Free Vibration Analysis of Size-Dependent, Functionally Graded, Rectangular Nano/Micro-plates based on Modified Nonlinear Couple Stress Shear Deformation Plate Theories

Free Vibration Analysis of Size-Dependent, Functionally Graded, Rectangular Nano/Micro-plates based on Modified Nonlinear Couple Stress Shear Deformation Plate Theories

K. Khorshidi ^a,b*, A. Fallah ^a

^a Department of Mechanical Engineering, Arak University, Arak,  Iran

^b Institute of Nanosciences & Nanotechnolgy, Arak University, Arak,  Iran

1. Introduction 

The potential and applications of nano-/micromaterials in the development of technologies such as electronics, energy, environmental remediation, nano-/microsystem, medical and health, future transportation, etc. are important factors that encourage scientists to choose it for future projects. Today, scientists and engineers can reduce production costs, energy consumption, and maintenance with the aid of nanotechnology and its integration with other technologies. Also, using nano-/microtechnology has increased the durability of engineering structures.

Generally, size-dependent material models can be developed based on size-dependent continuum theories like classical couple stress theory [1], nonlocal elasticity theory [2], and strain gradient theory [3]. Couple stress theory is one of the higher-order continuum theories that contains material-length scale parameters and can cover the size effects of nano-/microstructures.

Functionally graded materials (FGMs) are heterogeneous composite materials in which the material properties vary continuously from one surface to the other surface. This is achieved by gradually varying the volume fraction of mixture materials. The merit of using these materials is that they can survive high thermal gradient environments. FGMs were first used as thermal barrier materials for aerospace structural applications and fusion reactors. Recently, they have been developed for general application as structural components in high-temperature environments [4]. Typically, an FGM is a mixture of ceramic and metal for the purpose of thermal protection against large temperature gradients. The ceramic material provides high-temperature resistance due to its low thermal conductivity, while the ductile metal prevents fracture due to its greater toughness. Because of the wide use of nano-/microplates in engineering applications, the study of functionally graded (FG), rectangular nano-/microplates has received considerable attention in recent years.

Matsunaga [5] analyzed the natural frequencies and buckling stresses of plates made of FG materials by taking into account the effects of transverse shear, normal deformations, and rotatory inertia. By expanding the power series of displacement components, a set of FG plates was derived using Hamilton’s principle. Salehipour et al. [6] have developed a model for static and vibrating FG nano-/microplates based on the modified couple stress and three-dimensional elasticity theories. Ansari et al. [7] investigated the size-dependent vibrational behavior of FG, rectangular, Mindlin microplates, including geometrical nonlinearity. In their work, the FG Mindlin microplate was considered to be made of a mixture of metal and ceramic according to a power-law distribution. Kim and Reddy [8] have presented analytical solutions of a general third-order plate theory that accounts for the power-law distribution of two materials through thickness- and microstructure-dependent size effects. Thai and Vo [9] proposed a size-dependent model for the bending and free vibration of an FG plate based on the modified couple stress theory and sinusoidal shear deformation theory. Shaat et al. [10] developed a new Kirchhoff plate model using a modified couple stress theory to study the bending behavior of nanosized plates, including surface energy and microstructure effects. Lou and He [11] studied the nonlinear bending and free vibration responses of a simply supported, FG microplate lying on an elastic foundation within the framework of the modified couple stress theory, the Kirchhoff/Mindlin plate theory, and von Karman’s geometric nonlinearity. He et al. [12] developed a new, size-dependent model for FG microplates by using the modified couple stress theory. Based on the strain gradient elasticity theory and a refined shear deformation theory, Zhang et al. [13] developed an efficient, size-dependent plate model to analyze the bending, buckling, and free vibration problems of FG microplates resting on an elastic foundation. Lou et al. [14] proposed a unified higher-order plate theory for FG microplates by adopting the modified couple stress theory to capture size effects and using a generalized shape function to characterize the transverse shear deformation. Thai and Kim [15] developed a size-dependent model of the bending and free vibration of an FG Reddy plate. Gupta et al. [16] presented an analytical model for the vibration analysis of partially cracked isotropic and FG, rectangular plates based on a modified couple stress theory. Li and Pan [17] developed a size-dependent, FG, piezoelectric microplate model based on the modified couple stress and sinusoidal plate theories. Nguyen et al. [18] studied the size-dependent behaviours of FG microplates using a novel quasi-3D shear deformation theory based on modified couple stress theory. Lei et al. [19] presented a size-dependent FG microplate model based on a modified couple stress theory requiring only one material-length scale parameter. Jandaghian and Rahmani [20] investigated the free vibration analysis of FG, piezoelectric-material, nanoscale plates based on Eringen's nonlocal Kirchhoff plate theory under simply supported–edge conditions. Şimşeka and Aydınc [21] considered the static bending and forced vibration of an imperfect FG microplate carrying a moving load based on Mindlin plate theory and the modified couple stress theory. Thai and Choi [22] presented an analytical solution for size-dependent models for the bending, buckling, and vibration of FG Kirchhoff and Mindlin plates based on modified couple stress theory. Khorshidi et al. [23], investigated the free vibrations of size-dependent, FG, rectangular plates with simply supported–boundary conditions based on nonlocal, exponential shear deformation theory using a Navier-type solution. Khorshidi and Fallah [24] analyzed the buckling response of FG, rectangular nanoplates with all edges simply supported based on nonlocal, exponential shear deformation theory according to Navier-type solutions. Khorshidi and Khodadadi [25] used a new, refined trigonometric shear deformation plate theory to study the out-of-plane vibration of rectangular, isotropic plates with different boundary conditions. Reddy and Kim [26] adopted a higher-order shear deformation theory to develop a size-dependent model for FG microplates. Simsek and Reddy [27] examined the bending and free vibration of microbeams based on various higher-order beam theories. Using first-order plate theory, Jung et al. [28, 29] investigated the buckling, static deformation, and free vibration of sigmoid, FG-material nano-/microplates embedded in a Pasternak elastic foundation.

