MiS Preprint Repository

The linearly elastic energy of a thin film ${\mathrm{\Omega }}_h$ of thickness $h$ with displacement $u$ is given by the functional $E^h(u)={\int }_{{\mathrm{\Omega }}_h}W(\mathrm{\nabla }u)dx$. We consider materials whose energy density $W$ is linearly frame indifferent and vanishes on two linearized wells which are compatible in the plane but incompatible in the thickness direction. We prove compactness of displacement sequences satisfying $E^h(u^{(h)})\le C{h}^2$ and show that the limiting two-dimensional displacements can only have two different interface directions. Our main result is the derivation of the $\mathrm{\Gamma }$-limit of the functionals $\frac{1}{h^2}{E}^h$ as $h\to 0$. It is given by a weighted sum over the lengths of the interfaces.