Document Type : Research Article
Authors
Arak University
Abstract
Keywords

Mechanics of Advanced Composite Structures 4 (2017) 127137 

Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
Free Vibration Analysis of SizeDependent, Functionally Graded, Rectangular Nano/Microplates 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
Paper INFO 

ABSTRACT 
Paper history: Received 20161217 Revised 20170207 Accepted 20170301 
In the present study, a vibration analysis of functionally graded rectangular nano/microplates was considered based on modified nonlinear coupled stress exponential and trigonometric shear deformation plate theories. Modified coupled stress theory is a nonclassical continuum mechanics theory. In this theory, a materiallength scale parameter is applied to account for the effect of nanostructure size that earlier classical plate theories are not able to explain. The material properties of the plate were assumed to vary according to a powerlaw form in the thickness direction. The governing equation of the motion of functionally graded, rectangular nano/microplates with different boundary conditions were obtained based on the RayleighRitz method using complete algebraic polynomial displacement and rotation functions. The advantage of the present RayleighRitz method is that it can easily handle the different conditions at the boundaries of moderately thick rectangular plates (e.g., clamped, simply supported, and free). A comparison of the results with those available in the literature has been made. Finally, the effect of various parameters, such as the powerlaw index, thicknesstolength scale parameter ratio h/l, and aspect ratio a/b, on the natural frequency of nano/microplates are presented and discussed in detail. 



Keywords: RayleighRitz Vibration Couple stress theory Functionally graded 

DOI: 10.22075/MACS.2017.1800.1094 
© 2017 Published by Semnan University Press. All rights reserved. 
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, sizedependent material models can be developed based on sizedependent continuum theories like classical couple stress theory [1], nonlocal elasticity theory [2], and strain gradient theory [3]. Couple stress theory is one of the higherorder continuum theories that contains materiallength 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 hightemperature 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 hightemperature 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 threedimensional elasticity theories. Ansari et al. [7] investigated the sizedependent 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 powerlaw distribution. Kim and Reddy [8] have presented analytical solutions of a general thirdorder plate theory that accounts for the powerlaw distribution of two materials through thickness and microstructuredependent size effects. Thai and Vo [9] proposed a sizedependent 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, sizedependent 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, sizedependent 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 higherorder 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 sizedependent 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 sizedependent, FG, piezoelectric microplate model based on the modified couple stress and sinusoidal plate theories. Nguyen et al. [18] studied the sizedependent behaviours of FG microplates using a novel quasi3D shear deformation theory based on modified couple stress theory. Lei et al. [19] presented a sizedependent FG microplate model based on a modified couple stress theory requiring only one materiallength scale parameter. Jandaghian and Rahmani [20] investigated the free vibration analysis of FG, piezoelectricmaterial, 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 sizedependent 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 sizedependent, FG, rectangular plates with simply supported–boundary conditions based on nonlocal, exponential shear deformation theory using a Naviertype 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 Naviertype solutions. Khorshidi and Khodadadi [25] used a new, refined trigonometric shear deformation plate theory to study the outofplane vibration of rectangular, isotropic plates with different boundary conditions. Reddy and Kim [26] adopted a higherorder shear deformation theory to develop a sizedependent model for FG microplates. Simsek and Reddy [27] examined the bending and free vibration of microbeams based on various higherorder beam theories. Using firstorder plate theory, Jung et al. [28, 29] investigated the buckling, static deformation, and free vibration of sigmoid, FGmaterial 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 RayleighRitz method based on minimizing the Rayleigh quotient. The novelty of the present paper is that the analytical solution was developed for sizedependent, 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 powerlaw index, thicknesstolength scale parameter ratio (h/l), and aspect ratio (a/b) on the natural frequencies of nano/microplates are presented and discussed in detail.
Consider a sizedependent, 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 powerlaw distribution of the volume fractions of two materials between the two surfaces. The top surface ( ) of the sizedependent plate is fully ceramic, whereas the bottom surface ( ) is fully metal. The plate regions are given by Eq. (1) as follows:
, , 
(1) 
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
, 
(2) 
, 
(3) 
, 
(4) 
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 powerlaw index and takes only positive values.
Figure 1. Rectangular plate geometry, dimensions, and coordinate system 
According to Eqs. (2) and (3), when the powerlaw 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:
(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.
Based on the assumptions mentioned above, the displacement field can be described as
, 
(5) 
, 
(6) 
, 
(7) 
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 midplane displacements; and and are the rotation functions. With the assumed linear von Karman strain, the displacementstrain field will be as follows [22]:
, 
(8) 
, 
(9) 
, 
(10) 
, 
(11) 
. 
(12) 
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 stressstrain relationship will be as follows:
, 
(13) 
, 
(14) 
, 
(15) 
, 
(16) 
(17) 
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
, 
(18) 
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
, 
(19) 
, 
(20) 
where is the length scale parameter, and are the components of the rotation vector related to the displacement field. These are defined as follows:
, 
(21) 
, 
(22) 
. 
(23) 
The kinetic energy of the FG nano/microplate is defined as follows:
, 
(24) 
where the dottop index contract indicates the differentiation with respect to the time variable.
In this section, the RayleighRitz method is employed to analyze the free vibration of the FG, rectangular nano/microplates using coupled stess theory. In the RayleighRitz method, the admissible trial displacement and rotation functions can be introduced as follows [34]:
, 
(25) 
, 
(26) 
, 
(27) 
, 
(28) 
, 
(29) 
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
. 
(30) 
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 RayleighRitz approach, the Lagrangian function of the system is given as
. 
(31) 
With the application of the RayleighRitz minimization method, the eigenvalue equation can be derived from Eq. (32).
, 
(32) 
where is the vector of generalized coordinates and contains an unknown, undetermined coefficient. Eq. (32) can be written in matrix form as below:
, 
(33) 
where
, , 
(34) 
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.
Table 1. Fundamental functions of the admissible trial displacement and rotation functions for different combinations of boundary conditions 

