Characterization of the vibration,stability and static responses of graphene-reinforced sandwich plates under mechanical and thermal loadings using the refined shear deformation plate theory

Abstract

An analytical investigation was carried out to assess the free vibration, buckling and deformation responses of simply-supported sandwich plates. The plates constructed with graphene-reinforced polymer composite (GRPC) face sheets and are subjected to mechanical and thermal loadings while being simply-supported or resting on different types of elastic foundation. The temperature-dependent material properties of the face sheets are estimated by employing the modified Halpin-Tsai micromechanical model. The governing differential equations of the system are established based on the refined shear deformation plate theory and solved analytically using the Navier method. The validation of the formulation is carried out through comparisons of the calculated natural frequencies, thermal buckling capacities and maximum deflections of the sandwich plates with those evaluated by the available solutions in the literature. Numerical case studies are considered to examine the influences of the core to face sheet thickness ratio, temperature variation, Winkler- and Pasternak-types foundation, as well as the volume fraction of graphene on the response of the plates. It will be explicitly demonstrated that the vibration, stability and deflection responses of the sandwich plates become significantly affected by the aforementioned parameters.

Introduction

Among the different types of carbon nanomaterials, graphene and its derivatives have received considerable attention in many different fields due to their exceptional chemical and physical properties. Graphene varieties such as graphene oxide (GO), reduced-graphene oxide (rGO) and graphene nanoplatelets (GNPs) are available as super lightweight nano-sized particles with large surface area and excellent thermo-electro-mechanical properties [1–3]. several studies have investigated the effect of carbon nanotube on mechanical and thermal properties of polymer composites [4,5], as well as the static, dynamic and stability analyses of structural components formed by such nanocomposites [6–16]. A summary of the recent studies on graphene-reinforced composites is provided below.

Zhang et al. [17] investigated the bending, buckling, and vibration responses of functionally graded GO-reinforced composite beams based on the first-order shear deformation theory (FSDT). The effects of distribution and weight fraction of GO and slenderness ratio and boundary condition of the beams on their mechanical response were studied through a set of comprehensive numerical analyses. Rafiee et al. [18] developed a mathematical model for evaluating the large deflection, post-buckling and free nonlinear vibration responses of GNPs/fiber/polymer laminated multiscale composite beams. The model was developed based on the Euler-Bernoulli beam theory, which included the von Kármán geometric nonlinearity. Polit et al. [19] studied the effect of porosity distribution on the bending and stability responses of functionally graded graphene-reinforced porous nanocomposite curved beams by incorporating a higher-order theory, also accounting for the effect of thickness stretching. The analytical solution was developed using the Navier’s approach.

In other studies, the nonlinear bending and vibration, buckling and post-buckling responses of functionally graded graphene-reinforced laminated plates subject to thermal environments were studied by Shen et al. [20–23]. The governing equations were derived by accounting for the geometric nonlinearity using the von Kármán strains and incorporation of a higher-order shear deformation plate theory. A two-step perturbation approach was incorporated to determine the nonlinear response of functionally graded GRC laminated plates. Kiani [24,25] analyzed thermal post-buckling and large amplitude free vibration behaviour of clamped and simply-supported composite laminated rectangular plates reinforced with GNPs in a thermal environment. The analysis was based on the non-uniform rational B-spline-based isogeometric finite element formulation. The resulting nonlinear eigenvalue problem was solved using a displacement-controlled strategy.

More recently, Gholami and Ansari [26] examined the free vibration and pre- and post-buckled responses of multilayer functionally graded graphene platelet-reinforced polymer composite rectangular plates under compressive in-plane mechanical loading. The parabolic shear deformation plate theory, von Kármán type nonlinearity and Hamilton’s principle in conjunction with the variational differential quadrature technique were used to achieve the weak form of the nonlinear equations of motion. The influences of GNPs distribution scheme, weight fraction, plate’s geometry and boundary conditions on the response were investigated. The buckling behaviour of trapezoidal corrugated multilayer functionally graded GNPs reinforced nanocomposites thin plates subjected to various mechanical loadings, including in-plane uniform shear, uniaxial compression and a combination of both was investigated by Yang et al. [27]. The unilateral and bilateral buckling responses were evaluated using an analytical and the Ritz methods, respectively. Fan et al. [28] presented and discussed the nonlinear dynamic behaviours of functionally graded GRC laminated plates resting on viscoelastic foundation under various loading conditions. The effects of ambient temperature and the interaction between the plate and its foundation were both considered. The load-deflection relationship was obtained using a two-step perturbation technique and the fourth-order Runge-Kutta numerical method. In another notable study, Liu et al. [29] used the three-dimensional theory of elasticity to investigate the static axisymmetric and asymmetric bending and free vibration of the multilayer annular plates reinforced with GNPs. A semi-analytical method, which combined the differential quadrature method and the state-space based differential quadrature method was incorporated in the analysis. Numerical results for the bending response and natural frequencies were presented.

The developments and progress made in advanced composite materials in recent years have created significant interest in their sandwich construction [30–38]. In modern sandwich composite plates, two thin face sheets made of fiber-reinforced composites or functionally graded materials or nanoparticle-reinforced composites are used to sandwich an appropriate structural foam or honeycomb. The resulting hybrid plates yield significantly stiffer and stronger responses compared to their monolithic 2 D thin geometry counterparts in a cost-effective manner. As a result, the development of effective solutions for depicting the free vibration, stability and static responses of such plates has attracted considerable interest. On the other hand, the recent advancements made in the development of more effective and accurate shear deformation theories that do not rely on the incorporation of a shear correction factor have also opened another avenue of interest [39–45].

In summary, the ever-increasing applications of advanced materials in primary structural components in recent decades have necessitated the development of more advanced and accurate theoretical models by which one could accurately predict the response of sandwich plates under various loading conditions. Therefore, the present study is conducted based on two main objectives. The first objective is to investigate the effect of inclusion of graphene on the natural frequencies, critical buckling capacity and maximum deflection of sandwich plates formed with graphene-reinforced polymer composite (GRPC) face sheets subjected to various mechanical and thermal loads. The second objective is to construct an admissible mathematical model of the problem and develop its solution based on a new refined shear deformation theory, which does not require any shear correction factors and compare its results against those obtained by the available solutions. It is worth mentioning that, to the best of author’s knowledge, the characterization of the vibration, stability and static responses of graphene-reinforced sandwich plates under mechanical and thermal loadings using the presented refined shear deformation plate theory is one of the novelties of this research and has not been conducted elsewhere.

Problem formulation

A symmetrical sandwich plate (symmetry with respect to mid-plane) made of a homogeneous core and two similar GRPC face sheets, as illustrated in Figure 1, is considered. The sandwich plate is within a thermal environment and is simply-supported on its edges and maybe resting on a Winkler or a two-parameter Pasternak foundations (the latter combines the Winkler springs and a shear layer), or be foundationless. The length, width and total height of the plate are designated as $a$ , $b$ and $h$ , respectively. The thickness of each face sheet is assumed as $h_f$ and thickness of the core is denoted by $h_c$ . Therefore, the total thickness of the plate is ${h=h_c+2h}_f$ .

Figure 1.

Coordinate system and geometry of sandwich plates with GRPC face sheets resting on elastic foundations.

It should be mentioned that the assumptions and limitations of the present formulation are: (1) The displacements are small in comparison with the plate thickness; (2) The thickness stretching effect is not considered; (3) The transverse normal stress is negligible in comparison with the in-plane stresses; (4) The graphene reinforcement is aligned in the x-direction and is uniformly distributed through-the-thickness direction of the GRPC face sheets and (5) the present analytical solution is applicable to plates with simply-supported edges.

Materials properties of the sandwich plates

The required effective material properties of GRPC face sheets can be expressed by the modified Halpin-Tsai micromechanical model, as follows [21]

$$E_{11}={\eta }_1\frac{1\mathrm{ }+\mathrm{ }2\left(\frac{a_G}{h_G}\right){\gamma }_{11}^G{V}_G}{1-{\gamma }_{11}^G{V}_G}{E}_M$$

(1a)

$$E_{22}={\eta }_2\frac{1\mathrm{ }+\mathrm{ }2\left(\frac{b_G}{h_G}\right){\gamma }_{22}^G{V}_G}{1-{\gamma }_{22}^G{V}_G}{E}_M$$

(1b)

$$G_{12}={\eta }_3\frac{1}{1-{\gamma }_{12}^G{V}_G}{G}_M$$

(1c)

where $a_G$ , $b_G$ and $h_G$ are the length, width and the effective thickness of the graphene sheet, respectively, and

$${\gamma }_{11}^G=\frac{\frac{E_{11}^G}{E_M}-1}{\frac{E_{11}^G}{E_M}+2\left(\frac{a_G}{h_G}\right)}$$

(2a)

$${\gamma }_{22}^G=\frac{\frac{E_{22}^G}{E_M}-1}{\frac{E_{22}^G}{E_M}+2\left(\frac{b_G}{h_G}\right)}$$

(2b)

$${\gamma }_{12}^G=\frac{\frac{G_{12}^G}{G_M}-1}{\frac{G_{12}^G}{G_M}}$$

(2c)

where $E_M$ and $G_M$ refer to Young’s modulus and shear modulus of the matrix, while $E_{11}^G$ , $E_{22}^G$ and $G_{12}^G$ indicate the longitudinal and transverse Young’s moduli and shear modulus of the graphene sheet, $V_M$ and $V_G$ are the volume fractions of the matrix and the graphene, respectively, which satisfy the condition $V_M+V_G=1$ . In order to account for the effects relating to the load transfer and interaction between the graphene and polymeric phases, the graphene efficiency parameters ${\eta }_j\left(j=1\text{,}2,3\right)$ are introduced and incorporated in the original Halpin-Tsai model as presented in equation (1). The values of ${\eta }_1$ , ${\eta }_2$ and ${\eta }_3$ are tuned by comparing the results from equation (1) against the ones from the molecular dynamics simulations [46].

