Mathematics Colloquium

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The many, elaborate wrinkle patterns of confined elastic shells

Speaker: Ian Tobasco, UIC

Location: Warren Weaver Hall 1302

Date: Monday, February 6, 2023, 3:45 p.m.

Synopsis:

A basic fact of geometry is that there are no length-preserving smooth maps from a spherical cap to a plane. But what happens if you try to press a curved elastic shell flat anyways? It wrinkles and forms a remarkable pattern of peaks and troughs, the arrangement of which is sometimes random, sometimes not. After a brief introduction to the mathematics of elastic sheets and shells, this talk will focus on a new set of simple, geometric rules we have discovered for predicting wrinkle patterns. These rules are the latest from an ongoing study of elastic confinement using the tools of Gamma-convergence and convex analysis. The asymptotic expansions they encode reveal a beautiful and unexpected connection between opposite curvatures — apparently, surfaces with positive or negative Gaussian curvatures form dual pairs of wrinkle patterns when confined. Our predictions match the results of experiments and simulations done with the physicists Eleni Katifori (U. Penn) and Joey Paulsen (Syracuse). Behind their analysis is a class of interpolation inequalities of Gagliardo-Nirenberg type, whose best prefactors encode the optimal patterns.

This talk will be mathematically self-contained, not assuming a background in elasticity.