## 4 Numerical Results and Discussion
### Comparison and Validation  
As first example, we assume a S-S FG curved nanobeam of width B=1 nm, length L = 10 nm, made of Al and SiC material with elastic moduli E\({}_{\text{Al}}\) = 70 GPa, E\({}_{\text{SiC}}\) = 427 GPa, [PERSON]'s ratios \(\
u_{\text{Al}}\) = 0.3, \(\
u_{\text{SiC}}\) = 0.17, densities \(\rho\) = 2702 \(\frac{kg}{m^{3}}\), \(\rho\) = 3100 \(\frac{kg}{m^{3}}\), respectively. The material and geometry of the nanostructure are here selected according to the reference work by [PERSON] and [PERSON] [17] for comparative purposes. Table 2 summarizes the main results of the first three natural frequencies for a varying dimensionless length-to-thickness ratio \(\frac{L}{h}\) between 10 and 50, as well as for different values of the nonlocal parameter \(\mu^{2}\), while keeping constant n = 1 and \(\alpha\;=\;\pi/2\). Based on Table 2, it is worth noticing a gradual decrease of each natural frequency for an increasing value of the nonlocal parameter, together with a gradual increase for an increased length-to-thickness ratio \(\frac{L}{h}\), in line with predictions by [PERSON] and [PERSON] [17]. The very good agreement between our results and predictions from the literature clearly confirms the reliability and accuracy of the proposed method to capture the vibration response of the problem. A further study compares the first natural frequency of the curved nanobeam for two different boundary conditions, namely S-S, and C-C, as well as for two different length-to-thickness ratios \(\frac{L}{h}\) (i.e., \(\frac{L}{h}\;=\;10,\;20\)), two different opening angles (i.e., \(\theta_{0}\;=\;\pi/3,\;\pi/6\)), and different nonlocalities. The outcomes in Table 3 clearly show that the natural frequency increases for an increased length-to-thickness ratio, \(\frac{L}{h}\), and an increased opening angle, \(\theta\), whereas it decreases for an increased nonlocal parameter \(\mu\). As also expected, a C-C boundary condition enables higher values of the natural frequency, compared to a S-S boundary condition, under the same geometrical and non-local assumptions, \(\frac{L}{h}\), \(\theta\), and \(\mu\). The satisfactory agreement between our results and predictions in a previous study [57] confirms, once again, the capability of the proposed method for correctly reproducing the vibration response of composite nanobeams. Furthermore, we continue the investigation by analyzing the possible effect of some geometrical and mechanical parameters of the nanobeam on its global response, namely, the opening angle \(\theta_{0}\), the weight fraction \(g^{*}_{GPL}\), the total number of layers \(N_{L}\), the nonlocal parameter \(\mu\), the dimensionless length-to-thickness ratio of the GPL, \(\frac{L_{GPL}}{R_{GPL}}\), and the foundation parameters, \(k_{1},\ k_{2}\). All the computations are performed for a polymer matrix with elastic properties \(E_{matrix}=1.01\ TPA\), \(v_{matrix}=0.186\), and density \(\rho_{matrix}=1062.5\ kg/m^{3}\), and a dispersed GPL reinforcement with the following properties \(E_{GPL}=3\ GPa\), \(v_{GPL}=0.34\), \(\rho_{GPL}=1200\ kg/m^{3}\). In what follows, we consider a nanobeam of thickness \(h=5\ nm\) and length \(L=20\hbar\), whereas the GPL reinforcement has length \(l_{GPL}=2.5\ nm\), thickness \(h_{GPL}=0.15\ nm\), and weight \(w_{GPL}=1.5\ nm\). Table 4 collects the results in terms of the first natural frequency for increasing values of the nonlocal parameter \(\mu\), and weight fraction \(g^{*}_{GPL}\), as well as for different boundary conditions (C-C, C-S, and S-S, respectively), while keeping constant the total number of layers \(N_{L}=12\), the opening angle \(\theta_{0}=\frac{\pi}{6}\) Rad, and the foundation parameters \(k_{1}=k_{2}=0\). Based on the results listed in Table 4, a clear increase in the natural frequency is shown for increasing weight fractions, \(g^{*}_{GPL}\), and fixed values of \(\mu\) and boundary condition, due to an expectable global increase in stiffness of the reinforced material.