# 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
### Discretization of the Ideal FGM in Layers  
The simplest discretization would consist in a FGM monolayer. For such a material, the thermal strain is equal to:  
\[\varepsilon_{T,3}(T)=\alpha(T)(T-T_{0})=[(\alpha_{1}(T)-\alpha_{2}(T))\,C_{1}+ \alpha_{2}(T)](T-T_{0}) \tag{23}\]  
As determined by our design methodology, the mean thermal strain of the interlayer \(\varepsilon_{T,m3}\) must be equal to the target strain. One therefore obtains:  
\[ \begin{split}\frac{1}{\delta_{3}}\int_{\delta_{2}}^{\delta_{2}+ \delta_{3}}\varepsilon_{T,3}(T)dy=\varepsilon_{0}\\ C_{1}=\frac{\delta_{3}\varepsilon_{0}-\int_{\delta_{2}}^{\delta_{2} +\delta_{3}}\alpha_{2}(T)(T-T_{0})dy}{\int_{\delta_{2}}^{\delta_{2}+\delta_{3 }}[\alpha_{1}(T)-\alpha_{2}(T)](T-T_{0})dy}\end{split}  \tag{24}\]  
If more than one FGM layer is desired, the process is analogous. For each \(i\)-th layer the condition on the mean value of the thermal strain must be satisfied. The equations to determine the layer concentration \(C_{i}\) in the case of \(N\) total layers are therefore the following:  
\[ \begin{split}\frac{1}{y_{i}-y_{i-1}}\int_{y_{i-1}}^{y_{i}} \varepsilon_{T,i}(T)dy=\epsilon_{0}\rightarrow\begin{cases}C_{1}&= \frac{(y_{1}-\delta_{2})\varepsilon_{0}-\int_{y_{m-1}}^{y_{1}}\alpha_{2}(T)(T- T_{0})dy}{\int_{y_{m}}^{y_{1}}[\alpha_{1}(T)-\alpha_{2}(T)](T-T_{0})dy}\\ C_{1}&=\frac{(y_{i}-y_{i-1})\varepsilon_{0}-\int_{y_{m-1}}^{y_{1}} \alpha_{2}(T)(T-T_{0})dy}{\int_{y_{m-1}}^{y_{m}}[\alpha_{1}(T)-\alpha_{2}(T)]( T-T_{0})dy}\\ C_{N}&=\frac{(\delta_{2}+\delta_{3}-y_{N-1})\varepsilon_{0}-\int_{y_{m-1}}^{ y_{2}+\delta_{3}}\alpha_{2}(T)(T-T_{0})dy}{\int_{y_{m-1}}^{x_{2}+\delta_{3}}[ \alpha_{1}(T)-\alpha_{2}(T)](T-T_{0})dy}\end{cases}\end{split}  \tag{25}\]  
where \(y_{i}\) is the spacial coordinate at which the \(i\)-th layer ends. Moreover, it is appropriate to size the different layers in a way that the stresses are distributed evenly. Therefore the choice of the \(y_{i}\) is not arbitrary. Since the stresses depend on the difference between the total and the thermal strain, this constraint corresponds to the condition of equal absolute maximum of the thermal strain of the different FGMs:  
Figure 7: Ideal W concentration function of the FGM derived using Equation (11). The temperature field is derived from Equation (22), with: \(T_{wall}=111.56\)\({}^{\circ}\)C; \(q=0.5\) MWm\({}^{-2}\); \(a=1.502\times 10^{-2}\) Wm\({}^{-1}\)K\({}^{-2}\); \(b=13.98\) Wm\({}^{-1}\)K\({}^{-1}\); \(T_{0}=20\)\({}^{\circ}\)C. \(T_{wall}\) is the computed average temperature at the interface steel/water in the FEM model (Figure 4). The material properties of AISI316L and tungsten are taken from [6]. The FGM is made of a W/AlSI316L mixture. The interlayer thickness results to be \(\delta_{3}=2.7\) mm. \[ \begin{split}\max[\varepsilon_{TJ}(T)]&=\max[\varepsilon_{TJ+1 }(T)]\\ \alpha_{i}(\,T(y_{i})\,)[T(y_{i})-T_{0}]&=\alpha_{i+1} \,(T(y_{i+1})\,)[T(y_{i+1})-T_{0}]\end{split}  \tag{26}\]  
where \(\alpha_{i}\) is the secant CTE of the \(i\)-th layer. Equations (25) and (26) allow to find all the \(C_{i}\) and \(y_{i}\) to size the interlayer and can be solved iteratively. Figure 8 shows the discretization of \(C(y)\) up to four layers. Figure 8: Discretization of the ideal concentration of the FGM in a number of layers, following Equations (25) and (26). It should be highlighted that the amount of layers would not impact substantially the stresses inside the steel heat sink and the tungsten armor. This is because, by construction, each layer will have the same exact total strain, whether one chooses a monolayer or more. The difference will be in the stresses due to the thermal gradient inside the FGM, which depend on the difference between the thermal strain and \(\varepsilon_{0}\). The higher is the number of layers, the closer the FGM will be to the ideal condition where \(\varepsilon_{T,3}(T)=\text{cost}=\varepsilon_{0}\). One also notices that due to the condition of equal stress among the FGMs, the layers with a higher steel content are thinner than the others.