Determining the Advanced Frequency of Composited Functionally Graded Material Plates Using Third-Order Shear Deformation Theory and Nonlinear Varied Shear Coefficients

Abstract

The shear effect is usually considered in the numerical calculation of thick composited FGM plates. The characteristics that have the greatest effect on thickness are displacement type, shear correction coefficient, material property and temperature. For the advanced frequency study of thick composited functionally graded material (FGM) plates, it is interesting to consider the extra effects of the nonlinear coefficient c₁ term of the third-order shear deformation theory (TSDT) of displacement on the calculation of varied shear correction coefficients. The values of nonlinear shear correction coefficients are usually functions of c₁, the power-law exponent parameter and environment temperature. Numerical frequency computations are calculated using a simple homogeneous equation, and are investigated using TSDT and the nonlinear shear correction coefficient for thick composited FGM plates. Results for natural frequencies are found via the functions of length-to-thickness ratio, the power-law exponent parameter, c₁ and environment temperature. This novel study in advanced frequency aims to determine the effects of the TSDT and nonlinear shear correction on thick FGM plates under free vibration.

1. Introduction

There are some steady-state studies on composited FGM plates, but they do not usually include the variables of time and frequency. In 2020, Ghumarea and Sayyad [1] presented a fifth-order shear and normal deformation theory of displacement to obtain a steady transverse displacement for elastic-foundation FGM plates under hygro-thermo-mechanical loading. In 2019, Belardi et al. [2] presented a first-order shear deformation theory (FSDT) of displacement to obtain middle-plane deflection as well as circumferential and radial rotations for rectilinear orthotropic composite circular plates under transversal loads. There are some vibration studies on these plates, usually including the variables of time and frequency. In 2019, Khayat et al. [3] presented a third-order shear deformation theory (TSDT) of displacement to obtain the frequency and phase velocity of elastic-foundation FGM plates under harmonic wave propagation. Only one direction of mode vibration at a time was considered to obtain increasing numerical results in relation to frequency; this was a purely preliminary frequency study on FGM plates. In 2019, Tu et al. [4] presented the eighth unknown higher-order shear deformation theory of displacement to obtain the frequency of free vibrations of FGM plates under thermal environments. Two directions of mode vibration at a time were considered to obtain decreasing and increasing numerical results of natural frequencies; this fundamental frequency study on FGM plates did not consider the shear correction coefficient effect of shear stresses. In 2017, Duc et al. [5] presented the FSDT of displacement to obtain the static response and frequency results of free vibration for carbon nanotube FGM plates with elastic foundations. Also, only one direction of mode vibration at a time was considered to obtain the decreasing and increasing numerical results of the frequencies. This was a purely preliminary frequency study on nanotube FGM plates. In 2016, Bui [6] presented the TSDT of displacement to obtain the static bending result and frequency result of free vibration for FGM plates under environments with varying temperatures. Only one direction of mode vibration at a time was considered to obtain the decreasing and increasing numerical results of the frequencies. This can also be considered a purely preliminary frequency study on FGM plates. In 2013, Thai and Kim [7] presented a simple higher-order shear deformation theory (HSDT) of displacement to obtain the frequency of free vibrations of FGM plates by considering only one direction of mode vibration at a time. In 2013, Ungbhakorn and Wattanasakulpong [8] presented the TSDT of displacement to obtain the frequency of free vibrations of FGM plates by considering only one direction of mode vibration at a time. In 2013, Jha et al. [9] presented a higher-order shear/shear–normal deformation theory (HOST/HOSNT) of displacement to obtain the frequency of free vibrations of FGM plates by considering two directions of mode vibration at a time. This might also be considered a fundamental frequency study on FGM plates, although it did not consider the shear correction coefficient effect of shear stresses. In 2005, Kim [10] presented the TSDT of displacement to obtain the frequency of free vibrations of FGM plates by considering two directions of mode vibration at a time. This might also be considered a fundamental frequency study on thick FGM plates, although it did not consider the shear correction coefficient effect of shear stresses.

New theories, solutions, and methods that have been focused on within the last five years are as follows. In 2017, Tornabene et al. [11] presented a HSDT and numerical method to obtain the natural frequencies of FGM-composited sandwich shells. In 2019, Bacciocchi and Tarantino [12] presented a FSDT and numerical method to obtain the natural frequencies of FGM-composited plates. In 2024, You et al. [13] presented a FSDT and numerical method to obtain the buckling results of FGM carbon-nanotube-reinforced composite (CNTRC) conical and cylindrical shells. In 2022, Hua et al. [14] presented a HSDT and numerical method to obtain guided-wave analytical results for composited FGM visco-elastic polymer shells. In 2019, Moita et al. [15] presented a HSDT and numerical method to obtain static results for composited FGM plates and shells. In 2023, Hua et al. [16] presented a HSDT and numerical method to obtain wave-propagation results for composited graphene platelet (GPL) shells. In 2024, Wang and Ma [17] presented a beam theory and numerical method to obtain the free vibration of FGM-composited porous beams. In 2022, Civalek et al. [18] presented a FSDT and numerical method to obtain the free vibration of CNTRC plates. In 2018, Akgöz and Civalek [19] presented a Winkler–Pasternak elastic foundation model and numerical method to obtain the natural frequencies of micro-beams. In 2021, Ramteke and Panda [20] presented a HSDT and numerical method to obtain the natural frequencies of FGM-composited porous structures.

The author has some experience in investigations into and preliminary studies of the vibration frequency of thick composited FGM plates and shells, without considering the effects of nonlinear coefficient terms on the varied shear correction coefficient calculation. In 2020, Hong [21] presented a TSDT of displacement to obtain the frequency of free vibrations of FGM spherical shells by considering two directions of mode vibration at a time. In 2019, Hong [22] presented a TSDT of displacement to obtain the frequency of free vibrations of FGM plates by considering two directions of mode vibration at a time. This is an advanced study of the vibrations of thick FGM plates that considers the simultaneous effects of the TSDT of displacements, the nonlinear shear correction coefficient of shear stresses and the two directions of mode vibrations in time. The shear effect is usually considered in the numerical calculation of thick FGM plates. Mostly, displacement type, shear correction coefficient, material property and temperature affect thickness. To study the advanced frequencies of thick FGM plates, it is interesting to consider the extra effect of nonlinear coefficient $c_1$ term of the TSDT of displacements on the calculation of varied shear correction coefficient $k_{\alpha }$. The contribution of this study is to provide advanced frequency results for the free vibration analysis of thick FGM plates.

