Bending, Buckling and Vibrations Analysis of the Graphene Nanoplate Using the Modified Couple Stress Theory

Majid ESKANDARI SHAHRAKI*, Mahmoud SHARIATI**, Naser ASIABAN***, Jafar ESKANDARI JAM**** *Aerospace Engineering, Ferdowsi University of Mashhad, Mashhad, Iran, E-mail: mjdeskandari@gmail.com **Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran, E-mail: mshariati44@gmail.com ***Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran, E-mail: naser.asiaban@mail.um.ac.ir ****Mechanical Engineering, Malek-Ashtar University of Technology, Tehran, Iran, E-mail: jejam@mail.com


Introduction
The atomic and molecular scale test is known as the safest method for the study of materials in small-scales. In this method, the nanostructures are studied in real dimensions. The atomic force microscopy (AFM) is used to apply different mechanical loads on nanoplates and measure their responses against those load in order to determine the mechanical properties of the nanoplate. The difficulty of controlling the test conditions at this scale, high economic costs and time-consuming processes are some setbacks of this method. Therefore, it is used only to validate other simple and low-cost methods.
Atomic simulation is another solution for studying small-scale structures. In this method, the behavior of atoms and molecules is examined by considering the intermolecular and interatomic effects on their motions, which eventually involves the total deformation of the body. In the case of large deformations and multi atomic scale the computational costs is too high, so this method is only used for small deformation problems.
Given the limitations of the aforementioned methods for studying nanostructures, researchers have been looking for simpler solutions for nanostructures. Modeling small-scale structures using continuum mechanics is another solution to this problem. There are a variety of sizedependent continuum theories that consider size effects, some of these theories are; micromorphic theory, microstructural theory, micropolar theory, Kurt's theory, non-local theory, modified couple stress theory and strain gradient elasticity. All of which are the developed notion of classical field theories, which include size effects.

Modified couple stress theory
In 2002 Yang et al. [1] proposed a modified couple stress model by modifying the theory proposed by Toppin [2], Mindlin and Thursten [3], Quitter [4] and Mindlin [5] in 1964. The modified couple stress theory consists of one material length scale parameter for projection of the size effect, whereas the classical couple stress theory has two material length scale parameters. In the modified couple stress theory, the strain energy density in the three-dimensional vertical coordinates for a body bounded by the volume V and the area Ω [6], is expressed as the follows: where: ij  and ij  are the symmetric parts of the curvature and strain tensors; i  and i u are the displacement and the rotational vectors, respectively.
ij  , the stress tensor, and , ij m , the deviatory part of the couple stress tensor, are defined as: where:  and  are the lame constants; ij  is the Kronecker delta and l is the material length scale parameter. From Eqs.
(3) and (6) it can be seen that ij  and ij m are symmetric.

Fig. 1 A schematic of the nanoplate and axes
The displacement equations for the Mindlin's plate are defined as [8]: x y u x y z t u x y t z u x y z t v x y t z u x y z t w x y t where: x  and y  are the rotations of the normal vector around the x and y axis respectively, and w is the midpoint displacement of the plate in the z-axis direction. The strain and stress tensors, the symmetric part of the curvature tensor, and the rotational vector for the Mindlin's plate is obtained as follows:   For the sake of simplification, the coefficient of each variable in the above equation is named from 1 F to 15 F and this equation can be rewritten as shown below:   x y x y where: x P is the Axial force along the x axis; y P is the Axial force along the y axis; xy P is the shear force in the xy plane, and

( )
, q x y is the out-of-plane force.

Virtual work of the external forces
In these kind of problems, the virtual work of three kinds of external forces are included in the solutions, if the middle-plane and the middle-perimeter of the plate are shown as Ω and Γ respectively, these virtual works are [8]: 1. The virtual work done by the body forces, which is applied on the volum 2. The virtual work done by the surface tractions at the upper and lower surfaces Ω. 3. The virtual work done by the shear tractions on the lateral surfaces, ,, x y z f f f are the body forces, ( ) ,, x y z c c c are the body couples, ( ) ,, x y z q q q are the forces acting on the Ω plane, ( ) ,, x y z t t t are the Cauchy's tractions and ( ) ,, x y z S S S are surface couples the Variations of the virtual work is expressed as: Given that in this study only the external force z q was applied, virtual work becomes: the variation of kinetic energy is obtained as: where: ρ is the density.
Finally using the Hamilton's principle, it can be said that [9]: where: T is the kinetic energy; U is the strain energy, and w is the work of the external forces.

