JCAMECH. Refined plate theory for free vibration analysis of FG nanoplates using the nonlocal continuum plate model

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1 JCAMECH Vol. 8, No., Jun 7, pp -6 OI:.59/jcamch Rfind plat thory for fr viration analyi of FG nanoplat uing th nonlocal continuum plat modl M. Goodarzi,, M. Nikkhah Bahrami, V. Tavaf partmnt of Mchanical Enginring, Ahvaz Branch, Ilamic Azad Univrity, Ahvaz, Iran partmnt of Enginring, Scinc and Rarch Branch, Ilamic Azad Univrity, Thran, Iran partmnt of Mchanical Enginring at Univrity of South Carolina, Columia, USA Rcivd: Apr. 7, Accptd: 7 Jun. 7 Atract In thi articl, th fr viration havior of nanocal FG rctangular plat i tudid within th framwork of th rfind plat thory (RPT) and mall-cal ffct ar takn into account. Uing th nonlocal laticity thory, th govrning quation ar drivd for ingl-layrd FG nanoplat. Th Navir mthod i mployd to otain clodform olution for rctangular nanoplat auming that all dg ar imply upportd. Th rult ar uquntly compard with valid rult rportd in th litratur. Th ffct of th mall cal on natural frqunci ar invtigatd conidring variou paramtr uch a apct ratio, thickn ratio, and mod numr. It i hown that th RPT i an accurat and impl thory for th viration analyi of nanoplat, which do not rquir a har corrction factor. Kyword: Small cal, rfind plat thory, viration analyi, FGM nanoplat.. Introduction Ovr th pat two dcad, many rarchr hav mployd th nonlocal laticity thory for th invtigation of th viration havior and uckling rpon of nanotructur. Such nanotructur includ nanotu[-5], nanorod[6], nanoring[7]and nanoplat [8-]. Th nonlocal laticity thory wa introducd y Eringn [5]. H modifid th claical continuum mchanic for taking into account mall cal ffct. In thi thory, th tr tat at a givn point dpnd on th train tat at all point in th domain, whil in th local thory, th tr tat at any givn point dpnd only on th train tat at that point. Graphn i a truly two-dimnional atomic crytal with xcptional lctronic and mchanical proprti. Corrponding Author. Tl.: ; Fax: Addr: mz.goodarzi@gmail.com

