1. Introduction
Technology’s insatiable thirst to provide materials with greater and greater strength-to-weight ratios is what has drawn attention to GPL-reinforced composites. In recent years, graphene platelet has taken the place of its rival at the top of the table of optimal reinforcements due to its larger load-transfer surface and significant advantages in the nanocomposite manufacturing process compared to another pioneering reinforcement, e.g., carbon nanotube (CNT). For instance, Rafiei et al. [
1], in their experimental work, highlighted that GPL-reinforced composites may exhibit 10 times the strength and also 1.3 times the Young’s modulus when compared to CNT-reinforced composites. On the other hand, functionally graded materials (FGMs), owing to their controllability over mechanical properties in the required directions, are evaluated as the state-of-the-art materials. Today, the combination of two concepts of GPL-reinforced composite materials and functionally graded material, namely functionally graded graphene-reinforced composite (FG-GPLRC) materials, have unanimously been considered as one of the most promising and most interesting research topics [
2].
On the basis of the first-order shear deformation theory, Song et al. [
3] presented the free and forced vibration analysis on functionally graded GPL-reinforced plates. The vibration frequencies of FG-GPLRC plates were investigated by Guo et al. [
4] by employing the element-free IMLS-Ritz method. Zhao et al. [
5] proposed an FEM-based analysis around the free vibration and bending behavior of composite trapezoidal plates made of GPLRC layers. The bending response of functionally graded reinforced graphene nanoplatelet (GNP) quadrilateral plates was obtained by Guo el al. [
6] with the aid of the element-free IMLS-Ritz method. Gholami and Ansari [
7] took the von Kármán-type nonlinearity into account to investigate the nonlinear stability and free vibration of FG-GPLRC plates subjected to compressive in-plane mechanical loads. Wu et al. [
8] presented a numerical study on the parametric instability of FG-GPLRC plates under periodic uniaxial mechanical load and a uniform thermal load via the generalized quadrature method. Reddy et al. [
9] used the finite element model on the basis of first-order shear deformation theory assumptions to probe the vibratory features of thin/moderately thick/thick composite plates made of GPL-reinforced plies. Gao et al. [
10] estimated the effective elastic modulus of the GPL-reinforced composite media with the accordance of the assumption of closed-cell cellular solids under Gaussian random field scheme to obtain the vibration frequency of functionally graded GPL-reinforced porous plates. Yang et al. [
11] employed the Chebyshev–Ritz solution method to derive buckling loads and natural frequencies of porous GPL-reinforced laminated plates modeled with FSDT assumptions. Functionally graded GPL-reinforced laminated composite plate that was undergoing in-plane excitations and electrical voltage was subjected to free vibration and nonlinear aeroelastic analysis by Lin et al. [
12] in a high-order shear deformation model. Gholami and Ansari [
13] developed a numerical analysis around the nonlinear vibration behavior of thick and moderately thick FG-GPLRC rectangular plates on the basis of assumptions of a higher-order shear deformation model. By employing Mindlin’s plate model and the phase-field approach, Torabi and Ansari [
14] studied the vibration behavior of graphene-platelet-reinforced multilayer composite plates with the consideration of stationary crack. Within a higher-order shear deformation model, the analysis of variance on the natural frequencies of composite plates made of GPL-reinforced plies was presented by Pashmforoush [
15]. Ansari et al. [
16] proposed a numerical approach on the basis of variational differential quadrature (VDQ) and the finite element method (FEM) to study the postbuckling response and free vibration of buckled FG-GPLRC plates in an HSDT model. Zhao et al. [
17] adopted the small parameter perturbation method to obtain the free/forced vibration response of rotating FG-GPLRC plates under the action of rub-impact and thermal shock. Thai and Phung-Van [
18] employed a moving Kriging (MK) using a naturally stabilized nodal integration (NSNI) within the framework of a higher-order shear deformation model to obtain the free vibration characteristics of functionally graded GPL-reinforced plates of complicated shapes. Exploiting a quasi-3D plate model, Jafari and Kiani [
19] highlighted the free vibration characteristics of thick composite plates made of functionally graded GPL-reinforced materials. Shi et al. [
20] performed static and free vibration investigation of functionally graded porous skew plates with GPL reinforcements, utilizing a three-dimensional elasticity model. Through a Ritz formulation, Kiani and Zur [
21] planned a frequency analysis on functionally graded graphene-platelet-reinforced skew plates resting on point supports. Regarding the assumptions of the first-order shear deformation theory (FSDT) and the modified couple stress theory (MCST), Abbaspour et al. [
22] formulated active control of vibration of GPL-reinforced composite micro-plates with piezoelectric face sheets.
