The physics of crêpes: Elasto-gravity control of soft folding

Abstract

Like a crêpe 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.

Figures (4)

• Figure 1: Soft folding: a) Multi folded shape of a crêpe of radius $R \approx 13~\text{cm}$ and $\ell_{eg} \approx 0.86~\text{cm}$. b) Elastomeric circular membrane of radius $R = 7.5~\text{cm}$ and $\ell_{eg} = 1.99~\text{cm}$ wrapped in two configurations: i) a symmetric half-fold that forms a stable loop, ii) an asymmetric, partial fold that is mechanically unstable and unravels upon release. (scale bars are $2$ cm)
• Figure 2: Folding stability: a) Schematic of the soft folding geometry. b) Three representative folding configurations at different gap values $\delta$, for a strip $L=30$ cm long and $t=40$$\mu$m thick ($\ell_{eg}= 3.71 ~\text{cm}$). c) Fold length and height as functions of the gap for various sheet thicknesses; all lengths are rescaled by the elasto-gravity length $\ell_{eg}$. d) Dimensions of the fold at criticality (length, height, and gap) as functions of $\ell_{eg}$. In c) and d), solid lines represent theoretical predictions, while shaded regions indicate the range expected from frictional effects.
• Figure 3: Stacking folds: a) Snapshots of multi-folding states for rectangular sheets with $\ell_{eg}=3.71~\mathrm{cm}$ and $\ell_{eg}=2.4~\mathrm{cm}$, respectively. b) Snapshots of multi-folding states for a circular soft sheet (for $R=7.5~\text{cm}$ and $\ell_{eg}=4.8~\mathrm{cm}$), where the sector angle $\alpha$ is halved at each fold. c) Maximum number of times a rectangular strip (width $w=6~\mathrm{cm}$ and length $L=60~\text{cm}$) can be folded as a function of the rescaled length $L/\ell_{eg}$. d) Maximum fold count for circular sheets (crêpes) against rescaled radius $R/\ell_{eg}$. In a) and c), contour lines show the geometric prediction; the gray band indicates the admissible range for the fold to be stable.
• Figure 4: Dynamics of unfolding: a) Chronophotography of the unfolding dynamics for $\ell_{eg}=2.2~\text{cm}$ ($E =\, 884~\text{kPa}$, $h =1.13~\text{mm}$), the solid line reports the tip trajectory and the color modulates the instantaneous speed. Scale bars are $1~\text{cm}$. b) Peak flipping speed as a function of elasto-gravity length [inset: full kinematics of the sample shown in b), red point is the maximal speed].