The material properties of the matrix and graphene sheets are considered temperature-dependent in the present article. According to the Schapery model [47], the longitudinal and transverse thermal expansion coefficients of the GRPC face sheets can be given by

$${\alpha }_{11}=\frac{V_G{E}_{11}^G{\alpha }_{11}^G+V_M{E}_M^{}{\alpha }_M^{}}{V_G{E}_{11}^G+V_M{E}_M^{}}$$

(3a)

$${\alpha }_{22}=\left(1+{\nu }_{12}^G\right)V_G{\alpha }_{22}^G+\left(1+{\nu }_M\right)V_M{\alpha }_M-{\nu }_{12}{\alpha }_{11}$$

(3b)

where ${\alpha }_{11}^G$ , ${\alpha }_{22}^G$ and ${\alpha }_M$ are the thermal expansion coefficients, ${\nu }_{12}^G$ and ${\nu }_M$ are the Poisson’s ratios of the graphene sheet and the isotropic matrix, respectively. The Poisson’s ratio and mass density of the GRPC face sheets can also be obtained using the basic rule-of-mixture as follows:

$${\nu }_{12}=V_G{\nu }_{12}^G+V_M{\nu }_M$$

(4)

$$\rho =V_G{\rho }_G+V_M{\rho }_M$$

(5)

where ${\rho }_G$ and ${\rho }_M$ represent the densities of the graphene and matrix, respectively.

The four-variable shear deformation plate model and the constitutive equations

In this study, the four-variable shear deformation plate model proposed by Zaoui et al. [42] is utilized in order to establish the kinematic relations of the plate. According to this theory, the displacement field of a sandwich plate can be represented by

$$u\left(x,y,z,t\right)=u_0\left(x,y,t\right)-z\frac{\partial w_0\left(x,y,t\right)}{\partial x}+g_{1}f\left(z\right)\int \theta \left(x,y,t\right)dx$$

(6a)

$$v\left(x,y,z,t\right)=v_0\left(x,y,t\right)-z\frac{\partial w_0\left(x,y,t\right)}{\partial y}+g_{2}f\left(z\right)\int \theta \left(x,y,t\right)dy$$

(6b)

$$w\left(x,y,z,t\right)=w_0\left(x,y,t\right)$$

(6c)

where $u_0$ , $v_0$ and $w_0$ represent the mid-plane displacement components of the sandwich plate in the orthogonal x, y and z directions, respectively, $g_1$ and $g_2$ are geometric dependent coefficients, $\theta$ represents the rotation of the mid-plane of the sandwich plate and $f\left(z\right)$ is the shear strain shape function, which is represented by a sinusoidal function in the form of

$$f\left(z\right)=\sin \left(\frac{\pi z}{h}\right)$$

(7)

It can be seen that the kinematic relations of the sandwich plate presented in equation (6) include four unknowns ( $u_0$ , $v_0$ , $w_0$ and $\theta$ ). For the aforementioned sandwich plate, the linear strain-displacement relationships derived from the kinematic equations (6a)–(6c), within the confines of the linear, small-strain elasticity theory, can be expressed as

$${\epsilon }_x=\frac{{\partial u}_0\left(x,y,t\right)}{\partial x}-z\frac{{\partial }^2{w}_0\left(x,y,t\right)}{\partial x^2}+g_{1}f\left(z\right)\theta \left(x,y,t\right)$$

(8a)

$${\epsilon }_y=\frac{{\partial v}_0\left(x,y,t\right)}{\partial y}-z\frac{{\partial }^2{w}_0\left(x,y,t\right)}{\partial y^2}+g_{2}f\left(z\right)\theta \left(x,y,t\right)$$

(8b)

$${\gamma }_{xy}=\frac{{\partial u}_0\left(x,y,t\right)}{\partial y}+\frac{{\partial v}_0\left(x,y,t\right)}{\partial x}-2z\frac{{\partial }^2{w}_0\left(x,y,t\right)}{\partial x\partial y}+g_{1}f\left(z\right)\frac{\partial }{\partial y}\int \theta \left(x,y,t\right)dx$$

$$\mathrm{ }+g_{2}f\left(z\right)\frac{\partial }{\partial x}\int \theta \left(x,y,t\right)dy$$

(8c)

$${\gamma }_{xz}=g_1\left(\frac{\partial }{\partial z}f\left(z\right)\right)\int \theta \left(x,y,t\right)dx$$

(8d)

$${\gamma }_{yz}=g_2\left(\frac{\partial }{\partial z}f\left(z\right)\right)\int \theta \left(x,y,t\right)dy$$

(8e)

The integral terms used in the above equations may be resolved by a Navier-type method and can be expressed as follows [40]

$$\frac{\partial }{\partial y}\int \theta \left(x,y,t\right)dx=A^{\mathrm{\prime }}\frac{{\partial }^2\theta \left(x,y,t\right)}{\partial x\partial y}$$

(9a)

$$\frac{\partial }{\partial x}\int \theta \left(x,y,t\right)dy=B^{\mathrm{\prime }}\frac{{\partial }^2\theta \left(x,y,t\right)}{\partial x\partial y}$$

(9b)

$$\int \theta \left(x,y,t\right)dx=A^{\mathrm{\prime }}\frac{\partial \theta \left(x,y,t\right)}{\partial x}$$

(9c)

$$\int \theta \left(x,y,t\right)dy=B^{\mathrm{\prime }}\frac{\partial \theta \left(x,y,t\right)}{\partial y}$$

(9d)

in which the coefficients $A^{\prime }$ and $B^{\prime }$ are defined according to the type of solution used, which in this case would be for Navier’s method. Therefore, $A^{\prime }$ , $B^{\prime }$ , $g_1$ , and $g_2$ are expressed as follows

$$A^{\mathrm{\prime }}=-\frac{1}{{\lambda }^2},\mathrm{ }{B}^{\mathrm{\prime }}=-\frac{1}{{\mu }^2},\mathrm{ }{g}_1={\lambda }^2,\mathrm{ }{g}_2={\mu }^2$$

(10)

where $\lambda =m\pi /a$ , $\mu =n\pi /b$ presents the eigen frequency, $m$ and $n$ are the half-wave numbers. Substitution of equation (9) in equation (8) would yield the following linear strain expressions

$$\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\end{array}\right\}=\left\{\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right\}+z\left\{\begin{array}{c}{k}_x^b\\ k_y^b\\ k_{xy}^b\end{array}\right\}+f\left(z\right)\left\{\begin{array}{c}{k}_x^s\\ k_y^s\\ k_{xy}^s\end{array}\right\},\mathrm{ }\left\{\begin{array}{c}{\gamma }_{yz}\\ {\gamma }_{xz}\end{array}\right\}=\frac{df\left(z\right)}{dz}\left\{\begin{array}{c}{\gamma }_{yz}^s\\ {\gamma }_{xz}^s\end{array}\right\}$$

(11)

where

$$\left\{\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right\}=\left\{\begin{array}{c}\frac{{\partial u}_0\left(x,y,t\right)}{\partial x}\\ \frac{{\partial v}_0\left(x,y,t\right)}{\partial y}\\ \frac{{\partial u}_0\left(x,y,t\right)}{\partial y}+\frac{{\partial v}_0\left(x,y,t\right)}{\partial x}\end{array}\right\},\mathrm{ }\left\{\begin{array}{c}{k}_x^b\\ k_y^b\\ k_{xy}^b\end{array}\right\}=\left\{\begin{array}{c}-\frac{{\partial }^2{w}_0\left(x,y,t\right)}{\partial x^2}\\ -\frac{{\partial }^2{w}_0\left(x,y,t\right)}{\partial y^2}\\ -2\frac{{\partial }^2{w}_0\left(x,y,t\right)}{\partial x\partial y}\end{array}\right\},$$

$$\mathrm{ }\left\{\begin{array}{c}{k}_x^s\\ k_y^s\\ k_{xy}^s\end{array}\right\}=\left\{\begin{array}{c}{g}_1{A}^{\mathrm{\prime }}\frac{{\partial }^2\theta \left(x,y,t\right)}{\partial x^2}\\ g_2{B}^{\mathrm{\prime }}\frac{{\partial }^2\theta \left(x,y,t\right)}{\partial y^2}\\ \left(g_1{A}^{\mathrm{\prime }}+g_2{B}^{\mathrm{\prime }}\right)\frac{{\partial }^2\theta \left(x,y,t\right)}{\partial x\partial y}\end{array}\right\},\mathrm{ }\left\{\begin{array}{c}{\gamma }_{yz}^s\\ {\gamma }_{xz}^s\end{array}\right\}=\left\{\begin{array}{c}{g}_2{B}^{\mathrm{\prime }}\frac{\partial \theta \left(x,y,t\right)}{\partial y}\\ g_1{A}^{\mathrm{\prime }}\frac{\partial \theta \left(x,y,t\right)}{\partial x}\end{array}\right\}$$

