Document Type: Research Paper
Authors
^{1} Young Researchers and Elite Club,Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
^{2} School of Mechanical Engineering, Shahid Rajaee Teacher Training University (SRTTU), Tehran, Iran
Abstract
Keywords

Mechanics of Advanced Composite Structures 3 (2016) 123135 

Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
Static and Free Vibration Analyses of Functionally Graded Nanocomposite Plates Reinforced by Wavy Carbon Nanotubes Resting on a Pasternak Elastic Foundation
R. Moradi Dastjerdi^{ a,}^{*}, G. Payganeh ^{b}, S. Rajabizadeh Mirakabad ^{b}, M. Jafari MofradTaheri ^{b}
^{a }Young Researchers and Elite Club,Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
^{b }School of Mechanical Engineering, Shahid Rajaee Teacher Training University (SRTTU), Tehran, Iran
Paper INFO 

ABSTRACT 
Paper history: Received: 20160527 Revised: 20160723 Accepted: 20160826 
In this study, static and free vibration analyses of functionally graded (FG) nanocomposite plates, reinforced by wavy singlewalled carbon nanotubes (SWCNTs) resting on a Pasternak elastic foundation, were investigated based on a meshfree method and modified firstorder shear deformation theory (FSDT). Three linear types of FG nanocomposite plate distributions and a uniform distribution of wavy carbon nanotubes (CNTs) were considered, in addition to plate thickness. The mechanical properties were by an extended rule of mixture. In the meshfree analysis, moving least squares (MLS) shape functions were used for approximation of the displacement field in the weak form of a motion equation, and the transformation method was used for imposition of essential boundary conditions. Effects of geometric dimensions, boundary conditions, the type of applied force, and the waviness index, aspect ratio, volume fraction, and distribution pattern of CNTs were examined for their effects on the static and frequency behaviors of FG carbon nanotube reinforced composite (CNTRC) plates. Waviness and the distribution pattern of CNTs had a significant effect on the mechanical behaviors of FGCNTRC plates, even more than the effect of the CNT volume fraction.




Keywords: Static Free vibration Wavy carbon nanotube Nanocomposite plates MeshFree 


