Exact solution for functionally graded and layered magneto-electro-elastic plates

Transcription

1 Inernaional Journal of Engineering Science 43 (2005) Exac soluion for funcionally graded and layered magneo-elecro-elasic plaes E. Pan *,. Han Deparmen of Civil Engineering, ASEC, Universiy of Akron, Akron, OH , Unied Saes Received 22 Augus 2003; received in revised form 23 November 2003; acceped 23 November 2004 Absrac In his paper, an exac soluion is presened for he mulilayered recangular plae made of funcionally graded, anisoropic, and linear magneo-elecro-elasic maerials. While he edges of he plae are under simply suppored condiions, general mechanical, elecric and magneic boundary condiions can be applied on boh he op and boom surfaces of he plae. The funcionally graded maerial is assumed o be exponenial in he hickness direcion and he homogeneous soluion in each layer is obained based on he pseudo-sroh formalism. or mulilayered plae srucure, he propagaor marix mehod is employed so ha only a 5 5 sysem of linear algebraic equaions needs o be solved. The exac soluion is hen applied o wo funcionally graded (exponenial) sandwich plaes made of pieoelecric atio 3 and magneosricive Coe 2 O 4, under mechanical and elecric loads on he op surface. While he numerical resuls clearly show he influence of he exponenial facor, magneo-elecro-elasic properies, and loading ypes on induced magneo-elecric-elasic fields, hey can also serve as benchmarks o numerical mehods such as he finie and boundary elemen mehods. Ó 2005 Elsevier Ld. All righs reserved. Keywords: uncionally graded maerial; Magneo-elecro-elasic plae; Propagaor marix; Pseudo-Sroh formalism * Corresponding auhor. Tel.: ; fax: address: pan2@uakron.edu (E. Pan) /$ - see fron maer Ó 2005 Elsevier Ld. All righs reserved. doi:11/j.ijengsci

2 322 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) Inroducion Smar or inelligen maerials such as he pieoelecric and pieomagneic ones are currenly inensively invesigaed, due o heir abiliy of convering energy from one form o he oher (among magneic, elecric and mechanical energies). I is also observed ha, composies made of pieoelecric/pieomagneic maerials can exhibi he magneoelecric coupling ha is no presen in he single-phase pieoelecric or pieomagneic maerial [1 3]. Recenly, Pan [4] derived an exac closed-form soluion for he simply suppored and mulilayered plae made of anisoropic pieoelecric and pieomagneic maerials under a saic mechanical load, and Pan and Heyliger [5] solved he corresponding vibraion problem. On he oher hand, Li and Dunn [] carried ou a sudy on he micromechanics of magneo-elecro-elasic composie maerials. More recenly, Wang and Shen [7] sudied he wo-dimensional (2D) inclusion problem in magneo-elecro-elasic composie maerials, and Gao e al. [8] solved he crack problem in 2D magneo-elecro-elasic solids. The funcionally graded maerial (GM) srucure has araced wide and increasing aenions o scieniss and engineers. GM plays an essenial role in mos advanced inegraed sysems for vibraion conrol and healh monioring. While He e al. [9] invesigaed he GM plaes wih inegraed pieoelecric sensors and acuaors for he acive conrol purpose, Liew e al. [10] analyed he pos buckling of pieoelecric GM plaes subjeced o hermo-elecro-mechanical loading. The ime-dependen sress analysis in GM elasic cylinders was carried ou by Yang [11], and he effec of he iner-diffusion reacion on he compaibiliy in PZT/PNN GM maerials was analyed by Xu e al. [12]. Almajid and Taya [13] and Almajid e al. [14] also invesigaed he displacemen and sress fields in pieocomposie plaes wih funcionally graded microsrucure for poenial applicaions in he pieoelecric bimorph. More recenly, a hree-dimensional (3D) exac closed-form soluion was derived for anisoropic elasic [15] and pieoelecric [1] GM plaes under simply suppored edge condiions. In his paper, we presen an exac soluion for a mulilayered recangular plae made of anisoropic and funcionally graded magneo-elecro-elasic maerials. The plae is simply suppored along is edges, and boh mechanical and elecric loads are applied on he op surface. The homogeneous soluion is obained based on he pseudo-sroh formalism [17 19], and he propagaor marix mehod [20,21] is employed o rea he mulilayered case. In he numerical analysis, wo funcionally graded (exponenial) sandwich plaes made of pieoelecric atio 3 and magneosricive Coe 2 O 4 are analyed. Numerical resuls clearly show he influence of he exponenial facor, magneo-elecro-elasic properies, sacking sequence, and loading ypes on he induced magneo-elecric-elasic fields, which should be of ineres o he design of smar srucures. urhermore, hese numerical examples could also serve as benchmarks o numerical mehods such as he finie and boundary elemen mehods. 2. Problem saemen and basic formulaions Le us assume ha here is an N-layered recangular plae made of anisoropic and funcionally graded magneo-elecro-elasic maerials and ha is four sides are simply suppored. The dimensions of he layered plae are L x L y H wih H being he hickness. Each layer can be homo-

