Document Type : Research Paper
Authors
^{1} Young Researchers and Elite Club,Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
^{2} Department of Mechanical Engineering, BuAli Sina University, Hamedan
Abstract
Keywords

Mechanics of Advanced Composite Structures 4 (2017) 5973 

Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
Free Vibration and Buckling Analyses of Functionally Graded Nanocomposite Plates Reinforced by Carbon Nanotube
R. MoradiDastjerdi ^{a}^{*}, H. MalekMohammadi ^{b}
^{a }Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
^{b }Department of Mechanical Engineering, BuAli Sina University, Hamedan, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 20161005 Revised 20161118 Accepted 20161119 
This paper describes the application of refined plate theory to investigate free vibration and buckling analyses of functionally graded nanocomposite plates reinforced by aggregated carbon nanotube (CNT). The refined shear deformation plate theory (RSDT) uses four independent unknowns and accounts for a quadratic variation of the transverse shear strains across the thickness, satisfying the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The motion equations are derived from Hamilton’s energy principle and Navier’s method is applied to solve this equation. The material properties of the functionally graded carbon nanotube reinforced composites (FGCNTRCs) are assumed to vary along the thickness and estimated with the Mori–Tanaka approach. Effects on the natural frequency and critical buckling load of the FGCNTRC plates by CNT volume fraction, CNT distribution, CNT cluster distribution, and geometric dimensions of the plate are investigated. Effects of loading conditions on the critical buckling load are also examined.




Keywords: Mori–Tanaka approach Refined plate theory Aggregated carbon nanotubes Free vibration Buckling 

