ASYMPTOTIC HOMOGENIZATION MODELING OF MAGNETO-ELECTRIC SMART COMPOSITES

Transcription

1 THE 9 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS ASMPTOTIC HOMOGENIZATION MODELING OF MAGNETO-ELECTRIC SMART COMPOSITES D. A. Hadoz, A. L. Kaamarov 2*, S. Joh, A.V. Georgades, 3 Deparme of Mechaca Egeerg ad Maeras Scece ad Egeerg, Cyprus Uversy of Techoogy, Lemesos, Cyprus 2 Deparme of Mechaca Egeerg, Dahouse Uversy, Hafax, NS, Caada 3 Research U for Naosrucured Maeras Sysems, Cyprus Uversy of Techoogy, Lemesos, Cyprus * Correspodg auhor (Aex.Kaamarov@da.ca Keywords: Pezo-mageo-hermo-easc smar compose; Asympoc homogezao mehod; Effecve properes Absrac Comprehesve mcromechaca modes for he aayss of pezo-mageo-hermo-easc smar compose srucures wh orhoropc cosues are preseed. The frs asympoc homogezao mode s derved o he bass of he dyamc force ad herma baace as we as Maxwe s equaos. Subsequey geera reaos caed u ce probems are derved. They ca be used o deerme he effecve easc, pezoeecrc, pezomagec, herma expaso, deecrc permvy, magec permeaby, mageoeecrc, pyroeecrc ad pyromagec coeffces. The aer hree ses of coeffces are parcuary eresg he sese ha hey represe produc or cross-properes; hey are geeraed he macroscopc compose va he eraco of he dffere phases, bu may be abse from some of he cosues hemseves. The derved reaos perag o he u-ce probems ad he resua effecve coeffces are very geera ad hey are vad for ay 3D geomery of he u ce. Subsequey, he sae feaures of a secod quas-sac mode are oued. The modes deveoped are usraed by aayzg praccay mpora mageoeecrc amaes ad 3D ewor-reforced composes. 2 Iroduco Aded by rapd echoogca advacemes he fabrcao processg ad characerzao of sesors, acuaors ad ove maera sysems, he corporao of smar composes ad, more recey, smar aocomposes ew egeerg appcaos has spared a reewed eres deveopg ew mcromechaca modes for predcg her effecve properes a he desg sage. A speca cass of smar composes, hose composed of pezoeecrc ad pezomagec cosues, has recey araced aeo due o her eresg properes ad her sgfca poea for ew appcaos. Such maeras exhb he so-caed produc properes whch are mafesed he cosodaed compose bu o he dvdua cosues. Mos mpora amog hese produc properes are mageoeecrcy, pyroeecrcy ad pyromagesm. As Na e a [] show, hese properes ca be coveey preseed he form: Magec Mechaca Mageoeecrc Effec = Mechaca Eecrc Therma Mechaca Pyroeecrc Effec = Mechaca Eecrc Therma Mechaca Pyromagec Effec = Mechaca Magec Thus, appyg a eecrc fed o a pezoeecrcpezomagec compose geeraes mechaca sra he pezoeecrc phase. Ths deformao s he rasferred o he pezomagec phase whch, ur, geeraes a magec fed. Therefore, overa, a eecrc fed duces a magec fed ad vce-versa. Ths s mageoeecrcy. Smar pheomea udere pyroeecrcy ad pyromagesm. The creased eres such smar srucures ecessaes a eed furher deveopme of he

