An exponential-trigonometric higher order shear deformation theory (HSDT) for bending,free vibration,and buckling analysis of functionally graded materials (FGMs) plates

Abstract

Two assumptions have been made based on by this proposed theory, which come from recently developed exponential–trigonometric shape function for transverse shear deformation effect and a simple higher order shear deformation theory for plate, based on a constraint between two rotational displacements of axis parallel to the plate midplane, about the axes x, y Cartesian coordinates system, which caused fewer unknown number. For the application of this method, a displacement field extended as only bending membrane for transverse displacement is used, a governing equations of motion as a result are determined according to Hamilton’s principle, and simplified using Navier analytical solutions, as well as the transverse shear stresses effect that satisfied the stress-free boundary conditions on the simply supported plate free faces as a parabolic variation along the thickness are taken into account. A functionally graded materials plates are chosen for the parametric study, where the plates are functionally graded continuously in materials through the plate thickness as a function of power law or exponential form. The aim of this study is to analyze the bending, free vibration as well as the buckling mechanical behaviors, where the results are more focused on the investigation of different parameters such as the volume fraction index, geometric ratios, frequency modes, in-plane compressive load parameters and material properties effects on the deflection, stresses, natural frequencies, and critical buckling load, which are validated in terms of accuracy and efficiency with other plate theories results found in the literature.

Introduction

The FGMs characterized by attractive mechanical properties, in which varied continuous gradation of materials, allows for a structurural arrangement to move from complete ceramics on a surface to the entire metal on the other one, which becomes a solution to avoid the problem of inhomogeneous stresses distribution (stresses concentration) and the displacement discontinuity, which generate interface problems in composite materials with fiber. Consequently, FGMs are alternative materials widely used in several industries such as aerospace, nuclear reactor, optical, civil engineering, biomechanical, automotive, electronic, chemical, and mechanical industries, ^{1 –3} and recently, FGMs find their applications in micro- and nanodevices. ^{4 –7} The investigation on FGMs under bending (static), buckling (static), and vibration (dynamic) permits to study the mechanical behaviors of these structures, by various developed mathematical and analytical models. Consequently, Zenkour ⁸ studied the bending problem with transverse load on isotropic inhomogeneous rectangular plate using two-dimensional (2-D) trigonometric and three-dimensional (3-D) elasticity solutions, based on small-strain linear elasticity theory and the Ritz method with Chebyshev displacement functions. The vibration problem of FGMs plates with multitude boundary conditions are studied by Uymaz and Aydogdu. ⁹ Carrera et al. ¹⁰ evaluated the thickness stretching effect in single-layered and multilayered FGMs structures, with various plate and shell models and grading variable rates are implemented using Carrera’s unified formulation. Wu and Li ¹¹ used the Reissner’s mixed variational theorem using third-order shear deformation theory (TSDT) for the static analysis of simply supported multilayered FGMs plates under mechanical loads. A new hyperbolic higher order shear deformation theory (HSDT) has been used in the investigation of buckling and free vibration analysis of thick FGMs sandwich plates by El Meiche et al. ¹² A new exact closed-form procedures for free vibration analysis of thick FGMs rectangular plates with simply supported two opposite edges based on the Reissner–Mindlin first shear deformation theory (FSDT) and the Reddy’s TSDT have been implemented by Hosseini-Hashemi et al. ^13,14 Thai and Choi ¹⁵ have extended the four-variable refined theory of Shimpi ¹⁶ to the buckling analysis of FGMs plates subjected to in-plane loading. Then, Bachir Bouiadjra et al. ¹⁷ analyzed the thermal buckling of thick FGMs rectangular plates with the same plate theory. A hyperbolic HSDT predicting bending and free vibration responses of FGMs plates, including a composed transverse displacement to bending, shear, and thickness stretching parts, has been presented by Belabed et al. ¹⁸ A novel quasi-3-D trigonometric shear deformation plate theory with novel displacement field including undetermined integral variable terms and stretching effect to analyze the free vibration of FGMs plates resting on elastic foundation has been proposed by Abualnour et al. ¹⁹ Analytical solutions of the static governing equations for FGMs plates subjected to transverse bi-sinusoidal and distributed loads are obtained by a new trigonometric HSDT developed by Mantari et al. ^20,21 Zenkour ²² analyzed bending responses of FGMs plates and symmetric and nonsymmetric FGMs sandwich plates with a refined TSDT in the presence of transverse shear and normal deformations. Meziane et al. ²³ developed a refined shear deformation theory to study free vibration and buckling of exponentially graded materials (E-FGMs) sandwich plates under various boundary conditions. An inverse trigonometric shear deformation theory to predict bending, buckling, and vibration of simply supported isotropic and sandwich FGMs plates has been proposed by Nguyen et al. ²⁴ Taibi et al. ²⁵ studied the thermomechanical behavior in static bending and thermal buckling of nonsymmetrical thick FGMs sandwich plates resting on two-parameter foundation with a new refined plate theory. Nguyen ²⁶ studied the effect of various well-known parameters on the bending, buckling, and vibration of FGMs plates using a new hyperbolic HSDT. Sofiyev ²⁷ used a higher order shear deformation shell theory to investigate dynamic instability of E-FGMs single-layer and sandwich cylindrical shells under static and time-dependent periodic axial loadings. Barati et al. ²⁸ have employed a refined four-variable plate theory to examine the buckling behavior of functionally graded piezoelectric plates with porosities. A new hyperbolic HSDT based on the 3-D elasticity theory is used to study the static, free vibration, and buckling of simply supported FGMs sandwichs plates on elastic foundation by Akavci. ²⁹ Zaoui et al. ³⁰ presented a new quasi-3-D hybrid-type sinusoidal and parabolic HSDT to investigate the free vibration of FGMs plates resting on elastic foundation. A new shear deformation plate theory was developed by Meksi et al. ³¹ to illustrate the bending, buckling, and free vibration responses of FGMs sandwich plates. Younsi et al. ³² analyzed bending and free vibration with 2-D and quasi-3-D hyperbolic HSDT of FGMs plates, with displacement field including undetermined integral terms. Guerroudj et al. ³³ developed a quasi-3-D hybrid-type trigonometric and polynomial HSDT and analyzed the free vibration of FGMs plates on elastic foundation with thickness stretching effects. An hybrid-type quasi-3-D HSDT, which includes both shear deformation and thickness stretching effect for static and dynamic analysis of FGMs beams and with only three unknown, has been presented by Meradjah et al. ³⁴ A new hybrid-type quasi-3-D HSDT for FGMs shells studied bending analysis with six unknowns and stretching effect, under transverse load has been formulated by Mantari. ³⁵

The aim of this investigation is to analyze the bending, free vibration, and buckling mechanical behaviors of square and rectangular FGMs plates as power (P-FGMs) and exponential (E-FGMs) function, characterized the materials properties and distributions, of the following metallic and ceramics couple combinations, aluminum/alumina (Al/Al₂O₃ ), aluminum/zirconia (Al/ZrO₂ ), and aluminum/silicon carbide (Al/Sic), continuously varied through the plate thickness. This theory is a combination of recently developed exponential–trigonometric shape function for transverse shear deformation effect and developed displacement fields of four unknowns. Analytical solutions for equations of motion are obtained based on Hamilton’s principle and Navier-type solutions that satisfied the simply supported boundary conditions, and the stress-free boundary conditions on the plate free surfaces. Parametric studies of the volume fraction index, geometric ratios such as side-to-thickness ratio and aspect ratio, frequency modes, in-plane compressive load parameters, and material properties effects are evaluated on the deflection, stresses, natural frequencies, and critical buckling load using nondimensional relations. Numerical examples are compared and validated using several and different theories found in the literature.

Materials and methods

Theoretical formulation

In this study, FGMs plates are considered with length (a), width (b), and uniform thickness (h), the evolution of the thickness follows the coordinate z-axis perpendicular to the plate midplane defined by Cartesian coordinates system (x, y) as shown in Figure 1. Three geometric ratios characterized the plates are used, as the side-to-thickness ratio (a/h) (defined thick plate with lower ratio value, moderately thick plate with medium ratio value and thin plate with highest ratio value), the aspect ratio (a/b) (defined square plate (a/b = 1), and rectangular plate (a/b ≠ 1)) and through thickness materials distribution (z/h). The plates are made as a couple mixture of one metal (Al) and one ceramic (Al₂O₃ , ZrO₂ , or Sic) as shown in Table 1, according to a defined volume fraction, and as a result to material properties, which varied continuously through the plates thickness, estimated by continuous model, which neglects the microstructure and takes into account the continuous distribution of the FGMs, the material properties are described from homogeneous plate theories, after homogenized their effective modules, such as the Young’s modulus, the Poisson’s ratio, and the mass density. ²⁶ The Young’s modulus (E(z)) has been estimated by the power law in equation (1), ^36,37 described the distribution profile of P-FGMs plate, or according to an exponential form in equation (2), ^38,39 described the distribution profile of E-FGMs plate, which can be written as follows:

$E\text{(}z\text{)}=(E_c-E_m)V_c(z)+E_c$

1

$E(z)=E_m{e}^{p\left(\frac{1}{2}+\frac{z}{h}\right)}$

2

where c and m designate the ceramic and metal plate parts, respectively. V_c (z) is the volume fraction of the ceramic material given by the equation below as

$V_c(z)={\left(\frac{1}{2}+\frac{z}{h}\right)}^P$

3

where p indicates a positive volume fraction index specifies the distribution profile of the material in the FGMs plates thickness z ∈[−h/2, h/2], when (p = 0) or (p→∞), respectively, the plate is completely from ceramic, homogeneous and stiffer or completely from metal and extremely softer, else the plate is FGMs.

Figure 1.

The FGMs plate geometric model.

Table 1.

Metal and ceramics material properties.

Materials		Material properties
		Young’s modulus (GPa)	Mass density (kg/m3)	Poisson’s ratio
Metal	Aluminium (Al)	70	2702	0.3
Ceramic	Zirconia (ZrO2 )	151	3000	0.3
	Alumina (Al2O3 )	380	3800	0.3
	Silicon carbide (SiC)	420	3210	0.3

The Poisson’s ratio ν(z) is considered constant, ⁴⁰ because it has no significant effect on the FGMs plates, and the mass density ρ(z) is estimated using the power law as

$\rho (z)=\left({\rho }_c-{\rho }_m\right)V_c(z)+{\rho }_c$

4

Kinematics and deformations

The displacement field of this HSDT can be presented following similar procedures as given by Nguyen, ²⁶ as

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

5a

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

5b

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

5c

where u, v, w, u₀ , v₀ , and w₀ are axial displacement function follows the Cartesian coordinate axes (x, y and z), the displacement field has five unknowns, three of them are displacements of the plate midplane u₀ , v₀ , w₀ , and two displacements θ_x and θ_y are rotations of axes parallel to the midplane around y and x rectangular Cartesian coordinates system, respectively. The rotational displacements are reduced to a single unknown by making ${\theta }_x=-\partial \phi \left(x,y,t\right)/\partial x$ and ${\theta }_y=-\partial \phi \left(x,y,t\right)/\partial y$ . ⁴¹ The function f(z) represents the exponential–trigonometric shape function for transverse shear deformation distribution along the thickness (h), developed by Zaoui et al., ⁴² which is shown in equation (6a). The shear deformation effect becomes less effective and even neglected in a relatively large region, for thin plates with larger (a/h) ratio. Furthermore, the transverse shear stresses through the thickness are presented in term of the shape function derivative g(z) and have a parabolic pace satisfying the stress-free boundary conditions at the plate faces (z = ±h/2).

