Nonlocal continuum model for vibration of single-layered graphene sheets based on the element-free kp-Ritz method

Engineering Analysis with Boundary Elements 56 (2015) 90–97

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

Nonlocal continuum model for vibration of single-layered graphene sheets based on the element-free kp-Ritz method Yang Zhang a,b,c, Z.X. Lei b,c, L.W. Zhang d,n, K.M. Liew b,c, J.L. Yu a a

CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, China Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong Special Administrative Region c City University of Hong Kong Shenzhen Research Institute Building, Shenzhen Hi-Tech Industrial Park, Nanshan District, Shenzhen, China d College of Information Technology, Shanghai Ocean University, Shanghai 201306, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 18 September 2014 Received in revised form 11 December 2014 Accepted 29 January 2015

In this paper, an implementation of the kp-Ritz method with the nonlocal continuum model as an element-free computational framework has been performed to investigate the free vibration behavior of a single-layered graphene sheet (SLGS). The nonlocal continuum model, which combines the Eringen nonlocal constitutive equation with the classical plate theory, has the ability to take the small scale effect into account. The study has shown that the element-free kp-Ritz method is an efficient approach for solving nonlocal continuum model for the SLGS vibration. The accuracy of the kp-Ritz method has been validated through comparison with the MD computation. The values of nonlocal parameter used in the study have been derived by matching the frequency of nonlocal continuum model with that of the MD model for different sizes and boundary conditions. The simulated results have illustrated that the value of nonlocal parameter depends on the sizes and boundary conditions of SLGS. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Element-free kp-Ritz method Single-layered graphene sheet Nonlocal elasticity theory Free vibration analysis

1. Introduction Graphene sheets (GSs) and its deviation are regarded as promising for applications in nanotechnology [1–5], such as graphene transistors, gas detection, solar cells, and diagnosis devices. Given its distinctive and outstanding property of high elasticity [6], it could be used in manufacturing components with high strength to weight ratio such as building material or aircraft components. It could be rolled into carbon nanotubes (CNTs) and carbon nanocones (CNCs) which have attracted increasing attention. Many researchers have studied the mechanical properties of CNTs and CNCs such as free vibration [7–10], buckling and post-buckling behaviors [11–13] using the experimental and theoretical methods. It is rather difficult to conduct experiments with nano-scaled size specimens. Therefore, theoretical modeling approaches play an increasingly important role to investigate the mechanical properties of nanostructures. In general, there are three main approaches for modeling nanostructures: (a) atomistic modeling [14,15], (b) hybrid atomistic-continuum mechanics [16–19], and (c) continuum mechanics [20]. The atomistic method is computationally expensive and limited to small scale systems. The hybrid atomisticcontinuum mechanics saves computation resources but increases complexity of the analysis procedure. The continuum mechanics

n

Corresponding author. E-mail address: [email protected] (L.W. Zhang).

http://dx.doi.org/10.1016/j.enganabound.2015.01.020 0955-7997/& 2015 Elsevier Ltd. All rights reserved.

approach is less computationally expensive than the former two approaches and it has relatively simply formulations. A single-layered graphene sheet (SLGS) is a monolayer of graphite consisting of a repetitive honeycomb lattice in which carbon atoms bond covalently with their neighbors. The fullerenes CNC and CNTs are viewed as deformed graphene sheets [21–23]. Thus studying the SLGS is a fundamental issue in nano-scale studies. Vibration of nanostructures is of great importance in nanotechnology. Therefore, understanding the vibration behavior of SLGS is the key issue for many graphenebased devices such as oscillators and sensor devices. The classical continuum model is limited to identify the small-scale effect in analyzing nanostructures. Therefore, modified continuum models, which incorporates nonlocal elasticity theory into continuum mechanics, have been proposed to capture the precise mechanical behavior of small-scaled material [24,25]. The nonlocal elasticity theory of Eringen [26,27] can take into account of the small-scale effect resulting from its idea that the strain at every point in the body instead of one point contribute to the stress at a reference point. Peddieson et al. [28] first applied the nonlocal elasticity theory to study the small-scale effect in nanoscale structures. Ansari et al. [29] and Pradhan et al. [30] investigated the vibration analysis of SLGS by incorporating Eringen's nonlocal elasticity equation into the classical plate theory and the solutions are computed using a generalized differential quadrature method. Wang [31] developed two nonlocal continuum mechanics models to study the wave propagation in carbon nanotubes (CNTs). The classical and shear deformation beam and plate theories are reformulated using the nonlocal differential constitutive relations of Eringen and

