Nonlinear response of tensed membranes

We study the response of elastic membranes under tension$ T$, to localized normal forces $F$. Focusing on simple geometries, characterized by translational or radial symmetries, we calculate the membrane shape for a range of values of F and T by numerically solving the appropriate FvK equations. We find that the linear regime, where membrane displacement is proportional to F, vanishes in the asymptotic limit$F/T<<1$, and characterize scaling properties of the resulting nonlinear response. We discuss the relevance of our results to the puzzling scaling behavior of the length of radial wrinkles, recently found in ``drop on membrane'' experiments (Huang\textit{ et al. }Science 2007).