3 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) geneous or funcionally graded wih exponenially varying maerial properies. A Caresian coordinae sysem is aached o he plae wih is origin being a one of he four corners on he boom; he plae is in he posiive -region. Layer j is bounded by is lower inerface (or surface) a = j and upper inerface (or surface) a = j+1 wih is hickness h j = j+1 j. Obviously, he boom surface is a = 1 (=0) and he op surface a = N+1 (=H). Wihou loss of generaliy, exernal loads (mechanical, elecric or magneic) will be applied on he op surface of he N-layered plae. or an anisoropic and linearly magneo-elecro-elasic maerial, he coupled consiuive equaions for each layer can be wrien as r i ¼ C ik c k e ki E k q ki H k D i ¼ e ik c k þ e ik E k þ d ik H k ð1þ i ¼ q ik c k þ d ik E k þ l ik H k where r i, D i and i are he sress, elecric displacemen and magneic inducion (i.e., magneic flux), respecively; c i, E i and H i are he srain, elecric field and magneic field, respecively; C ij, e ij and l ij are he elasic, dielecric and magneic permeabiliy coefficiens, respecively; e ij, q ij and d ij are he pieoelecric, pieomagneic and magneoelecric coefficiens, respecively. Apparenly, various uncoupled cases can be reduced from Eq. (1) by seing he appropriae coupling coefficiens o ero. or a funcionally graded maerial wih exponenial variaion in he -direcion, he maerial coefficiens in Eq. (1) can be described by C ik ðþ ¼C 0 ik eg ; e ik ðþ ¼e 0 ik eg ; l ik ðþ ¼l 0 ik eg e ik ðþ ¼e 0 ik eg ; q ik ðþ ¼q 0 ik eg ; d ik ðþ ¼d 0 ik eg ð2þ where g is he exponenial facor characeriing he degree of he maerial gradien in he -direcion, and he superscrip 0 is aached o indicae he -independen facors in he maerial coefficiens. I is obvious ha g = 0 corresponds o he homogeneous maerial case. or an orhoropic solid, wih ransverse isoropy being a special case, he maerial coefficiens in Eq. (1) can be wrien as C 11 C 12 C e q 31 C 22 C e q 32 C e ½CŠ ¼ q C ; ½eŠ ¼ 33 0 e 24 0 ; ½qŠ ¼ 0 q Sym C e q C ð3þ ½eŠ ¼ 4 e e ; ½dŠ ¼ 4 d d ; ½lŠ ¼ 4 l l ð4þ 0 0 e d l 33

4 324 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) The exended srain (using ensor symbol for he elasic srain c ik ) displacemen relaion is c ij ¼ 0:5ðu i;j þ u j;i Þ ð5þ E i ¼ / ;i ; H i ¼ w ;i where u i, / and w are, respecively, he elasic displacemen, elecric poenial, and magneic poenial. The equaions of equilibrium, including he balance of he body force and elecric charge and curren, can be wrien as: r ij;j þ f i ¼ 0 D j;j f e ¼ 0 j;j f m ¼ 0 ðþ where f i, f e, and f m are, respecively, he body force, elecric charge densiy, and elecric curren densiy (or magneic charge densiy as compared o he elecric charge densiy). 3. Sroh-ype general soluions or a simply suppored and GM plae, we seek he soluion of he exended displacemen vecor u in he form [4] u x a 1 cos px sin qy u y a 2 sin px cos qy u u ¼ e s a 3 sin px sin qy ð7þ / 5 4 a 4 sin px sin qy 5 w a 5 sin px sin qy where p ¼ np=l x ; q ¼ mp=l y ð8þ wih n and m being wo posiive inegers. I is noed ha soluion (7) represens only one of he erms in a double ourier series expansion when solving a general boundary value problem. Therefore, in general, summaions for n and m over suiable ranges are implied whenever he sinusoidal erm appears. Subsiuion of Eq. (7) ino he general srain displacemen relaions (5), he consiuive equaions (1), and finally ino he equaions of equilibrium () wih ero force and densiies, yields he following eigenequaion ½Q gr þ sðr R þ gtþþs 2 TŠa ¼ 0 ð9þ where superscrip denoes he ranspose of he marix. Also in Eq. (9), a ¼½a 1 ; a 2 ; a 3 ; a 4 ; a 5 Š ð10þ

