Document Type : Research Paper
Authors
^{1} Department of Mechanical Engineering, Arak University, Arak, 3815688349, Iran  Institute of Nanosciences & Nanotechnology, Arak University, Arak, 3815688349, Iran
^{2} Department of Mechanical Engineering, Arak University, Arak, 3815688349, Iran
Abstract
Keywords
Main Subjects
Free Vibration Analysis of SizeDependent MultiLayered Graphene Sheets Based on Strain Gradient Elasticity Theory
^{a} Department of Mechanical Engineering, Arak University, Arak, 3815688349, Iran
^{b} Institute of Nanosciences & Nanotechnology, Arak University, Arak, 3815688349, Iran
KEYWORDS 

ABSTRACT 
Navier method; ESDT; Multilayered graphene sheets; Strain Gradient Theory. 
This study investigates the sizedependent free vibration analysis of multilayered graphene sheets based on exponential shear deformation theory (ESDT), which considers the effects of rotary inertia and transverse shear deformations. In order to capture the effects of length scale parameter on the vibrational behavior of the structure, modified strain gradient elasticity theory is utilized. An elastic multipleplate model is assumed in which the nested plates are coupled with each other through the van der Waals interlayer forces. The governing equations of motion are derived by implementing Hamilton’s principle and then are solved with the Navier approach. To verify the present model, results in specific cases are compared with the available papers in the literature and excellent agreement is seen. Finally, the effects of various parameters such as aspect ratio, thickness ratio, Winkler modulus, shear modulus, and size effects on the natural frequencies of a multilayered graphene sheet are presented and discussed in detail. 




