A study on large amplitude vibration of multilayered graphene sheets

8/9/2019 A study on large amplitude vibration of multilayered graphene sheets

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A study on large amplitude vibration of multilayered graphene sheets

E. Jomehzadeh a ,b , , A.R. Saidiba Shahid Bahonar University of Kerman, Young Researchers Society, Kerman, Iranb Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

a r t i c l e i n f o

Article history:Received 5 July 2010Received in revised form 25 October 2010Accepted 29 October 2010Available online 26 November 2010

Keywords:Small scaleNonlinear vibrationMultilayered graphene sheetNonlocal continuum

a b s t r a c t

In the present article, large amplitude vibration analysis of multilayered graphene sheets is presented andthe effect of small length scale is investigated. Using the Hamiltons principle, the coupled nonlinear par-tial differential equations of motion are obtained based on the von Karman geometrical model and Erin-gen theory of nonlocal continuum. The solutions of free nonlinear vibration, based on the harmonicbalance method, are found for graphene sheets with three different boundary conditions. For numericalresults single, double and triple layered graphene sheets with both armchair and zigzag geometries areconsidered. The results obtained herein are compared with those available in the literature for linearvibration of multilayered graphene sheets and an excellent agreement is seen. Also, the effects of numberof layers, geometric properties and small scale parameter on nonlinear behavior of graphene sheet arediscussed in details.

2010 Elsevier B.V. All rights reserved.

1. Introduction

Graphene is dened as a at one-atom-thick of carbon tightly

packed into a two-dimensional honeycomb lattice. It is observedto have promising mechanical and electrical properties. Grapheneis the thinnest known material in the universe and the strongestever measured. Its charge carriers exhibit giant intrinsic mobilityand can travel for micrometers without scattering at roomtemperature. Graphene can sustain current densities six ordersof magnitude higher than that of copper, shows record thermalconductivity and stiffness, and reconciles such conicting quali-ties as brittleness and ductility. Electrons whiz through grapheneat rates far beyond those achieved in other materials. Theseexceptional properties of graphene sheets have led to multipleusages in the eld of nanoelectronics, nanodevices, nanosensors,nanooscillators, nanoactuators, nanoresonators, nanocompositesand nanooptomechanical systems [1,2] . Due to its lightness andstiffness, there is also fast-growing interest in graphene as a basematerial for nano-electro-mechanical systems (NEMS) for sensingapplications [3,4] .

Since the graphene exhibits the highest stiffness to inertia ratio[4] , it can be used in high frequency nanomechanical devices. Thingraphene resonators have recently been demonstrated on smallscales and have shown extremely high stiffness to weight ratio[3,5] . Graphene-based resonators offer low inertial masses, ultra-high frequencies, and in comparison with nanotubes, low-resis-tance contacts that are essential for matching the impedance of

external circuits [1] . All these characteristics have made graphenea playground for researchers including theoretical and high energyphysicist looking to lend graphenes exceptional properties to

tomorrows ultrasmall technology.Due to the vast computational expenses of nano-structuresanalyses when using atomic lattice dynamics and molecular dy-namic simulations, there is a great interest in applying continuummechanics for analysis of such structures. Many studies have beencarried out for vibration analysis of nano-structures such as nano-tubes and graphene sheets or nano-plates. Lima and He [6] devel-oped a von Karman type nonlinear model for ultra-thin, elasticallyisotropic lms withsurface effect. Nanoscale vibrational analysis of a multilayered graphene sheet embedded in an elastic mediumwas investigated by Behfar and Naghdabadi [7] and the corre-sponding natural frequencies and the associated modes weredetermined. Kitipornchai et al. [8] used the continuumplate modelfor mechanical analysis of graphene sheets. He et al. [9] describedvan der Waals interaction between the layers of multilayeredgraphene sheets by an explicit formula. Based on the continuummechanics and a multiple-elastic beam model, Fu et al. [10] inves-tigated the nonlinear free vibration analysis of embedded carbonnanotubes. Liew et al. [11] studied the vibration behavior of multi-layered graphene sheets embedded in an elastic matrix based on acontinuum-based plate model. Pradhan and Phadikar [12] pre-sented classical and rst order shear deformation plate theoriesfor vibration analysis of nano-plates. Their approach was basedon the Navier solution and for a nano-plate with all edges simplysupported. Ke et al. [13] investigated the nonlinear free vibrationof embedded double-walled carbon nanotubes based on the Erin-gens nonlocal elasticity theory and von Karman geometric nonlin-earity using differential quadrature method. Dong and Lim [14]

0927-0256/$ - see front matter 2010 Elsevier B.V. All rights reserved.doi: 10.1016/j.commatsci.2010.10.045

Corresponding author. at: Shahid Bahonar University of Kerman, YoungResearchers Society, Kerman, Iran. Tel.: +98 341 2111763; fax: +98 341 2120964.