DOI: 10.22075/MACS.2016.496 
© 2017 Published by Semnan University Press. All rights reserved. 
Carbon nanotubes (CNTs), a new type of advanced material, have attracted a great deal of interest from researchers. Because of their extremely attractive mechanical, electrical and thermal properties, CNTs show promising application in polymer composites as a potential reinforcement and multifunctional element [1,2]. The introduction of CNTs into a polymer matrix may therefore greatly improve mechanical properties, such as tensile strength and elastic modulus, of the resulting nanocomposites [3]. Molecular dynamics (MD) is one technique that can be used to study CNTs. Han and Elliott [4] successfully used the MD method to determine the elastic modulus of composite structures under CNT reinforcement, and they investigated the effect of CNT volume fraction on mechanical properties of nanocomposites. They also investigated the effect of CNT waviness on the elastic properties and mechanical behavior of carbon nanotube reinforced composites (CNTRCs). Alian et al. [5] used a multiscale modeling technique to determine the effective elastic moduli of nanocomposite reinforced by agglomerated carbon nanotubes. Their results showed that the effective elastic properties of the nanocomposite decreased by increasing in CNT volume that is located in CNT clusters. The significant effect of CNT waviness on the load transfer and active constrained layer damping behavior of the short fuzzy fiberreinforced composite has been investigated [68]. Wuite and Adali [9] used a multiscale analysis to study the effects of volume, diameter and distribution of CNTs on deflection and static behavior of CNTRC beams. Formica et al. [10] presented the vibration behavior of CNTRC plates by employing an equivalent continuum model based on the Mori–Tanaka approach. They found that the improvement in modal properties achieved a maximum when the carbon nanotubes were uniformly aligned along the loading direction. Vodenitcharova and Zhang [11] used the Airy stressfunction method to experimentally and computationally investigate pure bending and bendinginduced local buckling of a nanocomposite beam reinforced by a single walled carbon nanotube (SWCNT).
Functionally graded materials (FGMs) are inhomogeneous composites characterized by smooth and continuous variations in both compositional profile and material properties. Such excellent qualities allow them to be fabricated as different structures in accordance with various service requirements. To obtain the required optimum performance, the gradient variation of material properties can be achieved by gradually changing the volume fraction of the constituent materials. Reddy [12] presented static and dynamic analyses of the FGM plates based on third order shear deformation theory and by using the theoretical formulation and finite element models. Zenkour[13] presented a two dimensional solution to study the bending, buckling, and free vibration of simply supported FG ceramic–metal sandwich plates. Cheng and Batra [14] used first and third order shear deformation theories to report deflections of a simply supported functionally graded polygonal plate. Also, Cheng and Batra [15] studied the buckling and steady state vibrations of a simply supported functionally graded polygonal plate based on Reddy’s plate theory. Amabili et al. [16] compared Von Kárman, and first (FSDT) and third order shear deformation theories for nonlinear vibration analysis of rectangular laminated composite plates with different boundary conditions, revealing that FSDT (with shear correction factor of and the higherorder shear deformation theory give practically coincident results. Khorshidi et al. [1718] analyzed vibration behaviour of laminated composite and functionally graded plates in contact with a bounded fluid using the Rayleigh–Ritz method and Fourier series. Also, vibrational behavior of single and multidirectional FG annular plates and laminated curved panels was investigated using threedimensional elasticity theory and generalized differential quadrature method (GDQM)[1922].
Using the concept of FGM, CNTs can be distributed in certain grading profiles through certain directions to improve the mechanical properties and to reinforce the composite structures. The composites, which are reinforced by CNTs with grading distribution, are called functionally graded carbon nanotubereinforced composites (FGCNTRCs). Shen [23] suggested that the interfacial bonding strength can be improved with the use of a graded distribution of CNTs in the matrix. He investigated postbuckling of functionally graded nanocomposite cylindrical shells reinforced by CNTs subjected to axial compression in a thermal environment, and showed that the linear functionally graded reinforcements can increase the buckling load. He estimated mechanical properties with a micromechanical model in volume fraction form with CNT efficiency parameters. Mehrabadi et al.[24] discussed mechanical buckling behavior of FG nanocomposite plates reinforced by SWCNTs based on the firstorder shear deformation theory (FSDT) and mindlin plate theory. However, the rule of mixture is not applicable when straight CNTs are oriented randomly in the matrix. In these cases the Mori–Tanaka approach [25] is one of the best known analytical approaches to accurately determine the effective material constants of composite materials. Yas and Heshmati [26] used the Mori–Tanaka approach to study the vibrational properties of FGnanocomposite beams reinforced by randomly oriented straight CNTs under the action of moving load. Sobhani Aragh et al. [27] presented vibrational behavior of continuously graded CNT–reinforced cylindrical panels based on the Eshelby–Mori–Tanaka approach. They used the 2D GDQM to discretize the governing equations and to implement the boundary conditions. Pourasghar et al. [28] and MoradiDastjerdi et al. [29] performed a free vibration analysis of FG nanocomposite cylinders reinforced by randomly oriented straight and locally aggregated CNTs, based on both threedimensional theory of elasticity, and meshfree methods. Both teams estimated material properties of FG CNTRCs with the Eshelby–Mori–Tanaka approach. Finally, vibrational behavior of single and multidirectional nanocomposite FGCNTRC thick plates, sandwich curved panels and annular plates resting on a Pasternak elastic foundation, were investigated using threedimensional elasticity theory and GDQM [3034].
Since FSDT violates the equilibrium conditions on the top and bottom surfaces of the plate, a shear correction factor is required to compensate for the error because of a constant shear strain assumption throughout the thickness. The shear correction factor not only depends on the material and its geometric properties, but also on its loading and boundary conditions. Although the FSDT provides a sufficiently accurate description of response for thin to moderately thick plates, it is not convenient to use because of the difficulty in determining the correct value of the shear correction factor. To avoid the use of a shear correction factor, many refined shear deformation plate theories (RSDTs) have been developed including the sinusoidal shear deformation plate theory (SSDT) [3536], RSDT [3738], and hyperbolic shear deformation plate theory (HSDT) [3941]. RSDT is based on an assumption that the inplane and transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The motion equation can be derived from Hamilton’s energy principle and Navier’s method solves this equation. MoradiDastjerdi et al. [42] used an RSDT with only four independent unknowns, and presented the free vibration analysis of sandwich plates with FG randomly oriented CNTRC face sheets resting on an elastic foundation. Khorshidi et al. [4344] used nonlocal elasticity theory based on exponential shear deformation theory, for free vibration and buckling analyses of the FG rectangular nanoplates. They also used refined trigonometric shear deformation plate theory to study the outofplane vibration of the rectangular isotropic plates with different boundary conditions [45].
Although several studies of the free vibration or buckling of FG and FG nanocomposite plates have been carried out based on a variety of plate theories, no studies can be found applying these analyses to aggregated CNT reinforced plates. In this study, the RSDT is developed to investigate the free vibration and buckling analyses of simply supported functionally graded nanocomposite plates reinforced by aggregated singlewalled carbon nanotubes (SWCNTs). The applied nanocomposite is assumed a mixture of CNTs (randomly oriented and locally aggregated into some clusters) that are embedded in a polymer. The material properties of the nanocomposite plates are assumed to vary along the thickness of plate and estimated though the Mori–Tanaka method because of its simplicity and accuracy even at a high volume fraction of inclusions. Effects on the natural frequency and critical buckling load of the FGCNTRC plates by CNT volume fraction, CNT distribution, CNT cluster distribution, and geometric dimensions of the plate are investigated. Effects of loading conditions on the critical buckling load are also examined.
Consider a CNTRC is made from a mixture of SWCNT (that randomly oriented and locally aggregated into some clusters) and matrix which is assumed to be isotropic. Many studies have been published each with a different focus on mechanical properties of polymer nanotube composites. However, the common theme has been enhancement of Young’s modulus. In this section, the effective mechanical properties of the CNT reinforced composite that straight CNTs are oriented randomly, or locally aggregated in to some clusters, are obtained based on the Eshelby–Mori–Tanaka approach. The resulting effective properties for these CNT reinforced composites are isotropic, despite the CNTs being transversely isotropic.
2.1 Composites reinforced with randomly oriented, straight CNTs
In this section, the effective mechanical properties of composites with randomly oriented nonclustered CNTs (as shown in Fig. 1) are studied. The orientation of a straight CNT is characterized by two Euler angles α and β, as shown in Fig. 1. When CNTs are completely randomly oriented in the matrix, the composite is isotropic, and its bulk modulus K and shear modulus G are derived as [46]:

