Three new hybrid quasi-3D and 2D higher-order shear deformation theories for free vibration analysis of functionally graded material monolayer and sandwich plates with stretching effect

Abstract

The present investigation brings to the readers three new hybrid higher-order shear deformation theory (HSDT) models and analyses the functionally graded material (FGM) plates. The major objective of this work is to develop three HSDTs in a unique formulation by polynomial–hyperbolic–exponential and polynomial–trigonometric forms, propose the three new HSDT models, investigate the effect of thickness stretching by considering a quasi-three-dimensional theory and analyse the free vibration of isotropic and FGM monolayer and sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) plates to demonstrate the models ability. Therefore, the Hamilton’s principle is exploited to develop equations of motion based on a displacement field of only five unknowns, of which three of them distinguished the transverse displacement membranes through the plate thickness (bending, shear and stretching displacements). In addition, the analytical solutions are found by applying the Navier approach for a simply supported boundary conditions type. The theory also considered that transverse shear deformation effect satisfied the stress-free boundary conditions on the plate-free surfaces without any requirement of shear correction factors. The used mechanical properties followed the power law and the Mori–Tanaka scheme distributions through the plate thickness. The determined results explained the effects of different non-dimensional parameters, and the proposed HSDTs predict the proper responses for monolayer and sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) FGM plates in comparison with other different plates’ theories solutions found in the literature references, thus the reliability and accuracy of the present approach are ascertained. It is obtained that the present formulations of polynomial–hyperbolic–exponential and polynomial–trigonometric forms can be further extended to all existing HSDTs models, for numerous problems related to the shear deformable effect.

Keywords

quasi-3D and 2D HSDTs FGM monolayer and sandwich plates free vibration stretching effect analytical model

Introduction

Functionally graded materials (FGMs) are a new generation of the composite materials (ceramic/metal) that were developed by a Japanese material scientists group from National Aerospace Laboratories in 1984, as a means of preparing thermal barrier materials.¹ These materials have been given great popularity in the design, fabrication and development research fields, after the first national project entitled ‘Research on the Basic Technology for the Development of FGMs for Relaxation of Thermal-Stress’, because of their considerable thermal and mechanical capabilities.^{2

–12} Their richer compositions of ceramic qualified them to be used in sectors of extremely high temperature, such as aeronautical structures, space aircraft and nuclear enclosures. The used FGM plates and shells avoid several problems in classical composite materials, especially during dynamic or cyclic loadings, such as the delamination problem. However, they are subjected to the vibration problem. As a result, these problems have been treated by many analytical and numerical studies based on different plate theories that were developing in three phases, which are the classical plate theory (CPT), the first-order shear deformation theory (FSDT) that requires a shear correction factor, as well the higher-order shear deformation theory (HSDT) that includes a shear deformation effect, such as the sinusoidal higher-order shear deformation theory (SSDT), which was used by a two-dimensional (2D) HSDT derived by Matsunaga.¹³ He analysed the free vibration problem by a method of power series expansion of displacements components, and a set of fundamental dynamic equations, for rectangular FGM plates with simply supported edges. Belabed et al.¹⁴ presented a hyperbolic HSDT with the stretching effect to predict free vibration responses of FGM plates. They indicated that the thickness stretching effect for thick plates is important. It is noticed that Talha and Singh¹⁵ presented an HSDT for an investigation of the free vibration problem and made a special modification in the transverse displacement in conjunction with finite element models. The obtained results employed a continuous isoparametric Lagrangian finite element with 13 degrees of freedom per node. They used the FGM plates with different boundary conditions. A new HSDT developed by Ait Atmane et al.¹⁶ studied free vibration resting on Winkler–Pasternak elastic foundations analysis of FGM plates. Whereas the higher-order shear and normal deformable plate theory studied by Qian et al.¹⁷ used the meshless local Petrov–Galerkin method. The theory investigated both of the free and forced vibrations analyses of a thick rectangular elastic FGM plate. They used only the Mori–Tanaka homogenization technique to calculate the effective material modules. Younsi et al.¹⁸ developed a new 2D and quasi three-dimensional (quasi-3D) hyperbolic HSDT for the analyses of free vibration problem of FGM plates, and the used displacements field included undetermined integral terms. A new quasi-3D hyperbolic HSDT for the free vibration analysis of functionally graded plates is developed by Hebali et al.¹⁹ Zaoui et al.²⁰ studied the free vibration of FGM plates resting on elastic foundations based on quasi-3D hybrid-type HSDT. A new SSDT developed by Thai and Thuc²¹ analysed the vibration of FGM plates. It can be seen that Neves et al.²² studied a quasi-3D SSDT for the free vibration problem analysed for FGM plates. A new HSDT developed by Thai and Kim²³ analysed the free vibration problem of FGM plates with simply supported edges. Abedalnour et al.²⁴ developed a new quasi-3D trigonometric HSDT and a new displacement field that introduced undetermined integral variables for the free vibration analysis of FGM plates with simply supported edges. A quasi-3D SSDT developed by Neves et al.²⁵ analysed the free vibration problem of the FGM plates. A new quasi-3D hyperbolic HSDT was developed by Neves et al.²⁶ for the free vibration analysis of FGM plates with simply supported edges. Thai et al.²⁷ proposed a new inverse tangent shear deformation theory for the free vibration analysis of laminated composite and sandwich plates. In addition, two new shear deformation theories for the free vibration analysis of FGM made of isotropic and sandwich plates are presented by Thai et al..²⁸ Nguyen-Xuan et al.²⁹ presented a new fifth-order shear deformation theory for composite sandwich plates and the free vibration analysis of rectangular and circular plates investigated for different boundary conditions. Thai et al.³⁰ derived a quasi-3D shear deformation theory for free vibration analysis of multilayer functionally graded graphene platelet-reinforced composite microplates. A new shear and normal deformations theory for the free vibration of FGM isotropic and sandwich plates is presented by Thai et al..³¹ Based on an HSDT, Thai et al.³² presented a non-classical model for the free vibration analysis of FGM isotropic and sandwich microplates. Nebab et al.³³ used an HSDT to predict the free vibration of the FGM plate. A novel quasi-3D HSDT constructed from a novel seventh-order shear deformation is proposed by Nguyen et al.³⁴ to investigate the free vibration responses of rectangular and circular FGM microplates. Bennoun et al.³⁵ developed a new quasi-3D HSDT for the free vibration analysis of FGM sandwich plates. Zaoui et al.³⁶ proposed a new hybrid 2D and quasi-3D HSDT (exponential–trigonometric), for the free vibration analysis of FGM plates, resting on elastic foundations. An HSDT was studied by Belkhodja et al.³⁷ for the free vibration analysis of FGM plates with simply supported edges. A new hyperbolic HSDT is presented for the free vibration analysis of FGM sandwich plates by El Meiche et al.³⁸

The present study developed three new hybrid quasi-3D and 2D HSDTs (polynomial–hyperbolic–exponential) and (polynomial–trigonometric) for the free vibration problems analysis of square and rectangular FGM monolayer and sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) plates with simply supported edges. The selected displacements field included the transverse shear deformation effect that satisfied the stress-free boundary conditions on the plate free surfaces, and only five unknowns, which three of them characterized the bending, shear and thickness stretching transverse displacement membranes through the plate thickness. The mechanical properties are continuously varied through the plate thickness as the power-law (P-FGMs) and the Mori–Tanaka scheme (MT-FGMs) distributions. Moreover, the produced five equations of motion were obtained from Hamilton’s principle and were solved by the Navier approach for simply supported boundary conditions. Furthermore, a richer study investigated several parameters effects such as the geometric ratios (the side-to-thickness ratio, the aspect ratio as well the thickness ratio for symmetric and non- symmetric, with hardcore as well as softcore sandwich plates), the volume fraction index, the frequency modes and the materials properties on the natural frequencies. Finally, analytical solutions were obtained and numerical results were validated by comparisons with others plates’ theories solutions found in the literature references to verify the accuracy and the efficiency of the present theories.

The theoretical formulation

An accurate and efficient theoretical formulation is achieved for the FGMs by a developed architecture of design, processing and evaluation.¹ Thus, the used architectural design is a monolayer (single-layer) and sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) plates of the dimensions indicated as length (a), width (b) and uniform thickness (h) as shown in Figures 1, 2 and 3, respectively. The thickness evolution follows the z-coordinate (z-axis) perpendicular to the rectangular Cartesian coordinates (0, x, y) defined in the plate median plane,³⁷ and all in-plane edges of the plate are parallel to x and y axes as well as the bottom and top plate faces are at the plate extremities z = ±h/2, with respect to the coordinates h1 and h4.

Figure 1.

The FGM plate geometric model.³⁷ FGM: functionally graded material.

Figure 2.

The FGM sandwich plate geometric model (type A). FGM: functionally graded material.

Figure 3.

The FGM sandwich plate geometric model (type B). FGM: functionally graded material.

The sandwich plates (symmetric as well as non-symmetric, with hardcore as well as softcore) are composed of three elastic isotropic homogeneous and anisotropic microscopically heterogeneous layers (Figures 2 and 3): Layer 1 is a bottom face layer (z∈[h1, h2]); layer 2 is a median core layer (z∈[h2, h3]); and layer 3 is a top face layer (z∈[h3, h4]), where h2 and h3 are the vertical coordinates of the two median interfaces (Figures 2 and 3).

Therefore, the plates were characterized by geometric ratios such as the side-to-thickness ratio (a/h), the aspect ratio (a/b)³⁷ and the thickness ratio. The ratio (a/h) specifies three plate thicknesses that are a thick plate (with a low ratio values such as (a/h = 2, $\sqrt{10}$ , 5)), moderately thick plate (a/h ≈ 10) and a thin plate (with high ratio values such as (a/h = 20)). It is necessary to know that the transverse shear deformation effect proportionally varies with the plate thickness value.³⁷ Moreover, the ratios (a/b = 1) and (a/b ≠ 1) define a square plate and a rectangular plate, respectively.³⁷ The thickness ratio is denoted by the combination of three numbers i-j-k, the sum of the three numbers is X, each number is assumed to be taken from X as ratio as i/X, j/X, k/X, X represent the number of small thicknesses that the plate is divided to, h1 and h2 are known. The plate in-plane always must be in the median of the plate, if it is noticed that the in-plane pass in a small thickness of the plate and it divided again, then the other small thicknesses will be divided too, all the new smallest thicknesses must have the same thickness. h1 and h2 will be taken from the new division, initiated by the bottom layer to top as (1-0-1), (2-1-2), (1-1-1), (1-2-1) for symmetric sandwich plates as well as (2-2-1), (2-1-1) for non-symmetric sandwich plates and so on. There are two types of sandwich plates (symmetric as well as non-symmetric, with hardcore as well as softcore): Type A is composed of homogeneous core layer and FGMs face layers (Figure 2), and vice versa for type B (Figure 3).

For structural and functional uses, FGM concept can be applied to several materials.¹ However, the studied FGM plate was made from a graded mixture of only two different materials, a metal (aluminium: Al) and a ceramic (alumina: Al₂O₃ or zirconia: ZrO₂), of mechanical properties grouped in Table 1.

Table 1.

The material properties of metal and ceramics.

