M.Goodarzi et al. 789

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1 Journal of Sold Mechancs Vol. 8, No. (06) pp Thermo-Mechancal Vbraton Analyss of FG Crcular and Annular Nanoplate Based on the Vsco-Pasterna Foundaton M. Goodarz, *, M. Mohammad, M. Khooran, F. Saad Department of Engneerng, College of Mechancal Engneerng, Ahvaz Branch, Islamc Azad Unversty, Ahvaz, Iran Department of Mechancal Engneerng, Shahd Chamran Unversty of Ahvaz, Iran Receved 3 July 06; accepted 8 September 06 ABSTRACT In ths study, the vbraton behavor of functonal graded (FG) crcular and annular nanoplate embedded n a Vsco-Pasterna foundaton and coupled wth temperature change s studed. The effect of n-plane pre-load and temperature change are nvestgated on the vbraton frequences of FG crcular and annular nanoplate. To obtan the vbraton frequences of the FG crcular and annular nanoplate, two dfferent sze dependent theores are utlzed. The materal propertes of the FGM nanoplates are assumed to vary n the thcness drecton and are estmated through the Mor Tanaa homogenzaton technque. The FG crcular and annular nanoplate s coupled by an enclosng vscoelastc medum whch s smulated as a Vsco- Pasterna foundaton. By usng the modfed stran gradent theory (MSGT) and the modfed couple stress theory (MCST), the governng equaton s derved for FG crcular and annular nanoplate. The dfferental quadrature method (DQM) and the Galern method (GM) are utlzed to solve the governng equaton to obtan the frequency vbraton of FG crcular and annular nanoplate. The results are subsequently compared wth vald result reported n the lterature. The effects of the sze dependent, the n-plane pre-load, the temperature change, the power ndex parameter, the elastc medum and the boundary condtons on the natural frequences are nvestgated. The results show that the sze dependent parameter has an ncreasng effect on the vbraton response of crcular and annular nanoplate. The temperature change also play an mportant role n the mechancal behavor of the FG crcular and annular nanoplate. The present analyss results can be used for the desgn of the next generaton of nanodevces that mae use of the thermal vbraton propertes of the nanoplate..all rghts reserved. Keywords : Crcular and annular nanoplate; Functonal graded nanoplate; Modfed stran gradent theory; Modfed couple stress theory. INTRODUCTION T HE concept of FGMs was frst consdered n Japan n 98 durng a space plane project, thereafter FGMs, due to ther specfc changng n ther materal propertes, were developed for a wde range of applcatons, such as automotve ndustres, space vehcles, bomedcal materals, reactor vessels, mltary applcatons, semconductor ndustry and general structural elements n hgh thermal envronments [-3], and wde research efforts n many * Correspondng author. Tel.: E-mal address: m.goodarz.au@gmal.com (M.Goodarz).. All rghts reserved.