E-mail address: [email protected] (E. Jomehzadeh).

Computational Materials Science 50 (2011) 10431051

Contents lists available at ScienceDirect

Computational Materials Science

j o u rna l homepa ge : www.e l s ev i e r. com/ loca t e / commat sc i
http://dx.doi.org/10.1016/j.commatsci.2010.10.045mailto:[email protected]://dx.doi.org/10.1016/j.commatsci.2010.10.045http://www.sciencedirect.com/science/journal/09270256http://www.elsevier.com/locate/commatscihttp://www.elsevier.com/locate/commatscihttp://www.sciencedirect.com/science/journal/09270256http://dx.doi.org/10.1016/j.commatsci.2010.10.045mailto:[email protected]://dx.doi.org/10.1016/j.commatsci.2010.10.045


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studied the nonlinear free vibrations of a nano-beam with simplysupports boundary conditions based on nonlocal elasticity theory.Murmu and Pradhan [15] developed a single elastic beam modelfor thermo-mechanical vibration of a single-walled carbon nano-tube embedded in an elastic medium basedon the nonlocal elastic-ity theory. Duan and Wang [16] characterized the bending andstretching of the circular graphene sheet using von Karman platetheory. Pradhan and Phadikar [17] presented a vibration analysisof multilayered graphene sheets embedded in polymer matrix byemploying the nonlocal continuum mechanics. A nonlocal platemodel was developed by Ansari et al. [18] to study the vibrationalcharacteristics of multilayered graphene sheets with differentboundary conditions embedded in an elastic medium using niteelement method. Sadeghi and naghdabadi [19] introduced a hybridatomistic-structural element for modeling nonlinear behavior of graphene sheets. Pradhan and Kumar [20] investigated the smallscale effect on the vibration analysis of orthotropic single layeredgraphene sheets embedded in an elastic medium. Both Winkler-type and Pasternak-type foundation models were employed tosimulate the interaction between the graphene sheet and sur-rounding elastic medium. Nonlinear free vibration of single-walledcarbon nanotubes basedon the Timoshenko beam model was stud-ied byYang et al. [21] . An elastic continuumapproach for modelingthe nonlinear vibration of double-walled carbon nanotubes underharmonic excitation was investigated by Hawwa and Qahtani[22] . Shen et al. [23] studied nonlinear vibration behavior for asimply supported single layer graphene sheet in thermal environ-ments in order to obtain the nonlocal parameter. Recently, Jomeh-zadeh and Saidi [24] developed a Navier solution for vibrationanalysis of nano-plates using the three-dimensional nonlocal elas-ticity theory.

Practically, most of the structures in mechanics such as graph-ene resonators are linear up to displacements on the order of itsthickness [3] . If the oscillations of the graphene resonator resultin amplitudes which are not very small, then the linear treatmentmay be too inaccurate for the purpose in view. In such cases, theaccuracy can often be improved sufciently by carrying out furtherapproximations via geometrically nonlinear model. In this paper, alarge amplitude vibration analysis of multilayered graphene sheetsis presented based on an orthotropic nano-plate model. By consid-ering the small scale effect in constitutive relations and using thevon Karman nonlinear model, the governing equations of motionare obtained in form of coupled nonlinear partial differential equa-tions. The nonlinear natural frequencies are presented for orthotro-pic nano-plates with three different boundary conditions includingall edges simply supported, all edges clamped and two edges sim-ply supported two edges clamped. The effects of nonlocal parame-ter, number of layer and aspect ratio on the nonlinear vibrationbehavior of multilayered zigzag and armchair graphene sheetsare discussed in details.

2. Constitutive relations of nonlocal continuum

According to nonlocal elasticity theory, the stress at a referencepoint X is considered to be a function of the strain eld at every

point X 0 in the body. The nonlocal stress tensor r at point X canbe expressed as

r Z V K j X 0 X j; sr 0 X 0dX 0 1where r 0 is the classical stress tensor and K (j X 0 X j) is the Kernelfunction represents the nonlocal modulus. While the constitutiveequations of classical elasticity is an algebraic relation betweenstress and strain tensors, that of nonlocal elasticity involves spatialintegrals which represent weighted averages of contributions of thestrain of all points in the body to the stress at the given point. Erin-gen [25] showed that it is possible to represent the integral consti-tutive relation in an equivalent differential form as

1 l r 2r r 0 2where l = (e0 a0)2 is the nonlocal parameter, a 0 an internal charac-teristic length and e0 a constant. Also, r

2

@ 2

@ x2 @ 2

@ y2 is the Laplacianoperator.

