# Engineering of a FGM Interlayer to Reduce the Thermal Stresses Inside the PFCs
## 1 Introduction
In other words, one engineers the interlayer, in its composition and thickness, in such a way that the armor would freely expand in the same fashion as the heat sink. A substantial difference of our approach from the mentioned optimization studies is that it allows to solve the thermoelastic problem for the separated armor and heat sink, which can be carried out analytically, reducing substantially the complexity from the formulation of a single body made of three different materials. After one computes separately the strain field experienced by the PFM and the heat sink, in the case that they both can deform freely due to the application of the temperature field, the FGM is then designed to make the two total strain fields match, while satisfying itself the same requirement. Such an optimization would allow to minimize, and ideally reduce to zero, the secondary stresses due to the mismatch of the CTE. In such a way, when the component is brought up to temperature the total deformation is equal to the one that its parts would experience if they were detached from one another. As explained in the following sections, this translates at least in a condition on the mean thermal strain of each sub-component. Another difference of our design approach from the optimization studies listed before, where the geometry of the problem is a given, is that only the thickness of the heat sink and of the armor are inputs. The first is derived such that the thermal stresses in the heat sink detached from the rest are below the allowable load. The latter is imposed by the plasma-wall interaction processes. The interlayer thickness is then found to be a key engineering parameter for the reduction of the thermal stresses in the component, and will be determined such that no additional stresses will arise due to the CTE mismatch between the PFM and the heat sink. The paper is organized in two main sections. In the first we derive the formulation for computing the total strain of freely supported bodies having a geometry relevant for the PFCs. Two cases will be described, both concerning solids having a thin thickness in the direction in which the thermal gradient is applied. The first consists in a hollow cylinder loaded homogeneously on the outer circumference, while the second is a plate loaded from one side. These examples with exact analytical solutions allow to understand the relevant engineering parameters. The substantial difference among the two instances is in the fact that the cylinder would not experience bending due to the axisymmetrical geometry and thermal boundary conditions, while the plate will have a curvature due to the monotonic nature of the temperature field. As we will see, if the bending occurs we cannot always find the matching of the \"free\" total strain of the PFM and of the heat sink, for all the possible choices of materials for the armor and the structural part of the PFC. In the second part of the paper, a design study is presented where the developed methodology is applied, specifically for the thermomechanical conceptual design of a steel-based PFC with an engineered W/steel FGM interlayer. Such component has a heat sink made of an AISJ316L plate in which square channels for the water flow have been machined. In this design study, bending must be avoided for the above mentioned reason, through appropriate kinematic conditions at the boundary that an actual fixing system will have to best satisfy. In this section, we will also address the differences from the simple ideal case, such as the need of a FGM discretization and the inequality of the total strain components in the two main directions of the plate due to the presence of the coolant channels.