Mechanics of Materials and Structures

Transcription

1 Journal of Mchanics of Matrials and Structurs MIXED PIEZOELECTRIC PLATE ELEMENTS WITH CONTINUOUS TRANSVERSE ELECTRIC DISPLACEMENTS Erasmo Carrra and Christian Fagiano Volum 2, Nº 3 March 27 mathmatical scincs publishrs

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3 JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Vol. 2, No. 3, 27 MIXED PIEZOELECTRIC PLATE ELEMENTS WITH CONTINUOUS TRANSVERSE ELECTRIC DISPLACEMENTS ERASMO CARRERA AND CHRISTIAN FAGIANO This papr proposs mixd finit lmnts, FEs, with an a priori continuous transvrs lctric displacmnt componnt z. Th Rissnr Mixd Variational Thorm RMVT) and th Unifid Formulation UF) ar applid to th analysis of multilayrd anisotropic plats with mbddd pizolctric layrs. Two forms of RMVT ar compard. In a first, partial, form P-RMVT), th fild variabls ar displacmnts u, lctric potntial Φ and transvrs strsss σ n. Th scond, full, form F-RMVT) adds z as an indpndnt variabl. F-RMVT allows th a priori and complt fulfillmnt of intrlaminar continuity of both mchanical and lctrical variabls. W trat both quivalnt singl-layr modls ESLM), whr th numbr of variabls is pt indpndnt of th numbr of layrs, an layrwis modls LWM), in which th numbr of variabls dpnds in ach layr. According to th UF th ordr N of th xpansions assumd for th u, φ, σ n and z filds in th plat thicnss dirction z as wll as th numbr of th lmnt nods N n hav bn tan as fr paramtrs. In most cass th rsults of th classical formulation which ar basd on Principl of Virtual Displacmnts PVD) ar givn for comparison purposs. Th supriority of th F-RMVT rsults, with rspct to th P-RMVT and to PVD ons, is shown by fw xampls for which thr-dimnsional solution is availabl. In particular, th F-RMVT rsults to b vry ffctiv for th valuation of intrlaminar continuous z filds. 1. Introduction In rcnt yars pizolctric matrials hav bn intgratd with structural systms to build smart structurs which ar th candidats for nxt gnration structurs of arospac vhicls as wll as for som advancd products in th automotiv and ship industris. Pizolctric matrials ar, in fact, capabl of altring th rspons of th structurs through snsing and actuation [Tirstn 1969]. By intgrating th surfac bondd and mbddd actuators in structural systms, th dsird localizd strains may b inducd in th structurs thans to th application of an appropriat voltag to th actuators. Such an lctromchanical coupling allows closd-loop control systms to b built up, in which pizomatrials play th rol of both th actuators and th snsors. An intllignt structur can thrfor b built in which, for instanc, thrmomchanical dformations or vibrations can b rducd by using appropriat control laws. For dtails s [Chopra 1996; 22] and th rlatd litratur. In ordr to succssfully incorporat actuator/snsors in a structurs, th mchanical intraction btwn th pizolctric layrs and th hosting structur must b compltly undrstood, that is, an Kywords: pizolctric plats, finit lmnts, mixd mthod, transvrs continuity, unifid formulation. This wor has bn carrid out in th framwor of STREP EU projct CASSEM undr contract NMP-CT

