EFFICIENT HIGHER ORDER ZIG-ZAG THEORY FOR COUPLED MAGNETO-ELECTRO-ELASTIC COMPOSITE LAMINATES

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1 8 H INERNAIONAL CONFERENCE ON COMPOSIE MAERIALS EFFICIEN HIGHER ORDER ZIG-ZAG HEORY FOR COUPLED MAGNEO-ELECRO-ELASIC COMPOSIE LAMINAES J. Lee, J.-S. Kim, M. Co * Scool o Mecaical ad Aerospace Egieerig, Seoul Natioal Uiversity, Seoul, Korea, Departmet o Itelliget Mecaical Egieerig, Kumo Natioal Istitute o ecology, Seoul, Korea Divisio o WCU Multiscale Mecaical Desig, Scool o Mecaical ad Aerospace Egieerig, Seoul Natioal Uiversity, Seoul, Korea. * mco@su.ac.r Keywords: Zigzag plate teory, Piezoelectric, Piezomagetic, Mageto-electro-elastic lamiates. Itroductio Multilayered structures made o piezoelectric ad piezomagetic materials ave bee used widely due to teir special properties o covertig te mecaical eergy ito electrical or magetic eergy ad vice versa [-]. For te piezoelectric materials, various studies ave bee carried out to aalyze ad to desig suc multilayered smart structures. Furtermore, or te accurate predictio o static ad dyamic beaviors, various coupled termo-electroelastic aalysis cotaiig termal eects tat are sigiicat i multipysics problems ave bee carried out. Elasticity solutio as bee proposed [4] ad iger-order zigzag models ave bee reported [,6]. I te same cotext, te aalysis o mageto-electroelastic (MEE) materials as bee icreasigly demaded recetly due to teir uique caracteristics. For te aalysis o te multilayered rectagular MEE plates, Pa [] obtaied te aalytical solutio uder te static siusoidal load based o te quasi-stro ormalism ad te propagator matrix metod. Moreover, Pa ad Heyliger [] solved te cylidrical bedig problem o MEE plates wit simply-supported edge coditio. e vibratio aalysis o MEE plates as also bee carried out usig a layerwise-type approximatio by Ramirez et al. [7]., I additio, Aigeri et al. [8] studied te ree vibratio beavior o MEE beam based o te membrae-type iite elemet model. However, eve toug te previous aalyses as bee reported or egieerig applicatios, oe o te major drawbacs o tem is tat te umber o uows is depedet upo te umber o layers, wic meas tose models are ot eiciet ad tus tey ave limitatios to be applied to te large-scale, sesig ad actuatig problem (i.e., dyamic aalysis o te multilayered, coupled MEE plates). us, more eiciet teory wic also cotais te accuracy are required. Amog te various studies o te multilayered plate structures, a eiciet iger order plate teory (EHOP) proposed by Co ad Parmerter [9] is te best perormer i displacemetbased zigzag teories [] ad recommeded or te aalysis o te MEE plates sice umerous studies ave bee veriied te accuracy ad eiciecy o te EHOP by aalyzig te ully coupled piezoelectric composite plates [7] as well as te covetioal composite lamiates. is teory reduces te ow variables usig te top/bottom boudary coditios ad te trasverse sear stress cotiuity coditios. I tis study, te multilayered MEE plates are cosidered to carry out te ully coupled magetoelectro-elastic aalysis built upo te EHOP. e displacemet ield, electric potetial ad magetic potetial are assumed as a tird order zigzag uctios. e umber o uows is reduced eectively by applyig te top/bottom coditios ad trasverse directio lux cotiuity coditios. For te practical usage o te preset metod, iite elemet discretizatio based o te beam-type model is applied. o ivestigate te dieret resposes o te elastic, electric ad