© 2016 Published by Semnan University Press. All rights reserved. 
The extraordinary and outstanding characteristics of carbon nanotubes (CNTs) have broadly attracted researchers’ attention since their discovery in the 1990s [1]. Conventional fiberreinforced composite materials are normally made of stiff and strong fillers with microscale diameters embedded into various matrix phases. The discovery of CNTs may lead to a new ways to improve the properties of the resulting composites by the changing reinforcement phases for nanoscaled fillers [2]. Carbon nanotubes are considered as a potential candidate for the reinforcement of polymer composites, which provide a good interfacial bonding between CNTs and the polymer, and proper dispersion of the individual CNTs in the polymeric matrix can be guaranteed [3]. The CNT/polymer structures may also be supported by an elastic foundation. These kinds of plates are mainly used in concrete roads, rafts, and mat foundations for buildings, and reinforced concrete pavements for airport runways. To describe the interaction between the plate and the foundation, various kinds of foundation models have been proposed. The simplest one is the Winkler or oneparameter model, which regards the foundation as a series of separated springs without coupling effects between them. This model was improved by Pasternak by adding a shear spring to simulate the interactions between the separated springs in the Winkler model. The Pasternak model is widelyused to describe the mechanical behavior of structurefoundation interactions [4].
On the other hand, the mechanical properties of CNTRCs decrease if the volume fraction of CNTs rises beyond certain limit [5]. Therefore, due to the high cost of CNTs, the modeling of CNTRCs incorporates the concept of functionally graded materials (FGMs) to effectively and efficiently make use of the CNTs. FGMs are classified as novel composite materials with gradient compositional variation. The concept of FGMs can be utilized for the management of a material's microstructure, so that the mechanical behavior of a structure made of such materials can be improved. The composites, which are reinforced by CNTs with grading distribution, are called functionally graded carbon nanotube reinforced composites (FGCNTRCs). Several works on FGCNTRC structures were carried out in the wake of new research using FGMs. For example, Shen [6, 7] suggested that the interfacial bonding strength could be improved through the use of a graded distribution of CNTs within the matrix. He examined postbuckling and the nonlinear bending behavior of FGCNTRC cylindrical shells and plates, respectively, and demonstrated that the linear FG reinforcements can increase the material’s mechanical behaviors. Zhu et al.[8] evaluated bending and performed free vibration analyses of thintomoderately thick FGCNTRC plates, using the finite element method (FEM) based on the firstorder shear deformation theory (FSDT). Centered on a threedimensional theory of elasticity, Alibeigloo [9] discussed static analysis of FGCNTRC plates imbedded in piezoelectric layers using three cases of CNT distribution. Malekzadeh et al. [10]studied the free vibration behaviors of quadrilateral laminated thintomoderatelythick FGCNTRC plates, using the FSDT and the differential quadrature method (DQM). Alibeigloo and Liew [11] presented the bending behaviors of FGCNTRC rectangular plates with simply supported edges, subjected to thermomechanical loads based on a threedimensional theory of elasticity. MoradiDastjerdi et al. [12] used the Navier method and a refined plate theory to investigate the free vibration analysis of simply supported sandwiched plates with FGCNTRC face sheets resting on a Pasternak elastic foundation. They used straight and randomly oriented CNTs in the FGCNTRC face sheets.
In addition, some forms of meshfree methods were used to analyze FGCNTRC structures. For example, MoradiDastjerdi et al. [1314] presented static and dynamic analyses of FG nanocomposite cylinders reinforced by straight CNTs carried out by a meshfree method based on an MLS shape function. Additionally, they reported the effects of orientation and aggregation of CNTs on the axisymmetric natural frequencies of FGnanocomposite cylinders. In this work, they used the Eshelby–Mori–Tanaka approach to estimate the mechanical properties [15]. Lie et al. [16] studied free vibration analysis of FGCNTRC plates, using the elementfree kpRitz method based on FSDT. Finally, in two related works, Zhang et al. [17, 18] used an elementfree based improved moving least squaresRitz (IMLSRitz) method and FSDT to study the buckling behaviors of FGCNTRC plates resting on a Winkler foundation and the nonlinear bending of the same type of plates resting on a Pasternak elastic foundation.
In all the abovementioned studies concerning FGCNTRC structures, they assumed that the CNTs were straight and did not consider the effects of the CNT aspect ratios or waviness, while CNT curvature (waviness index) and CNT aspect ratios dramatically decreased the modulus of elasticity. However, some researchers have studied the effects of FGCNTRC structures. For example, Martone et al. [19] presented the reinforcement effects of CNTs with different aspect ratios in an epoxy matrix. They showed that progressive reduction of the nanotubes’ effective aspect ratios occurred because of the increasing connectedness between the nanotubes with an increase in their concentration. In addition, they investigated the effects of nanotube curvature on the average contact number between tubes by means of the waviness that accounts for the deviation from straight particles’ assumptions.
Based on a threedimensional theory of elasticity, Jam et al. [20] investigated the effects of CNTs aspect ratios and waviness on the vibrational behavior of nanocomposite cylindrical panels, and their results indicated that the distribution pattern and volume fraction of CNTs have a significant effect on the natural frequencies of nanocomposite cylindrical panels. MoradiDastjerdi et al. [21, 22] also studied the effects of CNTs’ waviness and aspect ratios on the vibrational and dynamic behaviors of FGCNTRC cylinders. They used a new version for the rule of mixture to show the effect of CNT waviness on the reinforcement behaviors of the nanocomposites. They considered different distribution patterns and waviness conditions with variable aspect ratios, and they reported significant effects on the natural frequency and stress wave propagation of nanocomposite cylinders. Finally, Shams et al. [23] investigated the effects of CNT waviness and aspect ratios on the buckling behavior of FGCNTRC plates subjected to inplane loads. They employed a reproducing kernel particle method (RKPM) based on modified FSDT.
In this study, a meshfree method, based on FSDT, was developed to consider the effects of a twoparameter elastic foundation, CNT waviness, and CNT aspect ratios, on the static and free vibration analyses of nanocomposite plates reinforced by wavy CNTs. Material properties were estimated by a micromechanical model. Micromechanics’ equations cannot capture the scale difference between the nano and microlevels. In order to overcome this difficulty, the efficiency parameter was defined and estimated by matching the Young's moduli for the NTRCs obtained by the extended rule of mixture to those obtained by MD simulation. In the meshfree method, MLS shape functions were used for approximation of the displacement field in the weak form of a motion equation, and the transformation method was used for imposition of essential boundary conditions. This meshfree method did not increase the calculations against the elementfree Galerkin (EFG) method. Four linear types of CNT distributions were considered: uniform distribution, and three kinds of FG distributions. In addition, the other variables examined consisted of plate thickness, and the effects of geometric dimensions, boundary conditions, applied force, waviness index, aspect ratio, volume fraction, and the distribution pattern of CNTs. All of these variables were investigated for their effects on the deflection, stress distributions, and natural frequencies of FGCNTRC plates.
In this paper, FGCNTRC plates based on the Pasternak elastic foundation were considered with length a, width b, and thickness h (Figure 1). The FGCNTRC plates were made from a mixture of wavy SWCNTs in an isotropic matrix. The wavy SWCNT reinforcement were either uniformly distributed (UD) or functionally graded (FG) within the plate thickness. To obtain mechanical properties of CNT/polymer composites, a new rule of mixture equation assumed that the fibers were wavy and had uniform dispersion within the polymer matrix. This equation cannot consider the length of fibers, so it can be modified by incorporating an efficiency parameter ( ) to account for the nanotube aspect ratio (AR) and waviness (w) [19]. The effective mechanical properties of the CNTRC plates were obtained based on a micromechanical model according to [6]
(1) 

