Crumple Theory: We Can Learn a Lot From How Paper Crumples

That idea formed the basis of experiments performed several years ago by another of the paper’s authors, former Harvard physicist Shmuel M. Rubinstein, who is now at the Hebrew University of Jerusalem, and his students. As Rycroft explains, Rubenstein and his team crumpled a thin sheet repeatedly and measured the total length of creases on the sheet, which they called “mileage.” That research is described in this 2018 paper.

“They found that the growth of mileage is strikingly reproducible, and each time the accrual of new mileage would get a little less, because the sheet is progressively getting weaker,” Rycroft says.

That finding stumped the physics community, and Rycroft and Harvard doctoral candidate Jovana A Andrejevic wanted to understand why crumpling behaves that way.

“We found that the way to make progress was not to focus on the creases themselves, but rather to look at the undamaged facets that are outlined by the creases,” Rycroft says.

Crumple theory
The total length of the creases on a crumpled sheet of paper is called its “mileage.” Repeated crumpling produces less new mileage as the paper becomes weaker.

Flavio Coelho/Getty Images

“In the experiment, thin sheets of Mylar, a thin film that crumples similarly to paper, were systematically crumpled several times, developing some new creases with each repetition,” Andrejevic, the 2021 paper’s lead author, explains via email. “In between crumples, the sheets were carefully flattened and their height profile scanned using an instrument called a profilometer. The profilometer makes measurements of the height map across the surface of the sheet, which allows us to calculate and visualize the locations of creases as an image.”

Because creasing can be messy and irregular, it generates “noisy” data that can be tough for computer automation to make sense of. To get around that problem, Andrejevic hand-traced the crease patterns on 24 sheets, using a tablet PC, Adobe Illustrator and Photoshop. That meant recording 21,110 facets in total, as this recent New York Times article details.

Thanks to Andrejevic’s labors and image analysis, “we could look at the distributions of facet sizes as the crumpling progressed,” Rycroft explains. They found that the size distributions could be explained by fragmentation theory, which looks at how objects ranging from rocks, glass shards and volcanic debris break up into small pieces over time. (Here’s a recent paper from the Journal of Glaciology that applies it to icebergs.)

“That same theory can accurately explain how the facets of the crumpled sheet break up over time as more creases form,” Rycroft says. “We can also use it to estimate how the sheet becomes weaker after crumpling, and thereby explain how the accumulation of mileage slows down. This allows us to explain the mileage results — and the logarithmic scaling — that were seen in the 2018 study. We believe that the fragmentation theory provides a perspective on the problem and is especially useful to model the accumulation of damage over time,” Rycroft says.

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