## 4 Numerical Results and Discussion
### Comparison and Validation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{k_{2}(\frac{N}{m})}\)} & \multirow{2}{*}{\(\mathbf{\omega_{m}}\)} & \multicolumn{4}{c}{\(\mathbf{k_{1}(\times 10^{16}\ \frac{N}{m^{3}})}\)} \\ \cline{3-6}  & & **3** & **3.5** & **4** & **4.5** & **5** \\ \hline
10 & m = 1 & 53.6075 & 54.0087 & 54.3011 & 54.5226 & 54.6958 \\  & m = 2 & 100.9296 & 102.9847 & 104.6266 & 105.9049 & 106.8881 \\  & m = 3 & 139.5596 & 142.0667 & 144.4766 & 146.7824 & 148.9747 \\
11 & m = 1 & 53.7027 & 54.0769 & 54.3521 & 54.5621 & 54.7271 \\  & m = 2 & 102.6704 & 104.3787 & 105.7134 & 106.7410 & 107.5308 \\  & m = 3 & 144.1770 & 146.4964 & 148.7037 & 150.7876 & 152.7349 \\
12 & m = 1 & 53.7911 & 54.1410 & 54.4004 & 54.5996 & 54.7571 \\  & m = 2 & 104.1203 & 105.5133 & 106.5874 & 107.4125 & 108.0505 \\  & m = 3 & 148.4306 & 105.50309 & 152.4963 & 154.3126 & 155.9658 \\
13 & m = 1 & 53.8735 & 54.2012 & 54.4461 & 54.6354 & 54.7858 \\  & m = 2 & 105.3043 & 106.4268 & 107.2889 & 107.9545 & 108.4744 \\  & m = 3 & 152.2553 & 154.0912 & 155.7658 & 157.2676 & 158.5904 \\
14 & m = 1 & 53.9504 & 54.2578 & 54.4894 & 54.6695 & 54.8133 \\  & m = 2 & 106.2589 & 107.1599 & 107.8544 & 108.3958 & 108.8240 \\  & m = 3 & 155.5631 & 157.0872 & 158.4328 & 159.6003 & 160.5981 \\
15 & m = 1 & 54.0224 & 54.3113 & 54.5305 & 54.7020 & 54.8397 \\  & m = 2 & 107.0251 & 107.7501 & 108.3140 & 108.7588 & 109.1154 \\  & m = 3 & 158.2723 & 159.4621 & 160.4807 & 161.3424 & 162.0661 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Variation of the first three natural frequencies (GHz) with the elastic parameters of the Pasternak foundation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{\mu}\) (\(\mathbf{mm}\))} & \multicolumn{4}{c}{\(\mathbf{g^{*}_{GPL}\) = 0.5\%}} & \multicolumn{4}{c}{\(\mathbf{g^{*}_{GPL}\) = 1\%}} \\ \cline{2-7}  & \multicolumn{3}{c}{**Boundary Condition**} & \multicolumn{3}{c}{**Boundary Condition**} \\ \cline{2-7}  & **S-S** & **C-S** & **C-C** & **S-S** & **C-S** & **C-C** \\ \hline
0.0 & 2.2360 & 6.9765 & 8.9273 & 2.3185 & 7.3241 & 9.3883 \\
0.5 & 2.2358 & 6.9751 & 8.9258 & 2.3182 & 7.3227 & 9.3867 \\
1.0 & 2.2349 & 6.9709 & 8.9214 & 2.3173 & 7.3183 & 9.3821 \\
1.5 & 2.2336 & 6.9640 & 8.9140 & 2.3159 & 7.3111 & 9.3743 \\
2.0 & 2.2316 & 6.9544 & 8.9037 & 2.3139 & 7.3010 & 9.3634 \\
2.5 & 2.2292 & 6.9421 & 8.8905 & 2.3113 & 7.2880 & 9.3495 \\
3.0 & 2.2262 & 6.9271 & 8.8744 & 2.3082 & 7.2723 & 9.3326 \\
3.5 & 2.2226 & 6.9096 & 8.8555 & 2.3046 & 7.2539 & 9.3128 \\
4.0 & 2.2186 & 6.8895 & 8.8339 & 2.3004 & 7.2328 & 9.2900 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Variation of the first natural frequency for FG curved nanobeams with different nonlocal parameters, weight fractions, and boundary conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{\theta_{0}}\) (Rad)} & \multirow{2}{*}{**Pattern of Multilayer**} & \multicolumn{3}{c}{**Boundary Condition**} \\ \cline{3-5}  & & **S-S** & **C-S** & **C-C** \\ \hline \(\pi/12\) & Pattern 1 & 2.5058 & 5.7430 & 6.9190 \\  & Pattern 2 & 2.3910 & 5.5456 & 6.7041 \\  & Pattern 3 & 2.6163 & 5.9357 & 7.1293 \\  & Pattern 4 & 2.4923 & 5.7200 & 6.8935 \\ \(\pi/6\) & Pattern 1 & 2.4283 & 7.4648 & 9.5336 \\  & Pattern 2 & 2.3173 & 7.3183 & 9.3821 \\  & Pattern 3 & 2.5356 & 7.6117 & 9.6868 \\  & Pattern 4 & 2.4174 & 7.4543 & 9.5225 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Variation of the first natural frequency for a FG curved nanobeam with different patterns and boundary conditions, and two opening angles. As a further parametric study, we evaluate the sensitivity of the vibration response of the composite curved nanobeam reinforced with GPLs for different combinations of the opening angle and nonlocal parameters (\(\theta_{0},\mu\)).