Free Vibration Characteristics of Bidirectional Graded Porous Plates with Elastic Foundations Using 2D-DQM

Abstract

This manuscript develops for the first time a mathematical formulation of the dynamical behavior of bi-directional functionally graded porous plates (BDFGPP) resting on a Winkler–Pasternak foundation using unified higher-order plate theories (UHOPT). The kinematic displacement fields are exploited to fulfill the null shear strain/stress at the bottom and top surfaces of the plate without needing a shear factor correction. The bi-directional gradation of materials is proposed in the axial (x-axis) and transverse (z-axis) directions according to the power-law distribution function. The cosine function is employed to define the distribution of porosity through the transverse z-direction. Equations of motion in terms of displacements and associated boundary conditions are derived in detail using Hamilton’s principle. The two-dimensional differential integral quadrature method (2D-DIQM) is employed to transform partial differential equations of motion into a system of algebraic equations. Parametric analysis is performed to illustrate the effect of kinematic shear relations, gradation indices, porosity type, elastic foundations, geometrical dimensions, and boundary conditions (BCs) on natural frequencies and mode shapes of BDFGPP. The effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions. The proposed model can be used in designing BDFGPP used in nuclear, marine, aerospace, and civil structures based on their topology and natural frequency constraints.

1. Introduction

Nowadays, many studies have been performed on the design and manufacture of plates for use in different loading conditions (i.e., tremendously high loading and high-heat and -moisture environmental conditions) and a wide range of engineering applications such as aerospace, nuclear reactors, marine, automobiles, electronics, and biomedical structures [1]. Functionally graded materials (FGMs) with enhanced properties are new materials invented in 1984 [2] whose constituents are changed continuously within the body through spatial directions [3]. FGMs gain unique characteristics that do not exist in classical composites such as elimination of high stress concentrations, delamination, crack formation, and moisture sensitivity [4]. FGMs can sustain a temperature of 2000 K and temperature gradient of 1000 K across a cross-section of <10 mm [5]. Certainly, it is difficult to discover the exact gradation in FGMs as it requires an experimental control, which leads to some imperfect products [6].

Some engineering applications may need enhanced composite materials with properties that vary continuously in more than one direction [7]. Lieu et al. [8] evaluated the mechanical responses of BDFG plates with variable thickness. Wu and Yu [9] derived 3D solutions for a free vibration of BDFG plates with assorted boundary conditions (BCs) using finite annular prism methods. Hashemi and Jafari [10] considered the nonlinear vibration responses of in-plane 2D-FG plates with temperature-dependent properties. Ghatage [11] introduced a comprehensive review on the modelling, simulation, and analysis of multi-directional FG structures. Pham et al. [12] studied the free vibration of 2D-FG rectangular plates in the fluid medium using isogeometric analysis (IGA). Chen et al. [13] evaluated the nonlinear responses of BDFG Kirchhoff plates with geometrical imperfections including von Kármán’s nonlinear strain under external force. Mohammadian [14] obtained closed-form analytical solutions of the nonlinear vibration of damped 2D-FG beams using a cubic-quintic nonlinear model. Abo-Bakr et al. [15,16] found the optimum weight of 2D-FG macro/micro-beams under a constraint of maximum frequency and critical buckling load using a multi-objective particle swarm optimization technique. Karamanli et al. [17] investigated numerically mechanical responses of multi-directional FG modified strain gradient microplates in the framework of 3D shear deformation theory. Patpatiya et al. [18] investigated the state of the art of polyjet 3D printing of polymers, multi-material structures, and FGMs, with an emphasis on its applications in a range of industrial domains. Attia and Shanab [19,20] evaluated the natural frequencies and dynamic responses of 2D-FG nanobeams under a moving harmonic load with couple stress and surface energy.

The accuracy of the obtained results in the study of beams, plates, and shell theories is dependent on the selection of the displacement and shear functions. Although the classical Kirchhoff–Love plate theory ignores the influence of shear strain, it may be used in the analysis of very thin structures with acceptable accuracy, [21]. Reddy [22] implemented Mindlin’s plate theory to consider shear deformation, rotary inertia, and lamination scheme effects. In 1991 Touratier [23] utilised a classic Kirchhoff–Love plate theory with a cosine shear stress distribution to examine the mechanical responses of laminated/sandwiched composite plates. Based on unified shear theories, Assie and Mahmoud [24] studied the buckling response of BDFG porous plates embedded in an elastic environment. Assie et al. [25,26] used Mindlin’s theory to investigate the viscoelastic response of laminated plates under transient loads and optimize its weight according to a rigidity constraint. Malikan et al. [27] studied thermal buckling of functionally graded piezomagnetic micro- and nanobeams presenting the flexomagnetic effect using Timoshenko shear deformation theory.

Brischetto et al. [27] studied the free vibration of FG plates and cylinders using a 3D elasticity model. Vu et al. [28] explored the mechanical responses of FG plates resting on elastic foundation using a refined quasi-3D logarithmic shear deformation theory. Raissi [29] examined stresses in adhesive layers of a sandwich plate under to different temperature environments using layerwise plate theory. Based on a hybrid quasi-3D plate theory, Van [30] studied the bending and vibration responses of BDFGM plates resting on Pasternak foundations. Considering the Reddy and Touratier shear plate theories, Daikh et al. [31,32] explored the static and vibration responses of sigmoidal FG/multilayer FG carbon-reinforced composite nanoplates. Using a new logarithmic shape function, quasi-3D HSDT, Li et al. [33] evaluated the vibration behavior of FGM plates resting on elastic foundations described by a Winkler/Pasternak/Kerr model. Duc and Minh [34] exploited Shi’s third-order shear theory to study the free vibration of cracked FG plates. Tran et al. [35] studied the natural frequencies and vibrations of FG FOSDT plates in the thermal environment. Sadgui and Tati [36] analyzed the buckling and vibration response of FG plates using a trigonometric shear deformation plate theory. Naghsh et al. [37] investigated the instability of sandwich tapered plates under uniform/non-uniform in-plane loadings utilizing higher-order zigzag shear theory. Under different moisture conditions, Brischetto and Torre [38] developed a 3D stress analysis of multilayered FG plates utilizing an exact layerwise approach. Melaibari et al. [39,40] exploited hyperbolic sine function shear deformation theory to study the dynamic behavior of randomly oriented FG laminated plates and shells with different geometries.

Porosities and voids may arise inside materials through the manufacturing process of FGMs owing to the large differences in solidification temperatures between material constituents during the process of sintering [41,42,43]. On the basis of classical and first-order shear deformation plate theories, Kim et al. [44] studied the mechanical response of FG porous microplates. Esmaeilzadeh and Kadkhodayan [2] investigated the dynamic response of porous 2D-FG plates reinforced by stiffeners under a dynamic moving load. Babaei et al. [45] examined static and dynamic behaviors of FG porous elliptical sector plates based on the 3D finite element method (FEM). Esmaeilzadeh et al. [46] presented the nonlinear dynamic response of BDFG porous nanoplates using a nonlocal strain gradient model. Li et al. [47] exploited FOSDT as well as IGA in analyzing the bending, buckling, and vibration of porous 2D-FG plates. Katiyar and Gupta [48] examined the dynamic response of FG porous refined non-polynomial trigonometric higher-order shear plates resting on elastic foundations under different thermal environments. Ansari et al. [49] studied the free vibration of post-buckled arbitrary-shaped FG porous nanocomposite plates using the Reddy plate theory. Using a power law with a uniform porosity distribution, Abdollahi et al. [50] investigated the aeroelastic response of honeycomb sandwich plates. Akbaş et al. [51,52] examined the transient vibration behaviors of FG porous beams with viscoelastic cores under pulse loads. Sah and Ghosh [53] presented the dynamic and buckling responses of multi-directional FG sandwich porous plates.

According to the previous articles surveyed, the free vibration response of BDFG unified higher-order shear plates with porous materials placed on elastic foundations has not been considered. Thus, this paper aims to illustrate this topic in comprehensively. This analysis will be limited to the one- or two-dimensional gradation functions. The following sections will present the problem formulations, kinematic and constitutive relations, porosity distribution functions, and equations of motion (Section 2). The solution methodology, two-dimensional differential integral quadrature method, and formulations of stiffness/mass matrices will be presented in Section 3. Section 4 presents the model validations and discusses the influences of kinematic fields, porosity distribution, gradation indices, aspect ratio, slenderness ratio, elastic foundation, and boundary conditions on the natural frequencies and mode shapes. Section 5 presents the main features and conclusions.