5 pc 0 13 pe 0 31 pq 0 31 C qc 0 23 qe 0 32 pq 0 32 C R ¼ pc 0 55 qc ; T ¼ C 0 33 e 0 33 q pe 0 15 qe Sym e 0 33 d pq 0 15 qq l ðc 0 11 p2 þ C 0 q2 Þ pqðc 0 12 þ C0 Þ ðc 0 p2 þ C 0 22 q2 Þ Q ¼ ðc 0 55 p2 þ C 0 44 q2 Þ ðe 0 15 p2 þ e 0 24 q2 Þ ðq 0 15 p2 þ q 0 24 q2 Þ Sym e 0 11 p2 þ e 0 22 q2 d 0 11 p2 þ d q2 5 l 0 11 p2 þ l 0 22 q2 We remark ha when he gradien coefficien g = 0, Eq. (9) is reduced o he eigenequaion for he corresponding homogeneous case [4]. We now express he exended racion vecor as r x b 1 cos px sin qy r y b 2 sin px cos qy r ¼ e ðsþgþ b 3 sin px sin qy ð13þ D 5 4 b 4 sin px sin qy 5 b 5 sin px sin qy E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) which is differen from he corresponding homogeneous case [4]. y virue of he consiuive relaions (1) and he displacemen expression (7), we obain ð11þ ð12þ b ¼ð R þ stþa ð14þ where b ¼½b 1 ; b 2 ; b 3 ; b 4 ; b 5 Š ð15þ Similarly, he in-plane sresses and elecric and magneic displacemens can be expressed as r xx c 1 sin px sin qy r xy c 2 cos px cos qy r yy c 3 sin px sin qy D x ¼ e ðsþgþ c 4 cos px sin qy ð1þ D y c 5 sin px cos qy 4 7 x 5 4 c cos px sin qy 7 5 y c 7 sin px cos qy

6 32 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) where 2 4 c 1 c 2 c 3 c 4 c 5 c c C 0 11 p C0 12 q C0 13 s e0 31 s q0 31 s 3 C 0 q C0 p C 0 12 p C0 22 q C0 23 s e0 32 s q0 32 s ¼ e 0 15 s 0 e0 15 p e0 11 p d0 11 p 0 e s e0 24 q e0 22 q d0 22 q q 0 15 s 0 q0 15 p d0 11 p l0 11 p q 0 24 s q0 24 q d0 22 q l0 22 q These exended sresses (Eq. (13)) should saisfy he equaions of equilibrium (assuming ero body force and ero elecric and magneic charge densiies). We remark ha Eq. (9), derived for a simply suppored plae, resembles he Sroh formalism [17,18]. or he corresponding homogeneous case, his equaion was named as pseudo-sroh formalism because of is similariy o he Sroh formalism [4]. In he presen pseudo-sroh formalism, an ineresing feaure is observed: Tha is, if s is a complex (or purely imaginary) eigenvalue, hen is complex conjugae is also an eigenvalue. Wih aid of Eq. (14), Eq. (9) can now be recas as a linear eigensysem N a ¼ s a ð18þ b b where " # T 1 R T 1 N ¼ ð19þ Q RT 1 R RT 1 gi Depending upon he given maerial propery, he 10 eigenvalues of Eq. (18) may no be disinc. Should repeaed roos occur, a sligh change in he maerial consans would resul in disinc roos wih negligible error [22] so ha for all maerial siuaions, he simple and unified soluion given below can sill be used. Thus, le us assume ha he firs five eigenvalues have posiive real pars (if he roo is purely imaginary, we hen pick up he one wih posiive imaginary par) and he reminders have opposie signs o he firs five. We disinguish he corresponding 10 eigenvecors by aaching a subscrip o a and b. Then he general soluion for he exended displacemen and racion vecors (of he -dependen facor) are derived as u A 1 he s i A 2 he s i K 1 ¼ ð20þ 1 he ðsþgþ i 2 he ð sþgþ i K 2 where A 1 ¼½a 1 ; a 2 ; a 3 ; a 4 ; a 5 Š; A 2 ¼½a ; a 7 ; a 8 ; a 9 ; a 10 Š 1 ¼½b 1 ; b 2 ; b 3 ; b 4 ; b 5 Š; 2 ¼½b ; b 7 ; b 8 ; b 9 ; b 10 Š ð21þ he s i¼diag½e s1 ; e s2 ; e s3 ; e s4 ; e s5 Š and K 1 and K 2 are wo 5 1 consan column marices o be deermined. a 1 a 2 a 3 a 4 a ð17þ

