Surface Effects on Free Vibration Analysis of Nanobeams Using Nonlocal Elasticity: A Comparison Between Euler-Bernoulli and Timoshenko

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1 Journal of Solid Mechanic Vol. 5, No. 3 (3) pp Surface Effect on Free Vibration Analyi of Nanobeam Uing Nonlocal Elaticity: A Comparion Between Euler-Bernoulli and Timohenko Sh. Hoeini Hahemi,,*, M. Fakher, R. Nazemnezhad School of Mechanical Engineering, Iran Univerity of Science and Technology, Narmak, , Tehran, Iran Center of Excellence in Railway Tranportation, Iran Univerity of Science and Technology, Narmak, Tehran, Iran Received 3 June 3; accepted Augut 3 ABSTRACT In thi paper, urface effect including urface elaticity, urface tre and urface denity, on the free vibration analyi of Euler-Bernoulli and Timohenko nanobeam are conidered uing nonlocal elaticity theory. To thi end, the balance condition between nanobeam bulk and it urface are conidered to be atified auming a linear variation for the component of the normal tre through the nanobeam thickne. The governing equation are obtained and olved for Silicon and Aluminum nanobeam with three different boundary condition, i.e. Simply-Simply, Clamped-Simply and Clamped-Clamped. The reult how that the influence of the urface effect on the natural frequencie of the Aluminum nanobeam follow the order CC<CS<SS while thi i not the cae for Silicon nanobeam. On the other hand, the influence of the nonlocal parameter i oppoite and follow the order SS<CS<CC. In addition, it i een that conidering rotary inertia and hear deformation ha more effect on the urface effect than the nonlocal parameter. 3 IAU, Arak Branch. All right reerved. Keyword: Surface effect; Nonlocal elaticity; Free vibration; Nanobeam; Euler-Bernoulli theory; Timohenko theory INTRODUCTION I N order to tudy the mechanical behavior of nanotructure, the urface effect and nonlocal elaticity are two important field which are invetigated by reearcher eparately, or imultaneouly. The urface of a olid i a region with mall thickne which ha different propertie from the bulk. If the urface energy-to-bulk energy ratio i large, for example in the cae of nanotructure, the urface effect cannot be ignored []. On the other hand, the nonlocal elaticity theory which i initiated in the paper of Eringen and Eringen and Edelen [-5] expree that the tre at a point i a function of train at all point in the continuum. To account for the effect of urface/interface on mechanical deformation, the urface elaticity theory i preented by modeling the urface a a two dimenional membrane adhering to the underlying bulk material without lipping [6, 7]. There are many tudie related to the wave propagation, tatic, buckling and free linear and nonlinear vibration analyi of nanobeam and carbon nanotube (CNT) baed on different beam theorie [8-]. For example, Wang et al. [9] tudied the urface buckling of a bending beam uing the urface elaticity theory. The correponding buckling wave number wa analytically obtained in their work. They alo reported that urface with * Correponding author. Tel.: ; Fax: addre: hh@iut.ac.ir (Sh. Hoeini-Hahemi). 3 IAU, Arak Branch. All right reerved.