Boundary Conditions 
Fundamental Functions 

SSSS 

SCSS 

SCSC 

SSSF 

SFSF 

SCSF 

CCCC 

SSCC 

SCCC 

CFCF 

SSFF 

CFSF 

CFFF 

SFCS 

CFCC 

SFCC 

FFCC 

CFCS 

CSFF 

SFFF 

FFFF 
1 
1 
1 
1 
1 
In this section, the natural frequency parameters are obtained from the RayleighRitz 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 lengthtothickness ratio a/h on the first two nondimensional natural frequency parameters for simply supported, FG, rectangular nano/microplates with a/b = 1 and different powerlaw 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 RayleighRitz method, have greater values than those reported by Thai and Choi [22]. This is because in the RayleighRitz 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 firstorder 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, powerlaw 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.
Table 2. Comparison of nondimensional natural frequency of an FG nano/microplate with all edges simply supported 

TSDT 
ESDT 
Ref. [22] 
TSDT 
ESDT 
Ref. [22] 
TSDT 
ESDT 
Ref. [22] 

5 
0 
5.48130 
5.48140 
5.38710 
4.99490 
4.99510 
4.87440 
5.66210 
5.66230 
5.58180 

12.1209 
12.1214 
11.6717 
10.9028 
10.9035 
10.7905 
12.2316 
12.2316 
11.9931 

11.2690 
11.2692 
11.1311 
11.2779 
11.2784 
4.0451 
11.2364 
11.2364 
11.1666 

23.9416 
23.9418 
23.7023 
23.8609 
23.8615 
23.6723 
23.8985 
23.8987 
23.7146 

10 
0 
6.21220 
6.21220 
5.93010 
5.39620 
5.39630 
5.26970 
5.13090 
5.13120 
5.09030 

