# Engineering of a FGM Interlayer to Reduce the Thermal Stresses Inside the PFCs
## 2 Analytical Methods for the FGM Engineering  
In both cases that we will discuss, the design rationale is the same. First, the total strain experienced by the heat sink will be computed in the case where the body is left free to expand, ideally detached from the rest, due to the effect of the temperature field \(T\). In both instances, \(T=T(y)\) is a continuous monotonic function depending only on the thickness. We will assume that this scalar function is known a priori, in order to decouple the thermal from the thermomechanical problem. Actually, the temperature is not an arbitrary function, but instead is the solution of the Fourier equation, and it will depend on the boundary conditions, the FGM concentration and its thickness. Without loosing generality, we will assume that \(T(y)\) is given. In the real case, one can easily come up with an iterative algorithm, converging on both the FGM thickness and concentration. However, this numerical effort is outside the scope of this work and instead we will focus on identifying the main parameters that drive the thermal stresses due to the CTE mismatch. In both cases that will be analyzed, one considers the [PERSON]'s module \(E\) and the [PERSON]'s module \(\
u\) of the materials to be constant with temperature. The approach used in this paper to derive the formulation of the total strain is analogous to the work found in [14,15] for one-dimensional structures. After the heat sink deformation is computed, it will then become the target value of the total strain. The armor is thus required to have the same strain field and the FGM thickness is determined to satisfy this very constraint. Finally, the ideal FGM composition is derived by imposing, everywhere in the interlayer, the thermal strain to be equal to the target strain. In such a way, it is directly derived from the constitutive relation (\(\epsilon=\epsilon_{T}+\epsilon_{M}\)) that the FGM would be also stress-free since the mechanical strain \(\epsilon_{M}\) would be zero. The concentration function \(C\) is correlated to the thermal strain \(\epsilon_{T}\) in the FGM, by adopting a linear mixture law for the secant CTE which is commonly used to model fusion-relevant FGMs [16,17]:  
\[\alpha(T,C)=[\alpha_{1}(T)-\alpha_{2}(T)]C+\alpha_{2}(T) \tag{1}\]  
where \(\alpha\) is the CTE of the FGM, while \(\alpha_{1}\) and \(\alpha_{2}\) are the CTE of the two components of the FGM. The function \(C\) is the volume concentration function of the material 1, in our case W.  
By both using the definition of the thermal strain, \(\epsilon_{T}=\alpha(T,C)(T-T_{0})\), in the constraints that we imposed in the proposed methodology and assuming a stress-free temperature \(T_{0}\), the function \(C\) is obtained. The modeling of the CTE using Equation (1) connects with already produced W/steel FGMs, showing good agreement with the measured CTE [18]. Furthermore, the CTE of fabricated W/copper FGMs [19], relevant for fusion applications, follow the linear mixture law, justifying the choice of such a model for the engineering of a FGM interlayer. Anyhow, from a mathematical point of view, our methodology is not limited to this specific model of the FGM CTE, but theoretically it could use any function \(\alpha(T,C)\), as long as, at fixed \(T\), the inverse function of \(F_{T}(C):=\alpha(T,C)\) exists.