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PLATES, LAMINATES AND SHELLS Asymptotic Analysis and Homogenization

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Series on Advances in Mathematics tor Applied Sciences - Vol. 52

PLATES, LAMINATES AND SHELLS Asymptotic Analysis and Homogenization

T Lewinski Warsaw University of Technology, Poland

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

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Lewinski, T. Plates, laminates, and shells : asymptotic analysis and homogenization / T. Lewinski, J.J. Telega. p. cm. - (Series on advances in mathematics for applied sciences : vol. 52) Includes bibliographical references and index. ISBN 9810232063 1. Elastic plates and shells. 2. Homogenization (Differential equations) I. Telega, J6zef Joachim. II. Title. HI. Series. QA935.L39 1999 531\382--dc21 99-40193 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore by Regal Press (S) Pte. Ltd.

To the blessed memory ofPawelek Telega

and to

Ewa and Hanna for their patience and understanding

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Preface Introduction of composite materials into many fields of engineering practice has made the stress analysis complex due to the presence of two levels of analysis. For instance, the structure as a whole may affect failure phenomena at the material level and these microdefects do have influence on the overall response of the whole structure. It is clear that the entire analysis has to be branched into two analyses: at the macro- and microlevel. A systematic, albeit formal, technique of such an analysis is provided by the multiscale expansion method. Its correctness is lucidly justified by the homogenization theory. The theorems of homogenization apply to both media and structures of regular as well as irregular (random) layup. However, in the regular (or periodic) case the homogenization methods are constructive and thus applicable. Broad applications of the homogenization theory in investigation of effective properties of composite materials are the subject of the monographs by Bensoussan et al. (1978), Sanchez-Palencia (1980), Bakhvalov and Panasenko(1984) and Jikov et al. (1994); cf. also the comprehensive review paper by Ponte Castafieda and Suquet(1998). The effective properties characterize a hypothetic material with smeared-out inhomogeneities. From the mathematical standpoint the homogenization is equivalent to investigating G-limits of operators or T-limits of functional which govern a given physical problem. In recent years new methods of finding these limits have been developed, which has made it possible to examine the overall behaviour of the composite materials in the physically and geometrically nonlinear range, cf. Miiller (1987), Bouchitte" and Suquet (1991), Sanchez-Palencia and Zaoui (1987), Messaoudi and Michaille (1994) and Sab (1994). It is also feasible to consider cracks, both unilateral and bilateral, with or without friction, cf. Telega (1990b). The dynamic overall behavior is less investigated. The homogenization methods apply also to the plate and shell problems. The best results are obtained if the departure point is three-dimensional and an appropriate passage to a limit transforms the initial formulation to a two-dimensional one with the elastic moduli being smeared-out. Such an ingenious approach was proposed by Caillerie (1984) and Kohn and Vogelius (1984) in the papers on elastic plates. The monograph of Kalamkarov (1992) was inspired by this averaging method. The aim of the present book is to synthesize applications of the homogenization methods to plate, shell and laminate problems in a possibly broad perspective, by encompassing var ious averaging methods, whether starting from the three-dimensional or two-dimensional mathematical models. The analysis is confined to the carefully selected both practically important and variationally consistent structural models. In almost all these models the dual formulations are given more or less explicitly. The averaging by homogenization applied in the present book starts either from the threedimensional or two-dimensional settings. In the latter case the homogenization process can go in various manners, depending on the assumed scaling, which leads to the formulae of various applicability ranges. One of the aims of the book is to investigate these discrep-

Vlll

Preface

ancies and reveal physical meaning of the homogenization formulae. On the other hand an emphasis is put on the mathematical rigour. The majority of the results found by the two-scale expansion technique are justified by the r-convergence method. If available, the theorems on correctors are reported. The homogenization based on the two-dimensional models yields much simpler formu lae than those obtained from the three-dimensional setting. But simplicity is not the only motivation for starting from the two-dimensional models. The other argument follows from the theory of relaxation of the layout optimization problems of plates and shells. The layout optimization problems and the homogenization theory are linked with each other since the relaxed optimization problems are solvable and admit the presence of the composite domains, see Kohn and Strang (1986). This interesting interrelation is explained in Chapter VI by the fundamental layout problem for thin plates: mix two plate materials to form the stiffest plate of a given volume. The other optimum design problem: find the stiffest plate of varying thickness and given volume has found its partial solution with the help of the theory of Young measures. Designers who use composite materials have to reckon wim appearing the cracks. In fibrous composite laminates some cracks form regular patterns, which justifies using the homogenization method for predicting decay of stiffnesses and interrelate it with crack density parameters. In Chapter III some selected problems of such type are discussed in detail. The homogenization results turn out to be realistic only if an appropriate scaling is used. Some homogenization results, found by appropriate scalings, compare favourably with available experimental data. The material for the present book has been collected during the joint work of the authors since 1985. A large part of the work was supported by the State Committee for Scientific Research (KBN, Poland) through the grants No. 3 P404 013 06 and No. 7 T07A 016 12 as well as through the Statutory Projects at the Faculty of Civil Engineering, Warsaw University of Technology. Moreover, the material for this book was partly collected by the first author during his stay at the Essen University (Germany) as an Alexander von Humboldt fellow. The support of the Alexander von Humboldt Foundation is gratefully acknowledged. The present book would never be completed without stimulating discussions with our colleagues: W. Bielski, A. Galka, B. Gambin, G. Jemielita, S. Jemiolo, A. M. Othman, S. Tokarzewski, R. Wojnar, J. Bojarski, S. Bytner and S. Olszewski and without a cre ative atmosphere at the Institute of the Fundamental Technological Research of the Polish Academy of Sciences and the Institute of Structural Mechanics of the Faculty of Civil En gineering, Warsaw University of Technology. Moreover, we would like to express our sincere thanks Mrs M. Rejmund and to Ms I. Malicka for their dedicated work of word-processing and editing the whole manuscript and to Mrs B. Sobolewska for drawing the figures. We also express our thanks to the editors, and especially to Ms E.H.Chionh, for the invaluable editorial remarks and understanding, when the subsequent deadlines had gone.

Preface

IX

Last but not least we wish to express our gratitude to Professor Nicola Bellomo, the Editor-in-Chief of the Series on Advances in Mathematics for Applied Sciences, for in cluding our book into this collection. Tomasz Lewinski and Jdzef Joachim Telega Warsaw, March 1999

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Contents CHAPTER I. MATHEMATICAL PRELIMINARIES Introduction

1

1. Function spaces, convex analysis, variational convergence 1 1.1. Function spaces: IP and Sobolev spaces 1 1.1.1. Lebesgue spaces V 1 1.1.2. Sobolev spaces and trace operators 4 1.2. Elements of convex analysis and duality, minimization theorems, multivalued mappings 14 1.2.1. Convex sets and functions 14 1.2.2. Minimization theorems 21 1.2.3. Normal integrands, integral functionals and Rockafellar's theorem . 25 1.2.4. Quasiconvexity and yt-quasiconvexity 27 1.2.5. Elements of the duality theory 30 1.2.6. Set-valued maps 34 1.3. Variational convergence of sequences of operators and functionals 35 1.3.1. G-convergence 36 1.3.2. //-convergence and the energy method 37 1.3.3. Two-scale convergence 45 1.3.4. T-convergence 47 1.3.5. r-convergence of sequence of nonconvex functionals convex in highest-order derivatives: non-uniform homogenization 49 1.3.6. T-convergence and duality 52 1.3.7. Convergence of sets in Kuratowski's sense 64 1.4. Two approximation results 65 1.5. An augmented Lagrangian method for problems with unilateral constraints . 78 CHAPTER II. ELASTIC PLATES Introduction 2. Three-dimensional analysis and effective models of composite plates 2.1. Equilibrium problem of a periodic plate 2.2. Family of problems (PE) 2.3. Asymptotic analysis. Effective moduli and local problems 2.4. Case of transverse symmetry 2.5. Centrosymmetry of the periodicity cell 2.6. On computing effective stiffnesses 2.7. Case of moderately thick periodicity cells 2.8. Case of thin periodicity cells. Derivation by imposing Kirchhoff's constraints 2.9. Case of transversely slender periodicity cells of constant thickness

85 88 88 90 94 102 103 104 105 109 112

xii

Contents 2.10. r-convergence and justification of three models of thin, transversely inhomogeneous and anisotropic plates with constant thickness 2.10.1. Basic relations 2.10.2. Justification of the effective plate model of Sec. 2.8 by passing to zero: e —> 0 and then e —► 0 2.10.3. Justification of the effective plate model of Sec. 2.9 by passing to zero: e —> 0 and next e —* 0 2.10.4. Justification of the effective plate model of Sec. 2.3 by passing to zero: e —> 0 and e —> 0 simultaneously 2.11. Effective stiffnesses of longitudinally homogeneous plates

3. Thin plates in bending and stretching 3.1. Kirchhoff type description 3.2. Asymptotic homogenization. In-plane scaling approach 3.3. Refined scaling approach 3.4. Variational formulae for effective stiffnesses 3.5. Correctors 3.6. Variational formulae for effective compliances. Dual effective potential . . . 3.7. Transversely symmetric plates periodic in one direction 3.8. Ribbed plates. Bending problem 3.8.1. Formula of Francfort and Murat for stiffnesses 3.8.2. Ribbed plates of higher rank with the stronger phase taken as an envelope 3.8.3. Formula of Lurie-Cherkaev-Fedorov for stiffnesses 3.8.4. Formula of Francfort-Murat-type for compliances 3.9. Ribbed plates. Plane elasticity problem 3.9.1. Formula of Francfort and Murat for stiffnesses 3.9.2. Formula of Francfort and Murat-type for compliances 3.10. Plates periodic with respect to a curvilinear parametrization. Non-uniform homogenization 3.11. Effective bending stiffnesses of plates with quadratic inclusions 3.12. Perforated plates 3.13. Plates stiffened with rigid inclusions 4. Nonlinear behavior of plates 4.1. Von K£rmdn equations 4.2. Homogenization 4.3. Bifurcation and homogenization of perforated von Kdrmdn plates 4.3.1. Homogenization of perforated von Ka^man plates 4.3.2. Bifurcation of von K£rm£n plates: basic results 4.3.3. Bifurcation points of the homogenized plate and the linearized problem 4.3.4. Bifurcating branches of perforated and homogenized plates

116 116 120 136 138 151 153 153 157 164 166 168 172 173 175 176 179 181 182 183 183 185 188 190 195 200 203 203 205 209 209 213 221 223

Contents 5. Moderately thick transversely symmetric plates 5.1. Reissner-Hencky model 5.2. The in-plane scaling-based asymptotic homogenization 5.3. The refined scaling analysis 5.4. Justification of the refined scaling approach 5.4.1. Basic relations and auxiliary results 5.4.2. T-convergence of the sequence {Jc — Lc}c>0 5.5. Dual homogenization 5.6. Orthotropic plates periodic in one direction 5.6.1. Effective stiffnesses according to the in-plane scaling approach .. 5.6.2. Effective stiffnesses according to the refined scaling approach 5.6.3. Effective torsional stiffness of plates of step-wise varying thickness 5.6.4. Formula of Tartar-Francfort-Murat type for effective stiffnesses of ribbed plates. In-plane scaling approach 5.7. Other linear and nonlinear models of plates with moderate thickness. Homogenization study 5.7.1. Reissner's model 5.7.2. A refined theory of moderately thick plates undergoing moderately large deflections 5.7.3. Homogenization study

xiii 235 235 236 239 242 243 247 254 261 262 263 266 269 270 271 275 278

6. Sandwich plates with soft core 6.1. Hoff's theory 6.2. Effective stiffnesses in the periodic case 6.3. Reissner's approximation and relevant homogenization formulae

287 287 291 293

7. Comments and bibliographical notes

294

Chapter III. ELASTIC PLATES WITH CRACKS Introduction

303

8. Unilateral cracks in thin plates 8.1. Cracking modes 8.2. Periodic layout of cracks. Homogenization 8.3. Justification: T-convergence

304 304 305 310

9. Unilateral cracks in plates with transverse shear deformation 9.1. Admissible cracking modes 9.2. Periodic layout of cracks. Homogenization 9.3. Justification: variational inequality (9.2.1) and the energy method

315 315 319 323

10. Part-through the thickness cracks 10.1. Two-layer description 10.2. In-plane scaling and effective model 10.3. The study of convergence

339 339 345 349

xiv

Contents 10.4. 10.5. 10.6. 10.7. 10.8.

Dual homogenization Passage to classical models of cracked plates Refined scaling and effective model Plates with aligned cracks Cracks of arbitrary position. Three-dimensional local analysis 10.8.1. Asymptotic analysis 10.8.2. Justification by T-convergence

11. Stiffness loss of cracked laminates 11.1. Two-dimensional model of transversely symmetric laminates in stretching and in-plane shearing 11.2. Modelling the unilateral crack within the internal layer 11.3. Regular crack systems 11.4. Moderately thick laminate weakened by transverse cracks of high density. Model (/i,Zo) 11.5. Thin laminate with transverse cracks of high density. Model (h0, lo) 11.6. Thin laminate weakened by transverse cracks of arbitrary density. Model (ho, I) 11.7. Justification by T-convergence and duality 11.7.1. Moderately thick laminate 11.7.2. Refined scaling and T-convergence 11.8. Application of the augmented Lagrangian method to solving local problems with unilateral constraints 11.9. Case of aligned parallel intralaminar cracks. Effective characteristics according to the (h0,1) approach 11.10. Degradation of effective stiffnesses of laminates with aligned parallel cracks. The refined scaling approach-model (h0,1) 11.11. Degradation of effective stiffnesses of laminates [0° /90^J S . Comparison with experimental results and with other analytical predictions 11.11.1. Scope of the section 11.11.2. [07903] 5 glass/epoxy laminate tested by Highsmith and Reifsnider (1982) 11.11.3. [0790°], glass/epoxy laminate tested by Ogin et al. (1985) 11.11.4. [0^/90°] s graphite/epoxy laminates tested by Groves (1986) 11.11.5. [0790°], glass/epoxy laminates tested by Smith and Wood (1990) 11.11.6. [0790°], carbon/epoxy laminate tested by Smith and Wood (1990) 11.12. Stress distribution around crack tips 11.13. Crack spacing as a function of the averaged applied stress 12. Comments and bibliographical notes

358 363 364 369 378 378 384 388 388 394 396 397 401 403 409 410 411 432 434 442 448 448 449 455 455 458 458 459 462 464

Contents

xv

Chapter IV. ELASTIC-PERFECTLY PLASTIC PLATES Introduction

469

13. Mathematical complements, homogenization of functionals with linear growth 13.1. Functional setting: spaces Wh\ W2-\ LD, DV, DD, and HB 13.2. Convex functions and functionals of a measure 13.3. General homogenization theorems for functionals with linear growth 13.4. r-convergence and Dirichlet boundary conditions, relaxation 13.4.1. Thin Kirchhoff plates made of Hencky material 13.4.2. Moderately thick plates, refined scaling 13.4.3. On von KaYmdn plates made of Hencky material

469 470 478 483 494 494 496 497

14. Homogenization of plates loaded by forces and moments 14.1. Thin Kirchhoff plates loaded by boundary forces and moments 14.2. Three models of thin, transversely inhomogeneous and anisotropic plates with constant thickness 14.2.1. Basic relations 14.2.2. Derivation of the effective plate model by passing to zero: e -♦ 0 and next £ -> 0 14.2.3. Derivation of the second effective plate model: £ —► 0 and next e -» 0 14.2.4. Derivation of the third effective plate model: e —► 0 and £ —> 0 simultaneously

498 498 516 516 521 529 530

15. Comments and bibliographical notes

532

Chapter V. ELASTIC AND PLASTIC SHELLS Introduction

535

16. Linear and nonlinear models of elastic shells 16.1. Theory of shells with transverse shear deformation 16.2. Koiter's version of thin shell model 16.3. Budiansky-Sanders-Koiter version of a thin shell model 16.4. Thin shell model with moderately large rotations around tangents 16.5. The models of Mushtari-Donnell-Vlasov and Mushtari-Marguerre

535 535 541 545 547 548

17. Homogenization of stiffnesses of thin periodic elastic shells. Linear approach 17.1. Koiter's shell. Asymptotic analysis and the convergence theorem 17.1.1. Asymptotic analysis 17.1.2. Justification by the T-convergence method 17.2. Dual homogenization 17.3. Effective stiffnesses of ribbed orthotropic cylindrical shells 17.3.1. Geometrical and material characteristics of a cylindrical shell 17.3.2. Strong formulations of the local problems (Py)

551 551 551 557 560 567 567 568

xvi

Contents 17.3.3. Case of stiffeners along the generating lines 17.3.4. Case of circumferential stiffening 17.4. Shallow shells of periodic structure: effective properties

568 569 572

18. Homogenized properties of thin periodic elastic shells undergoing moderately large rotations around tangents

574

19. Perfectly plastic shells

575

20. Comments and bibliographical notes

578

Chapter VI. APPLICATION OF HOMOGENIZATION METHODS IN OPTIMUM DESIGN OF PLATES AND SHELLS Introduction

581

21. Mathematical complements 21.1. Alternative representation of second and fourth order tensors 21.2. ^-transformation 21.3. Fourier representation of V-periodic functions 21.4. Examples of quasiconvex and quasiaffine functions 21.4.1. A quasiaffine function of the strain tensor n 21.4.2. A quasiaffine function of two strain tensors 21.4.3. An aggregate form of the previous results 21.4.4. A quasiaffine function of the stress tensor m 21.4.5. A quasiaffine function of the stress tensor n 21.4.6. A quasiaffine function of two stress tensors 21.4.7. An aggregate form of the two previous results 21.5. Harmonic mean as a lower bound for effective energy 21.6. Elements of the theory of Young measures

582 582 588 593 594 594 595 596 596 597 598 598 599 599

22. Two-phase plate in bending. Hashin-Shtrikman bounds 22.1. Lower bound for the Kelvin modulus 22.2. Upper bound for the Kelvin modulus 22.3. Lower bound for the Kirchhoff modulus 22.4. Upper bound for the Kirchhoff modulus 22.5. Attainability of Hashin-Shtrikman bounds. The Francfort - Murat construction

606 606 608 609 611

23. Two-phase plate. Hashin-Shtrikman bounds for the in-plane problem 23.1. The upper and lower bounds for the Kelvin modulus 23.2. The upper and lower bounds for the Kirchhoff modulus 23.3. Attainability of Hashin-Shtrikman bounds 23.4. Summary of the main results

616 616 617 619 621

612

Contents

xvii

24. Explicit formulae for effective bending stiffnesses and compliances of ribbed plates 24.1. First rank ribbed structure 24.2. Second rank ribbed structure with soft phase taken as a core 24.3. Second rank ribbed structure with the strong phase taken as a core

622 622 624 629

25. Explicit formulae for effective membrane stiffnesses and compliances of ribbed plates 25.1. First rank ribbed plates 25.2. Second rank ribbed plates

632 632 633

26. Thin bending two-phase plates of minimum compliance 26.1. Ill-posedness of the initial formulation 26.2. Relaxation 26.3. Bounding the potential W* by the translation method of Cherkaev-Gibianskii 26.4. Attainability of the translation bound 26.4.1. Regime (2): CM e [Ca.Ci] 26.4.2. Regime (3): CM < C2 26.4.3. Regime (1): CM > 0 26.5. Physical interpretation of the relaxed problem 26.6. Primal formulation of the relaxed problem 26.7. On the shape design 26.8. Square clamped plates of minimum compliance 26.9. Optimal perforated plates of small volume

634 634 637 641 646 646 648 649 651 653 661 666 669

27. Minimum compliance problem for thin plates of varying thickness: application of Young measures

671

28. Thin shells of minimum compliance 28.1. Setting of the problem 28.2. Relaxation 28.3. Primal formulation of the relaxed problem 28.4. On the in-plane minimum compliance problem of two-phase plates

678 678 681 682 684

29. Truss-like Michell continua 688 29.1. Structures of minimum weight. Discrete versus continuum formulations .. 688 29.2. Dual formulation 691 29.3. A symmetric cantilever problem 694 30. Comments and bibliographical notes

696

References

703

Index

733

CHAPTER I

MATHEMATICAL PRELIMINARIES

Introduction Throughout the book rather advanced mathematical tools are used for the study of plates, laminates and shells. These tools comprise, grosso modo, functional analysis, modern variational, asymptotic and homogenization methods. Therefore to facilitate the reading of the book, in the first chapter we have gathered most of the relevant mathematical results. They include not only rather standard ones, though scattered throughout the literature, but also such that are either new or unattainable like those contained in Sections 1.3.2, 1.3.5, 1.3.6 and 1.4.

1. Function spaces, convex analysis, variational convergence As already mentioned, the aim of this section is to introduce basic mathematical notions, definitions and results, which will be of primal importance throughout the book. 1.1. Function spaces: LP and Sobolev spaces In the present section we introduce the notion of weak (weak-*) convergence and most important, from our point of view, results concerning Lebesgue and Sobolev spaces. 1.1.1. Lebesgue spaces LP We start with an abstract result. Definition 1.1.1. Let X be a Banach space, X* its dual and {•, •) the bilinear canonical pairing over X x X*. (a) We say that {v„}„gN C X (N is the set of natural numbers) converges weakly t o w e X and denote vn —>■ v in X ,

as n — ► oo

if W 6 X*

{vn,v') —> (v,v")

as n —♦ oo .

(b) We say that {v^}nen C X' converges weak-* to v* e X* and denote v* -^ v* in X',

as n —> oo

2

Mathematical preliminaries

if Vu £ X

(v,v^) —> (v, v")

as n —> oo .

□

Weafc convergence in Lebesgue spaces IP Let us first introduce the notion of V spaces. Definition 1.1.2. (i) Let 1 R is said to be in LP (ft) if

n n

(ii) Let p = oo and fl C R be an open set. A measurable function / : Q —* R is said to be in L°° if ||/|U~(n) = inf{a : |/(x)| < Q a.e. infi} < oo .

D

Sometimes we shall simply write II/IILW

= 11/11* = ll/llo,

and similarly for L°°(fi). We recall that for 1 < p < oo, the dual space of V(Q.) is Lq(Q.) where 1/p + 1 / 9 = 1 . The dual space of L°°(Q.) contains strictly L^fi). Let us pass now to the notion of weak convergence in 1^(0,). Case 1. Let 1 • / in W(Q.) if / fm{x)g(x)dx -> / f(x)g(x)dx ,

as

m-»oo

for every g G £ ? (fi). Case 2. Let p = 00, then fm -^ f in L°°(fi) if / fm(x)g{x)dx -> / f(x)g(x)dx , n n for every 5 e /^(ft). We also have the following result. Theorem 1.1.3. Let Q C R n be an open set. Case 1. Let p = 1 and let /

m

-/

in

L\fl).

as m -> 00

Function spaces, convex analysis, variational convergence

3

Then there exists K > 0 such that ll/mllii f

in

V((l).

Then there exists K > 0 such that \\fm\\u
in L"(fi) .

/

in L°°(n).

Case 3. Let p = oo. (i)Let m

-/

Then there exists K > 0 such that ll/mlli-^*. (ii) Conversely, if ||/m||z,°° < K, then there exists / G L°°(n) and a subsequence {/m } of {/m} such that fm, - /

in L~(n).

□

Case J (ii) is known in the literature as the Banach-Alaoglu theorem. Corollary 1.1.4. Let xE, e = - , be a sequence of characteristic functions. Then there exist 71

6 £ L°°(Q, [0,1]) and a subsequence {x€'} of {xc} such that Xe> ->• 6» in L°°(fi)

weak-*

as e' -> 0. D Throughout the whole book periodic homogenization will often be performed and the following theorem will play an important role. Theorem 1.1.5. Let Y = h (cn.bi) and let / e W{Y), 1 < p < oo. Extend / by t=i

periodicity from Y to R n . Let Q, C R n be an open set and fc(x) = f(x/e) ,

16(1,

e = 1/m, m e IM.

4

Mathematical preliminaries

Then ft ^ (f) = ^rjf(y)dy

in LT{{1), as e - 0

Y

if 1 • (/)

in L°°(fi)

weak-* ,

as e -> 0

if p = oo. Here |V| = meas Y = measure Y.

□

Let us provide two examples. (a) Riemann-Lebesgue lemma. Let y = (0,27r), /(y) = sin y, and Q C R. Then sin(x/e) —>■ 0 in L°°(n)

weak-* ,

as £ —> 0 .

(b) Let y = (0,1), 0 < A < 1, a,P e R, fi C R, and ,V,A _ / ~\0

Q

ny>

for

for

J/ e (°. A ). y€(A,l).

Then / £ (x) ->■ AQ + (1 - A)/3 in Z,°°(fi)

weak-* ,

as e -» 0 .

D

Suppose now that fie R n is a domain and /(x,y), x € H, y G Y, is a y-periodic function with respect to the second argument and sufficiently regular. Then \imjf{x,x/e)dx

= J(f(x,y))dx

,

(1.1.1)

where g(x) := (/(x,y)) =

-r—lf{x,y)dy.

1.1.2. Sobolev spaces and trace operators Let fi c R n be an open set and / € D'(fi) a distribution on fi. By a = (a^,..., a„) e n M we denote a multi-index and set y

/ = a dx? a , 1 ... x£» _.. = ( g l ) Q ' - - ( 9 r . r / .

(1.1-2)

where |a| = ai + ... + a„

(length of a ) .

(1.1.3)

Function spaces, convex analysis, variational convergence

5

Here dk = 7;—. For a = 2 we write: dxk

Obviously, these notations are also used for regular functions on fi. We set (1 < p < oo) Wm-P(n) = {v e Lp(n)\&*v e L»{n)

V a € N n , | a | <m} ,

(1.1.4)

where &*v denotes the matrix of the ath derivative, in the sense of distributions, of the function v, that is Vy?eD(f2),

f{dPv)ipdx = {-l)lal n

fv^ipdx,

(1.1.5)

n

where D(fi) is the space of C°° functions with compact support in fl. The space W m ' p (fi) is endowed with the norm m

ll^lkm,p (n) = ( ^ | | 5 a v | | ^ ) 1 / p

if

l
(1.1.6)

p = oo .

(1.1.7)

l«|=0

||v||w">.°° =

max {||3°'v|U°'>}

if

0<|a|<m

Basic properties (i) The space Wm-P(Q) is a Banach space, separable if 1 < p < oo and reflexive if 1 < p < oo. The reflexivity is equivalent to weak compactness of the unit ball (and closed bounded sets). (ii) If 1 < p < oo, the C°°(Q) functions are dense in Wm'p(Q) endowed with its norm. (iii) If fi is bounded, then Wi,oo(0,) is the space of Lipschitz functions. (iv) For p = 2 the space W m ' 2 (fi) = Hm(tl) ,

(1.1.8)

is a Hilbert space with the scalar product

(«,«)-» ^2 ({&'u)&'vdx.

(1.1.9)

|a|<mn

It is convenient to introduce the following notations: IMIm,P,n = IMIw-.p(n) ,

l
(1.1.10)

6

Mathematical preliminaries IMUnHMImAil.

(1.1.11)

For 1 < p < oo we introduce the spaces: W™-"{Q) = D(fi) ,

(1.1.12)

mp

where D(fi) denotes the closure in W ' (fl). Forp = 2 we write: W0m'2(ft) = /f0m(fi) .

(1.1.13)

In general we have MC(fi) C Wm'p{Sl) and the inclusion is strict. For m £ N, by H~m(Cl) we denote the dual space of the space //"(fi), endowed with the norm: „,n = sup{K/,i;)| : t; e H?{n) , \\v\\mfi = 1} .

(1.1.14)

It can be shown that each element / € H~m(Q) can be written in the form: / = £ 0 % .

/*eL2(ft),

\<*\<m.

(1.1.15)

\a\<m

Such a representation is not uniquely determined. The space H~m(S}) is a space of distributions. We observe that, in general, the dual space (Hm(fl))' of /fm(fi) is not a space of distributions. Regularity of the boundary T = dCl From practical point of view the most important are domains Q, c R n of class Cm(0 < m < oo) or C m Q . Let O be an open set of R n . The functions of class Cm-a{0){m € N, 0 < a < 1) are defined by:

f v e Cm(0), v € C"' Q (0) <=> <

V\\C"'"(0) = IMIm,oo,0 + max sup IJ9|="X r x.veO

J\d*v(x)

\-+

- d"v(y)\ ;—^-^ < +oo ,

|x-1/|tt

where | • | denotes the Euclidean norm in R n . The case m = 0, a = 1 corresponds to Lipschitz functions. It is well known that C°,l(0) coincides with the space W 1,00 (0) of functions with bounded generalized derivatives. Let us characterize more precisely the regularity of the boundary T = 3f2 of fi C R". To this end we assume that: (i) Cl is bounded.

Function spaces, convex analysis, variational convergence

7

(ii) There exist a > 0 and /? > 0 and afinitenumber of maps a^,\
(1.1.16)

r=l

with the conditions: {(xI.x^lM?") <x\ < a r ( x r ) + £ , | x T | < a } c f i , l < r < M ,

(1.1.17)

{(i;,5 r )|a r (S r ) - 0 < x\ < aT(xT), l^l < a} C R n \ft,1 < r < M. (1.1.18) Here ?" = (z!;, •.. ,xTn) and the condition \xr\ < a has the following meaning: |Jy| < a, 2 < i < n. The functions a,., 1 < r < M are of class C m or Cm,Q on the set A r of points a? satisfying |x^| < a, 2 < i < n, or |?"| < a. Imbedding theorems and traces of functions on the boundary Assume now that fi is at least of class C0,1. Defining the space C°°(fl) of restrictions to Q of the functions from the space C°°(R n ), we have the following density theorem: c°°(Ti) = wm-p{ri),

(i.i.i9)

where the closure is to be taken in the sense of the space Wm'p(Q). We now quote the Sobolev and Rellich-Kondrashov imbedding theorems. Theorem 1.1.6. Let il c R" be a bounded domain with Lipschitz boundary and let 1 < p < oo. Case 1. If 1 < mp < n, then

wm-p(n)cL»'{n),

-1 = 1 - p*

p

n

and the imbedding

wm*(n) c Lq(n) is compact for every 1 < q < p*. Case 2. If mp = n, then Wm'p(fi) C L"(ft)

for every

and the imbedding is compact. Case 3. If mp > n, then Wm-P(fi) c C{U) and the imbedding is compact.

1 < q < oo

8

Mathematical preliminaries

Note that if a space Wl,p is replaced by W01,p, then no regularity of the boundary is required. If fl is of class C0,1, one can define the surface measure dT on T = d£l by r Jvdr

M

= ^ J

r v(r,

M ^ ) Vrt^,

l + ^(ftar(?))W,

Or{^))

(1.1.20)

\ where {ipr}^=l is a partition of unity subordinate to the covering { a ^ A r ) } ^ of I\ We observe that the partial derivatives (ftOr) of Lipschitz functions are in L°°(Ar). Conse quently the r.h.s. of Eq. (1.1.20) is well defined. The existence of a surface measure on T permits one to define the spaces Lq(T) , 1 < q < oo. We set IMk,.r=IMI«
(1.1.21)

We are now in a position to formulate the first trace theorem. Theorem 1.1.7. Let ft c FT be a domain class of C 0 1 . Case 1. For 1 0 such that: VVeC°°(n),

|M| 0 .,.r
0 such that: V«eC°°(n),

|M| 0 .,.r
(1.1.22)

(1.1.23)

Case 3. For p > n, there exists a constant K > 0 such that: Vt, G C°°(n) , |M|o.oo.r < K\\v\\1Jtfl .

(1.1.24)

□ Corollary 1-1.8. By density result (1.1.19), the inequalities (1.1.22) - (1.1.24) remain valid for any v e Wl'p(£l). Consequently one can define the trace operator. For instance, if p = 2 then the trace mapping, denoted by tr, is an element of:

treL(H1(n),L"(r)), (the space of linear and continuous operators mapping Hl(£l) into Lq(T)) with

9= 2 +

dh>2

V<j for n = 2.

for n>2

\

d.1.25) J

Remark 1.1.9. In practice, we shall often use the notation trv = v\r or even trv = v.

□ □

Function spaces, convex analysis, vanational convergence

9

Another surface measure, often used in vanational analysis, is the so-called Hausdorff measure. Definition 1.1.10. Let O C FT, 0 < k < oo and 0 < 8 < oo. We set OO

OO

H<(0) = gf inf{£(diam O ^ I O c | J O, , diam O, < 6} , 3=1

(1.1.26)

>=1

and

fft(0) = limtf«(0)=supff£(0), s->o

(1.1.27)

s

where u>k is the measure of the unit ball in R*. The number Hk(0) dimensional Hausdorff measure of the set O.

is called the kD

In the specific case of f2 C R 3 , O = dfi, /^(dfi) stands for the surface, 2-dimensional Hausdorff measure of the boundary dQ. of fi. If fi C R" is of class C 0,1 , the outward unit normal n = ( n i , . . . , n„) is defined almost everywhere on T = dtt for the surface measure dr. One can then define the normal derivative 5n = -— for the functions of C°°(Cl) by: dn

v 6 C~(fi) =^dnv = £ > f t r , = ^

= X > | ^

t=i

t=i

(1.1.28)

*

Obviously, the last implication is also valid for v e C"(fi). By using (1.1.25) we conclude that for v = C°°(fi) one has dv_ <J2\\div\\0,q,r, dn o.«,r i=i

(1.1.29)

1<
On account of (1.1.19), (1.1.29) and applying Theorem 1.1.7 we infer that — is a continon uous linear operator from the space W2,p(£l) into the space Lq(T) with: (n - l)p
n —p 1 < q < oo q = oo

for

1 n.

(1.1.30)

Moreover, there exist positive constants K'p , K'q and K', independent of v e M/2p(f2) such that dv dn

< K'\\v\\2,Pln o.Q.r [ < K'\\v\\2,n,n

for for

\
and and

q=

(n-l)p

n —p 1 < q < oo ,

(1.1.31)

10

Mathematical preliminaries dv_ dn

(1.1.32)

for p>n

o,e»,r

for every u€VK 2 ' p (n). By definition of the spaces W™'p(f2), it is evident that

w^p(n) = {v e wl'p{n)\v = o on r } , W 2 ' p (fi) = {v

W2'p(Q)\v = f^ = 0 on an Suppose now that the real numbers p and q satisfy: e

1 < p < n , l
1< q 1< q

and and

and

(1.1.33) T}

-+ - < 1+ p q n

n

(1.1.34)

(1.1.35)

Then for u 6 W lp (f2) and v G W ^ f i ) we have the Green formula: I vdiudx = — / udivdx + / utm

n

(1.1.36)

r

It can be verified that in all the three cases specified in (1.1.35), the product uv belongs to Fractional Sobolev spaces on F Assume now that £1 is of class C°°. For 0 < s < 1 one can define the spaces H"(T) by:

u € HS(T) «• <

//Jj^j^^)^)<+».

0.1.37)

U r s

The space # ( r ) is a Hilbert space for the scalar product: (w.v)//«(r) =

uvdT+ r

r (u(x) - u(y))(v(x) - v(y)) / / •dT(i)dr(») • \x - j/ln-J+is r r

It can be shown that for v € H1^), the trace mapping tr e L(H1(Q), L ? (r)) with q as in (1.1.25), has the space Hl/2(T) as the range. It means that (i)\/veH1{Q),trve H^2{T), l 1 ( i i ) V s e H (T),3veH (n),trv = g. From (ii) it follows that

i/1/2(r) c V(r), with gas in (1.1.25).

Function spaces, convex analysis, variational convergence

11

More precisely, the point (ii) means that there exist a lift operator R (not necessarily unique): ReL(i/1/2(r),ff'(n)),

d.i.38)

such that Vg€Hl/2(T),tr(Rg)

= g.

(1.1.39)

= {trv\veHm{n)}.

(1.1.40)

Similarly, we define: VmeM\{0},

Hm-i(T)

Remark 1.1.11. The space Hm~ 5 ( r ) , and more generally the space Hs(r) (s > 0), can be defined in an intrinsic manner. Then, however one has first to introduce the space i / s ( R n ) by using the Fourier transformation for s > 0. □ One can also define the following norm: g e

ff"H(T)

=► | M l m - i , r = inf{|M| ra , n |u G // m (fi) ,trv = g}.

Equipped with this norm the space Hm~* (T) is a Banach space. More precisely, the space Hm~ 2 (T) is a Hilbert space with the scalar product defined by f,ge

ff"-J(r)

=*> ( / , 5 L - i , r = («.«)m.n ,

where u and v are the unique elements of the space (Hm(fl) D HQ(Q))L tr u — f ,

(1.1.41) satisfying:

tr v = g .

We recall that Hm-i(r)

= (Hm(Q) n H^tt))1

(in tfm(ft)) .

We set H°{T) = L2{T) .

(1.1.42)

Identifying the space L2(T) with its dual, we define the topological dual of Hm~i (F) :

//-"•+5(r) = (// m -3(r))\

meM\{o}.

(1.1.43)

| M | m _ i , r < 1} ,

(1.1.44)

Equipped with the norm: t;6/rm+J(r)=HM|_m+.,r = max{{v,ip)\veHm-Hr),

Mathematical preliminaries

12

this space is a Banach space. In fact, the space H m+* (T), the dual of a Hilbert space is also a Hilbert space since the norm (1.1.44) is generated by the scalar product:

u,i; e tf-m+3(r) => («,«)_ m+ . i r = (v,^) m -j.r.

where, by the Riesz theorem,

/) for all ueX,

and ||/|| = IWI •

(1.1.46)

□ Three important inequalities Theorem 1.1.13. (PoincarS's inequality). Let ft be a bounded open set and 1 < p < oo. There exists a constant K = K(p, ft) such that

\\v\\m,Pfl
(1-1.47)

|a|=m

for every v e W™'p(ft). Particularly, for m = 1 one has H i * < KIIVulU, ,

(1.1.48)

W01,p(ft).

for every v 6 If n = 1, ft = (0,1) and u € 7/^(0,1), then

IMIw < -IMU> G where v = —• ax Remark 1.1.14. Note that the constant 1/n is the best possible and it is attained whenever v(x) = sin7nr. Theorem 1.1.15. (Poincard-Wirtinger inequality). Let ft be a connected domain of class C1 and let 1 0 such that

||«-(u)||j>
(u) = ^-Judx, n

(1.1.49) D

Function spaces, convex analysis, variational convergence

13

Let us pass now to Korn's inequality. Let $7 be a domain in R3. Define 11= {v e H\Q)3\v = a + b x x ,x € fi} ,

(1.1.50)

where a,b e R3. The set K is afinite-dimensional(six-dimensional), closed linear subspace of Hl(fl)3 = [H^Q)}3. Let V be a closed subspace of Hl(fy3 such that H^(Q)3 C V C tf^fi)3. Let Kv = TZnV and let Qv be the orthogonal complement of Tlv in V, i.e., V = Hv © Qy. The inequality 3

J2

r e^vfoivjdx

> K\\v\\iM

formyvGQv,

(1.1.51)

is called Korn's inequality. Here K > 0 is a constant independent of w G Qv and e,(*)-(,)=(g +g)/2.

d.1.52)

The inequality 3

r

J2 / hj(v)eij(v)

+ twjdx > ^,||w||J in

forany Hl(Q)3 ,

(1.1.53)

defines the coerciveness of strains. Let us provide an example of Q v Let 90 = ToUTi with meas T0 > 0. Then Tly = {0} and V = Qv = {ve // 1 (n) 3 |v = 0 o n r 0 } .

(1.1.54)

Theorem 1.1.16. Let Q C R3 be a domain with a Lipschitz boundary. Then both the coerciveness of strains and Korn's inequality hold. □ Remark 1.1.17. For more details on Lebesgue spaces the reader is referred to any standard textbook on functional analysis like Alexiewicz (1969), Edwards (1965) or Yosida (1978) as well as to more specialized books (Adams, 1975; Kufner et al., 1977). Sobolev spaces are investigated in Adams (1975), Brezis (1983), Kufner et al. (1977), cf. also Dautray and Lions (1990), Lions and Magenes (1968), Ciarlet and Rabier (Appendix, 1980). Theorem 1.1.5 is a slight modification of Dacorogna's Theorem 1.5 (1989, Chap. 2), who also provided a detailed proof, cf. also Appendix to the paper by Duvaut (1979) in the case of periodic L2-functions. The properties of the space Hj^ are studied by Lions and Magenes (1968). The regularity of the boundary, including weaker assumptions and singular cases is studied by Adams (1975), cf. also NeCas (1976), Nazarov and Plamenevskii (1991) and Movchan and Movchan (1995). Definition 1.1.10 of the Hausdorff measure follows Giusti

14

Mathematical preliminaries

(1984), cf. also Part II of the book by Morel and Solomini (1995). Riesz' Representation Theorem 1.1.12 is proved in standard textbooks on functional analysis, already mentioned. For more details on Poincare' and Poincar6-Wirtinger inequalities the reader is referred to Brezis (1983) and Morrey (1966). Korn's inequality was proved by Duvaut and Lions (1976) and NeCas and HlavaCek (1981). 1.2.

Elements of convex analysis and duality, minimization theorems, multivalued mappings

The present section is intended as a brief introduction to convex analysis and select ed modern variational methods, including the theory of duality. For details the reader is referred to Castaing and Valadier (1977), Ekeland and Temam (1976), Hiriart-Urruty and Lemarechal (1996), Ioffe and Tihomirov (1979), Moreau (1974), Rockafellar (1970), Rockafellar and Wets (1998), cf. also Panagiotopoulos (1993), Smith (1985), Struwe (1990). Topological vector space and locally convex spaces are expounded in Yosida (1978, Chap. I), cf. also RoubiCek (1997). We recall that Banach spaces fall within this class of spaces. 1.2.1.

Convex sets and functions

Let V be a real topological vector space and / : V —» R = [-oo, +oo]. Particularly, V may be a finite-dimensional space. In this specific case the strong and weak topologies coincide. The effective domain of / is dam f = {v £ V\f{v) < oo} .

(1.2.1)

The epigraph of / is epif={(v,r)e

VxF|/(v)
(1.2.2)

The set [wi.wj] = {v € V\v = Xvx + (1 - X)v2, A e [0,1]} , is said to be the interval joining the points vi and v2. A subset C of the space V is said to be convex if it contains the interval joining any two of its points. The empty set is assumed to be convex by definition. Let C C V. The intersection of all convex sets which contain C is a convex subset of the space V. This set is called the convex hull of the set C, and it is denoted by coC. The intersection of all closed convex sets which contain C is a closed convex set of V, which is called the convex closure of C and denoted by WC. Proposition 1.2.1. The closure of the convex hull of a set C coincides with its convex closure, coC = coC. □

Function spaces, convex analysis, variational convergence

15

If {vi,..., vk} is a finite set of points of V, then every point v € V which can be represented in the form *: V = ^A,V{ ,

i=l

where Xt > 0, i = l,...,k,

£^Aj = 1, is called a convex combination of the points

Vi,...,Vk.

Proposition 1.2.2. (Caratheodory's theorem). Let C c R n . Then every point of the set coC is a convex combination of no more than n + 1 distinct points of the set C. □ A result from the theory ofHilbert spaces Let V be a Hilbert space with the scalar product (•, •), for instance the space L2(J7) or 2 L (Q)n. For an arbitrary set C C V we denote C 1 = {v|u±v V u e C } , where ul.v means that (it, v) = 0. It is known that C1 is a closed subspace of V. To derive the homogenized complementary potential we shall frequently exploit the fol lowing result. Proposition 1.2.3. Let Vi, V2 C V be closed subspaces of the Hilbert space V, then (V 1 -rV 2 ) 1 = ViJ-nv2J-. Proof. Since Va C V, + V2{a = 1,2) therefore (Vi + V2)1 C V£. Hence we conclude that

(v, + v 2 ) x c v,x n yf . To obtain the inverse inclusion we take u e V^ n V2X. Then u e ^ ( Q = 1,2), which means that uLVa. Hence u±.Vi + V2 and consequently Vi1 n V^ C (Vi + V2) 1 . □ We recall that Vi © V2 denotes the orthogonal sum of subspaces Vi, V2. The function / i s convex if for every u,v € V, A e [0,1], f[Xu+{\-X)v\ < A/(u)+(lX)f(v) (with the convention (+oo)-i-(-oo) = +00). Convex functions assuming value (-00) are very special. Let / : C —» R, C C V. With / we can associate the function / defined by 7(V) = \{{V) *VtCr> Jv ' [ +00 if v g C .

(1.2.3)

We observe that the function / is defined on the whole space V. This function is convex if and only if the set C is convex and the function / is convex on C. The convexity of / on C means that for each u, v € C the following inequality is satisfied: /[Au + ( l - A ) « ] < A / ( u ) + ( l - A ) / ( t ; )

VAe[0,l].

(1.2.4)

16

Mathematical preliminaries Let C be a subset of V. The indicator function Ic is denned by: i i \ Ic{v)

=

J° \+oo

if veC, if v*C.

(12 5)

-

The set C is convex if and only if the function Ic is convex. In this way the study of convex sets can be reduced to the investigation of convex functions. The inequality which follows is often used in proving homogenization theorems, see Dal Maso(1993). Theorem 1.2.4. (Jensen's inequality). Let ft c R" be a bounded open set, u e L}(Q) and / : R —► R be convex, then

1

mh[x)dx) - w\Jf{u{x))d-

,x.

h

D

' 'ii

/

Pedregal (1997) provides a general approach to Jensen's inequality in terms of paramet rized measures, see also Sec. 21.6. The next proposition will prove to be useful in the study of minimum compliance prob lems. Proposition 1.2^5. If {/i}*ez is a family of convex functions defined on a vector space V with values in R, then the function / = sup ft iei

is also convex.

□

Let V be a Hausdorff locally convex topological vector space, for instance a Banach space or a finite-dimensional space. By V we denote the space of continuous linear functionals on V. This space is called the topological dual of V. The value of v* G V* on v G V will be denoted by v'(v) or {v", v). The triple (V, V, (•, •)) is an example of the so-called dual pair. Thus for each v € V, v ^ 0, there exists v' £ V such that (v",v) ^ 0 and for each v" e l ^ , « ' / 0, there exists v e V such that (v', v) ^ 0. By o(V, V) we denote the weak topology on V generated by the duality between V and V. For instance, if V = L^fi), then V = L«(fi) where 1/p + l/q = 1. Then the dual of Ll(il) is L°°(n) and vice versa. In contrast, the normed dual of L°°(fi) is larger than L\n), cf. Rockafellar (1976), Yosida (1978). Lower semicontinuity A function / : V —> R is lower semicontinuous on V (l.s.c.) if WveV

f{v) < lim mff(v) .

(1.2.6)

For instance, the indicator function Ic of a set C C V is l.s.c. if and only if C is closed.

Function spaces, convex analysis, variational convergence

17

A convex l.s.c. function is said to be proper if it is not the constant +00 and if it does not assume the value —00. The set of all such functions defined on V is denoted by r 0 (V). Continuity of convex functions The following theorem is very useful in proving homogenization theorems. Theorem 1.2.6. Every l.s.c. convex function defined on a Banach space is continuous in interior points of its effective domain. □ In fact, this theorem holds in barrel spaces. Polar functions Let V be a topological vector space and / : V —» R. The function /* : V" -+ R denned by f(v')=sup{(v',v)-f{v)\veV} is called the conjugate, or polar, or dual function of / . The function /** : V —* R defined by f"(v)

=sup{(v',v)

- f"{v')\v'

eV'}

is called the biconjugate or bipolar of / . The function C / : V -» R defined by Cf = sup{g < f\g convex} is called the (lower) convex envelope of / . Theorem 1.2.7. Let / : V -* R U {+00}, then (i) /* is convex and lower semicontinuous. (ii) If / is convex and lower semicontinuous, then /* ^ +00. (iii) In general

/"

and if / is convex and lower semicontinuous, then

r = cf = f. In particular if / takes only finite values then /** = Cf. (iv) In general

r•=r• (v) / < g implies /* > g'.

□

Mathematical preliminaries

18 Example 1.2.8. The function

IZ(v')=sup{{v',v)\veC} is known as the support function of C. In the case of three-dimensional associated plasticity where V = V* = E^ and E^ stands for the space of symmetric 3 x 3 matrices, the support function has clear physical meaning: it is the density of plastic dissipation provided that C stands for the elasticity convex, see Sees. 13 and 14. Infimal-convolution Let V be a vector space, / i and f2 two functions from V to R. The infimal-convolution (or inf-convolution) is the function denoted by f\Of2, from V to R, defined by (/id/aXi;) = inf{/,(v - u)+f2(u)\u

e V} .

(1.2.7)

Hence we conclude that (/iD/ 2 )(v) = ( / 2 0 / i ) ( « ) = i n f ^ i / i H / a M i / i , v2 €V,Vl+v2

= v}.

(1.2.8)

Examples 1.2.9. 1°. Let Ca C V, a = 1,2. By Ica we denote their indicator functions. We calculate: (IClOIC2)(v)

= i n f { / C l M + ICi(v2)\vi +v2 = v} _ J 0 if there exist va e Ca, a = 1,2 such that V\ + v2 = v , \ +oo otherwise.

Consequently we obtain Ic^ki

= Ac+Cj) •

2°. Let v e V and , . ( 0 if u = ti, {v}{ i ^-i-oo otherwise. We find (/ { „}0/)(u) = inf{/ {w} (v,) + f(u -

Vl)\Vl

eV}=

f(u - v) .

3°. Let C C V and / : V -> R. One has (/<:□/)(«) = i n f ^ i ; , ) + / ( „ - u,)l«i e K} = inf{/(« - «,)|«i e C} . For instance, if V is a normed space and f(v) = \\v\\, then (/ C a||.||)(t;) = i n f { | | w - U l | | : t ; 1 e C } .

D

Function spaces, convex analysis, variational convergence

19

Proposition 1.2.10. Let V be a vector space and / i , f2 : V —» R convex functions. Then / i d / 2 is also a convex function. D Theorem 1.2.11. Let V be a Hausdorff locally convex space and /i and f2 two functions on V. Then ( / i a / 2 ) * = / 1 , t / 2 *,

(1.2.9)

(with the convention that (-oo)+(+oo) = - c o ) . Remark that / „ takes the value (-oo) if and only if fa is the constant (+oo). □ We shall say that the inf-convolution fiOf2 is exact at v, if (f\Of2)(v) G R implies that there exist v\ and v2 such that Vi + v2 = v and f\(vi) + f2(v2) = (f\^f2)(v). Theorem 1.2.12. Let V be a Hausdorff locally convex space and fx, f2 G r0(V). Suppose there exists v^ G V such that /* and f2 are finite at VQ, and that /* is continuous at VQ. Then f\ of2 is exact and l.s.c. □ Theorem 1.2.13. Let fi, f2 € r 0 ( V ) . Suppose there exists vo 6 V such that /i and f2 are finite at vQ, and one of them (say / i ) is continuous at v0. Then one has (/i + /2)* = / , * n / 2 , and the inf-convolution /i*0/ 2 is exact.

(1.2.10) D

Subdifferentiation Let V be a topological vector space, / : V —* R, and v0 e V such that /(t; 0 ) € R. Then v' G V* is said to be a subgradient of / at v0 if for every v G V, / ( v ) - /(vo) > (u*, u — v). The set of all subgradients of / at VQ is called the subdifferential and denoted by df(vo). We observe that if / : V —» (-oo, +00] is not the constant +00, one can define a subgradient v* at v0 by the formula Vi>,

/(v) >/("o) + ( " * . " - v o ) •

Then /(u 0 ) = +00 => 3/(u 0 ) = 0. Proposition 1.2.14. Let V be a Hausdorff locally convex space, / : V —► R, v0 G V such that f(v0) G R and t;* G V*. Then the following properties are equivalent: (i) v' G df(v0), (.ii)r(v') + f(v0) = (v',v0), mf'(v') + f{vQ)<(v',v0). Consequently df(v0) is closed and convex. □ Theorem 1.2.15. Let V be a Hausdorff locally convex space, /1 and / 2 two convex l.s.c. proper functions, and vt G V such that /] and / 2 are finite at i»i and /1 is continuous at v\.

20

Mathematical preliminaries

Then for every v e V, d{fi + h)(v)=dMv)

+ df2(v) .

(1.2.11) □

We recall that for } \ , f2 being only proper convex functions we have d(h + f2)(v) D dMv) + df2(v2). Obviously at least for convex functions subdifferentiability is a generalization of differ entiability. To this end we recall the definition of the Gateaux derivative. Let / : V -> R. The limit A-0+

A

if it exists, is called the derivative of / in the direction v and is denoted by f'(u; v). If there exists u' € V* such that VveV,

f'{u;v) = (u*,v),

then / is said to be Gdteaux differentiable at u and u* is called the Gdteaux derivative of / at u. Then we write u* = f'(u) or u' = Gf{u). In contrast to the subdifferential, which may consist of more than one elements, the Gateaux derivative is unique. Deeper interrelationship is provided by the following proposition. Proposition 1.2.16. Let / : V —> R be a convex function. If / is Gateaux differentiable at u e V, then it is subdifferentiable at u and df(u) = {f'{u)}. Inversely, if at u G V the function / is continuous, finite and possesses a unique subgradient, then / is Gateaux D differentiable at u and df(u) = {/'(")}• The subdifferential as a multivalued mapping. Let now V = R n and let / : R n —> R be a convex function. Proposition 1.2.14 states that df(x) is a closed and convex set, thus df : x —> df(x) is a multifunction. Proposition 1.2.17. The subdifferential mapping is monotone in the sense that, for all Xi and x2 in R n {x\-x\,x2-x{)

>0

forall

x'a€ df(xa) , a = 1,2 .

(1.2.12)

□ n

In (1.2.12), (x*,x) stands for the scalar product in R . In fact, we have more than monotonicity. Proposition 1.2.18. The subdifferential mapping is maximal monotone in the sense that (x*-T7,x-£) > 0

V77 6 d / ( £ ) ,

implies

x'edf{x).

(1.2.13)

□

Function spaces, convex analysis, variational convergence

21

Particularly, if / is differentiable, then TJ = — . Remark 1.2.19. Propositions 1.2.17 and 1.2.18 can be extended to proper, lower semicontinuous functions defined on Hilbert spaces, cf. BrSzis (1973, Chap. II). 1.2.2.

Minimization theorems

Let (V, r) be a topological space and let / : V —> R be a function. Definition 1.2.20. We say that / is r-coercive if for every t 6 R there exists a r-compact (and r-closed) subset Ct of V such that {veV\f{v)
(1.2.14)

The next two propositions are due to Buttazzo (1989). Proposition 1.2.21. Assume that (i) / is r-lower semicontinuous, (ii) / is r-coercive. Then / admits a minimum point on V.