4 422 ERASMO CARRERA AND CHRISTIAN FAGIANO appropriat us of pizolctric matrials, rquirs an accurat dscription of both th lctrical and mchanical filds in th constitutiv layrs. Early mchanical modls wr dvlopd by Crawly and d Luis [1987], L [199] and Mitchll and Rddy [1995], among othrs. Mor rcnt wors ar [Yang and Batra 1995; Wang t al. 1997; Vidoli and Batra 2; Batra and Vidoli 22]. A rcnt assssmnt of classical and rfind thoris with displacmnts and lctrical variabls for plats can b found in [Ballhaus t al. 25]. Equivalnt singl-layr ESL) and layrwis LW) thoris hav bn compard in th framwor of th application of th Principl of Virtual Displacmnt PVD) applications it is intndd that th numbr of indpndnt variabls is pt indpndnt by th numbr of th layrs in th ESL modls). Numrous bnchmar, xact solution analyss hav bn conducd for pizolctric plats; som ar givn in [Hyligr and Saravanos 1995]. Howvr, ths bnchmar solutions ar rstrictd to simpl gomtris and spcial boundary conditions. Th tratmnt of mor ralistic problms would rquir th us of fficint computational tools such as th finit lmnt mthod FEM). Th prsnt papr focuss on FEM lctromchanical two-dimnsional modlings of smart structurs with mbddd pizo layrs. Finit lmnt studis wr conductd by Robbins and Rddy [1991]. A finit lmnt that accounts for a first ordr shar dformation thory FSDT) dscription of displacmnt and layrwis form of th lctric potntial was dvlopd in [Shih t al. 21]. Th numrical, mmbran and bnding bhavior of th FEs basd on FSDT was analyzd in [Auricchio t al. 21] in th framwor of a suitabl variational formulation. Th third-ordr thory was applid by Thornbugh and Chattopadhyay [22] to drivd finit lmnts that account for lctromchanical coupling. Similar lmnts hav mor rcntly bn considrd in [Shu 25]. Extnsion of th third-ordr Ambartsumian zigzag multilayrd thory [Carrra 23a] to th finit analysis of lctromchanical problms has bn proposd by Oh and Cho [24]. An xtnsion of numrically fficint plat/shll lmnts basd on mixd intrpolation of tnsorial componnts MITC) to pizolctric plats has rcntly bn providd by Kögl and Bucalm [25a; 25b]. W also mntion th rviw paprs [Saravanos and Hyligr 1999; Bnjddou 2; Wang and Yang 2]. Our contributions to th application of th Rissnr Mixd Variational Thorm RMVT) to multilayrd mad structurs startd with [Carrra 1995; 1996; 21], and hav includd closd-form solution analyss [Carrra 1999a; 1999b] and FE applications [Carrra and DMasi 22a; 22b], showing th RMVT is a vry suitabl tool to provid quasi- dscription of strss and strain filds in anisotropic laminatd structurs. Th RMVT was also mployd in th framwor of Unifid Formulation UF), dalt with in dtail in [Carrra 21]. Th main fatur of UF is that it allows on to formulat both ESLM and LW modls in trms of a fw fundamntal nucli whos forms do not dpnd on ithr th ordr of th xpansion N usd for th various variabls in th thicnss dirction) or on by th numbr of nods of th lmnt N n. Th Muraami zigzag Function MZZF) [Carrra 21] was usd to rproduc th zigzag form of displacmnt fild in th ESLM cas. A classical formulation, basd on PVD, was dvlopd for comparison purposs. A first application of RMVT to pizolctric plats was providd in [Carrra 1997], whr an MITCtyp plat lmnt was xtndd to nonlinar dynamic analysis of pizolctric, composit plat. Th UF formulation was applid, in th PVD framwor, to pizolctric plats in [Ballhaus t al. 25]; attntion was rstrictd to analytical closd form solutions. RMVT closd form solutions wr prsntd in [D Ottavio and Kröplin 26], whil xtnsion to shll has bn providd in [Carrra t al. 25]. Finit lmnt applications hav also bn providd rcntly [Carrra and Boscolo 26].

5 PIEZOELECTRIC PLATES WITH CONTINUOUS TRANSVERSE ELECTRIC DISPLACEMENTS 423 All ths RMVT wors hav bn rstrictd to th a priori fulfillmnt of th intrlaminar continuity of th mchanical variabls transvrs normal and shar strss filds), that is, th continuity of transvrs normal componnt z of th lctric displacmnt vctor was not a priori guarantd. This form of RMVT is hrin rfrrd to as th partial form, or P-RMVT. RMVT has also bn applid in [Garcia Lag t al. 24a] to dvlop LW pizolctric plat lmnts in th static cas. Th transvrs componnt of lctric displacmnt z was considrd as an assumd variabl. W rfr to such an xtnsion as full RMVT applications, namly F-RMVT. Garcia Lag and his coauthors rstrictd thir attntion to th quadratic distribution of displacmnts mchanical and lctrical) and transvrs strss unnowns, and tratd only a layrwis modl. Ths rstrictions hav not allowd us to analyz th faturs of th a priori assumption of intrlaminar continuous transvrs lctric displacmnt. Hr w compars P-RMVT and F-RMVT in th framwor of UF, xtnding th analysis of [Carrra and Boscolo 26] to includ th normal lctrical displacmnt z as an assumd a priori variabl. A numbr of nw finit lmnts ar drivd and systmatically compard to thos basd on P-RMVT and PVD. ESLM and LW variabl dscription analyss ar compard to availabl solutions. Up to forth-ordr xpansions in th thicnss plat/layrs hav bn implmntd. Th papr is organizd as follows. Sction 2 givs th ncssary prliminaris. Sction 3 introducs th two RMVT forms for pizolctric continua along with variationally consistnt constitutiv quations. Th UF for finit lmnt applications ar drivd in Sction 3, and th FE matrics thmslvs in Sction 5. Sction 6 contains numrical rsults and discussion. 2. Prliminaris Figur 1 shows th gomtry and th coordinat systm of a laminatd plat with N l layrs, including pizolctric layrs. Th rfrnc systm is dnotd by x, y, z; th corrspondnt plat dimnsions ar dnotd by a, b, h, th last of which is th thicnss. Th matrial proprtis of a pizolctric continuum can b xprssd in diffrnt forms; w us th so-calld -form [Ida 1996]. Th rlvant nrgy is thn th lctric Gibbs nrgy G 2, which tas th form G 2 = 1 2 ɛt C ɛ T ɛ 1 2 T ε ɛ, 1) x z y pz x, y)= ^p zsin x y a sin a ) ) t = h a a b = Figur 1. Gomtry of Pizolctric Plat

