About lamination upper and convexification lower bounds on the free energy of monoclinic shape memory alloys in the context of T (3)-configurations and R-phase formation

摘要

This work is concerned with different estimates of the quasiconvexification of multi-well energy landscapes of NiTi shape memory alloys, which models the overall behavior of the material. Within the setting of the geometrically linear theory of elasticity, we consider a formula of the quasiconvexification which involves the so-called energy of mixing.We are interested in lower and upper bounds on the energy of mixing in order to get a better understanding of the quasiconvexification. The lower bound on the energy of mixing is obtained by convexification; it is also called Sachs or Reu lower bound. The upper bound on the energy of mixing is based on second-order lamination. In particular, we are interested in the difference between the lower and upper bounds. Our numerical simulations show that the difference is in fact of the order of 1% and less in martensitic NiTi, even though both bounds on the energy of mixing were rather expected to differ more significantly. Hence, in various circumstances it may be justified to simply work with the convexification of the multi-well energy, which is relatively easy to deal with, or with the lamination upper bound, which always corresponds to a physically realistic microstructure, as an estimate of the quasiconvexification. In order to obtain a potentially large difference between upper and lower bound, we consider the bounds along paths in strain space which involve incompatible strains. In monoclinic shape memory alloys, three-tuples of pairwise incompatible strains play a special role since they form so-called T (3)-configurations, originally discussed in a stress-free setting. In this work, we therefore consider in particular numerical simulations along paths in strain space which are related to these T (3)-configurations. Interestingly, we observe that the second-order lamination upper bound along such paths is related to the geometry of the T (3)-configurations. In addition to the purely martensitic regime, we also consider the influence of adding R-phase variants to the microstructure. Adding single variants of R-phase is shown to be energetically favorable in a compatible martensitic setting. However, the combination of several R-phase variants with compatible or incompatible martensite yields significant differences between the bounds considered.