Y. Zhang et al. / Engineering Analysis with Boundary Elements 56 (2015) 90–97

von Karman nonlinear strains by Reddy [32], who derived the equations of equilibrium of the nonlocal beam theories and presented the virtual work statements in terms of the generalized displacements for further use in finite element model. Arash et al. [33] studied the wave propagation in graphene sheets using the nonlocal finite element method. Although the element-free method has been employed to investigate the free vibration characteristics of SWCNTs [7], so far, the application of the element-free method for the free vibration analysis of SLGS described by nonlocal-continuum model has not been reported in the previous literature. The element-free method, having the advantage of independence on meshes compared with the finite element method (FEM), is promising in dealing with problems which are difficult to be solved using the FEM, such as large deformation problems and crack propagation problems. In the present study, the element-free kp-Ritz method [23,34–42] is employed to investigate the free vibration behavior of SLGS described by the nonlocal continuum model. The results have demonstrated that the kp-Ritz method is efficient in capturing the free vibration behavior of SLGS with consideration of the small scale effect.

91

Wðx; y; z; tÞ ¼ wðx; y; tÞ:

ð5Þ

where U, V and W are the displacement components in the x, y and z directions, respectively, and (u, v, w) are the displacement components at the mid-plane (z¼ 0). Subsequently, the strain–displacement relations can be obtained as

εxx ¼

∂u ∂2 w z 2 ; ∂x ∂x

ð6Þ

εyy ¼

∂v ∂2 w z 2 ; ∂y ∂y

ð7Þ

εzz ¼ 0; γ xy ¼

ð8Þ

∂u ∂v ∂2 w þ  2z ; ∂y ∂x ∂xy

ð9Þ

γ xz ¼ γ yz ¼ 0:

ð10Þ

Using the principle of virtual work, the equilibrium equation can be obtained as 2. Nonlocal continuum model 2.1. Nonlocal elasticity theory The nonlocal elasticity theory was first proposed by Eringen [26,27] who attributed it to the atomic theory of lattice dynamics and experimental measurements on phonon dispersion. According to Eringen [26,27], the stress field at a point x depends on both the strains at x and stains at all other points of the continuum. Sharing the equilibrium relation without the body
force (σij;j ¼ 0) and the strain–displacement relationship εij ¼ 1=2 ui;j þ uj;i with the classical elasticity theory, the linear nonlocal elasticity theory has its distinguished constitutive equation for homogenous and isotropic elastic solids. It can be written as Z   σ ij ðxÞ ¼ Kðx0 x; τÞC ijkl εkl ðx0 ÞdΩðx0 Þ; ð1Þ Ω

where C ijkl is the elastic modulus tensor. Depending on the distance jx0  xj (in Euclidean norm) and material constant τ, the kernel function Kðjx0  xj; τÞ is the nonlocal modulus which weights the strains in the domain. The material constant depends on internal characteristics lengths (such as length of C–C bonds, lattice parameter) and external characteristics lengths (such as wavelength and crack length). Generally, τ can be expressed as e0 a=l, in whiche0 , a and l are the material constant, internal and external characteristic lengths, respectively. The constitutive relation can be simplified to a partial differential form as ð1  ðe0 aÞ2 ∇2 Þσ ij ¼ C ijkl : εkl

ð2Þ

M x;xx þ 2M xy;xy þ M y;yy ¼ ρ0 w;tt  ρ2 ðw;xxtt þ w;yytt Þ; ð11Þ  T R h=2  T where  M x ; M y ; M xy ¼  h=2 σ xx ; σ yy ; σ xy z dz,    R ρ0 ; ρ2 T ¼ h=2h=2 ρ 1; z2 dz, in which ρ denotes the material density. The stress–strain relationship of the nonlocal plate can be described as 8 9 8 9 2 32 εxx 3 σ = σ = E=ð1  v2 Þ vE=ð1  v2 Þ 0 > > < xx > < xx > 76 7 σ yy  ðe0 aÞ2 ∇2 σ yy ¼ 6 4 vE=ð1 v2 Þ E=ð1  v2 Þ 0 54 εyy 5; > > :σ > ; :σ > ; γ xy xy xy 0 0 G ð12Þ where E, v and G denote the elastic modulus, Poisson's ratio and shear modulus, respectively. Therefore, the nonlocal moments can be written as  2  ∂ w ∂2 w M x  ðe0 aÞ2 ∇2 M x ¼  D þ v ; ð13Þ ∂x2 ∂y2 M y ðe0 aÞ2 ∇2 M y ¼  D

 2  ∂ w ∂2 w þ v ; ∂y2 ∂x2

M xy  ðe0 aÞ2 ∇2 M xy ¼  Dð1  vÞ

∂2 w : ∂x∂y

ð14Þ

ð15Þ

Substituting Eqs. (13)–(15) into Eq. (11), the following governing equation for vibration problem can be obtained: h i