6 424 ERASMO CARRERA AND CHRISTIAN FAGIANO whr ɛ T = {ɛ xx, ɛ yy, ɛ zz, ɛ xz, ɛ yz, ɛ xy } is th strain tnsor w us bold lttrs for arrays and T to dnot transposition), T = { x, y, z } is th lctric fild vctor, C is th stiffnss matrix calculatd at constant, is th pizolctric matrix that coupls lctrical and mchanical filds, and ε ɛ = {ε xx, ε yy, ε zz } is th prmittivity matrix calculatd at ɛ-constant. Th constitutiv quations will b writtn out in Sction 3 in a form suitabl for th F-RMVT application. Gomtrical rlations. Th strain-displacmnt gomtrical subscript G) rlations in th linar cas ar ɛ pg = D p u, ɛ ng = D np + D nz )u. 2) Th suprscript is th layr indx. Strains hav bn split into in-plan subscript p) and out-of-plan subscript n, for normal ) componnts: ɛ p = {ɛ xx, ɛ yy, ɛ xy }, ɛ n = {ɛ xz, ɛ yz, ɛ zz }, whil u = {u x, u y, u z } is th vctor of th displacmnt componnts. Th diffrntial matrics ar givn xplicitly by x x z D p = y, D np = y, D nz = z. 3) y x z Th lctric fild is rlatd to th lctric potntial by th gradint rlation = [ x y z ] Φ. 4) Th lctric potntial Φ bing a scalar, on obtains by sparating in-plan and normal componnts th quality ) = D p + D z Φ, 5) whr D T p = [ x y ], D T z = [ z ]. 6) 3. Variational statmnts for pizolctric continua Th classical variational tool most oftn usd to dvlop FEs, is th principl of virtual displacmnts PVD), which, for a pizolctric continuum, can b writtn N l =1 A δɛ pg σ pc + δɛ ng σ nc δ G C) d dz = δl. 7) Hr δ dnots virtual variations, A is th layr domain in th thicnss dirction, dnots th rfrnc surfac of th layr, and δl dnots th virtual variation of th wor mad by applid loadings. Th in-plan and out-of-plan strss componnts ar σ T p = {σ xx, σ yy, σ xy }, σ T n = {σ xz, σ yz, σ zz }.

7 PIEZOELECTRIC PLATES WITH CONTINUOUS TRANSVERSE ELECTRIC DISPLACEMENTS 425 Th lctrical wor is obtaind via th lctrical displacmnt vctor: = { x, y, z }. A subscript C will dnot strss and lctrical displacmnts from th constitutiv law, and a subscript G strains and lctrical filds from th gomtrical rlation. Th PVD allows on to assum two indpndnt filds for u and Φ. Th rmaining variabls ar obtaind from th constitutiv law of th pizolctric layrs. Th RMVT was proposd in [Rissnr 1984] for purly mchanical problms. A critical rviw on its us was givn in [Carrra 21]. A main fatur of th RMVT is that it allows on to assum two indpndnt filds for displacmnts u and transvrs strsss σ n. This allows th a priori fulfillmnt of th ncssary continuity quilibrium) conditions of transvrs normal and shar strsss at ach layr intrfacs. In th static cas, for pur mchanical problms RMVT stats that N l =1 δɛ pg σ pc + δɛ ng σ nm δσ nm ɛ ng ɛ nc )) d dz = δl. 8) A Th scond trm in th intgrand forcs th compatibility of transvrs strain obtaind by th matrial s constitutiv law which ar diffrnt from thos rlatd to PVD applications) and by th gomtric rlation. Th subscript M dnots thos variabls which ar assumd in a givn modl. By introducing th lctrical wor, w can writ th RMVT for pizolctric continua as N l =1 δɛ pg σ pc + δɛ ng σ nm δ G C δσ nm ɛ ng ɛ nc )) d dz = δl. 9) A This form of th RMVT will b calld th partial xtnsion of RMVT to pizolctric continua, or P-RMVT. A full xtnsion of th RMVT can b obtaind by introducing th transvrs componnts of lctric displacmnt z as additional variabls. Th RMVT thn assums th following full form, or F-RMVT: N l =1 A h δɛ pg σ pc + δɛ ng σ nm δ pg pc δ ng nm + δσ nm ɛ ng ɛ nc ) T nm ng nc )) d A dz = δl. 1) Th lctrical displacmnt and lctrical fild vctors hav bn split into in-plan and normal componnts as for th strsss σ and strains ɛ): p = { x, y }, n = { z }, p = {E x, y }, n = { z }. Th constitutiv quations of th -layr ar convnintly writtn as σ pc = C pp ɛ pg + C pn ɛ nc ppt pg npt nc, σ nm = C pnt ɛ pg + C nn ɛ nc pnt pg nnt nc, pc = pp ɛ pg + pn ɛ nc + ε pp pg + ε pn nc, 11) nc = np ɛ pg + nn ɛ nc + ε T pn pg + ε nn nc, whr w hav introducd th following arrays:

8 426 ERASMO CARRERA AND CHRISTIAN FAGIANO Stiffnss matrics: C 11 C 12 C 16 C 13 C 55 C 45 C pp = C 12 C 22 C 26, C pn = C 23, C nn = C 45 C ) C 16 C 26 C 66 C 36 C 33 Pizolctric matrics: pp = [ Prmittivity matrics: ], pn = [ ], np = [ ], nn = [ 33 ]. 13) [ ] [ ] ε pp = ε11 ε 12, ε pn ε 12 ε =, ε nn 22 = [ ]. ε 33 14) Application of th F-RMVT rquirs on to xprss th in-plan strsss σ pc, th normal strains ɛ nc, th normal lctric fild nc and th in-plan lctric displacmnts pc in trms of th rmaining variabls. Thus th constitutiv quations 11) can b solvd as follows: σ pc = Ĉ spm ɛ pg + Ĉ snm σ nm + Ĉ sp pg + Ĉ sn nm, ɛ nc = Ĉ dpm ɛ pg + Ĉ dnm σ nm + Ĉ dp pg + Ĉ dn nm, pc = Ĉ f pm ɛ pg + Ĉ f nm σ nm + Ĉ f p pg + Ĉ f n nm, E nc = Ĉ pm ɛ pg + Ĉ nm σ nm + Ĉ p pg + Ĉ n nm. 15) Th matrics abov ar obtaind from by thos in 11) by mans of th rlations C dpm = C nn 1 C pn C 1 ) nn C 1 nn + ε 1 nn) np nn C 1 ) nn C pn, C dnm = C nn 1 C 1 ) nn C 1 nn + ε 1 nn) nn C nn 1), C dp = C nn 1 pn C 1 ) nn C 1 nn + ε 1 nn) nn C 1 pn + ε pn), C dn = C ) nn 1 nn C 1 nn + ε 1, nn) C pm = nn C nn 1 nn + ε 1 nn) np nn C 1 ) nn C pn, C nm = nn C nn 1 nn + ε 1 nn) nn C nn 1), C p = nn C nn 1 nn + ε 1 nn) nn C 1 pn + ε pn), C n = nn C nn 1 nn + ε 1, nn) C spm = C pp + C pn C dpm np C pm, C snm = C pn C dnm np C nm, C sp = C pn C dp pp np C p, C sn = C pn C dn np C n, C f pm = pp + pn C dpm + ε np C pm, C f nm = pn C dnm + ε pn C nm, C f p = pn C dp + ε pp + ε pn C p, C f n = pn C dn + ε pn C n.

9 PIEZOELECTRIC PLATES WITH CONTINUOUS TRANSVERSE ELECTRIC DISPLACEMENTS 427 It must b notd that Ĉ dnm = Ĉ dnm, Ĉ spm = Ĉ spm, Ĉ f p = Ĉ f p, Ĉ n = Ĉ n, Ĉ dpm = Ĉ snm, Ĉ dn = Ĉ nm, Ĉ dp = Ĉ f nm, Ĉ pm = Ĉ sn. 4. Unifid formulation for plat lmnts Th unifid formulation is a tchniqu that allows on to handl in a unifid mannr a larg varity of plat modlings and finit lmnts. In this formulation, th finit lmnt matrics ar writtn in trms of a fw fundamntal nucli, which do not formally dpnd on: th xpansion N usd in th z-dirction, th numbr of th nod N n of th lmnt, or th variabls dscription LW or ESL). Th unnown variabls u, σ n, Φ and z ar xprssd in trms of th layr thicnss coordinat: u x,y,z),ϕ x,y,z),σ n x,y,z), n x,y,z)) = F b z) u b x,y),ϕ b x,y),σ nb x,y), nb x,y)) + F r z) u r x,y),ϕ r x,y),σ nr x,y), nr x,y)) + F t z) u t x,y),ϕ t x,y),σ nt x,y), nt x,y)). 16) Th subscript t and b dnot th linar part of th thicnss xpansion t and b will b usd to dnot top- and bottom-layr variabl valus in layrwis cass), whil subscript r rfrs to highr-ordr trms: r = 2,..., N 1. In compact form, u x,y,z),ϕ x,y,z),σ n x,y,z), n x,y,z)) = F τ z) u x,y),ϕ x,y),σ n x,y), n x,y)) τ. 17) Hr u x, y), φ x, y), σ n x, y), n x, y)) τ ar two-dimnsional unnowns, th F τ z) ar th bas functions of th xpansion, and th summation convntion ovr rpatd indx has bn adoptd. Th bas functions could b, in gnral, diffrnt for ach variabl. Diffrnt choics for F τ z) will lad to diffrnt plat/shll thoris. Th choics mad in our study ar brifly discussd blow; dtaild dscriptions can b found in th wors citd.. Layr-wis lmnts. Th thicnss functions ar givn by combinations of Lgndr polynomials P j as F t = P ζ ) + P 1 ζ ) 2, F b = P ζ ) P 1 ζ ), F r = P r ζ ) P r 2 ζ ), r = 2, 3,..., N, 18) 2 for ζ = z /2h, whr z is th local layr thicnss coordinat and h is th layr thicnss, so 1 ζ 1. As mntiond, t and b dnot top and bottom; that is, th chosn functions hav th proprtis { 1 : Ft = 1, F b =, F r =, ζ = 19) 1 : F t =, F b = 1, F r =, Thans to ths proprtis th intrlaminar continuity of th assumd variabls can b asily lind in th assmbly procdur from layr-lvl matrics to multilayr-lvl matrics. Th rsulting lmnts will b dnotd by th acronyms LFM1 to LFM4, in which L mans layrwis, FM stats that F-RMVT has bn mployd, and th digit is th ordr of th xpansion. Particular cass of P-RMVT and PVD will also b usd in th numrical analysis; ths applications will b dnotd by LPM1 to LPM4 and LD1 to LD4, rspctivly.

