. Artce. SCIENCE CHINA Physcs, Mechancs & Astronomy Octoer 13 Vo. 56 No. 1: 1844 1853 do: 1.17/s11433-13-534-1 The twoscae asymptotc error anayss for pezoeectrc proems n the quas-perodc structure FENG YongPng 1,*, DENG MngXang 1 &GUANXaoFe 3 1 Schoo of Mathematcs Informaton Scences, Key Laoratory of Mathematcs Interdscpnary Scences of Guangdong Hgher Educaton Insttutes, Guangzhou Unversty, Guangzhou 516, Chna; Department of Mathematcs, Unversty of Caforna, San Dego 993, USA; 3 Department of Mathematcs, Tong Unversty, Shangha 9, Chna Receved Octoer 9, 1; accepted May 3, 13; pushed onne August 19, 13 Appcatons for pezoeectrc effect have grown rapdy, pezoeectrc materas pay mportant roes n countess areas of modern fe. By means of twoscae method couped oundary ayer, some new nds of twoscae asymptotc expansons for soutons to the eectrca potenta the dspacement n quas-perodc structure under couped pezoeectrc effect are derved, the homogenzaton constants of pezoeectrc materas are presented. The couped twoscae reaton etween the eectrca potenta the dspacement s set up, some mproved asymptotc error estmates are anayzed. twoscae method, pezoeectrcty, quas-perodc structure, homogenzaton constants PACS numers:.3.mv, 77.65.Bn, 77.84.Lf,.6.C Ctaton: Feng Y P, Deng M X, Guan X F. The twoscae asymptotc error anayss for pezoeectrc proems n the quas-perodc structure. Sc Chna-Phys Mech Astron, 13, 56: 1844 1853, do: 1.17/s11433-13-534-1 1 Introducton Snce the dscovery of the pezoeectrc effect y the Cure rothers n 188, many research actvtes have een focused on understng the orgns of the pezoeectrc effect deveopng materas wth desrae pezoeectrc characterstcs for appcatons as sensors actuators. Despte the sgnfcant progress made n enhancng the coupng characterstcs etween the eectrca mechanca propertes n pezoeectrc materas, monothc pezoeectrc materas generay exht mtatons such as rtteness mted range of couped propertes. Accordng to the macro-mechanca approach, the heterogeneous structure of the composte can e repaced y a homogeneous medum wth ansotropc propertes. The homogenzaton of the near equatons of pezoeectrcty wth the *Correspondng author ema: ffyyppmath@163.com poarzaton gradent was frsty gven y Teega Bytner 1. For contrutons y other experts to the pezoeectrc homogenzaton, the readers are referred to ref.. Modeng approaches to predctng the eectromechanca response of porous pezoeectrc materas have een deveoped y Ront Venatesh 3. Utzng a modfed cues mode whch comnes the exact souton of an epsoda pore n an nfnte pezoeectrc sod wth an effectve medum approxmaton, Dunn Taya presented an anaytca framewor for descrng the couped ehavor of pezoeectrc materas wth cosed pores 3- type n refs. 4,5. In ref. 6, Zheng Du presented an expct unversay appcae estmate for the effectve propertes wth mut-phase compostes of ncuson dstruton. Fang hs co-worers studed the consttutve experments on pezo/ferroeectrcs under couped oads deformaton fracture n ref. 7. Jayachran Guedes dscussed the optmzaton mode numerca framewor to the optma mcrostructure of ferro- c Scence Chna Press Sprnger-Verag Bern Hedeerg 13 phys.scchna.com www.sprngern.com