7 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) Eq. (20) is a general soluion for a simply suppored magneo-elecro-elasic GM plae and reduces o he soluion for he corresponding homogeneous plae case. I should be furher noiced ha new GM hin plae models could also be reduced from his soluion by expanding he exponenial erm in erms of a Taylor series [23]. This is paricularly convenien since one need only o replace he diagonal exponenial marix wih is Taylor series expansion [24,25]. Wih Eq. (20) being served as a general soluion for a GM magneo-elecro-elasic plae, soluions for he corresponding mulilayered GM plae can be obained using he coninuiy condiions along he inerface and he boundary condiions on he op and boom surfaces of he plae. To handle a mulilayered srucure wih relaively large numbers of layers, we employ he propagaor marix mehod, insead of he convenional approach [2,27]. The propagaor marix mehod was developed exclusively for layered srucures and possesses cerain meris (for a brief review, see [21]), as can be observed in he nex secion. 4. Soluion of mulilayered GMsysem rom he general soluion (20), we solve he column coefficien marices K 1 and K 2 for layer j " # 1 K 1 A 1 he s ð j Þ i A 2 he s ð j Þ i u ¼ ð22þ K 2 j 1 he ðs þgþð j Þ i 2 he ð s þgþð j Þ i j where he subscrip j indicaes layer j and s * are he eigenvalues of layer j. Le = j and j+1 in Eq. (22), we find ha he column coefficien marices K 1 and K 2 can be expressed by he displacemen u and racion on eiher he lower inerface or he upper inerface of layer j. In oher words, we have K 1 ¼ A 1 A 1 2 u A 1 he s h j i A 2 he s h j 1 i u ¼ ð23þ K 2 j 1 2 j j 1 he ðs þgþh j i 2 he ð s þgþh j i j jþ1 where h j = j+1 j is again he hickness of layer j. Therefore, we can finally express he displacemen u and racion on he upper inerface by hose on he lower inerface of layer j as u A 1 he s h j i A 2 he s h j 1 i A 1 A 2 u 1 he ðs þgþh j i 2 he ð s þgþh j ð24þ i 1 2 jþ1 ¼ j j j Table 1 Maerial coefficiens of he pieoelecric atio 3 (C 0 ij in 109 N/m 2, e 0 ij in C/m2, e 0 ij in 10 9 C 2 /(N m 2 ), and l 0 ij in 10 Ns 2 / C 2 ) C 0 11 ¼ C0 22 C 0 12 C 0 13 ¼ C0 23 C 0 33 C 0 44 ¼ C0 55 C 0 ¼ 0:5ðC0 11 C0 12 Þ e 0 31 ¼ e0 32 e 0 33 e 0 24 ¼ e e 0 11 ¼ e0 22 e 0 33 l 0 11 ¼ l0 22 l

8 328 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) Assuming ha he displacemen u and racion are coninuous across he inerfaces, Eq. (24) can be applied repeaedly so ha one can propagae he physical quaniies from he boom surface = 0 o he op surface = H of he mulilayered GM plae. Consequenly, we have u ¼ P N ðh N ÞP N 1 ðh N 1 ÞP 2 ðh 2 ÞP 1 ðh 1 Þ u ð25þ H 0 Table 2 Maerial coefficiens of he magneosricive Coe 2 O 4 (C 0 ij in 109 N/m 2, q 0 ij in N/(Am), e0 ij in 10 9 C 2 /(N m 2 ), and l 0 ij in 10 Ns 2 /C 2 ) C 0 11 ¼ C0 22 C 0 12 C 0 13 ¼ C0 23 C 0 33 C 0 44 ¼ C0 55 C 0 ¼ 0:5ðC0 11 C0 12 Þ q 0 31 ¼ q0 32 q 0 33 q 0 24 ¼ q e 0 11 ¼ e0 22 e 0 33 l 0 11 ¼ l0 22 l Proporional acor in Maerial Properies ig. 1. Variaion of he GM proporional coefficien for g = 10, 5,0,5,10 ( in m and g in m 1 ). or 2 [0.2,0.3], he coefficien is he exponenial facor e g( 0.2). The proporional coefficien in 2 [0,0.1] is obained via symmeric requiremen.