 D∇4 w ¼ 1  ðe0 aÞ2 ∇2 ρ0 w;tt  ρ2 ðwxx;tt þ wyy;tt Þ ; ð16Þ 3

2.2. Classic plate theory The classical plate theory (CPT) neglects the shear deformation and rotational inertia by assuming that the straight line vertical to the mid-plane remains straight and vertical to the mid-plane after deformation. According to CPT, the displacement can be expressed as ∂w ; ∂x

ð3Þ

∂w ; ∂y

ð4Þ

Uðx; y; z; tÞ ¼ uðx; y; tÞ  z Vðx; y; z; tÞ ¼ vðx; y; tÞ  z

where D ¼ Eh =12ð1  v2 Þ is the bending rigidity of the plate. It is worth noting that the value of bending rigidity is generally not computed as the above form when using the nonlocal elasticity plate model to study the vibration behavior of SLGS [43,44].

3. The element-free kp-Ritz method 3.2. Weak form According to [33], a weak form of Eq. (16) is proposed to reduce the differentiability of the shape function used in the element-free kp-Ritz method. Multiplying the original strong form equation by

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Y. Zhang et al. / Engineering Analysis with Boundary Elements 56 (2015) 90–97

the virtual displacement δw, we obtain Z n h i

o δw D∇4 w þ 1  ðe0 aÞ2 ∇2 ρ0 w;tt  ρ2 ðwxx;tt þ wyy;tt Þ dx dy ¼ 0: Ω

ð17Þ

Eq. (17) is then integrated by part, i.e. R Ω Dðδw;xx w;xx þ 2δw;xy w;xy þ δw;yy w;yy Þdx dy Z þ ðρ0 δww;tt þ ρ2 ðδw;x w;xtt þ δw;y w;ytt ÞÞdx dy Ω Z þ ðe0 aÞ2 ðρ0 ðδw;x w;xtt þ δw;y w;ytt Þ

The two-dimensional shape function is expressed as

ψ I ðxÞ ¼ Cðx; x  xI ÞΦa ðx  xI Þ;

ð20Þ

where Φa ðx  xI Þ is the kernel function and Cðx; x  xI Þ is the correction function, which is used to satisfy the reproduction Table 3 Non-dimensional fundamental frequency for SSSS SLGS (width¼10 nm, (e0a)2 ¼1.85 nm2) with different scaling factor. Number of nodes

Ω

þ ρ2 ðδw;xx w;xxtt þ 2δw;xy w;xytt þ δw;yy w;yytt ÞÞdx dy Z þ Dðδww;xxx nx  δw;x w;xx nx þ 2δww;xyy nx

66 88 10  10 12  12 14  14 16  16 18  18

Γ

 2δw;x w;xy ny þ δww;yyy ny  δw;y w;yy ny Þds Z  ðρ2 þ ðe0 aÞ2 ρ0 Þðδww;xtt nx þ δww;ytt ny Þds Γ Z þ ðe0 aÞ2 ρ2 ððδww;xxxtt  δw;x w;xxtt Þðnx þ ny Þ

dmax

MD result [29]

2.3

2.5

2.7

2.9

3.1

1.049 1.024 1.017 1.008 1.004 0.999 0.999

1.038 1.011 1.005 0.999 0.997 0.994 0.994

1.016 0.995 0.993 0.992 0.991 0.989 0.989

0.989 0.980 0.983 0.986 0.987 0.989 0.989

0.963 0.969 0.980 0.986 0.988 0.991 0.991

1.0 1.0 1.0 1.0 1.0 1.0 1.0

Γ

þ 2δww;xyytt nx þ2δw;x w;xytt ny Þds ¼ 0:

ð18Þ

Eq. (18) should be discretized to form algebraic equations. Gauss integration is commonly used to evaluate the mass matrix and stiffness matrix in the element-free method, however, it is computationally expensive. As an alternative to Gauss integration, a direct nodal integration approach has been proven to be more efficient [45,46]. Thus, in the present study, a direct nodal integration approach is applied to construct the algebraic equations. 3.1. The kernel particle shape functions Construction of the kernel particle shape functions is briefly reviewed here [47,48]. For a domain discretized by a set of nodes xI , I ¼1… NP, displacement approximations are expressed in the discrete form uh ¼

NP X

Table 4 Frequency (THz) of different modes with different nodes for SSSS SLGS (width¼ 10 nm, dmax ¼ 2.3, (e0a)2 ¼ 1.85 nm2). Number of nodes

66 88 10  10 12  12 14  14 16  16 18  18

ð19Þ

I¼1

where ψ I ðxÞ and uI are the shape function and nodal parameter associated with node I, respectively. Table 1 Material properties of graphene sheet [49]. Young's Modulus (E) Poisson ratio (υ) Density (ρ) Thickness (h)

1.06 TPa 0.25 2250 kg/m3 0.34 nm

Table 2 Values of calibrated nonlocal parameters for different sizes of SLGS.

