The physics of cr\^epes: Elasto-gravity control of soft folding

TL;DR

This paper uncovers how a single length scale, the elasto-gravity length, governs the shape, stability, and dynamics of reversible folds in elastic sheets under gravity, providing a unified physical framework.

Contribution

It introduces the elasto-gravity length as the key parameter controlling soft folding behavior, supported by experiments and theory, unifying shape, stability, and dynamics.

Findings

Fold shapes and stability thresholds collapse when scaled by $\, ext{ell}_{eg}$.

Unfolding speed scales as $\, ext{sqrt}(g \, ext{ell}_{eg})$, indicating gravity-controlled dynamics.

Maximum number of reversible folds is limited by $\, ext{ell}_{eg}$.

Abstract

Like a cr\^epe resting on a plate, a thin elastic sheet can fold smoothly under its own weight, forming reversible shapes without creases or imposed hinges. Such soft folds arise from a balance between elastic bending and gravity, yet their stability, packing limits, and dynamics remain poorly understood. Here we show that these behaviors are governed by a single physical length scale, the elasto-gravity length $\ell_{eg}$. Using experiments and heavy-elastica theory, we demonstrate that $\ell_{eg}$ sets the characteristic fold geometry, determines when a fold becomes unstable and unfolds, and limits how many reversible folds can be stacked in rectangular and circular sheets. In particular, when lengths are rescaled by $\ell_{eg}$, fold shapes and stability thresholds collapse across materials and thicknesses. We further show that unfolding follows a universal speed scaling $v \sim \sqrt{g\,\ell_{eg}}$, revealing a gravity-controlled time scale for the release of stored bending energy. Together, these results establish a unified physical framework for reversible folding, compact storage, and gravity-assisted deployment of thin elastic sheets.