2. Mathematical Formulation

2.1. Displacement Field and Constitutive Relations

The displacement field of unified plate theory can be represented by

$$u\left(x,y,z\right)=u_o\left(x,y\right)-z\frac{\partial w_b}{\partial x}-F\left(z\right)\frac{\partial w_s}{\partial x}$$

(1)

$$v\left(x,y,z\right)=v_o\left(x,y\right)-z\frac{\partial w_b}{\partial y}-F\left(z\right)\frac{\partial w_s}{\partial y}$$

(2)

$$w\left(x,y,z\right)=w_b\left(x,y\right)+w_s\left(x,y\right)$$

(3)

in which $u_o,v_o,w_b\left(\mathrm{bending}\mathrm{part}\right)\mathrm{and}{w}_s\left(\mathrm{shear}\mathrm{part}\right)$ are the displacements defined with respect to the midplane. $F\left(z\right)$ is a shape shear distribution function through the thickness directions that satisfies a zero shear at the top and bottom surfaces as [23,54,55,56,57]

$$F\left(z\right)=z-\left(\frac{h}{\pi }\right)\sin \left(\frac{\pi z}{h}\right)\mathrm{Touratier}$$

(4)

$$F\left(z\right)=z\left(1-e^{-2{\left(\frac{z}{h}\right)}^2}\right)\mathrm{Karama\; et\; al.}$$

(5)

$$F\left(z\right)=\frac{4z^3}{3h^2}\mathrm{Reddy}$$

(6)

$$F\left(z\right)=z\left(-\frac{1}{4}+\frac{5}{3}\frac{z^2}{h^2}\right)\mathrm{Thai\; and\; Kim}$$

(7)

$$F\left(z\right)=\frac{\frac{h}{\pi }\sinh \left(\frac{\pi z}{h}\right)-z}{\cosh \left(\frac{\pi }{2}\right)-1}\mathrm{Taibi\; et\; al.}$$

(8)

The unified higher order shear theories are proposed to describe adequately the variation in transverse shear strain within the thickness of the plate. Based on Equations (1)–(3), the normal and shear strains can be represented by

$${\epsilon }_x={\epsilon }_x^0-z\frac{{\partial }^2{w}_b}{\partial x^2}-F\left(z\right)\frac{{\partial }^2{w}_s}{\partial x^2}, {\epsilon }_y={\epsilon }_y^0-z\frac{{\partial }^2{w}_b}{\partial y^2}-F\left(z\right)\frac{{\partial }^2{w}_s}{\partial y^2}$$

(9)

$${\gamma }_{xy}={\gamma }_{xy}^0-z\left(2\frac{{\partial }^2{w}_b}{\partial \mathrm{x}\partial \mathrm{y}}\right)-F\left(z\right)\left(2\frac{{\partial }^2{w}_s}{\partial \mathrm{x}\partial \mathrm{y}}\right),$$

(10)

$${\gamma }_{yz}=G\left(z\right)\frac{\partial w_s}{\partial y}, {\gamma }_{xz}=G\left(z\right)\frac{\partial w_s}{\partial x}$$

(11)

$$\begin{gathered} {\epsilon }_x^o=\frac{\partial u_o}{\partial x},{\epsilon }_y^o=\frac{\partial v_o}{\partial y} and{\gamma }_{xy}^o=\frac{\partial u_o}{\partial y}+\frac{\partial v_o}{\partial x} \\ F\left(z\right)=z-f\left(z\right)\mathrm{and}G\left(z\right)=1-F^{\prime }\left(z\right)=f^{\prime }\left(z\right) \end{gathered}$$

(12)

By considering isothermal conditions and neglecting the transverse normal strain $\left({\epsilon }_z=0\right)$, the constitutive relations for 2D shear deformation plate theory are

$$\left[\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\\ {\tau }_{yz}\\ {\tau }_{xz}\end{array}\right]=\left[\begin{array}{ccccc}{Q}_{11}& Q_{12}& 0& 0& 0\\ Q_{12}& Q_{22}& 0& 0& 0\\ 0& 0& Q_{66}& 0& 0\\ 0& 0& 0& Q_{44}& 0\\ 0& 0& 0& 0& Q_{55}\end{array}\right]\left[\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\\ {\gamma }_{yz}\\ {\gamma }_{xz}\end{array}\right]$$

(13)

in which the stiffness coefficients are

$$Q_{11}=Q_{22}=\frac{E}{1-v^2},Q_{12}=\frac{\nu E}{1-v^2}$$

(14)

$$Q_{44}=Q_{55}=Q_{66}=\frac{E}{2\left(1+\nu \right)}$$

(15)

The Young’s modulus and Poisson’s ratio are $E$ and $\nu$, respectively. Properties of the material including porosity and voids are graded in both axial and transverse directions by power law as

$$P\left(x,z,\varphi \right)=[P_m+P_{cm}{\left(\frac{1}{2}+\frac{z}{h}\right)}^{n_z}(\frac{x}{a}{)}^{n_x}]\left[1-\Phi \left(z\right)\right]$$

(16)

$$P_{cm}=P_c-P_m$$

(17)

where $P$ signifies a generic graded material property such as $E,\nu ,\mathrm{and}\rho$ (density). $n_z$ and $n_x$ are the graded indices through the z- and x-directions. Subscripts $c$ and $m$ represent a ceramic and a metal, respectively. The porosity distribution function $\Phi \left(z\right)$ through the thickness has the following distributions:

$$\Phi \left(z\right)=\varphi \mathit{cos}\left(\frac{\pi }{h}z\right)\mathrm{Type}1\left(\mathrm{center}\mathrm{enhanced}\right)$$

(18)

$$\Phi \left(z\right)=\varphi \mathit{cos}\left(\frac{\pi }{2}\left(\frac{z}{h}+\frac{1}{2}\right)\right)\mathrm{Type}2\left(\mathrm{top}\mathrm{enhanced}\right)$$

(19)

$$\Phi \left(z\right)=\varphi \mathit{cos}\left(\frac{\pi }{2}\left(\frac{z}{h}-\frac{1}{2}\right)\right)\mathrm{Type}3\left(\mathrm{bottom}\mathrm{enhanced}\right)$$

(20)

where the porosity parameter is $\varphi$.

2.2. Governing Equations of Motion

The governing equations of motion are derived based on Hamilton’s Principles as

$${{\displaystyle \int }}_0^T\delta \left(U+V+U_{ef}-K\right)dt=0$$

(21)

The virtual work of applied loads, $\delta V$ is evaluated by

$$\delta V={{\displaystyle \int }}_A\left(N_x^o\frac{\partial \overline{w}}{\partial x}\frac{\partial \delta \overline{w}}{\partial x}+N_{xy}^o\left(\frac{\partial \overline{w}}{\partial x}\frac{\partial \delta \overline{w}}{\partial y}+\frac{\partial \overline{w}}{\partial y}\frac{\partial \delta \overline{w}}{\partial x}\right)+N_y^o\frac{\partial \overline{w}}{\partial y}\frac{\partial \delta \overline{w}}{\partial y}\right)dA$$

(22)

in which

$${\overline{N}}^o=N_x^o\frac{{\partial }^2\overline{w}}{\partial x^2}+2N_{xy}^o\frac{{\partial }^2\overline{w}}{\partial x\partial y}+N_y^o\frac{{\partial }^2\overline{w}}{\partial y^2}$$

(23)

where $w=\overline{w}=w_b+w_s$.

The variation in potential energy of the Winkler–Pasternak elastic foundation (displayed in Figure 1) is

$$\delta U_{ef}={{\displaystyle \int }}_A[K_w(w_b+w_s)-K_p{\mathsf{\nabla }}^2\left(w_b+w_s\right)]\delta \left(w_b+w_s\right)dA$$

(24)

where: ${\mathsf{\nabla }}^2\left(w_b+w_s\right)=\frac{{\partial }^2\left(w_b+w_s\right)}{\partial x^2}+\frac{{\partial }^2\left(w_b+w_s\right)}{\partial y^2}$.

The virtual kinetic energy $\left(\delta K\right)$ is prescribed by

$$\delta K={{\displaystyle \int }}_V\rho \left(x,z,\varphi \right)[\dot{u}\delta \dot{u}+\dot{v}\delta \dot{v}+\dot{w}\delta \dot{w}]dV$$

(25)

Manipulating Equation (9) leads to the following:

$$\begin{gathered} \delta K={{\displaystyle \int }}_A\left[{\mathrm{I}}_o\left(x\right)\left({\dot{u}}_o\delta {\dot{u}}_o+{\dot{v}}_o\delta {\dot{v}}_o+\left({\dot{w}}_b+{\dot{w}}_s\right)\delta \left({\dot{w}}_b+{\dot{w}}_s\right)\right)-{\mathrm{I}}_1\left(x\right)\left({\dot{u}}_o\frac{\partial \delta {\dot{w}}_b}{\partial x}+\frac{\partial {\dot{w}}_b}{\partial x}\delta {\dot{u}}_o+{\dot{v}}_o\frac{\partial \delta {\dot{w}}_b}{\partial y}+\frac{\partial {\dot{w}}_b}{\partial y}\delta {\dot{v}}_o\right)- \\ {\mathrm{J}}_1\left(x\right)\left({\dot{u}}_o\frac{\partial \delta {\dot{w}}_s}{\partial x}+\frac{\partial {\dot{w}}_s}{\partial x}\delta {\dot{u}}_o+{\dot{v}}_o\frac{\partial \delta {\dot{w}}_s}{\partial y}+\frac{\partial {\dot{w}}_s}{\partial y}\delta {\dot{v}}_o\right)+{\mathrm{I}}_2\left(x\right)\left(\frac{\partial {\dot{w}}_b}{\partial x}\frac{\partial \delta {\dot{w}}_b}{\partial x}+\frac{\partial {\dot{w}}_b}{\partial y}\frac{\partial \delta {\dot{w}}_b}{\partial y}\right)+{\mathrm{J}}_2\left(x\right)\left(\frac{\partial {\dot{w}}_b}{\partial x}\frac{\partial \delta {\dot{w}}_s}{\partial x}+\frac{\partial {\dot{w}}_s}{\partial x}\frac{\partial \delta {\dot{w}}_b}{\partial x}+ \\ \frac{\partial {\dot{w}}_b}{\partial y}\frac{\partial \delta {\dot{w}}_s}{\partial y}+\frac{\partial {\dot{w}}_s}{\partial y}\frac{\partial \delta {\dot{w}}_b}{\partial y}\right)+{\mathrm{K}}_2\left(x\right)\left(\frac{\partial {\dot{w}}_s}{\partial x}\frac{\partial \delta {\dot{w}}_s}{\partial x}+\frac{\partial {\dot{w}}_s}{\partial y}\frac{\partial \delta {\dot{w}}_s}{\partial y}\right)\right]dA \end{gathered}$$

(26)

The virtual strain energy $\delta U:$

$$\delta U={{\displaystyle \int }}_V\left({\sigma }_x\delta {\epsilon }_x+{\sigma }_y\delta {\epsilon }_y+{\tau }_{xy}\delta {\gamma }_{xy}+{\tau }_{xz}\delta {\gamma }_{xz}+{\tau }_{yz}\delta {\gamma }_{yz}\right)dV$$

(27)