88 10  10 12  12 14  14 16  16 18  18 20  20 22  22 24  24

Number of nodes

1.85 2.68 3.59 6.94 7.55 7.56 7.57 7.59 7.6

0.81 1.28 1.33 6.61 3.75 1.85 0.74 0.14 0.01

2

3

4

5

0.0602 0.0598 0.0593 0.0591 0.0589 0.0588 0.0588

0.135 0.134 0.131 0.129 0.128 0.126 0.126

0.136 0.135 0.132 0.129 0.128 0.126 0.126

0.186 0.182 0.181 0.179 0.178 0.177 0.177

0.250 0.234 0.229 0.224 0.220 0.215 0.215

MD result [29]

dmax 2.3

2.5

2.7

2.9

3.1

1.139 1.042 1.022 1.011 1.006 0.999 0.999 0.999 0.999

1.009 0.981 0.981 0.982 0.985 0.987 0.987 0.987 0.987

0.945 0.946 0.957 0.964 0.972 0.979 0.979 0.979 0.979

0.910 0.924 0.941 0.953 0.962 0.973 0.973 0.973 0.973

1.048 1.019 1.009 1.002 0.998 0.995 0.993 0.993 0.993

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Table 6 Frequency (THz) of different modes with different nodes for CCCC SLGS (a¼b¼ 10 nm, dmax ¼ 2.3, (e0a)2 ¼ 0.81 nm2).

Width of square SLGS Nonlocal parameter (SSSS) Nonlocal parameter (nm) (nm2) (CCCC) (nm2) 10 15 20 25 30 35 40 45 50

1

Table 5 Non-dimensional fundamental frequency for CCCC SLGS (width ¼ 10 nm, (e0a)2 ¼ 0.81 nm2) with different scaling factor. Number of nodes

ψ I ðxÞuI ;

Mode sequence numbers

88 10  10 12  12 14  14 16  16 18  18 20  20 22  22 24  24

Mode sequence numbers 1

2

3

4

5

0.131 0.119 0.117 0.116 0.115 0.115 0.114 0.114 0.114

0.230 0.227 0.221 0.217 0.215 0.214 0.213 0.213 0.213

0.272 0.247 0.232 0.227 0.222 0.220 0.216 0.215 0.213

0.322 0.309 0.301 0.296 0.293 0.291 0.289 0.289 0.289

0.409 0.387 0.372 0.362 0.356 0.352 0.346 0.345 0.345

Y. Zhang et al. / Engineering Analysis with Boundary Elements 56 (2015) 90–97

Hð0Þ ¼ ½1; 0; 0; 0; 0; 0; T :

conditions NP X I¼1

93

ψ I ðxÞxpI yqI ¼ xp yq for p þ q ¼ 0; 1; 2:

ð21Þ

The correction function is expressed in a linear combination of polynomial basis functions T

Cðx; x  xI Þ ¼ H ðx  xI ÞbðxÞ;

ð28Þ

For this 2-D plate problem, kernel function Φa ðx  xI Þ is defined as

Φa ðx xI Þ ¼ Φa ðxÞ U Φa ðyÞ;

ð29Þ

in which x  x I ; a

ð22Þ

Φa ðxÞ ¼ φ

bðxÞ ¼ ½b0 ðx; yÞ; b1 ðx; yÞ; b2 ðx; yÞ; b3 ðx; yÞ; b4 ðx; yÞ; b5 ðx; yÞ ;

ð23Þ

HT ðx  xI Þ ¼ ½1; x xI ; y yI ; ðx  xI Þðy  yI Þ; ðx  xI Þ2 ; ðy  yI Þ2 ;

ð24Þ

where φðxÞ is the weight function. The cubic spline function is chosen as the weight function which is given by 82 9 2 3 for 0 r jzI jr 12 > > < 3 4zI þ 4zI =

T

where H is a vector of the quadratic basis and bðxÞ is a coefficient function of x, and y is to be determined. Then, the shape function can be written as

ψ I ðxÞ ¼ bT ðxÞHðx  xI ÞΦa ðx xI Þ:

ð25Þ

Substituting Eq. (25) into Eq. (21), we can obtain coefficient bðxÞ as bðxÞ ¼ M  1 ðxÞHð0Þ;

ð26Þ

where NP X

MðxÞ ¼

Hðx  xI ÞHT ðx  xI ÞΦa ðx  xI Þ;

