Modeling of the Through-the-Thickness Electric Potentials of a Piezoelectric Bimorph Using the Spectral Element Method

Transcription

1 Snsors 214, 14, ; do:1.339/s Artcl OPEN ACCESS snsors ISSN Modlng of th hrogh-th-hcknss Elctrc Potntals of a Pzolctrc Bmorph Usng th Spctral Elmnt Mthod Xngjan Dong *, Zhk Png, Hongxng Ha and Gang Mng Insttt of Vbraton Shock & Nos, Stat Ky Laboratory of Mchancal Systm and Vbraton, Shangha Jao ong Unvrsty, Shangha 224, Chna; E-Mals: z.png@sjt.d.cn (Z.P.); hhx@sjt.d.cn (H.H.); gmng@sjt.d.cn (G.M.) * Athor to whom corrspondnc shold b addrssd; E-Mal: donxj@sjt.d.cn; l.: (xt. 832). Rcvd: 1 Dcmbr 213; n rvsd form: 3 Fbrary 214 / Accptd: 12 Fbrary 214 / Pblshd: 2 Fbrary 214 Abstract: An ffcnt spctral lmnt (SE) wth lctrc potntal dgrs of frdom (DOF) s proposd to nvstgat th statc lctromchancal rsponss of a pzolctrc bmorph for ts actator and snsor fnctons. A sblayr modl basd on th pcws lnar approxmaton for th lctrc potntal s sd to dscrb th nonlnar dstrbton of lctrc potntal throgh th thcknss of th pzolctrc layrs. An qvalnt sngl layr (ESL) modl basd on frst-ordr shar dformaton thory (FSD) s sd to dscrb th dsplacmnt fld. h Lgndr orthogonal polynomals of ordr 5 ar sd n th lmnt ntrpolaton fnctons. h valdty and th capablty of th prsnt SE modl for nvstgaton of global and local rsponss of th pzolctrc bmorph ar confrmd by comparng th prsnt soltons wth thos obtand from copld 3-D fnt lmnt (FE) analyss. It s shown that, wthot ntrodcng any hghr-ordr lctrc potntal assmptons, th crrnt mthod can accratly dscrb th dstrbton of th lctrc potntal across th thcknss vn for a rathr thck bmorph. It s rvald that th ffct of lctrc potntal s sgnfcant whn th bmorph s sd as snsor whl th ffct s nsgnfcant whn th bmorph s sd as actator, and thrfor, th prsnt stdy may provd a bttr ndrstandng of th nonlnar ndcd lctrc potntal for bmorph snsor and actator. Kywords: spctral lmnt mthod; pzolctrc bmorph; lctrc potntal; sblayr; pcws lnar