(1) 
where subscripts m and r are referred to matrix and CNT respectively, f is volume fraction and also,
(2) 

(3) 
(4) 

(5) 
Figure 1. Representative volume element (RVE) with randomly oriented, straight CNTs.
k_{r}, l_{r}, m_{r}, n_{r}, and p_{r} are the Hill’s elastic moduli for the reinforcing phase (CNTs). As mentioned before, the CNTs are transversely isotropic and have a stiffness matrix given below (Hill’s elastic moduli):
(6) 

(7) 
where E_{L}, E_{T}, E_{Z}, G_{TZ}, G_{ZL}, G_{LT}, , and are material properties of the CNT reinforced composite which can be determined from the inverse of the rule of mixture.
So, the effective Young’s modulus E and Poisson’s ratio of the composite is given by:
(8) 

(9) 
2.2 Effect of CNT aggregation on the properties of the composite
The CNTs were arranged within the matrix to introduce clustering. Because of a large aspect ratio (usually >1000), a low bending rigidity, and Van der Waals forces, CNTs have a tendency to bundle or cluster together making it quite difficult to produce fullydispersed CNT reinforced composites. The effect of nanotube aggregation on the elastic properties of randomly oriented CNTRCs is presented in this section. Shi et al. [46] derived a two parameter micromechanics model to determine the effect of nanotube agglomeration on the elastic properties of a randomly oriented CNTRC (Fig. 2). It is assumed that a number of CNTs are uniformly distributed throughout the matrix and that other CNTs appear in cluster form because of aggregation, as shown in Fig. 2. The total volume of the CNTs in the representative volume element (RVE), denoted by V_{r}, can be divided into the following two parts:
(10) 
where denotes the volumes of CNTs inside a cluster, and is the volume of CNTs in the matrix and outside the clusters. The two parameters used to describe the aggregation are defined as:
(11) 
where V is the volume of RVE, V_{cluster} is the volume of clusters in the RVE. is the volume fraction of clusters with respect to the total volume V of the RVE, η is the volume ratio of the CNTs inside the clusters over the total CNT inside the RVE. When , there is uniform distribution of nanotubes throughout the entire composite without aggregation; with a decreasing , the agglomeration degree of CNTs becomes more severe. When , all nanotubes are located in the clusters. The case means that the volume fraction of CNTs inside the clusters is equal to that of CNTs outside the clusters, so all CNTs are located and randomly oriented as in Fig. 1. Thus, we consider the CNTreinforced composite as a system consisting of spherically shaped clusters in a matrix. We first estimate the effective elastic stiffness of the clusters and the matrix respectively, and then calculate the overall property of the whole composite system. The effective bulk modulus K_{in} and shear modulus G_{in} of the cluster can be calculated with Prylutskyy et al. [47]:
(12) 

(13) 
Figure 2. RVE with functionally graded Eshelby cluster model of aggregation of CNTs.
and the effective bulk modulus K_{out} and shear modulus G_{out} of the matrix outside the cluster can be calculated by:
(14) 

(15) 
Finally, the effective bulk modulus K and the effective shear modulus G of the composite are derived from the MoriTanaka method as follows:
(16) 

(17) 
with
(18) 

(19) 

(20) 
The effective Young’s modulus E and Poisson’s ratio of the composite can be calculated in the terms of K and G by Eqs. (8) and (9).
3. Refined Plate Theory
Consider a rectangular FGCNTRC plate with thickness h, and edges parallel to axes x and y, as shown in Fig. 3. The volume fractions of CNTs or , are varied along the thickness of the plate as following:
(21) 

