A Hyperbolic Shear Deformation Theory for Natural Frequencies Study of Functionally Graded Plates on Elastic Supports

Abstract

This study presents a hyperbolic shear deformation theory for free vibration of functionally graded plates on elastic foundations. The field of displacements is chosen based on the assumptions that axial and transverse displacements consist of components due to bending and shear. The components of the axial shear displacements give rise to the parabolic variation in the shear strain through the thickness, such that the shear stresses vanish on the plate boundaries. Therefore, the shear correction factor is not necessary. The material properties of the functionally graded plate are assumed to vary through the thickness according to the power law of the volume fraction of the constituents. The elastic foundation is modeled as a Pasternak foundation. The equations of motion are derived using Hamilton’s principle. The analytical solutions were established from Navier’s approach, and the results obtained are found to be in good agreement with the solutions of three-dimensional elasticity and with the solutions of the various plate theories. The effects of the power law index, the thickness ratio, and the foundation parameters on the natural frequency of the plates were also evaluated.

1. Introduction

Once the development of functional graded materials was introduced, the concept was to eliminate the interface problem that existed in traditional composite materials, and to replace it with a gradually progressive interface. The growing interest in this type of material has led to the development of different types of functionally graded materials. These materials were first designed as thermal barriers for aerospace structures, heat exchanger tubes, and fusion reactors where extremely high temperature and a large thermal gradient exist. Currently, functionally graded materials are exploited in mechanical engineering applications including nuclear, industrial and civil engineering [1,2,3,4,5,6].

In recent years, many functional gradient (FG) structures that have been applied in the different engineering fields have led to the development of various plate theories, to accurately predict the bending, buckling and vibration behaviors of functionally graded (FG) plates [7,8,9,10,11]. The classical plate theory (CPT), which is the simplest model, satisfying the Love–Kirchhoff hypothesis where the effects of transverse shear and normal deformation are neglected, is only valid for thin functionally graded structures. This theory is employed by Mohammadi et al. [12] to present an analytical method for buckling analysis of thin functionally graded (FG) rectangular plates, where the governing equations are derived using the principle of minimum total potential energy. Based on Love’s first approximation shell theory, Ebrahimi and Najafizadeh [13] studied the natural frequency characteristics of two-dimensional functionally graded circular cylindrical shells. Using the generalized differential–integral quadrature approach, the equations of motion and boundary conditions are discretized to solve the issue. The transverse vibrations and stability of a moving skew thin plate made of functionally graded ceramic–metallic material were investigated by Ruan and Wang [14]. On the framework of Love’s shell theory, the free vibration of porous laminated rotating circular cylindrical shells on simply supported boundary conditions was analysed by Ghasemi and Meskini [15].

To examine thick and moderately thick functionally graded structures, researchers developed the First-Order Shear Deformation Theory (FSDT). In this model, the in-plane displacements vary linearly through the thickness and require a shear correction factor to correct for the unrealistic variation in stresses and transverse shear strains through the thickness. By applying this theory, Bouazza et al. [16] examined the buckling of functionally graded plate under thermal loads. Two different types of thermal loadings, namely uniform temperature rise and linear temperature rise along the thickness, are applied to the plate. Golmakani and Alamatian [17] investigated nonlinear bending response of the moderately thick, radially functionally graded, solid/annular sector plates exposed to uniform and non-uniform transverse forces and supported by two-parameter elastic foundation. Ardestani et al. [18] used the first-order shear deformation theory to study concentrically and eccentrically functionally graded stiffened plates using the replicating kernel particle technique (RKPM), under uniformly distributed loads for both simply supported and clamped boundary conditions. For the bending and free vibration analysis of functionally graded plates supported by either the Winkler or Pasternak elastic foundations, a novel basic first-order shear deformation plate theory based on neutral surface position was established by Meksi et al. [19]. In order to describe the bending and dynamic behaviors of functionally graded plates, a novel first-order shear deformation theory was developed by Bellifa et al. [20]. This theory has the fewest unknowns when compared to the conventional first-order and other higher-order shear deformation theories. To analyse the dynamic behavior of functionally graded beams, Hadji et al. [21] derived a novel first-order shear deformation theory. Mantari and Granados [22] employed a novel first shear deformation theory to study a bending and free vibration of functionally graded plates supported by an elastic foundation. Only four unknowns make up this theory, which is even fewer than the traditional first-order shear deformation theory.

Although FSDT offers a fairly accurate description of the response of thin-to-moderately thick plates, it is impractical to use due to the difficulty of determining the precise value of the shear correction factor. To avoid the use of the correction factor, high-order shear strain theories (HSDT) have been developed by other researchers. Mantari and Soares [23] developed an analytical solution to the static analysis of functionally graded plates by using a new trigonometric higher-order theory, in which the stretching effect is included. Nguyen et al. [24] proposed a new inverse trigonometric shear deformation theory for static, vibration and buckling analyses of isotropic and FG sandwich plates. Abdelbari et al. [25] studied the free vibration behavior of simply supported functionally graded plates on elastic foundation based on higher-order shear deformation plate theory, in which natural frequencies were obtained by the Navier type solution method. A new hyperbolic shear and normal deformation plate theory, developed by Akavci [26], is utilised to define bending stresses, natural frequencies and buckling loads of the simply supported, functionally graded sandwich plates on elastic foundation. Benchohra et al. [27] presented a novel quasi-3D sinusoidal shear deformation theory to examine mechanical behavior of FG plates. To analyse static and free vibration responses of a functionally graded doubly curved shells, Rachid et al. [28] proposed a new formulated 2D and quasi-3D higher-order shell theory for considering the effect of transverse shear and shell thickness stretching. These theories can more accurately predict the behaviors of moderately thick functionally graded plates. They are based on a nonlinear distribution of the displacement field for a better representation of deformations and transverse shear stresses without resorting to the use transverse shear correction factors. In addition, the higher-order shear deformation theory allows you to take into account a possible warping of the cross section of the plate during deformation [29,30,31,32,33].

Many HSDTs based on non-polynomial functions have been developed recently. Meftah et al. [34] calculated natural frequencies of thick FG plates on elastic foundations by proposing a refined non-polynomial four-variable shear strain theory. A dynamic analysis of functionally graded (FG) plates by using a simple and original efficient 4-unknown quasi-3D hybrid type theory, which includes both shear deformation and thickness stretching effects, was presented by Ait Sidhoum et al. [35]. Taleb et al. [36] proposed a new refined hyperbolic shear deformation theory for free vibration investigation of simply supported functionally graded plates under thermal loads. Zaoui et al. [37] developed a new shear deformation function to analyse the free vibration of FGM plates on elastic foundations, using two-dimensional (2D) and quasi- three-dimensional (quasi-3D) shear deformation theories.

