Self-consistent homogenization for the thermo-mechanical analysis of Nb3Sn filaments and wires

The superconducting magnet structure of the future International Thermonuclear Experimental Reactor (ITER) consists of three main systems: the Central Solenoid coil (CS), the 18 Toroidal Field coils (TF), and the 6 Poloidal Field coils (PF). Because of the high magnetic field and current required, the CS and TF coils will be manufactured using Nb3Sn based cables, with the cable-in-conduit-conductor technology. The cables are formed by multi-stage cabling of superconducting strands, with the final stage consisting of 6 bundles twisted around a central cooling channel. For electrical and thermal stability the Nb3Sn compound is distributed into fine filaments (up to about 50 micrometers diameter) and embedded in a resistive matrix to form the elementary strand. Nb3Sn formation requires a solid state diffusion reaction at high temperature, which causes Sn gradient inside the filaments. It is well known that the critical parameters vary with composition (Sn content) and strain state. In this work the relation between compositional variations and strain is investigated: Nb3Sn wires are studied taking into consideration non-homogeneous filaments. A finite element discretization fine enough to take into consideration Sn gradient would result in a number of unknowns which is far beyond the capacity of nowadays computers. Therefore a thermo-mechanical model is developed, based on a self consistent homogenisation, suitably developed to deal with the material non-linearity and the coupling between the thermal and mechanical field. In this way the equivalent homogeneous properties are obtained and the analysis of the wires becomes possible. An appropriate unsmearing technique gives finally the strain state in the real, not homogenized materials.