Robert Kohn: The Mathematics of Wrinkles and Folds

Submitted by Tony I Garcia on

The Department of Applied Mathematics is pleased to host this series of colloquium lectures, funded in part by a generous gift from the Boeing Company. This series will bring to campus prominent applied mathematicians from around the world.

Speaker: Robert Kohn

Date: October 24th, 2019, 4pm

Location: (SMI 205)

Title: The Mathematics of Wrinkles and Folds

Abstract: The wrinkling and folding of thin elastic sheets is very familiar: our skin wrinkles; a crumpled sheet of paper has folds; and a flat sheet stretched over a round surface must wrinkle or fold. 

What kind of mathematics is relevant? The stable configurations of a sheet are local minima of a varational problem involving its elastic energy -- which consists of a nonconvex membrane energy (favoring isometry) plus a small coefficient times bending energy (penalizing curvature). The bending term is a singular perturbation; its small coefficient is the sheet thickness squared. The patterns and defects seen in thin sheets arise from energy minimization -- but not in the same way that minimal surfaces arise from area minimization. Rather, the analysis of wrinkles and folds involves the asymptotic character of minimizers as the sheet thickness tends to zero. 

What kind of methods are useful? It has been fruitful to focus on the energy scaling law, in other words the dependence of the minimum energy upon the thickness of the sheet. Optimizing within an ansatz gives an upper bound. A key mathematical challenge is to obtain ansatz-free lower bounds. When the lower and upper bounds are close to agreement they demonstrate the adequacy of the ansatz, and the underlying arguments help to explain why certain configurations are preferred.

A current frontier is the study of wrinkling due to geometric incompatibility. Such wrinkling occurs, for example, when a flat sheet is wrapped around a sphere or a curved shell is flattened by placing it on water. My talk will include some problems of this type, including dramatic recent progress by Ian Tobasco on wrinkling driven by geometric incompatibility in a regime involving "asymptotic isometry."

Youtube: Watch the talk online here