However, these theories are not widely used compared to HSDTs based on polynomial functions, except for the case of high-order sinusoidal theory (SSDT). Furthermore, most studies based on non-polynomial functions are limited to the analysis of linear problems. The development of numerical models based on non-polynomial functions is therefore necessary to also evaluate the accuracy and efficiency of HSDTs based on non-polynomial functions. Therefore, the objective of this work is to present an analytical study of the vibrational behavior of simply supported functionally graded structures resting on Winkler–Pasternak type foundations through a high-order hyperbolic theory. The specificity of this theory is that it uses a field of displacement with only four unknowns, while fulfilling the condition of zero shear stress on the free boundaries that does not require the consideration of the shear correction factor. The equilibrium equations are obtained by using Hamilton’s principle. These governing equations are then solved by Navier’s method. Therefore, fundamental frequencies are obtained by solving the eigenvalue problem.

2. Analytical Method

2.1. Basic Assumptions

Consider a rectangular functionally graded plate of length $(a)$, width $(b)$ and thickness $(h)$ resting on elastic supports of the Winkler–Pasternak type, as shown in Figure 1. The model is limited to linear elastic behavior.

The displacement field of the plate theory used is chosen based on the following assumptions:

• The displacements are small compared to the thickness of the plate and therefore the stresses involved are infinitesimal;
• The transversal normal stress ${\sigma }_z$ is negligible compared to the stresses in the plane ${\sigma }_x$ and ${\sigma }_y$;
• The transverse displacement $w$ includes two components of bending $w_b$ and shear $w_s$. These components are functions of coordinates $(x,y)$ and time only. They can be written as:

  $$w(x,y,z,t)=w_b(x,y,t)+w_s(x,y,t);$$

  (1)
• The planar displacements $U$ and $V$ consist of components of extension, bending and shear:

  $$U=u+u_b+u_s \mathrm{and} V=v+v_b+v_s;$$

  (2)
• The bending components $u_b$ and $v_b$ are assumed to be similar to the displacements given by classical plate theory. Therefore, the expressions for $u_b$ and $v_b$ can be written as:

  $$u_b=-z\frac{\partial w_b}{\partial x} \mathrm{and} v_b=-z\frac{\partial w_b}{\partial y};$$

  (3a)
• The shear components $u_s$ and $v_s$ give rise, together with $w_s$, to the parabolic variations in the shear strains ${\gamma }_{xz}$, ${\gamma }_{yz}$ and hence to the shear stresses ${\sigma }_{xz}$, ${\sigma }_{yz}$ through the thickness of the plate so that the shear stresses are zero on the upper and lower plate boundaries. Therefore, the expression for $u_s$ et $v_s$ can be given by:

  $$u_s=f(z)\frac{\partial w_b}{\partial x} \mathrm{and} v_s=f(z)\frac{\partial w_s}{\partial y};$$

  (3b)

2.2. Kinematics

Based on the assumptions made in the previous section, the displacement field can be obtained using Equations (3) as:

$$\begin{array}{l}U(x,y,z,t)=u(x,y,t)-z\frac{\partial w_b}{\partial x}+f(z)\frac{\partial w_s}{\partial x}\\ V(x,y,z,t)=v(x,y,t)-z\frac{\partial w_b}{\partial y}+f(z)\frac{\partial w_s}{\partial y}\\ W(x,y,z,t)=w_b(x,y,t)+w_s(x,y,t)\end{array}$$

(4)

The linear deformations associated to the displacements can be obtained as follows:

$$\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\end{array}\right\}=\left\{\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\epsilon }_z^0\end{array}\right\}+z\left\{\begin{array}{c}{k}_x^b\\ k_y^b\\ k_{xy}^b\end{array}\right\}+f\left\{\begin{array}{c}{k}_x^s\\ k_y^s\\ k_{xy}^s\end{array}\right\},\hspace{0.17em}\hspace{0.17em}\left\{\begin{array}{c}{\gamma }_{yz}\\ {\gamma }_{xz}\end{array}\right\}=g\left\{\begin{array}{c}{\gamma }_{yz}^s\\ {\gamma }_{xz}^s\end{array}\right\}$$

(5)

where:

$$\left\{\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial u}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\end{array}\right\},\hspace{0.17em}\hspace{0.17em}\left\{\begin{array}{c}{k}_x^b\\ k_y^b\\ k_{xy}^b\end{array}\right\}=\left\{\begin{array}{c}-\frac{{\partial }^2{w}_b}{\partial x^2}\\ -\frac{{\partial }^2{w}_b}{\partial y^2}\\ -\frac{{\partial }^2{w}_b}{\partial x\partial y}\end{array}\right\}, \left\{\begin{array}{c}{k}_x^s\\ k_y^s\\ k_{xy}^s\end{array}\right\}=\left\{\begin{array}{c}-\frac{{\partial }^2{w}_s}{\partial x^2}\\ -\frac{{\partial }^2{w}_s}{\partial y^2}\\ -2\frac{{\partial }^2{w}_s}{\partial x\partial y}\end{array}\right\},\hspace{0.17em}\hspace{0.17em}\left\{\begin{array}{c}{\gamma }_{yz}^s\\ {\gamma }_{xz}^s\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial w_s}{\partial y}\\ \frac{\partial w_s}{\partial x}\end{array}\right\},$$

(6)

$$g(z)=1-f^{\prime }(z)$$

$f(z)$ is the shape function determining the distribution of transverse shear strain and shear stress through the plate thickness. The shape functions $f(z)$ are chosen such that the boundary conditions at the top and bottom surfaces of the plate are satisfied, therefore a shear correction factor is not needed. In this study, the hyperbolic function used is presented in the following equation [38]:

$$f(z)=z-h\sinh (z/h)+z\cosh (1/2)$$

(7)