2. Formulation for the Advanced Nonlinear $k_{\alpha }$

The properties $P_i$ of individual constituent material for two-material thick composited FGM plates as shown in Figure 1, are functions of environment temperature T under free stress assumed can be represented in the following form [21],

$$P_i=P_0(P_{-1}{T}^{-1}+1+P_{1}T+P_2{T}^2+P_3{T}^3),$$

(1)

where $P_0,\hspace{0.17em}{P}_{-1},\hspace{0.17em}{P}_1,\hspace{0.17em}{P}_2$ and $P_3$ are the temperature coefficients.

The time dependence of nonlinear displacements $u$, $v$ and $w$ at any point ($x$, y, $z$) of thick composited FGM plates are assumed in the nonlinear coefficient $c_1$ term of the third order of z for TSDT equations [23], as follows,

$$\begin{gathered} u=u^0\left(x,y,t\right)+z{\psi }_x\left(x,y,t\right){-c}_1{z}^3({\psi }_x+{\displaystyle \frac{\partial w}{\partial x}}), \\ v=v^0\left(x,y,t\right)+z{\psi }_y\left(x,y,t\right){-c}_1{z}^3({\psi }_y+{\displaystyle \frac{\partial w}{\partial y}}), \\ w=w\left(x,y,t\right), \end{gathered}$$

(2)

where $u^0$ and $v^0$ are displacements in the $x$ and $y$ axes direction, respectively. $w$ is the transverse displacement in the $z$ axis direction of the middle plane of the plates. ${\psi }_x$ and ${\psi }_y$ are the shear rotations. t is the time. The coefficient for $c_1=4/(3{h^{*}}^2)$ is given in the TSDT approach, and $h^{*}$ is the total thickness of FGM plates.

The normal stresses (${\sigma }_x$ and ${\sigma }_y$) and shear stresses (${\sigma }_{xy}$, ${\sigma }_{yz}$ and ${\sigma }_{xz}$) in the thick composited FGM plates under temperature difference $∆T$ for the subscript (k)th layer are presented in the following equations [24,25],

$$\begin{gathered} {\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\sigma }_{xy}\end{array}\right\}}_{(k)}={\left[\begin{array}{ccc}{\overline{Q}}_{11}& {\overline{Q}}_{12}& {\overline{Q}}_{16}\\ {\overline{Q}}_{12}& {\overline{Q}}_{22}& {\overline{Q}}_{26}\\ {\overline{Q}}_{16}& {\overline{Q}}_{26}& {\overline{Q}}_{66}\end{array}\right]}_{(k)}{\left\{\begin{array}{c}{\epsilon }_x-{\alpha }_x\Delta T\\ {\epsilon }_y-{\alpha }_y\Delta T\\ {\epsilon }_{xy}-{\alpha }_{xy}\Delta T\end{array}\right\}}_{(k)}, \\ {\left\{\begin{array}{c}{\sigma }_{yz}\\ {\sigma }_{xz}\end{array}\right\}}_{(k)}={\left[\begin{array}{cc}{\overline{Q}}_{44}& {\overline{Q}}_{45}\\ {\overline{Q}}_{45}& {\overline{Q}}_{55}\end{array}\right]}_{(k)}{\left\{\begin{array}{c}{\epsilon }_{yz}\\ {\epsilon }_{xz}\end{array}\right\}}_{(k)}, \end{gathered}$$

(3)

where ${\alpha }_x$ and ${\alpha }_y$ are the coefficients of thermal expansion, ${\alpha }_{xy}$ is the coefficient of thermal shear, and ${\overline{Q}}_{ij}$ is the stiffness of FGM plates without considering the stack sequence. ${\epsilon }_x$, ${\epsilon }_y$ and ${\epsilon }_{xy}$ are in-plane strains, and not negligible ${\epsilon }_{yz}$ and ${\epsilon }_{xz}$ are shear strains.

Simpler forms of ${\overline{Q}}_{ij}$ and ${\overline{Q}}_{i^{*}{j}^{*}}$ for FGM plates are used and given as follows [26],

$$\begin{array}{l}{\overline{Q}}_{11}={\overline{Q}}_{22}={\displaystyle \frac{E_{fgm}}{1-{{\nu }_{fgm}}^2}},\\ {\overline{Q}}_{12}={\displaystyle \frac{{\nu }_{fgm}{E}_{fgm}}{1-{{\nu }_{fgm}}^2}},\\ {\overline{Q}}_{44}={\overline{Q}}_{55}={\overline{Q}}_{66}={\displaystyle \frac{E_{fgm}}{2(1+{\nu }_{fgm})}},\\ {\overline{Q}}_{16}={\overline{Q}}_{26}={\overline{Q}}_{45}=0,\end{array}$$

(4)

in which ${\nu }_{fgm}={\displaystyle \frac{{\nu }_1+{\nu }_2}{2}}$ is the Poisson’s ratios of the FGM plates. $E_{fgm}=(E_2-E_1){({\displaystyle \frac{z+h^{*}/2}{h^{*}}})}^{R_n}+E_1$ is the Young’s modulus of the FGM plates. $R_n$ is the power-law index. $E_1$ and $E_2$ are the Young’s modulus. ${\nu }_1$ and ${\nu }_2$ are the Poisson’s ratios of the FGM constituent material 1 and 2, respectively. The stiffness integrations with z are expressed in the following equations [27].

$$\begin{array}{l}(A_{i^s{j}^s},B_{i^s{j}^s},D_{i^s{j}^s},E_{i^s{j}^s},F_{i^s{j}^s},H_{i^s{j}^s})={\displaystyle {\int }_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}{\overline{Q}}_{i^s{j}^s}}(1,z,z^2,z^3,z^4,z^6)dz,\\ (A_{i^{*}{j}^{*}},B_{i^{*}{j}^{*}},D_{i^{*}{j}^{*}},E_{i^{*}{j}^{*}},F_{i^{*}{j}^{*}},H_{i^{*}{j}^{*}})={\displaystyle {\int }_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}{k}_{\alpha }{\overline{Q}}_{i^{*}{j}^{*}}}(1,z,z^2,z^3,z^4,z^5)dz,\\ (i^s,j^s=1,2,6), (i^{*},j^{*}=4,5),\end{array}$$

(5)

in which $k_{\alpha }$ is the advanced shear correction coefficient.