2 Snsors 214, Introdcton Pzolctrc matrals gnrat lctrc potntals n rspons to mchancal strsss, and convrsly, prodc mchancal movmnts n rspons to lctrc potntals. hrfor, pzolctrc matrals can b sd both as actators and snsors, thy transform lctrcal nrgy nto mchancal nrgy, and vc vrsa. o achv practcally manngfl actaton and snsng capablts, a pzolctrc bmorph consstng of two pzolctrc layrs s wdly sd [1,2]. A broad rang of lctromchancal applcatons hav bn rportd, sch as lctroacostc transdcrs [3,4], mdcal dvcs [5], mcrocantlvr bosnsors [6], and atomc forc mcroscop (AFM) cantlvrs [7]. Howvr, bfor pzolctrc bmorphs can b tlzd n all ths applcatons, t s frst ncssary to nvstgat both th global rsponss and th local rsponss,.g., th dflcton and th dstrbton of th lctrc potntal across th thcknss. hr hav bn many thors and modls dvlopd for analyzng pzolctrc bmorph strctrs wth mphass on approxmatng th mchancal dsplacmnt and lctrc potntal. By carryng ot xact 3-D analytcal soltons for th smply spportd pzolctrc plat [8,9], t s shown that th dstrbton of th lctrc potntal across th thcknss s narly qadratc. hs mpls that th assmpton of lnar dstrbton of th lctrc potntal across th thcknss adoptd by many nmrcal modls [1,11] cannot addrss ths nonlnar lctrc potntal. Snc xact 3-D analytcal soltons ar not avalabl for mor gnral cass of loadng and bondary condtons, th ntrodcton of th fnt lmnt (FE) mthod s dsrabl. A consdrabl amont of ltratr has bn pblshd on th FE analyss of pzolctrc smart strctrs [12 14]. Among ths works, th smplst and oftn sd modl s th qvalnt sngl layr (ESL) modl n whch th dsplacmnt and stran fnctons ar assmd to b contnos throgh th thcknss. hr ar two man knds of thors sd for ESL modls. On s th classcal lamnatd plat thory (CLP) [15,16], and th othr on s th shar dformaton thory, whch branchs ot nto frst-ordr shar dformaton thory (FSD) [17,18] and hghr ordr shar dformaton thory (HSD) [19,2]. h ESL modl s smpl and capabl of prdctng th global rsponss of th bmorph, bt t dos not accont for th nonlnar dstrbton of th lctrc potntal across th thcknss. o ovrcom ths shortcomng, th FE modl sng th layr-ws thory [21 24] or th sblayr thory [2,25 28] has bn rcommndd. In th lattr cas, th pzolctrc layr s dvdd nto approprat nmbr of thn sblayrs. For ach of ths sblayrs, a lnar lctrc potntal dstrbton across th plat thcknss s assmd. It s frthr xpctd that th qadratc dstrbton of th lctrc potntal across th plat thcknss can b accratly approachd wth mor sblayrs adoptd. Gnrally, accratly smlaton of th local rsponss of th pzolctrc bmorph strctrs wold nvtably lad to a vry dns FE msh whn sng th FE mthod. Hnc, convntonal FE smlaton bcoms comptatonally vry nffcnt. A mor ffcnt mthod s th spctral lmnt (SE) mthod whch combns th gomtrc flxblty of FE mthod wth th hgh accracy of th psdo spctral mthod. hs mthod was frst prsntd by Patra n th md 198s [29]. In fact, th SE mthod and FE mthod ar closly rlatd and blt on th sam das. h man dffrnc btwn thm s that SE mthod ss orthogonal polynomals, sch as Lgndr and Chybysv polynomals, n th shap fnctons. h SE mthod rslts natrally n dagonal mass matrcs whch s a dstnct advantag ovr tradtonal FE mthod spcally for transnt analyss. Morovr, to hav

3 Snsors 214, an accrat smlaton wth th convntonal FE mthod, a msh wth a larg nmbr of lmnts and dgrs of frdom (DOFs) s nvtably ndd. h SE mthod, n whch th polynomal ordr s ncrasd and th msh sz s dcrasd, can b sd to ovrcom ths problm. h SE mthod has bn wdly appld to many ngnrng problms rlatd to acostcs, fld dynamcs and ssmology [3 35]. Rcntly, th SE mthod has bn xtnsvly sd to nvstgat th wav propagaton problms for th prpos of damag dtcton n strctrs [36,37]. Howvr, accordng to th athors bst knowldg, th SE mthod has not bn prvosly sd for accratly modlng of th throgh-th-thcknss lctrc potntals for pzolctrc bmorphs. For th prpos of accratly rprsntng th mchancal dsplacmnt and th lctrc potntal, a rasonabl choc s to s th ESL modl for th mchancal varabls and th layr-ws thory or th sblayr thory for th lctrc varabls. In th prsnt work, w attmpt to combn th mrts of th SE mthod and th sblayr modl. Mor spcfcally, th mchancal varabls,.., th dsplacmnts, ar dscrbd basd on FSD. h lctrcal varabls,.., th potntals, ar dscrbd sng th sblayr modl. SE mthod s thn tlzd to ddc th govrnng qatons. Lgndr orthogonal polynomals ar adoptd n th ntrpolaton fncton to mprov th accracy. o valdat th ffctvnss and th capablty of th prsnt modl, nmrcal smlatons for a smply spportd pzolctrc bmorph wth two dffrnt load cass,.., a nform prssr load appld to th top srfac and a nform potntal appld to th top and bottom srfacs, ar carrd ot. h rslts obtand by th prsnt approach ar thn compard to thos comng from th copld 3-D FE smlatons sng ABAQUS. h comparsons show th good accracy and ffcncy of SE mthod for modlng of th throgh-th-thcknss lctrc potntals of th pzolctrc bmorph. 2. Mathmatcal Formlaton 2.1. Constttv Rlatonshps, Dsplacmnt and Stran A pzolctrc bmorph mad of two dntcal PZ-4 pzolctrc layrs, whch has bn nvstgatd by Frnands [1], s consdrd hr. h PZ-4 layr s assmd to bhav n a lnar orthotropc mannr wth small dsplacmnts and strans. As dpctd n Fgr 1, both pzolctrc layrs hav th sam thcknss.5 h and ar pold n th sam drcton. h x-y plan of th coordnat systm x-y-z concds wth th mddl plan of th bmorph, and th z axs s dfnd normal to th mddl plan followng th rght-hand rl. hs work ams to nvstgat th problm of a smply spportd pzolctrc bmorph ndr a nform prssr load or an appld lctrc potntal n th framwork of lnar thory of pzolctrcty. Assmng th PZ-4 layrs work ndr sothrmal condtons, th pyrolctrc ffcts and thrmomchancal coplngs ar not takn nto accont. Consqntly, a lnar constttv rlatonshp addrssng both th drct and convrs pzolctrc ffcts s tlzd for th analyss of th pzolctrc bmorph, whch can b wrttn as: whr σ and x y z yz zx xy and stran vctor, rspctvly. E Ex Ey Ez σ cε E D ε ge x y z yz zx xy (1) ε rprsnt strss vctor, th lctrc fld vctor, D Dx Dy Dz, th

