Surface energies emerging in a microscopic, two-dimensional two-well problem

Kitavtsev G., Luckhaus S., Rüland A.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol.147, no.5, pp.1041-1089, 2017 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 147 Issue: 5
  • Publication Date: 2017
  • Doi Number: 10.1017/s0308210516000433
  • Journal Name: Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1041-1089
  • Keywords: Discrete-to-continuum limit, Geometric rigidity theorems, Solid-solid phase transformation, Surface energies
  • Middle East Technical University Northern Cyprus Campus Affiliated: No

Abstract

In this paper we are interested in the microscopic modelling of a two-dimensional two-well problem that arises from the square-to-rectangular transformation in (two-dimensional) shape-memory materials. In this discrete set-up, we focus on the surface energy scaling regime and further analyse the Hamiltonian that was introduced by Kitavtsev et al. in 2015. It turns out that this class of Hamiltonians allows for a direct control of the discrete second-order gradients and for a one-sided comparison with a two-dimensional spin system. Using this and relying on the ideas of Conti and Schweizer, which were developed for a continuous analogue of the model under consideration, we derive a (first-order) continuum limit. This shows the emergence of surface energy in the form of a sharp-interface limiting model as well the explicit structure of the minimizers to the latter.