Selected Title s i n Thi s Serie s

21 Miche l Fortin , Editor , Plate s an d shells , 199 9

20 Kat i e Coughlin , Editor , Semi-analyti c method s fo r th e Navier-Stoke s equations , 199 9

19 Raji v Gupt a an d Kennet h S . Wil l iams , Editors , Numbe r theory , 199 9

18 Serg e D u b u c an d Gille s Deslauriers , Editors , Splin e function s an d th e theor y o f

wavelets, 199 9

17 Olg a Kharlampovich , Editor , Summe r schoo l i n grou p theor y i n Banff , 1996 , 199 8

16 Alai n Vincent , Editor , Numerica l method s i n flui d mechanics , 199 8

15 Frangoi s Lalonde , Editor , Geometry , topology , an d dynamics , 199 8

14 Joh n Harna d an d Ale x Kasman , Editors , Th e bispectra l problem , 199 8

13 Miche l C . Delfour , Editor , Boundaries , interfaces , an d transitions , 199 8

12 Pete r C . Greiner , Victo r Ivrii , Lui s A . Seco , an d Catherin e Sulem , Editors ,

Partial differentia l equation s an d thei r applications , 199 7

11 Lu c Vinet , Editor , Advance s i n mathematica l sciences : CRM' s 2 5 years , 199 7

10 Donal d E . Knuth , Stabl e marriag e an d it s relatio n t o othe r combinatoria l problems : A n

introduction t o th e mathematica l analysi s o f algorithms , 199 7

9 D . Levi , L . Vinet , an d P . Winternitz , Editors , Symmetrie s an d integrabilit y o f

difference equations , 199 6

8 J . Feldman , R . Froese , an d L . M . Rosen , Editors , Mathematica l quantu m theor y II :

Schrodinger operators , 199 5

7 J . Feldman , R . Froese , an d L . M . Rosen , Editors , Mathematica l quantu m theor y I :

Field theor y an d many-bod y theory , 199 4

6 Guid o Misl in , Editor , Th e Hilto n Symposiu m 1993 : Topics i n topolog y an d grou p

theory, 199 4

5 D . A . Dawson , Editor , Measure-value d processes , stochasti c partia l differentia l

equations, an d interactin g systems , 199 4

4 Hersh y Kisi levsk y an d M . R a m Murty , Editors , Ellipti c curve s an d relate d topics ,

1994

3 Rem i Vaillancour t an d Andre i L . Smirnov , Editors , Asymptoti c method s i n

mechanics, 199 3

2 Phil i p D . Loewen , Optima l contro l vi a nonsmoot h analysis , 199 3

1 M . R a m Murty , Editor , Thet a functions : Fro m th e classica l t o th e modern , 199 3

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Volume 21

c CR M R PROCEEDING S &

LECTURE NOTE S Centre d e Recherches Mathematique s Universite d e Montrea l

M

Plates an d Shell s

Michel Forti n Editor

The Centr e d e Recherche s Mathematique s (CRM ) of th e Universite d e Montrea l wa s create d i n 196 8 to promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Amon g it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l programs , an d publishing. Th e CR M is supporte d b y th e Universit e d e Montreal, th e Provinc e o f Quebe c (FCAR) , an d th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h th e Institu t de s Science s Mathematiques (ISM ) of Montreal , whos e constituen t members ar e Concordi a University , McGil l University , th e Universite d e Montreal , th e Universit e d u Quebe c a Montreal, an d th e Ecol e Polytechnique . Th e CR M may b e reached o n th e We b a t www.crm.umontreal.ca .

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American Mathematica l Societ y Providence, Rhod e Islan d US A

https://doi.org/10.1090/crmp/021

T h e product io n o f thi s volum e wa s suppor te d i n par t b y th e Fond s pou r l a Format io n de Chercheur s e t l 'Aid e a l a Recherch e (Fond s F C A R ) an d th e Natura l Science s an d Engineering Researc h Counci l o f C a n a d a (NSERC) .

1991 Mathematics Subject Classification. P r imar y 73Kxx ; Secondar y 65-XX .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Plates an d shell s / Miche l For t in, editor . p. cm . — (CR M proceeding s & lectur e note s ; v. 21 )

Includes bibliographica l references . ISBN 0-8218-0950- 4 (alk . paper ) 1. Shells (Engineering ) 2 . Plates (Engineering ) 3 . Elastic plates and shells . I . Fortin, Michel .

II. Series . TA660.S5P56 199 9 624.1776—dc21 99-2284 4

C1P

Copying an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r edu -cational an d scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y services tha t collec t fee s fo r deliver y o f document s an d provide d tha t th e customar y acknowledg -ment o f th e sourc e i s given . Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution, fo r advertisin g o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercial us e o f materia l shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permission@ams.org .

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and submitte d t o th e America n Mathematica l Societ y i n camer a read y form b y th e Centr e d e Recherche s Mathematiques .