For this advanced thick composited FGM plates study, it is interesting to consider the extra effects of nonlinear coefficient term of TSDT displacements on the calculation of an advanced varied shear correction coefficient. The advanced modified shear correction factor $k_{\alpha }$ expression [28] can be derived based on the energy equivalence principle. Let the total strain energy U due to transverse shears ${\sigma }_{xz}$ and ${\sigma }_{yz}$ be defined in the following form by Whitney in 1987 [25] along the lengths of $a$ and $b$ of FGM plates, as shown in Figure 1.

$$U={\displaystyle \frac{1}{2}}{\int }_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}{\int }_0^b{\int }_0^a\left({\sigma }_{xz}{\epsilon }_{xz}+{\sigma }_{yz}{\epsilon }_{yz}\right)dxdydz,$$

(6)

The shear forces $Q_y$ and $Q_x$ are defined in strains form, as follows

$$\left\{\begin{array}{c}{Q}_y\\ Q_x\end{array}\right\}=\left[\begin{array}{cc}{A}_{44}& 0\\ 0& A_{55}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_{yz}\\ {\epsilon }_{xz}\end{array}\right\},$$

(7)

and also defined in the stresses form, as follows

$$\left\{\begin{array}{c}{Q}_y\\ Q_x\end{array}\right\}={\int }_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}\left\{\begin{array}{c}{\sigma }_{yz}\\ {\sigma }_{xz}\end{array}\right\}dz,$$

(8)

The process used to describe $k_{\alpha }$ derivation is shown in Figure 2. Assuming an orthotropic material, ${\int }_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}{\overline{Q}}_{55}dz={\int }_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}{\overline{Q}}_{44}dz=QBAR$, with a constant shear through the thickness of the FGM plates, the total strain energy can be rewritten and in terms of the shear correction coefficient $k_{\alpha }$.

$$\begin{gathered} U={\displaystyle \frac{1}{2}}{\int }_0^b{\int }_0^a\left({\displaystyle \frac{{Q_x}^2}{A_{55}}}+{\displaystyle \frac{{Q_y}^2}{A_{44}}}\right)dxdy={\displaystyle \frac{1}{2{ k}_{\alpha }}}{\int }_0^b{\int }_0^a({\displaystyle \frac{{Q_x}^2}{{\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}{\overline{Q}}_{55}dz}}+{\displaystyle \frac{{Q_y}^2}{{\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}{\overline{Q}}_{44}dz}})dxdy \\ ={\displaystyle \frac{1}{2{ k}_{\alpha } QBAR}}{\int }_0^b{\int }_0^a({Q_x}^2+{Q_y}^2)dxdy, \end{gathered}$$

(9a)

Also, the total strain energy U can be re-expressed in another form in terms of stresses ${\sigma }_{xz}$ and ${\sigma }_{yz}$.

$$\begin{gathered} U={\displaystyle \frac{h^{*}}{2}}{\int }_0^b{\int }_0^a{\int }_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}({\displaystyle \frac{{{\sigma }_{xz}}^2}{{\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}{\overline{Q}}_{55}dz}}+{\displaystyle \frac{{{\sigma }_{yz}}^2}{{\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}{\overline{Q}}_{44}dz}})dzdxdy \\ ={\displaystyle \frac{h^{*}}{2 QBAR}}{\int }_0^b{\int }_0^a{\int }_{-{\displaystyle \frac{h^{*}}{2}}}^{{\displaystyle \frac{h^{*}}{2}}}({{\sigma }_{xz}}^2+{{\sigma }_{yz}}^2)dzdxdy, \end{gathered}$$

(9b)

Thus, the $k_{\alpha }$ expression can be derived and obtained, as listed in section Appendix A. The values of nonlinear $k_{\alpha }$ are usually functions of $c_1$, $R_n$ and T.

3. Numerical Results

The two-material thick composited FGM plates with individual constituent materials are used to study the free vibration frequency results with the effects of environment temperature and a nonlinear varied shear correction coefficient. Under the boundary condition of four sides simply supported, no thermal loads ($∆T$ = 0), no in-plane distributed forces and no external pressure load are assumed for the numerical calculation. The free vibration frequency ${\omega }_{mn}$ with mode shape numbers $m$ and $n$ in the subscripts for four sides simply supported boundary condition can be obtained by assuming that density integrations $I_1=I_3=J_1=0$ and stiffness integrations $B_{ij}=E_{ij}=0$, $A_{16}=A_{26}=0$, $D_{16}=D_{26}=0$ and $A_{45}=D_{45}=F_{45}=0$ under the following time sinusoidal displacement $u^0$, $v^0$, w and shear rotations ${\psi }_x$, ${\psi }_y$ forms with amplitudes $a_{mn}$, $b_{mn}$, $c_{mn}$, $d_{mn}$ and $e_{mn}$ [27].

$$\begin{gathered} u^0=a_{mn}\cos (m\pi x/a)\sin (n\pi y/b)\sin ({\omega }_{mn}t), \\ {v}^0=b_{mn}\sin (m\pi x/a)\cos (n\pi y/b)\sin ({\omega }_{mn}t), \\ w=c_{mn}\sin (m\pi x/a)\sin (n\pi y/b)\sin ({\omega }_{mn}t), \\ {\psi }_x=d_{mn}\cos (m\pi x/a)\sin (n\pi y/b)\sin ({\omega }_{mn}t), \\ {\psi }_y=e_{mn}\sin (m\pi x/a)\cos (n\pi y/b)\sin ({\omega }_{mn}t), \end{gathered}$$

(10)

Equation (10) can be substituted into dynamic equilibrium differential equations referred in the published paper [22] under free vibration without external loads $f_1=f_2=\cdots =f_5=0$. Thus, the simply homogeneous equation can be obtained by assuming that matrix elements in (row and column) with (1,3)-(1,5), (2,3)-(2,5), (3,1)-(3,2), (4,1)-(4,2) and (5,1)-(5,2) are neglected in the fully homogeneous equation. The determinant of the coefficient matrix in simply homogeneous equation vanishes to obtain non-trivial solution of amplitudes can be represented using a simple five-degree polynomial equation; then, ${\omega }_{mn}$ can be found.

3.1. Advanced Computational Values of $k_{\alpha }$

The composited FGM SUS304/Si₃N₄ material is used to implement the numerical computation under environment temperature T with free stress assumed. The mechanical properties of constituents SUS304/Si₃N₄ are shown in

Table 1. The mechanical properties of constituents SUS304/Si₃N₄ [29].

