## 4 Results and Discussion  
In order to verify the correctness of the derivation in this paper, we chose an SS-SS isotropic homogeneous shell to make a comparison without considering the small-scale effect (\(l=0\)). The system parameters used are as follows: \(\rho=2300\) kg/m\({}^{3}\), \(\
u=0.3\), \(E=1.06\) TPa, \(R=2.32\) nm, \(L/R=5\). As can be seen in Table 3, very good agreement was achieved. Considering the scale effect, dimensionless natural frequencies of an isotropic homogeneous cylindrical nanoshell were calculated, and are compared with the literature in Table 4. Here the parameters used are the same as those in Table 3. It was found that the present results match those given by [PERSON] et al. [42] well, bespeaking the validity of the present study. To make further comparisons, we considered a homogeneous cylindrical shell with the C-C boundary condition. The comparison result is listed in Table 5. It can be seen that the present results agree well with those in the literature [64]. Hereinafter, we conducted free vibration and buckling analyses of 3D-GrF cylindrical microshells. If not specified, the geometrical and material properties of the 3D-GrF microshell are:  
\[ \begin{array}{l}E_{1}=1.02\text{ TPa},\ \ \rho_{1}=2300\text{ kg/m}^{3},\ \ \
u=0.3\\ l=15\ \upmu\text{m},\ \ m=1,\ \ h=15\ \upmu\text{m},\ \ R/h=40,\ \ L/R=2 \end{array} \] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(h/R\) & \((m,n)\) & Present & [PERSON] et al. [42] \\ \hline \multirow{3}{*}{0.02} & (1,1) & 0.1955 & 0.1955 \\  & (2,2) & 0.2575 & 0.2575 \\  & (3,3) & 0.3067 & 0.3067 \\ \hline \multirow{3}{*}{0.05} & (1,1) & 0.1963 & 0.1963 \\  & (2,2) & 0.2869 & 0.2869 \\ \cline{1-1}  & (3,3) & 0.4586 & 0.4586 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of dimensionless natural frequency \(\Omega\) of an SS–SS isotropic homogeneous cylindrical nanoshell (\(\kappa_{0}=0\), \(l=h\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(h/R\) & \((m,n)\) & [PERSON] & [PERSON] et al. [42] & [PERSON] and [PERSON] [63] \\ \hline \multirow{3}{*}{0.02} & (1,1) & 0.1954 & 0.1954 & 0.1968 \\  & (2,2) & 0.2532 & 0.2532 & 0.2563 \\  & (3,3) & 0.2772 & 0.2772 & 0.2773 \\ \hline \multirow{3}{*}{0.05} & (1,1) & 0.1959 & 0.1959 & 0.2004 \\  & (2,2) & 0.2623 & 0.2623 & 0.2633 \\ \cline{1-1}  & (3,3) & 0.3220 & 0.3220 & 0.3158 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of dimensionless natural frequency \(\Omega(\Omega=\omega R\sqrt{\rho/E})\) of a simply supported–simply supported (SS–SS) isotropic homogeneous cylindrical shell (\(\kappa_{0}=0\), \(l=0\)). Based on the modified couple stress theory and classical continuum theory, Figure 6 shows the natural frequency versus dimensionless length scale parameter for the SS-SS 3D-GrF microshell, where \(n=3\) and \(\kappa_{0}=0.2\). One can see that the natural frequency from the modified couple stress theory increased while that from the classical theory did not change with length scale parameter. This is because size effect tends to increase the stiffness of the 3D-GrF microshell. Nevertheless, the classical shell theory fails to incorporate this effect and thus produces inaccurate results. Figure 5: Variation of natural frequency \(\omega\) (MHz) against circumferential wave number \(n\) of 3D-GrF microshell (3D-GrF-U). Figure 6: Natural frequency \(\omega\) (MHz) versus dimensionless length scale parameter for SS-SS 3D-GrF microshell: (**a**) 3D-GrF-I; (**b**) 3D-GrF-II; (**c**) 3D-GrF-U. Figure 7 gives the variation of natural frequency against the foam coefficient for different foam distributions. The results show that the increase in the foam coefficient led to a decrease in the natural frequencies of 3D-GrF cylindrical microshells. When the foam coefficient was small, the 3D-GrF-I microshell had the highest natural frequency while the 3D-GrF-II one had the lowest natural frequency. With the increase in the foam coefficient, however, the natural frequency of the 3D-GrF-U microshell dropped faster than that of the 3D-GrF-II one.