D

The last proposition is very general: the space V may be nonreflexive and the functional / nonconvex. We also observe that / may be a sum of two functional: / = / i + hSpecifically, f2 may be the indicator function of a set C C V. Then the minimization problem reads: find inf{/,(«) + Ic(v)\ veV}

= int{Mv)\v

€ C} .

In dual Banach spaces, in particular in reflexive spaces, the coerciveness with respect to the weak-* topology is phrased in the following proposition. Proposition 1.2.22. Let V = W be the dual of a Banach space W, and let r be its weak-* topology. Then a function / : V —» R is r-coercive if and only if lim f(v) = +oo. MI-~

(1.2.15)

Corollary 1.2.23. The condition (1.2.15) is equivalent to the following: there exists a function $ : R —» R such that lim $(s) = +oo

and

f(v) > $(||i>||).

□

s—»+oo

We observe that the last two propositions apply to the following dual pairs: (L 1 (0), L°°(Q,), (•r)z,'(n)xi,°°(n)) {L°°(i}),Ll(Q,), (•,-)z,°°(n)xL'(ft))' (M 1 (fi),Co(n), (-.^M'tnjxCofn))' w n e r e M'(fi) stands for the space of bounded measures on Q, see Sec. 13.1.

22

Mathematical preliminaries

The proposition which follows is due to Cea (1971) and is, in fact, a corollary from Propositions 1.2.21 and 1.2.22. Proposition 1.2.24. Let V be a reflexive Banach space and / : V - » R a weakly lower semicontinuous functional. If C C V is a bounded and weakly closed set then there exists at least one minimizer of / in the set C. D The following proposition, due to Ekeland (1974, 1979, 1990), is useful in proving the existence of minimizing sequences, cf. also Ekeland and Temam (1976 Chap. I), Benaouda and Telega (1997). Proposition 1.2.25. Let (V, d) be a complete metric space, and / : V —» R U {+00} a lower semicontinuous function, ^ 00, bounded from below. Let 77 > 0 be given, and a point ueV such that /(u) < i n f / + »j. Then there exists some point v e V such that f(v)
+ T,,

d(u,t;) f(v)-r]d(v,w)

(1.2.16) .

(1.2.17) (1.2.18)

□ Stronger version is formulated in the following form. Proposition 1.2.26. Let (V, d) be a complete metric space, and / : V —> R U {+00} a l.s.c. function, ^ +00, bounded from below. For any 77 > 0, there is some point v 6 V with f(v)<mif VuieV,

+ V,

(1.2.19)

f(w)>f{v)-j]d(v,w).

(1.2.20) D

This relies on the fact that there always exists a point u with f(u) < inf / + 77. The inequality (1.2.19) then proceeds from (1.2.16) whilst (1.2.20) from (1.2.18). The following theorem is also due to Ekeland (1974,1979). Theorem 1.2.27. Let / be a Gateaux differentiable function on a Banach space V, bounded from below and satisfying the following condition: (A) whenever f'{un) —» 0 in V and {/(un)} is bounded, then either f'(un) = 0 for some n or the sequence {un} has a cluster point in V. Then the function / attains its minimum: 3 u e V : / ( u ) = inf{/(u)|ue V}

and f'{u) = 0 .

□

Function spaces, convex analysis, variational convergence

23

We recall that in the last theorem / ' stands for the Gateaux derivative of / . For a large class of linear problems the following existence and uniqueness theorem is most appropriate, see Ciarlet (1988), Yosida (1978). Theorem 1.2.28 (Lax-Milgram lemma). Let V be a Banach space with norm || • ||, let L : V —> R be a continuous linear form and l e t a : V x V — » R b e a symmetric and continuous bilinear form that is V-elliptic (coercive) in the sense that there exists a constant K > 0 such that a(v,v) > K\\v\\2

for all

veV.

Then the problem: find

u £V such that a(u, v) = L(v) for all

v e V ,

has one and only one solution, which is also the unique solution of the equivalent mini mization problem: find

u e V such that J(u) = m({J{v)\v 6 V} ,

where J(v) =-a{v,v)

- L(v) .

D

A large class of unilateral contact problems without friction can be formulated in the form of the following variational inequality, cf. Telega (1987, 1988) ueC:

a(u,v-u)>{f,v-u)

for all

veC

(1.2.21)

where a(u, v) is a bilinear form on a Hilbert space V, more precisely, a : V x V —> R is continuous and linear in each of the variables u, v. The proof the theorem which follows is given in Kinderlehrer and Stampacchia (1980). Theorem 1.2.29. Let a(u, v) be a coercive bilinear form on V, C C V closed and convex and f £ V. Then there exists a unique solutions to the problem (1.2.21). In addition, the mapping / —> u is Lipschitz, that is, if Ui,u2 are solutions to the problem (1.2.21) corresponding to / i , f2 e V, then ||«i-U2||v<(l/ff)||/i-/2||v..

(1-2-22)

□ We observe that the last theorem reduces to the Lax-Milgram lemma provided that C = V. The mapping / —> u is linear if C is a subspace of V. If the bilinear form a(u, v) is

24

Mathematical preliminaries

symmetric, i.e. a(u,v) = a(v,u) for all u, v £ V, then the problem (1.2.21) is equivalent to the convex minimization problem \a{u,u) where f(v) =

- f(u) = inf{^o(t;,i;) - f(v)\v g C} ,

(1.2.23)

(f,v).

Brezzi 's theorem Let now V and S be real Hilbert spaces, and let a(-, •) and b(-, •) be continuous bilinear forms on S x S and S x V respectively. The search of the saddle point on V x S of the functional C(V,T)

= ^a{T,T) + b(r,v) - (f,v) - (g,r) ,

(1.2.24)

is equivalent to the following problem, see Brezzi (1974) Find (u, a) e V x S

such that

a(a, T) + b(r, u) = {g, r) 6(CT, V)

= {/, v)

for all

r € 5 ,

for all

T

eV

(1.2.25)

.

Here / and g are given functions in V* and 5* respectively. Let So = {T G S| 6(T, V) = 0 for all v 6 V}. One version of Brezzi's theorem was formulated by Arnold and Falk (1987). Theorem 1.230. Suppose there is a constant 7 > 0 such that O(T,T)>7||T|||

for all

reS,

and lnf

Then for all (f,g) Moreover,

€ V

SU

P 11 11 11 11

^ 7 •

x S', there is a unique solution (u,a)

\\u\\v + \W\\s
£ V x S of (1.2.24).

+ \\g\\s.),

where K depends only on 7 and bounds for the bilinear forms a(-, •) and />(•, •).

□

Carathiodory function and the last general existence theorem Let f2 C R n be an open set and let / : Q x R m x ¥\N -► R U {+00}. Then / is said to be a Carathiodory function if (i) f(x, ■, •) is continuous for almost every i £ ( l , (ii) / ( • , u, £) is measurable in x for every (u, £) G R m x R N .

Function spaces, convex analysis, variational convergence

25

Theorem 1.231. Let Q be a bounded open set of R n with Lipschitz boundary. Let / : 0, x R m x R n m —► R U {+00} be a Carathiodory function satisfying the coercivity condition

f(x,u,i)>a(x)

+ p\i\",

for almost every x G fi, for every (u,£) £ R m x R n m and for some a e L1^), and p > 1. Assume that f(x, u, •) is convex. Let J(u)=

0 >0

f(x,u(x),S7u(x))dx. n

Assume that there exists u £ tto + W/01,p(fl)m such that J(u) < +00, then inf{J(«)| u e «o +

W^(Q)m}

attains its minimum.

O

The proof of the last theorem can be found in Dacorogna (1989). For mixed boundary conditions where tto is prescribed on To C dU with meas To > 0 the minimization problem means evaluating inf{J(u)\u e Wl-"{Q.)m ,u = uQ

on

T0} .

(1.2.26)

Under the assumptions of Theorem 1.2.31 the minimization problem (1.2.26) is solvable provided that u 0 £ W 1_ 5-P(r 0 ). 1.2.3. Normal integrands, integral functionals and Rockafellar's theorem Let us first recall the notion of a Borel subset in R". Borel subsets are sets obtained from open and closed sets by performing the operation of talcing countable unions, countable intersections, completion and any countable combination of these operations. A function / with values in R is called the Borel function if / _ 1 (F) is a Borel subset for any closed set F. For instance, continuous functions are Borel. In the calculus of variations of great importance are normal integrands. Definition 1.232. Let B be a Borel subset in R m . A function / : Q. x B -> R is said to be a normal integrand if (i) for almost every 1 e fi C R " the function f(x, •) is lower semicontinuous on B, (ii) there is a Borel function / : Q. x B —» R such that f(x, •) = f(x, •) for almost every x eil. It is convenient to call / a proper integrand if f(x, •) is proper for every x e Q, i.e., if f(x, £) > —00 for all £ and f(x, £) ^ +00. Furthermore, / is said to be a convex integrand if f(x, £) is convex in £ for each x £ fi, i.e., if epi f(x, ■) is convex-valued. We observe that any Carathe\)dory function is a normal integrand. For more details the reader is referred to Ekeland and Temam (1976, Chap. VIII). An alternative definition

26

Mathematical preliminaries

can be found in Rockafellar (1976, p. 173), see also Ioffe and Tihomirov (1979). The definition given by Rockafellar involves the measurabihty of the epigraph multifunction epi / : fi — R m + 1 defined by epi f(x, ■) = {(£, a) € J T x R | / ( x , {) < a} . Measurable multifunctions will be introduced in Section 1.2.6. Let p be a non-negative, a-finite measure on (O, A) where (O, A, p) is a measure space. For instance, we may have p. = dx; if O stands for the boundary of a Lipschitz domain S l c R " then p denotes a surface measure, cf. Sec. 1.1.2. For any normal integrand / on fi x R m and any measurable function u : fi —► R m , we have f(x, u(x)) measurable in x, and therefore the integral Jf(u)=

ff(x,u{x))dx n

,

(1.2.27)

has a well-defined value in R under the following convention: if neither the positive nor the negative part of the function x —> / ( x , u ( x ) ) is summable, we set J(u) = +co. In particular, then, J/{u) < +oo => f(x,u(x))

< +oo

a.e.

Jj is called the integral functional associated with the integrand / . In practice, we are concerned with the restriction of Jj to a linear space X of measurable functions u : fi —» R m . Obviously, Jj is a convex functional on X, if / is a normal convex integrand. Among the linear spaces X of interest, besides the space of all measurable functions, are the various Lebesgue spaces, the space of constant functions, and in the case of topological or differentiable structure on f2, spaces of continuous or differentiable functions. According to Rockafellar (1976), in their role in the theory of integral functionals these spaces fall into two different categories, distinguished by the presence or absence of a certain property of decomposability. We shall say that X, a linear space of measurable functions u : O —► R m , is decom posable if O can be expressed as the union of a sequence of measurable subsets Ok(k = 1,2,....), such that for every O* and bounded measurable function Ui : O* —* R m , and every v.2 € X, the (measurable) function f Ui (x) u(x) = { { u 2 (x)

for

x € Ofc ,

for

x e 0/Ofc ,

(1.2.28)

belongs to X. Since p is a-finite, the sets O* can always be chosen with p(Ok) finite. The space of all measurable functions, the Lebesgue spaces and Orlicz spaces, are all decomposable. However, the space of constant functions and spaces of continuous or dif ferentiable functions furnish examples of nondecomposability.

Function spaces, convex analysis, variational convergence

27

The importance of the concept of decomposability results from the following celebrated theorem due to Rockafellar (1976). Theorem 1.233. Let / be a normal integrand on O x R m , and let X be a linear space of measurable functions u : O —» R m . For the relation ™jxjf(x, u(x)Mdx) = y"[uinfm/(x, u)}fi(dx) o o

(1.2.29)

to hold, it is sufficient that X be decomposable and that the first infimum not be +oo. (These conditions are superfluous in the case where X is the space of all measurable func tions.) □ We observe that a similar theorem can be formulated for a maximization problem, since sup / / = - inf / ( — / ) • In such form Rockafellar's theorem will be used in the case where X is a Lebesgue space. An extension of this theorem to convex functionals defined on a space of measures will be given in Section 13.2. Calculation of conjugate functionals Let n be a bounded open set in R n , and / a non-negative normal integrand on Q x R m . We define a non-negative functional by J(u) = I f(x,u(x))dx ,

where u e LP(fi) m , 1 < p < oo. Proposition 1.2.34. Let J be a functional (1.2.30) defined on L"(fi) m , 1 < p < oo, and let UQ e L°°(Q)m be an element such that J(tto) < oo. Then for all it* G L»(fi) m we have J'(u')=

ff"(x,um{x))dx. ft

(1.2.30) □

We recall that - + - = 1, and if p = 1 then q — oo and vice versa. For the proof of p q the last proposition the reader is referred to the book by Ekeland and Temam (1976, Chap. IX). 1.2.4.

Quasiconvexity and A-quasiconvexity

In optimal design problem the stored energy function may be nonconvex, cf. Chap. VI. Then the notion of quasiconvexity introduced by Morrey in 1952 proves very useful, cf. Morrey (1966). Here we follow Dacorogna (1982, 1989), cf. also Murat (1987).

Mathematical preliminaries

28

Definition 1.235. A Borel measurable and locally integrable function / : R nm — ► R - is said to be quasiconvex if

Jf(A + VV{x))dx > \D\f(A),

(1.2.31)

D

for every bounded domain D c R n , for every A G R nm and for every

)m = Suppose that D - is a hypercube, (p is D-periodic (assumes equal values on the opposite faces of D) and set a = A + S7(p. Then we readily get

since

^

=

\D\jdx

=

W\I^A

D

+ Vip)dx = A +

Wu* ®ndx

=A

-

dD

D

Here n = (r^) stands for the outward unit normal to D and / ip ® ndx = 0 . D

In the case of convex functions the inequality (1.2.31) is nothing but Jensen's inequality, seeTh. 1.2.4. We observe that the quasiconvexity condition (1.2.31) uses an infinite number of test functions

|,|A|),

(1.2.33)

for every (z, u, A) G n x R m x R nm , where a is increasing with respect to |u| and \A\ and locally integrable in x. Then the function / is said to be quasiconvex if ff(x0,Uo,A

+ V |I>|/(z 0 ) uo.il),

(1.2.34)

D

for every cube D c ft, for every (x0, «o, A) G f2 x R m x R"m and for every (p G Wo'°°(D)m. The importance of the notion of quasiconvexity for the calculus of variations can be summarized in the form of the following result due to Morrey (1966), cf. also Dacorogna (1982,1989).

Function spaces, convex analysis, variational convergence

29

Theorem 1.2.36. Let / : R n m —» R be continuous. A necessary and sufficient condition for / to be lower semicontinuous with respect to weak-* convergence in W 1,00 (fl) m , i.e., lim inf ff(Vuk(x))dx

> /f{Vu{x))dx

D k

,

(1.2.35)

D

whenever u —>■ u weak-* in W

1,00

m

(fi) , is that / is quasiconvex.

□

SverSk (1992) formulated a necessary and sufficient condition for quasiconvexity. Let us consider practically important case where A e E 3 is the space of 3 x 3 matrices. By Z we denote the set of integers. Propositions 1.2.37. A continuous function / : E 3 —> R is quasiconvex if and only if /

f(A + VV(x))dx>f(A),

(1.2.36)

[o,ip

for each A € E 3 and each smooth function

R 3 periodic with respect to Z 3 , i.e., such that for each x £ R 3 and each k 6 Z 3 we have tp(x + k) =
(*)<

v." ->• u in L°°(n) m weak-* , m " du" {au"), = J2 ^"iJkQT f u n d e d i n L2W / ( « " ) -^ / in L°°{n)

(i = 1, •••,«) ,

weak-*,

where a ^ 6 R are constants. Our aim now is to find for which / we have / > / ( « ) provided that the hypothesis (H) holds. We observe that weak-* convergence in L°° can be replaced by weak convergence in W{j) < oo). However, in that case, in order to ensure that f(u) is a distribution one has to impose a growth condition at infinity on / (e.g. |/(TJ)| < a + P\r)\p with 0 > 0) while in L°° the only requirement will be that / is continuous. Definition 1.2.38. A function / : R m —► R is said to be 4-quasiconvex if

ff(b + £(x))dx > Jf(b)dx = \D\f(b) , D

D

for every b e R m , for every hypercube D c R " and for every £ € L(D) where L(D) = {$€ L°°{D)m\ U{x))dx

= 0 and £ e Ker a} ,

(1.2.37)

Mathematical preliminaries

30

(by £ € Ker a we mean that y ^ a y t ^ " =0).

o

Remark 1.239. In the Russian edition of the book by Dacorogna (1982) the set L(D) is replaced by Li(D) = {£ e L(D)| £ is £>-periodic} . By D-periodicity of a function £ is meant here the periodicity of £ in R" with the period D. Theorem 1.2.40. (Necessary condition). Suppose that lim inf I'f[u"(x))dx > f f(u{x))dx , holds (i.e., / > f{u)) for every sequence A-quasiconvex.

{U"}^ 6 N

(1.2.38)

satisfying hypothesis (Tt). Then / is □

Theorem 1.2.41. (Sufficiency condition). Suppose that {u"}„erg,u satisfy hypothesis (Tt)and v? -ueKera.

(1.2.39)

If / is A-quasiconvex, then (1.2.38) holds for every bounded open set Cl c Rn. □ Remark 1.2.42. (i) A Borel measurable and locally integrable function / : R nm —» R is said to be quasiaffine if / and —/ are quasiconvex. (ii) Examples of quasiconvex functions, which are not convex, are provided by Dacorogna (1989), Gibianskii and Cherkaev (1984) and Lurie and Cherkaev (1997). These authors also give examples of quasiaffine functions. (iii) By Qf is denoted the quasiconvex envelope of /: Qf = sup{
Function spaces, convex analysis, variational convergence

31

To expound the Rockafellar theory of duality we follow Bkeland and Temam (1976), see also Laurent (1972). The primal problem means evaluating (P)

mf{J(zL)\u€V},

where V and V" constitute a pair of dual linear topological spaces with canonical bilinear form (•, •) : V x V — R. Let X and X* be another pair of dual Hausdorff linear topological vector spaces with the canonical bilinear form (•, -)x'%xWe introduce a function $ : V x X - t R such that *(u,0) = J ( u ) .

(1.2.40)

Consider the perturbed minimization problem (P p )

m{{4>(u,p)\ueV}.

We observe that for p = 0 the problem P0 coincides with P. The dual problem means evaluating (/»)

suP{-$'(0,P')\P' e

X'}.

Proposition 1.2.43. (i) - o o < sup P' < inf P < +oo. (ii) If the problem P is nontrivial, then supP* < i n f P < + o o . (iii) If the problem P* is nontrivial, then - o o < s u p P * < inf P . (iv) If the problems P and P* are nontrivial, then - o o **(w,0)|we V} .

Here $** denotes the polar function of $* or the so-called r-regularization of $, i.e., $** e T(V x X). By r ( V x X) we denote the space of all convex lower semicontinuous functions

32

Mathematical preliminaries

defined on V x X. We recall that T0{V x X) is the subset of proper functions belonging

tor(v xx). Since we always have $*** = $*, therefore the dual problem P'" of P" coincides with P'. The problem P" coincides with P($"(u, 0) = $(u, 0) V u e V) provided that $** = $, i.e., if er0(VxX).

(1.2.41)

Until the end of this section we assume that (1.2.41) is satisfied. For p e X w e define the marginal Junction by h{p) = inf Pp = inf{$(ti,p)|u e V} .

(1.2.42)

Lemma 1.2.44. Under the assumption (1.2.41) the function h : X —> Ft is convex.

□

We observe that in general h # r0{X) though $ e T0(V x X). Lemma 1.2.45.

(i) (ii)

Vp*ex*

h V ) = **(o,P').

supP* =sup{-/i*(p*)|p* G X*} =

ft**(0).

□

Hence we conclude that the inequality sup P' < inf P is equivalent to /i**(0) < /i(0). Definition 1.2.46. (i) The problem P is called normal if /i isfiniteand lower semicontinuous at zero. (ii) The problem P is said to be stable if /i(0) isfiniteand h is subdifferentiable at zero. Proposition 1.2.47. Under the assumption (1.2.41) the following conditions are equivalent: (i) P and P' are normal and possess solutions. (ii) P and P" are stable. (iii) P is stable and has a solution. D The following result provides a stability criterion. Proposition 1.2.48. Let $ be a convex function, inf P < oo and assume that there exists uo 6 V such that p— ► $(uo,p) isfiniteand continuous at O g X Then the problem P is stable.

(1.2.43) D

Extremality relations Proposition 1.2.49. If P and P* possess solutions and if -oo < infP = supP* < oo,

(1.2.44)

33

Function spaces, convex analysis, variational convergence

then all solutions u of P and all solutions p* of P* are interrelated by the extremality relation: $(u,0)+*(0,p*) = 0 ,

(1.2.45)

(0,p*)€d$(w,0).

(1.2.46)

which is equivalent to

Inversely, if u £ V and p* G X* satisfy (1.2.45) then u is a solution to P and p* is a solution to P and (1.2.44) is fulfilled. □ Practically important specific case Let A : V —> X be a continuous linear operator, i.e., A £ L(V, X) and let A* be the conjugate of A, A* £ L(X', V) defined by, cf. Yosida (1978, Chap. VII) (Aw, x')XxX.

= (v, A'x*)VxV.

for all v £ D(A) and all x" G D(A'). similarly D(A') is the domain of A*. Let the functional J be given by

,

(1.2.47)

Here D(A) denotes the domain of the operator A;

J(u) = G{Au) + F{u).

(1.2.48)

The perturbed functional (u, p) may be assumed in the form *(u,p) = G(Au + p) + P ( u ) .

(1.2.49)

It can easily be shown that the dual problem P* means evaluating sup{-G*(p*) - P*(-A*p*)|p* e X'} .

(1.2.50)

We observe that: (i) if F and G are. convex then $ is also convex; (ii) if F € r 0 (V) and G G r0(A") then $ £ r 0 (V x X). The condition (1.2.43) can be written as follows: there exists an element «o G V s u c n m a t F(u0) < +oo , G(Au 0 ) < +oo and G is continuous at A«o •

n

9 s n

The extremality condition (1.2.45) yields F(u) + F'(-A'p')

= ("A* P',u)v.xV

G(Au) + G'(p') = (p*, Au)x.xX

, .

(1.2.52) (1.2.53)

These relations are obviously equivalent to -A"p* £dF{u), p* G dG{Au) .

(1.2.54) (1.2.55)

Remark 1.2.50. Ekeland and Temam (1976) assume the perturbed functional in the form $(u,p) = G ( A u - p ) + F ( u ) .

(1.2.56)

34

Mathematical preliminaries

1.2.6. Set-valued maps As we already know, the subdifferential is a multivalued (set-valued) mapping. In plas ticity, with a point x of a body is associated a so-called elasticity convex, being a closed and convex set of plastically admissible stresses (or moments in the case of plastic plates). Therefore in the present section we shall present basic notions related to set-valued map pings, cf. Rockaffellar (1976), Aubin and Cellina (1984), Aubin and Frankowska (1990). Let X and Z be two sets. A set-valued map F from X to Z is a map that associates with any x € X a subset F(x) of Z. The subsets F(x) are the images or the values of F. Here we content ourselves with the case where X = Q C R" and (fi, A) is a measurable space. Also, the values F(x) are in thefinite-dimensionalspace R m , which usually will be identified with the space of symmetric n x n matrices. We set dom F={xe gphF =

n\F(x) ^ 0} , {{x,z)\zeF(x)}.

We shall denote by F~l : R m —> Q the multifunction given by F-1{z) =

{xen\z&F(x)}.

A multifunction F : Q —> R m is said to be a multivalued mapping with closed-values if F(x) is a closed subset of R m for every x e 0.. Such a multifunction is said to be measurable, relative to the cr-field A, if for each closed set C C R m the set F~X(C) is measurable, i.e., belongs to A. Here

F-'(C)= u F-1{z) = {xen\F(x)nC^ R m , the following prop erties are equivalent: (a) F is measurable. (b) F'^IC) is measurable for all open sets C. (c) F _ 1 (C) is measurable for all compact sets C. (d) F~l (C) is measurable for all closed balls C. (e) dist (z, F(x)) is a measurable function of x € fl for each z € R m , where dist (z, F(x)\ = min{\z - win'" ■ w 6 F(x)}. □ With a measurable closed-valued multifunction one can associate a function. Indeed, we have the following theorem. Theorem 1.2.52. If F : fl —> Rm is a measurable closed-valued multimapping, there exists at least one measurable selection, i.e., a measurable function / : dom F —* R m such that f(x) 6 F(x) for all x G dom F. □ The next issue is the problem of defining continuity of set-valued maps. In the case of single valued maps from Q to R m , continuous functions are characterized by two equivalent properties:

Function spaces, convex analysis, variational convergence

35

(i) for any neighborhoodN(f(x u )) of f(x0), there exists a neighborhood A/"(xo) of XQ such that /(Af(x„)) C AA(/(i 0 )); (ii) for any sequence of elements {X P } P € N converging to x0, the sequence f(xp) converges to f(x0). These two properties can be adapted to the case of set valued-maps from f2 to R m : they become: (a) for any neighborhood Af(F(x0)) of F(x0), there exists a neighborhood M{x0) of x0 such that F(Af(z 0 )) C Af{F(x<>)); (b) for any sequence of elements {i?}^ converging to xQ and for any 20 6 F(x0), there exists a sequence of elements zp £ F(xp) that converges to z0. In the case of multivalued mappings, these two properties are no longer equivalent. We call upper semicontinuous maps those that satisfy property (a), lower semicontinuous maps are those that satisfy property (b). Obviously, continuous multivalued mappings are the ones that satisfy both properties (a) and (b). The most famous continuous selection theorem is due to Michael, cf. Aubin and Cellina (1984). Theorem 1.2.53. Let F from Q into the closed convex subsets of R m be lower semicon tinuous. Then there exists / : fi —» R m , a continuous selection from F. D 1.3.

Variational convergence of sequences of operators and functionals

This section is intended as a brief introduction to the mathematical theory of homoge nization. More precisely, we shall introduce the notion of G- and //-convergence of se quences of operators as well as the notion of T-convergence of sequences of functionals. With a sequence of functionals one can associate the sequence of conjugate functionals, which in our case will represent the functionals involved in the complementary energy prin ciple. Thus naturally arises the problem of interrelationship between the T-convergence of the primal sequence of functionals and the T-convergence of the sequence of conjugate functionals. The dual homogenization will play an important role throughout the whole book. To derive effective models in the subsequent chapters of the book we shall often use the powerful method of two-scale asymptotic expansions. This method can be justified rigorously by using the method of two scale convergence. The books by Sanchez-Hubert and Sanchez-Palencia (1992, 1993) may serve as a good introduction to homogenization (and asymptotic methods), cf. also Persson et al. (1993). As an application of the T-convergence theory we shall find the T-limit of a sequence of nonconvex functionals, yet convex with respect to the highest order derivatives. This case of nonuniform homogenization includes geometrically nonlinear elastic plates with a periodically nonuniform microstructure as well as geometrically linear and non-linear elastic shells. To perform the homogenization of elastic-plastic plates we shall also need the notion of convergence in Kuratowski's sense of sequence of sets.

Mathematical preliminaries

36

1.3.1. G-convergence The notion of G-convergence was introduced by Spagnolo (1968), cf. also Attouch (1984), Dal Maso (1993) and Appendix A by Allaire to the book by Hornung (1997). The Gconvergence is a notion of convergence associated with sequences of symmetric, secondorder operators. The G means Green because this type of convergence corresponds to the convergence of inverse operators and thus to the convergence of the associated Green functions. For the sake of simplicity, the notion of G-convergence is introduced in the case of the following equation:

d i v ^ w w i. n u£ = 0 on

3

9H.

Here fi c R n is a bounded open set and e > 0 a "small" parameter intended to tend to zero. More precisely, e belongs to the set £ = {e = l/m\m G N\{0}}. By £', £"..., we denote infinite subsequences of £. The matrices Az belong to the following set o f n x n symmetric matrices: M.(a, 0,0) = {A(x) e Z,°°(fi, E?) : a|£| 2 < A(x)t ■ £ < /3|£|2 for any £ e R n and a.e. x in Q}. At this moment, we impose no periodicity assumption. Definition 1.3.1. The sequence of matrices Ac(x) is said to G-converge to a limit AQ(X), as e tends to 0, if, for anyright-handside / e L2(fi) in (1.3.1), the sequence of solutions ue converges weakly in HQ (Q.) to a limit u° which is the unique solution of the homogenized equation associated with AQ : - div (AoVvP) = f o n

in fi , *o

uu = 0 on ail. The set Ms(a,(3,Q) is compact with respect to the T-convergence, as stated in the next theorem. Theorem 1.3.2. (i) If a sequence Ae G-converges its G-limit is unique. (ii) For any sequence Ac in M,(a, 0, Q,), there exists a subsequence e' and a homogenized limit A0, belonging to M„(a, 0, SI) such that A£/ G-converges to A0. (iii) The G-limit of a sequence Ac is independent of the term / and the boundary condition on dfl. □ Similar properties will also hold for the /^-convergence. The properties of //-limit will also be satisfied by the G-limit.

37

Function spaces, convex analysis, variational convergence

Remark 1.3.3. For a general study of G-convergence the reader is referred to the books by Attouch (1984) and Dal Maso (1993). These authors define G-convergence in terms of inverse operators. D 1.3.2.

//-convergence and the energy method

The //-convergence method was introduced by Murat (1977/78) and Tartar (1977). The abridged English version of these fundamental contributions has recently been published by Murat and Tartar (1997). We observe that" //" stems from "homogenization". The Hconvergence is a generalization of the G-convergence to the case of nonsymmetric prob lems. Thus, symmetric problems, which are of main interest for us, are also covered by this theory. As previously fi is an open set of R n . By u CC Si we denote a bounded open set of Si such that u7 C Si. By a, 0, a', 0' we denote strictly positive real numbers such that 0 < a < 0 < +00, 0 < a' < 0' < +oo. We set

M(a,0,Q)={AeL°°(n,En):

a|£| 2 < (A(x)t,t)

for any { £ F

n

< /3|£|2

and a.e. x in Si} ,

where E n is the space of n x n matrices. Definition of the H-convergence Definition 1.3.4. A sequence Ae, e G £, of elements of M{a, 0, Si) //-converges to an element AQ A0 of M{a,'0', M{a,' 0', ffl){Ae H~l{u) the solution uc of

—>• A0) if and only if, for any w CC Si and any / in

—div (A £ Vu £ ) = /

in

u , (1.3.4)

u< e // 0 V), is such that if —>■ u°

weakly in

c

Q

AeVu

->• A0Vu

HQ(UI) ,

weakly in

L2(u)n

(1.3.5) ,

for e £ £, where u° is the solution of -div ( A ) V O = /

in

u,

This definition can be extended to higher order elliptic equations. Let us consider the fourth-order plate equation1 (Si C R 2 ) :

(Df^(x)w^),a0

= g in A,

2

w e tf0 (fi) , 1

Small Greek indices (except for e) run over the set {1,2}. If repeated at different levels, they imply summation.

Mathematical preliminaries

38 Q2

where w aa = -^—5—. Here De enjoys the usual symmetry property: D?0x,i = D*"*0 = axaoxp Z)fQAM and satisfies the following condition KM2

< Dfx»(x)pa0PXli

< Ki\p\2 ,

(1.3.8)

2

for any p € E 2 and a.e. x in Q. ; here \p\2 = Y_, PapPap and g € H~2(£l). In this case a,P=l

//-convergence means that for any w CC SI and g in H~2(u), the solution of (Df^(x)w^),a0 w e //02(w),

= g in u>,

(1.3.9)

is such that we —^ u;0 weakly in HQ(UI) , D£V2w;£ ^ D 0 V 2 w° weakly in

L 2 (w,E 2 )

(1.3.10)

as e —♦ 0, where w° solves the homogenized plate problem: (Dfx"(x)u,°^),a0 w° G H2{u) .

= g in ui ,

We observe that in the general case, the homogenized coefficients A$, DQ depend on the macroscopic variable x € ft.

(1.3.11) M

may still

Example 13.5. Suppose that in (1.3.7), only D\lu, D2222, D\122 = Dfu and D] 212 = £>i22i _ £>2ii2 __ £>2i2i ^ different from zero (orthotropic plate). We further assume that Df0X,i € L°°(a, b) are functions of the variable Xi only. By w° we denote the solution to

(D?X)^ = 9

in

«>0 e H2(Q),

«.

(1312)

where (D1111)-1

_ , (2,1111)-! ,

£,1122(£,1U1)-1 _ .

D1122(D1111)-1

£,2222 _ ( D 1 1 2 2 ) 2 ( L ) 1 1 1 1 ) - 1 _ , £,2222 _ £>£1212 _ , £,1221

weak-* in L°°(a, b) as e —> 0. We have, see Bonnetier and Vogelius (1987).

(D1122)2(L)1111)-1

(1.3.13)

Function spaces, convex analysis, variational convergence

39

Proposition 1.3.6. Let D"^ and DQ * denote orthotropic tensors such that (1.3.8) and (1.3.13) hold. If xtf and w° £ H$(Sl) denote the solutions to (1.3.7) and (1.3.12), respec tively, then w£ ->• w°

weakly in

H%(Q) .

Proof. It is sufficient to prove that w£ —^ w for sonic subsequence of the original se quence. By using (1.3.8) we conclude that

IKIk(fi) < K\\g\\LHa) ,

\\M<#\\LHa) < K\\g\\mn)

,

provided that g 6 L 2 (fi). Here M£Q/J = DfP^Kx^vf) stand for the components of the moment tensor and naj3(we) = —w'- Consequently we may extract a subsequence e' such that wc —*• w .,,. Mfa0 -± Ma0

in . in

Hg(D.) and , ,2 _ L (fi).

d-3.14)

d2Ma0 Moreover we have M Q " ag = ——-— = g. We shall now verify that oxaax0 Ma0 = Dfx"KXfi(w)

in

Q,

(1.3.15)

from which it follows that w = w° (the unique solution to (1.3.12)). (i) Let a = 1,0 = 2. The definition of M\? and the fact that D ^ is independent of x2 yields M^ = d2(2Dl?12di(-w£'))

.

(1.3.16)

Recalling that the imbedding H$(£l) —> tf'(fi) is compact, from (1.3.14)i we infer that drf'^diw

in

L2{i1) .

(1.3.17)

Combining it with (1.3.13)4 we obtain D^'dnif

->• Dl2l2d,w

L2(i1),

in

(1.3.18)

and consequently d2{2D1c2Ud1(-w£'))--d2(2D10212di{-w))

in

Passing to the limit in (1.3.16) we get M 1 2 = 2D^K12(W)

as desired.

,

//"'(ft) .

40

Mathematical preliminaries

(ii) The cases a = 0 = 1 and a = f3 = 2 are more involved. The local property of the //-limit implies that it suffices to prove (1.3.15) for any rectangle R = (ai, 6i) x (a2,62) contained inft,see below. Since dnM}} = g - dapMf = g- 2dnMl? - d^Mf , it follows that 11

11

9iM£V = fgdx - 2d2M]? - ^ I' Mfdx + fc£'(x2) in R . ai

(1.3. 19)

ai

The first three terms on the right-hand side of (1.3.19) are bounded in the space L'daub^-H-^M))Next, integration of (1.3.19) with respect to X\ gives MtV, and since these are bounded in L2(R), it follows that kf '(x2) are bounded in H-2{a2M)- The relation (1.3.19) thus implies that d\M]} are bounded in L2((ai,&i); H~2(a2,b2)). We conclude that M\} are bounded in the space S(R) = {Me L2(R)\ 3,M e ^ ( K M ; / / - 2 ^ , ^ ) ) } ,

(1.3.20)

equipped with the natural norm. By using Theorem (5.1) in Lions (1969) we conclude that S(R) is compactly imbedded in L2((ai,6i); H~2(a2,62))- Consequently we have M]>^MU

in L2[{aub,)-H-2{a2M)\-

0-3.21)

Further we have (D^)-lMfx

= K „ ( « 0 + a22[0£1P(^11)-1(-^")] ,

(1.3.22)

and M22 = DlJ22{D\},nYlMl}

+ d22[(D2222 - {Dl™)2{D\},u)-'){-w*"]\

. (1.3.23)

We recall that £>2212 = 0. By virtue of (1.3.13) and (1.3.21) we get (D*,1,11)-^1,! -

(D\m)-lMu

and D^iD^Y'Ml), 2

inL^KM;//- ^^)).

- Dll22(DlulylMu

(1.3.24)

Function spaces, convex analysis, variational convergence

41

Due to (1.3.13) and (1.3.14) we get Dl},22{D\},n)-lw'" 22

[Df

22 2

- D0122{D0lu)-lw u

e

, 2222

- {D\} ) {D]}, )-'\w

->■ [D

-

(1.3.25)

{Dl0l22)2(Dl0

w

in L2{R). A combination of (1.3.22) - (1.3.25) now leads to ( ^ 1 1 1 ) - ' M 1 1 = dn(-w)

+ d22[D»22(D^)-\-w)}

,

(1.3.26)

and M22 = D^iD^y'M11

+ ^[{Df22

- {Dll22)2(Dl0lu)-l){-w))

. (1.3.27)

Multiplying (1.3.26) by D\ln we obtain the expression for Mu. Next, substituting (1.3.26) into (1.3.27) we obtain M22. Recalling that DQ * are independent of x2, we get the desired constitutive relation for M 1 1 and M22. D Properties of the H-convergence and H-limit Similarly to the G-convergence, //-convergence means, in essence, the convergence of the inverse operators [—div (A £ grad ) ] " \ which are bounded linear operators from H~x{i1) to Z/o(fi), when both spaces H~l(Q) and Hl(Q) are equipped with their weak topologies. The underlying topology satisfies the property of uniqueness of the H-limit and the //-limit is local as the following proposition shows. Proposition 1.3.7. (i) A sequence Ac , e G £, of elements of M(a, 0, fi) has at most one //-limit. (ii) Let Ae and Bc , e 6 £, be two sequences in M{a, (3, Q.) that satisfy At —»• A0 ,

Be —>• B 0 i

and are such that Ae = Bc on an open set w C fi. Then A0 = Bo on w.

□

Lemma 1.3.8 (compensated compactness). Let Q be an open subset of R " and q€, vc, e e £, such that

f
l(f -> q° weakly in L 2 (Q) n , [ divq e -+ divq0 strongly in //~'(fi) ,

iVe//>(«),

\v£-±v°

weakly in

H\Q) .

Then ( g £ , W ) -^{q°,Vv°)

weak-*

in

D'(fi) ,

Mathematical preliminaries

42

i.e., for any tp G Co°(fi) we have /\q c ,Wv c )ipdx ->

f(q°,Vv°)
Here (•, •) stands for the scalar product in R".

D

We observe that the product (g*, Vwc) is that of two weakly and not strongly convergent sequences. This phenomenon is known as compensated compactness. Applying the last lemma we formulate the next one. Lemma 13.9. Let fl be an open subset of R n . Let Ac belong to M(a, (3,0.) for e € £. Assume that, for e &£,

' ue e H\Q) , u'-^u° weakly in H\n) , ' qc = AcVuc ->• q° weakly in L 2 (fi) n , - d i v ( A £ V w £ ) - » divg 0 strongly in // _ 1 (fi) , 'vc eHl(Q), E v ^v° weakly in H\n) , ' tf = ATcVve ->■ 77° weakly in L 2 (fi) n , -div ( A ^ V u £ ) - + -div77° strongly in // _ 1 (fi) . Then {q0,Vv°)

= (Vu°,Ti0)

a.e.in

SI.

D

Here A f stands for the transpose of the matrix At. Obviously, for symmetric matrices ATE = Ac. The following two theorems are of primal importance. Theorem 13.10. Assume that Ac, e G £, belong to M(a,{3,SI) AQ e M{a',0',Sl). Assume that

and //-converges to

'ueeHl(Sl), < -div(A£Vu£) = /£

in

SI,

uc^u°

weakly in H\Sl) ,

fc -> f°

strongly in

//-'(fi),

for e S £. Then A£Vue

->■ A0Vu°

weakly in

(A £ Vu £ ,Vu £ ) ^ ( A 0 V u ° , V u ° )

L2(Sl)n ,

weak-*

As previously, (•, •) denotes the scalar product in R n .

in

D'(ft) . □

Function spaces, convex analysis, variational convergence

43

Remark 1.3.11. (i) Actually, it can be shown that the energy (A c Vu e , Vu £ ) converges weakly in L/oc(f2). (ii) The boundary conditions have no influence on the //-limit. Theorem 1.3.12. Let Ac, e € £, belong to M{a, (3, fi). There exists a subset £' of £ and a matrix AQ e Mia,

—, fi) such that Ac //-converges to AQ for e € £'. a a This theorem shows the sequential compactness of M(a, 0, Q) for the topology induced by the //-convergence. Remark 1.3.13. By introducing the so-called corrector matrix it is possible to approximate Vu e in the strong topology of a suitable space, cf. Murat (1977/78), Tartar (1977), Murat and Tartar (1997). Remark 1.3.14. The energy method is a constructive proof for the compactness theorem of //-convergence, cf. Murat and Tartar (1997), Allaire (1997). This method, attributed to Tartar, has nothing to do with any kind of energy. It is sometimes more appropriately called the oscillating testfunction method, but it is most commonly referred to as the energy method. Denseness of periodic composites Dal Maso and Kohn (1991) provided a general characterization of //-limits and their approximation by periodic composites. At the moment of writing the book this important result has not yet been published. Therefore, we reproduce here the seminar given by Kohn during the 1st Workshop on Composite Media and Homogenization Theory in 1991, cf. also Francfort and Milton (1987). These results are limited to the scalar case. However, they can immediately be extended to the vector case (linear elasticity). The results which will now be presented are important, for instance, in optimal design problems. Consider a mixture of two isotropic materials with conductivities at and a 2 , respectively. The volume fractions are 9 and 1—0. We set Ac(x) = alXc(x) + a2{\ -

Xc{x))

,

a.e. x e il ,

(1.3.28)

where Xe{x) stands for the characteristic function of the first phase. Here fi C R" is a fixed domain. Let us introduce now the family of divergence operators L£u = div(AcVu),

(1.3.29)

and consider all possible //-limits Ao(x) AJ^Ao(x). Two fundamental problems arise naturally:

(1.3.30)

44

Mathematical preliminaries

(1) characterize all possible //-limits {-Ao(x)} = £ (the notation Ah is preserved for periodic or non-uniformly periodic homogenization). (2) Characterize all possible limits {Ao{x)} = CVl provided that vol {x € Q.\Ae(x) = ai}

=vi.

The answer to the first problem reads £ = {A(x)\A(x) e G a.e. x e Q,} , where G is a closed set of tensors. In fact, G is a closure of effective moduli of periodic composites, called G-closure of ai and a2. The answer to the second problem is £„, = {A(x)\A{x) e G9(X) a.e. xeQ.} , where G$ is a closed set of tensors and 0 < 9(x) < 1,

= v\. Ge is called the n Ge-closure of a.\ and a.2. More precisely, Ge = the closure of effective moduli of periodic composites with the volume fraction 9. The last function is obviously the weak-* limit of the sequence of the characteristic functions {\c }ooLet us pass to the proof of the above assertions. First, however, we recall a result from real variables analysis, cf. Dal Maso (1993). Lemma 1.3.15. Consider a set 5 C Ll(f{n, F m ) such that it is: (a) translation invariant or

/ e S =>/(• +a) e S

Vo,

l

(b) closed under strong L convergence on compact sets, (c) S is decomposable, i.e. for any Borel set B, and / i , / 2 € 5 the function H

>~{f2(x)

on

R»\B,

also belongs to S. Then S={/|/(i)€5a.e.} for some closed subset S C R m and 5 = { constant elements of S}.

□

Let us return to the first problem. Now we have S = H- closure of {Ae{x)I} .

(1.3.31)

The local character of //-convergence implies that we may assume that AE(x) is defined on R n . The set 5, given by (1.3.31), is translation invariant, closed for the strong L1 convergence on compact sets and decomposable since //-convergence is local. To prove

Function spaces, convex analysis, variational convergence

45

periodicity we take A0 e G. By the previous step there exist Ae(x)I —>■ A0. Consider now Ae restricted to the unit cube Y and then extended by periodicity. Let usfindthe effective conductivity of periodic composites with this fine scale structure: {AA,«

= iaiifMvKt

+ W , £ + V
,

Y

where Y = (0, l) n , £ e R n and H^r(Y) = {»€ //'(V))|v is y-periodic} . We claim that Ae — ► A0 as e —> 0. Indeed, (Ae£,£) is determined by solving cell problems: div[Ac{y)(£ + V<^E)] = 0 ,

^-periodic.

Then ipc tends to a solution associated to A0, say <po, as £ —> 0. Moreover, we have lim (Ac£, $) = lim [energy of cell problem for ACI] = energy of cell problem associated to {(//-limit of ACI) = A 0 } . To justify the answer to the second problem we take S = closure of {Ae(x)I, Xc(x)} , in the topology of (//-convergence) xa(L°°(tt), L^fi)). This set is translation invariant, closed for the strong topology and decomposable. Now we have S = {(Ao, 6) = closure of associated set for periodic composites} , and Gg = slice of S at given $ . 1.3.3. Two-scale convergence Two-scale convergence was introduced by Nguesteng (1989) and developed by Allaire (1992). In the present section we shall expound the essential points of this method. To start with we observe that the two-scale convergence method is confined to periodic homogenization problems. The method of two-scale asymptotic expansions will often be used in our book. This is a formal method which enables us to find the homogenized problem. Let £ denote the size of periodic heterogeneities, £ is a small parameter which tends to zero in the asymptotic process. Assume that the sequence of solutions of the considered partial differential equation with microperiodically oscillating coefficients is denoted by uc . The two-scale asymptotic expansion is an ansatz of the form: u*(x) = «<0) (x, - ) + £«(1) (x, - ) + £2u(2) ( i , - ) + ...

(1.3.32)

Mathematical preliminaries

46

x where each function u'*'(x, y)(y = —) in this series depends on two variables: the macro scopic (or slow) variable x and the microscopic (or fast) variable y e Y. Here Y is a so-called basic cell. Substituting (1.3.32) into the equation satisfied by ue and identifying powers of e leads to a chain of equations for each term u ( , ) ( i , y). Next, averaging with respect to y yields the homogenized equation for u' 0 '. From the mathematical point of view, the method of two-scale asymptotic expansions is only formal since, a priori, there is no reason for the ansatz (1.3.32) to hold true. Hence the need for a second step, a rigor ous justification of the homogenized problem. The two-scale convergence combines these two steps into a single one. One may say that the two-scale convergence is a rigorous justification of the first term of the ansatz (1.3.32) for any bounded sequence ue. Let us pass to the definition. To this end we introduce the space C™r(Y) of infinitely differentiable function in R n which are periodic of period Y. By D ( n ; C^.(Y)), we de note the space of infinitely differentiable functions with compact supports in £1 and with values in the space C™T(Y). Definition 1.3.16. A sequence of functions uc in L2(Q) is said to two-scale converge to a limit u ( 0 ) (i,y) belonging to L 2 (fi x Y) if, for any function tp(x, y) in D(fi; C£r(Y)), it satisfies: u£(x)ip (x,-)

lim

dx

n

=

]Y\

hiW(x,y)
Y

(1.3.33) D

The compactness theorem which follows, justifies this notion of two-scale convergence. Theorem 13.17. From each bounded sequence uc in L 2 (fi), one can extract a subsequence, and there exists a limit u'°'(x, y) € L2(f2 x Y) such that this subsequence two-scale con verges to u' 0 '. D The next two propositions provide more information on the two-scale limit. Proposition 1.3.18. Let uc be a sequence of functions in L 2 (fi) which two-scale converges to a limit u' 0 ' e L 2 (fi x Y). (i) Then, uc converges weakly in L?(Q) to u(x) = (u ( 0 ) (x,y)>,

(1.3.34)

where (•) means averaging over Y and limoll«£Hl*(n) > ll«(0)llL2(nxy) > IMIi»(n) •

d-3-35)

(ii) Assume, further, that u' 0 ' (x, y) is smooth and that l™0ll«*lly(iD = ll« (0, lli» (n xy)-

(»-3.36)

Function spaces, convex analysis, variations! convergence

47

Then | | u £ ( x ) - u ( 0 ) (x,-)

||| 2 ( n ) - ^ 0

asc^O.

(1.3.37) D

Until now only bounded sequences in L 2 (fi) have been considered. The second proposition examines the case of a bounded sequence in H'(fi). Proposition 13.19. Let ue be a bounded sequence in / / ' (fl). Then, up to a subsequence, uc two-scale converges to a limit u € Hl(£l), and Vu £ two-scale converges to V x u(x) + Vyu{1){x, y), where the function u (1) (x, y) belongs to L2(ft, H^r(Y)/F{). □ As an exercise, the reader is advised to study the two-scale convergence of a bounded sequence in H2(n). 1.3.4.

T-convergence

This type of convergence pertains to sequences of functionals. For instance, such sequences are generated by the variational principle of the total potential energy of solids or structures like plates or shells. A detailed presentation of the theory of T-convergence is provided by Attouch (1984) and Dal Maso (1993). Attouch (1984) prefers to use the notion of epi-convergence, which in fact is a special case of T-convergence. In our specific case these notions coincide. Definition 1.3.20. Let (X, r) be a metrisable topological space and {G £ } £> o a sequence of functionals from X into R - the extended reals. (a) The r(r)-limit inferior, denoted also by Gj, is the functional on X defined by GAu) = T(r) - lim inf Gc{u) = min lim inf G £ (u £ ) . (b) The r(r)-limit superior, denoted also by Gs, is the functional on X defined by Gs(u) = T(T) - lim supGc(u)

= min lim supG€(ue)

£—0

{Ujiu}

.

£—0

(c) The sequence {G£}£>o is said to be r(r)-convergent if Gt = Ga ; we then write G = r(r)-limGE. £—0

Theorem 1.3.21. Let (X, T) be a topological space with a countable base for r. Then there exists a subsequence {G£'}£<>o such that the limits G, and Gs exist and Gi(u) = Ga(u), for every u 6 X. rj Properties. Let G£ : (X,T) —► R be a sequence of r(r)-convergent functionals and let G = T(T) lim Gc. Then the following properties hold: £—»0

Mathematical preliminaries

48

(i) The functional Gt and Ga are r-lower semicontinuous (r-l.s.c). (ii) If the functional G £ are convex, then Gs = T(T) - lim supG £ is also a convex func£—0

tional. Hence the limit G = T(T) — lim Gt is a r-closed (T-1.S.C.) convex functional. £-•0

(iii) If $ : X —> R is a r-continuous functional, called a perturbation functional, then T(T)

- lim (G£ + $ ) = T ( T ) - lim G£ + $ = G + $ .

(iv) V {u€ -^ u\ , G(u) < lim inf GAuE) , u € X ; £—»0

G(u) = r ( r ) - lim Gc{u)<* < V u € X 3 a t - ^ u , such that G(u) > lim sup G £ (u £ ) £—0

Further characterization is given by the convergence of minima. Theorem 1.3.22. Let G = T ( T ) - lim Gc, and suppose that there exists a T-relatively compact subset X0 C X such that inf G £ = inf G£ (Ve > 0). Then inf G = lim (inf G £ ). XQ

X

X

e—*0

X

Moreover, if {u£}£>o is such that Gc{uc) — inf Gc —> 0, then every r-cluster point of the sequence {ue : e —► 0} minimizes G on X. Remark 13.23. From a practical point of view the following sufficient condition of exis tence of compact set X0 is very useful. If (X, \\ ■ ||) is a Banach space with T-relatively compact balls, then a sufficient condition of existence of the compact set X0 is that the sequence {G £ } £> o satisfies the condition of equi-coercivity lim supG £ (u £ ) < +00 =$■ lim sup||u £ || < +00 .

(1.3.38)

□ The study of two-dimensional plate models obtained from three-dimensional ones will involve loading functionals dependent on a small parameter e. Hence the need for pertur bation functionals which depend on e. Let X be a general topological space. Definition 1.3.24. ^Ve say that a sequence {$£}£>o is continuously convergent in X to a function $ : X —> R if for every u € X and for every neighborhood V of $(u) in R there exist £0 and W € Af(u) such that $ £ (v) € V for every e < e0 and for every v G U. □ We recall that N(u) stands for the set of all open neighborhoods of u in X. Remark 13.25. It is clear that continuous convergence is stronger than pointwise conver gence. Moreover, continuous convergence is stronger than T-convergence. Proposition 13.26. Suppose that {$ £ } £> o is continuously convergent to a functional <J>, and that <E>£ and $ are everywhere finite on X. Then T(r) - lim inf(G£ + $ £ ) = T(T) - lim inf G £ + $ , £-•0

£-»0

Function spaces, convex analysis, variational convergence

49

T(r) - lim sup(G £ + $c) = T(T) - lim supG, + $ . e->0

£-.0

In particular, if {G £ } £> o V—converges to G in X, then {G€+®E}€>o T-converges inX.

toG+$ □

The following nonstandard diagonalization lemma is due to Attouch (1984). Lemma 13.27. Let {a^plm = 1,2,...; p = 1,2,...} be a doubly indexed family in R. Then, there exists a mapping m —► p{m), increasing to +oo, such that lim supOm,p(m) < lim sup (lim s u p a m p ) . m—*oo

1.3.5.

p—*oo

m—»oo

r-convergence of sequence of nonconvex functionals convex in highest-order derivatives: non-uniform homogenization

The classes of homogenized models of geometrically nonlinear elastic plates as well as geometrically linear and nonlinear elastic shells, studied in the present book, can properly be modelled by T-limits of sequences of functionals of the following type (or possibly a straightforward extension), see Bielski and Telega (1999) f[x,-,u(x),e{u(x));w(x),Vw(x)y2w(x)]dx.

