A quantitative rigidity result for the cubic-to-tetragonal phase transition in the geometrically linear theory with interfacial energy (Q2882512)

The present paper is devoted to the cubic-to-tetragonal phase transition in a shape memory alloy. The authors consider a geometrically linear elasticity framework. In this framework, in [\textit{G. Dolzmann} and \textit{S. Müller}, Meccanica 30, No. 5, 527--539 (1995; Zbl 0835.73061)], it was shown that the only stress-free configurations are (locally) twins (i.e., laminates of just two of the three martensitic variants). However, configurations with arbitrarily small elastic energy are not necessarily close to these twins. In this paper, an interfacial energy is taken into account, and a (local) lower bound on the elastic plus interfacial energy in terms of the Martensitic volume fractions is established. The introduction of an interfacial energy introduces a length scale, and thus, together with the linear dimensions of the sample, a non-dimensional parameter. The authors' lower Ansatz-free bound has an optimal scaling in this parameter. It is the scaling predicted by a reduced model introduced and analyzed in [\textit{R. V. Kohn} and \textit{S. Müller}, `` Branching of twins near an austenite-twinned-martensite interface'', Phil. Mag. A 66, No. 5, 697--715 (1992; \url{doi:10.1080/01418619208201585})] with the purpose to describe the microstructure near an interface between austenite and twinned martensite. The optimal construction features branching of the martensitic twins when approaching this interface. Note that the present work is an extension the authors' paper [Commun. Pure Appl. Math. 62, No. 12, 1632--1669 (2009; Zbl 1331.82064)].