## 4 Results and Discussion
At last, the 3D-GrF-U microshell had the lowest natural frequency among the three types of foam distribution, except in the C-F boundary condition. The variations of natural frequency against the length-to-radius ratio for different foam distributions are illustrated in Figures 8-10, where \(n=3\) and \(\kappa_{0}=0.2\). It was found that the natural  
Figure 7: Variation of natural frequency \(\omega\) (MHz) against foam coefficient \(\kappa_{0}\) under various boundary conditions (\(n=3\)): **(a)** C–C; **(b)** C–F; **(c)** C–SS; **(d)** F–F; **(e)** F–SS; **(f)** SS–SS. Tables 6-8 show the natural frequency versus radius-to-thickness ratio for different foam distributions, where \(n=3\) and \(\kappa_{0}=0.2\). It is clear that natural frequencies decreased as the radius-to-thickness ratio rose for all the boundary conditions. Table 9 lists the variation of buckling load against circumferential wave number for the SS-SS 3D-GrF microshell, where \(L/R=3\) and \(\kappa_{0}=0.2\). Results show the buckling loads initially decreased and then increased with the circumferential wave number. It can be seen that the critical buckling load \(P_{\rm cr}\) occurred at \(n=3\). Therefore, mode (\(m=1\), \(n=3\)) is chosen as a representative mode in the next buckling studies. The buckling load against radius-to-thickness ratio for the SS-SS 3D-GrF microshell is listed in Table 10, where different foam distributions are taken into account. It is shown that the larger radius-to-thickness ratio led to the lower buckling load. One can also see that the buckling loads evaluated by the modified couple stress theory were higher than those evaluated by the classical theory, showing that the classical theory underestimated the buckling loads of 3D-GrF microshells. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(R\)/\(h\) & SS–SS & C–SS & C–C & C–F & F–F & F-SS \\ \hline
20 & 4.237 & 4.818  
Figure 11 examines the critical buckling load against the foam coefficient for SS-SS 3D-GrF microshells with different foam distributions. One can see that the higher foam coefficient led to the lower critical buckling load. Additionally, the 3D-GrF-I microshell had the highest critical buckling load. This shows that compact inner and outer surfaces and sparse mid-plane can lead to higher structural stiffness of 3D-GrF microshells. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\boldsymbol{n}\)} & \multicolumn{2}{c}{**3D-GrF-I**} & \multicolumn{2}{c}{**3D-GrF-II**} & \multicolumn{2}{c}{**3D-GrF-U**} \\ \cline{2-6}  & **CT** & **MCST** & **CT** & **MCST** & **CT** & **MCST** \\ \hline
30 & 63.509 & 109.735 & 61.314 & 107.690 & 62.528 & 108.682 \\
40 & 42.734 & 62.244 & 41.860 & 61.433 & 42.296 & 61.775 \\
50 & 32.374 & 42.365 & 31.961 & 41.984 & 32.133 & 42.109 \\
60 & 26.157 & 31.940 & 25.942 & 31.744 & 26.007 & 31.780 \\
70 & 21.996 & 25.638 & 21.879 & 25.532 & 21.893 & 25.529 \\
80 & 19.006 & 21.446 & 18.941 & 21.389 & 18.930 & 21.366 \\
90 & 16.747 & 18.461 & 16.713 & 18.432 & 16.689 & 18.400 \\
100 & 14.978 & 16.227 & 14.962 & 16.215 & 14.931 & 16.178 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Buckling load \(P\) (\(10^{6}\) N) against radius-to-thickness ratio \(R/h\) for SS-SS 3D-GrF microshell (\(n=3\), \(x_{0}=0.2\)). Figure 11: Critical buckling load \(P_{\rm cr}\) against foam coefficient \(x_{0}\) for SS-SS 3D-GrF microshell (\(n=3\)). Figure 12 depicts the buckling load against length-to-radius ratio for SS-SS 3D-GrF microshells with different foam distributions. With the increase of length-to-radius ratio, buckling load decreased initially and then increased. The lowest value of buckling loads occurred at nearby \(L/R=3.5\) for all types of foam distribution.