The coarsening of folds in hanging drapes

TL;DR

This paper analyzes how the pattern of folds in a hanging elastic sheet coarsens with height, identifying the energy scaling laws and optimal configurations through mathematical bounds and extending previous physics-based models.

Contribution

The authors derive the scaling law for the correction to elastic energy due to finite sheet thickness, and compare self-similar and alternative constructions for fold coarsening.

Findings

Self-similar coarsening achieves optimal energy scaling in certain regimes.

Other configurations involving lateral spreading outperform self-similar patterns elsewhere.

The analysis confirms and extends physics literature on fold coarsening in hanging drapes.

Abstract

We consider the elastic energy of a hanging drape -- a thin elastic sheet, pulled down by the force of gravity, with fine-scale folding at the top that achieves approximately uniform confinement. This example of energy-driven pattern formation in a thin elastic sheet is of particular interest because the length scale of folding varies with height. We focus on how the minimum elastic energy depends on the physical parameters. As the sheet thickness vanishes, the limiting energy is due to the gravitational force and is relatively easy to understand. Our main accomplishment is to identify the "scaling law" of the correction due to positive thickness. We do this by (i) proving an upper bound, by considering the energies of several constructions and taking the best; (ii) proving an ansatz-free lower bound, which agrees with the upper bound up to a parameter-independent prefactor. The coarsening of folds in hanging drapes has also been considered in the recent physics literature, using a self-similar construction whose basic cell has been called a "wrinklon." Our results complement and extend that work, by showing that self-similar coarsening achieves the optimal scaling law in a certain parameter regime, and by showing that other constructions (involving lateral spreading of the sheet) do better in other regions of parameter space. Our analysis uses a geometrically linear F\"{o}ppl-von K\'{a}rm\'{a}n model for the elastic energy, and is restricted to the case when Poisson's ratio is zero.