14.2254 
14.2256 
14.0893 
12.7138 
12.7139 
12.6460 
14.6621 
14.6625 
14.6464 

12.9139 
12.9143 
12.6360 
12.6693 
12.6693 
12.4128 
12.7405 
12.7409 
12.7302 

29.6572 
29.6576 
29.4588 
29.1949 
29.1953 
29.1174 
30.0109 
30.0113 
29.6008 

20 
0 
6.35950 
6.35960 
6.09970 
5.55870 
5.5590 
5.38800 
6.56880 
6.56910 
6.38370 

15.1465 
15.1466 
15.0319 
13.4209 
13.4210 
13.3192 
15.8744 
15.8748 
15.7108 

13.4285 
13.4287 
13.1786 
13.2017 
13.2020 
12.8871 
13.3998 
13.3999 
13.3030 

32.5074 
32.4947 
32.4952 
32.2374 
31.6689 
31.6689 
31.6012 
32.6132 
32.6133 
Table 3. Comparison of the fundamental nondimensional natural frequency parameter for SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SSCC, SFCC, and FFCC, FG, square nano/microplates for different aspect ratios and powerlaw index values 

ESDT 
TSDT 
a/b 
B.Cs. 

7.03770 
6.70340 
6.74270 
7.03750 
6.70330 
6.74260 
0.2 
SSSS 
9.77880 
7.85750 
7.92220 
9.77850 
7.85740 
7.92210 
0.5 

12.9404 
12.8693 
12.9143 
12.9402 
12.8691 
12.9141 
1 

7.05570 
6.72190 
6.76450 
7.05530 
6.72160 
6.76440 
0.2 
SCSS 
9.38490 
5.76980 
8.46440 
9.38480 
5.76960 
8.46420 
0.5 

14.7465 
14.0618 
14.3214 
14.7463 
14.0617 
14.3211 
1 

7.07760 
6.74540 
6.79230 
7.07760 
6.74500 
6.79180 
0.2 
SCSC 
15.6274 
14.8164 
15.6684 
15.6274 
14.8162 
15.6683 
0.5 

18.1387 
17.3800 
17.7942 
18.1385 
17.3798 
17.7941 
1 

1.12700 
1.09390 
1.10290 
1.12500 
1.09340 
1.10260 
0.2 
SSSF 
8.72160 
8.34450 
8.24750 
8.72150 
8.34450 
8.24740 
0.5 

8.72160 
8.24750 
8.34450 
8.72150 
8.24750 
8.34410 
1 

0.27100 
0.24870 
0.25730 
0.27090 
0.24870 
0.25680 
0.2 
SFSF 
1.78370 
1.53100 
1.70780 
1.78350 
1.53900 
1.70760 
0.5 

7.51420 
7.19320 
7.28030 
7.51360 
7.19290 
7.28010 
1 

2.79750 
2.71570 
2.77240 
2.79740 
2.71550 
2.77230 
0.2 
SCSF 
4.40560 
4.25520 
4.32510 
4.40540 
4.25510 
4.32500 
0.5 

9.21430 
9.04810 
9.17290 
9.21430 
9.04800 
9.17270 
1 

15.0082 
14.4513 
14.8819 
15.0081 
14.4513 
14.8817 
0.2 
CCCC 
15.9753 
15.3828 
15.7562 
15.9753 
15.3826 
15.7562 
0.5 

23.9918 
2.35750 
23.4679 
23.9911 
2.35710 
23.4676 
1 

10.5243 
10.1382 
10.3519 
10.5238 
10.1382 
10.3518 
0.2 
SSCC 
11.4301 
10.9700 
11.2156 
11.4295 
10.9700 
11.2156 
0.5 

17.3995 
16.8890 
17.2510 
17.3994 
16.8890 
17.2500 
1 

2.85220 
2.76970 
2.82770 
2.85210 
2.76970 
2.82760 
0.2 
SFCC 
5.04160 
4.88120 
4.98250 
5.04160 
4.88110 
4.98240 
0.5 

13.1621 
12.7617 
13.0365 
13.1615 
12.7611 
13.0364 
1 

2.50770 
2.42680 
2.48490 
2.50750 
2.42670 
2.48450 
0.2 
FFCC 
3.14900 
3.05610 
3.12160 
3.14700 
3.05580 
3.12170 
0.5 

5.29150 
5.18080 
5.25770 
5.29150 
5.18070 
5.25760 
1 

Table 4. Comparison of the fundamental nondimensional natural frequency parameter for CFCS, CFCF, SSFF, CFSF, CFFF, SCCC, CFCC, SFCS, CSFF, SFFF, and FFFF, FG, square nano/microplates for different aspect ratios and powerlaw index values. 

