Abstract
The Kohn-Müller model for the formation of domain patterns in martensitic shape-memory alloys consists in minimizing the sum of elastic, surface and boundary energy in a simplified scalar setting, with a nonconvex constraint representing the presence of different variants. Precisely, one minimizes
among all u:(0,l)×(0,h)→ ℝ such that ∂ y u = ± 1 almost everywhere. We prove that for small ε the minimum of J ε, β scales as the smaller of ε1/2β1/2 l 1/2 h and ε2/3 l 1/3 h, as was conjectured by Kohn and Müller. Together with their upper bound, this shows rigorously that a transition is present between a laminar regime at ε/l≫ β3 and a branching regime at ε/l≪ β3.
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Communicated by R. E. Caflisch
PACS 64.70.Kb, 62.20.-x, 02.30.Xx
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Conti, S. A lower bound for a variational model for pattern formation in shape-memory alloys. Continuum Mech. Thermodyn. 17, 469–476 (2006). https://doi.org/10.1007/s00161-006-0013-x
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DOI: https://doi.org/10.1007/s00161-006-0013-x