2 magetic quatities, various layups ad dieret loadigs are cosidered by comparig tem wit te results reported i te literatures. Formulatio A tree-dimesioal multilayered mageto-electroelastic plate is sow i Fig.. e costitutive equatios or a liear, aisotropic ad magetoelectro-elastic solid ca be writte as [-] C e E q H ij ijl l ij ij D e E d H B q d E H i il l i i i il l i i () were σ ij, D i ad B i are te stress, electric displacemet ad magetic iductio, respectively. ε l, E ad H are te strai, electric ield ad magetic ield, respectively. C ijl, γ i ad μ i are te elastic, dielectric ad magetic permeability coeiciets, respectively. e ij, q ij ad d i are te piezoelectric, piezomagetic ad magetoelectric coeiciets, respectively. e piezoelectric ad magetostrictive layers are made o BaiO ad CoFe O 4, respectively. Correspodig equilibrium equatios ca be writte as, D, B () ij, j i, i i, i e strai-displacemet, electric potetial ad magetic potetial relatiosip ca be expressed as ollows: ij ui, j uj, i, Ei, i, Hi, i () Fig.. Geometry ad coordiates o magetoelectro-elastic plate. I tis study, a ully coupled iger-order zigzag teory is proposed or eiciet modelig. e displacemet ield is assumed by superimposig zigzag liear ield to te globally varyig ield as ollows: (, ) ( ) ( ) ( ) N (, ) ( ) ( ) ( ) N s r ( x)( z z) ( z z) u x z u x x z x z ( x) z S ( x)( z z ) ( z z ) u x z w x r x z r x z (4) were (z-z ) is a Heaviside uit step uctio. u ad w represet te i-plae displacemet ad te out-o-plae displacemet rom te reerece plae, respectively. χ are te rotatio o te ormal about x axis. For te i-plae displacemet, a cubic varyig ield is applied. Wereas, or te out-oplae displacemet, a quadratic ield is assumed i order to express te pysical beavior o te displacemet ield accurately. I Eq. (4), te beamtype model wic cotais x, z ad o y axis is cosidered or te simplicity o te problem. Similar to te costructio o te displacemet ield, te electric ad magetic potetial are obtaied by superimposig liear zigzag ield oto te global cubic smoot ield as ollows: ( xz, ) ( x) ( xz ) ( xz ) N s ( x) z ( x)( z z ) ( z z ) ( xz, ) ( x) ( xz ) ( xz ) N s ( x) z ( x)( z z ) ( z z ) () (6) e uow variables i Eqs. (4-6) ca be reduced by applyig boudary coditios o top/bottom surace ad itroducig te cotiuity coditios o trasverse stresses, trasverse electric ad magetic displacemet betwee eac layer. e top/bottom boudary coditios ca be expressed as

3 EFFICIEN HIGHER ORDER ZIG-ZAG HEORY FOR COUPLED MAGNEO-ELECRO-ELASIC COMPOSIE LAMINAES z z () ( ) p, p z z () ( ), z z () ( ), z z (7) were is a ticess o te multilayered plate. For te simplicity, egieerig stress ad strai otatios are applied. e cotiuity coditios are applied as ollows:, zz zz zz zz m m m m D D, B B zz zz zz zz m m m m (8) It must be oted tat te stress ad displacemet ields wic obtaied rom Eqs. (-8) are statically admissible stress ield ad iematically admissible displacemet ield, respectively, wic idicates te ields satisy proper ticess directio static ad iematic coditios. e equatios ca oly be solved via te eigevalue problem usig te expoetial uctios. us, i tis study, by igorig te i-plae strai cotributio to te trasverse stress, trasverse electric displacemet ad trasverse magetic iductio i te costitutive equatios, oe ca solve te Eqs. (-8) witout usig te expoetial uctios ad derivig te eigevalue problem. e reduced costitutive equatios o -t layer are as ollows, wic oly used i applyig te cotiuity coditios. ( ) ( ) ( ) ( ) ( ) ( ) ( ) C e E q H ( ) ( ) ( ) ( ) ( ) ( ) ( ) C e E q H ( ) ( ) ( ) ( ) ( ) ( ) ( ) D e E d H ( ) ( ) ( ) ( ) ( ) ( ) ( ) B q d E H (9) From te Eqs. (-9), te layer depedet variables, S, ϕ s ad ψ s, are determied i terms o te primary variables o te reerece plae as ollows: s (,:) ( ) ( ) 7 8 ( ) ( ) K9 K s (,:) ( ) ( ) M M7 M8 ( ) ( ) M9 M s (,:) ( ) ( ) P P7 P8 ( ) ( ) P9 P (,:) ( ) ( ), 7, 8, ( ) ( ) ( ) L9, L, L r K K K S L L L () were te coeiciets o eac primary variable are omitted or te limited space. e vector wic cotais te top/bottom boudary coditio is expressed as, () ( ) () ( ) () ( ) p p () e umber o te primary variables also reduces by applyig Eq. (7) to te Eqs. (4-6) ad Eq. (). N S S S N S Fig.. Zigzag ield coiguratios. (I-plae displacemet ield) Virtual Wor Priciple ad Costitutive Equatios e virtual wor or a tree-dimesioal magetoelectro-elastic problem is stated as W DE V S DE BH BH dv pu ds ()

