Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments

Computational Materials Science 48 (2010) 680–685

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Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments Le Shen a, Hui-Shen Shen a,b,*, Chen-Li Zhang a a b

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 23 February 2010 Received in revised form 27 February 2010 Accepted 4 March 2010 Available online 30 March 2010 Keywords: Graphene sheet Nonlinear vibration Nonlocal plate model

a b s t r a c t Nonlinear vibration behavior is presented for a simply supported, rectangular, single layer graphene sheet in thermal environments. The single layer graphene sheet is modeled as a nonlocal orthotropic plate which contains small scale effects. The nonlinear vibration analysis is based on thin plate theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties are assumed to be temperature-dependent and are obtained from molecular dynamics simulations. The small scale parameter e0a is estimated by matching the natural frequencies of graphene sheets observed from the MD simulation results with the numerical results obtained from the nonlocal plate model. The results show that with properly selected small scale parameters and material properties, the nonlocal plate model can provide a remarkably accurate prediction of the graphene sheet behavior under nonlinear vibration in thermal environments. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Graphene sheet is a flat monolayer of carbon atoms tightly packed into a two-dimensional (2D) honeycomb lattice [1]. Motivated by its extraordinary physical, chemical and mechanical properties, considerable effort has been invested to explore its fundamental properties as well as various potential applications [2–4]. Studying single layer graphene sheets (SLGSs) is a fundamental issue in nanoscale studies because fullerenes and carbon nanotubes are viewed as deformed graphite sheets. Most studies on SLGSs have focused on their material properties [5–10]. However, very few theoretical studies have been done on the vibration behavior of these 2D atomic structures. Among those, Behfar and Naghdabadi [11] studied the vibration of simply supported, multi-layered graphene sheets embedded in polymer matrices. They used an anisotropic plate model for the graphene sheet and added the effect of other layers and the polymer molecules through van de Waals interactions. They derived the natural frequencies and associated vibration mode shapes of this composite system. Liew et al. [12] used a continuum-based plate model to study the vibration of simply supported, double-layered and triplelayered graphene sheets embedded in an elastic matrix. They stud-

* Corresponding author at: Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China. Tel./fax: +86 21 34206197. E-mail address: [email protected] (H.-S. Shen). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.03.006

ied the effect of van der Waals interactions in vibration modes with different wave numbers and deduced that the effect of surrounding medium can be neglected in lower-order resonant frequencies. Sakhaee-Pour et al. [13] used a structural molecular mechanics approach to study the vibration of SLGSs. They considered cantilever, bridged and fully clamped boundary conditions to obtain natural frequencies of structural models. They concluded that the natural frequencies of SLGSs are independent of chirality and aspect ratio. Although the classical continuum models are efficient in SLGSs vibration analysis through their relatively simple formulae [11,12], the small scale effect on the vibration behavior of SLGSs can not be accounted for. Eringen [14] proposed a nonlocal elasticity theory. The major difference between the classical local theory and the nonlocal theory lies in that the former theory assumes that the stress state at a given point is uniquely determined by the strain state at the same point, whereas in the latter theory, the stress state at a given point is considered as a function of the strain state of all points in the body. Thus, the theory of nonlocal continuum mechanics contains information about the long range forces between atoms, and the internal length scale is introduced into the constitutive equations simply as a material parameter. A nonlocal thin plate model is recently used in solving linear free vibration of single- and multi-layered graphene sheets embedded in polymer matrix [15,16]. However, to the best of the authors’ knowledge, there is no literature covering nonlinear vibration response of SLGSs. This is the problem studied in the present paper, for the case when all four edges of the graphene sheet are assumed to be simply supported with no in-plane displacements.

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L. Shen et al. / Computational Materials Science 48 (2010) 680–685

We propose here a nonlocal plate model to study nonlinear vibration behavior of SLGSs in thermal environments. The governing equations are based on classical thin plate theory with a von Kármán-type of kinematic nonlinearity and contain small scale effects. The thermal effects are also included and the material properties are assumed to be anisotropic, temperature-dependent and are obtained from molecular dynamics (MD) simulations. The small scale parameter e0a is estimated by matching the natural frequencies of graphene sheets observed from the MD simulation results with the numerical results obtained from the nonlocal plate model. The numerical illustrations show both linear and nonlinear vibration response of SLGSs under different sets of environmental conditions.

