Home

Science

The Latest Wrinkle in Crumple Theory

The Latest Wrinkle in Crumple Theory

A piece of crumpled paper, in all of its creased fragmentations, suffers from “geometric frustration.” Who among us can’t sympathize? In a sense, creases happen when a thin sheet of material gets claustrophobia. “New creases form if a sheet doesn’t comfortably fit into its confined area,” said Jovana Andrejević, a Ph.D. student in applied physics at Harvard and the lead author of a new paper detailing the latest advances in paper crumpling."

“The sheet is stressed, so something needs to happen to relieve that stress,” Ms. Andrejević said. She was speaking from her childhood home outside Chicago, where she is living temporarily with her parents and her twin sister, Nina, who is pursuing a Ph.D. in materials science at the Massachusetts Institute of Technology. “The formation of a crease is how the stress is relieved,” Ms. Andrejević said. “The role of the creases is effectively to protect as much of the sheet as possible from further damage.” Those protected areas — the “facets” — and how they break into smaller and smaller fragments when recrumpled are the focus of the new study. Published on Friday in Nature Communications, the investigation builds on a 2018 study by some of the same researchers. The earlier study showed how paper crumpling — a seemingly random, disordered and complex process — displays a surprising amount of mathematical order. This result represented “a remarkable reduction in complexity,” the researchers noted in their 2018 paper. Crumpled paper is a proxy for much more than our pandemic angst. Similar dynamics are at play, for instance, in the wrinkling of graphene sheets for high-performance batteries; the flexibility of wearable electronic devices and artificial skin, and the folding of Earth’s crust.

The 2018 study showed that the cumulative total length of all creases — the “mileage” — on a crumpled sheet served as a predictor of how the sheet would behave when crumpled again and again. Although the researchers crumpled a sheet as many as 70 times, after a few cycles it was difficult to see any difference, from one crumple to the next. But by tracking and analyzing the mileage, they noticed that a sheet never stopped forming creases, although it did so at a logarithmic rate, adding fewer creases with each recrumpling. Perhaps the most unexpected aspect of the original finding was the element of “universality,” according to Chris Rycroft, an applied mathematician and principal investigator on both papers. Two separate sheets, crumpled the same way, develop visually distinct crease patterns but rack up comparable overall mileage. “It’s very surprising the mileage is largely independent of the crease pattern,” he said.

The 2018 study showed that the cumulative total length of all creases — the “mileage” — on a crumpled sheet served as a predictor of how the sheet would behave when crumpled again and again. Although the researchers crumpled a sheet as many as 70 times, after a few cycles it was difficult to see any difference, from one crumple to the next. But by tracking and analyzing the mileage, they noticed that a sheet never stopped forming creases, although it did so at a logarithmic rate, adding fewer creases with each recrumpling. Perhaps the most unexpected aspect of the original finding was the element of “universality,” according to Chris Rycroft, an applied mathematician and principal investigator on both papers. Two separate sheets, crumpled the same way, develop visually distinct crease patterns but rack up comparable overall mileage. “It’s very surprising the mileage is largely independent of the crease pattern,” he said.

In the summer of 2019, Ms. Andrejević and Dr. Rycroft developed the facet analysis into a theory while visiting the Lawrence Berkeley National Laboratory in Berkeley, Calif., where Dr. Rycroft holds a visiting position. During hikes on the weekend, they found inspiration in the natural world, looking at eroding rockslides and pebbled beaches. Around the same time Ms. Andrejević delved into the scientific literature on fragmentation theory, which explores the physical principles explaining how materials (rock, glass, volcanic debris, meteorites) break into smaller and smaller pieces. The theory provides a way of characterizing the resulting fragments. For example, as fragmentation progresses, materials — regardless of their starting point — quickly tend toward a predictable “steady-state” distribution of fragment sizes. Ms. Andrejević and her collaborators wondered whether the evolution of crumpling paper could be described by the same principles and statistical trends. After hand-tracing the facets of the scanned sheets on her tablet, Ms. Andrejević colored each segment by area, and then sorted them by size. A familiar order-amid-disorder emerged. Comparing one sheet to the next, she noticed that, despite their differing crease patterns, the size distributions of facets were similar. Furthermore, the size distribution of facets fit perfectly with the predictions of fragmentation theory.

After hand-tracing the facets of the scanned sheets on her tablet, Ms. Andrejević colored each segment by area, and then sorted them by size. A familiar order-amid-disorder emerged. Comparing one sheet to the next, she noticed that, despite their differing crease patterns, the size distributions of facets were similar. Furthermore, the size distribution of facets fit perfectly with the predictions of fragmentation theory. This provided a theoretical underpinning for the behavior seen in the experimental study from 2018. “We were very excited about this result because it supports the idea that there is some universality across diverse disordered systems,” Ms. Andrejević said. Video