Homogenization of von Karman Plates Excited by Piezoelectric Patches

Transcription

1 Hoffmann, K.-H.; Botkn, N. D.: Homognzaton of von Karman Plats 579 ZAMM Z. Angw. Math. Mch. 8 2) 9, 579 ±59 Hoffmann, K.-H.; Botkn, N. D. Homognzaton of von Karman Plats Ectd by Pzolctrc Patchs A modl dscrbng vbraton of nonlnar von Karman thn plats ctd by actuators mad of pzolctrc cramcs s consdrd. Th modl contans strong oscllatng coffcnts du to th pzolctrc actuators. A procdur of homognzaton basd on th so-calld two-scal convrgnc s appld to th modl. Ths ylds a nonlnar systm of quatons wth constant coffcnts. Th unqu solvablty of th rsultng systm s provd. Th convrgnc of all solutons of th orgnal systm to th soluton of th rsultng systm as th numbr of pzolctrc actuators gos to nfnty s provd. Ky words: nonlnar von Karman thn plats, pzolctrc actuators, homognzaton, two-scal convrgnc MC 99): 35B27, 73K, 73R5. Introducton Th problmof homognzaton of partal dffrntal quatons dscrbng vbraton of nonlnar thn plats ctd by actuators mad of pzolctrc cramcs s [] and [2]) s consdrd. It s assumd that th numbr of th actuators gos to nfnty whras thr dmnson tnds to zro. A procdur of homognzaton basd on th thory of two-scal convrgnc studd n [3, 4, 5, 6] s usd. pcfc faturs of th problmconsdrd ar: tm dpndnt quatons, th apparanc of th forth spatal drvatvs n th frst quaton dscrbng vrtcal dsplacmnts of th plat, and nonlnarts typcal for von Karman systms. W apply a rsult of [6] about two-scal convrgnc of th scond drvatvs of subsquncs of squncs boundd n L 2 ;T; H 2, whch nabls us to handl a wak formulaton of th problm. Rsults of [7] and [8] ar usd. Computr smulatons dmonstrat a good appromaton of solutons of th orgnal quatons by solutons of th homognzd quatons whnvr th numbr of pzolctrc patchs s suffcntly larg. 2. Notaton R 2 s th doman occupd by th plat. K t; s a prscrbd dstrbuton of th voltag ovr th whol plat. Pl s th doman occupd by th lth pzopatch. P :ˆ m s th doman occupd by all pzopatchs. Pl lˆ B :ˆ n P :ˆ ; Š ; Š hg :ˆ g y dy C # C R 2 H m # H# m =R Q :ˆ ;T C T Q C Q s th doman occupd by th bas matral. s th unt squar. s th man valu of a functon. s th subspac of prodc functons,.. g y ;y 2 ˆg y ;y 2 ˆg y ;y 2, for all y ;y 2 2R 2. s th complton of C # for th normof Hm ; t holds: D a uj ˆ yˆ Da uj and yˆ Da uj ˆ y2ˆ Da uj for y2ˆ a ˆ a ; a 2 ; a ; a 2 ; a a 2 m. s th quotnt spac. s th subspac of all functons whch vansh and at t ˆ T along wth all drvatvs. C ;T Q; C# s th spac of nfntly dffrntabl functons from Q nto C # whch vansh att ˆ, and t ˆ T along wth all drvatvs. HT 2 ;T; L 2 s th subspac of all functons from H 2 ;T; L 2 whch vansh at t ˆ T along wth th frst tm drvatv. X 2 :ˆ L ;T; H 2 L ;T; H L ;T; H X :ˆ L ;T; H L ;T; L 2 L ;T; L 2 L s th st of all functons ;u ;u 2 that ar lmts of Galrkn appromatons n problm ) w.r.t. wak * topology of X 2.