$f\left(z\right)=\frac{\pi h}{{\pi }^4+h^4}\left(e^{\left(hz/\pi \right)}\left({\pi }^2\text{sin}\hspace{0.17em}\left(\frac{\pi z}{h}\right)+h^2\text{cos}\hspace{0.17em}\left(\frac{\pi z}{h}\right)\right)-h^2\right)$

6a

$g(z)=-\frac{\partial f(z)}{\partial z}$

6b

Applying the linear elasticity theory, the linear strain field is produced from the displacement field of equation (5) by derivation as follows:

$\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{yz}\\ {\gamma }_{xz}\\ {\gamma }_{xy}\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial u_0}{\partial x}\\ \frac{\partial v_0}{\partial y}\\ 0\\ 0\\ \frac{\partial u_0}{\partial y}+\frac{\partial v_0}{\partial x}\end{array}\right\}-z\left\{\begin{array}{c}\frac{{\partial }^2{w}_0}{\partial x^2}\\ \frac{{\partial }^2{w}_0}{\partial y^2}\\ 0\\ 0\\ 2\frac{{\partial }^2{w}_0}{\partial x\partial y}\end{array}\right\}-\left\{\begin{array}{c}f(z)\frac{{\partial }^2\phi }{\partial x^2}\\ f(z)\frac{{\partial }^2\phi }{\partial y^2}\\ -g(z)\frac{\partial \phi }{\partial y}\\ -g(z)\frac{\partial \phi }{\partial x}\\ 2f(z)\frac{{\partial }^2\phi }{\partial x\partial y}\end{array}\right\}$

7

where the FGMs plates stress–strain relations are presented as

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

8

where Q(z) are the stiffness coefficients, written as

$Q_{11}\left(z\right)=Q_{22}\left(z\right)=\frac{E\left(z\right)}{1-{\nu }^2\left(z\right)}$

9a

$Q_{12}\left(z\right)=\nu \left(z\right)Q_{11}\left(z\right)$

9b

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

9c

Equations of motion

The equations of motion appropriate to the displacement field and the constitutive equations are determined for deformable bodies using the Hamilton’s principle, which comes as

$0={\displaystyle \underset{0}{\overset{T}{\int }}\left(\delta U+\delta V-\delta K\right)}dt$

10

where T denotes a period of time and δU, δV, and δK are the variation of the plate strain energy, work done by external load, and kinetic energy, respectively.

The strain energy (energy of internal load) variation is calculated by the equations below as

$$\begin{gathered} \delta U={\displaystyle \underset{V}{\int }\text{(}{\sigma }_x\hspace{0.17em}\delta {\epsilon }_x+{\sigma }_y\hspace{0.17em}\delta {\epsilon }_y+{\sigma }_z\hspace{0.17em}\delta {\epsilon }_z+{\tau }_{xy}\hspace{0.17em}\delta {\gamma }_{xy} \\ +{\tau }_{yz}\hspace{0.17em}\delta {\gamma }_{yz}+{\tau }_{xz}\hspace{0.17em}\delta {\gamma }_{xz}\text{) }dV} \end{gathered}$$

11a

$$\begin{gathered} \delta U={\displaystyle \underset{A}{\int }[N_x\frac{\partial \delta u_0}{\partial x}-M_x^b\frac{{\partial }^2\delta w_0}{\partial x^2}-M_x^s\frac{{\partial }^2\delta \phi }{\partial x^2}+N_y\frac{\partial \delta v_0}{\partial y} \\ -M_y^b\frac{{\partial }^2\delta w_0}{\partial y^2}-M_y^s\frac{{\partial }^2\delta \phi }{\partial y^2}}+N_{xy}\left(\frac{\partial \delta u_0}{\partial y}+\frac{\partial \delta v_0}{\partial x}\right) \\ -2M_{xy}^b\frac{{\partial }^2\delta w_0}{\partial x\partial y}-2M_{xy}^s\frac{{\partial }^2\delta \phi }{\partial x\partial y}+S_x\frac{\partial \delta \phi }{\partial x}+S_y\frac{\partial \delta \phi }{\partial y}]dA \end{gathered}$$

11b

where A denotes a section and N, M, and S are the stresses resultants defined as

$$\begin{gathered} \left[\begin{array}{ccccc}{N}_x& N_y& 0& 0& N_{xy}\\ M_x^b& M_y^b& 0& 0& M_{xy}^b\\ M_x^s& M_y^s& 0& 0& M_{xy}^s\\ 0& 0& S_y& S_x& 0\end{array}\right] \\ ={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}\left\{\begin{array}{c}1\\ z\\ f(z)\\ g(z)\end{array}\right\}\left\{\begin{array}{ccccc}{\sigma }_x& {\sigma }_y& {\tau }_{yz}& {\tau }_{xz}& {\tau }_{xy}\end{array}\right\}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}dz \end{gathered}$$

12

The variation of work performed by external loads is written as

$\delta V=-{\displaystyle \underset{A}{\overset{\hspace{0.17em}}{\int }}\left(\overline{N}+q\right)}\delta w_{0}dA$

13

where $\overline{N}$ is an in-plane compressive load; it is assumed that the plate is subjected to axial compressive load in two directions, as mentioned below

$\overline{N}=N_x^0\frac{{\partial }^2{w}_0}{\partial x^2}+2N_{xy}^0\frac{{\partial }^2{w}_0}{\partial x\partial y}+N_y^0\frac{{\partial }^2{w}_0}{\partial y^2}$

14

where $N_x^0=-N_0^{\hspace{0.17em}},\hspace{0.17em}{N}_y^0=-\gamma N_0^{\hspace{0.17em}},\hspace{0.17em}{N}_{xy}^0=0$ and γ is the nondimensional load parameter, defined as three in-plane load types: axial compression-tension load (γ = 0), uniaxial compression load (γ = 1), or biaxial compression load (γ = −1), and q is the transverse mechanical sinusoidal load, which follows the plate transversal direction.

The kinetic energy variation is determined by the following integral as

$\delta K={\displaystyle \underset{A}{\overset{\hspace{0.17em}}{\int }}{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}\left(\frac{\partial u}{\partial t}\frac{\partial \delta u}{\partial t}+\frac{\partial v}{\partial t}\frac{\partial \delta v}{\partial t}+\frac{\partial w}{\partial t}\frac{\partial \delta w}{\partial t}\right)}}\rho \left(z\right)dAdz$

15a

$\begin{array}{l}\delta K={\displaystyle \underset{A}{\overset{\hspace{0.17em}}{\int }}[I_0\left(\frac{\partial u}{\partial t}\frac{\partial \delta u}{\partial t}+\frac{\partial v}{\partial t}\frac{\partial \delta v}{\partial t}+\frac{\partial w}{\partial t}\frac{\partial \delta w}{\partial t}\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}-I_1\left(\frac{\partial u_0}{\partial t}\frac{{\partial }^2\delta w_0}{\partial t\partial x}+\frac{{\partial }^2{w}_0}{\partial t\partial x}\frac{\partial \delta u_0}{\partial t}+\frac{\partial v_{.0}}{\partial t}\frac{{\partial }^2\delta w_0}{\partial t\partial y}+\frac{{\partial }^2{w}_0}{\partial t\partial y}\frac{\partial \delta v_0}{\partial t}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-I_2\left(\frac{{\partial }^2{w}_0}{\partial t\partial x}\frac{{\partial }^2\delta w_0}{\partial t\partial x}+\frac{{\partial }^2{w}_0}{\partial t\partial y}\frac{{\partial }^2\delta w_0}{\partial t\partial y}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-J_1\left(\frac{\partial u_0}{\partial t}\frac{{\partial }^2\delta \phi }{\partial t\partial x}+\frac{{\partial }^2\phi }{\partial t\partial x}\frac{\partial \delta u_0}{\partial t}+\frac{\partial v_{.0}}{\partial t}\frac{{\partial }^2\delta \phi }{\partial t\partial y}+\frac{{\partial }^2\phi }{\partial t\partial y}\frac{\partial \delta v_0}{\partial t}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+J_2\left(\frac{{\partial }^2{w}_0}{\partial t\partial x}\frac{{\partial }^2\delta \phi }{\partial t\partial x}+\frac{{\partial }^2\phi }{\partial t\partial x}\frac{{\partial }^2\delta w_0}{\partial t\partial x}+\frac{{\partial }^2{w}_0}{\partial t\partial y}\frac{{\partial }^2\delta \phi }{\partial t\partial y}+\frac{{\partial }^2\phi }{\partial t\partial y}\frac{{\partial }^2\delta w_0}{\partial t\partial y}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+K_2\left(\frac{{\partial }^2\phi }{\partial t\partial x}\frac{{\partial }^2\delta \phi }{\partial t\partial x}+\frac{{\partial }^2\phi }{\partial t\partial y}\frac{{\partial }^2\delta \phi }{\partial t\partial y}\right)]dA\end{array}$

15b

where I_i , J_i , and K_i are the moments of inertia expressed as

$\left(I_0,I_1,I_2,J_1,J_2,K_2\right)={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}\rho \left(z\right)\left[1,z,z^2,f\left(z\right),zf\left(z\right),f^2\left(z\right)\right]dz}$

16

The equations of motion are found by substituting equations (11), (13), and (15) in equation (10), integrating by part every term, than collecting the coefficients δu₀ , δv₀ , δw₀ , and δφ as follows

$\delta u_0:\frac{N_x}{\partial x}+\frac{N_{xy}}{\partial y}=I_0\frac{{\partial }^2{u}_0}{\partial t^2}-I_1\frac{{\partial }^3{w}_0}{\partial t^2\partial x}-J_1\frac{{\partial }^3\phi }{\partial t^2\partial x}$

17a

$\delta v_0:\hspace{0.17em}\frac{N_{xy}}{\partial x}+\frac{N_y}{\partial y}=I_0\frac{{\partial }^2{v}_0}{\partial t^2}-I_1\frac{{\partial }^3{w}_0}{\partial t^2\partial y}-J_1\frac{{\partial }^3\phi }{\partial t^2\partial y}$

17b

$\begin{array}{l}\delta w_0:\frac{{\partial }^2{M}_x^b}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}^b}{\partial x\partial y}+\frac{\partial M_y^b}{\partial y^2}+\left(\overline{N}+q\right)=I_0\frac{{\partial }^2{w}_0}{\partial t^2}\\ +I_1\left(\frac{{\partial }^3{u}_0}{\partial t^2\partial x}+\frac{{\partial }^3{v}_0}{\partial t^2\partial y}\right)-I_2\left(\frac{{\partial }^4{w}_0}{\partial t^2{\partial }^{2}x}+\frac{{\partial }^4{w}_0}{\partial t^2{\partial }^{2}y}\right)\\ -J_2\left(\frac{{\partial }^4\phi }{\partial t^2{\partial }^{2}x}+\frac{{\partial }^4\phi }{\partial t^2{\partial }^{2}y}\right)\end{array}$