Materials	${\mathit{P}}_{\mathit{i}}$	${\mathit{P}}_0$	${\mathit{P}}_{-1}$	${\mathit{P}}_1$	${\mathit{P}}_2$	${\mathit{P}}_3$
SUS304	$E_1$(Pa)	201.04$\times {10}^9$	0	3.079$\times {10}^{-4}$	−6.534$\times {10}^{-7}$	0
	${\nu }_1$	0.3262	0	−2.002$\times {10}^{-4}$	3.797$\times {10}^{-7}$	0
	${\rho }_1$(Kg/m3)	8166	0	0	0	0
Si3N4	$E_2$(Pa)	348.43$\times {10}^9$	0	−3.70$\times {10}^{-4}$	2.16$\times {10}^{-7}$	−8.946$\times {10}^{-11}$
	${\nu }_2$	0.24	0	0	0	0
	${\rho }_2$(Kg/m3)	2370	0	0	0	0

[29]. FGM 1 at the lower position is SUS304, and FGM 2 at the upper position is Si₃N₄, which is used for the free vibration frequency computations. Firstly, the calculated values of nonlinear $k_{\alpha }$ are usually functions of $c_1$, $R_n$ and T in the thick FGM plates with stiffness integration $B_{ij}\ne 0$. Including the nonlinear effect of $c_1$ on $k_{\alpha }$ for $a/b=1$, $R_n$ values from 0.12 mm to 12 mm, $h_1=h_2$, the advanced computational values of $k_{\alpha }$ under T = 1 K, 100 K, 300 K, 600 K and 1000 K are shown in

Table 2. (a) Nonlinear varied $k_{\alpha }$ vs. $c_1$ and $R_n$ under under T = 1 K [27]. (b) Nonlinear varied $k_{\alpha }$ vs. $c_1$ and $R_n$ under under T = 100 K [28]. (c) Nonlinear varied $k_{\alpha }$ vs. $c_1$ and $R_n$ under under T = 300 K. (d) Nonlinear varied $k_{\alpha }$ vs. $c_1$ and $R_n$ under under T = 600 K. (e) Nonlinear varied $k_{\alpha }$ vs. $c_1$ and $R_n$ under under T = 1000 K.

(a)
${\mathit{c}}_{\mathbf{1}}$ (1/mm2)	$h^{*}$(mm)	$k_{\alpha }$
		$\hspace{0.17em}{R}_n=0.1$	$\hspace{0.17em}{R}_n=0.2$	$\hspace{0.17em}{R}_n=0.5$	$\hspace{0.17em}{R}_n=1$	$\hspace{0.17em}{R}_n=2$	$\hspace{0.17em}{R}_n=5$	$\hspace{0.17em}{R}_n=10$
92.592598	0.12	−0.323869	−0.324963	−0.365392	−0.541369	−2.399161	0.802957	0.518229
0.925925	1.2	−0.323870	−0.324963	−0.365392	−0.541370	−2.399165	0.802958	0.518229
0.231481	2.4	−0.323869	−0.324963	−0.365392	−0.541370	−2.399165	0.802958	0.518229
0.037037	6	−0.323869	−0.324962	−0.365392	−0.541370	−2.399163	0.802957	0.518229
0.009259	12	−0.323870	−0.324962	−0.365392	−0.541370	−2.399163	0.802957	0.518229
0	0.12	0.915601	0.992033	1.175883	1.340146	1.396886	1.249938	1.099855
0	1.2	0.915601	0.992030	1.175884	1.340146	1.396886	1.249938	1.099855
0	2.4	0.915601	0.992030	1.175884	1.340146	1.396886	1.249938	1.099855
0	6	0.915600	0.992028	1.175884	1.340146	1.396886	1.249938	1.099855
0	12	0.915600	0.992027	1.175884	1.340146	1.396886	1.249938	1.099855
(b)
${\mathit{c}}_{\mathbf{1}}$ (1/mm2)	${\mathit{h}}^{\mathbf{*}}$(mm)	${\mathit{k}}_{\mathit{\alpha }}$
		$\hspace{0.17em}{\mathit{R}}_{\mathit{n}}\mathbf{=}\mathbf{0.1}$	$\hspace{0.17em}{R}_n=0.2$	$\hspace{0.17em}{R}_n=0.5$	$\hspace{0.17em}{R}_n=1$	$\hspace{0.17em}{R}_n=2$	$\hspace{0.17em}{R}_n=5$	$\hspace{0.17em}{R}_n=10$
92.592598	0.12	−0.448521	−0.456089	−0.539418	−0.922718	9.852672	0.682434	0.491249
0.925925	1.2	−0.448522	−0.456090	−0.539419	−0.922719	9.852635	0.682434	0.491249
0.231481	2.4	−0.448522	−0.456089	−0.539419	−0.922719	9.852635	0.682434	0.491249
0.037037	6	−0.448522	−0.456089	−0.539418	−0.922718	9.852679	0.682434	0.491249
0.009259	12	−0.448522	−0.456089	−0.539418	−0.922718	9.852675	0.682434	0.491249
0	0.12	0.899095	0.957858	1.091129	1.200860	1.232039	1.126363	1.021824
0	1.2	0.899095	0.957858	1.091129	1.200860	1.232039	1.126363	1.021824
0	2.4	0.899095	0.957858	1.091129	1.200860	1.232039	1.126363	1.021824
0	6	0.899095	0.957858	1.091129	1.200860	1.232039	1.126363	1.021824
0	12	0.899095	0.957858	1.091129	1.200860	1.232039	1.126363	1.021824
(c)
${\mathit{c}}_{\mathbf{1}}$ (1/mm2)	$h^{*}$(mm)	$k_{\alpha }$
		$\hspace{0.17em}{R}_n=0.1$	$\hspace{0.17em}{R}_n=0.2$	$\hspace{0.17em}{R}_n=0.5$	$\hspace{0.17em}{R}_n=1$	$\hspace{0.17em}{R}_n=2$	$\hspace{0.17em}{R}_n=5$	$\hspace{0.17em}{R}_n=10$
92.592598	0.12	−0.821563	−0.861922	−1.181502	−4.392330	1.474843	0.583927	0.463616
0.925925	1.2	−0.821565	−0.861923	−1.181503	−4.392341	1.474844	0.583927	0.463617
0.231481	2.4	−0.821565	−0.861923	−1.181503	−4.392341	1.474844	0.583927	0.463617
0.037037	6	−0.821564	−0.861924	−1.181502	−4.392332	1.474843	0.583927	0.463617
0.009259	12	−0.821564	−0.861924	−1.181503	−4.392332	1.474843	0.583927	0.463617
0	0.12	0.898426	0.956500	1.087890	1.195721	1.226106	1.121959	1.019033
0	1.2	0.898426	0.956498	1.087891	1.195721	1.226106	1.121959	1.019034
0	2.4	0.898426	0.956498	1.087891	1.195721	1.226106	1.121959	1.019034
0	6	0.898425	0.956496	1.087891	1.195721	1.226106	1.121958	1.019033
0	12	0.898426	0.956495	1.087891	1.195721	1.226106	1.121958	1.019033
(d)
${\mathit{c}}_{\mathbf{1}}$ (1/mm2)	$h^{*}$(mm)	$k_{\alpha }$
		$\hspace{0.17em}{R}_n=0.1$	$\hspace{0.17em}{R}_n=0.2$	$\hspace{0.17em}{R}_n=0.5$	$\hspace{0.17em}{R}_n=1$	$\hspace{0.17em}{R}_n=2$	$\hspace{0.17em}{R}_n=5$	$\hspace{0.17em}{R}_n=10$
92.592598	0.12	−0.778697	−0.814248	−1.096615	−3.535386	1.560072	0.589436	0.465330
0.925925	1.2	−0.778699	−0.814250	−1.096617	−3.535402	1.560071	0.589436	0.465330
0.231481	2.4	−0.778699	−0.814250	−1.096617	−3.535402	1.560071	0.589436	0.465330
0.037037	6	−0.778699	−0.814249	−1.096615	−3.535396	1.560071	0.589435	0.465330
0.009259	12	−0.778699	−0.814250	−1.096615	−3.535396	1.560071	0.589435	0.465330
0	0.12	0.899096	0.957861	1.091129	1.200860	1.232039	1.126363	1.021824
0	1.2	0.899095	0.957858	1.091129	1.200860	1.232039	1.126363	1.021824
0	2.4	0.899095	0.957858	1.091129	1.200860	1.232039	1.126363	1.021824
0	6	0.899095	0.957856	1.091129	1.200860	1.232039	1.126363	1.021824
0	12	0.899095	0.957856	1.091129	1.200860	1.232039	1.126363	1.021824
(e)
${\mathit{c}}_{\mathbf{1}}$ (1/mm2)	$h^{*}$(mm)	$k_{\alpha }$
		$\hspace{0.17em}{R}_n=0.1$	$\hspace{0.17em}{R}_n=0.2$	$\hspace{0.17em}{R}_n=0.5$	$\hspace{0.17em}{R}_n=1$	$\hspace{0.17em}{R}_n=2$	$\hspace{0.17em}{R}_n=5$	$\hspace{0.17em}{R}_n=10$
92.592598	0.12	−0.189321	−0.185985	−0.195625	−0.252506	−0.532897	1.590231	0.610227
0.925925	1.2	−0.189321	−0.185984	−0.195625	−0.252506	−0.532898	1.590231	0.610227
0.231481	2.4	−0.189321	−0.185984	−0.195625	−0.252506	−0.532898	1.590231	0.610227
0.037037	6	−0.189321	−0.185984	−0.195625	−0.252506	−0.532897	1.590231	0.610227
0.009259	12	−0.189321	−0.185984	−0.195625	−0.252506	−0.532897	1.590231	0.610227
0	0.12	0.932950	1.029296	1.276062	1.516531	1.616819	1.419804	1.206723
0	1.2	0.932949	1.029293	1.276062	1.516531	1.616820	1.419804	1.206723
0	2.4	0.932949	1.029293	1.276063	1.516531	1.616820	1.419804	1.206723
0	6	0.932948	1.029290	1.276063	1.516531	1.616819	1.419803	1.206723
0	12	0.932948	1.029290	1.276062	1.516531	1.616819	1.419803	1.206723

