## 3 Numerical Examples and Results
### Coupled Fluid-Structure-Thermal Analysis
#### 3.3.2 Parameter Study under Uneven Temperature Field  
An uneven temperature field is considered for the coupled fluid-structure-thermal analysis of the HCS plates. For the bottom and top face sheets, the temperature fields are treated as uniform due to their relatively thin thickness compared with the core layer. For the foam-filled corrugated core, the temperature field can be a linear or parabolic distribution.  
Therefore, the temperature field for the bottom face sheet is assumed to be \(\Delta T_{1}\) = 50, the temperature field for the top face sheet is assumed to be \(\Delta T_{3}\) = 300 and the temperature field for the foam-filled corrugated core is given.  
In linear distribution:  
\[\Delta T_{2}=\Delta T_{1}+\frac{\Delta T_{3}-\Delta T_{1}}{2}(1+\xi_{2})\]  
In parabolic distribution:  
\[\Delta T_{2}=\Delta T_{1}+\frac{\Delta T_{3}-\Delta T_{1}}{2}\frac{1}{2}(1+ \xi_{2})^{2}\]  
For the HCS plates under this uneven temperature field, the structural dynamic stabilities are mainly determined by the geometric parameters of the foam-filled corrugated core. Therefore, the effects of the inclined angle \(\theta\) and the corrugation thickness \(t\) on the dimensionless critical dynamic pressure under the linear or parabolic temperature field are investigated Figure 13 shows the critical dynamic pressure and the equivalent density against the inclined angle and corrugation thickness in a linear or parabolic temperature field. With the increase in the inclined angle, the equivalent density decreases, and the critical dynamic pressure increases for both linear and parabolic temperature distributions. It seems that increasing the inclined angle is a clever way to improve structural efficiency; however, as shown in Figure 8, the corrugated sandwich plate with a larger inclined angle has a smaller critical temperature change, which is more prone to occur thermal buckling. As for the corrugation thickness, with its increasing, the critical dynamic pressure improves, accompanied by the deterioration of mass and buckling characteristics.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Foam**Materials**} & & \multicolumn{2}{c}{\(\overline{\Delta T_{cr}}\)} & & \multicolumn{2}{c}{\(\overline{\Delta}\)} \\ \cline{2-7}  & **Base** & & & & & \\ \hline Empty & 2.35 & 2.39 & 2.35 & 338 & 306 & 193 \\ RC31 & 2.49 & 2.45 & 2.39 & 342 & 307 & 195 \\ RC51 & 2.64 & 2.52 & 2.41 & 346 & 310 & 197 \\ RC71 & 2.59 & 2.50 & 2.39 & 345 & 309 & 196 \\ AC & 2.43 & 2.45 & 2.33 & 346 & 310 & 198 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Effects of material combination of foam and base structures on critical temperature change and dimensionless critical dynamic pressure with temperature change \(\Delta T\) = 100.  
Meanwhile, for the given temperature field of bottom and top face sheets, the critical dynamic pressure obtained from the linear temperature distribution is smaller than that from the parabolic distribution. This means that the assumption of linear temperature distribution will lead to more conservative results for the HCS plates under aerodynamic pressure and uneven temperature fields.