2 M.Goodarz et al. 789 engneerng felds durng the recent years. Recently, FGMs are wdely used n mcro and nano-electromechancal systems (MEMS and NEMS) [ 8] and also atomc force mcroscopes (AFMs) [9]. So, analyss of the statc and dynamc behavor of FGM structures under dfferent actuaton s very mportant. Contnuum modelng of the nanostructure has also ncreasng deal of attenton of many researchers due to experments n nanoscale are dffcult [0] and molecular dynamc smulatons are hghly computatonally expensve. Because of dffcultes encountered n the expermental methods to predct the responses of nanostructures under dfferent loadng condtons as the sze of physcal systems s scaled down nto the nanoscale, theoretcal analyses have been more noteworthy. There are varous sze-dependent contnuum theores such as couple stress theory [], stran gradent elastcty theory [], modfed couple stress theory [3] and nonlocal elastcty theory [-9]. Among these theores, modfed stran gradent theory s one of the most appled theoretcal approaches for the nvestgaton of nano-mechancs due to ther computatonal effcency and the capablty to produce accurate results, whch are comparable to the atomstc model ones. The pull-nstablty of hydrostatcaly and electrostatcally actuated crcular mcroplate s analyzed by Mohammad et al. []. Wang et al. [30] presented Tmosheno nano-beams formulatons based on the modfed stran gradent theory. Ansar et al. [3,3] nvestgated the lnear and nonlnear vbraton characterstcs of FGM mcrobeams based on the stran gradent and Tmosheno beam theores. They found that the value of materal property gradent ndex plays a more mportant role n the vbratonal response of the FGM mcrobeams wth lower slenderness ratos. Recently, Sahman and Ansar [33] studed the free vbraton response of functonally graded hgher-order shear deformable mcroplates based on stran gradent elastcty theory. The nonlnear forced vbratons of a mcrobeam employng stran gradent elastcty theory were nvestgated by Ghayesh et al. [3]. Mohammad et al. [35] nvestgated the buclng of rectangular nanoplate under shear n-plane load and n thermal envronment. They showed that the crtcal shear buclng load of rectangular nanoplate s strongly dependent on the small scale coeffcent. Cvale and Agoz [36] analyzed the vbraton behavor of mcro-scaled sector shaped graphene surrounded by an elastc matrx. Murmu and Pradhan [37] employed the nonlocal elastcty theory for the vbraton analyss of rectangular sngle layered graphene sheets (SLGSs) embedded n an elastc medum. They have used both Wnler-type and Pasterna-type models for smulate the nteracton of the graphene sheets wth a surroundng elastc medum. They reported that the natural frequences of SLGSs are strongly dependent on the small scale coeffcents. Pradhan and Phadar [38] nvestgated the vbraton of embedded multlayered graphene sheets (MLGS) based on nonlocal contnuum models. In ther paper, they have shown that nonlocal effect s qute sgnfcant and needs to be ncluded n the contnuum model of graphene sheet. Y-Ze Wang et al. [39] studed the vbraton of double-layered nanoplate. In ther research, t s concluded thermal effect and nanoplate wth sotropc mechancal propertes. It has been reported that graphene sheets have orthotropc propertes [0]. Asencer and Aydogdu [] proposed levy type soluton for vbraton and buclng of nanoplate. In that paper, they consdered rectangular nanoplate wth sotropc property and wthout effect of elastc medum. Malezadeh et al. [] used the dfferental quadrature method (DQM) to study the thermal buclng of a quadrlateral nanoplates embedded n an elastc medum. Thermal vbraton analyss of orthotropc nanoplates based on nonlocal contnuum mechancs were studed by Satsh et al. [3] who consderate two varable refned plate theory for thermal vbraton of orthotropc nanoplate. In general, SLGSs are embedded n an elastc medum but they dd not consder effect of elastc medum n that paper. On the other hand, they represented vbraton frequency of rectangular nanoplate only for smply supported boundary condtons and they dd not represent vbraton frequency for other boundary condtons. Prasanna Kumar et al. [] represented thermal vbraton analyss of monolayer graphene sheet embedded n an elastc medum va nonlocal contnuum theory. In ther paper, they consdered smply support boundary condton and they dd not study other boundary condton. They nvestgated graphene sheet wth sotropc property. Farajpour et al. [5] studed axsymmetrc buclng of the crcular graphene sheets wth the nonlocal contnuum plate model. In that paper, the buclng behavor of crcular nanoplates under unform radal compresson s studed. Explct expressons for the buclng loads are obtaned for clamped and smply supported boundary condtons. It s shown that nonlocal effects play an mportant role n the buclng of crcular nanoplates. In that paper, ther results compared wth the results obtaned by molecular dynamc and t s observed that results predcted by nonlocal theory are n exactly match wth MD results. Thus the relablty of nonlocal theory and presented soluton s demonstrated. Mohammad et al [6] employed the nonlocal plate theory to analyze vbraton of crcular and annular graphene sheet. They founded that scale effect s less promnent n lower vbraton mode numbers and s hghly promnent n hgher mode numbers. It s cleared that the natural frequency s easly affected by the appled n-plane pre-load and temperature change. As a result, the effect of n-plane pre-load on the property of transverse vbraton of functonal graded crcular and annular nanoplate s one of the practcal nterestng subjects. Researches that nvestgated on the FG crcular and annular nanoplate are very lmted n number wth respect to the case of rectangular nanoplate. To the best

3 790 Thermo-Mechancal Vbraton Analyss of FG Crcular and Annular Nanoplate. nowledge of the authors, t s the frst tme that the modfed couple stress theory has been successfully appled to nvestgate the vbraton frequency of the FG crcular and annular nanoplate embedded n a Vsco-Pasterna elastc medum under thermal envronment. The nfluence of the surroundng elastc medum on the frequency vbraton of the FG crcular and annular nanoplate s nvestgated. In the present paper, the effect of the n-plane pre-load and temperature change on the vbraton frequency of FG crcular and annular nanoplate s nvestgated. The governng equaton of moton s derved usng the modfed couple stress theory. The DQM s utlzed to solve the governng equatons of FG crcular and annular nanoplate wth smply supported, clamped boundary condtons and mx of them. From the results, some new and absorbng phenomena can be observed. The present results would be useful to sutably desgn nano electro-mechancal system and mcro electro-mechancal systems (NEMS/MEMS) devces usng FG crcular and annular nanoplate. DIFFERENTIAL EQUATIONS FOR NANOPLATES A mono-layered crcular and annular nanoplate embedded n a Vsco-Pasterna foundaton s shown n Fg., n whch geometrcal parameters of outer radus a, nner radus b and thcness h s also ndcated. The FGM profle across the thcness drecton of the plate, made of ceramc and metal consttuent materals, may be assumed to follow a functon form as P-FGM plates as: E( z) Em ( Ec Em) z h h h ( z) m ( c m) - z h ( z) m ( c m) () where, the Ez ( ), (z) and (z) are the Young's module, the thermal expanson and the Posson's rato, respectvely. Here, the Krchhoff plate theory s consdered to study the mechancal behavor of FG annular and crcular nanoplate n the thermal envronment, because t s adequate n provdng accurate bendng results for the thn plate due to small thcness n comparson wth radus. On the bass of the Krchhoff plate theory, the dsplacement feld can be expressed as: w( r, t) uz, v=0, w=w(r,t) r () where w(r,t) s the dsplacement of the mddle surface of the nanoplate at the pont ( r,,0). Due to lacng materal length scales, the conventonal contnuum mechancs has not the capablty to predct the sze-dependent response of the nanostructures. So, the MSGT s used to nterpret the sze effect nto the vbraton analyss of a FG crcular nanoplate. In comparson wth the MCST, the MSGT contans two addtonal gradent tensors of the dlataton and the devatorc stretch n addton to the symmetrc rotaton gradent tensor. Three ndependent materal length scale parameters and two classcal materal constants for sotropc lnear elastc materals are utlzed to specfy these tensors. For a contnuum constructed by a lnear elastc materal occupyng regon wth nfntesmal deformatons, the stored stran energy U can be defned as: tensor () () () () m j j j j j j m U p m x dv (3) In whch the components of the stran tensor j, the dlataton gradent tensor, the devatorc stretch gradent () j, and the symmetrc rotaton gradent tensor () xj are gven as: [7]

