## 4 Numerical Results
The computed frequency-amplitude plot is compared with those of [PERSON] and [PERSON] [51] in Figure 7b, where two reference solutions were numerically obtained by applying the generalized DQM to FSDT and HSDT. The comparison reveals that the plot of the present method is positioned between HSDT and FSDT, but it is closer to FSDT because the present method is based on FSDT. Next, the sensitivity of the present phase-field crack model to the grid density was examined using the isotropic cylindrical panel taken from Table 1 in which an inclined central crack is included. The relative length \(a/S\) and the inclination angle \(\alpha\) of a crack were chosen as 0.7 and 45\({}^{\circ}\). The non-dimensional first frequency \(\hat{\omega}_{1}\) and its linear-nonlinear ratio \(\omega_{NL}/\omega_{L}\) at \(w_{\max}/h=1.0\) were computed for five grid densities and recorded in Table 3. It is clearly observed that the relative difference of \(\hat{\omega}_{1}\) uniformly decreases proportional to the grid density. Also, the relative fluctuation in \(\omega_{NL}/\omega_{L}\) becomes smaller in proportion to the grid density. The present method was also applied to non-porous FG-GPLRC cylindrical panels with a central crack to examine the influence of crack angle and length, and GPL distribution pattern on the fundamental frequency. The geometric dimensions and the aluminum material properties are the same as those of the first example given in Table 1, where the first frequencies are calibrated as \(\hat{\omega}_{1}=\omega_{1}S^{2}\sqrt{\rho_{m}/E_{m}}\). Meanwhile, the geometric dimensions and the material properties of GPLs are as follows: \(E_{GPL}=1.01\) TPa, \(\
u_{GPL}=0.186\), \(l_{GPL}=2.5\,\upmu\)m, \(w_{GPL}=t_{GPL}=1.5\,\upmu\)m, and \(\rho_{GPL}=10600\,\text{kg}/\text{m}^{3}\), respectively. It is observed from Table 4 that the non-dimensional fundamental frequency decreases proportionally to the crack length because the panel stiffness decreases as the crack length increases. Meanwhile, regarding the GPL distribution pattern, FG-X leads to the highest level while FG-O shows the lowest level. This relative order among the GPL distribution patterns is attributed to the fact that the panel stiffness becomes higher as GPLs become biased towards the top and bottom of the panel; see Figure 1b. Next, the nonlinear natural vibration of the clamped FG-GPLRC cylindrical panel with a central crack was profoundly examined by changing the major parameters. Figure 8a represents the effect of crack inclination angle on the variation in nonlinear-to-linear frequency ratio \(\omega_{NL}/\omega_{L}\) to the amplitude-thickness ratio \(w_{c}^{max}/h\). The mass fraction \(g_{GPL}^{*}\) and distribution pattern of GPLs are 0.4% and FG-U, and the relative crack length \(a/S\) is set at 0.3. It is seen that the frequency ratio \(\omega_{NL}/\omega_{L}\) becomes uniformly smaller in proportion to the crack inclination angle, but the decrease trend becomes smaller proportional to the crack angle. Thus, this trend reveals that the nonlinearity in free vibration decreases with increasing crack inclination angle. Figure 8b represents the effect of crack relative length on the nonlinear natural vibration of the FG-U GPLRC cylindrical panel when the crack inclination angle \(\alpha\) is zero. It is seen that the frequency ratio \(\omega_{NL}/\omega_{L}\) becomes slightly smaller with increasing crack relative length because the panel stiffness drops proportional to the crack relative length.