Feng Y P, et a. Sc Chna-Phys Mech Astron Octoer 13 Vo. 56 No. 1 1845 eectrc materas n ref. 8. Nchoas presented the stochastc modeng of mut-scae mut-physcs proems n ref. 9. More wors concernng the computatona methods of pezoeectrc proems are cted n refs. 1 13. Generay, most resuts are focused on expermenta research some speca pezoeectrc materas. In genera, composte materas are often made of very fne mcro-structures vary sharpy wthn a very sma perodc ce. Lmted y computng resources, many practca proems are st eyond the reach of drect smuatons. Few oundary vaue proems of pezoeectrcty of composte materas wth very fne perodcconfguratonshaveeen studed. The purpose of ths paper s to dscuss the homogenzaton constants of pezoeectrc materas the couped homogenzaton equatons, to present one new hgher order couped twoscae asymptotc expanson for pezoeectrc proem n quas-perodc structure usng twoscae method TSM couped oundary ayer method. The remanng part of ths paper s organzed as foows. In sect., some premnary resuts are refy shown. In sect. 3, some auxary emmas are provded to estash the exstence unqueness of ce proems. Approxmate soutons ased on the forma asymptotc expansons, reguartes of soutons asymptotc error estmates are gven n sect. 4. The twoscae asymptotc expansons of the dspacement u x the eectrca potenta Φ x Let Ω e a ounded doman of R n wth a Lpschtz oundary Γ. Ω = z T z Ω, where T s the suset of Z n consstng of a z such that z Ω, = {x :< x < 1, = 1,,..., n}, s a sma parameter. In, we ntroduce the foowng Sooev space, { } V = φ φ H 1,φs 1-perodc functon, φdξ =. Throughout ths paper, the Ensten summaton conventon on repeated ndces s adopted. In ths paper, we consder the foowng statonary pezoeectrc system n refs. 14 16, a x u e Φ = f, n Ω, Φ e x x u = ρ, n Ω, 1 Φ =Φ x, on Γ, u = u x, on Γ. Ths system has the foowng near consttutve reatons, σ u, Φ = a h hu e E Φ, D u, Φ = E Φ e u, 3 where u Φ denote the dspacement the eectrca potenta, { h u } {σ u, Φ } are the stran of dspacement u stress tensor correspondng to u, Φ, E D denote the -th component of the eectrca fed Φ the deectrc dspacement vector correspondng to u, Φ, {a h }, {e } { } are the 4th-order eastc tensor, 3rdorder pezoeectrc tensor nd-order deectrc tensor, respectvey, f ρ denote the ody force the densty of free eectrca charge, u h u x h, E Φ = Φ. 4 h u x = 1 In genera, {a h } satsfes the foowng coercve condton E 1,, { } satsfes the foowng postve defnte condton Pλ 1,λ n refs. 15,16. Ths means that the eements {a h } are ounded measurae functons ndependent of satsfy { a h = a h = a h, 1 η h η h a h xη hη η h η h, x Ω, 5 where {η h } s an artrary symmetrc matrx wth rea eements, 1 are postve constants ndependent of. The eements { } are ounded measurae functons satsfy { =, λ 1 η η xη η λ η η, x Ω, 6 where {η } s an artrary vector wth rea eements, λ 1 λ are constants greater than zero ndependent of. For tensor e we assume that e = e. In a smpe form, we rewrte the stress stran tensor as foows. u = 11,, 33, 3, 31, 1 T, 7 σu, Φ = σ 11,σ,σ 33,σ 3,σ 31,σ 1 T, 8 where T denotes the transpose of vector or matrx. Usng these reatons, eqs. 3 s equvaent to the foowng matrx operator dentty σu, Φ D u = S E, 9 where S s a 9 9 matrx gven y S = A e T e B,where A s a 6 6 symmetrc matrx dependng on the eastc tensor {a h }, e s a 3 6 matrx dependng on the pezoeectrc tensor {e }, B s a 3 3 matrx dependng on the eectrc permttvty { }, more owng to eqs. 5 6, A B are postve defnte matrces. Moreover, assumng that f, u, ρ Φ are suffcenty smooth vector-vaued or scaar-vaued functons n Ω. InΩ, et ξ = x/,thena h, e are 1-perodc functons are denoted y a h ξ, ξ e ξ.