10 428 ERASMO CARRERA AND CHRISTIAN FAGIANO Equivalnt singl-layr modl. In this cas th layrwis xpansion is prsrvd for th transvrs strsss, lctric potntial and lctric displacmnts, whil a Taylor-typ xpansion is usd for th displacmnt componnts: ux, y, z) = u τ x, y) z τ, τ =, N Th bas functions rlatd to displacmnts can b chosn as F b z) = 1, F r z) = z r, r = 1, N 1, F t z) = z N. Ths thoris will b dnotd with th acronyms EFMC1 to EFMC3, in which E mans quivalnt singl-layr, FM mans full mixd, and C that intrlaminar continuity conditions ar fulfilld for transvrs strsss, lctric potntial and transvrs lctric displacmnt. Th digit, as bfor, dnots th xpansion ordr. Rsults rlatd to P-RMVT application will b dnotd by EPMC1 to EPMC3. Whn th Muraami zigzag function is usd which allows th introduction of picwis continuous displacmnt filds in th thicnss plat dirction; s [Carrra 21]), th rsulting lmnts ar rfrrd to as EFMZC1 to EFMZC3 and EPMZC1 to EPMZC3 for th full and partial cass. Finit lmnt approximations. Finit lmnt approximations to th plat rfrnc surfac domain ar introducd by mans of isoparamtric dscriptions for th various fild variabls: ) ) u τ, Φ τ, σ nτ, nτ x, y) = N i x, y) q τi, g τi, f τi, d τi, i = 1, 2,..., N n, 2) whr th N i x, y) ar th shap functions, q τi th nodal unnown displacmnts, g τi th nodal unnown lctric potntials, f τi th nodal unnowns normal strsss and d τi th nodal unnown normal lctrical displacmnts. Th cass of 9, 8 and 4 nods ar considrd in th numrical implmntation rfrrd to as Q9, Q8 and Q4 finit lmnts [Carrra and DMasi 22b]. 5. Drivation of finit lmnt matrics This sction is dvotd to th fundamntal nucli of th F-RMVT finit lmnt matrics. Th RMVT and PVD matrics can b found in [Carrra and DMasi 22a; Carrra and Boscolo 26]. By starting from Equation 1), th fundamntal nucli ar drivd in svral stps: 1. Th constitutiv rlations 15) ar introducd in th F-RMVT statmnt at 1). 2. Th gomtric rlations ar usd to xprss strain in trms of displacmnts and lctric fild in trms of lctric potntial. 3. Th through-th-thicnss assumptions by mans of th Unifid Formulation ar introducd. 4. Th FE shap functions ar usd to liminat th in-plan plat coordinats by numrical intgration. 5. Matrix products ar mad, yilding th xplicit forms of th fundamntal nucli.

11 PIEZOELECTRIC PLATES WITH CONTINUOUS TRANSVERSE ELECTRIC DISPLACEMENTS 429 W omit th dtails for brvity. Th final form of th govrning quations is δq T τi : K τsi j uu q sj + K τsi uσ j δ f τi : K τsi σ u j q sj + K τsi σ σ j δgτi T : δdτi T : K τsi j u q sj + K τsi σ j τsi j K du q sj + dσ f sj + K τsi u j gsj + ud dsj = P uτ, f sj + K τsi σ j gsj + σ d dsj =, f sj + K τsi j gsj + d dsj = P τ, f sj + d gsj + dd dsj =. Th mchanical and lctrical loading trms on th right-hand sid ar 21) P uτ = K τsi up j p sj, P τsi j τ = K f sj. 22) Th xplicit forms of th fundamntal nucli thus obtaind ar K τsi uu j = D T p N i Ĉ spm Euu τs D ) p N j d, K τsi uσ j = D T p N i Ĉ snm Euσ τs N j + N i Dnp T Euσ τs N j + N i I T Eτ,zs uσ N j) d, u = ud = σ u = σ σ σ σ d u σ d du = dσ = = = = = = = = D T p N i Ĉ sp Eu τs D p N j ) d, D T p N i Ĉ sn Eu τs N j) d, Ni E σ u τs D np N j + N i E σ u τs,z I N j N i Ĉ dpm Eσu τs D p N j ) d, Ni Ĉ dnm Eσσ τs N j) d, Ni Ĉ dp Eσ τs D p N j ) d, Ni Ĉ dn Eσ τs N j) d, D T p N i Ĉ f pm Eu τs D p N j ) d, Ni D T pĉ f nm Eσ τs N j) d, Ni D T pĉ f p E τs D p N j ) d, Ni D T pĉ f n E τs D p N j + I N i E σ τ,zs N j) d, Ni Ĉ pm Edu τs D p N j ) d, Ni Ĉ nm Edσ τs N j) d,