Materials		Material properties
Materials		Young’s modulus (GPa)	Mass density (kg/m³)	Poisson’s ratio
Metal	Aluminium-1 (Al)₁	70	2702	0.3
	Aluminium-2 (Al)₂	70	2707	0.3
Ceramics	Zirconia (ZrO₂)	200	5700	0.3
Ceramics	Alumina (Al₂O₃)	380	3800	0.3

The mechanical properties of the FGMs

FGM compositions gradually change, resulting in a corresponding modification in the effective mechanical properties,¹ they although heterogeneous are idealized as continue through the plate thickness (z-axis), smoothly with respect to the spatial coordinates.³⁹ Contrary to a discrete model, the mechanical properties are assumed to be graded by simple continuous material distributions neglecting the microstructure of the plate.^37,40 The mechanical properties are the Young’s modulus, the Poisson’s ratio and the density, were described from homogeneous plate theories, after homogenized the FGM plate with their effective modules.^37,40

The Young’s modulus

The Young’s module (E(z)) can be expressed by the rule of mixture and follows the mathematical formulation that describes the power-law,⁴¹ and the Mori–Tanaka scheme,^42,43 according to the equation (1):

E^{(n)} (z) = (E_{c} - E_{m}) V_{c} ​^{(n)} (z) + E_{m},

where E⁽ⁿ⁾ , V⁽ⁿ⁾ , (n = 1,2,3) denotes the effective material property and the volume fraction function of layer (n), respectively. For the monolayer plate (n = 1), E_c and E_m are the Young’s moduli of the upper (ceramic) and lower (metal) FGM plate faces, respectively. The volume fraction of the ceramic material (V_c ) is defined in equations 2 (a–d):

For monolayer plate:

Type P-FGMs:

V_{c 1} (z) = {(\frac{1}{2} + \frac{z}{h})}^{P}

Type MT-FGMs:

V_{c 2} (z) = V_{c 1} (z) \frac{(3 - 3 ν)}{(3 - 3 ν) + (1 - V_{c 1} (z)) (\frac{E_{c}}{E_{m}} - 1) (1 + ν)}

For sandwich P-FGM plate:

Type A:

{\begin{cases} V_{c} ​^{1} (z) = {(\frac{z - h 1}{h 2 - h 1})}^{P}, \\ V_{c} ​^{2} (z) = 1, \\ V_{c} ​^{3} (z) = {(\frac{z - h 4}{h 3 - h 4})}^{P}, \end{cases}

Type B:

{\begin{cases} V_{c} ​^{1} (z) = 1, \\ V_{c} ​^{2} (z) = {(\frac{1}{2} + \frac{z - h 2}{h 3 - h 2})}^{P}, \\ V_{c} ​^{3} (z) = 0, \end{cases}

where the positive volume fraction index (p) specifies three plate types, namely the homogeneous ceramic plate (p = 0, extremely stiff), the FGM plate (p∈]0,∞[, there is the stiffer P-FGM plates (p∈]0,1[), the moderately stiff P-FGM plates (p = 1) as well as the softer P-FGM plates (p∈]1,∞[) and the metal plate (p→∞, extremely soft),³⁷ as well FGM distributions profiles through the plate thickness z∈[-h/2, h/2] only, follow the volume fractions (V_c ), continuously and gradually varied as a function of the position (z-coordinate), as presented in Figure 4 (a) and (b).³⁷

Figure 4.

Ceramic material volume fraction: (a) the power law profile $V_{c 1} (z) = {(z / h + 0.5)}^{P}$ along the thickness of P-FGM plate and (b) the Mori-Tanaka scheme profile along the thickness of MT-FGM plate $V_{c 2} (z) = V_{c 1} (z) . [(3 - 3 v) / [(3 - 3 v) + (1 - V_{c 1} (z)) [(E_{c} - E_{m}) - 1] (1 + v)]]$ . P-FGM: power-law functionally graded material; MT-FGM: Mori–Tanaka functionally graded material.

The Poisson’s ratio

When the Poisson’s ratios (ν (z)) of the ceramic and the metal are nearly equal, the Poisson’s ratio is considered constant because there is no significant difference between obtained results, and it has no significant effect on the FGM plate.^43,44

The density (the mass density)

The effective density (ρ(z)) is estimated only by the power law with Voigt’s mixtures rule⁴⁵ as:

ρ^{(n)} (z) = (ρ_{c} - ρ_{m}) V_{c} ​^{(n)} (z) + ρ_{m}

The kinematics, the strains, the stresses and the energies study of the present new HSDTs

The kinematics

The present developed HSDTs is a combination of both the procedure were developed for monolayer plates by Belabed et al.,¹⁴ with the used point displacements field extended as the bending (w_b ), the shear (w_s ) and the stretching (w_st ) transverse displacement membranes through z-axis, and with only five unknowns (significantly facilitated engineering analyses) as well as the procedures of sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) plates developed by El Meiche et al.,³⁸ written as:

u (x, y, z, t) = u_{0} (x, y, t) - z \frac{\partial w_{b}}{\partial x} - f (z) \frac{\partial w_{s}}{\partial x}

v (x, y, z, t) = v_{0} (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) + w_{s t} (x, y, t)

where u₀ and v₀ are, respectively, the axial displacements over the x and y Cartesian coordinates axes on the plate median plane. The developed new hybrid (polynomial–hyperbolic–exponential) and (polynomial–trigonometric) shape functions (f (z)) are presented in Table 2, with several other shape functions derived by others researchers illustrating the transverse shear strains and tangential stresses effect nonlinear distribution through the FGM plates thickness (h)³⁷ and specifies three plates theories displacements field that are the CPT with (f (z) = 0), the FSDT⁴² with (f (z) = z), else the HSDT. Numerical examples are given to show high accuracy of the proposed method in Tables 3 to 8, approved the novelty, simplicity and effectuality of these new HSDTs and shape functions.

Table 2.

Shape functions form of different higher-order shear deformation theories.

Models		Shape functions f(z) of shear strain
Polynomial functions	Ambartsumian⁴⁶	$\frac{z h^{2}}{8} (1 - \frac{4}{3} \frac{z^{2}}{h^{2}})$
	Kaczkowski⁴⁷	$\frac{5 z}{4} (1 - \frac{4}{3} \frac{z^{2}}{h^{2}})$
	Panc⁴⁸
	Reissner⁴⁹
	Levinson⁵⁰	$z (1 - \frac{4}{3} \frac{z^{2}}{h^{2}})$
	Murthy⁵¹
	Reddy⁵²
	Nguyen-Xuan et al.²⁹	$\frac{7}{8} z - \frac{2}{h^{2}} z^{3} + \frac{2}{h^{4}} z^{5}$
Trigonometric functions	Levy⁵³	$\frac{h}{π} sin (\frac{π}{h} z)$
	Stein⁵⁴
	Touratier⁵⁵
	Arya et al.⁵⁶	$sin (\frac{π}{h} z)$
	Thai et al.²⁷	$h arctan (\frac{2}{h} z) - z$
	Mantari et al.⁵⁷	$tan (m z) - m z {sec}^{2} (\frac{m h}{2}), m = \frac{1}{5 h}$
	Mantari et al.⁵⁷	$tan (m z) - m z {sec}^{2} (\frac{m h}{2}), m = \frac{π}{2 h}$
	Grover et al.⁵⁸	$z sec (\frac{r z}{h}) - z sec (\frac{r}{2}) (1 + \frac{r}{2} tan (\frac{r}{2})), r = 0.1$
	Grover et al.⁵⁸	${cot}^{- 1} (\frac{r h}{z}) - \frac{4 r z}{h (4 r^{2} + 1)}, r = 0.46$
	Nguyen et al.⁵⁹	$h {tan}^{- 1} (\frac{r z}{h}) - \frac{16 r z^{3}}{3 h^{2} (r^{2} + 4)}, r = 1$
Hyperbolic functions	Soldatos⁶⁰	$h sinh (\frac{z}{h}) - z cosh (\frac{1}{2})$
	El Meiche et al.³⁸	$\frac{\frac{h}{π} sinh (\frac{π}{h} z) - z}{cosh (\frac{π}{2}) - 1}$
	Akavci and Tanrikulu⁶¹	$\frac{3 π h}{2} tanh (\frac{z}{h}) + \frac{3 π z}{2} {sech}^{2} (\frac{1}{2})$
	Akavci and Tanrikulu⁶¹	$z sech (\frac{π z^{2}}{h^{2}}) - z sech (\frac{π}{4}) (1 - \frac{π}{2} tanh (\frac{π}{4}))$
	Mahi et al.⁶²	$\frac{h}{2} tanh (\frac{2 z}{h}) - \frac{4 z^{3}}{3 h^{2} {cosh}^{2} (1)}$
	Grover et al.⁶³	${sinh}^{- 1} (\frac{r z}{h}) - \frac{2 r z}{h \sqrt{r^{2} + 4}}, r = 3$
	Shi et al.⁶⁴	$\frac{h}{2} tanh (\frac{2}{h} z) + z \frac{1}{{cosh}^{2} 1}$
Exponential functions	Karama et al.⁶⁵	$z e^{- 2 {(\frac{z}{h})}^{2}}$
Exponential functions	Aydogdu⁶⁶	$z α^{- 2 {(z / h)}^{2} / ln α} = z (3^{- 2 {(z / h)}^{2} / ln 3}), \forall α 〉 0, α$
	Mantari et al.⁶⁷	$m^{- 2 {(z / h)}^{2}} z + y * z = {2.85}^{- 2 {(z / h)}^{2}} z + 0.28 z$
Combination functions	Mantari et al.⁶⁸	$sin (\frac{π}{h} z) e^{\frac{1}{2} cos (\frac{π z}{h})} + \frac{π}{2 h} z$
	Mantari et al.⁶⁹	$sinh (\frac{z}{h}) e^{m cosh (\frac{z}{h})} - \frac{z}{h} [cosh (\frac{1}{2}) + m {sinh}^{2} (\frac{1}{2})] e^{m cosh (\frac{1}{2})}, m = - 6, m = - 7$
	Mantari et al.⁷⁰	$z e^{m cos (\frac{n z}{h})} - z [1 - \frac{1}{2} n m sin (\frac{n}{2})] e^{m cos (\frac{n}{2})}, m = 1, n = 2.9$
	Thai et al.²⁸	${tan}^{- 1} [sin (\frac{π z}{h})]$
	Thai et al.²⁸	${sinh}^{- 1} [sin (\frac{π z}{h})]$
	Suganyadevi and Singh⁷¹	$\frac{h}{r} {tan}^{- 1} (\frac{r z}{h}) - \frac{z}{(r^{2} z^{2} / h^{2}) + 1}, r = 2.5$
	Singh and Singh⁷²	$tan (\frac{m z}{h}) + 2 z cosh (\frac{1}{2}), m = 5$
	Singh and Singh⁷²	$sin (\frac{π}{h} z) + \frac{π}{2 h} z$
	Zaoui et al.³⁶	$\frac{π h}{π^{4} + h^{4}} [e^{(h z / π)} [π^{2} sin (\frac{π z}{h}) + h^{2} cos (\frac{π z}{h})] - h^{2}]$
	Belkhodja et al.³⁷
	Present theory 1	$z \cdot ((\frac{25 z^{2}}{π (z^{2} h + h^{3})} + cosh (\frac{π z}{h}) + e^{{(\frac{z}{h})}^{2}}) - r 1)$ $r 1 = (\frac{3 π h e^{\frac{1}{4}} + 2 π h cosh (\frac{π}{2}) + π^{2} h sinh (\frac{π}{2}) - 2 π h + 26}{2 π h})$
	Present theory 2	$\frac{2 z}{13} \cdot (cosh (\frac{π z}{h}) + + e^{{(\frac{z}{h})}^{2}} + \frac{4 z^{2}}{π (z^{2} h + h^{3})} - r 2)$ $r2=0 .340334313$
	Present theory 3	$z - \frac{13}{16} z cos (\frac{π}{8}) - (\frac{h}{π}) sin (\frac{8 π z^{3}}{h (4 z^{2} - 9 h^{2})})$

Table 3.

Non-dimensional natural frequency ( $\bar{β}$ ) results of aluminium-1/zirconia (Al)₁ /ZrO₂ MT-FGM square plates, in the case of homogeneous ceramic plate $\bar{β} = ω h {\sqrt{ρ_{c} / E}}_{c}$ .

Theories	Ceramic		p = 1			a/h = 5
Theories	a/h = $\sqrt{10}$	a/h = 10	a/h = 5	a/h = 10	a/h = 20	p = 2	p = 3	p = 5
3D Vel and Batra⁷⁷	0.4658	0.0578	0.2192	0.0596	0.0153	0.2197	0.2211	0.2225
Quasi-3D HSDT Belabed et al.¹⁴	0.4659	0.0578	0.2192	0.0597	0.0153	0.2201	0.2214	0.2225
Present theory 2 (Quasi-3D)	0.4659	0.0578	0.2193	0.0597	0.0153	0.2201	0.2214	0.2225
Present theory 1 (Quasi-3D)	0.4660	0.0578	0.2193	0.0597	0.0153	0.2201	0.2214	0.2225
Quasi-3D Neves et al.⁷⁸	–	–	0.2193	0.0596	0.0153	0.2198	0.2212	0.2225
Quasi-3D Neves et al.⁷⁹	–	–	0.2193	0.0596	0.0153	0.2201	0.2216	0.2230

Table 4.

