# Engineering of a FGM Interlayer to Reduce the Thermal Stresses Inside the PFCs
## 4 Discussion and Conclusions  
In this paper we presented a comprehensive study on how to reduce the thermal stresses inside the PFCs, using a FGM interlayer to be interposed between the armor and the heat sink. The main design rationale is to develop the interlayer so that it ensures the kinematic continuity among all the sub-components, in a configuration where they deform into exactly the shape they would assume if they were detached from each other. Therefore, we first computed the shape that isostatically constrained bodies with geometries relevant for PFCs would assume only due the effect of a temperature field arising along their thickness. Such expansion is, under the assumption of thin solids, tightly linked to the mean value of the thermal strain field. When no bending occurs, as in the case of a thin hollow cylinder, we show that the thermal stresses due to the CTE mismatch are zero when the mean thermal strain of the armor and the heat sink is the same. Therefore, the first goal of the interlayer is in this case to bring the PFM up to a suitable temperature field such that it will expand as much as the heat sink. This does not necessarily imply the use of a FGM, also a poorly conductive interlayer (e.g., thermal break [25]) can be design following our design rationale. Moreover, a poorly conductive armor material could be beneficial to the thermal stresses in the PFC, as long as the loads due to the thermal gradient inside the PFM, which are worsened by a low thermal conductivity, are below the allowable stress. In any case, the thermal stresses in the heat sink would always benefit from a hotter W armor, since the matching of the thermal expansion would be improved. The major benefit of a FGM interlayer, compared to other solutions, is that its W concentration can be tailored to ideally eliminate the stresses inside itself. When bending occurs, as in the case of the plate, the proper matching of the shape of both the armor and the heat sink is achieved when the curvature that they would have if detached from each other is the same. We showed that such curvature depends only on the heat flux, which is imposed by the plasma, and on the material properties, specifically the ratio \(\frac{\pi}{\lambda}\). Therefore, a solution exists only if the PFM and the heat sink material have the same \(\frac{\pi}{\lambda}\). Unfortunately, this is not verified for the current candidates of PFC materials. Therefore, \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Armor Directly Joined to the Heat Sink (No Interlayer)**} \\ \hline Path & \(\overline{P_{m}+Q_{m}}\) [MPa] & \(\overline{P_{m}+P_{b}}\) [MPa] & \(\overline{Q}\) [MPa] \\ \hline \(\overline{AB}\) & 22.875 & 10.030 & 85.752 \\ \hline \(\overline{CD}\) & 169.52 & 18.821 & 223.31 \\ \hline \(\overline{EF}\) & 125.10 & 2.0706 & 201.74 \\ \hline \(\overline{GH}\) & 91.913 & 13.569 & 101.70 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Linearized stress in the steel heat sink having the armor directly joined (no interlayer). Paths are indicated in Figure 5. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Path**} & \multirow{2}{*}{\(\mathbf{T_{arg}}\) [\({}^{\circ}\)C]} & \multirow{2}{*}{\(\mathbf{S_{m}}\) [MPa]} & \multirow{2}{*}{\(\mathbf{S_{e}}\) [MPa]} & \(\overline{P_{m}+Q_{m}}\) & \(\overline{P_{m}+P_{b}}\) & \(\overline{P_{m}+P_{b}+Q}\) \\ \hline \(\overline{AB}\) & 72.034 & 127 & 146.05 & 0.156 & 0.0789 & 0.2513 \\ \hline \(\overline{CD}\) & 95.367 & 127 & 138.50 & **1.223** & 0.1481 & 0.6355 \\ \hline \(\overline{EF}\) & 88.458 & 127 & 140.75 & 0.8888 & 0.0163 & 0.5349 \\ \hline \(\overline{GH}\) & 90.707 & 127 & 140.02 & 0.6564 & 0.1068 & 0.3025 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Verification of the design criteria on the most critical sections. Case with the armor directly bonded to the heat sink. \(T_{arg}\) is the average temperature along the path; \(S_{m}\) is the allowable membrane stress; \(S_{e}\) is the allowable stress intensity. Conservatively \(K_{eff}\) is taken as 1.