3. Formulation

Consider a thin orthotropic nano-plate of total thickness h withdimension a b for modeling the single layer graphene sheet. Theorigin of the Cartesian coordinate system is located in the middleof the plate ( Fig. 1 ). Since the graphene sheet is assumed to havelarge amplitude motion, the von Karman type straindisplacementrelations are used as

e x e x0 z j xe y e y0 z j y 3c xy c xy0 z j xywhere

e x0e y0c xy08>: 9>=>;

@ u@ x 12 @ w@ x

2

@ v @ y 12 @ w@ y 2

@ u@ y @ v @ x @ w@ x @ w@ y8>>>>>: 9>>>=>>>;

;j xj yj xy8>: 9>=>;

@ 2 w@ x2

@ 2 w@ y2

2 @ 2 w@ x@ y8>>>: 9>>=>>;4

Here the time dependent variables u , v and w are the mid-plane dis-placement components in the x, y and z directions, respectively. Asit can be seen, the straindisplacement relations are nonlinear withrespect to transverse displacement. According to Hamiltons princi-ple, the equations of motion of theorthotropic nano-plate can be gi-ven by

dZ t 2t 1 Ldt 0 5where L is the Lagrangian and t is the time variable. Expressing theLagrangian parameter based on the von Karman theory and consid-ering the small scale effect from Eq. (2) in constitutive relations, thenonlinear equations of motion for a nano-plate can be obtained asfollows:

x

y

a

b

(a) (b)

Fig. 1. Geometry of a single layer graphene sheet. (a) discrete model (graphene sheet) and (b) continuum model (nano-plate).

1044 E. Jomehzadeh, A.R. Saidi / Computational Materials Science 50 (2011) 10431051


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@ N x@ x

@ N xy@ y 1 l r

2

I 0 u I 1@ w@ x 6a

@ N xy@ x

@ N y@ y 1 l r

2

I 0 v I 1@ w@ y 6b

@ 2M x@ x2 2

@ 2M xy@ x@ y

@ 2M y@ y2 1 l r

2

@ @ x

N x@ w@ x N xy

@ w@ y

1 l r2

@ @ y N xy

@ w@ x N y

@ w@ y 1 l r

2

P

1 l r2

I 0 wI 1@ u@ x

@ v @ y

I 2@ 2 w@ x2

@ 2 w@ y2 !! 6c

where a dot denotesdifferentiation with respect to time and P is theexternal load acting on the graphene sheet in z direction . The mod-eling of thegraphene sheet is based on the fact that theelastic mod-uli in two perpendicular orientations are different due to armchairor zigzag congurations, so the nano-plate is considered to beanisotropic. For such a case, the resultant forces ( N x,N y,N xy) andresultant moments ( M x,M y,M xy) can be denedin terms of strains as

N xN y

N xyM xM yM xy

2666666664

3777777775

A11 A12 A13 B11 B12 B13 A12 A22 A23 B12 B22 B23

A13 A23 A33 B13 B23 B33B11 B12 B13 D11 D12 D13B12 B22 B23 D12 D22 D23B13 B23 B33 D13 D23 D33

2666666664

3777777775

e x0e y0c xy0j xj yj xy

2666666664

3777777775

7

where Aij and Dij (i, j = 1, 2, 3) are called the stretching and bendingstiffness, respectively and Bij the stretchingbending coupling stiff-ness which is zero for the symmetric graphene sheet. For a gener-ally single layered anisotropic graphene sheet, the stiffnesscoefcients are expressed in terms of the reduced transformed coef-cients eQ ij as

Aij; Dij Z h=2h=2eQ ij1; z 2dz 8

In which the transformed coefcients are expressed in terms of material properties (Young modulus and Poisons ratio) as [23]

eQ 11eQ 12eQ 22eQ 13eQ 23eQ 33

26666666664

37777777775

c 4 2c 2 s2 s4 4c 2 s2

c 2 s2 c 4 s4 c 2 s2 4c 2s2s4 2c 2 s2 c 4 4c 2 s2

c 3 s cs3 c 3 s cs3 2csc 2 s2cs3 c 3 s cs3 c 3 s 2csc 2 s2c 2 s2 2c 2 s2 c 2 s2 c 2 s2