Non-dimensional fundamental frequency ( $\hat{ω}$ ) results of aluminium-1/alumina (Al)₁/Al₂O₃ P-FGM square plates.

$a / h$	Mode (m, n)	Theories	Power-law index p
$a / h$	Mode (m, n)	Theories	Ceramic	0.5	1	4	10
2	1(1,1)	Quasi-3D HSDT Belabed et al.¹⁴	0.9414	0.8248	0.7516	0.6056	0.5495
		Present theory 1 (Quasi-3D)	0.9411	0.8247	0.7514	0.6055	0.5494
		Present theory 2 (Quasi-3D)	0.9405	0.8242	0.7509	0.6063	0.5498
		Quasi-3D Matsunaga¹³	0.9400	0.8233	0.7477	0.5997	0.5460
		2D HSDT Thai and Kim²³	0.9297	0.8110	0.7356	0.5924	0.5412
		Present theory 1 (2D)	0.9295	0.8109	0.7355	0.5926	0.5412
		Present theory 2 (2D)	0.9295	0.8108	0.7354	0.5938	0.5420
	2(1,2)	Quasi-3D HSDT Belabed et al.¹⁴	1.7512	1.5498	1.4164	1.1147	0.9958
		Present theory 1 (Quasi-3D)	1.7503	1.5491	1.4156	1.1143	0.9954
		Present theory 2 (Quasi-3D)	1.7558	1.5456	1.4123	1.1136	0.9937
		Quasi-3D Matsunaga¹³	1.7406	1.5425	1.4078	1.1040	0.9847
		2D HSDT Thai and Kim²³	1.7233	1.5192	1.3844	1.0919	0.9807
		Present theory 1 (2D)	1.7219	1.5183	1.3834	1.0918	0.9802
		Present theory 2 (2D)	1.7198	1.5167	1.3818	1.0926	0.9800
5	1(1,1)	Present theory 2 (Quasi-3D)	0.2122	0.1826	0.1660	0.1409	0.1319
		Present theory 1 (Quasi-3D)	0.2122	0.1826	0.1660	0.1409	0.1318
		Quasi-3D HSDT Belabed et al.¹⁴	0.2121	0.1819	0.1640	0.1383	0.1306
		Quasi-3D Matsunaga¹³	0.2121	0.1819	0.1640	0.1383	0.1306
		2D HSDT Nguyen²⁹	0.2117	0.1807	0.1634	0.1378	0.1303
		2D Belkhodja et al.³⁷	0.2113	0.1808	0.1632	0.1378	0.1300
		Present theory 1 (2D)	0.2113	0.1807	0.1631	0.1378	0.1300
		2D HSDT Thai and Kim²³	0.2113	0.1807	0.1631	0.1378	0.1301
		Present theory 2 (2D)	0.2113	0.1807	0.1631	0.1379	0.1301
	2(1,2)	Present theory 1 (Quasi-3D)	0.4663	0.4044	0.3679	0.3048	0.2812
		Quasi-3D HSDT Belabed et al.¹⁴	0.4659	0.4041	0.3676	0.3047	0.2811
		Present theory 2 (Quasi-3D)	0.4658	0.4041	0.3676	0.3046	0.2811
		Quasi-3D Matsunaga¹³	0.4658	0.4040	0.3644	0.3000	0.2790
		2D HSDT Nguyen²⁹	0.4645	0.4004	0.3622	0.2981	0.2783
		2D Belkhodja et al.³⁷	0.4625	0.3990	0.3609	0.2980	0.2769
		2D HSDT Thai and Kim²³	0.4623	0.3989	0.3607	0.2980	0.2771
		Present theory 2 (2D)	0.4623	0.3989	0.3607	0.2980	0.2771
		Present theory 1 (2D)	0.4623	0.3989	0.3607	0.2979	0.2771
	3(2,2)	Present theory 1 (Quasi-3D)	0.6764	0.5896	0.5368	0.4382	0.4010
		Quasi-3D HSDT Belabed et al.¹⁴	0.6757	0.5890	0.5362	0.4381	0.4008
		Present theory 2 (Quasi-3D)	0.6754	0.5889	0.5360	0.4378	0.4005
		2D HSDT Thai and Kim²³	0.6734	0.5836	0.5286	0.4291	0.3974
		2D Belkhodja et al.³⁷	0.6694	0.5806	0.5258	0.4285	0.3946
		Present theory 1 (2D)	0.6689	0.5803	0.5255	0.4282	0.3948
		Present theory 2 (2D)	0.6688	0.5803	0.5254	0.4285	0.3948
		Quasi-3D Matsunaga¹³	0.6688	0.5803	0.5254	0.4284	0.3948
10	1(1,1)	Present theory 2 (Quasi-3D)	0.0578	0.0494	0.0449	0.0389	0.0369
		Present theory 1 (Quasi-3D)	0.0578	0.0494	0.0449	0.0389	0.0368
		Quasi-3D HSDT Belabed et al.¹⁴	0.0578	0.0494	0.0449	0.0389	0.0368
		Quasi-3D Matsunaga¹³	0.0578	0.0492	0.0443	0.0381	0.0364
		Present theory 1 (2D)	0.0577	0.0490	0.0442	0.0381	0.0364
		Present theory 2 (2D)	0.0577	0.0490	0.0442	0.0381	0.0364
		2D HSDT Thai and Kim²³	0.0577	0.0490	0.0442	0.0381	0.0364
		2D Belkhodja et al.³⁷	0.0577	0.0490	0.0442	0.0381	0.0364
	2(1,2)	Present theory 1 (Quasi-3D)	0.1382	0.1185	0.1078	0.0923	0.0868
		Present theory 2 (Quasi-3D)	0.1382	0.1185	0.1077	0.0923	0.0869
		Quasi-3D HSDT Belabed et al.¹⁴	0.1381	0.1184	0.1077	0.0923	0.0868
		Quasi-3D Matsunaga¹³	0.1381	0.1180	0.1063	0.0905	0.0859
		Present theory 2 (2D)	0.1377	0.1175	0.1060	0.0903	0.0857
		2D HSDT Thai and Kim²³	0.1377	0.1174	0.1059	0.0903	0.0856
		2D Belkhodja et al.³⁷	0.1377	0.1174	0.1059	0.0902	0.0856
		Present theory 1 (2D)	0.1377	0.1174	0.1059	0.0902	0.0856
	3(2,2)	Present theory 1 (Quasi-3D)	0.2124	0.1827	0.1661	0.1410	0.1319
		Present theory 2 (Quasi-3D)	0.2124	0.1827	0.1661	0.1409	0.1320
		Quasi-3D HSDT Belabed et al.¹⁴	0.2121	0.1825	0.1659	0.1409	0.1318
		Quasi-3D Matsunaga¹³	0.2117	0.1810	0.1634	0.1378	0.1303
		Present theory 2 (2D)	0.2115	0.1809	0.1633	0.1379	0.1302
		2D HSDT Thai and Kim²³	0.2113	0.1807	0.1631	0.1378	0.1301
		2D Belkhodja et al.³⁷	0.2113	0.1807	0.1632	0.1378	0.1300
		Present theory 1 (2D)	0.2113	0.1807	0.1631	0.1378	0.1300
20	1(1,1)	Present theory 1 (Quasi-3D)	0.0148	0.0127	0.0115	0.0100	0.0095
		Present theory 2 (Quasi-3D)	0.0148	0.0126	0.0115	0.0100	0.0095
		Quasi-3D HSDT Belabed et al.¹⁴	0.0148	0.0126	0.0115	0.0100	0.0095
		Present theory 2 (2D)	0.0148	0.0126	0.0113	0.0098	0.0094
		2D HSDT Thai and Kim²³	0.0148	0.0125	0.0113	0.0098	0.0094
		2D Belkhodja et al.³⁷	0.0148	0.0125	0.0113	0.0098	0.0094
		Quasi-3D Matsunaga¹³	0.0148	0.0125	0.0113	0.0098	0.0094
		2D HSDT Nguyen²⁹	0.0148	0.0125	0.0113	0.0098	0.0094
		Present theory 1 (2D)	0.0148	0.0125	0.0113	0.0098	0.0094
	2(1,2)	Present theory 2 (Quasi-3D)	0.0366	0.0313	0.0284	0.0247	0.0235
		Present theory 1 (Quasi-3D)	0.0366	0.0313	0.0284	0.0247	0.0234
		Present theory 2 (2D)	0.0366	0.0310	0.0280	0.0242	0.0232
		2D Belkhodja et al.³⁷	0.0365	0.0310	0.0279	0.0241	0.0231
		2D HSDT Nguyen²⁹	0.0365	0.0310	0.0279	0.0241	0.0231
		Present theory 1 (2D)	0.0365	0.0310	0.0279	0.0241	0.0231
	3(2,2)	Present theory 2 (Quasi-3D)	0.0579	0.0495	0.0450	0.0391	0.0370
		Present theory 1 (Quasi-3D)	0.0579	0.0495	0.0450	0.0390	0.0369
		Present theory 2 (2D)	0.0578	0.0491	0.0443	0.0382	0.0366
		2D Belkhodja et al.³⁷	0.0577	0.0490	0.0442	0.0381	0.0364
		2D HSDT Nguyen²⁹	0.0577	0.0490	0.0442	0.0381	0.0364
		Present theory 1 (2D)	0.0577	0.0490	0.0442	0.0381	0.0364

Table 5.

Non-dimensional fundamental frequency ( $\bar{ω}$ ) results of aluminium-1/alumina (Al)₁/Al₂O₃ P-FGM rectangular plates (b = 2a).

