## 5 Numerical Examples and Results
### Thermal Buckling Analysis  
This section provides results of the research based on comparative analysis of all shape functions (Table 1) and new proposed function. The obtained results for critical buckling temperatures of FG square and rectangular plates placed on an elastic foundation are totally in accordance with the results by authors in [55] which is applied trigonometric shear deformation plate theory and [25] which is applied HSDT based on the just one shape function. In Table 3, critical buckling temperatures (\(\Delta\mathrm{t}_{\mathrm{cr}}=\Delta\mathrm{Tcr}\cdot 10^{-3}\)) of FGM plates placed on an elastic foundation for case a linear temperature change through the plate thickness are presented. Values such as the power law index p, Winkler and Pasternak coefficient \(\mathrm{k}_{0}\), \(\mathrm{k}_{1}\) and thickness ratios a/h were varied during the analysis of the influence on the critical buckling temperature. By drawing an analogy between these findings where 13 different shape functions were applied, it can be clearly seen that the newly established shape function demonstrates quite similar results. From the findings, we can infer that critical buckling temperatures drops off with the rise of power law index p and ratio a/h. Additionally, Pasternak coefficient \(\mathrm{k}_{1}\) has a bigger effect than Winkler coefficient \(\mathrm{k}_{0}\) on the critical buckling temperatures. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{4}{c}{**Material Properties**} \\ \cline{2-5}  & \
\begin{tabular}{c} **Elasticity** \\ **Modulus E[GPa]** \\ \end{tabular} & **Poisson’s Ratio \(\mathbf{\gamma}\)** &
\begin{tabular}{c} **Thermal Expansion** \\ **Coefficient \(\alpha[^{\circ}\mathbf{C}^{-1}]\)** \\ \end{tabular} & **Density \(\rho\) [kg/m\({}^{3}\)]** \\ \hline Aluminum (Al) & \(\mathrm{E}_{\mathrm{m}}=70\) & \(\mathrm{\
u}=0.3\) & \(\mathrm{\alpha_{M}}=23\cdot 10^{-6}[^{\circ}\mathrm{C}^{-1}]\) & \(\mathrm{\rho_{m}}=2702\) \\ Alumina (Al\({}_{2}\)O\({}_{3}\)) & \(\mathrm{E}_{\mathrm{c}}=380\) & \(\mathrm{\
u}=0.3\) & \(\mathrm{\alpha_{C}}=7.4\cdot 10^{-6}[^{\circ}\mathrm{C}^{-1}]\) & \(\mathrm{\rho_{c}}=3800\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of functionally graded materials (FGM) constituents. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(p\)} & \multirow{2}{*}{**Source**} & \multicolumn{4}{c}{**\(\Delta\mathrm{t}_{\mathrm{cr}}\)**} \\ \cline{3-8}  & & \multicolumn{2}{c}{**k\({}_{0}\) = 0, k\({}_{1}\) = 0**} & \multicolumn{2}{c}{**k\({}_{0}\) = 10**, \(\mathrm{k}_{1}\) = 0**} & \multicolumn{2}{c}{**k\({}_{0}\) = 10**, \(\mathrm{k}_{1}\) = 10**} \\ \cline{3-8}  & & **a/h = 10** & **a/h = 20** & **a/h = 10** & **a/h = 20** & **a/h = 10** & **a/h = 20** \\ \hline \multirow{8}{*}{0} & [55] & 3.2276 & 0.833 & 3.3154 & 0.855 & 5.0479 & 1.2881 \\  & [25] & 3.2273 & 0.833 & 3.3151 & 0.855 & 5.0476 & 1.2881 \\  & Present study & 3.2274 & 0.8331 & 3.3151 & 0.855 & 5.0476 & 1.2881 \\  & No. 1 & 3.2273 & 0.833 & 3.3151 & 0.855 & 5.0476 & 1.2881 \\  & No. 2 & 3.2273 & 0.833 & 3.3151 & 0.855 & 5.0476 & 1.2881 \\  & No. 3 & 3.2276 & 0.833 & 3.3154 & 0.855 & 5.0479 & 1.2881 \\  & No. 4 & 3.2333 & 0.8334 & 3.3211 & 0.8554 & 5.0536 & 1.2885 \\  & No. 5 & 3.2273 & 0.833 & 3.3151 & 0.855 & 5.0476 & 1.2881 \\  & No. 6 & 3.2282 & 0.8331 & 3.316 & 0.855 & 5.0485 & 1.2882 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Critical buckling temperatures (\(\Delta\mathrm{t}_{\mathrm{cr}}\)) of FGM plates placed on an elastic foundation for case of linear temperature change across plate thickness (a/b = 1, m = n = 1, and T\({}_{\mathrm{m}}\) = 5\({}^{\circ}\)). Figures 4a and 5a clarify the impact of ratio a/b on the critical buckling temperature for case of linear and nonlinear temperature change through the plate thickness. It should be stressed that the rise of a/b ratio leads to the rise of critical temperature, e.g., curves show the harsh rise of a/b ratio functions.