# 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
### Mechanical Results
One observes that only the loads inside the interlayer are impacted by the different number of layers, while the contour inside the armor and heat sink is virtually unmodified. Therefore, during the design phase, the amount of divisions in the FGM is determined only by the desired stress inside the interlayer, with the minimum limit being the ideal FGM case. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Ideal FGM**} \\ \hline Path & \(\overline{P_{m}+Q_{m}}\) [MPa] & \(\overline{P_{m}+P_{b}}\) [MPa] & \(\overline{Q}\) [MPa] \\ \hline \(\overline{AB}\) & 15.891 & 10.362 & 86.761 \\ \hline \(\overline{CD}\) & 117.50 & 14.510 & 169.77 \\ \hline \(\overline{EF}\) & 53.460 & 1.4331 & 133.26 \\ \hline \(\overline{G}H\) & 47.635 & 9.6404 & 52.553 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Linearized stress in the steel heat sink with an ideal FGM. Paths are indicated in Figure 5.  
Figure 10: Contour of the von Mises stress in the component due to the heat load and the pressure. Case with an ideal FGM. Figure 11: Contour of the von Mises stress in the component due to the heat load and the pressure. Case with discretized FGMs: (**a**) One FGM layer. (**b**) Two FGM layers. (**c**) Three FGM layers. (**d**) Four FGM layers. One notices that the von Mises stresses both in the armor and the heat sink are virtually constant for all the different discretizations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{C_{i}}\) & \(\mathbf{y_{i}}\) & \(\mathbf{y_{i}}\) (Rounded) & \(\mathbf{\Delta t_{i}}\) \\  & & **[mm]** & **[mm]** & **[mm]** \\ \hline
**1 FGM layer** & \(C_{1}=79.798\%\) & - & - & \(\Delta t_{1}=2.7\) \\ \hline
**2 FGM layers** & \(C_{1}=65.958\%\) & \(y_{1}=1.107\) & \(y_{1}=1.1\) & \(\Delta t_{1}=1.1\) \\  & \(C_{2}=86.523\%\) & \(y_{2}=2.7\) & \(y_{2}=2.7\) & \(\Delta t_{2}=1.6\) \\ \hline
**3 FGM layers** & \(C_{1}=61.205\%\) & \(y_{1}=0.694\) & \(y_{1}=0.7\) & \(\Delta t_{1}=0.7\) \\  & \(C_{2}=76.660\%\) & \(y_{2}=1.575\) & \(y_{2}=1.6\) & \(\Delta t_{2}=0.9\) \\  & \(C_{3}=88.935\%\) & \(y_{3}=2.7\) & \(y_{3}=2.7\) & \(\Delta t_{3}=1.1\) \\ \hline
**4 FGM layers** & \(C_{1}=58.796\%\) & \(y_{1}=0.505\) & \(y_{1}=0.5\) & \(\Delta t_{1}=0.5\) \\  & \(C_{2}=71.073\%\) & \(y_{2}=1.107\) & \(y_{2}=1.1\) & \(\Delta t_{2}=0.6\) \\  & \(C_{3}=81.424\%\) & \(y_{3}=1.830\) & \(y_{3}=1.8\) & \(\Delta t_{3}=0.7\) \\  & \(C_{4}=90.140\%\) & \(y_{4}=2.7\) & \(y_{4}=2.7\) & \(\Delta t_{4}=0.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Discretization of the FGM interlayer up to four sub-layers (same data are also visually represented in Figure 8). For each layer, the concentration \(C_{i}\) as well as the thickness \(\Delta t_{i}\), rounded from the value derived trough the computation, are reported. As a sanity check of the proposed design methodology one carries out a last simulation having the heat sink, which was verified to be able to withstand the loads due to the thermal gradient, directly bonded to the tungsten arm without any interlayer. Therefore the tungsten slab, with thickness \(\delta_{1}=2\) mm, is for this case joined directly to the steel heat sink having the same geometry used up to now. Table 7 shows the linearized stresses in the AISI316L in this latter case. By comparing such values with the ones of the four FGM layers in Table 6, one concludes that with the engineered interlayer:  
* The linearized secondary stress \(\overline{Q}\) is reduced on average by more than 26%. The maximum reduction is of 48%, on the segment \(\overline{GH}\), while the minimum reduction is of 0.6%, on the segment \(\overline{AB}\). Most importantly, the most loaded section \(\overline{CD}\) has a reduction of more than 24% in the secondary stresses. * The equivalent primary plus secondary membrane stress \(\overline{P_{m}+Q_{m}}\) is reduced on average by more than 40%. The maximum reduction is of 52 %, on the segment \(\overline{EF}\), while the minimum reduction is of 28%, on the segment \(\overline{CD}\).