2.3. Constitutive Equations

Consider functionally graded plates made from a mixture of two materials, for example, a metal and a ceramic as shown in Figure 1. It is assumed that properties of the studied plate, such as the Young’s modulus and the density, vary throughout the thickness of the plate according to a power law distribution of the volume fraction of the two materials, as follows:

$$P(z)=P_m+(P_c-P_m){\left(\frac{1}{2}+\frac{z}{h}\right)}^p$$

(8)

where $P$ represents the effective material property, $p$ is the power law index and the subscripts $c$, $m$ represent the metallic and ceramic constituents, respectively. Since the effects of varying the Poisson’s ratio $\nu$ on the response of the functionally graded plates are very small, the Poisson’s ratio $\nu$ is generally assumed to be constant. The constitutive linear relations of a functionally graded plate can be written:

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

(9)

where:

$$c_{11}=c_{22}=\frac{E(z)}{1-{\nu }^2},\hspace{0.17em}\hspace{0.17em}{c}_{12}=\nu c_{11},\hspace{0.17em}\hspace{0.17em}{c}_{44}=c_{55}=c_{66}=\frac{E(z)}{2(1+v)}$$

(10)

2.4. Equations of Movement

The potential energy of the plate is given by:

$$\delta U_p=\frac{1}{2}{\displaystyle \underset{V}{\int }{\sigma }_{ij}}\delta {\epsilon }_{ij}dV=\frac{1}{2}{\displaystyle \underset{V}{\int }\left({\sigma }_x\delta {\epsilon }_x+{\sigma }_y\delta {\epsilon }_y+{\sigma }_{xy}\delta {\gamma }_{xy}+{\sigma }_{yz}\delta {\gamma }_{yz}+{\sigma }_{xz}\delta {\gamma }_{xz}\right)}dV$$

(11)

By substituting Equations (5) and (9) into Equation (11), and by integrating as a function of the thickness of the plate, the deformation energy of the plate can be rewritten as follows:

$$\begin{array}{l}\delta U_p=-{\displaystyle \underset{A}{\int }[\left(\frac{\partial N_x}{\partial x}+\frac{\partial N_{xy}}{\partial y}\right)\delta u+\left(\frac{\partial N_{xy}}{\partial x}+\frac{\partial N_y}{\partial y}\right)\delta v}+\left(\frac{{\partial }^2{M}_x^b}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}^b}{\partial x\partial y}+\frac{{\partial }^2{M}_y^b}{\partial y^2}\right)\delta w_b\\ \hspace{1em}\hspace{1em}+\left(\frac{{\partial }^2{M}_x^s}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}^s}{\partial x\partial y}+\frac{{\partial }^2{M}_y^s}{\partial y^2}+\frac{\partial Q_{yz}}{\partial y}+\frac{\partial Q_{xz}}{\partial x}\right)\partial w_s]dxdy\end{array}$$

(12)

where the resulting stresses N, M and Q are defined by:

$$\left(N_i,M_i^b,M_i^s\right)={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}(1,z,f){\sigma }_{i}dz},\hspace{1em}\left(i=x,y,xy\right) \mathrm{and} Q_i={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}g{\sigma }_i}dz,\left(i=xz,yz\right)$$

(13)

By substituting Equation (9) into Equation (13) and by integrating through the thickness of the plate, the stress resultants are related to the deformation by the following relations:

$$\left\{\begin{array}{c}{N}_x\\ N_y\\ N_{xy}\\ M_x^b\\ M_y^b\\ M_{xy}^b\\ M_x^s\\ M_y^s\\ M_{xy}^s\end{array}\right\}=\left\{\begin{array}{ccccccccc}{A}_{11}& A_{12}& 0& B_{11}& B_{12}& 0& B_{11}^s& B_{12}^s& 0\\ A_{12}& A_{22}& 0& B_{12}& B_{22}& 0& B_{12}^s& B_{22}^s& 0\\ 0& 0& A_{66}& 0& 0& B_{66}& 0& 0& B_{66}^s\\ B_{11}& B_{12}& 0& D_{11}& D_{12}& 0& D_{11}^s& D_{12}^s& 0\\ B_{12}& B_{22}& 0& D_{12}& D_{22}& 0& D_{12}^s& D_{22}^s& 0\\ 0& 0& B_{66}& 0& 0& D_{66}& 0& 0& D_{66}^s\\ B_{11}^s& B_{12}^s& 0& D_{11}^s& D_{12}^s& 0& H_{11}^s& H_{12}^s& 0\\ B_{12}^s& B_{22}^s& 0& D_{12}^s& D_{22}^s& 0& H_{12}^s& H_{22}^s& 0\\ 0& 0& B_{66}^s& 0& 0& D_{66}^s& 0& 0& H_{66}^s\end{array}\right\}\left\{\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\\ K_x^b\\ K_y^b\\ K_{xy}^b\\ K_x^s\\ K_y^s\\ K_{xy}^s\end{array}\right\}$$

(14a)

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

(14b)

where the stiffness terms are determined by the expressions (15) below:

$$\left\{A_{ij},B_{ij},D_{ij},B_{ij}^s,D_{ij}^s,H_{ij}^s\right\}={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{Q}_{ij}(1,z,z^2,f,zf,f^2)}dz,\hspace{1em}(i,j=1,2,6)$$

(15a)

$$A_{44}^s=A_{55}^s={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{Q}_{44}{(g(z))}^{2}dz}$$

(15b)

The deformation energy variation in the elastic foundation is expressed by the following formula:

$$\delta U_f={\displaystyle \underset{A}{\int }\left\{\left[K_w\left(w_b+w_s\right)-K_s{\nabla }^2\left(w_b+w_s\right)\right]\left(\delta w_b+\delta w_s\right)\right\}}dxdy$$

(16)

where ${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}$, $K_w$ and $K_s$ are the transverse and shear stiffness coefficients of the elastic foundation, respectively.