12 43 ERASMO CARRERA AND CHRISTIAN FAGIANO d = dd = K τsi up j = f = Ni E dφ τs,z I N j + N i Ĉ p Edφ τs D p N j ) d, Ni Ĉ n Edd τs N j) d, F 1 τ N i N j m s ) F1 s d, F 1 τ N i N j n s ) F1 s d. I is th unit matrix and I T = {,, 1}. Th following intgrals hav bn dfind: Eτs αβ = A F α τ Fβ s dz, E αβ τ, zs = A F α τ,z F β s dz, E αβ τs, z = A F α τ Fβ s,z dz, whr α and β can assum any of th valus u, σ, Φ, to dnot thicnss function usd for th rlatd variabls. Tabl 1 summarizs th dimnsions of th nucli. By varying th subscripts τ, s,, i, j ovr thir rangs on obtains th lmnt matrics; s [Carrra 23b]. uu [3 3] u [1 3] uσ [3 3] σ [1 3] u [3 1] [1 1] ud [3 1] d [1 1] f [1 1] up [3 3] σ u [3 3] du [1 3] σ σ [3 3] dσ [1 3] σ [3 1] d [1 1] σ d [3 1] dd [1 1] M τsi j uü [3 3] Tabl 1. Dimnsions of th fundamntal nucli. 6. Numrical rsults This sction shows th prformanc of th mixd FEs dvlopd on th basis of intrlaminar a priori continuous transvrsal lctric displacmnts z, comparing it with a mixd lmnts approach that dos not incorporat such continuity, as with on basd of PVD applications. Furthr comparisons ar givn with th rsults in [Garcia Lag t al. 24b] and with thr dimnsional solutions in [Hyligr 1994]. To compar th analysis with closd-form xact solutions, attntion has bn rstrictd to simply supportd squar plats. W rtain th rducd intgration tchniqu that was succssfully applid in [Carrra and DMasi 22b]. LW as wll as ESL analyss hav bn prformd for Q4, Q8 and Q9 lmnts. W considr four-layr plats, with th two innr layrs consisting of cross-ply [ /9 ] carbon fibr and th xtrnal sins mad of pizocramic matrial PZT-4. Th matrial proprtis ar shown in Tabl 2 on pag 432. Th two composit layrs hav thicnss h 2 = h 3 =.4h and th sins hav h 1 = h 4 =.1h. Th unit valu is assignd to th plat thicnss. A bisinusoidal distribution of transvrsal prssur with amplitud ˆp z = 1 is applid to th top surfac this coincids with a snsor configuration cas).

13 PIEZOELECTRIC PLATES WITH CONTINUOUS TRANSVERSE ELECTRIC DISPLACEMENTS 431 Figur 2 shows th in-plan displacmnt u y distribution in th thicnss dirction for th slctd plat lmnts z is th horizontal axis). Bttr rsults ar obtaind for th LFM and EFMZC analyss with rspct to ons basd on P-RMVT. Th numbr of lmnts for th plat sid N has bn placd to th right of th acronym. Layrwis analysis lads to much bttr rsults than ESL..4.2 EFMZC3-6l-Q9 EPMZC3-6l-Q9 LFM1-6l-Q9 LPM1-6l-Q9 LPM1-6l-Q8.4.2 EFMC1 EFMC4 z LFM1 LFM2 LFM4.4.2 EFMZC3 EFMZC1 z Figur 2. Prformanc of various FEs in prdicting th displacmnt u y a/2, ) vrsus z. Th a/ h ratio quals 4. Curvs labld show th xact solution rportd in [Hyligr 1994]; th rmaining curvs show th rsults obtaind from FE approachs basd on F-RMVT and P-RMVT uppr lft), ESL thory uppr right); LW thory lowr lft), and ESL thory incorporating Muraami s zigzag function lowr right).

14 432 ERASMO CARRERA AND CHRISTIAN FAGIANO Proprty PZT-4 Gr/Ep PVDF Proprty PZT-4 Gr/Ep PVDF E 1 GPa) C/m 2 ) E 2 GPa) C/m 2 ) E 3 GPa) C/m 2 ) ν C/m 2 ) ν C/m 2 ) ν ɛ 11 /ɛ G 23 GPa) ɛ 22 /ɛ G 13 GPa) ɛ 33 /ɛ G 12 GPa) ρ Tabl 2. Mchanical and lctrical matrial proprtis. Th sam conclusions can b drawn for th transvrsal normal strss valuation in Figur 3. Th us of LW lmnts with at last a parabolic distribution N = 2) in ach layr is rquird. Rmarabl improvmnts ar obtaind whn th Muraami zigzag function is usd. Data rlatd to th transvrsal lctrical displacmnt z, shown in Figur 4, ar of particular intrst. Various numbrs of nods for lmnts and FE mshs ar compard top lft). Thr ar difficultis whn crtain FEs ar usd to prdict z in th pizolctric layrs top right pan of figur); th rsults accuracy is vry much dpndnt on th choic of a modl, and th us of lmnts of typ LM2 at last) appars to b ncssary for corrct prdictions. This suggsts that th us of F-RMVT may b mandatory for th accurat computation of intrlaminar continuous z at a rasonabl computational cost, and that.4.2 LFM4 LFM3 LFM2 LFM1.4.2 EFMZC3 EFMC1 z Figur 3. Prformanc of various FEs in prdicting th transvrs normal strss σ zz a/2, b/2) vrsus z. Th ratio a/h quals 4.