(22) 
Where p ( ) is the volume fraction exponent and and are the values of CNT volume fraction in upper ( ) and downer surfaces ( ), respectively.
Fig. 4 shows the variation of CNT volume fraction along the thickness of a plate for different values of the volume fraction exponent, p. The effective Young’s modulus E and Poisson’s ratio are obtained from Eqs. (8), (9).
Figure 3. Schematic of the CNTRC plate.
Figure 4. Variation of properties along the thickness of cylinders for different values of p according to Eq. (21).
3.1 Basic assumptions
The assumptions of the present theory are as follows [37]:
a) The displacements are small in comparison with the plate thickness and thus the strains involved are infinitesimal.
b) The transverse displacement includes two components: bending and shear , and these components are functions of coordinates x, y only.
(23) 
c) The transverse normal stress is negligible in comparison with the inplane stresses and .
d) The displacements U in the xdirection, and V in the ydirection, consist of extension, bending, and shear components.
(24) 
The bending components and are assumed to be similar to the displacements given by classical plate theory. Therefore, the expression for and can be given as:
(25) 
In conjunction with , the shear components and give rise to the parabolic variations of shear strains , and hence to shear stresses , across the thickness of the plate in such a way that the shear stresses , are zero at the top and bottom faces of the plate. Consequently, the expression for and can be given as:
(26) 
3.2 Kinematics and constitutive equations
Based on the assumptions made in the preceding section, the displacement field can be obtained [37]
(27) 
where
(28) 
The strains associated with the displacements in Eq. (27) are:
,
,

(29) 
where
(30) 
For elastic and isotropic materials, the constitutive relations can be written as:
(31) 
where ( ) and ( ) are the stress and strain components, respectively. Using the material properties defined in Eq. (21), stiffness coefficients, , can be expressed as:
(32) 
3.3 Governing equations
Using Hamilton’s energy principle the motion equation of the isotropic plate is derived:
(33) 
where is the strain energy, work done by applied forces, and is the kinetic energy of the isotropic plate. Employing the minimum of the total energy principle leads to a general equation of motion and boundary conditions. Taking the variation of the above equation and integrating by parts:
(34) 
where represents the second derivative with respect to time and also are inplane prebuckling forces.
The equations of motion can be obtained by substitution of Eqs. (27) and (29) into Eq. (34) and by consideration of the following assumptions. The stress resultants , , and the mass moments of inertia are defined by:
(35.a) 

(35.b) 

(35.c) 
So, the equation of motion can be written as:
(36.a) 

(36.b) 

(36.c) 

(36.d) 
where
(37) 
Substituting Eq. (31) into Eq. (35) and integrating through the thickness of the plate, the stress resultants are given as:
(38) 
where
(39) 

(40) 

(41) 

, 
(42) 
and stiffness components are given as:
(i,j=1,2,6)

(43) 
3.4 Navier’s solution for simply supported rectangular plates
Rectangular plates are generally classified in accordance with the type of support used. The analytical solutions of Eq. (36) for simply supported FGCNTRC plates are used here. The following boundary conditions are imposed at the side edges [37]:
(44) 
The displacement functions that satisfy the equations of boundary conditions (Eq. (44)) are selected as the following Fourier series:
(45) 
where are the arbitrary parameters to be determined, is the eigen frequency associated with (m,n)^{th} eigen mode, and . Substituting Eq. (45) into equations of motion (Eq. (36)) we get the below eigenvalue equations for any fixed value of m and n:
(46) 
and the elements of the coefficient matrix k and M are given in Appendix A. To avoid trivial solution of equation (46), the following equations should be solved:
(47) 
or, with premultiplying Eq. (36) by , becomes:
(48) 
the natural frequencies ( ) can be derived by solving this equation.
For stability problems, the natural frequency vanishes and the obtained equations allow derivation of results that concern the buckling of a plate subjected to a system of uniform inplane compressive loads and . Assuming that there is a given ratio between these forces such that and ; , we get:
(49) 
where
, ,