2 M. Goodarzi t al. Th graphn ht ar widly ud in th micro lctro-mchanical ytm (MEMS), nano lctromchanical ytm (NEMS), and in dvic uch a ocillator, clock, and nor. Elctromchanical ronator ar NEMS dvic mad from upndd ingl- and multi-layrd graphn ht [6]. Furthrmor, potntial application hav n invtigatd for th SLGS a ma nor and atomitic dut dtctor [7]. Propr application of SLGS dpnd on a thorough undrtanding of thir mchanical proprti. Viration havior i on uch mchanical proprty that i of grat importanc from a dign prpctiv. Anari[8] otaind th natural frqunci of a multi-layrd graphn ht mddd in an latic mdium. Th viration analyi of th circular nanoplat wa invtigatd y rarchr[]. Thclaical plat thory (CPT) and firt-ordr har dformation thory (FST) wr dvlopd for th fr viration of nanoplat [9].Aghaaai and Rddy[9] rformulatd th third-ordr har dformation plat thory for th viration and nding of nanoplat. Anari t al.[8]invtigatd th virational charactritic of multi-layrd graphn ht uing th nonlocal finit lmnt modl. Aadi and Farhi [] tudid th fr viration of circular nanoplat taking into accountthir urfac proprti. A lvy typ mthod ha alo n ud in th viration and uckling analyi of nanoplat uing th nonlocal plat modl []. In thi papr th rfind plat thory i applid to otain th viration frquncy of th FG rctangular nanoplat. Th tranvr diplacmnt ha two (nding and har) componnt and th paraolic ditriution of th tranvr har train through th thickn of th plat i takn into account. In thi thory, it i aumd that th tranvr har train vary paraolically acro th thickn. Th har tr componnt atify th zro traction oundary condition on th top and ottom urfac of th plat without uing har corrction factor. In th prnt work, RPT ha n xtndd to ingl-layrd graphn ht. Th govrning quation ar drivd for th FG rctangular nanoplat ad on th nonlocal laticity thory. Explicit olution ar otaind for th natural frqunci of rctangular nanoplat with all dg imply upportd y applying th Navir mthod. Th natural frqunci calculatd y th propod thory ar compard with rult otaind from othr thori uch a th claical plat thory (CPT), firt ordr har dformation thory (FST), and third ordr har dformation thory (TST). Th ffct of th mall cal on natural frqunci ar alo tudidy conidring variou paramtr uch a apct ratio, thickn ratio, and mod numr.. Nonlocal laticity A mntiond in th prviou Sction, th nonlocal laticity thory, firt introducd y Eringn [], ha n widly ud for th analyi of nonlocal prolm [, -8]. According to thi thory, a tr-train rlationhip for a homognou latic olid i xprd a: ' l ij (, ) ij V whr, ij x x dv and l ij () ar th nonlocal and local tr tnor, rpctivly. Th intgration xtnd ovr th ntir ody volum,v. Th function i th nonlocal modulu, which contain th mall cal ffct. It i oviou that th nonlocal modulu ha th dimnion of. Thi function dpnd on two varial, namly ' x x ' x lngth ' x x and, a n from th aov quation. rprnt th ditanc twn point. i a matrial contant dfind y: x and l i L () whr, and dnot th intrnal and xtrnal charactritic lngth, rpctivly. Th valu of th paramtr i vital for th validity of nonlocal modl. Eringn [] otaind a valu of.9 for thi paramtr y matching th diprion curv ad on atomic modl. Rcntly, mot rarchr hav ud valu ranging from to nm for th nonlocal paramtr,,in th analyi of nanoplat. l i l i L It i difficult to apply Eq. () for olving nonlocal laticity prolm. Thrfor, th following diffrntial form of Eq. () i oftn ud [9] ij a C ij : () Th ymol : indicat th doul dot product, whil C and ar th fourth ordr laticity and train tnor, rpctivly. x y i th twodimnional Laplac oprator. Not that th claical rlationhip twn th tr and train tnor can otaind y tting a in th aov contitutiv quation. Th local tr tnor i a follow:. Govrning quation ad on RPT In th prnt work, w mploy th rfind plat thory for th viration analyi of rctangular nanoplat (Fig. ). According to thi thory, th tranvr har train vary paraolically ovr th plat thickn. Th har tr componnt mut atify th following condition:

3 Vol. 8, No., Jun 7 h xz yz at z () whr, h i plat thickn. Th diplacmnt fild can writtn a: w w U( x, y, z, t) u( x, y, t) z zf( z) x x w w V ( x, y, z, t) v( x, y, t) z zf( z) y y W( x, y, z, t) w ( x, y, t) w ( x, y, t) w ( x, y, t) (5),, T x y xy, T w w w,, x y xy u v u v,,, x y y x T w w w,, w x y xy, y x,, y T w w, T yz xz T w x (8), Th rultant tr N, M, and Q ar dfind y whr, Fig..A continuum plat modl of th nanoplat. h x, y, xy x, y, z M M M zdz h h x y xy x y z M, M, M,, f( z) zdz h h x, y, xy x, y, z N N N dz h h xz, yz xz, yz Q Q dz h h xz yz xz yz Q, Q, f( z) dz h (9 a) (9 ) (9 c) (9 d) (9 ) 5 z 5 z f( z), f( z) 5 h h (6) Uing Eq. (), th tr train rlationhip for a FG nanoplat i writtn a: in which, u and v dnot mid-plan diplacmnt of th plat in th x and y dirction, rpctivly. w, w,and rprnt th xtnion, nding, and har componnt of th tranvr diplacmnt, rpctivly. Th xtnion componnt w of th tranvr diplacmnt may rgardd a ngligil compard to othr diplacmnt componnt in mot ca. Uing th aov-mntiond diplacmnt fild, on can otain th train a follow: w () z zf z () f z (7) x x y y xy l i xy yz yz xz xz C C x C C y C66 xy C yz C55 xz () whr, in which, C i th fourth ordr laticity tnor. Th componnt of thi tnor ar dfind a: 5