Gc(u,w)=

(1.3.39)

n Here Q is a bounded and open set of R 2 , u G W1'"^)2,

w e W2'i(Q), p, q > 2 and

—-^ + —-^ ] /2. Obviously we can consider a domain Q C R 3 . Having in axp axaJ mind the aforementioned application to plates and shells, the present section is confined to two-dimensional domains. The paper by Bielski and Telega (1999) has been inspired by Braides' paper (1983), who found the r(L p (n))-limit of the following sequence of functionals: Gl(w)= ffl(x,-,w(x),Vw(x))dx,

(1.3.40)

n where Q C R n . Let us specify the assumptions on the integrand / appearing in (1.3.39). They are given by: {Ai) f = f(x,y,u,e;w,rj,p) : R 2 x R 2 x R 2 x E 2 x R x R 2 x E 2 -» [0,oo] , which is measurable and V-periodic in y, continuous in x, u, w and rj, and convex in e and P(A2) There exist y-periodic function a G L,' oc (R 2 ), increasing function g : R + —> R + which is continuous at 0 and such that g(0) = 0, and function b : R 2 —► R, continuous and non-negative, for which the following inequalities are satisfied: \f(x, y,u, e; w, T], p) - f{x', y, v!, e; w', rj', p)\ < g(\x - x'\ + \v! - u | +\w' -w\ + \t)' - r]\)(a(y) + f(x, y, u, e; w, r,, p)) ,

(1.3.41)

50

Mathematical preliminaries 0< f(x,y,u,e;w,ri,p) < b(x)[a(y) + \u\" + \e\" + \w\" + \f]\" + \p\")} , 2

2

(1.3.42)

2

for all x, x' G fi, y G Y, u, u', TJ, TJ' G R , and e, p G E . Here E stands for the space of symmetric 2 x 2 matrices. Obviously, the norms in (1.3.41) and (1.3.42) are Euclidean norms. Let us set T = s — (Z/(n) 2 x Wli9($l)). We are now in a position to formulate the homogenization theorem. Theorem 1.3.28. Under the assumptions (Ai) and (A2) the r(r)-limit of the sequence of functional {Ge}c>0 defined by (1.3.39) has the following form: r(r)-limG c {u,w) = f/h[x,u(x),e(u(x));w{x),

Vio(i),V2w(x)}dx , (1.3.43)

where u G W"'p(fi)2, w G W2-"{n) and h{x, £, e; ip, V, P) =

inf

{]yf / /[x>2/' £>c + ^{"(v))'* ^ V, P + V^(?/)]dj/ Y

\v € W£(Y)2

, i; G W%(Y)} ,

(1.3.44)

for all tp G R , £, TJ G R 2 and e,p€ E 2 . Here W^P(y)2 = {v

W l p (y) 2 | v is ^-periodic} , dv W£?(^) = {v G W 2 '«(y) 2 | » and are Y-periodic} , dya

, „ . .

I'dva

(1.3.45)

G

dvg\ ,„

,„0 .

(1.3.46)

d2v

Proof. Detailed proof has been given in Bielski and Telega (1999). Let us sketch the main points. (i) First we consider the simpler case where f(x,y,u,

e(u);w,Vw,V 2 w) = tp(y,e(u), V2w) .

Following Braides (1983) and exploiting (Ai) and (A2) we then prove that r ( r ) - l i m Lp (-,e{u),V2w(x)\ n

dx = hph(e{u{x)),V2w(x))dx n

,

where '(v(y)),p + V2yv(y)}dy\v€ W£(Y)2,v

G W%(Y)}.

Function spaces, convex analysis, variational convergence

51

(ii) Take now the following integrand f(x, y, u, e(u); w, Vw, V 2 w) = ip(x, y, e(«), V 2 w) . Following Braides (1983) once again we prove that there exists a function iph(x, e(u), V 2 w) such that r ( r ) - l i m hp{x, -,e{u),V2w)dx

=

n

,

n

where
, v G W%(Y)} .

Y

(iii) In the final step we prove (1.3.43), by using the integral representation of T-limits of sequences of integral functional due to Buttazzo and Dal Maso (1980). □ Remark 1.3.29. Let us fix x G fi, ip G R, and £, rj G F 2 . For each c, rj G E 2 we have = inf{T^T //)>(*,£.e + ev{v);, p + V2yw)dy

h(x,£,e;ip,V,P)

Y

\ve\V^p(Y)2 Indeed, since 0 G Wo,p(Y)2

,veW'-q(Y)},

(1.3.47)

and 0 G W02'"(y) therefore we can write

fh(x, £, e; ip, T), p) = —

/ / fc (x, £, c + 0; xp, rj, p + 0)dy Y

+ e » ; < M , P + V 2 H*/|« e W^(Y)2,

> M{±-Jfh[x,Z,e

v G W^V)} .

Y

On the other hand, Jensen's inequality yields fh(x, £, e; V,»?, p) = A [ i , t | ^ | y (e + e»(»))d»; V,»?, j p j / ( P + v J w ) d » l K

Y € + eV v

 Y

for each v G W01,P(V)2, v €

W^{Y).

2

( y> i M , P + V vv)dy

52

Mathematical preliminaries

1.3.6. T-convergence and duality With a sequence of functionals one can associate the sequence of conjugate or dual functionals. Suppose that the original sequence is T-convergent. In the present section we are going to study the interrelationship between the T-convergence of the primal sequence of functionals and the sequence of dual functionals. Here we follow the approach developed by Aze (1984), cf. also Az6 (1986) and Jikov et al. (1994). Let V and X be two separable Banach spaces and let V and X* be their topological duals. In general, these spaces are not necessarily reflexive. In practice, however, V is often a reflexive space whilst X is a space of perturbation parameters, cf. Sec. 1.2.5. Consider a sequence of functionals {G£}£>o from r0(V x X). This sequence is said to satisfy the assumption (A) if: (A) There exists r > 40 such that for each sequence {p£}£>o of elements from Br = {p 6 X'■HPII < r } . th ere exists a bounded sequence {f£}£>o such that lim supG£(t>*,p£) < e

+ 0O.

Werecallthatee£: = { - | n e N\{0}}. n Similarly, the sequence {G£}£>o satisfies the assumption (A*) if: (A*) There exists r* > 0 such that for each sequence {v€}c>0 C Br- = {v 6 V : \\v\\ < r*}, there exists a bounded sequence {p*}£>0 such that lim supG*(t;£,p*) < +oo. £

The assumption (A) is a universal qualification hypothesis whilst (A*) will be shown to play the role of a uniform coercivity hypothesis for the primal problems. Let usfirstprove two auxiliary lemmas. Lemma 1.3.30. Let {G £ } £>0 be a sequence of functionals from r0{V x X) satisfying (A). For A > 0 we set: GeA(v',P) = Gc(v',p) + ^\\v*\\2.

(1.3.48)

Let {"^1/jtHeN be a sequence of positive numbers bounded from above. Then, for any subsequence {n*} | lim sup (G1/nk,Al/k)'(v1/k,p'1/k) \ k [and {Vi/k} bounded

< +oo 1 >=> {p\/k} bounded. I

(1.3.49)

Proof. Let r > 0 be a constant appearing in (A) and let pi/jt 6 Br be such that r \ \p\,k 11 - 1 < (pj/fc. Pi/k)x-xx- Consider a sequence {pi/„} of elements from Br such that pi/ nt = pi/k. According to (A) to this sequence corresponds a sequence {v\,} bounded by a positive constant c such that: lim supGi/„(vJ /n ,pi /n ) < +oo .

53

Function spaces, convex analysis, variational convergence By using the definition of (Gi/nx\)* we get: (Gi/nk,xk)'(vi/k,p'l/k)

A + ^plK/nJ|2 >

+ Gi/nk(v'/nk,pi/nk)

(vl/k,v'1/nic)VxV.

+ {p'i/k>Pi/n,.)x'xX > - c l k / i H + r l l p J ^ H - 1 . The desired result follows by taking the limit superior with respect to k.

□

Corollary 1.3.31. If (A*) is satisfied, then we have lim supGWnk(v'k,pi//t) * ' " l'k ' and {pi/k} bounded

< +00 I k => {v*/k} bounded. J

(1.3.50)

Lemma 1.3.32. Let C be a convex set in V x X*. Then: go(Vy)xo(X\X)

=

g(s-V)xa{X',X)

3

Here (s - V) denotes the strong topology of the space V whilst the bar stands for the closure of C in the indicated topologies. Proof. The topology (s -V)x (w* - X*) is compatible with the duality a(V x X', V x X). The same pertains to the topology a(V, V) x a(X',X) and (1.3.51) follows. D Hence we conclude that for these two topologies the lower semicontinuous regularizations coincide. Let us pass to a result concerning the effect of duality on the T (strong x weak)convergence. Theorem 1.3.33. Let {Gc}£>0 be a sequence of functionals belonging to r 0 (V" x X) and satisfying (A). Then we have I > ; * x s) - lim inf Gc >G=>T(sx

to*) - lim sup (7; < G' .

(1.3.52)

Proof. We can assume that G > —00. Otherwise the result is evident. We begin by showing the following lemma. Lemma 1.3.34. Under the assumptions of Theorem 1.3.33 and G > -00 one has: For each B C X relatively strongly compact there exists K > 0 such that: W*eV\

VpeB,

Veef,

Ge{v',p)>-K(\\v,\\

+ l).

Proof. Suppose that the conclusion were false. Then there exists B cY, compact and such that VA: e N ,

3v'1/k e V ,

3p1/k e B ,

3nk ehi

Gi/„tK/t,p,/t)<-A(ll«r/*H + l)One may assume that the sequence {nk} is increasing.

(1.3.53)

relatively strongly

such that

54

Mathematical preliminaries

Case 1: {v\,k} is bounded. Since V is separable and B is relatively compact, therefore there exist subsequences still denoted by { n | , J , {pi/t} such that v1/k ->• v

and

p1/k -» p ,

as /t —> oo. Consequently we conclude that \immiGi/nk(v*/k,pi/k)

= -oo ,

and hence G(v',p) = —oo, which contradicts the assumption on G. Case 2: the sequence {v{,k} is unbounded. One may then assume, up to the extraction of a subsequence, that lim |\v\ /k \\ = +oo. Consider a sequence p"i/n A p, such that p\/n < r. According to the assumption (A), there exists a bounded sequence {v*,n} such that lim sup Gi/„(vJ/ n ,p"i/ n ) < +oo. We set v

l/k

7Z*

= v\/nk

>

Pl/fc = Pl/nk ■

Since the Banach space V is separable, therefore the bounded sequence {v',k} c V admits a subsequence (still denoted by v\,k) convergent to a certain v' S V* in the topology

a(V,V). We define Ci/k = hikV*\/k + U - ti/kWi/k.

^//t = *i/*Pi/t + (1 - ti/k)pi/k

,

where *i/* =

I V - vvk\

Forfcsufficiently large, £wk is well-defined and lim twt = 0. Moreover we have fc-»+oo

ll^/fc-^T/fcll = ' I A I K A - S i / * 11 = "Tr '

and thus £*A "^ ^*

^^

^V* —* P

as

^ —* + ° ° •

We also find Gl/nkiCl/k'Vl/k)

< tl/kG1/rlk(v'l/k,Pl/k)

+ (1 - i l A ^ l / n J ^ l A ' P l / * )

„ -v^(IK/tll + i) i i W i i and consequently lim inf Gi/n/t(£i/fc,77iA) = - o o ,

Function spaces, convex analysis, variational convergence

55

and G(v*,p) = - o o , which is impossible.

D

We now turn to the proof of Theorem 1.3.33. Since the space X is separable, therefore there exists a sequence {pk}k>l dense in it. By Xk we denote the subspace generated by {Pi. • • • ,Pk) and B denotes the unit ball in X. We set Bk = kB D Xk = {p € Xk : \\p\\ < k}. Bk is a strongly compact subset of X. For k > 1 and A > 0 we set: Ge,k,x(v\p) = G€(v',p) + IBk(p) + ±\\v'\\2 , GkAv\p)

= G(v',p) +

IBk(p)+±\\V\\2,

where Isk stands for the indicator function of BkIt will now be shown that VA:>1, VAX), V(v,p*)e Vx X* lim supG\lnXX{v,p') < GlA(v,p').

(1.3.54)

n

Obviously, this is equivalent to: limjnf [jnf^(G1/n(v',p)

+ ^\\v'\\2 - (v',v)v.xV

-

(p',p)x-*x)}

> inf [G(iAp) + ^ | M | 2 - ( ^ , t ; } - ( p * , p > ] .

(1.3.55)

P6St

On account of Lemma 1.3.34 there exists K > 0 such that W e T ,

Vp € Bk , Vn 6 M , G1/B.A(«*,p) > ^ I M | 2 - K(\\v'\\ + 1) . (1.3.56)

Consider now an extracted sequence {n;} for which the limit inferior is just a limit. The infimum in (vj,;, pi/;) on the left-hand side of (1.3.54) is then attained. Using now the sequence {5wn} appearing in the assumption (A) and associated with the sequence pj/ n = 0 and taking into account the estimate (1.3.55) we conclude that {vyt} is bounded. We can extract subsequences, still denoted by {vJ/(} and {pi/;}, such that v{/t ^ v* € V

and

pi/; A p € Bk

as I — ► +oo .

Recalling the definition of T(w* x s) - lim inf Gc and the semicontinuity of the norm one gets Inn inf [Gllnt{v\lhpul)

+ ^ | | ^ | | 2 ] > G(v',p) + ±\\v'\\2 ,

56

Mathematical preliminaries

which establishes (1.3.54)and consequently (1.3.53). Thus we conclude that for each k > 1 and for any A > 0 we have lim supG! /BiM («,p') < Glx(v,p')

< G'(v,p') .

n

Applying now diagonalization Lemma 1.3.27 we conclude that there exist X(n) — ► 0+, k(n) —> +oo such that lim supGI /Bit(B)iA(n) (t;,p') < G'(v,p') .

(1.3.57)

n

We may write Gi/n,k,\ = ^l/n.x + ^k with ^jt(v*,p) = /sfc(p). Hence, cf. Sec. 1.2 G 1 / n , M = (G; /n , A n*-)-*, where D denotes the inf-convolution. Let us show that G

l/n,t,A -

tT

l/n,A U V /fc

'

(1.J.58)

which represents the semicontinuous regularization. Indeed, G\,nkA = (Gi/ nA + ^t)* cannot assume the value —oo since in that case we would have GI/„,A + ** = +00. After (A) there exists {vi/n} such that lim supGi/„(i>J,n,0) < +00. Consequently, for n

n sufficiently large we have (Gi/n,A + **)(vj/n,0) < +00 and thus G ^ A + *jt is not identically equal to +00. We deduce that °l/i.,k,A -

lC7l/n,AUW*:i

~

^l/n.X^k

By Lemma 1.3.32 the last regularization can be taken in the sense of (s — V) x a(X*, X). Let us continue the proof of Theorem 1.3.33. We have: lim supG'l/nMn)Mn)(v,p')

< G'(v,p*) .

n

We may assume that G'(v,p') < +00 ; otherwise the result is evident. For each n € M we introduce the following set: U ; = {q* e X' : \(q'-p',Pi)x-xx

< - , 1 < i < n} . (1.3.59) n We recall that {Pt}i
<jj/neu;,

(GI/n.A(n)a**(«))(«l/».9l/») < SUP {Gl/„,*(n),A(„)(^P*) + K ~]

■ (13.60)

Function spaces, convex analysis, variational convergence

57

Simple calculation yields *;(Wip.) *v '

=

if

(^(p-,^) ^+00

" « = 0. otherwise.

(1.3.61)

Using the definition of inf-convolution we deduce the existence of rj[,n € X' such that G

Vn,A(n)(Ul/n- tf/J + *(»0<*(9l/» ~ tf/„. **(„)) < ^ P {
(1-3.62)

Since {ui/ n } is bounded, therefore, by Lemma 1.3.30 the sequence {r}'/n} Let us calculate G\ ,n A. We have G V „,A = G 1 / n +

7A

7x{v",p) = -\\v'\\2

with

ls

^so bounded.

.

Hence GI/».A = ( G ^ W

= G;/TID7A,

since G; / n A > - 0 0 (G 1/niA ^ +00). By Lemma 1.3.32 the semicontinuous regularization may be taken in the sense of (s V) x (w* - X'). Thus there exists (wi/n,p'l/n) eV x X* such that

\\Wl/n-ul/n\\
P;/new;,

( G Vn [:l 7A(n))( U; l/n.PVn) <

G

i

(L363)

t/n,A(„)( U l/«> tf/„) + ~ .

where W*n := {p' € X- : |
tf/n)Pi)|

P*/ n ) < (Gt /n l=l7l(„))(wi/n,Pv„) < GI /niA(n) (iti /n ,»7j /n ) + Hence, using Lemma 1.3.30 we deduce that {p\/n} is bounded, since the sequence {u>i/n} is bounded. Noting that

.,

.,

J^rlMI"2

if P* = 0,

[ +co

otherwise,

7A(V,P ) = < 2A"

and exploiting the definition of the inf-convolution we obtain Vi/n e V such that Gl/„(l>l/„,Pl /n ) + ^ W ^ ) " ^ " ~

W /n

' " 2 - ^/".MnJ^l/n.^'/n) + ~ •

(1-3.64)

Mathematical preliminaries

58

From (1.3.56) - (1.3.59) we get G

\/niVV^P\/n)

+ 2\(n)^Vl/n

~

Wl/n

< sup{G'l/nMn)Mn)(v,p*)

+ k d

^

- 'Jl/n. Xk(n))

^ ^'/n

+ ^ , -1-} + \ .

(1.3.65)

Taking the limit superior in n, by (1.3.57) we conclude that lim supG* /n (v 1/n ,p* /n ) n

< lim sup [sup {G'1/nMn)Mn)(v,p') fl

+ I , - 1 } + I ] < C(i;,p*) . il

TV

(1.3.66)

7i

It thus remains to show that v\/n -^ v and p\, —*• p" as n — ► oo. We observe that due to the assumption (A) there exist bounded sequences {v{/n}, {pi/„} such that lim supGi/n(iJJ/n,pi/„) < +oo . n

Hence we deduce that there exists K' > 0 such that: W 6 V, Vp* € X; Vn e N, G; /r >,p*) > -K'(\\v\\ + HP'II + 1).

(1.3.67)

Substituting (1.3.67) into (1.3.64) and noting that the sequence {p*/n} is bounded, we conclude that there exist positive constants K" and K'" such that:

- j n i h / » n + 1 ) + 2 ^ j i K » - ™i/nii2 ^ *'" • Since A(n) —* 0 and {wi/n} is bounded, therefore {^i/n} is bounded and Hvi/n-Wi/,,112 —> 2 0 as n — ► oo. Hence | |wi/„ — v| | < - and consequently v\/n —»v strongly in V as n —» oo. From (1.3.64) we get 11111 d n—»oo (
We set D=

t>i

UXk.

Obviously, D is dense in X. Let p € D, then beginning from a certain k(n) we have p € Xk{n). Let us introduce r'l/n € X^n) such that \\q'1/n - ij*/n - r j / n | | < d(gj /n - 7j*/n, ^(„) + ^ - W e h a v e <9l/n ~ V*/n,P)x-xX

= ( ? V „ - »?!*/« -

r

l/n-P)

Function spaces, convex analysis, variational convergence

59

Hence I"? faI/„-»?r/n.P> = 0 . n—*+oo

'

(1.3.68)

'

Moreover, from the definition of U* one has lim (q'1/n-p',p)=0.

(1.3.69)

From (1.3.67) and (1.3.68) we conclude that i™W/„-?',p) =0, for each p e D. Indeed, p j / n e W^ and thus for each p e D w e have ' " " ( P V' n - ^ / n' ' P ) = 0 -

n—>oo

Consequently we get Vp€£>,

lim ( p l / n - p * , p ) = 0 . n—»oo

'

Since {p[. } is bounded and D is dense, therefore p\,n —* p* weak-* as n —> c© and the theorem is proved. □ As a corollary we formulate the following result. Theorem 1.3.35. Let V and X be two separable Banach spaces. Let {G£}£>0, G be functional belonging to r 0 (V* x X), satisfying assumption (A) and G = r(wm x s ) - l i m G £ .

(1.3.70)

c—»0

Then G* = r ( s x w , ) - l i m G ! Proof. We conclude from Theorem 1.3.33 that r ( s x u / ) - l i m supG£* < G * . £-•0

The inequality G* < r ( s x w * ) - l i m infG! —

V

'

£-.0

£

results immediately from (1.3.69) and the definition of the conjugate (dual) functional.

□

We now pass to the central result of this section. This result concerns the T-convergence of sequence of marginal functional and of their conjugate functionals.

60

Mathematical preliminaries

Let G£ : V x X — ► Ft U {+00} be a sequence of functional from r 0 ( V x X),e G S. The marginal functional is given by, cf. the formula (1.2.42) h€(p) = inf

{G£(v',p)\v'eV}.

This functional is convex, not necessarily lower semicontinuous, cf. Sec. 1.2. The dual problem will involve the functional/i*(p*) = G*(0,p*). We observe that the T-convergence of G£ to G in a topology (T, 0 and G be functional belonging to r 0 (V* x X), satisfying the assumptions (A) and (A*) and G = T{w' x a ) - l i m G e .

(1.3.71)

Then (i) h = T(s) - lim hc, C-.0

(ii) h' =

r{w')-hmh'e,

(iii) G(-,0) = r K ) - l i m G £ ( - , 0 ) , (iv) if v"c is a minimizer of Pc up to &c and if p* is a minimizer of P£* up to Sc with e —> 0, then the sequences {v*} and {p*} are bounded and if v' and p* are limits of subsequences, one has: (a) the minimum of P is attained at D*, (b) the minimum of P' is attained at p*, (c) inf P = - inf P\ (d) inf P£ -> P and inf P£* -»inf P* as e —» 0. D Let us clarify the formulation of the primal and dual problems involved in the last theo rem. We have: (P£) inf{G £ (t;,0)| V eV}, (P£-) inf{G£*(0,p')|p' € X'} = inf{h'c(p')\ p' G X'} = -sup{-G £ '(0,p')|p'GX*}. We recall that the spaces V and X are not necessarily reflexive. Proof of Theorem 13.36. Step 1. Let p£ —> p strongly as E —> 0. We want to show that lim inf h£(pc) > h{p) .

Function spaces, convex analysis, variational convergence

61

To this end we take a sequence of real numbers {Mi/k} such that Mx/k j . lim inf hc(pE). There exists a subsequence {e*} and v\,k e V* such that Gek{v\/k,Pek)

< Ml/k .

On account of Corollary 1.3.31 and the assumption (A') the sequence {v[,k} is bounded in V. Since the space V is separable therefore there exists a subsequence, still denoted by v\,k, which converges to an element v' £ V" in the topology a(V', V). Thus we have lim inf he(pc) = lim inf M\/k > lim inf Gek(v\/k,pCk)

> G(v',p)

> h(p) .

Here we have used (1.3.70). Step 2. Let us show that there exists p £ —> p as £ —► 0 such that hc(pc) < h(p). We may assume that h(p) < +oo. Let {M\/k} be a sequence of real numbers such that M\/k [ h(p). There exists v',k G V" such that G{v\,k,p) < M\/k. According to (1.3.70), there exist: Pe,\/k -* P in the strong topology of X as £ —> 0 , and ^',1/jt "^ v\/k

m me

topology

a(V\ V)

as

e —► 0 ,

such that lim supG £ (v; i / J t ,p £ ,i / k ) < G(v'1/k,p) . Hence lim sup (lim sup/i £ (p £ ,i/ t )) < /i(p) • t

£

By diagonalization, there exists k(n) —» oo such that (e = 1/n) lim

Pe,l/*(n) -^ P .

SUp /l£(pe,l/fc(„)) < /l(p) £

as n —» oo. Thus (i) is proved. Step 3. The assumption (A) yields: for each sequence {p £ } £gi - withp £ € BT lim sup/i £ (p £ ) < +oo . £

We now apply Theorem 1.3.35 with V = V" = {0} and G£(j;*,p) = he(p), the lower semicontinuous regularization of ht. Since h = T(s)— lim/i £ , therefore we also have £—>0

h = r ( s ) - lim he. It is obvious that hc e r 0 ( X ) and he = /i*. Thus h' = T(w')~ and (ii) is shown.

lim/i*

62

Mathematical preliminaries

Step 4. We observe that (iii) results directly from (i) and (ii) combined with Theorem 1.3.35. Indeed, from Theorem 1.3.35 we deduce that G" = F(s x w') — limG*. Thus if we v

'

£—0

£

set kE(v) = inf {G*(u,p*)|p* e X'}, by using (i) and (ii) we have k = r ( s ) - lim k£ v

'

and

k' = T(w*)- lim Jfc* . v

£—0

'

£-.0

£

Here k't{v') = Gc{v',0) and k'(v') = G{v',0). Step 5. inf P = — inf P' results from the fact that h is lower semicontinuous as the strong T-limit of a sequence of functional. To terminate, it suffices to show that the sequence 5e is bounded, because the desired convergence results are then a consequence of variational properties of T-convergence. We observe that: (A*) implies lim sup (inf P*) < +oo, (A) implies lim sup (inf Pc) < +oo. £

The duality theory yields, cf. Sec. 1.2 infP£* > -infP r *. Hence, for e sufficiently small we obtain -co < inf P£ < +oo ,

—oo < inf P* < +oo .

Let {v'c} and {p*} be, up to <5£, minimizing sequences for the problems Pc and P*, respec tively. We have Ge («;, 0) < inf P€ + 6C ,

G'e (0, v'e) < inf P*c + St .

Hence we deduce, by using Lemma 1.3.30 and Corollary 1.3.31, that {v*} and {p*} are bounded. This completes the proof. D Remark 1.3.37. In applications the assumption (A) can easily be verified. The same, however, cannot be said about (A*). We are going to indicate a situation where (A) and (A') hold. Theorem 1.338. Under the following assumptions:

Vee£,

G = r(u>* x s) - lim Gc,

(1.3.72)

(A), V t i ' e V , Gc(v',0)>m(\\v'\\),

(1.3.73) (1.3.74)

where m is a coercive, convex and even function, the conclusions of Theorem 1.3.36 are verified. □

Function spaces, convex analysis, variational convergence

63

We shall need the following result. Lemma 1339. Under the assumption (A) one has: lim sup ( sup ft£(p)) < +00 . '

llp||
Proof. Suppose the assertion of the lemma is false. Then there would exist rik and pi/* € Br such thatfti/nk(pi//fc)> k. Thus we would have lim sup/ii/„t(pi/^) = +00, which on i/t

account of (A) is excluded.

□

The last lemma and (1.3.73) imply that beginning from a certain e the function ft£ is finite and continuous at 0, cf. Sec. 1.2. Consequently we haveft**> —00 and thus: G*(0,-) =

ft*^+co.

(1.3.75)

Proof of Theorem 13.38. It is sufficient to show that (A*) is verified. For v € V we set ke(v) =

m({G;(v,p')\p'eX'}.

We have k;(v') =

Ge(v\0)>m(\\v'\\).

Suppose that kc G r 0 (V), hence we would have kc(v) < m*(||u||). Since m is coercive, therefore there exist M € R and r* such that for each v £ V with ||u|| < r* one has ke{v) < M. If {vr}e>oisasequencewith||vf|| < r*, there exists {p*}t>0 such that lim supG"(v£,p*) < +00. According to Lemma 1.3.30, the sequence {p*}£>o is bounded and (A*) is then obviously satisfied. Thus it suffices to show that k€ e r 0 (V). We note that k€ ^ +00 becausefc*does not assume the value —00 and k€ > —00 because A;* is not identically equal to +00, at least for e sufficiently small, due to (A). It remains to show that kc is lower semicontinuous. This will be shown if one proves that for each r > 0 the functional /£(p') = inf{G*Kp*)||H|,p*) + ^ , p * ) with

TJ>(f,P )=/*(») = I + Q 0

if

llvll > r ,

otherw ' ise .

Obviously, i> is convex and lower semicontinuous in the topology (s - V) x (w* - X'). Thus, by Lemma 1.3.32 it is lower semicontinuous in the topology a(V, V) x a(X', X).

64

Mathematical preliminaries

Moreover, ip is continuous at (0, p*) in (s - V) x (w* - X'), which is an admissible topology for the duality a{V x X', V x X). According to (1.3.72) there exist elements of the form (0, p*) which belong to dom G*. Then we get:

*;(v;P) =

(Gew)(v\p),

and Ze(p*) = inf{*e(u,p*)|t>e V}. Hence

rt(p) = *;(o,p) = (G £ D^)(O I P ) We observe that ^{v'p)

=

\+oo

otl otherwise .

Hence ll{p) =

m({Gc(v',p)+r\\v'\\\v'eV'}.

It is evident that I* ^ +oo and l"e > -oo, otherwise we would have I" = +oo and lc = +oo, which is not the case. According to Lemma 1.3.39, the functional /* is finite in a neighborhood of 0. Being lower semicontinuous on a Banach space this functional is continuous at 0. The proof is complete. □ 1.3.7. Convergence of sets in Kuratowski's sense Let {Ce}e>o be a sequence of subsets of a topological space V. Definition 13.40. The K- lower limit of the sequence {C£}£>o, denoted by K - lim inf Cc, £-•0

is the set of all points v e V with the following property: for every l i 6 A/"(v) there exists eo > 0 such that U D C£ / 0 for every e < e0- The K-upper limit, denoted by K lim supC£, is the set of all points v e V with the following property: for every U e Af(v), £—0

for every e0 > 0, £o € S, there exists £ < £o such that U PI C£ ^ 0. If there exists a set C C V such that C = K - lim infC£ = K- lim supG£ then we write C = K lim C£, and we say that the sequence {C£}£>0 converges to C in the sense of Kuratowski, or £—»0

K-converges to C (in V). □ The X-lower limit is sometimes denoted by Lim inf, and similarly for /(-upper limit, cf. Aubin and Frankowska (1990). Then C = Lim Ce. £—0

The above definition implies that K— lim infC£ C K— lim supC£ ,

Function spaces, convex analysis, variational convergence

65

hence {C£ } £>0 K-converges to C if and only if K- lim supC£ CCC K- lim inf C€ . £^0

E-0

Example 1.3.41. If C is a subset of V and C€ = C for every e e £, then {C£}£>0 Kconverges to C, the closure of C in V. □ The A'-convergence of a sequence of sets is equivalent to the T-convergence of the cor responding indicator functions. Indeed, we have the following result. Proposition 13.42. Let {C£}£>o be a sequence of subsets of V and let C = K- lim inf Ce ,

C" = K- lim supC £ .

Then Ic = T—lim sup/c, £ -o

Ic" = T-lim inf lr . £^o

In particular {Cc}c>o A"-converges to C in V if and only if {ICe} T-converges to Ic in V. □ The following statement reveals the connection between T-convergence of functions and ^-convergence of their epigraphs. Due to this connection, the T-convergence is sometimes called epi-convergence, cf. Attouch (1984). Theorem 1.3.43. Let {G€}e>o be a sequence of functions from V int R, and let Gi = T— lim inf Gt,

G, = V— lim sup Gc .

Then epi G, = K— lim sup epi Gc ,

epi Gs = K- lim inf epi G£,

where the A"-limits are taken in the product topology V x R. In particular, {Ge} Tconverges to G in V if and only if {epi G £ } £>0 /("-converges to epi G in V x R. D Remark 13.44. The notion of /("-convergence has been introduced after Dal Maso (1993, Chap. 4), cf. also Attouch (1984) and Aubin and Frankowska (1990). Remark 1.3.45. In Section 13.2 we shall provide an example of /("-convergence applicable to two-phase plastic composites. 1.4. Two approximation results Most of the homogenization results presented in this book will be justified by using Tconvergence. In turn, in the prevailing number of cases the density of continuous affine functions in the space Hl(Q) will be exploited. Also, for most problems involving second-

66

Mathematical preliminaries

order derivatives, we shall use the fact that C 1 -functions with piecewise constant second gradient are dense in i/ 2 (fl). In the last case fi is a two-dimensional bounded domain. Definition 1.4.1. (a) Let fi C R n (n < 3) be a bounded domain. A function u : fi —> R is affine if it is a restriction to fi of an affine function on R n . (b) A function u : ?l —► R is called piecewise affine if it is continuous and there exists a partition of fi on a set with zero measure and a finite number of open sets on which u is affine. □ Thefirstapproximation result is formulated in the following form. Proposition 1.4.2. Let Q, be a bounded domain in R n (n < 3) with a Lipschitz boundary andu £ Hl{£l). There exists a sequence {um}m£N of piecewise affine functions on Q such that um —» u in Hl(Q)

when m — ► oo .

Proof. It is sufficient to prove the assertion for a function u e C°°(fl), continuous on Cl. Details are given in the book by Ekeland and Temam (1976, Chap. X). □ Having in mind application to plate problems, we have formulated this nice approxima tion results for n < 3 only. It can be formulated for any finite n and spaces W lp (fi). The second result, this time concerning approximation of functions from the space W2,P(Q) (1 < p < oo, fl C R 2 ) by means of C'-functions with piecewise constant second gradient, seems not to be so well known. Here we shall give a detailed proof due to Descloux, cf. also Birman and Solomyak (1967), Neuman and Schmidt (1983), Orlov (1978), Powell and Sabin (1977) and Sablonniere (1989). Let fi C R 2 be a quadrangle represented in Fig. 1.4.1.

♦ *2

Here P2, P4, Pe and Ps are mid-points of the corresponding sides of Q.

Function spaces, convex analysis, variational convergence

67

Proposition 1.4.3. For each fk e R, 1 < k < 8, there exists one and only one function / such that feC1

(a)

(ft),

(b)

/,„, e V2,

1< i < 4,

(c)

/(fit) = } k ,

l
where P2 denotes the set of polynomials of order not exceeding 2.

D

Prior to providing the proof, an auxiliary result will be given. By using rotation and homothety one can refer the results which follow to Fig. 1.4.2.

Fig. 1.4.2. Lemma 1.4.4. Let / e C^ft) be such that /|„ t e V2 for 1 < k < 4. Then there exist constants a, 0,7,5, u>, a, b, c such that f(xi, x2) f(x1,x2) f(xi,x2) f{xx,x2)

= = = =

ax\ 7X, 7xf aij

+ wxiz 2 +W21I2 + wx\X2 + ux\X2

+ &%\ + a x i + bx2+c + /SX2 + axi + bx2 + c + 5x1 + axi +bx2 + c + 8x\ + axi + bx2 + c

on fti, on fi2, on n 3 , on n 4 .

(1.4.1)

Proof. We set o = 5,/(0,0),

6=

ft/(0,0),

c=/(0,0).

(1.4.2)

where 9j = d/dx\, &2 = d/x2. Let g{xux2)

= f(xux2)

- axi

-bx2-c.

(1.4.3)

Mathematical preliminaries

68 Hence 9\ak e V2 ,

dl9K (0,0) = d2ghk ( 0 , 0 ) = gK (0,0) = 0

for 1 < k < 4. Consequently there exist coefficients p*, <& and rk such that 5ln k (xi,x 2 ) = Pkxl + qkXiX2 + rkxl ,

1 < A; < 4 .

(1.4.4)

The continuity of g along the axes xi and x2 yields the following relations Pi = p 4 := a,

n = r2 : +/?,

p 2 = Pz := 7 ,

r 3 = r 4 := <5 .

(1.4.5)

Next, from the continuity of the normal derivative of g along the same axes one gets 9i =92 = 9 3 = 94 : = w .

(1.4.6)

On account of (1.4.3) - (1.4.6) we conclude the proof.

D

The following lemma can easily be verified. Lemma 1.4.5. Let / be defined by (1.4.1). Then / e Cl(£l).

D

Proof of Proposition 1.43 (for Fig. 1.4.2). Let X = {/GC1(fi)|/KeP2,

i<*<4}.

Lemmas 1.4.4 and 1.4.5 show that X is a vector space of dimension 8. The conditions (a), (b) and (c) of Proposition 1.4.3 are equivalent to

/ex,

f(Vk) = fk,

i
On account of a classical theorem of linear algebra it suffices to establish the following relation / a ,

f(Vk)

= 0,

l < f c < 8 = > / = 0.

(1.4.7)

Let / € X be such that f(Vk) = 0,1 < k < 8. Hence one immediately concludes that / vanishes on dQ, and that d\f(Vk) = difCPk) = 0 for k = 1,3,5,7. Thus we have /(l,0) = Q + O + C = 0 / ( 0 , l ) = /3 + b + c = 0 9 , / ( 1 , 0 ) = 2Q + O = 0

fl!,/(l,0)=w + 6= 0 a1/(0,l)=w + a = 0 52/(0,1) = 2/3 + 6 = 0 d2f(-l,0) = -w + b = 0

on

Qi ,

on fii, on f2i , on fii , on fii , on fii , on Q2 .

(1.4.8) (1.4.9) (1.4.10) (1.4.11) (1.4.12) (1.4.13) (1.4.14)

Equations (1.4.11) and (1.4.14) yield b = w = 0, Eqs. (1.4.12), (1.4.10), (1.4.8) and (1.4.13) imply that a = 0, a = 0, c = 0 and 0 = 0 respectively. We have thus established that a = 0 = u> = a = b = c = O; moreover (1.4.1) readily gives 7 = S = 0. The proof is complete. □

Function spaces, convex analysis, variational convergence

69

Later on we shall need certain results on quadratic one-dimensional splines. Let TV be a positive integer, h = —, xx = ih, 0 < i < N and Sfc = { / e C 1 [ 0 , l ] | / | ,

L.*,l

eV2,

i
(1.4.15)

We employ here the conventional notation and " h" used in this section has obviously noth ing in common with homogenized quantities. One immediately verifies the following re sult. Lemma 1.4.6. Let j3, a0, Q i , . . . , aN G R be given numbers. Then there exists one and only one element s G S\ such that s(x{) = a, 0 < i < N and s'(0) = p. □ We introduce the next definition. Definition 1.4.7. Let / G Cl[0,1]. An element s e Sh such that s(x<) = f{xi), and s'(0) = /'(0) is called the interpolant of / with respect to ShLemma 1.4.8. Let / G C 2 [0,1] and, for fixed i, p G P 2 be such that f{xt)

0
/ ( x ! + i ) = p(x i + i). Let c > Obe such that \f"(x)-p"(x)\
Vie[i„ii+1],

\f'(x) - p'(x)\
(1.4.16)

V x G [Xi, xt+1] , Vi6[i„i,+1|.

Proof. We set r(x) = f(x) - p(x). By applying Rolle's theorem we conclude that there exists £ G ( i „ xi+i) such that r'(£) = 0. We have r'(i) =

jr"{ri)dri.

Hence, by using (1.4.16) we conclude the assertion (a). Next we write X

iv)dv ■ Hence (b) readily follows. Let us consider the function ip : R —» R defined by 0

if x < -h and x > 2h , x+h if -h<x<0, h _2_ x(x-h)if0<x
\fh<x<2h.

(1.4.17)

70

Mathematical preliminaries

Properties off

veCHF),

^, M J + m , 6 V2,

VieZ

v(o)

= v(ft) = l ,

(1.4.18)

'2

ifare [-/i,0]u[/i,2/i], f"(x) = \ \ -^ifie[0,h],

(1.4.19)

where Z stands for the set of integers. Performing simple calculation, from (1.4.18) and (1.4.19) one concludes Lemma 1.4.9. Let / e C1 [0,1] be such that /(0) = /'(0) = 0 and, for 0 < x < 1, let s be the function given by s(x) = /(<%(x) + (f(h) - f(0)Mx -h) + (f(2h) - f(h) + /(O)V(x - 2/i) + ... + [f(Nh) - f((N - l)h) + f((N - 2)h) + ... + (-l)Nf(0)Mx - Nh). (1.4.20) Then (a)

s is the interpolant of / with respect to 5/,;

(b)

s"(x) = -j^f(h),

0<x
(1.4.21) m_1

(c)

(d)

2 s"(x) = -r-2{J{x2m) + /(x 2m _ 1 ) - 4 ^ [ / ( x 2 i + 1 ) - / ( i a ) ] } , n (=o for x and m such that /i < X2m_i < x < X2m < 1 ; s"(x) = - { - 3 / ( x 2 m + 1 ) + /(x 2m ) + 4 ^ [ / ( x 2 ( + 1 ) - /(ia)]}.

(1.4.22)

(1.4.23)

(=0

for x and m such that 2/i < x 2m < x < x 2m+ i < 1.

□

Later on in this section we shall need the following result. Proposition 1.4.10. Let / e C4[0,1] and denote by sh the interpolant with respect to 5/,. Then there exists a constant c, independent of / and h such that

H/ - Sh ||+/in/' - S'h\\ + h*\\f» - 4\\ < ch3(\\r\\ + nrii), where, for g 6 C°[0,1], we introduce the notation: \\g\\ = sup |
Proof. We note that if p is a polynomial of the second order on [0,1], then the interpolant of / + p is given by s/, + p. Thus, without loss of generality we may assume that /(0) =/'(0) = 0 .

(1.4.24)

Function spaces, convex analysis, variational convergence

71

On account of Lemma 1.4.8, it is sufficient to establish existence of a constant c\, indepen dent of / and h, such that ||/"-S^||
(1.4.25)

Due to (1.4.24), the proof is reduced to evaluation of the expressions (1.4.21) - (1.4.23). In the sequel of this proof, we denote by r a generic function which may depend on x and h and such that there exists a constant c2> independent of x, h and / , such that K*,/i)|< C 2 (||/"'|| + | | / , v | | ) . Case 1. For 0 < x < h, by (1.4.21), (1.4.24) and Taylor's formula one gets

S ;(l) =

' ^(? / " ( 0 ) + / l 3 r ) = / " ( 0 ) + /lr

This implies existence of C\ such that m^c | / " ( x ) - ^ ( x ) | < C ] / i | | / ' " | | .

(1-4.26)

To treat the remaining two cases related to (1.4.22) and (1.4.23), from the trapezoid rule with correction one deduces, cf. Davis and Rabinowitz (1967, p. 53), Davis and Rabinowitz (1975) 2h{f{a) +f{a + 2/i) + f(a + 4/i) + ... + f{a + 2mh)} a+2mh

=

h[f(a) + f{a + 2mh)}+

f f{x)dx + j [f'(a + 2mh) - f{a)] a

+h3r ,

0
+ 2mh
(1.4.27)

By applying successively the last formula with a = 0 and a = h/2, one gets h

2 / i £ [ / ( r a + 1 ) - /(* 2 ,)] = h[f(h) ~ /(0)] - \{f'(h)

- /'(0)] -

f(x)dx

X2m+l

+h[f(x2m+l)-f(x2m)}+

J f(x)dx+j{f'(x2m+1)-f'(x2m)}

+ h4r. (1.4.28)

X2m

Taking into account (1.4.24), we have h

h[fW - /(0)] - j[f'(h)

- /'(0)] -

Jf{x)dx b

= y / " ( 0 ) - j/"(0)

- jf'{0)

+ h*r = h*r .

(1.4.29)

72

Mathematical preliminaries

Case 2. For X2m-i < x < x2m, on account of (1.4.22), (1.4.28) with m being replaced by m — 1, (1.4.29) and setting 6 = X2m-i we obtain 4(x) = J;{/(6 + A) + /(6) - £[/i(/(6) - /(b - /i)) 6

+ Jf(x)dx + j(f'(b) - f'(b - h))}} + hr , b-h b

sl(x) = ~{f(b +

h)-f(b)+2f(b-h)-ljf(x)dx b-h

-~lf'(b)-f(b-h)}}

+ hr,

sl(x) = l{hf'{b) + jf"(b) + 2[/(6) - hf(b) + ^f"(b)} - -h[hf(b) - y/'(6) + jf"(b)} - ~f"(b)

+ h*r}

4{x) = f"(x) + h?r , a;2m_1 < x < x2m .

(1.4.30)

We recall that r is a generic function introduced earlier. Case 3. For x2m < x < x 2m +i. on account of (1.4.23), (1.4.28) and (1.4.29), putting b = x2m we get 4(x) = ^ { - 3 / ( 6 + h) + /(b) + |[/i(/(6 + h) - f(b)) b+h

+ Jf(x)dx+j(f'(b+h)

- f'(b))]} + hr ,

b

4(x) = ?-2{~f(b + h)-f(b) b+h

+ ljf(x)dx+™[f'(b

+ h)-f'(b)]}

+ hr,

b

sl{x) =

^{~2f(b)-hf'(b)

- y / " W + 2/(6) + hf'(b) + jf"(b) + ~f"(b)} 4(x) = ~f"(b)

+ hr ,

+ hr,

4{x) = f"(x) + hr , x2m < x < x2m+1 .

(1.4.31)

Function spaces, convex analysis, variational convergence

73

From (1.4.26), (1.4.30) and (1.4.31) we eventually obtain (1.4.25) and thus the proof is complete. □ Now we are in a position to pass to two-dimensional approximations. Let D = {(x 1 ,i 2 )|0 < x\,x2 < 1}. By TV we denote a positive integer and set h = 1/N. Next, we assume the following notations: Teh - the set of quadrangles [ih, (i + l)h] x \jh, (j + l)/i], 0 < i, j < N - 1, ■ Tth - the set of triangles obtained by division of the quadrangles according to Fig. 1.4.3. N(; - the set of the vertices and mid-points of the sides of the quadrangles of T^,

Uh =

{ueCl(D)\ulTeV2,TeTtll}.

-2"

o

Vx

Fig. 1.4.3. Let f £ Cl (D). We are going to construct an approximation u of / . To this end we set: Shj - shj{x\) - the interpolant of f{-,jh) with respect to Sh in the sense of Definition 1.4.7,1 <j
0
(1.4.32)

For 0 < i, j < N - 1, let Cl} = [ih, (i + l)/i] x [jh, (j + l)h\ and let uxj : Ctj -> Ft satisfy the relations:

74

Mathematical preliminaries u

vK

€ Vi;

A; = 1,2,3,4;

utj € C 1 (Cy) (fi* is defined in Fig. 1.4.4),

Uij(ih,jh) = shj(ih) uii((i+l)h,jh)

=

, uy(i + -/i, j/i) = s^((i + -)/i) ,

shj{{i+l)h),

Ui3(ih, (j + -)h) = Svi{{j + -)h) , uy((i + l)/i, (j + i)/i) = Stn+1 ((jf + i)fc) , Uij(ih, (j +

(1.4.33)

\)h)=ShjU(ih),

uy((i + l)/i, (j + l)/i) = s A j + i((i + l)/i). Due to Proposition 1.4.3, there exists one and only one function satisfying (1.4.33). %*1

P5

$

Pi

(/+1)A

"3

04)* —

—F* ■

n2

" 4

PA

n{ y* J

!p>

Pi

1 0

i i 1

*

i i

3

i

i

—

f

1 —

i 1

0 4 ) * ('+1>A

*'

Fig. 1.4.4. We define u : D —> R by setting (1.4.34)

u

\ctj =uij 0
= shi{xi) ,

0 < i! < 1 ,

0<j
0<x2
0
and u(ih, x2) = svi(x2) ,

(1.4.35)

Function spaces, convex analysis, variational convergence

75

Hence one concludes that u is continuous in D. It remains to prove the continuity of the first derivatives d\u and d2u of u. To this end, let us consider Fig. 1.4.5 and verify the continuity of d2u along the interval AC. *2

i

O+DA C

C

i-U

iJ

A jh-

P

B C

%.

i-\J-\

(j-l)n ■

('- \)h

h

(*\)h

*l

Fig. 1.4.5. By virtue of (1.4.35), one has d2ui-^j-l{A)=d2Ui-lj{A) &iUi-i,j-i{B)=d2Uijj-i(B) d2utJ_1{C)=d2utj{C)

, = d 2 Ui-i,j(5) = d2u,}{B) , .

(1.4.36)

Along BC the derivatives c^Uij-i and c^Uij are the polynomials of the first order in the variable xi. Hence, taking into account (1.4.36) we conclude that c^u is continuous along AC. □ The last lemma shows that the following definition is meaningful. Definition 1.4.12. The function u given by (1.4.34) is the interpolant of / with respect to Lemma 1.4.13. Referring to Fig. 1.4.4, let Vtj = {v£ C\Cxl)\vK

€ V2 , 1 < A; < 4} ,

0 < i,j < N - 1 .

Then there exists a constant c, independent of i,j,k and v, such that for v € Vtj one has IMIuo + frlMly.1 + fc2|MI«j.2 < cmax\v(Pk)\

,

76

Mathematical preliminaries

where IMIij,o=

max\v(x)\,

IM|t,;,i = max(max|9 1 v(i)|,max|a 2 v(x)|) , ||u||jj, 2 = max(sup \duv{x)\, sup \d12v(x)\, sup \d22v(x)\) . xecxj ieC(, zee*, Proof. It follows easily by using Proposition 1.4.3 and standard approach consisting in the passage to the reference element defined in Fig. 1.4.2. D We are now in a position to prove the following proposition. Proposition 1.4.14. Let / e C4(D) and denote by uh the interpolant of / with respect to Uh. (Def. 1.4.12). Then there exists a constant c, independent of h but dependent on / , such that 11/ - u h ||o + h\\f - uh\U + h2\\f - uh\\3 < ch3, where IMIo = max \v(x)\ ,

[|v||i = max(max

x€D

a=l,2

\dav(x)\),

x£V

Proof. In the sequel c will denote a generic constant, independent ofa,P,h, depend on / . We shall only establish the relation

but which may

Il/-«fc||i
(1.4.37)

For 0 < i, j < N - 1, the reader is referred to Fig. 1.4.4. Let Uy denote the restriction of u\ to Cij. By virtue of construction of u^ and due to Proposition 1.4.10, one has maxg \f(Pk) -

Uij(Pk)\

< ch3 .

(1.4.38)

It is not difficult to establish (for instance, by using Taylor's expression) the existence of a second-order polynomial g such that 11/ - fflluo + h\\f - g\\i,jA + h2\\f -

fl||Wi2

< ch3.

(1.4.39)

Here we have used notations of Lemma 1.4.13; g depends on i, j , ft but not on c. We have 11/ - Uftlly.1 < 11/ - g\\ij,i + I K - 9\\ij.i ■

(1-4.40)

We observe that u i ; - g 6 V^, where V^ is defined in Lemma 1.4.13. Due to this lemma one has I K - flllu.i ^ I max \uij(pk) ~ 9(Pk)\ < I max \uxj{Pk) - f(Pk)\ + \ max \f(Pk) - g(Pk)\ < ch2 ,

(1.4.41)

77

Function spaces, convex analysis, variational convergence

where (1.4.38) and (1.4.39) have been taken into account. From (1.4.39) - (1.4.41) we deduce that ll/-«fc||tj,i < c / l 2 . where c is independent of i, j . In this way we have obtained (1.4.37), which completes the proof. D Remark 1.4.15. The proof of Proposition 1.4.14 exploits the classical argument of "the inverse hypothesis" in the theory of finite elements, established by Lemma 1.4.13. We observe that in Proposition 1.4.14 it could not be difficult to provide explicit dependence of c on / . We pass to proving the second density result. Let ficR2bea bounded domain with Lipschitzian boundary, 1 < p < oo .

(1.4.42)

Without loss of generality, we may assume that fic {(xi,x2)\Q<xux2 < 1} .

(1.4.43)

Proposition 1.4.16. Let Q satisfy (1.4.42) and (1.4.43) and let / g W 2 p (fi). Then there exists uh e Uh such that lim||/-Uh||vv2.P(fi) = 0 .

(1.4.44)

h—*0

Proof. Due to Calder6n's extension theorem (see Stein, 1970, Chap. VI), it suffices to consider the case where fi = (0,1) x (0,1). We note that Uh C W 2 ' p (fi). From the density ofC°°(f2) in W2'P(Q.) we deduce that one may assume that/ e C°°(Cl). Proposition 1.4.14 then implies the existence of u^. 6 Uh such that \im\\f -uh\\wi.ooin)

=0.

n—*0

For an / g W2'P(Q) there exists a sequence {/n}neN C C°°(fi) such that ||/-/nl|w*.p(n) - » 0

as

n-»oo.

In turn, for each n there exists unh € Uh such that 11/ — Un,/illw2oo(n) —* 0 when

n —» oo .

Applying the diagonalization method we infer the existence of a mapping h —> n(h) with lim nlh) = +oo such that denoting Uh = un(h),h,

(1.4.45)

78

Mathematical preliminaries

we obtain 11/ - «fc||w».i>(n) < 11/ - /n(h) I |w».i>(n) + ||/n(h) ~ «/i||w».P(n) < 11/ - fn{h)\\w2-P(tl)

+ C||/„(fc) - Uh||w2.°o(n) ,

where c > 0 depends on fi only. The assertion is proved.

D

Remark 1.4.17. Sablonniere (1989) considered similar problem of approximation by using general triangular mesh. 7.5. An augmented Lagrangian method for problems with unilateral constraints Augmented Lagrangian methods combine ordinary Lagrangian techniques and penalty methods without suffering from the disadvantages of these methods: slow convergence for the former and possible ill-conditioning as rj —► oo for the latter. More precisely, aug mented Lagrangian methods converge without requiring that the penalty parameter rj tends to infinity. Augmented Lagrangian methods have many applications in the analysis of optimization problems with constraints, cf. Glowinski and Le Tallec (1989), Ito and Kunish (1990,1996) Telega and Galka (1998,1999). Ito and Kunish (1990) proposed the augmented Lagrangian algorithm to solve the mini mization problem (P) below. In Sections 8-11 we shall see that local problems for plates and laminates weakened by periodically distributed fissures are always of such a type pro vided that friction is neglected. In the present section we shall present the main results obtained by Ito and Kunish (1990). To this end, we first introduce the following spaces and mappings: V is a Hilbert space, V\ is a reflexive Banach space, continuously injected into V, H is a Hilbert lattice with inner product (•,•), a ( , •) : V x V —> R is a bilinear, symmetric, continuous and V-elliptic form with a(u,u) > Co| |u||y for some constant Co > 0, / : V —► R is a continuous linear functional, g : V\ —> H is a convex, continuous and Gateaux differentiable mapping. We recall (Yosida, 1978) that H is a Hilbert lattice if it is a Hilbert space and a partially ordered set (with order denoted by <) such that every nonempty finite subset of H has a greatest lower and a least upper bound (denoted by sup), and such that (i) u < v implies u + w < v + w for all u , v, w 6 H, (ii) u < 0 implies QU for all u € H, a G R + , (iii) |u| < |v| implies | |u|| w < ||v|| H f o r a l l u . v e //.where |u| = s u p ( u , - u ) and||-|| w denotes the norm in H. In practice, < stands for the pointwise almost everywhere ordering. We identify H with its dual and assume that A e H, A > 0, implies (A,u) > 0 for all u>0.

Function spaces, convex analysis, variauonal convergence

79

The problem under investigation means evaluating (P)

mm{-a{u,u)-l(u)\g{u)<0,

u 6 Vi} .

We need the following assumption: (A) there exists (u, A)| G Vi x H such that u is a solution to (P), A > 0 and (a)

(X,g(u))=0,

(b)

a(u,v) - l{v) + (X,g'(u)v) = 0 for all v G Vi.