(50) 
, 
(51) 
4. Results and Discussions
In this section, first the accuracies of applied methods are examined in the calculations of the nanocomposite modulus, free vibration, and buckling, by comparing obtained results with reported corresponding results in the literatures. Second, the effects of plate dimensions, CNT volume fraction, orientation, aggregation, and their variation patterns are investigated regarding the frequency and critical buckling load parameter of FGCNTRC plates.
4.1 Validation of models
First, the MoriTanaka approach that is applied for calculation of the nanocomposite modulus is examined. As defined before, the parameters μ and η are indicators of the volume fractions of clusters, and CNTs in the clusters, respectively. Fig. 5 shows Young’s modulus of a CNTreinforced composite for various value of µ when η=1 that is compared with the experimental data (by Odegard et al. [48]). This figure shows that at full dispersion of the randomly oriented CNTs, μ=1, Young’s modulus has the biggest values. Young’s modulus was decreased by increasing the CNTs aggregation (decreasing of the μ) or decreasing the CNTs volume fraction. Also, it can be seen that the aggregation state of η=1 and μ=0.4 has nearly the same Young’s modulus as the experimental data. These results are in agreement with an argument proposed by Barai and Weng [49].
In the following simulations, CNTRC plates are considered made of Poly (methyl methacrylate, referred as PMMA) as matrix, with CNT as fibers. PMMA is an isotropic material with , and . The (10, 10) SWCNTs are selected as reinforcements. The adopted material properties for SWCNT are:
, , , and [23].
In this state the effects of distributions and orientations of the CNTs on the Young’s modulus of a CNTRC are examined. Fig. 6 shows Young’s modulus of alignment, randomly oriented and locally aggregated CNTRCs for various values of volume fraction of the CNTs. This figure shows that alignment orientation of CNTs estimatesvery high values for effective Young’s modulus despite Fig. 5 showing the experimental data has the same values with μ=0.4 and η=1. Also it can be seen that randomly oriented or fully dispersed, μ=η=1, nanotubes have more stiffness than other aggregated states, μ=0.4, 0.7, 0.9. After verification of the MoriTanaka approach, free vibration analysis is performed. First normalized frequency parameters ( ) of isotropic FGM plates are presented for various values of volume fraction exponent, p, and ratio of length to thickness, a/h, in Table 1. The normalized natural frequency is then defined as:
(51) 
where
(52) 
the subscript m is used for metal in the applied FGM plate. The comparisons show that the results agree very well with other available solutions.
Figure 5. Comparison of the Young's modulus of CNTreinforced composite at different degree of aggregation with the experimental data from Odegard et al. [48].
Figure 6. Comparison of the Young's modulus of CNTRC at different degree of aggregation with the randomly oriented and aligned CNTs.
Table 1. Comparison of the first frequency parameters of square isotropic FGM plates.
a/h 
Theory 
p=0 
p=1 
p=4 
p=10 
2 
[50] 
0.9400 
0.7477 
0.5997 
0.5460 
[41] 
0.9300 
0.7725 
0.6244 
0.5573 

Present 
0.9304 
0.7360 
0.5928 
0.5417 

5 
[50] 
0.2121 
0.1640 
0.1383 
0.1306 
[41] 
0.2113 
0.1740 
0.1520 
0.1369 

Present 
0.2113 
0.1631 
0.1378 
0.1301 

10 
[50] 
0.05777 
0.04427 
0.03811 
0.03642 
[41] 
0.05770 
0.04718 
0.04210 
0.03832 

Present 
0.05769 
0.04419 
0.03807 
0.03637 
Finally, a comparison is carried out for buckling analysis of a simply supported FGM plate with a/b=1, a/h=10 and the ratio of transverse load to axial load of, (uniaxial compressive pressure). Critical buckling load parameter is defined as and listed in Table 2 for the first mode of FGM plates. These values are compared with results of Bodaghi and Saidi [51] and Thai and Choi [52]. The results agree well with previous results for various values of p.
Table 2. Comparison of the critical buckling load parameters of square isotropic FGM plates with a/h=10.
Theory 
p=0 
p=1 
p=2 
[51] 
1437.361 
702.304 
534.441 
[52] 
1437.389 
702.251 
534.835 
present 
1437.390 
702.251 
534.837 
4.2 Free vibration analysis of CNTRC plates
First, simply supported FGCNTRC square plates are considered. In these plates the volume fraction of randomly oriented CNT, f_{r}, varies from zero to 0.4 according to Eq. (21) along the thickness of the plate. Fig. 7 shows the first natural frequency parameters, , that are calculated by the following equation based on the mechanical properties of CNT for various values of p and b/h.
(53) 
This figure shows that by increasing the ratio of width to thickness plates, b/h, or decreasing the volume fraction exponent, p, gives an increase . Also, Table 3 lists various modes of frequency parameters, , for the same plate with a/h=10. This table shows that UDCNTRC plates have more values of frequency parameters than FG plates and shows that and have the lowest and the highest values of frequency parameters, respectively.
Consider simply supported FGCNTRC square plates with randomly or aggregated CNT, a/h=10 and f_{r} =0→0.4. Table 4 shows the first natural frequency parameters of these plates. This table shows that randomly oriented and state of μ=η=1, have the highest frequency values and closest values with their material properties. Also, it can be concluded that the parameter of η has more effect than μ on the frequency, and states that are near to fullydispersed have more frequency values. Fig. 8 illustrates variation of versus μ for various values of η in UDCNTRC square plate with a/h=10 and f_{r} =0.4. Frequency parameters increase as μ increases or especially as η decreases. This behavior was seen for mechanical properties of the nanocomposites as well [29]. Fig. 9 shows the first natural frequency of the same plates with η=1 and various values of μ and a/h. This figure shows that frequency parameters are increased by increasing μ or decreasing the ratio of length to thickness, a/h. As another example, consider FGCNTRC square plates with a/h=10, f_{r} =0→0.4 and CNT aggregation state of μ=0.5 and η=1.
Figure 7. First frequency parameters versus b/h for FGCNTRC square plates with fully dispersed CNT and f_{r} =00.4.

