Moradi Dastjerdi, R., Payganeh, G., Rajabizadeh Mirakabad, S., Jafari Mofrad-Taheri, M. (2016). Static and Free Vibration Analyses of Functionally Graded Nano-composite Plates Reinforced by Wavy Carbon Nanotubes Resting on a Pasternak Elastic Foundation. Mechanics of Advanced Composite Structures, 3(2), 123-135. doi: 10.22075/macs.2016.474

R. Moradi Dastjerdi; G. Payganeh; S. Rajabizadeh Mirakabad; M. Jafari Mofrad-Taheri. "Static and Free Vibration Analyses of Functionally Graded Nano-composite Plates Reinforced by Wavy Carbon Nanotubes Resting on a Pasternak Elastic Foundation". Mechanics of Advanced Composite Structures, 3, 2, 2016, 123-135. doi: 10.22075/macs.2016.474

Moradi Dastjerdi, R., Payganeh, G., Rajabizadeh Mirakabad, S., Jafari Mofrad-Taheri, M. (2016). 'Static and Free Vibration Analyses of Functionally Graded Nano-composite Plates Reinforced by Wavy Carbon Nanotubes Resting on a Pasternak Elastic Foundation', Mechanics of Advanced Composite Structures, 3(2), pp. 123-135. doi: 10.22075/macs.2016.474

Moradi Dastjerdi, R., Payganeh, G., Rajabizadeh Mirakabad, S., Jafari Mofrad-Taheri, M. Static and Free Vibration Analyses of Functionally Graded Nano-composite Plates Reinforced by Wavy Carbon Nanotubes Resting on a Pasternak Elastic Foundation. Mechanics of Advanced Composite Structures, 2016; 3(2): 123-135. doi: 10.22075/macs.2016.474

Static and Free Vibration Analyses of Functionally Graded Nano-composite Plates Reinforced by Wavy Carbon Nanotubes Resting on a Pasternak Elastic Foundation

^{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

In this study, static and free vibration analyses of functionally graded (FG) nanocomposite plates, reinforced by wavy single-walled carbon nanotubes (SWCNTs) resting on a Pasternak elastic foundation, were investigated based on a mesh-free method and modified first-order 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 mesh-free 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 FG-CNTRC plates, even more than the effect of the CNT volume fraction.

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 Mofrad-Taheri ^{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: 2016-05-27

Revised: 2016-07-23

Accepted: 2016-08-26

In this study, static and free vibration analyses of functionally graded (FG) nanocomposite plates, reinforced by wavy single-walled carbon nanotubes (SWCNTs) resting on a Pasternak elastic foundation, were investigated based on a mesh-free method and modified first-order 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 mesh-free 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 FG-CNTRC plates, even more than the effect of the CNT volume fraction.

The extraordinary and outstanding characteristics of carbon nanotubes (CNTs) have broadly attracted researchers’ attention since their discovery in the 1990s [1]. Conventional fiber-reinforced 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 nano-scaled 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 one-parameter 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 widely-used to describe the mechanical behavior of structure-foundation 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 (FG-CNTRCs). Several works on FG-CNTRC 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 post-buckling and the nonlinear bending behavior of FG-CNTRC 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 thin-to-moderately thick FG-CNTRC plates, using the finite element method (FEM) based on the first-order shear deformation theory (FSDT). Centered on a three-dimensional theory of elasticity, Alibeigloo [9] discussed static analysis of FG-CNTRC plates imbedded in piezoelectric layers using three cases of CNT distribution. Malekzadeh et al. [10]studied the free vibration behaviors of quadrilateral laminated thin-to-moderately-thick FG-CNTRC plates, using the FSDT and the differential quadrature method (DQM). Alibeigloo and Liew [11] presented the bending behaviors of FG-CNTRC rectangular plates with simply supported edges, subjected to thermo-mechanical loads based on a three-dimensional theory of elasticity. Moradi-Dastjerdi et al. [12] used the Navier method and a refined plate theory to investigate the free vibration analysis of simply supported sandwiched plates with FG-CNTRC face sheets resting on a Pasternak elastic foundation. They used straight and randomly oriented CNTs in the FG-CNTRC face sheets.