2 9 Sh. Hoeini-Hahemi et el. poitive urface elatic modulu may buckle under compreion, while urface with negative urface elatic modulu are poible to wrinkle irrepective of the ign of urface train. The nonlocal continuum mechanic i uitable for modeling ubmicro- or nano-ized tructure becaue it avoid enormou computational effort when compared with dicrete atomitic or molecular dynamic imulation. Many reearcher have applied the nonlocal elaticity concept for the wave propagation [,], bending, buckling, and vibration [3-3] analye of beam-like element in micro- or nano electromechanical ytem. For example, Reddy [7] applied the Eringen nonlocal elatic contitutive relation to derive the equation of motion of variou kind of beam theorie (i.e., Euler Bernoulli, Timohenko, Reddy and Levinon) and propoed analytical and numerical olution on tatic deflection, buckling load, and natural frequencie of nano-beam. It can alo be mentioned to the Ref. [3-34] which they ued the nonlocal elaticity theory for analyi of the plate-like tructure. From literature, it can be found that mot of the work conidered the influence of the urface effect and the nonlocal effect eparately. There are a few paper in which both urface and mall cale effect on tatic and dynamic behavior of nanotructure and CNT are taken into account. Lee and Chang [35] tudied the urface and mall-cale effect on the free vibration analyi of a non-uniform nano-cantilever Euler-Bernoulli beam. Wang and Wang [36] conidered the influence of the urface effect on the free vibration behavior of imply upported Kirchhoff and Mindlin nanocale plate uing the nonlocal elaticity theory. Lei et al. [37] invetigated the urface effect on the vibrational frequency of double-walled carbon nanotube uing the nonlocal Timohenko beam model. The influence of the urface and mall-cale effect on electromechanical coupling behavior of piezoelectric nanowire i alo tudied in Ref. [38, 39]. It i reported in literature that the urface effect include the urface elaticity, the urface tre and the urface denity. In addition, ome literature introduce relation for atifying the balance condition between the nanotructure bulk and it urface. Mot of the tudie examine only the urface elaticity and tre effect and there i no work focuing on the influence of the urface denity and atifying the balance condition a well a the urface elaticity and tre effect. For example, Ghehlaghi and Haheminejad [] conidered the urface elaticity and tre on the nonlinear free vibration of imply upported Euler-Bernoulli nanobeam without atifying the balance condition. The imilar ituation can be found in Ref.[4] that the nonlinear free vibration of non-uniform nanobeam in the preence of the nonlocal effect a well a the urface elaticity and tre effect i tudied uing differential quadrature method (DQM). The other imilar work i the one done by Mahmood et al. [4]. In thi work,the tatic behavior of nonlocal Euler-Bernoulli nanobeam i conidered uing finite element approach. In the preent work, for conidering the urface and mall cale effect on the free vibration analyi of nanobeam with different end condition in detail, a comprehenive analytical model propoe to tudy all the urface effect, including the urface elaticity, the urface tenion and the urface denity, on the free vibration of nanocale Euler-Bernoulli and Timohenko beam, made from aluminum and ilicon, uing nonlocal elaticity. The urface denity i introduced into the governing equation auming a linear variation through the nanobeam thickne for the component of normal tre, σ zz. Thi alo atifie the balance condition between the nanobeam bulk and it urface. An exact olution i ued to obtain the natural frequencie of nanobeam. Latly, the urface and mall cale effect on the natural frequencie of nanobeam are examined for different boundary condition, nanobeam length and mode number. It i found out that conidering the urface denity and atifying the balance condition between the nanobeam bulk and it urface have coniderable influence on the natural frequencie of nanobeam. PROBLEM FORMULATIONS To obtain the governing equation of nanobeam in the preence of the urface effect via nonlocal elaticity, a nanobeam with length L ( x L), width b (-.5b y.5b) and height H=h (-h z h) i conidered.. Nonlocal elaticity theory and urface effect.. Nonlocal elaticity theory Nonlocal elaticity i firt conidered by Eringen [-5] aume that the tre field at a point x in an elatic continuum not only depend on the train field at the point but alo on train at all other point of the body. For homogeneou and iotropic elatic olid, the contitutive relation can be expreed in a differential form a: 3 IAU, Arak Branch