(2) 

(3) 

(4) 

(5) 
where
(6) 

, 
(7) 
and where , , , , and are the elasticity modulus, shear modulus, Poisson's ratio, effective reinforcement modulus, the average number of contacts per particle, and density, respectively, of the carbon nanotubes. , , and are corresponding properties for the matrix. and are the fiber (CNT) and matrix volume fractions, and they are related by . The equation showed the CNT efficiency parameters, and the efficiency parameters can be computed by matching the elastic modulus of the CNTRCs observed during the molecular dynamic MD simulation results with the numerical results obtained from the new rule of mixture in Eqs. (1)–(5).
The average number of contacts, , for the nanotubes is dependent on their aspect ratio [19]
(8) 
where the waviness index w has been introduced to account for the CNTs’ curvature within the real composite. According to the literature [19], the variation of the excluded volume due to nanotubes curvature was investigated by introducing the waviness parameter w. The accuracy of this method can be demonstrated through comparison with available literature data.
The profile of the fiber volume fraction variation has important effects on plate behaviors. In this paper, three linear types (FGV, FG, and FGX) were assumed for the distribution of CNT reinforcements along the thickness in FGCNTRC plates. In addition, a UD of CNTs within the nanocomposite plate of the same thickness, referred to as UDCNTRCs, were considered as a comparator.
These distributions along the plate’s thickness were presented as follows (see Figure 2):
For type V : 
(9) 
For type X: 
(10) 
For type O : 
(11) 
For UD: 
(12) 
Figure 1. Schematic of the plate resting on a twoparameter elastic foundation.
Figure 2. Variation of the nanotubes’ volume fraction along the thickness of the plate for different CNT distributions.
where
(13) 
and is the mass fraction of the nanotubes.
Based on the FSDT, the displacement components can be defined as
(14) 
where u, v and w are displacements in the x, y, z directions, respectively. The variables u_{0}, v_{0} and w_{0} denote midplane displacements, while and represent rotations of normal to the midplane about the yaxis and xaxis, respectively. The kinematic relations can be obtained as follows:
(15) 
where
(16) 
The linear constitutive relations of a FG plate can be written as

(17) 
in which denotes the transverse shear correction coefficient, which is suggested as for homogeneous materials. For FGMs, the shear correction coefficient is taken to be
[24].
also where
(18) 
In addition, by considering the Pasternak foundation model, total energy of the plate is as

(19) 
where f is the applied load, and k_{w} and k_{s} are coefficients of the Winkler and the Pasternak foundations, respectively. If the foundation was modeled as the linear Winkler foundation, the coefficient k_{s} in Eq. (19) is zero.
In these analyses, moving least square shape functions introduced by Lancaster and Salkauskas [25] were used for approximation of the displacement vector in the weak form of the motion equation. Displacement vector d can be approximated by the MLS shape functions as follows:
(20) 
where N is the total number of nodes; is the virtual nodal values vector, and is the MLS shape function of the node located at X(x,y) = X_{i}, and they are defined as follows:
(21) 
and
(22) 
Where is the cubic Spline weight function, is the base vector, and is the moment matrix. The vector and moment matrix can be defined as follows:
(23) 

(24) 
By using the MLS shape function, Eq. (15) can be written as
(25) 
in which
(26) 
In addition, for the elastic foundation, and can be defined as
(27) 
Substitution of Eqs. (17) and (25) into Eq. (19) leads to
(28) 
in which the components of the extensional stiffness , bendingextensional coupling stiffness , bending stiffness , transverse shear stiffness and also and , are introduced into the mass matrix. They are defined as
(29) 
and
(30) 
where , , and are the normal, coupled normalrotary and rotary inertial coefficients, respectively, defined by
(31) 
The arrays of the bendingextensional coupling stiffness matrix, , are zero for symmetric laminated composites.
Finally, using a derivative with respect to the displacement vector, , the Eq. (28) can be expressed as
(32) 
in which, , , and are the mass matrix, stiffness matrix, and force vector, respectively, which are defined as
(33) 

(34) 

