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

Sergio Conti

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

$\displaystyle J_{\varepsilon,\beta}(u)=\beta\Vert u_0\Vert^2_{H^{1/2}((0,h))}+ ... ...(0,h)} \vert\partial_x u\vert^2 + \varepsilon \vert\partial_y\partial_y u\vert$

among all $u:(0,l)\times(0,h)\to \mathbb{R}$ such that $\partial_y u=\pm 1$ almost everywhere. We prove that for small $\varepsilon$ the minimum of $J_{\varepsilon, \beta}$ scales as the smaller of $\varepsilon^{1/2}\beta^{1/2}l^{1/2}h$ and $\varepsilon^{2/3}l^{1/3}h$ , 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 $\varepsilon/l\gg \beta^{3}$ and a branching regime at $\varepsilon/l\ll \beta^{3}$ .

Download preprint: pdf (160 Kb)

Published in: Cont. Mech. Thermod. 17, pp. 469-476, 2006.

Link to Published paper

BibItem