a/h	Mode (m, n)	Theories	Power-law index p
a/h	Mode (m, n)	Theories	Ceramic	0.5	1	5	10
5	1(1,1)	FSDT Hosseini et al.⁸⁰	3.4409	2.9322	2.6473	2.2528	2.1677
		2D TSDT Hosseini et al.⁸⁴	3.4412	2.9347	2.6475	2.2272	2.1407
		Present theory 1 (2D)	3.4413	2.9347	2.6476	2.2268	2.1406
		Present theory 2 (2D)	3.4413	2.9347	2.6476	2.2275	2.1408
		2D HSDT Thai and Thuc²¹	3.4416	2.9350	2.6478	2.2260	2.1403
		2D Belkhodja et al.³⁷	3.4417	2.9350	2.6480	2.2269	2.1401
	2(1,2)	FSDT Hosseini et al.⁸⁰	5.2802	4.5122	4.0773	3.4492	3.3094
		2D TSDT Hosseini et al.⁸⁴	5.2813	4.5180	4.0781	3.3938	3.2514
		Present theory 1 (2D)	5.2814	4.5181	4.0782	3.3931	3.2511
		Present theory 2 (2D)	5.2815	4.5181	4.0782	3.3945	3.2515
		2D HSDT Thai and Thuc²¹	5.2822	4.5187	4.0787	3.3914	3.2506
		2D Belkhodja et al.³⁷	5.2824	4.5188	4.0792	3.3932	3.2501
	3(1,3)	FSDT Hosseini et al.⁸⁰	8.0710	6.9231	6.2636	5.2579	5.0253
		2D TSDT Hosseini et al.⁸⁴	8.0749	6.9366	6.2663	5.1425	4.9055
		Present theory 2 (2D)	8.0751	6.9368	6.2664	5.1439	4.9057
		Present theory 1 (2D)	8.0752	6.9369	6.2665	5.1411	4.9051
		2D HSDT Thai and Thuc ²¹	8.0772	6.9384	6.2678	5.1378	4.9044
	4(2,1)	FSDT Hosseini et al.⁸⁰	9.7416	8.6926	7.8711	6.5749	5.7518
		2D TSDT Hosseini et al.⁸⁴	10.1164	8.7138	7.8762	6.4074	6.0954
		Present theory 2 (2D)	10.1167	8.7140	7.8764	6.4094	6.0957
		Present theory 1 (2D)	10.1169	8.7143	7.8766	6.4054	6.0949
		2D HSDT Thai and Thuc²¹	10.1201	8.7167	7.8787	6.4010	6.0942
10	1 (1,1)	FSDT Hosseini et al.⁸⁰	3.6518	3.0983	2.7937	2.3998	2.3197
		2D TSDT Hosseini et al.⁸⁴	3.6518	3.0990	2.7937	2.3916	2.3110
		Present theory 1 (2D)	3.6518	3.0991	2.7937	2.3913	2.3109
		2D Belkhodja et al.³⁷	3.6518	3.0991	2.7938	2.3913	2.3108
		2D HSDT Thai and Thuc²¹	3.6519	3.0991	2.7937	2.3912	2.3108
		Present theory 2 (2D)	3.6525	3.0995	2.7941	2.3917	2.3115
		2D HSDT Nguyen²⁹	3.6533	3.0996	2.7946	2.3911	2.3118
	2(1,2)	FSDT Hosseini et al.⁸⁰	5.7693	4.8997	4.4192	3.7881	3.6580
		2D TSDT Hosseini et al.⁸⁴	5.7694	4.9014	4.4192	3.7682	3.6368
		Present theory 1 (2D)	5.7696	4.9016	4.4193	3.7675	3.6365
		2D HSDT Thai and Thuc²¹	5.7697	4.9016	4.4194	3.7673	3.6365
		2D Belkhodja et al.³⁷	5.7697	4.9016	4.4194	3.7674	3.6364
		Present theory 2 (2D)	5.7711	4.9026	4.4203	3.7684	3.6381
		2D HSDT Nguyen²⁹	5.7731	4.9031	4.4216	3.7671	3.6388
	3(1,3)	FSDT Hosseini et al.⁸⁰	9.1876	7.8145	7.0512	6.0247	5.8086
		2D TSDT Hosseini et al.⁸⁴	9.1880	7.8189	7.0515	5.9765	5.7575
		Present theory 1 (2D)	9.1884	7.8192	7.0517	5.9747	5.7568
		2D HSDT Thai and Thuc²¹	9.1887	7.8194	7.0519	5.9742	5.7566
		Present theory 2 (2D)	9.1924	7.8219	7.0543	5.9770	5.7606
	4(2,1)	FSDT Hosseini et al.⁸⁰	11.8310	10.0740	9.0928	7.7505	7.4639
		2D TSDT Hosseini et al.⁸⁴	11.8315	10.0810	9.0933	7.6731	7.3821
		Present theory 1 (2D)	11.8321	10.0815	9.0937	7.6704	7.3811
		2D HSDT Thai and Thuc²¹	11.8326	10.0818	9.0940	7.6696	7.3808
		Present theory 2 (2D)	11.8386	10.0859	9.0980	7.6742	7.3872
20	1(1,1)	FSDT Hosseini et al.⁸⁰	3.7123	3.1456	2.8352	2.4425	2.3642
		2D Belkhodja et al.³⁷	3.7123	3.1458	2.8353	2.4402	2.3618
		2D TSDT Hosseini et al.⁸⁴	3.7123	3.1458	2.8352	2.4403	2.3619
		2D HSDT Thai and Thuc²¹	3.7123	3.1458	2.8353	2.4401	2.3618
		Present theory 1 (2D)	3.7124	3.1458	2.8353	2.4401	2.3618
		2D HSDT Nguyen²⁹	3.7127	3.1455	2.8355	2.4401	2.3622
		Present theory 2 (2D)	3.7140	3.1469	2.8363	2.4419	2.3641
	2(1,2)	FSDT Hosseini et al.⁸⁰	5.9198	5.0175	4.5228	3.8939	3.7681
		2D Belkhodja et al.³⁷	5.9199	5.0180	4.5228	3.8881	3.7621
		2D HSDT Thai and Thuc²¹	5.9199	5.0180	4.5228	3.8881	3.7621
		Present theory 1 (2D)	5.9199	5.0180	4.5228	3.8881	3.7621
		2D TSDT Hosseini et al.⁸⁴	5.9199	5.0180	4.5228	3.8884	3.7622
		2D HSDT Nguyen²⁹	5.9209	5.0176	4.5234	3.8880	3.7629
		Present theory 2 (2D)	5.9240	5.0208	4.5255	3.8924	3.7678
		1
	3(1,3)	FSDT Hosseini et al.⁸⁰	9.5668	8.1121	7.3132	6.2903	6.0843
		2D TSDT Hosseini et al.⁸⁴	9.5669	8.1133	7.3132	6.2760	6.0690
		2D HSDT Thai and Thuc²¹	9.5671	8.1135	7.3133	6.2753	6.0688
		Present theory 1 (2D)	9.5671	8.1135	7.3134	6.2753	6.0688
		Present theory 2 (2D)	9.5777	8.1207	7.3202	6.2866	6.0836
		1
	4(2,1)	FSDT Hosseini et al.⁸⁰	12.4560	10.5660	9.5261	8.1875	7.9166
		2D TSDT Hosseini et al.⁸⁴	12.4562	10.5677	9.5261	8.1636	7.8909
		2D HSDT Thai and Thuc²¹	12.4565	10.5680	9.5263	8.1624	7.8905
		Present theory 1 (2D)	12.4566	10.5680	9.5263	8.1623	7.8905
		Present theory 2 (2D)	12.4745	10.5802	9.5379	8.1813	7.9153

Table 6.

Non-dimensional natural fundamental frequency parameter ( $\bar{ω}$ ) results of aluminium-2/alumina (Al)₂/Al₂O₃ P-FGM square symmetric and non-symmetric sandwich plates (type A: with homogeneous softcore).

a/h	p	Theories	Symmetric				Non-symmetric
a/h	p	Theories	1-0-1	2-1-2	1-1-1	1-2-1	2-2-1
5	0	Quasi 3D Akavci et al.⁷³	0.8538	0.8538	0.8538	0.8538	0.8538
		Quasi 3D Bessaim et al.⁸¹	0.8529	0.8529	0.8529	0.8529	0.8529
		3D Li et al.⁸²	0.8529	0.8529	0.8529	0.8529	0.8529
		2D Akavci et al.⁷³	0.8494	0.8494	0.8494	0.8494	0.8494
		Present theory 3 (2D)	0.8493	0.8493	0.8493	0.8493	0.8493
	0.5	Quasi 3D Bessaim et al.⁸¹	1.3877	1.3328	1.2899	1.2288	1.2563
		Quasi 3D Akavci et al.⁷³	1.3868	1.3312	1.2885	1.2287	1.2560
		2D Akavci et al.⁷³	1.3801	1.3250	1.2827	1.2235	1.2485
		Present theory 3 (2D)	1.3830	1.3286	1.2858	1.2248	1.2505
		3D Li et al.⁸²	1.3789	1.3206	1.2805	1.2258	1.2453
	1	Quasi 3D Bessaim et al.⁸¹	1.5237	1.4613	1.4088	1.3330	1.3694
		Quasi 3D Akavci et al.⁷³	1.5221	1.4578	1.4050	1.3307	1.3676
		Present theory 3 (2D)	1.5182	1.4567	1.4045	1.3287	1.3627
		2D Akavci et al.⁷³	1.5143	1.4506	1.3983	1.3249	1.3587
		3D Li et al.⁸²	1.5090	1.4333	1.3824	1.3213	1.3420
	5	Quasi 3D Akavci et al.⁷³	1.6658	1.6218	1.5671	1.4735	1.5267
		Present theory 3 (2D)	1.6591	1.6200	1.5690	1.4766	1.5225
		3D Li et al.⁸²	1.6587	1.5801	1.5028	1.4267	1.4601
		2D Akavci et al.⁷³	1.6568	1.6129	1.5588	1.4665	1.5160
		Quasi 3D Bessaim et al.⁸¹	1.5237	1.6257	1.5737	1.4810	1.5301
	10	Quasi 3D Akavci et al.⁷³	1.6761	1.6442	1.5944	1.5002	1.5544
		Quasi 3D Bessaim et al.⁸¹	1.6754	1.6471	1.6006	1.5084	1.5574
		3D Li et al.⁸²	1.6728	1.6091	1.5267	1.4410	1.4831
		Present theory 3 (2D)	1.6684	1.6412	1.5957	1.5041	1.5497
		2D Akavci et al.⁷³	1.6671	1.6350	1.5858	1.4928	1.5435
10	0	Quasi 3D Akavci et al.⁷³	0.9296	0.9296	0.9296	0.9296	0.9296
		Quasi 3D Bessaim et al.⁸¹	0.9290	0.9290	0.9290	0.9290	0.9290
		3D Li et al.⁸²	0.9290	0.9290	0.9290	0.9290	0.9290
		2D Akavci et al.⁷³	0.9278	0.9278	0.9278	0.9278	0.9278
		Present theory 3 (2D)	0.9278	0.9278	0.9278	0.9278	0.9278
	0.5	Quasi 3D Bessaim et al.⁸¹	1.5771	1.5310	1.4885	1.4179	1.4404
		Quasi 3D Akavci et al.⁷³	1.5771	1.5307	1.4883	1.4181	1.4409
		Present theory 3 (2D)	1.5753	1.5294	1.4870	1.4163	1.4364
		2D Akavci et al.⁷³	1.5741	1.5279	1.4857	1.4157	1.4356
		3D Li et al.⁸²	1.5735	1.5259	1.4846	1.4166	1.4342
	1	Quasi 3D Bessaim et al.⁸¹	1.7281	1.6863	1.6420	1.5630	1.5843
		Quasi 3D Akavci et al.⁷³	1.7280	1.6853	1.6408	1.5624	1.5843
		Present theory 3 (2D)	1.7261	1.6845	1.6403	1.5613	1.5793
		2D Akavci et al.⁷³	1.7246	1.6820	1.6377	1.5596	1.5776
		3D Li et al.⁸²	1.7223	1.6744	1.6305	1.5579	1.5704
	5	Quasi 3D Akavci et al.⁷³	1.8452	1.8436	1.8181	1.7486	1.7595
		Quasi 3D Bessaim et al.⁸¹	1.8447	1.8446	1.8203	1.7514	1.7597
		Present theory 3 (2D)	1.8423	1.8425	1.8185	1.7497	1.7541
		3D Li et al.⁸²	1.8420	1.8261	1.7896	1.7267	1.7273
		2D Akavci et al.⁷³	1.8414	1.8397	1.8144	1.7452	1.7515
	10	Quasi 3D Akavci et al.⁷³	1.8420	1.8544	1.8378	1.7757	1.7816
		Quasi 3D Bessaim et al.⁸¹	1.8411	1.8549	1.8397	1.7788	1.7816
		3D Li et al.⁸²	1.8402	1.8399	1.8081	1.7481	1.7478
		Present theory 3 (2D)	1.8388	1.8527	1.8379	1.7770	1.7762
		2D Akavci et al.⁷³	1.8383	1.8504	1.8339	1.7722	1.7737

Table 7.