Conducting experimental studies on nanoscale structures is not economically justified and is very difficult. Mathematical modeling is a way to overcome this issue. Molecular dynamics (MD) and continuum mechanics (CM) approaches are the most widely used types of mathematics-based modeling. Although MD modeling is much more accurate, its limitations, i.e., computationally expensive costs and time-consuming simulation, present it as a non-optimal choice for practical applications, and this is the point on which the reason and justification for more adoption of CM modeling, despite its lower accuracy, are based on.
When it comes to studying nanostructures, the limitation of classical CM models in considering size effects produces significant errors. For this reason, various size-dependent models have been released so far. Eringen’s nonlocal theory [
23] is one of the well-known and popular continuum mechanics theories that has the ability to include nano-scale effects with appropriate accuracy. Employing Eringen’s nonlocal theory can enable researchers to predict the static/dynamic behavior of a nanostructure without exploiting a large number of equations. By correlating nonlocal theory with different plate theories, such as the classical plate theory (CPT), first-order shear deformation theory (FSDT) [
24], and higher-order shear deformation theory (HSDT) [
25,
26], various nonlocal models for nanoplate analyses have been extended.
For instance, in the framework of Kirchhoff and the Mindlin plate theories, behaviors of isotropic nanoplates are probed by Lu et al. [
27] through a size-dependent nonlocal model. References [
28,
29,
30,
31] refer to more development of this model via analytical approaches. Karami et al. [
32,
33,
34] investigated the dynamic behavior of functionally graded graphene-nanoplatelet-reinforced doubly curved polymer composite nanoshells based on a nonlocal model. Wave dispersion was also discussed in detail. Furthermore, Pradhan and Phadikar [
35] dealt with free vibration analysis of nano-plates on the basis of a couple of classical plate theory (CLPT) and nonlocal FSDT models. This model was also employed in studies highlighted in [
36,
37]. Panyatong et al. [
38] developed a second-order shear deformation model to perform an analytical study on the free vibration characteristics of the functionally graded (FG) nanoplates surrounded by an elastic medium based on Eringen’s nonlocal elasticity. In the framework of a nonlocal four-variable plate model, Barati and Shahverdi [
39] used the homotopy perturbation method to present new numerical solutions of nonlinear vibration of a porous nanoplate rested on a nonlinear elastic foundation. Further, Aghababaei and Reddy [
40] obtained analytical solutions of free vibration of a simply supported nanoplate with the accordance of the assumptions of a nonlocal third-order shear deformation model. In this regard, based on a higher-order shear deformation theory of plates, Daneshmehr et al. [
41] utilized the generalized differential quadrature method (GDQM) to calculate the free vibration frequencies of nanoplates, considering small scale effects with the aid of the nonlocal model. An isogeometric-based finite element method was implemented by Natarajan et al. [
42] to compute the fundamental frequency of nanoplates made of functionally graded materials. Size dependency was considered via a nonlocal model. Cutolo et al. [
43] formulated free vibrations and buckling of a functionally graded thick nanoplate placed on a Winkler–Pasternak foundation based on third-order shear deformation theory and nonlocal elasticity formulation. Based on the assumptions of simple inverse hyperbolic shear deformation theory and nonlocal elasticity theory, Pun-Van et al. [
44] mathematically modeled the isogeometric approach on free vibrations of GPLRC. Xie et al. [
45] proposed a novel nonlocal higher-order theory to obtain accurate vibration properties of 2D functionally graded nanoplates.