In addition, some forms of mesh-free methods were used to analyze FG-CNTRC structures. For example, Moradi-Dastjerdi et al. [13-14] presented static and dynamic analyses of FG nanocomposite cylinders reinforced by straight CNTs carried out by a mesh-free method based on an MLS shape function. Additionally, they reported the effects of orientation and aggregation of CNTs on the axisymmetric natural frequencies of FG-nanocomposite 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 FG-CNTRC plates, using the element-free kp-Ritz method based on FSDT. Finally, in two related works, Zhang et al. [17, 18] used an element-free based improved moving least squares-Ritz (IMLS-Ritz) method and FSDT to study the buckling behaviors of FG-CNTRC 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 above-mentioned studies concerning FG-CNTRC 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 FG-CNTRC 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 three-dimensional 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. Moradi-Dastjerdi et al. [21, 22] also studied the effects of CNTs’ waviness and aspect ratios on the vibrational and dynamic behaviors of FG-CNTRC 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 FG-CNTRC plates subjected to in-plane loads. They employed a reproducing kernel particle method (RKPM) based on modified FSDT.

In this study, a mesh-free method, based on FSDT, was developed to consider the effects of a two-parameter 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 micro-levels. 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 mesh-free 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 mesh-free method did not increase the calculations against the element-free Galerkin (EFG) method. Four linear types of CNT distributions were considered: uniform distribution, and three kinds of FGdistributions. 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 FG-CNTRC plates.

2. Material Properties of FG-CNTRC Plates

In this paper, FG-CNTRC plates based on the Pasternak elastic foundation were considered with length a, width b, and thickness h (Figure 1). The FG-CNTRC 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 (FG-V, FG-, and FG-X) were assumed for the distribution of CNT reinforcements along the thickness in FG-CNTRC plates. In addition, a UD of CNTs within the nanocomposite plate of the same thickness, referred to as UD-CNTRCs, 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 two-parameter 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.

3. Governing Equations

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 y-axis and x-axis, 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.

4. Mesh-Free Numerical Analysis

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 , bending-extensional 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 normal-rotary and rotary inertial coefficients, respectively, defined by

(31)

The arrays of the bending-extensional 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, bending-extensional, 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

5. Results and Discussions

In the following simulations, static and vibration behaviors of the nanocomposite plates are characterized as FG plates reinforced by wavy CNTs. The polymer (methyl-methacrylate), 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 [20-22].

In this work, the static and free vibration analyses were presented to investigate the mechanical characteristics of FG-CNTRC plates using several numerical examples. The plates were assumed to be resting on two-parameter elastic foundations, and the developed mesh-free method was used. At first, the convergence and accuracy of the mesh-free 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 mesh-free 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 non-dimensional 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(1-υ_{m}^{2}) was the reference bending rigidity of the plate. In addition, the non-dimensional 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 mesh-free method in a central non-dimensional deflection of the plate with h/ac= 0.02, is shown in Figure 3. The applied mesh-free 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 FG-CNTRC 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 mesh-free 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 non-dimensional deflection , for different numbers of nodes in each direction (data comparision with Akhares et al. [28]).

Table 2. Comparison of the central non-dimensional deflection in simply supported square plates subjected to uniformly distributed load.

Non-dimensional 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 non-dimensional deflection , in clamped square plates reinforced by straight CNTs.

b/h

UD

FG-X

Mesh-Free

(35×35)

Zhu

et al. [8]

Mesh-Free

(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 mesh-free method in a non-dimensional fundamental frequency , for plates resting on a Winkler-Pasternak 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 non-dimensional 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 non-dimensional 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

FG-X

Mesh-Free

(31×31)

FEM

(31×31)

Zhu

et al. [8]

Mesh-Free

(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 FG-CNTRC 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 non-dimensional fundamental frequency of the applied mesh-free 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 FG-CNTRC 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 non-dimensional 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 FG-CNTRC plates. Additionally, in more cases, the FG-X and FG-O types of CNTRC plates have the minimum and maximum values for non-dimensional deflection, respectively.

Table 6. Central non-dimensional deflections , in clamped square FG-CNTRC plates, with the values K_{wv }= 100, K_{sv }= 10, f_{0v }= v-1e5, and h/a = 0.1

w

AR

UD

FG-V

FG-X

FG-O

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 FG-X 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 non-dimensional 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 FG-CNTRC 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 non-dimensional 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 FG-CNTRC plates, subjected to uniform distributed load, are investigated.

Figures 6 and 7 demonstrate the non-dimensional deflection of FG-CNTRC plates versus the Winkler coefficient K_{w}, and shear coefficient K_{s}, respectively. These plates are square and clamped using an FG-X 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 X-CNTRC plates, with the values AR = 1000, K_{w }= 100, K_{s }=10, f_{0 }=-1e5, and h/a = 0.1.