4 M. Goodarzi t al. E( z) E( z) E( z) C, C, C, C G( z), C G( z), C G( z) N z h E( z) ( Et E), h () N z h G( z) ( Gt G), h whr, E and E ar Young moduli of top and t ottom layr; G t and G ar har moduli of top and ottom layr of th FG nanoplat, rpctivly M li M F, F F, Q a Q H, H Q a Q H, H ( c ) ( d ) ( ) whr, N li N F, F F, M li M F, 6, F F ( a) ( ) N x M x M x N N y M M y M M y, N xy M xy M xy Q yz Q yz Q, Q Qxz Qxz,, () whr, F, H, tc., ar plat tiffn dfind y ij ij 6 F , F 6, F ,,, 6,,,, F 6,,, H 5 55,, 6, 6, 6, 6 6,,,, 6, 6, 6, 6,, F 6, F 6,,, 6 6 6, 6, 6, ,,,,, 5, 5, H,,,,,, 5 H ( a) ( ) h n n ij Cij z dz (i,j=,,6, n,,,,,6) (5 a) h, 5 ij ij (i,j=,,6) ij h (5 ), 5 ij ij (i,j=,,6) ij h (5 c ) 6, (i,j=,,6) 6 ij ij ij ij 6h 9h (5 d ) h ij ij ij ij,, C, z, z dz (i,j=,5) (5 ) h, 5 5 ij ij (i,j=,5) ij h (5 f ), ij ij (i,j=,5) ij ij 6 h h (5 g ) W now mploy th Hamilton principl to driv th govrning quation. Th analytical form of th principl can xprd a [] t ( U V T ) dt (6) 6

5 Vol. 8, No., Jun 7 whr, rprnt a variation with rpct to x and y. U, V, and T dnot th train nrgy of dformation, th potntial nrgy of xtrnal forc, and th kintic nrgy of th plat, rpctivly. Uing Eq. (6), and umming th cofficint of u, v, w, wand w, w may otain th following govrning quation u: v: N N x xy u I x y t y x t Ny Nxy v I (7 a) (7 ) M M w : I I x xy y t t t t M x xy y w w w w M M Q w : x xy y x y t t t 8 t M x xy y Qxz yz w w w I w I (7 c) (7 d) w : Q Q xz yz w w w I x y t t t (7 ) whr, th inrtia I and I ar dfind y h,, I I z dz h (8) u v v u u u u : Ch C h +C66h I li I x xy xy y t t in which, i th ma of th nanoplat dnity. Sutitution of Eq.(a-) and(5a-g) into Eq. (7a) yild th following govrning quation in trm of th diplacmnt: (9 a) v u u v v v v : Ch C h +C66h I li I y xy xy x t t (9 ) w w w w w w w w : - ( ) I I x x y y t t t t a 66 wa w w w li I li I t t t t (9 c) w w w w : - ( 66) 8 x x y y 5 wa w 5 wa w wa w w C55h C h I 6 x x 6 y y t t t I 8 w w w w I i w l t t t t 8 t (9 d) a li I 7

6 M. Goodarzi t al. wa 5 w wa 5 w wa w w wa : C55h C 55h C h C h I x 6 x y 6 y t t t wa w w li I t t t (9 ) It hould notd that whn th nonlocal paramtr i t to zro, ( l i ), Eq. (9a-) rduc to that of th claical quation [].. Analytical olution for th imply upportd rctangular nanoplat W mploy th Navir mthod to otain th clod form olution aociatd with dtrmining th natural frqunci of th rctangular nanoplat. Lt u now conidr th imply upportd oundary condition along all th dg of rctangular graphn ht. Th oundary condition ar of th form: At dg x = and x = a v w w w w y w y w y N M M, xx xx xx At dg y = and y = u w w w w x w x w x N M M, yy yy yy ( a) ( ) Th following xprion of th diplacmnt which automatically atify th aov oundary condition ar aumd. i t w W in( x)in( y) ( a ) m n m n mn i t w W in( x)in( y) m n mn i t w W in( x)in( y) m n mn u U co( x) in( y) i t mn v V in( x) co( y) i t mn m n ( ) ( c) ( d) ( ) In th aov xprion, m a and n, Wmn, Wmn, Wmn, Umn, Vmn ar cofficint, and i th natural frquncy of th nanoplat. Sutituting Eq.(a-) into Eq. (9a-), on can otain a ytm of quation in th following matrix form: whr, U, V, W, W, W Hr, T mn mn mn mn mn ij and nanoplat a follow: () () ij ar dfind for th rctangular,,,, 66 66,, ( ), 66 ( ),,,,, ( ) 6, 6, 6, 6,,, 66 55,,, I I( l i ) ( ) ( l ) ( ) ( i I I ) I ( li ) ( ) I ( ) () 8