1 e LðW; WÞ L 21 ðFÞ  e L 23 ðN T Þ ¼  e 2 where I1 = qh, I3 = qh3/12 and q is the mass density, and

@4 @4 @4 e L 11 ðÞ ¼ D11 4 þ 2ðD12 þ 2D66 Þ 2 2 þ D22 4 @X @X @Y @Y @4 @4 @4     e L 21 ðÞ ¼ A22 4 þ ð2A12 þ A66 Þ 2 2 þ A11 4 @X @X @Y @Y 2 2 2 @ @ @ e L 14 ðM T Þ ¼ 2 ðM Tx Þ þ 2 ðM Txy Þ þ 2 ðM Ty Þ @X@Y @X @Y  2 2  @ @ 1 e L 23 ðNT Þ ¼ 2 ðA12 NTx þ A22 NTy Þ þ 2  A66 NTxy 2 @X@Y @X

2. Theoretical development

þ

Consider an SLGS modeled as an orthotropic thin plate with length Lx, width Ly and constant thickness h. The plate is exposed to elevated temperature. As usual, the coordinate system has its origin at the corner of the plate, as shown in Fig. 1. Let U; V and W be the plate displacements parallel to a right-hand set of axes (X, Y, Z), where X is longitudinal and Z is perpendicular to the plate. Let F (X, Y) be the stress function for the stress resultants defined by N x ¼ F ;YY ; N y ¼ F ;XX and N xy ¼ F ;XY , where a comma denotes partial differentiation with respect to the corresponding coordinates. It has been reported that the material properties at nanoscales are size-dependent [7]. In order to incorporate the small scale effect, continuum plate models need to be refined. This may be accomplished by using the nonlocal continuum theory. In the theory of nonlocal elasticity, the constitutive relations of nonlocal elasticity for 3D problems are expressed as [14]

ð1  s2 L2x r2 Þrij ¼ C ijkl ekl

ð1Þ

where s = e0a/Lx, rij and eij are the stress and strain tensors, and Cijkl is the elastic module tensor of classical isotropic elasticity, e0 is a material constant, and a and Lx are the internal and external characteristic lengths, respectively. The distinct difference between the classical and nonlocal elasticity theories lies in the presence of small scale parameter e0a in the nonlocal theory. For carbon nanotubes the characteristic length a may be taken as the length of the C–C bond, i.e. a = 0.142 nm. Applying Eq. (1) to classical thin plate theory, the motion equations of SLGSs, including small scale effects and thermal effects, have readily been derived and can be expressed in terms of a stress function F, and a transverse displacement W. They are

i  h € þ I r2 W € e L 11 ðWÞ  e L 14 ðM Þ ¼ 1  s2 L2x r2 e LðW; FÞ  I1 W 3 T

ð2Þ

ð3Þ

@2 @Y 2

ðA11 NTx þ A12 NTy Þ

@2 @2 @2 @2 @2 @2 e LðÞ ¼ 2 2 þ 2 2 @X@Y @X@Y @Y @X 2 @X @Y 2 2 @ @ r2 ðÞ ¼ 2 þ 2 @X @Y

ð4Þ

In the above equations, the superposed dots indicate differentiation with respect to time. The geometric nonlinearity in the von Kármán sense is given in terms of e L() in Eqs. (2) and (3). Small scale parameter e0a is included in Eq. (2). The nonlocal plate model described by Eqs. (2)–(4) reduces to the local plate model when the small scale parameter e0a vanishes. It is assumed that the effective Young’s moduli E11 and E22, shear modulus G12 and thermal expansion coefficients a11 and a22 of an SLGS are temperature-dependent, whereas Poisson’s ratio m12 depends weakly on temperature change and is assumed to be a constant. In Eqs. (2)–(4), the thermal forces N T and moments M T caused by elevated temperature are defined by

2

NTx

6 T 6 Ny 4 NTxy

MTx

3

7 MTy 7 5¼ MTxy

Z

2 þh=2

h=2

Ax ðTÞ

3

6 7 4 Ay ðTÞ 5ð1; ZÞDT dZ Axy ðTÞ

ð5Þ

where DT = T  T0 is temperature rise from some reference temperature T0 at which there are no thermal strains, and

2

3 2 Ax ðTÞ Q 11 6 7 6 4 Ay ðTÞ 5 ¼ 4 Q 12 Axy ðTÞ Q 16

Q 12 Q 22 Q 26

3 s2 c2   7 a11 ðTÞ 76 2 c2 5 Q 26 54 s a22 ðTÞ 2cs 2cs Q 66 Q 16

32

ð6Þ

in which a11 and a22 are the thermal expansion coefficients in the longitudinal and transverse directions, and Q ij are the transformed elastic constants, defined by

Fig. 1. A single layer graphene sheet.