15 PIEZOELECTRIC PLATES WITH CONTINUOUS TRANSVERSE ELECTRIC DISPLACEMENTS LPM3-6l-Q8 LFM3-8l-Q4 LFM3-1l-Q4 LFM3-9l-Q4 LPM3-8l-Q4 LPM3-1l-Q4 LPM4-6l-Q9 z.2 LFM3-1l-Q4 LFM2-1l-Q4 LFM1-1l-Q4 EDMZC3-1l-Q4 EFMC4-1l-Q N LFM3 Exact- a/ h [Hyligr 1994] / / LFM LPM LD LFM LPM LD LFM LPM LD EFMZC EFMC Figur 4. Top: prformanc of various FEs in prdicting th transvrs lctric displacmnt z a/2, b/2) vrsus z, with a/h = 4. Lft: LW lmnts with various numbr of nods pr lmnt; right: LW and ESL lmnts for th Q4 cas.) Bottom lft: Convrgnc analysis for Q4 lmnts. Bottom right: Dpndnc of z a/2, b/2, h) 1 13 on th ratio a/h, for a [12 12] msh and Q4 lmnt. P-RMVT and ESL rsults may b unaccptabl. Sinc th lctric charg Q ovr a pizolctric patch is obtaind by intgrating th z distribution ovr th patch s surfac, wrong z valus lad to wrong Q valus, potntially rndring th closd-loop control compltly maninglss.) Not that th accuracy obtainabl with LFM2 is comparabl with what w gt with LPM4, confirming that th us of P-RMVT is advantagous as far as computational ffort is concrnd. For th sa of

16 434 ERASMO CARRERA AND CHRISTIAN FAGIANO compltnss, Figur 4 shows th convrgnc rat of th Q4 lmnts; thy ar consistnt with thos found for of pur mchanical problms in our arlir wor. Various plat thicnss ratio valus ar considrd in th tabl at th bottom right of Figur 4, showing th importanc of UF as a tool to stablish an assssmnt of simplifid, classical and advancd FEs for pizolctric plat analysis. Ths rsults ar confirmd in th valuation of th lctrical voltag distribution vrsus, shown in Figur 5. Th largst discrpancis among th thoris ar xprincd in th valuation of lctrical displacmnts..4.2 LFM1-6l-Q9 EPMZC3-6l-q9 EFMZC3-6l-Q9 LPM1-6l-Q9.4.2 EPMZC3-6l-Q9 EFMZC3-6l-Q9 z LFM1-6l-Q9 LPM1-6l-Q9 LPM2-6l-Q9.4.2 LFM1 EFMZC3 EFMC4 EFMC1 z Figur 5. Top: prformanc of various FEs in prdicting th transvrs lctric potntial Φa/2, b/2) vrsus z, with a/h = 4, a [6 6] msh and a Q9 lmnt.

17 PIEZOELECTRIC PLATES WITH CONTINUOUS TRANSVERSE ELECTRIC DISPLACEMENTS 435 a/ h a/ h xact / 3,3 / LFM LFM LPM LPM EFMZC LD EPMZC LFM EFMZC LPM EPMZC LD EFMC LFM EPMC LPM EFMC LD EPMC Tabl 3. Evaluation of u z a/2, b/2, ) 1 11 ; msh [6 6] and Q9 lmnts. Th xact data ar tan from [Hyligr 1994]. Tabl 3 compars our rsults, for both mchanical and lctrical variabls, with th thr-dimnsional xact solution and th rsults of Garcia Lag t al. [24b]. A squar plat is considrd with a lay-up [ /9 /] for th intrnal layrs; two pizolctric layrs of PVDF matrials s Tabl 2) ar usd as xtrnal sins. As in this last rfrnc, th pa valu of th applid prssur is 3 Pa. Th rlativ rrors ar displayd in Tabl 4. Th supriority of th full implmntation of RMVT is still rmarabl. 7. Concluding rmars Th papr xtnds th Unifid Formulation and th Rissnr Mixd Variational Thorm to th dvlopmnt of finit lmnts for th static analysis of pizolctric plats with a priori continuous transvrs lctrical displacmnt componnts z. Th following main conclusions can b drawn. 1) It has bn confirmd that UF is a valuabl tool in th hirarchical analysis of pizolctric plats using th finit lmnt mthod. Th implmntd FEs, in fact, can provid vry accurat dscriptions of both mchanical and lctrical filds. 2) FEs with intrlaminar continuous z appar to b vry suitabl for pizolctric plat analysis. Bttr rsults ar obtaind with rspct to th othr FEs hrin compard. 3) In ordr to prsrv computational fforts, th us of th proposd lmnts would sm to b mandatory if accurat valuations of z and th rlatd lctric charg ar rquird. Futur dvlopmnts should b dirctd towards considring th analysis of pizolctric plat with localizd patchs as snsors and/or actuators. Othr plat lay-ups and th ffct of additional boundary conditions and gomtris should b xamind. Th cas of imposd z at th intrfac should in particular b analyzd. Rfrncs [Auricchio t al. 21] F. Auricchio, P. Bisgna, and C. Lovadina, Finit lmnt approximation of pizolctric plats, Int. J. Numr. Mthods Eng. 5:6 21),