2 rgorous mcromechaca modes aowg he adequae predco of her effecve properes. Noeworhy amog he assocaed aayca modes are he wors of Harshe e a. [2], Huag e a [3], N e a [4], Bchur e a [5], Bravo-Casero e a [6], Hadoz e a [7-9] ad a few ohers. A mahemacay rgorous approach ha ca be effecvey apped for he modeg ad aayss of mageo-eecrc composes s he muscae asympoc homogezao mehod. May probems hermoeascy ad pezoeascy have bee soved usg hs echque, see, e.g., Kaamarov [0,]. The obecve of hs paper s o usrae he deveopme of a asympoc homogezao mode for he deermao of he effecve coeffces, cudg he produc properes, of 3D pezoeecrc-pezomagec composes made up of orhoropc cosues [7]. The wor w be usraed by meas of he praccay mpora hc amaes [8]. Dyamc Asympoc Homogezao Mode 2. Probem Formuao Fg. shows a geera 3D smar compose srucure represeg a homogeeous sod occupyg doma G wh boudary G ha coas a arge umber of perodcay arraged u ces wh characersc dmeso ε. 2 σ,, u,, x y x y = ρ ( 2 x ( x, y, D H ( x, y, = J ( x, y, (2 ( x, y, B E ( x, y, (3 ( x y ( x y T,, q,, C ρ (4 Here, σ ad u represe mechaca sresses ad dspacemes, H ad B are, respecvey, he magec duco ad magec fed, D s he eecrc dspaceme, J s he free coduco curre, C s he specfc hea capacy, T s he emperaure fed, ad q s he hea fux vecor. To compee he aayss, Eqs. ( - (4 mus be compmeed wh he approprae cosuve equaos show beow: u σ ( x, y, = C ( x, y, e E ( x, y, (5 x - Q y H x,y, -θ y T x,y, u D,, e,, ε E,, ( x y = ( x y ( x y ( x y ξ ( x y λ H,, T,, (6 x 3 u B,, Q,, E,, ( x y = ( x y λ ( x y H ( x, y, T ( x, y, µ η (7 Smar perodc compose Fg.. 3D perodc smar compose The macroscopc behavour of hs smar compose srucure s modeed by he foowg boudary vaue probems, descrbg dyamc force ad herma baace ad Maxwe s equaos. x x 2 ( x y ( x y J,, = Σ Ε,, (8 T q,, -V,, ( x y = ( x y (9 where E s he eecrc fed vecor, C, e, Q, ad θ are he esors of he easc, pezoeecrc, pezomagec ad herma expaso coeffces respecvey. As we, ε, λ, µ, ξ, η, Σ ad K represe, respecvey, he deecrc permvy, he mageoeecrc, he magec permeaby, he pyroeecrc, he pyromagec, he eecrca coducvy ad herma coducvy esors. The

3 ASMPTOTIC HOMOGENIZATION MODELING OF MAGNETOELECTRIC SMART COMPOSITES depedece of he maera coeffces ad he fed varabes o y = x /ε s a refeco of he src perodcy of he maera properes (wh perod ε; he fed varabes however, are, auray characerzed by boh a perodc ad a o-perodc compoes, see [7-]. Varabes x are ofe referred o as he sow or macroscopc varabes, whe her couerpars, y, are referred o as he fas or mcroscopc varabes. 2.2 Asympoc Homogezao Modeg ad U Ce Probems Aayss begs by asympocay expadg he depede fed varabes o powers of ε. For exampe, he mechaca sress s expressed as: ( 2 ( x y, ( x y, ( x y, σ, = σ, εσ, O ε (0 Subsequey, subsug expressos such as (0 o he cosuve equaos (5-(9 ad comparg erms wh he same powers of ε yeds expressos for he -h erm of each asympoc expaso. For exampe, ( ( ( u u ( ( ( σ = C ee QH θt, ( = 0,, 2, These asympoc expasos are he subsued o he goverg equaos (-(4 o oba, afer comparg e powers of ε, a se of dfferea equaos for he -h erm of each expaso. For exampe, he erms of he magec fed expaso, sasfy he foowg dfferea equaos [7]: ( ( ( H H D ( Ο( ε : ε e = e J e, = 0,, 2, (2 I woud aso o be amss o meo ha he process, urs ou ha he frs erms