## 5 Results and Discussion
### Free Vibration  
Firstly, metal-ceramic functionally graded materials are calculated to verify the validity of current methods. A simply supported Al/Al\({}_{2}\)O\({}_{3}\) functionally graded square plate with a total number of layers \(N_{L}=10\) is considered. The material properties of Al and Al\({}_{2}\)O\({}_{3}\) are shown in the following equations. Al: \(E_{1}=70.0\times 10^{9}\) N/m\({}^{2}\), \(\
u_{1}=0.3\), \(\rho_{1}=2702\) kg/m\({}^{3}\)Al\({}_{2}\)O\({}_{3}\): \(E_{2}=380\times 10^{9}\) N/m\({}^{2}\), \(\
u_{2}=0.3\), \(\rho_{2}=3800\) kg/m\({}^{3}\)  
The effective material properties of the functionally graded plates are determined by Equations (45) and (46), which follow the same procedure as when bending is considered. \(n\) is the volume fraction index. The dimensionless natural frequency of the FG-GRC plates is defined by the following equation. \[\omega_{c}=\omega h\sqrt{\rho_{2}/E_{2}} \tag{47}\]  
As shown in Table 3, the results of RPT and IRPT are in good agreement with those of higher-order shear deformation plate theory [50]. Next, we will analyze the FG-GRC plates. The material parameters of the epoxy matrix materials and GPLs are the same as those in the bending analysis. Where the dimensionless natural frequencies are defined as shown below. \[\omega_{c}=\omega h\sqrt{\rho_{\text{M}}/E_{\text{M}}} \tag{48}\]  
in which \(\omega_{c}\) and \(\omega_{\text{M}}\) are the natural frequencies of FG-GRC square plates and pure epoxy square plates, respectively. The dimensionless natural frequencies and relative frequency increase \((\omega_{c}-\omega_{\text{M}})/\omega_{\text{M}}\) of FG-GRC plates are given in Table 4. Each distribution mode of the FG-GRC square plates was calculated. The results of the calculations for the pure epoxy plates are also included in Table 4. The dimensionless natural frequencies and relative frequency increases in RPT \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline _a/h_ & & \(n\) & & & & \\ \hline  & & 0.0 & 0.5 & 1.0 & 4.0 & 10.0 \\ \hline
2 & Ref. [50] & 0.9400 & 0.8232 & 0.7476 & 0.5994 & 0.5460 \\  & RPT & 0.9297 & 0.8126 & 0.7366 & 0.5919 & 0.5393 \\  & IRPT & 0.9420 & 0.8267 & 0.7529 & 0.6053 & 0.5479 \\
5 & Ref. [50] & 0.2121 & 0.1819 & 0.1640 & 0.1383 & 0.1306 \\  & RPT & 0.2113 & 0.1811 & 0.1634 & 0.1375 & 0.1293 \\  & IRPT & 0.2122 & 0.1829 & 0.1662 & 0.1406 & 0.1310 \\
10 & Ref. [50] & 0.05777 & 0.04917 & 0.04426 & 0.03811 & 0.03642 \\  & RPT & 0.05769 & 0.04914 & 0.04427 & 0.03795 & 0.03611 \\  & IRPT & 0.05777 & 0.04950 & 0.04495 & 0.03878 & 0.03656 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dimensionless fundamental frequencies of Al/Al\({}_{2}\)O\({}_{3}\) functionally graded t square plates. and IRPT were compared. The role played by the effect of thickness stretching in the free vibration process of FG-GRC plates can be seen by comparing the results of RPT and IRPT calculations. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline _a/h_ & _r, s_ & **Theories** & \begin{tabular}{c} **Pure** \\ **Epoxy** \\ \end{tabular}  & \begin{tabular}{c} **UD** \\ \end{tabular}  & \begin{tabular}{c} **FG-O** \\ \end{tabular}  & \ \begin{tabular}{c} **FG-X** \\ \end{tabular}  & \begin{tabular}{c} **FG-A** \\ \end{tabular}  \\ \hline  
\begin{tabular}{c}  
Among the four distribution modes, GPLs symmetrically distributed UD-type, FG-O-type, and FG-X-type FG-GRC plates showed little difference in the calculated results under the two plate theories (RPT and IRPT). The calculated results for the FG-A-type FG-GRC plate with asymmetric distribution of GPLs are significantly different under RPT and IRPT. Compared with the IRPT results, the RPT results without considering the effect of thickness stretching are significantly lower, which is caused by the more significant anisotropy in the transverse displacement direction of the FG-A-type distribution than the other three GPLs symmetrically distributed FG-GRC plates.