9 where A 1 he s h ki A2 he s h ki A 1 A 1 2 P k ðh k Þ¼ 1 he ðs þgþh ki 2 he ð s þgþh ki k 1 2 k ðk ¼ 1;...; NÞ ð2þ is called he propagaing marix or propagaor. Eq. (25) is a surprisingly simple relaion, and for given boundary condiions, can be solved for he unknowns involved. As examples, he mechanical and elecric loads will be discussed. In he firs example, we assume ha, on he op surface ( = H) he -direcion racion componen is applied, i.e., r ¼ r 0 sin px sin qy ð27þ which may represen one of he erms in he double ourier series soluion for a general loading case (uniform or poin loading). All oher racion componens on boh surfaces are assumed o be ero. Thus, Eq. (25) is reduced o u ¼ M 1 M 2 u H M 3 M ð28þ where he four submarices M j are he muliplicaions of he propagaor marices in Eq. (25). Applying he racion boundary condiion (27), he lef-hand side of Eq. (28) on he op surface is expressed as u ¼½u x ; u y ; u ; /; w; 0; 0; r 0 sin px sin qy; 0; 0Š ð29þ H E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) (a) (b) -4E-012-2E E-012 4E-012 u x 0 2E-012 4E-012 E-012 8E-012 1E-011 u ig. 2. Variaion of he elasic displacemen componens u x and u (m) along he hickness direcion in GM magneoelecro-elasic // for facor g = 10, 5,0,5,10 (m 1 ) caused by a surface load on he op surface. u x in (a) and u in (b).

10 330 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) The unknown exended displacemens on boh surfaces of he mulilayered GM plae can hus be solved from Eq. (28). In he second example, he racion vecor on he boom surface is again assumed o be ero, bu we apply an elecric poenial / o he op surface ( = H), given as, / ¼ / 0 sin px sin qy ð30þ or his case, he lef-hand side of Eq. (28), i.e. he boundary values on he op surface, becomes u ¼½u x ; u y ; u ; / 0 sin px sin qy; w; 0; 0; 0; D ; 0Š ð31þ H (a) (c) φ φ (b) (d) -3E E-00-2.E E E-00-2E E-00 ψ -3E E-00-2E E-00-1E-00-5E-007 ψ ig. 3. Variaion of elecric poenial / (V) and magneic poenial w (C/s) along he hickness direcion in GM magneo-elecro-elasic plaes for facor g = 10, 5,0,5,10 (m 1 ) caused by a surface load on he op surface. / in (a) and w in (b) for //, and / in (c) and w in (d) for //.

11 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) ollowing he same procedures, all he unknowns on he op and boom surfaces can be found from Eq. (28). In order o obain he exended displacemen and racion vecors a any deph, say k k+1 in layer k, we propagae he soluion from he boom of he layered plae o he -level, i.e., u ¼ P ð k 1 ÞP k 1 ðh k 1 ÞP 2 ðh 2 ÞP 1 ðh 1 Þ u ð32þ 0 Wih he exended displacemen and racion vecors a a given deph being solved, he corresponding in-plane quaniies can be evaluaed using Eqs. (1) and (17). Similar soluions can also be obained for oher boundary condiions. Therefore, for an anisoropic, magneo-elecro-elasic, and GM mulilayered recangular plae, we have derived he exac soluion based on he pseudo-sroh formalism and he propagaor marix mehod. In he nex secion, we apply our soluion o invesigae he response of he sandwiched GM plae under mechanical and elecric loads. 5. Numerical analyses In he numerical calculaion, he layered GM plae is made of hree layers. The -independen maerial coefficiens are pieoelecric atio 3 and magneosricive Coe 2 O 4, lised respecively, in Tables 1 and 2 [4]. I is obvious ha he pieoelecric atio 3 and magneosricive Coe 2 O 4 are boh ransversely isoropic wih heir symmery axis along he -axis. The hree layers have equal hickness of 0.1 m (wih a oal hickness H = 0.3m) and he horional dimensions of he plae are L x L y =1m 1 m. Two sandwich plaes wih sacking sequences atio 3 /Coe 2 O 4 /atio 3 (a) (b) σ xx σ ig. 4. Variaion of sress componens r xx and r (N/m 2 ) along he hickness direcion in GM magneo-elecro-elasic // plae for facor g = 10, 5,0,5,10 (m 1 ) caused by a surface load on he op surface. r xx in (a) and r in (b).