(a–e), respectively. Nonlinear $c_1=92.592598$/mm² decreases to $c_1=$ 0.0092592/mm², and varied values of $k_{\alpha }$(from −0.323869 to −2.399161, then 0.518229 under T = 1 K; from −0.448521 to 9.852672, then 0.491249 under T = 100 K; from −0.821563 to 1.474843, then 0.463616 under T = 300 K; from −0.778697 to −3.535386, then 0.465330 under T = 600 K; and from −0.189321 to 1.590231, then 0.610227 under T = 1000 K) first increase and then decrease with $R_n$ values from 0.1 to 10. For linear $c_1=0$/mm², varied values of $k_{\alpha }$ (from 0.915601 to 1.396886, then 1.099855 under T = 1 K; from 0.899095 to 1.232039, then 1.021824 under T = 100 K; from 0.898426 to 1.226106, then 1.019033 under T = 300 K; from 0.899096 to 1.232039, then 1.021824 under T = 600 K; and from 0.932950 to 1.616819, then 1.206723 under T = 1000 K) also first increase and then decrease with $R_n$ values from 0.1 to 10. Thus, the values of $k_{\alpha }$ are independent of $h^{*}$. The author also produced a self-referred published paper under T = 1 K [27] and T = 100 K [28] conditions to verify correctness since there were no other papers published focused on the values of nonlinear $k_{\alpha }$.

3.2. Values of Non-Dimensional Frequency Parameters $f^{*}$, ${\omega }^{*}$ and $\mathsf{\Omega }$

Thus, advanced computational values of $k_{\alpha }$ for $a/b=1$ are used in frequency calculations of the free vibration ($∆T$ = 0), including the effects of the nonlinear coefficient $c_1$ term. The frequency parameter $f^{*}={\omega }_{11}{h}^{*}\sqrt{{\rho }_2/E_2}$ value is shown in

Table 3. (a) $f^{*}$ for SUS304/Si₃N₄. (b) ${\omega }^{*}$ for SUS304/Si₃N₄. (c) $\mathsf{\Omega }$ for SUS304/Si₃N₄. (d) Comparison of frequency $f^{*}$ for SUS304/Si₃N₄ and Al/ZrO₂. (e) Comparison of frequency ${\omega }^{*}$ for SUS304/Si₃N₄. (f) Comparison of frequency $\mathsf{\Omega }$ for SUS304/Si₃N₄.