Figure 8. First frequency parameters versus μ for FGCNTRC square plates with aggregated CNT, a/h=10 and f_{r} =0.4. 
Table 3. frequency parameters of FGCNTRC square plates with fully dispersed CNT, a/h=10 and f_{r} =00.4.
(m,n) 
p=0.01 
p=0.1 
p=0.4 
p=1 
p=2.5 
p=10 
p=100 
f_{r}=0.2 
(1,1) 
1.6843 
1.5623 
1.2708 
0.9246 
0.5399 
0.2425 
0.1964 
1.1062 
(1,2) 
4.0294 
3.7427 
3.0527 
2.2263 
1.3013 
0.5759 
0.4471 
2.6450 
(2,2) 
6.1980 
5.7633 
4.7113 
3.4431 
2.0145 
0.8815 
0.6628 
4.0666 
(1,3) 
7.5622 
7.0364 
5.7597 
4.2146 
2.4674 
1.0730 
0.7929 
4.9605 
Table 4. frequency parameters of FGCNTRC square plates with a/h=10 and f_{r} =00.4.
(1,1) 
p=0.01 
p=0.1 
p=1 
p=10 
f_{r}=0.2 
Randomly 
1.6843 
1.5623 
0.9246 
0.2425 
1.1062 
1.6834 
1.5616 
0.9244 
0.2425 
1.1059 

0.2002 
0.2008 
0.1986 
0.1520 
0.2039 

0.2731 
0.2735 
0.2615 
0.1685 
0.2759 

1.5848 
1.4714 
0.8755 
0.2389 
1.0463 
Table 5. frequency parameters of FGCNTRC square plates with a/h =10, μ=0.5, η=1 and f_{r} =00.4.
(m,n) 
p=0.01 
p=0.1 
p=0.4 
p=1 
p=2.5 
p=10 
p=100 
f_{r}=0.2 
(1,1) 
0.2002 
0.2008 
0.2016 
0.1986 
0.1774 
0.1520 
0.1327 
0.2039 
(1,2) 
0.4778 
0.4790 
0.4811 
0.4743 
0.4225 
0.3612 
03143 
0.4863 
(2,2) 
0.7332 
0.7352 
0.7385 
0.7285 
0.6559 
0.5528 
0.4796 
0.7463 
(1,3) 
0.8935 
0.8959 
0.9000 
0.8882 
0.8013 
0.6725 
0.5826 
0.9094 
Table 5 shows various modes of frequency parameters, , for various values of p. By comparing results of Table 5 and those of Table 3, it can be concluded that aggregation of CNTs sharply decreases frequency parameters in all modes.
In all of the above FGCNTRC plates, volume fraction of CNT was changed but FGCNTRC plates can also be made by changing of the volume fraction of clusters. Consider CNTRC square plates with a/h=10, f_{r} =0.2 and η=1. Volume fraction of clusters of CNTs, μ, varies from zero to 0.4 according to Eq. (21) along the thickness of plate. Table 6 shows various modes of frequency parameters, , for various values of volume fraction exponent of clusters, p. This table shows that increasing p decreases frequency parameters in all modes. Fig. 10 illustrates versus η for distributions of clusters in the CNTRC square plates with a/h=10 and f_{r} =0.4. Fig. 10 shows these UD and FG plates have similar values of for big values of η. Finally, consider a UDCNTRC square plate with a/h=10, f_{r} =0.2, η=1 and μ=0.5. Fig. 11 shows mode shapes of the plate at mode numbers of (1,1), (2,1), (1,2) and (3,1).
Figure 9. First frequency parameters versus a/h for UDCNTRC square plates with aggregated CNT and f_{r} =0.4.