ð27Þ

I¼1

1 0.5

1 0

-2 0

×1

.0e 5 -9

10 0

5

10 9 .0e-

15

4

> :3 0

4zI þ 4z2I  43 z3I

-2.5 0

×1 5 .0e -9

×1

10 0

5

10 9 .0e-

15

1 2

for

dI ¼ dmax cI ;

ð32Þ

in which dmax is a scaling factor ranging from 2.0 to 4.0. Distance cI is chosen by searching for enough nodes to avoid the singularity of matrix M. Eventually, the shape function can be expressed as

ψ I ðxÞ ¼ HT ð0ÞM  1 ðxÞHðx  xI ÞΦa ðx  xI Þ:

0

-1

-1

-2

-2

0

×1

.0e

×1

-9

5 10 0

5

15

10

0

×1

e-9

.0e

×1.0

1

1

0

0

0

-1

-1

-1

-1

-2 5 -9

×1

.0e

10 0

5

15

10

×1

0e-9

5

.0e

×1.

-9

10 0

5

15

10 9 .0e-

10 0

5

10 9 .0e-

15

×1

-2

0

×1

.0e

×1

-9

5

1

-2

0

ð33Þ

1

0

0

0

ð31Þ

> ;

where zI ¼ ðx  xI Þ=dI , dI is the size of the support of node I, calculated by

1

-2

o jzI jr 1

otherwise

1

-1

-1

φz ðzI Þ ¼

ð30Þ

-9

5 10 0

5

15

10

0

×1

5

.0e

e-9

×1.0

-9

10 0

5

10 9 .0e-

15

×1

Fig. 1. First eight vibration mode shapes of SLGS with SSSS boundary condition.

1

1

1

0

0

0

0

-1

-1

-1

-1

-2

×1

.0e

5

-9

10

0

5

15

10

0

×1

9

5

.0e

0e-

×1.

-2

-2

-2

0

1

10

-9

0

5

0

15

10

×1

9

.0e

0e-

×1.

5

-9

10

0

5

15

10

0

×1

.0e

9

0e-

×1.

1

1

1

0

0

0

0

-1

-1

-1

-1

-2

×1

5

.0e

-9

10

0

5

15

10 9

0e-

×1.

0

×1

5

.0e

-9

10

0

5

15

10 9

0e-

×1.

-9

10

0

5

15

10 9

0e-

×1.

1

-2

-2

0

5

-2

0

×1

5

.0e

-9

10

0

5

10

15

e-9

0

×1.

Fig. 2. First eight vibration mode shapes of SLGS with CCCC boundary condition.

0

×1

5

.0e

-9

10

0

5

10 9 0e×1.

15

94

Y. Zhang et al. / Engineering Analysis with Boundary Elements 56 (2015) 90–97

3.2. Discrete system equations For the SLGS discretized by a set of nodesxI , I ¼1… NP, approximations of the displacements are expressed as wh0 ¼

NP X

ψ I wI eiωt :

ð34Þ

I¼1

The discretized equation is obtained by substituting Eq. (34) into Eq. (18)  ω2 ½MW þ ½K W ¼ 0;

ð35Þ

where ω ¼ 2π f is the angular frequency, and ½M ¼ ½M L  þ ½M NL ;

ð36Þ

R ½M L  ¼ Ω ½N L T GL ½N L dΩ;

R ½M NL  ¼ Ω ðe0 aÞ2 ½N NL T GNL ½N NL dΩ ; ð37Þ

2

3

2

3

ρ0 0 0 ΨI 6 7 6 7 ½N L ðIÞ ¼ 4 Ψ I;x 5; GL ¼ 4 0 ρ2 0 5 ; Ψ I;y 0 0 ρ2 2

Ψ I;x

3

6Ψ 7 6 I;y 7 6 7 6 7 ½N NL ðIÞ ¼ 6 Ψ I;xx 7; 6 7 6 Ψ I;xy 7 4 5

Ψ I;yy

R ½K ¼ Ω ½BT R½BdΩ;

2

ρ0

6 0 6 6 0 GNL ¼ 6 6 6 0 4 0

ð38Þ

0

0

0

ρ0

0

0

0

ρ2

0

0

2 ρ2

0

0

0

R ¼ diag fD; 2D; Dg :

0

0

3

0 7 7 7 0 7; 7 0 7 5

ð39Þ

ρ2

Fig. 4. Comparison of MD and present nonlocal results for SLGS with CCCC boundary condition.

ð40Þ

Fig. 3. Comparison of MD and present nonlocal results for SLGS with SSSS boundary condition.

Fig. 5. Relationship between width and nonlocal parameter of SLGS with SSSS boundary condition.

