# Curvature-driven wrinkling of thin elastic shells

## TL;DR

This paper develops a rigorous mathematical framework using $mma$-convergence to analyze the energy and pattern formation in thin elastic shells undergoing wrinkling, revealing the interplay of ordered and disordered regions.

## Contribution

It introduces a novel variational approach to characterize wrinkling patterns in elastic shells, including the effects of order, disorder, and defect measures, advancing understanding of shell mechanics.

## Key findings

- Derived a limiting variational problem for shell displacement.
- Characterized patterns via boundary value problems and defect measures.
- Established partial regularity and uniqueness results for pattern formation.

## Abstract

How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on the wrinkling patterns of floating shells, we develop a rigorous method via $\Gamma$-convergence for answering this question to leading order in the shell's thickness and other small parameters. The observed patterns involve "ordered" regions of well-defined wrinkles alongside "disordered" regions whose local features are less robust; as little to no tension is applied, the preference for order is not a priori clear. Rescaling by the energy of a typical pattern, we derive a limiting variational problem for the effective displacement of the shell. It asks, in a linearized way, to cover up a maximum area with a length-shortening map to the plane. Convex analysis yields a boundary value problem characterizing the accompanying patterns via their defect measures. Partial uniqueness and regularity theorems follow from the method of characteristics on the ordered part of the shell. In this way, we can deduce from the principle of minimum energy the leading order features of stamped elastic shells.