2 58 ZAMM Z. Angw. Math. Mch. 8 2) 9 d j ;u :ˆ 2 u j u j j s th n-plan stran tnsor. W us notaton: d j :ˆ d j ;u ; d w j :ˆ d j w ;u w ; ^d j :ˆ d j ^; ^u tc. j u :ˆ 2 u j u j s th lnar part of th n-plan stran tnsor. kfk :ˆ kfk L2 s L 2 normof a functon. f; g :ˆ f; g L2 s L 2 scalar product. W assum summaton ovr rpatng ndcs. For ampl, k a k 2 ˆk a kk a k :ˆ P2 k a kk a kˆkk 2 H : aˆ 3. Problm sttng Consdr a systmof nonlnar quatons dscrbng oscllatons of a thn plat ctd by patchs mad of a pzolctrc cramc s [2]). For smplcty, assum that th plat occups a rctangular doman and th pzolctrc patchs occupy rctangular domans Pl s Fg. ). It s assumd that th patchs form a prodc structur of th prod so that th objct s compltly dfnd by. Dnot P :ˆ m Pl and B :ˆ n P. lˆ Fg.. Plat wth patchs Pl Unt cll scald from to Equatons dscrbng th modl rad ~q tt dv ~mr tt D ~g ~t b ˆFv l t DI Pl Gv l a a I Pl ; ~t ab ˆGv l t I Pl ; whr ~t ab a 2 u j u j j : Hr and u a ; a ˆ ; 2, ar vrtcal and longtudnal dsplacmnts, D s th Laplac oprator, v l t s th voltag appld to th lth pzolctrc patch Pl, I Pl s th ndcator functon of th lth patch. Th ndcs a and b run ovr f; 2g, th nd l runs from to m, whr m s th numbr of pzopatchs. ummaton ovr rpatng ndcs s assumd. Th coffcnts F and G ar constant; ~q, ~m, ~g, and ~`jab ar dscontnuous pcws constant functons dfnd as follows: ~q ˆ qp ; 2 P ; ~m ˆ mp ; 2 P ; q B ; 2 B ; m B ; 2 B ; ~g ˆ gp ; 2 P ; `Pjab ~`jab ˆ ; 2 P ; g B ; 2 B : `Bjab ; 2 B : Nonzro componnts of `Pjab and `Bjab ar dfnd as follows: Ew `w ˆ s 2 ; `w22 ˆ Ews w w s 2 ; `w6ˆj a6ˆb ˆ ; `w22 ˆ `w22 ; `w2222 ˆ `w ; w ˆ P; B : w 2 s w Hr, E P and E B ar th lastc modul; s P and s B ar th Posson ratos. Indcs P and B pont out to th pzolctrc and bas matrals, rspctvly. It s assumd that v l 2 H ;T, q P > ; q B > ; m P > ; m B > ; g P > ; g B > ; and nd ab d ab `wjab d jd ab Nd ab d ab ; w ˆ P; B 2 for any symmtrc d j 2 R 2 2, whr n and N ar postv constants. Assum that th controls v l ar bng chosn as follows. A dstrbuton K t; 2H ;T; L 2 of th voltag ovr th whol plat s prscrbd and w st v l t ˆmas Pl K t; d : 3 Pl E w

3 Lt K t; ˆ K t; ; 2 B ; 4 v l t ; 2 Pl : It s obvous that K! K n H ;T; L 2. On can rwrt ) as follows: q tt dv m r tt D g t a b ˆ FD K t; I a K t; I ; 5 whr q u t b t ab ˆ a a 2 u j u j : j K t; I ; a ˆ ; 2 ; P Hr q; m; g, I, `jab ar prodc functons dfnd on as follows: I y ˆI D y ; q y ˆI D y q P I D y q B ; m y ˆI D y m P I D y m B ; g y ˆI D y g I D y g B ; `jab y ˆI D y `Pjab I D y `Bjab ; 6 whr I D s th ndcator functon of D s Fg. rght). It follows from 2) that nd ab d ab `jab y d j d ab Nd ab d ab 7 for any y 2 and any symmtrc d j 2 R 22. Boundary and ntal condtons ar ˆ ; j tˆ ˆ ; t j tˆ ˆ ; u a ˆ ; u a j tˆ ˆ u a ; u at j tˆ ˆ u a ; a ˆ ; 2 : 8 Not that 5) should b suppld wth th followng ntrfac condtons: ˆ ; gdš gd ˆ ; u a Š ˆ ; t ab n b ˆ ; a ˆ ; 2 ; 9 that hold on th boundary btwn P and B bcaus of th ntgraton by parts whn drvng 5) froma wak formulaton. Hr, Š dnots th jump of a functon on th boundary btwn P and B. W do not pay any attnton to ths condtons bcaus w wll go back to th wak formulaton. Dfnton : W say that functons 2 L 2 ;T; H 2 ; u a 2 L 2 ;T; H ; a ˆ ; 2 ; forma wak soluton to systm 5) ± 9) f th followng qualty holds: for all Hoffmann, K.