4 Snsors 214, lctrc dsplacmnt vctor, c, th lastc coffcnt matrx, g, th dlctrc coffcnt matrx, and, th pzolctrc strss coffcnt matrx. Fgr 1. Gomtry of a pzolctrc bmorph. y z - h/2 h/2 b x a An ESL modl adoptng th FSD s adoptd to dscrb th mchancal dsplacmnt. h dsplacmnt fld of a pzolctrc bmorph basd on FSD taks on th form [17,18]: x, y, z, t ( x, y, t) z ( x, y, t) v x, y, z, t v( x, y, t) z ( x, y, t) w x, y, z, t w( x, y, t) (2) whr, v, w dnot th dsplacmnts of an arbtrary pont on th md plan z, and dnot th rotatons of a transvrs normal abot th y and x axs, rspctvly. In th FSD, th transvrs shar strans ar assmd to b constant wth rspct to th thcknss coordnat. h constant stat of transvrs shar strans across th thcknss s a gross approxmaton of th tr stran fld, whch s at last qadratc throgh th thcknss. W dfn: U v w (3) U v w (4) whr U s th dsplacmnt vctor, and U s a gnralzd dsplacmnt vctor. hn Eqaton (2) can b wrttn n matrx form as: whr: U ZU (5) 1 z Z 1 z (6) 1 h nfntsmal stran componnts assocatd wth th dsplacmnts ar gvn by:

5 Snsors 214, whr L s th drvaton oprator dfnd as: ε LU (7) x z y L (8) y z x z y x 2.2. Approxmatons for Dsplacmnts h Lgndr polynomals basd SE mthod can b dscrbd as follows: th bmorph s frstly dscrtzd sng a st of non-ovrlappng rctanglar lmnts, as n th tradtonal FE mthod. Each rctanglar lmnt, dnotd by, s thn mappd to a rfrnc lmnt, dnotd by rf : 1,1 1,1, sng an nvrtbl local mappng. h dscrtzaton procdr s llstratd n Fgr 2. Sbsqntly, a st of nods, dnotd by, j, ar dfnd n th local coordnat systm of th rfrnc lmnt rf as roots of th followng polynomal xprsson: 2 (1 ) P ( ) 2 N (1 ) P ( ) N (9) whr P N s th N-th ordr Lgndr polynomal. In fact, th nods ar th 2-D Gass-Lobatto- Lgndr (GLL) ponts. In contrast to th classcal FE mthod, th dstrbton of nods s rrglar, as shown n Fgr 2. In th crrnt formlaton, th 5-th ordr Lgndr polynomal s chosn, hnc 36 nods can b spcfd n th rfrnc lmnt rf, as dpctd n Fgr 2. Fgr 2. Dscrtzaton of a plat and an xampl of spctral lmnt. h 1-D shap fnctons at th 1-D GLL ponts ar dfnd as [36]: 2 1 (1 ) PN ( ) h( ) for 1,, N 1 N( N 1) P ( ) N (1)