Here g' denotes the Gateaux derivative of g. In view of the ellipticity of the bilinear form a(-, •) the solution u of (P) is necessarily unique. The element A is referred to as a Lagrange multiplier for (P). Now we are in a position to define a family of augmented Lagrangian problems associ ated with (P): (P)„,A

min L„{u, A) ,

where Lv(u, A) = ia(«,ti) - l(u) + l ( | | s u p ( 0 , A + V9(u))\\2H ~ \WO and A G //, 7) > 0, r? G R + . We note that L^u, A) can be written in an alternative way. Let g : V1 x H x R + - . H be defined by g(u,X,ri) = sup (g(u), — J . It can easily be shown that L,(u,A) = \a{u,u) - l(u) + (X,g(u,X,r,)) + |||p(u, A,77)||2„ .

(1.5.1)

We now pass to presentation and examination of the augmented Lagrangian algorithm to solve (P). It consists of a sequence of unconstrained optimization problems (P)v„,\n whose solutions (provided that they exist) converge to the solution of (P). The Algorithm (1) (2) (3) (4) (5)

Choose Ai G //, A! > 0, and 7? > 0. Put n = 1. Solve (P)„,A„ for ti,. Put An+i = An + T)g(un, K, V) = sup(0, An + T)g(un)). Put n = n + 1 and return to (3).

A stopping criterion must be added to the algorithm for actual computations.

80

Mathematical preliminaries

Prior to studying the convergence of this algorithm we require two technical lemmas. Lemma 1.5.1. Let A £ H, A > 0, 77 > 0, 77 € R + and assume that u —> Lv(u, A) is radially unbounded, i.e. Lv(u, A) —> oo as ||u|| v —» oo. Then there exists a unique solution uv of {P)n,\ which satisfies a(uv,v) - l{v) + (sup(0,A

+ TW(U,))10,(U»

= 0,

(1.5.2)

for all v £ V]. Lemma 1.5.2. Let A 6 H, 77 > 0,77 € R + . Then the function u->||sup(0,A + 77
TO(«))

,\€H

and sup(0, A + 7?ff(/iu + (1 - n)v)) < sup(0, n(\ + r]g{u))) + sup(0, (1 - M)(A + 773(7;))) = sup(0, A + T)g(u)) + (l-(j.)sup(0, A + 773(1))). Hence we deduce that ||sup(0,A + 77
Q

Proof of Lemma 1.5.1. Inviewoftheellipticityofa(-,),themappingu —> -a(u,u)-l(u) from Vi to R is strictly convex. Since also u —> (I/277)11 sup(0, A + r)9(u))\\2H is convex, it follows that u —* Lv{u, A) is strictly convex. Moreover, u -* sup(0, A + rjg(u)) is continuous (Baiocchi and Capelo, 1984, Th. 19.8) and thus u —* Ln(u, A) is continuous

Function spaces, convex analysis, variational convergence

81

as well. By assumption this mapping is also radially unbounded. Since Vi is reflexive, the existence of a unique solution uv of (P)Vt\ follows. Thus characterization (1.5.2) can be proved by differentiation of Lv(u, A) with respect to u. Theorem 1.5.3. Let hypothesis (A) be satisfied and assume that u —> Lv(u, A) is radially unbounded from V\ to R for every A > 0. Then coll" - "nil2, + ±\\K+i

- A||2„ R be given by J(u) = -a(u,u)

-l{u)

,

and set 6(u, A) = sup(0, A + r]g(u)) for A £ H and u e V. We note that 6(un, An) = A n+ i. With (A) we find J(u„) - J(u) = -a{un,un)

- l{un) - -a{u,u) + l(u)

= -a(un,un)

- a(un,u) + -TCL(U, U) 4- a(u, un — u) + l(u - un)

= ^a{un-u,un-u)

+ {X,g'(v.){u-un))

.

By using (1.5.2) we obtain J(u) - J{un)=-a(v.,u)

-l(u)

= -a(u -un,u-

- -a(un,un)

+ l(un)

un) - {6(un, Xn),g'(un)(u

- un)} .

Adding these two expressions we get 0 = a(u - un,u - u) + (X,g'{u){u - u)) - (9(un,\n),g'(un){u

- un)> .

(1.5.3)

Next we define G : Vi x H -> R G(u, A) = 1 [ | | sup(0, V g(u) + X)\\l - \\X\\l] , for 7] > 0. It is easily seen that G(u, A) < 0

for all

A e H , X>0.

In fact, by (A) and since H is a Hilbert lattice we conclude that sup(0,A + w(u)) < A,

(1.5.4)

Mathematical preliminaries

82

and ||sup(0,A +

W(ii))||„-||A||„<0.

Hence (1.5.4) follows. Moreover, for every v G Vi we find C u (u, X)v = (sup(0, ij0(u) + A), g'(u)v) . By Lemma 1.5.2 the mapping u —* G(u, A) is convex, consequently we obtain the follow ing subdifferential inequality G(u, An) > G(un, An) + (8(un, Xn), g'(un)(u - un)> .

(1.5.5)

Next we find G(un,Xn) + {e{un,Xn),g'(un)(u-un)) + (K,g(un,K,r)))

= ^\\g{un,Xn,7])\\l

+ (0(un,Xn),g'{un){u-un))

= -||
+ (K ~ A,g(un, Xn,rj)) + {Xg{un, Xn,rj)} + (0(un, Xn),g'(un)(u - u„)) = 2^\\V9(un,

A„,T/)

+ An - A||2 - — ||An - A||2 +

+ (Xn+1,g'(un)(u - un)) = ±\\9(un,Xn)

(X,g{un,Xn,V))

- X\\l - ^ | | A n - A||2„

+ (^, 9{un, Xn, 77)) + a{u - Un, u - un) + (X,g'(u)(u-un))

,

(1.5.6)

where for the last equality we used (1.5.3). From (1.5.4) - (1.5.6) combined with the equality (X,g(u)) = 0 we obtain -^l|A„+i-A||2„-^||An-A||2„+a(u-U,w-u) +(A,s(u n , An,r,) + g'(u)(u - un) - g(u)) < 0 .

(1.5.7)

The convexity of g implies g'{u)(u - u„) + g{un, Xn, r?) - g(u) > g'{u){u - u„) + g{un) - g(u) > 0 . (1.5.8) Combining (1.5.7) and (1.5.8) and using the fact that A > 0 we obtain ^ P n + 1 - A||2„ - ^ | | A n - A||2 + a(u This is the desired estimate in view of the ellipticity of a(-, •).

-un,u-un)<0. □

Remark 13.4. The assumptions on the radial unboundedness of u — ► Lv(u, A) and on the reflexivity of V\ are only required in the proof of Lemma 1.5.1 and can be replaced by the

Function spaces, convex analysis, variational convergence

83

assumption of existence of a solution of (P)n,x- If V\ = V, then the radial unboundedness of u —> L^u, A) is implied by the V-ellipticity of a(-, •). □ The first-order necessary optimality condition associated with (P)v,x„ of the algorithm has the following form: a(un, v) - l(v) + (sup(0, An + T]g(un)), g'{un)v) = 0 for all v € Vi. By definition of A n+ i this condition is equivalent to a{un,v)-l(v)

+ (\n+ug'(un)v)=0

for

all

v e Vi .

(1.5.9)

Corollary 13.5. Under the assumptions of Theorem 1.5.3, the sequence {An}nem is bound ed in H and un —» u in V with °° 1 c o ^ l l u - un\\l < - | | A , - A||2„ . n=l

(1.5.10)

'

If, moreover, g'{un)v —» g'(u)v in / / , for every v € Vj, then every cluster point A of {An} satisfies (A)(b) with A replaced by A and A > 0. Proof. Estimate (1.5.10) follows directly from Theorem 1.5.3. The second assertion is a consequence of (1.5.9). Finally, by step (4) of the algorithm (A, v) > 0 for each v > 0 and hence A > 0. D The correspondence between (P) and (P)„\ is formulated as Corollary 1.5.6. Under the assumptions of Theorem 1.5.3, u is the unique solution of the constrained problem (P) if and only if u is the solution of the unconstrained problem

(pkxProof. The result follows immediately from Theorem 1.5.3 with Ai = A.

□

Variable stepsize analysis In the augmented Lagrangian algorithm with variable stepsize, step (4) is replaced by (4')

put A n +i — An + ar]g{un, Xn, TJ) ,

where

a e (0,1] .

For numerical calculations it may be essential to take a < 1 to obtain good performance of the algorithm. Theorem 1.5.7. Assume that (A) is satisfied and that un —> Lv(u, A) is radially unbounded from Vi to R for every A > 0. Then we have coll" - un\\2v + V-(l - oOHSK,X n ,vW H + 2 ^ H A " + i - A||* -2^l|An-A||2«'

(,

-5-U)

84

Mathematical preliminaries

for all n G N, and 00

1

cb$3llfi - u^v + ^

- «)Mff<"». ^ H - l ^ w l l A l ~ X|l« •

n=l

(1512)

'

Proof. The proof is left to the reader as an exercise, cf. also Ito and Kunisch (1990).

D

Remark 1.5.8. The same authors (Ito and Kunisch, 1996) developed the augmented Lagrangian algorithm for a more general class of convex optimization problems: find min{/(u) + R is a lower semicontinuous, continuously differentiable, convex functional, A G L( V, H) and

R is a proper, lower semicontinuous convex functional. Moreover, V, H are real Hilbert spaces and K is a closed convex subset of V.

CHAPTER n

ELASTIC PLATES

Introduction The subject of this chapter is a statical analysis of a linearly elastic plate of properties periodic in two orthogonal longitudinal directions. Such a plate can be constructed by repeating the basic periodicity cell Z along the directions of the sides of the reference rect angle Z = (0,/*) x (0,/!), see Fig. 2.0. Due to highly oscillating elastic properties, the solution of the elasticity problem is also oscillating. The aim of the analysis is to extract from this solution those parts which describe the overall plate response. It is the two-scale expansion method which makes it possible to extract the terms representing the overall plate behavior. This method has already been successfully applied to the overall analysis of periodic three-dimensional composites. Here this method should be appropriately mod ified to take into account that the transverse dimension of the plate is much smaller than its in-plane dimensions. This has two consequences: distribution of displacements is al most linear across the thickness and the in-plane stresses assume much greater values than other stress components. The latter property means that the plane-stress state prevails. It turns out that both these features can be disclosed by the two-scale expansion method if an appropriate scaling of the loading is adopted.

Fig. 2.0.

Let the plate thickness be scaled by the small parameter e and the dimensions l\,h- by the small parameter e. A perfect effective plate model is constructed byfixingthe ratio e/e

86

Elastic plates

and passing to zero with e, cf. Caillerie (1982). This is equivalent to a simultaneous passage to the zero limit with e and e, which can be performed formally by the asymptotic expansion method (Sections 2.2, 2.3) or rigorously by the T-convergence technique (Section 2.10.4). Both approaches lead to the Caillerie-Kohn-Vogelius plate model of periodic plates made from cells Z of arbitrary shape. Just this effective plate model will further be treated as a reference model and all other averaging methods will be better or worse approximations of this model, depending on the shape of the basic cell Z. This model is, however, difficult to apply since it involves three-dimensional local problems. Difficulties that arise at any attempts to solve these local problems analytically justify further steps towards rational simplifications. The first simplified model refers to the case of the basic cell Z being a thin plate. This model can be derived from by two methods. The first one was originated in 1976 by Duvaut and Metellus who homogenized the thin plate equations of Kirchhoff, without making any reference to the three-dimensional plate pattern. The second modelling consists in impos ing Kirchhoff s constraints on the solutions of the Caillerie-Kohn-Vogelius local problems. This purely two-dimensional homogenization leads to relatively simple formulae for effec tive stiffnesses. In the case of plates with straight ribs these formulae can be put in one compact and algebraic formula, called the formula of Francfort and Murat, cf. Sec. 3.8.1. It will turn out clear in Chapter VI that this algebraic formula plays a crucial role in prov ing the extremal properties of ribbed plates. This shows that the homogenization results do not only provide us with rational averaging formulae for plates of repeated lay-up, but they yield complete characterizations of the sets of effective stiffness tensors of all plates constructed by mixing given constituents in fixed proportions. This creates a link between homogenization and optimization, a link that makes the homogenization results of major significance. The second simplified model of periodic plates refers to the case of the basic cell Z being a slender column, see Sections 2.9 and 2.10.3. This model was proposed by Caillerie (1984) for the case of constant thickness and by Kohn and Vogelius (1984) for the case of periodically varying thickness. Thus the analytical formulae for effective stiffnesses concern the periodic plates com posed of very thin or very slender cells. It turns out that these former formulae (of Duvaut and Metellus) can be improved by admitting the transverse shear deformation within pe riodicity cells. This can be performed by two different methods leading to the same final results. In the first method one imposes Hencky-Reissner constraints on the solutions to the three-dimensional basic cell problems of Caillerie-Kohn-Vogelius, see Section 2.7. The second method takes the Hencky-Reissner two-dimensional setting as a point of departure. The homogenization is based on a special scaling, called refined, which preserves relations between all length scales involved. There are at least two such length scales. The first is the length of the basic cell. The second is the quantity ( D / i / ) 1 / 2 , where D and H represent the bending and shearing stiffnesses, respectively; see Sections 5.3 - 5.5. If a thin plate is periodic with respect to a curvilinear parametrization, then homogeniza tion is nonuniform and leads to an effective model with slowly varying effective stiffnesses.

87

Introduction

This topic is discussed within the Kirchhoff framework in Section 3.10. The homogenization analysis of periodic plates undergoing moderately large deflections is performed in Section 4.2 under the von K&imin assumptions and generalized in Section 5.7 to the case of transverse shear deformable plates. The influence of openings on the bifurcation analysis of thin plates is discussed in Section 4.3. The Kirchhoff and Hencky-Reissner modelling cannot be applied to sandwich plates with stiff faces. The bending stiffness of the faces is taken into account in the model of Hoff. The formulae for effective stiffnesses of such sandwich plates of repeated lay-up (usually the core has a periodic structure) are derived in Section 6. In the present chapter the following conventions are adopted. The small Latin indices, like: i, j , k, I, m, n, s... run over 1, 2, 3; the small Greek indices, like a, 0, A, n,... (expect for e) assume the values 1, 2. The summation convention applies to the indices at different levels, e.g. 2

3

A°% = 5 > < % , M»ajk = £ M % f c . 0=i

j=\

In the expressions like Aal3hP or M'jajh the summation does not hold. The three Cartesian orthogonal systems will be used: (XJ), (z*), (&). The partial deriva tives are denoted shortly as follows

&- =

()

a^-()"

"

a^- ( ) "-

(01)

We also introduce the following notation: x = (xux2), x = (xi,x2,x3)

z = (zuz2), ,

2/= (2/1,2/2) ,

z = (zuz2,z3),

(0.2)

y = (1/1,1/2,1/3) •

The symmetrized gradients are denoted as follows Cy(w) = j j K j + fj,i) ,

4-(«) = 2 ^ -

+V

e y («) = ey-(w) , r

>J'

4-( ) = 2 ^

(0.3)

+ v

>^'

where v = («i, v2, v3); vt depend on either x or z or y. Let v = (f 1, v2) and w be functions of x, z or y. We set ea/3{v) = ^(va,P + Vp,a) i eZ

a0(v) = 2

C

^

+ V

^

;

a/?(«) = 9 ^ 1 / 5 + VP\c) i

Kap{w) = -W
'ap{w)

^a0H

=

~W^

= ~W\a0 ■

'

(0,4)

88

Elastic plates

The quantity e will be a small parameter scaling the plate thickness and should not be mistaken for the operators (0.3), (0.4). The rescaled plane and spatial periodicity cells are denoted by Y and y and are paramet rized by (ya) and (yi) respectively. The averages of a function / denned on these cells are denoted by

(/> = Y\jf{v)dy•

^fy= W\If{y)dy'

(0 5)

'

y

Y

where \Y\ =areaY, \y\ =\o\y.

2.

Three-dimensional analysis and effective models of composite plates

The aim of this section is to put forward a derivation of these effective models of peri odic plates which start from the three-dimensional setting. We consider two methods of modelling: (a) the process of reduction of the transverse dimension is indissolubly bond ed with the process of smearing-out the stiffnesses; (b) both processes being subsequently performed. A central role is played by model (Pn) with three-dimensional local problems. This model is derived by two different methods: by asymptotic expansions and then indepen dently derived and justified by T-convergence with all dimensions of the basic cell tending to zero. Other models discussed are approximations to model (Pn) and concern plates composed of very slender or very thin periodicity cells. These approximate models are not derived here by an asymptotic expansion method; the method of T-convergence is here self-explanatory. All convention given in the Introduction to the Chapter apply here. 2.1.

Equilibrium problem of a periodic plate

We consider a three-dimensional elastic body occupying the closure of a domain B lying between two surfaces of Z-periodic shape; Z = (0, Zf) x (0, Zf) : B = {x | x = (x, x3) ,

i = (xa) 6 ^ ,

xj(x) < i 3 < xj(x)} .

Here ft is a plane, open reference domain parametrized by Cartesian coordinates (xQ)(a = 1,2); x3-axis is directed normal to this plane and the (ij) system forms a left-handed Cartesian system with orthonormal basis (ei,e2,e 3 ). The functions x* are assumed to be Z-periodic, i.e. xf(x\ +mll, Xi + nl\) = x3t(xi,X2) ,

m,n 6 N .

They determine the upper (+) and lower (—) faces of B : r ± = {x | x G Q,

x3 = xf(x)} .

Three-dimensional analysis and effective models of composite plates

89

Let r = dQ, be the boundary of fi. The cylindrical surface T = {x | x 6 T,

xj(x) < x 3 < xj(x)}

is referred to as a lateral surface of B. If the in-plane dimensions of Q are much greater than max |xj" — x j |, then such a body could be called a plate with varying thickness. The plate is composed of cells Z Z = {x\x

e Z,

X3 (x) < x 3 < xj(x)} ,

except for a boundary layer around T. The elastic moduli C^ of the plate material are assumed to be Z-periodic functions in x, hence index Z at the core letter C . Within the framework of linear elasticity the stresses <7t; and strains ty, both referred to the Cartesian system (XJ), are interrelated by Hooke's law

a« = Cxt\x)ekl .

(2.1.1)

The strain-displacement relations are given by eij{w) = w(ij)

(2.1.2)

where w = (wi). The plate is subject to surface loads rf(x) on the r ± faces, i.e., o1J(x,x$)nf{x)

(2.1.3)

= r'±(x) ;

here n * = (nf) are versors normal to T±. The body forces b' = 6'(x) are Z-periodic in x. Along T the plate is clamped. Thus the space of kinematically admissible displacement fields is given by V0(B) = {ve

Hl{Bf

I v = 0 on r 0 } .

(2.1.4)

The equilibrium problem reads: findw 6 V0(B) such that a'je,j{v)dx= (Pz)

'

ri+{x)v,(x,x^)dT+ (2.1.5)

r+

- frl{x)vt(x,xJ)dT_

l

+ Ib {x)vi(x)dx

V v G V0{B)

Here axi depend upon to according to (2.1.1) and (2.1.2). We assume that the matrix Cz = (C% ) is positive definite: 3m > 0 such that V 7 G E3,

Cikllim

> m£(7„)2

(2.1.6)

Elastic plates

90 for almost every x € B. Moreover, the following symmetry conditions hold gijki

= cm

=

cm

=

Qijik

( 2.i.7)

The area elements dT± are given by dT± = (Gf (x))1/2dx ,

dx = dxxdx2

(2.1.8)

with G|(x) = l + (x3±1)2 + (x3±2)2;

(2.1.9)

here (•),<* = d/dxa. 2.2. Family of problems (Pe) Problem (Pz) involves three quantities: h = max | x j - X3 |, l\ and Zf which are small in comparison with the global dimensions of B. The presence of small quantities makes the problem intractable by usual analytical and discretized methods. Thus it is reasonable to take advantage of these parameters being small and use an asymptotic method. In the asymptotic method to be used we consider a family of problems (Pc) such that for a certain e = £0, {Pe0) = {Pz)- A common feature of all problems (P€) is that the periodically cells for (P€) remain homothetic to the original cell Z. Thus we substitute xf(x)

— x f (x) = £C± ( ^ ) , Xf{x) = EoC* ( U

lza ~» ela , l*a = e0la ; Z^eY

2

, Z = e0Y .

Here ~» means replacement and Y = (0,1]) x (0, l2). The functions c* are Y-periodic. Let y = {y I y = (y.1/3), y = (2/1,2/2) e Y , c~{y)
c+{y)}.

Thus replacements (2.2.1) mean Z — Zc = ey ,

ZC0=Z

.

(2.2.2)

y represents a rescaled cell of periodicity, cf. Fig. 2.2.1. Scaling (2.2.1) replaces domain B with domain Bc. To compensate for diminishing of the transverse dimensions of Bc when e —> 0 one must scale the loading r°±(x) - r£°(x) = e2p%(x) , r?a(x) = r£(x) , r 3 (x) - r f (x) = £3<7±(x) , 4<"3(x) = r 3 (x) , Q

3

2 3

6°(x) -~ e6 ( - ) , 6 (x)-* e 6 ( - ) ,

(2.2.3)

Three-dimensional analysis and effective models of composite plates

91

Fig. 2.2.1. Rescaled cell of periodicity where p j , q± are e-independent functions defined on fi; ba(-,y3), b3(-, 2/3) are K-periodic. The unknown displacement field is denoted by w£ — {w\) and the strains ef^ = e^{wc) and stresses a'J depend on e. The boundary conditions (2.1.3) assume the form o?(x, x f ( x ) ) n f (x) = e2pa±(x) ,

a?(x, x f ( x ) ) n f (x) = e3q±(x) , (2.2.4)

where ne± represents the unit vector outward normal to Fc± at (x, x ^ ) . Assume that func-

"fM-(M!))">(D' (2.2.5)

**&) = [*<£, *c% ±1],

df To make the scaling complete one should assume the following constitutive relation a?(x)=Ci>"(^)4(*):

(2-2-6)

Cf'(*) = C ^ < ( - , ^ ) ;

(2.2.7)

— J are K-periodic and C,ikl ( _ iJ/3j are defined for j/3 € (c~(y),c+(y)). Hence (fi^iy) are defined for y € y.

Elastic plates

92

The variational equilibrium equation (2.1.5) yields: fai'eijWdx

= f[e2j^vQ(x,ec+)

+ e3q+v3(x,ec+)]dT%

r« +

BC

+ / [e2ptva(x,ec')

e3q-v3{x,ec~)]drc_

+

H.

+ f[eba ( * ) va{x) + e2b3 ( * ) v3(x)]dx

(2.2.8)

Be

for all v vanishing on T £ . The problem of finding wc, e£, ac satisfying (2.2.8) and (2.2.6) will be called (Pe). The asymptotic method applied in the sequel requires reformulation of problem (P £ ) so that the local variable y € y would play the role of an independent variable. To this end we shall extrapolate the fields involved in [Pt) to the product domain Q. x y 3 {x, y), as follows. It is assumed that stresses and displacements can be expressed in terms of new functions ^(x,x3)

= a? (x; -,—) £

wci(x,x3)

= wti[x;

-,— j ,

,

£

(2.2.9)

Vi{x,x3)=Vi(x;

-,— J ,

such that o? \x; ;— ,J

u>i(x; -, — ,J ,

Vi(x;-,—

,J

are V-periodic functions. Let us rewrite (2.2.9) as follows f[x, X3) = f(xuX2,

t/l, 2/2, 2/3)L=x„/£, y3=i 3 /e

(2.2.10)

for / = a'J, u\, v\. Derivatives of / are denoted by ^=f,a dxa

l f = / V dyi

(2-2.11)

Hence ^T

= [f,a+-£f\X=*le-

(2-2.12)

Using (2.2.10)-(2.2.12) one can express the integrand on the l.h.s. of (2.2.8) as follows 1
Three-dimensional analysis and effective models of composite plates

93

Thus one can write Eq. (2.2.8) in the form c + (f)

£

I I lar (x' 7) ^a (x' 7) + \*< (*• 7) ** (x' 7)]d:rd (?)

Hc-(f) = £2/

[ P ? ( I ) « „ (x, ~£ , c + ) G | Q

+ }£(*)«„ (x, p C " )

G£ Q ]

dx

n +e3J

[q+(x)v3 (x, -e,c+) G$ (?-) + q~(x)v3 (x, ^ c " ) G! (*-)] dx

n

+e

'/ / V (") «■ (*■ ?) + £t* (?) * (*• 7 ) ] « ( ? )

(i2 13

- »

«c-(f)

Now we set j / = - , 2/3 = — or y = — and treat y as an independent variable. Integrating s e e both sides over Y one arrives at Ae(wc,i>) = F€{v)

(2.2.14)

where Ac{wc,i>) = eho I -< a?(w€)vt,a

*«(») = e2f[ho n

-< bQ(y)va(x,y)

y

+ pa_(x)(va(x,y,c-)(G-(y))1')}dx

+ ^{wc)vA]

>- dx ,

(2.2.15)

+pa+(x){va(x,y,c+)(G+(y))h e3J[h0^b3(y)v3(x,y)y

+ n

+

+ q4x){v3(x,y,c )(G+(y))^)

+ q-(x){v3(x,y,c-)(G-(y))h\dx

.

(2.2.16)

The notations (•} and -< • >- have been adopted in the Introduction. The quantity ho = \y\/\Y\

(2.2.17)

represents the average thickness of the rescaled cell y. The stress-displacement relations take the form *?'(*, y, 2/3) = "&, !&)&(*. y, wO

(2.2.18)

94

Elastic plates

where 2

2

«*J = K,0 + ™0,c, +£ ~«\0 + ™0\c) '" "'

^,3 = ™3,a + ~(™a|3 + ™3| J >

(2.2.19)

«33 = "*3|3

and w £ H(Q x y), where

H(n xy) = {v = (v,(x,y)) | v{x,-) e w(y)M-,y) € tf0W}, W(y) = {v € [i/'O 7 )] 3 !^ assumes equal values on the opposite lateral faces of y) and// 0 1 (fi) 3 = [i/ 0 1 (^)] 3 Now we are ready to formulate the boundary value problem on the product domain ilxy. It will be called (7*): find we e H(
Asymptotic analysis. Effective moduli and local problems

The solution wc of problem (T*) is sought in the form wc = u ( 0 ) (z) + eu (1 >(z; y) +
(2.3.1)

where ti (0) G H^(Q.f and iM € H(Q. x y), p = 1,2,.... Consequently, the stresses assume the form of a similar expansion a? = alj + eo? + e2o? + ... ,

(2.3.2)

l

where a J = rf{x, y), and ^ = C ^

o« = CPuu%l)

+ C^U<°>,

+&<**%■,

p=l,2,....

(2.3.3)

Substituting (2.3.2) into (2.2.14) and equating the terms of the same order with respect to e results in the reformulation of problem (Vc) to the following sequence of problems: Find u<°> € H£(n)3 and «<*> € H{U x y) such that

/

< al03viU >■ dx = 0 ,

(2.3.4)

n

/

-< < X a + °ii>i\j ydx

= 0,

(2.3.5)

n

ho I -< a?vita n

+ a'2jviU >- dx = F2(v) ,

(2.3.6)

Three-dimensional analysis and effective models of composite plates

^o J -< <7J°Vi,a + o3vAj ydx = F3{v) , n

95

(2.3.7)

I -< a™vi>a + (rlj+lv,i, ydx = 0, n for each v e H(Q x y), where F2(v) = J\pa+(x)(va(x,y,c+)(G+(y))^ n + j£.(x){va{x,y,c-)(G-(y))l)]dx

(2.3.8)

(2.3.9) + ho J ■< ba{y)vQ(x,y) y dx , n

j[q+(x){v3(x,y,c+)(G+(y))h

F3(v) = n

+ ho j ^ b3(y)v3(x,y) > dx . (2.3.10) n The sequence of problems given by (2.3.4) - (2.3.8) can be subsequently solved, thus enabling a construction of the solution (2.3.1) and (2.3.2) of problem (V*). To make this chapter self-contained, it is indispensable to follow at least first steps of the solution process. Step 1. Substitute the representation (2.3.3)] into (2.3.4) and take i>i =
■< [ C ^ u g + C^ujj^Wiu >-= 0 .

(2.3.11)

1

Hence u' ' can be expressed as «l1, = 9 i M ( s ) » i 0 i + « t W .

(2-3.12)

where 0 ^ e W{y) satisfies (for (j/3) = {kl)) {Vy)

| ay(eikl),w)

= - < Cijklwt]j y

Vw e W(y)

(2.3.13)

and the bilinear form ay(-, •) is defined by ay(u,v) =-< C^'d/K^i, y,

u,ve W{y) .

(2.3.14)

According to standard theorems problem (Vy) is uniquely solvable provided that, cf. Sec. 1.2.2 -< 0<*O y= 0 .

(2.3.15)

Elastic plates

96

Note that the function ©<3« =

_ 0k) t (e M) 'k )=- { h6y-y^k)'

(2.3.16)

where 2/3 = 2 / 3 - ^ 2 / 3 ^ ,

(2.3.17)

C*jkhG(ff + C'j30 = 0 ,

(2.3.18)

satisfies

along with (2.3.15). Thus 0 (3/3) solves problem (V],). Substituting (2.3.16) into (2.3.12) one obtains ull) = e^(y)u^0

u^ = e3a0)(y)u(^

- y3w,a + uAx) ,

+ u3(x) , (2.3.19)

where w(x) = u3 (i). Step 2. Substitute relation (2.3.19) into (2.3.3)i taking into account (2.3.18). One finds

a'j = AT^Z , Amm

=

cijUa(W

}

a(jM =

(2.3.20)

Qjton)

tfhgm)

+

(2.321)

Now let us substitute (2.3.20) into (2.3.5) with va = wa(x), v3 = 0 and wa S Hi (Q). Then /

V w £ Hi (Q) 2 ,

-< of > w0iC,dx = 0

(2.3.22)

where -< af >-= AfXfiu^ x

Af "=^Af

Xll

y

,

(2.3.23)

.

(2.3.24) 0)

We shall prove that solution to the equation (2.3.22) is trivial: u^ = 0, o^" = 0 and hence a% = 0 . To this end we have to concentrate on properties of the tensor Ay. Let us take (kl) = (a/3) and w = ©(A(i> in (2.3.13). One easily finds an identity ay(&{a0), e (J *»)+ -< C ^ e ' * " ' >-= 0 .

(2.3.25)

Combining this identity with (2.3.24) and (2.3.21) one obtains Af* =-< V^a^a^ which, by (2.1.7), implies the following symmetries

y,

(2.3.26)

Three-dimensional analysis and effective models of composite plates Moreover, one can prove that (Ay estimate Af^ia^x^

97

M

) is positive definite. By (2.3.26) and (2.1.6) one can

> c]T -< a^0)a\^] y 7 ^ 7 ^ = c ^ x 7^7,., y> 0

(2.3.28)

with % = o^7ofl •

(2-3.29)

On the other hand, the equality in (2.3.28) can only be attained if 7^ = 0. Then (7^) = 0. Taking into account that (0™u) = 0, one finds (7AM) = 7A,, = 0. Hence Ay is positive definite, which implies that ix° = 0 is the only solution to problem (2.3.22). Consequently, Eqs. (2.3.19) reduce to u[l) = -V3W.O + ua(x) ,

u{3l) = 1x3(1) .

(2.3.30)

Step 3. Let us substitute (2.3.30) into (2.3.3)2 for p = 1. One obtains a? = C>uu$ + C**uk,a - h ^ w ^ .

(2.3.31)

By inserting diis formula into Eq. (2.3.5), taking into account that 0% = 0 and choosing Vi = ip(x)wi(y),

y=0

Vwe W(y).

(2.3.32)

By linearity of the problem (2.3.32) its solution can be represented in the form u(2) =

e«»(y)u w - S^(y)w,a0 + n(x) ,

(2.3.33)

where rj = (rjQ) is at this moment undetermined while 0 (j/3) are solutions to problem (Py) subject to condition (2.3.15) and the fields S{a0) 6 W(y) satisfy (V2y) I ay&a0\w)

= -^y3Ci>Q0wx{j^

VweW(y).

(2.3.34)

They are also normalized as follows -< S ( Q / 3 ) y= 0 .

(2.3.35)

Due to the last condition the local fields S(Q^> are uniquely determined as solution to (Py). The proof is similar to the proof of well posedness of the problem (Vy). Substituting relation (2.3.16) into (2.3.33) one finds U(a] = Q{20\y)u^,p

- Si"0)(y)w^0 0)

- y3U3,a

+ T]a(x) ,

uf = e ^ f o K , - ^ (y)wn0 + r,3(x) .

(2.3.36)

98

Elastic plates

Substitution of (2.3.33) into (2.3.31) results in o? = AT0(y)up,a

- I?r0(y)w,aP

,

(2.3.37)

where A 0 is defined by (2.3.21) and Egae = cOHgWJ) ^ eWJ) = S M )

+ h6£50)

.

( 2 3 38)

Note that thefieldv3 contributes to u3 but does not affect a?. Step 4. One can easily prove that X o\> V= 0 .

(2.3.39)

Let us substitute w{ — Simy3 into (2.3.13). Hence .< A?3" y= 0 .

(2.3.40a)

The same substitution into (2.3.34) results in -<E?3a0y=O.

(2.3.40b)

Hence, by (2.3.37), one obtains (2.3.39). Further, the following notation will be used Ma0 = h0^afy,

Ma0 = h0< y3af

y,

Q° = h0 -< of >- .

(2.3.41)

Step 5. According to (2.3.1) and (2.3.19) we have wa = e{ua{x) - y3wi
w3 = w(x) + eu3(x) + 0(e2) .

(2.3.42)

The choice of test functions Vi involved in (2.3.6) - (2.3.7) should be compatible with representations (2.3.42). Thus we assume va = e[wa(x) - y3v>a] ,

v3 = v(x) .

First, let us substitute (2.3.43) into (2.3.6), assuming that wa S Taking into account (2.3.39) one finds jMa0wa,0dx

f(Ma0v,a0

= fpa{x)wa{x)dx

(2.3.43) HQ(Q)

,

+ Qav
and v €

HQ(Q).

(2.3.44)

,

(2.3.45)

Three-dimensional analysis and effective models of composite plates

99

where the loadings are given by pa(x) =ho^ a

b°(y) y +{(G^)pa+(x) +

+ <(C-)*)p°(x),

a

m (x) = <(c - -< 2/3 >)(G+)i)p +(x)

(2.3.46)

a

a

+ ((c"- -< y3 y)(G^)p _(x)-

-< y3b (y) y .

Now let us set va = 0 and v3 — v{x) in (2.3.7). One obtains lQav,adx = jq{x)v(x)dx , n n

(2.3.47)

q(x) = ((G + )i) 9 + (i) + ((G_)i)
(2.3.48)

where If v £

HQ(Q),

then, on combining formulae (2.3.45) and (2.3.47), one finds JMal3Ka0(v)dx

= f{qv - mav,a)dx ,

(2.3.49)

where /cQa(i>) = — v Q£ = —-5—5—. Let us define the effective moduli by OXpOXa

E?*

=-< £^ A * V , x

F / A " =-< y3AfA" x

Df »=^y3Ef »y

>- ,

•

(2.3.50)

The stress and couple stress averages involved in Eqs (2.3.44) and (2.3.49) are interrelated with e(u),n(w) by

A*°* = /^F-f A * e A » + hoDf^K^iw)

,

(2.3.51)

where Ay is defined by (2.3.24). Equations (2.3.51) follow from (2.3.37) and (2.3.41). The variational equations (2.3.44) and (2.3.49) along with constitutive relationships (2.3.51) form thefirsthomogenized problem (Ph.

Find(u,u>) e (tfoHfi))2 x H$(Q) = V#(fi) such that Eqs. (2.3.44), (2.3.49), (2.3.51) hold for each (w, v) e V#(fi) .

Step 6. We shall prove that problem {Phom) is well posed. Notefirstthat tensors (2.3.50) can be written as follows

Efx" =-< C^'a'f ' e ^ ' y ,

Ffx» =< &"<£?>a™ y , (2 3 52)

Elastic plates

100

where a§0) and e^f have been defined by Eqs. (2.3.21)2 and (2.3.38)2. The identities (2.3.52) can be proved similarly as the identity (2.3.26) proved previously. Thus the sym metry properties (2.1.7) readily imply Efx" Note, that

= F^0

,

Dfx" = D$T0 .

(2.3.53a)

Efx" £ E$T0 , Ffx" £ FX,M0 ,

(2.3.53b)

in general. The elastic potential of the homogenized plate is expressed by Wy(e, K) = {Afx»ea0eXli

+ Efx»KXliea0

+

D°0X"Ka0^)/2, (2.3.54)

KaP = na0(w) .

(2.3.55)

+ ff^K^t^

where eQ/j = ea0(u) ,

By virtue of representations (2.3.26), (2.3.52) one can rearrange the potential Wy to the form Wy(e, *) = \* Cvk'%%t y , (2.3.56) where 7y = <#"> (y)eXfl + e ^ (y)KXli.

(2.3.57)

Due to (2.1.6) the potential Wy is non-negative. We shall prove more, that Wy = 0 implies e = 0 and K = 0. Let Wy = 0. Then by (2.1.6) % = 0. In particular %0 = 0. Hence (%0) = 0. Due to V-periodicity of © (,j) and S ( ' J ', we have (7a/?) = «a/3 + hK-c.13 = 0 .

(2.3.58)

Hence ta0 = 0 and na0 = 0. Thus there exists a positive constant c such that

Wy(e, K) > c J2((^0)2 + ta/3)2) •

(2.3.59)

a,/?

Therefore, the problem (Phom) is uniquely solvable. Thus the solution (we,
,

XL* = £0Ua , M *

(-O.0 = Cc,0(un) ,

Wh = W ,

ina0 = £OCa0 ,

h

«S/3 =

(2-3-60)

K

o0

Three-dimensional analysis and effective models of composite plates Nf <

A

= (EO)2Ma0 ,

Mf

A

Ftfx» = ho(eo)2F§0^

= (EO)3Ma0 , 2

" = Mo< * ,

101

£-f*" = h«{ev) Ef » , ,

(2.3.61)

x

(2.3.62)

Dg* = V £ o ) 3 l C

Q = (£o)3q = (Gl)rl + (Gl)rl + e0h0 x 6 3 X , pa = (e0)2pa = (Gl)rl + (Gl)rl + e0h^ ■< ba y , ma =

e0((c--^y3>)Gl)ro_

(2.3.63)

+ E0{(c+- X j/3 ^)GIK - £o -< hba >- ■ The last three formulae express the equivalent loadings applied to the original, Z-periodic plate. Its effective membrane (An), reciprocal (En,Fn) and bending {Dn) stiffnesses are given by (2.3.62). The membrane forces Nh and moments M/, are interrelated with membrane strains eag{uh) and changes of curvature K,Qp(w>') by N? = Aa^ex,{uh) Mf

+ Etfx»KX,{wh) ,

= F?fx»ex,(uh) + D^K„(wh)

.

(2.3.64)

The effective problem for the original Z- periodic plate problem {Pz) reads Find (uh,wh) e H^{Q)2 x tf02(fi) = V£(fi) such that J\Nf(u\wh)ea0{w)

+

Mf{uh,wh)^{w)\dx

(Pn)

I

{pawa + qw- maw
V (w, w) € V£(ft)

where relations N^(uh, wh), M£0(uh, wh) are expressed by Eqs. (2.3.64). The displace ment fields uh, wh represent averaged in-plane displacement and transverse deflection of the original plate. Remark 23.1. Note that by Eqs. (2.3.2), a'0j = 0, (2.3.37) and (2.3.57) one can rewrite a? = eiA^e^iu)

+ Ey^K^w)]

+ 0(e2) ,

$ = e% + 0(e2)

(2.3.65)

Thus by Eqs. (2.3.24), (2.3.50), (2.3.51) one finds Ao ■< c^4j >-= e2Wa0ta() + Ma!)Ka0\ + 0(e3) .

(2.3.66)

Such equivalence between average of the internal work of stresses and internal work of av eraged stress - and couple stress resultants means that Hill's consistency criterion, written in the three-dimensional case as -<
(2.3.67)

102

Elastic plates

2.4. Case of transverse symmetry Assume now in addition that the geometry and elastic properties of the plate are sym metric with respect to the plane x3 = 0, i.e. C%kl(x,x3) = C%kl(x, -x3) and x3(x) = —x3 (x). Moreover the planes x3 = const are assumed to be planes of material symmetry, i.e. C**T = Cf«

=

0

.

( 2 A 1 )

The above assumptions can conveniently be put in the form

C O « ( ^ ) = c « « ( p - :_*1\

C3afr

t

c+{y) = -c-(y) = c(y), G+(»)=G_(y)=G(y),

=

Q3336

0 )

=

Q

(2.4.2)

yeY y3=y3.

Under the assumptions (2.4.2) thefieldsBo and E3 are even functions of y3, while the fields B3a0) and HiQ/3) are odd in y3. Hence fields AJSa0 are even in y3 and E2ia0 are odd in y3, see (2.3.21), (2.3.38). Consequently Ev = 0 ,

(2.4.3)

Fv = 0

or constitutive relations (2.3.51) decouple and problem {Phom) (Section 2.3) is decomposed into two problems: (i) the in-plane problem:findu e (HQ(CI))2 such that (■f/iom)

Ao f A°0X,1ea0(u)e^(w)dx = fpawadx V w € (/#(ft)) 3 , n n where p a = (G*)(p° + p°) + /i„ -< ba y ;

(2.4.4) (2.4.5)

(ii) the bending problem: find w G #£ (ft) such that Ao JDfx,1Ka0{w)K^{v)dx

= f(qv - rhQv7Q)dx V « £ i/02(ft) ,

(2.4.6)

(^

n where q=(Gi)(q++q.) + h,^^y, ma = (cG^){p% -pa_) . The underlying constitutive relationships have the form

Ma0 = M S ^ e ^ t * ) .

Ma0 =

hoDf^K^w)

(2.4.7)

(2.4.8)

Similarly, Eqs. (2.3.64) concerning the original plate decouple and problem {Pn) sim plifies to membrane and bending problems.

Three-dimensional analysis and effective models of composite plates 2.5.

103

Centrosymmetry of the periodicity cell

Apart from assumptions (2.4.2) assume in addition that the planes xa =const are planes of material symmetry. Consequently the plate material is orthotropic with respect to the system {xi). Let us shift the local coordinate system (j/i) to the centre of the cell, Fig. 2.5.1. Moreover, we assume that planes ya = 0 are planes of symmetry of the y cell, i.e. C" u (yi,ifi,,ifa) = Ci^(±yuiy2,±y3) GiyuVi)

=G(±yu±y2)

,

c(yuy2)

, =

c(±yu±y2)

(2.5.1)

Fig. 2.5.1. A centrosymmetric cell of periodicity Thus the cell y is composed of eight identical segments made of the orthotropic material. Such symmetry, called centrosymmetry, results in the following simplification of the final expansion of the solution (2.3.1) and (2.3.2) of the problem (P £ ) u% = e(uQ - y3w,a) + £ 2 [ e i V ) e v ( u ) + H ^ W " ) ] + 0(s3) , u3 = w + e'lei^e^u)

+ E^K^V)

Here ua, w, r]3 are functions of x, while 6J

M

+ %] + 0(e3)

(2.5.2)

.

', E\Xl1' depend on ( — j . The displacement

field (ua,w) are solutions to {P^m) and ( P ^ ) , Sec. 2.4, while 773 solves a subsequent effective problem. The result (2.5.2) differs from results found in Section 2.3 in that now we have (2.5.3) Va = 0 , V3 = 0 . The stresses are given by the formula a « = e[Ayo0ea0(u)

+ E^a0Ka0(w)}

+ 0(e2)

which does not differ from that valid in the general case, see (2.3.65).

(2.5.4)

104

Elastic plates

2.6. On computing effective stiffnesses The computational problem of finding the effective stiffnesses consists in solving the local problems (Py) (Eq. (2.3.13)) and (f£) (Eq. (2.3.34)). The unknown functions G{a0) and S ( Q "' can be approximated by the Galerkin method. Let {4>a)a=i be a basis in N-dimensional subspace Wu(y) of W(y). Assume addition ally that -< 4>a y= 0 to get rid of constants up to which solutions of problems (Py) are determined. We represent approximate solutions as

eia0) = e[a0)a<pa(y),

4 Q " = ^0)aUv),

a=i,...,N

(2.6.1)

+ Q«a0) = 0

(2.6.2)

and take trial functions as w = 4>b(y)- One finds K^e(^)a

+ Phi{a0) = 0 ,

K£z£0)a

where K% =< C^'cPa^j

Pb
y ,

(2.6.3)

Let us represent solutions to the algebraic equations (2.6.2) in the form e[a0)a = -k%P«a0) ,

E[a8)a = -kfxQl{a0)

(2.6.4)

l

where k = K~ . Substituting (2.6.4) into (2.3.24) and (2.3.50) one readily finds: ^ySaff

_

Q-,6a0 .

_pHyt) foob p>(a0)

E**

= < y3C^Sal3 y -Ql^kgP^

F** = -< fcC** >-

,

(2.6.5)

-Q^k^P^,

Dyia0 = -< (y3)2&5ae y -Q^kgQl^

.

We see that Galerkin's method does not violate symmetry properties (2.3.27) and (2.3.53). The main difficulty in implementing the above algorithm lies in forming the basis (<j>a). First, 4>a should assume the same values at opposite lateral boundaries of y. This property can be realized by identifying appropriate degrees of freedom, if one uses thefiniteelement method. We can, however, encounter difficulties in satisfying the condition -< 4>a y= 0 it is easy to satisfy if one uses the Fourier trigonometric representations and nontrivial to fulfil if one uses thefiniteelement method. The non-homogeneities of local problems (Py) can be shifted to the displacement bound ary conditions if both local problems are combined into the following form: Find Ua (PZ

such that 0)

= rmn{ay(v£0\v{?0))

ay(u^\u^ ) (

0)

l

v
|

+ y0ea) e W(y)};

(2.6.6)

Three-dimensional analysis and effective models of composite plates

105

here a,a,/3 = 1,2. The rescaled effective stiffnesses are given by Afx* = ay(uW\u™) , <

X

al3

Ff» = ay(u^\u[^) , Dfx» = ay^A™)

" = ay(u[ \«^>),

•

(2 6 7)

' '

Note that nonhomogeneity in (Py) follows from imposing displacements along lateral edges of y. In this way we enforce stretching, in-plane shearing, bending as well as torsion ofy. This algorithm can be computerized with ease in the case where y is centrosymmetric (Sec. 2.5), since symmetry conditions stabilize one quarter of y subjected to boundary displacements. 2.7.

Case of moderately thick periodicity cells

The modelling presented in Section 2.3 applies to thin plates of arbitrary periodicity cells Z. The assumption of thinness has the form: £n = maxlij" — xjl/diam (fi)
ca9 = C33a0/C3333

,

(2.7.1)

and the planes y3 = const are planes of material symmetry, i.e. (2.4.1) holds. The solutions to problems (Py) can be decomposed as follows 0(a»

=

QW + &a0]

g"^ = s + = ( a / J )

(2.7.2)

t

with e (tW?) = (0,0, - J c°0dy3) , The local fields 0 ( a / 5 ) and 3 find 9 (Py)

(Q

3 ( Q / ? ) = (0,0, - J y3c^dy3)

.

(2.7.3)

are solutions to the following modified local problems: ^ € W(y) such that

a(eia0),v)+ -= o vv G w(y).

{21A)

Find S e W(y) such that (Pi)

a(S{a0\v)+^y3C^vM5y=O

V»GVV(y).

(2 7 5)

' '

106

Elastic plates

Here The decomposition (2.7.2) follows from the following identities Ca0vvi}j = Ca^6v^s + c^C 3 3 '^,,,,

The problems (Py) can be interpreted as plate-type problems if y has the shape of a plate. Thus the Hencky theory of plates can be applied. According to this theory we represent the solutions of (Py) as follows

e(-*>(y) = 7**>(v) +

feZw»)(y) i

e

5 ^ ( V ) = ^ ( » ) + 2/3*1*%) , Similar constraints are imposed on the trial fields

^ ( V ) = X^(») •

v\(y) = ux{y) + y3
a0

a

r ( v ) = *"(»),

v3(y) = v{y).

a0)

(2.7.9)

a0)

The unknown functions: Zx \ X t>, O{ , Qx , y^, ux, (V) , ,^>3333o(Af<) °\(\pL) — ° "71* + L **3|3 '

-33 ,o337a~(V) , /"r3333'r(A/i) °2(A,«) — ^ "7|i + ° "3|3

n "7 U\\ U-'.IO)

are negligible. Eliminating the underlined quantities one can approximate the in-plane stress-typefieldsas follows: =a/3 /~
u

Hence

-a/3 na0kl~:(>>v) ^ A a ^ — f V ) ( 1 1 i n 2(AM) — ° ~/fc|f ~ ° "7|« • ^ • ' • » U

a

<(v> : = c^ejjf 0 * &#«&$ - <^0^c3333, aa0

na0kl-(W

~ fja0-,6~(^)

_ - j»0-A^3333

which introduces simplifications to the problems (Py) and to the formulae for effective stiffnesses. The kinematical assumptions (2.7.8) as well as stress assumptions (2.7.11), (2.7.12) make it possible to reduce the transverse dimension of y. Prior to formulating these re duced problems let us define the stiffnesses: c + (»)

/^a0Xn^ fiaPXp^ £)a0Xn\T _ c+(»)

/

Qa0Xtily\

e-(v)

Hx» = k J CX3^(y)dy3 , C-(V)

1 1/3

(h)2

Jfc = 1 ;

dy3 (2.7.13)

Three-dimensional analysis and effective models of composite plates

107

and the bilinear forms: 6(u,v) = (Ax^suMsvMll) , dH(u,v) = (D^u^v^} g2(u, v) = (Ha0ulav}g)

eH(u,v) = {E^u^v^) ,

fli(u,u)

= (H u0vla) ,

g3{u, v) = (Ha0uav0)

,

,

ali

(2.7.14)

,

2

where u, v e H^,T{Y) , u,v£ H^Y) and k is a transverse shear correction factor. We recall that angular brackets (•) mean here the averaging over Y. Let us define the rescaled membrane stress resultants N

iU = AX^T^0) + E**Z!$ + Ax»"0 ,

N

2&0) = ^ W # ! | f + EX^SS>{°0) + Ex^0 ,

moments:

M

Ha0) = E**Tffi

+ DX^SZ§0)

M

2(a0) = EX^6U(°0) + Dx^s^0) and transverse shear forces:

+ Ex^0 , + Dx^0 ,

(2.7.15)

(2.7.16)

Substituting (2.7.8) - (2.7.11) into (2.7.4) and (2.7.5) we reduce the transverse dimension and formulate the following two-dimensional problems find X<*« e HHvjY)

= H^{Y)2

x H^Yf

x H^Y)

(/?) such that (NX0)ux\, + < ^ , V A | M + < 3 W f | A + ^ = ° v(tt,v>,«))€ffHjr).

(2 7,8)

-

Here X M ) = ( r ( « « ] Z(a/3)i £(<*)) j

^ M ) = (yM^Wl^W)) _

( 2 ? 1Q)

The local problems (P£) are well posed. One can prove that fields T(a0), Ul°0), X
The homogenized problem assumes the form (Phom), Section 2.3, with effective stiffnesses given above. The density of the elastic energy of the effective plate is given by Wh(e,n) = (Afx»ea0e^

+ EZ0X»KXliea0 + Pf3^Ka0eXll

+ Dah0X,iKa0K^)l2 . (2.7.21)

108

Elastic plates

To prove the symmetry conditions required jia0\n

_

2A/iQ/3

f*

jja&Xii _ p\fu*0

n

x

Kf "

n

0aX

=K

f\a/3X^i _

n

0a x

» = K " ,

A\na0

n

,— _

n

K =

i-\«\

A,E,F,D,

one should form appropriate identities following from the variational equation of (.PjJ) and combining them with (2.7.20). Finally one arrives at

Afx" = (A^^-bil**®,!*™) - [eH{Zix"\T(a0)) + e„(T(A">, Z(a0))] - dH{Z{a0\ Z(A">) - g3{Z(Xfi), Z{a0)) pw

= {pa0x^

_

~b{u(a0),T^)

+ eH(&aP),Z^)}

- dH(&°0\ &W) -

EXtux0 = {Ea0X») - b(T(a0),UM) a

+ e w (Z( «,Lr

(V)

[e„(&af3\T^)

-

g3(&a0\Z^)

- [ ^ ( T ^ , *<*">) (a0

)] - dH{Z \^x^)

+ g2(x{a0\x^), Dfx» = (&**) - b(UM,U{a0))

-g3{ZW\&x*)

- [e„(LT(V), *<<>«)

+ eH{*™, U(aP))} - da(*M,*™)

-

g3(*{X"\*{a0))

+ S 2 (x M ) ,X ( A " ) ).

(2-7.23)

Due to symmetry of the bilinear forms b(-, •), g2{-, ■), g3(-, •)> dn(-, •), and e«(-, •), the stiffnesses (2.7.20) satisfy symmetry conditions (2.7.22). Note moreover, that the effective elastic potential WA can be rearranged to the following form Wn = H Ca0X»%0%» y+A^

Ca303%3%3 y)/2 ,

(2.7.24)

where iap = a}$(y)exll + e%(v)KXll,

2 7 a 3 = aXfi(y)e^ + e*{y)KXll,

(2.7.25)

with <% = W)

+ T ( 3g + foZgg, ,

<& = &*

aV = Z™ + X™ ,

(2.7.26)

#

(2-7-27>

= *™ + Xla" •

The tensors Ca0X>1 and Ca303 are positive definite. Hence Wn is non-negative. Assume that Wn = 0. Hence 7a0 = 0 and %3 = 0 for all y G y and consequently (7a/3) = 0, (7 Q3 ) = 0. Due to periodicity of X'f0'1 one concludes that eQ/3 + y3Kap = 0 for

Three-dimensional analysis and effective models of composite plates

109

almost every y3 g (c~(y), c+{y)). This implies that ea0 = 0, ttap = 0. Thus WA defines a positive definite quadratic form of (e, K). The effective stiffnesses of the original plate of Section 2.1 are given by

AfA" = e0AfA"

pa0^

_

£2pa0^

(2.7.28)

The homogenized constitutive relations assume the form ttaffX/i

_ paP^n

H/x

h A*

.

C

rjQ/SAfi^/i A*.