1846 Feng Y P, et a. Sc Chna-Phys Mech Astron Octoer 13 Vo. 56 No. 1 Remar.1 The exstence unqueness of proem 1 for suffcenty smooth f, ρ, Φ u are easy foowed from propertes of {a h }, { } {e }, Fredrchs s nequaty, Korn s nequaty generazed Lax-Mgram s emma. Inspred y deas n refs. 19 1, we consder that Φ x u x are the functons of x, ξ at the same tme, suppose that Φ x u x coud e exped formay as foows. Φ x =Φ x u x = u x <α>= <α>= <α>= H α ξd αφ x G α ξd α u x, 1 <α>= N α ξd α u x M α ξd αφ x, 11 where α = α 1,...,α, α = 1,...,n, = 1,...,, 1, α denotes the dmenson of vector α. The scaar-vaued functon Φ x vector-vaued functon u x aredefned on Ω; The scaar-vaued functons H α ξ, vector-vaued functons G α ξ M α ξ, matrx-vaued functons N αm ξ w e determned on 1-square. For α = = 1, they are soutons to the foowng parta dfferenta equatons. e ξ H α 1 ξ ah ξ h M α1 ξ e α1 ξ =, n, ξ H α 1 ξ e ξ M α1 ξ α1 ξ =, n, M α 1 ξdξ =, H α 1 ξdξ =, M α1 ξ, H α1 ξ s 1-perodc n ξ, ah ξ h N α1 mξ a α1 mξ e ξ G α 1 mξ =, n, e ξ N α1 mξ e mα1 ξ ξ G α 1 mξ =, n, N α 1 mξdξ =, G α 1 mξdξ =, N α1 mξ, G α1 mξ s 1-perodc n ξ. 1 13 For α = =, H α α ξ, M α α ξ, G α α ξ N α α mξ are the soutons to the foowng parta dfferenta equatons. H α1 α e ah h M α1 α = ê α1 α e α1 H α a α M α1 a α M α1 H α1 e α e α1 α, n, H α1 α e M α1 α = ˆ α1 α α1 H α e α M α1 e α M α1 H α1 α α1 α, n, α1 α H α3 α 1 H α α 3 e α1 α M α3 dξ 1 eα1 M α α 3 e α 1 M α α 3 e α1 α H α3 e α 1 H α α 3 a α1 α M α3 dξ 1 aα1 M α α 3 a α 1 M α α 3 M α1 α ξ, H α1 α ξ s 1-perodc n ξ, dξ =, dξ =, G α1 α e m ah h N α1 α m = â α mα 1 e α1 G α m a α N α1 m a α N α1 m G α1 m e α a α mα 1, n, G α1 α m e N α1 α m = ˆd α mα 1 α1 G α m e α N α1 m e α N α1 m G α1 m α e α mα 1, n, α 1 α G α3 m α 1 G α α 3 m e α1 α N α3 mdξ N α α 3 m e α 1 N α α 3 m dξ =, 1 eα1 e α1 α G α3 m e α 1 G α α 3 m a α1 α N α3 mdξ 1 aα1 N α α 3 m a α 1 N α α 3 m dξ =, N α1 α mξ, G α1 α mξ s 1-perodc n ξ, 14 15
Feng Y P, et a. Sc Chna-Phys Mech Astron Octoer 13 Vo. 56 No. 1 1847 where â h = ê = ˆ = ˆd = G h a h a m m N h e dξ, 16 H e a m m M e dξ, 17 H ξ e m ξ m M dξ, 18 G e e m m N dξ, 19 where {â h }, {ê },{ˆ } { ˆd } are caed homogenzaton constants. It shoud e mentoned that there exst two formuas 19 17 to compute the homogenzaton constants of {e }, we w prove the equvaent of these two formuas n the next secton. For α = 3, N αm ξ, M α ξ, G α ξ H α ξ are otaned y sovng the foowng parta dfferenta equatons. e H α1 α ah h M α1 α = e α1 α H α3 α e α1 H α α a α1 M α α a α1 M α α e α1 H α α a α1 α M α3 α, n, H α1 α e M α1 α = α1 α H α3 α α1 H α α e α1 M α α H α α e α1 M α α α1 e α1 α M α3 α, n, α α 1 H α1...α 1 α 1 H α e α α 1 M α1..α 1 dξ 1 eα1 M α e α 1 M α dξ =, e α1 α H α3...α 1 e α 1 H α a α1 α M α3...α 1 dξ 1 aα1 M α a α 1 M α dξ =, M α1 α ξ, H α1 α ξ s 1-perodc n ξ, G α1 α e m ah h N α1 α m = e α1 α G α3 α m e α1 G α α m a α1 N α α m a α1 N α α m e α1 G α α m a α1 α N α3 α m, n, G α1 α m e N α1 α m = α1 α G α3 α m α1 G α α m e α1 N α α m G α α e α1 N α α m m α1 e α1 α N α3 α m, n, α1 α G α3...α 1 m α 1 G α1...α m e α1 α N α3...α 1 m 1 eα1 N α m e α 1 N αm dξ =, e α1 α G α3...α 1 m e α 1 G αm a α1 α N α3...