## 5 Numerical Results  
The stiffness matrices and load vector introduced in the previous section were numerically integrated by applying seven Gauss points to each triangle and dividing the thickness into 50 uniform segments for the trapezoidal rule. The uniform temperature increment \(\Delta T\) and the convergence tolerance \(\delta\) in Equation (46) are set by 10 and \(1.0\times 10^{-3}\), respectively, while the shear stabilization parameter \(\vartheta\) and the iteration compensation factor are set by 0.1 and 1.0 unless otherwise stated. A square plate with an inclined central crack is considered, where the sides \(a\) and \(b\) and the thickness \(h\) are set by 1.0 m and 0.04 m. Three kinds of boundary conditions (BCs), i.e., clamped (C), simply-supported (S), and immovable simply-supported (IMM S) boundary conditions, are used. The clamped boundary condition is implemented as  
\[\text{C}:\;u_{0}=v_{0}=\omega_{0}=\vartheta_{x}=\vartheta_{y}=0 \tag{46}\]  
while two simply-supported BCs are enforced as  
\[\text{S}:\;v_{0}=\omega_{0}=\vartheta_{y}=0 \tag{47}\]  
\[\text{IMM S}:\;u_{0}=v_{0}=\omega_{0}=\vartheta_{y}=0 \tag{48}\]  
at \(\theta=0\) or \(\theta_{0}\). The components \(v_{0}\) and \(\vartheta_{y}\) in Equation (47) and \(\vartheta_{y}\) in Equation (48) are replaced with \(u_{0}\)\(\vartheta_{x}\) and \(\vartheta_{x}\) for the sides at \(y=0\) or \(L\). The immovable simply-supported BC is classified into the fully immovable (FIMM S) or the partially immovable (PIMM S) BC, depending on whether all sides are immovable or only the opposite two sides are immovable. First, the convergence of the critical buckling temperature parameter (CBTP) \(\lambda_{T}=10^{3}\alpha\,\Delta T_{cr}\) to the NEM grid density is tested using an intact isotropic plate with the following material properties: \(E=70\) GPa, \(\
u=0.3\) and \(\alpha=23\times 10^{-6}/\text{K}\). The fully immovable simply-supported condition (FIMM S) is enforced at all sides of the plate. As recorded in Table 1, the CBTPs \(\lambda_{T}\) were computed by increasing the NEM grid density from \(11\times 11\) to \(23\times 23\), and the relative differences \(\lambda_{T}^{\text{rel}}\) were calculated as \(|\lambda_{T}-\lambda_{T}|/\lambda_{T}\times 100\%\)  
Figure 5: A flowchart for thermal buckling and postbucking analysis of the cracked FG-PLRC plate. with the \(\hat{\lambda}_{T}\) at a grid density \(23\times 23\). The value of \(\lambda_{T}\) shows a convergence with the uniform decrease in the relative difference \(\lambda_{T}^{rel}\) such that the grid density of \(21\times 21\) produces a relative difference of less than \(3.0\%\). Hence, this grid density was selected for all of the numerical experiments in this study. The critical temperature parameters \(\lambda_{T}\) were also computed for three different thickness ratios \(a/h=10\), \(20\), and \(100\), and compared with the reference solutions, as shown in Table 2. [PERSON] et al. [19] applied the DQM to the first-order SDT, while [PERSON] et al. [52] and [PERSON] et al. [53] applied the local Kriging meshless and element-free \(kp\)-Ritz methods to the first-order SDT, respectively. It turns out that this method agrees well with the three reference methods, with the peak relative difference equal to \(-1.775\%\) from [PERSON] et al. [53] at \(a/h=10\). Next, the thermal postbuckling path was computed with an intact isotropic plate under a uniform temperature rise that is partially immovable and simply-supported. The nonlinear Equation (38) was solved by increasing the calibrated central deflection \(w_{\text{max}}/h\) from zero to unity with a uniform increment of \(0.2\). The geometric dimensions of the plate are \(a=b=0.2\) m and \(a/h=100\), and the elastic properties are \(E=1.0\,\text{MPa}\), \(\
u=0.15\) and \(\alpha=1.0\times 10^{-6}/\text{K}\), respectively.