In terms of stress resultants, virtual strain energy ($\delta U$) is obtained by

$$\begin{gathered} \delta U={{\displaystyle \int }}_A\left[N_x\delta {\epsilon }_x^0+N_y\delta {\epsilon }_y^0+N_{xy}\delta {\gamma }_{xy}^0-M_x^b\frac{{\partial }^2\delta w_b}{\partial x^2}-M_y^b\frac{{\partial }^2\delta w_b}{\partial y^2}-M_{xy}^b\left(2\frac{{\partial }^2\delta w_b}{\partial x\partial y}\right)-M_x^s\frac{{\partial }^2\delta w_s}{\partial x^2}-M_y^s\frac{{\partial }^2\delta w_s}{\partial y^2} \\ -M_{xy}^s\left(2\frac{{\partial }^2\delta w_s}{\partial x\partial y}\right)+S_{yz}^s\frac{\partial \delta w_s}{\partial y}+S_{xz}^s\frac{\partial \delta w_s}{\partial x}\right]dA \end{gathered}$$

(28)

in which the stress resultants in terms of generalized displacements $\mathrm{are}\mathrm{evaluated}$ by

$$\left[\begin{array}{c}\begin{array}{c}{N}_x\\ N_y\\ N_{xy}\end{array}\\ M_x^b\\ M_y^b\\ M_{xy}^b\\ M_x^s\\ M_y^s\\ M_{xy}^s\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{A}_{11}\\ A_{12}\\ 0\end{array}\\ B_{11}\\ B_{12}\\ 0\\ B_{11}^s\\ B_{12}^s\\ 0\end{array}\begin{array}{c}\begin{array}{c}{A}_{12}\\ A_{22}\\ 0\end{array}\\ B_{12}\\ B_{22}\\ 0\\ B_{12}^s\\ B_{22}^s\\ 0\end{array}\begin{array}{c}\begin{array}{c}0\\ 0\\ A_{66}\end{array}\\ 0\\ 0\\ B_{66}\\ 0\\ 0\\ B_{66}^s\end{array}\begin{array}{c}\begin{array}{c}{B}_{11}\\ B_{12}\\ 0\end{array}\\ D_{11}\\ D_{12}\\ 0\\ D_{11}^s\\ D_{12}^s\\ 0\end{array}\begin{array}{c}\begin{array}{c}{B}_{12}\\ B_{22}\\ 0\end{array}\\ D_{12}\\ D_{22}\\ 0\\ D_{12}^s\\ D_{22}^s\\ 0\end{array}\begin{array}{c}\begin{array}{c}0\\ 0\\ B_{66}\end{array}\\ 0\\ 0\\ D_{66}\\ 0\\ 0\\ D_{66}^s\end{array}\begin{array}{c}\begin{array}{c}{B}_{11}^s\\ B_{12}^s\\ 0\end{array}\\ D_{11}^s\\ D_{12}^s\\ 0\\ H_{11}^s\\ H_{12}^s\\ 0\end{array}\begin{array}{c}\begin{array}{c}{B}_{12}^s\\ B_{22}^s\\ 0\end{array}\\ D_{12}^s\\ D_{22}^s\\ 0\\ H_{12}^s\\ H_{22}^s\\ 0\end{array}\begin{array}{c}\begin{array}{c}0\\ 0\\ B_{66}^s\end{array}\\ 0\\ 0\\ D_{66}^s\\ 0\\ 0\\ H_{66}^s\end{array}\right]\left[\begin{array}{c}\begin{array}{c}\partial u_o/\partial x\\ \partial v_o/\partial y\\ \left(\partial u_o/\partial y\right)+\left(\partial v_o/\partial x\right)\end{array}\\ -{\partial }^2{w}_b/{\partial }^{2}x\\ -{\partial }^2{w}_b/{\partial }^{2}y\\ -2{\partial }^2{w}_b/\partial x\partial y\\ -{\partial }^2{w}_s/{\partial }^{2}x\\ -{\partial }^2{w}_s/{\partial }^{2}y\\ -2{\partial }^2{w}_s/\partial x\partial y\end{array}\right]$$

(29)

$$\left[\begin{array}{c}{S}_{yz}^s\\ S_{xz}^s\end{array}\right]=\left[\begin{array}{cc}{A}_{44}^s& 0\\ 0& A_{55}^s\end{array}\right]\left[\begin{array}{c}\partial w_s/\partial y\\ \partial w_s/\partial x\end{array}\right]$$

(30)

and the rigidity terms are

$$\begin{gathered} \left[A_{\mathrm{ij}}\left(x\right),B_{\mathrm{ij}}\left(x\right),D_{\mathrm{ij}}\left(x\right),B_{\mathrm{ij}}^s\left(x\right),D_{\mathrm{ij}}^s\left(x\right),H_{\mathrm{ij}}^s\left(x\right)\right]={{\displaystyle \int }}_{-h/2}^{h/2}{Q}_{\mathrm{ij}}\left(x,z\right)\left[1,z,z^2,F\left(z\right),zF\left(z\right),{\left(F\left(z\right)\right)}^2\right]dz \\ \mathrm{ij}=11,12,22,66 \end{gathered}$$

(31)

$$A_{\mathrm{ij}}^s\left(x\right)={{\displaystyle \int }}_{-h/2}^{h/2}{Q}_{\mathrm{ij}}\left(x,z\right){(G\left(z\right))}^{2}dz,\mathrm{ij}=44,55$$

(32)

where $Q_{\mathrm{ij}}\left(x,z\right)\mathrm{and}E\left(x,z\right)$ are defined by Equations (4) and (5).

Substituting Equations (22)–(24), (27) and (28) for $\delta V,\delta K\mathrm{and}\delta U,\mathrm{respectively}$ into Equation (21), the equations of FGM porous plates are:

$$\begin{array}{cc}\delta u_o:& \frac{\partial N_x}{\partial x}+\frac{\partial N_{xy}}{\partial y}=I_o\left(x\right){\ddot{u}}_o-I_1\left(x\right)\frac{\partial {ẅ}_b}{\partial x}-J_1\left(x\right)\frac{\partial {ẅ}_s}{\partial x}\end{array}$$

(33)

$$\begin{array}{cc}\delta v_o:& \frac{\partial N_{xy}}{\partial x}+\frac{\partial N_y}{\partial y}=I_o\left(x\right){\ddot{\nu }}_o-I_1\left(x\right)\frac{\partial {ẅ}_b}{\partial y}-J_1\left(x\right)\frac{\partial {ẅ}_s}{\partial y}\end{array}$$

(34)

$$\begin{gathered} \begin{array}{cc}\delta w_b:& \frac{{\partial }^2{M}_x^b}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}^b}{\partial x\partial y}+\frac{{\partial }^2{M}_y^b}{\partial y^2}+{\overline{N}}^o+K_P{\mathsf{\nabla }}^2\left(w_b+w_s\right)-K_w\left(w_b+w_s\right) \\ =I_o\left(x\right)\left({\ddot{w}}_b+{\ddot{w}}_s\right)+I_1^{\prime }\left(x\right){\ddot{u}}_o+I_1\left(x\right)\left(\frac{\partial {\ddot{u}}_o}{\partial x}+\frac{\partial {\ddot{v}}_o}{\partial y}\right)-I_2^{\prime }\left(x\right)\frac{\partial {\ddot{w}}_b}{\partial x}-I_2\left(x\right){\mathsf{\nabla }}^2{\ddot{w}}_b \\ -J_2^{\prime }\left(x\right)\frac{\partial {\ddot{w}}_s}{\partial x}-J_2\left(x\right){\mathsf{\nabla }}^2{\ddot{w}}_s\end{array} \end{gathered}$$

(35)

$$\begin{gathered} \begin{array}{cc}\delta w_s:& \frac{{\partial }^2{M}_x^s}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}^s}{\partial x\partial y}+\frac{{\partial }^2{M}_y^s}{\partial y^2}+\frac{\partial S_{yz}^s}{\partial y}+\frac{\partial S_{xz}^s}{\partial x}+{\overline{N}}^o+K_P{\mathsf{\nabla }}^2\left(w_b+w_s\right)-K_w\left(w_b+w_s\right) \\ =I_o\left(x\right){\ddot{(w}}_b+{\ddot{w}}_s)+J_1^{\prime }\left(x\right){\ddot{u}}_o+J_1\left(x\right)\left(\frac{\partial {\ddot{u}}_o}{\partial x}+\frac{\partial {\ddot{v}}_o}{\partial y}\right)-J_2^{\prime }\left(x\right)\frac{\partial {\ddot{w}}_b}{\partial x}-J_2\left(x\right){\mathsf{\nabla }}^2{\ddot{w}}_b \\ -K_2^{\prime }\left(x\right)\frac{\partial {\ddot{w}}_s}{\partial x}-K_2\left(x\right){\mathsf{\nabla }}^2{\ddot{w}}_s\end{array} \end{gathered}$$

(36)

where the boundary conditions are:

$$\begin{array}{cc}\delta u_o:& \left(N_x{\overline{n}}_x+N_{xy}{\overline{n}}_y\right)\delta u_0=0\end{array}$$

(37)

$$\begin{array}{cc}\delta v_o:& \left(N_{xy}{\overline{n}}_x+N_y{\overline{n}}_y\right)\delta v_o=0\end{array}$$

(38)

$$\begin{gathered} \begin{array}{cc}\delta w_b:& \left(\left(M_{x,x}^b+M_{xy,y}^b-I_1\left(x\right){\ddot{u}}_o+I_2\left(x\right)\frac{\partial {\ddot{w}}_b}{\partial x}+J_2\left(x\right)\frac{\partial {\ddot{w}}_s}{\partial x}\right){\overline{n}}_x \\ +\left(M_{xy,X}^b+M_{y,y}^b-I_1\left(x\right){\ddot{v}}_o+I_2\left(x\right)\frac{\partial {\ddot{w}}_b}{\partial y}+J_2\left(x\right)\frac{\partial {\ddot{w}}_s}{\partial y}\right){\overline{n}}_y+P\left(\overline{w}\right)\right)\delta w_b=0\end{array} \end{gathered}$$