-H.; Botkn, N. D.: Homognzaton of von Karman Plats 58 T h q j tt m r rj tt g D Dj t ab b j a FK t; I Dj GK t; I a j a q u a y att t ab y ab GK t; I y aa d dt h q j t ; j ; u ay at ; u a y a ; m r rj t ; r rj ; d ˆ j 2 H 2 T ;T; H \ L 2 ;T; H 2 ; y a 2 H 2 T ;T; L 2 \ L 2 ;T; H : Proposton : Lt m and u m a b Galrkn appromatons computd usng th abov dfnton of solutons. Thn: a) If 2 H 2 ; 2 H, u a 2 H ; u a 2 L 2, and K 2 H ;T; L 2, thn thr sts a constant C ndpndnt of ;m such that k m k 2 H 2 for any t 2 ;TŠ. km t k 2 H kum a k2 H kum at k2 L 2 C

4 582 ZAMM Z. Angw. Math. Mch. 8 2) 9 b) If ˆ ; ˆ ; u a ˆ ; u a ˆ, K 2 H2 ;T; L 2, and Kj tˆ ˆ, thn thr sts a constant C ndpndnt of ; msuch that k m t k 2 H 2 km tt k2 H kum at k2 H kum att k2 L 2 C 2 for any t 2 ;TŠ. Rmark : Th proof of th scond part of th proposton s basd on th formal tm dffrntaton of quatons dfnng m and u m a. Th trm K t s 3) and 4)) arss on th rght-hand-sd of ths quatons. Th assumpton K 2 H 2 ;T; L 2 s ncssary to stmat trms contanng K t. Th rqurmnts ˆ ; ˆ ; u a ˆ ; u a ˆ, and Kj tˆ ˆ provd th so-calld compatblty condtons s.g. [9]), whch guarants that th formally dffrntatd quatons dfn tm drvatvs of m and u m a. For ampl, on of th compatblty condtons rads g = D 2 H. Ths can b satsfd by vry spcal choc of usng th structur of th dscontnuous functon g =. To avod such dffcults, w st ˆ. mlar argumnts plan th choc of th othr ntal condtons to b homognous. P r o o f : Th proof of th frst part of th proposton can b found n [2]. Prov th scond part. Lt m ˆ Pm ˆ a m t w ; u m a ˆ Pm ˆ b m a t h ; a ˆ ; 2 ; whr fw g ˆ, fh g ˆ ar bass of H2 and H, rspctvly, and th coffcnts am t, bm a t satsfyng th ntal condtons a m ˆ; d=dt am ˆ; bm a ˆ; d=dt bm a ˆ ar found from ), that s q m tt j m r m tt rj g D m Dj t m ab m b j a d ˆ FK t; Dj d P q u m att y a t m ab P GK t; m a j a d ; y ab d ˆ P GK t; y aa d for all t 2 ;TŠ, j 2 spanfw ;...; w m g; y a 2 spanfh ;...; h m g: Hr t m j ˆ j m ; ;j ˆ ; 2 : j Formal dffrntaton w.r.t. t ylds q w m tt j m rw m tt rj g Dw m Dj _t ab m m b j a t m ab Hr ˆ FK t t; Dj d P q _t m j v m att y a _t ab m ˆ `jab 2 P GK t t; m a j a d y j P P GK t t; y aa j GK t; w m a j a w m b j a ; ;j ˆ ; 2 : nc th compatblty condtons ar fulflld, t holds w m ˆ m t and v m a ˆ um at. W st j ˆ w t, y a ˆ v at and sumup th quatons. Ths ylds th followng qualty: 2 d dt ˆ 3 2 kqw m t k 2 kmrw m k 2 kgdw m k 2 kqv m _t ab m wm a w m b d FK t t; P GK t w m a w m a d t at k2 d dt Dwm d ` abj _tm ab _tm j d P GK t t; d dt ` aaj _tm j d d t m ab wm a w m b d GK t; w m a w m a d ; 3! P P

5 Hoffmann, K.-H.; Botkn, N. D.: Homognzaton of von Karman Plats 583 whr ` abj s th nvrs of `abj, that s ` abpq`jpqs j ˆ s ab 4 for any symmtrc s 2 R 2 2. Not that n=n _t ab m _tm ab ` abj _tm ab _tm j =n _t ab m _tm ab 5 bcaus of 2) and 4). Intgraton of 3) ovr t gvs kqw m t ˆ k 2 kmrw m t k 2 kgdw m k 2 kqv m at k2 t m ab wm a w m b d 3 t 2 t FK tt t; Dw m d 2 P P 2 GK t w m a w m a d 2 P ` abj _tm ab _tm j d _t ab m wm a w m b d 2 FK t t; Dw m d P Usng th nqualts of chwartz and Gaglardo-Nrnbrg ylds P GK t t; ` aaj _tm j d 2 t GK tt t; ` aaj _tm j d P GK t; w m a w m a d : 6 kw m a w m b kkrw m k 2 L 4 Ckrwm kkw m k H 2 Ckwm k H 2 : Th fnal nqualty s tru bcaus of th boundnss of krw m k du to ). Usng that kt m ab k2 C du to ), w obtan from 6) kw m k 2 H 2 kwm t k 2 H