(35) 
in which, , , and are the stiffness matrices of the extensional, bendingextensional, and bending, respectively, while and are the stiffness matrices that represented the Winkler and Pasternak elastic foundations. They are defined as
(36) 

(37) 
For numerical integration, the problem domain is discretized to a set of background cells with gauss points inside each cell. Then, the global stiffness matrix is obtained numerically by sweeping all gauss points.
Imposition of essential boundary conditions in the system of Eq. (32) was not possible because MLS shape functions do not satisfy the Kronecker delta property. In this work, the transformation method is used for imposition of essential boundary conditions. For this purpose, a transformation matrix is formed by establishing the relationship between nodal displacement vector and virtual displacement vector .
(38) 
is the transformation matrix that is a (5N × 5N) matrix defined as
(39) 
where is an identity matrix of size 5. By using Eq. (38), the system of linear Eq. (32) can be rearranged to
(40) 
where
(41) 
Now the essential BCs. can be enforced within the modified equations system in Eq. (40) easily, like the FEM.
For a static problem, the mass matrix is eliminated, and Eq. (40) is changed to
(42) 
Therefore, the stress and displacement fields of the plate can be derived, solving this equation system. Additionally, in the absence of external forces, Eq. (40) is simplified as follows:
(43) 
Thus, the natural frequencies and mode shapes of the plate are determined by solving this eigenvalue problem.
Table 1. Comparisons of Young's moduli for polymer/CNT composites reinforced by (10,10) SWCNT at T_{0} = 300 K [7].
MD 

Extended rule of mixture 

E_{1} (GPa) 
E_{2} (GPa) 

E_{1} (GPa) 
η_{1} 
E_{2} (GPa) 
η_{2} 

0.12 
94.6 
2.9 

94.78 
0.137 
2.9 
1.022 
0.17 
138.9 
4.9 

138.68 
0.142 
4.9 
1.626 
0.28 
224.2 
5.5 

224.50 
0.141 
5.5 
1.585 
In the following simulations, static and vibration behaviors of the nanocomposite plates are characterized as FG plates reinforced by wavy CNTs. The polymer (methylmethacrylate), referred to as PMMA, was selected as the matrix material. The relevant material properties for CNTs and PMMA are as follows [7]:
, and
for PMMA. For (10,10) SWCNTs
, , , and
and the material properties of the nanocomposite are derived from Eqs. (1)–(6) with respect to [26] and the values in Table 1 [14]. The accuracy of this method was investigated by comparison with experimental results [2022].
In this work, the static and free vibration analyses were presented to investigate the mechanical characteristics of FGCNTRC plates using several numerical examples. The plates were assumed to be resting on twoparameter elastic foundations, and the developed meshfree method was used. At first, the convergence and accuracy of the meshfree method on the static and vibrational behaviours of the plates were examined by a comparison between the results and reported results in the literature concerning homogeneous FGMs and straight CNTRC plates. Then, new meshfree results on the static and free vibration characteristics of the wavy CNTRC plates on the elastic foundation were reported.
In all examples of CNTRC plates, the foundation parameters were presented in the nondimensional form of K_{w }= k_{w}a^{4}/D and K_{s}=k_{s}a^{2}/D, in which D = E_{m}h^{3}/12(1υ_{m}^{2}) was the reference bending rigidity of the plate. In addition, the nondimensional deflections and the natural frequencies of the CNTRC plates are defined as [27]
(44) 

(45) 
where, f_{0} is the value of applied (concentrated or uniform) load, and q is the central deflection.
5.1. Validation of models
In the first example, consider a simply supported homogeneous square plate under uniformly distributed load f_{0}. The convergence of the developed meshfree method in a central nondimensional deflection of the plate with h/ac= 0.02, is shown in Figure 3. The applied meshfree method has an excellent convergence and agreement with the exact results reported by Akhras et al. [28] in the bending analysis of the plate. The deflections of this plate for various values of h/a (=0.1, 0.05, 0.02 and 0.01) are listed in Table 2. The accuracy of the applied method was evident by comparison with exact [28] and other reported results[27,29,30].Figure 3 and Table 2 results show that by using an 11 × 11 node arrangement, the applied method provided more accuracy than FEM.
The bending analysis of the FG nanocomposite plate reinforced by straight CNT and without an elastic foundation was also validated. Consider a square clamped FGCNTRC plate with a CNT volume fraction , and values of b/h at 50, 20, and 10. Table 3 shows good agreement between the results of the applied meshfree method, and the FEM reported results by Zhu et al. [8] for both UD and FG nanocomposite reinforced by straight CNTs.
Figure 3. Convergence of the central nondimensional deflection , for different numbers of nodes in each direction (data comparision with Akhares et al. [28]).
Table 2. Comparison of the central nondimensional deflection in simply supported square plates subjected to uniformly distributed load.
Nondimensional Deflection 
Method 
h/a 
4.7864 
Present 
0.1 
4.7910 
Exact [28] 