(39)

$$\begin{array}{cc}\frac{\partial \delta w_b}{\partial x}:& \left(M_x^b{\overline{n}}_x+M_{xy}^b{\overline{n}}_y\right)\frac{\partial \delta w_b}{\partial x}=0\end{array}$$

(40)

$$\begin{array}{cc}\frac{\partial \delta w_b}{\partial y}:& \left(M_{xy}^b{\overline{n}}_x+M_y^b{\overline{n}}_y\right)\frac{\partial \delta w_b}{\partial y}=0\end{array}$$

(41)

$$\begin{gathered} \begin{array}{cc}\delta w_s:& \left(\left(M_{x,x}^s+M_{xy,y}^s+M_{xz}^s-J_1\left(x\right){\ddot{u}}_o+J_2\left(x\right)\frac{\partial {\ddot{K}}_b}{\partial x}+J_2\left(x\right)\frac{\partial {\ddot{w}}_s}{\partial x}\right){\overline{n}}_x \\ +\left(M_{xy,X}^s+M_{y,y}^s+M_{yz}^s-J_1\left(x\right){\ddot{v}}_o+J_2\left(x\right)\frac{\partial {\ddot{w}}_b}{\partial y}+K_2\left(x\right)\frac{\partial {\ddot{w}}_s}{\partial y}\right){\overline{n}}_y+P\left(\overline{w}\right)\right)\delta w_s=0\end{array} \end{gathered}$$

(42)

$$\begin{array}{cc}\frac{\partial \delta w_s}{\partial x}:& \left(M_x^s{\overline{n}}_x+M_{xy}^s{\overline{n}}_y\right)\frac{\partial \delta w_s}{\partial x}=0\end{array}$$

(43)

$$\begin{array}{cc}\frac{\partial \delta w_s}{\partial y}:& \left(M_{xy}^s{\overline{n}}_x+M_y^s{\overline{n}}_y\right)\frac{\partial \delta w_s}{\partial y}=0\end{array}$$

(44)

$$P\left(\overline{w}\right)=\left(N_x^o\frac{\partial \left(\overline{w}\right)}{\partial x}+N_{xy}^o\frac{\partial \left(\overline{w}\right)}{\partial y}\right){\overline{n}}_x+\left(N_{xy}^o\frac{\partial \left(\overline{w}\right)}{\partial x}+N_y^o\frac{\partial \left(\overline{w}\right)}{\partial y}\right){\overline{n}}_y$$

(45)

Substituting Equations (2), (3) and (11) into Equation (13), the governing equations of motion in terms of displacements are

$$\begin{gathered} \begin{array}{cc}\delta u_o:& {\mathit{A}}_{\mathbf{11}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}+\left({\mathit{A}}_{\mathbf{12}}\left(\mathit{x}\right)+{\mathit{A}}_{\mathbf{66}}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}\mathit{\partial }\mathit{y}}+{\mathit{A}}_{\mathbf{66}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}+{\mathit{A}}_{\mathbf{11}}^{\prime }\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}}+{\mathit{A}}_{\mathbf{12}}^{\prime }\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }\mathit{y}} \\ -\left({\mathit{B}}_{\mathbf{12}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{B}}_{\mathbf{66}}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }\mathit{x}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-\left({\mathit{B}}_{\mathbf{12}}^{\mathit{s}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{B}}^{\mathit{s}}{}_{\mathbf{66}}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }\mathit{x}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-{\mathit{B}}_{\mathbf{11}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{3}}} \\ -{\mathit{B}}_{\mathbf{11}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{3}}}-{\mathit{B}}_{\mathbf{11}}^{\prime }\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}-{\mathit{B}}_{\mathbf{12}}^{\prime }\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-{\mathit{B}}_{\mathbf{11}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}-{\mathit{B}}_{\mathbf{12}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}} \\ ={\mathit{I}}_{\mathit{o}}\left(\mathit{x}\right){\ddot{\mathit{u}}}_{\mathit{o}}-{\mathit{I}}_{\mathbf{1}}\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathsf{ẅ}}_{\mathit{b}}}{\mathit{\partial }\mathit{x}}-{\mathit{J}}_{\mathbf{1}}\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathsf{ẅ}}_{\mathit{s}}}{\mathit{\partial }\mathit{x}}\end{array} \end{gathered}$$

(46)

$$\begin{gathered} \begin{array}{cc}\delta v_o:& {\mathit{A}}_{\mathbf{22}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}+\left({\mathit{A}}_{\mathbf{12}}\left(\mathit{x}\right)+{\mathit{A}}_{\mathbf{66}}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}\mathit{\partial }\mathit{y}}+{\mathit{A}}_{\mathbf{66}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}+{\mathit{A}}_{\mathbf{66}}^{\prime }\left(\mathit{x}\right)\left(\frac{\mathit{\partial }{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }\mathit{y}}+\frac{\mathit{\partial }{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}}\right) \\ -\left({\mathit{B}}_{\mathbf{12}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{B}}_{\mathbf{66}}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}\mathit{\partial }\mathit{y}}-\left({\mathit{B}}_{\mathbf{12}}^{\mathit{s}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{B}}_{\mathbf{66}}^{\mathit{s}}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}\mathit{\partial }\mathit{y}}-{\mathit{B}}_{\mathbf{22}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{3}}} \\ -{\mathit{B}}_{\mathbf{22}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{3}}}-\mathbf{2}{\mathit{B}}_{\mathbf{66}}^{\prime }\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }\mathit{x}\mathit{\partial }\mathit{y}}-\mathbf{2}{\mathit{B}}_{\mathbf{66}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }\mathit{x}\mathit{\partial }\mathit{y}}={\mathit{I}}_{\mathit{o}}\left(\mathit{x}\right){\ddot{\mathsf{\nu }}}_{\mathit{o}}-{\mathit{I}}_{\mathbf{1}}\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathsf{ẅ}}_{\mathit{B}}}{\mathit{\partial }\mathit{y}}-{\mathit{J}}_{\mathbf{1}}\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathsf{ẅ}}_{\mathit{s}}}{\mathit{\partial }\mathit{y}}\end{array} \end{gathered}$$

(47)

$$\begin{gathered} \begin{array}{cc}\delta w_b:& {\mathit{B}}_{\mathbf{11}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{3}}}+{\mathit{B}}_{\mathbf{22}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{3}}}+\left({\mathit{B}}_{\mathbf{12}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{B}}_{\mathbf{66}}\left(\mathit{x}\right)\right)\left(\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}+\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}\mathit{\partial }\mathit{y}}\right)+\mathbf{2}\left({\mathit{B}}_{\mathbf{12}}^{\prime }\left(\mathit{x}\right)+{\mathit{B}}_{\mathbf{66}}^{\prime }\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}\mathit{\partial }\mathit{y}}+ \\ \mathbf{2}{\mathit{B}}_{\mathbf{66}}^{\prime }\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}+\mathbf{2}{\mathit{B}}_{\mathbf{11}}^{\prime }\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}+{\mathit{B}}_{\mathbf{11}}^{\prime \prime }\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}}+{\mathit{B}}_{\mathbf{12}}^{\prime \prime }\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }\mathit{y}}-{\mathit{D}}_{\mathbf{11}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{4}}}-{\mathit{D}}_{\mathbf{22}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{4}}}-{\mathit{D}}_{\mathbf{11}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{4}}}- \\ {\mathit{D}}_{\mathbf{22}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{4}}}-\mathbf{2}\left({\mathit{D}}_{\mathbf{12}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{D}}_{\mathbf{66}}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-\mathbf{2}\left({\mathit{D}}_{\mathbf{12}}^{\mathit{s}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{D}}_{\mathbf{66}}^{\mathit{s}}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-\mathbf{2}\left({\mathit{D}}_{\mathbf{12}}^{\prime }\left(\mathit{x}\right)+\mathbf{2}{\mathit{D}}_{\mathbf{66}}^{\prime }\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }\mathit{x}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}- \\ \mathbf{2}\left({\mathit{D}}_{\mathbf{12}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)+\mathbf{2}{\mathit{D}}_{\mathbf{66}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }\mathit{x}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-\mathbf{2}{\mathit{D}}_{\mathbf{11}}^{\prime }\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{3}}}-\mathbf{2}{\mathit{D}}_{\mathbf{11}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{3}}}-{\mathit{D}}_{\mathbf{11}}^{\prime \prime }\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}-{\mathit{D}}_{\mathbf{11}}^{{\mathit{s}}^{″}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}-{\mathit{D}}_{\mathbf{12}}^{\prime \prime }\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}- \\ {\mathit{D}}_{\mathbf{12}}^{{\mathit{s}}^{″}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}+{\overline{\mathit{N}}}^{\mathit{o}}+{\mathit{K}}_{\mathit{P}}{\mathsf{\nabla }}^{\mathbf{2}}\left({\mathit{w}}_{\mathit{b}}+{\mathit{w}}_{\mathit{s}}\right)-{\mathit{K}}_{\mathit{w}}\left({\mathit{w}}_{\mathit{b}}+{\mathit{w}}_{\mathit{s}}\right)={\mathit{I}}_{\mathit{o}}\left(\mathit{x}\right)\left({\ddot{\mathit{w}}}_{\mathit{b}}+{\ddot{\mathit{w}}}_{\mathit{s}}\right)+{\mathit{I}}_{\mathbf{1}}^{\prime }\left(\mathit{x}\right){\ddot{\mathit{u}}}_{\mathit{o}}+{\mathit{I}}_{\mathbf{1}}\left(\mathit{x}\right)\left(\frac{\mathit{\partial }{\ddot{\mathit{u}}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}}+\frac{\mathit{\partial }{\ddot{\mathit{v}}}_{\mathit{o}}}{\mathit{\partial }\mathit{y}}\right)- \\ {\mathit{I}}_{\mathbf{2}}^{\prime }\left(\mathit{x}\right)\frac{\mathit{\partial }{\ddot{\mathit{w}}}_{\mathit{b}}}{\mathit{\partial }\mathit{x}}-{\mathit{I}}_{\mathbf{2}}\left(\mathit{x}\right){\mathsf{\nabla }}^{\mathbf{2}}{\ddot{\mathit{w}}}_{\mathit{b}}-{\mathit{J}}_{\mathbf{2}}^{\prime }\left(\mathit{x}\right)\frac{\mathit{\partial }{\ddot{\mathit{w}}}_{\mathit{s}}}{\mathit{\partial }\mathit{x}}-{\mathit{J}}_{\mathbf{2}}\left(\mathit{x}\right){\mathsf{\nabla }}^{\mathbf{2}}{\ddot{\mathit{w}}}_{\mathit{s}}\end{array} \end{gathered}$$