4 M.Goodarz et al ( s) w w w w w w w xr ( ), ( ), ( ) r z 3 z r r r r r r r r r r r 3 3 () z w w w () () () z w w w rrr ( ), (3 ) 3 r r r 3 5 r r r r r 5 r r r r r () () () w w () () () w w rrz rzr zrr ( ), z z z ( ) 5 r r r 5 r r r 3 () () () z w w w () w w rzz zzr zrz (3 ), ( ) 3 zzz 5 r r r r r 5 r r r () Based on the components of the nematc parameters, whch are effectve on the stran energy densty of the structure, the consttutve equaton for a lnear sotropc elastc materal can be expressed by [7] as: Fg. FG crcular and annular nanoplate embedded on a Vsco- Pasterna foundaton. 3 ( s) w w w w w mr ( z) l( ), p r z( z) l0( ), 3 r r r r r r r r 3 w w () z( z) l w w w pz ( z) l0 ( ) ( ), rrr 3 r r r 5 r r r r r () () () ( zl ) w w rrz rzr zrr ( ) 5 r r r 3 () () () z( z) l w w w () ( z) l w w r r r (3 ), ( ) 3 zzz 5 r r r r r 5 r r r () () () ( zl ) w w z z z ( ) 5 r r r 3 () () () z( z) l w w w w z w rzz zzr zrz (3 ), 3 rr z, 5 r r r r r r r r E( z) w ( z) w rr ( )( ( )) z z z T ( ( z) ) r r r E( z) w w z ( z) ( z)( ( z)) T ( ( z) ) r r r (5) The parameters λ and μ appeared n the consttutve equaton of the classcal stress σ, denote the Lame constants, respectvely whch are gven as: [8] ( z) E( z) E( z) ( z), ( z) ( ( z)) ( ( z) ) (6)

5 79 Thermo-Mechancal Vbraton Analyss of FG Crcular and Annular Nanoplate. The correspondng components of the classcal and nonclasscal stresses can be evaluated by substtutng the components of the stran tensor, the dlataton gradent tensor, the devatorc stretch gradent tensor, and the symmetrc rotaton gradent tensor n Eq. (5). So, the stran energy based on Eq. (3) and netc energy of mcroplate T can be wrtten as: s 3 w w w p w w w Mrw, rr M Yr ( ) M ( ) r 3 r r r r r r r r r r s da 3 A w w w w w w T z w Pz ( ) T ( ) rrz M rrr M 3 r r r r r r r r r r r r w w T I I t rt A da (7) where A denotes the area occuped by the md-plane of the crcular FG nanoplate. Moreover, I and I are represented as the followng form: h/ h/ I ( z) dz, I ( z) z dz h/ h/ (8) In Eq. (7), bendng moments, couple moments, other hgher-order resultants force and hgher-order moments caused by hgher-order stresses effectve on the secton are expressed as follows: h/ E() z z w w Mrr ( )( ), z z T dz h/ ( ) r r r h/ E( z) z w w M z ( z)( ) T dz, h/ ( ) r r r h/ E() z w w Nrr z ( z)( ) T dz h/ ( ) r r r, h/ E( z) w w N z ( z)( ) T dz, h/ ( ) r r r w w w w Yr ( z) l ( ) dz, P ( z) l ( ) dz r r r r h/ h/ z 0 h/ r h/ r p w w M r z ( z) l ( ) dz, T r r h/ 0 h/ r rrz h/ 5 r r h/ r h/ h/ 3 dz rrr 3 5 r r 5 r r h/ r h/ r r r h/ r r h/ r r r ( zl ) w w ( ) dz ( z) l w w z ( z) l w w w T ( ), M ( ) dz z z ( z) l w w w M (3 ) dz r (9) The value of the wor done by the external forces appled on the plate can be expressed as: W qw( r, t) fw( r, t) da ext A (0) Here q and f are the dstrbuted external force and the reacton force of elastc foundaton, respectvely. The reacton force of elastc foundaton are expressed as:

