## 4 Numerical Results and Discussion
### Comparison and Validation
Table 7 summarizes the results obtained for the first three modes of vibration of the nanobeam with a \"Pattern 3\" reinforcement, while assuming a weight fraction \(g^{*}_{\text{GPL}}~{}=~{}1\%\), a total number of layers \(N_{L}~{}=~{}12\), and a null value for the foundation constants \(k_{1}~{}=~{}k_{2}~{}=~{}0\). The numerical results in Table 7 indicate a meaningful decrease of the natural frequencies for larger opening angles and increasing nonlocal parameters within the curved nanobeam, due to decreased stiffness of the material. The sensitivity of the natural frequencies of the composite curved nanobeam is also plotted in Figure 3 for varying nonlocal parameters, reinforcement distributions (i.e., for Patterns 1,2,3,4), and opening angles. As visible in Figure 3, the natural frequency decreases monotonically for an increased nonlocal parameter due to the reduced stiffness of the material. Moreover, Patterns 2 and 3 always yield the lowest and highest results, independently of the selected opening angle for the curved nanobeam. Results based on Patterns 1 and 4, instead, always fall within the lower-bound and upper-bound values (as provided by Patterns 2 and 3), and become even closer to each other for increasing opening angles. For each selected reinforcement distribution, we also observe a gradual shift of the curves downwards for increasing opening angles, in line with results of Table 7.  
The effect of the weight fraction on the vibration responses of the curved nanobeam is evaluated accordingly in Figure 4 for all the reinforcement distributions. Moreover, Figure 4 shows the variation of the first natural frequency of the curved nanobeam vs. the weight fraction for different opening angles \(\theta_{0}\) ranging between 0.5 rad and 2 rad, under the following assumptions: \(N_{L}~{}=~{}12\), \(\mu~{}=~{}1~{}nm\), \(k_{1}~{}=~{}k_{2}~{}=~{}0\). Based on the plots of Figure 4, it is worth observing a gradual increase of the natural frequency for increasing weight fractions, with a clear shift of the curves downwards for increasing opening angles. As also visible from a comparative evaluation of Figure 4a-d, the reinforcement distribution can affect the response, while noticing the highest sensitivity for a pattern of type 3. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\theta_{0}\) (Rad) & \(\omega_{m}\) & & & & \(\mu\) (\(nm\)) & & \\ \hline  & & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \(\frac{\pi}{12}\) & m = 1 & 2.6176 & 2.6163 & 2.6125 & 2.6061 & 2.5972 & 2.5859 \\  & m = 2 & 10.4115 & 10.3910 & 10.3302 & 10.2313 & 10.0974 & 9.9329 \\  & m = 3 & 22.9623 & 22.8610 & 22.5649 & 22.0960 & 21.4861 & 20.7716 \\ \(\frac{\pi}{6}\) & m = 1 & 2.5369 & 2.5356 & 2.5319 & 2.5257 & 2.5171 & 2.5061 \\  & m = 2 & 10.3316 & 10.3112 & 10.2509 & 10.1527 & 10.0199 & 9.8564 \\  & m = 3 & 22.8853 & 22.7843 & 22.4891 & 22.0217 & 21.4138 & 20.7012 \\ \(\frac{\pi}{4}\) & m = 1 & 2.4064 & 2.4053 & 2.4017 & 2.3958 & 2.3877 & 2.3773 \\  & m = 2 & 10.1994 & 10.1793 & 10.1197 & 10.0228 & 9.8916 & 9.7302 \\  & m = 3 & 22.7573 & 22.6568 & 22.3632 & 21.8983 & 21.2936 & 20.5849 \\ \(\frac{\pi}{3}\) & m = 1 & 2.2318 & 2.2307 & 2.2274 & 2.2219 & 2.2143 & 2.2047 \\  & m = 2 & 10.10164 & 9.9967 & 9.9382 & 9.8429 & 9.7140 & 9.5555 \\  & m = 3 & 22.5790 & 22.4793 & 22.1878 & 21.7264 & 21.1261 & 20.4227 \\ \(\frac{5\pi}{12}\) & m = 1 & 2.0194 & 2.0184 & 2.0155 & 2.0105 & 2.0037 & 1.9949 \\  & m = 2 & 9.7847 & 9.7654 & 9.7082 & 9.6151 & 9.4891 & 9.3342 \\  & m = 3 & 22.3512 & 22.2524 & 21.9638 & 21.5067 & 20.9122 & 20.2157 \\ \(\frac{\pi}{2}\) & m = 1 & 1.7765 & 1.7756 & 1.7730 & 1.7686 & 1.7626 & 1.7549 \\  & m = 2 & 9.5067 & 9.4879 & 9.4323 & 9.3418 & 9.2193 & 9.0687 \\  & m = 3 & 22.0750 & 21.9774 & 21.6921 & 21.2404 & 20.6529 & 19.9646 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Variation of the first three natural frequencies (GHz) with the opening angle and nonlocal parameter.