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2

L. Shen et al. / Computational Materials Science 48 (2010) 680–685

Q 11

3

2

c4

7 6 6 6 Q 12 7 6 c2 s2 7 6 6 6 Q 7 6 s4 6 22 7 6 7¼6 6 6 Q 16 7 6 c3 s 7 6 6 7 6 3 6 4 Q 26 5 4 cs Q 66 c2 s2

2c2 s2

s4

4c2 s2

c 4 þ s4 2c2 s2

c 2 s2 c4

4c2 s2 4c2 s2

2c2 s2

c 2 s2

ðc2  s2 Þ2

3

72 3 7 Q 11 7 76 Q 7 76 12 7 6 7 3 3 3 2 2 7 cs  c s cs 2csðc  s Þ 74 Q 22 5 7 c3 s  cs3 c3 s 2csðc2  s2 Þ 7 5 Q 66

3. Solution procedure

ð7Þ

Having developed the theory, we are in a position to solve Eqs. (2) and (3) with boundary conditions (11). Before proceeding, it is convenient first to define the following dimensionless quantities

where

Q 11 Q 12

Q 22

E22 ðTÞ ¼ ; ð1  m12 m21 Þ

Q 66 ¼ G12 ðTÞ

ð8aÞ

ð8bÞ

where h is the skew angle with respect to the plate X axis. Note that the graphene sheet is modeled as a single layer plate, no extension-flexural coupling exists, and Bij is all zero-valued. The reduced stiffness matrices ½Aij  and ½Dij  (i, j = 1, 2, 6) are functions of temperature, in the present case, determined through relationship 

A ¼A ;

D ¼D

ð9Þ

Z

þh=2

Q ij ðTÞð1; Z 2 ÞdZ

ð10Þ

h=2

Note that for the armchair graphene sheet h = 0°, and for the zigzag graphene sheet h = 90°, and therefore, A16 ; A26 ; D16 and D26 are zero-valued. It is not aware the realistic boundary conditions for an actual SLGS in practice. In the most previous theoretical studies the boundary conditions are usually assumed to be simply supported [11,12,15,16], and partial or fully clamped [13]. In the present study, we assume that all four edges are simply supported with no in-plane displacements, so that the boundary conditions are

X ¼ 0; Lx : W ¼0 2

M x ¼ D11

@ W

@X 2 Z Ly Z

W ¼0 @2W  D22 þ M Ty ¼ 0 @X @Y 2 ( " # Z Lx Z Ly Z Lx Z Ly 2 2 @V  @ F  @ F dYdX ¼ A22 2 þ A12 2 @Y @X @Y 0 0 0 0 9 !2 =   1 @W dY dX  A12 NTx þ A22 NTy  ; 2 @Y M y ¼ D12

@2W 2

p

W¼

W ½D11 D22 A11 A22 1=4

c14 ¼

  1=2 D22 ; D11

c5 ¼ 

A12 ; A22

c18 ¼ I3

E22 ; qD11

ðM x ; My Þ D11 ½D11 D22 A11 A22 1=4

c22 ¼

;

A12 þ A66 =2 ; A22

ðcT1 ; cT2 Þ ¼ 

;

L2x

ðATx ; ATy Þ

p2 ½D11 D22 1=2

^t ¼ pt E22 Lx q

1=2

ð12Þ

in which ATx and ATy are defined by

"

ATx

#

DT ¼ 

ATy

Z

h=2



h=2

Ax Ay



DT dZ

ð13Þ

h i € þ c r2 W € L11 ðWÞ ¼ ð1  s2 p2 r2 Þ c14 b2 LðW; FÞ  c17 W 18

ð14Þ

1 L21 ðFÞ ¼  c24 b2 LðW; WÞ 2

ð15Þ

where

L11 ðÞ ¼

@4 @4 @4 þ 2c12 b2 2 2 þ c214 b4 4 @x4 @x @y @y

L21 ðÞ ¼

@4 @4 @4 þ 2c22 b2 2 2 þ c224 b4 4 4 @x @x @y @y

LðÞ ¼

@2 @2 @2 @2 @2 @2  2 þ @x2 @y2 @x@y @x@y @y2 @x2 @2 @2 þ b2 2 @x2 @y

ð11aÞ

r2 ðÞ ¼

ð11bÞ

The boundary conditions of Eq. (11) become

x ¼ 0;