12 332 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) (called //) and Coe 2 O 4 /atio 3 /Coe 2 O 4 (called //) are invesigaed. While he middle layer is homogeneous, boh he op and boom layers are funcionally graded wih he symmeric exponenial variaion shown in ig. 1. As can also be observed from ig. 1, five differen exponenial facors, i.e., g = 10, 5,0,5,10 (m 1 ), were sudied. or boh he mechanical and elecric loadings, he invesigaed mode is fixed a m = n = 1 (i.e., p = p/l x, q = p/l y in Eqs. (27) and (30)) and he horional coordinaes are fixed a (x,y) = (0.75L x,l y ). The ampliudes in Eqs. (27) and (30) are, respecively, r 0 = 1 N/m 2 and / 0 = 1 V. I has been checked ha under he mechanical load, he resuls corresponding o he exponenial facor g = 0 are exacly he same as hose in Pan [4]. (a) (c) -8E-011-4E E-011 8E E-010 D x -8E-012-4E E-012 8E E-011 D x (b) (d) -1.2E-011-8E-012-4E E-0274E-012 8E E-011 D -8E-013-4E E-013 8E E-012 D ig. 5. Variaion of elecric displacemen componens D x and D (C/m 2 ) along he hickness direcion in GM magneo-elecro-elasic plaes for facor g = 10, 5,0,5,10 (m 1 ) caused by a surface load on he op surface. D x in (a) and D in (b) for he // case, and D x in (c) and D in (d) for he // case.

13 5.1. Mechanical load E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) ig. 2a and b shows, respecively, he variaion of he elasic displacemens u x (= u y ) and u along he -direcion in he GM // plae. Variaion of hese elasic displacemens in he corresponding // plae is similar o hose in ig. 2a and b, wih only sligh difference in he magniude (magniude of he elasic displacemen in // is generally larger han ha in //). I is observed ha on he op and boom surfaces, he magniude of he horional displacemen u x increases wih decreasing exponenial facor g. urhermore, on he op surface he horional (a) (c) -4E E-010 8E E-009 x -4E-009-2E E-0094 E-009 x (b) (d) -8E-011-4E E-011 -E-010-4E-010-2E E-010 4E-010 E-010 ig.. Variaion of magneic inducion componens x and (Wb/m 2 ) along he hickness direcion in GM magneo-elecro-elasic plaes for facor g = 10, 5,0,5,10 (m 1 ) caused by a surface load on he op surface. x in (a) and in (b) for he // case, and x in (c) and in (d) for he // case.

14 334 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) displacemen is posiive, while on he boom surface hey are negaive. As for he verical displacemen u, is value increases wih decreasing g in he whole plae. ig. 3a and b shows, respecively, he variaion of he elecric poenial / and magneic poenial w along he -direcion in he // plae. The resuls in he corresponding // plae are ploed in ig. 3c and d. I is observed ha in boh plae models, he magniudes of he elecric and magneic poenials decreases wih increasing facor g. I is furher noiced ha even hough hese poenials are coninuous across he inerfaces, heir slopes are no (i.e., ig. 3a a = 0.1 m, and ig. 3d a = 0.2 m). (a) (c) -1E E-011 2E-011 3E-011 u x -8E-013-4E E-013 8E-013 u x (b) (d) -4E E-011-3E E-011-2E E-011-1E-011 u -4E E-012-3E E-012-2E E-012-1E-012-5E-013 u ig. 7. Variaion of he elasic displacemen componens u x and u (m) along he hickness direcion in GM magneoelecro-elasic plaes for facor g = 10, 5,0,5,10 (m 1 ) caused by an elecric poenial on he op surface. u x in (a) and u in (b) for he // case, and u x in (c) and u in (d) for he // case.