α 1 mdξ 1 aα1 N α m a α 1 N αm dξ =, N α1 α mξ, G α1 α mξ s 1-perodc n ξ. 1 u x, Φ x wth oundary condton of eq. 1 presents one new proem, caed homogenzaton proem correspondng to eq. 1. For suffcenty smooth f, u, Φ ρ, the homogenzaton souton u x, Φ x s suffcenty smooth nsde Ω, the homogenzaton proem s defned as foows. L 1 u, Φ = {â h h u Φ } ê = f, n Ω, x L u, Φ = { ˆd u Φ ˆ } = ρ, n Ω, x x u x = u x, on Ω, Φ x =Φ x, on Ω. Remar. Each of eqs. 1 15 1 has a unque souton n V n Vowng to the propertes of postve defnteness of {a h } { }. The eements of the matrces N α1 α mξ, M α1 α ξ, G α1 α ξh α1 α ξ are pecewse
1848 Feng Y P, et a. Sc Chna-Phys Mech Astron Octoer 13 Vo. 56 No. 1 smooth scaar-vaued or vector-vaued functons eongng to Cper 1 = {φξ φ H1, φ s 1-perodc n ξ}. In the next secton, we w prove the exstence unqueness of these ce proems. Remar.3 It shoud e ponted out that the perodc functons N αm, G α, M α H α defned n ths paper are dfferent from those of some tradtona homogenzaton methods n refs. 17 18 n vew of the dfferent oundary condtons on the unt ce. Moreover, y means of the tradtona homogenzaton method, t s very dffcut to defne N α, G α, M α H α for <α>= 3. Under some speca assumptons, Cao L.. has otaned the hgher order asymptotc expansons y means of homogeneous oundary condtons of N α on for eastc proem n ref. 19. Remar.4 We may prove that L 1 L are symmetrc postve defnte operators, there exsts a unque souton to the proem gven y eq.. Remar.5 In aove formuas foowng deducton, D αu x D αφ x denote u x Φ x x α1 x α, D 1 α u x D α Φ x denote x α1 x α 1 u x x α1 x α x Φ x x α1 x α x x, respectvey. The scaar-vaued functon M s denotes the s-th component of vector-vaued functon M, the scaar-vaued functon N h denotes the h-th row th coumn eement of matrx-vaued functon N, the vectorvaued functon N denotes the -th coumn of matrx-vaued functon N. It s easy to see that M α ξ G α ξ pay the mportant roe n the twoscae forma asymptotc eqs. 1 11. From those tems we coud now more aout the couped effects homogenzaton effects etween the eectrca fed the mechanca fed. And the mechancs ehavor of ody s caused not ony y macroscopc condtons, such as the geometry of the structure, nterna eectrc resource oundary condtons, ut aso y the asc confguraton of composte matera. For L 1, we defne L-order asymptotc expanson of souton u, Φ forx Ω sted as foows. Φ L x =Φ x L u L x = u x L H α ξd α Φ x α = α >= G α ξd α u x, 3 L N α ξd αu x α = L M α ξd αφ x. 4 α =, Φ L do not satsfy the ound- In order to match the Generay speang, u L ary condton on Ω for u, Φ. oundary condtons of the souton u, Φ, we need to construct a proper oundary ayer proem n =Ω\Ω so as to mprove the approxmaton of the asymptotc expanson u L, Φ L. We defne a oundary ayer proem as foows. a u e Φ = f, n, Φ e x x u = ρ, n, ν a u e Φ = ν a u L e Φ L, on Ω, ν Φ e x u = ν Φ L e x u L, on Ω, Φ x =Φ x, u x = u x, on Ω. 5 Up to now, the L-order twoscae approxmate souton ũ,l, Φ,L ofsoutonu, Φ s gven as foowng formuas. u, Φ u L ũ,l, Φ,L, Φ = L, x Ω, 6 u, Φ, x Ω1. 3 Exstence unqueness of ce proems the oundary ayer proem In ths secton, we prove the exstence unqueness of those ce proems the oundary ayer proem. To egn wth, we estash the unqueness exstence of the soutons to the ce proems gven y eqs. 1 15 1. For the sae of smpcty, et n = 3nthefoowng text. Lemma 3.1 Under the assumptons of eqs. 5 6, each of ce proems eqs. 1 15 1 contans a unque souton n V 3 V. Proof Frsty, we prove the exstence unqueness of the wea souton to the proem gven y eq. 1. The wea formuaton of proem 1 s to fnd a souton par M α1, H α1 nv 3 V, such that a M α1, H α1, v,φ = F v,φ, v,φ V 3 V, 7 where the near form a s defned y a M α1, H α1, v,φ = h M α1 a h vdξ Hα1 φ H α1 e v φ e M α1 dξ, 8
