# Engineering of a FGM Interlayer to Reduce the Thermal Stresses Inside the PFCs
## 1 Introduction  
One of the main functions of the Plasma-Facing Components (PFCs) in a fusion reactor is to exhaust the plasma heating power, which is the thermal power continuously fed to the fuel to sustain the fusion reactions. Due to the small heat exchange area, the deposited power density in steady state can reach tens of MWm\({}^{-2}\) in the regions where the plasma touches the wall (e.g., divertor and limiters), or be of the order of 1 MWm\({}^{-2}\) where the load is mainly due to radiation [1, 2]. In their most general description, actively cooled PFCs are made of an armor and a heat sink. The first is the part of the component exposed directly to the plasma, made of a Plasma-Facing Material (PFM), having good compatibility with the plasma environment. Among the possible candidates, tungsten (W) is currently one of themost promising choices [3]. The armor is bonded to the heat sink, which is the part of the PFCs where the coolant channels are located. For water-cooled PFCs, the heat sink material must be different from the PFM, since the commercially available W has a ductile-to-brittle transition temperature above 400 \({}^{\circ}\)C [4], which is incompatible with the coolant operative range. The fusion relevant heat sink materials are usually copper-based or iron-based [5]. When a thermal load is applied to a PFC, a temperature field builds up within the body. If such temperature field results in a congruent thermal strain field, meaning that it satisfies both the continuity equations in the medium and the external kinematic constraints at the boundary, then no stresses are generated. The simplest example is a uniform temperature field applied to an isostatically constrained body made of a homogeneous and isotropic material. If, conversely, the temperature leads to a non-congruent thermal strain field, then thermal stresses, also called secondary stresses, are generated. In other words, to prevent the solid domain from breaking apart, internal forces arise opposing the differential thermal expansion of the several parts of the body. A major contribution to the thermal stresses is due to the intrinsic bimetallic nature of the PFCs, which leads to a mismatch of the Coefficient of Thermal Expansion (CTE) between the PFM and the heat sink. Indeed, W has a CTE which is 3-4 times less than that of copper or iron [6]. Therefore, even if a homogeneous temperature field were applied to an isostatically constrained component, secondary stresses would arise since the heat sink would want to expand much more than the armor. In order to mitigate this contribution to the thermal stresses, a promising solution consists in adopting an interlayer made of a Functionally Graded Material (FGM) [7], to be interposed between the PFM and the heat sink. FGMs are composites which can vary in space their thermophysical properties, such as the CTE, by changing the local mixture ratio of its metallic components. The most relevant FGMs for fusion application are W/steel and W/copper [8; 9]. By tailoring the W concentration in the volume of the FGM, one can engineer the thermal expansion of the component in such a way that the thermal stresses are reduced inside the PFC. In some works, different trends (linear, quadratic, exponential, etc.) of the spacial W content in the FGM interlayer, the so-called concentration function, were tested in a \"trial-and-error\" fashion in order to find the one minimizing the stresses [10]. Other studies [11; 12; 13], not specific to PFCs, optimize the FGM interlayer by minimizing numerically a cost function. In this work, we instead propose a design methodology according to which the FGM concentration function is derived analytically from the governing equations of thermoelasticity. Specifically, the design of the interlayer is carried out by keeping constant the total strain that each sub-component (armor, FGM and heat sink) would experience if it were left free to expand, hypothetically detached from the rest.