(12)

For linear thermoelastic materials, the stress field is defined as a linear function of the mechanical and thermal strain fields

$$\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ \begin{array}{c}{\tau }_{yz}\\ {\tau }_{xz}\\ {\tau }_{xy}\end{array}\end{array}\right\}=\left[\begin{array}{cc}\begin{array}{ccc}{Q}_{11}& Q_{12}& 0\\ Q_{21}& Q_{22}& 0\\ \begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}{Q}_{44}\\ 0\\ 0\end{array}\end{array}& \begin{array}{cc}0& 0\\ 0& 0\\ \begin{array}{c}0\\ Q_{55}\\ 0\end{array}& \begin{array}{c}0\\ 0\\ Q_{66}\end{array}\end{array}\end{array}\right]\left(\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ \begin{array}{c}{\gamma }_{yz}\\ {\gamma }_{xz}\\ {\gamma }_{xy}\end{array}\end{array}\right\}-\left\{\begin{array}{c}{\alpha }_{11}\mathrm{\Delta }T\\ {\alpha }_{22}\mathrm{\Delta }T\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\right\}\right)$$

(13)

In the above equation, $\mathrm{\Delta }T=T-T_0$ is the temperature differential, where $T$ and $T_0$ are the final and reference temperatures, respectively. Moreover, $Q_{ij}\left(i,j=12\text{,}456\right)$ are the reduced material stiffness coefficients compatible with the plane-stress conditions and are defined as follows [33]

$$\begin{array}{l}{Q}_{11}=\frac{E_{11}}{1-{\nu }_{12}{\nu }_{21}},\mathrm{\hspace{1em}}{Q}_{22}=\frac{E_{22}}{1-{\nu }_{12}{\nu }_{21}},\mathrm{\hspace{1em}}{Q}_{12}=\frac{{{\nu }_{21}E}_{22}}{1-{\nu }_{12}{\nu }_{21}},\mathrm{ }\\ Q_{44}=G_{23},\mathrm{\hspace{1em}}{Q}_{55}=G_{13},\mathrm{\hspace{1em}}{Q}_{66}=G_{12}\end{array}$$

(14)

The equations of motion of the aforementioned sandwich plate under thermo-mechanical loadings derived on the basis of the stationary potential energy [48] are mathematically represented as

$$\delta u_0:\frac{\partial N_x}{\partial x}+\frac{\partial N_{xy}}{\partial y}=I_0{\ddot{u}}_0-I_1\frac{\partial {\ddot{w}}_0}{\partial x}+J_0{g}_1{A}^{\mathrm{\prime }}\frac{\partial \ddot{\theta }}{\partial x}$$

(15a)

$$\delta v_0:\frac{\partial N_{xy}}{\partial x}+\frac{\partial N_y}{\partial y}=I_0{\ddot{v}}_0-I_1\frac{\partial {\ddot{w}}_0}{\partial y}+J_0{g}_2{B}^{\mathrm{\prime }}\frac{\partial \ddot{\theta }}{\partial y}$$

(15b)

$$\delta w_0:\frac{{\partial }^2{M}_x^b}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}^b}{\partial x\partial y}+\frac{{\partial }^2{M}_y^b}{\partial y^2}+q\left(x,y\right)-K_W{w}_0+K_s{{\nabla }^{2}w}_0+N_x^0\frac{{\partial }^2{w}_0}{\partial x^2}$$

$$\mathrm{ }+N_y^0\frac{{\partial }^2{w}_0}{\partial y^2}+2N_{xy}^0\frac{{\partial }^2{w}_0}{\partial x\partial y}$$

$$\mathrm{ }=I_0{\ddot{w}}_0+I_1\left(\frac{\partial {\ddot{u}}_0}{\partial x}+\frac{\partial {\ddot{v}}_0}{\partial y}\right)-I_2{{\nabla }^2\ddot{w}}_0+J_1\left(g_1{A}^{\mathrm{\prime }}\frac{{\partial }^2\ddot{\theta }}{\partial x^2}+g_2{B}^{\mathrm{\prime }}\frac{{\partial }^2\ddot{\theta }}{\partial y^2}\right)$$

(15c)

$$\delta \theta :-g_1{A}^{\mathrm{\prime }}\frac{{\partial }^2{M}_x^s}{\partial x^2}-\left(g_1{A}^{\mathrm{\prime }}+g_2{B}^{\mathrm{\prime }}\right)\frac{{\partial }^2{M}_{xy}^s}{\partial x\partial y}-g_2{B}^{\mathrm{\prime }}\frac{{\partial }^2{M}_y^s}{\partial y^2}+g_1{A}^{\mathrm{\prime }}\frac{\partial S_{xz}^s}{\partial x}+g_2{B}^{\mathrm{\prime }}\frac{\partial S_{yz}^s}{\partial y}$$

$$\mathrm{ }=-J_0\left(g_1{A}^{\mathrm{\prime }}\frac{\partial {\ddot{u}}_0}{\partial x}+g_2{B}^{\mathrm{\prime }}\frac{\partial {\ddot{v}}_0}{\partial y}\right)+J_1\left(g_1{A}^{\mathrm{\prime }}\frac{{\partial }^2{\ddot{w}}_0}{\partial x^2}+g_2{B}^{\mathrm{\prime }}\frac{{\partial }^2{\ddot{w}}_0}{\partial y^2}\right)$$

$$\mathrm{ }-J_2\left({\left(g_1{A}^{\mathrm{\prime }}\right)}^2\frac{{\partial }^2\ddot{\theta }}{\partial x^2}+{\left(g_2{B}^{\mathrm{\prime }}\right)}^2\frac{{\partial }^2\ddot{\theta }}{\partial y^2}\right)$$

(15d)

in which $I_0$ , $I_1$ , $I_2$ , $J_0$ , $J_1$ and $J_2$ are the mass moments of inertia defined by

$${(I}_0,I_1,I_2,J_0,J_1,J_2)={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\left(1,z,f\left(z\right),z^2,zf\left(z\right),f^2\left(z\right)\right)\rho dz$$

(16)

The terms $K_W$ and $K_s$ denote the stiffnesses of the normal (Winkler) springs and the shear layer, respectively (see Figure 1). The force and moment components to be used in equation (15) can be obtained by incorporating the following constitutive relations

$$\begin{array}{l}\left\{\begin{array}{ccc}{N}_x& N_y& N_{xy}\\ M_x^b& M_y^b& M_{xy}^b\\ M_x^s& M_y^s& M_{xy}^s\end{array}\right\}={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\left({\sigma }_x,{\sigma }_y,{\tau }_{xy}\right)\left\{\begin{array}{c}1\\ z\\ f\left(z\right)\end{array}\right\}dz,\mathrm{ }\\ \left(S_{xz}^s,S_{yz}^s\right)={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\left({\tau }_{xz},{\tau }_{yz}\right)\frac{df\left(z\right)}{dz}dz\end{array}$$

(17)

By substituting equation (11) in equation (13) and the obtained results into equation (17) one would obtain the resultant forces and moments in the following matrix form

$$\begin{array}{l}\left\{\begin{array}{c}{N}_x\\ N_y\\ \begin{array}{c}{N}_{xy}\\ M_x^b\\ \begin{array}{c}{M}_y^b\\ M_{xy}^b\\ \begin{array}{c}{M}_x^s\\ M_y^s\\ M_{xy}^s\end{array}\end{array}\end{array}\end{array}\right\}=\end{array}\left[\begin{array}{ccccccccc}{A}_{11}& A_{12}& 0& B_{11}& B_{12}& 0& B_{11}^s& B_{12}^s& 0\\ A_{12}& A_{22}& 0& B_{12}& B_{22}& 0& B_{12}^s& B_{22}^s& 0\\ 0& 0& A_{66}& 0& 0& B_{66}& 0& 0& B_{66}^s\\ B_{11}& B_{12}& 0& D_{11}& D_{12}& 0& D_{11}^s& D_{12}^s& 0\\ B_{12}& B_{22}& 0& D_{12}& D_{22}& 0& D_{12}^s& D_{22}^s& 0\\ 0& 0& B_{66}& 0& 0& D_{66}& 0& 0& D_{66}^s\\ B_{11}^s& B_{12}^s& 0& D_{11}^s& D_{12}^s& 0& H_{11}^s& H_{12}^s& 0\\ B_{12}^s& B_{22}^s& 0& D_{12}^s& D_{66}^s& 0& H_{12}^s& H_{22}^s& 0\\ 0& 0& B_{66}^s& 0& 0& D_{66}^s& 0& 0& H_{66}^s\end{array}\right]$$

$$\begin{array}{l}\times \left\{\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ \begin{array}{c}{\gamma }_{xy}^0\\ k_x^b\\ \begin{array}{c}{k}_y^b\\ k_{xy}^b\\ \begin{array}{c}{k}_x^s\\ k_y^s\\ k_{xy}^s\end{array}\end{array}\end{array}\end{array}\right\}-\left\{\begin{array}{c}{N}_x^T\\ N_y^T\\ \begin{array}{c}0\\ M_x^{bT}\\ \begin{array}{c}{M}_y^{bT}\\ 0\\ \begin{array}{c}{M}_x^{sT}\\ M_y^{sT}\\ 0\end{array}\end{array}\end{array}\end{array}\right\}\end{array}$$

$$\mathrm{ }\left\{\begin{array}{c}{S}_{yz}^s\\ S_{xz}^s\end{array}\right\}=\left[\begin{array}{cc}{A}_{44}^s& 0\\ 0& A_{55}^s\end{array}\right]\mathrm{ }\left\{\begin{array}{c}{\gamma }_{yz}^s\\ {\gamma }_{xz}^s\end{array}\right\}$$