kvm at k2 k_t ab m k2 C: 7 Takng nto account m b ˆ ` abj _tm j a 2 m a w m b w m a m b ; w obtan m a C k _tm ab k km a w m b k C k _t ab m k km a k 2 L 4 kwm a k 2 L 4 C k _t ab m k km k 2 H 2 kwm k 2 H 2 C usng ), 7) and th contnuous mbddng H 2 W 4. Th nqualty of Korn mpls kv m a k2 H C; whch complts th proof of proposton. & Proposton 2: Lt L b th st of all lmt ponts n wak * topology of X 2 ) of all possbl Galrkn appromatons n. Thn: a) L 6ˆ;and ach ;u ;u 2 2L s a waksoluton of. b) If th frst assumpton of proposton holds, thn thr sts a constant C ndpndnt of such that k k 2 L ;T; H 2 k t k2 L ;T; H ku a k2 L ;T; H ku at k2 L ;T; L 2 C 8 for any ;u ;u 2 2L. c) If th scond assumpton of proposton holds, thn thr sts a constant C ndpndnt of such that k t k2 L ;T; H 2 k tt k2 L ;T; H ku at k2 L ;T; H ku att k2 L ;T; L 2 C 9 for any ;u ;u 2 2L. P r o o f : Nonmptnss of L follows from ). Th proof that ach ;u ;u 2 2L s a wak soluton of ) can b found n [2]. It follows from ) and 2) that all ncssary tm drvatvs n 8) and 9) st, and that 8) and 9) hold. &

6 584 ZAMM Z. Angw. Math. Mch. 8 2) 9 4. Homognzaton Th followng holds for any ;u ;u 2 2L du to Proposton 2: k k 2 L ;T; H 2 ku k2 L ;T; H ku 2 k2 L ;T; H C; wth C ndpndnt from. Thrfor, th squnc ;u ;u 2 contans a wak * convrgng subsqunc n X2.Now w drv quatons that dfn lmt functons of such subsquncs ffctv quatons). W wll show that ths quatons hav a unqu soluton undr assumptons of th scond part of Proposton, and that th coffcnts of th ffctv quatons ar ndpndnt of th choc of subsquncs. Ths ylds that th squnc ;u ;u 2 2L convrgs wak * n X 2 to th soluton of th ffctv systm. mlar argumnts show that t ;u t ;u 2 t convrgs wak * n X 2, and tt ;u tt ;u 2 tt convrgs wak * n X to tm drvatvs of th soluton of th ffctv systm. Thn, usng Corollary 4 of mon [] and Thorm6. of Lons [], w conclud that ;u ;u 2 and t ;u t ;u 2t convrgs strongly n C ;TŠ; H 2 s C ;TŠ; H s C ;TŠ; H s for any postv ral s. In partcular, and t convrg unformly on ;TŠ. Now w apply two-scal convrgnc to drvaton of th ffctv quatons. For th ponrng works on twoscal convrgnc for tm-ndpndnt problms w rfr to [3] and [4]. Two-scal convrgnc for tm-dpndnt problms was consdrd n [5]. Ths rsults wr gnralzd n [6]. Lt us rproduc th dfnton of two-scal convrgnc of functons dpndng on addtonal paramtrs Dfnton 6.8 of Hallr [6]): Lt v 2 L 2 Q ; v 2 L 2 Q. It s sad that v 2! scal v,f lm T v t; y t; ; = d dt ˆ T v t; ; y y t; ; y dy d dt! for all y 2 C ;T Q; C#. It s provd Thorm6.5 of Hallr [6]) that all proprts of two-scal convrgnc hold, f th tst functons n th abov dfnton ar rplacd by mor gnral tst functons of th form: y t; ; y ˆf t; g y s t; ; y, whr f 2 L Q, g 2 L #, and s 2 C Q; C# not ncssary vanshs). Thorm6.2 of Hallr [6] s a gnralzaton of rsults of [4, 5] about two-scal convrgnc of boundd functonal squncs. Lt kv k L2 ;T; H m C wth C ndpndnt of and m ˆ ; 2. Thn thr st j, v t; 2L 2 ;T; H m, and v t; ; y 2L 2 Q; H# m such that v j wak! * v n L ;T; H m ; D k v 2 j! scal D k v ; k 2 ;m ; D m v 2 j! scal D m v D m y v: Du to th abov rsult, thr st j, t; 2L 2 ;T; H 2, u a t; 2L 2 ;T; H, t; ; y 2 L 2 Q; H# 2, and u a t; ; y 2L 2 Q; H# such that j u j a wak! * n L ;T; H 2 ; j 2! scal ; r j 2! scal r ; D j 2! scal D D y ; wak! * ua n L ;T; H ; uj a 2! scal u a ; u j a b 2! scal u ab u ayb : Morovr, from 8) and from th compact mbddng H 2 W q, for any q>, w conclud that f j g s rlatv compact n C;T ; Wq for any q> s []). o, w can assum that j! n C ;TŠ; Wq for any q>. To obtan ffctv quatons dfnng and u a, st j t; ˆh t; 2 f t; ; = ; y a t; ˆc a t; q a t; ; = ; whr h t; ; c a t; 2C T Q, and f t; ; y ; q a t; ; y 2C ;T Q; C#. ubsttutng ths functons nto ) ylds w omt th nd j for brvty) T q h tt 2 f tt Š m r rh tt 2... Š g D Dh D y f 2... Š t ab b h a 2... Š FK t; I Dh D y f 2... Š GK t; I a h a 2... Š q u a c att q. attš t ab c ab q ayb.. Š GK t; I. c aa q aya.. Š d dt n q h t ; h ; u ac at ; u a c a ; Š m o r rh t ; r rh ; Š d ˆ : 2

7 Hoffmann, K.-H.; Botkn, N. D.: Homognzaton of von Karman Plats 585. Th symbol... dnots trms wth th multplrs and, whras th symbol.. dnots trms wth th multplrs. Ths trms appar whn applyng dffrntal oprators of ) to th functons f t; ; = and q a t; ; =. For ampl, th trm... n th frst ln of 2) s qual to r f tt = r y f tt, whr r and r y dnot th gradnts wth rspct to th scond and th thrd varabls of f t; ; =. Consdrng q y h tt t; 2 f tt t; ; y Š ; m y rh tt t; 2... Š ; g y Dh t; D y f t; ; y 2... Š ; `jab y b t; h a t; 2... Š ; I y Dh t; D y f t; ; y 2... Š ; I y a t; h a t; 2... Š ;. q y c tt t; q tt t; ; y Š ; `jab y c ab t; q ayb t; ; y.. Š ;. I y c aa t; q aya t; ; y.. Š. as tst functons, on can pass to th two-scal lmt n 2) takng nto account that 2... and.. convrg unformly to, K! K n H ;T; L 2, and a! a n C ;TŠ; L q for any q>. nc th last systm contans two tst functons, th lmtng systm wll b splt nto two ons. Th frst, ffctv, systmdfnng th lmt functons and u a rads T fq y h tt m y rrh tt g y D D yš Dh `jab y 2 u j u j j u yj u jy b h a FK t; I y Dh GK t; I y a h a q y u a c att `jab y 2 u j u j j u yj u jy c ab GK t; I y c aa g dy d dt fq y h t ; h ; u ac at ; u a c a ; Š m y r rh t ; r rh ; Šg dy d ˆ : 2 Th so-calld cll quatons dfnng aulary functons and u a rad t fg y D D yš Dy f FK t; I y D y f `jab y 2 u j u j j u yj u jy q ayb GK t; I y q aa g dy d dt ˆ : 22 Hr y s th ndpndnt varabl, whras s tratd as a paramtr. nc th systm 22) s lnar, th suprposton prncpl ylds t; ; y ˆN y D M y FK t; ; u t; ; y ˆN mn y 2 u m n u nm n m M y GK t; ; whr N; M 2 H# 2 and N mn;m 2 H# ar unknown functons. ubsttutng thm n 22) and som computaton yld t fdg y D y NŠ FK t; g y D y M I y Šg D y f dy d dt ˆ ; T d mn ;u `jab y d m d jn 2 GK t; `jab y j d ab I q ayb dy d dt ˆ : Bcaus of th suprposton prncpl on can sk N; M; N mn, and M sparatly. Takng th tst functons of th form f t; ; y ˆf t; f 2 y, q a t; ; y ˆq a t; q2 a y, w obtan g y D y N D y f 2 dy ˆ ; 25 g y D y M I y D y f 2 dy ˆ ; 26

8 586 ZAMM Z. Angw. Math. Mch. 8 2) 9 `jb y `j2b y `jb y `j2b y 2d m d jmn q 2 b dy ˆ ; 2d m d jmn q 2 b dy 2d b I y 2d 2b I y q 2 y b dy ˆ ; q 2 2y b dy ˆ for all f 2 2 H# 2 ; q2 a 2 H # ; a ˆ ; 2; m; n ˆ ; ; 2; 2 ; ; 2. Not that th cass 2; and ; 2 ar quvalnt. Proposton 3: Th quaton 25 s quvalnt to th followng on: g y D y N ˆh=g ; for a:: y 2 : 29 Th quaton 29 has a unqu soluton N 2 H# 2 =R. Th quaton 26 s quvalnt to th followng on: g y D y M I y ˆ h=g hi=g ; for a:: y 2 : 3 Th quaton 3 has a unqu soluton M 2 H# 2 =R. P r o o f : Lt us sktch th proof of 29). Th clam 3) can b handld smlarly. Not that th quaton 25) s quvalnt to th followng on: g y D y N CŠ D y f dy ˆ ; 8f 2 H# 2 ; 3 whr C s an arbtrary constant. Ths holds bcaus Thn th quaton D y f dy ˆ for all f 2 H# 2. LtC ˆ g y D y N dy. D y f ˆ g y D y N C s solvabl n H# 2 =R bcaus g y D y N CŠ dy ˆ. ubsttutng th soluton nto 3) ylds g y D y N ˆC for a:: y2 : Dvdng ths quaton ovr g y, ntgratng ovr, and takng nto account that D y Ndyˆ, w obtan that C ˆh=g, whch provs 29). [2] for mor dtals f ncssary. & Proposton 4: ystms 27 ; 28 ar unquly solvabl: N mn ;M 2 H# =R. P r o o f : Lt us consdr th blnar formdfnd on H# =RŠ2 H# =RŠ2 that corrsponds to both 27) and p N; L ˆ dy @L b dy j b j a Th form s symmtrc and contnuous on H# =RŠ2 H# =RŠ2. It follows from 7) that p N; N n r j r j dy ; whr r j : Modfyng argumnts of th proof of Korn's nqualty s [3]), w obtan p N; N C N j N j dy ˆ C knk 2 H : # =RŠ2 Th lnar forms m mn L ˆ a `jab d m d jn dy ; m L ˆ b d ab I b corrspondng to 27) and 28) ar dfnd and contnuous on H# =RŠ2. Applyng th La-Mlgram lmma complts th proof. & dy

9 Hoffmann, K.-H.; Botkn, N. D.: Homognzaton of von Karman Plats 587 Aftr substtutng th functons, u a ; a ˆ ; 2; s 23) and 24)) nto th ffctv systm 2), w obtan T hq h tt hm rrh tt hg D y NŠ D Dh `jab y d m d jmn d mn b h a FK t; hgd y M I Dh d GK t; `jab j d ab I b h a d dt fhq h t ; h ; hm r rh t ; r rh ; g d ˆ ; 32 T hq u a c att `jab y d m d jmn d mn ;u GK t; `jab j d ab I y ab d dt hq u a c at ; u a c a ; d ˆ : P Thus, th followng systmwth constant coffcnts s obtand: T f^qh tt ^mrrh tt ^gd Dh ^t ab b h a FK t; ^I Dh GK t; ^J ab b h a g d dt f^q h t ; h ; ^m r rh t ; r rh ; g d ˆ ; 33 T f^qu a c att ^t ab c ab GK t; ^J ab c ab g d dt ^q u a c at ; u a c a ; d ˆ : P Th classcal formrads ^q tt ^m D tt ^g a ^t ab b ˆF ^I DK t; G ^J ab b ^t ab ˆ G a b K b K t; ; whr ^t ab ˆ ^`jab u j u j j : Th boundary and ntal condtons ar ; ; j tˆˆ ; t j tˆˆ ; u a ; u a j tˆˆ u a ; u at j tˆˆ u a : Comparng 32) and 33) and usng 29) and 3), w obtan plctly ^q ˆhq ; ^m ˆhm ; ^g ˆh=g ; ^I ˆh=g hi=g : Morovr, comparng 32) and 33) gvs ^`mnab ˆ `jab y d m d mn jmn y ; 35 ^J ab ˆ `jab j y d ab I y : 36 Conjctur: Th tnsor ^`mnab s postv dfnt,.., thr s a postv constant n such that ^`mnab d j d ab nd j d j for any symmtrc d 2 R 22. Unfortunatly, w wr not abl to succd n formal provng ths proprty. On th othr hand, numrous computatons show ts valdty for a wd rang of valus of s B ; s P, and E B =E P th aulary functons dpnd on th rato E B =E P ) that covr all ralstc matrals. Furthr, th proprty 37) s assumd to b vald. Th nt thorm follows mmdatly from th homognzaton procdur

10 588 ZAMM Z. Angw. Math. Mch. 8 2) 9 Thorm : Lt k ;u k ;u k 2 b a squnc of waksolutons of 5 convrgng to som ; u ;u 2 n th wak * topology of X 2 not that such a squnc always sts du to Proposton 2). Thn ; u ;u 2 s a waksoluton to problm 34. Dfnton 2: W say that functons ; u ;u 2 forma strong soluton to systm 34) f th followng holds: ;u ;u 2 2C ;TŠ; H 2 C ;TŠ; H C ;TŠ; H ; t ;u t ;u 2t 2C ;TŠ; H C ;TŠ; L 2 C ;TŠ; L 2 ; 38 and tt ;u tt ;u 2tt 2L ;T; H L ;T; L 2 L ;T; L 2 ; j tˆ ˆ 2 H 2 ; tj tˆ ˆ 2 H ; u a j tˆ ˆ u a 2 H ; u atj tˆ ˆ u a 2 L 2 ; ^q tt ; j ^m r tt ; rj ^g D; Dj ^t ab b ; j a F ^I K; Dj G^J ab K b ; j a ˆ; ^q u att ; y a ^t ab ; y ab G^J ab K; y ab ˆ 39 4 for any j 2 H 2, y a 2 H, and a.. t 2 ;TŠ. Lmma : If K 2 L ;T; H ; thn any strong soluton ;u ;u 2 of 34 posssss th proprty 2 L ;T; H 3 ; u ;u 2 2 L ;T; H 2 : 4 P r o o f : Th proof s just smlar to th on of Lmma 5.2 of [7]. Namly, w can rwrt 4) as follows: ^`jab j u ; ab y ˆ ^q u att ; y a ^`jab j ; ab y G ^J ab K b ; y a : 42 Usng ntgraton by parts and th proprts 38), on can prov that ; j ; ab y s from C ;TŠ; L q L q for any q>2. Thrfor, ; j ; ab y s from C ;TŠ; H s H s for any <s< bcaus H s L 2= s n two dmnsons. Th othr trms on th rght-hand sd ar from L ;T; L 2 L 2. Thrfor s.g. []), u and u 2 ar from L ;T; H 2 s, so thr sts a constant C> such that Hnc, k j u k H s C; t2 ;TŠ : kd j ;u b kˆk j u b 2 j b kk j u k L4 k b k L4 k b k 3 L 6 C k j u k H s kk H 2 kk3 H 2 C 2 ; 43 for t 2 ;TŠ, f w tak <s<=2. Now, rwrt 39) as follows: ^g D; Dj ˆ ^q tt ; j ^m r tt ; rj ^`jab d j ;u b ; j a F^I rk; rj G^J ab K b ; j a : Not 38) and 43) mply that all trms on th rght-hand-sd ar from L ;T; H. Thrfor s.g. []), s from L ;T; H 3. Th last rsult mpls that th rght-hand-sd of 42) s from L ;T; L 2 L 2. Thrfor u and u 2 ar from L ;T; H 2. Thus 4) s provd. & Lmma 2: If K 2 L ;T; H ; thn th strong soluton of 34 s unqu whnvr t sts. P r o o f : Assum that thr ar two strong solutons ;u a and 2 ;u 2 a. Thn th dffrnc ˆ 2 ; u a ˆ u a u2 a satsfs th quatons ^q tt ; j ^m r tt ; rj ^g D ; Dj ^`jab d j b d 2 j 2 b ; j a ^J ab K b ; j a ˆ ; ^q u att ; y a ^`jab d j d2 j ; ab y ˆ for any j 2 H 2, y a 2 H, and for a.. t 2 ;TŠ. W st j ˆ t h t h =2h, y a ˆ u a t h u a t h =2h and ntgrat from h to t h. Th passag to th lmt as h! gvs ^qk t k 2 ^mkr t k 2 ^gkd k 2 ^qku at k 2 ^`abj ab u ; j u C t kd j b d 2 j 2 b k 2 kk b k 2 kr t k 2 kr a b 2 a 2 b k 2 ku at k 2 dt : 46

11 Hoffmann, K.-H.; Botkn, N. D.: Homognzaton of von Karman Plats 589 Usng 38), 4) and mbddng thorms, w obtan that kd j b d 2 j 2 b k 2 kr a b 2 a 2 b k 2 C ku j b u 2 j 2 b k 2 k j b 2 2 j 2 b k 2 k a b g 2 a 2 b g k 2 C ku j b 2 b k 2 k 2 b u j u 2 j k 2 k j b 2 b k 2 k a b g 2 b g k 2 k b g a 2 a k 2 C ku j k 2 L 4 k b 2 b k 2 L 4 k2 b k 2 L ku j u 2 j k 2 k j k 2 L 4 k b 2 b k 2 L 4 k a k 2 L k b g 2 b g k 2 k b g k 2 L 4 k a 2 a k 2 L 4 C kk 2 H 2 ku ak 2 H ; t 2 ;TŠ : 47 It s asy to s that kk b k 2 kkk 2 L 4 k b k 2 L 4 CkKk2 H k k 2 H 2 Ck k 2 H 2 ; t 2 ;TŠ : 48 From 46), 47), 48), 2), and th nqualty of Korn, w gt kk 2 H 2 k t k 2 H ku ak 2 H ku atk 2 t C kk 2 H 2 k t k 2 H ku ak 2 H ku atk 2 dt : Takng nto account that ˆ; u a ˆ, w obtan ˆ ; u a ˆ, whch provs Lmma 2. Th nt thormstats a mor usful rlaton btwn 5) and 34) than Thorm. Thorm 2: Lt ˆ ; ˆ ; u a ˆ ; u a ˆ, K 2 H2 ;T; L 2 \ L ;T; H ; and Kj tˆ ˆ ; thn 34 has a unqu strong global soluton ;u ;u 2. Th sts L shrnkto ;u ;u 2 n wak * topology of X 2. That s, any squnc ;u ;u 2 2L convrgs to ;u ;u 2 n wak * topology of X 2 as!. P r o o f : Usng Proposton 2, w conclud that thr sts a squnc k ;u k ;u k 2 2L k convrgng to som trpl ;u ;u 2 n wak * topology of X 2 as k!. Thorm clams that ;u ;u 2 s a wak soluton of 34). Morovr, ; u ;u 2 2X 2 ; t;u t ;u 2t 2X 2 ; tt; u tt ;u 2tt 2X 49 du to Proposton 2. Usng Thorm6. of [], w obtan that ;u ;u 2 posssss th proprts 38). Thrfor, ;u ;u 2 s a strong soluton of 34). It s unqu du to Lmma 2. Assum that L dos not shrnk to ;u ;u 2. Thn on can fnd a subsqunc k ;u k ;u k 2 2L k sparatd from ;u ;u 2 n th wak * topology of X 2. Thr sts a sub-subsqunc kl k l ;u ;u k l 2 2Lk l that convrgs to som *;u* ;u* 2 n th wak * topology of X 2. Th sam argumnts as abov yld that *;u* ;u* 2 s a strong soluton of 34). Hnc, *;u* ;u* 2 ˆ ;u ;u 2, whch s a contradcton. & Rmark 2: It s suffcnt to assum that 2 H 2 \H3 ; 2 H2 ; u a 2 H \H2 ; u a 2 H, K 2 H 2 ;T; L 2 \ L ;T; H for th stnc of strong global solutons of 34). Th prov s basd on th tm dffrntaton of a rlatonshp dfnng Galrkn appromatons m ;u m ;um 2 for 34). Ths ylds an quaton that dfns th tm drvatvs w m ˆ m t ;vm ˆ um t ;vm 2 ˆ um 2t, f th ntal valus for wm ;v m ;vm 2 ar chosn proprly. Du to constant coffcnts of 34), th abov assumptons nsur compatblty condtons s.g. [9]) that mak possbl to fnd such approprat ntal valus for w m ;v m ;vm 2. Thus, an stmat lk 2) holds for m ;u m ;um 2. Ths s suffcnt to prov that any lmt pont of f m ;u m ;um 2 g s a strong global soluton of 34). 5. mulaton Th smulaton was don wth th followng paramtrs: s th unt squar, Pl s th =2 dg squar cntrd w.r.t. th corrspondng structural cll s Fg. ). Th othr valus ar: q P ˆ 2 ; q B ˆ ; g P ˆ 5 ; g B ˆ 3 ; m P ˆ : ; m B ˆ : ; E P ˆ 2 ; E B ˆ ; s P ˆ :4 ; s B ˆ :2 ; ˆ =6 ; F ˆ 5 ; G ˆ 5 ; K t; ; 2 ˆsn 5t cos 5 2 : Th ntal valus wr qual to zro. Th Bognr-Fo-chmt fnt lmnts s Carlt [4]) wr appld. Th numbr of fnt lmnts was qual to Each structural cll s Fg. ) occups 4 4 lmnts. Each pzopatch occups 2 2 lmnts. Each fgur blow shows th soluton of 5), th soluton of 34), and thr dffrnc at tms ndcatd.

12 59 ZAMM Z. Angw. Math. Mch. 8 2) Fg. 2. t ˆ :7 Fg. 3. t ˆ :4 Rfrncs Banks, H. T; mth, R. C; Wang,.: mart matral structurs: modlng, stmaton and control. Wly, Chchstr Hoffmann, K.-H.; Botkn, N. D.: Oscllatons of nonlnar thn plats ctd by pzolctrc patchs. ZAMM ), 495 ±53. 3 Ngutsng, G.: A gnral convrgnc rsult for a functonal rlatd to th thory of homognzaton. IAM J. Math. Anal ) 3, 68 ± Allar, G.: Homognzaton and two-scal Convrgnc. IAM J Math. Anal ) 6, 482 ±58. 5 Allar, G: Homognzaton of th unstady toks quatons n porous mda. In: Bandl, C. t al. ds.): Progrss n partal dffrntal quatons: calculus of varatons, applcatons. Ptman Rsarch Nots n Mathmatcs rs 267. Longman Hghr Educaton, Nw ork 992, pp. 9 ±23. 6 Hallr, H.: Vrbundwrkstoff mt Formgdachtnslgrung ± Mkromchansch Modllrung und Homognsrung. Dssrtaton. TU-Munchn, Munchn Pul, J.-P.; Tucsnak, M.: Global stnc for th full Karman systm. Appl. Math. Optmz ), 39 ±6. 8 Horn, M. A.; Lascka, I.: Global stablzaton of a dynamc von Karman plat wth nonlnar boundary fdback. Appl. Math. Optmz ), 57 ±84. 9 Wloka, J.: Partll Dffrntalglchungn. Tubnr, tuttgart 982. mon, J.: Compact sts n th spac L p ;T; B. Ann. Mat. Pura Appl., IV. r ), 65 ±96. Lons, J. L.; Magns, E.: Non-homognous boundary valu problms and applcatons. Vol. I. prngr Vrlag, Brln ± Hdlbrg ±Nw ork Botkn, N. D.: Homognzaton of an quaton dscrbng lnar thn platsctd by pzopatchs. Commun. Appl. Anal ) 2, 27 ±28. 3 Zdlr, E.: Nonlnar functonal analyss and ts applcatons. IV: Applcatons to mathmatcal physcs. prngr-vrlag, Nw ork ±Brln ±Hdlbrg ±London ±Pars ±Tokyo Carlt, P. G.: Th fnt lmnt mthod for llptc problms. North-Holland, Amstrdam 978. Rcvd August 2, 998, rvsd Jun 28, 999, accptd Novmbr 23, 999 Addrss: Prof. Dr. Karl-Hnz Hoffmann, Dr. Nkola D. Botkn, tftung casar, Frdnsplatz 6, D-53 Bonn, Grmany, -mal: hoffmann@casar.d, botkn@casar.d