(a)
$a/h^{*}$	$R_n$	$\mathbf{Present} \mathbf{Solution} f^{*}$
		$h^{*}=1.2$ mm$, c_1=4/(3{h^{*}}^2)$, Nonlinear Varied $k_{\alpha }$
		T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.001707	0.001072	0.001856	0.001219	0.001388
	1	0.001085	0.001912	0.001196	0.001253	0.002444
	2	0.001128	0.001910	0.001151	0.002200	0.001543
	10	0.004561	0.001836	0.001866	0.004803	0.002482
8	0.5	0.001653	0.002817	0.002974	0.001940	0.003718
	1	0.001723	0.003037	0.003077	0.003345	0.002314
	2	0.001800	0.001836	0.001883	0.002042	0.002452
	10	0.003197	0.002037	0.002110	0.003614	0.002793
10	0.5	0.003427	0.003523	0.003713	0.004119	0.004651
	1	0.002150	0.002203	0.002311	0.002820	0.004904
	2	0.002872	0.002294	0.004005	0.007483	0.003061
	10	0.002479	0.002509	0.002597	0.004550	0.003445
(b)
$a/h^{*}$	$R_n$	$\mathbf{Present} \mathbf{Solution} {\omega }^{*}$
		$h^{*}=1.2$ mm$, c_1=4/(3{h^{*}}^2)$, Nonlinear Varied $k_{\alpha }$
		T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.034605	0.021141	0.035339	0.023160	0.029544
	1	0.022002	0.037707	0.022780	0.023799	0.052017
	2	0.022863	0.037654	0.021917	0.041778	0.032853
	10	0.092434	0.036204	0.035525	0.091197	0.052815
8	0.5	0.085749	0.142172	0.144959	0.094311	0.202585
	1	0.089396	0.003001	0.149956	0.162600	0.126102
	2	0.093391	0.092694	0.091803	0.099286	0.133605
	10	0.165847	0.102802	0.102862	0.175707	0.152172
10	0.5	0.277821	0.277824	0.282792	0.312856	0.395915
	1	0.174300	0.173750	0.175997	0.214193	0.417443
	2	0.232849	0.180933	0.304987	0.568351	0.260551
	10	0.200992	0.197894	0.197787	0.345595	0.293273
(c)
$a/h^{*}$	$R_n$	Present Solution  $\mathsf{\Omega }$
		$h^{*}=1.2$ mm$, c_1=4/(3{h^{*}}^2)$, Nonlinear Varied $k_{\alpha }$
		T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.098593	0.060234	0.100685	0.065986	0.084176
	1	0.062687	0.107433	0.064904	0.067808	0.148203
	2	0.065139	0.107282	0.062445	0.119031	0.093604
	10	0.263356	0.103151	0.101216	0.259833	0.150476
8	0.5	0.244309	0.405064	0.413005	0.268703	0.577187
	1	0.254701	0.436718	0.427241	0.463265	0.359280
	2	0.266082	0.264097	0.261559	0.282879	0.380655
	10	0.472516	0.292895	0.293067	0.500611	0.433556
10	0.5	0.791543	0.791553	0.805706	0.891361	1.128005
	1	0.496600	0.495035	0.501435	0.610259	1.189342
	2	0.663413	0.515498	0.868943	1.619294	0.742339
	10	0.572648	0.563822	0.563517	0.984639	0.835567
(d)
${\mathit{c}}_{\mathbf{1}}$ (1/mm2)	$h^{*}$ (mm)	$f^{*}$
		$\mathbf{Present} \mathbf{Solution}, a/h^{*}$= 10,  T =  300 K, Nonlinear Varied $k_{\alpha }$, for SUS304/Si3N4			Jha et al. 2013 [9], $\mathbf{for} \mathbf{Al}\mathbf{/}{\mathbf{ZrO}}_{\mathbf{2}}\mathbf{,}$ $R_n=0.5$
		$R_n=0.5$	$R_n=1$	$R_n=2$
0.925925	1.2	0.003713	0.002311	0.004005	-
0.333333	2	0.007991	0.004971	0.005131	-
0.013333	10	0.053758	0.055573	0.057577	-
0.009259	12	0.070666	0.073052	0.075691	-
0.006802	14	0.089049	0.092056	0.095384	0.0839
(e)
${\mathit{c}}_{\mathbf{1}}$ (1/mm2)	$h^{*}$(mm)	${\omega }^{*}$
		Present Solution, $a/h^{*}$= 10,  T =  300 K, Nonlinear Varied  $k_{\alpha }$			Kim 2005 [10] Forced Vibration, $h^{*}$ = 200 mm, $∆\mathit{T}$ = 0	Duc et al. 2017 [5] CNTRC, FSDT
		$R_n=0.5$	$R_n=1$	$R_n=2$	$R_n=2$	UD type
0.925925	1.2	0.282792	0.175997	0.304987	-	-
0.333333	2	0.608525	0.378548	0.390709	-	-
0.013333	10	4.093280	4.231468	4.384090	4.1165	3.99244
0.009259	12	5.380705	5.562391	5.763298	-	-
0.006802	14	6.780427	7.009404	7.262784	-	-
(f)
${\mathit{c}}_{\mathbf{1}}$ (1/mm2)	$h^{*}$ (mm)	$\mathsf{\Omega }$
		$\mathbf{Present} \mathbf{Solution}, a/h^{*}$ = 10, T = 300 K$, \mathbf{nonlinear} \mathbf{varied} k_{\alpha }$			Ungbhakorn and Wattanasakulpong 2013 [8] $∆\mathit{T}$ = 400 K$, R_n=1$
		$R_n=0.5$	$R_n=1$	$R_n=2$
0.925925	1.2	0.805706	0.501435	0.868943	-
0.333333	2	1.733756	1.078525	1.113175	-
0.053333	5	4.123597	4.262512	4.414177	-
0.037037	6	5.420403	5.603161	5.803643	5.359
0.013333	10	11.662205	12.055917	12.490754	-