7 Vol. 8, No., Jun 7 For a nontrivial olution, th dtrminant of th cofficint matrix in Eq. () mut qual to zro. Thi provid an quation for dtrmining th natural frqunci of th nanoplat... Validation and comparionof th rult otaind For th validation of th rult otaind, comparion ar madtwn th rult and tho otaind from variou thori for th orthotropic plat. In l i th prnt work, th oundary condition along all th four dg ar aumd to imply upportd. Th matrial proprti of th orthotropic nanoplat ar otaind from th Rf. []. In Tal, th natural frqunci calculatd uing th RPT thory ar compard with tho calculatd uing CPT, FST, and TST []for iotropic graphn ht with th following matrial and gomtrical proprti: Th non-dimnional natural frqunci of th FG quar nanoplat for diffrnt nonlocal paramtr calculatd y RPT ar litd in Tal. Th matrial proprti of th FG nanoplat (Tal ) ar taulatd in th Tal []. Th comparion of non-dimnional natural frqunci otaind y variou thori a prntd in Tal and, th maximum and minimum valu ar orvd for CPT and RPT, rpctivly. On can aily find from Tal that non-dimnional natural frquncy dcra with incraing nonlocal paramtr. Furthr, thi dcra i mor nitiv at highr mod numr. l i In thi Tal, w aum that th nanoplat i iotropic, which man that matrial proprti at a givn point ar th am in all dirction. 5. Rult and dicuion 5.. Effct of apct ratio on viration of nanoplat Th non-dimnional natural frqunci in Fig. (-) for iotropic and FG nanoplat ar dfind a: h C Th variation in non-dimnional natural frquncy of th nanoplat with th nonlocal paramtr i hown in Fig. for variou apct ratio a. Plat lngth i takn to nm. In thi Sction, w aum that th nanoplat i FG. Th Young modulu and Poion ratio of th FG nanoplat wr prntd in th prviou Sction. It i orvdin Fig. that natural frquncy dcra with incraing apct ratio from to a it do alo with incraing nonlocal paramtr from nm to nm whil thi dcra i mor rapid for than it i with. Furthr, th diffrnc twn RPT and CPT com mor ignificant whn apct ratio dcra from to. a a 5.. Comparion of natural frqunci of iotropic and FG nanoplat In ordr to comparth natural frqunci of iotropic nanoplat with tho of FG on, w hav plottd non-dimnional natural frquncy vru nonlocal paramtr for oth iotropic and FG ca uing CPT and RPT (S Fig.). Th lngth of th quar nanoplat i nm.thmatrial proprti of iotropic graphn htar E 6 Gpa,.5, 5 kg m. It i clar from thi Figur that th natural frqunci of FG nanoplat ar alway mallr than thir iotropic countrpart for oth CPT and RPT. Alo, it i orvd that th rult otaind y RPT ar alway mallr than tho of CPT. 5.. Effct of highr mod and thickn ratio ahon viration of nanoplat To tudy th influnc of highr mod on th viration charactritic of rctangular nanoplat, th variation in non-dimnional natural frquncy with th nonlocal paramtr i hown in Fig.. Th curv ar plottd for diffrnt mod numr.th lngth of th quar FG nanoplat i nm. It i orvd that th mall lngth cal xhiit a highr ffct for highr mod. Thi phnomnon idu to th incraing intraction twn atom at mallr wavlngth (highr mod numr). Furthr, th gap twn th two curv (RPT and CPT) incra with an incra in mod numr; in othr word, th diffrnc twn th natural frqunci calculatd y RPT and CPT incra with incraing mod numr. Effct of thickn ratio ah on th nondimnional natural frquncy of FG nanoplat for variou nonlocal paramtr i hown in Fig.5. It i found that natural frqunci dcra with incraing thickn ratio from to Comparion of natural frqunci otaind y CPT and RPT In thi Sction, w conidr a quar FG nanoplatwith a lngthof nm. To invtigat th diffrnc twn th two thori (RPT and CPT), w dfin prcnt 9

