Analysis Seminar

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The cost of crushing: curvature-driven wrinkling of thin elastic shells

Speaker: Ian Tobasco, Univ of Michigan

Location: Warren Weaver Hall 1302

Date: Thursday, October 18, 2018, 11 a.m.

Synopsis:

How much energy does it cost to stamp a thin elastic shell flat? Motivated
by recent experiments on the wrinkling patterns formed by thin shells
floating on a water bath, we develop a rigorous method via Gamma-convergence
for evaluating the cost of crushing to leading order in the shell's
thickness and other small parameters. The experimentally observed patterns
involve regions of well-defined wrinkling alongside totally disordered
regions in which no single direction of wrinkling is preferred. Our goal is
to explain the appearance and lack thereof of such “wrinkling domains". The
basic mathematical objects that emerge in the limit are (linearly) short
maps from the mid-shell into the plane, and defect measures which describe
the wrinkling patterns. To solve for the limiting shape, one must maximize
the total area covered in the plane subject to a shortness constraint; to
solve for the optimal patterns, one must minimize the total defect subject
to a curvature constraint. We analyze these limiting problems using convex
duality, and obtain a boundary value-like problem that completely
characterizes optimal defect measures. Optimal defect measures are not in
general unique. Nevertheless, in some cases their restrictions to certain
sub-domains are uniquely determined, and explicit formulas exist.