(48)

$$\begin{gathered} \begin{array}{cc}\delta w_s:& {\mathit{B}}_{\mathbf{11}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{3}}}+{\mathit{B}}_{\mathbf{22}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{3}}}+\left({\mathit{B}}_{\mathbf{12}}^{\mathit{s}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{B}}_{\mathbf{66}}^{\mathit{s}}\left(\mathit{x}\right)\right)\left(\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}+\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}\mathit{\partial }\mathit{y}}\right)+\mathbf{2}\left({\mathit{B}}_{\mathbf{12}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)+{\mathit{B}}_{\mathbf{66}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}\mathit{\partial }\mathit{y}}+ \\ \mathbf{2}{\mathit{B}}_{\mathbf{66}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}+\mathbf{2}{\mathit{B}}_{\mathbf{11}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}+{\mathit{B}}_{\mathbf{11}}^{{\mathit{s}}^{″}}\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathit{u}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}}+{\mathit{B}}_{\mathbf{12}}^{{\mathit{s}}^{″}}\left(\mathit{x}\right)\frac{\mathit{\partial }{\mathit{v}}_{\mathit{o}}}{\mathit{\partial }\mathit{y}}-{\mathit{D}}_{\mathbf{11}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{4}}}-{\mathit{D}}_{\mathbf{22}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{4}}}-{\mathit{H}}_{\mathbf{11}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{4}}}- \\ {\mathit{H}}_{\mathbf{22}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{4}}}-\mathbf{2}\left({\mathit{D}}_{\mathbf{12}}^{\mathit{s}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{D}}_{\mathbf{66}}^{\mathit{s}}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-\mathbf{2}({\mathit{H}}_{\mathbf{12}}^{\mathit{s}}\left(\mathit{x}\right)+\mathbf{2}{\mathit{H}}_{\mathbf{66}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{4}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-\mathbf{2}\left({\mathit{D}}_{\mathbf{12}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)+\mathbf{2}{\mathit{D}}_{\mathbf{66}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }\mathit{x}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}- \\ \mathbf{2}\left({\mathit{H}}_{\mathbf{12}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)+\mathbf{2}{\mathit{H}}_{\mathbf{66}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }\mathit{x}\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-\mathbf{2}{\mathit{D}}_{\mathbf{11}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{3}}}-{\mathit{H}}_{\mathbf{11}}^{{\mathit{s}}^{\prime }}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{3}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{3}}}-{\mathit{D}}_{\mathbf{11}}^{{\mathit{s}}^{″}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}-{\mathit{D}}_{\mathbf{12}}^{{\mathit{s}}^{″}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{b}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}-{\mathit{H}}_{\mathbf{11}}^{{\mathit{s}}^{″}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}- \\ {\mathit{H}}_{\mathbf{12}}^{{\mathit{s}}^{″}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}+{\mathit{A}}_{\mathbf{44}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{y}}^{\mathbf{2}}}+{\mathit{A}}_{\mathbf{55}}^{\mathit{s}}\left(\mathit{x}\right)\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }{\mathit{x}}^{\mathbf{2}}}+{\mathit{A}}_{\mathbf{55}}^{{\mathit{s}}^{\prime }}\frac{\mathit{\partial }{\mathit{w}}_{\mathit{s}}}{\mathit{\partial }\mathit{x}}+{\overline{\mathit{N}}}^{\mathit{o}}+{\mathit{K}}_{\mathit{P}}{\mathsf{\nabla }}^{\mathbf{2}}\left({\mathit{w}}_{\mathit{b}}+{\mathit{w}}_{\mathit{s}}\right)-{\mathit{K}}_{\mathit{w}}\left({\mathit{w}}_{\mathit{b}}+{\mathit{w}}_{\mathit{s}}\right)={\mathit{I}}_{\mathit{o}}\left(\mathit{x}\right){\ddot{(\mathit{w}}}_{\mathit{b}}+ \\ {\ddot{\mathit{w}}}_{\mathit{s}})+{\mathit{J}}_{\mathbf{1}}^{\prime }\left(\mathit{x}\right){\ddot{\mathit{u}}}_{\mathit{o}}+{\mathit{J}}_{\mathbf{1}}\left(\mathit{x}\right)\left(\frac{\mathit{\partial }{\ddot{\mathit{u}}}_{\mathit{o}}}{\mathit{\partial }\mathit{x}}+\frac{\mathit{\partial }{\ddot{\mathit{v}}}_{\mathit{o}}}{\mathit{\partial }\mathit{y}}\right)-{\mathit{J}}_{\mathbf{2}}^{\prime }\left(\mathit{x}\right)\frac{\mathit{\partial }{\ddot{\mathit{w}}}_{\mathit{b}}}{\mathit{\partial }\mathit{x}}-{\mathit{J}}_{\mathbf{2}}\left(\mathit{x}\right){\mathsf{\nabla }}^{\mathbf{2}}{\ddot{\mathit{w}}}_{\mathit{b}}-{\mathit{K}}_{\mathbf{2}}^{\prime }\left(\mathit{x}\right)\frac{\mathit{\partial }{\ddot{\mathit{w}}}_{\mathit{s}}}{\mathit{\partial }\mathit{x}}-{\mathit{K}}_{\mathbf{2}}\left(\mathit{x}\right){\mathsf{\nabla }}^{\mathbf{2}}{\ddot{\mathit{w}}}_{\mathit{s}}\end{array} \end{gathered}$$

(49)

The superscripts $\prime \mathrm{and}\prime \prime$ indicate the first derivative and second derivatives, respectively, w.r.t. $\mathit{x}.$ To have nontrivial solutions (Eigenfrequencies, ${\omega }_i$) for the free vibration analysis, it is essential to ignore the in-plane forces, ${\overline{N}}^o$ in Equations (46)–(49). The boundary conditions used are presented in Appendix A.

3. Solution Methodology

The derived equations of motion (Equation (15a–d)) are very complicated to be solved analytically since they are partial differential equations with variable coefficients. Thus, the efficient differential-integral quadrature method (DIQM) is modified to solve derived equations of motion. Major applications of DQM and DIQM in science and engineering are discussed in [58,59,60,61,62,63]. The applicability of DQM in the analysis of 1D problems, 2D problems, and the discretization of stress resultants and governing equations can be referred to in [24].

Using these terminologies, the stress resultants (Equations (29) and (30)) can be discretized using DIQM as

[24]

$$\left[\begin{array}{c}\begin{array}{c}{\mathit{N}}_x\\ {\mathit{N}}_y\\ {\mathit{N}}_{xy}\end{array}\\ {\mathit{M}}_x^b\\ {\mathit{M}}_y^b\\ {\mathit{M}}_{xy}^b\\ {\mathit{M}}_x^s\\ {\mathit{M}}_y^s\\ {\mathit{M}}_{xy}^s\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\mathcal{A}}_{11}\circ {\mathbb{D}}_x\\ {\mathcal{A}}_{12}\circ {\mathbb{D}}_x\\ {\mathcal{A}}_{66}\circ {\mathbb{D}}_y\end{array}\\ {\mathcal{B}}_{11}\circ {\mathbb{D}}_x\\ {\mathcal{B}}_{12}\circ {\mathbb{D}}_x\\ {\mathcal{B}}_{66}\circ {\mathbb{D}}_y\\ {\mathcal{B}}_{11}^s\circ {\mathbb{D}}_x\\ {\mathcal{B}}_{12}^s\circ {\mathbb{D}}_x\\ {\mathcal{B}}_{66}^s\circ {\mathbb{D}}_y\end{array}\begin{array}{c}\begin{array}{c}{\mathcal{A}}_{12}\circ {\mathbb{D}}_y\\ {\mathcal{A}}_{22}\circ {\mathbb{D}}_y\\ {\mathcal{A}}_{66}\circ {\mathbb{D}}_x\end{array}\\ {\mathcal{B}}_{12}\circ {\mathbb{D}}_y\\ {\mathcal{B}}_{22}\circ {\mathbb{D}}_y\\ {\mathcal{B}}_{66}\circ {\mathbb{D}}_x\\ {\mathcal{B}}_{12}^s\circ {\mathbb{D}}_x\\ {\mathcal{B}}_{22}^s\circ {\mathbb{D}}_y\\ {\mathcal{B}}_{66}^s\circ {\mathbb{D}}_x\end{array}\begin{array}{c}\begin{array}{c}-({\mathcal{B}}_{11}\circ {\mathbb{D}}_{xx}+{\mathcal{B}}_{12}\circ {\mathbb{D}}_{yy})\\ -({\mathcal{B}}_{12}\circ {\mathbb{D}}_{xx}+{\mathcal{B}}_{22}\circ {\mathbb{D}}_{yy})\\ -2{\mathcal{B}}_{66}\circ {\mathbb{D}}_{xy}\end{array}\\ -({\mathcal{D}}_{11}\circ {\mathbb{D}}_{xx}+{\mathcal{D}}_{12}\circ {\mathbb{D}}_{yy})\\ -({\mathcal{D}}_{12}\circ {\mathbb{D}}_{xx}+{\mathcal{D}}_{22}\circ {\mathbb{D}}_{yy})\\ -2{\mathcal{D}}_{66}\circ {\mathbb{D}}_{xy}\\ -({\mathcal{D}}_{11}^s\circ {\mathbb{D}}_{xx}+{\mathcal{D}}_{12}^s\circ {\mathbb{D}}_{yy})\\ -({\mathcal{D}}_{12}^s\circ {\mathbb{D}}_{xx}+{\mathcal{D}}_{22}^s\circ {\mathbb{D}}_{yy})\\ -2{\mathcal{D}}_{66}^s\circ {\mathbb{D}}_{xy}\end{array}\begin{array}{c}\begin{array}{c}-({\mathcal{B}}_{11}^s\circ {\mathbb{D}}_{xx}+{\mathcal{B}}_{12}^s\circ {\mathbb{D}}_{yy})\\ -({\mathcal{B}}_{12}^s\circ {\mathbb{D}}_{xx}+{\mathcal{B}}_{22}^s\circ {\mathbb{D}}_{yy})\\ -2{\mathcal{B}}_{66}^s\circ {\mathbb{D}}_{xy}\end{array}\\ -({\mathcal{D}}_{11}^s\circ {\mathbb{D}}_{xx}+{\mathcal{D}}_{12}^s\circ {\mathbb{D}}_{yy})\\ -({\mathcal{D}}_{12}^s\circ {\mathbb{D}}_{xx}+{\mathcal{D}}_{22}^s\circ {\mathbb{D}}_{yy})\\ -2{\mathcal{D}}_{66}^s\circ {\mathbb{D}}_{xy}\\ -({\mathcal{H}}_{11}^s\circ {\mathbb{D}}_{xx}+{\mathcal{H}}_{12}^s\circ {\mathbb{D}}_{yy})\\ -({\mathcal{H}}_{12}^s\circ {\mathbb{D}}_{xx}+{\mathcal{H}}_{22}^s\circ {\mathbb{D}}_{yy})\\ -2{\mathcal{H}}_{66}^s\circ {\mathbb{D}}_{xy}\end{array}\right]\left[\begin{array}{c}\begin{array}{c}\mathit{U}\\ \\ \\ \\ \\ \\ \mathit{V}\end{array}\\ \\ \begin{array}{c}{\mathit{W}}_{\mathit{b}}\\ \\ \\ \\ \\ \\ {\mathit{W}}_{\mathit{s}}\end{array}\end{array}\right]$$