a/h	p	Theories	Symmetric				Non-symmetric
a/h	p	Theories	1-0-1	2-1-2	1-1-1	1-2-1	2-2-1	2-1-1
5	0	Quasi 3D Akavci et al.⁷³	1.6790	1.6790	1.6790	1.6790	1.6790	–
		Quasi 3D Bessaim et al.⁸¹	1.6772	1.6772	1.6772	1.6772	1.6772	–
		3D Li et al.⁸²	1.6771	1.6771	1.6771	1.6771	1.6771	–
		2D Akavci et al.⁷³	1.6702	1.6702	1.6702	1.6702	1.6702	–
		Present theory 3 (2D)	1.6701	1.6701	1.6701	1.6701	1.6701	–
	5	Quasi 3D Akavci et al.⁷³	0.9001	0.9416	1.0017	1.1202	1.0657	–
		Quasi 3D Bessaim et al.⁸¹	0.8985	0.9403	1.0005	1.1194	1.0642	–
		2D Akavci et al.⁷³	0.8953	0.9365	0.9958	1.1133	1.0531	–
		Present theory 3 (2D)	0.8943	0.9357	0.9954	1.1133	1.0528	–
		3D Li et al.⁸²	0.8909	0.9336	0.9980	1.1190	1.0561	–
								–
	10	Quasi 3D Akavci et al.⁷³	0.8771	0.9045	0.9562	1.0743	1.0228	–
		Quasi 3D Bessaim et al.⁸¹	0.8754	0.9031	0.9549	1.0734	1.0209	–
		2D Akavci et al.⁷³	0.8725	0.8998	0.9508	1.0677	1.0093	–
		Present theory 3 (2D)	0.8714	0.8989	0.9502	1.0676	1.0090	–
		3D Li et al.⁸²	0.8683	0.8923	0.9498	1.0729	1.0095	–
10	0	3D Li et al.⁸²	1.8268	1.8268	1.8268	1.8268	1.8268	1.8268
		2D SSDPT Touratier⁵⁵	1.8245	1.8245	1.8245	1.8245	1.8245	1.8245
		2D HSDT El Meiche et al.³⁸	1.8245	1.8245	1.8245	1.8245	1.8245	1.8245
		2D Akavci et al.⁷³	1.8245	1.8245	1.8245	1.8245	1.8245	1.8245
		2D PSDPT Reddy⁵²	1.8245	1.8245	1.8245	1.8245	1.8245	1.8245
		Present theory 3 (2D)	1.8245	1.8245	1.8245	1.8245	1.8245	1.8245
		FSDPT^83,85,75	1.8244	1.8244	1.8244	1.8244	1.8244	1.8244
	0.5	3D Li et al.⁸²	1.4461	1.4861	1.5213	1.5767	1.5493	1.5084
		2D SSDPT Touratier⁵⁵	1.4444	1.4842	1.5193	1.5745	1.5520	1.5126
		2D PSDPT Reddy⁵²	1.4442	1.4841	1.5192	1.5745	1.5520	1.5125
		Present theory 3 (2D)	1.4442	1.4841	1.5192	1.5745	1.5472	1.5064
		2D HSDT El Meiche et al.³⁸	1.4442	1.4841	1.5192	1.5746	1.5471	1.5064
		FSDPT^83,85,75	1.4417	1.4816	1.5170	1.5727	1.5500	1.5104
	1	3D Li et al.⁸²	1.2447	1.3018	1.3552	1.4414	1.3976	1.3351
		2D SSDPT Touratier⁵⁵	1.2434	1.3002	1.3534	1.4393	1.4079	1.3489
		2D PSDPT Reddy⁵²	1.2432	1.3001	1.3533	1.4393	1.4079	1.3489
		Present theory 3 (2D)	1.2432	1.3001	1.3533	1.4394	1.3957	1.3334
		2D HSDT El Meiche et al.³⁸	1.2431	1.3000	1.3533	1.4394	1.3956	1.3333
		FSDPT^83,85,75	1.2403	1.2973	1.3507	1.4372	1.4056	1.3464
	5	2D Akavci et al.⁷³	0.9462	0.9820	1.0448	1.1740	1.1090	–
		2D SSDPT Touratier⁵⁵	0.9463	0.9821	1.0448	1.1740	1.1474	1.0745
		2D PSDPT Reddy⁵²	0.9460	0.9818	1.0447	1.1740	1.1473	1.0743
		Present theory 3 (2D)	0.9459	0.9818	1.0446	1.1740	1.1090	1.0306
		2D HSDT El Meiche et al.³⁸	0.9457	0.9817	1.0446	1.1740	1.1088	1.0303
		3D Li et al.⁸²	0.9448	0.9810	1.0453	1.1757	1.1098	1.0294
		FSDPT^83,85,75	0.9426	0.9787	1.0418	1.1716	1.1447	1.0716
	10	2D SSDPT Touratier⁵⁵	0.9288	0.9433	0.9952	1.1346	1.0415	1.0456
		2D Akavci et al.⁷³	0.9286	0.9432	0.9956	1.1232	1.0612	–
		2D PSDPT Reddy⁵²	0.9284	0.9430	0.9955	1.1231	1.1053	1.0386
		Present theory 3 (2D)	0.9283	0.9429	0.9955	1.1231	1.0610	0.9921
		2D HSDT El Meiche et al.³⁸	0.9281	0.9428	0.9954	1.1231	1.0608	0.9918
		3D Li et al.⁸²	0.9273	0.9408	0.9952	1.1247	1.0610	0.9893
		FSDPT^83,85,75	0.9251	0.9396	0.9926	1.1207	1.1026	1.0358

Table 8.

	Theories	Mode (m, n)
	Theories	(1,1)	(1,2)	(2,2)	(1,3)	(2,3)	(1,4)	(3,3)	(2,4)	(3,4)	(4,4)
1-2-1	3D Akavci et al.⁷³	1.3051	3.1700	4.9385	6.0705	7.7061	9.7841	10.2870	11.2747	13.6465	16.7650
	2D El meiche et al.³⁸	1.3025	3.1573	4.9098	6.0287	7.6415	9.6847	10.1782	11.1464	13.4665	16.5069
	Present theory 3 (2D)	1.3025	3.1572	4.9096	6.0283	7.6410	9.6837	10.1772	11.1452	13.4647	16.5042
	2D Akavci et al.⁷³	1.3024	3.1569	4.9090	6.0275	7.6397	9.6816	10.1749	11.1425	13.4607	16.4984
	2D Zenkour⁷⁵	1.3024	3.1569	4.9085	6.0262	7.6360	9.6712	10.1619	11.1232	13.4176	16.3982
2-2-1	2D Zenkour⁷⁵	1.2678	3.0738	4.7807	5.8702	7.4400	9.4255	9.9044	10.8426	13.0826	15.9940
	3D Akavci et al.⁷³	1.2509	3.0406	4.7399	5.8287	7.4033	9.4058	9.8908	10.8436	13.1336	16.1481
	2D Akavci et al.⁷³	1.2439	3.0180	4.6968	5.7698	7.3181	9.2817	9.7564	10.6881	12.9227	15.8550
	Present theory 3 (2D)	1.2439	2.9768	4.6963	5.6224	7.2759	8.9872	9.7548	10.5344	12.8791	15.8516
	2D El meiche et al.³⁸	1.2438	3.0170	4.6946	5.7666	7.3132	9.2744	9.7485	10.6789	12.9101	15.8376
	FSDT Nguyen et al.⁷⁶	1.2436	3.0163	4.6932	5.7648	7.3110	9.2719	9.7460	10.6764	12.9084	15.8383

The thickness stretching transverse displacement (w_st ) is defined as

\begin{array}{l} w_{s t} (x, y, t) = {\begin{cases} 0 & For 2D HSDT (ε_{z} = 0) . (5a) \\ g (z) φ (x, y, t) & For 3D and quasi - 3D HSDT (ε_{z} \neq 0) . (5b) \end{cases} \end{array}

where an additional term of transverse displacement component ϕ accounts for the normal deformation effect (stretching effect), and the function g (z) describes thickness stretching effect distribution through the plate thickness and also the transverse shear stress distribution through the same thickness that satisfied and fulfilled the stress-free boundary conditions, as a parabolic variation, as well as it is set to zero at the plate extremities (top and bottom plates surfaces) z = ±h/2. ³⁷

g (z) = 1 - \frac{d f (z)}{d z} .

The strains

The linear strains field is determined by the linear elasticity theory application based on the displacements field (4) derived as

{\begin{matrix} ε_{x} \\ \begin{array}{l} ε_{y} \\ ε_{z} \end{array} \\ γ_{y z} \\ γ_{x z} \\ γ_{x y} \end{matrix}} = {\begin{matrix} \frac{\partial u_{0}}{\partial x} \\ \frac{\partial v_{0}}{\partial y} \\ \begin{array}{c} 0 \\ 0 \end{array} \\ 0 \\ \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} \end{matrix}} - z {\begin{matrix} \frac{\partial^{2} w_{b}}{\partial x^{2}} \\ \frac{\partial^{2} w_{b}}{\partial y^{2}} \\ \begin{array}{c} 0 \\ 0 \end{array} \\ 0 \\ 2 \frac{\partial^{2} w_{b}}{\partial x \partial y} \end{matrix}} - {\begin{matrix} f (z) \frac{\partial^{2} w_{s}}{\partial x^{2}} \\ \begin{array}{c} f (z) \frac{\partial^{2} w_{s}}{\partial y^{2}} \\ - g' (z) ϕ \end{array} \\ - g (z) (\frac{\partial w_{s}}{\partial y} + \frac{\partial ϕ}{\partial y}) \\ - g (z) (\frac{\partial w_{s}}{\partial x} + \frac{\partial ϕ}{\partial x}) \\ 2 f (z) \frac{\partial^{2} w_{s}}{\partial x \partial y} \end{matrix}}

and

g' (z) = \frac{d g (z)}{d z} = \frac{d^{2} f (z)}{d z^{2}} .

The stresses

The constitutive relations (stress–strain relations) described the linear mechanical behaviours of the FGM plates, and the linear stresses field are written in terms of the stiffness matrix as

{\begin{matrix} σ_{x} \\ σ_{y} \\ σ_{z} \\ τ_{y z} \\ τ_{x z} \\ τ_{x y} \end{matrix}}^{(n)} = {[\begin{matrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\ C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & G_{44} & 0 & 0 \\ 0 & 0 & \begin{array}{l} 0 \\ 0 \end{array} & 0 & G_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & G_{66} \end{matrix}]}^{(n)} {\begin{matrix} ε_{x} \\ ε_{y} \\ ε_{z} \\ γ_{y z} \\ γ_{x z} \\ γ_{x y} \end{matrix}}^{(n)}

where the stiffness matrix includes stiffness coefficients, which are defined for 2D HSDT (∊_z = 0) as

C_{11} = C_{22} = \frac{E (z)}{1 - ν^{2}}

10a

C_{12} = ν . C_{11}

10b

C_{33} = C_{13} = C_{23} = 0

10c

G_{44} = G_{55} = G_{66} = \frac{E (z)}{2 (1 + ν)}

10d

Hence, for 3D and quasi-3D HSDT (∊_z ≠ 0) are given as

C_{11} = C_{22} = C_{33} = \frac{(1 - ν) E (z)}{(1 - 2 ν) (1 + ν)}

11a

C_{12} = C_{13} = C_{23} = \frac{ν E (z)}{(1 - 2 ν) (1 + ν)}

11b

G_{44} = G_{55} = G_{66} = \frac{E (z)}{2 (1 + ν)}

11c

The energy principle

The equations of motion are determined for deformable bodies by the Hamilton’s principle that is formulated as

0 = \int_{0}^{t'} (δ U - δ K) d t

where t′ denotes a time period as well δU and δK are, respectively, the strain energy and kinetic energy variations of the FGM plates.

