# Engineering of a FGM Interlayer to Reduce the Thermal Stresses Inside the PFCs
## 2 Analytical Methods for the FGM Engineering
### Thin Plate Body with a One-Sided Thermal Load
We have extensively discussed in the previous case how it is possible to modify the W mean thermal strain \(\epsilon_{T,m1}\) by engineering the thickness of an interlayer interposed between the sub-components, and the same approach can be used for this instance. However, the equation concerning the radii of curvature cannot be generally satisfied. Let us semplify the description by considering the materials to have a constant thermal conductivity \(\lambda\) and CTE. In this linear approach, the temperature field is:  
\[T(y)\sim\frac{q}{\lambda}y+T_{m} \tag{17}\]  
where \(q\) is the impinging heat flux and \(T_{m}\) the mean temperature of the plate. Using Equation (15), one obtains:  
\[\left\{ \begin{array}{c}\epsilon_{0}=\epsilon_{T,m}=\alpha(T_{m}-T_{0})\\ \frac{1}{r}=\frac{\alpha}{\lambda}q\end{array} \right. \tag{18}\]  
The curvature depends only on the applied power density \(q\), fixed by the plasma, and on the material properties, namely the ratio \(\frac{\alpha}{\lambda}\) between the CTE and the thermal conductivity. Neither the geometry nor the temperature field can be used to tailor the bending of the plate. When the total strain of the \"free-to-deform\" bodies has a curvature term, an interlayer capable of bringing to zero the thermal stresses due to the CTE mismatch can be developed only if the PFM and heat sink material are such that:  
\[\frac{\alpha}{\lambda}\Big{|}_{1}\sim\frac{\alpha}{\lambda}\Big{|}_{2} \tag{19}\]  
Such a relation is not generally satisfied. Table 1 reports the values of \(\frac{\alpha}{\lambda}\) for several fusion-relevant materials. Unfortunately, it is possible to assess that there is no direct matching. Therefore, the total strain field of the armor and heat sink cannot be paired, at least generally, as was performed in the previous case. The design rationale that we propose to tackle this issue consists in constraining the PFCs in such a way that bending is not allowed. This is achievable, for example, by external kinematic conditions. It will then be necessary that a proper fixation system is designed to best achieve this condition in a real component in the reactor. Such an approach would result in the increase of the contribution to the thermal stresses due to the gradient, both in the armor and in the heat sink, because the free bending of the plate would be hindered. The benefit however lies in the possibility of reducing to zero the stresses due to the mismatch of the CTE. In such a way, the critical interface of joining between the PFM and the rest of the PFC, generally prone to detachments, does not experience any intensification of the thermomechanical stresses. With the absence of a curvature, the case of the plate degenerates into the previous one. The design rationale again becomes the matching, among the sub-components, of the total strain \(\epsilon_{0}\). As a first step, the thermomechanical design of the heat sink is carried out since it does not depend on the other sub-components, but only on \(T(y)\) which in our description is considered an input. Therefore, the geometry of the heat sink is determined by satisfying the design criteria while the temperature field is applied and no bending is imposed. The mean thermal strain of the heat sink \(\epsilon_{T,m2}\) is chosen as the target value of the total strain \(\epsilon_{0}\), which is kept constant in the whole PFC. The interlayer, in its thickness and composition, is designed by using the Equations (9) and (11) derived previously. In the same fashion as before, the ideal W concentration allows for the theoretical determination of a stress-free FGM. By hindering the bending of the plate, the engineering of a FGM interlayer able to eliminate the stresses due to the CTE mismatch is then possible.