Carbon nanotubes (CNTs), graphene sheets (GSs), and fullerenes are the principal elements of carbon nanostructures, which caused a considerable promotion in the world of nanotechnology. Graphene is a twodimensional atomic crystal with special electronic and mechanical properties. Graphene sheets are useful structures in polymer composites in order to strengthen them. Moreover, thanks to their extraordinary features, graphene sheets are widely utilized in various systems such as biomedical employment, nanomechanical devices, solar cells, etc. Multilayered graphene sheets that are held together by van der Waals interactions have received a great deal of attention to the scientific community, as the bending stiffness of the multilayered graphene sheet is usually more than singlelayered graphene sheet.
Experiments have shown that the smallscale effect has a major role in the analysis of smallscale structures and cannot be ignored [113]. Atomic simulations and continuum mechanics are common methods for modeling a structure in small scales. In contrary to atomic simulations, continuum mechanics are computationally expensive. Hence, researchers are interested in continuum mechanics to analyze the behavior of systems in small scales. Additionally, classical theories are not capable to capture the mechanical behavior response of micro and nanosized structures since they do not include any length scale parameters. Various sizedependent continuum models such as the modified couple stress theory [1417], the nonlocal elasticity theory [1822] and the strain gradient theory [23] can be used in smallscale systems. Strain gradient theory as a recognized theory for analysis of sizedependent behavior includes five independent length scale parameters and considers strain energy of the secondorder gradient of deformations. Lam et. al. [24] developed the strain gradient theory and assumed only three lengthscale parameters to catch the scale effects. Their modified strain gradient theory contains a new additional equilibrium equation as well as the classical equilibrium equations.
In recent years, reinforced structures like graphene sheets have been studied extensively [2530]. Ansari et. al. [31] investigated the vibrational characteristics of multilayered graphene sheets with various boundary conditions based on the Mindlin plate theory, and the Eringen nonlocal elasticity. In their study, the nested nanoplates interacted with each other through the van der Waals interactions between all the layers. Sobhy [32] studied a twovariable plate theory with a sinusoidal distribution for transverse shear stress in order to analyze the vibration of the orthotropic doublelayered graphene sheets in the hydrothermal environment. Wang et. al. [33] studied the vibration of doublelayered nanoplates in the thermal environment. They considered smallscale effects using the nonlocal continuum theory and concluded that nonlocal effects are more prominent at larger half wave numbers. Shafiei et. al. [34] conducted a study on the sizedependent vibration and buckling of multilayered graphene sheets using couple stress theory, which is a special case of strain gradient theory. They adopted a modified plate theory with only two variables with the help of the finite strip approach to formulate their model. Karličić et. al. [35] investigated the stability analysis and thermal vibration of multilayer graphene sheets (MLGS) embedded in an elastic medium based on the nonlocal KirchhoffLove plate theory. They applied Navier’s method to find the exact closedform solution for natural frequencies and critical buckling loads of the structure. Kitipornchai et. al. [36] analyzed the vibration of simplysupported MLGSs based on a continuum model. They concluded that vibration modes related to the classical natural frequencies are similar in amplitude and direction while vibration modes of resonant frequencies are different because of the van der Waals interactions. Hosseini Hashemi et. al. [37] studied the free vibration analysis of double viscoelastic graphene sheets coupled with a ViscoPasternak layer based on the classical plate theory. They employed the Kelvin–Voigt model and presented an exact solution for governing equations. He et. al. [38] investigated the stability of the multiwalled carbon nanotubes with an efficient algorithm. They assumed individual tubes as continuum cylindrical shells and considered van der Waals (vdW) interactions as a constant, which is independent of the tubes’ radius. Ansari et. al. [39] presented a numerical solution using the differential quadrature method for vibration analysis of embedded multilayered graphene sheets with various boundary conditions based on the Mindlin theory. Radić and Jeremić [40] utilized the Galerkin approach for buckling and vibration of Mindlin orthotropic doublelayered graphene sheets subjected to hygrothermal loading for seven different boundary conditions based on a nonlocal elasticity model. Nazemnezhad et. al. [41] studied the essence of nonlocal elasticity for vibration analysis of multilayer graphene sheets based on the sandwich model. They compared their results with Molecular Dynamic (MD) simulation and concluded that the van der Waals interactions result in the interlayer shear effect. Arefi et. al. [42] demonstrated the nonlocal dynamic behavior of threelayered nanoplates with piezomagnetic face sheets resting on Pasternak’s foundation based on firstorder shear deformation theory. In their research work, the Navier method was used to solve the governing equations.
In the present work, the exponential shear deformable plate theory (ESDPT) in conjunction with strain gradient elasticity is employed to study the effects of the material length scale parameters on the vibrational behavior of multilayered graphene sheets. The governing partial differential equations are obtained using Hamilton’s principle, and then the Navier solution is presented to analyze the free vibration of simply supported graphene sheets. The elastic foundation is considered to be Pasternak. To validate the presented method, our results are compared to those in the corresponding literature. The results of the present work can be used as benchmarks for future works.
Consider a multilayered graphene sheet embedded in an elastic medium with length a, width b, mass density ρ, and constant thickness h, as shown in Figure 1.
To study the motion of the graphene sheet, a coordinate system is located in the corner of the structure. The displacement field according to the exponential shear deformation plate theory for each layer of MLGS can be written as [20]

(1) 

(2) 

(3) 
where and for exponential (ESDT) and trigonometric theory (TSDT), respectively.
Figure 1. Rectangular multilayered nanographene sheet embedded in an elastic medium
In Eqs. (1)(3), , and denote the displacement of an arbitrary point in the , and directions for the ith sheet of the MLGS, respectively. and are the midplane displacements, and express the rotation functions.
The strain energy of the structure with volume can be written as [2]

(4) 
where the deformation functions can be defined as below:

(5) 

(6) 

(7) 

(8) 
in which is the displacement vector in the direction of i (i=x, y, z). , , and denote the strain tensor, the dilation gradient vector, deviatoric stretch gradient tensor and the symmetric rotation gradient tensor, respectively. Moreover, is the Kronecker delta and represents the permutation symbol, which can be expressed as follows:
indicates the classical stress tensor and , , are stress measures for the higher order stresses, which can be written as follows[2]:

(9) 

(10) 

(11) 

(12) 

(13) 
in which are material length scale parameters related to dilatation gradients, deviatoric stretch gradients, and rotation gradients respectively. Also, and are Lame constants that can be expressed in the following form:

(14) 

