# 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
However, such configuration is beneficial, at least for the heat sink, because, due to the prevented bending by the external kinematic constraints, the steel sub-component is axially compressed under the thermal gradient. Stresses normal to \(z\) in the plate will than lower at the interface, with a compressed FGM being the price to be paid. Equation (20) provides a value of the target strain of \(9.207\times 10^{-4}\) mm/mm. The interlayer thickness \(\delta_{3}\) is obtained in the same fashion of the thin hollow cylinder. Therefore one solves:  
\[\varepsilon_{T,m1} =\varepsilon_{0} \tag{21}\] \[\frac{1}{\delta_{1}(L+2\delta_{side})}\int_{\delta_{2}+\delta_{3} }^{\delta_{1}+\delta_{2}+\delta_{3}}\int_{0}^{L+2\delta_{side}}\alpha_{1}(T)(T -T_{0})dxdy =\varepsilon_{0}\]  
where the thickness of the whole heat sink \(\delta_{2}\) is in this case equal to \(L+2\delta_{hs}\). The temperature field is computed using a thermal FEM model. In such computations, the interlayer thickness is increased gradually up to the point that the mean thermal strain of the PFM is equal to \(\varepsilon_{0}\). The requirement is met for \(\delta_{3}\sim 2.68\) mm, corresponding to a W mean temperature of 222.74 \({}^{\circ}\)C. For our design the interlayer thickness is rounded to 2.7 mm. The whole geometry of the component is now completely defined. Subsequently, one proceeds in determining the concentration of the FGM. To do so, one consistently uses the condition of the mean thermal strain to be equal to \(\varepsilon_{0}\). However, as explained previously, a stronger condition can be put for the composite interlayer. For an ideal FGM, varying continuously its concentration in space, the function \(C\) can be tailored such that the thermal strain of this material would be constant and equal to the target value. Automatically, the constraint on the mean value would be satisfied. If the chosen manufacturing technique for the coating is the plasma spraying, the concentration of the FGM can vary only with the thickness \(C=C(y)\). Such a function is derived by solving Equation (11). For this purpose, a 1D approximation is required of the temperature field in the solid sub-domain having as boundaries the steel/water interface and the plasma-facing surface. We use:  
\[T(y)=\frac{b+\sqrt{b^{2}-2a(qy+c)}}{a}\ ;\ c=-\frac{a}{2}T_{wall}^{2}-bT_{wall} \tag{22}\]  
which is the analytical solution of the steady state heat equation of a plate loaded from one side and having a thermal conductivity \(\lambda(T)=aT+b\). The coefficients \(a\) and \(b\) are chosen to be the one of AISI316L [6]. \(q\) is the heat flux impinging on the plasma-facing surface and \(T_{wall}\) is the mean temperature at the interface between water and steel and the frame of reference is chosen such that \(y=0\) at this location. The resulting concentration profile is shown in Figure 7, together with the corresponding thermal strain that in this ideal case is constant in space and equal to \(\varepsilon_{0}\). However, a material with such continuously varying composition is very challenging to manufacture. Therefore, the function \(C(y)\) must be discretized in space. Figure 6: Principal components of the total strain tensor of the steel heat sink, when the temperature field is applied without any interlayer and armor.