# Engineering of a FGM Interlayer to Reduce the Thermal Stresses Inside the PFCs
## 2 Analytical Methods for the FGM Engineering
### Hollow Cylinder with Thin Thickness and a Homogeneous Thermal Load
To clarify this aspect, we will consider for now a case where the CTE is temperature independent, \(\alpha=\text{cost}\). We start by decomposing the monotone function \(T(y)\) in thickness into a constant part \(T_{m}=\frac{1}{\delta}\int_{0}^{\delta}Tdy\), and a thickness-dependent term \(T_{1}(y)\), whose integral over the thickness is zero. Therefore, we can write:  
\[T(y^{\prime})=T_{m}+T_{1}(y^{\prime}) \tag{6}\]  
\(y^{\prime}\) is a new abscissa along the thickness where by definition \(T(y^{\prime}=0)=T_{m}\). Let us now consider an elementary segment such as the one in Figure 1, and we consider it as if it were detached from the rest of the hollow cylinder. The thermal strain due to the temperature field \(T(y^{\prime})\) can also be decomposed into two terms: the first is a homothetic expansion \(\varepsilon_{T,m}=\alpha(T_{m}-T_{0})\) which causes an increase in the mean radius of the cylinder, plus a term depending on \(T_{1}(y^{\prime})\) that does not modify the volume of the cylinder but would distort the elementary element as shown in Figure 1. While the first contribution of the thermal deformation does not generate stresses as it automatically satisfies the continuity equations, the second tends to deform the segment in a non-congruent way and tangential stresses arise to \"bring the fibers back\" along the radii in such a way that the elementary segment satisfies the cylindrical symmetry. Such stresses are proportional to the thermal strain that they are opposing:  
\[\sigma_{t}=-\frac{E\alpha}{1-\
u}T_{1}(y^{\prime}) \tag{7}\]  
Figure 1: Deformation of an elementary segment of the hollow cylinder due to the temperature \(T\). In the case of a temperature independent CTE, the thermal field can be decomposed in a constant term \(T_{m}\) that generates a stress-free homotetic deformation, and a part \(T_{1}(y^{\prime})\) that generates the thermal stresses to satisfy the cylindrical symmetry. If \(T(y^{\prime})\) were linear, \(T_{1}(y^{\prime})\) would be equal to \((T_{\max}-T_{m})\frac{2y^{\prime}}{\delta}\), where \(T_{\max}\) is the maximum value of the temperature field. Therefore, the highest is the excursion between the maximum temperature value to the mean temperature, the more intense are the stresses. Such results can be generalized to the case with a temperature dependent CTE by decomposing the thermal strain field \(\varepsilon_{T}(y)\) in two terms, with an analogous procedure as previously performed in the linear case to the field \(T(y)\). The first contribution leads to a homotetic stress-free deformation, while the second is the one that generates the thermal stresses. Following the same approach, one obtains that in a general case the stresses depend on the quantity \((\varepsilon_{T,\max}-\varepsilon_{T,m})\), where \(\varepsilon_{T,\max}=\alpha(T_{\max})(T_{\max}-T_{0})\). Therefore, increasing \(\delta_{1}\) would reduce the contribution generated by the CTE mismatch, but would increase as well the loads due to the temperature gradient by bringing \(T_{\max}\) up. Moreover, it would be best to leave the armor thickness as a parameter determined only by the constraints due to the processes of plasma-wall interaction (PWI), such as the before mentioned erosion. To achieve a hot W layer without changing its size, one computes accordingly the thickness \(\delta_{3}\) of an interlayer, put between the PFM and the heat sink. The FGM geometry must then satisfy:  
\[ \begin{split}\varepsilon_{T,m1}&=\varepsilon_{T,m2} \\ \frac{1}{\delta_{1}}\int_{\delta_{2}+\delta_{3}}^{\delta_{1}+ \delta_{2}+\delta_{3}}\alpha_{1}(T-T_{0})dy&=\frac{1}{\delta_{2 }}\int_{0}^{\delta_{2}}\alpha_{2}(T-T_{0})dy\end{split}  \tag{8}\]  
In such a Equation, \(\delta_{1}\) is imposed by the PWI processes, \(\delta_{2}\) is an input pre-determined in such a way that the stresses due to the temperature gradient in the heat sink are below the maximum allowable value, and \(\delta_{3}\) is the only unknown.