(15) 
where is Young's modulus and is Poisson's ratio. Considering Hooke's Law for the stress field, the normal stress is assumed to be negligible in comparison with plane stresses and . Thus, stress and strain fields can be related as below

(16) 

(17) 

(18) 

(19) 

(20) 
where is the shear modulus of the structure. The kinetic energy and the work done by the external applied forces related to the ith layer of an Nlayered GS can be written as
, 
(21) 
, 
(22) 
where dotsuperscript denotes the differentiation with respect to the time variable t. Also, is shear modulus and is the pressure exerted on the ith layer through the van der Waals interaction forces and the surrounding elastic medium, which are defined as follows:

(23) 
In Eq. (23), indicates the pressure exerted on the ith layer due to the van der Waals interaction of the remaining layers and can be given as

(24) 
where the van der Waals coefficients C_{ij} represent the rising pressure to the ith layer from the jth layer and are given as follows [31]:

(25) 
where a_{cc} and h_{ij} denote the length of carboncarbon bond and the distance between two layers, respectively. Moreover, σ and ɛ are two parameters relating to the carboncarbon bond.
The van der Waals interaction coefficients C_{ij} are calculated using equation (25) and are expressed in Table 1 for a tenlayered GS. Note that the negative sign in Table 1 shows an attraction between two layers while the positive sign shows repulsion. According to this table, the van der Waals interaction coefficient C_{ij} between two adjacent layers is the largest coefficient, which indicates that the van der Waals interaction of two adjacent layers is the strongest interaction [31]. Moreover, it is seen that the van der Waals interaction decreases considerably as the distance between two layers increases.
It is worth mentioning that (P_{Winkler})_{i} in Eq. (23) is the Winkler foundation model which can be expressed as [32];

(26) 
in which K_{w} is the Winkler modulus, which depends on the material properties of the elastic medium. Implementing Hamilton’s principle, the governing equations of the Multilayered graphene sheets can be derived as follows [37]
, 
(27) 
where U, K, and W express strain energy, kinetic energy, and work done by external forces, respectively. Besides, is a variation operator. The governing equations for the ith layer of an Nlayered GS can be obtained by incorporating Eqs. (4), (21), and (22) into Eq. (27) and then using integrating by parts and setting the coefficients of and to zero as follows
, 
(28) 
, 
(29) 

(30) 


, 
(31) 
, 
(32) 
where the resultant loads and moment of inertias can be defined as

(33) 

(34) 

(35) 

(36) 

(37) 

(38) 
To solve the governing equations of the structure i.e. Eqs. (28) (32), the Navier approach is utilized. The admissible displacements and rotation functions that can satisfy all boundary conditions for a simply supported plate are defined as

(39) 

(40) 

(41) 

(42) 

(43) 
where ( ) are unknown coefficients, and denotes the natural frequencies. Substitution of Eqs. (39)  (43) into the Eqs. (28)  (32) leads to the following eigenvalue problem:
, 
(44) 
in which, [K] and [M] express the stiffness matrix and mass matrix of the system, respectively. Also is the vector of the unknown coefficients. Setting the determinant of the Eq. (44) to zero, eigenvalues and eigenvectors, which are related to the natural frequencies and mode shapes of the structure are obtained.
To validate the results, The eigen problem (44) was solved to obtain the natural frequencies of a graphene sheet. A comparison study was made between the present exponential shear deformation results and those reported by the Navier solution [39] for a rectangular nanoplate embedded in an elastic medium.
These results are listed in Table 2 for a square doublelayered GS based on the strain gradient, couple stress, and the classical model, respectively. The mechanical properties for each layer of graphene sheets are chosen from Ref. [39], in which length a=10 nm, width b=10 nm, and the initial interlayer separation between the two adjacent layers is assumed to be h=0.34 nm. The van der Waals interaction coefficients C12= C21=C are calculated using Eq. (25), and all the three material length scale parameters are equal to the amount of , that is . Besides, Young’s modulus, Poisson’s ratio, and density of each layer are assumed to be E=1.02 TPa, and , respectively. The natural frequencies of the modified strain gradient theory (MSGT), modified couple stress theory (MCST), and classical plate theory (CPT) with are obtained for m=1 and n= 1, 2, 3, in Table 2. As seen, a very good agreement can be observed. According to this table, there is a bit difference among results, which is due to the various shear deformation theories. These differences are created because function f(z) has different expansion through thickness in various theories.
From Table 2, it can be also seen that the natural frequency changes significantly as the mode number rises, which shows that the effect of the mode order on the natural frequency is not negligible. In another validation study, the fundamental frequency of a single layered graphene sheet was compared with that of our model for different values of length scale parameter and aspect ratio using MCST. As observed, again excellent agreement is seen.
Table 1. The van der Waals interaction coefficients (Gpa^{1}) for a tenlayered GS 

