## 4 Numerical Results
The (10,10) single-walled CNTs [52] were taken, and their orthotropic material properties are presented in Table 5. The effective material properties of the CNTRC structures are evaluated as  
\[E_{1}=\eta_{1}f_{CNT}E_{1}^{CNT}+f_{m}E_{m},\ \ \frac{\eta_{2}}{E_{2}}=\frac{f_{CNT}}{E_{ 2}^{CNT}}+\frac{f_{m}}{E_{m}} \tag{45}\]  
\[\frac{\eta_{3}}{G_{12}}=\frac{f_{CNT}}{G_{12}^{CNT}}+\frac{f_{m}}{G_{m}},\ \ \
u_{12}=f_{CNT}\
u_{12}^{CNT}+f_{m}\
u_{m} \tag{46}\]  
according to the modified linear rule of mixtures (MLRMs), where the CNT efficiency parameters \(\eta_{i}(i=1,2,3)\) are dependent of the CNT total volume fraction \(V_{CNT}^{*}\)[53]. Note that LRM accounts only for matrix-fiber deformation but does not consider nonlinearities such as fiber-matrix adhesion that might consume a large chuck of applied energy. Figure 9(a) represents the variation in \(\omega_{NL}/\omega_{L}\) to the relative central amplitude \(w_{c}^{max}/h\) of a cracked FG-X cylindrical panel with \(a/S=0.3\) and \(\alpha=30^{\circ}\), where the angle \(\alpha_{CNT}=90^{\circ}\) denotes that CNTs are aligned in the direction of cylinder axis. The frequency ratio \(\omega_{NL}/\omega_{L}\) shows a noticeable change to the CNT volume fraction \(V_{CNT}^{*}\), even though the extent of increase is not so large when compared with that shown in Figure 8(a). Figure 9(b) represents the variation for different GPL volume fractions \(V_{GPL}^{*}\), where the ratio \(\omega_{NL}/\omega_{L}\) and its dependence on \(V_{GPL}^{*}\) are shown to be smaller than those shown in Figure 9(a). Note, from the relation in Equation (3), that \(V_{GPL}^{*}=0.12\) equals 10.75% of the GPL mass fraction. Therefore, it is found that the dependence of frequency ratio \(\omega_{NL}/\omega_{L}\) on \(V_{GPL}^{*}\) and \(V_{CNT}^{*}\) becomes insensitive in proportion to the amount of GPLs and CNTs, and this trend is more apparent at CNTRC for the same volume fraction. This is because the nonlinearity intensity becomes saturated as the structural stiffness reaches the critical value owing to the increase in CNTs and GPLs, as will be seen later. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multicolumn{2}{c}{**Elastic Moduli (GPa)**} & \multicolumn{3}{c}{**Poisson’s Ratios**} & \multicolumn{3}{c}{**Shear Moduli (GPa)**} \\ \hline \(E_{1}^{cut}\) & \(E_{2}^{cut}\) & \(E_{3}^{cut}\) & \(\
u_{12}^{cnt}\) & \(\
u_{23}^{cnt}\) & \(\
u_{31}^{cnt}\) & \(G_{12}^{cnt}\) & \(G_{23}^{cnt}\) & \(G_{31}^{cnt}\) \\ \hline
5646.6 & 7080.0 & - & 0.175 & - & - & 1944.5 & - & - \\ \hline \end{tabular}
\end{table} 
Table 5: Material properties of (10,10) single-walled CNTs (\(1,2,3=x,y,z\)). Figure 9: Variation in nonlinear-to-linear frequency ratio \(\omega_{NL}/\omega_{L}\) (\(a/S=0.3\), \(\alpha=30^{\circ}\)) (**a**) to the GPL mass fraction \(g_{GPL}^{*}\) (FG-O) and (**b**) to the GPL distribution pattern (\(g_{GPL}^{*}=0.4\)%). Before the examination of the effect of higher \(V_{GPL}^{*}\) and \(V_{CNT}^{*}\) on the variation in \(\omega_{NL}/\omega_{L}\), the fundamental frequencies of GPLRC and CNTRC are compared for the FG-X cylindrical panel with \(a/S=0.3\) and \(\alpha=30^{\circ}\). From Table 6, it is found that the first frequencies of GPLRC are much higher than those of CNTRC, implying that GPLRC possesses much higher structural stiffness than CNTRC for the same volume fraction. This is because the material properties of CNTRC are axis-dependent such that the two lateral elastic moduli are much lower than the axial one. It is clearly found that the non-dimensional first frequency of CNTRC becomes higher when the CNTs are aligned in the circumferential direction (i.e., \(\alpha_{CNT}=0^{0}\)). Figure 11a represents the variation in \(\omega_{NL}/\omega_{L}\) with respect to the GPL mass fraction \(g_{GPL}^{*}\) for different GPL distribution patterns when \(w_{C}^{max}/h\) is 1.0.