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Exact solutions for the wrinkle patterns of geometrically-confined shells

Analysis, Dynamics, and Applications Seminar

Exact solutions for the wrinkle patterns of geometrically-confined shells
Series: Analysis, Dynamics, and Applications Seminar
Location: Online
Presenter: Ian Tobasco, Department of Mathematics, University of Illinois, Chicago

A basic fact of geometry is that there is no length-preserving map from any neighborhood of a sphere to the plane. But what happens if you force a thin elastic sphere, which likes to deform in an approximately isometric way, to reside nearby a plane? It wrinkles, and the wrinkles form a pattern we'd like to understand. This talk will present a recently derived method based on energy minimization for determining the wrinkle patterns formed when a curved shell is put onto a soft substrate. Perhaps surprisingly, the energetically optimal patterns can in many cases be solved for exactly and by hand. Further investigation of the solutions leads to a beautiful and unexpected connection between the patterns of negatively and positively curved shells. These theoretical results are obtained by rigorous analysis of the equations of elasticity in the vanishing thickness limit. The predicted patterns match the results of numerous experimental and numerical tests, done in collaboration with Eleni Katifori (U. Penn) and Joey Paulsen (Syracuse).

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