models, 260 polycrystalline thin ﬁlms, 16 power-law composites, 229 power-law materials, 216, 222, 275 primary solutions, 127 probability space, 304 pseudo-elasticity, 241 quasiconformal mappings, 107, 117 quasiconvex envelope, 23, 70, 72 quasiconvexiﬁcations, 201 quasiperiodic media, 307 random domain, 307 random integral functionals, 316 random process, 305 rank-2 sequential laminate, 4 rank-3 sequential laminate, 5 rank-one condition, 239 rate-independent memory, 237 rate-sensitivity exponent, 287 recoverable strain, 35, 33, 34 reference lattice, 45 reiterated stochastic convergence, 335 relaxation, 143, 155, 198–203, 208 relaxed compliance optimization, 171 relaxed formulation, 4 relaxed functional, 4 relaxed integrand, 205 relaxed problem, 201, 208 relaxed stored energy, 258 released thin ﬁlm, 22 scaled domain, 23 secant bounds, 221

secant moduli, 276 secant modulus tensor, 223 second level approximation, 280 second-order estimates, 216, 233– 235 second-order laminates, 207 second rank laminated composite plates, 152 self-adjoint problem, 7 self-consistent approximation, 216, 234 self-consistent estimates, 222, 233– 235 semicontinuous and subadditive envelope, 50 sequential laminates, 4 sequential-lamination envelope, 81 set Gθ , 3 several loadings, 91 Shape derivative, 7 shape design of membrane shells, 162 shape memory alloys, 237 shape sensitivity, 1, 7 shape-memory alloy, 15, 16, 17 shape-memory eﬀect, 17, 237 shape-memory ﬁlms, 18 signed distance function, 9 SMA/elastic multilayers, 35 smart materials, 237 soft device, 249 special functions of bounded variation, 52 speciﬁc entropy, 254 sputtered NiTi ﬁlms, 41 sputtering, 16, 18 St.Venant-Kirchhoﬀ-like form, 245 stability of the aﬃne solution, 73 stability to perturbations, 68 stationarity, 225–231 stationary variational estimate, 228 statistically homogeneous, 305

355 statistically periodic media, 306 stiﬀness, 178 stochastic calculus, 309 stochastic diﬀerential equations, 336 stochastic divergence, 311 stochastic gradient, 311 stochastic homogenization, 313 stochastic homogenization process, 317 stochastic integral, 336 stochastic Laplacian, 311 stochastic two-scale convergence in the mean, 327 Stoilow factorization , 117 strength, 178 stretching and shearing the ﬁlm, 17 strong interfacial energy, 39 structural optimization, 2, 7 structural variation, 81 subadditive ergodic theorem, 320 subadditive process, 320 suboptimal projects, 87 suboptimal solutions, 72 tangent modulus, 226 tangent second-order estimates, 225, 228, 231 tangent second-order method, 232 tetragonal martensite, 238, 246 tetragonal transformation, 19 thermal expansion, 94 thermodynamical evolution, 254 thin ﬁlms, 15, 20 thin shells in bending, 159 three-dimensional cantilever, 11 three-dimensional optimal structures, 85 three-point bounds, 216, 222 topology optimization, 170, 172 total free energy, 18 translated bounds, 93

translation bounds, 98 translation matrix, 156 translation method, 91, 155 translation operator, 317 translator, 96 trigonal martensite, 238 twins, 238 two-dimensional eﬀective energy, 21 two-phase shells, 159 uniform growth condition, 54 upper bound, 221 Van der Waals type interfacial energy, 20, 29 variants of martensite, 19 variational bound, 89, 277 variational bounding technique, 277 variational problems in elasticity, 66 vector variational problem, 209 viscoelastic material, 98 viscosity/capillarity-like model, 249 volume constraint, 197 volume-fraction models, 259 weak∗ -limit of the measures, 47 weak-star topology, 122 Weierstrass E-function, 72 Weierstrass-type test, 81 weighted sum, 2 Young measure, 256, 258 zero level set, 8