8 M. Goodarzi t al. diffrnc in non-dimnional natural frqunci calculatd uing RPT and CPT a follow: Tal : Comparion of th non-dimnional natural frqunci h C of an orthotropic quar plat. Mod numr m n Non-dimnional natural frquncy Exact[] Rddy[5] mn for an orthotropic plat in variou thori Sriniva[] CPT Prnt Tal : Matrial proprti of th nanoplat. Typ of Matrial Young' Modulu Poion' Ratio Gomtrical Proprti 6 Iotropic E. a=, a =, a h = E 765 Gpa, E 588 Gpa. a, a h FG CPT RPT Prcntag diffrnc = Th aov prcnt diffrnc vru apct ratio for variou mod numr and thickn ratio ar plottd in Fig. 6 and 7,rpctivly. Apct ratio ha a dcraing ffct on prcnt diffrnc and thi ffct CPT vanih aftr a crtain apct ratio; i.., th diffrnc twn th two thori com contant yond a crtain apct ratio. In addition, th diffrnc twn th two thori incra with incraing mod numra alrady mntiond aov. Finally, it i orvd that th prcnt diffrnc dcra with an incra in th thickn ratio from to. Tal : Comparion of th non-dimnional natural frqunci h G of an iotropic quar nanoplat. frqunci l i TST[] FST[] CPT Prnt

9 Vol. 8, No., Jun 7 Tal : Th non-dimnional natural frqunci h C of an FG nanoplat calculatd y RPT. Mod numr m n l i (nm) Fig.. Variation in non-dimnional frquncy with nonlocal paramtr for diffrnt apct ratio

10 M. Goodarzi t al. Fig.. Variation in non-dimnional frquncy with nonlocal paramtr for iotropic and FG nanoplat Fig..Variation in non-dimnional frquncy with nonlocal paramtr for diffrnt mod frqunci.

11 Vol. 8, No., Jun 7 Fig. 5. Variation in non-dimnional frquncy with thickn ratio for diffrnt nonlocal paramtr. 6. Concluion Bad on th two-varial rfind plat thory and th nonlocal plat thory, th mall cal ffct on th fr viration of FG rctangular nanoplat wa invtigatd. Th oundary condition along all th four dg wr aumd to imply upportd. Th govrning quation of th plat wr olvd uing th Navir mthod. Th ffct of nonlocal paramtr, thickn ratio, apct ratio, and mod numr on th natural frqunci of FG nanoplat wr invtigatd. Th following concluion may drawn from th finding of th prnt tudy: (). Natural frquncy dcra with apct ratio incraing from to. (). Th natural frqunci of FG nanoplat ar alway found to mallr than thir iotropic countrpart for oth CPT and RPT and th rult otaind y RPT ar alway found to mallr than tho of CPT. (). Th ffct of mall lngth cal i highr for highr mod. Furthrmor, th diffrnc twn th rult from th two thori incra with incraing mod numr. Fig. 6. Prcnt diffrnc in non-dimnional frquncy twn RPT and CLPT with apct ratio for diffrnt mod frqunci