3 Surface Effect on Free Vibration Analyi of Nanobeam Uing Nonlocal Elaticity 9 ( τ L ) σij Cijmn εmn () where e a / L i a material contant that depend on internal (a) and external (L) characteritic length, e i a material contant, i the Laplacian operator and C i the fourth-order elaticity tenor. The nonlocal contitutive relation in Eq.(), for the macrocopic tre take the following pecial relation for nanobeam [7] xx xz xx E, ( xx xz G xz e a ) () x x.. Surface effect At the micro/nanocale, the fraction of energy tored in the urface become comparable with that in the bulk, becaue of the relatively high ratio of urface area to volume of nanocale tructure; therefore the urface and the induced urface force cannot be ignored. The contitutive relation of the urface layer S + and S - given by Gurtin and Murdoch [6, 4] can be expreed a: u u u u, u,,,, 3 3, (3) where,,,, τ are reidual urface tenion under uncontrained condition, λ and μ Lame contant on the urface S + and S - which can be determined from atomitic calculation [43], are the urface the Kronecker delta and u are the diplacement component of the urface S + and S -. If the top and bottom layer α have the ame material propertie, the tre train relation of the urface layer, i.e. Eq.(3), can be reduced to the following relation for nanobeam E u, E, u S S xx x, x nx n, x (4) where n denote the outward unit normal. The equilibrium relation for the urface layer can be expreed in term of the urface and bulk tre component a [44] T u (5) i, i i where i x, n, m; x, m ; denote the denity of urface layer; T i the contact traction on the contact urface between the bulk material and the urface layer; m i the tangent unit vector; and u denote the acceleration of the i urface layer in the i- direction. In claical beam theory, the tre component σ zz i neglected. However, σ zz mut be conidered to atify the urface equilibrium equation of the Gurtin Murdoch model. Following Lu et al. [45], it i aumed that the bulk tre σ zz varie linearly through the nanobeam thickne. Therefore, zz zz zz z zz zz H (6) Subtituting of Eq. (4) and Eq. (5) into Eq. (6) yield z zz nx, x nx, x un un nx, x nx, x un un H z H uz, xx uz, xx uz uz uz, xx uz, xx uz uz (7) 3 IAU, Arak Branch

4 93 Sh. Hoeini-Hahemi et el.. Governing equation for nanobeam.. Timohenko beam theory The bending moment and vertical force equilibrium equation including rotary inertia, hear deformation and the urface effect can be expreed a follow [5] dm xx, xzd Q uxzda ux zd (8) dx A dq nx, xnzd q( x) uzda unnzd dx A (9) where τ xx and τ nx are nonzero membrane tree due to urface energy; q(x) i the tranvere ditributed force; Q and M are the tre reultant defined a follow: Q da M zda A xz A xx () Bulk tre train relation of the nanobeam can be expreed a [5] E ; G () xx xx zz xz xz where E i the elatic modulu, ν i Poion ratio and G i the hear modulu. Defining the diplacement field a Timohenko beam theory [5] u z ( x, t) ; u w( x, t) () x z where (x, t) and w(x, t) denote the rotation of cro ection and vertical diplacement of mid-plane at time t, repectively. So, the nonzero train are given by (, ) (, ) u x x t ; ( u u x z w x t xx z xz ) ( ( x, t) ) (3) x x z x x Subtituting Eq. () and Eq. (3) into Eq. (4) and Eq. (7), then ubtituting the reult into Eq. (), the contitutive relation are obtained a follow z w xx E( z ) ( w) x H x z w w ( w ) ; xz Gk ( ) zz H x x w w xx z( ) ; nx nz ; nx x x x (4) where k denote hear correction coefficient. Subtituting Eq. (4) into Eq. () and conidering Eq.(), the nonlocal contitutive relation can be written a: * I w I w ( ) M xxzd EI E I x x H x H t (5) w w ( ) Q n d kga b x nx z x x (6) 3 IAU, Arak Branch

5 Surface Effect on Free Vibration Analyi of Nanobeam Uing Nonlocal Elaticity 94 where, we have I * =bh +4h 3 /3 for rectangular cro ection. Subtituting Eq. (5) and Eq. (6) in Eq. (8) and Eq. (9) yield the nonlocal governing equation for Timohenko nanobeam in preence of the urface effect a follow w ( ) w w A b ( ) kga b (7) x t x x x 3 3 * w * I w I w I I ( ) ( ) kga EI E I 3 (8) x t x x H x H xt It can be een that Eq. (7) and (8) can be reduced to the governing equation of Timohenko nanobeam with only the nonlocal effect by etting the urface effect equal zero (i.e. E ); and in the cae of, Eq. (7) and (8) will be reduced to the governing equation of Timohenko nanobeam with only the urface effect. Finally, the reultant moment and hear force of a nanobeam cro ection, including the urface layer contribution and the nonlocal parameter, are given by 3 T * I w I w * w w M EI E I I I A b b (9) x H x H t xt t x 3 ( w ) w T ( ) w Q kga b A b () x x xt.. Euler-Bernoulli beam theory In the cae of Euler Bernoulli beam theory, the rotational diplacement of the cro ection i related to the lope of the vertical deflection and the hear deformation effect i ignored, i.e. w x [46, 47]. In addition, the rotational inertia effect are neglected. Thu, the governing equation of nonlocal Euler Bernoulli nanobeam in the preence of the urface effect can be obtained from Eq. (7) and Eq. (8) in term of the tranvere deflection a: 4 4 * I w w I w w EI E I b 4 ( A b ) () H x x H x t x t where the M E and Q E are given by E * I w I w w w M EI E I ( A b) b () H x H t t x E * I w I w w w Q EI E I b 3 ( A b ) (3) H x H xt x xt According to Eq. (9), (), () and (3), it can be expected that the nonlocal characteritic equation of nanobeam in preence of the urface effect will be different from the claic one which are tated in Ref. [46]. 3 SOLUTIONS FOR FREE VIBRATION OF NANOBEAMS In thi ection, the approach of olving governing equation of nanobeam i preented for three different boundary condition, imply-imply (SS), clamped-imply (CS) and clamped-clamped (CC). 3 IAU, Arak Branch

6 95 Sh. Hoeini-Hahemi et el. 3. Free vibration of timohenko nanobeam In order to olve the nonlocal governing equation of Timohenko nanobeam in the preence of the urface effect, we conider (x, t) and w(x, t) a follow: w( x, t) W ( x) e ( x, t) Φ( x) e it it (4) where, λ denote the vibration frequency. Conidering Eq.(4), the olution of Eq. (7) and Eq. (8) can be expreed a: W( x) c in( x) c co( x) c inh( x) c coh( x) (5) 3 4 Φ( x) c in( x) c co( x) c inh( x) c coh( x) (6) where 4 4 R Q Q kk, 6, ( ), ( ) (7) and Q k k k k, R k k k k k k k, k ( I I ) kga * * * I I k ( I I ) ( EI E I ), k3, k4, k5 kga H H k ( A b ), k ( A b ) kga b 6 7 (8) Subtituting Eq. (5) and Eq. (6) into Eq. (8) give c k c, c k c, c k c, c k c 5 a 6 a 7 b 4 8 b 3 (9) where k a ( ) ( k k k k k k ) , k b k k k k (3) The boundary condition of SS, CS and CC nanobeam are given, repectively, by W () M() ; W ( L) M( L) W () Φ() ; W ( L) M( L) W () Φ() ; W ( L) Φ( L) Conidering mentioned boundary condition, the characteritic equation of SS, CS and CC nanobeam are obtained, repectively, a: (3) in( L)inh( L) k in( L)coh( L) co( L)inh( L) a kb ( a b in( )inh( ) a b co( )co k k ) L L k k L h( L) (3) 3 IAU, Arak Branch

7 Surface Effect on Free Vibration Analyi of Nanobeam Uing Nonlocal Elaticity Free vibration of euler-bernoulli nanobeam In order to olve the nonlocal governing equation of Euler-Bernoulli nanobeam in the preence of the urface effect, i.e. Eq.(), it i aumed that i t w(x,t) W( x) e (33) Conidering Eq.(33), the olution of Eq. () can be expreed a: W( x) c in( x) c co( x) c inh( x) c coh( x) (34) 3 4 where ) 4 4 Q ( R A b,, ( ), ( ) Q (35) and * I Q ( EI E I ) H I (36) R b ( A b ) H According to the mentioned boundary condition, Eq. (3), the characteritic equation of Euler Bernoulli nanobeam with SS, CS and CC end condition are obtained, repectively, a: Sin( L) ( Sin( L)Coh( L) Sinh( L)Co( L) 3 ) ( ) ( )Sin( L)Sinh( L) o( L)Coh( L) C (37) A previouly mentioned, it i een that the nonlocal characteritic equation of nanobeam in the preence of the urface effect are different from the claic one, except for SS boundary condition. Now, the only unknown value in the characteritic equation i the natural frequency (i.e., ).Uing a computer code written in mathematica oftware, the characteritic equation are olved and the natural frequencie are obtained. 4 NUMERICAL RESULTS AND DISSCUSSION In order to invetigate the influence of the urface effect and the nonlocal parameter on the free vibration of Euler- Bernoulli and Timohenko nanobeam, in thi ection the numerical reult of nanobeam made of Aluminum (Al) and Silicon (Si) are preented. The relevant bulk material propertie are, E = 7GPa, ν =.3 and =7Kg/m 3 for Al [48] and E = GPa, ν =.4 and =37 Kg/m 3 for Si [49]. Furthermore, the urface material propertie are E =5.88N/m, τ =.98 N/m and = kg/m for Al [5]; and E =-.6543N/m, τ =.648 N/m, and =3.7-7 kg/m for Si [5]. It hould be noted that the value of the nonlocal parameter i et to be 4 nm for the cae conidering the nonlocal elaticity. In the following dicuion, NSF, SF, NF and CF tand for the natural frequency of the Euler-Bernoulli nanobeam with conidering both of the urface effect and the nonlocal parameter, the natural frequency with only 3 IAU, Arak Branch

8 97 Sh. Hoeini-Hahemi et el. the urface effect, the natural frequency with only the nonlocal parameter and the claical natural frequency of the Euler-Bernoulli nanobeam, repectively. Alo, T indicate the relevant reult for Timohenko nanobeam. 4. Comparion tudy To confirm the reliability of the preent formulation and reult, comparion tudie are conducted for the natural frequencie of the Euler-Bernoulli (EBT) and Timohenko (TBT) nanobeam. Firtly, the accuracy of the nonlocal natural frequencie i invetigated [7,4], then the validity of the natural frequencie of nanobeam with conidering the urface effect i tudied [5]. In Table and, fundamental non-dimenional natural frequencie, L A and ( L A ), of nonlocal nanobeam (without conidering the urface effect) are lited. In EI EI addition, the reult given by Reddy et al. [7] (Table.) and Malekzadeh and hojaee [4] (Table.) are provided for direct comparion. It i oberved that the preent reult agree very well with thoe given by Ref. [7, 4]. Table Comparion of non-dimenional fundamental natural frequencie ( L A ) of Euler-Bernoulli (EBT) and Timohenko EI (TBT) nanobeam (L=, E = 3 6, v =.3, =, E = τ = =) Beam type EBT TBT Thickne ratio h/l Nonlocal parameter µ Simply-Simply Preent Ref.[7] Clamped-Clamped Preent Ref.[7] Table Comparion of non-dimenional fundamental natural frequencie ( L A ) of Euler-Bernoulli (EBT) and Timohenko EI (TBT) nanobeam (L=, h=.l, E = 3 6, v =.3, =, µ=, E = τ = =) Beam type Simply-Simply Clamped-Simply Clamped-Clamped EBT TBT EBT TBT EBT TBT Preent Numerical[4] Table 3 Comparion of the firt four natural frequencie of the SS and CC Euler-Bernoulli (Thin nanobeam with L= nm) and Timohenko (Thick nanobeam with L=5 nm) nanobeam incorporating the urface effect when, µ= and b=h=3 nm. t (GHz) nd (GHz) 3 rd (GHz) 4 th (GHz) Beam type Preent Ref.[5] Preent Ref. [5] Preent Ref. [5] Preent Ref. [5] SS CC Thin Thick Thin Thick IAU, Arak Branch

9 Surface Effect on Free Vibration Analyi of Nanobeam Uing Nonlocal Elaticity 98 Alo, a comparative tudy for evaluation of the firt four natural frequencie between the preent olution, without conidering the nonlocal parameter (i.e., µ=), and the reult given by Liu and Rajapake [5] i carried out in Table 3 for Euler-Bernoulli (thin) and Timohenko (thick) nanobeam made of ilicon with SS and CC boundary condition. The reult given in Table 3 are obtained in the preence of the urface effect. Table 3 confirm the reliability of the preent formulation and reult. 4. Benchmark reult Firtly, in order to invetigate the influence of conidering the urface denity and atifying the balance condition between the nanobeam bulk and it urface, the fundamental natural frequencie of Al nanobeam, normalized with repect to the fundamental natural frequency without the urface and mall cale effect, i.e. CF and TCF, veru the nanobeam length are preented in Fig. (a) and (b) for two different cae. In Cae, the reult are obtained without conidering the urface denity and atifying balance condition while in Cae, the urface denity i conidered and the balance condition i atified. In Fig. (a), only the urface effect are conidered wherea in Fig. (b) the nonlocal parameter a well a the urface effect i conidered. Alo, b=h=.l. Comparion between the Cae curve with thoe of Cae indicate that there are ignificant difference between the reult of two cae, epecially in low length. It can be een that all of the Cae curve are located bellow the Cae curve, implying that conidering the urface denity and atifying the balance condition lead to decreaing the natural frequency of nanobeam. In other word, conidering the mentioned parameter make the nanobeam more flexible. The behavior i oberved in both of Fig. (a) and (b). Next, the urface and mall cale effect on the natural frequencie are examined in Fig. 3 and 4 for variou value of the nanobeam length. Fig.3 (a-d) how variation of frequency ratio of Al nanobeam veru the nanobeam length. In Fig. 3(a) and (b), the influence of the urface effect and the nonlocal parameter on the natural frequencie are eparately tudied while in Fig. 3(c) and (d), the influence of the urface effect and the nonlocal parameter are imultaneouly invetigated. It hould be noted here that in Fig. 3(a-c), the nanobeam cro ection i conidered to be dependent on the nanobeam length, b=h=.l, wherea in Fig. 3(d), the nanobeam cro ection i conidered to be contant, b=h=.5 nm. It i een from Fig. 3(a) that conidering the urface effect increae the natural frequencie of Euler-Bernoulli and Timohenko nanobeam made of Al while Fig. 3(b) illutrate that the nonlocal parameter how a decreaing effect on the natural frequencie. In addition, Fig. 3(a) how that, for both nanobeam type, the influence of the urface effect on the natural frequency of Al nanobeam follow the order CC<CS<SS, implying that uing the ofter boundary condition caue an increae in the influence of the urface effect. However, the ituation i revere when only the nonlocal effect i conidered. In thi cae, Fig. 3(b)diplay that the influence of the nonlocal parameter for nanobeam with tiffer boundary condition are more pronounced and follow the order SS<CS<CC. Comparion between Euler-Bernoulli and Timohenko frequency ratio curve in Fig. 3(a) and (b) how that the increaing influence of the urface effect and the decreaing influence of the nonlocal parameter are more pronounced for Euler-Bernoulli nanobeam than Timohenko one, indicating that regarding hear deformation and rotary inertia caue a reduction in the influence of the urface effect and the nonlocal parameter. Moreover, thee difference are more coniderable for nanobeam with tiffer boundary condition. Fig. 3(c) illutrate that the effect of the nonlocal parameter become dominant for horter nanobeam with tiffer end condition while it i not the cae for the influence of the urface effect. In other word, the influence of the urface effect become dominant for nanobeam with ofter boundary condition and for all value of the nanobeam length. A een from Fig. 3(a-c), by increaing the nanobeam length all frequency ratio curve gradually approach the claic olution line which entail an expected decreae in the influence of the urface and mall cale effect. However, in Fig. 3(d), increaing the nanobeam length increae the urface and mall cale effect. Thi i due to thi fact that ince in Fig. 3(a-c) the cro ection ize depend on the length of the nanobeam, i.e. b=h=.l, the urface energy-to-bulk energy ratio decreae by increaing the length of nanobeam and the internal length cale become much maller than the nanobeam ize while thi i the other way round for Fig. 3(d). Here, it i worth to note that the general concluion of ome literature [8,, 4] i that increaing the nanobeam length reult in decreaing the influence of the urface effect while thi concluion i not generally true becaue it depend on the relation between the nanobeam cro ection and length a i hown in Fig. 3(a-d). Similar finding a thoe found in Fig. 3(a-d) for nanobeam made of Al can be oberved in Fig. 4(a-d) for nanobeam made of Si. It i worth noting here that unlike Fig. 3(a-d) where the urface effect and the nonlocal parameter have oppoite influence on the fundamental natural frequencie of Al nanobeam, it can be oberved 3 IAU, Arak Branch

10 99 Sh. Hoeini-Hahemi et el. from Fig. 4(a-d) that the urface effect and the nonlocal parameter exhibit imilar influence on the fundamental natural frequencie of Si nanobeam. Thi effect can be attributed to the negative value of the Si urface elaticity. Latly,variation of the frequency ratio veru mode number for Al and Si imply upported nanobeam are preented in Fig. 5(a) and 5(b), repectively, when L=5 nm and b=h=.5 nm. From Fig. 5(a) and 5(b), it i oberved that at low mode number the NSF/CF curve i cloe to the SF/CF curve and away from the NF/CF curve. Thi implie that the influence of the urface effect on the natural frequencie i dominant at low mode number. A the mode number increae, the difference between the NSF/CF and SF/CF curve become more and the NSF/CF curve approache the NF/CF one. Thi indicate that by increaing the mode number the urface effect decreae and the nonlocal parameter effect become dominant. It i worthwhile to point out that the aluminum SF/CF curve doe not reach to the claic olution curve a the mode number increae. Thi behavior i alo oberved from Fig. (b) in Ref. [] for the nonlinear free vibration of Euler-Bernoulli nanobeam. Ref. [] conclude that thi behavior may be due to the nonlinear effect. But the imilar behavior i alo een from Fig. 5(a) for the linear free vibration of Al nanobeam with the urface effect. Therefore, the behavior i not due to the nonlinear effect. A final point to note i that the frequency ratio curve of both Euler-Bernoulli and Timohenko nanobeam made of Al and Si with only the nonlocal parameter effect (i.e., NF/CF and TNF/TCF) have the ame trend for all mode number wherea thi i not the cae for the frequency ratio curve with only the urface effect. At lower mode number, the frequency ratio curve of Euler-Bernoulli and Timohenko nanobeam with only the urface effect have the ame trend, indicating that conidering rotary inertia and hear deformation doe not change the influence of the urface effect on the natural frequency. While by increaing the mode, number the difference between the frequency ratio curve of Timohenko and Euler-Bernoulli nanobeam become more. It can be deduced from thi difference that conidering rotary inertia and hear deformation decreae the influence of the urface effect. Thi reduction i uch a way that at higher mode number the urface effect decreae the natural frequencie of the Al Timohenko nanobeam. Fig. Schematic of problem. (a) (b) Fig. Variation of the frequency ratio of Aluminum nanobeam veru the length with b=h=.l and conidering, (a) only urface effect, (b) both of the urface effect and nonlocal parameter. Cae: without urface denity and atifying balance condition, Cae: with urface denity and atifying balance condition. 3 IAU, Arak Branch

11 Surface Effect on Free Vibration Analyi of Nanobeam Uing Nonlocal Elaticity 3 (a) (b) (c) (d) Fig.3 Variation of the frequency ratio of Aluminum nanobeam veru the length with conidering, (a) urface effect, b=h=.l, (b) nonlocal parameter, b=h=.l, (c) both of the urface effect and nonlocal parameter b=h=.l, (d) both of the urface effect and nonlocal parameter, b=h=.5 nm. 3 IAU, Arak Branch

12 3 Sh. Hoeini-Hahemi et el. (a) (b) (c) (d) Fig.4 Variation of the frequency ratio of Silicon nanobeam veru the length with conidering, (a) urface effect, b=h=.l, (b) nonlocal parameter, b=h=.l, (c) both of the urface effect and nonlocal parameter b=h=.l, (d) both of the urface effect and nonlocal parameter, b=h=.5 nm. (a) (b) Fig.5 Variation of the frequency ratio of imply upported nanobeam veru mode number when, L=5 nm and b=h=.5 nm, (a) Aluminum, (b) Silicon. 3 IAU, Arak Branch

13 Surface Effect on Free Vibration Analyi of Nanobeam Uing Nonlocal Elaticity 3 5 CONCLUSIONS The influence of the urface and nonlocal effect on the vibrational behavior of Euler-Bernoulli and Timohenko nanobeam are tudied for three different boundary condition, SS, CS and CC. The nanobeam are conidered to be made of Al with poitive urface elaticity and Si with negative urface elaticity. A linear variation for the normal tre along the nanobeam height i aumed to atify the balance condition between the nanobeam bulk and it urface layer. Furthermore, the urface denity in addition to the urface elaticity and the urface tenion i introduced into the governing equation where it wa not conidered in the previou work. Following concluion are made from thi tudy:. Conidering the urface denity and atifying the balance condition between the nanobeam bulk and it urface caue ignificant difference in comparion with neglecting them.. The rotary inertia and hear deformation lead to a reduction in the influence of the urface effect on vibration frequencie of nanobeam. Thi i more ignificant in the cae of tiffer boundary condition than the ofter one. 3. The rotary inertia and hear deformation how notable effect on the influence of the urface effect on the natural frequencie at high mode number. 4. The rotary inertia and hear deformation do not how coniderable effect on the influence of the nonlocal parameter on the natural frequencie of imply-imply nanobeam. While in the cae of clamped-imply and clamped-clamped nanobeam, regarding hear deformation and rotary inertia caue a reduction in the decreaing effect of nonlocal parameter. 5. The urface-to-bulk ratio of nanobeam i an important parameter in determining the influence of the urface effect o that it can be aid: if an increae in the nanobeam length lead to decreaing the urfaceto-bulk ratio, the influence of the urface effect on frequency ratio will be diminihed. Otherwie, the influence of the urface effect on frequency ratio will increae a the nanobeam length increae. REFERENCES [] He L., Lim C., Wu B., 4, A continuum model for ize-dependent deformation of elatic film of nano-cale thickne, International Journal of Solid and Structure 4(3): [] Eringen A.C., 97, Nonlocal polar elatic continua, International Journal of Engineering Science ():-6. [3] Eringen A.C., Edelen D., 97, On nonlocal elaticity, International Journal of Engineering Science (3): [4] Eringen A.C., 983, On differential equation of nonlocal elaticity and olution of crew dilocation and urface wave, Journal of Applied Phyic 54(9): [5] Eringen A.C.,, Nonlocal Continuum Field Theorie, Springer. [6] Gurtin M., Murdoch A.I., 975, A continuum theory of elatic material urface, Archive for Rational Mechanic and Analyi 57(4):9-33. [7] Gurtin M., Weimüller J., Larche F., 998, A general theory of curved deformable interface in olid at equilibrium, Philoophical Magazine A 78(5):93-9. [8] Agharifard Sharabiani P., Haeri Yazdi M.R., 3, Nonlinear free vibration of functionally graded nanobeam with urface effect, Compoite Part B: Engineering 45(): [9] Wang G., Feng X., Yu S., 7, Surface buckling of a bending microbeam due to urface elaticity,europhyic Letter 77(4):44. [] Aadi A., Farhi B.,, Size-dependent longitudinal and tranvere wave propagation in embedded nanotube with conideration of urface effect, Acta Mechanica (-):7-39. [] Fu Y., Zhang J., Jiang Y.,, Influence of the urface energie on the nonlinear tatic and dynamic behavior of nanobeam, Phyica E: Low-Dimenional Sytem and Nanotructure 4(9): [] Ghehlaghi B., Haheminejad S.M.,, Surface effect on nonlinear free vibration of nanobeam, Compoite Part B: Engineering 4(4): [3] Guo J.-G., Zhao Y.P., 7, The ize-dependent bending elatic propertie of nanobeam with urface effect, Nanotechnology 8(9):957. [4] Hoeini-Hahemi S., Nazemnezhad R., 3, An analytical tudy on the nonlinear free vibration of functionally graded nanobeam incorporating urface effect, Compoite Part B: Engineering 5:99-6. [5] Liu C., Rajapake R.,, Continuum model incorporating urface energy for tatic and dynamic repone of nanocale beam, Nanotechnology 9(4):4-43. [6] Nazemnezhad R., Salimi M., Hoeini-Hahemi S., Sharabiani P.A.,, An analytical tudy on the nonlinear free vibration of nanocale beam incorporating urface denity effect, Compoite Part B: Engineering 43: IAU, Arak Branch

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15 Surface Effect on Free Vibration Analyi of Nanobeam Uing Nonlocal Elaticity 34 [49] Zhu R., Pan E., Chung P.W., Cai X., Liew K.M., Buldum A., 6, Atomitic calculation of elatic moduli in trained ilicon, Semiconductor Science and Technology (7):96. [5] Miller R.E., Shenoy V.B.,, Size-dependent elatic propertie of nanoized tructural element, Nanotechnology (3):39. 3 IAU, Arak Branch