(18)

where $A_{ij}$ , $B_{ij}$ , $D_{ij}$ and $H_{ij}$ are the equivalent sandwich plate’s stiffness, defined by

$$\left\{\begin{array}{cc}\begin{array}{c}{A}_{11}\\ A_{22}\\ \begin{array}{c}{A}_{12}\\ A_{66}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}{B}_{11}\\ B_{22}\\ \begin{array}{c}{B}_{12}\\ B_{66}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}{D}_{11}\\ D_{22}\\ \begin{array}{c}{D}_{12}\\ D_{66}\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}{B}_{11}^s\\ B_{22}^s\\ \begin{array}{c}{B}_{12}^s\\ B_{66}^s\end{array}\end{array}& \begin{array}{c}{D}_{11}^s\\ D_{22}^s\\ \begin{array}{c}{D}_{12}^s\\ D_{66}^s\end{array}\end{array}& \begin{array}{c}{H}_{11}^s\\ H_{22}^s\\ \begin{array}{c}{H}_{12}^s\\ H_{66}^s\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\left\{\begin{array}{c}{Q}_{11}\\ Q_{22}\\ \begin{array}{c}{Q}_{12}\\ Q_{66}\end{array}\end{array}\right\}\left(1,z,z^2,f\left(z\right),zf\left(z\right),f^2\left(z\right)\right)dz,$$

$$\mathrm{ }\left\{\begin{array}{c}{A}_{44}^s\\ A_{55}^s\end{array}\right\}={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\left\{\begin{array}{c}{Q}_{44}\\ Q_{55}\end{array}\right\}{\left(\frac{df\left(z\right)}{dz}\right)}^{2}dz$$

(19)

The resultant forces and moments due to the thermal loading ( $N_x^T=N_y^T$ , $M_x^{bT}=M_y^{bT}$ and $M_x^{sT}=M_y^{sT}$ ) are obtained by

$$\left[\begin{array}{ccc}{N}_x^T& M_x^{bT}& M_x^{sT}\\ N_y^T& M_y^{bT}& M_y^{sT}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\left[\begin{array}{cc}{Q}_{11}& Q_{12}\\ Q_{12}& Q_{22}\end{array}\right]\left[\begin{array}{c}{\alpha }_{11}\\ {\alpha }_{22}\end{array}\right]\left(\mathrm{\Delta }T\right)\left(1,z,f\left(z\right)\right)dz$$

(20)

The equations of motion represented by equation (15) can be expressed in terms of the four unknowns ( $u_0$ , $v_0$ , $w_0$ and $\theta$ ) by substituting for the resultants forces and moments from equation (18). Therefore, the governing equations of motion for a general sandwich plate resting on an elastic foundation, take the form

$$A_{11}\frac{{\partial }^2{u}_0}{\partial x^2}+A_{66}\frac{{\partial }^2{u}_0}{\partial y^2}+\left(A_{12}+A_{66}\right)\frac{{\partial }^2{v}_0}{\partial x\partial y}-B_{11}\frac{{\partial }^3{w}_0}{\partial x^3}-\left(B_{12}+{2B}_{66}\right)\frac{{\partial }^3{w}_0}{\partial x\partial y^2}$$

$$\mathrm{ }+g_1{A}^{\mathrm{\prime }}{B}_{11}^s\frac{{\partial }^3\theta }{\partial x^3}+\left(g_2{B}^{\mathrm{\prime }}{B}_{12}^s+\left(g_1{A}^{\mathrm{\prime }}+g_2{B}^{\mathrm{\prime }}\right)B_{66}^s\right)\frac{{\partial }^3\theta }{\partial x\partial y^2}$$

$$\mathrm{ }=I_0{\ddot{u}}_0-I_1\frac{\partial {\ddot{w}}_0}{\partial x}+J_0{g}_1{A}^{\mathrm{\prime }}\frac{\partial \ddot{\theta }}{\partial x}$$

(21a)

$$A_{22}\frac{{\partial }^2{v}_0}{\partial y^2}+A_{66}\frac{{\partial }^2{v}_0}{\partial x^2}+\left(A_{12}+A_{66}\right)\frac{{\partial }^2{u}_0}{\partial x\partial y}-B_{22}\frac{{\partial }^3{w}_0}{\partial y^3}-\left(B_{12}+{2B}_{66}\right)\frac{{\partial }^3{w}_0}{\partial x^2\partial y}$$

$$\mathrm{ }+g_2{B}^{\mathrm{\prime }}{B}_{22}^s\frac{{\partial }^3\theta }{\partial y^3}+\left(g_1{A}^{\mathrm{\prime }}{B}_{12}^s+\left(g_1{A}^{\mathrm{\prime }}+g_2{B}^{\mathrm{\prime }}\right)B_{66}^s\right)\frac{{\partial }^3\theta }{\partial x^2\partial y}$$

$$\mathrm{ }=I_0{\ddot{v}}_0-I_1\frac{\partial {\ddot{w}}_0}{\partial y}+J_0{g}_2{B}^{\mathrm{\prime }}\frac{\partial \ddot{\theta }}{\partial y}$$

(21 b)

$$B_{11}\frac{{\partial }^3{u}_0}{\partial x^3}+\left(B_{12}+2B_{66}\right)\left(\frac{{\partial }^3{u}_0}{\partial x\partial y^2}+\frac{{\partial }^3{v}_0}{\partial x^2\partial y}\right)+B_{22}\frac{{\partial }^3{v}_0}{\partial y^3}-D_{11}\frac{{\partial }^4{w}_0}{\partial x^4}$$

$$\mathrm{ }-2\left(D_{12}+{2D}_{66}\right)\frac{{\partial }^4{w}_0}{\partial x^2\partial y^2}-D_{22}\frac{{\partial }^4{w}_0}{\partial y^4}+g_1{A}^{\mathrm{\prime }}{D}_{11}^s\frac{{\partial }^4\theta }{\partial x^4}+g_2{B}^{\mathrm{\prime }}{D}_{22}^s\frac{{\partial }^4\theta }{\partial y^4}$$

$$\mathrm{ }+\left(g_1{A}^{\mathrm{\prime }}+g_2{B}^{\mathrm{\prime }}\right)\left(D_{12}^s+2D_{66}^s\right)\frac{{\partial }^4\theta }{\partial x^2\partial y^2}+q\left(x,y\right)-K_W{w}_0+K_s{{\nabla }^{2}w}_0+N_x^0\frac{{\partial }^2{w}_0}{\partial x^2}$$

$$\mathrm{ }+N_y^0\frac{{\partial }^2{w}_0}{\partial y^2}+2N_{xy}^0\frac{{\partial }^2{w}_0}{\partial x\partial y}$$

$$\mathrm{ }=I_0{\ddot{w}}_0+I_1\left(\frac{\partial {\ddot{u}}_0}{\partial x}+\frac{\partial {\ddot{v}}_0}{\partial y}\right)-I_2{{\nabla }^2\ddot{w}}_0+J_1\left(g_1{A}^{\mathrm{\prime }}\frac{{\partial }^2\ddot{\theta }}{\partial x^2}+g_2{B}^{\mathrm{\prime }}\frac{{\partial }^2\ddot{\theta }}{\partial y^2}\right)$$

(21c)

$$-g_1{A}^{\mathrm{\prime }}{B}_{11}^s\frac{{\partial }^3{u}_0}{\partial x^3}-\left(g_2{B}^{\mathrm{\prime }}{B}_{12}^s+\left(g_1{A}^{\mathrm{\prime }}+g_2{B}^{\mathrm{\prime }}\right)B_{66}^s\right)\frac{{\partial }^3{u}_0}{\partial x\partial y^2}$$

$$\mathrm{ }-\left(g_1{A}^{\mathrm{\prime }}{B}_{12}^s+\left(g_1{A}^{\mathrm{\prime }}+g_2{B}^{\mathrm{\prime }}\right)B_{66}^s\right)\frac{{\partial }^3{v}_0}{\partial x^2\partial y}-g_2{B}^{\mathrm{\prime }}{B}_{22}^s\frac{{\partial }^3{v}_0}{\partial y^3}+g_1{A}^{\mathrm{\prime }}{D}_{11}^s\frac{{\partial }^4{w}_0}{\partial x^4}$$

$$\mathrm{ }+\left(g_1{A}^{\mathrm{\prime }}+g_2{B}^{\mathrm{\prime }}\right)\left(D_{12}^s+2D_{66}^s\right)\frac{{\partial }^4{w}_0}{\partial x^2\partial y^2}+g_2{B}^{\mathrm{\prime }}{D}_{22}^s\frac{{\partial }^4{w}_0}{\partial y^4}-{\left(g_1{A}^{\mathrm{\prime }}\right)}^2{H}_{11}^s\frac{{\partial }^4\theta }{\partial x^4}$$

