Quantitative aspects of the rigidity of branching microstructures in shape memory alloys via H-measures

Simon, T.M.

Zusammenfassung

Based on a geometrically linearized theory, we describe the partition into twins observed in microstructures of shape memory alloys undergoing cubic-to-tetragonal transformations in an ansatz-free way using H-measures, a tool of microlocal analysis to describe the direction of oscillations and concentration effects of weakly convergent sequences. As an application, we give a 𝐵1,∞2/3-estimate for the characteristic functions of twins generated by finite energy sequences in the spirit of compactness for Γ-convergence. Heuristically, this suggests that the larger-scale interfaces, such as habit planes, can cluster on a set of Hausdorff-dimension 3−23. We provide evidence indicating that this fractional dimension is optimal. Furthermore, we get an essentially local lower bound for the blow-up behavior of the limiting energy density close to a habit plane.