## 5 Results and Discussion
### Static Bending  
A functionally graded square plate composed of Al/(ZrO\({}_{2}\))-1 is used as an example to verify the effectiveness of the current method. The material properties are shown below  
Al: \(E_{1}=70\) GPa, \(\
u_{1}=0.3\)  
(ZrO\({}_{2}\))-1: \(E_{2}=200\) GPa, \(\
u_{2}=0.3\)The plate has Al on the upper surface and \((\mathrm{ZrO_{2}})\)-1 on the lower surface with an intermediate gradient distribution. The effective material properties are shown below [46]:  
\[P(z)=P_{1}+(P_{2}-P_{1})\bigg{(}\frac{1}{2}+\frac{z}{h}\bigg{)}^{n}(n\geq 0) \tag{44}\]  
where \(P_{1}\) and \(P_{2}\) represent the material properties of Al and \((\mathrm{ZrO_{2}})\)-1, such as Young's modulus, density, etc., respectively. \(n\) is the volume fraction index. The functionally graded plate is divided into \(N_{L}\) layers of isotropic plates along the thickness direction. The total material properties of the functionally graded plates are superimposed on the material properties of each layer in the thickness direction, and the equivalent material property \(P_{\mathrm{eq}}^{(k)}\) for the \(k\)th layer is defined as follows [47]:  
\[P_{\mathrm{eq}}^{(k)}=\int_{z_{k}}^{z_{k+1}}\frac{P(z)}{z_{k+1}-z_{k}}\;\mathrm{ d}z\;,(k=1,2,\cdots,N_{L}) \tag{45}\]  
The dimensionless deflection is defined as follows and the uniformly distributed transverse load \(F_{0}\)\(=500\) kPa. \[w_{c}=100 wE_{1}h^{3}/\Big{[}12\Big{(}1-\
u_{1}^{2}\Big{)}F_{0}a^{4}\Big{]} \tag{46}\]  
The effects of different truncation levels \(R\) and \(S\) and \(N_{L}\) on the dimensionless deflection are given in Table 1. \(n\) is the volume fraction index. The results show that the dimensionless deflection is close to convergence when the truncation levels \(R\) and \(S\) of Equation (37) are taken as 10 for the different total numbers of layers. Therefore, the use of \(R\)\(=\)\(S\)\(=\) 10 for all subsequent bending deflection-related calculations balances accuracy and calculation costs. The dimensionless central deflection values for \(\mathrm{Al/}\left(\mathrm{ZrO_{2}}\right)\)-1 functionally graded square plates at \(a/h\)\(=\) 5, \(N_{L}\)\(=\) 10, and \(R\)\(=\)\(S\)\(=\) 10 are given in Table 2, where \(n\) is the volume fraction index. By comparing with the finite element method [48] and the meshless local Petrov-Galerkin method [49], it can be seen that RPT and IRPT are in good agreement with the available literature in terms of results. \begin{table}
\begin{tabular}{c c c c c} \hline  & & \(R\) = \(S\) = 1 & \(R\) = \(S\) = 10 & \(R\) = \(S\) = 20 \\ \hline \(N_{L}\) = 10 & RPT & 0.2402 & 0.2313 & 0.2312 \\  & IRPT & 0.2550 & 0.2462 & 0.2462 \\ \(N_{L}\) = 20 & RPT & 0.2407 & 0.2318 & 0.2317 \\  & IRPT & 0.2556 & 0.2467 & 0.2467 \\ \(N_{L}\) = 30 & RPT & 0.2408 & 0.2319 & 0.2318 \\  & IRPT & 0.2557 & 0.2468 & 0.2468 \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensionless central deflection values of \(\mathrm{Al/}\left(\mathrm{ZrO_{2}}\right)\)-1 functionally graded square plates at different truncation levels. (\(n\)\(=\)\(0.5\)). \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{\(n\)} & & \\ \hline  & 0 & 0.5 & 1 & 2 \\ \hline Ref. [48] & 0.1703 & 0.2232 & 0.2522 & 0.2827 \\ Ref. [49] & 0.1671 & 0.2505 & 0.2905 & 0.3280 \\ RPT & 0.1716 & 0.2313 & 0.2715 & 0.3140 \\ IRPT & 0.1830 & 0.2462 & 0.2883 & 0.3327 \\ \hline \end{tabular}
\end{table} 
Table 2: Dimensionless central deflection values of \(\mathrm{Al/}\left(\mathrm{ZrO_{2}}\right)\)-1 functionally graded square plates under a transverse uniform load. (\(R\)\(=\)\(S\)\(=\)\(10\), \(N_{L}\)\(=\)\(10\), \(a/h\)\(=\)\(5\)). Next, consider an FG-GRC plate. The epoxy is selected for the matrix material, and its material properties are shown below.