ð16Þ

p:

W ¼0

ð17aÞ ð17bÞ

ð11dÞ

F ;xy ¼ 0; Mx ¼ 0 #  2 Z p Z p (" 2 @2F 1 @W 2@ F 2 c24 b 2  c5 2  c24 @y @x 2 @x 0 0  2 þðc24 cT1  c5 cT2 ÞDT dx dy ¼ 0

ð11eÞ

W ¼0

ð17dÞ

F ;xy ¼ 0; My ¼ 0 #  2 Z p Z p (" 2 2 @ F 1 2@ F 2 @W   c b c b 5 @x2 @y2 2 24 @y 0 0

ð17eÞ

ð11cÞ

Y ¼ 0; Ly :

Nxy ¼ 0;

ðM x ; My Þ ¼

2

@ W  D12 þ M Tx ¼ 0 2 @Y (" # Z Ly Z Lx Lx @U @2F @2F A11 2 þ A12 2 dXdY ¼ @X @Y @X 0 0 0 0 9 !2  = 1 @W  A11 NTx þ A12 NTy dX dY ¼ 0  ; 2 @X

Nxy ¼ 0;

E22 L2x ; p2 qD11 L2x 2

Lx ; Ly

where Ax and Ay are given in detail in Eq. (6). The nonlinear Eqs. (2) and (3) may then be written in dimensionless form as

where Aij and Dij are the plate stiffnesses, defined by

ðAij ; Dij Þ ¼

c17 ¼ I1

b¼

;

½D11 D22 1=2 D12 þ 2D66 ; 12 ¼ D11   1=2 A11 ; 24 ¼ A22

c

s ¼ sin h

1

Y ; Ly

c

and

c ¼ cos h;

y¼p F

F¼

E11 ðTÞ ¼ ; ð1  m12 m21 Þ m21 E11 ðTÞ ¼ ; ð1  m12 m21 Þ



X ; Lx

x¼p

ð11fÞ

where Mx and My are, respectively, the bending moments per unit width and per unit length of the plate.

y ¼ 0;

ð17cÞ

p:

þðcT2  c5 cT1 ÞDTgdy dx ¼ 0

ð17fÞ

By virtue of the fact that DT is assumed to be uniform, the thermal coupling in Eqs. (2) and (3) vanishes, but terms in DT intervene in Eqs. (17c) and (17f).

L. Shen et al. / Computational Materials Science 48 (2010) 680–685

The initial conditions are assumed to be

Wjs^¼0

@W ¼ ¼0 ^ s^¼0 @s

ð18Þ

Applying Eqs. (14)–(16), (17a)–(17f), (18), the nonlinear vibration response of SLGSs in thermal environments is now determined by means of a two step perturbation technique, for which the small perturbation parameter has no physical meaning at the first step, and is then replaced by a dimensionless deflection at the second step. The essence of this procedure, in the present case, is to assume that

Wðx; y; ~t; eÞ ¼

X

ej wj ðx; y; ~tÞ; Fðx; y; ~t; eÞ ¼

j¼1

X

ej fj ðx; y; ~tÞ

ð19Þ

j¼0

where e is a small perturbation parameter, and the first term of wj ðx; y; ~tÞ is assumed to have the form

w1 ðx; y; ~tÞ ¼ w1 ð~tÞ sin mx sin ny

ð20Þ

where (m, n) is the vibration mode. Here we introduce an important parameter ~t ¼ e^t to improve perturbation procedure for solving nonlinear vibration problem. Substituting Eq. (19) into Eqs. (14) and (15) and collecting the terms of the same order of e, a set of perturbation equations is obtained. By using Eq. (20) to solve these equations step by step, we obtain asymptotic solutions, up to third-order, as

€ 1 ð^tÞ sin mx sin ny þ ðew1 ð^tÞÞ3 ½g 31 Wðx; y; ^tÞ ¼ e½w1 ð^tÞ þ g 1 w  sin 3mx sin ny þ g 13 sin mx sin 3ny þ Oðe4 Þ

