Three-Scale Bridging for the Non Linear Analysis of Composite

Up to now mainly single scale models have been used to study material and structural behaviour at macroscopic level; however many materials show different internal structures depending on the scale they are considered. Several large experimental equipments, such as particle accelerators and detectors or the future nuclear fusion machines, require the generation of a high magnetic field which is generated by means of superconducting (SC) coils, fed by currents of some tens of kA. Hence SC coils can be regarded as very good examples of hierarchical structures where lower levels take part in the global behaviour. According to the current design, the SC alloy is formed into fine filaments, which are embedded in a low-resistivity matrix of normal metal to make the elementary strand. After that more than one thousand strands are twisted together according to a multi-level twisting scheme to form the final cable and wind the coil. Since the superconducting filaments are strain sensitive, it is extremely important to know the strain field under operating conditions. Because of the scale separation between structural levels a spatial discretization - e.g. that of a finite element mesh - fine enough for the micro level would result in a huge number of elements and unknowns at the macro level. This would be numerically difficult to manage and, first of all, not necessary. In this work an alternative approach is proposed: the macroscopic behaviour is studied by means of a numerical homogenized constitutive relation. The FE tools of theory of asymptotic homogenisation are here extended for the piecewise linear analysis of the SC fibrous composite with non-linear, temperature dependent components. We account also for local material yielding at the stage of microanalysis. To recover the strain inside each single component a suitable unsmearing technique is then applied. The procedure has a general character and can be applied to any cable stage of the coil.