(2.7.29)

where e^ = eQ€Xli, K$M = KA^Let us examine consequences of transverse symmetry. Then E defined by (2.7.13) van ishes. Problems (Py) decouple into membrane and bending-type problems: find T(a0) g H^r(Y)2 such that

(Pkv) b(T^a0\u) + {A^u^})

(2.7.30)

Vu g / / ^ ( V ) 2

=0,

Find (*(Q/3), x(a/3)) 6 ^ r ( y ) 2 x H^r(Y) such that (j§y)

d H (* ( < *\ y>) +5i(¥>,X(Q/5)) + 93(* W,, ,vO + (0 7fo V 7 |^> = 0 , 92(x(Q/3),™) + ffi(*M,,H = 0 ,

(2.7.31)

■£>£)

r(<*«

andZ^>=O,CTw=O,JrP,=0. The non-zero effective stiffnesses are: ^O(3AM _ /^a/3Aji\ _ ^/ y(a/3) J.(A/J) \

f)f * = (£)«/>*«) - d/,(* (A *\ $ (a/3) ) - p 3 (* ( V ) , * (a/3) ) + .920 .(a/3)

(A;.)

(2.7.32)

The homogenized constitutive relations are decoupled: ■a/3 _

AaffX/i h C AM

NT = A

M°"

= za,a/3Au

h AM

'

(2.7.33)

2.8. Case of thin periodicity cells. Derivation by imposing Kirchhoff's constraints If periodicity cells are thin plates themselves one can neglect their transverse shear defor ,a(<»0) mations. Thus the transverse shear deformations associated with displacementfields© (a/?) and S in problems (Py) and (Py) respectively can be neglected, i.e. U

A|3

■eg? = o,

~(a0) ~(a/3) _ „ —A|3 "•" ^3|A ~ u '

110

Elastic plates

The same assumption concerns the trial fields: ^ | 3 + v3\\ = 0. The above requirements are fulfilled if displacement assumptions (2.7.8) are modified by setting

*r=-$»

A«P) = - -X,("» "A - ^|A '

*A

_

=

X|A

'

(2.8.1)

Consequently the displacement assumptions (2.7.8) are reduced to the form of Kirchhoff. As in Section 2.7 we assume that planes y3 = const are planes of material symmetry, i.e. (2.4.1) holds. Let us introduce the strain measures 1, e

a/?(«) = ^Va\0

+ V

K^M

0\^ '

= -«w

Next we define new bilinear forms eK(w,v)

= (E°^(y)KvXli(wya0(v))

,

dK(w,v)

= (D^iy^wKpiv))

,

(2.8.2)

and introduce the function spaces HK.pcriY) = H^iY)2

x H^(Y),

H^(Y)

= {v € H^(Y)

The space H^r(Y) has been defined in Section 1.3 and H^Y) The local problems (Py) are rearranged to the form: find Xf> (P°K,Y

| (t,) = 0} .

= W™(Y), cf. (1.3.46).

e HK,per{Y) such that

("%0)W + <

VA»>

V (U,W) £ HK^Y)

<2-8-3>

=° ,

where X{2a0) = (U(a0)

^ h

and quantities N^a/}), M^a0) are changed accordingly. Let us write problems (P£ y ) more explicitly: find ( T ( a « , X< a «) G HKMY) b(TW\u)

such that + (Aa^(y)e^(u))

+ iK(XW,u)

=0,

1

(P K,V)

eK(w,T^aP))

+ dK{X«*>,w)

+ {Ea/3X»(y)KvXtl(w)) = 0 V (n,to) Z HKj*r(Y)

Find (U(a0\X{a0))

(Plv)

b(U(al3\u) eK(w,UiaP))

6 HKiPer(Y)

+ eK{X^\u) + dK(X{a0\

(2.8.4)

.

such that

+ (E^iyy^u))

= 0,

(2.8.5)

w) + ( D ^ ^ ( y ) / c ^ H > = 0 V (u,w) £ HKiPer(Y)

.

Three-dimensional analysis and effective models of composite plates

111

Problems {Pfcy) alt uniquely solvable. The formulae (2.7.20) for effective stiffnesses assume the form (the subscript h is now replaced by h) : ^aPXn

__ i^X/ia0

_|_ ^A(i7ij>(a^) _

ga0k»

_ lgXpc.0

+

Dfx» = (DW

fiXiiyS

j((a0)\

_ gXn-yt J-aP1\ _

^X^S(j(a0)

- Dx>'",ix\°s)) ■

+ E^'Offi

Taking u = T^\ w = X in (PlKY) and u = UM, w = x(A,j) in {Ply) one obtains identities which combined with (2.8.6) lead to new formulae for effective stiffnesses ^a/3A^ _

M Q J J V \ _ ^lrp(a0)

rp(X^.)\ _ U

(Jfta/J) ^ < y ( V h

+ e , f ( ^ l , T M ) | - <**(*«*>, *<**>) , ^ w

ff*

=

^a0x^

= ( £a^)

_ 5(T(«flf #<*">) _ [g* (*w», t>(Art)

a0

M

_ b(rf \T ) - [eK(X^,U

(a0

la/)

+eK(X \T^)}

M

- dK(x ,X )

Dfx» = (&**) - b(U(a0),UM)

(2 8 7)

{a0)

-

)

, [eK(X^,tj{a0))

+eK(x{a0\U(a0)))-dK(x^),X(X"))By virtue of symmetry properties of bilinear formsfe(-,■), dx (•, ■) it is readily seen that the effective stiffnesses (2.8.7) satisfy symmetry conditions (2.7.22). The effective potential (2.7.24) (set h instead of ft) can be written in the form Wh =< Ca8X^a0^

>■ /2 ,

(2.8.8)

and 7a/J are given by (2.7.25)i with a

a0 ~

d

{c,d0) + 1{a]0) ~ y*A\a0

'

e

a0 ~ ^ ( a ^ )

+ U(a\0)

2/3X|Q(3 •

U-B-V)

As previously the effective elastic potential is strictly convex. The effective constitutive relations assume the form (2.7.29), where the subscript h should be replaced by h. The effective stiffnesses of the original plate are Ah = e 0 A ,

Fh = [e0)2Fh,

Eh = {e0)2Eh,

Dh = {e0)3Dh.

(2.8.10)

Elastic plates

112

Let us consider consequences of transverse symmetry. Problem (P^ Y) turns out to be the same as {Psy) of Section 2.7. Problem (Pf- Y) reduces to: find x(Q/S) G H^.(Y) ia0)

(PL Ksyi

dK(x

such that

a0X

,v)

+ (D

"(yWXlt(v))

=0

(2.8.11)

The homogenized constitutive relations have the form (2.7.33) with h replaced by h; Ah = Ah and Dh is given by Dfx» or

Dfx"

= e30 [(D^»)

- 4 ( X ( Q « , X™)}

= e30 dK(XW> - p<«», XM

,

- P(A"') ,

(2.8.12a) (2.8.12b)

where p(a0)

r

Hence p

(a/3)

Sa50

=

w u

\tTK

aK

are polynomials of second order in t/i, y2- In other words Df*

= {D^[6°5%

+ «*,(x<«*>)]## + < , ( x ( V ) ) ] > ,

(2.8.12c)

where D = {eo)3D represents the bending stiffness tensor of the original plate. 2.9.

Case of transversely slender periodic ity cells of constant thickness

We consider a plate of constant thickness with Z-periodic variation of elastic moduli. We assume in this section that the plate thickness |x;|" - £31 is much greater than in-plane dimensions Z*. The periodicity cell Z is said to be transversely slender. Planes x 3 = constant are not necessarily planes of material symmetry. Although this case is comprised by the asymptotic method of Sections 2.2, 2.3 it seems reasonable to derive approximate formulae for effective stiffnesses in order to avoid difficulties associated with solving the three-dimensional local problems (Py) of Section 2.3. The results of this section will be justified in Section 2.10.3 by the T-convergence method. Step 1. To find effective stiffnesses we first introduce a small parameter e by considering a sequence of plate problems with eV-periodic elastic moduli. Transverse geometry is kept e-independent. Thus we apply the scaling, cf. (2.2.1) - (2.2.3), x

$(x)

= x3 ~* x3 <

l

a^

Z-^eY,

el

<> 1% = £ok ,

Z = e0Y;

r«±(i)~.ri(ar),

(2.9.1) {

b*-^ b .

The elastic moduli are eY- periodic in x C f (s,is) -

C

u

( p x3) ,

and C ijW (-, x 3 ) are Y- periodic.

Cik\x,x3)

= C*H ( | , z 3 )

(2.9.2)

Three-dimensional analysis and effective models of composite plates

113

The first step of the algorithm is to perform homogenization of the material based on the scaling (2.9.1), (2.9.2). In the next step we reduce the third dimension either by the asymp totic method of Friedrichs-Dressler-Goldenveizer or by imposing stress-displacement con straints of Kirchhoff type. Let us pay attention to the first step. We shall outline briefly the process of smearing out elastic properties in i i , 12 directions. The variational equilibrium equation has the form / <Ti>eij(v)dx = / [p> Q (x, x%) + q+v3(x, x$)}dx n

B a

+ f\p _va{x,x3)

+ fb{ Q

+ q-V3{x,x3))dx

n

vt(x)dx

V» £ V0{B)

(2.9.3)

B

where a?' = C « H ( p i 3 ) e « ( t i ; e ) .

(2.9.4)

To find effective behavior of the plate considered we apply the method of two-scale expansions: w*

=wt-°\x,x3)

+ £iu(1)(a;,X3,?/) + £2ww(x,x3,y)

v = v^0){x,x3)

+ ev{1\x,x3,y)

+ e2v{2){x,x3,y)

+ ... , +

where y = x/e and «;'*'(x,x 3 , •),«'*'(a;, £3,-) are V-periodic functions for fc > 1. Expan sions of deformations read: ea(}(w*) = eQ/3(u,<°>) + w{"0) + 0(e) , e33{w<) = e33(wM) + 0{E),

2ea3(w<) = 2ea3(wM) + w^a .

(0

Let us substitute v = v '(x) into Eq. (2.9.3) and pass to zero with e. One obtains Jofa(vM)dx

= J\p°+v^(x,

x+) + PZv^(x,

x3)

(2.9.6)

n

B

+ q+v3{x,x^)

+ q-v3{x,x3)]dx+

/ (b'jv^dx

,

with

(2.9.7) ijkl

0% = C ekl(wM) Note that o?et](v)

+ CT^w^

+C^tug .

„(0h , „a/3 (1) , „ ? 3 „ 0 ) = a 0 %(«<°>) + a S ^ + ofv™ + 0(e) .

(2.9.8) (2.9.9)

Hence < ^ e t » > = ofaivW)

+ (ofv™

+ ofv™)

+ 0(e)

(2.9.10)

Elastic plates

114 Passing to zero with e in (2.9.3) and taking into account (2.9.6) one finds J(e?v§ + °?vU)dx = 0

(2.9.11)

B

forany«(1>(x,-)G^(^)3. By taking vw = <j>(x) ■ v(y), <j> e U(B) one can localize Eq. (2.9.11): (°?v% + ^ l )

=0

V v e HlAYf

.

(2.9.12)

Hence we may write v>? = e[kl)ekl(wM) ,

w «"

= #Wekl(wW) ,

(2.9.13)

where functions 9^ ', #(*'' are solutions to the following local problem find (Q(kt) ,tf<w>)e H^iYf

such that

QA

<[C " "eW + c*»«a [f + c * > a W ) = o,

(P?l))

for any v € H^iY)3

(2.9.14)

.

The solution of the above problem is determined up to an additive constant. Substituting representations (2.9.13) into (2.9.8) one finds:
(2.9.15)

cy M = c3kl + c t j V e ^ + c"^3t?[f.

(2.9.16)

with Hence ffjf= CyHcH(ti;«0') ,

C«*' = ( C f V

(2.9.17)

The vanational equilibrium equation (2.9.6) along with the constitutive relation (2.9.17) constitute the homogenized three-dimensional elasticity problem. Note that if x3 =const are planes of material symmetry (cf. conditions (2.4.1)) then problem (PY ) decouples into: - the in-plane problem: find 0 ( *'' e H^Y)2 such that

(pY) {[ca^e^

+ c^k'}val0) = o,

vV€
(2.9.18)

- and the transverse shear problem find dm e H^iY) (PY)

((Ca3X»ti\kl) + Coau)vm)

= 0,

such that V v3 e H^Y)

.

(2.9.19)

Three-dimensional analysis and effective models of composite plates

115

Consequently e^Q3) = 0, ^a0) = 0, i?(33) = 0. Hence Ca0j6

=

C

a

W

+

Cap\»Q(,c,0)

^

C a/333 =

£,a303 __ gc.303 , (ja3X3fl(03)

3

^33

C^^Q^J

+

£3333 _ r-3333 , £-33.^(33)

Ca30X

0

C

cx3a

=

(w

(2.9.20)

Q ^

=

and o? , af depend upon © ', while a? depends on tfC33'. Equations (2.9.6) and (2.9.17) constitute a 3D problem (f? D ). Step 2. Having found the problem (P2D) one can now impose the condition of the thick ness being small as compared with diam fi. This process of reduction of the transverse dimension can be performed in two ways: a) by imposing Kirchhoff constraints, b) by asymptotic analysis. Both approaches lead to the same plate equations with effective constitutive relations of the form (2.7.29) and with effective stiffnesses given by x

/

r\a0\n

3

1

3

h

C?*(*3)

x3 = x3- -{x^ +x3

dx3

(2.9.21)

2

L(* 3 ) J

1. since the conditions c* = const imply -< y3 >-= -(c+ + c ). The tensor Cu refers to the plane-stress state and is constructed as follows. Let us write the homogenized constitutive relations in the form a/3 'I) ~

IQ/?A/X ,

= c:f ) w

,

^M"

1

ra0k3^

"^

j

3

k

3

,

*h -w,, =O «(,

m3 ^M ^ , , -+r ^C i ,h 7*3,

(2.9.22)

3

where 7^3 = 2£^3, 733 = £33. Let us require that at = 0, which makes it possible to eliminate 7*3 : (2.9.23) 7*3 = - c t C f ^ V . with

C'„ = (C^' 3 )

c" = [C, Thus where

Aa/3A/i

(2.9.24)

a0 _ s^aflXn

(2.9.25)

s~ia0\ti

(2.9.26)

(~ia0i3~t) *~<j3\fi

In the case of planes x3 = const being planes of material symmetry, the formula for C^ reduces to ^a0\n

I,

s~ta0Xn

- - 1 ,

ra/333/ / ^i3333\-l/-.33A^ c

- ^c\ D

^

)

t,

(2.9.27)

which coincides with formula (2.7.6). Note that in this case tensor C^ depends only on e<«>. A rigorous justification of formulae (2.9.21) will be given in Section 2.10.3.

116 2.10.

Elastic plates

T-convergence

and justification

of three models of thin,

transversely

inhomogeneous and anisotropic plates with constant thickness The method of assessing effective stiffnesses presented in Sections 2.2, 2.3 as well as the formulation of the effective problem concern a general shape of periodicity cell Z. Thus these formulae can be viewed as reference formulae. Further models were concerned with specific shapes of Z. Results of Sections 2.7, 2.8 are applicable to flat cells, while results of Section 2.9 refer to cells Z that are transversely slender. The aim of this section is to explain the meaning of the effective models of Sections 2.2, 2.3, 2.8 and 2.9 approximate the overall properties of periodic plates. It turns out that these models describe a limit behavior at dimensions of Z tending to zero. To show it two small parameters e and e will be introduced, the latter controls the transverse dimensions of Z. In the present section the plate thickness will be assumed as constant: x3 and x3 are fixed constants. The plane x3 = 0 determines the middle plane of the plate; x3 = -x3. We scale the thickness X3 = —x3 = —ec ,

e>0,

and consider the plate with periodicity cells 4 = ^ x

(-ec,ec) ,

Y = (0,h) x (0,l 2 ) ■

The present section is arranged as follows. In Section 2.10.2 we prove that the effective model of Section 2.8 can be obtained by passing to zero: first e —► 0 and next e —» 0. In Section 2.10.3 we show that model of Section 2.9 follows also from passing to zero: first s —* 0 and next e —» 0. A simultaneous passage to zero: (e, e) —> (0,0) is considered in Section 2.10.4, which results in formulae of Section 2.3. The convergence proofs given here justify the asymptotic expansion methods used in Sections 2.1 - 2.9. 2.10.1. Basic relations We shall assume that the plate thickness is constant but do not assume any material sym metry; in particular (2.4.1) is not imposed here. The domain of the plate in its undeformed configuration is denoted by Be = fi x ( - e c , ec). Plate faces are T± = Q x {±ec} and the lateral boundary is: T ' = T x (—ec, ec), T = dSl. The elastic moduli are here scaled as follows, Cf\x) =i c « « ( ^ , ^ ) , (2.10.1) where C"jA:'( • , x 3 /e) are V-periodic; e > 0 scales the dimensions of the periodicity rectangle eY . To work with an e-independent domain of the plate we introduce new variables za = xQ , z3 = x3/e , (2.10.2) hence 23 G (—c, c) . Now we set Cf\z)

= C ^ ( ^ , z3) ,

C£W(z) = \ciH{z)

.

(2.10.3) (2.10.4)

Three-dimensional analysis and effective models of composite plates

117

As usual, we make the following assumptions, cf. (2.1.7) &*"&)

= C'ikl(y)

= Cw%)

(2.10.5)

for a.e. y € y = Y x (—c, c); C,jkl 6 L°°(^) ;

(2.10.6)

there exists a constant m > 0 such that 3

Ve 6 E3S

C>kl(y)etJekl

>m £

ei]it] ,

(2.10.7)

for a.e. y ey . The plate is subject to body forces 6 = (&') and tractions (p°/e,q + ) = g + , (p"/e, q~) = g~ on T^_ and Te_, respectively. The plate is assumed to be clamped on T e . Let 6(x) = b(xa, x3/e) belong to L2(Be)3 = \L2(Be)]3 while g+ = (g'+) and g~ = (g!_) are elements ofL 2 (Q) 3 . The space of kinematically admissible displacement fields is defined by V0(Be) = {v G H\Bef l

3

If v e H (Be)

| v = OonTe} .

(2.10.8)

then the strain tensor is given by

For fixed e > 0 and e > 0 the functional of the total potential energy assumes the form Ja(v)

= \ JcZk%(v)exu(v)dx

- Jb%dx

- J ~pa±vadT - Jq^dT

,

(2.10.9)

where fpa±vadT

= jp%vadT + LtvadT

..

and similarly for the integral of q± . The minimum principle of the total potential energy means evaluating Jec{v) = inf{ j„(t>) | * € V0(Be)} .

(2.10.10)

M

On account of (2.10.5) - (2.10.7), v exists and is unique. From now on until the end of this section we shall work with the domain B = Q. x (—c, c). To achieve this we proceed in the following fashion. Let v G Vo(Be), then the function v defined by va(zi) = -va(z0,ez3)

,

v3(zi) = v3(za,ez3)

,

(2.10.11)

118

Elastic plates

belongs to V0{B) = {v€H1{B)3\v

= OonT},

T = rx(-c,c).

(2.10.12)

Moreover we have eza0(v) = -exal3(v) ,

e*a3(v) = e*a3(v) ,

e$j(») = ee^v)

,

(2.10.13)

where

«•>-(£♦£)"• By taking into account (2.10.1)- (2.10.4), (2.10.11) and (2.10.13) in (2.10.9) we obtain the rescaled functional of the total potential energy J«(v) := Jes(eva,v3) = \ JciJkl(Q'e'(v))ti(Q'e'(v))kldz

- Le(v) ,

(2.10.14)

B

where v 6 V0(B) while Q e : E3S -» E3, is defined by (Qee)a0 = ea0 , (Qec)Q3 = \eai , (Qee)33 = ^ 3 3 •

(2.10.15)

The rescaled loading functional has the following form Le{v) = fe{ebava + b3v3)dz + I'(p%va + q+v3)dT + I'{pa_va + q~v3)dr . (2.10.16) B r+ r. We observe that the rescaled stress tensor
+ ^Ci^ef,

,

(2.10.17a)

where je(z,Qee) = j(^,z3,Qee)

= Jc"' w (^ ) 2 3 )(Q e e) 0 (Q e e) w .

(2.10.18)

Prior to passing to zero with the "small" parameters e and e we shall formulate a lemma playing a crucial role in our subsequent developments. Lemma 2.10.1. Let v" 6 V0(B) be a sequence weakly convergent to ve € V0(B) in Hl(B)3 as e —> 0. Suppose that there exists a constant K > 0, independent of e, such that Jj [j, z3 , Qeez(v"]\ dz
(2.10.19)

Three-dimensional analysis and effective models of composite plates

119

Thene^(v £ ) = 0 and *£ = « « ( * * ) - * s | J ,

v£3 = wl(za),

(2.10.20)

where uca G H*(Q) and wc3 e H$(Q). Proof. From (2.10.7), (2.10.18) and (2.10.19) we obtain m ||

ffe!(i)a)

||2

where

II Cee2(«K) ||2

= || (<,(«")) "2

+ ji II «.(«")) ||22(B), +1 || *(««) ||22(B) Hence

^'^(^lU ^^i. '■""

ie l l <&(«") i u , < * i ,

e

where /G does not depend on e. Consequently we have dvc g£=0,

< 3 (t> £ ) = 0 .

(2.10.21)

Now we readily obtain (2.10.20) by applying Theorem 1.4-1 and Theorem 3.3-1 due to Ciarlet (1990), cf. also Destuynder (1986a). a Suppose that va solves the following minimization problem (Pec)

Jet{if')

= al{Jet(v)\veV0{B)}.

(2.10.22)

In other words we may write Jet(v")

< JU(V)

VW G V0(B) .

For v = 0 we have l

-Jj\^,z3,Q°e>(v«)}dz
i

B

i e

J

< |L e (««)| < K2 (|| « « || i2(B)3 + || « « || t 2 ( r + ) 3 + || t , " | | l 2 ( r ) 3 ) .

(2.10.23)

Thus m 2

^(v")

'

||» < tf2 (|| « - || l2(B)3 + || r - || i 2 ( r + ) 3 + || v' "^(B.E?)

\\LHr_)3)

,

120

Elastic plates

where K2 > 0 depends onfi,b and g± but not on e. As in the previous case, for 0 < e < 1 we write

1 II «■(«") H2L2(,E3, < & (ll « " IL,(B)3 + II « " I U + „ + II « " || l2(r _ )3 ) 6 << K / 2 "II v „ , ,

Korn's inequalityfinallyyields

^II^II^^^II^IUw . M

Thus the sequence v is bounded and there exists a subsequence, still denoted by ue£, weakly convergent to v€ in Hl{B)3 and thus strongly convergent in L2(B)3 . The boundedness of the sequence v1* implies now the existence of a positive constant K appearing in (2.10.19). Lemma 2.10.1 introduces Kirchhoff's rescaled displacement fields. It is thus natural to introduce the following subspaces of the space H1 (B)3: VK(B) = = {v <= H'iBflvc

{veHl(B)3\e*3(v)=0}

= ua- z3~—, v3 = w,ua(E H\Sl), w £ H2(Q)} (2.10.24)

and V°(B) = {ve VK(B) I ua e H'0(Q),w € i/o2(")} •

(2.10.25)

2.10.2. Justification of the effective plate model of Sec. 2.8 by passing to zero: e— ► 0 and then e —» 0 The aim of this section is to put forward a rigorous justification of the homogenized plate model derived in Section 2.8.1 by imposing displacement constraints (2.8.1). Having this in mind we pass now to the study of T-convergence of the sequence of functionals {JM}e>o,£>o defined by (2.10.14). We shall first pass to zero with the thickness parameter (e — ► 0) and next with the parameter e characterizing the periodic structure of the plate. The result of this passage to the limit is formulated in the following form. Theorem 2.10.2. For anyfixede > 0 the sequence of functionals { Jec}e>o is T-convergent in the strong topology of the space L2(B)3 (weak topology of Hl {B)3) to the functional

Uv) = [ \^^^^dz

~ L^ if » e V"W

(2 1Q26)

^ +oo otherwise, where Cfx"{z)

= Ca0X"(—,z3)

,

Ca0Xli = Co0x,i - C^dijC'3^

,

(2.10.27)

121

Three-dimensional analysis and effective models of composite plates £(*>) = [plvadT + jq±v3dY

-I

M + P - K + («?+ +q~)w-

(Pa+ -

Va-)c~W

= J[{p% + V°-)ua + (q+ + q~)w + <™-^(pa+ ~ P°-)W ,

(2.10.28)

and c = (dij) is the inverse of the invertible (3 x 3)-matrix (Ci3j3), v = (ua — z3-—,w), uae H^(Q),w€ Hi{Q.). Proof. We divide it into several steps. (i) The sequence of functionals {Le}e>o is continuously convergent. In fact, for any se quence {ve}e>o C L2(B)3 strongly convergent to v 6 L2(B)3 we have \Le(ve) - L(v)\ -> 0 as e —> 0. By applying Proposition 1.3.26, we conclude that it suffices to study the T-limit of the sequence J^ defined by ■/«(«) = Jje(z,e>ee*(v))dz

,

v g V°{B) .

(2.10.29)

B

Since the space (Hl(B)3, L2) has a countable base, therefore the r-limit of the sequence of functionals {Ja}t>a exists, cf. Th. 1.3.21. We claim that it is given by the functional ■£(«) = \ jCf^{z)el0(v)e\li{v)dz

,

v € VK(B) .

(2.10.30)

B

(ii) It will now be shown that for any v e Hl (B)3\Vj<{B) we have fc(v) = +oo.

(2.10.31)

In fact, for any such v , at least one of the strain measures e^v) does not vanish. We may assume that e|3(v) jt 0 ; the two remaining cases can be treated similarly. There exists a sequence {«e}e>o C Hl(B)3 such that ve —> vir\L2(B)3 and\imfee(ve) e—»0

fe{v). On account of (2.10.7), || ez{ve) \\ . < const. Consequently, e|3(we) ->■ v 2 33( ) weakly in L (B) when e —» 0 and, by the lower-semicontinuity of the norm

e

0
> lim II QV(w e ) ||2

where m is a constant appearing in (2.10.7).

> lim ^ || ez{ve) f

= +oo ,

=

Elastic plates

122 1

(iii) Let c := C

. It means that (2.10.32) (2.10.33)

Now we are in a position to prove that for any u £ VK(B) we have (2.10.34) B

dw dw Let v = (ua - z3^-,w) 6 VK{B) and take a sequence ve = (ua - z3^-, w + e3£). aza aza where £ € H1(B). If v € V$(B), then £ is a function from the space V$(B). It is evident that ve —* v in L2(B)3 when e —» 0. Moreover OZ\

<£*(»)

Q e e 2 (v e ) = .

e2^dz-i

e2K dz2

e2^ dz2

dz3

The definition of the T-limit yields J>) <

e—0

\imf(Geez(ve))

= lim

sup

/[(GV(««)) y o y -

= lim

sup

jWe'iv'))^

- k

< lim

sup

f{ao0elJv)

+ 2e2^-a°3

0

oo

'^
U]k^ak'}dz w

ajV -

U3]3oao»]dz

^a

+ e ^ - a 3 3 - i c ^ ^ ^ l d * = lim

[[°a0e*a0(v)

sup a

-lc*Q0Xtio°^}dz

+

Km

sup

2

3

/7 2 e |f<7<' + 023

B

= (

IJcf^el^ei^dz

if a a = 0,

B +00

otherwise,

where C^JAM is the inverse of the matrix CfA/1, see step (v) below.

(2.10.35)

Three-dimensional analysis and effective models of composite plates

123

(iv) We shall prove the inverse inequality to that specified by (2.10.34). For any {ve}e>0 H\Bf we have Jl{v<) =

sup

/"[(GV(«e))ya« -

C

U^o^dz

B

>

sup aa'i33=0 =0

sup

/[(CTe'(t» e ))ya« -

Ujkl^ak'}dz

B

/V'4K) - ^

w

a'V]dz.

(2.10.36)

tTeCT

i3

(7 =0 e

X

Suppose that {v }e>o C H (B)3 is such that u e —► v in L2(B)3 strongly when e —> 0 and lim Jl(ve) = •/,?(«). Then, by using (2.10.36) we infer that e—*0

fe(v)>

sup

fwa0e^(v)-l
(21037)

(v) It remains to prove that the matrix C £ , given by (2.10.27), is the inverse of (c^/3A/J), where (c^ w ) = ( C £ ) _ 1 . Firstly, however, let us justify (2.10.27). For stresses and strains interrelated by the constitutive relationship:
,

3

and knowing that a* = 0 we get ff«* = Cfx»t^

,

(2.10.38)

with C t defined by (2.10.27). The derivation of this formula is similar to the derivation of formula (2.9.26) and hence is omitted here. We observe that (2.10.7) implies that both (Ca0XtL) and (Ci3>3) are invertible matrices. To prove that (Ca/JAM) is the inverse of (C^A,,), we calculate:

= C " * C ^ - C°0i%(C»klCktX»

~ 2 C ^ % 3 A „ - C>333C33A^)

where we have used (2.10.33) and the relations c = (C*3*3)"1, 5 3 = 0. Similarly we find

which concludes the proof.

D

124

Elastic plates

The properties of the matrix (C yW ) imply that there exists a positive constant M such that for a.e. y 6 y we have 3

(2.10.39)

for each t 6 E', Both sides of the last inequality represent (strictly) convex functions. Hence the following coercivity condition readily follows, cf. Th. 1.2.7(v) (F3)2

cyufoJTOT* >jji£ for each T € Ej and a.e. y € y. Thus the matrix

(C^A^)

is also coercive:

(TQ")2 ,

cWvJTT*" > — ^

(2.10.40)

(2.10.41)

o,/9=l

for each (Ta0) G E£ and a.e. y € 3>. Similarly we obtain ci]kl{y)TiiTkl < - T

(rj)2.

(2.10.42)

<J=1

Hence i

2

(T°0)2 .

CapMT*!* < - Y

(2.10.43)

Q,/3=l

The last relation readily yields i

Ca^(yK0eXli >mJ2 («<*)2 .

(2.10.44)

Q,/3=l

for each (eQ^) G E2. and a.e. y € y. It is worth noting that m is the constant appearing in (2.10.7). Problem (Pe) and its dual Having determined the T-limit of Jc (e > 0) we find it useful to study its mechanical properties. Towards this end we formulate the following minimization problem. Problem (Pe) {e > 0 and fixed) Find Jc(v<) = mi{fe (v) - L(v) | v e V£{B)} .

(2.10.45)

The property (2.10.44) and the continuity of the functional L ensure that the minimizer ve = (uca - z3dwc/dza, wc) exists, where uca 6 H^(Q) and wc € H$(Sl). Then we have

Three-dimensional analysis and effective models of composite plates

/ V ) = I JCfx»(z){e*al}(u*) + zzKUwS)WexMe) Z

or

125

^{™e)]dz

+

B

J £ V ) = Jw€[za,z3,ez{u£),K*{wc)\dz

,

(2.10.46)

where ue = {uca), ^0(w) = -cPw/dzadzp. Moreover

Nf = Af ^ ( u ' ) + Ef^^xif)

,

(2.10.47) (2.10.48)

where

F£a/3A"

p

= E^

and

Xfx"(z) = Xat3X»{zle),

X e {A, E, F, D},

with {z\,z2)=

/

CfA"(z)dz3

Z3

(2.10.49)

Note that A, E, D have already appeared, see (2.7.13); moreover eA„(«E) = e^(u £ ) ,

«v(w e ) = K^(W £ ) .

The potential VV£ has the form W,(za,e>K)=1-(Afx^a0eXll

+ 2Ef^ea0KXlI + Dfx>'Ka0K^); e,K 6 E* .(2.10.50)

Properties of the matrices A, E, F, D -T

(i) E = F , where "T" stands for the transpose of a matrix. (ii) Their components belong to L°°(fi). (iii) Xa0X" = X"aA" = Xx^, X e {A, E, F, D). (iv) Aal3X»eaf}eXli + Ea0X»{ta0PXli + evpQ/3) 2 2 + Da0X,ipappXll

> -m

^

provided that c = 1; e, p € E^ and m is the constant.

(€a^£a/J + PctfAtf) ,

(2.10.51)

Elastic plates

126

Proof. Thefirstthree properties are evident. To prove the last one, we calculate c

c

2

a(SX

/ C

/ (€<*0 +

»{eap + z3pa/3)(cAM + zzpxn)dzz > m ^ 2

= 2mc ^

2 2 («^) 2 + jTO:3 Yl

Q,/3=l

zzpa0fdzz

M*

C>,0=1

For c = 1 one obtains (iv).

D

It is instructive now to derive the dual problem (Pc)' or the maximum principle of the total complementary energy. Firstly, however, we shall formulate the dual problem (P«)* of (Pet)- To apply Rockafellar's theory of duality presented in Section 1.2.5, we set Av = Qee'{v) , v e V0(B) . Forp* €L2(B,E3S) we find {Av,p')L,xL,

=

J(Q°e*(v))iiP*dz B

= f\p'a0ezQ0(v)

+ -P'a3e*a3(v) +

J

6

Kv'^e'Mdz e

B

= j (ffpf^wd* = -J (crp*)".jiHdz B

+ J(CPp'FnjVidT

B

=
(2.10.52)

r±

We denote: () ; ; = d( )/dzy The last relation yields ( -div C o * A*p* = { ,„e , ^ \(Q p*)n

in B , onr±.

(2.10.53)

We assume the following notation: Q~e = (Qe)~ ; hence Q _ e Q e = J, where I is the identity matrix. Let us set j«(*.«) = 3e(z, Oee) = j ( ^ , z 3 ) Q e € ) ,

esEj.

For c* £ E3 we calculate &(*,£*) = SUP{6* : € - jer(*,«) I « € E*}

(2.10.54)

Three-dimensional analysis and effective models of composite plates = sup{e* : (Q-eQee)-je{z,Q'e)

127

| € e E3}

= s u p { ( ( Q - ) V ) : Q e e -jc(z,Qee)

| e e E3}

= sup{(Q-V) : £ - ; , ( * , £ ) | e e E ? } = Jt(z, Q" V ) = ^(Q-'e'Y'iCT'e'f because ( Q _ e ) T = Q" e and
(2.10.55)

One can easily verify that

( Q " V ) a 3 = ee'a3 ,

( Q - V ) < * = e«# ,

,

(CTV)33 = e V 3 3 .

(2.10.56)

Now we must find the conjugate functional of (—Le). By taking into account (2.10.53), one obtains (-Le)*(<7) = s u p { ( - A V , v) + Le(v) | v 6 V0(B)} = sup{{ div Q V , « ) -

[(Q'ayin^dT

+ Le{v) \ v G V 0 (B)}

r± e

aj

= sup{ f{(Q (T) .}va

+ {Q'trf'.jVs

+ e2bava + eb3v3}dz

B

- j[{&
= <

+ (Qe
+ fvJdT

1 aa0a + ~
0

+00

otherwise

I v € V0(B)}

(2.10.57) (2.10.58) (2.10.59)

because n = (0,0, ±1) on T±. Multiplying Eq. (2.10.58) by e > 0 and next adding to Eq. (2.10.58) we get 3

+ -u -a* ;33 + ;a T

e2b* = 0

in B

(2.10.60)

e

Let us set Se = {
- ISC(
(2.10.61) (2.10.62)

Now we are in a position to formulate the dual problem (Pee)* of (P M ) and exhibit their interrelationship.

128

Elastic plates

Problem (P«)* Find M O

= sup{c7K(<7) I a- e L 2 (B, E*)}.

Corollary 2.10.3. The solution er" of the dual problem (PM)* exists and is unique. More over, we have inf(/>„)• = min(PM)* = sup(PM)* = max(PM)* . (2.10.63) Proof. The properties of the complementary energy j* and the regularity of (&') and (P±> 9*) imply the existence and uniqueness of tra. The duality relation (2.10.63) results then from Proposition 1.2.49. D We proceed to the derivation of the dual problem (P£)*. Now the operator A involved in the theory of duality (Sec. 1.2.5) has the form A(u,w) = {Aiu, A2iu) = (e{u),n(w)) ,

(2.10.64)

where u € Wd(fi), w e HQ(Q). Proceeding similarly to the derivation of (2.10.53) one obtains A* = (AJ, AJ), where

Here N,M

AJJV = -divW in ft ,

(2.10.65)

AjM = —div divM in ft .

(2.10.66)

2

6 L (ft, Ej). Next one finds

VV*(zQ,e*,p*) = sup{e*:€-l-p*:p-VV E (z a ,€,p)|e,peE 2 } *a0f*^ *Xn i, O^E O-E — -lnc fe*a0 £ - 2 v a Q0A„ e + zea0\ve

«a0 «A/j ,. JE «a0-»A/t + a P a0\»P

.a/3 «AM\ P )

(2.10.67)

where

i>(za) =

a'

e

r x

=

Be\za),

e

a0\n

~ f Ajia(3

(2.10.68)

and Be{za) =

(2.10.69)

Further, we have: (-LY(-A'(N,M))=snP{{-MN,u)H-Hn)xHiim + (-KM,w)H_2mxHim 0

+ L{u,w) | (u,w) e H^Q)2 x

(diWV + p + + p _ = 0 inD'(ft), if < [divdivM + cdiv(p+ - p _ ) + (
+00, otherwise.

Him (2.10.70) (2.10.71)

Three-dimensional analysis and effective models of composite plates

129

H e r e p ± = (pj). We set e [ L 2 ( n , E s ) ] 2 |div/V e L2(n)2 ,divdivM € L 2 (Q), and Eqs (2.10.70), (2.10.71) are satisfied}.

5 = {(N,M)

(2.10.72)

We formulate the dual problem. Problem (Pe)' Find s u p { - fw?[za,

N(z), M(z)]dz

- IS(N, M)\N,Me

L2(Q, E<)} .

By applying the results presented in Section 1.2.5 we conclude that solution (Nc, Me) G 5 of (P*) exists and is unique. Further, the minimizers (uc,wc) € HQ(Q)2 X HQ(Q) and (NC,MC) are interrelated by the extremality condition, which now takes the form of Eqs. (2.10.47), (2.10.48). The second extremality condition means that in this case Ne satisfies Eq. (2.10.70) while Mc satisfies Eq. (2.10.71). Remark 2.10.4. For (a£) the weak form of Eqs. (2.10.57) - (2.10.59) is given by

B

B „22 e

fb\dz

+ fplvadr

+ e jq^vzdT

.

(2.10.73)

B

Caillerie (1984) has shown that by a proper choice of test functions in the last equation, after the passage to zero with e, the equilibrium equations assume the following form: diviV£ + p + + p_ = 0

in n ,

divM£ - Qc + c(p+ - p _ ) = 0 d i v Q £ + (
infi,

(2.10.74)

inQ.

Note that Eqs. (2.10.70), (2.10.71) are formally equivalent to Eqs. (2.10.74). Let us set

N^ = jaaJdz3-

a, 0 = 1,2, (2.10.75)

-C

c

M*

= jzzo°Jdz3

c

,

a

Q% = -e Ja Jdz3

,

130

Elastic plates

where (o£) is the solution of (Pes)'- Caillerie (1984) proves that for subsequences of N&, Ma and Q^, still denoted by the same subscripts, one has weakly in L2(Q) ,

NSf -* Nf

weakly in L2{0) ,

M<£ ->■ Mf

(2.10.76)

l

Q%^Q°

weak-* in H~ ((l),

when e —> 0. A similar convergence takes place when e and e tend to zero. Then the limits in (2.10.76) do not depend on e. □ Now we are in a position to formulate the dual T-convergence theorem. Theorem 2.10.5. For afixede > 0 the sequence of functionals: 0„(
(2.10.77)

B

T-converges in the weak topology of L2(B, E^), when e — ► 0, to the functional ge(N,M)

= -Jwc[za,N(z),M(z)}dz n

- IS(N,M),

(2.10.78)

where 5 is defined by (2.10.72). Moreover, for convergent subsequences we have: (i) a% ->• a?

weakly in L2{B) ,

C

N«0 _, fiat) =

weakly in L 2 (fi),

ffffdzi —C C

M

Zf -1- M f = [z3
weakly in L2(Q) ,

—c

when e — ► 0, where of = 0 for i = 1,2,3. (ii) inf P« —> inf P£ ,

sup P^ -* sup P£*,

when e —» 0 . Proof. The proof is based on Theorem 1.3.36. It is sufficient to show that for afixede > 0 the sequence of functionals: 4 2) («-P) = Jjt[z,Cr(e>{v)

+p)]dz , v e Hl(B)3 , p 6 L2(B, E]),

(2.10.79)

Three-dimensional analysis and effective models of composite plates

131

is T-convergent in the topology (s - L2(B)3) x (s - L2{B, E33)) to J?Hv,p) = \fc?»»(z)le>a/,(v)+pafi)

[ e y » ) +p^}dz , v € VK{B) ,

B

where p^ = 0. Details are left to the reader, cf. also Sec. 5.5.

□

Homogenization: e —> 0 The elastic moduli specified by (2.10.49) are ey-periodic. In order to derive an effective plate model one has to perform homogenization; according to our procedure a r-limit has to be found when e —» 0. The loading functional L, given by (2.10.28), still plays the role of a perturbation functional. Effective elastic potential and its properties Let us denote by VVA the elastic potential describing the effective plate behavior. By applying the general homogenization theorem, namely Theorem 1.3.28 one has Wh(e,p) = i n f { ( W ( y a , e » + £,«»(«) + p)} | (v,v) e HKiPer(Y)} , (2.10.80) where e,p € E 2 ; eva0{v), nya0(v) are defined by

()| a = d/dya, and HKxPeT(Y) is defined in Section 2.8; moreover W(y„,e,p) = \Aa^(ya)ea0eXll

+ Ea^{ya)ca0P^

+

\Da0X»(ya)pa0pXli.

Properties ofWh

(0 2

[(ea0)2 + (pa0)2} ,

VVh(e,p) < (W(yQ,e,p)> < M, ^

(2.10.81)

Q,/3=l

for each e, p G E 2 and c = 1. Here Mi is a positive constant. To prove this property it suffices to take v = 0 and v = 0 in (2.10.80). (H) VV„(e, p) > - £ , £

[(eQ/3)2 + (pa„)2] ,

(2.10.82)

for each e, p e E 2 and c = 1. In fact, by taking into account (2.10.51) one obtains Wh(e, p) = {W(jfa, e»(t>) + £,«»(«) + />)) > y <( |e"(v) + c|2 + !«»(*) + Pi2)) > ^ | ( k l 2 + IPI2) ,

forc= 1 ,

132

Elastic plates

because 2

2

o,/3=l

<*0=1

By (t), v) we have denoted a minimizer of the functional appearing on the r.h.s. of (2.10.80). Let us discuss now the problem of existence of (v, v). To this end we write Wh{e, p) = inf{J Y (v, v) + £{v, v) + mi\

(v, v) e

HK:Per)

where Jy(«,u) = (W(» a , £»(«),«»(«))),

l(v,v) = \{2A°^e%{v)eXll a

+E ^el0(v)Px,

+F ^ p ^ v )

mi = \{Aa^ea0eXli

+ £*»*%,*$»

+F

+ Ea^ea0pXll

Q/JA

"<»eA M + 2 D ^ P a ^ v A » > ,

+ Fa^pa0eXll

+ Da/3^pa0pXli)

■

The functional Jy is strictly convex and coercive on the space Hfc:Per(Y). In fact by using (2.10.82) we easily obtain JY(v,v)

> K(\\

v \\2HHY)2

+ || v \\2HHY))

,

(v,v)

€ HK^Y)

•

Here K is a positive constant. In the space H1 (Y) 2 , the set of rigid body motion is given by H={v\ va{yu y2) = aa3y0 + KQ] , (2.10.83) where (aa0) is a skew-symmetric matrix and (K a ) e R 2 . To satisfy the periodicity require ment, aap (a,0 = 1,2) should vanish. Thus in the space / / ^ . ( Y ) 2 , rigid body motions reduce to the set of constant vectors, i.e. to R 2 . It implies that in H^Y)2 the kernel of the operator ev is equal to the null vector. Similarly, in H^iY)2 the kernel of the operator K? reduces to zero. It is evident that the functional £ is continuous on the space Hl(Y)2 x H2(Y), and thus alsoon^K- lP er(V). Summarizing, we conclude that the following minimization problem: Jy{v,v)

+£(v,v)

+mi

= inf{J Y {v, v) + £(v, v) + m1\ [v,v) 6 HK
,

is uniquely solvable. We observe that for the minimization problem: find

in{{JY(v,v)

+ e(v,v) + ml I (v,v) e H^Y)2

x

H^Y)}

a minimizer exists and is determined up to a constant vector (K, K) € R 2 x R.

(2.10.84)

Three-dimensional analysis and effective models of composite plates

133

Effective constitutive relationships Due to linearity of the problem under investigation one can write v = T^a0)ea0 + U{a0)Pa0 , + xia0)Pa0,

v = X^ea0 where T{a0\ U^aB) € H^T(Y)2 relationships are given by

(2.10.85)

while X^a0\

(a0) x

(2.10.86)

£ H^.(Y).

The effective constitutive

C«o/3

M? where Pf*

= E^0

= 5T^ = F f ^ dp,'a/3

+4

W

PV ,

(2.10.88)

and

Af> = 1dcy6de P^, 0

'

fTW

,

a

= / f

#M h bf*

=

_ dp £ ^{de_ p, y

a

= -£™±-.

(2.io.89) (2.10.90)

The formulae above are fully equivalent to the formula (2.8.6) found previously. The sym metry properties (2.7.22) can easily be deduced from relations (2.10.89) and (2.10.90). The K-periodic functions T{a0\ U{a0\ Xia0), \(a0) are solutions to local problems (P% Y) °f Section 2.8. Those local problems are here derived as necessary conditions for the minimization problem involved on the r.h.s. of (2.10.80). As we know, this minimiza tion problem is a convex optimization problem. Thus local problems {Pf
= J}{v) = fw n

[—, e(u),K(u/)] dx ,

(2.10.91)

where u € Hl{9.)2 and w 6 H2(Q) is T-convergent in the strong topology of L2{Q)2 x Hl{£l) to the limit functional Jl(u,w)=

jWh[e(u),K{w)]dx. n

(2.10.92)

134

Elastic plates

Here VVh is given by (2.10.80) and v = (ua For any fixed e > 0 the problem

z3dw/dxa,w).

Ji{u*,wc)-L{u*,w')

(P £ )

= inf{J f '(u,w) - L(u,w)

| {u,w) G H^(Cl)2 x Hg(Sl)}

and the homogenized problem (P„)

Jl(u,w)

-

L(u,w)

= inf{ j£{u,w) - L(u,w)

| (u,w) G //^(fi) 2 x

H^Sl)}

are uniquely solvable. Moreover, we have (uc,wc) ->■ (u,ii;) weakly in if 1 (ft)2 x H2(Q) inf P e -»inf Ph .

(2.10.93)

Proof. T-convergence is left to the reader as an exercise. In fact, it can be performed along the same lines as for the scalar case or three-dimensional elasticity, cf. also Sec. 2.10.4 where a more complicated case will be studied. The existence and uniqueness of {u,w) is ensured by the properties of the effective potential VVh . Problem (P £ ) has already been discussed, cf. (2.10.45). Finally, (2.10.93) results immediately by an application of Theorem 1.3.22. Dual effective potential The effective complementary energy VVj| can be found as Fenchels's conjugate of VV/,: VV^(e*, p*) = sup{e' : e + p" : p - Wh(e, p) | e, p G E 2 } , where e*, p* G E 2 . By using (2.10.80) we find W'h(e',p') -W{ya,e*(v)

= sup{(e' : («*(») + e) + p' : (is»(«) + p) + €,«»(«) + p)) | e , p € E 2 , {v,v) € HKtVer(Y)}

, (2.10.94)

because (e* : ey{v)) = 0 and (p* : »»(«)> = 0 , where v G H^Y)2,

v G H^r(Y)

.

We set H(K) = [ev(/f^(y)2) x K'iH^iY))) K ( p „ f t ) = \Y\(W(yQ,pup2))

® (E2 x E 2 ) ,

, Pa e

L2(Y,E23).

Then we can write, cf. Sec. 1.2.1, W;(e',p')

= | F r 1 ( K + / H m ) * ( 6 * , p - ) = \Y\-l(K'UI

)(e',p')

,

(2.10.95)

Three-dimensional analysis and effective models of composite plates

135

where e*, p* are identified with elements of L2(Y, E£); moreover K-(n,m) = |y|<W-(j, 0 ,n,m)) ,

5 ^ ( 7 ) = [H(y)] x

and.cf. (2.10.68), (2.10.69), VV*(yQ,n,m) = -[n,m]B- 1 (7/ Q )[n,m] 7 ' . The symbol □ represents here the inf-convolution. Let us find the orthogonal complement of [H(Y)]-1 in the sense of L2: S^OO = \H(Y))± = \ey(H^{y?) while Is

t

x ""(^(Y)))1

n (E 2 x E 2 )^

is the indicator function of the set Sper(Y).

We calculate (E 2 x E 2 )^ = {(n, m) e L2(Y, Etf \ (n(y) : e> + {m(y) : p) = 0 , Ve, p e E 2 } = {(n,m) 6 L2(Y, Etf \ {n(y)} = 0 , (m(t/)) = 0} , (2.10.96)

[e"(//,Un2) x " ^ ( W = {(n,m) € L2(Y, Es2)2 \(n(y) : e » + m(y) : «»(»)) = 0 , V(t;, t>) 6 ^ r ( Y ) 2 x H^(Y)} . (2.10.97) Performing integration by parts in the following variational equations: Jn(y) : ey(v)dy = 0 ,

V» 6 H^Y)2

,

Y

Jm{y)

: K«(v)dy = 0 ,

V« e / ^ ( V ) ,

y

and taking into account (2.10.96) wefinallyobtain S^iY)

= {(n, m) e L2(y, E*)2 | (n(y)) = 0, <m(y)) = 0 , div„n = 0 , divydiv„m = 0 in Y ; m,, assume equal values and nfi and q opposite values on the opposite sides of Y] , (2.10.98)

where nn = ( n Q " ^ ) , mM = mQ/VaM0 , W drrvr (2.10.99) a0 dy0 ds Here r = (r a ) is the tangent unit vector and s denotes an abscissa on dY measured pos itively in the direction of T. We observe that notation such as (2.10.99) is typical for the

Elastic plates

136

mechanical setting. In general, mM and q are to be understood in the sense traces, cf. Temam (1985). According to the definition of inf-convolution, from (2.10.95) one obtains WU*',P') = |p|inf{K*(n 1 ,m 1 ) + /^ t r ( y ) (n 2 ,m 2 ) | £ ' = ^ + 1 1 2 , p* = mi + m 2 , (Tia.TrO e S^Y)} = inf{ jpjK'(e* " n 2,P* - ma) | (n 2 ,m 2 ) e

S^Y)}

= m({(W*(ya,n(y) + e*,m(y) + p*)> | (n,m) G S^Y)}

, (2.10.100)

because the set Sper(Y) of admissible generalized local stresses (n, m) is a linear space. Having derived the complementary effective potential VVJ , given by (2.10.100) one can formulate the dual homogenization theorem. Theorem 2.10.7. The sequence of functionals {££}E>o, given by (2.10.78) is T-convergent in the weak topology of L2(Q, E 2 ) x L2(fi, E 2 ) to the functional gk(N,M) = - fwh[N(xa),M{xa)}dx-Is(N,M)

.

(2.10.101)

n If (JVe, M £ ) is a solution of the problem (P*) and (Ph*)

gh(Nh, Mh) = sup{gh(N, M) I (N, M) G L 2 (n, E 2 ) 2 } ,

then {Ne, Mc) ->• (Nh, Mh) weakly in L2(fi, E^)2 , sup (Pc') - sup (Ph*) , when £ —► 0.

Proof. The proof follows by applying Theorem 1.3.36 and Theorem 2.10.6, cf. also Sec. 5.5. a 2.10.3. Justification of the effective plate model of Section 2.9 by passing to zero: E - > 0 and next e —> 0 For any fixed e > 0 the r-convergence when e —» 0 for a thin body Be with eV-periodic microstructure is a particular case of the homogenization of the equations of the threedimensional elasticity. It is thus sufficient tofindthe homogenized potential when e —» 0. Homogenized elastic potential For our particular case of a two-dimensional homogenization, this potential, denoted now by j 0 , is given by j0(ito,e) = i n f { ^ | c ^ Q , 2 / 3 ) ( e ? > ) Y

+e*i)(e«(«) + ««)<*!/ I v G H^Yf)

,

(2.10.102)

Three-dimensional analysis and effective models of composite plates

137

where e e E*, and H^r{Y)3=

{« 6 tf^y)3 |u, is ^-periodic , i= 1,2,3; (v) = 0} .

Now we have

It is evident that a minimizer v e H^.r{Y)3 of the minimization problem on the r.h.s. of (2.10.102) exists and is unique, cf. Sec. 2.10.2. Moreover, we may write v = 0 w >e, ; ,

0 t o ) e H^Y)3.

(2.10.103)

The homogenized constitutive equation has the form (2.9.18), which will be written as <% = | r

= C j f W , ,

(2.10.104)

where C ^ , already defined in Section 2.9, can be put in the form q ^ t e ) = {Ctjkt(ya,lfe)[e&(e<"»>)

+ S?%] [ e ^ e ' " ' ) 4- 6[5?}) ,

(2.10.105)

which can further be simplified since e^ 3 (0 ( m n ) ) = 0 . We note that 6^ m n ) are defined as in Section 2.9 and 9^ m n ) = ^mn\ see Sec. 2.9, problem (ff'>). The following properties of jo may easily be verified: (i) there exist positive constants m' < M' such that for a.e. z3 G ( - 1 , 1 ) m'\e\2<j0(z3,e)<M'\e\2, for each e € E3S (c = 1). (ii) h — h — h Now we are in a position to formulate the T-convergence theorem. Theorem 2.10.8. For an e > 0 and fixed, the sequence of functionals {•/«},.,„ is Tconvergent in the weak topology of H1(B)3 (strong of L2(B)3) to the functional ■/«'(«) = jJ0(z3,&e*(v))dz

,

v e H\B)3

.

B

The minimization problem Jle(ve) - Le(ve) = i n f { J » - Le(v) | v G V0(B)} ,

(P.)

is uniquely solvable and /

-> ve

weakly in

H\B)3,

when e -> 0 , where (Pa) is defined by (2.10.22).

inf(Pe£) -> inf(P e ), D

Elastic plates

138

Plate model: e tends to zero Now the situation is similar to that for the functional J^ when e —> 0 . The only difference is in the elastic moduli; in the present case they are specified by (2.10.105). Thus we can formulate the limit theorem. Theorem 2.10.9. The sequence of functionals {J\ — Le}e>0 is T-convergent in the strong topology of L2(B)3 to the functional J

hl") = \ J C^(z3)(ezQ0(v)el(v))dz

- L(v) ,

v e V&(B)

(2.10.106)

B

where C^ is given by (2.9.27).