Y. Zhang et al. / Engineering Analysis with Boundary Elements 56 (2015) 90–97

95

The results of non-dimensional fundamental frequencies for different number of nodes and different scaling factors are shown in Table 5. The simulated case is a SLGS with four edges clamped (CCCC). The relationships between the frequencies of different modes and number of nodes are illustrated in Table 6. To simulate the CCCC SLGS, the scaling factor is chosen to be 2.3 and the number of distributed nodes to be 24  24 according to Tables 5 and 6. The difference of the distributed nodes between SSSS SLGS and CCCC SLGS results from their individual treatments of imposing boundary conditions in procedure. 4.2. Free vibration studies of SLGS The first eight vibration mode shapes of SSSS SLGS and CCCC SLGS are depicted in Figs. 1 and 2, respectively. The non-dimensional expression of frequency from [50] is adopted

λ4 ¼

Fig. 6. Relationship between width and nonlocal parameter of SLGS with CCCC boundary condition.

Solving the eigenvalue problems described by Eq. (35), we can obtain the natural frequency of the free vibration and associated mode shapes.

ρhω2 a4 Dð1  ν2 Þ

where ω and D represent the natural frequency and bending rigidity, respectively and a denotes the length of SLGS. Here we consider a rectangular SLGS with its aspect ratio set at 1.5. The aspect ratio is defined as width/length. It is evident that the CCCC boundary conditions make SLGS stiffer compared to the SSSS boundary conditions from that the value of λ in Fig. 1 is relatively smaller than that in Fig. 2. It should be noted that the vibration modes of SSSS and CCCC SLGS are different. For SSSS SLGS, the projections of the vibration mode tend to be close to the corner while the projections of the vibration mode tend to be parallel to the side as for CCCC SLGS. In this paper, the MD results from [29] are used for comparison. Adjusting the value of nonlocal parameter, vibration frequencies agree well with the MD results for SSSS SLGS and CCCC SLGS with

4. Numerical results and discussion Free vibration analysis of SLGS is simulated by the element-free kpRitz method. Mechanical properties of the graphene sheet used in the present simulation are extracted from [49], as shown in Table 1. The samples used in the following simulations are rectangular singlelayered graphene sheets (SLGSs) with different widths. According to Ansari [29] and Arash [33], the chirality of graphene sheet has only a small effect on its vibration behavior. Therefore, we only considered the zigzag SLGS in the present study. 4.1. Convergence studies A square SLGS with width of 10 nm is selected to carry out the convergence study. Two types of boundary conditions are considered in present study, i.e. simply supported and clamped boundary conditions. Table 3 shows the non-dimensional fundamental frequencies for different number of nodes and different scaling factors. Boundary conditions of the square SLGS are simply supported on four edges (SSSS). In Table 4, the frequencies of different modes are given for different number of nodes and different scaling factors. After careful examine Tables 3 and 4, we found it is suitable to select number of distributed nodes to be 18  18 and set scaling factor to be 2.3 to simulate the SSSS SLGS. It is noted that the frequencies of the second and third modes are close to each other in Table 4. We can observe that the second and third modes are symmetrical. Theoretically, frequencies of the second and third modes are exactly the same because SLGSs is regarded as an isotropic material and the boundary conditions are symmetrical.

Fig. 7. Relationship between fundamental frequency and width for different nonlocal parameter of SLGS with SSSS boundary condition.

96

Y. Zhang et al. / Engineering Analysis with Boundary Elements 56 (2015) 90–97

Fig. 8. Relationship between fundamental frequency and width for different nonlocal parameter of SLGS with CCCC boundary condition. Fig. 10. Variation of fundamental frequency versus aspect ratio for different nonlocal parameters of SLGS with CCCC boundary condition.

Fig. 9. Variation of fundamental frequency versus aspect ratio for different nonlocal parameters of SLGS with SSSS boundary condition.

different widths, as shown in Figs. 3 and 4. Accordingly, the calibrated nonlocal parameter is displayed in Fig. 5 for different sizes of SSSS SLGS. The relationship between nonlocal parameter and width of a square CCCC SLGS is illustrated in Fig. 6. The curve