4.7866 
Ferreira et al. [29] 

4.7912 
Ferreira et al. [27] 

4.7700 
FEM (Reddy [30]) 

4.6274 
Present 
0.05 
4.6250 
Exact [28] 

4.6132 
Ferreira et al. [29] 

4.6254 
Ferreira et al. [27] 

4.5700 
FEM (Reddy [30]) 

4.5829 
Present 
0.02 
4.5790 
Exact [28] 

4.5753 
Ferreira et al. [29] 

4.5788 
Ferreira et al. [27] 

4.4960 
FEM (Reddy [30]) 

4.5765 
Present 
0.01 
4.5720 
Exact [28] 

4.5737 
Ferreira et al. [29] 

4.5716 
Ferreira et al. [27] 

4.4820 
FEM (Reddy [30]) 
Table 3. Comparison of the central nondimensional deflection , in clamped square plates reinforced by straight CNTs.
b/h 
UD 

FGX 

MeshFree (35×35) 
Zhu et al. [8] 
MeshFree (35×35) 
Zhu et al. [8] 

50 
0.1690 
0.1698 
0.1213 
0.1223 
20 
8.309×10^{2} 
8.561×10^{2} 
7.039×10^{3} 
7.290×10^{3} 
10 
1.353×10^{3} 
1.412×10^{3} 
1.261×10^{3} 
1.318×10^{3} 
In the second stage of validation, consider a simply supported FGM square plate as reported in Thai and Choi [4] in which the material properties of plate are varied as follows:
(46) 
where P is an indicator for the material properties of a plate that were used in place of the modulus elasticity, E, Poisson's ratio, , and density, . Additionally, n is the volume fraction exponent, and the subscripts b and t represent the bottom and top constituents, respectively. The convergence of the applied meshfree method in a nondimensional fundamental frequency , for plates resting on a WinklerPasternak elastic foundation with h/a = 0.2, K_{w }= 100, K_{s }= 100, and a volume fraction exponent of n = 1, are shown in Figure 4. This figure also shows that by using only a 5 × 5 node arrangement, the applied method had very good accuracy and agreement with the results reported by Thai and Choi [4] for FGM (n = 1) plates. The nondimensional fundamental frequencies for this plate are presented in Table 4, using various values of h/a that equal 0.05, 0.1, and 0.2, and including elastic foundation coefficients. This table reveals that the applied method had very good accuracy and agreement with the reported results, especially for thinner plates.
Figure 4. Convergence of the nondimensional fundamental frequency , for the FGM plates with n=1, h/a=0.2, K_{w}=100, K_{s}=100 for different numbers of nodes in each direction (data comparison with Thai and Choi [4]).
Table 4. Comparison of the fundamental frequency, , in simply supported square FGM plates.
K_{w} 
K_{s} 
h/a 
Method 
n=0 
n=1 
0 
0 
0.05 
Present 
0.0291 
0.0222 
Baferani et al. [31] 
0.0291 
0.0227 

Thai & Choi [4] 
0.0291 
0.0222 

0.1 
Present 
0.1135 
0.0869 

Baferani et al. [31] 
0.1134 
0.0891 

Thai & Choi [4] 
0.1135 
0.0869 

0.2 
Present 
0.4167 
0.3216 

Baferani et al. [31] 
0.4154 
0.3299 

Thai & Choi [4] 
0.4154 
0.3207 

100 
100 
0.05 
Present 
0.0411 
0.0384 
Baferani et al. [31] 
0.0411 
0.0388 

Thai & Choi [4] 
0.0411 
0.0384 

0.1 
Present 
0.1618 
0.1519 

Baferani et al. [31] 
0.1619 
0.1542 

Thai & Choi [4] 
0.1619 
0.1520 

0.2 
Present 
0.6167 
0.5857 

Baferani et al. [31] 
0.6162 
0.5978 

Thai & Choi [4] 
0.6162 
0.5855 
Table 5. Comparison of the fundamental frequency , in simply supported square plates reinforced by straight CNTs
b/h 
UD 
FGX 

MeshFree (31×31) 
FEM (31×31) 
Zhu et al. [8] 
MeshFree (31×31) 
FEM (31×31) 
Zhu et al. [8] 

