MiS Preprint Repository

We obtain a Gamma-convergence result for the gradient theory of solid-solid phase transitions, in the case of two geometrically linear wells in two dimensions. We consider the functionals $$I_{ϵ}[u]={\int }_{\mathrm{\Omega }}\frac{1}{ϵ}W(\mathrm{\nabla }u)+ϵ|{\mathrm{\nabla }}^{2}u{|}^2$$ where $u:\mathrm{\Omega }\subset R^2\to R^2$, W depends only on the symmetric part of $\mathrm{\nabla }u$, and W(F)=0 for two distinct values of F, say A and B. We show that, under suitable growth assumptions on W and for star-shaped domains $\mathrm{\Omega }$, as $ϵ\to 0$ $I_{ϵ}$ converges, in the sense of Gamma convergence, to a functional $I_0$. The limit $I_0$ is finite only on functions u such that the symmetric part of $\mathrm{\nabla }u$ is a function of bounded variation which takes only values A and B. On those functions, the energy concentrates on the jump set J of $\mathrm{\nabla }u$, with a surface energy depending on the normal $\nu$ to J, and is given by $$I_0[u]={\int }_{J}k(\nu )d{H}^1.$$ The interfaces can have, in general, two orientations.