The free vibration problem of plates can be solved using either the energy functional or the governing partial differential equations. Both can be taken by using standard analytical and numerical techniques. Among the techniques available are the finite element method [30], the boundary element method [31], the finite difference method [32], the differential quadrature method [33], the collocation method [34], the Galerkin method [35], and the Ritz method [36–39]. In this article, a modified couple stress theory according to the nonlinear exponential and trigonometric shear deformation theories was applied to analyze the free vibration of FG, rectangular nano-/microplates. The natural frequencies of the FG nano-/microplates were calculated using the Rayleigh-Ritz method based on minimizing the Rayleigh quotient. The novelty of the present paper is that the analytical solution was developed for size-dependent, FG, rectangular nano-/microplates using the modified nonlinear couple stress shear deformation theories for a combination of different boundary conditions (i.e., simply supported [S], clamped [C], and free [F]), as follows: SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SSCC, SCCC, CFCF, SSFF, CFSF, CFFF, SFCS, CFCC, SFCC, FFCC, CFCS, CSFF, SFFF, and FFFF. A comparison of the results with those available in the literature has been made. Finally, the effect of various parameters such as the power-law index, thickness-to-length scale parameter ratio (h/l), and aspect ratio (a/b) on the natural frequencies of nano-/microplates are presented and discussed in detail.

1. Governing Equation of Motion and Solution Procedure

Consider a size-dependent, rectangular nano-/microplate with uniform thickness h, length a, and width b made up of FG material as shown in Fig. 1. The properties of the nano-/microplate are assumed to vary through the thickness of the nanoplate according to a power-law distribution of the volume fractions of two materials between the two surfaces. The top surface ( ) of the size-dependent plate is fully ceramic, whereas the bottom surface ( ) is fully metal. The plate regions are given by Eq. (1) as follows:

where ,  and  are Cartesian coordinates. Poisson’s ratio of the plate ϑ is assumed to be constant for ceramic and metal throughout the analysis.

Young’s modulus and mass density are assumed to vary continuously through the plate thickness direction as

where the subscripts m and c represent the metallic and ceramic constituents, respectively;  is the plate density per unit area of the FG plate;  is the Young's modulus of the FG plate;  is the volume fraction; and g is the power-law index and takes only positive values.