10 
17.0010 
17.0189 
16.815 
18.5240 
18.5382 
18.278 
20 
21.5053 
21.541 
21.456 
24.8639 
24.8973 
24.764 
50 
23.6323 
23.6791 
23.697 
28.3400 
28.3891 
28.413 
Finally, consider square simply supported FGCNTRC plates with a CNT volume fraction , and values of b/h equal to 50, 20, and 10. Table 5 shows a good agreement between the nondimensional fundamental frequency of the applied meshfree method, FEM, and FEM reported results by Zhu et al. [8], for both UD and FG nanocomposites reinforced by straight CNTs.
5.2. Static analysis of FGCNTRC plates
Consider square clamped nanocomposite plates reinforced by wavy CNTs subjected to uniformed distribution load of f_{0 }= 1e5, resting on a Pasternak foundation with the values K_{w }= 100, K_{s }= 10, and h/a = 0.1. Table 6 shows the central nondimensional deflection , for the plates for different types of CNT distributions and various values of CNT volume fraction , aspect ratio, and waviness index. The deflection parameter was decreased by increasing the CNT volume fraction, CNT aspect ratio, and by decreasing the waviness index. Table 6 reveals that the CNT waviness had the biggest effect (even more than the CNT volume fraction), and CNT aspect ratio had the smallest effect (especially at its higher values) on the bending behaviors of FGCNTRC plates. Additionally, in more cases, the FGX and FGO types of CNTRC plates have the minimum and maximum values for nondimensional deflection, respectively.
Table 6. Central nondimensional deflections , in clamped square FGCNTRC plates, with the values K_{wv }= 100, K_{sv }= 10, f_{0v }= v1e5, and h/a = 0.1
w 
AR 
UD 
FGV 
FGX 
FGO 

0.12 
0 
100 
0.5470 
0.5825 
0.5010 
0.6369 
500 
0.4984 
0.5212 
0.4726 
0.5530 

1000 
0.4949 
0.5165 
0.4705 
0.5465 

0.425 
100 
0.9536 
0.9623 
0.9470 
0.9669 

500 
0.9221 
0.9183 
0.9235 
0.9046 

1000 
0.9172 
0.9102 
0.9198 
0.8928 

0.17 
0 
100 
0.3459 
0.3691 
0.3101 
0.4103 
500 
0.3145 
0.3295 
0.2929 
0.3544 

1000 
0.3122 
0.3263 
0.2915 
0.3500 

0.425 
100 
0.6653 
0.6595 
0.6427 
0.6765 

500 
0.6603 
0.6501 
0.6402 
0.6575 

1000 
0.6592 
0.6471 
0.6396 
0.6524 

0.28 
0 
100 
0.2864 
0.2936 
0.2512 
0.3259 
500 
0.2702 
0.2713 
0.2437 
0.2916 

1000 
0.2689 
0.2695 
0.2430 
0.2889 

0.425 
100 
0.6377 
0.6219 
0.5841 
0.6595 

500 
0.6360 
0.6141 
0.5826 
0.6439 

1000 
0.6356 
0.6118 
0.5823 
0.6399 
Despite increased CNT volume fractions from to , deflection can be reduced by using a proper CNT distribution. Also, Figure 5 shows variations in the normalized deflection of the plates versus the waviness index for the FGX type of distribution of CNTs, AR = 1000, and various values of CNT volume fractions. Increasing the CNT volume decreased the deflection values for the plates with straight or wavy CNTs. In most cases, the deflection was increased with an increased waviness index.
To investigate the effect of the type of applied force on bending behaviors of these plates, consider square clamped nanocomposite plates subjected to a uniformed distribution load or a concentrated force with the same values of f_{0 }= 1e5, with h/a = 0.1, AR = 1000, and . Table 7 shows the nondimensional deflection of these plates for elastic foundation parameters of K_{w }= 0 or 100, and K_{s }= 0 or 100, and for w = 0 or 0.425. The table shows that the concentrated forces and the wavy type of CNTs dramatically increased the deflection of the plate. Additionally, in wavy nanocomposite plates, UD distribution caused the largest value in the deflection parameter. Elastic foundations decreased the deflection of all plates.
The effect of essential boundary conditions on the deflection of square and rectangular FGCNTRC plates, subjected to uniform distributed force f_{0} = 1e5, with the values K_{w }= 100, K_{s }= 10, w = 0.425, AR = 1000, and , was investigated in Table 8, where C, S, and F represented clamped, simply supported, and free edges. The table shows that the clamped plates have the smallest values of deflection, while the simply supported plates have the largest ones. Evidently, the deflection parameter was dramatically increased by increasing the ratio of a/b from 1 to 3 because the plate nearly reveals the beam’s manners. Increasing the thickness of the plate increased the nondimensional deflections , because of their definition, but the deflection was decreased by increasing the plate’s thickness.
In the next example, the elastic foundation coefficients of FGCNTRC plates, subjected to uniform distributed load, are investigated.
Figures 6 and 7 demonstrate the nondimensional deflection of FGCNTRC plates versus the Winkler coefficient K_{w}, and shear coefficient K_{s}, respectively. These plates are square and clamped using an FGX type CNT distribution, and with the values f_{0} =1e5, w = 0.425, AR = 1000, , and h/a = 0.1. These figures revealed that increased foundation coefficients decreased the plates’ deflection, but the shear coefficient had a larger effect, particularly in the smaller values of the Winkler coefficient.
Figure 5. Central normalized deflections , versus the waviness index in clamped square XCNTRC plates, with the values AR = 1000, K_{w }= 100, K_{s }=10, f_{0 }=1e5, and h/a = 0.1.
Figure 6. Central nondimensional deflections , versus K_{w} in square clamped XCNTRC plates, with the values f_{0 } = 1e5, w = 0.425, AR = 1000, , and h/a = 0.1.
Figure 7. Central nondimensional deflections , versus K_{s} in square clamped XCNTRC plates, with the values f_{0} = 1e5, w = 0.425, AR = 1000, , and h/a = 0.1.
Table 7. Central nondimensional deflections , in clamped, square FGCNTRC plates with the values AR = 1000, , and h/a = 0.1
K_{w} 
K_{s} 
w 
Uniform distributed force 