The kinetic energy variation in the mass system is given by:

$$\begin{array}{l}\delta T=\frac{1}{2}{\displaystyle \underset{V}{\int }\rho \left(z\right)}\left(\dot{U}\delta \dot{U}+\dot{V}\delta \dot{V}+\dot{W}\delta \dot{W}\right)dxdydz\\ \hspace{1em}={\displaystyle \underset{A}{\int }\{\left(I_0\ddot{u}-I_1\frac{\partial {\ddot{w}}_b}{\partial x}+I_2\frac{\partial {\ddot{w}}_s}{\partial x}\right)\delta u}+\left(I_0\ddot{v}-I_1\frac{\partial {\ddot{w}}_b}{\partial y}+I_2\frac{\partial {\ddot{w}}_s}{\partial y}\right)\delta v\\ \hspace{1em}\hspace{1em}+[I_0\left({\ddot{w}}_b+{\ddot{w}}_s\right)+I_1\left(\frac{\partial \ddot{u}}{\partial x}+\frac{\partial \ddot{v}}{\partial y}\right)-I_3{\nabla }^2{\ddot{w}}_b+I_5{\nabla }^2{\ddot{w}}_s]\delta w_b\\ \hspace{1em}\hspace{1em}+\left[I_0\left({\ddot{w}}_b+{\ddot{w}}_s\right)+I_2\left(\frac{\partial \ddot{u}}{\partial x}+\frac{\partial \ddot{v}}{\partial y}\right)-I_5{\nabla }^2{\ddot{w}}_b-I_4{\nabla }^2{\ddot{w}}_s\right]\delta w_s\}dxdy\end{array}$$

(17)

where $\rho \left(z\right)$ is the density given by Equation (8). The terms $\left(I_i\right)$ are the inertia mass expressed by:

$$(I_0,\hspace{0.17em}\hspace{0.17em}{I}_1,\hspace{0.17em}{I}_2,\hspace{0.17em}{I}_3,\hspace{0.17em}\hspace{0.17em}{I}_4,\hspace{0.17em}{I}_5)={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}\left(1,\hspace{0.17em}z,\hspace{0.17em}f,\hspace{0.17em}{z}^2,\hspace{0.17em}{f}^2,\hspace{0.17em}zf\hspace{0.17em}\right)\rho (z)\hspace{0.17em}}\hspace{0.17em}dz\hspace{0.17em}$$

(18)

Hamilton’s principle is used herein to find the equations of motion [39,40]. The principle can be stated in the following analytical form:

$$0={\displaystyle \underset{0}{\overset{t}{\int }}\delta \left(U_p+U_f-T\right)}dt$$

(19)

By substituting Equations (12), (16) and (17) into Equation (19) and collecting the expressions as a function of the coefficients $\delta u,\hspace{0.17em}\delta v,\hspace{0.17em}\delta w_b\hspace{0.17em}\mathrm{and}\hspace{0.17em}\delta w_s$, the equations of motion of the plate are obtained as:

$$\delta u:\frac{\partial N_x}{\partial x}+\frac{\partial N_{xy}}{\partial y}=I_0\ddot{u}-I_1\frac{\partial {\ddot{w}}_b}{\partial x}+I_2\frac{\partial {\ddot{w}}_s}{\partial x}$$

(20a)

$$\delta v:\frac{\partial N_{xy}}{\partial x}+\frac{\partial N_y}{\partial y}=I_0\ddot{v}-I_1\frac{\partial {\ddot{w}}_s}{\partial y}+I_2\frac{\partial {\ddot{w}}_s}{\partial y}$$

(20b)

$$\begin{array}{l}\delta w_b:\frac{{\partial }^2{M}_x^b}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}^b}{\partial x\partial y}+\frac{{\partial }^2{M}_y^b}{\partial y^2}-K_w\left(w_b+w_s\right)+K_s{\nabla }^2\left(w_b+w_s\right)\\ \hspace{1em}\hspace{1em}=I_0\left({\ddot{w}}_b+{\ddot{w}}_s\right)+I_1\left(\frac{\partial \ddot{u}}{\partial x}+\frac{\partial \ddot{v}}{\partial y}\right)-I_3{\nabla }^2{\ddot{w}}_b+I_5{\nabla }^2{\ddot{w}}_s\end{array}$$

(20c)

$$\begin{array}{l}\delta w_s:\frac{{\partial }^2{M}_x^s}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}^s}{\partial x\partial y}+\frac{{\partial }^2{M}_y^s}{\partial y^2}+\frac{\partial Q_{xz}}{\partial x}+\frac{\partial Q_{yz}}{\partial y}-K_w\left(w_b+w_s\right)\\ \hspace{1em}+K_s{\nabla }^2\left(w_b+w_s\right)=I_0\left({\ddot{w}}_b+{\ddot{w}}_s\right)+I_2\left(\frac{\partial \ddot{u}}{\partial x}+\frac{\partial \ddot{v}}{\partial y}\right)-I_5{\nabla }^2{\ddot{w}}_b+I_5{\nabla }^2{\ddot{w}}_s\end{array}$$

(20d)

Equation (20) can be expressed in terms of displacements $(u,\hspace{0.17em}v,\hspace{0.17em}{w}_b,\hspace{0.17em}{w}_s)$ by replacing resultants of the forces given in Equation (14). Therefore, for functionally graded plates supported on elastic foundations, the equations of motion take the following form:

$$\begin{array}{l}{A}_{11}\frac{{\partial }^{2}u}{\partial x^2}+A_{66}\frac{{\partial }^{2}u}{\partial y^2}+\left(A_{12}+A_{66}\right)\frac{{\partial }^{2}v}{\partial x\partial y}-B_{11}\frac{{\partial }^3{w}_b}{\partial x^3}\\ \hspace{1em}\hspace{0.17em}-\left(B_{12}+2B_{66}\right)\frac{{\partial }^3{w}_b}{\partial x\partial y}-B_{11}^s\frac{{\partial }^3{w}_s}{\partial x^3}-\left(B_{11}^s+2B_{66}^s\right)\frac{{\partial }^3{w}_s}{\partial x\partial y^2}\\ \hspace{1em}\hspace{0.17em}=I_0\ddot{u}-I_1\frac{\partial {\ddot{w}}_b}{\partial x}+I_2\frac{\partial {\ddot{w}}_s}{\partial x}\end{array}$$

(21a)

$$\begin{array}{l}{A}_{66}\frac{{\partial }^{2}v}{\partial x^2}+A_{22}\frac{{\partial }^{2}v}{\partial y^2}+\left(A_{12}+A_{66}\right)\frac{{\partial }^{2}u}{\partial x\partial y}-B_{22}\frac{{\partial }^3{w}_b}{\partial y^3}\\ \hspace{1em}\hspace{0.17em}-\left(B_{12}+2B_{66}\right)\frac{{\partial }^3{w}_b}{\partial x^2\partial y}-B_{22}^s\frac{{\partial }^3{w}_s}{\partial y^3}-\left(B_{12}^s+2B_{66}^s\right)\frac{{\partial }^3{w}_s}{\partial x^2\partial y}\\ \hspace{1em}\hspace{0.17em}=I_0\ddot{v}-I_1\frac{\partial {\ddot{w}}_b}{\partial y}+I_2\frac{\partial {\ddot{w}}_s}{\partial y}\end{array}$$

