Geometry of Crumpled Paper

We measure the geometry of a crumpled sheet of paper with laser-aided topography and discuss its statistical properties. The curvature of an elastoplastic 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 $0.71\pm 0.01$ in contrast with previous results.

Figure 1

(a) Schematic diagram of the experimental apparatus. (b) Example of a crumpled paper measured with laser-aided topography and rendered with shadow lighting.

Figure 2

The curvature $C$ of a folded sheet plotted versus $F^{\prime }$, the force per unit length. The shapes corresponding to the folded (▽) and unfolded (○) curvatures are shown in the inset.

Figure 3

The probability distribution of the magnitude of the curvature $P(\overline{c})$ averaged over regions of $25{\mathrm{mm}}^2$ for each thickness of paper. The increase of the curvature for thicker sheets indicates that the uncrumpling process does not remove the quenched curvature of the ridges for thicker sheets.

Figure 4

The distribution of measured ridge lengths $P(\ell )$ (a) $h_p=0.156\mathrm{mm}$, (b) $h_p=0.207\mathrm{mm}$, and (c) $h_p=0.360\mathrm{mm}$ display. The dashed line is a fit to Eq. (1) and shows good agreement with and captures the existence of a peak. (d) Width of the log-normal distribution $\sigma$ versus the paper thickness $h_p$, the dashed line is a linear least squares fit.

Figure 5

(a) Histogram of the number of nearest neighbor ridges at a vertex $N_n$. The high frequency of zero occurrences indicates that many ridges dissipate into the paper. (b) Histogram of angle between ridges $\theta$ for $N_n=3$. (c) Hurst plot of the one-dimensional line cross sections of the crumpled paper. The least squares fit gives the Hurst exponent $H$. We observe that for small $w$, there exists a scaling usually associated with $1/f$ noise and that for larger intervals the scaling follows $H\approx 0.70$. (d) Hurst plot of the two-dimensional crumpled gives $H\approx 0.72$. ($h_p=0.207\mathrm{mm}$.)