Thermo-Mechanics of Superconducting Coils for Fusion Reactors

An analysis of a superconducting coil used in fusion devices reveals that a wide spectrum of coupled physical problems should be taken into account during its design. This is also a good example of a general analysis of a hierarchical structure. Due to the large number of repetitive superconducting strands, the single cable can be considered as a homogeneous body in meso-scale, each strand being at the same time a micro composite of several components. A bundle of cables makes in turn a substantial part of a larger, D shaped superconducting coil. The above problems are numerically analysed using various, specialised algorithms of FE method. The single strand has its internal structure. It is made of filaments of superconducting alloy embedded in a matrix of bronze. The detailed geometry of the microstructure depends on the strand type. At this stage the classical model of Kirchhoff-Bernoulli beam is applicable, for which the beam type stiffness is computed by means of homogenisation theory and beam-type kinematical hypothesis. A triplet of strands can be seen as a single beam, but in this case the Kirrchhoff-Bernoulli hypothesis is not sufficient to define the mechanics of the structure. The common mechanical work of the triplet is influenced by friction between strands, their pointwise contacts and the helicoidal geometry. We propose a mixed Kirchhoff-Timoshenko approach with a torsion-tension coupling to model the beam-like behaviour at this stage and at the higher order cabling stages. Thus we are performing a successive substitution of discrete models involving many beams (three at the level of the triplet, four at the level of quadruplet) with a single, continuous beam model, that can be qualitatively different and the parameters of which can be deduced from the preceding cabling stages. This recursive substitution allows to maintain the number of beam elements in the model around a reasonable optimum. Since the thermo-mechanical loading is applied at the macro level, the analysis of the macro structure is indispensable. On the other hand, the most important engineering phenomena are related to the micro components of the composite. In our approach we can recover the micro-structural effects from the results at the macro level via suitable unsmearing procedures. All along the hierarchically organised FE analysis of the structure we deal with coupled thermo-mechanical phenomena and local material yielding occurring at the level of the micro structure. The lecture is illustrated with some examples revealing the specific properties of the different beam models at several cabling stages and with an overview of the model of the bundle of strands.