A rigidity result for a perturbation of the geometrically linear three‐well problem

Abstract

We study a three-dimensional model for alloys that undergo a cubic-to-tetragonal phase transition in the martensitic phase. Any pair of the three martensitic variants can form a stress-free laminate. However, this laminate is only compatible on average with the remaining variant. The resulting local stresses favor a microstructure if all three variants are present (for instance, because of an externally imposed average strain).Next to the linearized elastic energy, the variational model features an interfacial energy between the three variants. This introduces a material length scale, which together with the sample size (which we mimic by periodic boundary conditions) gives rise to a nondimensional parameter Á.We rigorously establish the scaling of the minimal energy e per volume in case of externally imposed volume fractions of the martensitic variants in Á. More precisely, we show e Á 2=3 . The upper bound construction is achieved by a few patches of branched laminates; the lower bound relies on suitable interpolation inequalities. This is in the spirit of a celebrated work by Kohn and Müller and relies on techniques developed for domain branching in micromagnetics.We also prove a rigidity result in the sense that if the energy per volume e of a configuration is much smaller than Á 2=3 , the configuration is approximately a simple unbranched laminate. In particular, one of the three volume fractions has to be small. This is related to similar rigidity results by Dolzmann and Müller and by Kirchheim.

Nucleation Barriers for the Cubic‐to‐Tetragonal Phase Transformation

“…These results are global versions of corresponding results in a periodic setting in [4]. These results are global versions of corresponding results in a periodic setting in [4].…”

“…The following lemma is taken from lemma 3.1 and step 1 in the proof of proposition 2.4 in [4]. The following lemma is taken from lemma 3.1 and step 1 in the proof of proposition 2.4 in [4].…”

“…Notice that constructions of the austenite-to-martensite interface have been given in the literature (e.g., [4,5,18,19]). Notice that constructions of the austenite-to-martensite interface have been given in the literature (e.g., [4,5,18,19]).…”

“…They showed the energetic optimality of a self-similar configuration of two martensitic phases. The energetic scaling of an austenite-martensite mixture for a threedimensional model was recently addressed by Capella and Otto [4,5] for the cubicto-tetragonal phase transformation. The energetic scaling of an austenite-martensite mixture for a threedimensional model was recently addressed by Capella and Otto [4,5] for the cubicto-tetragonal phase transformation.…”

Deformation concentration for martensitic microstructures in the limit of low volume fraction

“…These studies suggest a transition between uniform structures and the formation of microstructures. Some of the results have been extended also to the vectorvalued case (see [8,9,10,24,7,6,32,33]). Similar phenomena have been found for a variety of variational models, with applications including pattern formation in ferromagnets (see [15,13,39,34]), in type-1-superconductors (see [14,12,17]), and in thin compressed films (see [4,30,31,5,3]).…”