2

26666666664

37777777775E 1=1 m12 m21 E 2=1 m12 m21

m21 E 1 =1 m12 m21E 1 =2 2m12

26664

37775

9

where s = sin h and c = cos h and h is 0 and 90 for armchair and zig-zag graphene sheets, respectively. Also, the inertia parameters(I 0 , I 1 , I 2) in Eq. (6) can be expressed in term of density of the graph-ene sheet, q , as follows:

I 0 ; I 1 ; I 2 Z h=2h=2 q1; z ; z 2dz 10As it can be concluded, the parameter I 1 is identically zero for a

symmetric graphene sheet with respect to z -axis.Raju et al. [26] showed that the effect of longitudinal or in-plane

inertia on large amplitude vibration of thin-walled structures isnegligible. Vanishing the in-plane inertia terms, it can be seen that

two rst Eq. (6) will be exactly satised if a stress function u isdened such as

N x @ 2u@ y2

; N y @ 2u@ x2

; N xy @ 2u@ x@ y 11

Substituting these relations into Eq. (6c) and expressing theresultant moments in terms of transverse displacement yields

D11 @ 4 w

@ x4 2D12 2D33 @ 4 w

@ x2 @ y2 D22 @ 4 w

@ y4 I 0 I 21 l r 2w r 2 w

1 l

r2

P

1 l

r2

@ 2 w

@ x2@ 2 u@ y2

2 @

2 w@ x@ y

@ 2 u@ x@ y

@ 2 w@ y2

@ 2 u@ x2

12Eq. (12) is a nonlinear fourth order partial differential equation

in terms of transverse displacement and stress function, and itneeds to be augmented with a compatibility equation. Eliminatingu and v fromEq. (4) and expressing the in-plane strain componentsin terms of stress function, one can obtained the compatibilityequation as

A 111@ 4u@ x4 2 A

112 2 A133

@ 4u@ x2 @ y2 A

122@ 4u@ y4

@ 2 w

@ x@ y !2 @ 2 w@ x2 @ 2 w@ y2 13where A 1ij represent the components of the inverse of stretchingstiffness matrix. Eqs. (12) and (13) are the starting equations foranalyzing the nonlinear vibration analysis of graphene sheets. Asit can be seen, these equations are two nonlinear partial differentialequations with a total degree of eight. The nonlinear behavior of graphene sheets can be considered as two kinds, one from the in-plane forces due to the nature of edge restrains and the other dueto the interaction of displacement components affected in the in-plane differential equations as well as compatibility conditions.

Also, the in-plane displacement components can be expressedin terms of the transverse deection and stress function by helpof Eqs. (4), (7) and (11) by the following relations:

u Z x

0 A 111

@ 2u@ x2 A

112

@ 2u@ y2 A

113

@ 2u@ x@ y

1

2@ w

@ x

2

!dx 14a v Z y0 A 112 @ 2u@ x2 A 122 @ 2u@ y2 A 123 @ 2u@ x@ y 12 @ w@ y 2 !dy 14b Thus, by determining the transverse displacement and stress

function, the in-plane displacements can also be dened.

4. Modeling of a multilayered graphene sheet (MLGS)

Let us consider a multilayered graphene sheet with N layers(Fig. 2 ). The pressure at a point between any two adjacent layersdepends on the difference of their deections at that point. Thus,the van der Waals interaction can be expressed as

P i XN

j1c ijw i w j 15

Fig. 2. A sample of the multilayered graphene sheet.

E. Jomehzadeh, A.R. Saidi / Computational Materials Science 50 (2011) 10431051 1045
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where P i is the external load in the form of van der Waals pressurebetween the ith layer and the jth layer and c ij is the coefcient of thevan der Waals interaction between the ith layer and the jth layer.This coefcient can be obtained using the van der Waals interactionpotential energy, as a function of the inter-atomic spacing using theLennardJones model as

U LJ

4e

rd

12 rd

6

16

where U LJ is the LennardJones potential, r and e are parameterschanging for different types of carboncarbon van der Waals inter-action and d is distance between interacting atoms that have theLennardJones interaction. The van der Waals force can be deter-mined by taking the derivative of the potential energy. Differentiat-ing the potential energy (16) with respect to distance d andexpressing the Taylor expansion of the results around the equilib-rium position, the coefcient of the van der Waals interaction canbe dened as [9]

c ij 4

ffiffiffi3p

9acc

!2

24 er 2

racc

8 3003 p256

X5

k0 1

k

2k

1

5k

racc

6 1

h12

ij

(35 p8 X2

k0 1

k

2k 12k

1h6ij) 17

Here acc is the carboncarbon bond length and hij is the distancebetween two layers.

