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- Title
Energy scaling laws for microstructures: from helimagnets to martensites.
- Authors
Ginster, Janusz; Zwicknagl, Barbara
- Abstract
We consider scalar-valued variational models for pattern formation in helimagnetic compounds and in shape memory alloys. Precisely, we consider a non-convex multi-well bulk energy term on the unit square, which favors four gradients (± α , ± β) , regularized by a singular perturbation in terms of the total variation of the second derivative. We derive scaling laws for the minimal energy in the case of incompatible affine boundary conditions in terms of the singular perturbation parameter as well as the ratio α / β and the incompatibility of the boundary condition. We discuss how well-studied models for martensitic microstructures in shape-memory alloys arise as a limiting case, and relations between the different models in terms of scaling laws. In particular, we show that scaling regimes arise in which an interpolation between the rather different branching-type constructions in the spirit of Kohn and Müller (Commun Pure Appl Math 47(4):405–435, 1994) and Ginster and Zwicknagl (J Nonlinear Sci 33:20, 2023), respectively, occurs. A particular technical difficulty in the lower bounds arises from the fact that the energy scalings involve various logarithmic terms that we capture in matching upper and lower scaling bounds.
- Subjects
SHAPE memory alloys; SINGULAR perturbations; MICROSTRUCTURE; INTERPOLATION
- Publication
Calculus of Variations & Partial Differential Equations, 2024, Vol 63, Issue 1, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-023-02604-5