6 Snsors 214, An mportant proprty of ths ntrpolaton fnctons s th dscrt orthogonalty xprssd as: whr j th 1-D ons: h ( j ) j dnots th Kronckr dlta. h 2-D shap fnctons ar constrctd as a tnsor prodct of (11) j(, ) h( ) hj( ) for, j 1,, N 1 (12) Fgr 3 shows two xampls of th 2-D shap fnctons whch ndcat that ach shap fncton has th val 1 at on nod and vansh at all othr nods. Fgr 3. Slctd shap fnctons for a 36-nod spctral lmnt. (a) 32 (, ) ; (b) 45 (, ). (a) (b) Coordnats x and y wthn ach may b nqly rlatd to and pon th nvrtbl mappng: 6 6 x(, ), y(, ) (, ) x, y (13) 1 j 1 j j j whr x j and y j dnot th coordnats of x and y, rspctvly, of th lmnt nods,. h j rf gnralzd dsplacmnts, v, w, and ovr an rfrnc lmnt ar dscrtzd by th 2-D shap fnctons as: whr j, v j, 6 6 (, ), v(, ), w(, ), (, ), (, ) (, ), v, w (14) w j, j and j lmnt nodal dsplacmnt vctor s xprssd as: 1 j 1 j j j j ar th nodal vals of th gnralzd dsplacmnts. h dscrt 11 q12 q66 whr q j s th dsplacmnt vctor of th nod, : j q q (15)

7 Snsors 214, Sbstttng Eqatons (14) nto Eqaton (15) ylds: whr wth: q j j vj wj j j (16) U N q (17) N s th dsplacmnt shap fncton matrx whch can b xprssd as: N Z N N N (18) N j j I 5 5 (19) whr I 5 5s a 5 5 dntty matrx. Sbstttng Eqaton (17) nto Eqaton (7) ylds: whr B s stran-dsplacmnt matrx whch can b wrttn as: 2.3. Approxmatons for Elctrc Potntal ε Bq (2) B LN (21) For th prpos of accratly modlng th dstrbton of th lctrc potntal across thcknss, ach layr of th pzolctrc bmorph s sbdvdd mathmatcally nto n thnnr sblayrs. As shown n Fgr 4, th sblayrs ar nmbrd n top-to-bottom ordr. h z coordnats of th top and bottom srfacs of th -th sblayr ar dnotd by z and z 1, rspctvly. In ach sblayr, th dstrbton of th lctrc potntal () z s assmd to b lnar across th thcknss sch that: whr N s th ntrpolaton fncton and at th top and th bottom srfacs of th 1 () z N Φ (22) Φ s a colmn matrx composd of th lctrc potntals th sblayr, whch can b xprssd as: N z z z z 1, h z z 1, z 1 z z (23) h 1 Φ (24) In ths way, th assmpton of lnar dstrbton of lctrc potntal across th thcknss s sd not n th whol pzolctrc layr, bt n ach sblayr nstad. As a rslt, th lctrc potntal s approxmatd as pcws lnar across th thcknss and t s xpctd that th qadratc dstrbton of th lctrc potntal across th bmorph thcknss can b approachd wth mor sblayrs adoptd. As mntond bfor, th mchancal dsplacmnt fld s approxmatd sng ESL modl basd on FSD and th pzolctrc bmorph s dscrtzd sng 2-D msh. o kp th compatblty,

8 Snsors 214, ach sblayr of th pzolctrc layr s also dscrtzd sng th sam msh. Consqntly, an lmnt potntal vctor Φ s thn ntrodcd n th spctral plat fnt lmnt, whch s dfnd as: 1 2n Φ (25) h srfac potntal of th sblayr,, s assmd to b constant ovr th lmnt and, 1, 2n ar tratd as lmntal DOFs, as llstratd n Fgr 2. Frthrmor, th top and bottom srfacs of th pzolctrc layrs ar always coatd wth mtallc coatngs of zro thcknss and th potntals on th lctrods shold b takn as ndpndnt of x, y. hs th prsnt mthod combns an ESL thory for th dsplacmnt and a pcws lnar approxmaton for th lctrc potntal. Undr th qas-lctrostatc approxmaton, th lctrc fld and th lctrc potntal n ach sblayr hav th followng rlatonshp: whr E () z s th lctrc fld of th -th sblayr, E () z B Φ (26) B s th lctrc fld-potntal matrx, gvn by: N B (27) Fgr 4. A sblayr modl for a pzolctrc bmorph Govrnng Eqatons By applyng Hamlton s prncpl, th lmntary dynamc qatons for th pzolctrc bmorph plat can b obtand: Φ K q K Φ F M q K q K F (28) whr M dnots th lmnt mass matrx; K pzolctrc coplng matrcs; appld forc; and, mchancal stffnss matrx; K and K th K th dlctrc prmttvty matrx; F th vctor of xtrnally F th vctor of xtrnally appld charg, rspctvly:

9 Snsors 214, whr 2n 1 1 h M N N J dd (29) 2n 1 1 h K B cb J dd (3) 2n 1 1 h K K B B J dd (31) 2n 1 1 h K B gb J dd (32) s F N P J dd (33) 1 1 F N q s J dd (34) 1 1 s th mass dnsty, P s s th srfac forc vctor, q s s th srfac charg dnsty vctor. J s th wll-known Jocoban matrx of th mappng (13) whch s dfnd as: x y J= ( xy, ) (, ) x y (35) h GLL ntgraton rl s thn sd to calclat th charactrstc matrcs and th nodal forc vctor n Eqaton (28) at th lmntal lvl [36]. In ths stdy, th ntrfac btwn th two PZ layrs s grondd. wo sts of lctrc bondary condtons ar consdrd,.., (1) snsor fncton 2 wth th top and bottom srfacs grondd and a nform prssr load of S 1, N/m appld to th ppr srfac, and Eqaton (2) actator fncton wth an lctrc potntal of V 5V appld to th top and bottom srfac of th bmorph. By applyng th lctrc bondary condtons, th DOFs for th lctrc potntal ar condnsd ot sch that Eqaton (28) s fnally of th form: p M q K Kp q F F a (36) whr K dnots th mchancal stffnss matrx ndcd by th lctromchancal coplng of PZ-4 layr, and F a dnots th mchancal forcs ndcd by th appld voltags of pzolctrc actators [2]. h lctrc potntal s thn rcovrd by th nvrs procss of th aformntond condnsaton. Assmblng all lmntary qatons, on can hav a global dynamc systm qaton: M q K Kp q F F a (37) whr M, K, K p, F and F a ar th assmbld contrparts of matrcs M, K, K p, F and F ; q s th global nodal dsplacmnt vctor. Snc th DOFs for th sblayr lctrc potntals a hav bn condnsd ot, ths approach wll not rslt n a larg nmbr of potntal DOFs. For th prpos of statc analyss, th govrnng qatons n Eqaton (37) rdcs to:

10 Snsors 214, Nmrcal Rslts K Kp q F F a (38) In ths scton, th drvd SE modl s convrtd nto a nmrcal cod and cas stds ar carrd ot to valdat th ffctvnss and th capablty of th prsnt modl for prdctng both th global rsponss and th local rsponss,.., th dflctons of th bmorph and th dstrbton of th lctrc potntal across th bmorph thcknss. A smply spportd rctanglar pzolctrc bmorph shown n Fgr 1, whch has bn nvstgatd by Frnands [1], s consdrd hr. h matral constants of PZ-4 ar gvn as: c GPa 25.6 (39) C/m (4) g 13.6 nf/m (41) h lngth a and wdth b of th bmorph ar 25 mm and 12.5 mm rspctvly. wo vals of slndrnss rato, S a / h 5 and S 5, whch rprsnt th thck and thn bmorph plat, rspctvly, ar consdrd. Unlss othrws statd, th ordr of Lgndr polynomal s chosn as 5, and th msh n Fgr 3 s sd n ths work. wo load cass corrspondng rspctvly to snsor fncton and actator fncton ar consdrd. o ovrcom th ll condton problm rsltd from th hg dffrnc of th lmnt vals of K and K n magntds, Eqaton (28) s rwrttn sng dmnsonlss varabls. Consqntly, th nmrcal rslts for th dflcton and th lctrc potntal ar gvn n dmnsonlss nts as: 2 c11 W, ( w, / E) for snsor fncton hs E W, ( w, / E) for actator fncton V (42) (43) 1 whr th amplfcaton factor E s takn as E 1 V/m. For th prpos of comparson, a copld 3-D analyss s carrd ot sng 2-nodd hxahdral 3-D pzolctrc lmnts (C3D2RE)