Feng Y P, et a. Sc Chna-Phys Mech Astron Octoer 13 Vo. 56 No. 1 1849 the near form F s gven y φ F v,φ = e α1 v α1 dξ. 9 Snce φ φ av,φ, v,φ = va h h v dξ c v 1, φ 1,, accordng to the Poncaré nequaty Korn s nequaty, the near form av,φ, v,φ s coercve on V 3 V. In vrtue of Poncaré nequaty, Korn s nequaty trace theorem of Sooev space, we have the foowng denttes. au,ψ, v,φ = ua h h v ψ φ dξ ψ φ e v e udξ c 1 u 1, v 1, c ψ 1, φ 1, c 3 ψ 1, v 1, c 4 φ 1, u 1, c 5 v 1, φ 1, u 1, ψ 1,. Therefore, y means of generazed Lax-Mgram s emma, the exstence unqueness of M α1, H α1 can e otaned. Next, we use the homogeneous method to prove the unqueness exstence of the souton to the proem defned y eq. 14. Set M α1 α, H α1 α V 3 V,we assume that M α1 α, H α1 α satsfes the foowng equatons. H α1 α e ah h M α1 α = ê α1 α e α1 α e α1 H α a α M α1 a α M α1 H α1 e α, n, H α1 α e M α1 α = ˆ α1 α α1 α α1 H α e α M α1 e α M α1 H α1 α, n, M α1 α dξ =, H α1 α dξ =, M α1 α ξ, H α1 α ξ s 1-perodc n ξ. 3 In vrtue of the generazed Lax-Mgram emma, Korn s nequaty Poncaré nequaty, there exst a unque souton par M α1 α, H α1 α of eq. 3. It s easy to see that one can fnd a unque vector ĉ 1 scaar ĉ, such that M α1 α M α1 α = ĉ 1, H α1 α H α1 α = ĉ, α1 α H α3 α 1 H α α 3 e α1 α M α3 dξ 1 eα1 M α α 3 e α 1 M α α 3 dξ =, e α1 α H α3 e α 1 H α α 3 a α1 α M α3 dξ 1 aα1 M α α 3 a α 1 M α α 3 dξ =, hod, the exstence unqueness of the souton to the proem defned y eq. 14 s proved. The exstence unqueness of the soutons to the ce proems gven y eqs. 13, 15, 1 coud e proved n a smar way. From eqs. 17 19, there are two dfferent formuas apped to cacuate the homogenzaton pezoeectrc constants of {e ξ}. In fact, eq. 19 s equvaent to eq. 17. Lemma 3. Under the assumptons of coercve postve defnte condtons of {a h } { }, eq. 17 s equvaent to eq. 19. Proof In fact, we have G m M e s a ht ht N m a m dξ =, 31 s H N m e s a ht ht M e dξ =, 3 H Gm s G m e sh sh N m e m H e s s M dξ =, 33 dξ =. 34 Based on the assumptons of {a h }, { }, {e } denttes eqs. 31 34, we can otan the foowng dentty H a m M e sm s = dξ e N m G m s dξ. 35 It s easy to see that eqs. 17 19 are equvaent. Now we estash the coercvty postve defnteness of homogenzaton constant {â h } {ˆ }, the unqueness of the souton to the homogenzaton proem defned y eq. the souton to the oundary ayer proem defned y eq. 5. Lemma 3.3 Under the assumptons of coercvty postve defnteness of {a h ξ} { ξ}, {â h } satsfes the coercve condton E λ 1, λ, {ˆ } meets the postve defnte condton P 1,. In terms of coercve condton of {â h } postve defnte condton of {ˆ }, Lemma 3.1, n a smar way, we have the foowng emmas. Lemma 3.4 There exsts a unque souton u, Φ tothe homogenzaton proem estashed y eq..
