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Nonlinear Analysis

Exact solutions for the wrinkle patterns of confined elastic shells

Ian Tobasco, University of Illinois at Chicago

Location:  zoom
Date & time: Wednesday, 05 May 2021 at 9:30AM - 10:30AM

Abstract: A basic fact of geometry is that there are no length-preserving maps from a sphere to the plane. But what happens if you confine a thin elastic shell, which prefers to be a curved surface but can deform approximately isometrically, to reside nearby a plane? It wrinkles, and forms a remarkable pattern of peaks and troughs, the arrangement of which is sometimes random, sometimes not depending on the shell. After a brief introduction to the mathematical modeling of thin elastic shells, this talk will focus on a new set of simple, geometric rules we have derived for wrinkle patterns via Gamma-convergence and convex analysis of the limit problem. Our rules govern the asymptotic layout of the wrinkle peaks and troughs --- for instance, negatively curved wrinkles tend to arrange along segments solving the minimum exit time problem, in the infinitesimally wrinkled limit. Positively curved shells can be understood more or less completely as well, through a hidden duality with their negatively curved counterparts. Our predictions for the wrinkle patterns of confined shells match the results of numerous experiments and simulations done with Eleni Katifori (U. Penn) and Joey Paulsen (Syracuse). Underlying their analysis is a certain class of interpolation inequalities of Gagliardo-Nirenberg type, whose best prefactors encode the optimal patterns.