Series on Advances in Mathematics for Applied Sciences, Volume 52 : Plates, Laminates and Shells - Asymptotic Analysis and Homogenization

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.

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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 < p < oo and let ft C Rn be an open set. A measurable function / : $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 < p < oo, then fm ->• / 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.°° =

max {||3°'v|U°'>}

if

0