## 4 Numerical Results  
The nonlinear free vibration formulae given in Section 3 for cracked porous cylindrical panels using the phase field theory and the crack rotation concept were coded in the framework of 2-D XNEM [19]. All the stiffness and mass matrices given in Equations (38)-(41) were numerically integrated using 7 Gauss integration points. First, the sensitivity of the present method to the density of 2-D NEM grid was investigated using a clamped intact (i.e., without an internal crack) aluminum cylindrical panel with the geometry dimensions of \(L/R=2\), \(S/L=1\), \(R/h=20\). In this study, two kinds of boundary conditions, simply supported (S) and clamped (C), were used, and which were implemented as  
\[\text{S}:\;v_{0}=\vartheta_{y}=0 \tag{43}\]  
\[\text{C}:\;u_{0}=v_{0}=\omega_{0}=\vartheta_{x}=\vartheta_{y}=0 \tag{44}\]  
at \(\theta=0\) or \(\theta_{0}\). The component \(v_{0}\) in Equation (44) was excluded when the clamped condition was specified for the side with \(y=0\) or \(L\). The isotropic material properties were \(E_{m}=70\) GPa, \(v_{m}=0.3\) and \(\rho_{m}=2707\) kg/m\({}^{3}\), and the first natural frequency was calibrated as \(\hat{\omega}_{1}=\omega_{1}L^{2}\sqrt{\rho_{m}/D_{m}}\) with the flexural rigidity \(D_{m}=E_{m}h^{3}/12(1-v_{m}^{2})\). The non-dimensional first frequencies \(\hat{\omega}_{1}\) were computed and are presented in Table 1, where \(\hat{\omega}_{1}^{rel}\) indicates the relative percentage difference with respect to the fundamental frequency computed with a grid density of \(25\times 25\). It was found that the relative difference uniformly decreases in proportion to the grid density such that it becomes less than 3.0% when the grid density goes up to \(21\times 21\). So, according to this convergence result, the grid density was set to \(21\times 21\) for the whole numerical experiment in this paper. Next, the accuracy of the present method was verified by comparing it with the phase field formulation of [PERSON] and [PERSON] [36]. A metal-ceramic functionally graded cylindrical panel with an internal crack, shown in Figure 3, is taken. The geometric dimensions and the boundary condition are the same as those from the above convergence test problem, and also, the previous aluminum is taken for the metal while alumina with \(E_{c}=380\) GPa and \(v_{c}=0.26\) is chosen for the ceramic. The power-law function of \(V_{c}(z)=(0.5-z/h)^{r}\) is adopted to identify the thickness-wise volume fractions of the ceramic and also the metal using the relation \(V_{m}(z)=1-V_{c}(z)\). The relative crack length \(a/S\) is set at 0.3, while the crack angle and the ceramic power-law index are taken as variables. By comparing with the reference solutions, the shear stabilization factor \(\vartheta\) in Equation (42) was set at \(\vartheta=0.3\times[1-0.1\times(1-\alpha/90)]\) to reflect the influence of crack angle on the stiffness matrix \(\mathbf{K}_{L,\sigma}\) in Equation (38). The phase field \(\phi(\mathbf{x})\) in the mass matrix \(\mathbf{M}\) in Equation (41) was set to 0 in order to prevent the over-reduction of mass due to an internal crack in the coarse NEM grid. The comparison in Table 2 reveals that the present results show good agreement with the reference solutions, with the maximum relative difference equal to 4.838%. Except for the case of \(\alpha=0\), all the present results lead to the relative differences being less than 1.0%. Meanwhile, the fundamental frequency uniformly decreases proportional to the ceramic power-law index \(r\) because the relative region occupied by a stiffer ceramic reduces in proportion to the value of \(r\). It is found that the first frequency uniformly increases with increasing crack inclination angle because the reduction in the circumferential stiffness of a panel shows that the circumferentially dominated free vibration becomes smaller as the crack inclination angle increases.