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

What do a sheet of paper being crushed into a ball and tossed into a wastebasket, the front end of a car deforming in a crash, and the Earth's crust gradually forming mountains over millions of years all have in common? They're all undergoing a physical process called crumpling, which occurs when a relatively thin sheet of material — one with a thickness that's much less than its length or width — has to fit into a smaller area.

And while it's easy to imagine crumpling as mere desultory disarray, scientists who've studied crumpling have discovered that it's anything but that. To the contrary, crumpling turns out to be a predictable, reproducible process governed by mathematics. The latest breakthrough in our understanding of crumpling is a paper recently published in Nature Communications, in which researchers describe a physical model for what happens when thin sheets are crumpled, unfolded and recrumpled.

"From an early age, everyone is familiar with crumpling a sheet of paper into a ball, unfolding it, and looking at the complicated network of creases that form," explains Christopher Rycroft, the paper's corresponding author. He's an associate professor in the John Al Paulson School of Engineering and Applied Sciences at Harvard University, and head of the Rycroft Group for scientific computing and mathematical modeling. "On the surface this seems like a random, disordered process, and you might think that it's difficult to predict anything at all about what happens."

"Suppose now that you repeat this process, crumple the paper again, and unfold it. You will get more creases," Rycroft writes in an email. "However, you won't double the number, because the existing creases already weakened the sheet and allow it to fold more easily the second time around."

Total Length of Creases = "Mileage"

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.

"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.

Why Does Crumple Theory Matter?

Gaining insights about crumpling is potentially really important to all sorts of things in the modern world. "If you are using a material in any structural capacity, it is critical to understand its failure properties," Rycroft says. "In many situations it is important to understand how materials will behave under repeated loading. For example, aircraft wings vibrate up and down many thousands of times over their lifetime. Our study of repeated crumpling can be viewed as model system for how materials are damaged under repeated load. We expect that some core elements of our theory, about how materials are weakened by fractures/creases over time, may have analogs in other material types."

And sometimes, crumpling might actually be utilized technologically. Rycroft notes that crumpled graphene sheets, for example, have been suggested as a possibility for making high-performance electrodes for Li-ion batteries. Additionally, crumple theory provides insights into all sorts of phenomena, from how insects' wings unfold and how DNA packs into a cell nucleus, as this 2018 New York Times article notes.

Why do some objects crumple, as opposed to simply breaking apart into a lot of little pieces?

"Paper and other materials which crumple are characteristically flexible and easy to bend, so they are not likely to break," Andrejevic explains. "However, hard materials like rock or glass do not bend easily, and thus break in response to a compressive force. I would say that crumpling and breaking are quite distinct processes, but there are some similarities we can recognize. For example, both crumpling and breaking are mechanisms of relieving stress in a material. The idea of creases protecting other regions of a sheet from damage refers to damage being localized to very narrow ridges in the sheet. In fact, the sharp vertices and ridges that form when a sheet crumples are localized regions of stretching in the sheet, which are energetically unfavorable. As a result, the sheet minimizes these costly deformations by confining them to very narrow regions, protecting the rest of the sheet as much as possible."

"Thin sheets which crumple prefer to bend rather than stretch, an observation that we can make readily with a sheet of paper by trying to bend or stretch it with our hands. In terms of energy, this means that bending costs far less energy than stretching. When a sheet is confined so that it can no longer stay flat, it will start to bend in order to conform to the changing volume. But after a certain point, it becomes impossible to fit the sheet into a small volume through bending alone."

Increasing the Understanding of Creases

There's a lot that still needs to be learned about crumpling. For example, as Rycroft notes, it's not clear whether different types of crumpling — using a cylindrical piston, for example, rather than your hand — results in a different type of crease pattern. "We'd like understand how general our findings are," he says.

In addition, researchers want to learn more about the actual mechanics of how creases form, and to be able to take measurements during the process, rather than just examining the end result.

"To get around this, we are currently developing a 3D mechanical simulation of a crumpled sheet, which can allow us to observe the entire process," Rycroft says. "Already, our simulation can create crease patterns that are similar to those seen in the experiment, and it provides us with a much more detailed view of the crumpling process."