ð21Þ

2 y2 ð0Þ x  b00 2 2   2 2 ð2Þ x ð2Þ y 2 ^ þ ðew1 ðtÞÞ B00  b00 þ f20 cos 2mx þ f02 cos 2ny þ Oðe4 Þ 2 2 ð0Þ

F ¼ B00

ð22Þ and

h

i

€ 1 ð^tÞg 3 þ ðew1 ð^tÞÞ3 g 2 sin mx sin ny þ Oðe4 Þ ¼ 0 ew1 ð^tÞg 1 þ ew ð23Þ

Note that in Eqs. (21)–(23) ~t is replaced by ^t and all coefficients in Eqs. (21)–(23) are functions of temperature with details being given in Appendix A.Multiplying Eq. (23) by (sin mx sin ny) and integrating over the plate area, one has 2

g3

d ðew1 Þ þ g 1 ðew1 Þ þ g 2 ðew1 Þ3 ¼ 0 d^t 2

ð24Þ

The solution of which may be written as [17]



xNL ¼ xL 1 þ

3g 2 2 A 4g 1

1=2 ð25Þ

where xL = [g1/g3]1/2 is the dimensionless linear frequency, and A ¼ W max =h is the amplitude to thickness ratio. 4. Numerical results and discussion Numerical results are presented in this section for simply supported SLGSs in thermal environments. The key issue is first to determine the material properties and effective thickness of SLGSs. However, a large variation of Young’s modulus E and Poisson’s ratio m, as well as effective thickness h was obtained and reported in the open literature. The wide dispersion of the mechanical properties of SLGSs can be attributed principally to the uncertainty associated to the thickness of these nanostructures. For the majority of models used, the assumed thickness

683

of the graphene layer is 0.34 nm. The 0.34 nm value provides in-plane Young’s modulus of the order of 1 TPa [6]. In contrast, Hemmasizadeh et al. [9] used a mixed MD–continuum mechanics model to obtain E = 0.939TPa and m = 0.19 with h = 0.1317 nm. Huang et al. [18] used the second generation Brenner potentials to calculate the in-plane Young’s modulus, Poisson’s ratio and thickness of SLGSs and to obtain E = 2.99 TPa and m = 0.397 with h = 0.0811 nm. Their value of Young’s modulus is found to be about three times as large as that of Hemmasizadeh et al. [9]. It has been reported that the material properties of single-walled carbon nanotubes (SWCNTs) are anisotropic, chirality- and sizedependent and temperature-dependent [19–21]. From these studies we believe that the elastic properties of SLGSs, such as Young’s moduli and shear modulus, are also anisotropic and temperature-dependent. Therefore, all material properties and effective thickness of an SLGS need to be carefully determined, otherwise the results may be incorrect. The MD simulations are first carried out, in which Newtonian equations of motion governed by interatomic interactions are solved numerically to determine the trajectories of a large number of atoms. A Velocity-Verlet algorithm is used to integrate the equations of motion and a basic time step of 0.5 fs is employed to guarantee good conservation of energy. In our MD simulations we use the LAMMPS code, in which the adaptive intermolecular reactive empirical bond order potential (AIREBO) is adopted to describe the short-range covalent C–C interactions, the long-range van der Waals interaction (LJ terms) and torsion interactions [22]. System temperature conversion is carried out by the Nose–Hoover feedback thermostat [23]. To begin the MD simulation, the SLGS is initially optimized and freely relaxed to reach the minimum energy configuration. The deformation of the SLGS is carried out in a quasi-static way by gradually increasing the applied load in a small increment and allowing the SLGS to relax fully until the next equilibrium configuration is reached. The perfect SLGSs subjected to uniaxial tensile force and tangential force are simulated under temperature varying from 300 K to 700 K. The bending tests are then carried out at 300 K and the effective thickness of SLGSs can be determined uniquely. Six types of armchair and zigzag SLGSs with three different values of aspect ratio are considered. From our MD simulation results the material properties of SLGSs under thermal environmental conditions T = 300, 500, 700 K are obtained numerically. Typical results are listed in Table 1. It is noted that the effective thickness for the armchair graphene sheets is 0.129–0.156 nm, while for the zigzag graphene sheets is 0.145–0.154 nm. The key issue for successful application of the nonlocal continuum mechanics models to SLGSs is to determine the magnitude of the small scale parameter e0a. In the most studies e0 is usually taken to be 0.39 proposed by Eringen [14]. For a single-walled carbon nanotube, the small scale parameter e0a is found to be less than 2.0 nm [24]. Small scale effects on the vibration behavior of graphene sheets were carried out analytically by assuming a range of values e0a = 0–2.0 nm, since its actual value is not known [15,16]. However, there are no experiments conducted to determine the value of e0a for SLGSs. In the present study, we give the estimation of parameter e0a by matching the natural frequencies of graphene sheets observed from the MD simulation results with the numerical results obtained from the nonlocal plate model. The natural frequencies (in GHz) for armchair and zigzag SLGSs with three different values of aspect ratio under thermal environmental condition T = 300 K are calculated and compared with MD simulation results in Table 2. It is shown that the nonlinear vibration analysis of SLGSs predicts significantly higher natural frequencies. It is found that the natural frequency of SLGSs is very sensitive to the small scale parameter e0a. Through comparison, we find that the natural frequencies obtained from the nonlocal plate model