The strain energy

The strain energy (energy of internal loads) variation is calculated in equations (13) as follows, without the stretching effect (ϕ = 0):

δ U = \int_{V} (σ_{x} δ ε_{x} + σ_{y} δ ε_{y} + σ_{z} δ ε_{z} + τ_{x y} δ γ_{x y} + τ_{y z} δ γ_{y z} + τ_{x z} δ γ_{x z}) d V

13a

\begin{array}{l} δ U = \int_{A} [N_{x} \frac{\partial δ u_{0}}{\partial x} - M_{x}^{b} \frac{\partial^{2} δ w_{b}}{\partial x^{2}} - M_{x}^{s} \frac{\partial^{2} δ w_{s}}{\partial x^{2}} + N_{y} \frac{\partial δ v_{0}}{\partial y} - M_{y}^{b} \frac{\partial^{2} δ w_{b}}{\partial y^{2}} - M_{y}^{s} \frac{\partial^{2} δ w_{s}}{\partial y^{2}} + N_{x y} (\frac{\partial δ u_{0}}{\partial y} + \frac{\partial δ v_{0}}{\partial x}) \\ - 2 M_{x y}^{b} \frac{\partial^{2} δ w_{b}}{\partial x \partial y} - 2 M_{x y}^{s} \frac{\partial^{2} δ w_{s}}{\partial x \partial y} + S_{x z}^{s} (\frac{\partial δ w_{s}}{\partial x} + \frac{\partial δ ϕ}{\partial x}) + S_{y z}^{s} (\frac{\partial δ w_{s}}{\partial y} + \frac{\partial δ ϕ}{\partial y}) - N_{z} δ ϕ] d A \end{array}

13b

where A, V and N, M, S denote, respectively, the section, the volume and the stresses resultants that are defined as in equations (14), without the stretching effect (N_z = 0):

[\begin{matrix} N_{x} & N_{y} & 0 & 0 & 0 & N_{x y}^{​} \\ M_{x}^{b} & M_{y}^{b} & 0 & 0 & 0 & M_{x y}^{b} \\ M_{x}^{s} & M_{y}^{s} & 0 & 0 & 0 & M_{x y}^{s} \\ 0 & 0 & 0 & S_{y z}^{s} & S_{x z}^{s} & 0 \\ 0 & 0 & N_{z}^{​} & 0 & 0 & 0 \end{matrix}] = \sum_{n = 1}^{3} ​ \int_{h_{n}}^{h_{n + 1}} {\begin{matrix} 1 \\ z \\ f (z) \\ \begin{array}{l} g (z) \\ g' (z) \end{array} \end{matrix}} {\begin{matrix} σ_{x} & σ_{y} & σ_{z} & τ_{y z} & \begin{matrix} τ_{x z} & τ_{x y} \end{matrix} \end{matrix}} d z

The kinetic energy

The kinetic energy variation is determined as in equations (15), without the stretching effect (ϕ=0):

δ K = \int_{- h / 2}^{h / 2} \int_{A}^{​} (\dot{u} δ \dot{u} + \dot{v} δ \dot{v} + \dot{w} δ \dot{w}) ρ (z) d A

15a

\begin{array}{l} δ K = \int_{A} [I_{0} ({\dot{u}}_{0} δ {\dot{u}}_{0} + {\dot{v}}_{0} δ {\dot{v}}_{0} + ({\dot{w}}_{b} + {\dot{w}}_{s}) (δ {\dot{w}}_{b} + δ {\dot{w}}_{s})) - I_{1} ({\dot{u}}_{0} \frac{\partial δ {\dot{w}}_{b}}{\partial x} + \frac{\partial {\dot{w}}_{b}}{\partial x} δ {\dot{u}}_{0} + {\dot{v}}_{0} \frac{\partial δ {\dot{w}}_{b}}{\partial y} + \frac{\partial {\dot{w}}_{b}}{\partial y} δ {\dot{v}}_{.0}) \\ + I_{2} (\frac{\partial {\dot{w}}_{b}}{\partial x} \frac{\partial δ {\dot{w}}_{b}}{\partial x} + \frac{\partial {\dot{w}}_{b}}{\partial y} \frac{\partial δ {\dot{w}}_{b}}{\partial y}) - J_{1} ({\dot{u}}_{0} \frac{\partial δ {\dot{w}}_{s}}{\partial x} + \frac{\partial {\dot{w}}_{s}}{\partial x} δ {\dot{u}}_{0} + {\dot{v}}_{0} \frac{\partial δ {\dot{w}}_{s}}{\partial y} + \frac{\partial {\dot{w}}_{s}}{\partial y} δ {\dot{v}}_{.0}) \\ + J_{2} (\frac{\partial {\dot{w}}_{b}}{\partial x} \frac{\partial δ {\dot{w}}_{s}}{\partial x} + \frac{\partial {\dot{w}}_{s}}{\partial x} \frac{\partial δ {\dot{w}}_{b}}{\partial x} + \frac{\partial {\dot{w}}_{b}}{\partial y} \frac{\partial δ {\dot{w}}_{s}}{\partial y} + \frac{\partial {\dot{w}}_{s}}{\partial y} \frac{\partial δ {\dot{w}}_{b}}{\partial y}) + K_{2} (\frac{\partial {\dot{w}}_{s}}{\partial x} \frac{\partial δ {\dot{w}}_{s}}{\partial x} + \frac{\partial {\dot{w}}_{s}}{\partial y} \frac{\partial δ {\dot{w}}_{s}}{\partial y}) \\ + J_{1}^{s} (({\dot{w}}_{b} + {\dot{w}}_{s}) δ \dot{ϕ} + \dot{ϕ} . δ ({\dot{w}}_{b} + {\dot{w}}_{s})) + K_{2}^{s} \dot{ϕ} . δ \dot{ϕ}] d A d z \end{array}

15b

where the dot-superscript convention indicates the differentiation with respect to the time variable (t), the terms I_i , J_i and K_i denote the moments of inertia that are expressed as in equations (16), without the stretching effect $(J_{1}^{s}, K_{2}^{s}) = 0$ :

(I_{0}, I_{1}, I_{2}, J_{1}, J_{2}, K_{2}, J_{1}^{s}, K_{2}^{s}) = \sum_{n = 1}^{3} \int_{h_{n}}^{h_{n + 1}} ρ (z) [1, z, z^{2}, f (z), z f (z), f^{2} (z), g (z), g^{2} (z)] d z

The equations of motion:

The five equations of motion are found by substituting the energies variations (13 and 15) in the Hamilton’s principle (12), then collecting the coefficients δu₀ , δv₀ , δw₀ and δ ϕ together, after integrating by parts every found term, they are appropriate to the five unknowns of the displacements field and the constitutive equations. The first equation of motion system is expressed in terms of displacements and strains as in equations (17), without the stretching effect (N_z = 0), (ϕ = 0), $(J_{1}^{s}, K_{2}^{s}) = 0$ :

\begin{array}{l} δ u_{0} : \frac{\partial N_{x}}{\partial x} + \frac{\partial N_{x y}}{\partial y} = I_{0} {\ddot{u}}_{0} - I_{1} \frac{\partial {\ddot{w}}_{b}}{\partial x} - J_{1} \frac{\partial {\ddot{w}}_{s}}{\partial x} \end{array}

17a

\begin{array}{l} δ v_{0} : \frac{\partial N_{x y}}{\partial x} + \frac{\partial N_{y}}{\partial y} = I_{0} {\ddot{v}}_{0} - I_{1} \frac{\partial {\ddot{w}}_{b}}{\partial y} - J_{1} \frac{\partial {\ddot{w}}_{s}}{\partial y} \end{array}

17b

\begin{array}{l} δ w_{b} : \frac{\partial^{2} M_{x}^{b}}{\partial x^{2}} + 2 \frac{\partial^{2} M_{x y}^{b}}{\partial x \partial y} + \frac{\partial^{2} M_{y}^{b}}{\partial y^{2}} = I_{0} ({\ddot{w}}_{b} + {\ddot{w}}_{s}) + I_{1} (\frac{\partial {\ddot{u}}_{0}}{\partial x} + \frac{\partial {\ddot{v}}_{0}}{\partial y}) - I_{2} \nabla^{2} {\ddot{w}}_{b} - J_{2} \nabla^{2} {\ddot{w}}_{s} + J_{1}^{s} \ddot{ϕ} \end{array}

17c

\begin{array}{l} δ w_{s} : \frac{\partial^{2} M_{x}^{s}}{\partial x^{2}} + 2 \frac{\partial^{2} M_{x y}^{s}}{\partial x \partial y} + \frac{\partial^{2} M_{y}^{s}}{\partial y^{2}} + \frac{\partial S_{x z}^{s}}{\partial x} + \frac{\partial S_{y z}^{s}}{\partial y} = I_{0} ({\ddot{w}}_{b} + {\ddot{w}}_{s}) + J_{1} (\frac{\partial {\ddot{u}}_{0}}{\partial x} + \frac{\partial {\ddot{v}}_{0}}{\partial y}) - J_{2} \nabla^{2} {\ddot{w}}_{b} - K_{2} \nabla^{2} {\ddot{w}}_{s} + J_{1}^{s} \ddot{ϕ} \end{array}

17d

\begin{array}{l} δ ϕ : \frac{\partial S_{x z}^{s}}{\partial x} + \frac{\partial S_{y z}^{s}}{\partial y} - N_{z} = J_{1}^{s} ({\ddot{w}}_{b} + {\ddot{w}}_{s}) + K_{2}^{s} \ddot{ϕ} \end{array}

17e

where $\nabla^{2} = (\partial^{2} / \partial x^{2}) + (\partial^{2} / \partial y^{2})$ is the Laplacian operator in 2D Cartesian coordinates system.

When the stresses field (10) is substituted into the first stresses resultants (15), the second ones (18) are presented as a function of strain as in equations (18), without the stretching effect (N_z = 0), (ϕ = 0), (P, P^a, T, T^a )=0:

{\begin{matrix} N_{x} \\ N_{y} \\ N_{x y} \\ M_{x}^{b} \\ M_{y}^{b} \\ M_{x y}^{b} \\ M_{x}^{s} \\ M_{y}^{s} \\ M_{x y}^{s} \\ S_{y z}^{s} \\ S_{x z}^{s} \\ N_{z} \end{matrix}} = [\begin{matrix} A_{11} & A_{12} & 0 & B_{11} & B_{12} & 0 & B_{11}^{s} & B_{12}^{s} & 0 & 0 & 0 & P \\ A_{12} & A_{22} & 0 & B_{12} & B_{22} & 0 & B_{12}^{s} & B_{22}^{s} & 0 & 0 & 0 & P \\ 0 & 0 & A_{66} & 0 & 0 & B_{66} & 0 & 0 & B_{66}^{s} & 0 & 0 & 0 \\ B_{11} & B_{12} & 0 & D_{11} & D_{12} & 0 & D_{11}^{s} & D_{12}^{s} & 0 & 0 & 0 & P^{a} \\ B_{12} & B_{22} & 0 & D_{12} & D_{22} & 0 & D_{12}^{s} & D_{22}^{s} & 0 & 0 & 0 & P^{a} \\ 0 & 0 & B_{66} & 0 & 0 & D_{66} & 0 & 0 & D_{66}^{s} & 0 & 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 & T \\ B_{12}^{s} & B_{22}^{s} & 0 & D_{12}^{s} & D_{22}^{s} & 0 & H_{12}^{s} & H_{22}^{s} & 0 & 0 & 0 & T \\ 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 & 0 & A_{44}^{s} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_{55}^{s} & 0 \\ P & P & 0 & P^{a} & P^{a} & 0 & T & T & 0 & 0 & 0 & T^{a} \end{matrix}] {\begin{matrix} \begin{matrix} \frac{\partial u_{0}}{\partial x} \\ \frac{\partial v_{0}}{\partial y} \\ \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} \end{matrix} \\ \begin{matrix} - \frac{\partial^{2} w_{b}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{b}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{b}}{\partial x \partial y} \end{matrix} \\ - \frac{\partial^{2} w_{s}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{s}}{\partial y^{2}} \\ \begin{matrix} - 2 \frac{\partial^{2} w_{s}}{\partial x \partial y} \\ (\frac{\partial w_{s}}{\partial y} + \frac{\partial ϕ}{\partial y}) \\ \begin{matrix} (\frac{\partial w_{s}}{\partial x} + \frac{\partial ϕ}{\partial x}) \\ ϕ \end{matrix} \end{matrix} \end{matrix}}

where the stiffness coefficients for quasi-3D HSDT (∊_z ≠ 0) areas in equations (19), without the stretching effect (P, P^a, T, T^a )=0:

\begin{array}{l} {\begin{matrix} A_{11} & B_{11} & D_{11} & B_{11}^{s} & D_{11}^{s} & H_{11}^{s} & 0 & P & P^{a} & T & T^{a} \\ A_{12} & B_{12} & D_{12} & B_{12}^{s} & D_{12}^{s} & H_{12}^{s} & 0 & 0 & 0 & 0 & 0 \\ A_{66} & B_{66} & D_{66} & B_{66}^{s} & D_{66}^{s} & H_{66}^{s} & A_{44}^{s} & 0 & 0 & 0 & 0 \end{matrix}} \\ = \sum_{n = 1}^{3} \int_{h_{n}}^{h_{n + 1}} [1, z, z^{2}, f (z), z f (z), {[f (z)]}^{2}, {[g (z)]}^{2}, g' (z), z g' (z), f (z) g' (z), {[g' (z)]}^{2}] {\begin{matrix} C_{11} (z) \\ C_{12} (z) \\ G (z) \end{matrix}} d z, \end{array}