in Fig. 6 reveals that the nonlocal parameter approaches zero when the size of SLGS is large enough. Table 2 shows the values of calibrated nonlocal parameter for different widths of SSSS SLGS and CCCC SLGS. From Table 2, it is found that the nonlocal parameter for CCCC SLGS is relatively smaller compared with that of the SSSS SLGS, which indicates that clamped boundary conditions have a better ability to limit the small scale effect than the simply supported boundary conditions. A comprehensive study about the influence of nonlocal parameter and size effect on the vibration behavior of SSSS SLGS is provided in Fig. 7. It can be concluded that: (a) the frequency decreases with the increase of width; (b) for the same width, the frequency decreases as the nonlocal parameter increases, which reveals that the small scale effect reflected by the nonlocal parameter makes SLGS less rigid; and (c) when the width of SLGS is relatively large, the frequency becomes less sensitive to the nonlocal parameter, which indicates that the curve in Fig. 5 can also be a convex curve. Besides, the corresponding information regarding the influence on the vibration behavior of CCCC SLGS is illustrated in Fig. 8, from which we can draw similar conclusions as the above. The results have demonstrated that these three features are not influenced by the boundary conditions. It is shown that the small scale effect is remarkable when the size of SLGS is relatively small. With increasing the size of SLGS, the nonlocal parameter has reduced its influence on the frequency of SLGS. Thus, a prominent feature can be observed from Figs. 7 and 8 that the fundamental frequencies seem to be independent of the nonlocal parameter for a larger width. Figs. 9 and 10 illustrate the effect of aspect ratio on the vibration behavior of SLGS for different boundary conditions. The study shows that the fundamental frequency decreases with the increase of aspect ratio.

Y. Zhang et al. / Engineering Analysis with Boundary Elements 56 (2015) 90–97

5. Conclusion The free vibration of a single-layered graphene sheet (SLGS) has been investigated using the element-free kp-Ritz method. The nonlocal continuum model that incorporates the nonlocal constitutive equation which takes the small scale effect into account has been employed for this study. The MD results have been used for comparison with the nonlocal solutions that computed by using the kp-Ritz method. Through fitting the nonlocal solutions to the MD results, the calibrated nonlocal parameter has been obtained for different sizes and boundary conditions of the SLGS. The results of the study indicated that both sizes and boundary conditions of the SLGS influence the magnitude of fundamental frequencies. Besides, the nonlocal parameter has significant influence on the fundamental frequencies of the SLGS. Acknowledgments The work described in this paper was fully supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. 9042047, CityU 11208914) and the National Natural Science Foundation of China (Grant nos. 11402142 and 51378448). References [1] Stankovich S, Dikin DA, Dommett GHB, Kohlhaas KM, Zimney EJ, Stach EA, et al. Graphene-based composite materials. Nature 2006;442(7100):282–6. [2] Bunch JS, van der Zande AM, Verbridge SS, Frank IW, Tanenbaum DM, Parpia JM, et al. Electromechanical resonators from graphene sheets. Science 2007;315(5811):490–3. [3] Schedin F, Geim AK, Morozov SV, Hill EW, Blake P, Katsnelson MI, et al. Detection of individual gas molecules adsorbed on graphene. Nat Mater 2007;6(9):652–5. [4] Chowdhury R, Adhikari S, Rees P, Wilks SP. Graphene-based biosensor using transport properties. Phys Rev B 2011;83:045401–8. [5] Yan JW, Liew KM, He LH. Ultra-sensitive analysis of a cantilevered singlewalled carbon nanocone-based mass detector. Nanotechnology 2013; 24:125703. [6] Lee C, Wei X, Kysar JW, Hone J. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Sci 2008;321(5887):385–8. [7] Yan JW, Liew KM, He LH. Free vibration analysis of single-walled carbon nanotubes using a higher-order gradient theory. J Sound Vib 2013;332 (15):3740–55. [8] Yang J, Ke LL, Kitipornchai S. Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Phys E: Low Dimens Syst Nanostruct 2010;42(5):1727–35. [9] Arash B, Wang Q. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comp Mater Sci 2012;51 (1):303–13. [10] Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, et al. Electric field effect in atomically thin carbon films. Science 2004;306 (5696):666–9. [11] Yan JW, Liew KM, He LH. Buckling and post-buckling of single-wall carbon nanocones upon bending. Compos Struct 2013;106:793–8. [12] Yan JW, Liew KM, He LH. Analysis of single-walled carbon nanotubes using the moving Kriging interpolation. Comput Methods Appl Mech Eng 2012; 229:56–67. [13] Yan JW, Liew KM, He LH. A mesh-free computational framework for predicting buckling behaviors of single-walled carbon nanocones under axial compression based on the moving Kriging interpolation. Comput Methods Appl Mech Eng 2012;247:103–12. [14] Ball P. Roll up for the revolution. Nature 2001;414(6860):142–4. [15] Baughman RH, Zakhidov AA, de Heer WA. Carbon nanotubes—the route toward applications. Science 2002;297(5582):787–92. [16] Bodily BH, Sun CT. Structural and equivalent continuum properties of singlewalled carbon nanotubes. Int J Mater Prod Technol 2003;18(4):381–97. [17] Li C, Chou T-W. A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 2003;40(10):2487–99.