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L. Shen et al. / Computational Materials Science 48 (2010) 680–685

Table 1 Elastic properties of SLGSs in thermal environments. G12 (TPa)

a11 (106/K)

a22 (106/K)

1.039 1.039 1.078

2.2 1.8 1.9

2.0 1.9 1.9

Armchair sheet II: Lx = 6.995 nm, Ly = 4.847 nm, h = 0.143 nm, m12 = 0.202,q = 5727 kg/m3 300 2.154 2.168 0.923 500 2.133 2.140 0.937 700 2.112 2.119 0.958

2.3 2.0 1.8

2.1 2.1 1.7

Armchair sheet III: Lx = 4.888 nm, Ly = 4.855 nm, h = 0.156 nm, m12 = 0.201, q = 5295 kg/m3 300 1.949 1.962 0.846 500 1.942 1.949 0.859 700 1.923 1.936 0.859

1.9 2.0 2.1

2.1 2.3 2.4

Zigzag sheet IV: Lx = 9.496 nm, Ly = 4.877 nm, h = 0.145 nm, m12 = 0.223, q = 5624 kg/m3 300 2.145 2.097 500 2.103 2.055 700 2.069 2.014

0.938 0.959 0.959

1.7 2.0 2.1

1.5 1.7 2.0

0.913 0.926 0.933

1.8 1.8 1.7

1.6 2.0 1.9

0.857 0.870 0.870

2.1 2.3 2.4

1.9 2.0 2.1

T (K)

E11 (TPa)

E22 (TPa)

Armchair sheet I: Lx = 9.519 nm, Ly = 4.844 nm, h = 0.129 nm, m12 = 0.197,q = 6316 kg/m 300 2.434 2.473 500 2.388 2.403 700 2.310 2.333

3

Zigzag sheet V: Lx = 7.065 nm, Ly = 4.887 nm, h = 0.149 nm, m12 = 0.204, q = 5482 kg/m3 300 2.067 2.054 500 2.040 2.027 700 2.013 2.007 Zigzag sheet VI: Lx = 4.855 nm, Ly = 4.888 nm, h = 0.154 nm, m12 = 0.205, q = 5363 kg/m3 300 1.987 1.974 500 1.974 1.968 700 1.961 1.948

and MD simulations can match very well if the small scale parameters are properly chosen, e.g. e0a = 0.67 nm for the armchair sheet I with Lx = 9.519 nm, Ly = 4.844 nm and h = 0.129 nm. It is clear that a large range of values for the small scale parameters e0a is possible due to different SLGSs, and these values will be used in all the following examples. Table 3 shows the effects of vibration amplitude as well as temperature change on the nonlinear to linear frequency ratios xNL/xL of the same six types of SLGSs. It can be seen that the temperature rise decreases the natural frequencies but increases the nonlinear to linear frequency ratios. It is found that the natural frequencies

Table 2 Comparisons of nature frequencies (in GHz) for SLGSs at 300 K.

a

Armchair sheet I

Armchair sheet II

Armchair sheet III

MD

Plate model

MD

Plate model

MD

Plate model

57.2

57.0 (0.67)a

76.3

76.4 (0.47)a

114.4

114.4 (0.27)a

Zigzag sheet IV

Zigzag sheet V

Zigzag sheet VI

MD

Plate model

MD

Plate model

MD

Plate model

66.8

66.3 (0.47)a

81.1

81.2 (0.32)a

114.4

114.2 (0.22)a

The number in brackets indicate the value of e0a (nm).