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Contents

Preface vi i

Derivation an d Justificatio n o f Plat e Model s b y Variationa l Method s Stephen M. Alessandrini, Douglas N. Arnold, Richard S. Falk, and Alexandre L. Madureira 1

Simple Triangular Shel l Elements for Large Strain Estimations of Sheet Meta l Forming Part s Jean-Louis Batoz, Ying Qiao Guo, and Frederic Mercier 2 1

Some Approximatio n Method s fo r Linea r Thi n Shel l Problem s Michel Bernadou 3 7

Numerical Analysi s o f Piezoelectri c Shell s Michel Bernadou and Christophe Haenel 5 5

Etude intrinsequ e d'u n problem e d e coqu e e n grande s deformation s M. Carrive and P. Le Tallec 6 5

Locking-Free Mixe d Stabilize d Finit e Elemen t Method s fo r Bending-Dominated Shell s Dominique Chapelle and Rolf Stenberg 8 1

Un elemen t d e coqu e simpl e a troi s nceud s en grande s rotation s e t elastoplasticite F. Dammak, S. Chamlal, A. Gakwaya, and G. Dhatt 9 5

Intrinsic Nonlinea r Model s o f Shell s Michel C. D elf our and Jiabin Zhao 10 9

Convergence o f th e Linea r P ( l , 1 ) an d P(2,1 ) Thi n Shell s t o Asymptoti c Shells Michel C. D elf our and Jean-Paul Zolesio 12 5

Two-Level Overlappin g Schwar z Method s fo r Plat e Element s o n Unstruc -tured Meshe s Usin g Non-Matchin g Coars e Grid s Qingping Deng and Xiaobing Feng 15 9

Large Elasto-Plasti c Deformation s o f Thi n Shell s wit h Applicatio n t o th e Stamping o f Anisotropic Shee t Meta l Part s J.-C. Gelin and L. Boubakar 17 1

vi CONTENT S

Justification o f Two-Dimensiona l Linea r Shel l Model s b y th e Us e of T- Convergence Theor y Karine Genevey 18 5

Numerical Simulatio n o f High-Rate Stampin g o f Tube s an d Sheet s S. F. Golovashchenko and N. M. Bessonov 19 9

An Intrinsi c For m o f the Stres s Resultan t Geometricall y Exac t Shel l Theor y Adrian Ibrahimbegovic 20 9

Numerical Inf-Su p Analysi s o f MIT C Plat e Bendin g Element s Alexander Iosilevich, Klaus- Jilrgen Bathe, and Franco Brezzi 22 5

Optimisation de s plaque s e t coque s E. Pagnacco and J. E. Souza de Cursi 24 3

Vibroacoustic Behavio r o f a Rectangula r Bo x M. A. Tournour and N. Atalla 25 5

Error Analysi s o f Mixe d Finit e Element s fo r Cylindrica l Shell s Geng Yang, Michel C. Delfour, and Michel Fortin 26 7

Preface

The 199 6 Summe r Semina r o f th e Canadia n Mathematica l Societ y too k plac e in Quebe c cit y fro m Jul y 2 2 to Jul y 27 , in the buildin g o f l'Ecole d'Architectur e a t the Universit e Laval . Th e scientifi c committe e consiste d o f

• Miche l Forti n (chair) , Departemen t d e mathematique s e t d e statistique , Laval,

• Mari o Fafard , Departemen t d e Geni e civil , Laval , • Miche l Delfour , Centr e d e recherches mathematiques , Montreal .

The objectiv e o f thi s Semina r wa s t o brin g togethe r bot h mathematician s an d engineers interested i n the theory or application o f plates and shells , or more gener -ally in the modelisatio n o f thin structures . Fro m thi s i t was hoped tha t ther e coul d emerge a better understandin g o f the problem: mathematician s coul d becom e mor e aware o f ho w actua l engineerin g application s wer e performe d an d engineer s woul d know bette r wha t informatio n o f theoretica l result s coul d provide .

Thin structure s ar e me t everywher e i n applications . Thei r approximatio n b y a lower-dimensional mode l i s natural an d almos t compulsor y . I t i s also full o f pitfall s and som e engineering catastrophe s ca n b e attribute d t o bad numerica l scheme s fo r shells.

From th e mathematica l poin t o f view , man y studie s tr y t o mak e cleare r th e convergence o f ful l three-dimensiona l model s t o model s o f plate s o r shells . I n par -ticular, th e underlyin g assumption s ar e mad e mor e explicit .

Numerical method s ar e th e ke y t o actua l computations . Thi n structure s ar e known t o nee d specia l methods , ofte n inspire d b y mixe d finite elemen t methods . The goa l i s to avoi d numerica l lockin g whil e providin g a goo d accuracy .

New applications , suc h a s th e simulatio n o f cras h i n th e automobil e industry , bring ne w challenge s a s model s an d metho d ar e stretche d t o thei r extrem e limit .

All thes e aspect s ar e reporte d i n thes e Proceeding s whic h giv e a snapsho t o f the stat e o f the ar t o f a fas t evolvin g subject .

Michel Forti n Montreal, Februar y 199 9

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