Ritz Method in Vibration Analysis for Embedded Single-Layered Graphene Sheets Subjected to In-Plane Magnetic Field

: Vibrations of single-layered graphene sheets subjected to a longitudinal magnetic ﬁeld are considered. The Winkler-type and Pasternak-type foundation models are employed to reproduce the surrounding elastic medium. The governing equation is based on the modiﬁed couple stress theory and Kirchhoff–Love hypotheses. The effect of the magnetic ﬁeld is taken into account due to the Lorentz force deriving from Maxwell’s equations. The developed approach is based on applying the Ritz method. The proposed method is tested by a comparison with results from the existing literature. The numerical calculations are performed for different boundary conditions, including the mixed ones. The inﬂuence of the material length scale parameter, the elastic foundation parameters, the magnetic parameter and the boundary conditions on vibration frequencies is studied. It is observed that an increase of the magnetic parameter, as well as the elastic foundation parameters, brings results closer to the classical plate theory results. Furthermore, the current study can be applied to the design of microplates and nanoplates and their optimal usage.


Introduction
The application of micro and nanoscale structures in the high tech industry as elements of nano-electromechanical systems (NEMS), micro-electromechanical systems (MEMS), resonators, sensors, energy storage systems, DNA detectors, drug delivery [1][2][3][4][5] has been attracted the interest of scientists. The above-mentioned applications of micro/nanoplates are associated with their excellent mechanical, electrical, magnetic and chemical properties. It results from the study of the vibrational characteristics of micro/nanoplates that the ways of their optimization play an important role. The experimental and theoretical investigations lead to a conclusion that when the thickness of the element is in a micro or nanoscale, a size dependence effect of a material appears and significantly affects the mechanical behavior of such objects [6][7][8] and the classical theory for a small-scale structure analysis can not be used. Thus, in the study of the micro and nano-elements, higher-order continuum theories have been applied, for example, the theory of micropolar elasticity by Cosserat and Cosserat [9], the couple stress theory by Mindlin and Tiersten [10], Toupin [11], Koiter [12], the nonlocal elasticity theory by Eringen [13], strain gradient theory by Lam et al. [6] and the modified couple stress theory proposed by Yang et al. [14].
Recently, the modified couple stress theory was employed to various small-scale plate and beam problems. For instance, Tsiatas in [15] proposed a new Kirchhoff plate model for the static analysis

Formulation of the Problem
The present study is based on the modified couple stress theory which contains only one additional material length scale parameter and a symmetric couple stress tensor. According to this theory, the strain energy of an orthotropic plate has the following form where σ ij , ε ij , m ij , χ ij are components of stress tensor, strain tensor, diviatory part of the couple stress tensor and symmetric curvature tensor, respectively. Moreover, in (1) components of strain tensor are defined as where vector U = (u x , u y , u z ) stands for the vector of displacements. The components of curvature tensor are with and e ijk stands for a permutation symbol. Based on the Kirchhoff-Love theory, displacements of the plate in x, y and z directions have the following form where w(x, y, t) is displacement of points of the middle plane in z direction and t denotes time.
The strain-displacement relations are expressed as The components of the rotation vector (4) taking into account (5) are as follows and components of the curvature tensor (3) take the form The constitutive relations for an orthotropic plate have the following form In the last expressions E 1 , E 2 are Young's modules, ν 1 , ν 2 are Poisson's ratios with ν 1 E 2 = ν 2 E 1 , G is shear modulus, l stands for the material length scale parameter. Thus, the strain energy of an orthotropic microplate can be expressed with respect to displacements in the following way where In (11) h denotes the thickness of the plate. Using Hamilton's principle, the governing equation describing vibration process of micro-scale plates resting on the two parameters foundation and subjected to transverse external load, can be obtained in the following form: where k W and k G x , k G y are the Winkler and shear modules of the elastic surrounding medium, respectively, q l is the external load, ρ is the density of the plate. Equation (12) is supplemented with the boundary conditions. The part of the boundary is supposed to be simply supported: and clamped: In (13) and (14) M n is a bending moment, whereas n stands for a normal vector to the boundary.

Influence of the Magnetic Field
The choice of an appropriate external load plays a crucial role in changing the vibration characteristics of the considered micro/nanoplate. The necessary effect can be achieved by exposing the plate to a magnetic field. In this paper the plate subjected to the in-plane uni-axial magnetic field [31][32][33][34], see Figure 1, defined by the vector of magnetic field strength is considered. Distributing vector of the magnetic field h is defined using Maxwell's relations: Substituting (15) into (16) gives the following form of the distributing vector: The current density J is defined as follows and can be expressed as Thus, the Lorentz force can be written as where η is the magnetic permeability. It should be pointed out that the transverse vibrations are studied, so we have taken into account the third coordinate f z . Formula (20) yields Reflecting Kirchhoff-Love hypotheses, the transverse component f z is recast to the following form Thus, the magnetic field produces the force expressed by the following formula

Application of the Ritz Method
The variational formulation of the considered problem using (10), (23) and carrying out the integration by parts yields the following functional: According to the Ritz method, the deflection of the plate is represented as In (25) c i stand for unknown coefficients, w i = gφ i , and g is shape function selected depending on the boundary conditions and shape of the plate, whereas φ i is a system of the power polynomials. For the rectangular plate with sides a and b, the shape function is taken as where p, q, r, s depend on the boundary conditions. In order to find minimum of the functional (24), we substituted (25) in (24), and the partial derivatives with respect to unknown coefficients are equated to zero, giving the following algebraic system of equations where elements of the matrices are defined as System (27) has nonzero solutions if its determinant is 0, which gives an equation for determining the frequency spectrum.