(21b)

$$\begin{array}{l}{B}_{11}\frac{{\partial }^{3}u}{\partial x^3}+\left(B_{12}+2B_{66}\right)\frac{{\partial }^{3}u}{\partial x\partial y^2}+B_{22}\frac{{\partial }^{3}v}{\partial y^3}+\left(B_{12}+2B_{66}\right)\frac{{\partial }^{3}v}{\partial x^2\partial y}\\ \hspace{1em}\hspace{0.17em}-D_{11}\frac{{\partial }^4{w}_b}{\partial x^4}-2\left(D_{12}+2D_{66}\right)\frac{{\partial }^4{w}_b}{\partial x^2\partial y^2}-D_{22}\frac{{\partial }^4{w}_b}{\partial y^4}\\ \hspace{0.17em}\hspace{1em}-D_{11}^s\frac{{\partial }^4{w}_s}{\partial x^4}-2\left(D_{12}^s+2D_{66}^s\right)\frac{{\partial }^4{w}_s}{\partial x^2\partial y^2}-D_{22}^S\frac{{\partial }^4{w}_S}{\partial y^4}\\ \hspace{1em}\hspace{0.17em}-K_w\left(w_b+w_s\right)+K_s{\nabla }^2\left(w_b+w_s\right)=I_0\left({\ddot{w}}_b+{\ddot{w}}_s\right)\\ \hspace{1em}\hspace{0.17em}+I_1\left(\frac{\partial \ddot{u}}{\partial x}+\frac{\partial \ddot{v}}{\partial y}\right)-I_3{\nabla }^2{\ddot{w}}_b+I_5{\nabla }^2{\ddot{w}}_S\end{array}$$

(21c)

$$\begin{array}{l}{B}_{11}^S\frac{{\partial }^{3}u}{\partial x^3}+\left(B_{12}^S+2B_{66}^S\right)\frac{{\partial }^{3}u}{\partial x\partial y^2}+\left(B_{12}^S+2B_{66}^S\right)\frac{{\partial }^{3}v}{\partial x^2\partial y}+B_{22}^S\frac{{\partial }^{3}v}{\partial y^3}\\ \hspace{1em}\hspace{0.17em}-D_{11}^S\frac{{\partial }^4{w}_b}{\partial x^4}-2\left(D_{12}^S+2D_{66}^S\right)\frac{{\partial }^4{w}_b}{\partial x^2\partial y^2}-D_{22}^S\frac{{\partial }^4{w}_b}{\partial y^4}\\ \hspace{1em}\hspace{0.17em}-H_{11}^s\frac{{\partial }^4{w}_s}{\partial x^4}-2\left(H_{12}^s+2H_{66}^s\right)\frac{{\partial }^4{w}_s}{\partial x^2\partial y^2}-H_{22}^S\frac{{\partial }^4{w}_s}{\partial y^4}\\ \hspace{1em}\hspace{0.17em}+A_{55}^s\frac{{\partial }^2{w}_s}{\partial x^2}+A_{44}^s\frac{{\partial }^2{w}_s}{\partial y^2}-K_w\left(w_b+w_s\right)+K_s{\nabla }^2\left(w_b+w_s\right)\\ \hspace{1em}\hspace{0.17em}=I_0\left({\ddot{w}}_b+{\ddot{w}}_s\right)+I_2\left(\frac{\partial \ddot{u}}{\partial x}+\frac{\partial \ddot{v}}{\partial y}\right)+I_5{\nabla }^2{\ddot{w}}_b+I_4{\nabla }^2{\ddot{w}}_S\end{array}$$

(21d)

3. Analytical Solutions for Rectangular Plates

Navier’s Approach

Given a rectangular simply supported plate having a length $(a)$ and a width $(b)$. The boundary conditions for simply supports in this study are:

$$v=w_b=w_s=N_x=M_x^b=M_x^s=0, \mathrm{on} \mathrm{the} \mathrm{edge} x=0,\hspace{0.17em}a$$

(22a)

$$u=w_b=w_s=N_y=M_y^b=M_y^s=0, \mathrm{on} \mathrm{the} \mathrm{edge} y=0,\hspace{0.17em}b$$

(22b)

Based on Navier’s approach, the following displacement expressions are chosen to satisfy the boundary conditions of Equation (22):

$$\begin{array}{l}u\left(x,y,t\right)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{U}_{mn}{e}^{i\omega t}\cos \alpha x\sin \beta y}}\\ v\left(x,y,t\right)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{V}_{mn}{e}^{i\omega t}\sin \alpha x\cos \beta y}}\\ w_b\left(x,y,t\right)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{W}_{bmn}{e}^{i\omega t}\sin \alpha x\sin \beta y}}\\ w_s\left(x,y,t\right)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{W}_{smn}{e}^{i\omega t}\sin \alpha x\sin \beta y}}\end{array}$$

(23)

where $\alpha =m\pi /a,\hspace{0.17em}\hspace{0.17em}\beta =n\pi /b,\hspace{0.17em}\hspace{0.17em}i=\sqrt{-1,}$$\left(U_{mn},V_{mn},W_{bmn},W_{smn}\right)$ are unknown coefficients to be determined and $\omega$ is the natural frequency. By substituting Equation (23) into Equation (21), the system of eigenvalue equations is presented by:

$$\left(\left[\begin{array}{cccc}{s}_{11}& s_{12}& s_{13}& s_{14}\\ s_{12}& s_{22}& s_{23}& s_{24}\\ s_{13}& s_{23}& s_{33}& s_{34}\\ s_{14}& s_{24}& s_{34}& s_{44}\end{array}\right]-{\omega }^2\left[\begin{array}{cccc}{m}_{11}& 0& m_{13}& m_{14}\\ 0& m_{22}& m_{23}& m_{24}\\ m_{13}& m_{23}& m_{33}& m_{34}\\ m_{14}& m_{24}& m_{34}& m_{44}\end{array}\right]\right)\left\{\begin{array}{c}{U}_{mn}\\ V_{mn}\\ W_{bmn}\\ W_{smn}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right\}$$

