The Geometry of Crumpled Paper

TL;DR

This paper investigates the detailed geometry and statistical properties of crumpled paper sheets, revealing how ridges form, break, and connect, with implications for understanding crumpling mechanics.

Contribution

It provides new quantitative measurements of ridge curvature, distribution, and network structure, and introduces a hierarchical model for ridge breaking during crumpling.

Findings

Ridge curvature scales linearly with applied force.

Curvature distribution follows an exponential form.

Ridge network shows incomplete connectedness and many ridges terminate without bifurcation.

Abstract

We measure the geometry of a crumpled sheet of paper with laser-aided topography and discuss its statistical properties. The curvature of an elasto-plastic fold scales linearly with applied force. The curvature distribution follows an exponential form with regions of high curvature localized along ridges. The measured ridge length distribution is consistent with a hierarchical model for ridge breaking during crumpling. A large fraction of the ridges are observed to terminate without bifurcating and the ridge network connectedness is not as complete as anticipated. The self-affinity of the surface is characterized by a Hurst exponent of $0.72\pm 0.01$ in contrast with previous results.