Concentrated force 

UD 
FGV 
FGX 
FGO 

UD 
FGV 
FGX 
FGO 

0 
0 
0 
0.3334 
0.3496 
0.3100 
0.3768 

3.4612 
3.4922 
3.3146 
3.6275 
0.425 
0.7595 
0.7435 
0.7335 
0.7505 

5.1815 
5.0946 
5.0442 
5.1396 

100 
100 
0 
0.2267 
0.2341 
0.2157 
0.2459 

2.5988 
2.6083 
2.5241 
2.6687 
0.425 
0.3648 
0.3611 
0.3589 
0.3626 

3.1911 
3.1607 
3.1464 
3.1730 
Table 8. Central nondimensional deflections , in CNTRC plates with K_{w }= 100, K_{s }= 10, f_{0} = 1e5, w = 0.425, AR = 1000, and
h/a 
b/a 
CCCC 

CSCF 

SSSS 

UD 
FGX 

UD 
FGX 

UD 
FGX 

0.05 
1 
0.5545 
0.5345 

0.9102 
0.8332 

1.4703 
1.4081 
3 
0.9980 
1.0477 

6.6235 
6.5770 

3.0439 
3.1287 

0.1 
1 
0.6592 
0.6396 

1.0440 
0.9706 

1.5315 
1.4710 
3 
1.1474 
1.1933 

6.6585 
6.6150 

3.1014 
3.1826 

0.2 
1 
1.0153 
0.9959 

1.4961 
1.4344 

1.7566 
1.7017 
3 
1.6705 
1.7029 

6.7670 
6.7315 

3.3130 
3.3814 
(a)

(b)

(c)

(d)

(e)

Figure 8. (a) , (b) , (c) , (d) , and (e) , square FGCNTRC plates clamped along the thickness, with the values K_{w }= 100, K_{s }= 10, f_{0} = 1e5, w = 0.425, AR = 1000, , and h/a = 0.1.
Finally, stress distributions of square nanocomposite plates clamped along the thickness of the plate are illustrated in Fig. 8 for plates with the values K_{w }= 100, K_{s }= 10, f_{0} = 1e5, w= 0.425, AR = 1000, , and h/a = 0.1. CNT distribution had a significant effect on the stress distribution, and the values of normal stresses were more than the shear stress values.
5.3. Free vibration analysis of FGCNTRC plates
Consider square clamped nanocomposite plates reinforced by wavy CNTs resting on a Pasternak foundation with the values K_{w }= 100, K_{s }= 10, and h/a = 0.1.
Table 9 shows the frequency parameter , for the plates with different types of CNT distribution, and various values of CNT volume fraction , aspect ratio, and waviness index. The frequency parameter was increased by increasing the CNT volume fraction and CNT aspect ratio, and by decreasing the waviness index. This table reveals that the CNT waviness had the largest effect (even more than the CNT volume fraction), while the CNT aspect ratio had the smallest effect, particularly in its larger values, on the frequencies of FGCNTRC plates. With the FGX and FGO CNT distribution types, the CNTRC plates showed the maximum and minimum values in the frequency parameter, respectively.
Table 9. Nondimensional fundamental frequency , in clamped square plates with the values K_{w }= 100, K_{s }= 10, and h/a = 0.1
w 
AR 
UD 
FGV 
FGX 
FGO 