The final governing equations of the plate after applying the buckling and external forces
Using Hamilton's principle, Eq. (48), and the Eqs. from (44) to (47), the governing equations of the plate including the buckling and external forces are obtained as follows:

Obtaining the general governing equation of the Mindlin's plate (including buckling, bending and vibrations)
Considering the following constants: 1 , 4 Simply-supported boundary conditions were also satisfied by the Navier's method according to the following equations:

The general equation matrix of a Mindlin's plane
After solving the governing equations and naming the coefficient of each variable, we have: Finally, the general equation matrix of the Mindlin's plate along with the auxiliary equations will be obtained as follows: In this study, graphene is chosen as the plate's material. A single-layer graphene plate has the following properties where: μ and λ are the lame's coefficients; E is the Young's modulus [10]. The value of the distributed force was considered to be 2 1N m . q = .

Results and discussion
Results were obtained using a computational program coded in the MATLAB software. The results have also been compared with the literature [11,12] and good agree-ments between results were observed. The plate's dimensional parameters are chosen as follows: a is plate's length; b is plate's width; h is plate's thickness; l is material length scale parameter Table 1 shows the Mindlin's nanoplate bending rate under sinusoidal load for different material length scale parameters to thickness l/h and length to width ratio a/b. As can be seen, as the length scale parameter to thickness ratio increases, the bending ratio decreases but it increases due to the increase in the plate's length to width ratio.  Table 2 compares the values of critical force for different nanoplates under a bi-axial surface loading for various length to thickness ratios. It was observed that, the Mindlin's nanoplate has the highest, and the Third-order nanoplate has the lowest critical force values.  Table 3 shows the dimensionless bending values of Mindlin's nanoplate under the uniform surface traction and sinusoidal load for material length scale to thickness and length to width ratios. As shown in the table, except for the classical theory l=0, the dimensionless bending values under sinusoidal load were higher than bending values obtained under the uniform surface traction. It was also found that with an increase in the material length scale parameter to thickness and length to width ratio of the nanoplate, the dimensionless bending value decreases. Fig. 3 shows the values of dimensionless critical force for Mindlin's nanoplate under a uniaxial force in the x-direction. It was found that this value increases due to an increase in length to thickness ratio of the nanoplate. Furthermore, when the effect of size parameter is neglected (classical theory), the value of dimensionless critical force becomes constant and reaches its lowest value, but with an increase in the size parameter the dimensionless critical force value increases.    (except for the classical theory l=0. It was observed that this value increases due to an increase in length to thickness ratio. Also, for the classical theory (neglecting the effect of size parameter) the dimensionless frequency reaches its lowest value, but with an increase in the size effect, the dimensionless frequency values increase.  Table 4 shows that the dimensionless frequency of different modes of Mindlin's nanoplate increases due to an increase in material length scale parameter to thickness ratio.
By comparing Tables 4 -7 it was found that with an increase in length to thickness ratio of the Mindlin's nanoplate, the vibration frequency decreases.

Conclusion
In this study, the bending, buckling and vibration of a graphene Mindlin's nanoplate were investigated using the modified couple stress theory. As observed in the tables and figures, the Mindlin's nanoplate bending rate under sinusoidal load, decreases with an increase in length to thickness ratio of the nanoplate, but, this value increases with an increase in the aspect ratio of the nanoplate. Furthermore, by comparing different nanoplates under uniform surface traction it was found that the Kirchhoff's nanoplate yields the lowest and the third-order nanoplate yields the highest values for bending.
The buckling analysis showed that the dimensionless critical force increases due to an increase in material length scale parameter to thickness ratio and decreases due to an increase in length to thickness ratio of the nanoplate. But when the size effect parameter is neglected (classical theory), the value of dimensionless critical force becomes constant and reaches its lowest value, but with an increase in the size parameter the dimensionless critical force value increases.
Analysis of frequencies of different modes showed that this value increases due to an increase in length to thickness ratio. Also, for the classical theory (neglecting the effect of size parameter) the dimensionless frequency reaches its lowest value, but with an increase in the size effect, the dimensionless frequency values increase. It was also found that the Mindlin's nanoplate yields the highest and the thirdorder nanoplate yields the lowest values for frequency.