N 
j=1 
j=2 
j=3 
j=4 
j=5 
j=6 
j=7 
j=8 
j=9 
j=10 
i=1 
0 
108.6 
1.8720 
0.1688 
0.0301 
0.0079 
0.0026 
0.0010 
0.0004 
0.0004 
i=2 
108.6 
0 
108.6 
1.8720 
0.1688 
0.0301 
0.0079 
0.0026 
0.0010 
0.0004 
i=3 
1.8720 
108.6 
0 
108.6 
1.8720 
0.1688 
0.0301 
0.0079 
0.0026 
0.0010 
i=4 
0.1688 
1.8720 
108.6 
0 
108.6 
1.8720 
0.1688 
0.0301 
0.0079 
0.0026 
i=5 
0.0301 
0.1688 
1.8720 
108.6 
0 
108.6 
1.8720 
0.1688 
0.0301 
0.0079 
i=6 
0.0079 
0.0301 
0.1688 
1.8720 
108.6 
0 
108.6 
1.8720 
0.1688 
0.0301 
i=7 
0.0026 
0.0079 
0.0301 
0.1688 
1.8720 
108.6 
0 
108.6 
1.8720 
0.1688 
i=8 
0.0010 
0.0026 
0.0079 
0.0301 
0.168 
1.872 
108.6 
0 
108.6 
1.8720 
i=9 
0.0004 
0.0010 
0.0026 
0.0079 
0.030 
0.1688 
1.8720 
108.6 
0 
108.6 
i=10 
0.000 
0.0004 
0.00105 
0.00265 
0.0079 
0.03012 
0.16889 
0.16889 
108.6 
0 
Table 2. Natural frequencies of a square doublelayerd GS with b= 10 nm using MSGT, MCST and CPT in the first nine modes ( , =0, ) 