D

Corollary 2.10.10. The constant two-dimensional elastic moduli are calculated by the formula (2.9.22), with i j = - c , x3 = c, x 3 = z3, x3 = z3. Remark 2.10.11. Dual formulations are left to the reader as an exercise. 2.10.4. Justification of the effective plate model of Section 2.3 by passing to zero: e - > 0 and e —♦ 0 simultaneously Now we have to deal with two small parameters which tend to zero simultaneously. The study of T-convergence of the sequence of functionals {JM}e>o, £>o is different from the previous two cases. In fact, our proof essentially exploits some tools elaborated in homogenization of microperiodic bodies. Prior to passing to the study of T-convergence we shall describe the macroscopic or effective (homogenized) elastic potential of the plate and derive its dual or the complementary energy density. We shallfrequentlyrefer to formulae of Sections 2.2 and 2.3. Here, however, the plate of constant thickness is considered, cf. Introduction to Sec. 2.10. Effective potential Due to the assumption of the thickness being constant, the space W(y) is defined by, see Sec. 2.2 W(y) = {ve \Hl(y)}31 v(-,vs) is y-periodic for2/3 e {-c,c)} ,

(2.10.107)

where y = Y x (-c,c). Let us define also the space

w(y) = {ve w{y) \ -< v y= 0},

(2.10.108)

where averaging over y is denoted by -< • >-, see Sec. 2.2. The effective elastic potential has now the following form jn(e, p) = inf{^ ■< C«"(i/)[c» (t>) + ey + &(*][<%,&) + eu +y3pki] y\ve w(y)}, (2.10.109)

Three-dimensional analysis and effective models of composite plates

139

where ho = vol ^/area Y = 2c; c, p e E , ; ei3 = 0, pa = 0; ej^v) = (vi\j + v^i)/2 and ( h = d( )/dyl. The properties of the elasticity tensor C immediately imply that the minimization prob lem on the r.h.s. of (2.10.109) is uniquely solvable in the space W(y) and up to a constant vector in the space W(y). Let us denote by v € W (y) the minimizes Properties of J-H For each e, p e E^ with e^ = 0, pt3 = 0 we have l-jnfap)

< y ^ ^ " ( y X ^ + l f c f t j K e H + WsPu) ^ < M(|e| 2 + |p| 2 ),

(2.10.110)

where M is the constant appearing in (2.10.39). 2. j „ ( e , p ) > ^

^ | e » ( f ,) +

e + y3p[2

v

> ^(|£|2

+

|p|2},

for c =

i ,

(2.10.111)

since

£ -< e^v) >= 0. D

The minimizer v depends linearly on e and p. Therefore we may write:

* = e(a0)(y)ea0 + s M) (y)pa/3 ■

(2.10.112)

Taking into account (2.10.112) in (2.10.109) we derive the effective constitutive relation ships tfa0

=

pn_t Oea0

M°0

=

p2L^ Opa0

(2.10.113)

that assume the form (2.3.51) with e\M(v) replaced with e ^ and H^W) replaced with p>M. The effective stiffnesses involved in (2.3.51) satisfy the expected symmetry conditions (Sec. 2.3). It is worth noting that property (2.10.111) of j n is equivalent to the positive definiteness of the matrix AH En (2.10.114) B« = cf. (2.3.62). The minimization problem on the r.h.s. of (2.10.109) is a convex optimization problem. The necessary condition for a minimum leads naturally to two local problems (Py) (see Sec. 2.3) for the determination of the functions e ( Q / 3 ) and 3{a0) ; here one should replace y3 with y3 in (Py), since -< y3 >-= 0. Dual effective potential To derive the dual elastic potential j ^ , which represents the density of the complemen tary energy of the plate, we shall apply the theory of duality outlined in Section 1.2.5.

140

Elastic plates

By definition of the conjugate function one has Jw(c*» P') = sup{e* : e + p' : p - jn(e, p) | e, pGE 2 } =

sup

{e* : e + p* : p -

...€E2

inf ~

fj(y, e»(v) + e + y3p)dy} • (2.10.115)

„ew(}>)|r|./

«i3=°.<>.3=0

J'

For fixed e, p G E, with e^ = 0, p i 3 = 0 we first consider the following minimization problem: (P (tiP) )

mi{Jj(y,

e*(v) + e + y3p)dy \ v G WQ>)} .

y

Let Av = e*{v), A •. H\yf one finds

- V(y) = L2{y, E3S). For any p* G V*(}>) = L2(}>, Es3)

(Aw,P">^();,Ej)x^(y,E5) = / P *

:

e"(v)dy

= - / (div„p*) • vdy + / p ' ^ u ^ d S = (A*p*, v)(//i(j,)3].xWi(>,)3 ay Hence

in?, on ay.

[p'u

(2.10.116)

Further we set J(*,P){y, A) = j(y, A + e + y3p) , G (£ ,,)(P)

= Jj(e,P)(y,p(y))dy,

A G E* ,

ve

L2(V, EJ)

.

By using Proposition 1.2.34, one has G(£,p)(p*) = / j ( V p ) ( y . p * ( y ) ) ^ . y

P*eL2(y,E«).

(2.10.117)

Here, for A* G Eg, we calculate J(Vp)(y- A') = sup{A' : A - jM(y, A) I A G E*} = sup{A* : A - j(y, A') | A' 6 E^} ,

(2.10.118)

where A' = A + e + y3p. Hence one readily obtains JU,p)(y, A*) = \c3ki{y)^A* where c = C~l.

- A* : (e + y3p) ,

(2.10.119)

Three-dimensional analysis and effective models of composite plates

141

If F = 0 then F*(-A*p*) = sup{ (-A*p*,u> - 0 | v e

where S°(y)

fo

.fp-eW)

(^ +oo

otherwise,

W(y)} '

= {P* ^ Z , 2 ^ , Eg) j div y p* = 0 in y, p V takes opposite values on the opposite faces of dY x (-c,c) , p V ^ O o n Y x {±c}}.

(2.10.121)

Taking into account (2.10.118) - (2.10.120) one can formulate the dual problem of (P(Ci p)):

(PMY

sup{- y j-M(y,p-(y))dy

\ p' e S°(y)} .

Our assumptions imply inf(P( € ,,)) = sup(P (€ ,,))*

= -M{J

j'{tJy,p'(y))dy

\ p* e S°(y)} .

(2.10.122)

y

Thus from (2.10.115) and (2.10.122) one obtains Jn(e',P')

=

sup

{€* : e + p* : p

«,3=°.P(3=o

+fto

inf

x j'{y,p'(y))

- p* : (e + y 3 p) ^ }

+(p* - /i 0 x y 3 p*(y) ^ ) : P + ho ■< j'{y,p'{y))

= inf{ho x f(y,p'(y))

>■ I P* e ^ ( y ) } ,

>}

(2.10.123)

where S'OO = (P* e S°(y)

I /JO ^ p*(y) y= e' , ho < y3p' y= p'} .

(2.10.124)

An equivalent form of (2.10.123) is given by Me',p')

= inf{/i0 -< j ' ( » , j U * + « • ( ! / ) ) >- |q* € 5 ( ^ ) } ,

(2.10.125)

where SQ>) = {?" e L2(y, E*) | div y q' = 0 in Y , X q' >-= 0 , fto X y3q' >-= p ' , q V takes opposite values on the opposite faces of dY x {~c,c) , q V = 0 o n r x {±c}} .

(2.10.126)

142

Elastic plates

We recall that e"3 = 0. To corroborate (2.10.125) and (2.10.126) we note that ho -< p' >-= e' <=> -< (p* - — c*) >-= 0 . Then, for q'{y) = p'(y) - — e* one has -< q' >-= 0. Now the proof is straightforward no and is left to the reader. □ Below, in the proof of a T-convergence theorem an important role will be played by the following lemma. Lemma 2.10.12. Assume that T € S(y) and let {vc}e>0 c Hl(B)3 be such that {CFez(vc)}<.>0 is bounded in L2(B, E3,) and v£ converges strongly to v 6 H\B)3. Then one has lim / i p i z a ) ^ (—,z 3 ) (Q£e*(«£))ydz = f^p' : Kz(w)dz , (2.10.127) B n where rp € D(fi), v = (u 0 - z3dw/dza, w) and p* e Ej appears in (2.10.126). Proof. Let us set

R* = JtKz.m^zJiCfe'Whdz

, 7? = (|,* 3 ) •

B

Integration by parts yields fie = - AV'CfTe) V'ufdz+ [ ^TjnjvldY s r±ur0

.

(2.10.128)

Since ^ = 0 on T and 7jjJ'rij = 0 on T ± , therefore the last integral vanishes. We know that divyT = 0 in y. After a rescaling y —» ( —, z3 J we obtain ( e 7^ ; Q + r 3 ; 3 ) ( ^ , 2 3 ) = 0

iaB,

d d d d because -— = e-— and 7— = 7—. For e > 0 the last relation is equivalent to dya dza dy3 dz3 TZ + \l?3 = Q in

B.

On the other hand we have

( O T J l =T$+

l

~T?3,

(Q
Thus (Q£T^

= o,

Three-dimensional analysis and effective models of composite plates

143

and consequendy c

c

Re = - I'/'^(Qer<)ywf
ci -c

-c

c

= -jiP;aj(T?% + -T^vl)dz . We observe that c

\RA = ||^(Q£T£)X(V£)||L>(B) =

\J^aj{Tf%+l-T?v\)dz\

n < Kx ,

< K\\Te\\L,\\
-c

where K\ is a positive constant independent of e. Finally we obtain c

c

lim/k = - lim ji>,a \Tfv%dz n

f-T*av£3dz

- lim (i>.a

-c

n

-c

c

= -ji>,aJ(T0a(y,,y3)) n -c

(up - z3~^j

dz = j ^ K ^ d z n

,

because ^ ; 3 = 0, -< T ^ = 0, h0 -< y3T >-= p*, and c

a

-c

D

r-convergence of the sequence { JM}e>o, £>o As we know from the proof of Theorem 2.10.2 the sequence of loading functionals {i e }e>o >s continuously convergent. The main result of T-convergence when e —► 0 and e —> 0 simultaneously is formulated in the form of the following theorem. Theorem 2.10.13. The sequence of functional {J^}e>o, £>o defined by (2.10.29) is Tconvergent in the strong topology of L2(B)3 (weak topology of Hl(B)3) to the limit functional J^ given by Jit(ua,v>) = Jjn(ez(u),Kz(w))dz n where uQ <= Hl{9),

w e

H2(ty.

Proof. We divide it into two major parts.

,

(2.10.129)

Elastic plates

144

I. We shall prove that for any v = (va, v3) e VK(B), va = ua — z3w;a, v3 = w, there exists a sequence {vE}£:>o C Hl(B)3 strongly convergent to v in L 2 (B) 3 and such that Jit{u,w) > lim supJ^(vc) ,

(2.10.130)

E—0

where J]c = J^e=e- Tacitly we assume that ve stands for wM where e = e. Step 1. We take v = (va, v3) in the form 2

Va = ^Va0Z0 8=1 V3

+ Zi{Pa0Z0 + aa)\ + Ca ,

2

y3 = Z3 ,

(2.10.131)

2

= 7) ^2 {-Paf}ZaZfi) - ^2aaza 2 „

+a

a,0=l 3

where e, p € E with ea = 0, pa = 0 and aQ, ca, a € R. Obviously, v belongs to VK{B). Let v be a function at which the functional appearing on the r.h.s. of (2.10.109) achieves a minimum value. For v specified by (2.10.131) we set t£ = va + eva ( ^ , z3) , The sequence < vt ( - , z3 j > ve ->v

ve3 = v3 + e2v3 ( ^ , z3) •

(2.10.132)

is bounded for a.e. z3 6 (-c, c). Hence we conclude that in L 2 (B) 3 when e -+ 0.

Next we calculate (€>eez(vc))a0 = ea0 + z3pa0 + ezae{v) (— , z3J , (Q£ez{vc))a3 = eza3{v) ( ^ ,z 3 ) ,

(2.10.133)

Taking into account (2.10.133) we obtain

B c

= / < / ±C*« ( ^ , * ) [«$(«) ( f , z3) + et] + 23py] f!

-c

x [4,(«) ( j ,2 3 ) + €« + z3pw] <*z3}
(2.10.134)

Three-dimensional analysis and effective models of composite plates

145

We recall that ti3 = 0, p& = 0 and 23 = 1/3. The integrand in (2.10.134) is an ey-periodic function and therefore tends weakly in Ll(B) to j«(e, p) when e —► 0, cf. Th. 1.1.5. Consequently we write Km-4(*>£) = I Jn{e,P)dz = J jH[eza0{u),Kz{w))dz n n

,

2

a2

where, on account of (2.10.131), ua = ^taffZa+CcW

= v3,Kza0(w) = - - — - — = pa0.

0=1

OZpOZa

Step 2. Let {HK}K€>C be a finite partition of the domain Q formed by polygonal sets. We set QSK = {z G fi I dis* (z, a n K ) > 5} , 8>0. We take a function u = (va, v3) 6 [C(B) 2 x C l (fi)] n V K ( B ) , given by Ua(z) = X ] t £ « e ^ 0=1 2

+

Z

3(Pa0Z0 + fla )] + C a > z3 = (*a, 3l) € fi X ( - C , c), 2

1

w3(zi,22) = ] T -^{-p%)zaz0

- Y^a*z* + °-K .

a,/3=l

a=l

(2.10.135) where eK, pK G E* with eg = 0 and p% = 0; a £ , c£, a* G R, tf G /C. We observe that u3 G C'(fi) and -— G C(Cl). Further, we assume that the function oza 2

UQ(ZI, Z2) = y ^ e ^ z ^ + c£, (z a ) G ft/c, being piecewise affine is continuous. Hence we p=\

conclude that va G C(Q) and for a fixed z 3 G (—c, c) it is also piecewise affine. The partition just introduced enables us to exploit the local character of the functionals Jle. Let ipsK G D(fiff) be such that 0 sK < 1 and ^ ( z a ) = 1 for (zQ) G ttsK. With every family of functions vK G W(3 ; ) we associate the following sequences

fee*:

(2.10.136)

2

t£*(*) = t;3(zQ) + £ £<(z / 3 )t,f (^,z 3 ) KeK £,<

It is evident that for e -» 0 one has t> -» v strongly in L 2 (B) 3 . Let us take t G (0,1). It is not difficult to show that t&e>(v<*

( | , z 3 ) ) = ttfK(za)\e*{vK)

( | , z 3 ) + e « + z 3 p*]

+ t ( l - < ( z a ) ) ( e * + z 3 p«) + (1 - t ) ^ ( < ( , ( * > £ ( f

,z3))

146

Elastic plates

where

and(zQ,z3) € ilK x (~c,c). For instance, let us find t{Qce(ve,5))a3. We calculate t(Qe(v

))a3-t-ea3(v

) - t-- ^ — + —

Ht(**+«f+«*g-**--f+,*£*,'^r) = tv: = tyfc [ea3(vK) (*f , 23) + 4 , + *3P*3] + *(1 - !&)(«& + *3P&) " ^ ~ ^ Y ^ ^ t " ^ " ^ 3 ' (lT ' 2 3 ) '

^2Q' Z^

e n

*

x

(_c'c)'

because e£3 = 0, p£, = 0 and &ipK/dz3 = 0. By using (2.10.7), the convexity of the function jc(z, •) and noting that tify + t(l — V'fc) + (1 - t) = 1, one gets c

S!K - c c

=£

/ A'(- • z3- *<&(*.) («(«*) ( - , *) +eK + z3pK)

+ t(l - <(^))(6^ + *3p") + (1 - t ) y ^ (V4.(^) ( 7 .*))]«**

^ £ / 1 / 4 7 •23'e(v'f) (7 '20 + ^ + ^ ^ 3 } ^ c

+ £ > KeK

/ ( l - < ) d z / I eK + z3pK \2dz3

nK

i dz ,

/s e,c

'

nK -c

because j > 0. Now let e tend to zero. By using Step 1, we arrive at lim sup J]c{tva) < T{ho

\ilK\ -< j[V, eV(vK) + eK + y3pK\ y

Three-dimensional analysis and effective models of composite plates

147

- tfK)dzJ | eK + z3pK \2dz3} ,

+ m(l - t)J(l

-c

fix

where \ilK\ = the Lebesgue measure of Qj<. Next, let t —> 1 and <5 —» 0, then lim sup lim s u P 4 ( t « d ) < Y >

0

|^ K | -< j l y . e " ^ ) + eK + y3pK] >} .

By applying Lemma 1.3.27, one can construct a mappings — ► (i(e),<5(e)) with (i(e), 5(e)) (1,0) such that setting vc = t(e)ve'S^\ we conclude that v£ — ► v strongly in L2{B)3 , and lim sup J]c{vc) < T {Ao |fi*| -< j[y, e»(t>*) + € « + ^ p * ] y} . By taking the infimum on the r.h.s. of the last inequality for vK running over W(y) one obtains J%{u,w) < lim s u p 4 ( « £ ) < jn[ez{u),Kz{w)]dz

=

Tl^lJnie^pX) = Jli(u,w),

n where (2.10.135) has been taken into account and Jtf stands for the T-limit superior. The properties of the effective elastic potential imply that the functional J^ is convex and finite on //'(O) 2 x H2(Q), thus it is also continuous on this space. By applying density Propositions 1.4.2,1.4.16, we conclude the proof of (2.10.130). II. The second part consists in proving that for any sequence {vc}£>o C H1(B)3 converg ing to v e VK{B) strongly in L2{B)3, the following inequality is satisfied: J^{u,w) < lim inf Jls,{vc) ,

(2.10.137)

where, as in the previous case, va = ua — z3-—, v3 = w,ua G //'(fi) and w G H2(Sl). As we know from Section 2.10.2, if v e Hl(B)3\VK{B),

then

c

lim inf J},(v ) = +oc , £—0

and inequality (2.10.137) is trivially satisfied. By using duality arguments we claim that lirnjnf Jxe£{ve) > Jn{u,w) = I' jn(ez{u),Kz{w))dz = sup{ f[N : e*{u) n n +M : K » - j'H(N(za),M(za))]dz \N, M € L2(Sl, Es2)} .

Elastic plates

148

Step 3. First we take N(za) = XK(za)NK

,

NK € Ej ,

where VK,7

*

x_ / 1

lZoj

-\o

if (*») e n „ ,

if

(za)enK.

Now {H/f }K€)C is afinitefamily of open disjoint sets such that fi = (Jfi^. K

Let T K G 50 1 ), if € /C. By applying Lemma 2.10.12 and recalling that; > 0 we write c

l

c

lim inf J cc(v ) > Um inf £

/ < ( * » ) { f\j ( f ,*3,Q<e> £ ))

- 7# ( ^ , 2 3 ) : ( Q £ 4 ( V £ ) y d * 3 + M : « » } d * c

= limjnf £ / < {M : K » + /[? ( ^ , 23, Q«e*(t;')) eK * riK - TKc : Qce*{ve)} dz3} dz ,

(2.10.138)

whereTK£(z)=T/f(^,23). We set jTKe

( ^ ,z 3 ,e) = j ( ^ ,23,Q<e) - TKe:

Fenchel's conjugate of jT j'TKe

(j

Q£e .

(2.10.139)

(—, z3, • J is calculated as follows:

, 23, e*) = sup{e* :c + T « t : Q£e - J ( j , z3, QFe)\ee

E3S}

= sup{(Q-£e* + TKc) : Q£e - J ( y , *3,Q£e) | e e E*} = sup{(CTV + TKt)

: e' - j ( ^ , z3, e') | c' e E*}

= j * ( ^ ,z 3 ,Q-v + r,fe) , where e* e E 3 .

(2.io.i40)

By applying Fenchel's inequality to j T l ( e I —, z3,-j at the point

e

[&e*(v (z)), —NK], where N% = 0, we get 2c UKt ( ^ ,z3,Qfe*(vn)

>

YCNK

■■ &ez(v')-j;K€

(j

-

2

3 , ^ K )

(2.10.141)

Three-dimensional analysis and effective models of composite plates

149

Taking into account (2.10.139) - (2.10.141) in (2.10.138) one obtains limjnf Jl{vE)

> lim inf ^

{ fipKM

: K*{W)

KefC c

dz3}dz.

(2.10.142)

It is evident that c

n

-c c

= J^KJ^N^<0(U n

- z3Vw)dz

-c

= jtsKNfe%0(u)dz nn

.

The sequence of eV-periodic functions j ' l

L {B)

(2.10.143)

is bounded in

and weakly convergent to <J*\y«,V3^NK

+ TKiy)\)

as e -> 0 .

(2.10.144)

We recall that z3 = y3, 2c = ho and N$ = 0. From (2.10.142) - (2.10.144) we obtain limjnf . / > ' ) > £ i f 4 l ^ % ( u ) + Keic -hejridz

M^Kl0(w)}dz

nK -< 3*[v, YCNK

+ T

*(y)l

v

> •

Passing to the supremum on the r.h.s. of the last inequality when TK runs over S(y), where p* = M, we arrive at lim inf J £ > E ) > J2

-

l^K[Nfeza0(u)

+

M^K'a0(w)}dz

U6KJn(NK,M)dz,

(2.10.145)

because s u p ( - / ) = - inf/. Recalling that N is a step function, we obtain limjnf J > £ ) > / X > k ( * a ) [ J V ( * / j ) : e*(u(z,)) +M(2Q)«z(w(2fl))]d2-

fj2i>KMJn(N(z0),M(20))dz. /C€/C

(2.10.146)

ISO

Elastic plates

The inequality 0 < TJV'jc < 1 implies Keic

0 < Y.^z^H(N{z0),M{z0))<JH{N{za),M{za))

,

because fa > 0. Thus we get

lim inf Jl(vF) > f^sK(za)[N(z0) n *eK +A*(Z/9) : /c2(w(z7))]d2 - Jj'n{N{za), n When 6 -» 0, ^

: e*(u(z,)) M(za))dz .

VJl tends to 1 for a.e. (z a ) e ft and (2.10.146) yields

tf€AC

limjnf 4 ( « £ ) > J[N{*a) ■■ ez(u(z0)) +M(* a ) : K*(Z0) - Jn(N(za), M{za))\dz . Step 4. For each N e L 2 (fi, E 2 ) there exists a sequence of step functions {NA}A€N L 2 (fi,E 2 ) such that iVx -> W

C

L2(f2, E 2 ) as A —> oo .

strongly in

Each function iV^ can be represented in the form

K(A)

where l, ) = [ l . ^(^)loi \0, YSA

if

(*»)») € SIK(A) , otherwise otherwis .

6A = 1/A, diam QK(A) < £4 and fl = (J fix(>i). By using the previous step one has K{A)

limjnf J E > £ ) > / [ A U z J : e*(ufo,)) + A*0O : «*(*,) - j ^ J V ^ . M ^ J d z . n Passing to the limit on the r.h.s. of the last inequality we eventually obtain limjnf 4 ( v £ ) > j[N{za) n which concludes the proof.

: e*(u(z0)) + M(za) : K'(Z„) - fH{N{za),M{za)))dz

, D

Three-dimensional analysis and effective models of composite plates

151

Remark 2.10.14. In the second part of the above proof M(za) can be approximated sim ilarly as N(za), since M also belongs to L2(f2, E,). The proof is not influenced by such an approximation. Remark 2.10.15. The reader will now be able to perform dual homogenization and formu late a counterpart of Theorem 2.10.5. □ Remark 2.10.16. The scaling (2.2.3)2 concerning the body forces adopted in Section 2.2 has balanced the fact that the volume tends faster to zero than the area. Consequently the body forces entered into the homogenized problem, see (2.3.44) - (2.3.46). In this section we have proceeded differently and have not treated body forces in such a special way. Thus, by necessity, they do not enter the functional L given by (2.10.28) and do not affect the homogenized solutions. The scaling (2.10.1) was proposed by Caillerie (1984). We observe that one can either scale the elastic moduli or the loading. Remark 2.10.17. The method of determination of the effective stiffnesses of plates pre sented in Sections 2.3 and 2.10.4 refers also to the case of hollow plates, widely used in the building industry as ceiling plates. In the special case of hollows going in one direction the local problems (2.3.13) and (2.3.34) are posed on a plane domain, which simplifies the numerical algorithm, see Fig. 2.10.1. If the openings have slender cross-sections and are located transversely symmetric with respect to the middle plane one can apply the simpli fied formulae of Section 3.7. In the case of transverse asymmetry one should use more complicated formulae derived in Lewinski (1995).

□ □m

Fig. 2.10.1. A plate hollowed in one direction

2.11. Effective stiffnesses of longitudinally homogeneous plates Consider a plate of constant thickness whose elastic moduli CJ'*' vary only in transverse direction. Thus c" and c + are constant and Cijki = C^'(y3)

ho = C+ - C ,

,

c°0 = c^{y3) , 1 _ (y3) = -(c + C+) .

(2.11.1)

The planes y3 =const are assumed to be planes of material symmetry, i.e. (2.4.1) holds. Under these conditions the decompositions (2.7.2) take place with 9 ( a W = 0,

~Xial3(y3) = 0.

(2.11.2)

Elastic plates

152 Thus by (2.7.7) we find ^3|3 l(«9) Q(«9)

^ _

=

W

i

—3|3 - (( Qc / 3 )

Q(or/J) M ) =_ o A

H

n

= H(Q/3) = 0

Substitution of (2.11.3) into (2.3.21) and (2.3.38) gives

c~ c+

^

=

^/fe5^^(BS)dw'

^

=

^

'

(2 !

' * - 4)

c~ c+

v

Of = ±J(y3)C°^(y3)dy3, c~

where y3 = y3 — -(c~ + c + ). These formulae coincide with the conventional formulae for the stiffnesses of transversely nonhomogeneous plates. Note that the tensor C defined by (2.7.3), representing the reduced moduli of the plane stress state, appears here without extra stress-type assumptions. Remark 2.11.1 In the case of laminated plates in which all laminae satisfy the conditions mentioned above the expressions (2.11.4) assume algebraic forms coinciding with those reported in classical books on laminated plates.

Thin plates in bending and stretching

3.

153

Thin plates in bending and stretching

3.1.

Kirchhoff type description

Consider once again the plate problem of Section 2.1 with two additional simplifications: the body forces will be omitted and planes x3 = const will be assumed as planes of material symmetry, i.e. relations (2.4.1) hold here. In the approach presented here, the asymptotic analysis is preceded by the Kirchhoff construction of a two-dimensional plate model. This modelling is based on: i) kinematic assumptions: wa{x, x3) = ua(x) - x3w Q , z 3 = x3 w3(x,x3) = w{x); *3 = \J[{xtf - (x3)2]dx/J(xt - x3)dx, z z accordingly to the definition of y3, see Eq. (2.3.17). ii) Stress assumptions: stress-strain relations have the form °a0 = Cf A %,, where

^ = ev(u);

Cf* = C?* - c^Cf00, cx/ = Cf^/Cf33.

(3.1.1) (3.1.2)

(3.1.3)

(3.1.4) (3.1.5)

The formula given by Eqs. (3.1.4), (3.1.5) determines so-called reduced moduli of the generalized plane-stress state. They follow from elimination of e33 by assumption: a33 = 0. As usual, the elasticity tensor Cz is assumed to be positive definite. Kinematic assumptions (3.1.1) are simultaneously imposed on trial kinematic fields v = i),Eq. (2.1.5). Thus ff«ey(i>) = aa0ea0{v) = aa0[ea0(v) + x3na0(v)},

(3.1.6)

where va = va(x) - x 3 v Q) v3 = v. Consequently, Eq. (2.1.5) assumes the form f[Na0eQ0{v) + Ma0Ka0{v))dx = J\pazva - m>, Q +
(3.1.7)

where the loadings are p?(x) = (G$(x)M(x) m?(x) = (C+(x))*(x+(x) -

I»K(I)

+ (Gz(x))lr°_(x), + (G-z(x))Hx3(x) - x°3)r°_(x), (3.1.8) +

(Gz(x))^3_(x).

154

Elastic plates

The membrane forces and moments Na0 =

f

aa0dx3,

M'

M1)

«JW x3oa0dx3

' " /

(3.1.9)

*s(*)

are interrelated with deformations by

M*0 =

Ff^e^-u)

+ Df^KXlt(w)

(3.1.10)

where Afx»{x) Ef^ix) Xll

Df (x)

*J(x)

1 *3

- /1 M ) >)2.

Cfx,i(x,x3)dx3

(3.1.11)

and F Q | 3 A ' ' = E"13^ = EXfWl13.

The equilibrium problem of a plate clamped along its edge F = dQ reads:

find (u,w) e v#(n) = [^oH")]2 x ^o(n) (PW)

such that Eqs. (3.1.7) and (3.1.10) are satisfied forall(w,v)e V#(fi).

One can show that the problem (P (/f) ) is uniquely solvable provided that p | , mg, gf are el ements of the space L2 (fi). The outline of the proof reads as follows. The positive definiteness condition (2.1.6) implies positivity of the energy density No0ea0(u) + Ma0Ka0(w). Consequently the form 2W(ea0,

Ka0)

= Afx»ea0eXll

+ Efx^Xliea0 + Ffx»eXllKa0 + DfX»KalSKXli

(3.1.12)

is non-negative: W > 0. Since the problem isfinite-dimensionalone can find cQ > 0 such that 2W(e, K) > ci ]T(e a / 3 ) 2 + c2 £(*o/j) 2 a,0

(3.1.13)

a,0

If we assume that all quantities are dimensionless then we can put a = c2. By applying the Korn inequalities we note that the bilinear form of the left-hand side of (3.1.7) is V£-elliptic. Since this form as well as the linear form on theright-handside of (3.1.7) are continuous, we invoke the Lax-Milgram lemma to conclude that the problem (pM) i s uniquely solvable.

Thin plates in bending and stretching

155

Let us pass now to the local formulation of the problem ( P ( K ) ) . We shall draw all local consequences of the variational equilibrium equation (3.1.7). First, note that Na0ea0(v) = Na0vafi by symmetry of N and hence Na0ea0(v)

= (Na0va),0

- NQ0,0va

.

(3.1.14)

Moreover Ma0Ka0{v)

= -(Ma0v,a)i0

- Ma0,a0 v + ( M ° % «) i 0

(3.1.15)

and consequently f\Na0ea0{v) n

+

Ma0Ka0(v)}dx

= - j[Na0va

+ Ma0
n

+ j{Nal3n0va

- Matsn0v,a

+ Mafnav)ds

.

(3.1.16)

r

The right-hand side of (3.1.7) can be put in the form

/ [ P X + {Q + <Mdx n

- [™yavds.

(3.1.17)

r

Here n = (na) represents a unit vector outward normal to T = dfi. The clamping condition implies v = 0 and v Q = 0 on T, and the contour integrals in (3.1.16) — (3.1.17) vanish. Then, equating these formulae and using the classical lemma of variational calculus leads to the local equilibrium equations in fi: -Nf

= p° ,

-M%

= q + m°Q .

(3.1.18)

Thus the local (or strong) formulation of the equilibrium problem of a clamped plate amounts to finding the fields w, v, ea0(v), Ka0{v), Na0, Ma0 satisfying - boundary conditions on T: v = 0,

v = 0,

— =0onr, an

(3.1.19)

- equilibrium equation (3.1.18) in fi, - constitutive relations (3.1.10). Remark 3.1.1. The Kirchhoff theory of plates admits other types of boundary conditions on T. To disclose them let us substitute the expressions dv v,a = ^-na on

dv + — ra , as

va = vnna + vTra ,

vn = vana ,

vT = vara ,

(3.1.20)

156

Elastic plates

into (3.1.16) and introduce the notation Nn = Naf)nan0

,

Mn = Ma0nan0 ,

NT = Na0nar0 , MT = MaPnaT0 ,

(3.1.21)

Q = M«%nQ + *£ . Here T represents a unit vector tangent to T; d/dr = d/ds. Q is called the effective Kjrchhoff force. Thus we find f[Na0ea0(v) nn

+ Ma0KQ0{v)}dx = - f[Nfva n

+ Ma0,a0v}dx

+ I (Nnvn + NTvr + Mn (- — j + Qv\ ds + LR, r

(3.1.22)

where LR = -J~{MTv)ds r

.

(3.1.23)

Assume that V has m corner points Ok at s = sk, k = 1,..., m, where n and r jump. Then m

LR = ~Y}MT{sk

- 0)v(sk - 0) - MT(sk^ + 0)v(sk^ + 0)]

(3.1.24)

fc=i

with so = sm. Assuming that v is continuous at sk we represent LR in the form m

LR = Y^Rkv(sk) ,

(3.1.25)

Rk = MT(sk + 0) - MT(a* - 0)

(3.1.26)

where

or Rk = [MT](sk). The theory of thin plates admits 24 = 16 types of boundary conditions, since one can assign the values of Nn or vn, NT or vT, Mn or (-dv/dn), Q or v. The most important boundary conditions are (i) free edge; Nn, Nr, Mn, Q are given, (ii) simply supported sliding edge; Nn, NT, Mn, v are given, (iii) immovable supported edge; vn,vT, Mn, v are given, (iv) clamped edge; vn, vT — dv/dn, v are given.

Thin plates in bending and stretching

157

At free corner points Ok, R-k = 0. If v(sk) = 0, then Rk are concentrated reactions and are a priori unknown. Assume that the plate is rectangular with corners at points Oi(0,0), 02(0., 0), 03(a, b), 04(0, b). If the field M12(x) is continuous in fi, then Rt = 2{-l)i-M12(Oi).

(3.1.27)

3.2. Asymptotic homogenization. In-plane scaling approach If dimensions of Z = (0, if) x (0, i|) are much smaller than diam(fl) the (P (/f) ) problem becomes intractable even by numerical methods. Thus it is thought helpful to use the twoscale asymptotic method in order to separate the local analysis within the periodicity cells from the global analysis of the plate behavior. The asymptotic analysis used in this section will be called in-plane scaling approach, since, in contrast to the scaling used in Section 2.2, the scaling concerns only the in-plane dimensions /* of the cell of periodicity. Instead of considering problem (P (/f) ) we shall deal with a family of problems (Pj ') indexed by e > 0. This family is constructed by replacing Z ~» eY,

lza ~» ela,

/* = e0la ,

*3(T)^XMI)=T3±(?)=£°C±(!)' (3.2.1a)

Gg(x)~ G± g ) = Gf ( ^ ) , Gf'(x,x 3 ) - &>« (*,x 3 ) = Ct

( f *,z 3 ) .

Consequently we replace Kx[x)

- K Q

= Kz ( ^ )

paz{x) ~» pa (x, | ) , Qz(x) ^* q (x,-j

; K =

A,E,D

m?(x) ~» rha (x, | ) , ,

(3.2.1b)

r*(x)-^r|(x),

where p°(x,y) = (G+{y))lr°(x) + (G_(t/))ir°(x), m°(x,y) = (G+(y))*(x+(y) -i§)r?(x) + (G_ (
(3.2.1c)

9(1,») = (G + (y))iri(i) + (G_(j/)) V (x). Thus moduli Ka/3X», K = A,E,D,are interrelated with moduli Ka0Xti (cf. (2.7.13)) by A(y)=E0A(y),

E(y) = (e0)2E(y) ,

J>(v) = (^o)3D(y) ■

(3.2.Id)

158

Elastic plates

The scaling (3.2.1a) will be named in-plane, since the transverse plate dimension remains intact under this scaling. The e-dependent counterpart of problem (P'K^) reads: find (u£,wc) e V$(Q) such that

(Pj*0)

j[Nf{ue,wc)ea0{v) + M^(ue,we)Ka0{v)]dx = n = \pa [x, - ) va - rha (x, - ) riQ + q (x, - J v]dx V

where Nf{u',w*)

(3.2.2)

(v,v)GV°(Q),

= Aa^ ( ? ) e A > e ) + Ea^ ( ? ) KXli(uf) ,

A f f V , uf) = E^x" ( ? ) eXll(u*) + Da^ ( ? ) K^V?) .

(3.2.3)

We see that now ( P ^ ) = (P(*>). We assume that p°, mQ, q e L2(fi x y). Let us proceed now to construct asymptotic process of solving the (P£(/f)) problem. We use a two-scale asymptotic expansion method. The solution ue, wc is represented in the form wc = w<®(x) + e2w& ( x , ? ) + £3W(3) ( x> £ )

(3.2.4)

+

The trialfieldsare expanded similarly

v^ = v^(x)+ev^(x^)+e^)(x,

?) + . . . , (3.2.5)

v = w«°'(i) + eV 2 ) (x, ? ) + eV3> ( ^ ? ) + . . . . It is assumed that (u<°\ w<°>) e V%, (v(0), v<°>) e V£,

uL'H^O.^Hx,-) 6

tf^r(y);

«;(2»(x,-),«(2)(^-) G HUX).

(3.2.6)

The spaces //£, r (y) have been defined in Sections 1.3 and 2.8. Thefirst-orderterms of the deformation measures assume the form ea0(vc) = £/> + 0(e) , 2

Kapivf) = K°a0 + 0 ( £ ) ,

e°a0 = eQ/3(u<°>) + e ^(u<"),

(3.2.7)

K°, = Kap(v^) + K%(w^) ,

where the strain measures evap, Kvaf3 are defined in Section 2.8. Thefirst-orderterms of the stress and stress couple resultants read

Nf = Nf + 0{e),

Mf = Mf + 0(e)

(3.2.8)

Thin plates in bending and stretching

159

where NS" Nf = = A*» Aa0X" g(-)) el e°„ + + E°^ Ea^ (Q Mf

= E°^

Q

K%,

(3.2.9)

e°„ + Da0x» ( - ) K ^ .

(3.2.10)

Further analysis is based on the averaging result (1.1.1) for Y-periodic functions. Let us substitute (3.2.8) and va = % , v = i/ 0) into Eq. (3.2.2) and pass to zero with e using (1.1.1). One finds = [(pOyW - mav^ + qv^)dx

f[Nfea0(v<-V) + MfKa0{v^)\dx

V («eV 0 >) e v$(tl), where

N? = (Nf), a

Q

< = «>);

(

a

p (x) = (p (x, I/)) , m (x) = {rha(x, y)) , q(x) = (q(x, y)) , and (•) means averaging over Y. The loadings pa,ma, q coincide with effective loadings (2.3.63) introduced in Section 2.3. Now let us substitute Eqs. (3.2.4) and the whole expansions given by Eqs. (3.2.5) (first two terms are sufficient) into Eq. (3.2.2), pass with e to 0 and combine equation thus obtained with Eq. (3.2.11). Then one arrives at / W e y ^ " ) + MfKlB(vW))dx = 0 .

(3.2.13)

Let us aassume s s u m e t' v£] = £a{x)va(y), u(2) = r}(x)v(y), £Q, V € D(Q) and (v, v) e HK,per{Y) where 2 2 U H (Y) . (3.2.14) KtPer(Y) } = -;,per\' — [HUY)} ["per\' l\ x ~ H "per Due to £Q, 7j being arbitrary functions of D(fi) onefindsthe equation (KPeva0(v) + MS0^))

=0

V(«, v) e HKiPer{Y) ,

(3.2.15)

for the determination of unknown functions (u(1)(x, •), w(2)(x, •)) € ///clPer(^)Since this problem is linear one can represent its solution as follows ttd) = T^\y)el

+ U^(V)KI

,

w™ = X^y)^

+ X^Hvhl

•

(3.2.16)

We recall that ^

= eAM(«(0)),

<

= KA/>(0)).

New functions in y 6 Y involved in Eqs. (3.2.16) are defined as follows.

(3.2.17)

160

Elastic plates

Let us introduce the bilinear forms

2

foru,v e [H^Y)} ,

b(u,v) =

(A^yy^uy^v)),

eK(w,v) =

(E°"»{y)K\ll(wWaf,{v)),

dK(w,v) =

(D°^(y)Kl(wK0(v))

(3.2.18a)

v,we H^.(Y). Note that eK(w, v) = (eQ)2eK(w, v) ,

b(u, v) = e0b(u, v) ,

(3.2.18b)

dK(w,v) = (e0)3dK(w,v) ,

cf. (2.7.14), (2.8.2). The newfieldsinvolved in Eqs. (3.2.16) are solutions to the following problems: find (T 0 ^, XQ"> e HKtPer(Y) such that b(Ta0\u)

(PL

+ {Aa^{VyXll{u))

+eK(XW,u)

= 0,

eK(w,TW>) + dK(Xe#>,w) + (Ea^(y)K%(w)) = 0 V (u,w) £ HKiPer(Y) ■ find (Uia0\X{a0)) b{U^\

(Ply)

(3.2.19a)

e HK:Per{Y) such that

u) + eK(XW\u)

+ {^^(yy^u))

eK(v>,Ul°V) + dK(XW\w)

= 0,

+ ^ ( ^ ( w ) ) = 0, V

(3.2.19b)

(u,w)£HK,per(Y).

Compare problems (P£,y) w ' ^ (Pfcy) of Section 2.8. On taking into account relations (3.2.18b) one concludes that functions T (a " } , x (a/J) coincide with those defined by (PK,Y)\ moreover ^ x
e a/3 — >->~rt )u. +

K

*&.n \ 1 "•Q^^A/i

_ WX,1fh -4- PX,1Kh a0 ~ "a/3 e Af» ^ ra0K\n <

„0 K

(3.2.21)

where

PZ = e%(UM),

P& = £$ + W])

(3.2.22)

!

here 5(aSf = A5a50 + «*£<$)• Thus ihe quantities (3.2.9) can be interrelated with defor mations (3.2.17) as follows N^ = Afx^l

+ E^^Kl,

MS0 = F^0X"el + Dfx"Kl,

(3.2.23)

Thin plates in bending and stretching

161

where Afx"

= Aa™S%

paPXfi _

ira0-t6 c V

+ Eaf*sW%

E^0X" = A"I*SR% + Ea^sP^

,

i r)a/3-y<5u/A»»

r^apXfi _ pa0-,S

pA(i , Qa0iS

, pX/i

(3.2.24)

According to the definition (3.2.12) the constitutive relations of the effective plate assume the form r«A

a/3

Ml

+ D:pffx^ K

(3.2.25)

where the effective stiffnesses are given by -taPX/i

paffXfi _

ipa0\n\

Do,0X»

a0\n\

= (ES

(£>a0AM)

=

(3.2.26)

Let us substitute (3.2.1a) into (3.2.24) and take into account (3.2.20). Then comparing (3.2.26) with (2.8.10), (2.8.6) we conclude that stiffnesses given by (3.2.26) coincide with stiffnesses determined by (2.8.10). Consequently the stiffnesses (3.2.26) possess the re quired symmetry and the matrix A" =

AhEh Fh Dh

is positive definite, cf. Sec. 2.8. The homogenized problem: find (t*<°>, w(0>) € V$(fi) such that Eq. (3.2.11) (Fh)

holds for each (u (0) , T/°>) 6 V%(ft), the effective constitutive relations being given by (3.2.25),

is well posed. The homogenized potential is given by Wh = (K0eha0

+ M?K%,)/2

,

(3.2.27)

or W„ = ((K0){e%)

+ (Mf)(K°a0))/2 h

.

(3.2.28)

h

Obviously, W/, is a strictly convex function of e and K . By making use of all identities produced by substituting u = r ( A f , ) or U{Xfl) and w = X (a/5) or x{a0) into Eqs. (3.2.19), (3.2.20) one can rearrange formula (3.2.28) to the form: a/3 r° Wh = « " C + AC^>/2

(3.2.29)

Equivalence of expressions (3.2.28) and (3.2.29) confirms that the homogenization process satisfies the consistency criterion (2.3.67) of Hill.

162

Elastic plates

Remark 3.2.1. By exploiting the properties of the elasticity tensor C it can be shown that the matrix My) E{y)

Mv) = F(V) D(y) is also positive definite. Problem (P£ ) is equivalent to the following minimization problem: find Jv(u',vf)

= in[{J£(v,v)\(v,v)

6 VJ?(0)} ,

where

Mv>v) = 2a'(v'v;

v v

' "> ~

Cc v v

(<)

and a'(v,v; v,v) = J[(Af^e„(v) + (Ef^ex^v)

+

Ef^KXli(v))ea0(v)

+ Dfx»KXll(v))Ka0(v)}dx

,

Cc(v, v) = / |pP (x, - j va - rha (x, - J v Q + q (x, - j v}dx . n We recall that A?"A"(x) = Aa0X,i (x, - V etc. It is not difficult to prove the following convergence result. Theorem 3.2.2. The sequence of functionals {JE}e>o is T-convergent in the weak topology of H^Cl)2 x ff2(fi) to, cf. (3.2.11) and (3.2.27) Jh(v, v) = / Wh(e(v), n(v))dx - Ch(v, v andCh{v,v) denotes the loading functional appearing on the r.h.s. of (3.2.11). Moreover, at least for a subsequence still denoted by {ue, WC}C>Q we have (uc,wc) -* ( u ^ . t u ^ ) weakly in H^(Q)2 x H$(Q), where Jh(vF>,wl0))=M{Mv,v)\(v,v)

6 V°(Q)} .

a

Thin plates in bending and stretching

163

Consider the transverse symmetry case. Then X1-"® = 0, U(a0) = 0. Problems (P£ y ) assume the form find T(a0) 6 H^.(Y)2 such that

(Pk KS,Y)

6(T
V u 6 //^(K)2

=0

find x(a0) G W^r(^) such that

(/*KS,Y)

dK(x(a/3),v) + (DapxnyWx,(v)) = o

v ve/^(r)

(3.2.30)

(3.2.31)

The homogenized stiffnesses are given by

DfA" = ( D ^ + D a ^/c^(x (AM) )>,

££*** = 0.

(3.2.32)

The alternative formulae will be given in the sequel. Let us consider the strong form of the local problems (P£s y ). By standard arguments of the variational calculus one arrives at: A. The strong formulation of ( 4 s y ) : find Tia0) € Cl(Y) such that T{a0) assume equal values on the opposite sides of Y and functions H&) = ^ W (2/)e^(T' Q «) + Aa0X»(y)

(3.2.33)

satisfy: (i) differential equations

*&)!„ = °

iny

'

(1134)

where ()|M = 5/5yM, (ii) boundary conditions: N

W)W\

and

N^wx,

assume equal values on the opposite sides of Y; /x and r are unit vectors normal and tangent to dY. (iii) Continuity conditions along line 7 of discontinuity of stiffnesses Aa0X,i(y):

X

A

assume equal values on both sides of 7; here the vectors v and T are outward normal and tangent to 7 and are common to both sides of 7. B. The strong formulation of (PKS.Y)- find x(a0) e ^{Y) such that (0 X

d\{a0) assume equal values and —^ OfJ.

opposite values on the opposite sides of Y.

164

Elastic plates

(ii) The functions

+ Da0X»(y),

< % = D**(y)K«s(XW)

(3.2.35)

satisfy: - differential equation <%|A M = 0

in Y ;

(3.2.36)

- boundary conditions: M$°® = M^0)fj,\fip assume equal values on the opposite sides of Y ; Q(a0) = < % I* ^ + |l( M (t%MA^)

(3.2.37) (3.2.38)

assume opposite values on the opposite sides of Y; - equilibrium of point reactions at the corners Ot of Y: * £ ( - I ) ' J » C ) ( Q O = °;

<3-2-39)

>=i

this condition is identically satisfied. - continuity conditions along line 7 of discontinuity of stiffnesses Da/3x>'(y): ^ and g
M
3.3. Refined scaling approach Within the framework of the in-plane scaling based approach of Section 3.2 thickness of the plate is viewed as e-independent. Thus this analysis runs counter to the asymptotic analysis of Section 2.2, based upon the scaling (2.2.1) according to which the transverse dimension of the periodicity cell goes to zero simultaneously with in-plane dimensions. The aim of this section is to show that the latter scaling applied to Kirchhoff plate equations leads to the homogenization formulae found by the in-plane scaling. Scaling (2.2.1) will be from now onward called refined scaling, since it preserves three-dimensional shape of the Z cell, see Fig. 1.1. Similarly to Section 2.2 let us assume the refined scaling defined by Eqs. (2.2.1) and (2.2.2). Consequently we replace (3.3.1) Efx»(x)

~» e2Ea0^ ( - ) ,

Dfx"(x)

— e2Da^

(-) ,

and the arrows become equalities for e = e0. Hence A = A/e0, E = E/(e 0 ) 2 , D = D/(s0)3.

Thin plates in bending and stretching

165

To compensate for the loss of stiffnesses for e —» 0 we scale the loadings q (x, *) - £ s+1 q (x, ^ ) ,

F ( i , ^ £ 2 3(f) Problem P
(P^)

+

Mf(tf,w*)Ka0(v)\dx

n

= /"[e* PQ (x, ^ ) «0 - £ s+1 m Q (x, ^ ) w,a +
(3.3.3)

V(w,t;)eVj?(n), where Nf{u£,w*)

= e i Q ^ ( - ) eAM(w£) + e 2 £ Q ^ ( - )

Mf{ue,wc)

= £ 2 £ < ^ ( - ) ex^u^+e3^^

KXtl(vf)

, (3.3.4)

(-)

« AM (W £ ).

By virtue of relations (3.3.1) - (3.3.4) solutions to problems (P^K)) of Section 3.3 and (A ) above are interrelated by uX = I —

£o

(3.3.5)

Moreover, the following homogeneity relation for the plate potential VV£ holds: ,2s-l

We(w,w) = ( - ) * W£

—)

u, I — |

w

(3.3.6)

Just this homogeneity relation shows that both in-plane and refined scaling approaches are equivalent in the considered case of Kirchhoff plate modelling. It will turn out that, if applied to different plate modelling, both scalings yield different results. Owing to the linearity of the model, the parameter s > 2 was indetermined. This param eter assumes the fixed value s = 3 if the asymptotic analysis is applied to the geometrically nonlinear von Karman equations of thin plates; this will be discussed in Section 4.

166

Elastic plates

3.4. Variational formulae for effective stiffnesses We say that strain measures eag and Kag are kinematically admissible and write c e K e (n),

K 6 KK(tl),

if they satisfy the compatibility equations. Thus Ke(Si) = {e = {e3a) € L2(ft, E 2 )| fs^e0adx n

=0,

V a € D(ft, E 2 ), div a = 0} ,

€ L2(ft, E 2 )| JM^nBadx n V M e D ( f i , E * ) , div div Af = 0} ,

£ « ( « ) = {K = M

=0 ,

fl being arbitrary, even multiconnected. Kinematically admissible strains €#» are associated with a displacement field u = (ua) such that eBa = -(ua,/3 + Ufl,a). Similarly kinematically admissible strains nag are associ ated with a scalarfieldw such that Kag = —wiag. Having in mind application to the determination of the homogenized potential we intro duce two spaces: K%T(Y) = {ey e L2(Y, E2)|e« € K.e{Y) and e" is associated with a y-periodic displacement field}, KJTW = {«" e L2{Y, E])\K? e Xis(y) and K" is associated with a V-periodic displacement field}. The local potential is given by W(e»,K>>) = ^ A ^ ( y ) ^ ^ + J ^ f o ) ^ +£<" 3 A '%KX/ 3 + D^iyW^J

,

(3.4.1)

and the homogenized potential (3.3.27) has the form W„(e*, Kh) = \[Afx^a0el

+

K0X"4A

+K0XtL4Ap + Df*"«0}

■

(3-4.2)

Thus we have W„(e\/c A ) = min{(W(€»,«»)>!€» € K^(Y), K" e /CJT(y) and <e") = e \ («») = Kh} .

(3.4.3)

Thin plates in bending and stretching

167

To prove (3.4.3) let us note that the formula (3.2.29) can be rearranged as follows Wh = (W(e°, K 0 ) ) ,

(3.4.4)

where e°, K° are given by (3.2.7), or e° = £h + ey{uw)

,

K° = Kh + K?{wm) ,

with(u (1) ,u/ 2 >) € HKtPer(Y). By definition of {u^\w(2) (3.4.4) can be represented by

(3.4.5)

),cf. Eq. (3.2.15), the expression

Wh = m i n ^ e " + e»(v) , « " + K»(V)))\(V,V)

€ HKjxr(Y)}

.

(3.4.6)

Let us define a set A = {(e,K)e

L2 (Y, E 2 ) 2 | e = e»(u), K = «»(«;) , ua = ihQ0y0 + aea0y0 + va ,

f = ^ < j v V + «W/° + " ,

(v,v) & HKyPer(Y)}

,

(3.4.7)

where a, ba £ F and eap stand for the components of Ricci's pseudotensor. Note that (c, n) £ A means that e = e" + ey(v) ,

K = K" + K » ,

(3.4.8)

where (v, v) £ /// C p e r (K). Thus potential Wh given by (3.4.6) can be put in the form Wh = min{(W(€,»s)>|(e,*s) € A} ,

(3.4.9)

and since (c) = (e w («)) = eh, (K) = (« w (w)) = nh we arrive at the representation (3.4.3). In the case of transverse symmetry the formula (3.4.3) splits into - formula for £>>,: 4t,Df**l

= m i n { ( ^ D ^ ^ ) | K» £ K%(Y),

<«») = «"} ,

(3.4.10)

- formula for Ah: ^ K ^ l

= m i n ^ ^ y l ^ ^ ) | e* £ ^ ( V ) , <€»> = € h } .

(3.4.11)

Remark 3.4.1. If fi is a connected domain and sa0, Ma0 involved in definitions of £ e ( f i ) , AC/c(fi) are twice differentiable, then div s = 0,

div div M = 0 ,

(3.4.12)

168

Elastic plates

provided that s, M are represented as follows Sn=«22(
S22 = KU{lfi)

n

M =e22(
M

22

= euM,

S12 = -K12{
M

12

,

= -daO) .