17c

$\begin{array}{l}\delta \phi :\frac{{\partial }^2{M}_x^s}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}^s}{\partial x\partial y}+\frac{\partial M_y^s}{\partial y^2}+\frac{\partial S_x}{\partial x}+\frac{\partial S_y}{\partial y}\\ =J_1\left(\frac{{\partial }^3{u}_0}{\partial t^2\partial x}+\frac{{\partial }^3{v}_0}{\partial t^2\partial y}\right)-J_2\left(\frac{{\partial }^4{w}_0}{\partial t^2{\partial }^{2}x}+\frac{{\partial }^4{w}_0}{\partial t^2{\partial }^{2}y}\right)\\ -K_2\left(\frac{{\partial }^4\phi }{\partial t^2{\partial }^{2}x}+\frac{{\partial }^4\phi }{\partial t^2{\partial }^{2}y}\right)\end{array}$

17d

The stresses resultants are obtained as a function of strains, and the transverse shear forces found from the constitutive equations, while substituting equation (7) into equation (8), then into equation (12), the following reduced and compact form is obtained

$\left\{\begin{array}{c}{N}_x\\ N_y\\ N_{xy}\\ M_x^b\\ M_y^b\\ M_{xy}^b\\ M_x^s\\ M_y^s\\ M_{xy}^s\\ S_{yz}^s\\ S_{xz}^s\end{array}\right\}=\left[\begin{array}{ccccccccccc}{A}_{11}& A_{12}& 0& B_{11}& B_{12}& 0& B_{11}^s& B_{12}^s& 0& 0& 0\\ A_{12}& A_{22}& 0& B_{12}& B_{22}& 0& B_{12}^s& B_{22}^s& 0& 0& 0\\ 0& 0& A_{66}& 0& 0& B_{66}& 0& 0& B_{66}^s& 0& 0\\ B_{11}& B_{12}& 0& D_{11}& D_{12}& 0& D_{11}^s& D_{12}^s& 0& 0& 0\\ B_{12}& B_{22}& 0& D_{12}& D_{22}& 0& D_{12}^s& D_{22}^s& 0& 0& 0\\ 0& 0& B_{66}& 0& 0& D_{66}& 0& 0& D_{66}^s& 0& 0\\ B_{11}^s& B_{12}^s& 0& D_{11}^s& D_{12}^s& 0& H_{11}^s& H_{12}^s& 0& 0& 0\\ B_{12}^s& B_{22}^s& 0& D_{12}^s& D_{22}^s& 0& H_{12}^s& H_{22}^s& 0& 0& 0\\ 0& 0& B_{66}^s& 0& 0& D_{66}^s& 0& 0& H_{66}^s& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& A_{44}^s& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& A_{55}^s\end{array}\right]\left\{\begin{array}{c}\frac{\partial u_0}{\partial x}\\ \frac{\partial v_0}{\partial y}\\ \frac{\partial u_0}{\partial y}+\frac{\partial v_0}{\partial x}\\ -\frac{{\partial }^2{w}_0}{\partial x^2}\\ -\frac{{\partial }^2{w}_0}{\partial y^2}\\ -2\frac{{\partial }^2{w}_0}{\partial x\partial y}\\ -\frac{{\partial }^2\phi }{\partial x^2}\\ -\frac{{\partial }^2\phi }{\partial y^2}\\ -2\frac{{\partial }^2\phi }{\partial x\partial y}\\ \frac{\partial \phi }{\partial y}\\ \frac{\partial \phi }{\partial x}\end{array}\right\}$

18

where A, B, D, B^s , D^s , H^s and A^s are FGMs plate stiffness parameters, given as

$\left(A,B,D,H^s,D^s,B^s,A_{\hspace{0.17em}}^s\right)={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}\left[1,z,z^2,f\left(z\right),zf\left(z\right),f^2\left(z\right),g^2\left(z\right)\right]}\cdot Q\left(z\right)dz$

The following equations of motion can be expressed and simplified in terms of displacement field by substituting equation (18) into equation (17), as follows

$\begin{array}{l}{A}_{11}^{\hspace{0.17em}}\frac{{\partial }^2{u}_0}{\partial x^2}+A_{66}^{\hspace{0.17em}}\frac{{\partial }^2{u}_0}{\partial y^2}+\left(A_{12}^{\hspace{0.17em}}+A_{66}^{\hspace{0.17em}}\right)\frac{{\partial }^2{v}_0}{\partial x\partial y}-B_{11}^{\hspace{0.17em}}\frac{{\partial }^3{w}_0}{\partial x^3}-\left(B_{12}^{\hspace{0.17em}}+2B_{66}^{\hspace{0.17em}}\right)\frac{{\partial }^3{w}_0}{\partial x\partial y^2}\\ \hspace{1em}-B_{11}^s\frac{{\partial }^3\phi }{\partial x^3}-\left(B_{12}^s+2B_{66}^s\right)\frac{{\partial }^3\phi }{\partial x\partial y^2}=I_0\frac{{\partial }^2{u}_0}{\partial t^2}-I_1\frac{{\partial }^3{w}_0}{\partial t^2\partial x}-J_1\frac{{\partial }^3\phi }{\partial t^2\partial x}\end{array}$

19a

$\begin{array}{l}{A}_{22}^{\hspace{0.17em}}\frac{{\partial }^2{v}_0}{\partial y^2}+A_{66}^{\hspace{0.17em}}\frac{{\partial }^2{v}_0}{\partial x^2}+\left(A_{12}^{\hspace{0.17em}}+A_{66}^{\hspace{0.17em}}\right)\frac{{\partial }^2{u}_0}{\partial x\partial y}-B_{22}^{\hspace{0.17em}}\frac{{\partial }^3{w}_0}{\partial y^3}-\left(B_{12}^{\hspace{0.17em}}+2B_{66}^{\hspace{0.17em}}\right)\frac{{\partial }^3{w}_0}{\partial x^2\partial y}\\ \hspace{1em}-B_{22}^s\frac{{\partial }^3\phi }{\partial y^3}-\left(B_{12}^s+2B_{66}^s\right)\frac{{\partial }^3\phi }{\partial x^2\partial y}=I_0\frac{{\partial }^2{v}_0}{\partial t^2}-I_1\frac{{\partial }^3{w}_0}{\partial t^2\partial y}-J_1\frac{{\partial }^3\phi }{\partial t^2\partial y}\end{array}$

19b

$\begin{array}{l}{B}_{11}^{\hspace{0.17em}}\frac{{\partial }^3{u}_0}{\partial x^3}+\left(B_{12}^{\hspace{0.17em}}+2B_{66}^{\hspace{0.17em}}\right)\frac{{\partial }^3{u}_0}{\partial x\partial y^2}+\left(B_{12}^{\hspace{0.17em}}+2B_{66}^{\hspace{0.17em}}\right)\frac{{\partial }^3{v}_0}{\partial x^2\partial y}-B_{22}^{\hspace{0.17em}}\frac{{\partial }^3{v}_0}{\partial y^3}-D_{11}^{\hspace{0.17em}}\frac{{\partial }^4{w}_0}{\partial x^4}\\ \hspace{1em}-D_{22}^{\hspace{0.17em}}\frac{{\partial }^4{w}_0}{\partial y^4}-2\left(D_{12}^{\hspace{0.17em}}+2D_{66}^{\hspace{0.17em}}\right)\frac{{\partial }^4{w}_0}{\partial x^2\partial y^2}-D_{11}^s\frac{{\partial }^4\phi }{\partial x^4}-D_{22}^s\frac{{\partial }^4\phi }{\partial y^4}\\ \hspace{1em}-2\left(D_{12}^s+2D_{66}^s\right)\frac{{\partial }^4\phi }{\partial x^2\partial y^2}+\overline{N}+q=I_0\frac{{\partial }^2{w}_0}{\partial t^2}+I_1\left(\frac{{\partial }^3{u}_0}{\partial t^2\partial x}+\frac{{\partial }^3{v}_0}{\partial t^2\partial y}\right)\\ \hspace{1em}-I_2\left(\frac{{\partial }^4{w}_0}{\partial t^2{\partial }^{2}x}+\frac{{\partial }^4{w}_0}{\partial t^2{\partial }^{2}y}\right)-J_2\left(\frac{{\partial }^4\phi }{\partial t^2{\partial }^{2}x}+\frac{{\partial }^4\phi }{\partial t^2{\partial }^{2}y}\right)\end{array}$

19c

$\begin{array}{l}{B}_{11}^s\frac{{\partial }^3{u}_0}{\partial x^3}+\left(B_{12}^s+2B_{66}^s\right)\frac{{\partial }^3{u}_0}{\partial x\partial y^2}+\left(B_{12}^s+2B_{66}^s\right)\frac{{\partial }^3{v}_0}{\partial x^2\partial y}-B_{22}^s\frac{{\partial }^3{v}_0}{\partial y^3}-D_{11}^s\frac{{\partial }^4{w}_0}{\partial x^4}\\ \hspace{1em}-D_{22}^s\frac{{\partial }^4{w}_0}{\partial y^4}-2\left(D_{12}^s+2D_{66}^s\right)\frac{{\partial }^4{w}_0}{\partial x^2\partial y^2}+A_{55}^s\frac{{\partial }^2\phi }{\partial x^2}+A_{44}^s\frac{{\partial }^2\phi }{\partial y^2}-H_{11}^s\frac{{\partial }^4\phi }{\partial x^4}\\ \hspace{1em}-2\left(H_{12}^s+2H_{66}^s\right)\frac{{\partial }^4\phi }{\partial x^2\partial y^2}-H_{22}^s\frac{{\partial }^4\phi }{\partial y^4}=J_1\left(\frac{{\partial }^3{u}_0}{\partial t^2\partial x}+\frac{{\partial }^3{v}_0}{\partial t^2\partial y}\right)\\ \hspace{1em}-J_2\left(\frac{{\partial }^4{w}_0}{\partial t^2{\partial }^{2}x}+\frac{{\partial }^4{w}_0}{\partial t^2{\partial }^{2}y}\right)-K_2\left(\frac{{\partial }^4\phi }{\partial t^2{\partial }^{2}x}+\frac{{\partial }^4\phi }{\partial t^2{\partial }^{2}y}\right)\end{array}$

19d

Analytical solution of Navier

The procedure of Navier solutions is used to obtain analytical solutions that are the partial differential equations in functions of displacement and expressed by double Fourier series for the shear deformation model, which is developed in this plate theory, satisfies the simply supported boundary conditions and equations of motion, written as

$u_0\left(x,y,t\right)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}{U}_{mn}{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{cos}{\left(\alpha x\right)}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{sin}\left(\beta y\right)\cdot e^{i\omega t}}}$

20a

$v_0\left(x,y,t\right)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}{V}_{mn}{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{sin}{\left(\alpha x\right)}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{cos}\left(\beta y\right)\cdot e^{i\omega t}}}$

20b

$w_0\left(x,y,t\right)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}{W}_{mn}{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{sin}{\left(\alpha x\right)}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{sin}\left(\beta y\right)\cdot e^{i\omega t}}}$

20c

$\phi \left(x,y,t\right)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}{\phi }_{mn}{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{sin}{\left(\alpha x\right)}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{sin}\left(\beta y\right)\cdot e^{i\omega t}}}$