he asympoc expasos for mechaca dspaceme ad emperaure are depede of y ad he frs erms he eecrc ad magec fed expasos are rroaoa ad ca hus be expressed as he grades of scaar feds: E H ( 0 ( 0 (, = ( x, ( x, y, E ~ ϕ x (3a 0 (,, H ~ ψ x y x, = x, (3b ( 0 ( Combg he asympoc expasos ( (as we as oher smar oes wh he dfferea equaos (2 ad he expressos (3a ad (3b resus a seres of four equaos, caed goverg reaos whch ca be used o deerme he four, ye uow fucos, ( ( u ( x, y,, T ( x, y,, ϕ ( x,y, ad ψ ( x,y, from whch, a he fed varabes ca be deermed. Oe of hese goverg reaos s gve beow: ( u φ ψ φ e ε λ Σ = e u ε λ E Η ξ Σ T E (4 I ca be ready observed from he Eq. (4 as we as he remag hree goverg reaos, ha, afer ag he Lapace Trasform (wh respec o me, he rgh-had sde coas erms whch are producs of a fuco of y ad a fuco of x. Ths separao of varabes eabes us o wre dow he souo of he four goverg reaos as foows: û ( x, û x, y, N y, M E ( x, (5 ( = N, H, M, T, ( x ( x ( x u, φ x, y, = Λ y, Z y, E x, Γ, H,, T, ( x ( x ( x û, ψ x, y, = Α y, Β y, E x, H ( x, T ( x, Ξ Π (6 (7 ( x y = T ( x, β ( y (8 ( T,,

4 Here, quaes wh a ha, for exampe ( x, y,, represe he Lapace rasform of û ( u ( (, y, x wh beg he Lapace parameer. We observe ha Eqs. (5-(8 coa 3 uow fucos, N, M, N, M, Λ, Ẑ, Γ,, Α, Β, Ξ, Π,ad β (whch s o a fuco of he Lapace varabe. Ther souos are obaed by bac-subsuo of Eqs. (5-(8 o he Lapace rasform of he goverg reaos o oba a se of 3 dfferea equaos caed u ce probems ha have he foowg form: τ ( y, C τ ; e y = χ ; θ χ, Q = d = ( y, e ν ; λ ν ( y, ξ ; ς ( y, Q ς ( y, λ ; ω (, µ ω (9 (20 (2 (22 y (23 ( y, ξ ; V V (, y (24 d y ε Σ (25 The quaes o he ef-had sdes of Eqs. (9- (25 deped o he maera parameers of he smar compose as we as oher fucos such as τ. These aer fucos are ur defed erms of he so-caed oca fucos such as N ha appear Eqs. (5-(8. ready oba: For exampe, oe ca m = m N, Λ, τ y, C y e Q Â, 2.3 Effecve Coeffces (26 Foowg he dervao of he u-ce probems, we are ow a poso o cacuae he frs erms he asympoc expasos of he mechaca sress, eecrc dspaceme, magec fed, curre desy ad hea fux. To hs ed, from he Lapace rasform of Eqs. (, (3a, (3b ad he remag asympoc fed expasos, couco wh Eqs. (5-(8, we oba he foowg expressos: û ( x y σ,, = C τ e τ E { } { } Q χ H θ χ T û x y = x D,, e d F ε λ ν ξ ν O T d G E I H û B,, Q E ( x y = { ς } { λ ς} { } H { } µ ω η ω J,, = Σ G E q ( x y { κ } T û F I H O T (, y, V V (27 (28 (29 (30 T x { } (3 We w ow appy he homogezao procedure, whch eas egrao over he voume of he u ce A ~ = A dv (32

5 ASMPTOTIC HOMOGENIZATION MODELING OF MAGNETOELECTRIC SMART COMPOSITES o oba he effecve Lapace rasform for sress, eecrc dspaceme, magec fed, free coduco curre ad hea fux. Comparg he resug expressos wh he orga cosuve equaos Eqs. (5-(9 yeds he effecve coeffces. Therefore, from Eq. (27 we oba he effecve easc coeffces, C (, pezoeecrc, ê, pezomagec, Q (, ad herma θ. They are gve by: expaso coeffces, Ĉ ~ ~ ê Q~ ~ θ = C τ = e τ (33a (33b = Q χ = θ χ (33c (33d Appyg a smar procedure o Eq. (28 we oba he effecve pezoeecrc, deecrc