# Engineering of a FGM Interlayer to Reduce the Thermal Stresses Inside the PFCs
## 3 Conceptual Design of a Steel-Based PFC with a Engineered W/Steel FGM Interlayer
### Engineering of the FGM Interlayer  
Once the steel heat sink has been sized, one proceeds in designing the interlayer. This part will be engineered so that the whole component would have a total strain, due to the secondary loads, as much as possible similar to the one of the heat sink without any armor. Due to the boundary conditions, thermal and kinematic, that we applied, the total strain of the steel sub-component in the \(z\) direction is constant on the whole plate. However, due to the dependence on both \(x\) and \(y\) of the temperature field, as shown in Figure 4, the total strain is not homogeneous in such directions. Figure 6 shows the principal total strains of the steel body, without any interlayer or armor, only due to the effect of the temperature field. One observes that the total strain along z is different from the mean total strain in the \(x\) direction. This means that the expansion of the heat sink is anisotropic. The rectangular surface of the steel plate stays rectangular after the application of the temperature field, but the ratio between homologous sides is not the same. Coupling an isotropic material, the interlayer, to such total strain field will unavoidably lead to additional stresses even in an ideal case. Apart from this effect, which cannot be mitigated, one engineers an interlayer that would not add any additional loads to the heat sink. This is done by designing the component in such a way that the total strain of both the armor and heat sink would be similar to the one of the steel body without the PFM. Therefore, as explained before, we impose that the mean thermal strain of the FGM and of the W to be equal of a target value \(\varepsilon_{0}\). In this study, this target parameter is chosen equal to the total strain \(\varepsilon_{x}\) of the steel body, averaged inside an elementary cell:  
\[\varepsilon_{0}=\frac{1}{L+2\delta_{side}}\int_{0}^{L+2\delta_{side}}\varepsilon _{x}dx \tag{20}\] \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{4}{c}{**No Interlayer and Armor**} \\ \hline Path & \(\overline{P_{m}+Q_{m}}\) [MPa] & \(\overline{P_{m}+P_{b}}\) [MPa] & \(\overline{Q}\) [MPa] \\ \hline \(\overline{AB}\) & 0.4259 & 7.21 & 72.034 \\ \hline \(\overline{CD}\) & 119.36 & 63.932 & 169.25 \\ \hline \(\overline{EF}\) & 68.823 & 70.384 & 165.9 \\ \hline \(\overline{GH}\) & 60.254 & 53.752 & 80.898 \\ \hline \end{tabular}
\end{table} 
Table 2: Linearized stress in the steel heat sink without armor and interlayer. Paths are indicated in Figure 5. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Path**} & \multirow{2}{*}{\(\mathbf{T}_{avg}\) [\({}^{\circ}\)C]} & \multirow{2}{*}{\(\mathbf{S}_{m}\) [MPa]} & \multirow{2}{*}{\(\mathbf{S}_{e}\) [MPa]} & \(\overline{P_{m}+Q_{m}}\) & \(\overline{P_{m}+P_{b}}\) & \(\overline{P_{m}+P_{b}+Q}\) \\ \cline{1-1}  & & & & \(S_{e}\) & \(\overline{K_{eff}S_{m}}\) & \(\overline{S_{m}+P_{b}}+\overline{Q}\) \\ \hline \(\overline{AB}\) & 72.034 & 127 & 146.05 & 0.003 & 0.0568 & 0.2080 \\ \hline \(\overline{CD}\) & 95.367 & 127 & 138.50 & 0.8618 & 0.5034 & 0.6120 \\ \hline \(\overline{EF}\) & 88.458 & 127 & 140.75 & 0.4890 & 0.5542 & 0.6202 \\ \hline \(\overline{GH}\) & 90.707 & 127 & 140.02 & 0.4303 & 0.4232 & 0.3534 \\ \hline \end{tabular}
\end{table} 
Table 3: Verification of the design criteria on the most critical sections. \(T_{avg}\) 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. Paths are indicated in Figure 5.  
Our choice results in the total deformation of the interlayer matching, on average, the one of the heat sink along \(x\). However, since this value is higher than \(\varepsilon_{z}\), some stresses along \(z\) due to the mismatch of the thermal expansion will arise. The heat sink will be tractioned by the interlayer and an analogous compression would occur inside the FGM.