Let us assume that all layers have the same thickness and effec-tive material constants. Applying Eqs. (12) and(13) to each layer of the multilayered graphene sheet by considering the van der Waalsinteraction, it can be obtained

D11@ 4w i@ x4 2D12 2D33

@ 4 w i@ x2 @ y2 D22

@ 4w i@ y4

I 0 I 21 l r2

w i r2

w i 1 l r2

XN

j1c ijw i w j

1 l r 2 @ 2 w i

@ x2@ 2u i@ y2 2

@ 2 w i@ x@ y

@ 2u i@ x@ y

@ 2w i@ y2

@ 2u i@ x2 ! 18a

A 111@ 4u i@ x4 2 A

112 2 A133

@ 4u i@ x2 @ y2 A

122@ 4u i@ y4

@ 2w i

@ x@ y !2 @ 2 w i@ x2 @ 2 w i@ y2 18b where i = 1, 2, . . . , N . Therefore, the large amplitude vibration of a N -layered graphene sheet can be described by 2 N coupled nonlinear

Eq. (18) .

5. Free vibration analysis of double layered graphene sheets(DLGSs)

Let us consider the large amplitude free vibration analysis of double layered graphene sheets. In this case, the following fournonlinear governing equations should be considered:

D11@ 4w1@ x4 2D12 2D33

@ 4 w1@ x2@ y2 D22

@ 4 w1@ y4

I 0 I 21 l r 2w1 r 2 w1 1 l r 2c 12 w1 w2

1 l

r2

@ 2 w1

@ x2

@ 2u 1

@ y2

2 @

2 w1

@ x@ y

@ 2u 1

@ x@ y

@ 2w1

@ y2

@ 2u 1

@ x2

! 19a

A 111@ 4u 1@ x4 2 A

112 2 A133

@ 4u 1@ x2 @ y2 A

122@ 4u 1@ y4

@ 2 w1

@ x@ y !2 @ 2 w1@ x2 @ 2 w1@ y2 19b D11

@ 4 w2

@ x4 2

D12

2D33

@ 4 w2

@ x2 @ y2 D22

@ 4 w2

@ y4

I 0 I 21 l r 2w2 r 2 w2 1 l r 2c 12 w2 w1 1 l r 2

@ 2 w2@ x2

@ 2u 2@ y2 2

@ 2 w2@ x@ y

@ 2u 2@ x@ y

@ 2w2@ y2

@ 2u 2@ x2 ! 19c

A 111@ 4u 2@ x4 2 A

112 2 A133

@ 4u 2@ x2 @ y2 A

122@ 4u 2@ y4

@ 2 w2

@ x@ y !2 @ 2 w2@ x2 @ 2 w2@ y2 19d where w1 and w2 are the transverse deections of bottom and toplayers of the graphene sheet, respectively. Two different boundaryconditions are considered for the DLGSs as simply supported andclamped which are both movable in-plane edges.

5.1. A simply supported DLGS (2-SSSS)

Let us consider a double layered graphene sheet with all edgesmovable simply supported with the following boundaryconditions:

w i M xi @ 2 u i@ x@ y R b=2b=2 @ 2 u i@ y2 dy 0 at x a2w i M yi @ 2 u i@ x@ y R a=2a=2 @ 2 u i@ x2 dx 0 at y b2

20where in this case the index i is 1 or 2. The movable in-plane edge isthe edge which is kept straight by a distribution of normal stressesand therefore the resultant stresses on the edge is zero.

The nonlinear free vibration response of double layered simplysupported graphene sheets can be obtained by introducing the fol-lowing admissible function for transverse deection of each layer:

w i hW it cos np x

acos

mp yb 21

where n and m are the numbers of half cosine waves in the x and ydirections, respectively and W i(t ) is a function of time only. It isobvious that Eq. (21) satises the rst two boundary conditions inEq. (20) of each edge. Substituting Eq. (21) into the right side of Eqs. (19b) and (19d) , the general solutions for the stress functionu i can be obtained as

u 1 Eh2

32 n2

m2

a2

b2 m4a4 cos2np x=a n4b

4 cos2mp y=bW 1t 2

22a u 2

Eh2

32 n2 m2 a2b2 m4a4 cos2np x=a n4b

4 cos2mp y=bW 2t 2

22b These functions exactly satisfy two last conditions of Eq. (20)(in-plane boundary conditions). Introducing the transverse dis-placement w i and the stress function u i fromEqs. (21) and (22) intothe basic Eqs. (19a) and (19c) and then using the Galerkin method,themodal equations of theproblem canbe obtained. In according tothe Galerkin procedure, the integral

