# Analysis Seminar

#### 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 minimize

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.