As the literature survey demonstrates, the vibrational behavior of functionally graded GPL-reinforced multilayer thick nanoplates through a nonlocal quasi-3D model has not been explored so far, and this is what motivated us to plan the current research. To perform a numerical study, effective mechanical properties of GPLRC layers are estimated based on a modified Halpin–Tsai micromechanical model and the rule of mixtures. In order to obtain the effects of non-uniform shear strains through the thickness, thickness stretching effects, and size-dependent effects, governing equations are derived via a nonlocal quasi-3D model and are solved with extending a Navier solution method. Comparative studies confirm the accuracy of the results and provide the credibility to perform parametric studies. The rest of the article is allocated to parametric studies around the effects of number of layers, nonlocal parameters, length-to-thickness ratio, GPL weight fraction, and distribution pattern of GPLs.
2. Problem Statement
In this section, the basics of the GPLRC nanoplate are provided. The methods of the evaluation of the material properties are provided, and, also, the functionally graded patterns of the GPLs are introduced.
Herein, an
-layer functionally graded graphene-reinforced nanoplate with
-length,
-width, and
-height is under free vibration study. To evaluate the deformations, a right-handed coordinate system that has its origin at the corner of the plate is located in the middle surface of the plate so that the axes
, and
are through the length, width, and thickness directions.
Figure 1 provides the schematic of the plate.
The volume fraction of GPLs through the
-th layer, which is highly dependent on the scattering patterns of GPL across the thickness direction of the nanoplate, plays a key role in estimating the mechanical properties of the
-th layer. The impacts of GPLs distribution pattern on the free vibration characteristics of multilayer FG-GPLRC nanoplate are evaluated by considering four patterns of GPLs distribution, which are achieved by functionally arranging the layers reinforced with different values of GPL’s volume fraction.
Figure 2 provides the patterns.
Based on the first pattern, the volume fraction of GPLs is considered the same in all layers, besides in the cases of the non-uniform pattern; the highest volume fraction of GPLs is allocated to the outer layers, the middle layer, and the upper layer, respectively, whose mathematical expressions in terms of the total volume fraction of GPLs across the plate,
, take the following form:
For the
k-th layer, which is reinforced with randomly oriented and uniformly dispersed GPLs, the effective Young’s modulus based upon the modified Halpin–Tsai scheme can be read as follows [
4]
where
is the volume fraction of GPLs in
k-th layer. Moreover,
stands for the Young’s modulus of the polymer matrix and
and
are defined as
denotes the elasticity modulus of the GPLs, and the effects of the size and geometry of the nanoscale reinforcements are included in
and
according to the following relations
In Equation (4), , and symbolize the average length, width, and thickness of the GPLs, respectively.
On the basis of rule of mixtures, the effective mass density (
and Poisson’s ratio (
are acquired as
It is worth highlighting that the parameters related to matrix and GPLs are separated by applying subscripts and GPL.
7. Analytical Solution
In the current study, the motion equations of an FG-GPLRC nanoplate subjected to simply supported at all edges are solved exploiting Navier’s solution technique. Aiming to implement this technique, compatible with the simply supported boundary conditions and derived governing equations, the unknown displacement functions are expanded as the following formula:
In above, and and signify to the frequency of the FG-GPLRC nanoplate, which endures and as half waves through the length and width of it. Moreover, , and are the unknown coefficients that should to be determined.
Applying the above-stated expansions to the governing equations, one can obtain
where stiffness and inertia matrices are symbolized with
and
, respectively, and have the following nonzero elements:
Finally, non-trivial solution of Equation (21) will be the frequencies, and, based upon them, the responding mode shapes are achieved.