20d

where ω is the natural frequency of the plate free vibration, U_mn , V_mn , W_mn , and φ_mn are undetermined parameters, $\alpha =m\pi /a$ , $\beta =n\pi /b$ and $\sqrt{i}=-1$ , m and n defined the first three modes (m, n) of frequency as 1(1,1), 2(1,2), and 3(2,2). The transverse load q is also represented by double Fourier series as

$q\left(x,y\right)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}{Q}_{mn}{\hspace{0.17em}}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{sin}{\left(\alpha x\right)}_{\hspace{0.17em}}^{\hspace{0.17em}}\text{sin}\left(\beta y\right)}}$

21a

where the coefficient Q_mn is given below for some typical loads as

$Q_{{\hspace{0.17em}}_{mn}}^{\hspace{0.17em}}=\frac{4}{ab}\hspace{0.17em}\hspace{0.17em}{\displaystyle \underset{0}{\overset{a}{\int }}{\displaystyle \underset{0}{\overset{b}{\int }}\text{sin}\hspace{0.17em}\hspace{0.17em}\left(\alpha x\right)\hspace{0.17em}\text{sin}\hspace{0.17em}\hspace{0.17em}\left(\beta y\right)}}\hspace{0.17em}dx\hspace{0.17em}dy$

21b

$=\{\begin{array}{c}{q}_0\hspace{1em}\hspace{1em}\text{For} \text{sinusoidal} \text{distributed} \text{load.\hspace{1em}\hspace{1em}(21c)}\\ \frac{16\hspace{0.17em}\hspace{0.17em}{q}_0}{mn{\pi }^2}\hspace{1em}\text{For} \text{uniformly} \text{distributed} \text{load.\hspace{1em}\hspace{1em}(21d)}\end{array}$

Substituting equations (20) (21a), and (21b) into equation (19), the following equivalent system is obtained

$\left(\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ a_{12}& a_{22}& a_{23}& a_{24}\\ a_{13}& a_{23}& a_{33}+\lambda & a_{34}\\ a_{14}& a_{24}& a_{34}& a_{44}\end{array}\right]-{\omega }^2\left[\begin{array}{cccc}{m}_{11}& 0& m_{13}& m_{14}\\ 0& m_{22}& m_{23}& m_{24}\\ m_{13}& m_{23}& m_{33}& m_{34}\\ m_{14}& m_{24}& m_{34}& m_{44}\end{array}\right]\right)\left\{\begin{array}{c}{U}_{{\hspace{0.17em}}_{mn}}^{\hspace{0.17em}}\\ V_{{\hspace{0.17em}}_{mn}}^{\hspace{0.17em}}\\ W_{{\hspace{0.17em}}_{mn}}^{\hspace{0.17em}}\\ {\phi }_{{\hspace{0.17em}}_{mn}}^{\hspace{0.17em}}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ Q_{{\hspace{0.17em}}_{mn}}^{\hspace{0.17em}}\\ 0\end{array}\right\}$

22

where

$\begin{array}{l}{a}_{11}={\alpha }^2{A}_{11}+{\beta }^2{A}_{66},\hspace{1em}\\ a_{12}=\alpha \beta \left(A_{12}+A_{66}\right),\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_{11}=m_{22}=I_0,\\ a_{13}=-{\alpha }^3{B}_{11}-\alpha {\beta }^2\left(B_{12}+2B_{66}\right),\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}{m}_{13}=-\alpha I_1,\\ a_{14}=-{\alpha }^3{B}_{11}^s-\alpha {\beta }^2\left(B_{12}^s+2B_{66}^s\right),\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{m}_{14}=-\alpha J_1,\\ a_{22}={\alpha }^2{A}_{66}+{\beta }^2{A}_{22},\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_{23}=-\beta I_1,\\ a_{23}=-{\beta }^3{B}_{22}-{\alpha }^2\beta \left(B_{12}+2B_{66}\right),\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{m}_{24}=-\beta J_1\\ a_{24}=-{\beta }^3{B}_{22}^s-{\alpha }^2\beta \left(B_{12}^s+2B_{66}^s\right),\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{m}_{33}=I_0+\left({\alpha }^2+{\beta }^2\right)I_2,\\ a_{33}={\alpha }^4{D}_{11}+2{\alpha }^2{\beta }^2\left(D_{12}+2D_{66}\right)+{\beta }^4{D}_{22},\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}{m}_{34}=\left({\alpha }^2+{\beta }^2\right)J_2,\\ a_{34}={\alpha }^4{D}_{11}^s+2{\alpha }^2{\beta }^2\left(D_{12}^s+2D_{66}^s\right)+{\beta }^4{D}_{22}^s,\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{m}_{44}=K_2\left({\alpha }^2+{\beta }^2\right),\\ a_{44}={\alpha }^4{H}_{11}^s+2{\alpha }^2{\beta }^2\left(H_{12}^s+2H_{66}^s\right)+{\beta }^4{H}_{22}^s+{\alpha }^2{A}_{55}^s+{\beta }^2{A}_{44}^s,\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\lambda =-\left({\alpha }^2+\gamma {\beta }^2\right)N_0\end{array}$

23

The equation system (22) represents the general form to analyze the problem of bending, free vibration, and buckling mechanical behaviors in the FGMs plate subjected to transverse loads (q) and compressive loads (N₀ ) in the plane.

Results and discussion

Consider a FGMs plate, varied in the gradation of materials as a power function with a power law (P-FGMs), or as exponential function (E-FGMs), which characterized the materials distribution, and depending on a varied volume fraction index (p), defined stiffer FGMs plate with lower index (p) values, and softer FGMs plate with highest index (p) values, of varied dimensions shown in Figure 1, the distributed materials varied as the following metallic and ceramics couple combinations, aluminum/alumina (Al/Al₂O₃ ), aluminum/zirconia (Al/ZrO₂ ), and aluminum/silicon carbide (Al/Sic), of material properties collected in Table 1 and subjected to variable loads. The results of the present parameters variation effected on the nondimensional deflections, the normal and tangential stresses, the natural frequencies, and the critical buckling load of FGMs plates are compared and validated using several numerical examples of different theories found in the literature, using nondimensional relations, listed as

$$\begin{gathered} \begin{array}{l}\overline{u}=\frac{100E_c{h}^3}{q_0{a}^4}u\begin{array}{c}\left(0,\frac{b}{2},z\right)\end{array},\overline{w}=\frac{100E_c{h}^3}{q_0{a}^4}w\begin{array}{c}\left(\frac{a}{2},\frac{b}{2}\right)\end{array},\\ {\overline{\sigma }}_{xx}\left(z\right)=\frac{h}{q_{0}a}{\sigma }_{xx}\begin{array}{c}\left(\frac{a}{2},\frac{b}{2},z\right)\end{array},{\overline{\sigma }}_{xy}\left(z\right)=\frac{h}{q_{0}a}{\sigma }_{xy}\left(0,0,z\right),{\overline{\sigma }}_{xz}\left(z\right)=\frac{h}{q_{0}a}{\sigma }_{xz}\left(0,\frac{b}{2},z\right)\\ \\ {\widehat{N}}_{cr}=\frac{N_{cr}{a}^2}{D_{11}^{\hspace{0.17em}}-B_{11}^2/A_{11}^{\hspace{0.17em}}},{\overline{N}}_{cr}=\frac{N_{cr}{a}^2}{E_m{h}^3},N_{cr}=N_0\\ \widehat{\omega }=\frac{a\omega }{\left(\frac{a}{h}\right)}\sqrt{\frac{{\rho }_c}{E_c}}=\omega h\sqrt{\frac{{\rho }_c}{E_c}}\hspace{0.17em},\hspace{0.17em}\overline{\omega }=a\omega \left(\frac{a}{h}\right)\sqrt{\frac{{\rho }_c}{E_c}}=\omega \frac{a^2}{h}\sqrt{\frac{{\rho }_c}{E_c}}=\widehat{\omega }{\left(\frac{a}{h}\right)}^2\\ \overline{\beta }=\left(\frac{a}{h}\right)\frac{\omega b}{{\pi }^2}\sqrt{\frac{12\left(1-{\nu }_c^2\right){\rho }_c}{E_c}}=\widehat{\omega }\frac{ab}{h^2{\pi }^2}\sqrt{12\left(1-{\nu }_c^2\right)}=\overline{\omega }\frac{b}{a{\pi }^2}\sqrt{12\left(1-{\nu }_c^2\right)}\end{array} \end{gathered}$$

24

Numerical examples and bending analysis results

The present theory accuracy is evaluated for bending problems analysis of Al/Al₂O₃ FGMs plate, only subjected to sinusoidal load (q), and neglecting the in-plane compressive load as well as the kinetic energy (omitting the matrix (M)) in equation (22).

P-FGMs plates

Example 1. The present theory validity in predicting bending responses is verified in Table 2, for stiffer, softer, moderately thick, and square Al/Al₂O₃ P-FGMs plate, graded in materials as a power function with a power law of equation (1), by examining the nondimensional values of in-plane axial displacement $\overline{u}$ , deflection $\overline{w}$ (transverse displacement) value in the plate center, in-plane normal stress ${\overline{\sigma }}_{xx}\left(z\right)$ , in-plane shear stress ${\overline{\sigma }}_{xy}\left(z\right)$ , and transverse shear stress ${\overline{\sigma }}_{xz}\left(z\right)$ . The obtained nondimensional results are validated with solutions given by different theories as the HSDT, TSDT, quasi-3-D theory, as well as those predicted by sinusoidal shear deformation theory (SSDT). It should be noted that the results showed good agreement with different theories solutions, especially with HSDT of Thai and Kim ⁴⁵ and Wu and Li, ¹¹ where solutions are very close to each other, even if the index (p) values grows larger, where the values of $\overline{u}$ and $\overline{w}$ becomes more significant for the FGMs plates, unlike ${\overline{\sigma }}_{xx}\left(z\right)$ values which decreased, as well as ${\overline{\sigma }}_{xy}\left(z\right)$ and ${\overline{\sigma }}_{xz}\left(z\right)$ values present a peak of minimum and maximum value for (p∈ [2,4]), respectively. The rate of error is independent of this value profile. For instance, the nondimensional values $\overline{u}$ , $\overline{w}$ , ${\overline{\sigma }}_{xx}\left(z\right)$ , ${\overline{\sigma }}_{xz}\left(z\right)$ and ${\overline{\sigma }}_{xy}\left(z\right)$ produce a maximum rate of error of 0.102%, 0.041%, 0.116%, 1.926% and 0.018%, which is negligible compared with Thai and Kim ⁴⁵ theory results as (p = 8) and (p = 4), respectively. The visualization of nondimensional values in terms of through thickness distribution (z/h), is shown in Figure 3(a) to (d), for ceramic homogeneous, stiffer FGMs (p = 1) and softer FGMs (p = 20) plate, compared with those implemented by Nguyen. ²⁶ Again, a good correlation between results is found in all cases except for ${\overline{\sigma }}_{xz}\left(z\right)$ , confirming the present theory accuracy in predicting bending responses of FGMs plates. It can be observed that $\overline{u}$ becomes more significant for the softer FGMs plate bottom part (from (z/h = 0.16) toward (z/h = −0.5)). The fact that the bending compressed the plate ceramic top part and stretched the metal bottom part, $\overline{u}$ and ${\overline{\sigma }}_{xy}\left(z\right)$ are of negative values on the top ceramic part from (z/h = 0.12) and (z/h = 0), for (p = 20), (p = 1) and (p = 0), respectively. However, an inverse mechanical behavior is observed for ${\overline{\sigma }}_{xx}\left(z\right)$ , where the negative values are obtained in the plate bottom part. The stresses ${\overline{\sigma }}_{xx}\left(z\right)$ and ${\overline{\sigma }}_{xy}\left(z\right)$ are sensitive to the index (p) variation. Thus, the current ${\overline{\sigma }}_{xz}\left(z\right)$ results are slightly overestimated, visualized as a small difference between paces (see Figure 2(d)); this might be due to an important reason of the presence of various displacement models and their corresponding number of unknowns, the shape function accuracy, the boundary conditions and the significant thickness stretching effect, for thick plates, which is presented in 3-D and quasi-3-D theories, which provide good result prediction, as considered by Wu and Chiu. ⁴⁴ The maximum rate of error is achieved for (p = 2) and (p = 8) as 9.928%, 1.892% and 3.003%, which is even insignificant, compared with results implemented by Wu and Chiu, ⁴⁴ quasi-3-D Carrera et al. ¹⁰ and HSDT Nguyen, ²⁶ respectively.