m 
n 
Method 






1 
1 
ESDT 
0.1543 
2.4194 
0.1017 
2.4152 
0.0663 
2.4137 
TSDT 
0.1543 
2.4195 
0.1018 
2.4152 
0.0663 
2.4138 

Navier[39] 
0.1895 
2.6888 
0.1165 
2.6846 
0.0690 
2.6830 

1 
2 
ESDT 
0.3898 
2.4446 
0.2534 
2.2949 
0.1648 
2.0820 
TSDT 
0.3899 
2.4447 
0.2534 
2.2951 
0.1648 
2.0821 

Navier[39] 
0.4742 
2.7237 
0.2911 
2.6979 
0.1725 
2.6877 

1 
3 
ESDT 
0.7794 
2.5471 
0.5043 
2.3321 
0.3267 
2.1873 
TSDT 
0.7794 
2.5471 
0.5043 
2.3321 
0.3267 
2.1874 

Navier[39] 
0.9499 
2.8454 
0.5823 
2.7446 
0.3450 
2.6877 
The natural frequencies of MSGT, MCST, and CPT with for a doublelayered graphene sheet are listed in Table 4 for different foundation parameters and aspect ratios.
Furthermore, Figs 25 are plotted for a better understanding of the effect of different parameters on natural frequencies using MSGT, MCST, and CPT. As seen in Table 4, The natural frequency values decline with a growth in the aspect ratio. This is because the graphene sheet considered here is simply supported at all edges, and a decrease in width at a constant length leads to the decrease of degrees of freedom (DOF). Consequently, this causes a decline in stiffness and natural frequency.
Table 5 shows the natural frequencies for double, triple, and fivelayered square nanographene sheets when b = 10 nm and .
As can be seen in Table 5, the natural frequency ( ) is independent of the van der Waals interactions. However, all other higher natural frequencies from to depend on the van der Waals interactions for different numbers of layers calculated by CPT, the MSGT, and the MCST. This happens because the equations of motion are coupled to each other by the van der Waals interaction coefficients although the effect of the van der Waals interaction does not exist in the fundamental frequency ( ).
Figure 2 depicts the impacts of smallscale parameter on the frequency ratio of a doublelayered nanographene sheet. What stands out here is that the frequency ratio is more significant in lower values of thicknessto smallscale parameter ratio. This is due to the fact that when the structure is smaller, the influence of size effects is more notable. Figure 3 demonstrates the variation of fundament frequency of a doublelayered nanographene sheet using MSGT, MCST, and CPT. As plotted, MSGT has the highest and CPT has the lowest eigenvalue, indicating that strain gradient tensors rise the overall stiffness of the structure. It can be also concluded that when structure thickness is more than ten times of the smallscale parameter, results obtained by MSGT and MCST converge to those of CPT.
Table 3. Comparison study for natural frequencies of a graphene sheet with b= 10 nm and h=0.34 nm 

Method 
a/b 
l 

0.5h 
h 
2h 

Ref [44] 
1 
1.0528 
1.1974 
1.6537 
Present 
1.0461 
1.1861 
1.6208 

Ref [44] 
2 
1.0236 
1.0915 
1.3286 
Present 
1.0141 
1.0832 
1.3014 

Ref [44] 
3 
1.0191 
1.0744 
1.2718 
Present 
1.0086 
1.0624 
1.2596 
Table 4. Natural frequencies using MSGT, MCST and CPT for various Winkler modulus , shear modulus , and aspect ratios (h = 0.34 nm, b = 10 nm, , m=1 ). 















n 

a/b 
1014 
1017 
1014 
1017 
1014 
1017 
1014 
1017 
1014 
1017 
1014 
1017 
1 
0 
1 
0.9884 
0.9884 
2.6754 
2.6738 
0.9892 
0.9892 
2.6757 
2.6756 
0.9908 
0.9908 
2.6782 
2.688 
2 
0.2328 
0.2406 
2.6660 
2.6653 
0.1308 
0.1443 
2.6730 
2.6722 
0.2689 
0.2564 
0.2635 
1.5305 

5 
1 
0.1648 
0.1531 
2.6811 
2.6805 
0.2534 
0.2460 
2.6880 
2.6874 
0.3883 
0.3835 
2.7042 
2.7036 

2 
0.0663 
0.0256 
2.6806 
2.6799 
0.1017 
0.0998 
2.6818 
2.6817 
0.1534 
0.1407 
2.6842 
2.6835 

2 
0 
1 
1.5629 
1.5628 
2.6676 
2.6670 
1.5658 
1.5658 
2.7458 
2.7451 
2.7974 
2.7968 
3.4675 
3.4675 
2 
0.1655 
0.1540 
2.666 
2.6652 
0.6722 
0.6695 
2.6729 
2.6728 
1.2052 
0.2564 
2.6892 
2.6885 

5 
1 
0.5487 
0.5454 
2.7184 
2.7177 
0.8518 
0.8496 
2.7951 
2.7945 
1.3140 
1.3425 
2.9388 
2.9388 

2 
0.1648 
0.1531 
2.6812 
2.6805 
0.2534 
0.2527 
2.6881 
2.6880 
0.3883 
0.3835 
2.7042 
2.7036 

3 
0 
1 
2.2102 
2.2102 
3.4104 
2.6591 
1.6533 
1.6522 
3.4104 
3.4104 
2.69021 
2.6895 
3.7736 
3.7731 
2 
0.8661 
0.8640 
2.7818 
2.7811 
0.3040 
0.3034 
3.1167 
3.1161 
0.6652 
0.6624 
3.4675 
3.4675 

5 
1 
1.1565 
1.1550 
3.4104 
2.8847 
1.8220 
1.8211 
3.4104 
3.4104 
1.2635 
1.2036 
3.8507 
3.8503 

2 
0.3267 
0.3209 
2.8854 
2.6894 
0.5043 
0.504 
3.2094 
3.2088 
0.7775 
0.7751 
2.9233 
2.9227 
Table 5. Comparison of natural frequencies (THz) between double, triple and eightlayered square graphene sheets with width b = 10 nm, m=1, , , and ). 