Table 3 Comparisons of nonlinear to linear frequency ratios for SLGSs in thermal environments. T (K)

xL (GHz)

xNL/xL W=h ¼ 0

W=h ¼ 0:5

W=h ¼ 1

W=h=1.5

W=h ¼ 2

W=h ¼ 2:5

1.0798 1.1289 1.2663

1.1720 1.2719 1.5355

1.2900 1.4485 1.8477

1.4274 1.6479 2.1844

1.0634 1.0831 1.1274

1.1377 1.1788 1.2688

1.2342 1.3011 1.4436

1.3482 1.4431 1.6413

Armchair sheet III: Lx = 4.888 nm, Ly = 4.855 nm, h = 0.156 nm, m12 = 0.201, q = 5295 kg/m3 300 114.4 0 1.0137 1.0536 400 105.2 0 1.0161 1.0629 500 94.7 0 1.0198 1.0770

1.1170 1.1366 1.1661

1.2002 1.2324 1.2805

1.2993 1.3456 1.4141

Zigzag sheet IV: Lx = 9.496 nm, Ly = 4.877 nm, h = 0.145 nm, m12 = 0.223, q = 5624 kg/m3 300 66.3 0 1.0185 400 57.7 0 1.0240 500 47.7 0 1.0347

1.0719 1.0927 1.1324

1.1556 1.1986 1.2788

1.2634 1.3328 1.4592

1.3897 1.4877 1.6627

Zigzag sheet V: Lx = 7.065 nm, Ly = 4.887 nm, h = 0.149 nm, m12 = 0.204, q = 5482 kg/m3 300 81.2 0 1.0150 400 72.7 0 1.0185 500 61.7 0 1.0255

1.0587 1.0722 1.0984

1.1278 1.1561 1.2102

1.2179 1.2643 1.3513

1.3249 1.3911 1.5135

Zigzag sheet VI: Lx = 4.855 nm, Ly = 4.888 nm, h = 0.154 nm, m12 = 0.205, q = 5363 kg/m3 300 114.2 0 1.0134 400 104.9 0 1.0158 500 93.9 0 1.0197

1.0527 1.0619 1.0767

1.1152 1.1346 1.1655

1.1971 1.2291 1.2794

1.2948 1.3409 1.4125

Armchair sheet I: Lx = 9.519 nm, Ly = 4.844 nm, h = 0.129 nm, m12 = 0.197, q = 6316 kg/m 300 57.0 0 1.0205 400 43.8 0 1.0337 500 29.1 0 1.0728

3

Armchair sheet II: Lx = 6.995 nm, Ly = 4.847 nm, h = 0.143 nm, m12 = 0.202, q = 5727 kg/m3 300 76.4 0 1.0162 400 66.2 0 1.0214 500 52.7 0 1.0333

L. Shen et al. / Computational Materials Science 48 (2010) 680–685

are decreased with increase in the aspect ratio of the SLGS and the nonlinear to linear frequency ratios are increased with increase in the non-dimensional vibration amplitude of the SLGS. The results show that the armchair sheets will have lower natural frequencies and slightly higher nonlinear to linear frequency ratios than those of zigzag sheets when the two sheets have the same aspect ratio and under the same thermal environmental condition. 5. Conclusions Nonlinear vibration response of SLGSs in thermal environments has been presented on the basis of a nonlocal plate model. We considered the small scale effects in two ways, that is, the nonlocal stress incorporating the small scale parameter e0a was introduced into the governing equations, and the size-dependent and temperature-dependent material properties obtained by MD simulations. The results reveal that vibration amplitude as well as temperature change has a significant effect on the nonlinear vibration response of both armchair and zigzag graphene sheets. Acknowledgment The support for this work, provided by the National Natural Science Foundation of China under Grant 10802050, is gratefully acknowledged. Appendix A In Eqs. (24) and (25)



 1 4 m þ 2c12 m2 n2 b2 þ c214 n4 b4  c14 R11 cT1 m2 þ cT2 n2 b2 DT R11 ð2c224  c25 Þðm4 þ c224 n4 b4 Þ þ 2c5 c224 m2 n2 b2 g2 ¼ c224  c25

g1 ¼

g 3 ¼ c17 þ c18 ðm2 þ n2 b2 Þ

ðA:1Þ

685

and

R11 ¼ 1 þ s2 p2 ðm2 þ n2 b2 Þ

ðA:2Þ

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