permvy, mageoeecrc, ad pyroeecrc coeffces. They ê, λ, ad are, respecvey, ( ε (, ξ ad are gve beow Eqs. (34a-(34d. The aer wo represe wo of he aforemeoed produc properes. ê~ = e ( y d F dv (34a ε = ε d ( y, G ( y, dv ~ λ ~ ξ = λ ν Î (34b y dv (34c y dv (34d = ξ ν Ô Smary, appyg he homogezao procedure o Eq. (29 ad comparg wh he correspodg cosuve equao (7, we oba he effecve pezomagec, Q, mageoeecrc, λ (, magec permeaby, µ (, ad pyromagec, η ( coeffces. They are gve Eqs. (35a- (35d. Q ~ λ = Q ς = λ ς y y, dv (35a (35b µ ~ = µ ( y ω dv (35c η~ = η ω (35d Homogezao of Eq. (30 yeds he effecve eecrca coducvy, Σ ad hree ew produc properes, F (, Ĩ (, Õ (. These aer properes reae he free coduco curre o, respecvey, he mechaca deformao, he magec fed, ad he emperaure. They are abse from he orga cosuve equao (8. ~ Σ F~ ~ I Ô ~ = Σ Ĝ = F (36a y, dv (36b y, dv (36c = { I } y, dv (36d = { O } Fay, he effecve herma coducvy s obaed from Eq. (3 ad s gve by:

6 V = V y V dv (37 Before cosg hs Seco s worhwhe o oe ha boh he u-ce probems ad he effecve coeffces are soved erey he doma of he u ce. As such, hey are depede oy o he maera ad geomerca mae-up of he u ce ad oce deermed hey ca be apped o ay boudary-vaue probem perag o he gve geomery. 3 Exampes ad Mode Vadao 3. Exampe The deveoped mode s apped o he praccay mpora exampes of he smar compose amaes. The frs exampe peras o he amae show he Fg. 2. x x 3 boh he axs of symmery ad he pog ad magezao dreco. Le us ow proceed o he compuao of he effecve coeffces of hs srucure. Le us beg wh he effecve easc coeffces. As Fg. 2 shows, for he srucure uder cosderao, he oy varao maera properes aes pace he y dreco. Cosequey, he frs u ceprobem Eq. (9 ca be reduced o: ( y, C ( y m m (38a τ m where τ s defed Eq. (26. Iegrag (38a yeds: m ( (38b τ y, C y O m m m where O ( are he cosas of egrao. To cacuae he effecve easc coeffce C we assume = = m =. Thus: ( (38c τ y, C y O As we, Eq. (26 reduces o: Pog ad Magezao x 3 dreco x 2 y U Ce ( y N y, Λ y, τ y, C y e y m ( = m ( ( Q Â y, (38d pezomagec pezoeecrc Fg. 2. Smar compose amae ad s u ce We w assume ha he amae cosss of aerag amae of pezoeecrc (Barum Taae ad pezomagec (Coba Ferre maera. The voume fraco of he pezoeecrc maera s ϑ ad ha of he pezomagec s (- ϑ. The easc, pezoeecrc, ec. properes of hese maeras ca be foud [8]. The maeras are rasversey soropc wh he x 3 dreco beg y 2 Referece o he maera properes [8] w revea ha for he rasversey soropc BaTO 3 ad CoFe 2 O 4 (aroud x 3 axs e ad Q are boh zero. Furhermore, C 2 ad C 3 are boh zero. Cosequey, Eq. (38d reduces o: N ( y, ( = (38e τ y, C y From Eqs. (38c ad (38e we oba: y C ( y N y, O (38f

