MiS Preprint Repository

The nonlinear elastic energy of a thin film of thickness $h$ is given by a functional $E^h$. Recently, Friesecke, James and Müller derived the $\mathrm{\Gamma }$-limits, as $h\to 0$, of the functionals $h^{-\alpha }{E}^h$ for $\alpha \ge 3$. We study the local invertibility of almost minimizers of these functionals and prove an upper bound for the diameter of preimages. We also prove that almost minimizers are in fact invertible almost everywhere on subdomains which have positive distance from the boundary, and we provide an upper bound for this distance. Then we give two examples which show that the bounds derived in the positive results are optimal. In particular, for all $\alpha \ge 3$ there exist almost minimizers which are two to one on a connected set of nonzero volume.