(24)

where the stiffness and mass matrix coefficients in Equation (24) are given in Equation (25). The natural frequencies of the functionally graded plate can be determined by taking the determinant of the system (28) equal to zero:

$$\begin{array}{l}{s}_{11}={\alpha }^2{A}_{11}+{\beta }^2{A}_{66}, s_{12}=\alpha \beta (A_{12}+A_{66}), s_{22}={\alpha }^2{A}_{66}+{\beta }^2{A}_{22}\\ s_{13}=-{\alpha }^3{B}_{11}-\alpha {\beta }^2(B_{12}+2B_{66}), s_{14}=-\alpha ({\alpha }^2{B}_{11}^s+{\beta }^2(B_{12}^s+2B_{66}^s))\\ s_{23}=-{\alpha }^2\beta \hspace{0.17em}(B_{12}+2B_{66})-{\beta }^3{B}_{22}, s_{24}=-\beta ((B_{12}^s+2B_{66}^s){\alpha }^2+{\beta }^2{B}_{22}^s)\\ s_{33}={\alpha }^4{D}_{11}+{\beta }^4{D}_{22}+2{\alpha }^2{\beta }^2(D_{12}+2D_{66})+k_w+k_s({\alpha }^2+{\beta }^2),\\ s_{34}={\alpha }^4{D}_{11}^s+2{\alpha }^2{\beta }^2(D_{12}^s+2D_{66}^s)+{\beta }^4{D}_{22}^s+k_w+k_s({\alpha }^2+{\beta }^2)\\ s_{44}={\alpha }^4{H}_{11}^s+{\beta }^4{H}_{22}^s+2(H_{12}^s+2H_{66}^s){\alpha }^2{\beta }^2+{\alpha }^2{A}_{55}^s+{\beta }^2{A}_{44}^s+k_w+k_s({\alpha }^2+{\beta }^2)\\ m_{11}=m_{22}=I_0, m_{13}=-\alpha \hspace{0.17em}{I}_1, m_{14}=-\alpha \hspace{0.17em}{I}_2, m_{23}=-\beta \hspace{0.17em}{I}_1\\ m_{24}=-\beta \hspace{0.17em}{I}_2, m_{33}=I_0+I_3({\alpha }^2+{\beta }^2)\\ m_{34}=I_0+I_5({\alpha }^2+{\beta }^2),m_{44}=I_0+I_4({\alpha }^2+{\beta }^2)\end{array}$$

(25)

4. Analysis of the Free Vibration of Different Types of Plates

In this section, the results of the analysis of the free vibration of the functionally graded plates are presented. Several numerical calculations have been established for homogeneous and functionally graded plates sustained on simple and elastic supports. Plates (Al/Al₂O₃) composed of an aluminum alloy (as metal) and alumina (as ceramic) are examined. To simplify calculations, the Poisson’s ratio is assumed constant at any point in the plate thickness. The Young’s modulus, the density of the ceramic and the metal used are listed in

Table 1. Physical properties of the materials used.

Materials	Properties
	$\mathit{E}$ (Gpa)	$\mathit{\rho }$(kg/m3)	$\mathit{\nu }$
Metal: Al	70	2702	1/3
Ceramic: Al2O3	380	3800	1/3
Ceramic: ZrO2	200	5700	1/3

.

Comparative studies are carried out for a wide range of plate and foundation parameters. For convenience, non-dimensional parameters are used to present the numerical results in tables and graphs:

$${\overline{K}}_w=\frac{k_w{a}^4}{D},\hspace{0.17em}{\overline{K}}_s=\frac{k_s{a}^2}{D},\hspace{0.17em}D=\frac{E{h}^3}{12(1-{\nu }^2)}$$

$$\tilde{\omega }=\omega a^2\sqrt{\rho h/D},\hspace{0.17em}\widehat{\omega }=\omega h\sqrt{\rho /E},\hspace{0.17em}\overline{\omega }=\frac{\omega a^2}{h}\sqrt{\rho /E}$$

(26)

4.1. Study of Simply Supported Isotropic Plates

The first example is established for a square homogeneous plate. The first eight dimensionless fundamental frequency modes for different geometry ratios $(a/h)$ are calculated and compared to those given by Hosseini-Hashemi et al. [41], based on a five-variable TSDT theory, and those presented by Thai and Choi [42], using a third-order theory in

Table 2. Comparison of the first eight non-dimensional frequencies $\tilde{\omega }$ of simply supported homogeneous isotropic square plate. $(b=a,p=0,{\overline{K}}_w={\overline{K}}_s=0)$.

$\mathit{h}/\mathit{a}$	Method	Modes
		1	2	3	4	5	6	7	8
0.01	Hosseini et al. [41]	19.7320	49.3032	49.3032	78.8421	98.5169	98.5169	128.0024	128.0024
	Thai and Choi [42]	19.7320	49.3032	49.3032	78.8421	98.5169	98.5169	128.0024	128.0024
	Present	19.7320	49.3031	49.3031	78.8421	98.5169	98.5169	128.0024	128.0024
0.1	Hosseini et al. [41]	19.0653	45.4869	45.4869	69.8093	85.0646	85.0646	106.7350	106.7350
	Thai and Choi [42]	19.0653	45.4869	45.4869	69.8093	85.0646	85.0646	106.7350	106.7350
	Present	19.0653	45.4869	45.4869	69.8091	85.0642	85.0642	106.7342	106.7342
0.2	Hosseini et al. [41]	17.4523	38.1883	38.1883	55.2543	65.3135	65.3135	78.9865	78.9865
	Thai and Choi [42]	17.4523	38.1883	38.1883	55.2543	65.3135	65.3135	78.9865	78.9865
	Present	17.4522	38.1876	38.1876	55.2524	65.3104	65.3104	78.9812	78.9812
0.3	Hosseini et al. [41]	15.5745	31.6413	31.6413	44.0236	51.1314	51.1314	60.6549	60.6549
	Thai and Choi [42]	15.5744	31.6413	31.6413	44.0236	51.1314	51.1314	60.6549	60.6549
	Present	15.5742	31.6396	31.6396	44.0190	51.1244	51.1244	60.6436	60.6436
0.4	Hosseini et al. [41]	13.8136	26.5910	26.5910	36.1319	41.5668	41.5668	48.8370	48.8370
	Thai and Choi [42]	13.8136	26.5908	26.5908	36.1319	41.5668	41.5668	48.8370	48.8370
	Present	13.8131	26.5877	26.5877	36.1242	41.5550	41.5550	48.8194	48.8194

. It can be seen that the results of the established model correlate well for all cases ranging from very thin to very thick plates.