7 ASMPTOTIC HOMOGENIZATION MODELING OF MAGNETOELECTRIC SMART COMPOSITES If we ow egrae Eq. (38f whe cosderg a he same me he perodcy of N we arrve a: ϑ ϑ 0 O C C e m (38g where he superscrps (m ad (e refer o he pezomagec ad pezoeecrc cosue respecvey. To cacuae C we mae use of Eq. (33a whch, for he probem a had, reduces o: C ( = C ( y τ ( y, dy (39a 0 Thus, Ĉ ~ C C = (39b ( m ( e ϑc ( ϑ C The as sep he procedure s o ae he verse Lapace rasform of Eq. (39b o yed: C ~ C C ( δ m e ϑc ϑ C = (39c ( where δ( s he Drac fuco. Iegrag hs expresso over he ere specrum of me yeds: = = C C C = C ( d = ( m e = = ϑ C ( ϑ C δ C C = ( m ( e ϑ C ( ϑ C (39d C ~ ~ e Q ~ 5 µ µ 2 µ 3 ( = κe κ2e κ3e κ4δ( µ 4 µ 5 µ 6 ( = κ5e κ6e κ7e κ8δ( µ 7 µ 8 µ 9 ( = κ e κ e κ e κ δ( (39d where κ, =,2,...,2 ad µ, =,2,...,9 are ow fucos whch deped o he aforemeoed maera parameers of he pezoeecrc ad he pezomagec cosues of he u ce. Overa, s eresg o oe ha eve hough he orga cosues boh exhb rasverse soropy wh respec o x 3 axs, he effecve macroscopc compose exhbs cass mm2 symmery as show beow [8]: [ C ] C ~ C ~ C ~ C ~ C ~ C ~ 2 C ~ C ~ C ~ ( = C ~ ( C ~ ( C ~ 0 66 The deermao of he effecve pezoeecrc, pezomagec ad herma expaso coeffces foows from he secod u ce probem Eq. (9 ad he u ce probems Eq. (20 ag o accou he defos (33b-(33d. Foowg he same mehodoogy as he oe empoyed for he deermao of he effecve easc coeffces, yeds ypca expressos for ẽ 3, Q ad 32 θ : 22 ( m ( e m e ϑ C e3 ϑ e3 C 3 = ( m ( e ϑ C ( ϑ C e (4a The compuao of he remag easc coeffces proceeds much he same sraghforward fasho wh he oabe excepo of he coeffce C. For hs coeffce, he procedure, 55 hough sraghforward, s agebracay edous. Deas ca be foud [8]. I he process, we oba o oy C bu aso he effecve pezoeecrc 55 coeffce ẽ 5 ad he pezomagec coeffce Q. 5 The resug expressos are oo eghy o be reproduced here bu have he geera form [8]: ( e ( m Q3 Q ϑ ϑ 3 ( m ( m ϑ C C2 Q3 ϑ C2 C Q3 C C ( m ( e 32 = ϑ 2 ( ϑ 2 ( m ( e ϑ C ( ϑ C Q C C C C x ϑ C Q ( ϑ C Q ( m ( e 3 3 C C (4b

8 ( e m 22 θ22 ϑ θ ϑ ( e ( m ( m ( e ϑ C2θ e3 ϑ C2 θ e3 22 = e3 e3 ( 2 e m m e ϑ θ e ϑ θ e 3 3 ( m ( e e 3 e3 ϑ e3 ϑ e3 θ (4c The remag effecve pezoeecrc, herma expaso ad pezomagec coeffces w be preseed graphcay he seque. The cacuao of he effecve deecrc permvy, magec permeaby, ad eecrca ad herma coducvy proceeds much he same maer by sarg from he approprae u ce probems. Some of hese coeffces w be gve graphcay he seque. Of parcuar eres hs wor are he effecve produc properes. The mehodoogy for obag hem s of course he same as above ad cosders he approprae u ce probems ad he expressos Eqs. (34c, 34(d, 35(c, 35(d, (36b- 36d. For exampe, λ ( e ( m ( m ( e ϑ e 3 Q3 C ϑ e3 Q3 C C C x ( m ( e e ϑ 3 C ϑ e3 C ( m ( e C ( C ϑ ϑ ( m ( e ϑ Q3 C ϑ Q3 C 33 C C (42a = α σ α σ2 α σ3 α δ = α σ4 α σ5 α σ6 α δ ( σ7 e σ8 e σ9 e ( F e e e I e e e λ = α α α α δ (42c where α, =,2,...,2 ad σ, =,2,...,9 are ow fucos whch deped o C 55, e 5, Q 5, ε ad µ parameers of he pezoeecrc ad he pezomagec cosues of he u ce. The quaes Eq. (42c woud he have o be egraed he maer of Eq. (39d. The resug expressos are oo eghy o be reproduced here. Fg. 3 shows he varao of he effecve easc properes of he compose vs. he voume fraco of Barum Taae. I s see ha mos of he effecve coeffces o-eary decrease wh a crease he voume fraco of he pezoeecrc phase because he correspodg properes of hs cosue are ower ha hose of s couerpar. Effecve Easc Properes x 0 2 GPa x Voume Fraco of BaTO3 Fg. 3. Effecve easc coeffces vs. voume fraco of BaTO 3 c c 2 c 3 c 22 c 23 c 33 c 44 c 66 ( e m m e ϑ e3θ C ϑ e3 θ C ( e ( m C C ( m ( e ϑ e 3 C ϑ e3 C 3 ξ 3 = x C C ( m ( e ϑ θ C ϑ θ C ( m ( e ϑ C ( ϑ C (42b (C/m 2 Effecve Pezoeecrc Propery Fg. 4. Effecve pezoeecrc coeffces vs. voume fraco of BaTO 3 ẽ 3 ẽ 32 ẽ ẽ ẽ Voume Fraco of BaTO3