(3 4 13>

' '

In such a case the sets of kinematically admissible strain measures can be defined as follows /C'c(n) = {ee L2{Q., E 2 )| J[ellK72{)]cte = 0 VyeD(fi)},

(3.4.14)

/CK(fi) = {« 6 L 2 (n, E 2 )| [[nueniip) + n22eu{
(3.4.15)

Here the primes indicate that additional regularity assumptions are imposed. 3.5. Correctors In Theorem 3.2.2 the convergence of (uc, w€) to (u(0',ti;(0)) is only a weak one. Conse quently the first gradient of u€a and second gradients of we are, in general, poorly approxi mated. To achieve strong convergence we make the following assumptions: u{°] e // 3 (fi), T{0i)

{a0)

x

e

tyi,oo(y)

(

2

e w '°°{Y),

w'°> e // 4 (fi), Ujfh) e vyi,oo(y) s

(3.5.1) (3.5.2) (3.5.3)

^(a/3) g jy2,oo(y)

Let us introduce the functions v* = u£ - u<°> - eu^ , (1)

(3.5.4)

/ ^ ' - / I - E V *

2

(1)

2

(2)

where u , u/ ' are given by (3.2.16). The functions eu and e w are called correc tors. As we shall see, the functions u (0) + ew(1) and u>(0) + £2u;(2) provide much better approximations of it' and vf respectively than u' 0 ', u/°> alone. Below we tacitly assume that loading functions are square integrable. Theorem 3.5.1. Under the assumptions (3.5.1) - (3.5.3) we have vc—»0

strongly in Hl{Q)2,

zc—>0

strongly in # 2 (ft),

when e - » 0 . Proof. From (3.2.16), (3.5.1)! and (3.5.2) we conclude that v% -> 0 strongly in L2{0) as e —» 0 ; similarly z? —> 0 in Hl{Q). It is thus sufficient to show that a t (v £ ,z £ ; vc,ze)

Thin plates in bending and stretching

169

converges to zero as e —► 0. We calculate, cf. Remark 3.2.1 ac{v£,z^; vc,zc) = a (u - «<°) - eu^\w - wM - eV 2 >; u£ - «<°1 - eu<-l\ w£ - w<°) - + euM.wW + e V ) + A£ , (3.5.5) £

£

c

where A£=a£(u^+eu^,w^

+ e2w^; u<°> W \ w < ° > + eV 2 >) .

(3.5.6)

We recall that the loading functional Cc is defined in Remark 3.2.1. Let us assume for the moment that limAe = £ h (u ( 0 ) ,iu ( 0 ) ),

(3.5.7)

where £h() has been defined in Remark 3.2.1. Then, by virtue of (3.5.5), we obtain \\ma£(v£,z£;v£,z£)

= 0<

£—*0

and the theorem follows because l i m £ £ ( u W ) = £fc(u(0),w(0>) . £-•0

In this way it remains to prove (3.5.7). To this end we calculate eQ/3(u<°> + euW) = ea0(uW) + eea0(u^) Ka0(wW + e2wW) = Ka0(wM) + e2Ka0(^2))

+ eva0(u^(y)) ,y = x/e + <M2)(V))

We recall that both tt(1) and w(2) are functions of x and x/e. Hence A£ = / { [ / ^ ( e v ( u « » ) + « * > < » ) + e ^ u ' 1 ' ) ^ ) ) +

££^(KA><°>)

+ eea0(u^)

+

+ e2^(w™)

+ ^( W ( 2 »)(^))](e^(«'(0)\

ei0(u^)(-))

+ D*"(/c v ( W (°')+£ 2 % ( W ( 2 ») + + S2Ka0(wM) +

*^(wm)(*))Mv>'„h

Kl0(w^))}dx

= A\+eJ{[Af^{ex,{u^)

e

+ e%{u(!))(£))

,V = x/e.

170

Elastic plates

+ E^^wM)

+ <> (2) )(f ))]ea„(u)

+ Ef^(Ka0(w^)

+ ^( W ( 2 ')(^))e v (u' 1 »)}dx

+ e2J[Af^ex,{u^)ea0{u^) + Ef(ex,(uM) + 2Df(KXli(wM)

+

+ Ef^KXli(w^)(ea0(u^)

+ e^u'1')^))

eyXll(uM)(t))Ka0(wW)

+

Kl^^K^w^dx

+ e3j[Ef^KXli(w^)ea0(u^) + e4 JDf^Ka0{w^)KXll(w^)dx

Ef0x"eXll(u^)Ka0(w^)}d2

+ ,

where A\ = j[Af^{exM°]) n

+ ^> (1 >)(^))(e a ,(u<°>) + ^ ( u " ) ) ! ^ ) )

+ Ef(KX»(wW)

+ ^(^ 2 ))(^))(e Q / 3(« ( 0 ) ) + 0 ( 1 ) ) ( f ) )

+ Ef^(ex,(u^)

+ e%(uW)^))(Ka0(WM)

+ Df*»(Ka0(wW)

+ ^(«,< 2 »)(^))(/c A ,( W (°») + «j!>< a >)(±))]dx .

(3.5.8)

The integrands associated with s, e2, e 3 and e4 are functions bounded in L1^). remains to find the limit of A ' when e —> 0. Equation (3.2.16) yields

Thus it

<0(™{2)) = Kya0(X^)eXll(uW)

+ 0(2>)(^))

+ ^0(xMWMO))

■

In accordance with (3.5.2) and (3.5.3) we infer that eva0(um) € L2{Q) and Kya0(wm) e L2(fl). Also, we observe that Ka0{wW) = X(A">KQ/3(eA>(0))) +

X(A*W«A>(0)))

•

Hence, on account of (3.5.1) and (3.5.3) we deduce that na0(w^) £ L2(Q). Similarly, (3.5.1) and (3.5.3) yield eQ/3(«(1») e L2(fi). Consequently, by using (3.2.9) - (3.2.13) and Theorem 1.1.5 we find limAf = J±J[A°^(y)(ea0(uW)

+ e > " ) ) ) ( e ¥ ( « 0 ) ) + e A >< 1 )))

Thin plates in bending and stretching

171

+ Da^(y)(Ka0(w^)

= J\K0eag(u^)

+ /&(u; ( 2 ) ))(KA,ry o ) ) + < > ( 2 ) ) ) ] ))ds

[Nfea0(u^)

+ MfKa0{w^)\dx

= Ch(ul°\wM)

Thus the theorem has been proved.

□

Remark 3.5.2. In the last theorem the strong convergence takes place in H1 (Q) 2 x H2(11). Since w(0) 6 HQ(CI) and w(0> G H$(il), a natural question arises: how to construct correc tors leading to strong convergence in 77Q(Q) 2 X HQ(Q.)1 To this effect we introduce cut-off functions mc satisfying the following properties (i) (ii)

m,eD(ll), mc(x) = 0 if

d(x, T) = (distance from x to Oil) <e ,

(iii) mc(x) = 1 if c((:r, T) > 2e , (iv) e | Q '|D Q m £ (x)| < ca , VQ, where ca depends on a but is independent of e . Here D1 = V = gradient, D2 = V 2 (|Q| = 2), etc. Such functions me exist, provided F is smooth enough. In our case it would be sufficient to take m £ e C 2 ^ ) and to take the a's with |Q| = 1 and \a\ = 2. Obviously the above conditions still have to be satisfied. The correctors are now defined by «<" = emcu^

,

w?> = e2mew{2) •

(3.5.10)

We are in a position to formulate the second result on correctors. Theorem 3.5.3. Under the assumption (3.5.1) - (3.5.3), if u i " and wf] are defined by (3.5.10), then vc = uc - u ( 0 ) - ui1] — 0 e

z* = w - w

{0)

- wi

2)

-»0

in Hi (Q)2 in H${ty

2

strongly, strongly.

Proof. The proof runs precisely along the lines of Theorem 3.5.1.

□

172

Elastic plates

3.6. Variational formulae for effective compliances. Dual effective potential We say that local membrane forces n a/J are statically admissible, and write n € 5f er (F) if they satisfy: ST(Y)

= {ne L\Y, E$)| < n Q " e * » > = 0 V

Here /i represents a unit vector outward normal to dY. Let r represent a unit vector tangent to dY. Let us define m,, = ma0nafi0

Am

9 = namae}0 + -^— ,

,

m , = ma/Vc.7/j

(3.6.1)

for m = (TnQ") e L2(F, EJ); s parametrizes d r and <9/<9s = d/dr. Statically admissible local moments are such that m e S?T(Y), where S r ( ^ = {m € L2(F, E5)| ( m ^ f o , ) ) = 0 V

eA„(«£) = a w Q Wf» + e w (^) Mf , (3.6.2) «A M (^ £ )

= ex^0 ( | ) A/^ + d

w

( | ) Jtff .

Consequently the dual potential assumes the form W(N,M)

= \[aa0Xll(y)Nal3N^

+

+ eal3xMMa0Nx»

ea0Xll(y)M^Na0

+ da^(y)M^Ma0}

.

(3.6.3)

Its effective counterpart is given by

W'h(Nh,Mh) = lia^NfN^

+

e^M^K0

+ CxX'W0 + dU.MJ?Mf).

(3.6.4)

The following variational formula Wh(Nh,Mh)

= inf{(W*(n + Nh,m + Mh))\n e S^Y) m € SF(Y), (n) = 0, (m) = 0} h

h

h

h

, (3.6.5) h

determines the effectiveflexibilitiesa \ e , f , d . The tensors e and f e Q/3An = f\iw,0- Formula (3.6.5) can be expressed as follows Wh{Nh,Mh)

= inf{(W(n,m))|n 6 ST(Y), m € S?T(Y), (n) = Nh, (m) = Mh} .

are linked by:

(3.6.6)

173

Thin plates in bending and stretching

The formula (3.6.5) is a particular case of the analogous formula for the effective dual po tential for thin periodic shells. The formula for shells will be rigorously derived in Section 17.2 by using the duality theory. Therefore a less complicated derivation of Eq. (3.6.5) is omitted here. In the case of transverse symmetry the formula (3.6.6) splits into - formula for dh A C < 3 A / X " = i n f ^ m ^ c W ^ m ^ ) | m e ST(Y),

<m) = Mh}

(3.6.7)

(n) = Nh} .

(3.6.8)

- formula for ah Nf*UX* 3.7.

= ^ { ( t i ^ A ^ X " ) |n € S n n

Transversely symmetric plates periodic in one direction

Assume that the moduli A"0^, D00^ are /i-periodic functions in z/i and independent of y2. The aim of this section is to derive the formulae for D^x>i. The formulae for A^3^ will be given without derivation. Our task is to solve {P^S,Y) m ' ^ s f r o n g formulation, cf. Sec. 3.2. The functions M^ J g) (a« are ^-independent, hence Eq. (3.2.37) reduces to -£nll(2/i)

= 0.

d{yy?

(3.7.1)

Conditions (3.2.28) imply
2

(0) =

tPXlaff) d(y\

(3.7.2)

-(h

and (3.2.35) give dy(Q«

(3.7.3)

dy\ By (3.7.1)-(3.7.3) we get K

n(X

«, 2 (x

) (aW

+

Dlul(yi)

{a0)

) = ^{x )

(BIW)

D1111(l/i

(3.7.4)

=0

where (3.7.5) o

Note that Eq. (3.7.4) is sufficient to evaluate the effective stiffnesses. The complete char-

174

Elastic plates

acterization of x'Q"' is not necessary. By applying the formula (3.2.32)2 one finds D

iin

((D"ii)-i)-i

=

,

nu

/D \ ~\Dnn/ /DU22\

h

1222 _

Dnn

/ni222v

= {Dnn}

h

'

_ /Dll*D2™\

(Dun)2\ _ ^^_j_j

^ /^1122\ / ^ ! ! \

+

^

nllii

(3-7.6)

/D12u\2 J D„„ .

The remaining stiffnesses can be found using symmetries (2.7.23). Assume now that the plate material is orthotropic with orthotropy axes coinciding with ylt y2. Then Dnu = D2221 = 0 and formulae (3.7.6) reduce to pim

((£>ii")-i)-i ,

=

£,1112

=

D222i =

0 (

n22 )ii 2 2 \

/D \ \\ riim D2222 //'

fc

h

?2 22 2 = ( D 2 2 2 2 ) _ / ( ^ \ + £,1212

/g^\ 2 D nn

(3.7.7)

(£,1212)

=

Tofindsimilar formulae for effective compliances let us introduce functions: f(x) = (x- 1 )- 1 , j(x) = (x),

5(x,j,)=(|)/(x),

(3.7.8)

Note that Dlin=f(D»»), D™2=j(D™2), D1hl22 = g{Dnn,Dn22), D2222 = h{Dnn,Dn22,D2222) In the orthotropic case the inverted constitutive relations have the form «n = duuM11 + dunM22 , «22 = d22\\Mn + d2222M22 , K12 = 2di2l2M12 .

(3.7.9)

(3.7.10)

The effective compliances for the bending problem are given by ^1111 = ^(^ 2 2 22 ,dll 22 ,rfllll) , C

'2222

=

^n 2 2 = ^(^2222, ^1122) ,

/ f a ) . d1212 = /(dl 2 l 2 ) •

(3.7.11)

Thin plates in bending and stretching

175

These formulae follow from (3.7.9) by a direct algebraic computation. Let us report now the counterparts of formulae (3.7.6) for effective in-plane stiffnesses; the derivation will be omitted to save space, cf. Section 5.6.1 where a short outline of a similar derivation is reported. Define

J1212

■ A1212 - (A 1112 ) 2 , d22 = (

1 1

du =

d2 I '

A = dnd22 - (di2)2 ;

jr),

(^1212^1122 _ ^1112^1222\ \

di = (

-=2

(3.7.12)

/ (^1111^1222 _ ^1112^1122']

d2 = '

) ,

d

\
"

/ '

\

d?

The effective stiffnesses read ,.1111 _ ^22

Ah

~~K'

,1122 _ Ah

.1112 _

Ah

(d^di

di2

-~~&'

— d\2d2)

2212 _

A:

—

. 1 2 1 2 _ ^11

Ah

> '

h Ah

~A ' (^11^2 -

di2di)

A

—

[4111W 41222-12 , 41212/41122x2 / x2222 _ K 1 \ A ) + "■ (A I

^2222 _ ^h

o n4 I I I 2 L

A1122 42212] J\ J\ ]

[

dn(d2)2-2dUdld2\

For the case of orthotropy the effective membrane stiffnesses A^x,i are given by A\nl

= f{Ann)

,

A1212 = f(A1212)

A2222 = h(Auu,An22,

Alhl22=g(Ann,A22u),

,

A2222) ,

A"12 = Aj?22 = 0 .

(3 7 14)

' '

Let &a0\)i be compliances for the membrane problem. The effective compliances in the orthotropic case are given by a

a

llll

=

M a 2 2 2 2 , ^1122, O l l l l ) ,

1112 = °2122 = ° >

a

1122 = #( a 2222, O1122) ,

°2222 = / ( a 2 2 2 2 ) ,

(3.7.15)

af 2 1 2 = j ( a i 2 1 2 ) •

Let us emphasize an analogy between formulae in the orthotropic case for: (a) effective bending stiffnesses and effective compliances due to stretching, (b) effective bending compliances and effective stretching stiffnesses. The mutual formulae can be found by replacing indices: (1,2) —► (2,1). 3.8.

Ribbed plates. Bending problem

Let us consider a transversely symmetric plate made from strips of constant alternate prop erties. Our aim is to find effective bending stiffnesses D J J * \ This problem has already

176

Elastic plates

been solved and its solution is given by (3.7.6). These formulae, however, do not take into account that D"0^ are piece-wise constant. Moreover these formulae are not explicitly dependent on the direction of stiffeners. The objective of this section is to derive the for mulae for effective stiffnesses DffXl1 as explicit functions of direction of stiffeners and area fractions 6a of both constituents. 3.8.1.

Formula of Francfort and Murat for stiffnesses

Consider a transversely symmetric plate made from strips of stiffnesses Di and Di with area fractions Q\ and 02 respectively; #i + 02 = 1. Assume here that the quadratic form / ( « ) = Ka0(D?x»

- D?»)KXll

,

(3.8.1)

is positive definite, which will be briefly written: D? > D\ and called the "ordered case". The directions of ribs n = (na) is not correlated with anisotropy directions of Da, see Fig. 3.8.1.

interface 7

0

«i

h

yt

Fig. 3.8.1. Ribbed plate of first rank The effective tensor Dh is determined by the following implicit formula of Francfort and Murat 9,(D 2 - A , ) " 1 = (D2 - D,)-1

- 02rD ,

(3.8.2)

where T D = (naUffD^^nxn,,) and operation ( ) Let

_1

'rn ,

(r„) 0/9A/1 =

nan0nxn,,

(3.8.3)

is understood as follows. M«() =

ry**^

,

Kx>i

= dXlia0MQ0

(3.8.4)

Then we write d = D_1

(3.8.5)

Thin plates in bending and stretching

177

Formula (3.8.2) is compatible with (3.2.32)2 and with variational formula (3.4.4). Because of algebraic nature of formula (3.8.2), its direct (algebraic) derivation is easier than a rig orous derivation based on variational fonnula (3.4.4). Thus we have found it appropriate to report below this easiest method of derivation. We shall use only some selected facts resulting from the very definition of Dh given in the previous sections. Proof of (3.8.2). We refer to Section 3.7 and omit indices (a/3) assigned to x(Q/3'- Accord ing to (3.7.4) Kyn(x) is a piece-wise constant function. Consequently the strain tensor ny 1

2

within Y is a constant tensor K in phase 1 and a constant tensor K within phase 2; we omit index y. Upon averaging we find (3.8.6)

K

a0 - 01 Ka0 + # 2 KQ/3 ,

where Ka0 are referred to the ea ® e# basis. Similarly fO.0

M?

= 0i Ma0 + 92Mro/3

Ma0 = Dfx"

Ka0

(3.8.7)

(do not sum over a !).

The continuity conditions (3.2.41), (3.2.42) on both sides of 7 (see Fig. 3.8.1) imply

Ixh = 0,

d\ drdn

dx

0,

= 0,

dn

-" IT

h

d2x dr2

0.

(3.8.8)

17

We observe that here I7 does not denote partial differentiation with respect to y~,. Hence 1 ^nr

2 NaT 1

1

2

(3.8.9)

where Knr — K.<xt)n T

«TT =

Ka/jT'V3

(3.8.10)

Let us represent both these tensors K in the basis generated by (n, r ) k=Knn n ® n+ nnr (n®r

+ T ® n)+ KTT r
(3.8.11)

Hence 1

2

-

r, i1

2

,

^

« n n )n ® n ,

(3.8.12a)

u

(3.8.12b)

or 1

2

The constant k will be specified in the sequel.

178

Elastic plates

Let us define the homogenized stiffnesses by postulating a linear relation between M/, and Kh: Mf = Dfx»KhXll .

(3.8.13)

By (3.8.6) and (3.8.13) we have Dfx^l

= OtD?* kx, +e2D?x» h, ■

(3.8.14)

02«v=<-0i<W

(3-8.15)

A

(3.8.16)

Let us substitute

into (3.8.14). Hence (Dfx" - Df

")«^ = 6,(Dfx» - 0?*) KX» ■

On the other hand (3.8.6) and (3.8.12b) imply «£/? = ft Kap +02[nap -k nan0\ , or K

a& =lic,0 -k62nan0

.

(3.8.17)

The continuity condition Maenan0

=Ma0nan0

,

(3.8.18)

along with (3.8.6)2 and (3.8.12b) implies (Df* - Dfx") k^ nQn0 = {D^n^n^k

,

(3.8.19)

which determines A; involved in (3.8.12b). Let us define paP =

(DfX» _ D«P^ ^

( 3g 2 Q )

Upon substituting k given by (3.8.19) into (3.8.17) one finds

Combining (3.8.16) with (3.8.20) one obtains (Dfx"-Df^)Kl

= 9lP^.

(3.8.22)

Thin plates in bending and stretching

179

Let us invert (3.8.20) and (3.8.22) K A „= (D2 - A f e p ^

< = *i(Ai - Dh)-X^p°e

,

(3.8.23)

nxn)1nan0

(3.8.24)

and insert these expressions into (3.8.21). One arrives at the formula e,(D2 - D^1

a 0

= (D2 - D,)-x'

- § ^

0

equivalent to (3.8.2) - (3.8.3). This completes the proof.

D

Note that relations (3.8.22) are invertible, since condition (3.8.1) implies that the quadrat ic form /i(«) = KaniD?* -

(3.8.25)

D?*)K»

is positive definite; it means that the strongest phase is stronger than the homogenized phase. This intuitive property can be deduced from (3.4.4). Note that K^D^iy)^

< K%Df^

.

(3.8.26)

Hence if #i ^ 0 we can estimate < mm{(Kya0Dfx"Kl)

<0Df^l

h

x

= K a0Df »Kl

|K» e KK(Y) , («»> = «"}

,

(3.8.27)

which completes the proof. Remark 3.8.1. The ordering assumption D2-Dx > 0 implies that the matrix D2 - Dx is invertible and the formula (3.8.24) makes sense. The same formula still holds if we assume only that the matrix D2 — D^ is invertible. 3.8.2. Ribbed plates of higher rank with the stronger phase taken as an envelope Assume that both stiffness tensors D„ are isotropic. Thus they possess the following rep resentations Dfx» = 2kaI?0Xfi + 2^1 fx»

, a € {1,2} ,

(3.8.28)

where ja0X„ = l5aP5X„ _

ja/tt/i

=

ItfaXgffv

D

^

+

ga^ffX _ J^jV)

(3.8.29)

Thus we have £im=D2222

= A.ff + Mffi

= k a

- ^ ,

D™ = »a.

(3.8.30)

180

Elastic plates

The condition D2 > D\ means that k2 > ki ,

n2 > Hi ■

(3.8.31)

Due to the isotropy of D2 one notes that the quantity n^nsD^^ricnp

= k2 + /i 2

(3.8.32)

is independent of n and the formula of Francfort and Murat (3.8.2) reduces to e1(D2-Dhy1

(D2~Dl)-1

=

9,(D 2 - Dhyl

k2 + fi2 r„

(3.8.33)

= (D2 - D O " 1 - 0 2 r D ,

where

rD = k

2

1 + fi2 rn,

and ^ "

_

2/i2 \ n a npuxrin

M2. -rtan^nxn^ . ^2 + H2

The formula (3.8.33) holds irrespective of the isotropy of D\. That is why this formula turns out to be very helpful in evaluating effective stiffnesses of so-called ribbed plates of higher rank. To be more specific, let us consider a ribbed plate of 2nd rank made by stacking together material 2 with the homogenized material of stiffness D^ determined by (3.8.33). The area fractions of materials "2" and " h" are a2 and a\ respectively. Let m be the direction of stiffeners, see Fig. 3.8.2.

Fig. 3.8.2. Ribbed plate of second rank

Thin plates in bending and stretching

181

On using formula (3.8.33) one finds the effective tensor Dhh of the new composite plate: - Dhh)~l = (D2 - D O ' 1 - - ^ — r m . (3.8.34) k2 + ^2 Let us multiply both sides of (3.8.34) by #i and combine this identity with (3.8.33). One finds the formula for effective stiffnesses of a ribbed plate of second rank: Ql (£> 2

a,0,(D 2 - Dhh)~l = (D2 - DO" 1 - r ^ — r n , m , k2 + fi2

(3.8.35)

rn,m = o2Tn + Q\a2 r m . Note that ai#i means the resulting area fraction of material "1". This process of mixing material "2" with ribbed plates of subsequent ranks can be easily continued. We shall prove in Chapter VI that three subsequent laminations (or introducing ribs) done in an appropriate way lead to an isotropic effective plate of maximal stiffness. The idea of subsequent laminations is crucial in understanding the extremal properties of optimal plate designs. 3.8.3. Formula of Lurie-Cherkaev-Fedorov for stiffnesses Let us consider once again the problem of Section 3.8.1 but without assumption (3.8.1). We shall derive the following formula for Dh. nKnu[DKWi"'}enpnl, where • = «i/i + #2/2 ,

[/]« = 0i/ 2 + fc/i •

(3.8.37)

First, let us solve (3.8.6), (3.8.12b) with respect to k: 1

2

K

a0= KQ/3 + 92knan0 ,

na0= K^0 - 9iknan0 .

(3.8.38)

Upon substituting (3.8.38) into (3.8.6) one finds Mf = 0 , D f *"(*$, + 02fcn,nM) + 62Df A "(^ M - 0,fcn,nM) , or Mf

= {Da^)6KhXli + 9,92k(DfXti - D?X)1)nxn» .

(3.8.39)

Let us multiply (3.8.38)i by £>f v and (3.8.38)2 by D f v . Upon subtracting these identi ties we find MXti - M2A" = - {Dp00 - £>AMa/3)<3 + k{DXlusl3]gnan0 .

(3.8.40)

182

Elastic plates

Let us multiply both sides of (3.8.40) by n^n^ and take into account (3.8.18). Hence 0 k = nxnJE^T "v '

~ !DP00'-^ )**, . r

(3.8.41)

Now it is sufficient to substitute A; given by (3.8.41) into (3.8.39) and we arrive at (3.8.36). The denominator of (3.8.41) is always positive definite, irrespective of materials being ordered or not. 3.8.4. Formula of Francfort-Murat-type for compliances Let us consider the same problem as in Section 3.8.1. Our aim is tofinda counterpart of formula (3.8.2) for the compliance tensors: d", dh. The following set of equations is at our disposal, cf. Sec. 3.8.1 Mf

= 0XM? + 62Mf ,

(3.8.42)

^ A X " = * K°0 +*2 Lp , Ka(3= dl0XliM^ (do not sum over a) , (Mf - Mf)nan0

(3.8.43) (3.8.44)

=0,

(3.8.45)

< 3 A / X " - iZexpMp = k nan0 .

(3.8.46)

The derivation runs as follows. Let us define Ncff = M°0

_ Ma0

(3.8.47)

f 1

P*0 = ( C v ~ ZfiiJM?

■

(3.8.48)

Equation (3.8.42) can be rearranged to the form Mf

= Mf + 6iNa0 .

(3.8.49)

Functions (3.8.42) -(3.8.44) imply dha0xM" = dlt^M*

- 62M?) + 62dl0XllM^ ,

(3.8.50)

hence «3AM - dl^M^

= 92 ■ pa0 .

(3.8.51)

Thus we can express M^\M 2 A "intermsofp Mj?=92[{d1-d!>)-1]**patl, a0

Our task now is to express N

M^ = [(d1-d2)-1}"MpPa0-

in terms of p. We use (3.8.46) and (3.8.49) to get

< . * . ( * * " + M 2") - d^xM*

=

k n n

°P

(3.8.52)

Thin plates in bending and stretching

183

or Poff + < / J V ^ A " = k nQn0

and NX»

Equation (3.8.45) implies nxN^n^

=

DXvaP{k

naH0

_

pa/j)

(3 g 5 3 )

— 0, hence d.

(3.8.54)

We go back to (3.8.53) and find Nx» = I^Pafi

T

* -

,

n.nlDf^nl

(3.8.55)

" D>

(3 8 56)

•

-"

Now we insert (3.8.55), (3.8.52) into (3.8.49) to obtain e2[{d1 - d " ) - 1 ] ^ " = [(d1 - d 2 ) " 1 ] ^ + 0lTfafi ,

(3.8.57)

or e2(dl - d " ) - 1 = (d 1 - d 2 ) - 1 + ^ r

d

.

(3.8.58)

The ordering assumption (3.8.1) decides that inversion operations in (3.8.57) make sense. In the case of material " 1 " being isotropic one can compute operations in (3.8.56) to find - i - r * " 0 " = n V n V - hs^S1* 2/i-i

2

+ 5a"S0X)

- ^ Z J f l ( < 5 ^ _ A " ) ( f - nan0) . M +Mi 3.9.

(3.8.59)

Ribbed plates. Plane elasticity problem

We consider the same composite plate as in Section 3.8. Our aim is to find closed formulae for effective in-plane stiffnesses A^fXli and compliances a j j ^ . 3.9.1. Formula of Francfort and Murat for stiffnesses Our considerations will be confined to the ordered case: the matrix A2 — A\ is assumed to be positive definite; A„ represent the in-plane stiffness tensors of strips; the area fractions are denoted by 9a; the lamination direction is determined by a versor n , see Fig. 3.8.1 which remains valid here. On analyzing problem (PKS,Y)< see Sec. 3.2.A, we conclude that e ^ ( T ) (indices (a/3) 1

2

are suppressed here) are piece-wise constants; they will be denoted by ea0, eQ/3; the indices

184

Elastic plates

1 and 2 refer to materials "1" and "2". Let us refer these strains to the (n, r ) basis; n, T being normal and tangent to the discontinuity line 7, see Fig. 3.8.1. ^a0

(3.9.1)

^nn

where

^

la ,

«rr~

^

<, _ dTn

U ,

^ f„r-

^

8TT

\a +

fYT1

s¥T*

^

(3.9.2)

\„ ■

fYT^

fYT1

Note that Tnll = Tnh and TT|l = TT,a, hence - ^ | , = - ^ | 2 and - ^ | , = - ^ | 2 . Conse quently 1 2 1 ., . dTn dTT , i= ia$ - ta0= « nanp + -l{naT0 + npTa), k = - dn dn and we can write 1 2 «o/3 -

e

a / ? = Pa^/3 + P/j7l Q ,

1 pa = - ( K T l a + / r a J

(3.9.3)

The quantities p\, pi will be found in the sequel. Moreover, the following relationships, that can be inferred from (i) - (iii) of part A of Section 3.2, are at our disposal: (3.9.4)

Za0 ~ ^1 eaP + ^ 2 ea0 i

N? =

0lNat,+87Nafi,

(3.9.5) (3.9.6)

Na0no

=Na0na

(3.9.7)

Proceeding similarly to Section 3.8 onefinds,see Eqs. (3.8.16), (3.8.17) (Afx" - A**)!^

= 9,Ma0 ,

(3.9.8)

where Na0

=

( A a/3A„

_ ^ 0 ^

^

^

( J g

g)

and e

a0 = <3 - *a(nQP0 + n0Pa).

(3.9.10) 2

Tofindpa we use (3.9.7), substitute there (3.9.6) and eliminate e by (3.9.3) to obtain Ma0na=G0ap<,, G* = 2Afx%na.

(3.9.11) (3.9.12)

Thin plates in bending and stretching

185

Let g be the matrix inverse to G ; then Pa = ga^a0na

gx0G0a = 5J .

,

(3.9.13)

Now let us substitute (3.9.13)! into (3.9.10) to obtain <£/s =€a/3 -h{nag^nc

+ n0gaynt,)Af"'1.

(3.9.14)

On inverting (3.9.8) and (3.9.9) and substituting into (3.9.14) one finds <M(A2 - Ah)~%^

= \{A2 - A,)-%0Xtl

- 02(rA)a/3A„ ,

(rA W v = - (nag/}xn^ + npgaxn^ + nag0linx + n0gailnx) .

(3.9.15) (3.9.16)

In the case of A2 being isotropic, Eq. (3.9.11) can be explicitly inverted. Assume that A?*

= 2k2lfXl1 + 2n2lfXt' ,

(3.9.17)

where l„ are given by (3.8.29). Then tensor T\ assumes the form

- Tianpuxn^] -

-nanpnxn^ \ . ^2 + M2

(3.9.18)

J

3.9.2. Formula of Francfort and Murat-type for compliances Let us consider the same ribbed plate as in the previous section. We shall prove that the effective compliance tensor ah is determined by 0 2 ( Ol - a")" 1 = (a 1 - a 2 )" 1 + fcr. ,

(3.9.19)

where pa/3A/i _ _

.a,0^

+ - Aa0ap

A6^{n6nph^

+ ra^/i^ + ngr^h^ + nan^hSp) , (3.9.20)

where hxpH13" = 5"x ,

H0» = 2 Aa0X,inanx

(3.9.21)

or h = H\ Indeed, the following relations are at our disposal: Nf = 6, Na0 + 92 Nc0 , 4 = «X^'

a0

l^'oce^N* a0

{N -N )na

= O,

(do not sum over a) 2

ea0- ta0=naP0

(3922)

+ n0Pa.

186

Elastic plates

Let us define pa& = (Aapxv - Ac**) NX» ,

Na0 =Na0- Na0 ■

(3.9.23)

Hence =Na0 + 9iNa0 ,

Nf

Nf

= 62[(al - a " ) - 1 ] ^ " ^ ,

(3.9.24)

Na0= [(a1 - a2)->\a0X» pXli .

(3.9.25)

The main problem is to express N in terms of p. Tofindthis relation wefirstwrite aapw {NXll+ Nx")~ la0x„NXti = nap0 + n0pa , which gives aQ/3A„ N^ + pa0 = nap0 + n0pa , and N<#

=A^^(nAp M + n^Px -

PXll)

•

(3.9.26)

Since NaXna = 0 wefindthe set of equations Hp"plt = ^

(3.9.27)

with

H<* = 2 Aa0X,inQnx

,

q0 =Aa0X»pXlina .

Let /i = if" 1 or hxpH^ = <5£. Thus PA = **> A Q / 3 w p w n a .

(3.9.28)

Now we return to (3.9.26) to find Na0 = T00^ - pXfi,

(3.9.29)

with T given by (3.9.20). Upon substituting (3.9.25), (3.9.29) into (3.9.24) one obtains (3.9.19). D In the case when a 1 is isotropic, i.e.,

al

= i J l + 2^ 1 2 '

(3 9 30)

--

Thin plates in bending and stretching

187

and A= 2fci/i + 2/xi/ 2 ,

(3.9.31)

one can express the tensor T a as follows 1 pa^

=

-ncfipnxn^

L60»nxna - ^'~

+ J/J^^ M l

+ Sa,nxn0

( ^ - nanp){6^

+

S^n^)

- nxn") - l{5aX60» + < W A ) .

(3.9.32)

Remark 3.9.1 Let us recall all the Francfort-Murat formulae derived in Sections 3.8 and 3.9:

0,(x2 - x,)- 1 = (x2 - x,)- 1 - 02rx, e2(x1-x")-1 = (x 1 -x 2 )- 1 + e1rx,

(3.9.33)

for X G {A, D}, x e {a, d}, x = X - 1 . Despite the disclosed analogies between formulae for Ah, dh, ah and D^ (see Sec. 3.7) we see no similarity between the tensors T^ and T^ and between Ta and T ^ . This lack of visible similarity can be justified as follows. The analogy between formulae: Ah(A) and dh(d) has involved changing indices 1 <-+ 2. Such a change becomes unclear if the axes ya are deviated from directions of orthotropy, which is the case here. One can, however, disclose analogies between formulae for Xh and Xh in the case of isotropic constituents. To this end let us recall the projection tensors of Hill for plane elasticity: £ap\» - 2^0^nxTla

+ ^ " ^ a + <WnAn0 + SaxtinUp) - nanpnxn^

(3.9.34)

and for plate theory: £

?/3v

= n n n

a 0 >
£D = Tn.

(3.9.35)

Let us introduce the tensors (n„)Q<3A^ = {Sa0 - nan0)(6x,i

- n A n^) ,

Iapx^ = -(<5aA<W + <W<5/3A)

(3.9.36)

and the ratios °« =

k

lTT^

■

<3-9-37>

Then one can express the tensors T x and T x as follows

rx = ^-(sx-a2rn),

r x = 2/i 1 (5 x -j-a,n n ).

These formulae (for X = A) have been discovered by Norris.

(3.9.38)

188

Elastic plates

3.10. Plates periodic with respect to a curvilinear parametrization. Non-uniform homogenization Assume that domain fl is parametrized by a curvilinear coordinate system (£ a ). Thus f2 can be viewed as an image of a certain domain Qo: Q = $(^0).

t2 / /

«i

«0

,P» '> Fig. 3.10.1. Periodicity with respect to a curvilinear parametrization The mapping $ is determined by relations: xa = xa(£l, £2). The metric tensor is defined by 9a0 = ra ■ r0, ra = dr/df?, r = [^(f), x2(£)], £ = (£°). The Christoffel symbols are denoted by Va0 and the covariant derivative assumes the form /a||/? = fo.,0

O0J-I >

(3.10.1)

where () i/3 = d( )/d£0 ; in this section comma means differentiation with respect to £0. Similarly to Section 3.8 we focus attention on the bending (anti-plane) problem of a transversely symmetric plate. The only unknown is a scalar function w representing de flection of the plate. Since scalars are invariants, the same quantity represents deflection of the plate with the middle plane parametrized by (f a ). Consider the bending stiffness tensor of the form

D€ =

Da0X»U,ii\ra®r0<S>rx<

(3.10.2)

with D^^it;, •) being y0-periodic functions, Y0 = (0, h) x (0, / 2 ). The domain Q is formed by curvilinear rectangles Y = $(Y0). On the other hand the functions Da0Xti(-,^/e) deter mine slow variations of stiffnesses. Assume that the plate is subject to transverse loading q = q(£), the quantity being eindependent. Moreover, assume for simplicity that the plate is clamped along T ~ dil.

Thin plates in bending and stretching

189

Thus w€ = 0 and -^- = 0 on T . (3.10.3) an Here ws represents the unknown deflection; e indicates that this deflection depends on the size of periodicity cells. The moments M°" are interrelated with changes of curvature «a/3(w£) = -VJC\\al}

(3.10.4)

by the constitutive relationship Ma0 =

p*^ L i \

K^w^

(3 1Q 5 )

Let us define the bilinear form deK(w, v) = jDa0X" no

U, ^\ ^(w)Kal3(v)^g

<%,

g = det(gQ/3) , (3.10.6)

for w, v e // 2 (fio) and the linear form !K{V)= jqv^dE,.

(3.10.7)

no The equilibrium problem reads: KS) (He

find we e Hi (ft0) such that

dFK(wc,v) = fK(vc)

v^e// 0 2 (fi)

(3.10.8)

To find the effective characteristics of the plate considered we discuss the behavior of w€, M"0, Kap{wc) as e —» 0. To this end we represent wc and ve by the two-scale asymptotic form z*(t) = z<°>(£) + e V 2 ' ( ^ , y ) + £ 3 « (3) K,y) + • • • l„=f

(3-10.9)

with z = w or v and assume that z<°» e H20(Q0) ,

z^^^eH^Yo);

k>2,z

= w,v.

Hence we have **/>{*) = Ka0(z[o)) + Kyal){z(2)) + 0 ( e i = t / e With KyaJz)

=

-Zlag.

(3.10.10)

Elastic plates

190

We shall use the following averaging lemma:. Let F(£, •) be V^-periodic. Then, cf. the relation (1.1.1)

flo

with

flo

{F}=

(3.10.11)

F{u)dy

\kJ

-

By using the same homogenization technique as in Section 3.2 one finds: (a) the local problem: findx^K,-) (PKS,YO)

€ / £ , ( % ) such that

(D^i^yM^)

+ SiSlWapiv)) =0

Vv<

6

(3.10.12) H^(Y0)

(b) the homogenized stiffnesses (3.10.13) (c) the homogenized problem findty'0' e / f ^ n o ) such that (3.10.14)

(PH) flo

flo

where Mf

=

Dfx»KXlI(wM)

(3.10.15)

Justification of (a) - (c) will be given in Chapter V in a broader context of periodic shells. A new approach to non-periodic homogenization is offered by the method of 6 — 2 conver gence developed by Alexandre (1997). 3.11.

Effective bending stiffnesses of plates with quadratic inclusions

The subject of this section is a transversely symmetric plate stiffened (or weakened) by quadratic inclusions of doubly periodic layout, see Fig. 3.11.1. The cell Y consists of the inclusion Vi = (-r/2,r/2) x (-r/2,r/2) and the matrix y2 = Y\Yu Y = (s/2,s/2) x (s/2,s/2). The distribution of bending stiffnesses is assumed as follows Da/3x»{y) = and DQ

M

Dfx" XDtf*

foryGF 2 , fory€Ylt

(3.11.1)

are constants; A is a positive parameter. For A = 1 the plate is homogeneous.

Thin plates in bending and stretching

191

♦*2

1"

Y

*> ICN

[??• lv:--:~T: !":•.•.■.:•>:•:

■1|<S

~Ti

r_ r 2 2

Fig. 3.11.1. A plate with quadratic inclusions. Geometry of the basic cell Solutions to the local problems (P^S,Y) °^ ^ ec - ^-^ c a n ^ f ° u n d by ^ with trigonometric basis functions. We represent the (a/3) solution

xM=Exl*«j),

Galerkin method

(3.11.2)

with the help of ^ - p e r i o d i c functions 4>a taken from the set {Cm,n{yi,y2),

Sm,n(2/1,2/2)},

(3.11.3)

where Cm,n(2/) = Cm(y 1 ) c „ ( y 2 ) ,

Sm,n(y)

= sm(2/1 )s„(2/2),

and Cn(t) = cos(27rnt) ,

sn(t) = sin(27rnt) .

We have put s = 1 for simplicity. On substituting (3.11.2) into (3.2.31) and choosing v = (pi, one finds * W Q / 3 ) o + D(ba0) = 0 ,

(3.11.4)

where a, 6 6 { 1 , . . . , 7 V } and A * = dK(<pa,4>b) ,

Z ^ " ' = (D*>*(y)«*,(&)) .

(3.11.5)

Elastic plates

192 The homogenized stiffnesses are given by (3.2.32)2, which results in

(3.11.6) The main problem is to solve (3.11.4). Assume that D0 is isotropic or a0Xii

D:

= D v8a08Xht +

-{6aX60» + Sa"S0X)

(3.11.7)

and that D and v are constants. The only parameter that makes difference between the matrix and inclusion properties is A. To be specific let us assume N = 12. We choose the basis functions as follows 01 = C\fi ,

02 = Co,l ,

03 = C\tl ,

0 4 = C2,2 ,

05 — C2,l ,

06 = Cl,2 ,

07 = C*2,o ,

08 = Co,2 ,

09 = Si,i ,

010 = Sl,2 ,

011 = ^2,1 ,

012 = 52,2 •

v = 0.3 /t-0.0001 N=S4 0.6

0.8

Fig. 3.11.2. Effective bending stiffnesses of plate of Fig. 3.11.1 versus the ratio r/s. The squares " □ " represent results of Grigoliuk and Filshtinskii (1970). A = 0.0001 for a) - c), A = 10.0 for d) - f).

193

a

3 00

3 &

Thin plates in bending and stretching

a.

r " > Z

Note that the function 12. Note moreover, that due to symmetries of the periodicity cell the basis functions: Cmd/^Snd/s-a) are redundant. All components of matrices (3.11.5) can be explicitly expressed in terms of v and A. One observes that for A 3> 1 the convergence as N increases becomes poor, see Tables 3.11.la-c.

I

Table 3.11.1

p p p p

1.000 1.357 1.755 1.969 2.354 |

1.000 1.322 1.559 1.847 2.121 |

1.000 1.298 1.547 1.684 1.801 |

1.000 1.297 1.489 1.605 1.767 |

1.000 1.285 1.464 1.585 1.650

ON LA ■ LA 0 0 i O LA

t > © 00 \0 i © p © - J LA \ 0 -O 

N) S ■

2 bo bo

i OO OO OO

^- A > J OOi

p p p p

— 00 LA UJ i | 0 0 00 00 OO tO *>. LA tO i

LAdvLALAQLAK)N>N)

N r* r- r- r- P P P P u)\p^l>jQbobobobo

> bo bo bo bo

bo

0.800 0.801 0.805 0.839

o

p o p ©

0.803 0.804 0.808 0.841

r/s = 1 / 3

2

84

8

O-P->S(O--JO<»OO

0.090 0.090 0.093 0.110 0.167 0.142 0.099 0.074 0.070

p p p p p p p p p

oo oo

Ov — *

0.089 0.089 0.092 0.111 0.167 0.141 0.094 0.077 0.047 | o o o o o o o o o

0 0 OO O^ >-* ^

o o o o o o o o o

p p p p p p p p p

60

O

2

40

0.089 0.089 0.092 0.112 0.167 0.142 0.083 0.061 0.061 |

^

24

O091 0.091 0.094 0.112 0.167 0.140 0.089 0.048 0.006 |

OO * .

12

QJ£OO£.O\*-^5^5^5

4

0.127 0.101 0.127 0.101 0.127 0.103 0.133 0.117 0.157 0.167 0.142 0.146 0.132 0.054 0.194 0.044 [ 0.306 I 0.053 |

r/s = 1/3 0.362 0.362 0.362 0.369 0.417 0.653 1.131 1.667 2.147 |

0.361 0.361 0.362 0.368 0.417 0.649 1.052 1.366 1.849 |

0.361 0.361 0.362 0.368 0.417 0.645 1.005 1.294 1.512 |

2

0.366 0.363 0.363 0.369 0.417 0.669 1.223 2.144 6.603 |

0.361 0.361 0.362 0.368 0.417 0.642 0.981 1.194 1.392

- .- P P o p p p p

s

2 4 4 0 6 0 8 4

s

0.366 0.366 0.366 0.371 0.417 0.674 1.821 12.070 | 114.338 |

12

•^■ONLA^ — ONONONON

4

r^r^r^PPPPPP

lCF 10- 2 lO" 1 10° 101 102 103 104 105

0.807 0.804 0.812 0.844

£

3

0.821 O H ! 0.822 0.811 0.824 0.814 0.850 0.846

W - - p p p p p O ^-<7s^-0\"**-U)L0UiL0 -4 -O — W - J O lO N W

x

N

84

(O

v = 1/6

^^o.^!?

1

c) D\2n/D

13

^^€9,?.^°°°

RF3 10- 2 10" * 10° 101 102 10 3 104 105

24

ts>

x

N

60

p p p p p p p p p

i/=l/6

1.000 1.374 1.947 5.261 I 37.473 I

40

O s W ^ p p p p p p

b) Dl™/D

-OLA^-^- — © O O © • ^ S) vO U i » 00 00 00 00 •vj o s - ^ - J < - J LA LA LA

101 102 103 104 10 5

0.853 0.854 0.856 0.873

12

t-

^,^i%9.^,
HF 10~2 10"1 10°

4

p p p p p p p p p

3

^

N

x

r/s = 1 / 3

O

v = 1/6

JS* fO H - p O O © O

a)Dlln/D

194

Elastic plates

The algorithm works fairly well for very small A, which means that results of Dffx,i tend uniformly to the case of quadratic openings - see Fig. 3.11.2, where the results of Filshtinskii and Grigoliuk are additionally depicted. The latter results were found for circular holes and here they are put by recalculating the r/s ratio from the condition of the square and circular areas being equal. Let us look lastly at global behavior of a clamped square plate with 25 inclusions, as in Fig. 3.11.3; s = a/5. Poisson's ratio v is constant and equals 1/6. The stiffness D in the matrix and inclusions equal D and XD respectively.

A///j

'///////MS///////////////////////;////////,

r

q □□□□ ri n □ □ □ □ □ □ □□ □ □ nun" □ E

r = s/l

s = a/5

IDDDDD '>///////////////////////////////////////// Fig. 3.11.3. Plate with quadratic inclusions The plate is subject to a constant uniformly distributed transverse loading q. The nondimensional deflection is defined by w = Dw/qa4. In case of A -C 1 the plate deflection grows rapidly within the inclusions, see Fig. 3.11.4a (A = 10~4). In the matrix region ti)(0' approximates w from the upper side fairly well. In case of A > 1 function w^ is a lower estimate of tv, see Fig. 3.11.4b.

Thin plates in bending and stretching

195

a)

1.0 10"

b)

+8.0 10"4

Fig. 3.11.4. The homogenized defections t/)'0' versus thefiniteelement approximation w. A = 10"4 and 10.0 for a) and b) respectively 3.12.

Perforated plates

In the present section we shall study homogenization problem for periodically perforat ed plates. The proof of homogenization Theorem 3.12.1 is based on penalization, which consists in replacing the voids with a soft material. Comments on the derivation of the effective dynamic plate model from the three-dimen sional dynamic equations of a thin perforated solid are given in Section 7. Let us consider a periodically perforated plate, clamped along T = dQ. Now the basic cell Y contains a finite number of holes. The voids are cylindrical and traverse all the

196

Elastic plates

thickness of the plate. The vertical axes of the voids are perpendicular to the mid-plane of the plate. The sum of holes in Y is denoted by Y° whilst the complementary part by Yc. Thus we have Y = Yc U y° . By Pit i = 1,... ,I(e) we denote the set of eY-cells which intersect Q. We observe that 1(e) is proportional to e~2, i.e., 1(e) = 0(e~2). The intersection of voids with 0. is denoted by 7/ and nc = n\ui*.

(3.12.1)

i

We make the following assumptions'. (Ai) Cle is connected. (A2) The holes 77 have regular boundaries and are locally located on one side of their boundary. (A3) Each of 77 does not intersect T. The bending stiffness moduli Da0x,i(y) are defined on Yc and are assumed to be Yperiodic. Particularly, homogeneous plates are not precluded, then the moduli D a " A/i are constants. The coefficients Da0X^(y) possess usual symmetry property and we assume that Da0X" e L°°(YC),

(3.12.2)

and that there exists a constant C > 0 such that VpeE*

Da0^(y)pa0PXli

> C\p\2

fora.e.2/erc.

(3.12.3)

We recall that \p\2 = ^ PapPapa,0=l

Let b 6 L2(Q) and If = b\w = XJA where ( ) =

*< *

1

if

if

i6ff, i3A2A)

Xea\*.

Obviously, the function Xe is eV-periodic. We set Vt = {v 6 H2(Q.c)\v = 0,

^ = 0 on r } . (3.12.5) on For anyfixede > 0 the functional of the total potential energy of the plate is given by Jt(w) = -ac(w,w) - / tfwdx = -ac(w,w) - /Xc(x)b(x)w(x)dx

,

(3.12.6)

Thin plates in bending and stretching

197

where ac(u,v) = IDfx»{x)Kali(u)KXli{v)te

-

u,veVc.

(3.12.7)

The minimum principle of the total potential energy means evaluating (P£)

J e K ) = mi{J€(w)\w G Vc} .

(3.12.8)

The assumptions (Ai) - (A2) and (3.12.3) imply the existence and uniqueness of we, cf. Sec. 1.2.2. Homogenization To obtain the effective plate model we let e tend to zero. Let r stand for the strong topology of H\Q). Theorem 3.12.1. The sequence of functional {Jc}c>o, defined by (3.12.6), is T(T)convergent to Jh{w)= fwh(n(w))dxn where w G

HQ(£1)

fobwdx, n

(3.12.9)

and

Wh(P) = inf { ^ | JD°0X»(y)(Kip(Z) + M K M ( 0 + Px»)dy\Z G H^Y')}

.

(3.12.10) HerepG E* and 9 = \YC\/\Y\. Proof. Since {xe}oo is a sequence of ey-periodic functions and \€ G L2(fl£), therefore we have, cf. Sec. 1.1.1 1 f

\YC\

Xe -"■ Ty-J xYc{y)dy = -^ = e, Y

where xYc stands for the characteristic function of the set Yc. A solution £ of the minimization problem appearing on the r.h.s. of (3.12.10) exists and is unique up to arigidmotion or a polynomial of thefirstorder. Suppose now that the holes are replaced by soft inclusions with the elastic moduli £ T ? D " / V ( I ) (77 > 0). The loading is still applied to fi only. The functional of the total potential energy now takes the form J?(«) = Je(vc) + TJ j

D^(x)Ka0(vo)KXll(vo)dx

,

(3.12.11)

Elastic plates

198

where v = (vc, v°) e H$(Q,), vc ~ v^, v° = «|n\n< • For afixed77 > 0 the homogenization (e — ► 0) yields the limit functional J*(v) = fwJl(K(v))dx - febvdx , n n

(3.12.12)

where v £ H$(Cl) and W^(p) = i n f { — JDf^(y)(Kl0(a

+ pa0)K,(O

+ px,)dy\t e H^Y)}

.

(3.12.13) Here ^(y)

if

y € rc,

*"**(y)

if

2/ G Y°.

(3.12.14)

To complete the proof it is thus sufficient to show that for anyfixedp G E2. one has W1(p)^Wk{p)

as r ? ^ 0 .

(3.12.15)

To corroborate (3.12.15), let us denote by f a solution to the minimization problem on the r.h.s. of (3.12.13). Then

Wj(p) = 2j7f JDfx"(y)K^)

+ P*0)(KU?) + P»)dy

Y

= 2j^i JDa0X,i{y)K0{Q) + Pa0)(KWc) + px,)dy Yc

y°

< 2j^[ Jna0X>(y)P»0P*dy+^-jDfx>>(y)pa0pXlldy ^Cdpf + T/lpn^lpl2,

(3.12.16)

since 0 < 77 < 1. The positive constant d is independent of 77 and we have set f (G. S ) € H^(Y), Q =tfYC,$ = ^ o . From (3.12.16) we conclude that

=

ll"^)+PllWj)< 0 is independent of 77. Consequently, the sequence Q is bounded in H^r(Y) and there exists a subsequence, still denoted by £?. s u c h m a t Q^lc

in # 2 (y c )

weakly as 77 -> 0 .

Thin plates in bending and stretching

199

Similarly

\\^y(Co) +

P\\lHYo,Bi)
and

vJDf^K^

+ P^M^+Px^dy^O

a s ^ O ,

Hence

w » - 2j7[ f

Da0X

"(yWSc) + pcMM) + px,)dy

Yc

= Wh(p) as r / ^ 0 ,

(3.12.17)

since the functionals involved in (3.12.17) are convex, finite, lower semicontinuous and equi-coercive with respect to r\. Being convex andfinitethey are continuous. This com pletes the proof. □ Corollary 3.12.2. The effective (homogenized) moduli are now given by

D^" = | i | JDa0X"(y)(Kva0(^) + M / W ) ( K W " ° ) + *Mdy

• (3-12.18)

Y'

Indeed, D~lSv<' =

— and due to linearity of the problem we may set £ = — papip^a^.

OPyjOPi,p

The K-periodic functions ip^s) € H^r(Y) are solutions to Wh{r6) = i n f { ^ jD°^{y)(K%{0

+ / $ ( K J k ( 0 + l£)dy\£ € H^(Ye)}

,

Yc

where / ^ | are the components of the identity tensor, i.e., Ia0 = 6*6$ =tf0A<W'because the coordinate system is Cartesian. More precisely, one should consider the identity tensor with the components given by

-(5 Q A<W

+ <W<W- However, due to the symmetry of the

tensor D, it suffices to deal with Pj0. Remark 3.12.2. To derive the dual principle P* one can exploit the theory of duality outlined in Section 1.2.5. Similarly, to perform dual homogenization one can use Az6's theory presented in Section 1.3.6.

200

Elastic plates

3.13. Plates stiffened with rigid inclusions Assume now mat voids are filled with rigid inclusions. We retain the notations of the previous section. The essential difference is that now instead of voids we deal with rigid inclusions. The assumptions (A\) - (A 3 ), (3.12.2) and (3.12.3) are assumed to be satisfied. Obviously, dT? are boundaries of rigid inclusions. As before, the loading b is of class L 2 (fi). For any fixed e > 0 the space of kinematically admissible transverse displacements is given by I ? = {w G ff£(n)|«|n\n. £ ^ }

(3-13.1)

where 11 stands for the set of rigid displacements. Since K(W) = 0 if and only if w(x) = aiXi + a2x2 + b, where ai, a2, b £ R, therefore the set V, coincides with the space of first order polynomials in xx and x 2 . The functional of the total potential energy is given by G e M = ^ IDfx,1{x)K,av(vj)KXli{w)dx n«

- fbwdx. n

(3.13.2)

The minimum principle of the total potential energy means evaluating (Q e )

Gc{wc) = inf{G e (u;)|ti; e VT€) .