15 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) ig. 4a and b shows, respecively, he variaion of he sress componens r xx (=r yy ) and r along he -direcion for he // plae. The resuls in he corresponding // plae are very similar o hose shown in ig. 4a and b (for boh curve shapes and magniudes). While he normal sress r does no change much for differen facor g (ig. 4b), he horional normal sress r xx is very sensiive o he exponenial facor g. Near he inerfaces or surfaces (ig. 4a), is magniude can be doubled by varying g from 10 o 10 (m 1 ). urhermore, he horional sress componen is disconinuous across he inerface, as expeced. The variaions of he elecric displacemens D x (= D y ), D and magneic inducions x (= y ), along he -direcion are ploed, respecively, in igs. 5a d and a d. While igs. 5a,b and a,b are for he // plae, igs. 5c,d and c,d for he // plae. I is observed, from hese figures, ha because of he maerial propery disconinuiies in differen layers, he horional elecric displacemen and magneic inducion are disconinuous across he inerfaces. I is also ineresing o noe ha he magniude of horional elecric displacemen (magneic inducion) is very small in magneosricive Coe 2 O 4 (pieoelecric atio 3 ) layers, due o he fac ha for magneosricive Coe 2 O 4 (pieoelecric atio 3 ) maerial, he pieoelecric (pieomagneic) coefficiens are ero. urhermore, he similariy among he response curves should be also noiced: D x in ig. 5a vs. x in ig. c; D in ig. 5b vs. in ig. d; D x in ig. 5c vs. x in ig. a; and D in ig. 5d vs. in ig. b Elecric load ig. 7a and b shows, respecively, he variaions of he elasic displacemens u x (= u y ) and u along he -direcion for he // plae, whils ig. 7c and d is he corresponding elasic displacemens for he // plae. In conras o he mechanical load case, we observe here clearly ha he (a) (b) 0 1E-00 2E-00 3E-004 E-00 ψ -1E-007-5E E-008 1E E-007 2E-007 ψ ig. 8. Variaion of magneic poenial w (C/s) along he hickness direcion in GM magneo-elecro-elasic plaes for facor g = 10, 5,0,5,10 (m 1 ) caused by an elecric poenial on he op surface. w in (a) for he // case and w in (b) for he // case.

16 33 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) elasic displacemens in he wo sandwich plaes are compleely differen from each oher. In paricular, he magniude of he elasic displacemens in // is roughly one order larger han ha in //. The magneic poenial w along he -direcion in he // and // plaes are shown, respecively, in ig. 8a and b. I is ineresing ha for boh sandwich plaes, relaively large magneic poenial can be induced in he boom layer when an elecric poenial is applied on he op surface. urhermore, in he boom layer, he magniude of he magneic poenial in // is larger han ha in //. (a) (c) σ xx σ xx (b) (d) σ σ ig. 9. Variaion of sress componens r xx and r (N/m 2 ) along he hickness direcion in GM magneo-elecro-elasic plaes for facor g = 10, 5,0,5,10 (m 1 ) caused by an elecric poenial on he op surface. r xx in (a) and r in (b) for he // case, and r xx in (c) and r in (d) for he // case.

17 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) (a) (b) -8E-010 -E-010-4E-010-2E E-010 4E-010 x -2E E-011 4E-011 E-011 8E-011 1E-010 ig. 10. Variaion of magneic inducion componens x and (Wb/m 2 ) along he hickness direcion in GM magneo-elecro-elasic // for facor g = 10, 5,0,5,10 (m 1 ) caused by an elecric poenial on he op surface. x in (a) and in (b). ig. 9a and b shows, respecively, he variaion of he sress componens r xx (=r yy ) and r along he -direcion for he // plae. The resuls for he corresponding // plae are ploed in ig. 9c and d. In conras o he mechanical load (ig. 4a and b); he sacking sequence now has a dramaic influence on he sress componens in erms of boh he curve shape and magniude (ig. 9a vs. ig. 9c and ig. 9b vs. ig. 9d). urhermore, he normal sress componen r can be grealy alered by varying he exponenial facor g (ig. 9b and d), as compared o he mechanical loading case. inally, ig. 10a and b shows, respecively, he variaions of magneic inducions x (= y ), along he -direcion in // case. I is observed ha he magneic inducion changes dramaically in he op and boom layers (magneosricive Coe 2 O 4 ), and i is almos ero in he middle layer (pieoelecric atio 3 ). In oher words, relaively large magneic field can sill be induced in he magneosricive layer even if an elecric load is applied.. Conclusions In his paper, an exac soluion is presened for a layered recangular plae made of anisoropic and funcionally graded magneo-elecro-elasic maerials. The plae is under simply suppored edge condiions and boh he mechanical and elecric loads are applied on he op surface of he plae. While he homogeneous soluion in each GM layer is derived based on he pseudo- Sroh formalism, he propagaor marix mehod is employed o handle he mulilayered srucures. In he numerical sudy, wo sandwich plaes are analyed, which are made of pieoelecric atio 3 and magneosricive Coe 2 O 4, wih maerial properies varying exponenially in he hickness direcion wihin he op and boom layers. I is observed ha, in general, differen