The reduction in the geometric ratio leads to an increase in the natural frequency of the isotropic plate. It should be noted that the natural frequencies are almost identical for the low geometric ratios. The relative difference between the results obtained in comparison with those of Thai and Choi [42] and Hosseini et al. [41] is almost zero, even less than 0.02 for all cases. The error decreases from 3.6 × 10⁻⁴% to 3.6 × 10⁻⁵ of frequency 1 to 8 for the ratio $h/a=0.4.$

4.2. Analysis of Simply Supported Functionally Graded Plates

In this example, the results presented in

Table 3. Comparison of non-dimensional fundamental frequency $(\overline{\omega }=\omega h\sqrt{\rho /E})$ of simply supported square functionally graded plate $(Al/A{l}_2{O}_3)$.

$\mathit{b}/\mathit{a}$	$\mathit{a}/\mathit{h}$	$\mathit{p}$	Theory
			Jin et al. [43]	Mantari [30]	Thai and Choi [42]	Present
1	10	0	0.1135	0.1137	0.1134	0.1134
		1	0.0870	0.0883	0.0868	0.0868
		2	0.0789	0.0806	0.0788	0.0788
		5	0.0741	0.0756	0.0740	0.0740
	5	0	0.4169	0.4183	0.4151	0.4150
		1	0.3222	0.3271	0.3205	0.3205
		2	0.2905	0.2965	0.2892	02892
		5	0.2676	0.2726	0.2665	0.2667
	2	0	1.8470	1.8543	1.8266	1.8265
		1	1.4687	1.4803	1.4452	1.4451
		2	1.3095	1.3224	1.2891	1.2891
		5	1.1450	1.1565	1.1319	1.1320
2	10	0	0.0719	0.0719	0.0717	0.0717
		1	0.0550	0.0558	0.0549	0.0549
		2	0.0499	0.0510	0.0498	0.0498
		5	0.0471	0.0480	0.0470	0.0470
	5	0	0.2713	0.2721	0.2705	0.2704
		1	0.2088	0.2121	0.2081	0.2081
		2	01888	0.1928	0.1882	0.1882
		5	0.1754	0.1789	0.1749	0.1750
	2	0	0.9570	1.3075	1.2904	1.2904
		1	0.7937	1.0371	1.0134	1.0134
		2	0.7149	0.9297	0.9066	0.9066
		5	0.6168	0.8248	0.8070	0.8071

are established for typical $A{l}_2{O}_3$ functionally graded plates placed on simple supports without elastic foundations. Non-dimensional fundamental frequencies for different values of shape ratio $(b/a)$, geometric ratio $(a/h)$ and material index $(p)$ are calculated and compared to the exact 2D solution proposed by Jin et al. [43], the refined theory presented by Mantari [30] and the third-order theory of Thai and Choi [42]. The same boundary conditions are considered. A good agreement is found for all cases ranging from thin to thick plates.

It should be noted that excellent agreement is found for all values of the power law index p, especially with Thai and Choi [42]. It is noticeable also that the natural frequencies of the plate in ceramic material (p = 0) are identical, and are greater than those of the other type of FGM for the same ratio $a/h.$ In addition, the increase in the geometric ratio $b/a$ leads to the decrease in the natural frequency. Whatever the geometric ratio $b/a$ of the plate, the natural frequency obtained by the present HSDT theory is reduced, respectively, by 23% and 34% when the power index goes from 1 to 5 in comparison with that of the homogeneous material.

It is quite clear that the frequency becomes smaller by increasing the volume fraction of the metal, which results in the increase in the power index. On the other hand, the effect of the power index p on the natural frequency is more important for thick plates in comparison to thin plates. For the aspect ratios $b/a=1$ and $h/a=2$ (thick plate), the natural frequency is fifteen times (15) and four (4) times higher compared to medium and thin plates ($h/a=5$ and 10), respectively.

By increasing the amount of the metal (aluminum) in the FGM mixture, the structure loses more stiffness compared to the same plate with a lower volume fraction of metal (the Young’s modulus of aluminum is low compared to the one of ceramics). The reduction in the overall rigidity of the plate leads to the reduction in the natural frequencies.

4.3. Investigation of Functionally Graded Plates on Elastic Supports

Based on the current theory, the natural frequencies of functionally graded plates on elastic foundations have been evaluated.

Table 4. Comparison of non-dimensional fundamental frequencies $(\overline{\omega }=\omega h\sqrt{\rho /E})$ of functionally graded $(Al/A{l}_2{O}_3)$ plates on elastic supports.

${({\overline{\mathit{k}}}_{\mathit{w}},{\overline{\mathit{k}}}_{\mathit{s}})}^{\mathit{b}}$	$(\mathit{h}/\mathit{a})$	Method	$\mathit{p}$
			0	1	2	5
(0,0)	0.05	Baferani et al. [44]	0.0291	0.0227	0.0209	0.0197
		Thai and choi [42]	0.0291	0.0222	0.0202	0.0191
		Present	0.0291	0.0222	0.0202	0.0191
	0.1	Baferani et al. [44]	0.1134	0.0891	0.0819	0.0767
		Thai and choi [42]	0.1134	0.0868	0.0788	0.0740
		Present	0.1134	0.0868	0.0788	0.0740
	0.2	Baferani et al. [44]	0.4154	0.3299	0.03016	0.2765
		Thai and choi [42]	0.4150	0.3205	0.2892	0.2667
		Present	0.4150	0.3205	0.2892	0.2667
(0,100)	0.05	Baferani et al. [44]	0.0406	0.0382	0.0380	0.0381
		Thai and choi [42]	0.0406	0.0378	0.0374	0.0377
		Present	0.0406	0.0378	0.0374	0.0376
	0.1	Baferani et al. [44]	0.1599	0.1517	0.1508	0.1515
		Thai and choi [42]	0.1597	0.1494	0.1478	0.1487
		Present	0.1597	0.1494	0.1478	0.1486
	0.2	Baferani et al. [44]	0.6080	0.5876	0.5861	0.5879
		Thai and choi [42]	0.6075	0.5753	0.5694	0.5722
		Present	0.6074	0.5752	0.5694	0.5722
(100,0)	0.05	Baferani et al. [44]	0.0298	0.0238	0.0221	0.0210
		Thai and choi [42]	0.0298	0.0233	0.0214	0.0204
		Present	0.0298	0.0232	0.0214	0.0204
	0.1	Baferani et al. [44]	0.1162	0.0933	0.0867	0.0821
		Thai and choi [42]	0.1161	0.0910	0.0836	0.0795
		Present	0.1161	0.0910	0.0836	0.0795
	0.2	Baferani et al. [44]	0.4273	0.3476	0.3219	0.2999
		Thai and choi [42]	0.4269	0.3381	0.3097	0.2901
		Present	0.4268	0.3381	0.3096	0.2901
(100,100)	0.05	Baferani et al. [44]	0.0411	0.0388	0.0386	0.0388
		Thai and choi [42]	0.0411	0.0384	0.0381	0.0384
		Present	0.0411	0.0384	0.0381	0.0384
	0.1	Baferani et al. [44]	0.1619	0.1542	0.1535	01543
		Thai and choi [42]	0.1617	0.1519	0.1505	0.1515
		Present	0.1617	0.1519	0.1504	0.1514
	0.2	Baferani et al. [44]	0.6162	0.5978	0.5970	0.5993
		Thai and choi [42]	0.6156	0.5852	0.5800	0.5834
		Present	0.6156	0.5852	0.5800	0.5834