Z A Ciw; u Wi dA 23



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can be computed over the area of the graphene sheet. C i (w ,u ) isthe nonlinear Eqs. (19a) or (19c) and W i is the corresponding spatialpart of admissible function (21) . A simple but lengthy calculation of the above integral leads to modal time differential equations of Dufngs type, for both layers, which can be written as

d2 W 1t dt 2 aW 1t bW 1t

3

cW 2t 0 24a d2 W 2t

dt 2 aW 2t bW 2t 3

cW 1t 0 24b where the parameters a , b and c are dened in terms of graphenesheet properties as

b 3p4 h2a4eQ

22 b4eQ 11 eQ

11

eQ 22 eQ

212 4qeQ 11 eQ 22 a2 b2

12 a2 b2

a2 h2p 2 b

2 h2p 2 25b

c 12c 12 a2 b

2

qh12 a2 b2

a2 h2p 2 b

2 h2p 2 25c

For linear vibration of the graphene sheet in which the termbW i(t )3 can be neglected, the corresponding linear natural fre-quency is given by x L

ffiffiffiffiffiffiffiffiffiffiffiffi a cp .

In order to nd the nonlinear natural frequencies of DLGSs, theharmonic balance (HB) method is employed. The HB method is ananalytical approach for solving nonlinear oscillators, in which theinitial conditions are generally simplied by setting velocity or dis-

placement to be zero.Here, the periodic solutions with frequency of x are consideredfor Eq. (24) . Introducing a new variable as s = x t , Eq. (24) can betransformed into

X W 001 aW 1 bW 31 cW 2t 0 26a

X W 002 aW 2 bW 32 cW 2t 0 26b

where X = x 2 and the superscript ( 0) denotes differentiation withrespect to s . Considering the initial velocity of the graphene sheetequal to zero, the ( K + 1)th HB solutions can be described as [27]

W K 1i XK 1

k1S ki cos2k 1s 27

Substituting the above equation into Eq. (26) , expanding resultsas Fourier series about s and letting the coefcients of {cos s ,cos3 s , . . .,cos(2 N + 1) s} be zero, it follows that:

CSki S i; x i 0 k 1; 2; . . . ;K 1

XK 1

k1S 2k 1i Ai 28

where Ai is the initial displacement of the ith layer. From the aboveequations, the frequency and parameter S can be obtained for eachstep. For example, rst step of the HB method have the followingsolution:

First step W i Ai cos ffiffiffiffiffixp t x2 a

34 bA

2i c 29

5.2. A two edges clamped two edges simply supported DLGS (2-CCSS)

It is assumed that the graphene sheet has clamped edges in xdirection and simply supported edges in y direction. The boundaryconditions for such a nano-plate may be identied as

w i @ w i

@ x @ 2 u i@ x@ y

R b=2

b=2@ 2 u i@ y2 dy 0 at x a2

w i M yi @ 2 u i@ x@ y R

a=2

a=2

@ 2 u i@ x2 dx 0 at y

b2

30Assume that a solution of transverse displacement can be

expressed in the following form:

w i hW it cos 2 np x

acos

mp yb 31

It is easy to show that the admissible function (31) exactlysatises the two rst boundary conditions (30) . Upon substitutingEq. (31) into the right side of Eqs. (19b) and (19d) , the generalsolution for the stress function can be obtained. Doing the sameprocedure as the previous section, a Dufngs equations with thedifferent coefcients can be obtained. Its coefcients are presentedin Appendix A .

5.3. A clamped DLGS (2-CCCC)

The boundary conditions for a clamped edges nano-plate withmovable in-plane edges may be identied as

w i @ w i

@ x @ 2 u i@ x@ y

R b=2

b=2@ 2 u i@ y2 dy 0 at x a2

w i @ w i@ y @ 2

u i@ x@ y R a=2a=2 @

2

u i@ x2 dx 0 at y b2 32

Let us consider the following relation for the transverse

displacement:

w i hW it cos 2 np x

acos 2

mp yb 33

Above admissible function exactly satises the two rst bound-ary conditions of clamped edges. Substituting Eq. (33) into theright side of Eqs. (19b) and (19d) , the general solution for the stressfunction is obtained. Doing the same procedure as the previouscases, the Dufngs equations can be obtained as Eq. (24) with dif-ferent coefcients.

