# 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  
Another relevant case for the PFCs can be found in a plate having a monotonic temperature field over its thickness. A one-sided thermal boundary condition corresponds to the real situation which occurs in a fusion reactor, since the PFCs are loaded only on the surface facing the plasma. This results in a hotter and a colder side of the component, which in turn leads to the bending of the module. Therefore, even if one considers the thickness \(\delta\) to be much smaller than the other dimensions of the plate, the total strain in the principal directions will not be constant anymore over \(y\). The constitutive equations in Cartesian coordinates (\(x\),\(y\),\(z\)), for a thin plate are obtained by considering the normal stress in the thickness direction to be negligible compared the other principal stresses (\(\sigma_{y}\sim 0\)):  
\[ \begin{cases}\varepsilon_{x}&=\frac{1}{E}(\sigma_{x}-v\sigma_{z})+\alpha(T-T_ {0})\\ \varepsilon_{z}&=\frac{1}{E}(\sigma_{z}-v\sigma_{x})+\alpha(T-T_{0})\end{cases}  \tag{13}\]  
According to the plate theory [21], the normal strain in the \(x\) and \(z\) direction have the following form:  
\[ \begin{cases}\varepsilon_{x}=\varepsilon_{0,x}+\frac{y}{r_{x}}\\ \varepsilon_{z}=\varepsilon_{0,z}+\frac{y}{r_{z}}\end{cases}  \tag{14}\]  
where \(\varepsilon_{0,x}\) and \(\varepsilon_{0,z}\) are the normal strains at \(y=0\) in the \(x\) and \(z\) directions, \(r_{x}\) is the radius of curvature of the plate in a plane parallel to the \(xy\) plane and \(r_{z}\) is the radius of curvature in a plane parallel to the \(zy\) plane. Analogously to the previous case, the temperature is a function of \(y\) only, and therefore \(\varepsilon_{x}\) and \(\varepsilon_{z}\) do not depend on \(x\) or \(z\). Moreover, the \(r_{i}\) can be considered constant over the thickness, being valid the hypothesis of a thin plate. In absence of external mechanical loads, both the integral of the principal stresses over the plate thickness and the bending moments are zero in the whole body. These equilibrium conditions, together with Equation (13), allow to obtain the unknown variables in Equation (14):  
\[ \begin{cases}&\varepsilon_{0}=\varepsilon_{0,x}=\varepsilon_{0,z}=\frac{1}{ \delta}\int_{-\frac{\delta}{2}}^{\frac{\delta}{2}}\alpha(T-T_{0})dy\\ &\frac{1}{r}=\frac{1}{r_{x}}=\frac{1}{r_{z}}=\frac{12}{\delta^{3}}\int_{- \frac{\delta}{2}}^{\frac{\delta}{2}}\alpha(T-T_{0})ydy\end{cases}  \tag{15}\]  
For a simpler representation of the integrals in Equation (15) the frame of reference is chosen with \(y=0\) at the mean plane of the plate. In an analogous fashion as the previous case, one notices that \(\varepsilon_{0}\) corresponds to the mean thermal strain of the body \(\varepsilon_{T,m}\). Additionally, one observes that the total strains in the \(x\) and \(z\) directions result to be equal to each other, and therefore the plate is bent to a spherical shell. This result is valid for the armor and the heat sink, since both are considered thin plates. If they were not bonded to each other, each sub-component would deform freely in the described fashion, even if having a different strain of the mean plane \(\varepsilon_{0}\) and curvature \(\frac{1}{r}\). In order to eliminate the stresses due to the CTE difference, one must engineer the component such that the total strain field of the PFM and the heat sink are matched. This would require the curvature of the sub-components to be equal, together with a condition on the mean thermal strains of the bodies:  
\[ \begin{cases}&\frac{1}{r_{1}}=\frac{1}{r_{2}}=\frac{1}{r}\\ &\varepsilon_{T,m1}=\varepsilon_{T,m2}+\frac{\Delta y}{r}\end{cases}  \tag{16}\]  
where the subscript \(1\) refers to the armor and \(2\) to the heat sink material, and \(\Delta y\) is the thickness between the location in the heat sink where \(\varepsilon_{T}=\varepsilon_{T,m2}\) and the point in the PFMwhere \(\epsilon_{T}=\epsilon_{T,m1}\).