9 ASMPTOTIC HOMOGENIZATION MODELING OF MAGNETOELECTRIC SMART COMPOSITES (0-9 Ns/VC Effecve MageoEecrc Coeffce (0-5 C/m 2 K Effecve Pyroeecrc Coeffce Fgs. 4, 5 ad 6 show, respecvey, varaos of he effecve pezoeecrc, mageoeecrc ad pyroeecrc coeffces vs. he voume fraco BaTO 3. As expeced, creasg he voume fraco of he pezoeecrc ayer resus a correspodg crease ( he absoue vaue sese he vaues of he effecve pezoeecrc coeffces. Fgs. 5 ad 6 usrae he srogy o-ear varao of he effecve mageoeecrc ad pyroeecrc produc properes wh he voume fraco of Barum Taae. For he parcuar exampe uder cosderao, hese produc properes maxmze hemseves a a Barum Taae voume fraco of aroud Voume Fraco of BaTO x 0-9 Fg. 5. Effecve mageoeecrc coeffces vs. voume fraco of BaTO 3 0 x Voume Fraco of BaTO3 ~ Fg. 6. Effecve pyroeecrc coeffce ξ 3 vs. voume fraco of BaTO 3 The pracca mporace of he prese wor es he fac ha he mode ca be used o aor he λ λ 33 effecve properes of a parcuar amae by chagg cera geomerc ad/or maera parameers of he u ce. I s hus of sgfca pracca mporace o egeerg desg. 3.2 Quassac Mode Comparso of Modes The mcromechaca mode preseed hs paper was derved o he bass of dyamc force baace ad he me-varyg form of Maxwe s equaos. I he absece of era forces ad free coduco curres s possbe o derve a smar mode based o sac equbrum ad he quas-sac approxmao of Maxwe s equaos [7]. Such a mode may be referred o as he quas-sac mode ad he pere goverg equaos have he form: x σ x, ε x = f o G wh u x, = 0 o G ε x D x, ε x = 0 o G wh D x, = 0 o G ε x B x, ε x = 0 o G wh B x, = 0 o G ε (43 Here, f represe body forces. Furhermore, he rroaoa eecrc ad magec feds may be wre dow as grades of eecrc ad magec poeas [7]: x φ x ψ E x,, H, ε x ε (44 Foowg he mehodoogy of Secos 2.2 ad 2.3 s possbe o derve a ew se of u ce probems ad dffere expressos for he effecve coeffces [7]. For exampe, wo of hese u ce probems are gve beow: τ C ; ƛ e Here, he fucos τ ad (45 ƛ y are gve by:

10 Nm Cm e Q R τ y = y y ƛ C L m = m e Q P ψ y y Γ (46a (46b For he sae of coveece we w assume ha he cosues are he same as Exampe. For Exampe 2 we w appy he quas-sac mode. As Fg. 7 shows, for he srucure uder cosderao, he oy varao maera properes aes pace y 3 dreco. Cosequey, he frs u ce-probem Eq. (45 reduces o: (47a dτ y dc y dy dy 3 3 The remag u ce probems ad assocaed defos ca be foud [7]. Oe observes ha he u-ce probems perag o he quas-sac mode are o fucos of he Lapace varabe sce he orga goverg ad cosuve equaos are o fucos of me. If we he appy he obaed resus o he amae cosdered Exampe, we w see ha he wo modes coform exacy wh oe aoher excep he case of sx effecve coeffces C ~, 55 ẽ, 5 Q, 5 ~ ε, ~µ ad ~ λ. No surprsgy, hese are he effecve coeffces ha coa he expoea erms he dyamc mode. The resus of he quas-sac mode aso agree wh he wors of oher researchers, see [6]. 