## 5 Numerical Results
The difference in the linear CBTRs between distribution patterns at \(w_{c}/h=0\) is continuously maintained up to \(w_{c}/h=1\). This is because the GPL distribution pattern affects plate buckling stiffness, and this relative difference does not change along the postbuckling path. The results show that the thermal buckling behavior of FG-GPLRC plates is significantly affected by the total amount and distribution pattern of the GPL. Figure 11a shows the influence of the length ratio \(b/a\) of two sides of the plate on the postbuckling path of the clamped FG-U cracked plate with \(g_{GPI}^{*}=0.3,c/a=0.3\) and  
Figure 10: Variation in thermal postbuckling paths (\(c/a=0.3\), \(\alpha=30^{\lx@math@degree}\), CCCC): (**a**) The GPL mass fraction \(g_{GPI}^{*}\) (FG-O); (**b**) The GPL distribution pattern (\(g_{GPI}^{*}=0.2\)%). Figure 9: Variation in thermal postbuckling paths (FG-U, \(g_{GPI}^{*}=0.3\)%, \(a/h=25\), CCCC): (**a**) The crack relative length \(c/a\) (\(\alpha=0^{\lx@math@degree}\)); (**b**) The crack angle (\(c/a=0.3\)). \(\alpha=45^{\circ}\). The value of \(b/a\) is set by changing the side length of \(b\) with the side length of \(a\), which is kept at 1.0 m. It can be seen that both the CBTR and the postbuckling path decreases as the length ratio \(b/a\) increases, but the decreasing slope to \(b/a\) becomes insensitive as the value of \(b/a\) becomes larger. This is because the plate buckling stiffness decreases as the size of the plate becomes larger in proportion to \(b/a\). Moreover, the relative difference in the CBTRs between the values of \(b/a\) is shown to be slightly increased. Figure 11b shows the influence of the boundary conditions on the CBTR and postbuckling path. The highest and lowest levels can be seen in the CCCC and SSSS (FIMM), respectively, and the intermediate level is shown at CSCS. This is because the clamped condition is stronger in the displacement constraint than in the simply-supported condition; therefore, the clamped condition produces a higher CBTR because the plate buckling stiffness becomes larger as the boundary condition becomes stronger. Meanwhile, the case of SSSS(PIMM) exhibits a higher level than SSSS(FIMM) owing to the above-mentioned even expansion of thermally induced deformation in almost any direction in the square plate. Its CBTR is similar to that of CSCS at \(w_{c}/h=0\), but approaches the case of CCCC as \(w_{c}/h\) becomes unity. The results imply that the thermal buckling strength is significantly dependent on the plate aspect ratio and the boundary constraint. Finally, thermal buckling and postbuckling of cracked square plates between GPLRC and CNTRC were compared without changing the underlying matrix material. (10,10) Single-walled CNTs [60] have orthogonal anisotropic material properties given in Table 5 and the thermal expansion coefficients \(\alpha_{11}=3.4584\times 10^{-6}/\)K and \(\alpha_{22}=5.1682\times 10^{-6}/\)K were used. The effective Young's and shear moduli of CNTRC plates were evaluated as \((1,2,3=x,y,z)\)  
\[E_{1}=\eta_{1}V_{cnt}E_{1}^{cnt}+V_{m}E_{m},\ \ \frac{\eta_{2}}{E_{2}}=\frac{V_{ cnt}}{E_{2}^{cnt}}+\frac{V_{m}}{E_{m}} \tag{49}\]  
\[\frac{\eta_{3}}{G_{12}}=\frac{V_{cnt}}{G_{12}^{cnt}}+\frac{V_{m}}{G_{m}} \tag{50}\]  
by introducing the CNT efficiency parameters \(\eta_{i}(i=1,2,3)\), which depend on the total CNT volume fraction \(V_{cnt}^{*}\)[61]. The effective Poisson ratios and thermal expansion coefficients are evaluated using the simplest linear rule of mixtures [62]. Figure 11: Variation in thermal postbuckling paths (FG-U, \(g_{GPL}^{*}=0.3,\ c/a=0.3,\ \alpha=45^{\circ}\)): **(a)** The aspect ratio \(b/a\) (CCCC); **(b)** The boundary condition. Figure 12a represents the variation in the CBTR of a clamped FG-U CNTRC plate with respect to the crack angle for different CNT alignments, where the highest level appears at \(\alpha_{cnt}=0^{\circ}\) and \(90^{\circ}\), the lowest level at \(\alpha_{cnt}=45^{\circ}\), and the intermediate level at \(\alpha_{cnt}=30^{\circ}\) and \(60^{\circ}\).