6 M.Goodarz et al. 793 Kww( r, t) The wnler foundaton f Kww( r, t) KG w( r, t) The Pasterna foundaton w( r, t) Kww( r, t) KG w( r, t) Cd The Vsco-Pasterna foundaton t () By usng Hamlton s prncple settng the coeffcents of w t t ( W ) dt 0 and tang the varaton of w, ntegratng by parts and s T ext equal to zero leads to the followng governng equaton: P P Mr, r M, r 3 P P M Mr, r P r z, r Mr, rr Yr, r Yr, rr Mr, rr Mr, rrr P 3 z, rr Trrz, rr r r r r r r T T 3M M M r r r r r r t r t rrz, r z, r rrr, rr M r r, r r, rr w w M rrr, rrr f q I 3 I () Also, the boundary condtons are obtaned as the followng form: M w 0 or -M rm M Y ry M rm rp T rt r T M rm M M 0 p p p r r r, r r r, r r, r r, rr z, r rrz rrz, r z rrr, r rrr, rr r r, r w p 0 or rmr ryr rmr, r rpz rtrrz Mrrr rmrrr, r Mr 0 r w p 0 or rm 0 r rm rrr r (3) By nsertng the Eq. (8), the Eq. (9) and the Eq. () nto the Eq. (), the governng equaton s obtaned as the followng form 6 5 l w 6 l0 l w l 8l w l0 A 6 A 5 C l B l0 B A B 5 r r 5r r 5r 5 r 3 l l0 l0 6l l w C B B A B A r r r r 5r 5r r l 8l0 l0 8l * w C B A B B N 0 N KG r r r r 5r r * l 8l0 l0 8l l N0 N KG w C B A B B A r r r r 5r 5r r r r r Kww Cd I I t t t r w w w () where the A, B and C are represented as the followng:

7 79 Thermo-Mechancal Vbraton Analyss of FG Crcular and Annular Nanoplate. h/ h/ m c m m c m h h/ h/ z h z h A z ( ) z dz, B ( ) dz, h h/ z h z h C E ( ) ( ) h/ mz Ec Em z m c m dz h h h/ z h z h D m ( c m ) dz, E mz ( c m ) z dz, h h N * h/ h/ h/ z h E ( ) / m Ec E h m h z h m ( c m ) Tdz h/ z h h m ( c m ) h (5) A FG crcular and annular s consdered to be embedded n a Vsco-Pasterna elastc medum. The geometrc propertes of the nanoplate are denoted by outer radus a, nner radus b, thcness h. For convenence and generalty we ntroduce the followng non-dmensonal parameters: w r r b l Bl Bl Bl K a W,, =, =, =, =, =, =, K a a b a h C C C C 0 W 0 W Cda C KGa N a 0 * N a A 0 a E t C C, K G, P, P, =, D, x, C D C C C C C D a D (6) Usng the above expressons, a non-dmensonal dfferental equaton for vbraton of FG crcular and annular nanoplate n the thermal envronment can be obtaned 6 5 W 6 0 w 8 w w * P P KG w * P P KG w W Ia W Ia W KWW C 0 C C (7) In order to obtan the governng equaton of the classcal FG crcular and annular plate, one can nsert the sze dependent parameters set equal to zero ( l 0 l l 0 ). Also, wth the assumpton l0 l 0 n the Eq. (7) the governng equaton of the crcular and annular plate wll be obtaned by the MCST. The Eq. (7) s rewrtten as the followng form: Md Cd Kd 0 (8) In the above equaton, the matrces M, C andk are the mass, damper and stffness matrx, respectvely. By defnng the new freedom vector and general soluton of the Eq. (7) as the followng form:

8 M.Goodarz et al. 795 Q d, W(r, )= Q e d (9) By usng the Eq. (9), we can rewrte the Eq. (8) as: 0 I I M K C A B Q 0, A, B 0 0 (0) In the above equaton, the s a complex number and the vbraton frequency of the FG crcular and annular nanoplate s the magnary part of the.the elements of the stffness, mass and damper matrx are gven n the Eq. (), respectvely. 6 5 d 6 0 d [ K] d 5 d 8 d 0 KW 5 5 d d d * P P KG * P P KG d d [ C] C Ia d Ia [ M] C d C d d () 3 SOLUTION PROCEDURE 3. Soluton FG crcular nanoplate by Galern method Eq. (7) s a sxth-order ordnary dfferental equaton n. If t s not mpossble to solve the governng equaton analytcally, t s very dffcult to obtan such a soluton. Hence, approxmate numercal method should be used. In the present study, Galern method s employed for the soluton of the governng equaton. The Galern method s a powerful and effcent numercal technque to solve the dfferental equatons. Snce ths numercal method provdes smple formulaton and low computatonal cost, t has been wdely used for the analyss of mechancal behavor of the structural elements at large and small scales, such as statc, dynamc and stablty problems [9, 50]. Furthermore, Galern s method s more general than the Raylegh Rtz method because no quadratc functonal or vrtual wor prncple s necessary. On the hands and to the best of authors nowledge, no satsfactory varatonal prncple has been reported for the axsymmetrc vbraton of FG crcular nanoplates yet. Usng the general procedure of the method yelds the followng: n Q Aj f j f d () j where f ( ) ( j,,..., n) are the basc functons whch must satsfy all boundary condtons but not necessarly j satsfy the governng equaton. A ( j,,..., n) are unnown coeffcents to be determned. The ntegraton extends j