ESDT 
TSDT 
a/b 
B.C 

* 

1.35310 
1.31750 
1.33920 
1.35290 
1.31710 
1.33890 
0.2 
CFCS 
5.05720 
4.89360 
4.99820 
5.05700 
4.89340 
4.99810 
0.5 

17.8132 
17.3941 
17.7298 
17.8126 
17.3940 
17.7295 
1 

0.65900 
0.69230 
0.65680 
0.65800 
0.69230 
0.65660 
0.2 
CFCF 
4.32290 
4.18700 
4.28060 
4.32290 
4.18690 
4.27970 
0.5 

19.8915 
16.8847 
19.7786 
19.8914 
16.8847 
19.7784 
1 

0.20740 
0.54510 
0.55020 
0.20700 
0.54510 
0.54950 
0.2 
SSFF 
1.40520 
1.36300 
1.37260 
1.40500 
1.36100 
1.37250 
0.5 

2.95410 
2.78790 
2.83330 
2.95380 
2.78790 
2.83340 
1 

0.44400 
0.42000 
0.43520 
0.44380 
0.41960 
0.43440 
0.2 
CFSF 
2.89690 
2.78050 
2.85190 
2.89660 
2.78050 
2.85180 
0.5 

11.8647 
11.4962 
11.7224 
11.8645 
11.4958 
11.7223 
1 

0.19600 
0.09900 
0.10260 
0.19550 
0.09840 
0.10220 
0.2 
CFFF 
0.67300 
0.64800 
0.66570 
0.67290 
0.64780 
0.66530 
0.5 

2.84280 
2.67270 
2.73150 
2.84260 
2.67270 
2.73140 
1 

14.9707 
14.4491 
14.7614 
14.9706 
14.4490 
14.7612 
0.2 
SCCC 
15.6274 
15.0347 
15.3964 
15.6270 
15.0343 
15.3964 
0.5 

20.3359 
19.7427 
20.2216 
20.3358 
19.7427 
2.22150 
1 

2.91680 
2.83320 
2.89300 
2.91670 
2.83310 
2.89300 
0.2 
CFCC 
5.98950 
5.81000 
5.93590 
5.98900 
5.89910 
5.93560 
0.5 

18.1564 
17.7509 
18.1155 
18.1563 
17.7508 
18.1151 
1 

1.23070 
1.19640 
1.21090 
1.22990 
1.19590 
1.21040 
0.2 
SFCS 
3.90480 
3.75220 
3.82210 
3.90450 
3.75200 
3.82210 
0.5 

12.5125 
12.2400 
12.4766 
12.5124 
12.2398 
12.4761 
1 

0.60808 
0.59630 
0.60450 
0.60870 
0.59610 
0.60400 
0.2 
CSFF 
1.78730 
1.73100 
1.74190 
1.78730 
1.73000 
1.74180 
0.5 

4.44620 
4.35090 
4.38030 
4.44600 
4.35090 
4.37990 
1 

0.08840 
0.08440 
0.08780 
0.08800 
0.08430 
0.08740 
0.2 
SFFF 
0.58860 
0.55230 
0.57920 
0.58840 
0.55210 
0.57890 
0.5 

2.81410 
2.04150 
2.17190 
2.81390 
2.04140 
2.17170 
1 

0.03810 
0.03430 
0.03750 
0.03790 
0.03400 
0.03750 
0.2 
FFFF 
0.34860 
0.32080 
0.33190 
0.34800 
0.32070 
0.33190 
0.5 