6. Free vibration analysis of triple layered graphene sheets(TLGSs)

For large amplitude free vibration analysis of graphene sheetswith three layers, there are six nonlinear governing equations asfollows:

D11@ 4 w1@ x4 2D12 2D33

@ 4w1@ x2 @ y2 D22

@ 4 w1@ y4

I 0 I 21 l r 2w1 r 2 w1 1 l r 2c 12w1 w2 c 13 w1 w3

1 l

r2

@ 2 w1

@ x2@ 2u 1@ y2

2 @ 2 w1

@ x@ y

@ 2u 1@ x@ y

@ 2w1

@ y2@ 2u 1@ x2 !

34a

a 12 a2b21 p 2 l a2 p 2l b

2

c 12 p 42eQ 12 a2 b2 4eQ 33 a

2 b2 eQ 22 a4 eQ 11 b

4

h3

12 a2 b2

a2 h2p 2 b

2h2p 2a2b2

a2 lp 2 b2lp 2qh

25a



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7/9

Armchair1:

a 9:519 nm ; b 4:844 nm ; h 0:129 nm ; q 6316 kg =m 3 ;e0 a0 0:67 nm

m12 0:197 ; E 1 2:434 TPa ; E 2 2:473 TPa ;G12 1:039 TPa37a

Armchair2:

a 6:995 nm ; b 4:847 nm ; h 0:143 nm ; q 5727 kg =m 3 ;e0 a0 0:47 nmm12 0:202 ; E 1 2:154 TPa ; E 2 2:168 TPa ; G12 0:923 TPa

37b Armchair3:


m12 0:201 ; E 1 1:949 TPa ; E 2 1:962TPa ; G12 0:846 TPa37c

Zigzag1:

a 9:496 nm ; b 4:877 nm ; h 0:145 nm ; q 5624 kg =m3

;e0 a0 0:47 nm

m12 0:223 ; E 1 2:145 TPa ; E 2 2:097 TPa ; G12 0:938 TPa37d

Zigzag2:


m12 0:204 ; E 1 2:067 TPa ; E 2 2:054 TPa ; G12 0:913 TPa37e

Zigzag3:

a 1:987 nm ; b 1:974 nm ; h 0:154 nm ; q 5363 kg =m 3 ;e0 a0 0:22 nmm12 0:205 ; E 1 1:987 TPa ; E 2 1:974 TPa ; G12 0:857 TPa

37f

Also, the van der Waals coefcients are considered as follows[9] :

c 12 108 :651 GPa =nm ; c 13 1:87206 GPa =nm 38The negative sign of van der Waals coefcient represents an

attraction between two layers and the positive sign representsrepulsion.

The fundamental linear natural frequencies for different typesof graphene sheets with three different boundary conditions arepresented in Table 2 . The results are obtained for single, doubleand triple layered graphene sheets.

Based on the results in this table, it can be concluded that therst natural frequency of the graphene sheets is independent of the number of layers. The results indicate that the increase of thenumber of the layers results in a higher linear natural frequenciesas all the layers of the graphene sheet vibrate as a whole. It can alsobe seen that the clamped graphene sheets have the higher linear

Table 2

Linear natural frequencies (THz) of graphene sheets.

Number of layer

1 2 3

SSSS Armchair1 0.0569 0.0569 2.5988 0.0569 1.8062 3.1827Armchair2 0.0763 0.0763 2.5923 0.0763 1.8020 3.1744Armchair3 0.1143 0.1143 2.5818 0.1143 1.7957 3.1610

Zigzag1 0.0453 0.0453 2.5972 0.0453 1.8049 3.1808Zigzag2 0.0426 0.0426 2.5948 0.0426 1.8032 3.1778Zigzag3 0.0491 0.0491 2.5799 0.0491 1.7929 3.1596

CCSS Armchair1 0.0642 0.0642 2.5999 0.0642 1.8063 3.1827Armchair2 0.0958 0.0958 2.5927 0.0958 1.8028 3.1747Armchair3 0.1695 0.1695 2.5843 0.1695 1.7998 3.1629


CCCC Armchair1 0.1126 0.1126 2.5999 0.1126 1.8083 3.1833Armchair2 0.1450 0.1450 2.5944 0.1450 1.8057 3.1758Armchair3 0.2127 0.2127 2.5871 0.2127 1.8040 3.1649


/

L

0 1 21

1.25

1.5

1.75

2

2.25

2.5

2.75

3

zigzag1-CCSSzigzag1-SSSSarmchair1-CCSSarmchair1-SSSSzigzag1-CCCCarmchair1-CCCC

Wmax

Fig. 3. The backbone curves for single layer graphene sheets.

