Multiscale analysis by Gamma-convergence of a one-dimensional nonlocal functional related to a shell-membrane transition

We study the asymptotic behavior of one-dimensional functionals associated with the energy of a thin nonlinear elastic spherical shell in the limit of vanishing thickness ( proportional to a small parameter) epsilon and under the assumption of radial deformations. The functionals are characterized by the presence of a nonlocal potential term and defined on suitable weighted functional spaces. The shell-membrane transition is studied at three different relevant scales. For each we give a compactness result and compute the Gamma-limit. In particular, we show that if the energies on a sequence of configurations scale as epsilon(3/2), then the limit configuration describes a ( locally) finite number of transitions between the undeformed and the everted configurations of the shell. We also highlight a kind of "Gibbs phenomenon" by showing that nontrivial optimal sequences restricted between the undeformed and the everted configurations must have energy scaling of at least epsilon(4/3).