$$\mathrm{ }-2\left(g_1{A}^{\mathrm{\prime }}{H}_{12}^s-\left(g_1{A}^{\mathrm{\prime }}+g_2{B}^{\mathrm{\prime }}\right)H_{66}^s\right)\frac{{\partial }^4\theta }{\partial x^2\partial y^2}-{\left(g_2{B}^{\mathrm{\prime }}\right)}^2{H}_{22}^s\frac{{\partial }^4\theta }{\partial y^4}+{\left(g_1{A}^{\mathrm{\prime }}\right)}^2{A}_{55}^s\frac{{\partial }^2\theta }{\partial x^2}$$

$$\mathrm{ }+{\left(g_2{B}^{\mathrm{\prime }}\right)}^2{A}_{44}^s\frac{{\partial }^2\theta }{\partial y^2}$$

$$\mathrm{ }=-J_0\left(g_1{A}^{\mathrm{\prime }}\frac{\partial {\ddot{u}}_0}{\partial x}+g_2{B}^{\mathrm{\prime }}\frac{\partial {\ddot{v}}_0}{\partial y}\right)+J_1\left(g_1{A}^{\mathrm{\prime }}\frac{{\partial }^2{\ddot{w}}_0}{\partial x^2}+g_2{B}^{\mathrm{\prime }}\frac{{\partial }^2{\ddot{w}}_0}{\partial y^2}\right)$$

$$\mathrm{ }-J_2\left({\left(g_1{A}^{\mathrm{\prime }}\right)}^2\frac{{\partial }^2\ddot{\theta }}{\partial x^2}+{\left(g_2{B}^{\mathrm{\prime }}\right)}^2\frac{{\partial }^2\ddot{\theta }}{\partial y^2}\right)$$

(21d)

Solution procedure

The Navier method is implemented to formulate the closed-form solutions of equation (21) for obtaining the temperature-dependent free vibration, buckling and deflection responses of simply-supported sandwich plates with GRPC face sheets resting on elastic foundation subject to mechanical and thermal loadings. The applied simply-supported boundary conditions are of the following form

$$v_0=w_0=\theta =N_x=M_x^b=M_x^s=0\mathrm{ }\mathrm{at}\mathrm{ }x=0,a,$$

(22a)

$$u_0=w_0=\theta =N_y=M_y^b=M_y^s=0\mathrm{ }\mathrm{at}\mathrm{ }y=0,b.$$

(22 b)

Here, on the basis of the Navier method, the solution of equation (21), which automatically satisfies the boundary conditions in equation (22), can be represented by

$$\left\{\begin{array}{c}{u}_0(x,y,t)\\ v_0(x,y,t)\\ \begin{array}{c}{w}_0(x,y,t)\\ \theta (x,y,t)\end{array}\end{array}\right\}=\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}\left\{\begin{array}{c}{U}_{mn}\cos \left(\lambda x\right)\sin \left(\mu y\right)e^{i\omega t}\\ V_{mn}\sin \left(\lambda x\right)\cos \left(\mu y\right)e^{i\omega t}\\ \begin{array}{c}{W}_{mn}\sin \left(\lambda x\right)\sin \left(\mu y\right)e^{i\omega t}\\ {\mathrm{\Theta }}_{mn}\sin \left(\lambda x\right)\sin \left(\mu y\right)e^{i\omega t}\end{array}\end{array}\right\}$$

(23)

where $U_{mn}, V_{mn}, W_{mn}, {\mathrm{\Theta }}_{mn}$ are unknown coefficients. The transverse distributed load, $q\left(x,y\right)$ , can also be represented by a double-Fourier sine series as

$$q\left(x,y\right)=\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{q}_{mn}\sin \left(\lambda x\right)\sin \left(\mu y\right)$$

(24)

In which

$$q_{mn}=\frac{4}{ab}{\int }_0^a{\int }_0^{b}q\left(x,y\right)\sin \left(\lambda x\right)\sin \left(\mu y\right)d\mathit{xdy}$$

(25)

It should be pointed out that $q_{mn}$ of a plate subjected to a uniformly distributed transverse load $q_0$ can be defined as

$$q_{mn}=\frac{16q_0}{mn{\pi }^2}(m=n=1\text{,}3,5,\dots )$$

(26)

Upon substitution of the displacement functions of equation (23) into the equations of motion (equation (21)), and rearranging the terms, one can obtain the closed-form solutions through the following matrix.

$$\left({\left[K\right]}_{4\mathrm{ }\times \mathrm{ }4}-{\omega }^2{\left[M\right]}_{4\mathrm{ }\times \mathrm{ }4}\right)\left\{\begin{array}{c}{U}_{mn}\\ V_{mn}\\ \begin{array}{c}{W}_{mn}\\ {\mathrm{\Theta }}_{mn}\end{array}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ \begin{array}{c}{Q}_{mn}\\ 0\end{array}\end{array}\right\}$$

(27)

For a deflection analysis, the natural frequency ( $\omega$ ) and the buckling capacities ( $N_x^0$ , $N_y^0$ , $N_{xy}^0$ ) in equation (27) are set to zero and the resulting simultaneous equations are solved for deflections of the sandwich plates. For vibration and buckling analyses, the determinant of the coefficient matrix of equation (27) is set to zero and the non-trivial solution is obtained. Also, the following dimensionless quantities for Pasternak type foundation is introduced in the numerical results

$$k_1=K_W\frac{E_0{h}^3}{b^4},\mathrm{ }{k}_2=K_s\frac{E_0{h}^3}{b^2}$$

(28)

Numerical results and discussion

In the present paper, the natural frequencies, buckling capacity and deflection of sandwich plates with GRPC face sheets subject to various mechanical and thermal loading scenarios are investigated using the four-variable shear deformation plate model presented above. In doing so, one should first establish the effective material properties of GRPCs, which is used to form the face sheets of the sandwich plates. Poly (methyl methacrylate) thermoplastic polymer (i.e., PMMA), is adopted as the matrix, and the material properties of it are provided in Table 1, where $T=T_0+\mathrm{\Delta }T$ and $T_0=300K$ (room temperature). Titanium alloy is adopted as the homogeneous core layer, and the material properties of it are assumed to be nonlinear functions of temperature as shown in Table 1. The zigzag graphene sheets with an effective thickness of $h_G=0.188\hspace{0.17em}\mathrm{nm}$ and ${\rho }_G=4118\hspace{0.17em}\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ , $a_G=14.76\hspace{0.17em}\mathrm{n}\mathrm{m}$ , $b_G=14.77\hspace{0.17em}\mathrm{n}\mathrm{m}$ are selected as reinforcement. The graphene efficiency parameters are presented in Table 2, and it is assumed that $G_{13}=G_{23}=0.5G_{12}$ [22]. The temperature-dependent material properties of zigzag graphene sheets are extracted from reference [46]. For $300\hspace{0.17em}K\le T\le 1000\hspace{0.17em}K$ the temperature-dependent thermomechanical properties of graphene sheets such as Young’s and shear moduli and thermal expansion coefficients are defined by the following relationships

$$E_{11}^G=\left(2.16637\mathrm{ }-\mathrm{ }0.00193T+2.93701\times {10}^{-6}{T}^2-1.51775\times {10}^{-9}{T}^3\right)TPa$$

$$E_{22}^G=\left(2.16868\mathrm{ }-\mathrm{ }0.00193T+2.85954\times {10}^{-6}{T}^2-1.45145\times {10}^{-9}{T}^3\right)TPa$$

$$G_{12}^G=\left(0.53514\mathrm{ }+\mathrm{ }8.22436\times {10}^{-4}T-1.2932\times {10}^{-6}{T}^2+5.78507\times {10}^{-10}{T}^3\right)TPa$$

$${\alpha }_{11}^G=\left(-3.83788\mathrm{ }+\mathrm{ }0.01416T-1.63355\times {10}^{-5}{T}^2+6.33589\times {10}^{-9}{T}^3\right)\times {10}^{-6}/K$$

$${\alpha }_{22}^G=\left(-3.73997\mathrm{ }+\mathrm{ }0.01296T-1.35033\times {10}^{-5}{T}^2+4.60392\times {10}^{-9}{T}^3\right)\times {10}^{-6}/K$$

$${\nu }_{12}^G=0.177$$

(29)

Table 1.

Material properties of PMMA and titanium alloy.

	PMMA [24]
Density	${\rho }_M=1150\mathrm{kg}/{\mathrm{m}}^3$
Poisson’s ratios	${\nu }_M=0.34$
Thermal expansion coefficients	${\alpha }_M=45\times \left(1+0.0005\Delta T\right)\times {10}^{-6}/K$
Young’s modulus	$E_M=\left(3.52-0.0034T\right)GPa$
	Titanium alloy [53]
Density	${\rho }_C=4429kg/m^3$
Poisson’s ratios	${\nu }_C=0.29$
Thermal expansion coefficients	${\alpha }_C=7.5788\times \left(1+6.638{\times 10}^{-4}T-3.147{\times 10}^{-7}{T}^2\right)\times {10}^{-6}/K$
Young’s modulus	$E_C=122.56\times \left(1-4.586{\times 10}^{-4}T\right)GPa$

Table 2.

The efficiency parameters for different volume fraction of graphene and various thermal environment.