11 z/h z/h Snsors 214, wth a msh sz of n ABAQUS and th rslts from th copld 3-D FE analyss ar takn as accrat Snsor Fncton 2 For ths cas a nform prssr load of S 1, N/m s appld to th ppr srfac and th bmorph s sd as a snsor wth th top and bottom srfacs grondd. h varatons of both th dflcton W and th lctrc potntal across th bmorph thcknss at th cntr of th bmorph plat ( x.5 a, y.5b ) for th slndrnss rato S 5 and S 5 ar shown n Fgrs 5 and 6, rspctvly. It can b obsrvd from Fgr 5a and Fgr 6a that th dflcton W stmatd by th prsnt mthod adoptng dffrnt nmbr of sblayrs s constant throgh th thcknss and t s a good approxmaton of th nonlnar dstrbton dscrbd by th copld 3-D analyss. h prsnt modl basd on FSD wth assmpton of nform dflcton throgh th thcknss cannot prdct th nonlnar varaton of W throgh th thcknss. h lctrc potntals ndcd by th dformaton of th bmorph throgh th drct pzolctrc ffcts ar shown n Fgrs 5b and 6b. It s obsrvd that th dstrbton of th lctrc potntal across th thcknss provdd by th prsnt approach wth mor than 2 sblayrs s n good agrmnt wth th nonlnar dstrbton prdctd by th copld 3-D analyss. Frthrmor, t s xpctd that wth mor sblayrs adoptd th qadratc dstrbton of th lctrc potntal across th bmorph thcknss can b accratly approachd wthot ntrodcng any hghr-ordr lctrc potntal assmptons. Howvr, th convntonal lnar lctrc potntal assmpton [38] wll rslt n an naccrat prdcton of th local lctrc potntal rspons for th cas of snsors. h crvs n Fgrs 5 and 6 ar symmtrcal wth rspct to th ntrfac btwn th two PZ layrs. It shold b hghlghtd that althogh th prsnt mthod cannot prdct accratly th dstrbton of W across th bmorph thcknss, t may b abl to provd good approxmat rslts for wth approprat nmbr of sblayrs for both thck and thn bmorph plats. Fgr 5. Bmorph snsor of S 5 ndr prssr load. (a) Dmnsonlss dflcton; (b) Dmnsonlss lctrc potntal. 3-D FE analyss (fll ln), prsnt modl wth n 2 (trangls) and prsnt modl wth n 1 (small crcls) W (a) (b)

12 z/h z/h z/h z/h Snsors 214, Fgr 6. Bmorph snsor of S 5 ndr prssr load. (a) Dmnsonlss dflcton; (b) Dmnsonlss lctrc potntal. 3-D FE analyss (fll ln), prsnt modl wth n 2 (trangls) and prsnt modl wth n 1 (small crcls) W x 1 6 (a) (b) 3.2. Actator Fncton o achv practcally manngfl actaton capablts and garant that th pzolctrc matral bhavs lnarly, an lctrc potntal of V = 5V s appld to th top and bottom srfacs of th bmorph wth ntrmdat lctrod grondd. h throgh-th-thcknss varatons of W and at th cntr of th plat for S 5 and S 5 ar shown n Fgrs 7 and 8, rspctvly. Fgr 7. Bmorph actator of S 5 ndr potntal load. (a) Dmnsonlss dflcton; (b) Dmnsonlss lctrc potntal. 3-D FE analyss (fll ln), prsnt modl wth n 2 (trangls) and prsnt modl wth n 1 (small crcls) W (a) (b)