19a

(A_{22}, B_{22}, D_{22}, B_{22}^{s}, D_{22}^{s}, H_{22}^{s}, A_{55}^{s}) = (A_{11}, B_{11}, D_{11}, B_{11}^{s}, D_{11}^{s}, H_{11}^{s}, A_{44}^{s})

19b

By substituting the stresses resultants (18) into the first equations of motion system (17), the following simplified system is found as in equations (20), without the stretching effect (N_z = 0), (ϕ = 0), (P, P^a, T, T^a ) = 0 $(J_{1}^{s}, K_{2}^{s}) = 0$ :

\begin{array}{l} δ u_{0} : A_{11}^{​} \frac{\partial^{2} u_{0}}{\partial x^{2}} + A_{66}^{​} \frac{\partial^{2} u_{0}}{\partial y^{2}} + (A_{12}^{​} + A_{66}^{​}) \frac{\partial^{2} v_{0}}{\partial x \partial y} - B_{11}^{​} \frac{\partial^{3} w_{b}}{\partial x^{3}} - (B_{12}^{​} + 2 B_{66}^{​}) \frac{\partial^{3} w_{b}}{\partial x \partial y^{2}} \\ - B_{11}^{s} \frac{\partial^{3} w_{s}}{\partial x^{3}} - (B_{12}^{s} + 2 B_{66}^{s}) \frac{\partial^{3} w_{s}}{\partial x \partial y^{2}} + P \frac{\partial ϕ}{\partial x} = I_{0} {\ddot{u}}_{0} - I_{1} \frac{\partial {\ddot{w}}_{b}}{\partial x} - J_{1} \frac{\partial {\ddot{w}}_{s}}{\partial x} \end{array}

20a

\begin{array}{l} δ v_{0} : A_{22}^{​} \frac{\partial^{2} v_{0}}{\partial y^{2}} + A_{66}^{​} \frac{\partial^{2} v_{0}}{\partial x^{2}} + (A_{12}^{​} + A_{66}^{​}) \frac{\partial^{2} u_{0}}{\partial x \partial y} - B_{22}^{​} \frac{\partial^{3} w_{b}}{\partial y^{3}} - (B_{12}^{​} + 2 B_{66}^{​}) \frac{\partial^{3} w_{b}}{\partial x^{2} \partial y} \\ - B_{22}^{s} \frac{\partial^{3} w_{s}}{\partial y^{3}} - (B_{12}^{s} + 2 B_{66}^{s}) \frac{\partial^{3} w_{s}}{\partial x^{2} \partial y} + P \frac{\partial ϕ}{\partial y} = I_{0} {\ddot{v}}_{0} - I_{1} \frac{\partial {\ddot{w}}_{b}}{\partial y} - J_{1} \frac{\partial {\ddot{w}}_{s}}{\partial y} \end{array}

20b

\begin{array}{l} δ w_{b} : B_{11}^{​} \frac{\partial^{3} u_{0}}{\partial x^{3}} + (B_{12}^{​} + 2 B_{66}^{​}) \frac{\partial^{3} u_{0}}{\partial x \partial y^{2}} + (B_{12}^{​} + 2 B_{66}^{​}) \frac{\partial^{3} v_{0}}{\partial x^{2} \partial y} - B_{22}^{​} \frac{\partial^{3} v_{0}}{\partial y^{3}} \\ - D_{11}^{​} \frac{\partial^{4} w_{b}}{\partial x^{4}} - D_{22}^{​} \frac{\partial^{4} w_{b}}{\partial y^{4}} - 2 (D_{12}^{​} + 2 D_{66}^{​}) \frac{\partial^{4} w_{b}}{\partial x^{2} \partial y^{2}} - D_{11}^{s} \frac{\partial^{4} w_{s}}{\partial x^{4}} - D_{22}^{s} \frac{\partial^{4} w_{s}}{\partial y^{4}} \\ - 2 (D_{12}^{s} + 2 D_{66}^{s}) \frac{\partial^{4} w_{s}}{\partial x^{2} \partial y^{2}} + P^{a} (\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}}) = I_{0} ({\ddot{w}}_{b} + {\ddot{w}}_{s}) + I_{1} (\frac{\partial {\ddot{u}}_{0}}{\partial x} + \frac{\partial {\ddot{v}}_{0}}{\partial y}) \\ - I_{2} (\frac{\partial^{2} {\ddot{w}}_{b}}{\partial^{2} x} + \frac{\partial^{2} {\ddot{w}}_{b}}{\partial^{2} y}) - J_{2} (\frac{\partial^{2} {\ddot{w}}_{s}}{\partial^{2} x} + \frac{\partial^{2} {\ddot{w}}_{s}}{\partial^{2} y}) + J_{1}^{s} \ddot{ϕ} \end{array}

20c

\begin{array}{l} δ w_{s} : B_{11}^{s} \frac{\partial^{3} u_{0}}{\partial x^{3}} + (B_{12}^{s} + 2 B_{66}^{s}) \frac{\partial^{3} u_{0}}{\partial x \partial y^{2}} + (B_{12}^{s} + 2 B_{66}^{s}) \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_{b}}{\partial x^{4}} - D_{22}^{s} \frac{\partial^{4} w_{b}}{\partial y^{4}} - 2 (D_{12}^{s} + 2 D_{66}^{s}) \frac{\partial^{4} w_{b}}{\partial x^{2} \partial y^{2}} + A_{44}^{s} \frac{\partial^{2} w_{s}}{\partial y^{2}} + A_{55}^{s} \frac{\partial^{2} w_{s}}{\partial x^{2}} \\ - H_{11}^{s} \frac{\partial^{4} w_{s}}{\partial x^{4}} - 2 (H_{12}^{s} + 2 H_{66}^{s}) \frac{\partial^{4} w_{s}}{\partial x^{2} \partial y^{2}} - H_{22}^{s} \frac{\partial^{4} w_{s}}{\partial y^{4}} + T (\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}}) + A_{44}^{s} \frac{\partial^{2} ϕ}{\partial y^{2}} + A_{55}^{s} \frac{\partial^{2} ϕ}{\partial x^{2}} \\ = I_{0} ({\ddot{w}}_{b} + {\ddot{w}}_{s}) + J_{1} (\frac{\partial {\ddot{u}}_{0}}{\partial x} + \frac{\partial {\ddot{v}}_{0}}{\partial y}) - J_{2} (\frac{\partial^{2} {\ddot{w}}_{b}}{\partial^{2} x} + \frac{\partial^{2} {\ddot{w}}_{b}}{\partial^{2} y}) - K_{2} (\frac{\partial^{2} {\ddot{w}}_{s}}{\partial^{2} x} + \frac{\partial^{2} {\ddot{w}}_{s}}{\partial^{2} y}) + J_{1}^{s} \ddot{ϕ} \end{array}

20d

\begin{array}{l} δ ϕ : P (\frac{\partial u_{0}}{\partial x} + \frac{\partial v_{0}}{\partial y}) - P^{a} (\frac{\partial^{2} w_{b}}{\partial x^{2}} + \frac{\partial^{2} w_{b}}{\partial y^{2}}) + (T - A_{44}^{s}) \frac{\partial^{2} w_{s}}{\partial x^{2}} + (R - A_{55}^{s}) \frac{\partial^{2} w_{s}}{\partial y^{2}} + T^{a} ϕ - A_{44}^{s} \frac{\partial^{2} ϕ}{\partial x^{2}} - A_{55}^{s} \frac{\partial^{2} ϕ}{\partial y^{2}} \\ = J_{1}^{s} ({\ddot{w}}_{b} + {\ddot{w}}_{s}) + K_{2}^{s} \ddot{ϕ} \end{array}

20e

Navier approach for simply supported plates

In general, plates are classified in accordance with the used support type. Thus, in the present study, the following boundary conditions form for simply supported plate edges is imposed at the plate side edges as:

v_{0} (0, y) = w_{b} (0, y) = w_{s} (0, y) = \frac{\partial w_{b}}{\partial y} (0, y) = \frac{\partial w_{s}}{\partial y} (0, y) = 0

21a

v_{0} (a, y) = w_{b} (a, y) = w_{s} (a, y) = \frac{\partial w_{b}}{\partial y} (a, y) = \frac{\partial w_{s}}{\partial y} (a, y) = 0

21b

N_{x} (0, y) = M_{x}^{b} (0, y) = M_{x}^{s} (0, y) = N_{x} (a, y) = M_{x}^{b} (a, y) = M_{x}^{s} (a, y) = 0

21c

u_{0} (x, 0) = w_{b} (x, 0) = w_{s} (x, 0) = \frac{\partial w_{b}}{\partial x} (x, 0) = \frac{\partial w_{s}}{\partial x} (x, 0) = 0

21d

u_{0} (x, b) = w_{b} (x, b) = w_{s} (x, b) = \frac{\partial w_{b}}{\partial x} (x, b) = \frac{\partial w_{s}}{\partial x} (x, b) = 0

21e

N_{y} (x, 0) = M_{y}^{b} (x, 0) = M_{y}^{s} (x, 0) = N_{y} (x, b) = M_{y}^{b} (x, b) = M_{y}^{s} (x, b) = 0

21f

The Navier approach for simply supported boundary conditions is used to find analytical solutions of equations (20). The Navier approach solutions are partial differential equations in terms of displacements functions that satisfy the equations of boundary conditions (21), expressed by the following double-Fourier series for the plate with shear deformation model as in equations (22), without the stretching effect (ϕ = 0):

{\begin{matrix} u_{0} (x, y, t) \\ v_{0} (x, y, t) \\ w_{b} (x, y, t) \\ w_{s} (x, y, t) \\ ϕ (x, y, t) \end{matrix}} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} e^{i ω t} {\begin{matrix} U_{m n} cos (λ x) sin (μ y) \\ V_{m n} sin (λ x) cos (μ y) \\ W_{b m n} sin (λ x) sin (μ y) \\ W_{s m n} sin (λ x) sin (μ y) \\ Φ_{m n} sin (λ x) sin (μ y) \end{matrix}}

where U_mn , V_mn , W_bmn , W_smn and Φ _mn are arbitrary determined parameters subjected to the conditions that the solution in (22) satisfies equations of motion (20), ω is the eigenfrequency associated with (m, n)th eigenmode of the plate for the free vibration analysis, λ = mp/a, µ = np/b.