(a), where ${\omega }_{11}$ is the fundamental first natural frequency obtained using subscript $m=n=1$, and ${\rho }_2$ is the density of FGM 2. For the SUS304/Si₃N₄ thick plate under free vibration with $h^{*}=1.2$ mm, the $f^{*}$ values under T = 1 K, 100 K, 300 K, 600 K and 1000 K with nonlinear varied $k_{\alpha }$ and $c_1$ effects are small in value, and a value no greater than 0.007483 occurs at a length-to-thickness ratio of $a/h^{*}$ = 10, $R_n$ = 2, T = 600 K. The frequency parameter ${\omega }^{*}=({\omega }_{11}{b}^2/{\pi }^2)\sqrt{I_s/D_s}$ value is shown in Table 3 (b), where $I_s={\displaystyle {\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}{\rho }_1}dz$, $D_s={\displaystyle {\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}{\overline{Q}}_1{z}^2}dz$, ${\rho }_1$ is the density of FGM 1, ${\overline{Q}}_1=E_1/(1-{{\nu }_1}^2)$. For the SUS304/Si₃N₄ thick plate under free vibration with $h^{*}=1.2$ mm, the ${\omega }^{*}$ values under T = 1 K, 100 K, 300 K, 600 K and 1000 K with nonlinear varied $k_{\alpha }$ and $c_1$ effects are small in value, and a value no greater than 0.568351 occurs at $a/h^{*}$ = 10, $R_n$ = 2, T = 600 K. The frequency parameter $\mathsf{\Omega }=({\omega }_{11}{a}^2/h^{*})\sqrt{{\rho }_1(1-{{\nu }_1}^2)/E_1}$ value is shown in Table 3 (c), and for SUS304/Si₃N₄ thick plate under free vibration with $h^{*}=1.2$ mm, the $\mathsf{\Omega }$ values under T = 1 K, 100 K, 300 K, 600 K and 1000 K with varied $k_{\alpha }$ and $c_1$ effects have values no greater than 1.619294 occuring at $a/h^{*}$ = 10, $R_n$ = 2, T = 600 K.

There are some results produced by other researchers in this area of research that are available for comparison, such as works by Duc et al. [5], Ungbhakorn and Wattanasakulpong [8], Jha et al. [9] and Kim [10]. It is interesting to compare the present solution of free vibration frequency values for $a/b=1$, with some other works shown in Table 3 (d–f). The values of $f^{*}$ vs. $h^{*}$ for SUS304/Si₃N₄ under $a/h^{*}$ = 10 and 300 K with nonlinear varied $k_{\alpha }$ and $c_1$ effects are shown in Table 3 (d), and the value $f^{*}$ = 0.089049 at $h^{*}$ = 14 mm, $R_n$ = 0.5 is in close to $f^{*}$ = 0.0839, with Al/ZrO₂ under no environmental temperature effect, as presented by Jha et al. in 2013 [9] when using HOSNT12 in which 12 degrees of freedom were considered for the model. The values of ${\omega }^{*}$ vs. $h^{*}$ for SUS304/Si₃N₄ under $a/h^{*}$ = 10 and 300 K with nonlinear varied $k_{\alpha }$ and $c_1$ effects are shown in Table 3 (e), with the value ${\omega }^{*}$ = 4.093280 at $h^{*}$ = 10 mm, $R_n$ = 0.5 is close to ${\omega }^{*}$ = 4.1165 with a $h^{*}$ = 200 mm forced vibration under uniform a temperature rise ($∆T$= 0), as presented by Kim in 2005 [10]. Furthermore, when compared, the present ${\omega }^{*}$ result is close to the analytical FSDT result ${\omega }^{*}$ = 3.99244 of the uniform distribution (UD) in CNTRC FGM plates resting on elastic foundations presented by Duc et al. in 2017 [5]. The values of $\mathsf{\Omega }$ vs. $h^{*}$ for SUS304/Si₃N₄ under $a/h^{*}$ = 10 and 300 K with nonlinear varied $k_{\alpha }$ and $c_1$ effects are shown in Table 3 (f), and the value $\mathsf{\Omega }$ = 5.420403 at $h^{*}$ = 6 mm, $R_n$ = 0.5 is close to $\mathsf{\Omega }$ = 5.359 with forced vibration under temperature rise ($∆T$= 400 K) presented by Ungbhakorn and Wattanasakulpong in 2013 [8]. The values of dimensionless natural frequency parameters are found in the functions of $a/h^{*}$, $R_n$, $c_1$ and T.

3.3. Natural Frequency ${\omega }_{mn}$ Values

Secondly, the natural frequency ${\omega }_{mn}$ values (unit 1/s) of free vibration ($∆T$= 0) according to mode shape when using subscript numbers $m$ and $n$ for the SUS304/Si₃N₄ FGM thick plate $a/b=1$ are calculated. The values of the fundamental first (subscript $m$ = $n$ = 1) natural frequency ${\omega }_{11}$ vs. $R_n$ with $h^{*}=1.2$ mm, nonlinear varied $k_{\alpha }$ and $c_1=0.925925$/mm² effects under T = 1 K, 100 K, 300 K, 600 K and 1000 K are shown in

Table 4. (a) Fundamental natural frequency ${\omega }_{11}$ for nonlinear varied $k_{\alpha }$, $c_1$, $h^{*}=1.2$ mm. (b) ${\omega }_{mn}$ vs. $m$ and $n$ under nonlinear varied $k_{\alpha }$, $c_1$,$\hspace{0.17em}{R}_n=0.5$ and T = 300 K [28].