(50)

$$\left[\begin{array}{c}{\mathit{S}}_{yz}^s\\ S_{xz}^s\end{array}\right]=\left[\begin{array}{cc}{\mathcal{A}}_{44}^s\circ \mathbb{D}y& 0\\ 0& {\mathcal{A}}_{55}^s\circ \mathbb{D}x\end{array}\right]\left[\begin{array}{c}{\mathit{W}}_{\mathit{s}}\\ {\mathit{W}}_{\mathit{s}}\end{array}\right]=\left[\begin{array}{c}{\mathcal{A}}_{44}^s\circ \mathbb{D}y\\ {\mathcal{A}}_{55}^s\circ \mathbb{D}x\end{array}\right]{\mathit{W}}_{\mathit{s}}$$

(51)

The kinetic terms in the right-hand side of Equations (33)–(36) can be represented by

$$\left\{u_0\left(x,t\right),v_0\left(x,t\right),w_b\left(x,t\right),w_s\left(x,t\right)\right\}=e^{-\omega t}\left\{U\left(x\right),V\left(x\right),W_b\left(x\right),W_s\left(x\right)\right\}$$

(52)

Finally, DQM discretization of Equations (33)–(36) can be put in matrix form

$$\left(\mathcal{K}-N_0\mathcal{N}\right)\mathit{X}={\omega }^2\mathcal{M}\mathit{X}$$

(53)

where the entries of matrices $\mathcal{K}$, $N_0,$ and $\mathcal{M}$ are the stiffness, load, and mass, respectively. For free vibration analysis, in-plane and transversal loads are neglected, and the governing equation reduces to a generalized eigenvalue problem

$$\left(\mathcal{K}-{\omega }^2\mathcal{M}\right)\mathit{X}=\mathbf{0}$$

(54)

where the element mass $\left[M\right]$ and stiffness $\left[\mathcal{K}\right]$ matrices are presented in Appendix B.

4. Numerical Results

This section is divided into two main subsections. The first one presents the validation of the present model with many previous peer works. The second subsection illustrates parametric studies to present the influence of the number of nodes of DQM on the solution conversion, effects of shear functions, boundary conditions, gradation indices, porosity types, and elastic foundation on the natural frequencies (eigenvalues) and associated mode shapes (eigenvectors) of bi-directional functionally graded porous unified plates. The material properties for metal and ceramic ingredients used in the analysis are presented in

Table 1. Material properties of BDFGM plates.

Property/ Material	Metal			Ceramic
	Aluminum $\left(\mathit{A}\mathit{l}\right)$	Stainless Steel $(\mathit{S}\mathit{u}{\mathit{s}}_3{\mathit{O}}_4)$	Coskun et al. [64]	Alumina $\left(\mathit{A}{\mathit{l}}_2{\mathit{O}}_3\right)$	Silicon Nitride $(\mathit{S}{\mathit{i}}_3{\mathit{N}}_4)$	Silicon Carbide (SiC)	Coskun et al. [64]
E(GPa)	70	201.04	1.44	380	348.43	420	14.4
$\nu$	0.3	0.24	0.38	0.3	0.3	0.3	0.3
$\rho \left(kg/m^3\right)$	2707	2370	1220	3800	8166	3800	12,200

, and the notation for boundary conditions of the plate are illustrated in Figure 2.

4.1. Model Validations

The variation in the first natural frequency for a 1D-FG SSSS square plate at different gradation indices through the thickness direction and different slenderness ratios ($\mathit{a}/\mathit{h}$) is presented in

Table 2. 1st nondimensional frequencies $\overline{\omega }=\omega h\sqrt{{\rho }_c/E_c}$ of isotropic $Al/A{l}_2{O}_3$ SSSS square plates.

$\left(\mathit{h}=1,{\mathit{n}}_{\mathit{x}}=0\right)$	$\mathit{a}/\mathit{h}$	${\mathit{n}}_{\mathit{z}}$
		0	0.5	1	4	10
Touratier	2	0.9308	0.8113	0.7364	0.5921	0.5417
Karama		0.9327	0.8127	0.7377	0.5924	0.5428
Reddy		0.9297	0.8105	0.7356	0.5924	0.5414
Thai & Kim		0.9297	0.8105	0.7356	0.5924	0.5414
Taibi		0.9294	0.8103	0.7354	0.5931	0.5417
Nguyen et al. [65]		0.9114	0.8099	0.7445	0.6165	0.5417
Matsunaga [66]		0.9400	0.8233	0.7477	0.5997	0.5460
Thai and Choi [67]		0.9265	0.8062	0.7333	0.6116	0.5644
Touratier	5	0.2113	0.1806	0.1631	0.1377	0.1301
Karama		0.2114	0.1807	0.1632	0.1377	0.1301
Reddy		0.2113	0.1806	0.1631	0.1378	0.1301
Thai&Kim		0.2113	0.1806	0.1631	0.1378	0.1301
Taibi		0.2113	0.1806	0.1631	0.1379	0.1302
Nguyen et al. [65]		0.2100	0.1808	0.1639	0.1401	0.1304
Matsunaga [66]		0.2121	0.1819	0.1640	0.1383	0.1306
Thai and Choi [67]		0.2112	0.1805	0.1631	0.1397	0.1324
Touratier	10	0.0577	0.0490	0.0442	0.0381	0.0364
Karama		0.0577	0.0490	0.0442	0.0381	0.0364
Reddy		0.0577	0.0490	0.0442	0.0381	0.0364
Thai&Kim		0.0577	0.0490	0.0442	0.0381	0.0364
Taibi		0.0577	0.0490	0.0442	0.0381	0.0364
Nguyen et al. [65]		0.0576	0.0490	0.0443	0.0383	0.0364
Matsunaga [66]		0.0578	0.0492	0.0443	0.0381	0.0364
Thai and Choi [67]		0.0577	0.0490	0.0442	0.0382	0.0366

, to validate the one-dimensional gradation with previous works published by Nguyen et al. [65], Matsunaga [66], and Thai and Choi [67]. As shown, for isotropic pure ceramics at ${\mathit{n}}_{\mathit{z}}=\mathbf{0}$, the present results are very close to the results obtained by Thai and Choi [67] and between the results of Nguyen et al. [65] and Matsunaga [66] at $\frac{\mathit{a}}{\mathit{h}}=\mathbf{2}$ and the results are identical with Thai and Choi [67] for $\frac{\mathit{a}}{\mathit{h}}>\mathbf{2}$. It is noted that, by increasing the gradation index through the thickness, the material constituent changes from fully ceramic to grade to fully metal; therefore, the stiffness decreased and hence the natural frequency also decreased. It is observed that among the different shear theories, the results from the Reddy, Karama, and Thai and Kim theories are identical; however, the Touratier and Taibi theories are overestimated and underestimated, respectively, compared to the other theories.

The validation of the 1D-FG plate model for higher modes is presented in

Table 3. The first three non-dimensional frequencies $\overline{\omega }=\omega (a^2/h)\sqrt{{\rho }_c/E_c}$ of isotropic $Al/A{l}_2{O}_3$ SSSS plates.