Figure 10. First frequency parameters versus η for UDCNTRC square plates with aggregated CNT, a/h =10 and f_{r} =0.4. 
Table 6. frequency parameters of UDCNTRC square plates with aggregated CNT, a/h=10, f_{r} =0.2, η=1 and μ=00.4.
(m,n) 
p=0.01 
p=0.1 
p=0.4 
p=1 
p=2.5 
p=10 
p=100 
f_{r}=0.2 
(1,1) 
0.1791 
0.1714 
0.1569 
0.1455 
0.1376 
0.1290 
0.1203 
0.1453 
(1,2) 
0.4271 
0.4091 
0.3746 
0.3468 
0.3268 
0.3058 
0.2863 
0.3462 
(2,2) 
0.6554 
0.6281 
0.5753 
0.5318 
0.4995 
0.4668 
0.4385 
0.5309 
(1,3) 
0.7986 
0.7655 
0.7013 
0.6478 
0.6073 
0.5672 
0.5336 
0.6466 
4.3 Buckling analysis of CNTRC plates
In this section, buckling of FGCNTRC plates is investigated. First, consider fully dispersed CNT reinforced nanocomposite plates under uniaxial compressive pressure ( ) with CNT volume fraction of, f_{r} =0→0.4. The critical buckling load parameters of these plates are listed in Table 7 for various values of plate dimensions (a/b and a/h) and volume fraction exponent (p). The results show that critical buckling load parameter is increased by increasing the ratios of a/b and a/h or decreasing p. When increasing the aspect ratio of the plates (a/b), they show the behavior of simply supported beams. Thus, the critical buckling load is increased by increasing the ratio of a/b. Also, by considering the definition of critical buckling load parameter, decreasing the plate thickness h, increases critical buckling load parameter.It was observed that in some cases, critical buckling happened at modes of (2, 1), (3, 1) or (4, 1).
(a) (1,1) 
(b) (2,1) 
(c) (1,2) 
(d) (3,1) 
Figure 11. The mode shapes of UDCNTRC square plates with μ=0.5, η=1, a/h=10 and f_{r} =0.2.
Second, buckling of FGCNTRC square plates under uniaxial compressive pressure ( ) with a/h=10, f_{r} =0→0.4 is investigated. Table 8 shows the critical buckling load parameters of the plates for various states of CNT distributions and various values of p. States of μ=η=1 and fully dispersion have the biggest and closest buckling parameters, especially at p=10. Also, the results reveal that the critical buckling load of the plates has a higher value when distribution of the CNT in polymer is better, as the stiffness of CNTRC plates is larger when CNT distribution is better.
Third, consider FGCNTRC plates as previously, but instead under biaxial compressive pressure ( ). Critical buckling load parameters of this third model of plates are shown in Table 9. Comparing the results of Tables of 8 and 9, shows that critical buckling load parameters of the plates under biaxial compressive load are almost half of the corresponding values of the plates under uniaxial compressive load.
Finally, consider UDCNTRC plates with f_{r} =0.2 and aggregation state of μ=0.5 and η=1. Table 10 shows critical buckling load parameters of these plates with various plate dimensions (a/b and a/h) and loading parameter ( ). It can be seen that the critical buckling load parameter is increased by increasing ratios of a/b and a/h, whereas it is decreased by increasing the loading parameter.
Table 7. Critical buckling load parameters of FGCNTRC plates with fully dispersed CNT, , and f_{r} =00.4.
a/b 
a/h 
p 

0.01 
0. 1 
0.4 
1 
2.5 
10 
100 
f_{r}=0.2 

1 
2 
294.8462^{a} 
260.8422^{a} 
186.7806^{a} 
113.2799^{a} 
48.7021^{a} 
7.8708^{a} 
1.8977^{a} 
120.7497^{a} 
5 
642.5812 
552.7512 
366.0923 
195.3273 
67.9488 
13.2080 
6.8846 
265.7493 

10 
736.8569 
630.0932 
411.1198 
214.7617 
72.5821 
14.5825 
9.3568 
305.4156 

30 
770.4270 
657.4098 
426.7051 
221.3030 
74.0840 
15.0505 
10.4791 
319.5893 

100 
774.4428 
660.6697 
428.5542 
222.0729 
74.2590 
15.1057 
10.6243 
321.2803 

1.5 
2 
354.5132^{b} 
314.8943^{b} 
229.1755^{b} 
144.0864^{b} 
66.6355^{b} 
10.8079^{b} 
2.2533^{b} 
145.0786^{b} 
5 
1153.5330^{a} 
1002.9824^{a} 
683.1757^{a} 
380.1545^{a} 
140.7012^{a} 
25.5806^{a} 
9.6911^{a} 
475.1888^{a} 

10 
1629.5324^{a} 
1399.5335^{a} 
923.2362^{a} 
489.7820^{a} 
169.0318^{a} 
33.1534^{a} 
18.2061^{a} 
674.3114^{a} 

30 
1858.4267^{a} 
1586.7252^{a} 
1031.3711^{a} 
535.9525^{a} 
179.8799^{a} 
36.4333^{a} 
24.7930^{a} 
770.7330^{a} 

100 
1888.6603^{a} 
1611.2839^{a} 
1045.3228^{a} 
541.7741^{a} 
181.2064^{a} 
36.8507^{a} 
25.8629^{a} 
783.5016^{a} 

2 
2 
396.9993^{c} 
352.7124^{c} 
258.4639^{c} 
166.0574^{c} 
80.3948^{c} 
13.7075^{c} 
2.6459^{c} 
162.5994^{c} 
5 
1567.6549^{b} 
1374.9253^{b} 
959.1718^{b} 
554.8582^{b} 
218.7951^{b} 
37.4486^{b} 
11.3034^{b} 
643.8058^{b} 

10 
2570.3249^{a} 
2211.0049^{a} 
1464.3691^{a} 
781.3092^{a} 
271.7952^{a} 
52.8321^{a} 
27.5384^{a} 
1062.9971^{a} 