3.3 Exampe 2 We w ow cosder a dffere amae whch he pog ad magezao drecos are perpedcuar o he erphase, see Fg. 7. x 3 x 2 Iegrag hs expresso wh respec o y 3 yeds: τ ( y C ( y O (47b where O3 are cosas of egrao. Leg = ad, = 3 resus : τ 3 ( y C ( y O 3 (47c Expresso (46a ow becomes: ( y ( y 3 3 dnm y3 dl y3 3 ( y3 C3m3 ( y3 e33 ( y3 dy3 dy3 τ = Q 33 3 dr 3 3 (47d For he rasversey soropc cosues uder cosderao, e 33, Q 33, C 323 ad C333 vash [8]. Thus, Eq. (47d reduces o: 3 3 dn ( y3 τ ( y = C ( y (47e dy 3 x y 3 Combao of Eqs. (47d ad (47e resus : 3 C55 C55 τ 3 ( y3 C55 ( y3 (47f ( m ( e ϑ C55 ( ϑ C55 Pog ad Magezao x 3 dreco pezomagec pezoeecrc Fg. 7. Smar compose amae (perpedcuar coecvy ad s u ce y Fay, appyg Eq. (33a o Eq. (47f yeds he effecve C 55 coeffce: C55 C55 C55 = ( m ( e ϑ C55 ( ϑ C55 (47g

11 ASMPTOTIC HOMOGENIZATION MODELING OF MAGNETOELECTRIC SMART COMPOSITES The remag effecve coeffces ca be obaed he smar maer. The resus of he quas-sac mode geera ad hs exampe parcuar coform o hose obaed by Bravo-Casero e a [6], who used he perodc ufodg homogezao echque. Before devg o he ex exampe, shoud be meoed here ha he effecve properes cacuaed are assocaed wh a hc amae (sce he uderyg modes are 3D modes. The prepoderace of uses of compose maeras oday however, s he form of h amaed paes ad shes. For exampe, he effecve easc coeffces shoud be cassfed o he famar exesoa, bedg ad coupg easc coeffces, (smar cosderaos appy o oher effecve coeffces. Cosequey, f oe s eresed aayzg h amaes, aoher paper of he auhors becomes pere, see Hadoz e a [9]. he sff reforcemes are embedded a sof marx. The u ce probem assocaed wh hs srucure s gve by he frs expresso Eq. (45. τ y are defed Eq. (46a The oca fucos ad, afer omg exra erms, are: m N τ = Cm ( y (48 Brefy, he procedure voved aayzg hs probem eas he cosderao of a smpe uce wh oy a sge arbrary oreed reforceme, performg a coordae rasformao whch reores oe axs o cocde wh he reforceme (a procedure whch reduces a herey 3D probem o a 2D oe ad fay obag he effecve easc coeffces of mufber u ces va superposo, see Hassa e a [2]. 3.4 Exampe 3 Ths exampe deas wh he mcromechaca aayss ad deermao of he effecve easc coeffces of ewor reforced composes such as he srucure show Fg. 8, see [2]. Effecve easc coeffce (MPa 3 E ~, FEM, Reforcemes ad Marx E ~ 3, FEM, Reforcemes oy E ~ 3, AHM Toa reforceme voume fraco Fg. 9. Varao of he effecve sffess moduus, E ~ 3, for he srucure of Fg. 8 [2] Fg. 8. 3D Grd-reforced compose srucure wh reforcemes arraged a rhombc fasho [2]. For he purposes of hs exampe, e us assume ha we are oy eresed deermg he effecve easc coeffces of he ewor srucure whch The resus for he effecve oug s Moduus E ~ 3 are show Fg. 9 aogsde correspodg vaues obaed va he Fe Eeme Mehod (FEM. Two FEM pos are show; oe egecs he (sof marx properes ad he oher s a compee FEM mode wh boh he marx ad he reforceme properes ae o cosderao. I s ready observed ha he resus from asympoc homogezao mode coform que we o her couerpars from he fe eeme echque.