18 436 ERASMO CARRERA AND CHRISTIAN FAGIANO σ zz a/2, b/2,.5) [Pa] σ yz a/2,, ) [Pa] σ xx a/2, b/2,.5) [Pa] Exact LFM %) %) %) LPM %) %) %) LD %) %) %) Litratur %) %) %) z a/2, b/2,.5) 1 11 [C/m 2 ] Φa/2, b/2, ) 1 3 [V] σ yy a/2, b/2, ) [Pa] Exact LFM %) %) %) LPM %) %) %) LD %) %) %) Litratur %) %) %) u z a/2, b/2,.5) 1 11 [m] u x, b/2,.5) 1 12 [m] u y a/2,,.5) 1 12 [m] Exact LFM %) %) %) LPM %) %) %) LD %) %) %) Litratur %) %) %) Tabl 4. Comparison of prsnt analysis with rspct to availabl rsults: msh [6 6], a/h = 4, Q8 lmnt + mans top valu). Th litratur valus ar from [Garcia Lag t al. 24b] [Ballhaus t al. 25] D. Ballhaus, M. D Ottavio, B. Kröplin, and E. Carrra, A unifid formulation to assss multilayrd thoris for pizolctric plats, Comput. Struct. 83: ), [Batra and Vidoli 22] R. C. Batra and S. Vidoli, Highr-ordr pizolctric plat thory drivd from a thr-dimnsional variational principl, AIAA J. 4:1 22), [Bnjddou 2] A. Bnjddou, Advancs in pizolctric finit lmnt modling of adaptiv structural lmnts: a survy, Comput. Struct. 76:1-3 2), [Carrra 1995] E. Carrra, A class of two dimnsional thoris for multilayrd plats analysis, Mm. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur ), [Carrra 1996] E. Carrra, C Rissnr Mindlin multilayrd plat lmnts including zig-zag and intrlaminar strsss continuity, Int. J. Numr. Mthods Eng. 39: ), [Carrra 1997] E. Carrra, An improvd Rissnr Mindlin typ lmnt for th lctromchanical analysis of multilayrd plats including pizo layrs, J. Intll. Matr. Syst. Struct. 8:3 1997), [Carrra 1999a] E. Carrra, A Rissnr s mixd variational thorm applid to vibrational analysis of multilayrd shll, J. Appl. Mch. ASME) 66:1 1999), [Carrra 1999b] E. Carrra, A study of transvrs normal strss ffcts on vibration of multilayrd plats and shlls, J. Sound Vib. 225:5 1999), [Carrra 21] E. Carrra, Dvlopmnts, idas, and valuations basd upon Rissnr s mixd variational thorm in th modling of multilayrd plats and shlls, Appl. Mch. Rv. 54:4 21),

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20 438 ERASMO CARRERA AND CHRISTIAN FAGIANO [Shu 25] X. Shu, Fr-vibration of laminatd pizolctric composit plats basd on an accurat thory, Compos. Struct. 67:4 25), [Thornbugh and Chattopadhyay 22] R. P. Thornbugh and A. Chattopadhyay, Simultanous modling of mchanical and lctrical rspons of smart composit structurs, AIAA J. 4:8 22), [Tirstn 1969] H. F. Tirstn, Linar pizolctric plat vibrations, Plnum Prss, Nw Yor, [Vidoli and Batra 2] S. Vidoli and R. C. Batra, Drivation of plat and rod quations for a pizolctric body from a mixd thr-dimnsional variational principl, J. Elasticity 59:1-3 2), [Wang and Yang 2] J. Wang and J. Yang, High-ordr thoris of pizolctric plats and applications, Appl. Mch. Rv. 53:4 2), [Wang t al. 1997] J. Wang, Y.-K. Yong, and T. Imai, Finit lmnt analysis of th pizolctric vibrations of quartz plat rsonators with highr-ordr plat thory, pp in Procdings of th 1997 IEEE Intrnational Frquncy Control Symposium Orlando, FL), IEEE, Nw Yor, [Yang and Batra 1995] J. S. Yang and R. C. Batra, Mixd variational principls in non-linar lctrolasticity, Int. J. Non- Linar Mch. 3:5 1995), Rcivd 9 Oct 26. Accptd 29 Nov 26. ERASMO CARRERA: rasmo.carrra@polito.it Dpt. of Aronautics and Arospac Enginring, Politcnico di Torino, Corso Duca dgli Abruzzi, 24, 1129 Torino, Italy CHRISTIAN FAGIANO: C.Fagiano@tudlft.nl Dpt. of Aronautics and Arospac Enginring, Politcnico di Torino, Corso Duca dgli Abruzzi, 24, 1129 Torino, Italy Currnt addrss: Dpartmnt of Mchanics, Arospac Structurs and Matrials, Dlft Univrsity of Tchnology, P.O. Box 558, 26 GB Dlft, Nthrlands