9 796 Thermo-Mechancal Vbraton Analyss of FG Crcular and Annular Nanoplate. over the entre doman of the plate. The symbol Q ndcates a dfferental operator and s the rght-hand sde of Eq. (). Here, the boundary condtons (BC) for the FG crcular of constant thcness along the edge are assumed to be clamped. The boundary condtons are mathematcally wrtten as W dw d 0 at. In the Galern method, the lateral deflecton can be descrbed by a lnear combnaton of the basc functons for the numercal solutons of the problem under nvestgaton. The basc functons must satsfy all the above-mentoned boundary condtons. The chosen basc functon for W( ) are j f j Usng Eq. (), Eq. (3) and (0), one can obtan the followng system of lnear algebrac equatons A B * * B A Q 0 *, j 0 0 j M 0 j j C j I j d I d 0 K *, j 0 (3) () Here M,C and I are dfferental operators and are gven n the Eq. (). The Galern method transforms the vbraton problem nto a standard egenvalue problem. The vbraton frequency parameter s the egenvalue of Eq. () that can be found by usng standard egenvalue extracton technques. 3. DQM soluton In ths secton, for the soluton of Eq. (8) the dfferental quadrature method (DQM) [5] s employed. DQM s an effcent numercal method for the soluton of partal and ordnary dfferental equatons. DQ technque can be used to deal wth complcated problems reasonably well because ts mplementaton s very smple. In comparson wth other approxmate numercal methods such as the fnte elements method (FEM), the fnte dfference method, the boundary element method and the meshless methods, n ths approach, the problem formulaton becomes smpler and also low computatonal efforts are requred to obtan acceptable solutons. In addton, snce the strong form of the governng equatons and the related boundary condtons are dscretzed n ths method, they are free of the shear locng phenomenon that occurs n the FEM. In the revew paper of Bert and Mal [5], some other advantages and dsadvantages of the DQ technque are also reported. In recent years, many researchers used DQ approach for the global behavor analyss of the nanostructural elements, such as elastc buclng of sngle-layered graphene sheets [53, 5]. The DQM s based on a smple mathematcal concept that a partal dervatve of a functon wth respect to a space varable at a dscrete mesh pont (grd pont) n doman can be approxmated by tang a weghted lnear sum of the functonal values at all grd ponts n the whole doman. Accordng to DQ method, the partal dervatves of a functon f() r as an example, at the pont ( r ) are expressed as: [5,5] s d f () r s C f ( r ),,,..., n (5) dr n s rr j j j s where n s the number of grd ponts n the r drecton. C j represents the respectve weghtng coeffcent related to the Sth-order dervatve. Accordng to Shu and Rchard rule [5], the weghtng coeffcents n r drecton are determned as, If S namely, for the frst order dervatve,

10 M.Goodarz et al. 797 M ( r) C for j and, j,,..., n j () () ( r rj ) M ( rj ) (6) and n j j( j) C C for j and,,..., n (7) where M () () r s the frst order dervatve of Mr () and they can be expressed as: n n () ( ) ( j ), ( ) ( j ) j j( j) (8) M r r r M r r r If s, namely, for the second dervatves, the weghtng coeffcents are obtaned by usng the followng smple recurson relatonshp n n n 3 3 j j j j j j (9) C C C, C C C, C C C for, j,,... n One ey pont n the successful applcaton of the dfferental quadrature method s how to select the grd ponts. It has been shown that the grd pont dstrbuton whch s based on well accepted Gauss-Chebyshev Lobatto ponts [5], gves suffcently accurate results. Accordng to ths grd pont s dstrbuton, the grd pont dstrbuton n the drecton for annular and crcular FG nanoplate are gven as the followng form [5] ( ) cos,,,..., N cos,,,..., n ( N) n (30) The non-dmensonal computatonal doman of the nanoplate s 0. Usng the DQ method, Eq. () can be dscretzed as: 6 [ K] C W C W n n j j j j 5 j 5 j 8 n 0 CjWj KWW 5 5 j n j j C j C 0 n j CW 3 j 8 P P K 5 j n * G CW j j j * P P KG [ C] CW I a I a [ M] C W W n j CW j j (3) where Wj s the functonal value at the grd pont j. It should be noted that Eq. (3) s solved for nner grd ponts. The nner grd ponts are calculated by consderng the boundary condtons n the Eq. (3). The boundary condtons are ncorporated n the analyss by drect substtuton of them nto the dscrete governng equaton [5].