N 
Natural frequencis (THz) 
CPT 
MCST 
MSGT 

n=1 
n=2 
n=3 
n=1 
n=2 
n=3 
n=1 
n=2 
n=3 

2 

0.0256 
0.1531 
0.3209 
0.0812 
0.246 
0.0501 
0.1407 
0.3835 
0.7751 


2.6799 
2.6805 
3.4104 
2.6811 
2.6874 
3.4104 
2.6835 
2.7037 
3.4675 
3 

0.0256 
0.1531 
0.3209 
0.0813 
0.246 
0.5007 
0.1407 
0.3835 
0.7751 


1.8622 
1.8657 
1.8829 
1.8638 
1.8756 
1.9217 
1.8673 
1.8986 
2.0109 


3.2822 
3.2810 
3.2859 
3.2831 
3.2867 
3.3083 
3.2851 
3.3000 
3.3613 
5 

0.0256 
0.1531 
0.3209 
0.0812 
0.246 
0.0501 
0.1407 
0.3835 
0.7751 


1.4859 
1.1403 
1.1723 
1.1342 
1.1565 
1.2337 
1.1403 
1.1933 
1.3684 


1.8717 
1.8736 
2.1986 
1.8394 
1.9444 
2.2394 
1.5305 
2.2203 
2.3156 


2.2579 
2.4782 
3.3253 
3.0488 
2.5877 
2.4305 
2.9846 
3.0704 
3.1352 


3.0485 
3.6587 
3.4963 
3.6013 
2.6638 
3.1844 
3.1457 
3.3224 
3.6706 
The influences of aspect ratio and mode numbers on the vibrational behavior of a doublelayered nanographene sheet are shown in Figures 3 and 4. From Figure 4, it is observed that natural frequency increases by increasing of aspect ratio. Additionally, as seen, the impacts of size effects get less and less by decreasing of aspect ratio. This is because, for a constant length of the nanographene sheet, the width of nanographene sheet gets smaller by increasing the aspect ratio. On the other hand, it is obvious that the dynamic behavior of a nanographene sheet is considerably dependent on its dimensions. Hence, as this figure depicts, the size effects are more notable for higher aspect ratios. Figure 5 demonstrates the variations of mode numbers on natural frequencies. As seen, natural frequencies increase by rising mode number. This is because motions in higher modes need more energy and the structure needs to be stiffer.
In this paper, the vibration response of a multilayered nanographene sheet mounted on an elastic medium was studied based on the strain gradient elasticity theory using the exponential shear deformation theory. The model includes three material length scale parameters, which may effectively include the size effect. The model can also cover the modified couple stress plate model or the classical plate model, by setting two or all of the material length scale parameters equal to zero. In the exponential shear deformation theory, an exponential function is applied in terms of thickness coordinate to include the effect of transverse shear stress and rotary inertia.


Figure 2. Effects of length scale ratio on frequency ratio of a doublelayered nanographene sheet 



Figure 3. Effects of length scale ratio on fundament frequency of a doublelayered nanographene sheet using MSGT, MCST, and CPT 



Figure 4. Effects of aspect ratio on fundament frequency of a doublelayered nanographene sheet using MSGT 



Figure 5. Effects of mode number on natural frequencies of a doublelayered nanographene sheet using MSGT 
The nested nano plates react with each other through the van der Waals interaction between all layers. The equations of motion were derived using Hamilton’s principle. Comparison between the present results and those reported in the literature for simply supported multilayered nanographene sheets shows high stability and accuracy of the present Navier method. The presented results show the effect of variations of the aspect ratio a/b, Winkler modulus , shear modulus and size effects on the natural frequency of sizedependent multilayered nanographene sheets. The results of the present work can be used as benchmarks for future studies.
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