Figure 2.

Variation of nondimensional (a) in-plane displacement $(\overline{u})$ , (b) in-plane stress $({\overline{\sigma }}_{xx})$ , (c) in-plane shear stress $({\overline{\sigma }}_{xy})$ , and (d) transverse shear stress $({\overline{\sigma }}_{xz})$ through the aluminum/alumina (Al/Al₂O₃ ) plate thickness.

Figure 3.

Effect of (a) the volume fraction index (p) and (b) side-to-thickness ratio (a/h) on the nondimensional natural frequency ( $\overline{\omega }$ ) of the aluminum/alumina (Al/Al₂O₃ ) plates.

Table 2.

Nondimensional displacements and stresses of aluminum/alumina (Al/Al₂O₃ ) moderately thick (a/h = 10) square plates.

p	Theories	$\overline{u}(-h/4)$	$\overline{w}$	${\overline{\sigma }}_{xx}\left(h/3\right)$	${\overline{\sigma }}_{xy}\left(-h/3\right)$	${\overline{\sigma }}_{xz}\left(h/6\right)$
1	SSDT (Zenkour 43 )	0.6626	0.5889	1.4894	0.6110	0.2622
	Quasi-3-D (Carrera et al. 10 )	0.6436	0.5875	1.5062	0.6081	0.2510
	Quasi-3-D (Wu and Chiu 44 )	0.6436	0.5876	1.5061	0.6112	0.2511
	HSDT (Mantari et al. 20 )	0.6398	0.5880	1.4888	0.6109	0.2566
	TSDT (Wu and Li 11 )	0.6414	0.5890	1.4898	0.6111	0.2599
	HSDT (Thai and Kim 45 )	0.6414	0.5890	1.4898	0.6111	0.2608
	HSDT (Nguyen 26 )	0.6401	0.5883	1.4892	0.6110	0.2552
	Present	0.6410	0.5889	1.4893	0.6111	0.2631
2	SSDT (Zenkour 43 )	0.9281	0.7573	1.3954	0.5441	0.2763
	Quasi-3-D (Carrera et al. 10 )	0.9012	0.7570	1.4147	0.5421	0.2496
	Quasi-3-D (Wu and Chiu 44 )	0.9013	0.7571	1.4133	0.5436	0.2495
	HSDT (Mantari et al. 20 )	0.8957	0.7564	1.3940	0.5438	0.2741
	TSDT (Wu and Li 11 )	0.8984	0.7573	1.3960	0.5442	0.2721
	HSDT (Thai and Kim 45 )	0.8984	0.7573	1.3960	0.5442	0.2737
	HSDT (Nguyen 26 )	0.8961	0.7567	1.3947	0.5439	0.2721
	Present	0.8978	0.7573	1.3954	0.5442	0.2770
4	SSDT (Zenkour 43 )	1.0941	0.8819	1.1783	0.5667	0.2580
	Quasi-3-D (Carrera et al. 10 )	1.0541	0.8823	1.1985	0.5666	0.2362
	Quasi-3-D (Wu and Chiu 44 )	1.0541	0.8823	1.1841	0.5671	0.2362
	HSDT (Mantari et al. 20 )	1.0457	0.8814	1.1755	0.5662	0.2623
	TSDT (Wu and Li 11 )	1.0502	0.8815	1.1794	0.5669	0.2519
	HSDT (Thai and Kim 45 )	1.0502	0.8815	1.1794	0.5669	0.2537
	HSDT (Nguyen 26 )	1.0466	0.8818	1.1766	0.5664	0.2593
	Present	1.0493	0.8818	1.1783	0.5668	0.2586
8	SSDT (Zenkour 43 )	1.1340	0.9750	0.9466	0.5856	0.2121
	Quasi-3-D (Carrera et al. 10 )	1.0830	0.9738	0.9687	0.5879	0.2262
	Quasi-3-D (Wu and Chiu 44 )	1.0830	0.9739	0.9622	0.5883	0.2261
	HSDT (Mantari et al. 20 )	1.0709	0.9737	0.9431	0.5850	0.2140
	TSDT (Wu and Li 11 )	1.0763	0.9747	0.9477	0.5858	0.2087
	HSDT (Thai and Kim 45 )	1.0763	0.9746	0.9477	0.5858	0.2088
	HSDT (Nguyen 26 )	1.0719	0.9744	0.9444	0.5852	0.2117
	Present	1.0752	0.9750	0.9466	0.5857	0.2129

HSDT: higher order shear deformation theory; TSDT: third-order shear deformation theory; SSDT: sinusoidal shear deformation theory; 3-D: three-dimensional.

The ${\overline{\sigma }}_{xz}\left(z\right)$ maximum peak is located in the median plane for homogeneous plates, or trends to move slightly toward the top surface, and will possess an asymmetrical profile across the FGMs plate thickness for larger values of index (p). Thus, ${\overline{\sigma }}_{xz}\left(z\right)$ becomes less important than ${\overline{\sigma }}_{xx}\left(z\right)$ and ${\overline{\sigma }}_{xy}\left(z\right)$ as 8 and 4, 12 and 2, 23 and 4 times lower for the indexes (p = 0), (p = 1), (p = 20), respectively.

E-FGMs plates

Example 2. For this example in Table 3, the deflection $\overline{w}$ results are presented for stiffer, thick, square, and rectangular Al/Al₂O₃ E-FGMs plates, which is a FGMs plate, graded in materials as an exponential function with exponential form of equation (2). A comparison between the calculated results and those given by HSDT, ^21,26,45 3-D, and quasi-3-D solutions ^8,46 are carried out. It is clear that the present theory provide good results as compared to the HSDT ^21,26,45 and slightly overestimates $\overline{w}$ results with 3-D and quasi-3-D solutions, ^8,46 which might be due to different conditions as mentioned earlier for Table 2. It can be seen that, for the E-FGMs plates, $\overline{w}$ values becomes more important with the decreases of both the index (p) and the ratio (a/h) values, as well as the increases of ratio (b/a) values. Thus, the maximum rate of error is 2.315%, 2.306%, and 0.64%, which is insignificant obtained for very thick (a/h = 2), square plate as well (p = 1), (p = 1.5), and (p = 0.1), compared with Mantari et al., ²¹ Thai et Kim, ⁴⁵ and Nguyen, ²⁶ respectively.

Table 3.

Nondimensional deflection $(\overline{w})$ of aluminum/alumina (Al/Al₂O₃ ) E-FGMs square plates.

a/h	b/a	Theories	p
			0.1	0.3	0.5	0.7	1	1.5
2	1	3-D (Zenkour 8 )	0.5769	0.5247	0.4766	0.4324	0.3727	0.2890
		Quasi-3-D (Zenkour 8 )	0.5731	0.5181	0.4679	0.4222	0.3612	0.2771
		Quasi-3-D (Mantari and Soares 46 )	0.5776	0.5222	0.4716	0.4255	0.3640	0.2792
		HSDT (Mantari et al. 21 )	0.6363	0.5752	0.5195	0.4687	0.4018	0.3079
		HSDT (Thai and Kim 45 )	0.6362	0.5751	0.5194	0.4687	0.4011	0.3079
		HSDT (Nguyen 26 )	0.6211	0.5615	0.5073	0.4579	0.3921	0.3014
		Present	0.6251	0.5645	0.5093	0.4592	0.3925	0.3008
	2	3-D (Zenkour 8 )	1.1944	1.0859	0.9864	0.8952	0.7727	0.6017
		Quasi-3-D (Zenkour 8 )	1.1880	1.0740	0.9701	0.8755	0.7494	0.5758
		Quasi-3-D (Mantari and Soares 46 )	1.1938	1.0790	0.9748	0.8797	0.7530	0.5785
		HSDT (Mantari et al. 21 )	1.2776	1.1553	1.0441	0.9431	0.8093	0.6238
		HSDT (Thai and Kim 45 )	1.2775	1.1553	1.0441	0.9431	0.8086	0.6238
		HSDT (Nguyen 26 )	1.2569	1.1367	1.0275	0.9284	0.7965	0.6153
		Present	1.2615	1.1396	1.0289	0.9285	0.7952	0.6125
	3	3-D (Zenkour 8 )	1.4430	1.3116	1.1913	1.0812	0.9334	0.7275
		Quasi-3-D (Zenkour 8 )	1.4354	1.2977	1.1722	1.0580	0.9057	0.6962
		Quasi-3-D (Mantari and Soares 46 )	1.4419	1.3035	1.1774	1.0626	0.9096	0.6991
		HSDT (Mantariet al. 21 )	1.5341	1.3874	1.2540	1.1329	0.9725	0.7506
		HSDT (Thai and Kim 45 )	1.5340	1.3873	1.2540	1.1329	0.9719	0.7506
		HSDT (Nguyen 26 )	1.5115	1.3671	1.2360	1.1169	0.9587	0.7414
		Present	1.5163	1.3699	1.2371	1.1167	0.9569	0.7380
4	1	3-D (Zenkour 8 )	0.3490	0.3168	0.2875	0.2608	0.2253	0.1805
		Quasi-3-D (Zenkour 8 )	0.3475	0.3142	0.2839	0.2563	0.2196	0.1692
		Quasi-3-D (Mantari and Soares 46 )	0.3486	0.3152	0.2848	0.2571	0.2203	0.1697
		HSDT (Mantari et al. 21 )	0.3602	0.3259	0.2949	0.2668	0.2295	0.1785
		HSDT (Thai and Kim 45 )	0.3602	0.3259	0.2949	0.2668	0.2295	0.1785
		HSDT (Nguyen 26 )	0.3575	0.3235	0.2927	0.2649	0.2280	0.1775
		Present	0.3598	0.3255	0.2945	0.2664	0.2292	0.1782
	2	3-D (Zenkour 8 )	0.8153	0.7395	0.6708	0.6085	0.5257	0.4120
		Quasi-3-D (Zenkour 8 )	0.8120	0.7343	0.6635	0.5992	0.5136	0.3962
		Quasi-3-D (Mantari and Soares 46 )	0.8145	0.7365	0.6655	0.6009	0.5151	0.3973
		HSDT (Mantari et al. 21 )	0.8325	0.7534	0.6819	0.6173	0.5319	0.4150
		HSDT (Thai and Kim 45 )	0.8325	0.7534	0.6819	0.6173	0.5319	0.4150
		HSDT (Nguyen 26 )	0.8285	0.7498	0.6787	0.6145	0.5296	0.4135
		Present	0.8319	0.7528	0.6813	0.6167	0.5313	0.4146
	3	3-D (Zenkour 8 )	1.0134	0.9190	0.8335	0.7561	0.6533	0.5121
		Quasi-3-D (Zenkour 8 )	1.0094	0.9127	0.8248	0.7449	0.6385	0.4927
		Quasi-3-D (Mantari and Soares 46 )	1.0124	0.9155	0.8272	0.7470	0.6404	0.4941
		HSDT (Mantari et al. 21 )	1.0325	0.9345	0.8459	0.7659	0.6601	0.5154
		HSDT (Thai and Kim 45 )	1.0325	0.9345	0.8459	0.7659	0.6601	0.5154
		HSDT (Nguyen 26 )	1.0281	0.9305	0.8424	0.7628	0.6576	0.5137
		Present	1.0319	0.9338	0.8453	0.7653	0.6594	0.5149

HSDT: higher order shear deformation theory; 3-D: three-dimensional.