185 Feng Y P, et a. Sc Chna-Phys Mech Astron Octoer 13 Vo. 56 No. 1 Remar 3.1 From defntons of homogenzaton constants of â h, ˆ ê, t s easy to prove that these constants are consstent wth the defnton of the tradtona homogenzaton method. So the homogenzaton souton par u, Φ are unque consstent wth the tradtona homogenzaton method. Lemma 3.5 There exsts a unque souton u, Φ to the oundary ayer proem gven y eq. 5. 4 Man convergence theorems asymptotc error theorems In ths secton, we present some pror estmates reguaton resuts for oundary ayer proem L-order twoscae approxmate souton par ũ,l, Φ,L. For convenence, et n = 3 =Ω\ Ω. Frsty, we present a pror estmate resut for souton to the oundary ayer proem gven y eq. 5. Theorem 4.1 Let u, Φ e the souton to eq. 5. Assumng that a h ξ, e ξ ξ L satsfy the E 1, Pλ 1,λ condtons, f H L Ω, ρ H L Ω, u H L1 Ω Φ H L1 Ω, L 1, the foowng hods. u 1, Φ 1, c f L,Ω1 u L1/,Ω1 ρ L,Ω1 Φ L1/,Ω1, where c s a postve constant ndependent of,u, Φ, f,ρ u, Φ. Proof Settng H 1,Ω = {v H 1, v Ω = }. It s easy to show that u, Φ u, Φ H 1,Ω 3 H 1,Ω. The varatona formua of eq. 5 s to fnd u, Φ, such that u, Φ u, Φ H 1,Ω 3 H 1,Ω, va h h u Φ e vdx Φ φ e u x x φ dx = ν v a u L Φ L e ds ν φ Φ L e x u L ds f v ρφdx, v,φ H 1,Ω 3 H 1,Ω, where ν s the -th component of unt outer vector norma to the surface. By vrtue of the condtons E 1,, Pλ 1,λ, Korn s nequaty, Poncaré nequaty, trace theorem of Sooev space reguarty of souton of eq., there may e a unque souton to the aove varatona proem. We arrve at c 1 u u 1,Ω1 Φ Φ 1,Ω1 c 1 u u 1, Φ Φ 1,Ω 1 u u a h h u u dx Φ Φ Φ Φ dx x x Φ e Φ u x u dx Φ Φ e u x u dx fu u ρφ Φ dx u a h h u u Φ Φ Φ dx x x Φ e u u Φ e Φ u dx ν u u a u L e Φ L ds ν Φ Φ Φ L e x u L ds c f,ω1 ρ,ω1 Φ 1,Ω1 u 1,Ω1 Φ L 1,Ω u L 1,Ω u u 1,Ω1 Φ Φ 1,Ω1 c 3 f L,Ω1 ρ L,Ω1 Φ L1/,Ω1 u L1/,Ω1 u u 1,Ω1 Φ Φ 1,Ω1. Therefore, u u 1,Ω1 Φ Φ 1,Ω1 c 4 f L,Ω1 ρ L,Ω1 Φ L1/,Ω1 u L1/,Ω1. Tang nto consderaton the reguarty of souton to the proem defned y trace theorem of Sooev space eads to u 1,Ω1 Φ 1,Ω1 c f L,Ω1 ρ L,Ω1 Φ L1,Ω1 u L1,Ω1, we compete the proof of Theorem 4.1. Aso we have the foowng reguarty theorem for suffcenty smooth functons f, u, Φ ρ. Theorem 4. Let u, Φ e the wea souton to the oundary ayer proem defned y eq. 5. If a h, e,
Feng Y P, et a. Sc Chna-Phys Mech Astron Octoer 13 Vo. 56 No. 1 1851 C Ω, ξ a h, ξ, ξ e L Ω, f H L1 Ω, ρ H L1 Ω, u H L Ω Φ H L Ω, then u, Φ H 3 H u,ω1 Φ,Ω1 c 1 f L1,Ω1 ρ L1,Ω1 Φ L,Ω1 u L,Ω1, And a αl1 h h M α1 α L e αl1 α L H α1 α L 1 H α1 α e αl1 H α1 α L e L αl1, F = a αl hα L1 M α1 α L h e αl α L1 H α1 α L. where c 1 c,csapostve constant ndependent of. Now, we gve the convergence theorem for the twoscae method n sect., then derve the proof of ths resut. Theorem 4.3 Suppose that Ω s a ounded doman wth Lpschtz oundary, u, Φ s the souton to the proem gven y eq. 1, u, Φ s the souton to the proem gven y eq. 5, ũ,l, Φ,L s the asymptotc L-order souton defned y 6. If a h, e, C Ω, f H L Ω, ρ H L Ω, u H L1 ΩΦ H L1 Ω, then ũ,l ũ,l u 1,Ω Φ Φ,L 1,Ω c 1 L 1, L, 36 u 1,Ω Φ Φ,L 1,Ω c 1/, L = 1, 37 hod, where c 1 c are postve constants ndependent of. Proof Introduce the foowng two operators L 1 u, Φ = a h x hu e Φ, L u, Φ = Φ e x x u. If x Ω, then we have the foowng denttes whch hod n the sense of dstrutons, where L 1 u ũ,l, Φ Φ,L = L 1 u u L Φ L = L 1 F 1 L1 u m m x α1 x αl x αl1 L F L u m m x α1 x αl1 x αl L 1 F 1 L1 Φ x α1 x αl x αl1 L F L Φ x α1 x αl1 x αl O L1, 38 F 1 m = a αl1 hα L N α1 α L 