shows the dimensionless fundamental frequencies of rectangular plates with a geometric ratio $(h/a)$ varying from 0.05 to 0.2, a material index varying from 0 to 5, and different values of the elastic parameters which are compared with the results of Baferani et al. [44] and Thai and Choi [42]. From this table, it can be noted that a very good agreement between the solutions confirms the accuracy of the current theory in predicting the behavior in free vibration of functionally graded plates on elastic foundations.

The findings indicate that the Winkler and Pasternak foundation parameters have increasing effects on the dimensionless frequency, with the Pasternak parameter having a more significant influence than the Winkler parameter. The effect of the power law index on the dimensionless frequency is very interesting. It is observed that if a plate is supported by just the Winkler foundation, increasing the power law index decreases the dimensionless frequency. The situation is different if the plate is supported by the Pasternak foundation, regardless of the presence/absence of the Winkler foundation.

The Pasternak coefficient has a remarkable effect on the evolution of the dimensionless frequency. In the case of pure ceramic (p = 0) for a geometric ratio h/a = 0.2, the increase in the dimensional frequency is of the order of 31.7%, 2.7% and 32.5%, respectively, in the presence of the foundation parameter Winkler–Pasternak, and in comparison with the rigid foundation. The proportion of the increase in the dimensionless frequency is greater for a ductile material (p important). For a power index p = 5, this increase is of the order of 53.4%, 8.06% and 54.28%. This observation is identical for all the geometric ratios. For this purpose, the dimensionless frequencies are more important in the presence of flexible elastic supports in comparison with rigid supports.

The dimensionless frequency obtained in the presence of the Pasternak foundation is twelve times higher for a ceramic plate in comparison with the Winkler foundation, where it is just seven times higher for a ductile plate.

4.4. Parametric Study

In the following illustrations, a parametric study has been established in order to analyze the influence of different parameters on natural frequencies of functionally graded plates. Figure 2 presents the variation in the dimensionless natural frequency of a simply supported functionally graded plate as a function of the power index $(p)$ and different values of the width–thickness ratio $(a/h)$. It can be noticed on this figure that the natural frequency decreases with the increase in the material index and increases with the increase in the thickness ratio $(a/h)$. For high values of the gradient index $(p)$ and the same width–thickness ratio $(a/h)$, the natural frequency does not vary significantly. From these results, it can be observed that the more the gradient index $(p)$ increases, the more the natural frequency decreases, and this is due to the decrease in the plate’s stiffness. It can also be seen that for a fixed value of the gradient index $(p)$, as the shape ratio $(a/b)$ increases, the nominal frequency increases.

In Figure 3, the effect of elastic foundation parameters on the free vibration response of $(Al/A{l}_2{O}_3)$ functionally graded plates is presented. This figure reveals that the dimensionless fundamental frequency increases as the foundation parameters $({\overline{K}}_s,{\overline{K}}_w)$ increase. In comparison with the Winkler parameter $({\overline{K}}_w)$, the Pasternak foundation parameter $({\overline{K}}_s)$ has a dominant effect on the increase in the non-dimensional natural frequency. This behavior is because the inclusion of the foundation parameters increases the stiffness of the plate, which leads to an increase in the frequency.

In the following example, the influence of the type of material used on the variation in fundamental frequencies of functionally graded plates supported by elastic foundations is examined as shown in Figure 4. Three materials were used: one homogeneous material $(Al)$, and two functionally graded materials $(Al/A{l}_2{O}_3,\hspace{0.17em}Al/Zr{O}_2)$. It is noted that the plates made with $(Al/A{l}_2{O}_3)$ have higher natural frequency values than the plates made with $(Al/Zr{O}_2)$ or a homogeneous material; this is due to the high stiffness of the material $(Al/A{l}_2{O}_3)$ compared to the other two materials. Thus, the Winkler parameter is not very significant on the evolution of the natural frequency.

Figure 5 shows the variation in the fundamental frequency as a function of foundation parameters $({\overline{K}}_w,{\overline{K}}_s)$ and different values of the power index $(p)$. It is observed that the non-dimensional frequency decreases when the power index increases, and that the Pasternak foundation parameter $({\overline{K}}_s)$ has more effect than the Winkler parameter $({\overline{K}}_w)$. The increase in the power index leads to a decrease in the stiffness of the plate.

5. Conclusions

Through this work, the free vibration of functionally graded structures resting on simple or elastic supports has been studied. The methodology used is based on the presentation of a mathematical model using high-order shear deformation theories. This model uses a hyperbolic warping function that describes an adequate variation in strains and shear stresses and respects the boundary conditions at the lower and upper surfaces of functionally graded plates. The accuracy of the present theory is verified by comparing results obtained with those of HSDT solutions available in the literature. Thus, the presented HSDT produces results with good precision in comparison with the FSDT and other HSDTs with a higher number of unknowns. Therefore, this model can be used as a reference to verify the effectiveness of approximate numerical methods. Moreover, in industrial applications, the study of the vibration behavior is very important to choose appropriate material parameters in order to ensure that these FGM plates do not affect the structural integrity once subjected to severe external conditions. As a perspective, it would be very interesting to study the influence of manufacturing defects, namely porosity, on the overall response of functionally graded plates under different stresses and support conditions.