Surface energies emerging in a microscopic, two-dimensional two-well problem
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 Ham...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
14 August 2017
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| In: |
Proceedings. Section A, Mathematics
Year: 2017, Volume: 147, Issue: 5, Pages: 1041-1089 |
| ISSN: | 1473-7124 |
| DOI: | 10.1017/S0308210516000433 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1017/S0308210516000433 Verlag, lizenzpflichtig, Volltext: https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/surface-energies-emerging-in-a-microscopic-twodimensional-twowell-problem/0420D60A4C84010A8C87FE1B4CF0C5E7 
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| Author Notes: | Georgy Kitavtsev, Stephan Luckhaus, Angkana Rüland |
MARC
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| 520 | |a 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. | ||
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