According to Eqs. (2) and (3), when the power-law index g approaches zero or infinity, the plate is fully ceramic or metal, respectively. According to the following assumptions, the displacement field of the proposed plate theory is given as follows:

1. The displacement components u and v are the in-plane displacements in the x and y directions, respectively, and w is the transverse displacement in the z-direction. These displacements are small in comparison with the plate thickness.
2. The in-plane displacement u in the x direction and v in the y direction each consist of two parts.

(a)         A displacement component similar to displacement in classical plate theory.

(b)        A displacement component due to shear deformation, which is assumed to be exponential in exponential shear deformation theory and trigonometric in trigonometric shear deformation theory with respect to the thickness coordinate.

1. The transverse displacement w in the z direction is assumed to be a function of the x and y coordinates.

Based on the assumptions mentioned above, the displacement field can be described as

where, for exponential shear deformation plate theory, , and for trigonometric shear deformation plate theory,  [23–25]. Also , , and  are the displacement in the , , and  directions, respectively;  and  are the mid-plane displacements; and  and  are the rotation functions. With the assumed linear von Karman strain, the displacement-strain field will be as follows [22]:

In the Eqs. (8–12),  are normal strains and  are shear strains. Considering Hooke's law for stress fields, the normal stress  is assumed to be negligible in comparison within plane stresses  and . Thus, the stress-strain relationship will be as follows:

where  is the shear modulus of the plate.

In the modified couple stress theory, the strain energy of a linearly elastic continuum body on volume ∀ is defined by a function of both strain tensor and curvature tensor as

where ,  and  are the components of the stress, normal strains, and shear strain tensors, respectively [1]. Also,  are the components of the deviatoric part of the symmetric couple stress tensor, and  are the components of the symmetric curvature tensor defined by

where  is the length scale parameter, and  are the components of the rotation vector related to the displacement field. These are defined as follows:

The kinetic energy of the FG nano-/microplate is defined as follows:

where the dot-top index contract indicates the differentiation with respect to the time variable.

In this section, the Rayleigh-Ritz method is employed to analyze the free vibration of the FG, rectangular nano-/microplates using coupled stess theory. In the Rayleigh-Ritz method, the admissible trial displacement and rotation functions can be introduced as follows [34]:

where , , ,  and  are the generalized constant coefficients of the admissible trial functions;  is the natural frequency of the plate;  is the imaginary number;  is the order of approximation; and  are the fundamental functions. The fundamental functions of the moderately thick, FG, rectangular nano-/microplates that satisfy the geometric boundary conditions are introduced as

The fundamental functions for different boundary conditions of the moderately thick, FG, rectangular nano-/microplates that were considered in the present study are listed in Table 1. In the Rayleigh-Ritz approach, the Lagrangian function of the system is given as

With the application of the Rayleigh-Ritz minimization method, the eigenvalue equation can be derived from Eq. (32).

where  is the vector of generalized coordinates and contains an unknown, undetermined coefficient. Eq. (32) can be written in matrix form as below:

where

 is the stiffness matrix, and  is the mass matrix. This eigenvalue problem is solved to obtain the natural frequency parameters and vibration modal shapes of the FG, rectangular nano-/microplate.

1. Numerical Results and Disscusions

In this section, the natural frequency parameters are obtained from the Rayleigh-Ritz method, presented here, and expressed in dimensionless form as . Numerical calculations have been performed for different combinations of boundary conditions (SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SSCC, SCCC, CFCF, SSFF, CFSF, CFFF, SFCS, CFCC, SFCC, FFCC, CFCS, CSFF, SFFF, FFFF). In the numerical calculations, Poisson’s ratio  has been used. The FG nano-/microplate is made up of the following material properties: , , ,  and the small scale parameter is .

Table 2 shows a comparison study of the nondimensional natural frequency parameters ( ) for a simply supported, FG, square nano-/microplate with those reported based on Mindlin plate theory by Thai and Choi [22]. The effect of the length scale parameter  and length-to-thickness ratio a/h on the first two nondimensional natural frequency parameters for simply supported, FG, rectangular nano-/microplates with a/b = 1 and different power-law indices are shown in Table 2.