Numerical examples and free vibration analysis results

In this part, the present theory accuracy is also evaluated for free vibration analysis of both Al/Al₂O₃ and Al/ZrO₂ FGMs plates as the power law. Thus, the vibration problem is achieved by omitting the transverse load (q) and neglecting the in-plane compressive load in equation (22).

Example 3. The nondimensional natural frequencies $\overline{\beta }$ results are presented in Table 4, for different homogeneous, stiffer, softer, thick, moderately thick, thin, and square Al/ZrO₂ FGMs plates, and compared with those generated by 3-D theory ⁹ based on 3-D plate model, and with HSDT. ²⁶ It should be noted that the obtained results are in good agreement with the reference solutions even for thin (a/h = 100), stiffer ceramic homogeneous plates, where $\overline{\beta }$ becomes important. The maximum rate of error between the present model and Uymaz and Aydogdu ⁹ is 2.977% of insignificant value, for (a/h = 5, p = 1), this error is reduced in comparison with the Nguyen, ²⁶ in the order of 0.755% for (a/h = 2, p = 10).

Table 4.

Nondimensional fundamentals frequencies ( $\overline{\beta })$ of aluminum/zirconia (Al/ZrO₂ ) square plates.

a/h	Theories	p
		0	0.1	0.2	0.5	1	2	5	10
2	3-D (Uymaz and Aydogdu 9 )	1.2589	1.2296	1.2049	1.1484	1.0913	1.0344	0.9777	0.9507
	HSDT (Nguyen 26 )	1.2571	1.2259	1.2010	1.1443	1.0882	1.0325	0.9771	0.9540
	Present	1.2538	1.2239	1.1983	1.1412	1.0856	1.0322	0.9748	0.9468
5	3-D (Uymaz and Aydogdu 9 )	1.7748	1.7262	1.6881	1.6031	1.4764	1.4628	1.4106	1.3711
	HSDT (Nguyen 26 )	1.7723	1.7241	1.6850	1.6003	1.5245	1.4629	1.4084	1.3726
	Present	1.7687	1.7220	1.6827	1.5979	1.5217	1.4605	1.4059	1.3690
10	3-D (Uymaz and Aydogdu 9 )	1.9339	1.8788	1.8357	1.7406	1.6583	1.5968	1.5491	1.5066
	HSDT (Nguyen 26 )	1.9330	1.8783	1.8342	1.7402	1.6593	1.5994	1.5500	1.5095
	Present	1.9318	1.8785	1.8341	1.7398	1.6583	1.5986	1.5492	1.5083
20	3-D (Uymaz and Aydogdu 9 )	1.9570	1.9261	1.8788	1.7832	1.6999	1.6401	1.5937	1.5491
	HSDT (Nguyen 26 )	1.9824	1.9257	1.8799	1.7830	1.7006	1.6417	1.5945	1.5524
	Present	1.9821	1.9267	1.8806	1.7833	1.7004	1.6415	1.5943	1.5521
50	3-D (Uymaz and Aydogdu 9 )	1.9974	1.9390	1.8920	1.7944	1.7117	1.6522	1.6062	1.5620
	HSDT (Nguyen 26 )	1.9971	1.9398	1.8935	1.7957	1.7129	1.6544	1.6079	1.5653
	Present	1.9971	1.9410	1.8944	1.7962	1.7129	1.6543	1.6078	1.5652
100	3-D (Uymaz and Aydogdu 9 )	1.9974	1.9416	1.8920	1.7972	1.7117	1.6552	1.6062	1.5652
	HSDT (Nguyen 26 )	1.9993	1.9418	1.8955	1.7975	1.7147	1.6562	1.6098	1.5671
	Present	1.9993	1.9431	1.8964	1.7981	1.7147	1.6562	1.6098	1.5671

HSDT: higher order shear deformation theory; 3-D: three-dimensional.

Example 4. The first three modes of nondimensional free vibration fundamental frequencies $\overline{\omega }$ are computed in Table 5, for different homogeneous, stiffer, softer, thick moderately thick, thin, square, and rectangular Al/Al₂O₃ FGMs plates under various frequency modes, and the results are in good agreement, compared with HSDT solutions of Nguyen, ²⁶ the maximum rate of error between both theories is 0.71%, a negligible error, signaled for thick, softer FGMs (p = 10), square plate with third mode. It is to highlight that $\overline{\omega }$ values increases with the decrease of the index (p) values and the increase of both ratios (a/h), (a/b) and frequency modes values, which are presented clearly in Figure 3, where the highest $\overline{\omega }$ values are obtained from Figure 3(a), illustrated for moderately thick plates; for stiffer FGMs plate and highest frequency mode, the $\overline{\omega }$ values reduction are greater for stiffer FGMs plates (p < 2), then $\overline{\omega }$ pace has a trend to stabilize as the plate becomes softer; and are obtained from Figure 3(b) for thin, stiffer homogeneous ceramic plate, the reason is that the thin plate with big ratio (a/h) values, make the shear deformation effect becomes less effective, as well as the stiffer plate with the small index (p) values, makes the ceramic volume fraction and the elasticity modulus values increase, and consequently the plate becomes ceramic as the ceramic content increases. $\overline{\omega }$ values increase greatly for the ratio range (a/h∈[2,10]), then $\overline{\omega }$ pace has a trends to stabilize, which is even proved in Figure 4 as a 3-D diagram presented an interaction between the index (p), the ratio (a/h), and fundamental frequency $\overline{\omega }$ , using the current theory. Again, $\overline{\omega }$ values increases with the decrease of the index (p) values and the increase of the ratio (a/h) values.

Figure 4.

Effect of the side-to-thickness ratio (a/h) and the volume fraction index (p) on the nondimensional fundamental frequency ( $\overline{\omega }$ ) of aluminum/alumina (Al/Al₂O₃ ) square plates.

Table 5.

Nondimensional natural frequencies ( $\overline{\omega })$ of aluminum/alumina (Al/Al₂O₃ ) square plates.

a/b	a/h	Mode	Theories	p
				0	0.5	1	2	5	10
0.5	5	1(1,1)	HSDT (Nguyen 26 )	3.4464	2.9380	2.6509	2.3971	2.2260	2.1432
			Present	3.4417	2.9350	2.6480	2.3955	2.2269	2.1401
		2(1,2)	HSDT (Nguyen 26 )	5.2932	4.5258	4.0860	3.6859	3.3919	3.2574
			Present	5.2824	4.5188	4.0792	3.6820	3.3932	3.2501
		3(2,2)	HSDT (Nguyen 26 )	11.6113	10.0109	9.0538	8.1181	7.2951	6.9568
			Present	11.5624	9.9754	9.0213	8.0967	7.2931	6.9233
	10	1(1,1)	HSDT (Nguyen 26 )	3.6533	3.0996	2.7946	2.5371	2.3911	2.3118
			Present	3.6518	3.0991	2.7938	2.5365	2.3913	2.3108
		2(1,2)	HSDT (Nguyen 26 )	5.7731	4.9031	4.4216	4.0105	3.7671	3.6388
			Present	5.7697	4.9016	4.4194	4.0090	3.7674	3.6364
		3(2,2)	HSDT (Nguyen 26 )	13.7855	11.7519	10.6036	9.5884	8.9042	8.5729
			Present	13.7665	11.7399	10.5913	9.5798	8.9048	8.5608
	20	1(1,1)	HSDT (Nguyen 26 )	3.7127	3.1455	2.8355	2.5773	2.4401	2.3622
			Present	3.7123	3.1458	2.8353	2.5771	2.4402	2.3618
		2(1,2)	HSDT (Nguyen 26 )	5.9209	5.0176	4.5234	4.1104	3.8880	3.7629
			Present	5.9199	5.0180	4.5228	4.1100	3.8881	3.7621
		3(2,2)	HSDT (Nguyen 26 )	14.6131	12.3983	11.1785	10.1482	9.5645	9.2471
			Present	14.6075	12.3964	11.1750	10.1457	9.5649	9.2433
1	5	1(1,1)	HSDT (Nguyen 26 )	5.2932	4.5258	4.0860	3.6859	3.3919	3.2574
			Present	5.2824	4.5188	4.0792	3.6820	3.3932	3.2501
		2(1,2)	HSDT (Nguyen 26 )	11.6113	10.0109	9.0538	8.1181	7.2951	6.9568
			Present	11.5624	9.9753	9.0213	8.0967	7.2931	6.9233
		3(2,2)	HSDT (Nguyen 26 )	16.8351	14.5888	13.2140	11.8101	10.4647	9.9360
			Present	16.7347	14.5144	13.1447	11.7609	10.4499	9.8655
	10	1(1,1)	HSDT (Nguyen 26 )	5.7731	4.9031	4.4216	4.0105	3.7671	3.6388
			Present	5.7697	4.9016	4.4194	4.0090	3.7674	3.6364
		2(1,2)	HSDT (Nguyen 26 )	13.7855	11.7519	10.6036	9.5884	8.9042	8.5729
			Present	13.7665	11.7399	10.5913	9.5798	8.9048	8.5608
		3(2,2)	HSDT (Nguyen 26 )	21.1728	18.1033	16.3438	14.7435	13.5677	13.0296
			Present	21.1289	18.0747	16.3152	14.7232	13.5672	13.0017
	20	1(1,1)	HSDT (Nguyen 26 )	5.9209	5.0176	4.5234	4.1104	3.8880	3.7629
			Present	5.91993	5.0180	4.5228	4.1100	3.8881	3.7621
		2(1,2)	HSDT (Nguyen 26 )	14.6131	12.3983	11.1785	10.1482	9.5645	9.2471
			Present	14.6075	12.3964	11.1750	10.1457	9.5649	9.2433
		3(2,2)	HSDT (Nguyen 26 )	23.0925	19.6126	17.6865	16.0419	15.0685	14.5550
			Present	23.0786	19.6066	17.6776	16.0356	15.0692	14.5458

HSDT: higher order shear deformation theory.