$T\left[K\right]$	$V_G$	Efficiency parameters
		${\eta }_1$	${\eta }_2$	${\eta }_3$
300	3%	2.929	2.855	11.842
	5%	3.068	2.962	15.944
	7%	3.013	2.966	23.575
	9%	2.647	2.609	32.816
	11%	2.311	2.260	33.125
400	3%	2.977	2.896	13.928
	5%	3.128	3.023	15.229
	7%	3.060	3.027	22.588
	9%	2.701	2.603	28.869
	11%	2.405	2.337	29.527
500	3%	3.388	3.382	16.712
	5%	3.544	3.414	16.018
	7%	3.462	3.339	23.428
	9%	3.058	2.936	29.754
	11%	2.736	2.665	30.773

In the following, the variations of different configurations of the sandwich plate (i.e., as per GRPC face sheets types) are stated:

The (1-1-1) configuration: in this configuration, the sandwich plate is made of three equal thickness layers (i.e., $h_f=h_c=1\hspace{0.17em}\mathrm{m}\mathrm{m}$ ).

The (1-2-1) configuration: in this configuration, the thickness of the core is twice of the face sheets (i.e., $h_f=1\hspace{0.17em}\mathrm{mm}$ and $h_c=2\hspace{0.17em}\mathrm{mm}$ ).

The (1-4-1) configuration: in this configuration, the thickness of the core is four times of the face sheets (i.e., $h_f=1\hspace{0.17em}\mathrm{mm}$ and $h_c=4\hspace{0.17em}\mathrm{mm}$ ).

The (2-1-2) configuration: in this configuration, the thickness of the core is half of the face sheets (i.e., $h_f=1\hspace{0.17em}\mathrm{mm}$ and $h_c=0.5\hspace{0.17em}\mathrm{mm}$ ).

Free vibration analysis

The free vibration responses of the aforementioned square sandwich plates with GRPC face sheets subject to different temperatures obtained based on the abovementioned solution are compared against some numerical results available in the literature and the results are presented in Tables 3 and 4. All plates examined in these two tables had GRPC face sheets containing 7% (by volume) graphene. Note that the natural frequency values in this paper have been presented by the following relation: $\mathrm{\Omega }=\frac{\omega a^2}{h}\sqrt{\frac{{\rho }_0}{E_0}}$ , as this parameter was used to present the results by other researchers, whose results will be used throughout for comparison. Therefore, hereafter, the statement “natural frequency” refers to this parameter. It should be noted that ${\rho }_0$ and $E_0$ are the reference values of ${\rho }_C$ and $E_C$ at $T=300K$ . Table 3 presents the comparison of the natural frequencies of the plates subject to various temperatures. Table 4 presents the influence of foundation support on the natural frequencies, starting with a foundationless plate (i.e., $\left(k_1,k_2\right)=\left(0\text{,}0\right)$ ), a plate supported on Winkler type foundation ((i.e., $\left(k_1,k_2\right)=\left(100\text{,}0\right)$ ), and a plate resting on Pasternak type foundation (i.e., $\left(k_1,k_2\right)=\left(10\text{,}010\right)$ ). It can be seen that the present results match well with those reported in Refs. [21,25].

Table 3.

Comparison of the first six natural frequencies ( $\mathrm{\Omega }=\frac{\omega a^2}{h}\sqrt{\frac{{\rho }_0}{E_0}}$ ) of square GRPC plates under different temperatures ( $b/h=10$ , $h=2\hspace{0.17em}\mathrm{mm}$ , $V_G=7\%$ ).

$T\left[K\right]$	Source	${\mathrm{\Omega }}_{11}$	${\mathrm{\Omega }}_{12}$	${\mathrm{\Omega }}_{21}$	${\mathrm{\Omega }}_{22}$	${\mathrm{\Omega }}_{13}$	${\mathrm{\Omega }}_{31}$
300	Shen et al. [21]	28.0982	64.9584	65.0364	95.6904	116.1739	116.3356
	Kiani [25]	28.0794	64.8676	64.9378	95.4811	116.8946	117.0263
	Present	28.0855	64.8455	65.0837	95.6278	115.9986	116.4499
400	Shen et al. [21]	21.9591	54.8037	56.4669	84.0640	101.1662	105.3867
	Kiani [25]	21.9430	55.8754	57.6844	83.8599	102.5441	103.4163
	Present	21.9515	54.7910	57.6719	84.0228	101.1841	105.4034
500	Shen et al. [21]	15.0773	45.2749	52.0752	74.6301	88.3236	97.9272
	Kiani [25]	15.0636	47.9939	49.4060	74.4269	91.9083	93.9191
	Present	15.0680	45.2891	52.0856	74.6098	88.4061	98.0032

Table 4.

Comparison of the first six natural frequencies ( $\mathrm{\Omega }=\frac{\omega a^2}{h}\sqrt{\frac{{\rho }_0}{E_0}}$ ) of square GRPC plates resting on elastic foundations under temperature of 300 K with various values of $(k_1,k_2)$ ; ( $b/h=20$ , $h=2\hspace{0.17em}\mathrm{mm}$ , $V_G=7\%$ ).

$(k_1,k_2)$	Source	${\mathrm{\Omega }}_{11}$	${\mathrm{\Omega }}_{12}$	${\mathrm{\Omega }}_{21}$	${\mathrm{\Omega }}_{22}$	${\mathrm{\Omega }}_{13}$	${\mathrm{\Omega }}_{31}$
(0,0)	Shen et al. [21]	29.5854	72.7627	73.0926	112.3927	140.7286	141.4999
	Present	29.5811	72.7445	73.0714	112.3420	140.6734	141.4383
(100,0)	Shen et al. [21]	30.9790	73.3379	73.6652	112.7646	141.0252	141.7949
	Present	30.9745	73.3193	73.6437	112.7136	140.9697	141.7331
(10,010)	Shen et al. [21]	33.5605	76.1127	76.4282	115.6587	143.9198	144.6742
	Present	33.5554	76.0926	76.4053	115.6057	143.8615	144.6098

Moreover, the influence of plate configurations (i.e., the core to face sheet thickness ratio ${(h}_c/h_f)$ and temperature (i.e., $T=300K$ and $T=400K$ ) on the natural frequencies of the sandwich plates with GRPC face sheets with $b/h=20$ and $V_G=9\%$ are examined with the results presented in Table 5. In the present study, the thickness of each GRPC face sheet is identical and $h_f=1\hspace{0.17em}\mathrm{mm},$ whereas the total thickness of the sandwich plates is taken to be $h=2.5, 3, 4$ and 6 mm; therefore, the examined core to face sheet thickness ratios are taken to be $h_c/h_f=$ 0.5, 1, 2 and 4 [49,50]. In other words, the plates have the same planar dimensions and only their thicknesses are different. As expected, it can be observed that the natural frequencies of the sandwich plates with GRPC face sheets decreases with an increase in the temperature, as the stiffness of the materials become adversely affected by the applied temperature. As an example, the first natural frequency of (1–1-1) sandwich plates is reduced by approximately 56%, when the temperature rises from $T=300K$ to $T=400K$ . This is due to the fact that the rise in temperature reduces the stiffness of the sandwich plates, in turn reducing the frequencies of the panel.

Table 5.

The first six natural frequencies ( $\mathrm{\Omega }=\frac{\omega a^2}{h}\sqrt{\frac{{\rho }_0}{E_0}}$ ) of square sandwich plates with GRPC face sheets under different temperatures ( $b/h=20$ , $V_G=9\%$ ).

Configuration	$T\left[K\right]$	${\mathrm{\Omega }}_{11}$	${\mathrm{\Omega }}_{12}$	${\mathrm{\Omega }}_{21}$	${\mathrm{\Omega }}_{22}$	${\mathrm{\Omega }}_{13}$	${\mathrm{\Omega }}_{31}$
(1-1-1)	300	7.4375	18.3181	18.3905	28.9872	35.8581	36.0407
	400	3.2873	13.2412	15.0712	23.9008	29.6426	32.0970
(1-2-1)	300	6.8221	16.8286	16.8886	26.6607	33.0144	33.1668
	400	3.7227	12.9092	14.1313	22.5446	28.1136	29.8165
(2-1-2)	300	8.1113	19.9294	20.0103	31.4689	38.8720	39.0738
	400	2.8034	13.7053	16.1535	25.4615	31.3639	34.5547
(1-4-1)	300	6.3867	15.7643	15.8085	24.9729	30.9425	31.0552
	400	4.0971	12.8287	13.5621	21.7495	27.1986	28.2616

However, this margin of reduction in the frequency as a result of the applied temperature is reduced when the change in higher frequencies is considered. Moreover, it can also be observed that as expected, the natural frequencies of the sandwich plates decrease with an increase in the core to face sheet thickness ratio $h_c/h_f$ . This would have been expected, since in sandwich plates a relatively small change in the core thickness would result in a significant change in plate’s stiffness (for instance, stiffness of 1-4-1 configuration is almost larger than 1-1-1 configuration). It is also worth mentioning that by increasing the core to face sheet thickness ratio, the decreasing effect of the temperature rise on the natural frequency is decreased.