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13 Vol. 8, No., Jun 7 [5] A. C. Eringn, On diffrntial quation of nonlocal laticity and olution of crw dilocation and urfac wav, Journal of Applid Phyic, Vol. 5, No. 9, pp. 7-7, 98. [6] R. Nazmnzhad, S. Hoini-Hahmi, Fr viration analyi of multi-layr graphn nanorion incorporating intrlayr har ffct via molcular dynamic imulation and nonlocal laticity, Phyic Lttr A, Vol. 78, No., pp. 5-,. [7] H. G. Craighad, Nanolctromchanical ytm, Scinc, Vol. 9, No. 596, pp. 5-6, Nov,. ng [8] R. Anari, B. Arah, H. Rouhi, Viration charactritic of mddd multi-layrd graphn ht with diffrnt oundary condition via nonlocal laticity, Compoit Structur, Vol. 9, No. 9, pp. 9-9,. [9] J. N. Rddy, Nonlocal nonlinar formulation for nding of claical and har dformation thori of am and plat, Intrnational Journal of Enginring Scinc, Vol. 8, No., pp , //,. [] A. Aadi, B. Farhi, Siz dpndnt taility analyi of circular ultrathin film in latic mdium with conidration of urfac nrgi, Phyica E: Low-dimnional Sytm and Nanotructur, Vol., No. 5, pp. -7, //,. [] A. C. Eringn,, Nonlocal Continuum Fild Thori, Springr Nw York, [] M. Aydogdu, Axial viration analyi of nanorod (caron nanotu) mddd in an latic mdium uing nonlocal laticity, Mchanic Rarch Communication, Vol., pp. -,. [] M. Mohammadi, A. Farajpour, M. Goodarzi, H. Mohammadi, Tmpratur ffct on viration analyi of annular graphn ht mddd on vico-patrnak foundation, Journal of Solid Mchanic, Vol. 5, No., pp. 5-,. [] M. Mohammadi, M. Goodarzi, M. Ghayour, S. Alivand, Small cal ffct on th viration of orthotropic plat mddd in an latic mdium and undr iaxial in-plan pr-load via nonlocal laticity thory, Journal of Solid Mchanic, Vol., No., pp. 8-,. [5] M. Mohammadi, A. Farajpour, M. Goodarzi, Numrical tudy of th ffct of har inplan load on th viration analyi of graphn ht mddd in an latic mdium, Computational Matrial Scinc, Vol. 8, pp. 5-5,. [6] A. Hadi, A. Ratgoo, A. anhmhr, F. Ehani, Str and train analyi of functionally gradd rctangular plat with xponntially varying proprti, Indian Journal of Matrial Scinc, Vol.,. [7] M. Z. Njad, A. Hadi, Eringn' non-local laticity thory for nding analyi of idirctional functionally gradd Eulr Brnoulli nano-am, Intrnational Journal of Enginring Scinc, Vol. 6, pp. -9, 6. [8] M. Hoini, M. Shihaz, K. N. Tahan, A. Hadi, Str analyi of rotating nano-dik of varial thickn mad of functionally gradd matrial, Intrnational Journal of Enginring Scinc, Vol. 9, pp. 9-5, 6. [9] Ö. Civalk, Ç. mir, Bnding analyi of microtuul uing nonlocal Eulr Brnoulli am thory, Applid Mathmatical Modlling, Vol. 5, No. 5, pp. 5-67, 5//,. [] S. R. Ami, A. Farajpour, Viration charactritic of doul-pizolctricnanoplat-ytm, Micro & Nano Lttr, IET, Vol. 9, No., pp. 8-85,. [] R. Shimpi, H. Patl, A two varial rfind plat thory for orthotropic plat analyi, Intrnational Journal of Solid and Structur, Vol., No., pp , 6. [] B. Akgöz, Ö. Civalk, Fr viration analyi of axially functionally gradd taprd Brnoulli Eulr microam ad on th modifid coupl tr thory, Compoit Structur, Vol. 98, pp. -,. [] R. Aghaaai, J. Rddy, Nonlocal thirdordr har dformation plat thory with application to nding and viration of plat, Journal of Sound and Viration, Vol. 6, No., pp , 9. [] S. Sriniva, A. Rao, Bnding, viration and uckling of imply upportd thick orthotropic rctangular plat and laminat, Intrnational Journal of Solid and Structur, Vol. 6, No., pp. 6-8, 97. [5] J. Rddy, A rfind nonlinar thory of plat with tranvr har dformation, 5

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