$\left(\mathit{h}=1,{\mathit{n}}_{\mathit{x}}=0\right)$	$\mathbf{Mode}\left(\mathit{m},\mathit{n}\right)$	${\mathit{n}}_{\mathit{z}}$
		0.5	1	2	5	8
Touratier	1(1,1)	3.1430	2.8353	2.5771	2.4402	2.3924
Karama		3.1430	2.8353	2.5771	2.4401	2.3924
Reddy		3.1429	2.8352	2.5771	2.4403	2.3925
Thai&Kim		3.1429	2.8352	2.5771	2.4403	2.3925
Taibi		3.1430	2.8353	2.5771	2.4404	2.3926
Nguyen et al. [65]		3.1457	2.8358	2.5785	2.4423	2.3933
Hosseini et al. [68]		3.1456	2.8352	2.5777	2.4425	2.3948
Reddy [69]		3.1458	2.8352	2.5771	2.4403	2.3923
Thai and Vo [70]		3.1458	2.8353	2.5771	2.4401	2.3922
Touratier	2(1,2)	5.0135	4.5228	4.1100	3.8881	3.8108
Karama		5.0136	4.5229	4.1101	3.8879	3.8108
Reddy		5.0134	4.5228	4.1100	3.8884	3.8110
Thai&Kim		5.0134	4.5228	4.1100	3.8884	3.8110
Taibi		5.0135	4.5228	4.1101	3.8887	3.8113
Nguyen et al. [65]		5.0179	4.5244	4.1136	3.8936	3.8134
Hosseini et al. [68]		5.0175	4.5228	4.1115	3.8939	3.8170
Reddy [69]		5.0180	4.5228	4.1100	3.8884	3.8107
Thai and Vo [70]		5.0180	4.5228	4.1100	3.8881	3.8105
Touratier	3(1,3)	8.1062	7.3133	6.6432	6.2753	6.1476
Karama		8.1066	7.3137	6.6434	6.2749	6.1474
Reddy		8.1061	7.3132	6.6433	6.2761	6.1481
Thai&Kim		8.1061	7.3132	6.6433	6.2761	6.1481
Taibi		8.1061	7.3133	6.6435	6.2769	6.1488
Nguyen et al. [65]		8.1133	7.3176	6.6527	6.2896	6.1547
Hosseini et al. [68]		8.1121	7.3132	6.6471	6.2903	6.1639
Reddy [69]		8.1133	7.3132	6.6433	6.2760	6.1476
Thai and Vo [70]		8.1135	7.3133	6.6432	6.2753	6.1471

. As seen, the obtained results are very close to those obtained by Reddy [69], Nguyen et al. [65], Hosseini et al. [68], and Thai and Vo [70] for the first, second, and third modes under the same conditions.

Figure 3 presents the validation of the porosity model with that developed by Coskun et al. (2019). Figure 3a illustrates the variation in the first natural frequency with porosity coefficients for the three porosity models. It is noted that by increasing the porosity coefficient the natural frequency increases in porosity Type 3, decreases in porosity Type 2, and approximately remains constant in porosity Type 1. Therefore, the effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The influence of the gradation index through thickness on the natural frequency for the different porosity models is portrayed in Figure 3b. As seen, the natural frequencies for all porosity models decreased as the gradation index increased from 0 to 2, then the reverse effect dominated (i.e., the natural frequency increased linearly by increasing the gradation index from 2 to 10). The obtained results for different porosity coefficients and gradation indices are confirmed with those obtained by Coskun et al. [64].

The effects of the foundation parameters on the first natural frequency of 1D-FG SSSS plate are presented in

Table 4. The dimensionless frequencies $\overline{\omega }=\omega h\sqrt{{\rho }_m/E_m}$ of $Al/A{l}_2{O}_3$ square plates. $K_w=\frac{k_w{a}^4}{(E_m{h}^3/12\left(1-{\nu }^2\right))}$, $K_p=\frac{k_p{a}^2}{(E_m{h}^3/12\left(1-{\nu }^2\right))}$,$a/h=5,b=a,h=17.6\mu m,n_x=0$ with SSSS boundary conditions.

	$\left({\mathit{K}}_{\mathit{w}},{\mathit{K}}_{\mathit{p}}\right)$	${\mathit{n}}_{\mathit{z}}$
		0	0.5	1	2
Touratier	$\left(0,0\right)$	0.4151	0.3548	0.3205	0.2892
Karama		0.4153	0.3550	0.3207	0.2893
Reddy		0.4150	0.3548	0.3205	0.2892
Thai&Kim		0.4150	0.3548	0.3205	0.2892
Taibi		0.4151	0.3548	0.3205	0.2893
Vu et al. [28]		0.4199	0.3603	0.3282	0.3001
Baferani et al. [71]		0.4154	0.3606	0.3299	0.3016
Touratier	$\left(0,100\right)$	0.6075	0.5856	0.5753	0.5695
Karama		0.6076	0.5857	0.5753	0.5695
Reddy		0.6075	0.5856	0.5753	0.5694
Thai&Kim		0.6075	0.5856	0.5753	0.5694
Taibi		0.6075	0.5856	0.5753	0.5695
Vu et al. [28]		0.6134	0.5916	0.5823	0.5776
Baferani et al. [71]		0.6080	0.5932	0.5876	0.5861
Touratier	$\left(100,0\right)$	0.4270	0.3700	0.3381	0.3096
Karama		0.4271	0.3701	0.3382	0.3097
Reddy		0.4269	0.3700	0.3381	0.3097
Thai&Kim		0.4269	0.3700	0.3381	0.3097
Taibi		0.4269	0.3700	0.3381	0.3097
Vu et al. [28]		0.4309	0.3745	0.3445	0.3189
Baferani et al. [71]		0.4273	0.3758	0.3476	0.3219
Touratier	$\left(100,100\right)$	0.6157	0.5949	0.5852	0.5754
Karama		0.6158	0.5950	0.5853	0.5754
Reddy		0.6156	0.5949	0.5852	0.5754
Thai&Kim		0.6156	0.5949	0.5852	0.5754
Taibi		0.6156	0.5949	0.5852	0.5754
Vu et al. [28]		0.6210	0.6004	0.5917	0.5876
Baferani et al. [71]		0.6162	0.6026	0.5978	0.5970

. By increasing the coefficient of $K_w$ and $K_p$ the natural frequency is increased due to the increase in the overall stiffness. The $K_p$ has a significant effect rather than $K_w$ on the natural frequency. As seen from

Table 5. The non-dimensional frequency $\overline{\omega }=\omega \frac{a^2}{{\pi }^2}\sqrt{{\rho }_{m}h/D}$, $D=(E_c{h}^3/12(1-v^2)$ of the Si3N4/SUS3O4 square BDFG SSSS plates $(a/h=10,n_z=1)$.

	${\mathit{n}}_{\mathit{x}}$
	0	2	5
Touratier	1.1355	0.8738	0.8191
Karama	1.1357	0.8739	0.8193
Reddy	1.1355	0.8737	0.8191
Thai&Kim	1.1355	0.8737	0.8191
Taibi	1.1355	0.8738	0.8191
Pham et al. [12]	1.1371	0.8726	0.8179
Li et al. [47]	1.1066	0.8604	0.8082

, the obtained results are very close to those obtained by Vu et al. [28] and Baferani et al. [71] at various gradation indices and elastic foundation parameters.

Table 5 is presented to verify the 2D gradation functions with previous work for the SSSS plate case. As shown, by fixing the gradation through the thickness direction $n_z=1$ and varying the material constituents through the axial direction, the natural frequency is deceased. By increasing the gradation index through the axial direction ${\mathit{n}}_{\mathit{x}}$ from 0 to 2, the fundamental frequency reduced by around 24%. The same results were obtained previously by Pham et al. [12] and Li et al. [47] with different plate theories and solution techniques.

4.2. Parametric Analysis

a. 

Convergence Study

To use a DQM in solving the proposed model, the convergence of solutions should be presented (see

Table 6. Convergence of DQM of the fundamental frequency $\left(\overline{\omega }=\omega \left(a^2/h\right)\sqrt{{\rho }_c/E_c}\right)$ for a $Al/A{l}_2{O}_3$ FG square plate for various boundary conditions.

BCs	${\mathit{N}}_{\mathit{D}\mathit{Q}}$
	8	10	12	14	16	18	20
CCCC	7.0292	6.9962	6.9898	6.9873	6.9859	6.9849	6.9841
SCSC	5.7122	5.6956	5.6933	5.6924	5.6920	5.6918	5.6915
CCCF	4.9200	4.7720	4.7020	4.6694	4.6544	4.6460	4.6392
CFCF	4.4474	4.3620	4.3200	4.3012	4.2927	4.2887	4.2899
SSSS	3.9122	3.9040	3.9027	3.9025	3.9025	3.9025	3.9025
SCSF	2.5560	2.5197	2.5173	2.5173	2.5173	2.5173	2.5173
SSSF	2.3402	2.3203	2.3188	2.3188	2.3189	2.3189	2.3189
SFSF	1.9330	1.9180	1.9171	1.9172	1.9172	1.9173	1.9173

and Figure 4). It is noted that the natural frequencies are overestimated in the case of small numbers of discretization points. The natural frequencies for all BCs are convergent after ${\mathit{N}}_{\mathit{D}\mathit{Q}}\ge 14$. Hence, ${\mathit{N}}_{\mathit{D}\mathit{Q}}=15$ is used to confirm the convergence of the solutions in the analysis.

b. 

Effect of Shear shape functions

Assessment of the first three natural frequencies $\overline{\omega }$ using transverse shear shape functions $F\left(z\right)$ in Equations (4)–(8) for SSSS plates at $a/h=5,10,20,n_x=n_z=1$ without an elastic foundation is shown in

Table 7. Comparison of the non-dimensional frequencies $\overline{\omega }=\omega \left(a^2/h\right)\sqrt{{\rho }_c/E_c}$ of isotropic $Al/A{l}_2{O}_3$. SSSS square plates. $\left(a=1,n_x=n_z=1\right)$.