Figure 6. Central non-dimensional deflections , versus K_{w} in square clamped X-CNTRC plates, with the values f_{0 } = -1e5, w = 0.425, AR = 1000, , and h/a = 0.1.

Figure 7. Central non-dimensional deflections , versus K_{s} in square clamped X-CNTRC plates, with the values f_{0} = -1e5, w = 0.425, AR = 1000, , and h/a = 0.1.

Table 7. Central non-dimensional deflections , in clamped, square FG-CNTRC plates with the values AR = 1000, , and h/a = 0.1

K_{w}

K_{s}

w

Uniform distributed force

Concentrated force

UD

FG-V

FG-X

FG-O

UD

FG-V

FG-X

FG-O

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 non-dimensional 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

FG-X

UD

FG-X

UD

FG-X

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 FG-CNTRC 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 FG-CNTRC 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 FG-CNTRC plates. With the FG-X and FG-O CNT distribution types, the CNTRC plates showed the maximum and minimum values in the frequency parameter, respectively.

Table 9. Non-dimensional fundamental frequency , in clamped square plates with the values K_{w }= 100, K_{s }= 10, and h/a = 0.1

w

AR

UD

FG-V

FG-X

FG-O

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. Non-dimensional fundamental frequency , in clamped square plates with the values AR = 1000, , and h/a = 0.1

K_{w}

K_{s}

w

UD

FG-V

FG-X

FG-O

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 non-dimensional 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. Non-dimensional 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

FG-X

UD

FG-X

UD

FG-X

UD

FG-X

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 FG-CNTRC 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.

6. Conclusions

In this paper, a mesh-free 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 FG-CNTRC plates.

The following results were obtained:

The developed mesh-free method had an excellent convergence and accuracy in both the static and free vibration analysis of FG-CNTRC plates.

The waviness and distribution pattern of CNTs had a significant effect on the static and vibrational behaviors of FG-CNTRC plates, even more than the effect of the CNT volume fractions.

The aspect ratio of CNTs had a small effect on the static and vibrational behaviors of FG-CNTRC plates, particularly at its larger values.

In most CNT distributions, FG-X and FG-O showed the best and worst reinforcement behaviors, respectively.

With the same values, a concentrated force led to more deflection than a uniform distributed force.

The shear coefficient of an elastic foundation had a larger effect on the mechanical behaviors of FG-CNTRC plates than the Winkler coefficient.

The values of normal stresses increased more than the values of shear stresses.

References

[1] Iijima S, Ichihashi T. Single-shell carbon nanotubes of 1-nm diameter. Nature, 1993; 363: 603-5.

[2] Thostenson ET, Ren Z, Chou T-W. Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol, 2001; 61: 1899–912.

[3] Fiedler B, Gojny FH, Wichmann MHG, Nolte MCM, Schulte K. Fundamental aspects of nano-reinforced composites. Compos Sci Technol, 2006; 66: 3115–25.

[4] Thai HT, Choi DH. A refined plate theory for functionally graded plates resting on elastic foundation. Compos Sci Technol, 2011; 71: 1850–1858.

[5] Meguid SA, Sun Y. On the tensile and shear strength of nano-reinforced composite interfaces. Mater Des, 2004; 25: 289–96.

[7] Shen HS. Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I: Axially-loaded shells. Compos Struct, 2011; 93: 2096-108.

[8] Zhu P, Lei ZX, Liew KM. Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory, Compos Struct, 2012; 94: 1450–1460.

[9] Alibeigloo A. Static analysis of functionally graded carbon nanotube-reinforced composite plate embedded in piezoelectric layers by using theory of elasticity. Compos Struct, 2013; 95: 612–22.

[11] Alibeigloo A, Liew KM. Thermoelastic analysis of functionally graded carbon nanotube-reinforced composite plate using theory of elasticity. Compos Struct, 2013; 106: 873–881.

[12] Moradi-Dastjerdi R, Payganeh G, Malek-Mohammadi H. Free Vibration Analyses of Functionally Graded CNT Reinforced Nanocomposite Sandwich Plates Resting on Elastic Foundation. J Solid Mech, 2015; 7: 158-172.

[13] Moradi-Dastjerdi R, Foroutan M, Pourasghar A. Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method. Mater Des, 2013; 44: 256–66.