18 338 E. Pan,. Han / Inernaional Journal of Engineering Science 43 (2005) exponenial facors will produce differen magniudes for he response curves. urhermore, he sacking sequences (// or //) and he boundary condiions (mechanical or elecric load) can have significan effecs on he induced magneic, elecric, and elasic fields. An ineresing example is ha, under he mechanical load, he normal sress componen r is very insensiive o he exponenial facor; however, under he elecric poenial, differen exponenial facors can produce compleely differen normal sress componens. While he numerical examples can serve as benchmarks for various numerical mehods, he special characerisics discussed in his paper should be useful o he design of smar srucures based on he pieoelecric and magneosricive GMs. References [1] G. Harshe, J.P. Doughery, R.E. Newnham, Theoreical modeling of muliplayer magneoelecric composies, In. J. Appl. Elecromagn. Maer. 4 (1993) [2] C.W. Nan, Magneoelecric effec in composies of pieoelecric and pieomagneic phases, Phys. Rev. 50 (1994) [3] Y. envenise, Magneoelecric effec in fibrous composies wih pieoelecric and pieomagneic phases, Phys. Rev. 51 (1995) [4] E. Pan, Exac soluion for simply suppored and mulilayered magneo-elecro-elasic plaes, J. Appl. Mech. 8 (2001) [5] E. Pan, P. Heyliger, ree vibraions of simply suppored and mulilayered magneo-elecro-elasic plaes, J. Sound Vib. 253(2002) [] J. Li, M.L. Dunn, Micromechanics of magneoelecroelasic composie maerials: average fields and effecive behavior, J. Inel. Maer. Sys. Sruc. 9 (1998) [7] X. Wang, Y. Shen, Inclusions of arbirary shape in magneoelecroelasic composie maerials, In. J. Eng. Sci. 41 (2003) [8] C. Gao, H. Kessler, H. alke, Crack problems in magneoelecroelasic solids, Par I: exac soluion of crack, In. J. Eng. Sci. 41 (2003) [9] X.Q. He, T.Y. Ng, S. Sivashanker, K.M. Liew, Acive conrol of GM plaes wih inegraed pieoelecric sensors and acuaors, In. J. Eng. Sci. 38 (2001) [10] K.M. Liew, J. Yang, S. Kiipronchai, Posbuckling of pieoelecric GM plaes subjec o hermo-elecromechanical loading, In. J. Solids Sruc. 40 (2003) [11] Y.Y. Yang, Time-dependen sress analysis in funcionally graded maerials, In. J. Solids Sruc. 37 (2000) [12] J. Xu, X. Zhu, Z. Meng, Effec of he inerdiffusion reacion on he compaibiliy in PZT/PNN funcionally gradien pieoelecric maerials, IEEE Trans. Compon. Packag. Technol. 22 (1999) [13] A.A. Almajid, M. Taya, 2D-elasiciy of GM pieo-laminaes under cylindrical bending, J. Inel. Maer. Sys. Sruc. 12 (2001) [14] A.A. Almajid, M. Taya, S. Hudnu, Analysis of ou-of-plane displacemen and sress field in a pieocomposie plae wih funcinonally graded microsrucure, In. J. Solids Sruc. 38 (2000) [15] E. Pan, Exac soluion for funcionally graded anisoropic elasic composie laminaes, J. Compos. Maer. 37 (2003) [1] Z. Zhong, E.T. Shang, Three-dimensional exac analysis of a simply suppored funcionally gradien pieoelecric plae, In. J. Solids Sruc. 40 (2003) [17] A.N. Sroh, Dislocaions and cracks in anisoropic elasiciy, Philos. Mag. 3(1958) [18] T.C.T. Ting, Anisoropic Elasiciy, Oxford Universiy Press, Oxford, 199. [19] T.C.T. Ting, Recen developmens in anisoropic elasiciy, In. J. Solids Sruc. 37 (2000) [20]. Gilber, G. ackus, Propagaor marices in elasic wave and vibraion problems, Geophysics 31 (19)

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