	Mode	$\mathit{a}/\mathit{h}$
		5	10	20
Touratier	1(1,1)	3.4964	3.8073	3.9025
Karama		3.4979	3.8078	3.9026
Reddy		3.4958	3.8072	3.9025
Thai&Kim		3.4958	3.8072	3.9025
Taibi		3.4960	3.8072	3.9025
Touratier	2(1,2)	6.6013	9.0607	9.5980
Karama		6.6013	9.0633	9.5987
Reddy		6.6013	9.0598	9.5977
Thai&Kim		6.6013	9.0598	9.5977
Taibi		6.6013	9.0602	9.5979
Touratier	3(1,3)	6.7222	9.0717	9.6068
Karama		6.7222	9.0742	9.6075
Reddy		6.7222	9.0708	9.6065
Thai&Kim		6.7222	9.0708	9.6065
Taibi		6.7222	9.0711	9.6067

. As indicated, the results of the Reddy and Thai–Kim functions are the same for any mode and slenderness ratio. All shear functions give identical values for the third natural frequency at a/h = 5. The Karama shear function gives overestimated natural frequencies for all modes and all slenderness ratios.

c. 

Effect of Slenderness ratio

The variation in fundamental frequency $\overline{\omega }$ of a clamped plate against gradation indices $n_z\mathrm{and}{n}_x$ at different side-to-thickness ratios ($a/h$) is illustrated in Figure 5 and Figure 6. As presented, the gradation indices in both the thickness and axial directions tend to reduce the natural frequencies, due to decreasing the percentage of ceramic phase in the constituent, which causes a reduction in the stiffness. The maximum natural frequency is noticed at $n_z=n_x=0$ and the minimum natural frequency is observed at $n_z=n_x=5$. The variation in natural frequency vs. the gradation indices can be described by exponential decaying functions. From the figures, increasing in a/h tends to increase the natural frequency, which indicates the increasing stiffness of structures relative to their density.

d. 

Effect of Boundary Conditions

The variation in the natural frequency of BDFG plates vs. the boundary conditions, shear distribution functions, and axial gradation index are presented in

Table 8. Comparison of the non-dimensional frequencies $\overline{\omega }=\omega \left(a^2/h\right)\sqrt{{\rho }_c/E_c}$ of BDFG $Al/A{l}_2{O}_3$ square plates. $\left(a/h=10,b=a=1,n_x=0.5\right)$.

	BC	${\mathit{n}}_{\mathit{z}}$
		0	0.5	1	2	5
Touratier	CCCC	8.7451	7.6046	7.0130	6.5025	6.1207
Karama		8.7490	7.6079	7.0156	6.5042	6.1204
Reddy		8.7434	7.6036	7.0121	6.5020	6.1229
Thai&Kim		8.7437	7.6037	7.0117	6.5024	6.1229
Taibi		8.7445	7.6036	7.0119	6.5033	6.1260
Touratier	SCSC	7.3299	6.3311	5.8215	5.3949	5.0984
Karama		7.3324	6.3330	5.8227	5.3957	5.0983
Reddy		7.3291	6.3310	5.8208	5.3945	5.0999
Thai&Kim		7.3296	6.3308	5.8209	5.3948	5.1000
Taibi		7.3302	6.3307	5.8208	5.3954	5.1017
Touratier	CCCF	5.8655	5.1114	4.7346	4.4128	4.1799
Karama		5.8677	5.1233	4.7478	4.4075	4.1814
Reddy		5.8552	5.1197	4.7431	4.4072	4.1877
Thai&Kim		5.8662	5.1297	4.7415	4.4160	4.1792
Taibi		5.8611	5.1204	4.7434	4.4163	4.1819
Touratier	CFCF	5.3721	4.6941	4.3482	4.0508	3.8393
Karama		5.3668	4.6960	4.3495	4.0529	3.8383
Reddy		5.3757	4.6942	4.3482	4.0508	3.8397
Thai&Kim		5.3663	4.6948	4.3483	4.0521	3.8397
Taibi		5.3738	4.6951	4.3466	4.0530	3.8411
Touratier	SSSS	5.1177	4.4128	4.0581	3.7699	3.5860
Karama		5.1184	4.4133	4.0586	3.7702	3.5859
Reddy		5.1175	4.4126	4.0579	3.7700	3.5866
Thai&Kim		5.1175	4.4126	4.0579	3.7700	3.5866
Taibi		5.1176	4.4126	4.0580	3.7702	3.5873
Touratier	SCSF	3.3319	2.8695	2.6379	2.4520	2.3386
Karama		3.3321	2.8697	2.6378	2.4520	2.3384
Reddy		3.3329	2.8695	2.6379	2.4521	2.3389
Thai&Kim		3.3323	2.8694	2.6377	2.4519	2.3389
Taibi		3.3308	2.8694	2.6380	2.4522	2.3392
Touratier	SSSF	3.0791	2.6508	2.4365	2.2649	2.1611
Karama		3.0793	2.6510	2.4366	2.2650	2.1610
Reddy		3.0791	2.6507	2.4364	2.2650	2.1613
Thai&Kim		3.0791	2.6507	2.4364	2.2650	2.1613
Taibi		3.0791	2.6508	2.4365	2.2651	2.1615
Touratier	SFSF	2.5710	2.2086	2.0274	1.8832	1.7968
Karama		2.5711	2.2087	2.0275	1.8832	1.7967
Reddy		2.5709	2.2085	2.0274	1.8832	1.7969
Thai&Kim		2.5709	2.2085	2.0274	1.8832	1.7969
Taibi		2.5710	2.2086	2.0274	1.8833	1.7971

. By fixing the gradation index and shear functions, the highest natural frequency is observed in the CCCC case and the minimum is recorded for SFSF boundary conditions. It is found that the natural frequencies for CCCF, CFCF, and SSSS boundary conditions are very close to each other. Based on this table, the natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions.

The variation in the first four mode shapes with gradation indices for a CCCC BDFG plate is shown in Figure 7. The second and third modes are approximately identical from a profile perspective because the plate is symmetric in boundary conditions (CCCC) and geometrical dimensions ($b=a=1$). For each profile of the four modes, no significant effect is noticed. However, there is a small deviation in peaks’ values and their spatial coordinates due to changing of the material gradations. Figure 8 illustrates the influence of different boundary conditions on the first three mode shapes of BDFG plates at $\frac{a}{h}=10,a=b,n_x=n_z=1$. As is clear, the first, second, and third mode shapes are varied completely according to boundary conditions. Therefore, the response of the structure is dependent significantly on the type of constraints.

e. 

Effect of Geometrical Dimensions

The influence of side ratio (b/a) on the natural frequency versus gradation indices is presented in Figure 9. As shown, the natural frequency is decreased by increasing the gradation indices for both thickness and axial directions and by increasing the side ratio (b/a). It is found that the influence of side ratio (b/a) on the natural frequency is decreased for higher side ratios. This means that, by increasing b/a from 1 to 1.5, the natural frequency is reduced by 24%; however, when increasing b/a from 1.5 to 2 the natural frequency is reduced by 10%.

f. 

Effect of porosity type

The dimensionless fundamental frequencies $\overline{\omega }$ of a clamped square BDFG porous plate against gradation indices, porosity types, and porosity values $\varphi$ are presented in Figure 10 and Figure 11. As concluded from these figures, the natural frequency is dependent on the coupled effects of porosity type, porosity value, and gradation indices. At gradation index $n_x$= 0, the natural frequency increases with increasing porosity coefficient for both porosity Types 1 and 2, and the natural frequency is reduced for porosity Type 3. However, at gradation index $n_x$ = 5, by increasing the porosity value, the natural frequency is increased for porosity Type 1 and is decreased for Type 2 and Type 3.

Figure 12 illustrates the variation in fundamental frequency vs. gradation indices for different porosity types at porosity value $\varphi =0.5$. As shown in Figure 12a, the natural frequency is decreased a little bit by increasing the gradation index $n_x$ from 0 to 0.5, then it increased by increasing the gradation index for all porosity types. However, in the case of variation in the gradation index $n_z$, the natural frequency is decreased by increasing $n_z$ from 0 to 1.5 for porosity types 1 and 2 increased for $n_z>1.5$. For Type 3, the natural frequency decreases in the range of $0<n_z<1$ and increases in the other domain as illustrated in Figure 12b.

g. 

Effect of Elastic foundation parameters

The variation in fundamental frequency of SSSS BDFG plates vs. the elastic foundation parameters $K_w\mathrm{and}{K}_P$ on are represented in Figure 13. As seen in this figure, the frequency is raised by raising both $K_w\mathrm{and}{K}_P$. The effect of $K_P$ is more substantial than the effect of $K_w$ on the frequency. At $n_z=n_x=1$, by raising $K_P$ from 0 to 100, the natural frequency is raised from 3.9 to 7.55 (raised by 93%). If $K_w$ is raised from 0 to 100, the natural frequency is raised from 3.9 to 4.2 (raised by 7%).

5. Conclusions

A mathematical model for analyzing the free vibration of BDFG porous plates resting on elastic foundations is derived in this manuscript. The BDFG material varied through the axial and transverse directions is depicted by a power function with three types of porosity. From parametric studies, it can be concluded that

➢

The natural frequencies for all BCs are convergent after ${\mathit{N}}_{\mathit{D}\mathit{Q}}\ge 14$.

➢

The Reddy and Thai–Kim shear functions are identical for any modes and slenderness ratios. All shear functions give identical values for the third natural frequency at a/h = 5. The Karama shear function gives overestimated natural frequencies for all modes and all slenderness ratios.

➢

The gradation indices in both thickness and axial directions tend to reduce the natural frequencies, due to the decreasing percentage of ceramic phase in the constituent, which causes a reduction in the stiffness.

➢

The greatest natural frequency is noticed in the CCCC case and the minimum is recorded for SFSF boundary conditions. It was found that the natural frequencies for CCCF, CFCF, and SSSS boundary conditions are very close to each other.

➢

It was found that the influence of the side ratio (b/a) on the natural frequency is decreased for higher side ratios.

➢

The natural frequency is dependent on the coupled effects of porosity type, porosity value, and gradation indices.

➢

The fundamental frequency is raised by raising the elastic foundation parameters $K_w\mathrm{and}{K}_P$. However, it is observed that the impact of $K_P$ is more significant than the impact of $K_w$ on the fundamental frequency.