## 4 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{\(\mathbf{b_{\mathrm{G}}/t_{\mathrm{G}}=10}\)} \\ \cline{2-6} Mode & m = 1 & m = 2 & m = 3 & m = 4 & m = 5 \\ \hline
1 & 0.6629 & 1.5822 & 2.9590 & 4.6511 & 6.5472 \\
2 & 1.5820 & 2.4285 & 2.9591 & 4.6515 & 7.1267 \\
3 & 1.7007 & 3.7128 & 3.7133 & 5.3141 & 8.0444 \\
4 & 2.9590 & 3.9271 & 4.8759 & 6.3494 & 9.2438 \\
5 & 3.1799 & 4.5122 & 4.8763 & 7.6830 & 9.2441 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{5}{c}{\(\mathbf{b_{\mathrm{G}}/t_{\mathrm{G}}=100}\)} \\ \cline{2-6} Mode & m = 1 & m = 2 & m = 3 & m = 4 & m = 5 \\ \hline
1 & 0.6763 & 1.6140 & 3.0187 & 4.7449 & 6.6789 \\
2 & 1.6139 & 2.4773 & 3.7873 & 5.4216 & 6.6806 \\
3 & 1.7350 & 2.4774 & 3.7881 & 6.4760 & 7.2672 \\
4 & 3.0186 & 3.7878 & 4.9743 & 6.4773 & 7.2707 \\
5 & 3.2440 & 4.0061 & 6.2109 & 7.8382 & 7.2708 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{5}{c}{\(\mathbf{b_{\mathrm{G}}/t_{\mathrm{G}}=100}\)} \\ \cline{2-6} Mode & m = 1 & m = 2 & m = 3 & m = 4 & m = 5 \\ \hline
1 & 0.6782 & 1.6186 & 3.0270 & 4.7585 & 6.6983 \\
2 & 1.6185 & 2.4842 & 3.7984 & 5.4362 & 7.2900 \\
3 & 1.7399 & 2.4843 & 3.7988 & 6.4957 & 7.2902 \\
4 & 3.0271 & 2.4844 & 4.9881 & 7.8604 & 8.2288 \\
5 & 3.2532 & 3.7983 & 6.2285 & 7.8609 & 8.2301 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Natural frequencies of the SSSS GPLs reinforced PMMA square plates for different GPLs width-to-thickness ratios (\(t_{\mathrm{G}}=1.5\) nm). \begin{table} \begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{\(\mathbf{a_{\mathrm{G}}/b_{\mathrm{G}}=1}\)} \\ \cline{2-6} Mode & m = 1 & m = 2 & m = 3 & m = 4 & m = 5 \\ \hline
1 & 0.6782 & 1.6183 & 3.0267 & 4.7581 & 6.6961 \\
2 & 1.6183 & 1.6184 & 3.0270 & 4.7584 & 6.6971 \\
3 & 1.7398 & 2.4839 & 3.7981 & 5.4348 & 6.6973 \\
4 & 3.0269 & 2.4840 & 4.9879 & 6.4941 & 7.2899 \\
5 & 3.2529 & 2.4841 & 6.2280 & 6.4945 & 8.2277 \\ \hline \hline \end{tabular} \begin{tabular}{c c c c c} \hline \hline  & \multicolumn{5}{c}{\(\mathbf{a_{\mathrm{G}}/b_{\mathrm{G}}=4}\)} \\ \cline{2-6} Mode & m = 1 & m = 2 & m = 3 & m = 4 & m = 5 \\ \hline
1 & 0.6783 & 1.6186 & 3.0272 & 4.7588 & 6.6978 \\
2 & 1.6186 & 2.4845 & 3.7991 & 4.7591 & 6.6985 \\
3 & 1.7400 & 3.7987 & 4.9885 & 5.4351 & 7.2911 \\
4 & 3.0273 & 4.0178 & 6.2289 & 5.4353 & 8.2296 \\
5 & 3.2534 & 4.6164 & 8.1301 & 6.4955 & 9.4578 \\ \hline \hline \end{tabular} 
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{5}{c}{\(\mathbf{a_{\mathrm{G}}/b_{\mathrm{G}}=10}\)} \\ \cline{2-6} Mode & m = 1 & m = 2 & m = 3 & m = 4 & m = 5 \\ \hline
1 & 0.6783 & 1.6187 & 3.0273 & 4.7592 & 6.6987 \\
2 & 1.6187 & 2.4846 & 3.0275 & 5.4354 & 7.2903 \\
3 & 1.7401 & 3.7987 & 3.7990 & 6.4980 & 8.2312 \\
4 & 3.0275 & 3.7988 & 4.9884 & 7.8609 & 8.2315 at \(\xi=0.1\) with FGM pattern UD used in all examples. The dimensionless frequency and wavenumber are \(\Omega=\sqrt{\rho_{M}\omega^{2}a^{2}/E_{M}}\) and \(\chi=ka\), respectively. Figure 4: Effect of GPLs weight fraction WG on the dispersion spectra of GPLs reinforced plates. Figure 5: The dispersion spectra of FG-GPLs reinforced plates with different GPLs distribution patterns. Figure 4 investigates the effect of weight fractions on the dispersion spectra of GPLs reinforced plates. The frequencies always increase with the increasing weight fraction, so the dispersion curves will move upward as shown in Figure 4. For every mode, the blue solid branch for \(W_{\rm G}=0\) is the lowest, the red dot dash line for \(W_{\rm G}=0.5\%\) is middle, while the black dash line for \(W_{\rm G}=1\%\) is the highest. Figure 5 shows the influences of different GPLs distribution patterns on the dispersion spectra of the FG plates. As demonstrated by the figure, the GPLs distribution patterns  
Figure 6: Effect of boundary condition on the dispersion spectra of GPLs reinforced plates. Figure 7: The dispersion spectra of GPLs-reinforced plates with different aspect ratios.