The results presented in Table 6 reveals the effects of foundation stiffness and the core to face sheet thickness ratio $h_c/h_f$ on the natural frequencies of the sandwich plates with $b/h=50$ , $V_G=9\%$ at $T=300K$ . Plates resting on no foundation (i.e., $\left(k_1,k_2\right)=\left(0\text{,}0\right)$ ), as well as those resting on two different foundation types are considered. The two foundations considered are (i) the Winkler elastic foundation with $\left(k_1,k_2\right)=\left(100\text{,}0\right)$ and (ii) the Pasternak elastic foundation with $\left(k_1,k_2\right)=\left(10\text{,}010\right)$ . As expected, the natural frequencies are increased with an increase in the supporting foundation’s stiffness. It can be seen that the first natural frequency of (1-1-1) sandwich plate with $\left(k_1,k_2\right)=\left(10\text{,}010\right)$ is 226.8% higher than those of the foundationless sandwich plate. Since the elastic foundation is assumed to be compliant, the introduction of elastic foundation will provide an added stiffness to the sandwich plate.

Table 6.

The first six natural frequencies ( $\mathrm{\Omega }=\frac{\omega a^2}{h}\sqrt{\frac{{\rho }_0}{E_0}}$ ) of square sandwich plates with GRPC face sheets resting on elastic foundations under temperature of 300 K with various values of $(k_1,k_2)$ ; ( $b/h=50$ , $V_G=9\%$ ).

Configuration	$(k_1,k_2)$	${\mathrm{\Omega }}_{11}$	${\mathrm{\Omega }}_{12}$	${\mathrm{\Omega }}_{21}$	${\mathrm{\Omega }}_{22}$	${\mathrm{\Omega }}_{13}$	${\mathrm{\Omega }}_{31}$
(1-1-1)	(0,0)	7.4941	18.6610	18.7371	29.8463	37.1699	37.3708
	(100,0)	15.4603	23.0430	23.1047	32.7634	39.5495	39.7384
	(10,010)	24.4943	37.8524	37.8900	50.1546	58.0173	58.1463
(1-2-1)	(0,0)	6.8684	17.1102	17.1727	27.3672	34.0956	34.2611
	(100,0)	14.0938	21.0743	21.1251	30.0041	36.2454	36.4011
	(10,010)	22.3070	34.5123	34.5434	45.7679	52.9738	53.0805

The effect of the volume fraction of graphene on the fundamental frequency of square sandwich plates with GRPC face sheets is demonstrated in Figure 2. All tested four configurations of the sandwich plates have $b/h=20$ at $T=300K$ . The results shown in the figure reveals the significant influence of graphene on the fundamental frequency of the sandwich plates. The sandwich plate whose face sheets contains 11% graphene produced the highest gain in the fundamental frequency. For instance, the fundamental frequency of (2-1–2) configuration containing 11% graphene in its face sheets reaches a maximum value of 8.21, an increase by 55.5% compared to (2-1-2) configuration containing 3% graphene. This significant enhancement in the fundamental frequencies of the sandwich plates with GRPC face sheets is due to the inherent stiffness and reinforcing effect contributed to the plate by graphene. Moreover, the results illustrated in the figure reveal that 9% volume fraction would be the optimal graphene volume content. Additionally, since the variation of change in the frequencies as a function of volume fraction attains somewhat of a plateau, one could expect that the inclusion of additional graphene would not improve the fundamental frequency of the plates by an appreciable margin.

Figure 2.

Fundamental frequency $\mathrm{\Omega }=\frac{\omega a^2}{h}\sqrt{\frac{{\rho }_0}{E_0}}$ of square sandwich plates with GRPC face sheets in thermal environment ( $b/h=20$ , $T=300K)$ .

Buckling analysis

In the following section, the effects of various parameters such as temperature, graphene volume fraction, core to face sheet thickness ratio $h_c/h_f$ , foundation stiffness and different loading on the critical mechanical and thermal buckling capacities of the aforementioned sandwich plate will be evaluated by examining the numerical results.

Buckling response under thermal load

Before presenting the numerical results of the buckling analyses, it is of paramount importance to establish the integrity of the presented formulation. For this purpose, the results obtained from the present study are compared with those obtained through the literature. Table 7 presents the critical buckling temperatures of square plates resting on elastic foundations subjected to a uniform temperature rise, which have $h=2mm$ , $T=300K$ , $V_G=7\%$ , compared with the results presented from Refs. [23,51]. As seen, the results obtained by the proposed formulation match very closely to those in the literature, thus validating the integrity of the proposed formulation.

Table 7.

Comparison of the critical buckling temperatures (in [K]) of square plates resting on elastic foundations under uniform temperature rise ( $h=2mm$ , $T=300K$ , $V_G=7\%$ ).

$(k_1,k_2)$	Source	$b/h=15$	$b/h=20$	$b/h=50$
(0,0)	Shen et al. [23]	453.858	392.830	316.262
	Mirzaei and Kiani [51]	454.189	392.930	316.255
	Present	457.8579	394.5225	316.3132
(100,0)	Shen et al. [23]	470.194	402.313	317.798
	Present	476.0628	404.3879	317.8595
(10,010)	Shen et al. [23]	503.018	421.107	320.835
	Present	514.5131	424.4008	320.9211

The variation of the critical buckling temperatures against graphene volume fraction for different configurations of sandwich plates is shown in Figure 3. The results reveal that the buckling strength increases as a function of increasing graphene volume fraction of GRPC face sheets. Moreover, increasing the amount of graphene volume fraction from 3% to 11% generated a maximum enhancement of approximately 7% in the critical buckling temperatures in (1-1-1) configured sandwich plate. Moreover, according to the results, the sandwich plates with lower core to face sheet thickness ratio ${(h}_c/h_f)$ exhibit higher range of critical buckling temperatures. More specifically, the critical buckling temperature of (1–4-1) sandwich plate is 15.5% more than (2–1-2) sandwich plate which contains 11% volume fraction of graphene. Additionally, the effects of foundation stiffness and core to face sheet thickness ratio ${(h}_c/h_f)$ on the critical buckling temperature of the sandwich plates with $b/h=50$ , $V_G=9\%$ at $T=300K$ are tabulated in Table 8. The results in this Table show that the critical thermal buckling temperatures of the sandwich plates increase as the foundation’s stiffness increases.

Figure 3.

Critical buckling temperatures (in [K]) of square sandwich plates with GRPC face sheets under uniform temperature rise ( $b/h=20$ , $T=300K)$ .

Table 8.

Critical buckling temperatures (in [K]) of square sandwich plates with GRPC face sheets resting on elastic foundations under uniform temperature rise ( $b/h=50$ , $V_G=9\%$ , $T=300K$ ).

$(k_1,k_2)$	Configuration
	(1-1-1)	(1-2-1)	(2-1-2)	(1-4-1)
(0,0)	322.8836	326.7197	320.6098	332.3618
(100,0)	369.0194a	382.5218a	361.1872a	403.1388a
(10,010)	508.6825a	554.3198a	482.8418a	627.4313a

a

Mode for sandwich plate is (m, n) = (1, 2).

Buckling response under mechanical load

In this section, the resulting mechanical buckling capacities are presented. Tables 9 and 10 present the effects of the core to face sheet thickness ratio ${(h}_c/h_f)$ , the temperature variation $(T=300, 400$ and $500K)$ and graphene volume fractions of 3% to 11% on the critical buckling capacity of variously configured sandwich plates with $b/h=20$ under uniaxial and biaxial compression, respectively. It is observed that the critical buckling capacity of the sandwich plates with GRPC face sheets decreases with increase in the applied temperature. Since the material properties for both core and face sheets are assumed to be temperature-dependent, the variation in the temperature would cause a reduction in the elastic moduli and strengths of the sandwich plates. Moreover, the critical buckling capacity of the sandwich plates increases with an increase in the core to face sheet thickness ratio ${(h}_c/h_f)$ . This was expected since the flexural rigidity of the plate increases as its thickness increases. Furthermore, the results presented in these tables show that the increase in the volume fraction of graphene results in an increase in the buckling capacity of the sandwich plates. This is due to the fact that the modulus of elasticity of graphene is much larger than that of the matrix; thus, the addition of graphene results in a significant increase in the maximum flexural rigidity of the plate, despite the relatively low volume fraction content of graphene. A more significant enhancement is observed in (2–1-2) configured plates at room temperature; the buckling capacity of these plates was enhanced by 166.3% when the graphene content was increased from 3% to 11%. Moreover, by increasing the temperature from $T=300K$ to $T=500K$ the enhancement in the critical buckling capacity is reduced from 166.3% to 136.7%.

Table 9.

Critical buckling capacities $N_{cr}$ (in kN) of square sandwich plate with GRPC face sheets subjected to uniaxial compression in thermal environments ( $b/h=20$ ).

$T\left[K\right]$	$V_G$	Configuration
		(1-1-1)	(1-2-1)	(2-1-2)	(1-4-1)
300	3%	60.5215	125.3314	39.6560	359.1412
	5%	89.5521	172.3044	60.3264	444.0628
	7%	122.3607	225.4225	83.7046	540.2013
	9%	146.0921	263.7958	100.6802	609.6724
	11%	153.0198	275.0204	105.6175	630.0185
400	3%	56.3340	117.0996	36.8607	337.2190
	5%	79.1775	154.0658	53.1176	404.0470
	7%	107.9508	200.6531	73.6133	488.3506
	9%	123.9348	226.5154	85.0315	535.1695
	11%	133.1793	241.5017	91.6115	562.3297
500	3%	56.0655	115.2032	36.8551	326.7349
	5%	75.3048	146.3476	50.5367	383.0484
	7%	100.9911	187.9431	68.8271	458.3228
	9%	116.7739	213.4927	80.0904	504.5843
	11%	126.8427	229.8193	87.2540	534.1772

Table 10.