(3.13.3)

For any e > 0, a solution wc exists and is unique. Indeed, for w € Vre we write jDfx»(x)Ka0{w)^{w)dx

> C\\K(w)\\2LHile) > CiHiull^n,,

(3.13.4)

since 2

^2

f Ka0(w)Ka0(w)dx

(3.13.5)

is a norm on Hi (fi) equivalent to the natural norm of this space. Moreover, the linear form L(v) = fbwdx n

(3.13.6)

is continuous on VJf. Here C and Cx are positive constants independent of e. Let us pass now to homogenization. Theorem 3.13.1. The sequence of functional {Gc}c>0 is F(w — H2(fl)) convergent to Gn(w) = fwttK{w))dx

- L(w) ,

weH${n),

(3.13.7)

Thin plates in bending and stretching

201

where >Vh(p) = i n f { ^ JDae^(y)(Kva0(v)+pa0)(K%(v)+pXli)dy\v

6 7/rjwP(p)} , (3.13.8)

Y"

where p e E , and Hr.perip) = {« € ^ ( J O K * - p)|yXK, € ft} . 1

(3.13.9)

2

Here p = - ^ PcffVaVpProof. The notation (v - p)|yNyc £ 72. means that on the part Y\YC occupied by the rigid inclusion (or inclusions) a function (v — p) coincides with a rigid displacement. Let us replace therigidinclusions with deformable ones for which the elastic moduli are given by \D$0x,i{y) with A being a positive number intended to tend to infinity. Then the functional of the total potential energy takes the following form G*M = Gt{w) + A I Dfex"(x)K.ap{w)KXll(w)dx .

(3.13.10)

For anyfixedA > OtheT(s- //'(n))-limitof {G*}£>0 isgivenby Gx{w) = [wZ\n{w))dx n

- L{w) ,

w e//02(fi) ,

(3.13.11)

where

WrAp) = i n f { ^ JDa^(y)(Kl0(v) + **)(«$>) + pXli)dy Yc

+

W\J

D 0X {y){

? *

<e{v)

+

" < * ) « » + P*)dv\°

e

"leriY)} ■ (3-13.12)

It thus remains to show that for anyfixedp 6 E, we have WIX(P) -> Wttp)

^

A-^+oo.

(3.13.13)

In fact, let vx be a minimizer of the minimization problem appearing on the r.h.s. of (3.13.12). Take v € H^.(Y) such that KV(V) + p = Ky(v - p) = 0, where p is the same as in (3.13.9). Such a function v can always be chosen provided that the inclusion is contained in Y, i.e. does not intersect dY. Then

ColMt;A) + Pll V . E j , + CiA||* V ) + /»H2»(ny.iEj) <

C3\\K*{V) +

p||^ ( y « i E j ) = C3 .

(3.13.14)

Elastic plates

202

The positive constants Co,..., C3 do not depend on A. Hence

CiMv*) + p||2L2(y,E;) < Cb||«V) + pllWc,Ej) +C 1 A||KV(^)+p|| 2 L 2 { n K c E ; ) 1. Consequently the sequence {wA} is bounded in H^er(Y) and there exists a subsequence which converges weakly to v € H^er(Y). Moreover, from (3.13.14) we conclude that

ll«» + PllW l E 2,
^ Mjnjnf ll"v(wA) +Plli»(y\y.Ej) ^ °

and thus «»(?; - p) = 0 on Y\YC. It means that v € / ^ ^ ( V ) . Eventually we find Urn W^(p)

= ~

JD^{y){Kl0{v)

+ Pa0)(K%(v) + pXtl)dy = Wh(p)

Yc

and the proof is complete.

D

Remark 3.13.2. Derivation of the dual problem Q'c as well as dual homogenization are left to the reader. Rigid inclusions make the problem interesting in itself.

Nonlinear behavior of plates

4.

203

Nonlinear behavior of plates

In this section we shall deal with two homogenization problems for von Karmaii plates. In Sec. 4.2, by exploiting the results of Sec. 1.3.5, we shall perform (two-dimensional) homogenization of such plates exhibiting a periodic structure. Next, in Sec. 4.3, perforated von Karman plates will be investigated. Thus the homogenization results of Sec. 3.12 will be extended to this nonlinear plate model. Also, we shall discuss the problem of convergence of the spectrum and of the bifurcating branches of the perforated von K£rmSn plate to the corresponding elements of the homogenized plate. Primarily, however, in Sec. 4.1 basic equations of von Karmin plate model will be introduced in a standard manner, cf. also Sec. 5.7. 4.1.

Von Kdrmdn equations

Let us consider a transversely symmetric thin plate of elastic moduli C*J'*' satisfying con ditions (2.4.2)i|2- Assume first that variability of C in x e fi is arbitrary and the plate is of constant thickness. We introduce the following notation: B = Q x {——, - ) , T = <9fi,

r = Four,, r 0 nr, = 0, T = rx (~, £), T = T 0 UT,, T0nT, = 0, T0 = r0 x {—, - ) , T ) = r i x ( - - , - ) , r ± = S l x { ± - } . The plate is subjected to transverse loading q(x) on T + , in-plane and transversely symmetric loading: t = {ta,0), ta(x,z) = ta(x, — z), 2 = i 3 , o n T i , Displacements w(x, z) vanish on To- The nonlinear strain tensor 7 i ; (w) is given by 1ij(w) = ey(u>) + -6klwkiiwij

(4.1.1)

where e^(it;) = W(ij)- Its variation Sfij is denoted by \j{w, Atj(w,

v) = eij{v) + -6ki{wijvkti

v):

+ wkiivi:j)

,

(4.1.2)

where v vanishes on T 0 . In the traditional notation we write ha = ^{Swij + 8wjti) + -5u{wijSwkii

+ wkj6wij).

(4.1.3)

Denoting by Stj the components of the second (symmetric) Piola-Kirchhoff stress tensor, the constitutive relationship is taken in the linear from: 5,;=^%,.

(4.1.4)

The functional of the total potential energy is now given by J{w) = i fcijk'lij{w)lkl{w)dx B

where x = {xa, x3) = (xa, z).

-

fq{x)w3{x, n

h/2)dx -

f ta(x)wa(x)dS T,

,

(4.1.5)

204

Elastic plates

The equilibrium problem means evaluating J(w) = inf{J(w)\w e Wl-\Bf

,

w = 0 on T 0 } .

(4.1.6)

The reader is advised to formulate an existence theorem, assuming that the loading func tional is continuous, cf. Sec. 1.2.2. We observe that the nonlinearity of the strain measure ~f{w) implies that the functional (4.1.5) is nonconvex. The condition of vanishing of thefirstvariation of J yields = Jq{x)v3(x, h/2)dx + JtavadS

(w)Akl(w,v)dx fi

B

,

(4.1.7)

T!

for each v vanishing on T 0 . To solve Eq. (4.1.7) one can use the Faedo-Galerkin method. All minimizers solving (4.1.6) satisfy Eq. (4.1.7), but not necessarily vice versa. Let us pass to a concise presentation of the von Karman plate model. This modelling is based on: (i) kinematic assumptions wa{x, x3) = ua{x) - zw a ,

w3(x, z) = w{x) .

(4.1.8)

(ii) Stress assumptions which replace (4.1.4) with S"* = C ^ - y ^

Sa3 = 2Ca3A37A3,

(

S 33 = 0,

(4.1.9)

where C is defined similarly to Cz, cf. Sec. 3.1. (iii) Strain assumption: nonlinear terms with respect to ua are neglected in the definition of the strain measure 7, hence it is assumed that ja0{wa, w) = ea0(u) + -w,awt0 + ZKa0(w) , 7a3(w/3, w) = 0 , where Ka0(w) = -wyOL0. Consequently

j

733(^0, w)=0,

Aa0(w^,w;Uy, v) = ea0{vj + ~{w%0viC, + wi0vi0) + zna0{v) , 1

(4.1.11)

A M = 0, and h/2

f f SijAij{wQ,w-va,v)dxdz= n -h/2

f[Na0ea0(v^) n

+ Na0wt0v,a + Ma0Kal}{v)\dx ,

(4.1.12)

where h/2

h/2 a0

N°~ ■* = j S dz, -h/2

M

a0

= j zSa0dz. -h/2

(4.1.13)

Nonlinear behavior of plates

205

Similarly, by using (4.1.8) the virtual work of the loading functional L reduces to jqvdx+ JNavadr n r,

L(va,v)=

,

(4.1.14)

where h/2 a

N = I tadz.

(4.1.15)

-h/2

According to (4.1.9) and (4.1.13) the constitutive relationships assume the following form Nafl = Aa0x»EXp{u,w) , AT" = D a/5A "K V M , (4.1.16) where Ea0(u, w) = ea0(u) + -wiQwi0 .

(4.1.17)

Under the hypotheses (i) - (iii) the variational Eq. (4.1.7) takes the form r

{Na0ea0{v) + Na0wi0va

+ M^KapWdx = L{v,v) , (4.1.18) n valid for v = (va) , v , dv/dn vanishing on r 0 . Here n = (na) denotes the unit outward normal vector to T. The equilibrium problem: find {u,w) satisfying (4.1.18), with Na0 and Ma0 given by (4.1.16), is solvable in the space //'(fi) 2 x H2(£l). Obviously (u, w) is to be such that«, w and dw/dn vanish on r 0 . /

'

■

Remark 4.1. Duvaut and Lions (1974) used the Faedo-Galerkin method to show the exis tence of solutions to a large class of boundary value problems, including unilateral bound ary conditions. Minimization methods were used by Ciarlet and Rabier (1980), cf. also Ciarlet (1997). 4.2. Homogenization Consider now a von Karmdn plate with an eY-periodic structure, cf. Sees. 2 and 3. It means that the elastic moduli Af^(x) are eY-periodic. We assume that Aa0Xfl ci > Co > 0 such that

= Aa0X» ( - ) ,

Df A"(x) = Da0x» Q

€ L°°{Y) and Da0^

V e € E2. V p 6 E2

€ ^{Y)

,

(4.2.1)

and that there exist constants

co|e|2 < A^(y)ea0eX)i

< C l |e| 2 ,

colpl2 < Da^(y)pa0p^

< c, |p| 2 .

206

Elastic plates

The functional of the total potential is now given by Je(u,w) =

y\Afx»(x)Ea0(u,w)EXll(u,w)

a + Dfx"{x)Ka0{w)Kx^w)}dx

- L(u, w) ,

(4.2.3)

where (u, w) £ Hl(Q)2 x H2(Q). We recall that in the two-dimensional case H2(Q) c W1,4(fi). The loading functional L is not necessarily of the form (4.1.14). It suffices to assume its continuity in the weak topology of //'(ft) 2 x H2(Q). Then this functional is a perturbation functional. We observe that the virtual work of boundary bending moments can readily be included into the functional L. We set V(ft) = {(«, w) 6 //'(ft) 2 x H2{n)\u = 0 , w = ^

on T0} .

(4.2.4)

For a fixed e > 0 the minimization problem: (Pt)

Jc(u',w<) = inf{J e (u,«;)|(u ) «;) e V(il)}

is solvable. Let us pass to finding the limit functional Jh = Gh — L. To this end we set G e («,w) = ±f[A?»(z)Ea/i(u,w)EXlt{u,w)

+ Df^(x)na0(w)^(w)]dx

. (4.2.5)

Now we are in a position to formulate the homogenization theorem. Theorem 4.2.1. Under the assumption (4.2.2) the sequence of functionals {G€}e>0 is T(r)convergent (T = s - (L2(ft)2 x Hl(Q))) to GH(U, w) = y[A?*Eat,(u, n

w)EXll(u, w) + Dfx>iKa0(w)KXll(w)}dx , (4.2.6)

where (u,w) G //'(ft) 2 x /72(ft). The homogenized elastic potential is given by Wh(Eh,Kh) = i n f { ^ 7 j y " A ^ " ( y ) ( £ ^

+ ev/3(v))(£^ + e^(v))dy|v e

H^Y)2}

Y

+ ™l{^\jD°^{y){Kha0 + 0 ) ) «

+ ^(v))dy\v € H^Y)}

,

(4.2.7)

Nonlinear behavior of plates where Eh,nh

207

€ E.]. The homogenized moduli are calculated from

A=

32

^ " d^k,=w\JAO*r{y){6'xSr,t+e-(*(AM)))dy • (4-2-8) D

=

^ = -£^F

W\J Da0°T^6°^

X

"

+
(4.2.9)

Y

respectively. The functions x ^ and 6 ( Q "' are solutions to the following local problems. Problem P^. Find X(OT) 6 H^r{Y)2 such that + e\Xii){X(°T)))ele[v)dy

JA^iyW^ v

= 0 , V v 6 H^Y)2

. (4.2.10)

Problem / £ . . Find e(ffT» 6 ^ r ( V ) 2 such that

y"z^^(j/)(^v + «{^)(e(aT)))C(«)di/ = o, v«€fl^(y). (4-2.ii) y

Moreover inf{A(«,w)| (u,w) € V(ft)} = lim(inf{J£(u,u,)| (u.io) 6 V(fi)}) , (4.2.12) and there exists a subsequence e' such that {uc\we') ->• (u, w)

weakly in

tf1^)2

x H2(Cl) as e' -> 0 .

The element (u, w) £ V(fi) is a minimizer of the homogenized plate problem (Ph)

Jh(u,w) = inf{J h (u,«;)|(u ) «;) £ V(fi)} .

Proof. Applying Theorem 1.3.28 we conclude that T(r) - limG£ = Gh and the ho£-•0

mogenized potential is given by (4.2.7), where E^0 = ea0 + -wtC,w^, e e E2,. In the local minimization problem appearing on the r.h.s. of (4.2.7) the macroscopic gradient Vw(x) = (w0(x)) plays the role of a vector parameter. Therefore it is reasonable to treat - (Vw) ® Vw as a part of Eh. Let us denote by (v,v) 6 {{^.(Y)2 x H^r(Y) a solution to the local minimization problem. We obviously have ^ = E^0)(y), and the formulae (4.2.9), (4.2.10) follow.

v = Kha0Q(a0),

(4.2.13)

208

Elastic plates

The local problems (4.2.10), (4.2.11) are equivalent to the two minimization problems involved in (4.2.7). The remaining part of the proof is based on Theorem 1.3.22 and Remark 1.3.23. First, we have tofinda T-relatively compact subset X0 C V(fi) such that M{Je{u,w)\(u,w)

e X0} = Je(uc,wc) ,

(4.2.14)

cf. also the problem Pc. We may write Je{ue,wc)

< Jc{u,w)

V(«,iw) G V(fi) .

Hence, for u — 0 and w = 0we get J£(u£,w£)<0,

(4.2.15)

and consequently ||u'lkn+IM|a,n
(4.2.16)

where c > 0 is a constant, independent of e. Indeed, the following result has been proved by Bielski and Telega (1996). Lemma 4.2.2. Let r] > 0 and K > 0 be given constants. There exists a constant ct = C\ (rj, K) such that 2

£

i

r

/ \ea0(u) + -w,ow,0\2dx > cSu\\ln

+ \\w\\ln] ,

(4.2.17)

aJ3=l Q

for each couple (tt,w) e V(Q) satisfying weTM,

IH| 2 , n + | H I ^ > 7 7 2 ,

(4.2.18)

where TM = {we //? 0 (fi)| | H | 2 f l < Mp(w)} .

(4.2.19)

Here H$0 = {w € H2(n) | w = ^

=0

on r 0 } ,

(4.2.20)

M > 0 is a given constant and p(w) = { I (w,aw,a)2dx}1/4 n is a seminorm on W1,4(fi). The set T}^ is a weakly closed one.

(4.2.21) □

Nonlinear behavior of plates

209

Let us continue the proof of Theorem 4.2.1. To prove (4.2.16) we use (4.2.2), (4.2.15), (4.2.17) and the continuity of the loading functional. On account of (4.2.16) we conclude that there exists a T-relatively compact set X0 C V(Q) such that (4.2.14) is satisfied. Thus, by Theorem 1.3.22, (4.2.12) follows. We can extract a subsequence {uc', we'}£>>0 such that {uc>,wc')-> {u,w)

weakly in Hl(£l)2 x H2(Q)

as e'— 0,

where (u,w) 6 V(il). Moreover we have lim Jc,(u£',w£') = \im{mi{Jc.(u,w)\{u,w) c'—»0

c'—O

€ V{B)\) = lim JMuc',we') . E'->0

Consequently (tt, w) is a minimizer of J^ on V(B). This completes the proof.

□

Remark 4.23. Ciarlet and Rabier (1980) defined the set TM as a subset of //^(fi). The formula (4.2.19) extends their definition provided that T0 constitutes only a part of T and measr0 > 0. Also, the characterization of TM due to Ciarlet and Rabier (1980, p. 69) remains valid in our more general case since the imbedding H2(Q) c W1'*^) is compact. D

4.3. Bifurcation and homogenization of perforated von Kdrmdn plates The main aim of the present section is to combine homogenization and bifurcation of a periodically perforated von Karman plate subjected to in-plane boundary loading. For such plate it is natural to study the relationships between the critical values of the homogenized plate and those characterizing the perforated one. The same pertains to the bifurcating branches. 4.3.1. Homogenization of perforated von Karman plates Consider now a von Kirman plate perforated similarly to the linear Kirchhoff plate, cf. Section 3.12. We assume that the perforated von Karman plate is made of a homogeneous material. The consideration which follow can readily be extended to periodic and even nonuniformly periodic elastic moduli. Obviously, the inequalities (4.2.2) hold now in the part Yc of the basic cell. The boundary T of fi is assumed to consist of four parts: Too, r 0 , I\ — _ — — dw and T2 and r = Too U r 0 U 1^ U T2. On Too the plate is clamped: w = -^— = 0 and the on in-plane displacement vanish: u = 0. On r 0 the plate is clamped: w = —— = 0 whilst on T] it is simply supported: w = 0. On T0 U 1^ the plate is subjected to a boundary loading described by the loading functional L\ («, w), which in the homogenization procedure plays the role of a perturbation functional. No boundary conditions are imposed on IY The functional of the total potential energy is given by Jl{u,w) = G€(u,w) - /Xc(x)q(x)w(x)dx - Ly(u,w). (4.3.1)

210

Elastic plates

Here Xt is defined by (3.12.4) and Gc is given by (4.2.5) with Q. being replaced by Cf. The minimum principle of the total potential energy means evaluating Jl(ue,vf)

{Qt)

= inf{Jre1(tt,«;)|(u,ii;) G V^Sf)} ,

where Vl(nc) = Uu,w)eH1{nc)2x

H2(Q£)\u = 0,

w = ~=0 on on r , } .

on Too;

dw _ w = - s - = 0 on r 0 , w = 0 (4.3.2) on Korn's type inequalities are preserved for the domain CF, cf. Oleinik et al. (1992), Duvaut (1977b) and Sees. 8.3 and 9.3. Consequently a solution (uc,w£) e Vi(fi£) exists since minimization methods still apply. Prior to performing homogenization we shall introduce two extension operators for func tions defined on Cle. Lemma 43.1. There exists an extension operator:

such that 3C>0, Ve>0, VveHl(nc)2, HPfrlkn < C||«||o,n. , l|e(P^)|| 0 ,n < C\\e(v)\\0,n< . The constant C does not depend on e.

D

The proof of the above lemma is left to the reader. We note, however, that similar problem is dealt with below and in Sec. 9.3. Let us introduce the functional space V, which in the present section will play an impor tant role V = V(Cl) = {we H2(Cl)\ w=0 ^ = 0

on TooL/ToUT!,

on rooUTo}.

(4.3.3)

c

The space V€ = V(fl ) is defined similarly. Lemma 43.2. There exists an extension operator: P £ : Ve->V such that BK >0, VoO.Vioer, ||PcHlo,n < fflMkn. , IIP'Hkn < A l M k n . , l|V 2 (P^)||o,n < K\\V2w\\0,v ■

D

Nonlinear behavior of plates

211

Before proceeding to the proof of the last lemma we comment on the homogenization Theorem 3.12.1. In this case V = H%{Q) and Vc is defined by (3.12.5). Let w € H${Q.) solve the following homogenized problem: Jh(w) = inf{Jh{w)\ w G Hfttt)} ,

(4.3.4)

with Jh given by (3.12.9). Then Pewc converges to w in H$(il) weakly. The proof of Lemma 4.3.2 exploits the following result, cf. Duvaut (1977b). Lemma 4.3.3. There exists an extension operator: P : H2{YC) -> H2(Y) such that 3/Ci>o,

Vw€#2(yc),

l|PHIo,yMo.vc , and the constant Ky is independent of w. Proof. Let V be the set of first order polynomials. Any function w € H2(YC) can be written as follows w = Wi +p ,

where p € V and wi±V in L2(YC). Then we have 3 / C > 0 , ||w 1 || 2 ,^
Ww£H2(Yc).

(4.3.6)

We recall that d^ = V2, \a\ = 2. Now we choose an extension of w to Y which leaves p invariant. To this end w\ is extended to Y by continuous lifting of the traces of w\ and — - on Y°, where /x is the inward unit normal vector to 0Y°. If this extension is denoted on by u>i, one has ^HdJ^lkyS^IKIkyc.

(4.3.7)

|c|=2

We set Pw = wi + p . Now the inequalities (4.3.5) readily follow.

(4.3.8) □

212

Elastic plates

Proof of Lemma 43.2. It suffices to show that: (i) if w G V€, one can find an extension we V of w such that there exists a constant K3 > 0 independent of e such that ||V 2 w||o, n <^ 3 ||V 2 «;||o, n . .

(4.3.9)

(ii) If w € Vc, there exists a constant K4 > 0 independent of e and w such that HH|o,n<
(4.3.10)

To prove the last two inequalities we observe that since fi£ is connected and the distribution of holes is periodic we may write ft=((j77)U^,

(4.3.11)

where A\ contains no hole. By using the homothetic transformation: y = x/e each domain T,c is mapped into the same domain Y° C R 2 . Here, for the sake of simplicity, we assume that Y° contains only one hole. We set v(y) = w(ey),

(4.3.12)

and apply Lemma 4.3.3. Hence we deduce the existence of extension w{x) to eY given by w{x)=v(x/e).

(4.3.13)

Here we tacitly assume that eY has been suitably shifted in the plane R 2 , cf. Sec. 9.3. Moreover we have

5 2 / (™.°B)2dx = 75 5Z / (^W)2
J(vW)2dy

= i{

a,0 yc

(w.°e?dx ■

'E/ a,0

(4.3.14)

€yc

d2v We recall that v\ag = -—-—. Summing over all cells eY covering Cl and noting that the dyady0 function w remains unchanged in a neighborhood of V we arrive at (4.3.9). To prove (4.3.10) we denote by w the extension of w € Vc given by the previous step. Since w e V therefore there exists a constant K5 independent of e and v and such that INIo.n<^EHat"«'llo.n.

(4-3-15>

|o|=2

provided that meas(roo U T0) > 0. From (4.3.9) and (4.3.15) we immediately get (4.3.10), D because IMkn* - ll^llo.nNow we are in a position to formulate the homogenization theorem.

Nonlinear behavior of plates

213

Theorem 4.3.4. The sequence of functionals { J\ }£>o is T-convergent in the weak topology o f / / 1 ^ ) 2 * #2(fi)to Jl{u,w)

= Gl,(u,w) -

feq(x)w(x)dx o.

- Li{u,w),

(u,w) G Vi(ft) ,

(4.3.16)

where 9 = \YC\/\Y\ and G\ has the form (4.2.6). The homogenized moduli Dfx», AfXfl are given by (4.2.9), (4.2.10), whilst the local problems have the form (4.2.10), (4.2.11) with Y being replaced by Yc. Let Jlh(u,w)

= inf{J,i(u,ii/)| (u,w) G V,(n)} . £

(4.3.17)

There is a subsequence {Pfii ', P*™*'}^^ weakly convergent in H (Q, ) x H2(Q) to {u,w) € Vi(fi), being the solution of (4.3.17). D 4.3.2.

l

2

Bifurcation of von Kirmdn plates: basic results

Prior to the discussion of bifurcation problems of perforated von Karman plates we intro duce the relevant operators and results on bifurcation of von Kirmdn plates without holes. For the sake of simplicity we assume that on To U Ti the plate is subjected to one-parameter in-plane loading, i.e., Na0na

= XNa on r 0 U r \ .

(4.3.18)

Let Vroo = { « e / / W | u = 0

on

r„o}.

(4.3.19)

fi,

(4.3.20)

Since diviV = 0

in

therefore the following variational equation is readily obtained INaPea0{v)dx

= A / NavadV ,

n

V v e Vroo .

(4.3.21)

r 0 ur,

Taking into account the constitutive equation (4.1.16)i we conclude that the solution N = (Na0) of (4.3.21) can be written in the following form Na0 = XNf

+ Sa0(w) ,

(4.3.22)

where Na0

=

Aa^e^(u0)

^

(4

3

23)

and u° G Vroo i s a solution to /Aa^eXli(u°)ea0(v)dx n

= j r 0 un

N°vadT ,

VveVr<xs.

(4.3.24)

214

Elastic plates

S — S(w) is given by Sa0(w) = Aa0X»[eXli{u{w)) + \w,xwtll} ,

(4.3.25)

and u(w) € VrM solves the following problem J Ao0X»eXll(u(w))eQ0(v)dx

= - J Aa0X" fXli(w)ea0(v)dx

,

V v € VVao . (4.3.26)

Here /a/3(io) = -wAW£. We observe that because wi0, € L"(fi) for eachfinitep, therefore The reader is advised tofindthe strong form of the variational equations (4.3.24), (4.3.26). Taking into account (4.3.22) in (4.1.18) we arrive at j D*0X»KX,{w)Ka0{v)dx + xJNf^-^-dx + jsa0(w)^-^dx n n n w € V , v&V .

= 0, (4.3.27)

We recall that the space V is now defined by (4.3.3). Equation (4.3.27) involves three operators. The first operator A: V -» V ,

(4.3.28)

j Da0X,1KXfl(w)Ka0{v)dx . n

(4.3.29)

is defined by (Aw,v)=

Since D = (Da0X,i) is positive definite, therefore A is an isomorphism of V on V*. The second operator B : V -* V , is defined by {Bw, v) = - j N£0w,0v:O,dx ,

(4.3.30)

Lemma 4.3.5. The operator B is linear, self-adjoint and compact from V into V". Proof. Indeed, w e V and thus w>a € L^ft) for anyfinitep. On the other hand, N£0 € L2(fi). Hence we conclude that N^w,0 € L 2 _ ' , (n), 77 > 0. Consequently B is compact. The remaining properties of B are evident. Remark 43.6. Let us recall the definition of completely continuous and compact operators, cf. Yosida(1978).

Nonlinear behavior of plates

215

Definition 4.3.7. Let X be a reflexive Banach space and let Xi be a Banach space. A mapping T : X - Xl , is called completely continuous if for each sequence {vn}n6N C X weakly convergent to v G X, the sequence {T(vn)}nen converges in the norm to T(v) in the space X\. □ It is evident that each completely continuous operator is continuous. To define a compact operator we require: (i) the continuity in the sense of norms, (ii) that the image of each set bounded in X be relatively compact in X\. We observe that due to the reflexivity of the space X, a completely continuous operator from X to Xi is compact. The converse statement is not true in general, the exception being the linear operators. □ The third operator C : V -» V* , is defined by (C{w), v) = - fsa0(w)wil3vtadx . (4.3.31) n Lemma 4.3.8. The operator C : V —> V" is completely continuous. It is homogeneous of degree 3 and is derivable from a positive potential Q : C(w) = ±
(4.3.32)

where Q{w) = [sal3{w)wt0Wiadx .

(4.3.33)

2

/ o E,) C 1 by The functional Q defined on Lr 22(fi,

Q(f) = 2Jsa0(f)fa0dx ,

(4.3.34)

n a0

is convex and positive. Here S (w) = Sa0(f{w)), }ap{w) = -w:awtp. Proof. For tp, £ e V, by (T(ip, 0,u{f, 0 ) w e denote the solution in L2(Q, Es2) x Vpoo o f the following problem divT = 0 in ft , TQ% = 0 n u =0

on r\roo , „„ r on Too,

(4.3.35) v

y

216

Elastic plates

It is evident that Ta0{ip, f) is bilinear in ip and £; moreover Sa*3(w)=Tal3(w,w).

(4.3.36)

We set Q( V, 0) < JT00^,

Ov,Aadx .

(4.3.37)

We have \Q(
,

(4.3.38)

where K is a positive constant. Moreover, the mapping ( Q(
(4.3.39)

By using (4.3.37) and (4.3.38) we conclude that Q(w) = Q{w, w, w, w) is Frechet contin uously differentiable on V and {Ql(w),9)=4Q{w,w,w,e)

= 4(C{w),9) .

(4.3.40)

It is not difficult to show that Q(w) and C(w) are completely continuous. To show that the functional (4.3.34) is positive we write /a/3 = -ea0(u(f))

+ aa0XllS^(f)

,

where a = A'1. Hence Jsa0(f)fa0dx n

= Jaa/l*Sal'(f)S*(f)dx n

- Jea0(u(f))Sa0(f)dx n

.

On account of (4.3.35)i, (4.3.36) and recalling the definition of S(f) we conclude that the last term vanishes. Since the matrix a is positive definite, the functional Q is positive. The convexity follows from the fact that it is quadratic. □ Remark 43.9. It is worth noting that the solutions of Q(w) = 0 are solutions to the Monge Ampere equation, cf. Mignot et al. (1981)

&w d2w f a2w y dx% dx\

\dxadx0) D

Nonlinear behavior of plates

217

Having introduced the operators A, B and C we may write the von Kantian equations in the form (PA)

weV

,

Aw-

XBw + C(w) = 0 .

(4.3.42)

An equivalent form is given by w - XBiw + Ci{w) = 0 ,

(4.3.43)

where Bi = A~lB and d = A~XC. If V is equipped with the scalar product (
(4.3.44)

Indeed, the properties of the operators A, B and C imply that the following general theorem applies, cf. Krasnosel'skii (1964). Theorem 4.3.10. Let H be a Hilbert space and N : H —> H a nonlinear completely continuous operator with N(0) = 0. It is assumed that N is the gradient of a weakly continuous functional F , F(0) = 0, and F is uniformly differentiable in a neighborhood of 0. Let N be Freshet differentiable at 0 and set N'(0) = D. Let D be a completely continuous self-adjoint operator. Then every characteristic number of the linear operator D is a bifurcation point of the nonlinear operator N. □ Remark 43.11. Consider the following nonlinear equation

(4.3.45)

where N(0) = 0. A number fi0 is said to be a bifurcation point if for any r],S > 0 there exists a characteristic number fi of the operator N such that \fi — /io| < 77 and to this characteristic number corresponds at least one eigenfunction

|M|<«5.

D

It is evident that the linearized problem (4.3.44) admits a countable family of eigenvalues Xj, j e (n,TO),n < 0, m > 0, and at least one of the integers is infinite: • •• < A-2 < A-i < 0 < Xi < X2 < . . . . The spectrum admits a positive part provided that there exists w such that (Bw, w) > 0. Then 1 (Bw.w) ,A „ ,,^ - r = max)--^-f , (4.3.46) Ai wev (Aw,w) and £(Aj) = {w € V\ Aw - X^Bw = 0} = [w e V\ (Aw,w) - A,(Bw,w) = 0} . (4.3.47)

Elastic plates

218

The eigenspace corresponding to thefirstpositive (or negative) eigenvalue is not necessarily of dimension 1, since the problem is of fourth order. Let us pass to the study of bifurcation points corresponding to the eigenvalues A-1 and Ai. We assume that Ai exists (Ai < +oo). If the negative part of the spectrum is empty, we set A-j = —oo. Theorem 43.12. (i) If A 6 (A-1, Ai) then the problem (Px) admits only the trivial solution w = 0. (ii) For A = Ai the set of solutions to (Px) consists of the cone of eigenvectors correspond ing to Ai, which verify Q(w) = 0. Proof. (i) Let A e (A-1, Ai) and let w be a solution to (Px). Then we have {Aw, w) - X(Bw, w) + (C(w), w) = 0.

(4.3.48)

By using (4.3.46) we get (Aw,w) - X(Bw,w) > I 1 - — 1 (Aw,w) > 0 , for A e [0, A^.and {Aw, w) - X(Bw, w) > f 1 - -— J (Aw, w) > 0 , for A 6 (A-!,0]. Since (C(w),w) =

~Q(w)>0,

therefore w = 0. (ii) If w is a solution to (4.3.44) for A = Ai then w satisfies {Aw, w) - Xi (Bw, w) = 0 ,

Q(w) = 0 .

(4.3.49)

Hence, by (4.3.46), we conclude that w e E(X\). Moreover, we know that Q(w) = 0 is equivalent to Sa0(w) = 0, which implies C(w) = 0. The solutions of (4.3.49) are thus also solutions to (Px). It remains to observe that if ty solves (4.3.49), then r)w ,77 g R, is also a solution. D Remark 43.13. (a) For A = A-1 and A-1finitethe part (ii) of the last theorem is formulated similarly. (b) If the equation of Monge Ampere with the boundary conditions: dw w = -r— = 0 on Too U T0 , w=0 on Tj , on admits only the solution w = 0, then for A = Xi the problem (PA) also admits only the solution w = 0.

Nonlinear behavior of plates

219

Theorem 4.3.14. Let A! be finite. Then the point (Ai,ti; = 0) is a bifurcation point for the von Karman plate considered. More precisely, there exists a bifurcating branch (Ai (R), wi (/?)) defined for every R > 0 such that AWl{R) - Xl{R)Bwl{R)+C(wl(R)) \wi(R)\\ = y/2R(l + O(R)) ,

=0,

limA,(fl) = A1

(4.3.50)

R—»0

where ||wi(i?)||2 = (Awi,wi). Proof. Let us introduce the manifold ER in V: T,R = {w€V\-(Aw,w)

+ -(C(w),w) = R},

R>0.

(4.3.51)

By using the Lagrange multiplier method to solve the maximization problem: Find (PR)

max{J(w)\ w G ER}

and denoting the relevant multiplier by n we have A = l//j. Here J(w) = -(Bw, w). We observe that the set EH is bounded and star-shaped with respect to the origin. Indeed, for each w £V, there exists unique t such that tw € ER. It can be shown that problem (PR) possesses a maximizer w € ER. TO this end one can use the direct method of the calculus of variations. Consequently, for each R > 0 there exists (Ai (R), w\ (R)) being a solution to (4.3.50)i. To prove (4.3.50)2 we observe that for each w e E j j one has 2R- Uc{w),w) = {Aw,w) = | H I 2 < 2R. On the other hand 0 < (C(w),w)< K\|w\\*<4KR\

K>0.

Thus, for each i » e E R w e obtain 2R - 2KR2 < | \w\ | 2 < 2R ,

(4.3.52)

and (4.3.50)2 follows. Let us pass to proving (4.3.50)3. Theorem 4.3.12 implies that \\(R) > X\. Let tR be such that tRwi G ER, ||WI|| = 1, Aw\ - X\Bw\ = 0. From the relation (4.3.52) we conclude that tR is of the order of 2R or t2R ~ 2R. Hence 1 > 1 X, - X,(R)

(Aw^w^R))

(Bwi(fl),u>i(fl)) + {CfaiR^MR))

+ 0{R)) 2R + AKR* -2R + 4KW-Tl{l ' which establishes the formula. Here we have exploited (4.3.50)i and (4.3.46).

Elastic plates

220

Remark 43.15. To prove the last theorem we made no assumption on the solutions of the Monge Ampere equation. In fact a vertical bifurcating branch is not precluded, i.e., Xi(R) = Ai in the case where (4.3.41) admits non-trivial solutions. Remark 4.3.16. Consider now the case of von Karman plate subjected additionally to transverse loading q. Written in the weak form the equilibrium equations consists of (4.3.21) and, cf. (4.1.18) f{Na0w,0v,a + Ma0Ka0(v))dx = jq{x)v(x)dx , V v 6 V . (4.3.53) n n We recall that now V is defined by (4.3.3). In this case Eq. (4.3.42) is to be replaced by (Px)

we V,

Aw- XBw + C(w) = q .

(4.3.54)

We have the following result, provided that previous assumptions are satisfied. Theorem 43.17. If Eq. (4.3.41) admits only null solution, then for each A the problem (Px) possesses at least one solution w. Proof. The solutions of (PA) are obviously the critical points of the functional J(v) = \{Av,v) - -(Bv,v)+Q(v)

- jqvdx . n This functional is lower semicontinuous. Thus it suffices to show that

(4.3.55)

lim J(v) —> +00 as ||i>|| —» oo . To obtain a contradiction, suppose that there exist K and a sequence {vn}nen, \\vn\\ —> +oo such that J{vn) < K. Taking V"

one easily shows that

1 2

W ' ^\\V- ^ ' O ^

2 X _ > '" '

Hence ^ - ^ , 0 < 0 ,

(4.3.56)

where £ is a weak limit of {£"}. Then Q(£) = 0, and since the equation of the Monge Ampere admits only null solution we conclude that f = 0 which contradicts (4.3.56). □

Nonlinear behavior of plates

221

4.3.3. Bifurcation points of the homogenized plate and the linearized problem In this section it is shown that the bifurcation points of the homogenized plate are the limits of the bifurcation points of the perforated plate when e — ► 0. For the perforated plates characterized by a small parameter e > 0 the relevant operators are denoted by Ac, Be,Cc. The space Vc = V(Q.C) is defined by (4.3.3) with SI being replaced by fi£. Consider the linearized problem for the perforated plate wc € VE ,

Aew£ - \cBewe = 0 , (Bcwc,wc) = ± 1 .

(4.3.57)

We observe that the operators A£ are uniformly coercive. Consequently the space V£ can be endowed with the norm \\v\\2 = {Acv,v).

(4.3.58)

Similarly to the previous section the problem (4.3.57) admits a countable family of eigen values {\Cj} , j € (n£, m£), where at least one of the two integers rf, mc is infinite, with ... < Aij. < Ai J+1 < ... < Ai, < 0 < A^ < A' < ... < A£ < ... . It is convenient to repeat the eigenvalues AE with their multiplicities. With each \\ is associated an eigenvector w\ normalized by {B€w\, w£) = sign A£. In this way the system {w\} constitutes a complete orthogonal system for the space Ve equipped with the norm \\v\\2 = (Acv,v). By E£(\) we denote the subspace of eigenvectors associated with the eigenvalue A. Similarly, the homogenized von Karmdn plate with smeared out holes is described by the operators Ah, Bh, Ch. These operators involve the homogenized moduli A^fXti, D^0Xl' given by Theorem 4.3.4. With the linearized eigenvalue problem is associated the system {(A*,«£)}, ie(no,mo),

t»feV,

Akw? = \!>Bkw? , (Bhw!*,w?) = sign A,''.

(4 3 59)

Here the space V is defined by (4.3.3). The following theorem specifies the spectral properties of the sequence {Ac — XBe}c>0 when e tends to zero. Theorem 4.3.18. Let {A| 0 } 6>0 (resp. {Ai^}£>0) remain bounded independently of e. Then there exists a subsequence e' of e such that V i e (0,fco](resp. V j g [-j0,0))

Af - A?

and P £ w\ -± w^ weakly in V as e' - » 0 .

222

Elastic plates

Proof. The extension operator P* and its properties are specified by Lemma 4.3.2. The proof of the last theorem is a consequence of the compactness of Bh and of the lemma, which follows, cf. Sec. 1.3.2. □ Lemma 43.19, Let {gc}e>o be a sequence in V strongly converging to gh. Consider the sequence C,e of the solutions in Ve of A'C = gc ■ Then P £ C e -'-C e

^

V weakly,

h

where £ is the solution of the homogenized problem h

A V = eg

.

C

Here 6 = \Y \/\Y\, cf. Sec. 3.12.

D

We also have the following result. Theorem 4.3.20. Let Xh be an eigenvalue of multiplicity p of the problem w£V, Ahw = XhBhw , Then, for every 77 > 0 there exist e0 > 0 such that for e < e0 , Af,..., Af+p_a lie in the interval (Ah -77, Xk + r)). Proof. For e sufficiently small there exists an eigenvalue \c in (Xh — 77, Xh + 77). To corroborate this statement suppose that it were not true. Then we could find a subse quence e' tending to zero such that (Ac> — A h B e ') -1 : VJ —> VC' has a norm bounded by AT/T?, K >Q. Moreover, for g € L2{Q) the solution £e' of

A*
llCllvi, < ^IMI0,fi« < *> • Hence P'V^C

in V weakly.

By using Lemma 4.3.19 we conclude that AhQ - XhBhC, = 9g . This contradicts the Fredholm alternative, since (Ah - Ahfl*)(V) n L2(fi) ={ge L2(fi)| fgtpdx = 0 , V

D

223

Nonlinear behavior of plates

4.3.4. Bifurcating branches of perforated and homogenized plates The results of Sec. 4.3.2 can be applied to the perforated von K£rm£n plate. Thus for each e > 0 the point (A,,0) is a bifurcation point: there exists a bifurcating branch (\\{R), Wi(R)) defined for R > 0, such that Acw\{R) - \\{R)Bcw$R)

+ Cc{w\(R)) = 0 ,

(Acw\(R), wl(R)) = 2R{1 + 0{R)),

lim \\(R) = X\ .

(4-3-61)

n—*0

The parameter X[(R) is the Lagrange multiplier associated with the maximization problem max.{{Bcv,v)\veT,%}

,

where ££R = {v e Ve\X-{A'v,v) + \(C*v,v) = R} . The passage with £ to zero yields the following result. Theorem 4.3.21. Let the homogenized problem Ahw - \Bhw admit a positive eigenvalue. Then the bifurcating branch (A* {R), w\(R)) converges in Fix V weakly to (Aj(fi), tuf (R)), being the non-trivial solution of Ahv$(R) - Af(rt)BX(rt) + C*(w? (/?)) = 0 . Moreover, (Bhv${R),v${R))

= max{(Bhv,v}\

veZhR},

where

Tk = {veV\±(A\v)

+ \(Chv,v) = R}.

Proof. Let us fix R and set w\(R) = w\. A priori estimates Since w\ G E^ and (Cew\, w$ > 0, we conclude that IKIk<2fl.

(4.3.62)

Tofindan estimate of Af (R) we write u

;

{Bcw\,w\)

Since {Aew\,w\) < 2R and (Ce(w\),v}\} < 4R, to estimate X\(R) from above it suffices to minorize {B£w[, w{).

224

Elastic plates

According to the assumption of the theorem, the problem (Ah — XBh) admits a positive eigenvalue Aj : {Ahrt,d)=0.

A $ - AfB\tf = 0 ,

Let tc be such that t ^ <= T,ER. We set $? = < K, ,

K2 < (AV^- ¥>f> < *3 ,

where K\ is independent of e. It can easily be shown that Ki

K2 + K
= Kl/K2

■

Since

and lim(BV^> = ^ , we deduce that OD

/V2 + K4/?

Thus we eventually obtain 0 < \\{R) < KG .

(4.3.64)

Passage to the limit (i) On account of (4.3.62) and (4.3.64) one can extract a subsequence, still denote by e, such that \im \$R) = n ,

l i m P £ ^ = wf

in V weakly .

(ii) Let R£ be the restriction operator from V to V£, FFv = v^. The dual operator, some times called the adjoint operator, R* : V* —» V has the following properties: 1. If / e W^V), R*(/) = /where -

2. R:(/,Q) = g-. a2/ 3

-

R

-(/^)

=

&^;-

w

{f{x)

for xeW ,

\o

for ien\ff.

225

Nonlinear behavior of plates 4. There exists a constant K > 0 such that |F^|U(Ve-,V) < K : for each £ > 0. When e tends to zero, we have lim r\l(Bew$ = Bhw? ,

in V strongly.

c—»0

Indeed, for v € V we write

Since Q

i• __^ N^ /\/- p0 in Z/(Q) weakly , Nf

and dip

(P£w,) -» - ^

in Lp(fi) strongly (for each p < oo)

dx/j

therefore ^— (P*™f) -^ KW- p^ dx0 dx0

^ Consequently,

dxa

Nf^(Pcw\)

j L

•" "(ty weakly, q < 2 .

converges to BhWi in W

1,q

(£l) weakly and hence

in V strongly. (iii) Similarly, it can be shown that t?.C£(wc) -► C'(wf)

in V strongly as e -* 0

We recall that

«c-M), w >-/^ W )g^

/^ u,ex:

-—dx , oxa

v&V

(4.3.65)

where S?0(wl) is the solution of the following problem: S £ € L2(0*, E°2) : jSf{w\)ea0{v)dx

S?»W)

= 0 , V v € Vr roo '

= ^ M „ ' ) + ig^I.

(4.3.66)

226

Elastic plates

The space V^M is defined by (4.3.19) with fi being replaced by Qe. We already knew that F^wi tends to w^ weakly in V, cf. step (i). Hence w

i,aw\,0

~ wi,<>wiJ3 ~* °

in i2 fi£

(

)

when

e -» ° •

Homogenization yields Sf(w\)

-± Sa0{w^

in L2(Q, E°2) weakly as e - 0 ,

where S(wi) is the solution of the homogenized problem: Sh{w1) € L2(fi, EJ) : Jsf(w^)ea0(v)dx

= 0,

V « e V^ (4.3.67)

Here u'10 is the solution of = J JVQt;Qdr , r0uri

JAf^ex^u^e^^dx n

V v e V^ -

(4-3.68)

and Eq. (4.3.23) is to be replaced by Na0 = Af^exii{u^

.

(4.369)

(iv) The previous steps imply that K[-K(R)Bcwt

+ Cc{w\)\ -> -uBhw^ + Ch{wt)

in V" strongly as e —» 0. Hence we conclude that Ahwt - vBhw* + C*(w?) = 0 • It thus remains to show that w\ ^ 0, w^ G £/j» and that (Bhw%, O

= max{(Bhw, w)\w € T.kR} .

(4.3.70) 1

This will imply that v = Aj (R) is the Lagrange multiplier corresponding to w }. Since R';4 £ K;I converges to AhWi in V strongly, we have l i m ( ^ X - ^ i ) = Iim(RM e i«f.PX) = {Ahw^,w^ . Passing to the limit in $Acwl,w*) + \{Ccwlw$

=R,

(4.3.71)

i (i4 V . «£> + \ (Chwl w?) = R.

(4.3.72)

we get

Nonlinear behavior of plates

227

Hence tyf 6 E^, and particularly wj" ^ 0. It remains to prove (4.3.70). Let (A, w) be the solution of Ahw° - XhBhw° + Ch(w°) = 0 , {B w°,w°) = sup{(Bhw,w)\w e £%}

(4.3.73)

h

We define Th e V by (Th, v) = A" /'Nfwapviadx n c is Similarly, T e V£* defined by

- fsa0{w°)w°0v_odx , n

- fsa0(wo)wo0v,adx

VveV.

,

V t, G V£ .

Here JV0Q" = 4£" v e;^(u 0 ), and u° is the solution of (4.3.24) with A being replaced by Aft. It is clear that R*T£ tends to Th in V* strongly when e -> 0. Let £c be the solution of A'C = T . It is obvious that Pc£e tends to £°, where £° is the solution of Ah(,° = Th . Hence £° = w°. There exists tc > 0 such that tc£c € T,CR:

|<^f,f > + fai?),?) = R ■

(4-3-74)

We observe that \hn(Ace,C> = ( ^ V . t u 0 ) ,

l i m < C m O = (C>°),ii, 0 ) .

C—*0

£-»0

It is not difficult to show that the positive solution of (4.3.74) tends to the positive solution of

%(AW,nP) + £{Ck(vP),vP)=R. We conclude that to = 1 and w° = wf. This completes the proof.

□

Remark 4.3.22. (i) The spectrum of (Ah — XBh) always has a positive or negative eigenvalue and the bifur cating branch at this point depends on e in the manner described by the last theorem. The geometry of the plate and the anisotropic properties of the homogenized moduli strongly influence this spectrum.

Elastic plates

228

(ii) The method based on maximization of the functional -{B,w,w}

over E# (or

- (Bh, w, w) over E^) is due to Berger (1967).

□

Kikuchi's method Berger's method just presented permits one to obtain only the bifurcation branch cor responding to the first positive or negative eigenvalue. We observe that the solution thus obtained realizes the minimum of the energy and is stable in the sense of Lyapunov. Our aim now is to show that the bifurcating branch in a neighborhood of a simple eigen value of the homogenized problem is the limit of the bifurcating branch (at the correspond ing simple eigenvalue) of the perforated plate. The approach used exploits an idea due to Kikuchi (1976), cf. also Kesavan (1974). It is based on a parametrization of the bifurcating branch. Let A/, be a simple eigenvalue of the problem (A h — \Bh) and wh an associated eigen vector:

( B V . fh) = sign A„ .

AV - AhBV = 0 ,

(4.3.75)

We introduce the following spaces Uh = {v 6 V\ (Bhv,
(4.3.76) (4.3.77)

By Fredholm's alternative the operator (Ah - XhBh) realizes an isomorphism between Uh and U'h. On account of Theorem 4.3.20 there exist 77 > 0 and e0 > 0 such that for each e, 0 < e < £0, the problem (Ae — XBe) admits only one simple eigenvalue A£, A£ € (Ah — V> ^h + v) associated with an eigenvector tpc such that Aeipc - A £ BV = 0 ,

( B V , iff) = sign Xc = sign Xh ,

(4.3.78)

and lim A£ = Xh ,

lim P V =
£—•0

£—>0

(4.3.79)

We also introduce the following spaces

uc = {v G v^ffv,
u; = {/

e

v;\(f, /> = 0}.

Thus we have the orthogonal decomposition of Vc for the scalar product generated by (A'Z,
teR.

Nonlinear behavior of plates

229

For S > 0 we set A/i(«) = {(A,ai)eRx Vc\w = t 0 and e0 > 0 such that for each e, 0 < e < e0. the bifurcating branch of the problem (P£) at (A£, 0) admits in the neighborhood Mc{&) of this point the following continuous parametrization: t - ( A £ ( t ) , vf{t) = t vc(t) assumes its values in Ue and (A£, wc) satisfies Acwc(t) - Xc{t)Bcwc(t) + C'lufft)) = 0 , A£(0) = A£,

wc(0)=0.

(4.3.80)

When £ tends to zero, for each t in (-<5,6), (\[{t), P'vc(t)) tends to (A(t), v(t)) i n R x V weakly. The function t —► (A(i),ty'1 + v(0 = ii/(t))isthe continuous parametrization of the bifurcating branch of the problem (p,) at (A&, 0) in the neighborhood A//,(<5) of this point:

/lV(() - X^i^B^it)

+ Ch{wh{t)) = 0 .

Moreover, the above representations give all non-zero solutions of (P£) (resp. (P/,)) in K(8) (resp. A/"„(c5)). □ Prior to passing to the proof we formulate four lemmas. We assume the following no tation: an assertion valid for e e [0,£o] will mean that it is true for the problems (P t ), e € (0, £0] as well as for the homogenized problem (Ph). To simplify our subsequent considerations we assume that Xh > 0, and consequently {Bc
IMIw»-«(n«) < #«IIHI«»(n«) • Proof. By using the properties of the extension operator P* we get, see Lemma 4.3.2 IMk>..(n«) < \\Pcw\\w.Hnn < K'q(n)\\Pew\\Hm) c

< Kq\\w\\HHnt) .

The nonlinear behavior of C is made precise by the next lemma.

D

Elastic plates

230

Lemma 4J.25. There exist a constant K > 0 independent of e e [0, e0] such that | | C » - C(v)\\v.

< K{\\w\\Ve + \\v\\Vc)2\\w - v\\Vt ,

V ^ £ K .

Proof. It suffices to consider the case e > 0. Recalling that, cf. (4.3.31) (Cc(w), 0 = I"Sf{w)w,0v,adx

, V w, v 6 Vc

we get

\\C(w)-C{v)\\v.

j,-S?{v)vJl)\\L>iav ■

Next we have

\\(Sf(w)w,0-Sf(v)v,0)\\w

< \\{(SfM -Sf (v))w,0]\\w + \\l(Sf(v)(w,d-v,0)}\\w)>-

By the previous lemma we get \\(Sf(v)(Wj3

- vfi))\\w

< K2\\[Sf(v)}\\L^n.,Ei)\\w

- v\\Vt .

Moreover l|Se(t;)|U»(n.1Ej) < K3\\v\We , \\Sc(w)-Se(v)\\Liin.,£i} < A'4||(w, 0 «', / 9-u a t;^)|| L , (n . iE j )
\\v\\v.)\\w-v\\v.,

since Vw ® Vw — Vv ® Vv = (Vw + Vv) ® (Vu; — Vv). This completes the proof. D To exhibit the nontrivial solutions of (P£) in the neighborhood of (A£, 0) we may set e w

= t
ve G Uc ■

We are going to show that these solutions are parametrized by t. Recalling that Ac
+ vc).

The Fredholm alternative implies that this problem admits a solution if and only if (\< - \€)(B*(t
Nonlinear behavior of plates

231

Hence A,

_^

{&(*? + *),
=

since {Beipc,
(4.3.81)

£

£

A = A £ + - ( C ( ^ + 0,¥> >Suppose that t is fixed. Below it is shown that Eq. (4.3.81)i admits a unique solution ve in a neighborhood of 0. To this end afixedpoint method is used. Then Eq. (4.3.81)2 yields A£ as a function of t. We set Htit(v) = ~CCW

+ v) + $Cc{tye + v),
(4.3.82)

Since {AE — XcBe) is an isomorphism between Ue and U*, Eq. (4.3.81 )i can be written as follows v = (A€ - \cB<)-xHc%t(v) = Tt,t{v).

(4.3.83)

Thefirstestimate of if is given by the following lemma. Lemma 4.3.26. Let fc > 1. There exists r(fc) > 0, independent of e G [0,e0]. such that each solution of (Pc) : (Xe, wc), vf = t 0 independent of e such that \\(A< - \t&)-i\\Uu.tUt) £

<M ,

im|L(v, ^ < M ,

£

||C (w )|| H . < M\\w'\\\ ■

By Lemma 4.3.25 and Eqs. (4.3.82), (4.3.83) we write IKHv. < M(W

+ ve\\2Vt + \\< - K$(\t\ + \WW.),

or \\v<\\Ve-M(\\t
232

Elastic plates

Hence, for |*| < r, ||v£||Ve < Jfcv, |A£ - A£| < r one has M[(l + fc)V +

T]

,,

It is thus sufficient to take for r(k) the positive root of 2M(l+k)2T2 + 2Mr -1 = 0 , and the lemma follows.

D

Rem