(a)
$a/h^{*}$	$R_n$	${\omega }_{11}$  (1/s)
		T   = 1 K		T   = 100 K		T   = 300 K	T   = 600 K		T   = 1000 K
5	0.5	0.029227		0.018034		0.030248	0.019096		0.020672
	1	0.018583		0.032166		0.019499	0.019623		0.036396
	2	0.019310		0.032121		0.018760	0.034448		0.022987
	10	0.078070		0.030884		0.030408	0.075196		0.036954
10	0.5	0.058662		0.059249		0.060513	0.064490		0.069255
	1	0.036803		0.037054		0.037661	0.044152		0.073021
	2	0.049166		0.038585		0.065263	0.117157		0.045576
	10	0.042439		0.042203		0.042323	0.071239		0.051300
(b)
$a/h^{*}$	${\omega }_{1n}$ (1/s)
	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
5	0.030248	0.012079	0.010355	0.005455	0.003399	0.007451	0.006401	0.005629	0.005048
10	0.060513	0.042673	0.035361	0.020371	0.015934	0.031821	0.014847	0.011815	0.006108
$a/h^{*}$	${\omega }_{2n}$ (1/s)
	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
5	0.019082	0.009523	0.010442	0.004176	0.007139	0.007156	0.006237	0.005555	0.005061
10	0.023112	0.030248	0.014602	0.012079	0.015355	0.010355	0.015735	0.005455	0.006460
$a/h^{*}$	${\omega }_{3n}$ (1/s)
	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
5	0.008481	0.007713	0.010970	0.009022	0.007734	0.006781	0.006089	0.006744	0.001734
10	0.027054	0.023712	0.012403	0.016924	0.009628	0.011476	0.015462	0.004309	0.006783
$a/h^{*}$	${\omega }_{4n}$ (1/s)
	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
5	0.010463	0.009661	0.008689	0.007765	0.007011	0.006711	0.002147	0.001856	0.001620
10	0.012677	0.019082	0.010558	0.009523	0.012944	0.010442	0.013141	0.004176	0.007080
$a/h^{*}$	${\omega }_{5n}$ (1/s)
	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
5	0.005210	0.007929	0.007340	0.006738	0.006999	0.002301	0.001972	0.001712	0.001502
10	0.010366	0.015779	0.014532	0.008480	0.008050	0.017151	0.011124	0.001000	0.008826
$a/h^{*}$	${\omega }_{6n}$ (1/s)
	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
5	0.006944	0.006693	0.006307	0.005852	0.005405	0.004941	0.004526	0.004152	0.003821
10	0.008856	0.008481	0.008063	0.007713	0.008185	0.010970	0.009856	0.009022	0.003542
$a/h^{*}$	${\omega }_{7n}$ (1/s)
	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
5	0.005915	0.005753	0.005469	0.005921	0.005027	0.004625	0.004279	0.003964	0.003679
10	0.012552	0.008211	0.011614	0.010954	0.010263	0.009585	0.006448	0.008361	0.007834
$a/h^{*}$	${\omega }_{8n}$ (1/s)
	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
5	0.005130	0.005009	0.006575	0.005036	0.004590	0.004284	0.004006	0.003746	0.003506
10	0.010707	0.010463	0.010102	0.009661	0.009180	0.008689	0.004282	0.007765	0.007361
$a/h^{*}$	${\omega }_{9n}$ (1/s)
	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
5	0.004509	0.007183	0.005173	0.004502	0.004205	0.003967	0.003743	0.003530	0.003328
10	0.005754	0.005626	0.005439	0.005199	0.004907	0.007963	0.007584	0.007228	0.006915

(a). The results of fundamental first natural frequencies are found in functions of the length-to-thickness ratio, power-law exponent parameter, $c_1$ and environment temperature. The values of natural frequency ${\omega }_{mn}$ vs. subscript $m$,$n$ = 1,2,…,9 with $R_n$ = 0.5, $h^{*}=1.2$ mm under nonlinear varied $k_{\alpha }$ and $c_1=0.925925$/mm² effects are shown in Table 4 (b). The results of dimensional natural frequencies are found in varied with mode values $m$, $n$ and functions of length-to-thickness ratio and $c_1$ for the given power-law exponent $\hspace{0.17em}{R}_n=0.5$ and environment temperature T = 300 K [28].

Figure 3 shows the values of ${\omega }_{1n}$ vs. $R_n$ in $h^{*}=1.2$ mm FGM plate for $a/h^{*}$ = 5, 10, respectively, with nonlinear varied $k_{\alpha }$ and $c_1=0.925925$/mm² effects under T = 300 K. Generally, the values of ${\omega }_{1n}$ oscillate and go to around 0.005/s, with subscript values of $n$ from 1 to 9 for $R_n$ = 0.5, 1 and 10. The greatest value of ${\omega }_{12}$= 0.047690/s is found, which then decreases to a value of ${\omega }_{19}$ = 0.005686/s for $a/h^{*}$ = 5, $R_n$ = 10. The greatest value of ${\omega }_{11}$ = 0.060513/s is found, which then decreases to a value of ${\omega }_{19}$ = 0.006108/s for $a/h^{*}$ = 10, $R_n$ = 0.5. Figure 4 shows the values of ${\omega }_{1n}$ vs. T in FGM plate for $a/h^{*}$ = 5, 10, respectively, under nonlinear varied $k_{\alpha }$, $c_1=0.925925$/mm² and $R_n$ = 0.5. Generally, the values of ${\omega }_{1n}$ oscillate and go to around 0.005/s with subscript values of $n$ from 1 to 9 for T = 300 K, 600 K and 1000 K. The greatest value of ${\omega }_{13}$= 0.050642/s is found, which then decreases to a value of ${\omega }_{19}$ = 0.005048/s for $a/h^{*}$ = 5, T = 300 K. The greatest value of ${\omega }_{11}$ = 0.069255/s is found, which then decreases to a value of ${\omega }_{19}$ = 0.011216/s for $a/h^{*}$ = 10, T = 1000 K.

4. Discussion

Advanced computational values of $k_{\alpha }$ greatly affect the numerical results of nondimensional frequency parameters $f^{*}$, ${\omega }^{*}$ and $\mathsf{\Omega }$. These values of frequency parameters are strongly dependent on the use of a simply homogeneous equation or fully homogeneous equation or an equation that cannot be obtained by assuming that matrix elements in (row and column) with (1,3)-(1,5), (2,3)-(2,5), (3,1)-(3,2), (4,1)-(4,2) and (5,1)-(5,2) are neglected or not in present studies. The frequency findings concerning the use of this simply homogeneous equation can be used as preliminary data and can be possibly studied in future research by using further fully homogeneous equations. To highlight the significance of this study and its applicability in future studies of advanced frequencies of thick composite materials. The effects of TSDT and nonlinear shear correction effects needed to be considered in the determinant of homogeneous equations for thick composite materials.

5. Conclusions

Advanced values of nonlinear varied $k_{\alpha }$ vs. $c_1$ and$\hspace{0.17em}{R}_n$ under T = 1 K, 100 K, 300 K, 600 K, and 1000 K are obtained. The values of $k_{\alpha }$ are independent of $h^{*}$. The numerical frequency computations are calculated using a simply homogeneous equation and investigated by using TSDT and nonlinear shear correction coefficients for thick composited FGM plates. Generally the values of ${\omega }_{1n}$ oscillate and go to around 0.005/s with subscript values of $n$ from 1 to 9 for $a/h^{*}$ = 5 and 10, $R_n$ = 0.5, 1 and 10. Results of natural frequencies are found in the functions of the length-to-thickness ratio, power-law exponent parameter, $c_1$ and T.