JFI ad hoc Seminar
Ian Tobasco
Math Department, University of Illinois, Chicago
A geometric theory of wrinkling for confined elastic shells
Thin elastic shells easily take on shapes wildly different from their own. Motivated by the puzzle of determining the wrinkle patterns exhibited by shallow shells floating on a water bath, we obtain a fully rigorous asymptotic expansion of the energy (elastic and otherwise) valid in the high-frequency limit. After renormalizing by the typical energy of wrinkles, we derive a coarse-grained model in which an elastically compatible pattern is assigned energy proportional to the difference between its intrinsic undeformed area and its projected area in the plane. Energetically optimal patterns therefore maximize their projected area. Surprisingly, this limiting model turns out to be explicitly solvable in a large variety of cases, including for shells whose (possibly non-constant) curvature is of a known sign. We demonstrate our methods with concrete examples, and offer comparisons with simulation and experiment. What results is an ansatz-free explanation for the geometry of wrinkle patterns in confined elastic shells.
photo courtesy Joey Paulsen, Syracuse University