## 6 Conclusions  
The thermal buckling and postbuckling of an FG-GPLRC plate with a central crack were numerically examined by developing an effective and reliable numerical method. The thermally induced large deformation was expressed by first-order SDT and von Karman geometric nonlinearity, and the arbitrary inclined crack was represented by making use of phase-field formulation and the virtual rotation of the crack. The singular displacement field was approximated by combining the regular L/I functions and the crack-tip singular  
Figure 12: CNTRC (FG-U, \(c/a=0.3\), CCCC): (**a**) Critical buckling temperature rise \(\Delta T_{cr}(K)\) (\(V_{cnt}^{*}=0.12\)); (**b**) Comparison of postbuckling paths with GLRC (\(\alpha=0^{\circ},\alpha_{cnt}=90^{\circ}\)). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Young’s Modulus (GPa)**} & \multicolumn{3}{c}{**Poisson’s Ratio**} & \multicolumn{3}{c}{**Shear Modulus (GPa)**} \\ \hline \(E_{1}^{cnt}\) & \(E_{2}^{cnt}\) & \(E_{3}^{cnt}\) & \(\
u_{12}^{cnt}\) & \(\
u_{23}^{cnt}\) & \(\
u_{31}^{cnt}\) & \(G_{12}^{cnt}\) & \(G_{23}^{cnt}\) & \(G_{31}^{cnt}\) \\ \hline
5646.6 & 7080.0 & - & 0.175 & - & - & 1944.5 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Elastic properties of (10,10) single-walled CNTs. functions in the framework of 2D XNEM. The developed numerical method was validated through benchmark testing, and the thermal buckling and postbuckling behavior of cracked FG-GPLRC plates with respect to the relevant parameters were investigated in depth. The numerical experiments led to the following major observations:  
* The PFF- and XNEM-based numerical method provides stable convergence and reasonable agreement with the reference solution for the thermal buckling problem, requiring two separate numerical analyses in sequence, with a peak relative difference of 10.619%. * CBTR decreases uniformly in inverse proportion to plate thickness, whereas the variation with respect to the GPL mass fraction \(g^{\star}_{GPL}\) is strongly affected by GPL distribution patterns, but the larger the value of \(g^{\star}_{GPL}\), the less pronounced the variation becomes. * The CBTR gradually decreases with the increase in the relative crack length \(c/a\) but becomes saturated as the value of \(c/a\) becomes sufficiently large. The effect of the crack angle on the CBTR and postbuckling path is not remarkable. * The magnitude of the CBTR in linear thermal buckling and postbuckling is relatively ordered as follows: FG-X > FG-U > FG-\(\Lambda\) > FG-O. However, it decreases proportionally to the length ratio \(b/a\), but the decreased slope to \(b/a\) becomes insensitive as \(b/a\) becomes sufficiently large. * The CBTR becomes larger as the boundary condition becomes stronger so that its magnitude is relatively ordered as follows: CCCC > CSCS > SSSS(FIMM). Meanwhile, the SSSS(PIMM) is similar to CSCS at \(w_{c}/h=0\) but approaches the case of CCCC as \(w_{c}/h\) becomes unity. * The CBTR in the CNTRC plate is smaller than that of the GPLRC plate, implying that the buckling stiffness of GPLRC is stronger than CNTRC for the same volume fraction. In addition, the buckling stiffness of CNTRC is remarkably affected by the crack inclination angle, which differs from that of GPLRC. The developed numerical method can be extended to predict the CBTR and the postbuckling path of cracked FG-GPLRC shell-like structures. And, the numerical results can be usefully used as a guideline for choosing the suitable values of the parameters associated with the FG-GPLRC plate. Further research can further expand the scope of the phase-field model and analyze the crack growth behavior of FG-GPLRC plates, together with the comparison with possible experimental data, which is of great significance for improving the service life of composite materials under extreme working conditions. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2023-00240618). Not applicable.