4 () () () () R R R R R z z z ( z z) ( z z) () () R R R z ( z z ) () () () () () () () () () V V V V V z z ( z z) ( z z) ( z z) F F F F F D z z z ( z z ) ( z z ) () () () F F F F D z z ( z z ) () () () () G G G G G B z z z ( z z) ( z z) () () () G G G G B z z ( z z ) () were p deotes te i-plae tractio wic did ot cosider i Eq. (7). e stress resultats ca be obtaied by itegratig te stresses i te troug-te-ticess directio as sow i Eq. (). e costitutive equatios o te lamiate ca be derived usig te Eqs. (-6) ad Eqs. (-) as ollows: R ε R ε 4 F 4 E G 4 H V ε F 6 4 E G 6 4 H (4) () were te resultats i te let-ad side are te colum vectors i Eq. (). e stiess matrices are calculated by itegratig te material costats i Eq. () alog te ticess directio. e strais, electric ield ad magetic ield i te rigt-ad side are te colum vectors wic are deied similar to tose i Eq. (). e sizes o te vectors ad matrix i Eq. (4) ad () are (6 ), (6 6) ad ( ), ( ), respectively. 4 Fiite Elemet Model o assess te validity o te preset metod, a iite elemet is developed or oe-dimesioal problems. e primary displacemet, magetic ad electric potetial uows are expressed i terms o odal values ad iterpolatio uctios as ollows: Nm u m m u w Q Y m w w m m,,,,, m, m (6) were is te umber o odes i a typical iite elemet. N m is a Lagragia iterpolatio uctio ad Q m, Y m are Hermite iterpolatio uctios. e compoets o te ield variables ad te odal displacemet vectors are give by () () () () N ε () () N ε ε () () () () N () () N () () () () () N E E E E E E E () () () N E E E E E E H H () () () () N H H H H H H () () () N H H H H H (7)

5 EFFICIEN HIGHER ORDER ZIG-ZAG HEORY FOR COUPLED MAGNEO-ELECRO-ELASIC COMPOSIE LAMINAES u u w w,,,,, εˆ ε ε ε E E H H ε ε ε E E H H B B B B B B B B u B B,, B, εˆ B u (8) were (), te superscript wit bracets, is te coeiciet o z. N deotes te umber o layers. [B] represets te strai-displacemet matrix, ad te details o te matrix are omitted or te limited space. e stiess matrix ca be costructed rom te virtual wor priciple as ollows: W ε Cεdx x u B CBdx x ε CB dx u x Ku u (9) For te multilayered MEE lamiates, te top/bottom boudary coditios are te exteral orces i te iite elemet problem. Fially, odal uows ca be obtaied by solvig te Eq. (9). Coclusios A udametal iger order zigzag teory o multilayered MEE plate is preseted i tis paper. Startig rom te ully-coupled tree-dimesioal ield equatios, te displacemets ad potetials are assumed as cubic ad quadratic zigzag uctios. e top/bottom boudary coditios ad te cotiuity coditios are employed i order to reduce te layer-depedet variables. For te applicatio o te preset metod, a iite elemet is developed or oe-dimesioal problems. e ial iite elemet goverig equatios cotai te ully coupled mageto-electro-elastic beaviors. o examie te couplig eects o MEE plate, te electrical ad magetic loadigs as well as te mecaical loadig are cosidered or various layup coiguratios. e preset metod ca be used as a eiciet tool or te aalysis o suc smart composite structures. Acowledgemets is wor was supported by Deese Acquisitio Program Admiistraio ad Agecy or Deese Developmet uder te cotract UD48JD. Reereces [] C.W. Na, Magetoelectric eect i composite o piezoelectric ad piezomagetic pases. Pysical Review B, Vol., No. 9, pp , 994. [] E. Pa Exact solutios or simply supported ad multilayered mageto-electro-elastic plates. Joural o Applied Mecaics, Vol. 68, pp 68-68,. [] E. Pa ad P.R. Heyliger, Exact solutios or mageto-electro-elastic lamiates i cylidrical bedig. Iteratioal Joural o Solids ad Structures, Vol. 4, pp ,. [4] K. Xu, A.K. Noor ad Y.Y. ag, reedimesioal solutios or coupled termoelectroelastic respose o multilayered plates. Computer Metods i Applied Mecaics ad Egieerig, Vol. 6, No. -4, pp -7, 99. [] M. Co ad J. O, Higer order zig-zag teory or ully coupled termo-electric-mecaical smart composite plates. Iteratioal Joural o Solids ad Structures, Vol. 4, No. -6, pp -6, 4. [6] J. O ad M. Co, A iite elemet based o cubic zig-zag plate teory or te predictio o termoelectric-mecaical beaviors. Iteratioal Joural o Solids ad Structures, Vol. 4, No. -6, pp 7-7, 4. [7] F. Ramirez, P.R. Heyliger ad E. Pa, Free vibratio respose o two-dimesioal magetoelectro-elastic lamiated plates. Joural o Soud ad Vibratio, Vol. 9, No. -, pp , 6.

6 [8] A.R. Aigeri, N. Gaesa ad S. Swaramai, Free vibratio beavior o multipase ad layered mageto-electro-elastic beam, Joural o Soud ad Vibratio, Vol. 99, No. -, pp 44-6, 7. [9] M. Co ad R.R. Parmerter, Eiciet iger order composite plate teory or geeral lamiatio coiguratios. AIAA Joural, Vol., No. 7, pp 99-6, 99. [] E. Carrera, Historical review o zig-zag teories or multilayered plates ad sells, Applied Mecaics Review (ASME), Vol. 6, No., pp 87-8.