1.22530 
1.17160 
1.21460 
1.22510 
1.17150 
1.21410 
1 

As the results show in Tables 2–4, the effect of the powerlaw index on dimensionless natural frequencies is very interesting. It is observed that increasing the powerlaw 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, higherorder deformation theory in which the frequency parameter decreases with an increase in the powerlaw index. On the other hand, Thai and Choi [22] analyzed sizedependent, 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 powerlaw index, the dimensionless natural frequency decreases first and then rises because an increase in the powerlaw 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 lengthtothickness 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 lengthtothickness 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.
Table 5. Comparison of the three first nondimensional natural frequency parameters for SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SCCC, CFCC, SSCC, and SFCC, homogeneous, square nano/microplates for different lengthtothickness ratios (g = 0, a/b = 1, ) 

ESDT 
TSDT 
a/h 
B.C 

Third mode 
Second mode 
First mode 
Third mode 
Second mode 
First mode 

19.0885 
17.3734 
7.51870 
19.0881 
17.3733 
7.51870 
5 
SSSS 
27.7967 
2404702 
10.0574 
27.7967 
24.4701 
10.0573 
8 

48.3788 
41.5901 
16.5255 
48.3785 
41.5901 
16.5254 
15 

63.4543 
54.2770 
21.4145 
63.4540 
54.2730 
21.4144 
20 

19.7053 
19.0885 
8.78180 
19.7049 
19.0883 
8.78180 
5 
SCSS 
28.9468 
28.6739 
12.0817 
28.9464 
28.6735 
12.0816 
8 

50.3357 
49.9104 
20.1517 
50.3354 
49.9971 
20.1517 
15 

65.9542 
65.4489 
26.1877 
65.9540 
65.4489 
26.1871 
20 

19.0952 
19.0816 
10.4078 
19.0949 
19.0814 
10.4071 
5 
SCSC 
30.4398 
30.0424 
14.8677 
30.4398 
30.0424 
14.8677 
8 

57.2944 
53.0401 
24.8475 
57.2942 
53.0401 
24.8474 
15 

76.2987 
68.8502 
32.9207 
76.2986 
68.8501 
32.9207 
20 

12.3624 
12.1604 
5.61230 
12.3623 
12.1597 
5.61190 
5 
SSSF 
19.4566 
17.8130 
7.75900 
19.4560 
17.8060 
7.75880 
8 

36.4812 
30.6762 
12.8755 
36.4799 
30.6762 
12.8753 
15 

48.6416 
401171 
16.7025 
48.6415 
40.1169 
16.7024 
20 

8.97430 
7.07110 
4.50870 
8.97390 
7.07060 
4.50840 
5 
SFSF 
14.3589 
9.86410 
6.15520 
14.3589 
9.86400 
6.15490 
8 

26.9229 
16.5391 
16.2031 
26.9227 
16.5390 
16.2031 
15 

35.8972 
21.4781 
13.2281 
35.8966 
21.4775 
13.2273 
20 

12.3628 
12.1609 
5.61270 
12.3625 
12.1599 
5.61260 
5 
SCSF 
19.4569 
17.8170 
7.73590 
19.4568 
17.8168 
7.73590 
8 

36.4815 
30.6764 
12.8759 
36.4815 
30.6763 
12.8754 
15 

48.6419 
40.1173 
16.7029 
48.6419 
40.1171 
16.7027 
20 

50.7356 
36.5427 
33.6441 
50.7359 
36.5426 
33.6435 
5 
CCCC 
27.0417 
19.6007 
19.2698 
27.0412 
19.5999 
19.2698 
8 

50.7359 
36.5427 
33.6441 
50.7354 
36.5425 
33.6437 
15 

67.4061 
48.5932 
44.6753 
67.4057 
48.5931 
44.6753 
20 

19.0906 
19.0848 
11.6717 
19.0905 
19.0884 
11.6715 
5 
SCCC 
30.5488 
30.5417 
16.9560 
30.5487 
30.5411 
16.9500 
8 

57.2544 
57.1773 
28.6683 
57.2543 
57.1772 
28.6675 
15 

82.0035 
76.3459 
37.7139 
82.0034 
76.3458 
37.7139 
20 

15.0734 
12.1604 
10.1567 
15.0731 
12.1604 
10.1566 
5 
CFCC 
22.2911 
19.4566 
14.9637 
22.2904 
19.4554 
14.9629 
8 

38.8576 
36.4812 
26.0918 
38.8571 
36.4809 
26.0903 
15 

50.9237 
48.6416 
34.2048 
50.9229 
48.6405 
34.2044 
20 

20.5762 
19.0885 
10.2617 
20.5756 
19.0878 
10.2615 
5 
SSCC 
30.5416 
30.1799 
14.4481 
30.5415 
30.1797 
14.1480 
8 

57.2654 
52.4612 
24.4775 
57.2644 
52.4607 
24.4789 
15 

76.3539 
68.7305 
31.9135 
76.3535 
68.7304 
31.9134 
20 

16.9181 
12.1604 
7.63680 
16.9177 
12.1697 
7.63620 
5 
SFCC 
19.6768 
19.4566 
10.8800 
19.6766 
19.4565 
10.8778 
8 

36.4812 
34.0332 
18.5534 
36.4805 
34.0324 
18.5530 
15 

48.6416 
44.5388 
24.2071 
48.6414 
44.5387 
24.2066 
20 

Table 6. Comparison of the three first nondimensional natural frequency parameter for SSCC, SFCC, FFCC, CFCS, SFCS, CSFF, CFCF, SSFF, CFSF, CFFF, SFFF, and FFFF, homogeneous, square nano/microplates for different lengthtothickness ratios (g = 0, a/b = 1, ) 

