Free Vibration Analysis of Size-Dependent Multi-Layered Graphene Sheets Based on Strain Gradient Elasticity Theory

1. Khorshidi ^a,b* , M. Karimi ^a , A.A. Alinezhad ^a

^a Department of Mechanical Engineering, Arak University, Arak, 38156-88349, Iran

^b Institute of Nanosciences & Nanotechnology, Arak University, Arak, 38156-88349, Iran

1.     Introduction

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 two-dimensional 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, nano-mechanical devices, solar cells, etc. Multi-layered 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 multi-layered graphene sheet is usually more than single-layered graphene sheet.

Experiments have shown that the small-scale effect has a major role in the analysis of small-scale structures and cannot be ignored [1-13]. 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 nano-sized structures since they do not include any length scale parameters. Various size-dependent continuum models such as the modified couple stress theory [14-17], the nonlocal elasticity theory [18-22] and the strain gradient theory [23] can be used in small-scale systems. Strain gradient theory as a recognized theory for analysis of size-dependent behavior includes five independent length scale parameters and considers strain energy of the second-order gradient of deformations. Lam et. al. [24] developed the strain gradient theory and assumed only three length-scale 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 [25-30]. Ansari et. al. [31] investigated the vibrational characteristics of multi-layered 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 two-variable plate theory with a sinusoidal distribution for transverse shear stress in order to analyze the vibration of the orthotropic double-layered graphene sheets in the hydrothermal environment. Wang et. al. [33] studied the vibration of double-layered nanoplates in the thermal environment. They considered small-scale 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 size-dependent vibration and buckling of multi-layered 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 multi-layer graphene sheets (MLGS) embedded in an elastic medium based on the nonlocal Kirchhoff-Love plate theory. They applied Navier’s method to find the exact closed-form solution for natural frequencies and critical buckling loads of the structure. Kitipornchai et. al. [36] analyzed the vibration of simply-supported 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 Visco-Pasternak 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 multi-walled 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 multi-layered 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 double-layered 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 multi-layer 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 three-layered nanoplates with piezomagnetic face sheets resting on Pasternak’s foundation based on first-order 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 multi-layered 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.

2.     Formulation

Consider a multi-layered 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]

where  and for exponential (ESDT)  and trigonometric theory (TSDT), respectively.

Figure 1. Rectangular multilayered nano-graphene sheet embedded in an elastic medium

In Eqs. (1)-(3), , and  denote the displacement of an arbitrary point in the ,  and  directions for the i-th sheet of the MLGS, respectively. and are the mid-plane displacements, and express the rotation functions.

The strain energy of the structure with volume  can be written as [2]

where the deformation functions can be defined as below:

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]:

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:

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

where  is the shear modulus of the structure. The kinetic energy  and the work done by the external applied forces  related to the i-th layer of an N-layered GS can be written as

where dot-superscript denotes the differentiation with respect to the time variable t. Also,  is shear modulus and  is the pressure exerted on the i-th layer through the van der Waals interaction forces and the surrounding elastic medium, which are defined as follows:

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

where the van der Waals coefficients C_ij represent the rising pressure to the i-th layer from the j-th layer and are given as follows [31]:

where a_cc and h_ij denote the length of carbon-carbon bond and the distance between two layers, respectively. Moreover, σ and ɛ are two parameters relating to the carbon-carbon bond.

The van der Waals interaction coefficients C_ij are calculated using equation (25) and are expressed in Table 1 for a ten-layered 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];

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 Multi-layered graphene sheets can be derived as follows [37]

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 i-th layer of an N-layered 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

where the resultant loads and moment of inertias can be defined as

3.     Solution Procedure

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

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:

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.

4.     Discussion

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 double-layered 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.

The natural frequencies of MSGT, MCST, and CPT with  for a double-layered graphene sheet are listed in Table 4 for different foundation parameters and aspect ratios.

Furthermore, Figs 2-5 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 five-layered square nano-graphene 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 small-scale parameter on the frequency ratio of a double-layered nano-graphene sheet.  What stands out here is that the frequency ratio is more significant in lower values of thickness-to- small-scale 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 double-layered nano-graphene 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 small-scale parameter, results obtained by MSGT and MCST converge to those of CPT.

The influences of aspect ratio and mode numbers on the vibrational behavior of a double-layered nano-graphene 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 nano-graphene sheet, the width of nano-graphene sheet gets smaller by increasing the aspect ratio. On the other hand, it is obvious that the dynamic behavior of a nano-graphene 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.

5.     Conclusion

In this paper, the vibration response of a multi-layered nano-graphene 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.

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 multi-layered 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 size-dependent multi-layered nanographene sheets. The results of the present work can be used as benchmarks for future studies.

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