Example 5. The first three nondimensional free vibration fundamental frequencies $\widehat{\omega }$ are calculated in Table 6, to verify the results accuracy, for different homogeneous, stiffer, softer, thick, moderately thick, thin, and square Al/Al₂O₃ FGMs plates under various frequency modes. The obtained results are compared with different theories as the quasi-3-D theory, ⁴⁷ TSDT, ¹⁴ FSDT, ¹³ as well the HSDT solutions, ^26,45 where the agreement is very well, the maximum rate of error is 0.111% and 0.705%, which is even negligible, for (p = 4, a/h = 10, mode 2) and (p = 10, a/h = 5, mode 3), compared with Hosseini et al., ¹⁴ Thai et Kim, ⁴⁵ and Nguyen, ²⁶ respectively. As long as the plate becomes thinner, the shear deformation effect becomes less significant, the results become less significant and almost identical, which presented more accuracy. It is to highlight that $\widehat{\omega }$ values increases with the decrease of the index (p) and (a/h) ratio values, as well as the increase of frequency modes values toward the third one.

Table 6.

The first three nondimensional frequencies ( $\widehat{\omega })$ of aluminum/alumina (Al/Al₂O₃ ) square plates.

a/h	Mode (m, n)	Theories	p
			0	0.5	1	4	10
5	1(1,1)	Quasi-3-D (Matsunaga 47 )	0.2121	0.1819	0.1640	0.1383	0.1306
		TSDT (Hosseini-Hashemi et al. 14 )	0.2113	0.1807	0.1631	0.1378	0.1301
		FSDT (Hosseini-Hashemi et al. 13 )	0.2112	0.1805	0.1631	0.1397	0.1324
		HSDT (Thai and Kim 45 )	0.2113	0.1807	0.1631	0.1378	0.1301
		HSDT (Nguyen 26 )	0.2117	0.1807	0.1634	0.1378	0.1303
		Present	0.2113	0.1808	0.1632	0.1378	0.1300
	2(1,2)	Quasi-3-D (Matsunaga 47 )	0.4658	0.4040	0.3644	0.3000	0.2790
		TSDT (Hosseini-Hashemi et al. 14 )	0.4623	0.3989	0.3607	0.2980	0.2771
		FSDT (Hosseini-Hashemi et al. 13 )	0.4618	0.3978	0.3604	0.3049	0.2856
		HSDT (Thai and Kim 45 )	0.4623	0.3989	0.3607	0.2980	0.2771
		HSDT (Nguyen 26 )	0.4645	0.4004	0.3622	0.2981	0.2783
		Present	0.4625	0.3990	0.3609	0.2980	0.2769
	3(2,2)	TSDT (Hosseini-Hashemi et al. 14 )	0.6688	0.5803	0.5254	0.4284	0.3948
		FSDT (Hosseini-Hashemi et al. 13 )	0.6676	0.5779	0.5245	0.4405	0.4097
		HSDT (Thai and Kim 45 )	0.6688	0.5803	0.5254	0.4284	0.3948
		HSDT (Nguyen 26 )	0.6734	0.5836	0.5286	0.4291	0.3974
		Present	0.6694	0.5806	0.5258	0.4285	0.3946
10	1(1,1)	Quasi-3-D (Matsunaga 47 )	0.0578	0.0492	0.0443	0.0381	0.0364
		TSDT (Hosseini-Hashemi et al. 14 )	0.0577	0.0490	0.0442	0.0381	0.0364
		FSDT (Hosseini-Hashemi et al. 13 )	0.0577	0.0490	0.0442	0.0382	0.0366
		HSDT (Thai and Kim 45 )	0.0577	0.0490	0.442	0.0381	0.0364
		HSDT (Nguyen 26 )	0.0577	0.0490	0.0442	0.0381	0.0364
		Present	0.0577	0.0490	0.0442	0.0381	0.0364
	2(1,2)	Quasi-3-D (Matsunaga 47 )	0.1381	0.1180	0.1063	0.0905	0.0859
		TSDT (Hosseini-Hashemi et al. 14 )	0.1377	0.1174	0.1059	0.0903	0.0856
		FSDT (Hosseini-Hashemi et al. 13 )	0.1376	0.1173	0.1059	0.0911	0.0867
		HSDT (Thai and Kim 45 )	0.1377	0.1174	0.1059	0.0903	0.0856
		HSDT (Nguyen 26 )	0.1379	0.1175	0.1060	0.0902	0.0857
		Present	0.1377	0.1174	0.1059	0.0902	0.0856
	3(2,2)	TSDT (Hosseini-Hashemi et al. 14 )	0.2113	0.1807	0.1631	0.1378	0.1301
		FSDT (Hosseini-Hashemi et al. 13 )	0.2112	0.1805	0.1631	0.1397	0.1324
		HSDT (Thai and Kim 45 )	0.2113	0.1807	0.1631	0.1378	0.1301
		HSDT (Nguyen 26 )	0.2117	0.1810	0.1634	0.1378	0.1303
		Present	0.2113	0.1807	0.1632	0.1378	0.1300
20	1(1,1)	TSDT (Hosseini-Hashemi et al. 14 )	0.0148	0.0125	0.0113	0.0098	0.0094
		FSDT (Hosseini-Hashemi et al. 13 )	0.0148	0.0125	0.0113	0.0098	0.0094
		HSDT (Thai and Kim 45 )	0.0148	0.0125	0.0113	0.0098	0.0094
		HSDT (Nguyen 26 )	0.0148	0.0125	0.0113	0.0098	0.0094
		Present	0.0148	0.0125	0.0113	0.0098	0.0094
	2(1,2)	HSDT (Nguyen 26 )	0.0365	0.0310	0.0279	0.0241	0.0231
		Present	0.0365	0.0310	0.0279	0.0241	0.0231
	3(2,2)	HSDT (Nguyen 26 )	0.0577	0.0490	0.0442	0.0381	0.0364
		Present	0.0577	0.0490	0.0442	0.0381	0.0364

HSDT: higher order shear deformation theory; TSDT: third-order shear deformation theory; FSDT: first-order shear deformation theory; 3-D: three-dimensional.

The three natural frequency types ( $\overline{\beta }$ , $\overline{\omega }$ and $\widehat{\omega }$ ) increase in values by increasing and decreasing the ratio (a/h) values (the trends of the results are shown in Tables 4 to 6), corresponding to the nondimensional relations of each one, which are proportional and inverse proportional to the ratio (a/h), respectively.

Numerical examples and buckling analysis results

The buckling response of Al/Al ₂ O ₃ and Al/SiC FGMs plates, as the power law, which depends on the volume fraction index (p), and subjected to different in-plane compressive load depends on the parameters (γ), has been evaluated by omitting transverse sinusoidal load (q) and the kinetic energy (omitting the matrix (M)). Thus, the plate is only subjected to in-plane compressive load N ₀ in equation (22).

Example 6. The present theory accuracy in predicting the critical buckling loads ${\overline{N}}_{cr}$ values are highlighted in Table 7, for different homogeneous, stiffer, softer, thick, moderately thick, thin, square, and rectangular Al/Al ₂ O ₃ FGMs plates, under various in-plane compressive loads. As it is observed from the table, the results are in good agreement, compared with HSDT ^15,26 solutions, where the maximum rate of error is 1.053% and 0.209%, insignificant error obtained for (γ = −1, b/a = 1, a/h = 5, p = 10) and (γ = −1, b/a = 1, a/h = 5, p = 2), respectively. It is to highlight that ${\overline{N}}_{cr}$ values increases with the decrease of both index (p) and parameters (γ) values as well the increase of both ratios (a/h) and (b/a) values, presented clearly in Figure 5, for rectangular plates, where it can be observed that the ${\overline{N}}_{cr}$ higher values are obtained from Figure 5(a) illustrated with thick plates, for stiffer FGMs plate with biaxial compression load, the ${\overline{N}}_{cr}$ values reduction are greater for stiffer FGMs plates (p < 2), then, ${\overline{N}}_{cr}$ pace has a trend to stabilize as the plate become softer; and are obtained from Figure 5(b) illustrated for uniaxial compression load, for stiffer homogeneous ceramic plate and highest ratio (a/h), for the reason mentioned before in example 4, and that’s way ${\overline{N}}_{cr}$ values increase slightly with the increase of the ratio (a/h) values, especially for (a/h∈]2,10[), then, ${\overline{N}}_{cr}$ pace has a trend to stabilize. Otherwise, the same mechanical behavior is observed for both $\overline{\omega }$ and ${\overline{N}}_{cr}$ for the variation of index (p), ratio (a/h), frequency modes (m, n), and parameter (γ), and consequently, the interaction between them is clear in Figure 6.

Figure 5.

Effect of (a) the volume fraction index (p) and (b) side-to-thickness ratio (a/h) on the nondimensional critical buckling load ( ${\overline{N}}_{cr}$ ) of aluminum/alumina (Al/Al₂O₃ ) plates.

Figure 6.

Effect of the critical buckling load ( ${\overline{N}}_{cr}$ ) on the nondimensional fundamental frequency ( $\overline{\omega }$ ) of aluminum/alumina (Al/Al₂O₃ ) plates.

Table 7.

Nondimensional critical buckling loads ( ${\overline{N}}_{cr})$ of aluminum/alumina (Al/Al₂O₃ ) plates.

