Surface Energies Emerging in a Microscopic, Two-Dimensional Two-Well Problem

TL;DR

This paper models surface energies in a microscopic two-dimensional two-well problem related to shape-memory materials, deriving a continuum limit that reveals surface energy emergence and minimizer structure.

Contribution

It introduces a discrete Hamiltonian framework for the two-well problem and derives a first-order continuum limit showing surface energy emergence.

Findings

Surface energy appears as a sharp-interface limit.

Explicit structure of minimizers is characterized.

The model connects discrete Hamiltonians with continuum surface energies.

Abstract

In this article we are interested in the microscopic modeling of a two-dimensional two-well problem which 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 analyze the Hamiltonian which was introduced in \cite{KLR14}. 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-dimensonal spin system. Using this and relying on the ideas of Conti and Schweizer \cite{CS06}, \cite{CS06a}, \cite{CS06c}, 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.