1 hm a αl1 N α1 α L m a αl1 h h N α1 α L m e αl1 α L G α1 α L 1 m e αl1 G α1 α L m e αl1 G α1 α L m F m = a α L hα L1 N α1 α L hm e αl α L1 G α1 α L m, F 1 = a αl1 hα L M α1 α L 1 h a αl1 M α1 α L, where L u ũ,l, Φ Φ,L = L u u L =, Φ Φ L e u u L = L 1 C 1 m L C m L 1 C 1 L C C 1 m = L1 u m x α1 x αl x αl1 Φ Φ L x L u m x α1 x αl1 x αl L1 Φ x α1 x αl x αl1 L Φ O L1, 39 x α1 x αl1 x αl e αl1 hα L N α1 α L 1 hm e αl1 N α1 α L m e αl1 h h N α1 α L m αl1 α L G α1 α L 1 m αl1 G α1 α L m αl1 G α1 α L m C m = e α L hα L1 N α1 α L hm αl α L1 G α1 α L m,, C 1 = e αl1 hα L M α1 α L 1 h e αl1 M α1 α L e αl1 h h M α1 α L αl1 α L H α1 α L 1 αl1 H α1 α L αl1 H α1 α L C = e αl hα L1 M α1 α L h αl α L1 H α1 α L. If x, then t hods L, 1 u ũ,l, Φ Φ,L = L 1 u u, Φ Φ = f f =, L u ũ,l, Φ Φ,L = L u u, Φ Φ = ρ ρ =. Next, we consder the oundary condton for u ũ,l, Φ Φ,L. It s easy to see that u ũ,l =, Φ Φ,L = f x Ω, ν a h h u ũ,l Φ Φ,L e = ν a h h u u L e Φ Φ L, x Ω,
185 Feng Y P, et a. Sc Chna-Phys Mech Astron Octoer 13 Vo. 56 No. 1 Φ ν Φ,L e u ũ,l x Φ = ν Φ L e u u L x, x Ω. In summary, u ũ,l, Φ Φ,L s the wea souton to the foowng oundary vaue proem. a x u ũ,l e Φ Φ,L = L 1 F 1 m L F m L 1 F 1 L F L1 u m x α1 x αl x αl1 L u m x α1 x αl1 x αl L1 Φ x α1 x αl x αl1 L Φ x α1 x αl1 x αl O L1 = F, ξ, x, n Ω, Φ Φ,L e x x u ũ,l = Introduce the Sooev space H 1 Ω = {vx vx H1 Ω, vx Ω = }. Settng the test functon vx,φx = u ũ,l, Φ Φ,L H 1Ω3 H 1 Ω, usng Green formuas, one otans, u ũ,l a h h u ũ,l dx Ω Φ Φ,L Φ Φ,L dx Ω x x u ũ,l Φ Φ,L e dx Ω x h u ũ,l Φ Φ,L e h dx Ω x = Ω Fu ũ,l CΦ Φ,L dx, where F s an asseme of F m, F, D S α u m DS α Φ, m = 1,, 3; = 1, ; S = L 1, L C s an dentty dependent of C m, C, DS αu m D S αφ m = 1,, 3; = 1, ; S = L 1, L. By means of Poncaré nequaty, Korn s nequaty, coercvty postve defnteness assumptons of {a h } { },wehave L 1 C 1 m L C m L 1 C 1 L C L1 u m x α1 x αl x αl1 L u m x α1 x αl1 x αl L1 Φ x α1 x αl x αl1 L Φ x α1 x αl1 x αl O L1 = C, ξ, x, n Ω, a x u ũ,l e Φ Φ,L =, n, Φ Φ,L e x x u ũ,l =, n, ν a u ũ,l e Φ Φ,L = ν a u u L e Φ Φ L, x Ω, ν Φ Φ,L e x u ũ,l = ν Φ Φ L e x u u L, x Ω, Φ Φ,L =, u ũ,l =, on Ω. 4 u ũ,l 1,Ω Φ Φ,L 1,Ω c 1 L 1, L. Usng the technque from refs. 18 19, for L = 1, eq. 37 s vad. Therefore, we fnsh the proof of Theorem 4.3. 5 Concusons By ntroducng specfc coupng factors, one new nd of hgher order twoscae asymptotc expansons for the soutons of the dspacement u the eectrca potenta Φ are presented. From these expansons we derve the homogenzaton constants homogenzaton equatons of pezoeectrc effect. The twoscae asymptotc expansons can e dvded nto two parts. The frst part s composed of the homogenzaton souton u x, Φ x on goa Ω, whch shows the homogenzaton nformaton of pezoeectrc effect; the second part s produced y a seres of sma scae scaar-vaued, vector-vaued matrx-vaued soutons N α ξ, M α ξ, G α ξ, H α ξ, α = α 1,...,α, α = 1,...,n, = 1,...,, 1 defned on the asc confguratons of composte matera caused y the dstruton of dfferent compostons nsde asc ce, whch shows the oca nformaton of materas nterreaton etween the eectrca fed the mechanca fed. To otan the hgher order approxmaton soutons to the eectrca potenta the dspacement of composte matera, we redefne the coupng oundary ayer proems to match the coupng effect to suppy a modfcaton for edge effect.