30 
3029.9260^{a} 
2587.5585^{a} 
1682.8889^{a} 
875.2117^{a} 
294.0531^{a} 
59.4851^{a} 
40.1106^{a} 
1256.4700^{a} 

100 
4817.9138^{c} 
4111.0404^{c} 
2668.1722^{c} 
1383.6709 
463.1451^{c} 
94.1032^{c} 
65.5928^{c} 
1998.5589^{c} 
^{a}Mode for plate is (m,n)=(2,1) ^{b}Mode for plate is (m,n)=(3,1) ^{c}Mode for plate is (m,n)=(4,1)
Table 8. Critical buckling load parameters of FGCNTRC square plates with a/h=10, , and f_{r} =00.4.

p=0.01 
p=0.1 
p=1 
p=10 
f_{r}=0.2 
Randomly 
736.8569 
630.0932 
214.7617 
14.5825 
305.4156 
666.4453 
577.1593 
204.3684 
14.3893 
292.8660 

10.4138 
10.4014 
9.8397 
5.5833 
10.3722 

19.3693 
19.3032 
17.0656 
6.8847 
19.0047 

652.4199 
558.9323 
192.5201 
14.1503 
273.2313 
Table 9. Critical buckling load parameters of FGCNTRC square plates with a/h=10, , and f_{r} =00.4.

p=0.01 
p=0.1 
p=1 
p=10 
f_{r}=0.2 
Randomly 
368.4285 
315.0466 
107.3809 
7.2912 
152.7078 
333.2226 
288.5797 
102.1842 
7.1946 
146.4330 

5.2069 
5.2007 
4.9199 
2.7916 
5.1861 

9.6846 
9.6516 
8.5328 
3.4423 
9.5023 

326.2099 
279.4662 
96.2600 
7.0751 
136.6156 
Table 10. Critical buckling load parameters of UDCNTRC plates with μ=0.5, η=0.7, and f_{r} =0.2.
a/b 
a/h 

0 
0.1 
0.2 
0.5 
1 
2 
5 
10 

1 
2 
108.2666^{a} 
105.6259^{a} 
103.1110^{a} 
83.7473 
62.8105 
41.8736 
20.9368 
11.4201 
5 
237.8406 
216.2187 
198.2005 
158.5604 
118.9203 
79.2802 
39.6401 
21.6219 

10 
273.2313 
248.3921 
227.6927 
182.1542 
136.6156 
91.0771 
45.5386 
24.8392 

30 
285.8653 
259.8776 
238.2211 
190.5769 
142.9327 
95.2884 
47.6442 
25.9878 

100 
287.3778 
261.2525 
239.4815 
191.5852 
143.6889 
95.7926 
47.8963 
26.1530 

1.5 
2 
130.1103^{b} 
126.9369^{b} 
123.9146^{b} 
111.8032^{a} 
91.6786^{a} 
67.4107^{a} 
37.5732^{a} 
21.6223^{a} 
5 
425.5896^{a} 
402.9251^{a} 
382.5524^{a} 
266.8472 
174.4771 
103.1001 
46.2898 
24.1298 

10 
603.4322^{a} 
571.2459 
482.6043 
329.3064 
215.3157 
127.2320 
57.1246 
29.7777 

30 
689.4420^{a} 
613.9956 
518.7204 
353.9504 
231.4291 
136.7536 
61.3996 
32.0062 

100 
700.8266^{a} 
619.2725 
523.1785 
356.9924 
233.4181 
137.9289 
61.9273 
32.2812 

2 
2 
145.8203^{c} 
142.2637^{c} 
138.8765^{c} 
126.5728^{b} 
86.6133 
48.1185 
20.6222 
10.5626 
5 
576.9331^{b} 
552.3828^{b} 
523.8876^{a} 
394.0031 
236.4019 
131.3344 
56.2862 
28.8295 

10 
951.3625^{a} 
864.8751^{a} 
792.8021^{a} 
529.7794 
317.8677 
176.5932 
75.6828 
38.7644 

30 
1123.9652^{a} 
1021.7872^{a} 
936.6377^{a} 
590.4315 
354.2589 
196.8105 
84.3474 
43.2023 

100 
1147.7093^{a} 
1043.3721^{a} 
956.4245^{a} 
598.2341 
358.9405 
199.4114 
85.4620 
43.7732 
^{a}Mode for plate is (m,n)=(2,1) ^{b}Mode for plate is (m,n)=(3,1) ^{c}Mode for plate is (m,n)=(4,1)
5. Conclusions
In this paper the effects of various parameters on the natural frequency and critical buckling load of simply supported FGCNTRC plates are investigated. The randomly oriented nanotubes were assumed to have aggregated into some clusters and the Mori–Tanaka approach was used to estimate the mechanical properties of nanocomposites. The motion equation was derived from Hamilton’s energy principle and Navier’s method solved this equation. The following results are obtained from these analyses:
Appendix
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