12 Cocusos Two comprehesve mcromechaca modes for he aayss of pezo-mageo-hermo-easc smar compose srucures are deveoped ad apped o exampes of pracca mporace. The frs mode s based o dyamc equbrum, Maxwe s equaos ad herma baace whe he secod mode empoys sac force baace ad he quas-sac approxmao of Maxwe s equaos. Boh modes derve geera reaos caed u ce probems. They ca be used o deerme he effecve easc, pezoeecrc, pezomagec, herma expaso, deecrc permvy, magec permeaby, mageoeecrc, pyroeecrc ad pyromagec coeffces. The aer hree ses of coeffces are parcuary eresg he sese ha hey represe he so-caed produc properes. The resus of he wo modes ad he use of he effecve coeffces are usraed by meas of hree praccay mpora exampes. The frs wo exampes pera o amaes cossg of rasversey soropc pezoeecrc ad pezomagec cosues. I s observed ha hs sace he resus from he wo modes coform very we o oe aoher wh he excepo of sx effecve coeffces; C ~ 55, ~ e55, Q ~ 55, ~ ε, ~µ ad ~ λ. As we, he resus of he quas-sac mode agree wh he wors of oher researchers. The hrd exampe peraed o he deermao of he effecve easc coeffces of 3D ewor reforced composes wh a perodc grd of sff reforcemes embedded a sof marx. I s ready see ha he resus from he asympoc homogezao modes coform que we o her couerpars from he fe eeme echque. Acowedgemes The auhors woud e o acowedge he faca suppor of he Cyprus Uversy of Techoogy ( s, 3 rd ad 4 h auhors, he Research U for Naosrucured Maeras Sysems (3 rd ad 4 h auhors ad he NSERC - Naura Sceces ad Egeerg Research Couc of Caada (2 d auhor. Refereces [] Na C-W, Bchur MI, Dog S, Vehad D, Srvasa G. Muferroc mageoeecrc composes: Hsorca perspecve, saus, ad fuure drecos. J. App. Phys 2008; 030( - 030(35. [2] Harshé G, Doughery JP, Newham RE. Theoreca modeg of muayer mageoeecrc composes. I. J. App. Eecromag. Maer. 993; 4: [3] Huag JH, Kuo W-S. The aayss of pezoeecrc/pezomagec compose maeras coag epsoda cusos. J. App. Phys. 996; 8(3: [4] N, Prya S, Khachaurya AG. Modeg of mageoeecrc effec poycrysae muferroc amaes fueced by he oreaos of apped eecrc/magec feds. J App Phys 2009; 05: 08394(-08394(4. [5] Bchur MI, Perov VN, Aver SV, Lvers E. Prese saus of heoreca modeg he mageoeecrc effec mageosrcvepezoeecrc aosrucures. Par I: Low frequecy eecromechaca resoace rages. J. App. Phys. 200; 07(5: ( (. [6] Bravo-Casero J, Rodrgues-Ramos R, Mechour H, Oero J, Saba FJ. Homogezao of mageoeecro-easc muamaed maeras. Q J Mechacs App Mah 2008; 6(3: [7] Hadoz AD, Georgades AV, Kaamarov AL, Joh S. Mcromechaca modeg of pezomageo-hermo-easc compose srucures: Par I Theory. Europea Joura of Mechacs 203; 39: [8] Hadoz AD, Georgades AV, Kaamarov AL, Joh S. Mcromechaca modeg of pezomageo-hermo-easc compose srucures: Par II Appcaos. Europea Joura of Mechacs 203; 39: [9] Hadoz DA, Georgades AV, Kaamarov AL. Dyamc modeg ad deermao of effecve properes of smar compose paes wh rapdy varyg hcess. Ieraoa Joura of Egeerg Scece 202; 56, [0] Kaamarov AL. Compose ad Reforced Eemes of Cosruco. 992; (Wey, New or. [] Kaamarov AL, Kopaov AG. Aayss, Desg ad Opmzao of Compose Srucures, 997; (Wey, New or. [2] Hassa EM, Georgades AV, Kaamarov AL. Asympoc homogezao ad fe eeme modeg of 3D grd-reforced orhoropc compose srucures. Ieraoa Joura of Egeerg Scece 20; 49(7: