at muatTheicave aodtes

ents o(NEM

tic andrials aDue toscalee, therials anistic s

tructur

response of nanoscale beams based on the GurtinMurdoch

eveloped a 3D FEn nanowires. Liut of the thin andong and Pan [21]eld in nano-

.ary and loading

efcient numerical model is not available at this moment for the

Contents lists available at SciVerse ScienceDirect

.e

Finite Elements in A

Finite Elements in Analysis and Design 74 (2013) 2229

developed by Lu et al. [11] by using the GurtinMurdoch surfaceE-mail addresses: wangbl2001@hotmail.com, wangbl@hitsz.edu.cn (B.L. Wang).theory. Assadi et al. [13] studied the size-dependent dynamic analysis of nanoscale plates. In the present paper, a nite elementmodel is developed to analyze the bending behavior of nanoplateswith consideration of surface residual stress and surface elasticity.The present FE model is based on the plate mathematical model

0168-874X/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nel.2013.05.007

n Corresponding author. Tel.: +86 755 26033490.dependent thin plate model by complementing Lim and He'smodel. Liu and Rajapakse [12] studied the static and dynamic

conditions are often used in NEMS. Such complicated structuralsystems cannot be studied by analytical models. However, anIt is essential to nd an efcient tool to analyze the mechanicsbehavior of nanoscale structures. Gurtin and Murdoch [8,9]proposed a modied continuum theory which incorporates thesurface/interface effects into the traditional continuum mechanics.This theory has been widely used to study the mechanics responseof nanoscale structures. For examples, Lim and He [10] proposed acontinuum model to analyze the bending behaviors of thin elasticnanoplate of nanoscale thickness. Lu et al. [11] proposed a size-

studied the mechanics of nanoscale inhomogenematrix by proposed a FE model. Feng et al. [19] dmodel to study the resonant properties of silicoet al. [20] proposed a Galerkin-type nite elementhick beam with the surface effect. In addition, Dproposed a BE method to analyze the stressinhomogeneities with the surface/interface effect

Nanoplates with complex geometry, boundmethod is difcult to apply to the analysis of NEMS with complexgeometries, due to the limit of the available computational power.

continuum FE model to take into account the surface elastic effect(based on the GurtinMurdoch theory). Tian and Rajapkse [18]

ities in an elasticNanoscale plates are key componfor nano-electromechanical systemimportance to understand the stabehaviors of these advanced matedesign and manufacture of NEMS.volume ratio, structures at nanodependent behavior [24]. Thereforconsidered for the analysis of matecale. Some researchers applied atomsize-dependent properties of nanosf actuators and sensorsS) [1]. Naturally, it isdynamic mechanical

nd structures for thetheir high surface-to-

show signicant size-surface effect must bed structures at nanos-imulation to study thees [57]. However, this

effect on the buckling and vibration behaviors of nanowires. Fanget al. [16] studied the inuence of the surface/interface effecton the dynamic stress of two interacting cylindrical nano-inhomogeneities under compressive waves based on the surface/interface elasticity theory.

Analytical solutions are impossible for the structures withcomplex geometry and boundary conditions. It is necessary todevelop a versatile numerical model, such as, the nite element(FE) method and the boundary element (BE) method. Wei et al.[17] proposed a kind of surface element for a two dimensional1. Introduction response of nanoplates by using the GurtinMurdoch theory.Wang and Feng [14,15] studied the inuence of the surfaceA nite element model for the bendingplates with surface effect

K.F. Wang, B.L. Wang n

Graduate School at Shenzhen, Harbin Institute of Technology, Harbin 150001, PR China

a r t i c l e i n f o

Article history:Received 19 December 2012Received in revised form10 April 2013Accepted 11 May 2013Available online 21 June 2013

Keywords:Nanoscale plateFinite element methodSurface residual stressSurface elasticity

a b s t r a c t

A continuum nite elemenis developed. Governing eqnite element technique.solutions. The results indfrequencies of the plate haof the plates. The present mbehaviors of nanoscale plaproperties.

journal homepage: wwwnd vibration of nanoscale

odel for the nanoscale plates considering the surface effect of the materialions for Kirchoff and Mindlin nanoplates are derived by using the Galerkinmodel is veried by comparing the results with available analytical

te that, depending on the boundary conditions, the deections anddramatic dependence on the residual surface stress and surface elasticityel is an efcient tool for the analysis of the static and dynamic mechanicalwith complex geometry, boundary and loading conditions and material

& 2013 Elsevier B.V. All rights reserved.

lsevier.com/locate/finel

nalysis and Design

K.F. Wang, B.L. Wang / Finite Elements in Analysis and Design 74 (2013) 2229 23elasticity theory. The accuracy and convergence of the present

Fig. 1. (a) Four-node plate element; (b) eight-node plate element.

nite element model are veried by comparing the results withthe available analytical solutions. The model is used to investigatethe inuence of residual surface stress and surface elasticity onbending and free vibration of nanoplates with different boundaryconditions.

2. Finite element formulation

The static equilibrium equations for the bulk of the platewithout considering body force are sij;j 0, where sij denotestresses of the bulk. According to Ref. [8], the surface stressessatisfy the following relations:

7i;s7i3 0 1

where 7i denote the surface stresses on the surface S7 . Using

sij;j 0 and Eq. (1) we can obtain the equilibrium equations ofplate with the surface effect [11]

Nni; q I ui 2a

Mn;N3 J u 2b

where I R h=2h=2 dz, J R h=2h=2 z2 dz, Nni Ni i i, Mn M h=2, and Nij

R h=2h=2 sij dz and Mij

R h=2h=2 sijz dz.

According to Refs. [4,8], linear constitutive equations for thesurface are

0 Css; 3 0u3; 3

where0 , Cs and

s are the receptivity, the residual surface

stresses, the surface elastic constants and surface strains. Both 0and Cs can be obtained from atomistic calculations.2.1. Static bending of Kirchhoff plate

According to the Kirchhoff plate theory, the displacementcomponents are u zu3; and u3 w. Using Eq. (2), we obtainedMnx;xx 2Mnxy;xy Mny;yy 2xxw;xx yyw;yy q 0 4

where fMn;xx Mn;yy Mn;xy gT D h2=2Cs, fw;xx w;yy

2w;xygT , and the material property matrices D and Cs are givenin Appendix A. For static bending of the Kirchhoff plate, applyingGalerkin's weighted residual method to Eq. (4) gives

AMnx;xx 2Mnxy;xy Mny;yy 2xxw;xx yyw;yy qw dA 0 5

Using Green's theorem, we get the weak form of Eq. (5) as

AMnxw;xx 2Mnxyw;xy Mnyw;yydA 2Axxw;xx yyw;yyw dA Aqw dA

ZSVnxnx Vnynyw dS

ZSMnxnx Mnxynyw;x dS

ZSMnxynx

Mnynyw;y dS 0 6

where Vnx Mnx;x Mnxy;y and Vnx Mnxy;x Mny;y.The boundary conditions are usually expressed in terms of

directions that are normal and tangent to the boundaries. Theseare the derivatives in the normal direction w=n and in thetangential direction w=T . Here n is the outward unit vectornormal to the boundary of the plate, whose components are nxand ny, T is the unit vector tangent to the boundary of the plate,whose x and y components are ny and nx. By these denitions,wn nxwx nywy, wT nywx nxwy and n2x n2y 1. The lasttwo boundary integrals in Eq. (6) can now be written asZ

SMnxnx MnxynynxwnnywT Mnxynx Mnynynyw;n nxw;T dS

ZSMnxn2x Mnyn2y 2Mnxynxnywn

Mnxnxny Mnynxny Mnxyn2xn2ywT dS

ZSMnnwn dS

ZSMnTwT dS 7

Finally, the weak form Eq. (6) can be rewritten as

ATDdAh2

2ATCsdA2Axxwxx yywyyw dA

Aqw dAZsVnn MnT ;T w ds

ZsMnnwn ds 0 8

where

w;xx w;yy 2w;xyh iT

9

Consider a four-node nite element with three nodal degrees offreedom per node, i.e.,w, x and y as shown in Fig. 1(a).

The element nodal displacement vector is

ue w1 x1 y1 ::: w4 x4 y4h iT

10

The displacement vector of the element and the vector ofelement curvatures are, receptivity, wNTue and BTue. Herethe shape function N and the geometry matrix B are given inAppendix A. Substituting Eq. (9) and the weighting functions(w-Ni and -BT) into Eq. (8), we obtainke kb ks k 11where

kb ABT DBdA 12a

ks h2 ABT CsBdA 12b2

N.

K.F. Wang, B.L. Wang / Finite Elements in Analysis and Design 74 (2013) 222924k 2xxAN;xxTN dA2yyAN;yyTN dA 12cThe element nodal force vector can be expressed as

qe R R

AqNTdA. Assembling the element stiffness and nodal force

vector, the global equilibrium equation of the system can beobtained as Ku q, where K, u and q denote, respectively, theglobal stiffness matrix, the nodal displacement vector and thenodal force matrix.

2.2. Dynamic behavior of Kirchhoff plate

For dynamic analysis, the deection is interpolated within aplate element as wNTueeit . The element mass matrix can beobtained as

me IANTNdA 13where I h. The same interpolation functions as static case areused to obtain the mass matrix. The global equilibrium equationsfor dynamic analysis can be obtained in following form:

M uKu q 14where M is global mass matrix. With the substitution ofut ueit , the free vibration eigenvalue problem can be obtainedfrom Eq. (14) as

K2Mu 0 15

2.3. Static bending of Mindlin plate

In the case of Mindlin plates, the shear deformation must beconsidered. The governing equations are [11]

Mnx;x Mnxy;yVx 0 16a

Mnxy;x Mny;yVy 0 16b

Vx;x Vy;y q 2xxwxx yywyy 0 16cApplying Galerkin's weighted residual method, the following

weighted residual equation can be obtained from Eq. (16):

AVx;xVy;yq2xxw;xx yyw;yywVx Mnx;x Mnxy;yx Vy Mnxy;x Mny;yydA 0 17

where w, x and y are the weight functions. Using Green'stheorem on the derivative terms, we can obtain the following:

AGTkGhGdA AbT DbdAh2

2AbT CsbdA

2Axxw;xx yyw;yyw dAAqw dA

ZSVnw dS

ZSMnnn dS

ZSMnTT dS 18

where

G G13 00 G23

" #

Mn;xx Mn

;yy Mn

;xy

n oT D b

h2

2Cs

b

Vx Vy T kGhG

b x;x y;y x;y y;x T

b x;x y;y x;y y;x T

G w;xx w;yyh iT

G w;xx w;yyh iT

Vn Vxnx Vyny, n nxx nyy and T nyx nxy. Here

nx and ny are the components of the outward unit vector normal toUsing Eqs. (18) and (19) and the sets of weighting functions, weobtained the following element stiffness matrix:

Ke kG kb ks k 21where

kG ABGTkGhBGdA 22a

kb ABbT DBbdA 22b

ks h2

2ABbTCsBbdA 22c

k 2xxANwTxxNwdA2yyANwTyyNwdA 22d

where BG, Bb and Nware shown in Appendix B. In order to avoidshear locking, the reduced-integration is used to the stiffness term.In present case, for an eight-node element, the shear stiffness(22a), the bending stiffness term (22b) and the surface stiffnessterm (22c) are obtained by using the 22 Gaussian integrationscheme. Finally, the element nodal force vector is qe AqNwTdA.

2.4. Dynamic analysis of Mindlin plate

In the dynamic analysis, the element mass matrix can beobtained by using the same interpolation functions as the staticcase. Therefore the element mass matrix can be expressed as

me AINwTNwdA JANx TNx Ny TNy dA 23

where Nx and Ny are shown in Appendix B. I h and J h3=12.

3. Numerical results and discussion

In order to assess the accuracy of the developed nite elementmethod, the static deection and frequency of a simply supportedplate made of Si is computed and compared with the analyticalresult. The results from the traditional model are also presented toget a quantitative assessment of the inuence of surface effects.It is should be noted that the surface elasticity constants andresidual surface tension can be determined by atomic simulations,which indicate that the elasticity constants and residual surfacestresses of some materials (i.e. FCC Al, diamond Si) can be eitherpositive or negative, depending on the crystallographic structureof the materials. In the present paper, the bulk and surface elasticconstants of Si (100) have been obtained by Shenoy [7] by usingthe embedded atom method, and the properties are: E 107 Gpa, 0:33, 0 2:7779 N=m, 0 4:4939 N=m and 0 0:6056the boundary of the nanoplate, k is the shear correction coefcient(k 2=12) and G is a shear module matrix.

A eight-node plate element is used with three degrees offreedom per node (w, x and y), as shown in Fig. 1(b). Thedisplacement is interpolated by using the shape function as

w x yn oT

Nue 19

where N is the matrix of interpolation functions, which is

N N1 0 0 N2 0 0 0 N1 0 0 0 N1 0 0 N2

264

375 20

where Ni (i1, 2,, 8) are shape functions, shown in Appendix B.For convenience, the sets of weighting functions (w, x and y) areexpressed as the columns of the matrix of interpolation functionsN=m.

Fig. 2. Deection of the Kirchhoff nanoplate with simply supported boundarycondition (h10 nm).

Fig. 3. Deection of the Mindlin nanoplate with simply supported boundarycondition (h10 nm).

Fig. 4. Deections of a cantilever Kirchhoff nanoplate considering surface elasticity

K.F. Wang, B.L. Wang / Finite Elements in Analysis and Design 74 (2013) 2229 253.1. Analytical solution for static bending

According to the Kirchhoff plate theory, for a nanoplate ofisotopic material property, the equilibrium equation can beexpressed as [11]

Def f4w202w qx; y 24where Def f Eh3=1212 Esh2=2 and Es 20 0. If thelateral loading applied on the plate is qx; y q0 sin mx=a sin ny=b, we can obtain the deection of the platefrom Eq. (24) as follows:

w q0 sin mx=a sin ny=bDef f 4m=a2 n=b22 202m=a2 n=b2

25

In the case of a uniformly distributed load q0 on the plate, wecan represent q0 in a double trigonometric series as

q0 16q02

3;5;:::

m 1

3;5;:::

m 1

1mn

sin mx=a sin ny=b 26

Using Eqs. (25) and (26), we can get

w 3;5;:::

m 1

3;5;:::

m 1

16q0 sin mx=a sin ny=bmn4Def f 4m=a2 n=b22 20m=a2 n=b2

27Eq. (27) is the solution of a plate with surface effect under a

load q0 uniformly distributed.According to the Mindlin plate theory, for a plate of isotopic

material property, the equilibrium equation can be expressedas [11]

Def f4w 1Def fkGh

2

qx; y 202w 28

where k is the shear factor. Following the above process for theKirchhoff plate, we can obtain the deection of a Mindlin platewith the surface effect under a uniformly distributed load q0 as

w 3;5;:::

m 1

3;5;:::

m 1

16q01 m=a2 n=b2 sin x=a sin y=bmn4Def f 1 20=k=G=h2m=a2 n=b22 20m=a2 n=b2

29

3.2. Finite element solution for static bending

We chose a square plate whose dimension is ab200 nm.The deections of the plate based on the Kirchhoff and Mindlinplate theories are computed by using 14 by 14 elements. Forcomparison, the classical solutions which neglect the surfacestress are also presented. Fig. 2 plots the deection of the simplysupported Kirchhoff plate under a uniformly distributed loadq0 1000 kN=m2. It can be seen that the nite element solutionagrees well with the analytical solution. The relative error iswithin 2%. Such accuracy is adequate for most practical applica-tions. It is also found that the surface effect reduces the deectionof the plate, this means that the surface effect makes the platestiffer.

Fig. 3 shows the deection of the simply supported Mindlinplate. Once again, the FEM solution agrees well with the analyticalsolution. The relative error is also less than 2%.

In order to show the effect of surface elasticity and residualsurface stress on the deection of plate with different boundaryconditions, the positive surface elasticity constants 0 2:7779 N=mand 0 4:4939 N=m and negative residual surface stress0 0:6056 N=m are assumed for the calculation. Figs. 4, 5 and 6show the deections of plate (at yb/2) with cantilever, simplysupported, and clamped boundary conditions, respectively. It is foundthat, for a positive residual surface stress, the cantilever nanoplate

exhibits a softer elastic behavior but the simply supported and and residual surface stress (h15 nm).

As shown in Fig. 5, an upward curvature occurs in the simplysupported nanoplate. This leads to a negative curvature and resultsin a negative distributed force which decreases the deection ofthe nanoplate. For the bending of nanoplates with all edgesclamped, both downward and upward curvatures occur in Fig. 6.Due to the upward curvature was dominant. Therefore, a positiveresidual surface stress (040) decreases the deection of clampednanoplate.

From Figs. 46, we know that a positive surface elasticityreduces the deections of the nanoplates, but the negative oneincreases the deections. Moreover, it is found that the cantileverplates are most signicantly inuenced by the surface effects,followed by the simply supported nanoplates, and clampednanoplates. Note that the inuence of surface elasticity andresidual surface stress on the deection of the Mindlin plates isquite similar to that for the Kirchhoff plates. Therefore, the resultsfor the Mindlin plates are not shown here.

In order to study the inuence of the shear deformation on the

K.F. Wang, B.L. Wang / Finite Elements in Analysis and Design 74 (2013) 222926Fig. 5. Deections of a simply supported Kirchhoff nanoplate considering surface

clamped nanoplates exhibit a stiffer elastic behavior (vice versa for anegative residual surface stress). The similar phenomenon has beenfound for the bending of nanowires [22]. This phenomenon can beexplained by LaplaceYoung equations sij s

ijninj 0 , where ni

is the unit vector normal to the surface. From LaplaceYoungequations, it can be seen that the signs of the curvature andresidual surface stress 0 during the static bending of the nanoplatesdetermine the stiffer or softer behavior of nanoplates. If the signs ofthe curvature and residual surface stress 0 are the same, it canresult in a positive distributed transverse force (which has the samedirection with the external load and will increase the deection ofthe bending nameplate). If the curvature and residual surfacestress 0 have an opposite sign, it will result in a negative distributedtransverse force. From Fig. 4, it can be seen that a positive residualsurface stress (040) increases the deection of the nanoplate. Thisis due to the fact that a downward curvature occurs in the cantilevernanoplate. This leads to a positive curvature and results in a positivedistributed force, which increases the deection of the nanoplate.In the sameway, we can explain that a positive residual surface stress(040) decreases the deection of nanoplates with simply sup-ported boundary conditions (vice versa for 0o0).

length and width of the nanoplate. It should be noted that Eq. (31)

elasticity and residual surface stress (h15 nm).

Fig. 6. Deections of a clamped Kirchhoff nanoplate considering surface elasticityand residual surface stress (h15 nm).can be obtained from our previous study [23] by neglecting thenonlocal effect ( 0).

Based on the Mindlin plate theory, the motion governingequation can be expressed as [11]

k2Ghx;x y;y 2w 202w I w 32abending deection of the plates, the solutions of the Kirchhoff andMindlin plates are plot in Fig. 7. It can be seen that the sheardeformation makes the plate softer. Therefore, the deection ofthe Mindlin plate is always larger than that of the Kirchhoff plate.However, the inuence of the shear deformation on the deectionof the plate decreases with decreasing thickness of the plate.

3.3. Analytical solution for free vibration

Based on the Kirchhoff plate theory, the motion governingequation can be expressed as [11]

Def f4w202wI w 30

where I h. Under the simply supported boundary condition, weobtained the natural frequency as

nm Def f 2n 2m2 202n 2m

I

s; n;m 1;2 31

where n n=a and m m=b, a and b are, respectively, theFig. 7. Deection of simply supported nanoplate with different values of thickness.

D x;xx 121x;yy

121 y;xy

h2

22u0 0x;xx 0 u0y;xy u0x;yy

k2Ghx wx J x 32b

D y;yy 121y;xx

121 x;xy

2

clamped boundary conditions, but decreases the fundamentalfrequencies of a cantilever nanoplate (and vice versa for a residualsurface stress).This trends were similar to the case of nanowirebending [22]. In addition, a comparison of Tables 1, 3, 4 withTable 5 suggests that, a negative surface elastic constant decreasesthe fundamental frequencies of nanoplates. As an additionalexample, for a simply supported nanoplate with the surface effect,the natural frequency has been derived as Eq. (31), and theclassical natural frequency for a simply supported nanoplate is2nm D2n 2m2=h. The difference is derived as

s2nm2nm

Esh2

22n 2m 20

" #2n 2m 34

For Si [100], Esh22n 2m=2o0 and 2040. When thickness his small, the surface effect will increase the frequencies. Asthickness h increases, Eq. (34) will be negative. At this situation,the frequency from the FEM with surface effect becomes smallerthan the classical result. Based on this, it is not difcult tounderstand why the frequencies from FEM with surface effectare smaller than the classical results for h15 nm and h20 nm,

Table 4Fundamental natural frequencies of cantilever Kirchhoff nanoplates.

Thickness(nm)

With surface effect(GHz)

No surface effect(GHz)

Different(%)

h5 0.4552 0.8965 49.23h10 1.5430 1.7929 13.94h15 2.5453 2.6894 5.36h20 3.4841 3.5858 2.84

Table 5Fundamental natural frequencies of Kirchhoff nanoplates with only considerationof the residual surface stress (h10 nm).

SS CC CF

0 0:6056 N=m 11.347 19.269 1.59240 0 N=m 10.227 18.458 1.79290 0:6056 N=m 8.8419 17.606 2.0561

K.F. Wang, B.L. Wang / Finite Elements in Analysis and Design 74 (2013) 2229 27

h2

2u0 0y;yy 0 u0x;xy u0y;xxk2Ghy wy J y 32c

The natural frequency of a simply supported nanoplate is

mn g1

g73

p

g83

p3g0

s33

Where g0, g1, g7 and g8 are shown in Appendix B. It should benoted that Eq. (33) can be obtained in our previous study [23] byneglecting the nonlocal effect ( 0).

3.4. Finite element solution for free vibration

In this subsection, the frequencies of nanoplates based on bothKirchhoff and Mindlin plate theories are computed by using1010 element mesh conguration. Table 1 lists the fundamentalfrequencies of simply supported nanoplate with varying thick-nesses. Comparing the FEM results with the analysis results, it canbe seen that the error is within 1%. Such accuracy is adequate formost practical situations. Comparing the fundamental frequenciesusing FEM with the results for classical theory, it found that thesurface effect has a substantial effect on the fundamental frequen-cies of thinner nanoplate. The inuence of the surface effect on thefundamental frequencies increases with the decreasing thicknessof nanoplate.

Table 2 shows the fundamental frequencies of nanoplate withdifferent aspect ratio (a/b). It can be seen that the inuence of thesurface effect on fundamental frequencies is more obvious if theaspect ratio of the nanoplate equals to 1.

The fundamental frequencies of nanoplate with clampedboundary condition are shown in Table 3. It is found that whenthe thickness is larger than 15 nm the surface effect may beneglected since the relative error is within 1%.

Table 4 lists the fundamental frequencies of cantilever name-plates. It is found that the fundamental frequencies calculated forcantilever Si (100) nanoplates are lower than those calculatedwithout the surface effect.

Comparing Tables 14, one can conclude that the inuence ofsurface effect on the fundamental frequencies of the nanoplatedepends on the signs of surface elasticity constants and residualsurface stress, nanoplate thickness and boundary conditions. Forexample, Table 5 shows the fundamental frequencies of nano-plates with only consideration of the residual surface stress. It isfound that a positive residual surface stress increases the funda-mental frequencies of nanoplates with simply supported and

Table 1Variation of fundamental natural frequencies of the Kirchhoff nanoplate withthickness of the plate for simply supported boundary condition.

Thickness(nm)

FEM(GHz)

Analytical results(GHz)

Results for classical theory(GHz)

Errors(%)

h5 8.5889 8.6501 5.1133 0.7h10 11.059 11.180 10.227 1.0h15 15.482 15.638 15.340 1.0h20 20.324 20.510 20.453 0.9Table 2Variation of fundamental natural frequencies of Kirchhoff nanoplate with aspectratio of the plate for simply supported boundary condition (h5 nm).

Aspectratio

FEM(GHz)

Analytical results(GHz)

Results for classical theory(GHz)

Errors(%)

a/b1 8.5889 8.6501 5.1133 0.7a/b1.25 10.132 10.215 6.5514 0.81a/b1.5 11.956 12.061 8.3091 0.87a/b2 16.437 16.591 12.783 0.93

Table 3Fundamental natural frequencies of Kirchhoff nanoplate with clamped boundarycondition.

Thickness(nm)

With surface effect(GHz)

No surface effect(GHz)

Different(%)

h5 11.701 9.2290 26.79h10 18.772 18.458 1.7h15 27.535 27.687 0.55h20 36.599 36.916 0.86in Table 3.

h30 25.296 25.505 25.494 0.82

K.F. Wang, B.L. Wang / Finite Elements in Analysis and Design 74 (2013) 222928Table 6 shows the fundamental frequencies of simply sup-ported Mindlin nanoplates of various values of thickness.

Once again, the relative errors between the FEM results and theanalysis results are less than 1%. The fundamental frequenciesshow a negligible dependency on the surface effect if the thicknessof the nanoplate is sufciently large (e.g., 425 nm). Note that theinuence of surface effect on the frequencies of the Mindlin plates

h35 28.197 28.232 28.237 0.12h40 30.358 30.577 30.588 0.72

Fig. 8. A T-shape Si nanoresonator.Table 6Fundamental natural frequencies of Mindlin nanoplate with simply supportedboundary condition.

Thickness(nm)

FEM(GHz)

Analytical results(GHz)

Results for classical theory(GHz)

Errors(%)

h25 22.176 22.362 22.312 0.83

is quite similar to that for the Kirchhoff plates. Therefore,the results for the Mindlin plates are not shown here.

3.5. Applications of the current model to a MEM/MEMS device

In this subsection, we use the present FEM to calculate thefundamental frequency of a T-shape nanoplate silicon nanoreso-nator, which is widely used in MEMS/NEMS [2426]. The thicknessof the plate is h2 nm and the FEM mesh is shown in Fig. 8.he device has relatively complicated geometries and boundaryconditions. First, consider the case that edge AB is clamped and theremaining edges are free. The fundamental frequency of theresonator is calculated as 41.593 GHZ. If neglecting surface effect,the fundamental frequency is calculated as 52.126 GHZ. Thissuggests that the surface effect decreases the frequency of theresonator. Next, consider the case that edge AB clamped, edge EFsimply supported and the remaining edges free. The fundamentalfrequencies of the plate calculated are 87.753 GHZ if the surfaceeffect is included in the model and 90.430 GHZ if the surface effectis ignored. In this boundary condition, the surface effect increasesthe frequency of the resonator.

4. Conclusion

A nite element model including the inuence of surfaceelastic and residual surface stress has been derived based on theweighted residual method. With the model developed, it ispossible to investigate the size dependence of the static anddynamic behaviors of nanoplates. Numerical results show that

where s xxc=a and t yyc=b. (xc ,yc) is the coordinate ofAppendix B

N1 14 s1t1s t 1N2 12 s21t1N3 14 t1s2 ts t 1N4 12 s 1t21N5 14 s 1t 1s t1N6 12 s21t 1N7 14 s1t 1st 1N8 12 s1t21

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

BGT N1x 0 N1

N2x 0

N1y N1 0

N2y N2

24

35certain of element.

B

1a22Ns2

1b22Nt2

2ab2Nst

8>>>>>:

9>>>=>>>;

;Acknowledgments

This research was supported by National Science Foundation ofChina (project ID 11172081) and Shenzhen Research InnovationFoundation, China (project ID JCYJ20120613150312764).

Appendix A

D D11 D12 0D21 D22 00 0 D66

264

375; Cs

Cs11 Cs12 0

Cs21 Cs22 0

0 0 Cs66

264

375

N N1 N2 N3 N4

N1 18 s1t1s2 s t2 t2; 18 bs1t12t 1; 18 as12s 1t1

N2 18 s 1t1s2s t2 t2; 18 bs 1t12t 1; 18 as1s 12t1

N3 18 s 1t 1s2s t2t2; 18 bs 1t1t 12; 18 as1s 12t 1

N4 18 s1t 1s2 s t2t2; 18 bs1t1t 12; 18 as12s 1t 1 the surface effect can increase or decrease the deection andfrequencies of the nanoplates, depending on the signs of surfaceelastic constants and residual surface stress, and the boundaryconditions of the nanoplate. For example, a positive residualsurface stress increases the deection of cantilever nanoplates,but decreases the deection of nanoplates with simply supportedand clamped boundary conditions (vice versa for a negativeresidual surface stresses). A positive surface elasticity reducesthe deections of nanoplates (and vice versa for a negative one).In addition, a positive residual surface stress increases the fre-quencies of nanoplates with simply supported and clampedboundary conditions, but reduces the frequencies of a cantilevernanoplate. The nite element model developed in this paper canbe used to study the static and dynamic behaviors of nanoplateswith complicated geometries, boundary and loading conditionsand material properties. It provides an efcient tool for theanalysis and design of nanoscale plate devices in nanotechnology.

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[4] R.E. Miller, V.B. Shenoy, Size-dependent elastic properties of nanosizedstructural elements, Nanotechnology 11 (3) (2000) 139147.

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K.F. Wang, B.L. Wang / Finite Elements in Analysis and Design 74 (2013) 2229 29

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A finite element model for the bending and vibration of nanoscale plates with surface effectIntroductionFinite element formulationStatic bending of Kirchhoff plateDynamic behavior of Kirchhoff plateStatic bending of Mindlin plateDynamic analysis of Mindlin plate

Numerical results and discussionAnalytical solution for static bendingFinite element solution for static bendingAnalytical solution for free vibrationFinite element solution for free vibrationApplications of the current model to a MEM/MEMS device

ConclusionAcknowledgmentsReferences