$\gamma$	b/a	a/h	Theories	p
				0	0.5	1	2	5	10
0	0.5	5	HSDT (Nguyen 26 )	6.7417	4.4343	3.4257	2.6503	2.1459	1.9260
			HSDT (Thai and Choi 15 )	6.7203	4.4235	3.4164	2.6451	2.1484	1.9213
			Present	6.7222	4.4248	3.4178	2.6467	2.1478	1.9201
		10	HSDT (Nguyen 26 )	7.4115	4.8225	3.7137	2.8911	2.4155	2.1911
			HSDT (Thai and Choi 15 )	7.4053	4.8206	3.7111	2.8897	2.4165	2.1896
			Present	7.4057	4.8209	3.7113	2.8897	2.4158	2.1892
		20	HSDT (Nguyen 26 )	7.6009	4.9307	3.7937	2.9585	2.4942	2.2695
			HSDT (Thai and Choi 15 )	7.5993	4.9315	3.7930	2.9582	2.4944	2.2690
			Present	7.5994	4.9315	3.7931	2.9581	2.4942	2.2689
	1	5	HSDT (Nguyen 26 )	16.1003	10.6670	8.2597	6.3631	5.0459	4.4981
			HSDT (Thai and Choi 15 )	16.0211	10.6254	8.2245	6.3432	5.0531	4.4807
			Present	16.0286	10.6303	8.2301	6.3491	5.0512	4.4769
		10	HSDT (Nguyen 26 )	18.6030	12.1317	9.3496	7.2687	6.0316	5.4587
			HSDT (Thai and Choi 15 )	18.5785	12.1229	9.3391	7.2631	6.0353	5.4528
			Present	18.5801	12.1240	9.3400	7.2632	6.0327	5.4515
		20	HSDT (Nguyen 26 )	19.3593	12.5652	9.6702	7.5386	6.3437	5.7689
			HSDT (Thai and Choi 15 )	19.3528	12.5668	9.6675	7.5371	6.3448	5.7668
			Present	19.3532	12.5670	9.6676	7.5371	6.3439	5.7664
1	0.5	5	HSDT (Nguyen 26 )	5.3934	3.5475	2.7406	2.1202	1.7167	1.5408
			HSDT (Thai and Choi 15 )	5.3762	3.5388	2.7331	2.1161	1.7187	1.5370
			Present	5.3778	3.5398	2.7343	2.1174	1.7183	1.5361
		10	HSDT (Nguyen 26 )	5.9292	3.8580	2.9710	2.3129	1.9324	1.7529
			HSDT (Thai and Choi 15 )	5.9243	3.8565	2.9689	2.3117	1.9332	1.7517
			Present	5.9246	3.8567	2.9691	2.3118	1.9326	1.7514
		20	HSDT (Nguyen 26 )	6.0807	3.9445	3.0350	2.3668	1.9953	1.8156
			HSDT (Thai and Choi 15 )	6.0794	3.9452	3.0344	2.3665	1.9955	1.8152
			Present	6.0795	3.9452	3.0345	2.3665	1.9954	1.8151
	1	5	HSDT (Nguyen 26 )	8.0501	5.3335	4.1299	3.1815	2.5230	2.2491
			HSDT (Thai and Choi 15 )	8.0105	5.3127	4.1122	3.1716	2.5265	2.2403
			Present	8.0143	5.3152	4.1151	3.1745	2.5256	2.2384
		10	HSDT (Nguyen 26 )	9.3015	6.0659	4.6748	3.6344	3.0158	2.7293
			HSDT (Thai and Choi 15 )	9.2893	6.0615	4.6696	3.6315	3.0177	2.7264
			Present	9.2901	6.0620	4.6700	3.6316	3.0163	2.7258
		20	HSDT (Nguyen 26 )	9.6796	6.2826	4.8351	3.7693	3.1718	2.8844
			HSDT (Thai and Choi 15 )	9.6764	6.2834	4.8337	3.7686	3.1724	2.8834
			Present	9.6766	6.2835	4.8338	3.7685	3.1720	2.8832
−1	0.5	5	HSDT (Nguyen 26 )	8.9890	5.9124	4.5676	3.5337	2.8612	2.5679
			HSDT (Thai and Choi 15 )	8.9604	5.8980	4.5551	3.5268	2.8646	2.5617
			Present	8.9630	5.8997	4.5571	3.5289	2.8638	2.5601
		10	HSDT (Nguyen 26 )	9.8820	6.4299	4.9516	3.8548	3.2206	2.9214
			HSDT (Thai and Choi 15 )	9.8738	6.4275	4.9481	3.8529	3.2219	2.9195
			Present	9.8743	6.4278	4.9484	3.8529	3.2210	2.9190
		20	HSDT (Nguyen 26 )	10.1345	6.5742	5.0583	3.9447	3.3255	3.0260
			HSDT (Thai and Choi 15 )	10.1324	6.5753	5.0574	3.9442	3.3259	3.0253
			Present	10.1325	6.5754	5.0574	3.9441	3.3256	3.0252
	1	5	HSDTa (Nguyen 26 )	26.4999	17.9424	13.9872	10.6421	7.9571	6.9626
			HSDT (Thai and Choi 15 )	26.2058	17.7704	13.8486	10.5589	7.9590	6.8970
			Presenta	26.2390	17.7911	13.8705	10.5810	7.9565	6.8893
		10	HSDTa (Nguyen 26 )	35.9559	23.6497	18.2704	14.1349	11.4447	10.2717
			HSDT (Thai and Choi 15 )	35.8416	23.5920	18.2206	14.1073	11.4583	10.2468
			Presenta	35.8499	23.5972	18.2250	14.1084	11.4473	10.2417
		20	HSDTa (Nguyen 26 )	39.5280	25.7197	19.8065	15.4190	12.8824	11.6857
			HSDT (Thai and Choi 15 )	39.4951	25.7100	19.7925	15.4115	12.8878	11.6779
			Presenta	39.4972	25.7113	19.7934	15.4112	12.8835	11.6761

^a Critical buckling occurs at (m, n) = (2,1).

HSDT: higher order shear deformation theory.

Example 7. The critical buckling load ${\widehat{N}}_{cr}$ effect on the fundamental frequency $\overline{\omega }$ values variation are presented in Table 8; for different homogeneous, stiffer, softer, moderately thick, and square Al/SiC FGMs plates under various frequency modes and in-plane compressive loads, the ${\overline{N}}_{cr}$ results are compared with HSDT ^26,15,48 and FSDT. ⁴⁹ It can be noticed that a close agreement between the results are found for all cases, the maximum rate of error is 5.233%, 4.931%, 5.101%, and 4.912%, which is insignificant for homogeneous ceramic plate under second frequency mode and biaxial compression load, in comparison with all theories, respectively. It is to highlight that ${\widehat{N}}_{cr}$ values increases with the decrease of both index (p) and parameter (γ) values, which are presented clearly in Figure 6, for rectangular, moderately thick plate; Figure 6(a) is illustrated for homogeneous ceramic plate and Figure 6(b) for softer FGMs plate. It can be seen that the same mechanical behaviors are observed for $\overline{\omega }$ and ${\overline{N}}_{cr}$ , and for any in-plane compressive load; which either the $\overline{\omega }$ and ${\overline{N}}_{cr}$ highest values are found while the other one is null (at zero coordinate), for rectangular, homogeneous plates and biaxial compression load; and the smallest values (characterized the wicked plates), where the plate does not support buckling, are those for FGMs plate and uniaxial compression load, the FGMs plate had a significant effect in buckling, and leads to a reduction of 36.81% in the $\overline{\omega }$ and 71% in the ${\overline{N}}_{cr}$ compared to an homogeneous ceramic plate, as well an inverse mechanical behavior is signaled for negative values of ${\overline{N}}_{cr}$ , for Figures 6(a-b). An inverse relationship regroups the $\overline{\omega }$ with ${\overline{N}}_{cr}$ values; When the first increases, the second decreases considerably, and vice versa.

Table 8.

Nondimensional critical buckling loads ( ${\widehat{N}}_{cr}$ $)$ of aluminum/silicon carbide (Al/SiC) moderately thick (a/h = 10) square plates.

$\gamma$	Mode	Theories	p
			0	0.5	1	2	5	10
0	1(1,1)	HSDT (Nguyen 26 )	37.4215	37.6650	37.7560	37.6327	36.8862	36.5934
		HSDT (Thai and Choi 15 )	37.3721	—	37.7143	37.6042	—	—
		HSDT (Bodaghi and Saidi 48 )	37.3714	—	37.7172	37.5765	—	—
		FSDT (Mohammadi et al. 49 )	37.3708	—	37.7132	37.7089	—	—
		Present	36.5501	37.6334	37.7178	37.6050	36.9000	36.5501
1		HSDT (Nguyen 26 )	18.7107	18.8325	18.8780	18.8163	18.4431	18.2967
		HSDT (Thai and Choi 15 )	18.6861	—	18.8572	18.8021	—	—
		HSDT (Bodaghi and Saidi 48 )	18.6860	—	18.8571	18.8020	—	—
		FSDT (Mohammadi et al. 49 )	18.6854	—	18.8566	18.8545	—	—
		Present	18.2750	18.8167	18.8589	18.8025	18.4500	18.2750
−1	2(1,2)	HSDT (Nguyen 26 )	72.3281	73.4526	73.8426	73.2827	69.9876	68.7244
		HSDT (Thai and Choi 15 )	72.0983	—	73.6437	73.1436	—	—
		HSDT (Bodaghi and Saidi 48 )	72.2275	—	73.6645	73.1587	—	—
		FSDT (Mohammadi et al. 49 )	72.0834	—	73.6307	73.6112	—	—
		Present	68.543	73.2757	73.6614	73.1491	70.0346	68.5429

HSDT: higher order shear deformation theory; FSDT: first-order shear deformation theory.

Conclusion

An HSDT is used for bending, free vibration, and buckling analyzes of FGMs plates. The theory is a combination of recently developed exponential–trigonometric shape function for transverse shear deformation effect and developed displacement fields of four unknowns, without thickness stretching effect, and simply supported boundary conditions, satisfying the stress-free boundary conditions on the plate free surfaces. Equations of motion are extracted from Hamilton’s principle and solved via Navier-type solution. The effects of several parameters as the volume fraction index, geometric ratios, frequency modes, in-plane compressive load parameters, and material properties on the deflection, stresses, natural frequencies, and critical buckling load are analyzed. The accuracy of the proposed model is checked by comparing the calculated results with other plate theories found in the literature. As a conclusion, this theory is appropriate, simple and accurate, given the following results:

The obtained results are in good agreement and closer to different HSDT results in many cases, which defined the theory convergence.

For the E-FGMs plates, $\overline{w}$ values becomes more important with the decrease of both the index (p) and the ratio (a/h) values, as well the increases of ratio (b/a) values. However, for the FGMs plates, $\overline{w}$ values increase as the index (p) values increase.

For the FGMs plates, $\overline{ u}$ values increases with the increase of the index (p) values. However, ${\overline{\sigma }}_{xx}\left(z\right)$ , $\overline{\beta }$ , $\overline{\omega }$ , $\widehat{\omega }$ , and ${\overline{N}}_{cr}$ values increase with the decrease of the index (p) values.

$\overline{\beta }$ , $\overline{\omega }$ and $\widehat{\omega }$ values increase by increasing and decreasing the ratio (a/h) values, corresponding to the nondimensional relations of each one, which are proportional and inversely proportional to the ratio (a/h), respectively. Thus, $\overline{\omega }$ and $\widehat{\omega }$ values increase with the increase of the frequency modes values.

${\overline{\sigma }}_{xz}\left(z\right)$ values become less important than ${\overline{\sigma }}_{xx}\left(z\right)$ and ${\overline{\sigma }}_{xy}\left(z\right)$ as 8 and 4, 12 and 2, 23 and 4 times lower, for the indexes (p = 0), (p = 1), (p = 20), respectively.

The same mechanical behaviors are observed for both $\overline{\omega }$ and ${\overline{N}}_{cr}$ , and for any in-plane compressive load; which either the $\overline{\omega }$ and ${\overline{N}}_{cr}$ , highest values are found while the other one is null (at zero coordinate), for rectangular, homogeneous plates, and biaxial compression load; and the smallest values, are those for FGMs plate and uniaxial compression load; the FGMs plate had a significant effect in buckling, and leads to a reduction of 36.81% in the $\overline{\omega }$ and 71% in the ${\overline{N}}_{cr}$ compared to an homogeneous plate.

An inverse relationship regroups the $\overline{\omega }$ with the ${\overline{N}}_{cr}$ values. When the first increases, the second decreases considerably, and vice versa. The $\overline{\omega }$ and ${\overline{N}}_{cr}$ values reduction are obvious for stiffer plates with (p < 2) and thick to moderately thick plate with (a/h∈]2,10[). Then, $\overline{\omega }$ and ${\overline{N}}_{cr}$ paces show a trend to stabilize after this values.