Solution by the Navier Method
According to the Navier method, the deflection of simply supported rectangular graphene sheet with length of the sides a and b is taken as in [24,35], and the linear frequency of the embedded small-scale plate effecting by in-plane uni-axial magnetic field can be calculated by the following formula

Results and Discussions
In this work we investigated the model studied in [31,32], but in a contrast to the mentioned works, we used the modified couple stress theory, which allowed us to obtain the new results. In order to establish the validity of the current work we considered several vibration problems, which represent a simplification of the current model by using the classical plate theory (the material length scale parameter is neglected) and vanishing the magnetic influence. For each considered problem analysis of convergence of the results was performed and it was made a conclusion about a sufficient amount of terms in the series expansion (25). In Table 1 frequencies of the isotropic square plate with various boundary conditions are presented. Calculations are performed for the following types of the boundary conditions: all edges are simply supported (SSSS), three edges are simply supported and one is clamped (SSSC), two opposite sides are simply supported, the other two ones are clamped (SCSC), two opposite sides are simply supported, one is clamped and one is free (SCSF), the sides except one are simply supported and one is free (SSSF). We assumed that k G x = k G y = k G [25]. In this study, the dimensionless frequency parameter and the dimensionless Winkler and shear parameters are defined as follows Analysis was performed for ν = 0.3, h/a = 0.01, l/h = 0. Here it should be noted that for isotropic case E = E 1 = E 2 , ν = ν 1 = ν 2 , G = E/(2(1 + ν)). The solution by the Ritz method proposed in the paper is denoted as RS. The orthotropic square plate with a free edge and others clamped edges are studied for the following mechanical and geometrical data [21]: The obtained results are reported in Table 2. In the next case study we considered the simply supported orthotropic rectangular graphene sheet with the following mechanical parameters [24]: and the geometrical parameters h = 100µm, h/a = 0.01, b/a = 1.5. Table 3 contains the values of dimensionless frequency Ω obtained for the magnetic parameter MP = 0, 25, 50, where the dimensionless magnetic parameter is introduced as follows and the dimensionless Winkler and shear modules KW = 100, KG = 10. The results are calculated by the Ritz (RS) and Navier (NS) methods. For a rectangular plate with mechanical properties (32) and h = 100µm, h/a = 0.01, b/a = 1.5, the effect of magnetic field was investigated versus the thickness ratio l/h. The results are presented for plates with all simply supported edges ( Figure 2) and clamped edges (Figure 3). In both cases, it can be seen an increase in the dimensionless frequency Ω (30), with an increase of both: the magnetic parameter as well as the material length scale parameter. The similar influence of the magnetic field was reported in [31,32] based on the nonlocal elasticity theory for simply supported boundary conditions. The largest values of frequency parameter are characteristic of the plate with clamped edges. Herewith an increase within the magnetic parameter leads to a decrease in the difference in the results between the classical and the modified couple stress theory. Thus, inter-atomic bonds of the micro/nanoplates influence the vibration behaviour less for higher values of the magnetic field strength.  To study the effect of the elastic foundation using the Winkler-type model, the Winkler modulus parameter KW is taken within the range of 0..400 as in [23,25]. The results are presented for the thickness ratio l/h = 0, 0.2, 0.4, 0.6, 0.8, 1, MP = 0, KG = 0 and simply supported boundary conditions ( Figure 4). Analysing the results it can be noticed that by an increase of the Winkler modulus, the value of the frequency increases for all values of l/h and the difference in results from the classical theory becomes smaller. Taking the shear modulus factor KG in the range 0-10 [23,25], we studied influence of the Pasternak-type foundation on the vibration frequency parameter ( Figure 5). Calculations are performed by varying l/h = 0, 0.2, 0.4, 0.6, 0.8, 1 for fixed MP = 0, KW = 100. Obtained results show an increase of the dimensionless frequency with an increase of the shear modulus. The small scale effect was investigated, changing the type of boundary conditions ( Figure 6). Here four types of symmetrical boundary conditions are considered: SSSS, CCCC, CSCS (x = −a/2; a/2 are clamped, y = −b/2; b/2 are simply supported), SCSC (x = −a/2; a/2 are simply supported, y = −b/2; b/2 are clamped). The graphene sheet is exposed to the magnetic field with the magnetic parameter MP = 10. We also consider that the Winkler modulus and the shear modulus of the surrounding elastic medium equal to 100 and 10, respectively. Analysis of results allows concluding that an increase in the clamped part of the boundary implies an increase in the frequency value and the greatest values are reached under CCCC boundary conditions. The difference in the results with an increase in the thickness ratio l/h is more significant for clamped boundary conditions.

Concluding Remarks
The small-scale analysis of a single-layered rectangular orthotropic plate exposed to a magnetic field is performed. The considered plate is embedded in an elastic medium modeled as the Winkler and the Pasternak foundation. The study is based on the modified couple stress theory and the Kirchhoff-Love hypotheses. The influence of the magnetic field is derived by the Lorentz force. The proposed approach uses the Ritz method. The variational formulation of the considered problem is given, which can be used for future investigation of plates with more complicated shape by changing the shape functions and a region of integration.
The results of the numerical investigation contain the analysis of an influence of the boundary conditions, the magnetic parameter, the parameters of elastic foundation on vibration frequencies. It is concluded that the frequency of a plate is sensitive to the material length scale parameter. The magnetic field plays an important role and significantly increases the vibration frequencies.
Simultaneously it is observed that for higher values of the magnetic field strength the effect of the length scale parameter decreases. Thus, the small-scale effect is dampened by an increase in magnetic field strength. An increase in the clamped part of the boundary leads to an increase in the frequency values and a more pronounced small-scale effect. The biggest values of frequencies appear for a plate with all clamped edges.

Conflicts of Interest:
The authors declare no conflict of interest.