Asymptotic self-similarity of minimizers and local bounds in a model of shape-memory alloys. (English) Zbl 1481.74601
Summary: We prove that microstructures in shape-memory alloys have a self-similar refinement pattern close to austenite-martensite interfaces, working within the scalar Kohn-Müller model. The latter is based on nonlinear elasticity and includes a singular perturbation representing the energy of the interfaces between martensitic variants. Our results include the case of low-hysteresis materials in which one variant has a small volume fraction. Precisely, we prove asymptotic self-similarity in the sense of strong convergence of blow-ups around points at the austenite-martensite interface. Key ingredients in the proof are pointwise estimates and local energy bounds. This generalizes previous results by one of us to various boundary conditions, arbitrary rectangular domains, and arbitrary volume fractions of the martensitic variants, including the regime in which the energy scales as \(\varepsilon^{2/3}\) as well as the one where the energy scales as \(\varepsilon^{1/2}\).
MSC:
| 74N15 | Analysis of microstructure in solids |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 74B20 | Nonlinear elasticity |
| 74G10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics |
Keywords:
scalar Kohn-Mueller model; martensitic phase transformation; microstructure; asymptotic self-similarity; local boundReferences:
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