A multilevel homogenised model for superconducting strand thermomechanics

In the present concept of ITER fusion reactor the toroidal field and the central solenoid coils are made of Nb₃Sn based strands with the cable-in-conduit-conductor (CICC) technology. It is well known that the critical parameters of the Nb₃Sn strand material are strain sensitive; experimental investigations on short samples of basic strands and subsize CICC cables already demonstrated significant effects of residual strain on the critical parameters. In this paper a method is proposed to analyse in detail the thermal strain induced by the cool down from the strand reaction temperature to the coil working conditions. The superconducting strand can be regarded as a very good example of a hierarchical structure, since there is a clear distinction between the micro-scale of the Nb₃Sn filaments, the meso-scale of the SC filament groups and the macro-scale of the strand, where it can be regarded as homogeneous. A constitutive relation for the homogenised micro- and meso-components is deduced from the knowledge of the respective internal structures, starting from an accurate description of the single representative cells. This two-scales homogenisation technique is associated with an efficient finite element procedure for computing effective material coefficients to be used with standard orthotropic 3D elements in structural codes. Finally the finite elements routines developed for the unsmearing process provide the real stress and strain values over each single material, which are essential to catch the local features needed for engineering design.