[14] Moradi-Dastjerdi R, Foroutan M, Pourasghar A, Sotoudeh-Bahreini R. Static analysis of functionally graded carbon nanotube-reinforced composite cylinders by a mesh-free method. J Reinf Plast Comp, 2013; 32: 593-601.

[15] Moradi-Dastjerdi R, Pourasghar A, Foroutan M. The effects of carbon nanotube orientation and aggregation on vibrational behavior of functionally graded nanocomposite cylinders by a mesh-free method. Acta Mech, 2013; 224: 2817-2832.

[16] Lei ZX, Liew KM, Yu JL. Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal environment. Compos Struct, 2013; 106: 128–138.

[17] Zhang LW, Lei ZX, Liew KM. An element-free IMLS-Ritz framework for buckling analysis of FG–CNT reinforced composite thick plates resting on Winkler foundations. Eng Anal Bound Elem, 2015; 58: 7–17.

[18] Zhang LW, Song ZG, Liew KM. Nonlinear bending analysis of FG-CNT reinforced composite thick plates resting on Pasternak foundations using the element-free IMLS-Ritz method. Compos Struct, 2015; 128: 165–175.

[19] Martone FG, Antonucci V, Giordano M, et al. The effect of the aspect ratio of carbon nanotubes on their effective reinforcement modulus in an epoxy matrix. Compos Sci Technol, 2011; 71: 1117–1123.

[20] Jam JE, Pourasghar A, Kamarian S. The effect of the aspect ratio and waviness of CNTs on the vibrational behavior of functionally graded nanocomposite cylindrical panels. Polymer Compos, 2012; 33: 2036-44.

[21] Moradi-Dastjerdi R, Pourasghar A, Foroutan M, Bidram M. Vibration analysis of functionally graded nanocomposite cylinders reinforced by wavy carbon nanotube based on mesh-free method. J Compos Mater, 2014; 48: 1901-13.

[22] Moradi-Dastjerdi R, Pourasghar A. Dynamic analysis of functionally graded nanocomposite cylinders reinforced by wavy carbon nanotube under an impact load. J Vib Control, 2016: 22, 1062-1075.

[23] Shams S, Soltani B. The Effects of Carbon Nanotube Waviness and Aspect Ratio on the Buckling Behavior of Functionally Graded Nanocomposite Plates Using a Meshfree Method. Polymer Compos, 2015; DOI 10.1002/pc.23814.

[24] Efraim E and Eisenberger M. Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. J Sound Vib, 2007; 299: 720–38.

[25] Lancaster P, Salkauskas K. Surface Generated by Moving Least Squares Methods. Math Comput,1981; 37: 141-58.

[26] Song YS, Youn JR. Modeling of effective elastic properties for polymer based carbon nanotube composites. Polym, 2006; 47: 1741–8.

[27] Ferreira AJM, Castro LMS, Bertoluzza S. A high order collocation method for the static and vibration analysis of composite plates using a first-order theory. Compos Struct, 2009; 89: 424–432.

[28] Akhras G, Cheung MS, Li W. Finite strip analysis for anisotropic laminated composite plates using higher-order deformation theory. Compos Struct, 1994; 52: 471–7.

[29] Ferreira AJM, Roque CMC, Martins PALS. Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method. Compos Part B, 2003; 34: 627–36.

[30] Reddy JN. Introduction to the finite element method. New York: McGraw-Hill; 1993.

[31] Baferani AH, Saidi AR, Ehteshami H. Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation. Compos Struct, 2011; 93: 1842–53.

References

[1] Iijima S, Ichihashi T. Single-shell carbon nanotubes of 1-nm diameter. Nature, 1993; 363: 603-5.

[2] Thostenson ET, Ren Z, Chou T-W. Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol, 2001; 61: 1899–912.

[3] Fiedler B, Gojny FH, Wichmann MHG, Nolte MCM, Schulte K. Fundamental aspects of nano-reinforced composites. Compos Sci Technol, 2006; 66: 3115–25.

[4] Thai HT, Choi DH. A refined plate theory for functionally graded plates resting on elastic foundation. Compos Sci Technol, 2011; 71: 1850–1858.

[5] Meguid SA, Sun Y. On the tensile and shear strength of nano-reinforced composite interfaces. Mater Des, 2004; 25: 289–96.

[7] Shen HS. Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I: Axially-loaded shells. Compos Struct, 2011; 93: 2096-108.

[8] Zhu P, Lei ZX, Liew KM. Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory, Compos Struct, 2012; 94: 1450–1460.

[9] Alibeigloo A. Static analysis of functionally graded carbon nanotube-reinforced composite plate embedded in piezoelectric layers by using theory of elasticity. Compos Struct, 2013; 95: 612–22.

[11] Alibeigloo A, Liew KM. Thermoelastic analysis of functionally graded carbon nanotube-reinforced composite plate using theory of elasticity. Compos Struct, 2013; 106: 873–881.

[12] Moradi-Dastjerdi R, Payganeh G, Malek-Mohammadi H. Free Vibration Analyses of Functionally Graded CNT Reinforced Nanocomposite Sandwich Plates Resting on Elastic Foundation. J Solid Mech, 2015; 7: 158-172.

[13] Moradi-Dastjerdi R, Foroutan M, Pourasghar A. Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method. Mater Des, 2013; 44: 256–66.

[14] Moradi-Dastjerdi R, Foroutan M, Pourasghar A, Sotoudeh-Bahreini R. Static analysis of functionally graded carbon nanotube-reinforced composite cylinders by a mesh-free method. J Reinf Plast Comp, 2013; 32: 593-601.

[15] Moradi-Dastjerdi R, Pourasghar A, Foroutan M. The effects of carbon nanotube orientation and aggregation on vibrational behavior of functionally graded nanocomposite cylinders by a mesh-free method. Acta Mech, 2013; 224: 2817-2832.

[16] Lei ZX, Liew KM, Yu JL. Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal environment. Compos Struct, 2013; 106: 128–138.

[17] Zhang LW, Lei ZX, Liew KM. An element-free IMLS-Ritz framework for buckling analysis of FG–CNT reinforced composite thick plates resting on Winkler foundations. Eng Anal Bound Elem, 2015; 58: 7–17.

[18] Zhang LW, Song ZG, Liew KM. Nonlinear bending analysis of FG-CNT reinforced composite thick plates resting on Pasternak foundations using the element-free IMLS-Ritz method. Compos Struct, 2015; 128: 165–175.

[19] Martone FG, Antonucci V, Giordano M, et al. The effect of the aspect ratio of carbon nanotubes on their effective reinforcement modulus in an epoxy matrix. Compos Sci Technol, 2011; 71: 1117–1123.

[20] Jam JE, Pourasghar A, Kamarian S. The effect of the aspect ratio and waviness of CNTs on the vibrational behavior of functionally graded nanocomposite cylindrical panels. Polymer Compos, 2012; 33: 2036-44.

[21] Moradi-Dastjerdi R, Pourasghar A, Foroutan M, Bidram M. Vibration analysis of functionally graded nanocomposite cylinders reinforced by wavy carbon nanotube based on mesh-free method. J Compos Mater, 2014; 48: 1901-13.

[22] Moradi-Dastjerdi R, Pourasghar A. Dynamic analysis of functionally graded nanocomposite cylinders reinforced by wavy carbon nanotube under an impact load. J Vib Control, 2016: 22, 1062-1075.

[23] Shams S, Soltani B. The Effects of Carbon Nanotube Waviness and Aspect Ratio on the Buckling Behavior of Functionally Graded Nanocomposite Plates Using a Meshfree Method. Polymer Compos, 2015; DOI 10.1002/pc.23814.

[24] Efraim E and Eisenberger M. Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. J Sound Vib, 2007; 299: 720–38.

[25] Lancaster P, Salkauskas K. Surface Generated by Moving Least Squares Methods. Math Comput,1981; 37: 141-58.

[26] Song YS, Youn JR. Modeling of effective elastic properties for polymer based carbon nanotube composites. Polym, 2006; 47: 1741–8.

[27] Ferreira AJM, Castro LMS, Bertoluzza S. A high order collocation method for the static and vibration analysis of composite plates using a first-order theory. Compos Struct, 2009; 89: 424–432.

[28] Akhras G, Cheung MS, Li W. Finite strip analysis for anisotropic laminated composite plates using higher-order deformation theory. Compos Struct, 1994; 52: 471–7.

[29] Ferreira AJM, Roque CMC, Martins PALS. Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method. Compos Part B, 2003; 34: 627–36.

[30] Reddy JN. Introduction to the finite element method. New York: McGraw-Hill; 1993.

[31] Baferani AH, Saidi AR, Ehteshami H. Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation. Compos Struct, 2011; 93: 1842–53.