97

[18] Li C, Chou T-W. Single-walled nanotubes as ultrahigh frequency nanomechanical oscillators. Phys Rev B 2003;68(7):3405. [19] Pradhan SC, Phadikar JK. Nonlinear analysis of carbon nano tubes. In: Proceedings of fifth international conference on smart materials, structures and systems Bangalore. Indian Institute of Science; 2008. p. 24–6. [20] Wang CM, Tan VBC, Zhang YY. Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes. J Sound Vib 2006;294(4–5):1060–72. [21] Tersoff J. Energies of fullerenes. Phys Rev B 1992;46(23):15546–9. [22] Shioyama H. Cleavage of graphite to graphene. J Mater Sci Lett 2001;20 (6):499–500. [23] Yan JW, Zhang LW, Liew KM, He LH. A higher-order gradient theory for modeling of the vibration behavior of single-wall carbon nanocones. Appl Math Model 2014;38(11–12):2946–60. [24] Wang Q, Varadan VK, Quek ST. Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models. Phys Lett A 2006;357(2):130–5. [25] Arash B, Wang Q. Vibration of single- and double-layered graphene sheets. J Nanotechnol Eng Med 2011;2(1):011012. [26] Eringen AC. Nonlocal polar field models. Berlin: Springer; 1976. [27] Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 1983;54(9):4703–10. [28] Peddieson J, Buchanan GR, McNitt RP. Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 2003;41(3–5):305–12. [29] Ansari R, Sahmani S, Arash B. Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys Lett A 2010;375(1):53–62. [30] Pradhan SC, Kumar A. Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Compos Struct 2011;93(2):774–9. [31] Wang Q. Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J Appl Phys 2005;98(12):124301-1–6. [32] Reddy JN. Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int J Eng Sci 2010;48(11):1507–18. [33] Arash B, Wang Q, Liew KM. Wave propagation in graphene sheets with nonlocal elastic theory via finite element formulation. Comput Methods Appl Mech Eng 2012;223–224:1–9. [34] Zhu P, Zhang LW, Liew KM. Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov– Galerkin approach with moving Kriging interpolation. Compos Struct 2014 298–314. [35] Zhang LW, Zhu P, Liew KM. Thermal buckling of functionally graded plates using a local Kriging meshless method. Compos Struct 2014;108:472–92. [36] Zhang LW, Lei ZX, Liew KM, Yu JL. Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels. Compos Struct 2014;111:205–12. [37] Zhang LW, Lei ZX, Liew KM, Yu JL. Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels. Comput Methods Appl Mech Eng 2014;273:1–18. [38] Zhang LW, Deng YJ, Liew KM. An improved element-free Galerkin method for numerical modeling of the biological population problems. Eng Anal Boundary Elem 2014;40:181–8. [39] Zhang L, Liew KM. An element-free based solution for nonlinear Schrödinger equations using the ICVMLS-Ritz method. Appl Math Comput 2014;249:333–45. [40] Lei ZX, Zhang LW, Liew KM, Yu JL. Dynamic stability analysis of carbon nanotube-reinforced functionally graded cylindrical panels using the elementfree kp-Ritz method. Compos Struct 2014;113:328–38. [41] Guo PF, Zhang LW, Liew KM. Numerical analysis of generalized regularized long wave equation using the element-free kp-Ritz method. Appl Math Comput 2014;240:91–101. [42] Zhang L, Liew KM. An improved moving least-squares Ritz method for twodimensional elasticity problems. Appl Math Comput 2014;246:268–82. [43] Wang Q. Effective in-plane stiffness and bending rigidity of armchair and zigzag carbon nanotubes. Int J Solids Struct 2004;41(20):5451–61. [44] Wang Q, Liew KM. Molecular mechanics modeling for properties of carbon nanotubes. J Appl Phys 2008;103(4):046103. [45] Beissel S, Belytschko T. Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 1996;139(1–4):49–74. [46] Zhao X, Liew KM. Geometrically nonlinear analysis of functionally graded plates using the element-free kp-Ritz method. Comput Methods Appl Mech Eng 2009;198(33–36):2796–811. [47] Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. Int J Numer Methods Fluids 1995;20(8–9):1081–106. [48] Chen JS, Pan C, Wu CT, Liu WK. Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 1996;139(1–4):195–227. [49] Saeid Reza A, Ali F, Mehdi B, Amir Hessam H. Thermal effects on the stability of circular graphene sheets via nonlocal continuum mechanics. Lat Am J Solids Struct 2014;11(4):704–24. [50] Sakiyama T, Huang M. FREE vibration analysis of rectangular plates with variable thickness. J Sound Vib 1998;216(3):379–97.