11 798 Thermo-Mechancal Vbraton Analyss of FG Crcular and Annular Nanoplate. The dervatves n the boundary condtons are also dscretzed by the DQ procedure. After mplementaton of the boundary condtons, the Eq. (3) can be wrtten n the followng form: 6 [ K] C W C W n n ˆ6 0 ˆ5 0 j j j j 5 j 5 j 8 ˆ n 0 CjWj KWW 5 5 j n 0 6 ˆ CW 3 j 5 5 j n * ˆ 0 P P K CW 5 j * n P P KG ˆ CW j 5 5 j [ C] CW I a I a [ M] C W W n ˆ j j C j C j G j j j (3) where n s ˆs ˆs s s j j j j nj n j C W, C C C C C for,,..., n and s,,...,6 (33) After employng the aforementoned soluton procedure, one obtans the followng system of lnear algebrac equatons A BQ 0 (3) where A and B are n n square matrces. The elements of these matrces are easly obtaned from the Eq. (0). Agan, t s observed that non-dmensonal buclng load can be determned from the egenvalues of a system of algebrac equatons and ths parameter s a complex number that the magnary part s the vbraton frequency of the FG crcular and annular nanoplate. NUMERICAL RESULTS AND COMPARISONS The combnaton of materals conssts of alumnum and ceramc SC. The Young s modulus, coeffcent of thermal expanson, the densty and Poson rato for alumnum are 6 3 Em 70 GPa, m 30, m 70 g m and m 0.3, respectvely. These parameters for the ceramc are 6 3 Ec 7 GPa, c 7. 0, c 300 g m andc 0.7, respectvely. These materal parameters are reported by Ke et al. [55]. The method of ths paper can deal wth all nds of combnatons of clamped and smply supported boundary condtons. Natural frequency parameters are lsted n tables. The exstng local plate model solutons are used for verfcaton of the accuracy of crcular and annular results. Followng four boundary condtons have been nvestgated n the vbraton analyss of the annular graphene sheets. SS: Annular graphene sheet wth smply supported outer and nner radus. CS: Annular graphene sheet wth clamped outer and smply supported nner radus. SC: Annular graphene sheet wth smply supported outer and clamped nner radus. CC: Annular graphene sheet wth clamped outer and nner radus.

12 M.Goodarz et al Valdaton of present computed results The present sze dependent model can easly reduce to the classcal crcular and annular plate model wth the sze effect beng neglected n the consttutve relatons. To confrm the relablty of the present formulaton and results, comparson studes are conducted for the natural frequences of the crcular plate wthout consderng sze dependent parameters. Table. tabulates the comparson of the vbraton frequency parameters for crcular plates. It s seen that the present results for all the two boundary condtons are n excellent agreement wth the classcal results. Table Comparson of the present results wth natural frequency parameters of classcal plate theory Boundary condton smply supported boundary condton clamped boundary condton Lessa and Narta [56] Km and Dcnson [57] References L and L [58] Zhou et al. [59] h Ca. Present Km and Zhou et al. Present Present Carrngton [60] Lessa [6] Dcnson [57] [59] (DQM) (Galern) Table. presents the comparson of the vbraton frequency parameters for the annular plates. The annular plate solutons n Table. were found to be n good agreement wth those non-dmensonal natural frequency values obtaned by prevous researchers [6, 59] who have used other technques. All the results are presented for the four boundary condtons. Table Comparson of the present results wth classcal plate theory for the lowest sx natural frequency parameters a h C, = b a 0.. References Boundary condton Charaverty et al. [6] Zhou et al. [59] Present CC SS SC CS For further valdaton, the present results are compared wth the frst three natural frequences of the FG annular mcroplate n Table 3. Snce there are no publshed results avalable for annular nanoplates n open lterature, the results of annular mcroplate are used for comparson. To obtan these results, the modfed couple stress theory s utlzed. From ths table, one can see that the present results are n good agreement wth the reported results n the lterature. To obtan the natural frequences of the FG annular plate, the boundary condton of annular mcroplate are assumed SS and CC. The other materal propertes of the annular mcroplate are reported by Ke et al. [55]. Table 3 Comparson of the present results wth MCST for the natural frequency parameters of annular FG mcroplate = ba 0.5, =0.5, =. Boundary References condton Frst Mode Second Mode Thrd Mode Ke et al. [55] Present Ke et al. [55] Present Ke et al. [55] Present CC SS A computer code s developed n MATLAB based on Eq. (0). As DQ results are senstve to lower grd ponts, a convergence test s performed to determne the mnmum number of grd ponts requred to obtan stable and accurate results for Eq. (0). The analyss s carred out by radus of nanoplate 0 nm, the sze dependent parameter 0.5 nm, the shear, Wnler and dampng coeffcents are assumed 0, 00 and 0, respectvely. Accordng to the Fg.

