## 5 Numerical Results
Next, the thermal buckling and postbuckling behaviors of the FG-GPLRC plate were parametrically investigated by including an inclined central crack. The key parameters were chosen. By the angle and length of the crack, the total mass and distribution pattern of the GPL, and the plate aspect ratio and the boundary condition. Figure 8a comparatively represents the variation in the CBTR of an FG-U GPLRC to the relative crack length for \(\alpha=0^{\lx@math@degree}\) when the GPL mass fraction \(g^{*}_{GPL}\) is 0.3%. The CBTR value has been shown to gradually decrease as the value of \(c/a\) becomes sufficiently large. This is because the structural stiffness decreases as the crack-induced damage becomes larger. Meanwhile, the value of the CBTR is highest at CCCC, lowest at SSSS, and intermediate at SCSC and CSCS. This is due to the fact that as the restraint of the plate becomes stronger, the buckling stiffness of the plate increases. The four-letter combination, C and S, represents the boundary conditions specified on the four sides (1), (2), (3), and (4) of the plate (see Figure 3). Figure 8b comparatively represents the variation in the CBTR with respect to the crack angle \(\alpha\) when the relative crack length \(c/a\) is 0.3. It can be observed that the variation in the CBTR is not remarkable because the thermally induced deformation develops evenly in almost any direction in the square plate. The detailed numerical results of CCCC and SSSS (FIMM) reveal that the CBTR slightly decreases until \(\alpha=45^{\lx@math@degree}\) and then increases up to \(\alpha=90^{\lx@math@degree}\). However, CSCS and SCSC show continuously decreasing and increasing variations owing to the non-symmetric boundary condition with respect to the \(45^{\lx@math@degree}\) diagonal line. The thermal postbuckling paths were computed by successively increasing the relative central deflection \(\overline{w}_{c}=w_{c}/h\) from 0 to 1.0 with a uniform increment \(\Delta\overline{w}_{c}=0.2\). In order to compensate for the magnitude change in the nonlinear stiffness matrix \(\mathbf{K}_{NL}\) during the nonlinear computation, the compensation factor \(\sigma\) in Equation (38) was introduced as a form of \(\sigma=\left(1+a_{1}\overline{w}_{c}^{2}\right)\). Two constants, \(a_{1}\) and \(a_{2}\), which are dependent on the GPL pattern, were determined through the preliminary experiment by referring to the following reference solutions: 6.0 and 2 for FG-U, 4.0 and 2 for FG-O, 5.75 and 2 for FG-\(\Lambda\), and 25.0 and 4 for FG-X, respectively. The simulation parameters were set by \(a=b=1.0\), \(a/h=25\), \(c/a=0.3\), and \(g^{*}_{GPL}=0.3\%\), unless otherwise stated. Figure 9a comparatively represents the paths of the clamped FG-U plates for different crack lengths \(c/a\), where intact indicates the case of \(c/a=0\). It can be clearly seen that the thermal postbuckling paths show a uniform increase with almost the same differences seen in the linear CBTRs between the crack lengths, except for the case of intact. Figure 9b shows the postbuckling paths for different crack angles when the relative crack length \(c/a\) is 0.3. The CBTR values uniformly increase without remarkable differences between the  
four crack angles, as for the linear CBTRs. The detailed numerical results indicate that the linear CBTR increases by 2.4 times when \(\overline{w}_{c}\) reaches unity in this case. Figure 10a comparatively represents the postbuckling paths for different GPL mass fractions when the crack relative length \(c/a\) and angle \(\alpha\) are 0.3 and \(30^{\lx@math@degree}\). Not only the CBTR but also the postbuckling path shows a uniform increase proportional to the GPL mass fraction. This is, of course, due to the fact that as the amount of GPL increases, plate buckling stiffness increases. Figure 10b shows the influence of the GPL distribution pattern on the postbuckling path when the GPL mss fraction \(g_{GPI}^{*}\) is 0.2%.