Critical buckling capacities $N_{cr}$ (in kN) of square sandwich plate with GRPC face sheets subjected to biaxial compression in thermal environments ( $b/h=20$ ).

$T\left[K\right]$	$V_G$	Configuration
		(1-1-1)	(1-2-1)	(2-1-2)	(1-4-1)
300	3%	30.2608	62.6657	19.8280	179.5706
	5%	44.7761	86.1522	30.1632	222.0314
	7%	61.1804	112.7113	41.8523	270.1007
	9%	73.0461	131.8979	50.3401	304.8362
	11%	76.5099	137.5102	52.8088	315.0093
400	3%	28.1670	58.5498	18.4304	168.6095
	5%	39.5888	77.0329	26.5588	202.0235
	7%	53.9754	100.3266	36.8067	244.1753
	9%	61.9674	113.2577	42.5158	267.5848
	11%	66.5897	120.7509	45.8058	281.1649
500	3%	28.0328	57.6016	18.4276	163.3675
	5%	37.6524	73.1738	25.2684	191.5242
	7%	50.4956	93.9716	34.4136	229.1614
	9%	58.3870	106.7464	40.0452	252.2922
	11%	63.4214	114.9097	43.6270	267.0886

Table 11 presents the critical buckling capacities of simply-supported foundationless sandwich plates and those resting on an elastic foundation. As can be seen, the buckling capacities for the sandwich plate resting on Winkler and/or Pasternak elastic foundations are much higher than the foundationless plates. This is due to the buckling mode changing from first mode in the foundationless plates (i.e., $\left(m,n\right)=\left(1\text{,}1\right)$ ) to mode $\left(m,n\right)=\left(2\text{,}1\right)$ in the plates resting on a foundation. It should also be stated that with the presence of an elastic foundation, the critical buckling capacity of all types of sandwich plates is increased at approximately the same rate.

Table 11.

Critical buckling capacities $N_{cr}$ (in kN) of square sandwich plate with GRPC face sheets resting on elastic foundations subjected to uniaxial and biaxial compression ( $b/h=50$ , $V_G=9\%$ , $T=300K$ ).

Load	$(k_1,k_2)$	Configuration
		(1-1-1)	(1-2-1)	(2-1-2)	(1-4-1)
Uniaxial	(0,0)	59.2060	106.7431	40.9075	246.5755
	(100,0)	140.7763a	252.5935a	97.3991a	578.0669a
	(10,010)	378.5972a	675.3863a	262.5525a	1529.3507a
Biaxial	(0,0)	29.6030	53.3716	20.4537	123.2878
	(100,0)	112.6210a	202.0748a	77.9193a	462.4535a
	(10,010)	302.8778a	540.3090a	210.0420a	1223.4805a

a

Mode for this sandwich plate is (m, n) = (2, 1).

Deflection analysis

For verification purposes, the dimensionless maximum central deflections obtained by the proposed solution in this study are compared with those obtained by the recently developed existing solutions. Table 12 reports the comparison of the dimensionless maximum central deflections of a simply-supported square PmPV composite plates reinforced with carbon nanotube (PmPV/CNT) subjected to a uniformly distributed load of $q_0=-1\times {10}^{5}N/m^2$ under temperature of $T=300K$ . The obtained results agree well with the results obtained by the closed-form solution of Wattanasakulpong and Chaikittiratana [52]. It should be noted that the material properties considered in this case study are listed in the aforementioned reference.

Table 12.

Comparison of the dimensionless maximum central deflections $\left(\overline{w}=\frac{1000D_{0}w}{a^4{q}_0}\right)$ of a square PmPV/CNT plate subjected to uniformly distributed load ( $a/h=10$ , $h=2mm$ , $T=300K$ , $q_0=-1\times {10}^{5}N/m^2$ ).

$(k_1,k_2)$	Source	$V_{\mathit{CNT}}=0.11$	$V_{\mathit{CNT}}=0.14$	$V_{\mathit{CNT}}=0.17$
(0,0)	Wattanasakulpong and Chaikittiratana [52]	0.7356	0.6475	0.4712
	Present	0.7205	0.6228	0.4628
(100,0)	Wattanasakulpong and Chaikittiratana [52]	0.6983	0.6182	0.4556
	Present	0.6967	0.6052	0.4528
(10,050)	Wattanasakulpong and Chaikittiratana [52]	0.4773	0.4387	0.3500
	Present	0.5103	0.4594	0.3658

After the validation of the proposed solution, the influences of temperature variation and elastic foundation on the deflection response of all the variously configured GRPC-reinforced sandwich plates are examined. The results presented in Table 13 show the comparison of the maximum dimensionless central deflection of the plates subjected to a uniformly distributed load. The results reveal that the deflections of the variously configured sandwich plates evaluated at room temperature are significantly less than those tested at higher temperatures. This is because the increase in temperature leads to a decrease in the stiffness in the face sheets of the plates. Moreover, it can be seen that as expected, the dimensionless maximum central deflection values of the sandwich plate resting on elastic foundations are lower than those of the foundationless sandwich plate.

Table 13.

Dimensionless maximum central deflections $\widehat{w}=-\frac{w}{h}$ of square sandwich plate with GRPC face sheets resting on elastic foundations subjected to uniformly distributed load ( $b/h=50$ , $V_G=9\%$ , $q_0=-1\times {10}^{5}N/m^2$ ).

$T\left[K\right]$	$(k_1,k_2)$	Configuration
		(1-1-1)	(1-2-1)	(2-1-2)	(1-4-1)
300	(0,0)	322.1771 × 10−3	317.6698 × 10−3	323.8420 × 10−3	309.4159 × 10−3
	(100,0)	82.3922 × 10−3	82.0213 × 10−3	82.5451 × 10−3	81.3481 × 10−3
	(10,010)	33.7870 × 10−3	33.7125 × 10−3	33.8208 × 10−3	33.5802 × 10−3
400	(0,0)	379.7260 × 10−3	369.433 × 10−3	383.2698 × 10−3	352.5023 × 10−3
	(100,0)	86.2698 × 10−3	85.448 × 10−3	86.5197 × 10−3	84.4782 × 10−3
	(10,010)	34.5010 × 10−3	34.164 × 10−3	34.5513 × 10−3	34.1691 × 10−3
500	(0,0)	402.7050 × 10−3	392.2888 × 10−3	406.5017 × 10−3	373.7063 × 10−3
	(100,0)	87.5476 × 10−3	86.9346 × 10−3	87.7951 × 10−3	85.8195 × 10−3
	(10,010)	34.7247 × 10−3	34.6129 × 10−3	34.7741 × 10−3	34.4124 × 10−3

Next, the effect of volume fraction of graphene and the core to face sheet thickness ratio $h_c/h_f$ on the dimensionless maximum central deflections of the sandwich plates are compared and the results are presented in the graph in Figure 4. As can be seen, the increase in the volume fraction of graphene from 3% to 11% improved (decreased) the value of deflection of the plates; however, the rate of change in the improvement becomes insignificant once the volume fraction surpasses 9%. Hence, it can be evidently deduced that graphene reinforcement plays a very important role in improving the overall stiffness of the sandwich plates.

Figure 4.

Dimensionless maximum central deflections $\widehat{w}=-\frac{w}{h}$ of square sandwich plates with GRPC face sheets subjected to uniformly distributed load ( $b/h=20$ , $T=300K$ , $q_0=-1\times {10}^{5}N/m^2$ ).

Summary and conclusion

This paper investigated the influence of the addition of graphene on the vibration, stability and static responses of sandwich plates. In addition, it presented an analytical solution for analyzing the natural frequencies, buckling capacity and elastic deformation of plates under various mechanical and thermal loading scenarios. Various configurations of the simply-supported sandwich plates formed with nanoparticle-reinforced face sheets, resting on different elastic foundations subject to the combined loads were investigated. Temperature-dependent material properties were considered for both the core and face sheets of the sandwich plates. The governing differential equations of motion were derived on the basis of a recently developed refined shear deformation plate theory and were subsequently solved analytically using the Navier method.

The results revealed that the natural frequencies and critical mechanical and thermal buckling capacities increased with the addition of nanoparticles, and that the plates experienced lower deflection when carrying a uniformly distributed load. Moreover, it was found that graphene volume contents beyond 9% could not further improve the natural frequencies, critical buckling and elastic deformation of the sandwich plates in a significant way. Moreover, the fundamental frequencies and buckling capacities of sandwich plates having a fixed volume fraction of graphene in their face sheets examined at lower temperatures were higher than those considered under higher temperatures. As also expected, the presence of an elastic foundation improved the buckling performances of the reinforced plates, resulting in the change of their buckling mode. As for the influence of nanoparticles on the natural frequencies, the plate with (2–1-2) configuration, which had the lowest thickness, exhibited the highest natural frequencies.

In conclusion, the refined shear deformation plate theory incorporated in this study, which includes a lower number of unknowns compared to the other pertinent theories that are involved with a greater number of unknowns, produced as accurate results. The future work will extend the formulation and assess the response of circular and skewed plates.