ESDT 
TSDT 
a/h 
B.C 

Third mode 
Second mode 
First mode 
Third mode 
Second mode 
First mode 

20.5762 
19.0885 
10.2617 
20.5756 
19.0878 
10.2615 
5 
SSCC 
30.5416 
30.1799 
14.4481 
30.5415 
30.1797 
14.1480 
8 

57.2654 
52.4612 
24.4775 
57.2644 
52.4607 
24.4789 
15 

76.3539 
68.7305 
31.9135 
76.3535 
68.7304 
31.9134 
20 

16.9181 
12.1604 
7.63680 
16.9177 
12.1697 
7.63620 
5 
SFCC 
19.6768 
19.4566 
10.8800 
19.6766 
19.4565 
10.8778 
8 

36.4812 
34.0332 
18.5534 
36.4805 
34.0324 
18.5530 
15 

48.6416 
44.5388 
24.2071 
48.6414 
44.5387 
24.2066 
20 

10.1397 
7.74920 
3.09180 
10.1395 
7.74880 
3.09160 
5 
FFCC 
13.6711 
12.3987 
4.38680 
13.6710 
12.3987 
4.38650 
8 

23.2475 
22.8323 
7.59510 
23.2474 
22.8321 
7.59480 
15 

30.9967 
29.6265 
9.96270 
30.9967 
29.6263 
9.96240 
20 

16.9181 
12.1604 
9.92720 
16.9177 
12.1604 
9.27710 
5 
CFCS 
20.0742 
19.4566 
14.6355 
20.0741 
19.4550 
14.6350 
8 

36.4812 
34.6875 
25.5669 
36.4810 
34.6873 
25.5664 
15 

48.6416 
45.3792 
33.5370 
48.6414 
45.3788 
33.5368 
20 

16.9181 
12.0543 
7.33010 
16.9175 
12.0537 
7.32940 
5 
SFCS 
19.4566 
17.1289 
10.4140 
19.4562 
17.1287 
10.4110 
8 

36.2810 
29.1811 
17.7667 
36.2807 
29.1804 
17.7661 
15 

48.6416 
38.0750 
23.1912 
48.6409 
38.0744 
23.1911 
20 

10.1397 
7.74920 
2.51770 
10.1395 
7.74920 
2.51760 
5 
CSFF 
12.3987 
10.8310 
3.60230 
12.3984 
10.8308 
3.60220 
8 

23.2475 
17.9114 
6.29800 
23.2474 
17.9114 
6.29700 
15 

30.9967 
23.2027 
8.28150 
30.9965 
23.2027 
8.28140 
20 

16.5349 
11.0593 
8.97390 
16.5346 
11.0581 
8.97350 
5 
CFCF 
16.3444 
14.3596 
14.3243 
16.3444 
14.3596 
14.3237 
8 

28.4434 
26.9226 
26.0234 
28.4434 
26.9226 
26.0234 
15 

37.2508 
35.8962 
32.7453 
37.2508 
35.8962 
32.7453 
20 

8.1157 
7.32850 
1.60520 
8.11560 
7.32850 
1.60500 
5 
SSFF 
11.7012 
9.77340 
2.27190 
11.7010 
9.77330 
2.27180 
8 

20.2377 
15.8028 
3.91690 
20.2374 
15.8020 
3.91660 
15 

26.4931 
20.3743 
5.12480 
26.4930 
20.3741 
5.12450 
20 

15.5240 
8.83300 
6.83580 
15.5240 
8.83300 
6.83580 
5 
CFSF 
14.3589 
12.6404 
9.76200 
14.3587 
12.6404 
9.76190 
8 

26.9229 
21.5324 
16.7452 
26.9226 
21.5320 
16.7449 
15 

38.8972 
28.0669 
21.8875 
35.8968 
28.0662 
21.8871 
20 

7.96570 
3.34000 
1.65540 
7.96540 
3.33970 
1.65510 