13 z/h z/h Snsors 214, Fgr 8. Bmorph actator of S 5 ndr potntal load. (a) Dmnsonlss dflcton; (b) Dmnsonlss lctrc potntal. 3-D FE analyss (fll ln), prsnt modl wth n 2 (trangls) and prsnt modl wth n 1 (small crcls) W (a) (b) Onc agan, th prsnt modl basd on FSD cannot prdct accratly th throgh-th-thcknss dstrbton of W. Smlar to th prvos obsrvaton, th constant dflcton W throgh th thcknss calclatd by th prsnt mthod adoptng dffrnt nmbr of sblayrs s a good approxmaton of th nonlnar dstrbton provdd by th copld 3-D analyss. It s notcd that as th sblayr nmbr ncrass a smallr dflcton s obtand whch s also pontd ot by Wang [2]. h lctrc potntals at th cntr of th plat ar plottd n Fgr 7b and Fgr 8b for th slndrnss rato S 5 and S 5, rspctvly. It can b obsrvd that th almost lnar dstrbton of across th thcknss prdctd by th prsnt mthod for both thck and thn bmorph plats s n xcllnt agrmnt wth th copld 3-D analyss, ndcatng that th nonlnar ndcd lctrc potntal s nsgnfcant compard to th xtrnally appld potntal. Consqntly, th convntonal lnar lctrc potntal assmpton [38] may b accrat nogh to calclat th local lctrc potntal rspons for th cas of actators. 4. Conclsons h prsnt work ams to dvlop an ffcnt SE modl wth lctrc potntal DOFs for th statc lctromchancal rspons of a pzolctrc bmorph. h approach s th combnaton of an ESL modl basd on FSD for th mchancal dsplacmnt wth a sblayr modl basd on th pcws lnar approxmaton for th lctrc potntal. 2-D GLL shap fnctons ar sd to dscrtz th dsplacmnts and thn th govrnng qaton of moton s drvd followng th standard SEM procdr. By applyng th lctrc bondary condtons, th DOFs for th lctrc potntal ar condnsd ot sch that th prsnt modl wll not rslt n a larg nmbr of potntal DOFs. Nmrcal smlatons basd on th prsnt modl ar carrd ot for two dffrnt load cass,.., a nform prssr load appld to th top srfac and a nform potntal appld to th top and bottom srfacs. o valdat th ffctvnss and th capablty of th prsnt modl for nvstgaton of both global and local rspons of th pzolctrc bmorph, th nmrcal rslts ths obtand ar

14 Snsors 214, compard to thos from 3-D analyss sng ABAQUS. h rslts ndcat that th dflcton W stmatd by th prsnt mthod s a good approxmaton of th nonlnar dstrbton prdctd by th copld 3-D analyss. It s frthr shown that th prsnt modl provds vry accrat prdcton for th lctrc potntal dstrbtons across th bmorph thcknss vn for rathr thck bmorph plat wthot ntrodcng any hghr-ordr lctrc potntal assmptons. It s also rvald that th convntonal lnar lctrc potntal modl s accrat nogh to prdct th local lctrc potntal rspons for th cas of actators. hs obsrvaton conssts wth th prvos fndngs proposd by Yang [39]. On of th lmtatons s that th dflcton W across th thcknss s constant. Nvrthlss, t s accrat nogh to nvstgat th global rspons of th pzolctrc bmorph. h prsnt work s mportant for rsarchrs to bttr ndrstand th nonlnar ndcd lctrc potntal for bmorph snsor and actator. An mportant xtnson of th prsnt rsarch s to stdy th vbraton charactrstcs of th pzolctrc bmorph basd on SE mthod. h nflnc of th ndcd stffnss matrx on th natral frqncs of th bmorph plat ndr varos lctrc bondary condtons s to b nvstgatd. h convrgnc stdy of th prsnt modl wth rspct to th ordr of th Lgndr polynomal s also a practcal and ntrstng problm to b condctd. Acknowldgmnts h rsarch was spportd by Natonal Scnc Fnd for Dstngshd Yong Scholars (grant nmbr ) and Natral Scnc Fondaton of Chna (grant nmbr 17239). Athor Contrbtons All athors contrbtd xtnsvly to th work prsntd n ths papr. XngJan Dong wrot th cod and draftd th manscrpt. Zhk Png analyzd and ntrprtd th data ndr th sprvson of Gang Mng. Hongxng Ha concvd th stdy and dtd th manscrpt. Gang Mng gav concptal advc and crtcally rvsd th manscrpt. All athors approvd th fnal vrson of th manscrpt for sbmsson. Conflcts of Intrst h athors dclar no conflct of ntrst. Rfrncs 1. Frnands, A.; Pogt, J. Analytcal and nmrcal approachs to pzolctrc bmorph. Int. J. Solds Strct. 23, 4, Wang, S.Y. A fnt lmnt modl for th statc and dynamc analyss of a pzolctrc bmorph. Int. J. Solds Strct. 24, 41, Wang, H.J.; Mng, Q.F. Analytcal modlng and xprmntal vrfcaton of vbraton-basd pzolctrc bmorph bam wth a tp-mass for powr harvstng. Mch. Syst. Sgnal Proc. 213, 36,

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