13 800 Thermo-Mechancal Vbraton Analyss of FG Crcular and Annular Nanoplate., present soluton s convergent. From the fgure t s clearly seen that eleventh number of grd ponts ( N ) are suffcent to obtan the accurate solutons for the present analyss. Fg. Convergence study and mnmum number of grd ponts ( N ) requred to obtan accurate results.. The results of present study Varaton of vbraton frequency versus compressve n-plane pre-load s shown for FG crcular nanoplate n the Fg. 3. The non-dmensonal parameter of elastc medum such as shear modulus parameter K G, Wnler modulus parameter K W and dampng modulus of damper C for the surroundng polymer matrx s gotten 0, 00 and 0 respectvely. The curves show that the vbraton frequency s senstve to the modulus of the surroundng elastc medum. It s also clearly shown that the vbraton frequency decreases by ncreasng the n-plane pre-load. Further, t s llustrated that the dampng modulus decreases the vbraton frequency; thus, the greatest and the smallest vbraton frequency are belong to the FG nanoplate based on the Pasterna medum and Vsco-Wnler medum, respectvely. To determne the effect of the aspect rato on the vbraton frequency of the FG nanoplate, the vbraton frequences of the FG annular nanoplate aganst the aspect rato are plotted n the Fg. for four dfferent boundary condtons. To obtan the results, the vbraton frequences are presented by consderng the power ndex parameter 0 and the nner radus 0 nm. Also, the sze dependent parameter s assumed 0.5h n the MSGT. From ths fgure, t s clearly shown that the non-dmensonal frequency ncreases wth the ncrease of the aspect rato. Also, the results show that the non-dmensonal frequency ncreases monotoncally by ncreasng the rgdty of the boundary condtons. In Fg., the gap between the curves ncreases wth ncreasng the aspect rato. Fg.3 Varaton of vbraton frequency wth the compressve preload for varous nd of elastc medum.

14 M.Goodarz et al. 80 Fg. Varaton of vbraton frequency wth the aspect rato for varous boundary condtons. The frst vbraton frequences of the crcular FG nanoplate are plotted n the Fg. 5. In these fgures, the dependency of the vbraton frequency versus the radus of the crcular FG nanoplate s shown for dfferent temperature changes. To obtan these results, t s assumed that the FG nanoplate embedded on a vsco-pasterna foundaton and the shear elastc, the Wnler elastc and the external dampng coeffcent are assumed 0, 00 and 00, respectvely. Also, the power ndex of FG materal and the sze dependent length of the crcular nanoplate are consdered 0, and 0.5h, respectvely. The vbraton frequences are obtaned by consderng the modfed stran gradent theory. From ths fgure, t s clear that the vbraton frequency of the crcular nanoplate s strongly depend on the radus of the crcular nanoplate and ths dependency s more for the larger temperature change. Also, the effect of the temperature change decreases wth decreasng the radus of the nanoplate. Furthermore, ths fgure shows that the temperature change has a decreasng effect on the vbraton frequency of the crcular nanoplate. Accordng to ths fgure, one can easly fnd that the natural frequency s strongly depended to the radus of the nanoplate and ths parameter has a decreasng effect on the natural frequency of the FG nanoplate. Moreover, ths fgure ndcates that there s crtcally temperature and radus for the FG nanoplate because the vbraton frequency becomes zero n a specfc radus and temperature change; thus, ths specfc radus and temperature change s called the crtcally parameters. Influence of the sze dependent on the frequences of nanoplate under the temperature change by consderng the modfed couple stress and the modfed stran gradent theory s studed. Fg. 6 shows the varaton n vbraton frequency wth sze dependent parameter under varous power ndex parameters. The nanoplate s assumed to be subjected to a temperature change of T 50. It can be observed that as the sze dependent parameter ncreases, the non-dmensonal frequency ncreases for all values of the power ndex parameter. The results of fgure ndcate that, by ncreasng the values of sze dependent parameter, the dfference between the MCST and the MSGT s ncreased for all values of power ndex parameters. It s also observed that as the power ndex parameter ncreases the vbraton frequency decreases. Furthermore, as can be seen from Fg. 6, the non-dmensonal frequency s senstve to the sze dependent parameter for small values of power ndex parameter. In the other words, t s seen that the non-dmensonal frequency for FG nanoplates wth hgher sze dependent parameter are strongly affected by the used elastcty theory (MSGT and MCST). Fg.5 Varaton of vbraton frequency wth the radus of crcular FG nanoplate for varous temperature changes.