5 
CFFF 
5.71810 
5.34400 
2.28760 
5.71790 
5.34350 
2.28730 
8 

10.0615 
10.0199 
3.88530 
10.0614 
10.0199 
3.88520 
15 

13.3599 
13.2335 
5.07200 
13.3599 
13.2334 
5.07170 
20 

3.54470 
1.48690 
0.76020 
3.54460 
1.48690 
0.75970 
5 
SFFF 
3.16780 
2.93870 
1.27670 
3.16750 
2.93840 
1.27650 
8 

5.57400 
5.55500 
2.15720 
0.68850 
0.53760 
0.06170 
15 

7.41210 
7.32740 
5.07200 
1.34340 
0.56070 
0.10610 
20 

1.44240 
0.60490 
0.45110 
1.44230 
0.60490 
0.45090 
5 
FFFF 
1.28630 
1.19490 
0.51920 
1.28640 
1.19480 
0.51910 
8 

2.26600 
2.26020 
0.87710 
2.26100 
2.27960 
0.87680 
15 

4.39310 
4.34430 
2.06430 
4.39290 
4.34430 
2.06410 
20 

The free vibration of sizedependent, rectangular, FG, nano/microplates was analyzed based on nonlinear shear deformation plate theories using modified couple stress theory. The modified couple stress theory contains one materiallength scale parameter, and it can also be degenerated to the classical FG, rectangular plate by setting the materiallength scale parameter equal to zero. Equations of motion for free vibration can be found through an implementation of the RayleighRitz method, which may satisfy any combination of boundary conditions, including: SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SSCC, SCCC, CFCF, SSFF, CFSF, CFFF, SFCS, CFCC, SFCC, FFCC, CFCS, CSFF, SFFF, and FFFF. Material properties were assumed to change continuously through the thickness according to a powerlaw distribution. A comparison of the present results with those reported in the literature for sizedependent, rectangular, FG nano/microplates illustrated the high accuracy of the present study. This research shows the effects of variations of the length scale parameter, lengthtothickness ratio, powerlaw index, and the aspect ratio as well as different boundary conditions on the free vibration of a sizedependent, rectangular, FG nano/microplate.
By looking into the present results, the following points may be concluded:
All analytical results presented here can be provided to other research groups of a reliable source to compare their analytical and numerical solutions.
Acknowledgments
The authors gratefully acknowledge the funding by Arak University, under Grant No. 95/8589.
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