Feng Y P, et a. Sc Chna-Phys Mech Astron Octoer 13 Vo. 56 No. 1 1853 We mproved the tradtona homogenzaton error form order O 1/ to asymptotc error O L 1 forl. There are many proems to e nvestgated n the further wor ased on the resuts. It woud e usefu to derve some numerca smuatons computatona agorthms for these systems. We emphasze that our detaed studes on the couped pezoeectrcty case can e used to dscuss other mutpe physca feds couped system such as photoeectrcty thermoeectrcty n quas-perodc compostes. Perhaps even more chaengng, more mportant couped ehavor appcatons woud e to predct the homogenzaton of nonnear stochastc couped systems. Ths wor was supported y the Natona Natura Scence Foundaton of Chna Grant Nos. 1814, 111613, 1117157, the Specazed Research Fund for the Doctora Program of Hgher Educaton of Chna Grant No. 144111. Ths wor was fnshed party durng vstng Unversty of Caforna, San Dego supported y Chna Schoarshp Counc from Juy 1 to Juy 13. The authors are gratefu to the anonymous referees for ther hepfu comments suggestons. 1 Teega J J, Bytner S. Pezoeectrcty wth poarzaton gradent: Homogenzaton. Mech Res Commun,, 9: 53 59 Ne S E C, Nshwa S, Kuch N. Desgn of pezoeectrc materas, pezoeectrc transducers usng topoogy optmzaton, Part II. Arch Comp Meth Eng, 1999, 6: 191 3 Ront K G, Venatesh T A. Eectromechanca response of porous pezoeectrc materas. Acta Mater, 6, 54: 463 478 4 Dunn M L, Taya M. Mcro-mechancs predctons of the effectve eectro-eastc modu of pezoeectrc compostes. Int J Sods Struct, 1993, 3: 161 175 5 Dunn M L, Taya M. An anayss of pezoeectrc composte materas contanng epsoda nhomogenetes. Proc Roy Soc London A, 1993, 443: 65 87 6 Zheng S, Du D X. An expct unversay appcae estmate for the effectve propertes of mut-phase compostes whch accounts for ncuson dstruton. J Mech Phys Sods, 1, 49: 765 788 7 Fang D N, Mao G Z. Expermenta study on eectro-magnetomechanca coupng ehavor of smart materas. J Mech Strength, 5, 7: 17 6 8 Jayachran K P, Guedes M, Heder C R. Stochastc optmzaton of ferroeectrc ceramcs for pezoeectrc appcatons. Struct Mutdsc Optm, 11, 44: 199 1 9 Nchoas Z. Stochastc modeng of mut-scae mut-physcs proems. Comput Methods App Mech Engrg, 8, 197: 3419 3419 1 Yamaguch M, Hashmoto K Y, Mata H. Fnte eement method anayss of dsperson characterstcs for 1-3 type pezoeectrc composte. Proc IEEE Utrasonc Symposum, 1987, 657 661 11 Baras S, Perre G, Banc F A. Perodc fnte eement formuaton for the desgn of - composte transducers. Proc IEEE Utrasonc Symposum, 1999, 957 96 1 W K, Cao W W. Fnte eement anayss of perodc rom - pezo-composte transducers wth fnte dmensons. IEEE Trans UFFC, 1997, 44: 1168 1171 13 Metzer A H, Tersten H F. IEEE Stard on Pezoeectrcty. New Yor: IEEE, 1988 14 Ernger A C. Theory of nonoca pezoeectrcty. J Math Phys, 1984, 5: 717 77 15 Yang J S, Batra R C. Conservaton aws n near pezoeectrcty. Eng Fract Mech, 5, 51: 141 147 16 Zhang F X. Modern Pezoeectrcty n Chnese. Beng: Scence Press, 1. 1 5 17 Jov V V, Kozov S M, Oen O A. Homogenzaton of Dfferenta Operators Integra Fuctonas. Bern: Sprnger, 1994. 1 18 Oen O A, Shamaev A S, Yosfan G A. Mathematca Proems n Eastcty Homogenzaton. Amsterdam: North Ho, 199. 5 149 19 Cao L. Mut-scae asymptotc expanson fnte eement methods for the mxed oundary vaue proems of second order eptc equaton n perforated domans. Numer Math, 6, 13: 11 45 Cu J Z, Shn T M, Wang Y L. The twoscae anayss method for the odes wth sma perodc confguratons. Struct Eng Mech, 1999, 76: 61 614 1 Feng Y P, Cu J Z. Mut-scae anayss for the structure of composte materas wth sma perodc confguraton under condton of couped thermo-eastcty. Acta Mech Sn, 3, 19: 585 59