The double-Fourier series of displacements form (22) are substituted into the equations of motion system (20), and after some derivations and simplifications, the following equivalent system is obtained as in equations (23), without the stretching effect (N_z = 0), (ϕ = 0), (P, P^a, T, T^a )=0 $(J_{1}^{s}, K_{2}^{s}) = 0$ :

([A] - ω^{2} [M]) {R} = {0}

23a

([\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{12} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{13} & a_{23} & a_{33} & a_{34} & a_{35} \\ a_{14} & a_{24} & a_{34} & a_{44} & a_{45} \\ a_{15} & a_{25} & a_{35} & a_{45} & a_{55} \end{matrix}] - ω^{2} [\begin{matrix} m_{11} & 0 & m_{13} & 0 & 0 \\ 0 & m_{22} & m_{23} & m_{24} & 0 \\ m_{13} & m_{23} & m_{33} & m_{34} & m_{35} \\ 0 & m_{24} & m_{34} & m_{44} & m_{45} \\ 0 & 0 & m_{35} & m_{45} & m_{55} \end{matrix}]) {\begin{matrix} U_{m n} \\ V_{m n} \\ W_{b m n} \\ W_{s m n} \\ Φ_{m n} \end{matrix}} = {\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}}

23b

where the matrix [A] is

\begin{array}{l} a_{11} = (λ^{2} A_{11}^{​} + μ^{2} A_{66}^{​}), \\ a_{12} = λ μ (A_{12}^{​} + A_{66}^{​}), \\ a_{13} = - (λ^{3} B_{11}^{​} + λ μ^{2} (B_{12}^{​} + 2 B_{66}^{​})), \\ a_{14} = - (λ^{3} B_{11}^{s} + λ μ^{2} (B_{12}^{s} + 2 B_{66}^{s})), \\ a_{22} = (λ^{2} A_{66}^{​} + μ^{2} A_{22}^{​}), \\ a_{23} = - (μ^{3} B_{22}^{​} + λ^{2} μ (B_{12}^{​} + 2 B_{66}^{​})), \\ a_{24} = - (μ^{3} B_{22}^{s} + λ^{2} μ (B_{12}^{s} + 2 B_{66}^{s})), \\ a_{33} = (λ^{4} D_{11}^{​} + 2 λ^{2} μ^{2} (D_{12}^{​} + 2 D_{66}^{​}) + μ^{4} D_{22}^{​}), \\ a_{34} = (λ^{4} D_{11}^{s} + 2 λ^{2} μ^{2} (D_{12}^{s} + 2 D_{66}^{s}) + μ^{4} D_{22}^{s}), \\ a_{44} = (λ^{4} H_{11}^{s} + 2 λ^{2} μ^{2} (H_{12}^{s} + 2 H_{66}^{s}) + μ^{4} H_{22}^{s} + λ^{2} A_{55}^{s} + μ^{2} A_{44}^{s}), \end{array}

24a

For the additional stretching effect

\begin{array}{l} a_{15} = - P λ, \\ a_{25} = - P μ, \\ a_{55} = (λ^{2} A_{44}^{s} + μ^{2} A_{55}^{s} + T^{a}), \end{array}

24b

As well the matrix [M] is

\begin{array}{l} m_{11} = m_{22} = I_{0}, \\ m_{33} = (I_{0} + I_{2} (λ^{2} + μ^{2})), \\ m_{34} = (I_{0} + J_{2} (λ^{2} + μ^{2})), \\ m_{44} = (I_{0} + K_{2} (λ^{2} + μ^{2})), \end{array}

24c

For the additional stretching effect

\begin{array}{l} m_{13} = - λ I_{1}, \\ m_{14} = - λ J_{1}, \\ m_{23} = - μ I_{1}, \\ m_{24} = - μ J_{1}, \\ m_{35} = m_{45} = J_{1}^{s}, \\ m_{55} = K_{2}^{s} . \end{array}

24d

Numerical examples, results and discussions

The present theory accuracy, novelty, simplicity and effectuality are evaluated for the free vibration analysis (the plate is not subjected to external loads) of both the square and rectangular, aluminium-1/alumina (Al)₁/Al₂O₃ and aluminium-1/zirconia (Al)₁/ZrO₂ P-FGM and MT-FGM monolayer plates in Tables 3 to 5 as well as square aluminium-2/alumina (Al)₂/Al₂O₃ P-FGM sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) plates in Tables 6 to 8 with simply supported edges. The obtained results were determined from the new developed hybrid HSDTs (polynomial–hyperbolic–exponential), (polynomial–trigonometric) and Belabed et al.,¹⁴ El meiche et al.³⁸ theoretical formulation models. Many numerical examples investigated the different non-dimensional parameters effects such as the volume fraction index (p), the geometric ratios (a/h, a/b), the thickness ratio, the frequency modes (m, n) and the materials properties on the non-dimensional free vibration fundamental and natural frequencies ( $\hat{ω}$ ), ( $\bar{ω}$ ) and ( $\bar{β}$ ). The results are compared with those generated by the FSDT,^{74,76,80,83,85} the 3D and quasi-3D HSDT^{13–14,73,77
–79,81–82} and the 2D HSDT.^{21,23,29,37,38,52,55,73,75,84}

For this study, the used matrices general form (23) analysed the free vibration problem as well the resulted non-dimensional mathematical relations are listed as:

For P-FGM and MT-FGM monolayer plates:

\bar{z} = \frac{z}{h}, \bar{β} = ω h \sqrt{\frac{ρ_{m}}{E_{m}}}, \hat{ω} = ω h \sqrt{\frac{ρ_{c}}{E_{c}}}, \bar{ω} = \frac{ω a^{2}}{h} \sqrt{\frac{ρ_{c}}{E_{c}}},

25a

For P-FGM sandwich plates

\bar{z} = \frac{z}{h}, \bar{ω} = \frac{ω a^{2}}{h} \sqrt{\frac{ρ_{0}}{E_{0}}},

25b

where ρ₀ = 1 kg/m³ and E₀ = 1 GPa

Example 1. The natural frequency ( $\bar{β}$ ) results are presented in Table 3, for homogeneous ceramic moderately stiff, softer, thick, moderately thick, thin and square (Al)₁/ZrO₂ MT-FGM plates, under the frequency modes ((m, n)=1(1,1), 2(1,2), 3(2,2)). It should be noted that the obtained results are in good agreement with all the mentioned theories solutions, especially with quasi-3D HSDT solutions of Belabed et al.¹⁴ even for thick and softer MT-FGM plates, where ( $\bar{β}$ ) becomes more important, and the produced maximum error between both the present theory 1, 2 solutions and Belabed et al.¹⁴ is 0.0456% for (a/h = 5, p = 1), which is negligible.

Example 2. The fundamental frequency ( $\hat{ω}$ ) results are computed in Table 4, for homogeneous ceramic, stiffer, moderately stiff, softer, thick, moderately thick, thin, square (Al)₁ /Al₂O₃ P-FGM plates, under the three frequency modes ((m, n) = 1(1,1), 2(1,2), 3(2,2)). The 2D HSDT 1 and 2 results are in very good agreement with all the presented theories solutions, especially with Belkhodja et al.³⁷ as well as Thai and Kim,²³ where the calculated maximum errors are, respectively, 0.0747% as well as 0.6683% and 0.7937%, these are an insignificant values obtained for (p = 0, a/h = 5,(m, n) = 3(2,2)) and (a/h = 20, p = 0.5,(m, n) = 1(1,1)). The 2D HSDT 1 results are identical to the others theories solutions, for thin and also moderately thin plates. The quasi-3-D HSDT 1 and 2 results are in well agreement with the others theories solutions, especially Belabed et al.,¹⁴ where the maximum error is 1.8453%, this is an insignificant value, obtained for (p = 4, a/h = 5, (m, n) = (1,1)). For thin plates, the quasi-3D HSDT results are very close to each other (the transverse shear deformation effect is not significant). Thus, the different theories solutions present more accuracy and convergence. It is to highlight that, the ( $\hat{ω}$ ) values increase with the index (p) and (a/h) ratio decrease, as well as the frequency modes increase.

Example 3. The fundamental frequency ( $\bar{ω}$ ) results are calculated in Table 5, for homogeneous ceramic, stiffer, moderately stiff, softer, thick, moderately thick, thin and rectangular (Al)₁/Al₂O₃ P-FGM plates under the frequency modes ((m, n) = 1(1,1), 2(1,2), 3(1,3), 4(2,1)). It can be observed that well agreement is presented between the both 2D HSDTs 1 and 2 solutions as well as the others theories, especially with Belkhodja et al.³⁷ where the maximum error computed is, respectively, 0.0308% and 0.1513% obtained for (p = 10, a/h = 5 and 20, mode 2), the maximum error in comparison with Hosseini et al.⁸⁴ is 0.0352% and 0.3083% obtained for (p = 5 and 10, a/h = 10 and 20, mode 4), as well the maximum error is, respectively, 0.0687% and 0.3133% in comparison with Thai and Thuc²¹ obtained for (p = 5 and 10, a/h = 5 and 20, mode 4), these errors are insignificant. It is to highlight that, the ( $\bar{ω}$ ) values increase with the index (p) decreases as well as the increase of both the ratio (a/h) and the frequency modes that is clear in Figure 5.

Figure 5.

Non-dimensional fundamental frequency ( $\bar{ω}$ ) variations for (Al)₁/Al₂O₃ P-FGM plates versus (a) the volume fraction index (p) and (b) the side-to-thickness ratio (a/h).

Example 4. The fundamental frequency ( $\bar{ω}$ ) results are calculated in Table 7, for homogeneous ceramic, stiffer, moderately stiff, softer, thick, moderately thick and square (Al)₂/Al₂O₃ P-FGM symmetric and non-symmetric sandwich plates type A, with homogeneous hardcore.

It should be noted that the obtained results are in very good agreement with all the mentioned theories solutions, especially with Akavci et al.⁷³ and El Meiche et al.³⁸ solutions even for thin and stiffer P-FGM plates, where ( $\bar{ω}$ ) becomes more important, and the produced maximum error between both former theories and the present 2D theory 3 solutions is negligible, obtained as 0.1261% and 0.0302% for (a/h = 5, p = 10, (1-0-1)) and (a/h = 10, p = 10, (2-1-1)), respectively.

Example 5. The fundamental frequency ( $\bar{ω}$ ) results are again calculated in Table 8, for stiffer, moderately thick and square (Al)₂ /Al₂O₃ P-FGM square symmetric and non-symmetric sandwich plates type A, with homogeneous hardcore, under 10 frequency modes ((m, n) = 1(1,1), 2(1,2), 3(2,2), 4(1,3), 5(2,3), 6(1,4), 7(3,3), 8(2,4), 9(3,4), 10(4,4)).

The 2D HSDT 3 results are in well agreement with all the presented theories solutions, especially with El Meiche et al.³⁸ and Akavci et al.⁷³ where the calculated maximum errors are, respectively, 3.0967% and 3.1729%. These are an insignificant values obtained for ((m, n) = (1,4)) and non-symmetric sandwich plates (2-2-1). The 2D HSDT 3 results are important for the frequency mode ((m, n) = (4,4)).

Example 6. The fundamental frequency ( $\bar{ω}$ ) results are presented also in Table 8, for stiffer, moderately stiff, thick, moderately thick and square (Al)₂/Al₂O₃ P-FGM square symmetric and non-symmetric sandwich plates type A, but with homogeneous softcore.

The 2D HSDT 3 results are in good agreement with all the presented theories solutions, especially with Akavci et al.⁷³ where the calculated maximum errors is 0.7513%, this is a negligible value obtained for (a/h = 5, p = 10, and non-symmetric sandwich plates (1-2-1)).

Conclusion

The developed three hybrid quasi-3D and 2D HSDTs analysed the free vibration problem of the isotropic, the MT-FGMs and P-FGMs, monolayer and sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) plates with simply supported edges, the theory investigated the displacements field of five unknowns, in which the transverse displacement membranes are the bending, the shear and the stretching through the plate thickness. The stretching displacement is in terms of the transverse shear deformation and stress that satisfied the stress-free boundary conditions on the plate free surfaces. The appropriate equations of motion are extracted from the Hamilton’s principle and are solved by the Navier approach. The study of several parameters effects such as the volume fraction index, the geometric ratios, the frequency modes, and the materials properties on the natural frequencies are analysed. The proposed model reliability and accuracy, novelty, simplicity and effectuality of these new HSDTs are ascertained by comparisons of the calculated results with others plate theories solutions found in the literature references. As a conclusion, these theories are appropriate, simple, accurate and effective, as well as it gave the following results:

It is obtained that the present formulations of polynomial–hyperbolic–exponential and polynomial–trigonometric forms can be further extended to all existing HSDTs models for numerous problems related to the shear deformable effect.

The obtained results are in good agreement with the different others quasi-3D and 2D HSDT solutions in many cases that confirms the theory convergence.In the P-FGM plates, the ( $\bar{β}$ ), ( $\bar{ω}$ ) and ( $\hat{ω}$ ) increase with the index (p) decreases.

The increase in the fundamental frequencies ( $\bar{ω}$ ) and ( $\hat{ω}$ ) is influenced by the increase in the frequency modes.

Furthermore, there is always a little difference between the quasi-3D results curve and the 2D results curves except for homogeneous ceramic plate (p = 0), where the quasi-3D results curve is underestimated in comparison with others ones and the reason is the stretching effect that is clear for softer and thick plates.

Footnotes

Acknowledgement

The authors thank Professor B Achour for their help and Editorial Office team and the reviewers for their valuable comments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Y Belkhodja

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