0.12 
0 
100 
16.4091 
15.9619 
17.0870 
15.3134 
500 
17.1010 
16.7771 
17.5304 
16.3236 

1000 
17.1542 
16.8447 
17.5649 
16.4092 

0.425 
100 
12.6534 
12.5975 
12.6980 
12.5673 

500 
12.8625 
12.8890 
12.8559 
12.9806 

1000 
12.8962 
12.9446 
12.8811 
13.0626 

0.17 
0 
100 
20.5661 
19.9943 
21.6430 
19.0273 
500 
21.4565 
21.0409 
22.1945 
20.3333 

1000 
21.5275 
21.1309 
22.2398 
20.4479 

0.425 
100 
15.1032 
15.1681 
15.3568 
14.9845 

500 
15.1631 
15.2817 
15.3910 
15.2004 

1000 
15.1757 
15.3169 
15.3990 
15.2597 

0.28 
0 
100 
22.2456 
22.0692 
23.7155 
20.9892 
500 
22.8240 
22.8535 
24.0347 
22.0557 

1000 
22.8716 
22.9211 
24.0630 
22.1479 

0.425 
100 
15.2542 
15.4474 
15.9190 
15.0170 

500 
15.2785 
15.5516 
15.9444 
15.2017 

1000 
15.2839 
15.5810 
15.9495 
15.2491 
In the next example, the effects of the elastic foundation coefficients were investigated on the square clamped nanocomposite plates, with values of AR = 1000, , and h/a = 0.1.
Table 10. Nondimensional fundamental frequency , in clamped square plates with the values AR = 1000, , and h/a = 0.1
K_{w} 
K_{s} 
w 
UD 
FGV 
FGX 
FGO 
0 
0 
0 
20.8588 
22.4490 
21.5944 
19.7404 
0.425 
14.1890 
14.3402 
14.4291 
14.2778 

100 
0 
24.9941 
24.6572 
25.6049 
24.0851 

0.425 
19.9084 
20.0160 
20.0734 
19.9774 

100 
0 
0 
21.0740 
20.6684 
20.8023 
19.9675 
0.425 
14.5020 
14.6499 
14.7370 
14.5890 

100 
0 
25.1741 
24.8396 
25.7806 
24.2717 

0.425 
20.1328 
20.2393 
20.2960 
20.2012 
Table 10 displays the nondimensional frequency parameters of these plates for elastic foundation parameter values of K_{w}= 0 or 100, K_{s}=0 or 100, and w = 0 or 0.425. This table reveals that the elastic foundation increased the frequency parameters of the plates, but the shear coefficient had a larger effect than the Winkler coefficient.
Table 11. Nondimensional fundamental frequency , in CNTRC plates with K_{w }= 100, K _{s}= 10, w = 0.425, AR = 1000, and
h/a 
b/a 
CCCC 

CSCF 

SSSS 

FFFF 

UD 
FGX 

UD 
FGX 

UD 
FGX 

UD 
FGX 

0.05 
1 
16.7546 
17.0581 

11.1890 
11.7203 

10.1382 
10.3576 

4.4405 
4.4398 
3 
11.5422 
11.3124 

3.8505 
3.8805 

6.6867 
6.6199 

3.1906 
3.1916 

0.1 
1 
15.1757 
15.3990 

10.4140 
10.8249 

9.8516 
10.0493 

4.4217 
4.4207 
3 
10.7021 
10.5348 

3.8350 
3.8630 

6.5908 
6.5289 

3.1893 
3.1902 

0.2 
1 
11.9603 
12.0720 

8.6479 
8.8533 

9.0085 
9.1504 

4.3500 
4.3482 
3 
8.7779 
8.7176 

3.3025 
3.3217 

3.6705 
3.6903 

3.1843 
3.1850 
Finally, the effects of the essential boundary conditions on the frequencies of square and rectangular FGCNTRC plates, with the values K_{w }= 100, K_{s }= 10, w = 0.425, AR = 1000, and , were investigated and are displayed in Table 11. This table reveals that the clamped plates had the largest values for the frequency parameters, while the free plates had the smallest values. The frequency parameter values for square plates were also more than the values for the rectangular plates.
In this paper, a meshfree method, based on FSDT, was developed to analyze static and free vibration of FG nanocomposite plates, reinforced by wavy CNTs resting on a Pasternak elastic foundation. Material properties of the CNTRC plates were assumed to be graded within the thickness of the plate and estimated by an extended rule of mixture. Therefore, the effects of the aspect ratios, waviness index, distribution pattern, volume fraction of CNTs, boundary conditions, and dimensions of the plates, were investigated regarding the static and vibrational behaviors of the FGCNTRC plates.
The following results were obtained:
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