Wmax

/

L

0 1 2 3 41

1.002

1.004

1.006

1.008

1.01

1.012

3-CCCC

3-CCSS2-CCCC3-SSSS2-CCSS2-SSSS

Fig. 4. The backbone curves for double and triple layer armchair graphene sheets.



8/9

natural frequencies than the other boundary conditions in all cases

since the boundary support is the stronger for the clamped graph-ene sheets.

To study the effects of physical properties of graphene sheets,the variation of nonlinear frequency ratio (nonlinear to linear fre-quencies) versus maximum amplitude of the bottom surface isinvestigated. Backbone curves (nonlinear to linear frequencyamplitude curve) are plotted for single, double and triple layeredgraphene sheets in Figs. 35 with all edges clamped, all edges sim-ply supported and two simply supported two clamped edgesboundary conditions. It can be found that the geometry nonlinear-ity makes graphene sheets have hardening behavior. i.e., the non-linear frequency ratio increases as the vibration amplitudeincreases. Also, it can be seen from Fig. 3 that zigzag graphenesheets have higher values of nonlinear to linear frequencies than

armchair graphene sheets. In addition, it can be concluded fromFigs. 4 and 5 that as the number of layer in MLGSs increases, the

nonlinear ratio increases. To study the effect of small length scaleon nonlinear behavior of graphene sheets, the backbone curves of a simply supported single layer armchair graphene sheet (type 1)are depicted in Fig. 6 for various values of nonlocal parameters. Itcan be seen that the nonlocal parameter has a considerable effecton the nonlinear vibration behavior of SLGSs. At a given vibrationamplitude, the nonlinear frequency ratio increases with the in-crease of the nonlocal parameter. During calculations, it has beenfound that the nonlocal parameter does not have a signicant ef-fect on nonlinear frequency ratio of multilayered graphene sheets.

In order to study the effect of van der Waals interaction on non-linear behavior of graphene sheets, the backbone curves of a sim-ply supported double layer graphene sheet are shown in Fig. 7for some values of van der Waals coefcient. It can be seen thatthe van der Waals interaction decreases the nonlinear behaviorof the graphene sheets.

8. Conclusion

The nonlinear free vibration analysis of multilayered graphenesheets has been studied based on the nonlocal elasticity theory.Considering the Eringen nonlocal theory and von Karman hypoth-esis, the nonlinear equations of motion have been obtained usingthe Hamiltons principle. For illustrative examples, the large ampli-tude vibration solutions of single, double and triple layered graph-ene sheets with simply supported or clamped boundary conditionshave been found using the harmonic balance method. The resultsin the form of linear and nonlinear frequencies have been pre-sented for different types of armchair and zigzag graphene sheets.It has been seen that the small length scale increases the nonlinearbehavior of the graphene sheets but the van der Waals interactiondecreases it. Also, it can be seen the van der Waals interactiondecreases the nonlinear behavior of DLGSs but increases the non-linear behavior of TLGSs.

Acknowledgment

The support for this work, provided by the Iran NanotechnologyInitiative Council, is gratefully acknowledged.

Wmax

/

L

0 1 2 3 41

1.002

1.004

1.006

1.008

1.01

1.012

3-CCCC

3-CCSS2-CCCC

3-SSSS2-CCSS2-SSSS

Fig. 5. The backbone curves for double and triple layer zigzag graphene sheets.

Wmax

/

L

0 1 2 3 41

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

=(1 10-9)2

=(0.75 10-9)2

=(0.5 10-9)2

=(0.25 10-9)2

=0

Fig. 6. The backbone curves of a simply supported single layer graphene sheets fordifferent nonlocal parameters.

Wmax

/

L

0 1 2 3 41

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

c 12 =0c 12 =10

16

c 12 =1017

Fig. 7. The backbone curves of a simplysupported double layer graphene sheets forsome van der Waals coefcients.



9/9

Appendix A

The Dufngs coefcients for a double layered CCSS graphenesheet

References

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a a2b236 a2 b

2

36 p 2 l a2 48 p 2 l b2

c 12 p 48eQ 12 a2 b2 16 eQ 33 a

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4

h3

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6

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8l 16 a4b8

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8l 32 a4 b8

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2 l 17 a10 b2

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212 2a2 b

2

eQ 12 eQ 66 a4

eQ 66 eQ 22 3p2a4 b2h2 36 a4 b

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48 p 2 a2 b4

l 3p 2 a4b2

4p 2 a2 b4

h2

36 a4 b4 qh A:3


A study on large amplitude vibration of multilayered graphene sheets

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