We propose a nonlinear elasticity model for vesicle membranes which is an Eulerian version of a model introduced by Pantz and Trabelsi.We describe the limit behavior of sequences of configurations whose energy goes to 0 in a fixed domain. The material is highly anisotropic and the analysis is based on some rigidity estimates adapted to this anisotropy. The main part of the paper is devoted to these estimates and to some of their consequences. The strongest form of these estimates are used in a second article to derive the thin-shell limit bending theory of the model.