15 80 Thermo-Mechancal Vbraton Analyss of FG Crcular and Annular Nanoplate. To see the effect of temperature change n dfferent power ndex parameter from the soluton for nondmensonal frequency FG annular nanoplate wth dfferent length are shown n Fg. 7. The varaton of nondmensonal natural frequency for FG annular nanoplate wth CC and SS boundary condton s plotted n Fg. 7. The aspect rato of nanoplate s consdered 0.. It s found that the non-dmensonal frequency ncreases wth ncrease of the rgdty of the boundary condton for all lengths. Smlar vbraton response as ths trend s observed for other boundary condtons. The results are shown that the vbraton frequency of the FG annular nanoplate by hgh power ndex parameter s lower than that the vbraton frequency of the FG nanoplate by low power ndex parameter. It s also observed that the effect of the temperature change on the non-dmensonal frequency s mportant for SS annular nanoplate wth hgh power ndex parameter n comparson wth the annular nanoplate wth rgdty boundary condton and small power ndex parameter. Fg.6 Varaton of vbraton frequency wth the sze dependent parameters of the FG nanoplate for varous power ndex parameter and two dfferent elastcty theores. Fg.7 Varaton of vbraton frequency wth the temperature change of the FG annular nanoplate for varous power ndex parameter, radus and boundary condtons. In order to show the dependency of the hgher vbraton frequency mode of the vscoelastc nanoplate on the foundaton type and sze dependent parameter, the sze dependent dfference percentage s ntroduced as follows: Vbraton frequency 0 Vbraton frequency 0 Sze dependent dfference p ercentage = 00 Vbraton frequency 0 To ths end, the sze dependent dfference percentage versus sze dependent parameter s plotted n the Fg. 8 for varous nd of elastc foundaton and three dfferent vbraton frequency modes. The aspect rato of the crcular annular nanoplate s assumed 0.; also, the outer and nner boundary condton of the annular nanoplate are consdered clamp. From ths fgure, t s clear that the vbraton frequency of the nanoplate s strongly depend on the sze dependent and the nd of the elastc foundaton and ths dependency s more for the hgher mode vbraton frequency. Also, the effect of the sze dependent on the frst vbraton frequency mode s more than that the thrd vbraton frequency mode. On the other hand, the effect of the elastc foundaton on the vbraton frequency s more mportant for the frst vbraton frequency mode n comparson wth the thrd vbraton frequency mode. Further, one can easly see that the effect of the sze dependent on the vbraton frequency mode for the nanoplate wth the Wnler foundaton s more mportant than the nanoplate wth Vsco-Pasterna foundaton.

16 M.Goodarz et al. 803 Fg.8 Varaton of sze dependent dfference percentage wth the sze dependent parameter for varous nd of elastc foundaton and three dfferent frequency mode. 5 CONCLUSIONS Ths study llustrates the sgnfcance of sze dependent effects on the vbraton behavor of FG crcular and annular nanoplate n the thermal envronment. Two dfferent sze dependent contnuum theores (MSGT and MCST) are utlzed to obtan the vbraton frequences of the FG annular and crcular nanoplate. To calculate the natural frequences, the dfferental quadrature method and the Galern method are used. The results are presented for clamp and smply supported FG crcular nanoplate. The vbraton frequences of the annular nanoplate wth four dfferent boundary condtons are also obtaned. From the results followng conclusons are notceable: The vbraton frequency of the FG crcular and annular nanoplate s strongly depended on the radus of the crcular nanoplate and ths dependency s more for the larger temperature change. The nfluence of temperature change reduces, by decreasng of radus. The non-dmensonal natural frequency decreases at hgh temperature case wth ncreasng the temperature change. The effect of temperature change on the non-dmensonal frequency vbraton becomes the opposte at low temperature case n compresson wth the hgh temperature case. The power ndex effect for the MSGT s much more than that for the MCST. The power ndex effect has a decreasng effect on the vbraton frequency of the FG crcular and annular effect. The sze dependent parameter has an ncreasng effect and ths parameter play an mportant role n the case of FG crcular and annular nanoplate wth smaller power ndex parameter. The dfferences between the vbraton responses of the FG nanoplate ncrease by ncreasng the sze dependent parameter. The effect of temperature on the frequency vbraton ncreases wth ncreasng the radus of annular nanoplate. The effect of temperature change s sgnfcant to predct the mechancal behavor of FG crcular and annular nanoplate wth hgh power ndex parameter and cannot be gnored. In the frst mode, the effect of the elastc foundaton on the vbraton frequency s more mportant n comparson wth the thrd vbraton frequency mode. The effect of the sze dependent on the frst vbraton frequency mode s more than that the thrd vbraton frequency mode. REFERENCES [] Sallese J.M., Grabns W., Meyer V., Bassn C., Fazan P., 00, Electrcal modelng of a pressure sensor MOSFET, Sensors and Actuators A: Physcal 9: [] Naban A., Rezazadeh G., Haddad-derafsh M., Tahmaseb A., 008, Mechancal behavor of a crcular mcroplate subjected to unform hydrostatc and non-unform electrostatc pressure, Mcrosystem Technologes : [3] Bao M., Wang W., 996, Future of mcroelectromechancal systems (MEMS), Sensors and Actuators A: Physcal 56: 35-.

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