# Closed Form Solutions for Thermal Buckling of Functionally Graded Rectangular Thin Plates
## 4 Numerical Results and Discussion  
Two types of FGM thin plates are considered in this section. Type-I FGM plate is temperature-dependent; it is made from stainless steel (SUS304) and Silicon nitride (Si\({}_{3}\)N\({}_{4}\)). The material properties are shown in Table 4. Type-II FGM plate is temperature-independent; it is made from aluminum as its bottom surface and aluminum oxide as its top surface. \(E_{\rm t}=70\) GPa, \(\alpha_{\rm t}=23\times 10^{-6}/\mathrm{K}\), \(E_{\rm b}=380\) GPa, \(\alpha_{\rm b}=7.4\times 10^{-6}/\mathrm{K}\), \(\
u=0.3\). Plates with different boundary conditions investigated below are shown in Figure 2.  
Firstly, the results of the present method are compared with those in the literature to validate the correctness of this method. Tables 5 and 6 compare the critical buckling temperature \(\Delta T_{\rm cr}\) of simply supported type-II FGM square plate based on different theories. Table 7 compares \(\Delta T_{\rm cr}\) of fully clamped type-II FGM rectangular plate. It can be seen that the present results coincide well with available results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Property** & **Material Type** & \(P_{\textbf{-1}}\) & \(P_{\textbf{0}}\) & \(P_{\textbf{1}}\) & \(P_{\textbf{2}}\) & \(P_{\textbf{3}}\) \\ \hline \multirow{2}{*}{\(E\)} & Si\({}_{3}\)N\({}_{4}\) & 0 & \(348.43\times 10^{9}\) & \(-3.070\times 10^{-4}\) & \(2.160\times 10^{-7}\) & \(-8.946\times 10^{-11}\) \\  & SUS304 & 0 & \(201.04\times 10^{9}\) & \(3.079\times 10^{-4}\) & \(-6.534\times 10^{-7}\) & 0 \\ \hline \multirow{2}{*}{\(\
u\)} & Si\({}_{3}\)N\({}_{4}\) & 0 & \(0.2400\) & 0 & 0 & 0 \\  & SUS304 & 0 & \(0.3262\) & \(-2.002\times 10^{-4}\) & \(3.797\times 10^{-7}\) & 0 \\ \hline \multirow{2}{*}{\(\alpha\)} & Si\({}_{3}\)N\({}_{4}\) & 0 & \(5.8723\times 10^{-6}\) & \(9.095\times 10^{-4}\) & 0 & 0 \\  & SUS304 & 0 & \(12.330\times 10^{-6}\) & \(8.086\times 10^{-4}\) & 0 & 0 \\ \hline \multirow{2}{*}{\(\rho\)} & Si\({}_{3}\)N\({}_{4}\) & 0 & \(2370\) & 0 & 0 & 0 \\  & SUS304 & 0 & \(8166\) & 0 & 0 & 0 \\ \hline \multirow{2}{*}{\(k\)} & Si\({}_{3}\)N\({}_{4}\) & 0 & \(9.19\) & 0 & 0 & 0 \\  & SUS304 & 0 & \(12.04\) & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The material properties of type-I FGM plate. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(n\) & **Method** & _a/h = 10_ & _a/h = 20_ & _a/h = 40_ & _a/h = 60_ & _a/h = 80_ & _a/h = 100_ \\ \hline \multirow{4}{*}{\(1\)} & Present & 782.016 & 197.813 & 49.600 & 22.056 & 12.409 & 7.943 \\  & CPT [15] & 794.377 & 198.594 & 49.648 & 22.066 & 12.412 & 7.943 \\  & TSDT [15] & 757.891 & 196.257 & 49.500 & 22.037 & 12.402 & 7.939 \\  & HDT5 [15] & 749.260 & 195.623 & 49.404 & 22.029 & 12.400 & 7.939 \\ \hline \multirow{4}{*}{\(5\)} & Present & 713.894 & 180.841 & 45.361 & 20.173 & 11.350 & 7.265 \\  & CPT [15] & 726.571 & 181.642 & 45.410 & 20.182 & 11.352 & 7.265 \\  & TSDT [15] & 678.926 & 178.528 & 45.213 & 20.144 & 11.340 & 7.260 \\  & HDT5 [15] & 669.402 & 177.805 & 45.166 & 20.134 & 11.337 & 7.259 \\ \hline \multirow{4}{*}{\(10\)} & Present & 733.217 & 185.863 & 46.629 & 20.737 & 11.667 & 7.468 \\  & CPT [15] & 746.927 & 186.731 & 46.682 & 20.747 & 11.670 & 7.469 \\ \cline{1-1}  & TSDT [15] & 692.519 & 183.141 & 46.455 & 20.703 & 11.657 & 7.462 \\ \cline{1-1}  & HDT5 [15] & 683.211 & 182.432 & 46.409 & 20.694 & 11.654 & 7.462 \\ \hline \end{tabular}
\end{table} 
Table 5: Critical buckling temperature \(\Delta T_{\rm cr}\) of SSSS type-II FGM square plate with different _a/h_ under a uniform temperature field (_a/b_ = 1). Figure 2: Plates with different boundary conditions. Tables 8 and 9 show \(\Delta T_{\rm cr}\) of type-I FGM thin plate under a uniform temperature field with different boundary conditions. The results indicate that fully clamped (CCCC) plates have the largest \(\Delta T_{\rm cr}\), and \(\Delta T_{\rm cr}\) decreases as the power law index \(n\) increases. Tables 10 and 11 give \(\Delta T_{\rm cr}\) of Type-I FGM thin plate under different temperature fields.