A lower bound for a variational model for pattern formation in shape-memory alloys

Conti, Sergio
April 2006
Continuum Mechanics & Thermodynamics;Apr2006, Vol. 17 Issue 6, p469
Academic Journal
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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