From the results shown in Table 2, it can be observed that the present results, which were obtained by the Rayleigh-Ritz method, have greater values than those reported by Thai and Choi [22]. This is because in the Rayleigh-Ritz method, the admissible trial displacement and rotation functions that can satisfy the different boundary conditions at all edges of the plate are in the form of a finite polynomial series. Reducing the number of series terms decreases the degree of freedom of the plate and increases the stiffness and frequency parameter, in contrast with what was reported in Thai and Choi’s work based on the Navier method (exact solution) [22]. Moreover, the different distribution of shear stress and rotary inertia in the thickness direction led to differences in the gained results, which are explained by the exponential, trigonometric, and first-order shear deformation plate theories. The results in Table 2 show that there is a good agreement between the present results and those of Thai and Choi [22].

Tables 3 and 4 show the effect of different boundary conditions, power-law index ( , 1, and 10) and aspect ratios (a/b = 0.2, 0.5, and 1) on the dimensionless natural frequency ( ) of FG, rectangular nano-/microplates using the exponential and trigonometric shear deformation plate theories. From the results presented in Tables 3 and 4, it can be observed that an increasing aspect ratio (a/b) leads to an increase in the dimensionless natural frequency parameters because decreasing the width of a plate with a constant length decreases the degrees of freedom of the plate and increases the stiffness.

As the results show in Tables 2–4, the effect of the power-law index on dimensionless natural frequencies is very interesting. It is observed that increasing the power-law index value initially decreases, reaches a minimum, and then increases the frequency. This is because decreasing or increasing the dimensionless natural frequency depends on the kind of material researchers choose to study. For example, Matsunga [5] presented the free vibration and stability of FG plates according to a 2D, higher-order deformation theory in which the frequency parameter decreases with an increase in the power-law index. On the other hand, Thai and Choi [22] analyzed size-dependent, FG, Kirchhoff and Mindlin plate models based on a modified couple stress theory. Their results show that the frequency parameter decreases first and then rises. This phenomenon could be due to the fact that the frequency parameter of FG materials are dependent on both Young’s modulus (Young’s modulus  plate rigidity) and density (density  plate softening). In the presented material research, and similar to the results reported by Thai and Choi [22], with an increase in the power-law index, the dimensionless natural frequency decreases first and then rises because an increase in the power-law index in this research’s material caused Young’s modulus and density to decrease. A reduction in the Young’s modulus, consequently, caused the plates rigidity and frequency parameter to decrease. However, a decrease in the density leads to an increase in the frequency parameter. So, first, the effect of the Young’s modulus is greater than the effect of density on the frequency parameter; consequently, the dimensionless natural frequency first decreases. But after reaching a minimum, the effect of the density becomes greater than the effect of Young’s modulus, and it causes the dimensionless natural frequency to increase. By comparing the obtained dimensionless natural frequencies of the different boundary conditions that are shown in Tables 2–6, it was found that the dimensionless natural frequencies increase as the degrees of freedom of the plate decrease (increasing the geometric constraints on the edges of the plate). Because of the decreased degree of freedom at each edge of the rectangular plate, the plate gets stiffer, leading to increased dimensionless natural frequencies.

Tables 5 and 6 show the effect of the different boundary conditions and length-to-thickness ratios (a/h = 5, 8, 15, and 20) on the 3 first dimensionless natural frequencies ( ) of homogeneous, rectangular nano-/microplates using the exponential and trigonometric shear deformation plate theories. From the results in Tables 5 and 6, it can be found that, with an increase in the length-to-thickness ratio (constant length and thickness decreases), the dimensionless natural frequency increases. From these results, it can be seen that if the thickness increases, the effective stiffness and effective mass of the plate increase, but the growth of the effective stiffness is greater than the effective mass, so the natural frequency of the nano-/microplate increases.

As shown in Table 2, it can be found that the dimensionless natural frequency of nano-/microplates according to couple stress theory is greater than the dimensionless natural frequency of the plate, due to classical linearly elastic continuum mechanics ( ). This is because the potential energy of linearly elastic continuum mechanics is only defined by a function of the strain tensor in the classical exponential and trigonometric shear deformation plate theories.