Cracking the complex case of crumpling

Thanks to Witten's research, scientists may soon better understand how blood cells shrink to pass through the tiniest capillaries, how a crumpling car fender absorbs the energy of a crash and how the earth's crust buckles to form a mountain range.

Crumpling is a ubiquitous phenomenon, occurring on scales ranging from the microscopic to the geological, but it is poorly understood. A blood cell's membrane folds and buckles, compacting the cell so it can pass through capillaries that are normally much too narrow. Mountains are formed when the earth's tectonic plates collide, thrusting up a crumpled landscape of hills and valleys.

Understanding these diverse phenomena requires answering a very fundamental question. Imagine crumpling a sheet of paper into a ball. Now squeeze it until it's 10 times smaller. How much energy does that take? How about if you squeeze hard enough to make it 100 times smaller? How hard do you have to squeeze? You can answer this question empirically for different materials, but until now, scientists had no clear understanding of how to predict quantitatively the amount of energy required.

Witten said that answering this question may aid, for example, in the design of better packing materials and of cars that can absorb the energy of a high-speed crash without crushing their occupants.

Witten, who has a crumpled sheet of Mylar mounted on his office wall, wants to understand the common elements of the various materials that crumple. He approaches the problem from a theoretical perspective: These materials seem, on the surface, to be very different, but they all crumple, so what makes them the same? Witten and fourth-year graduate student Alex Lobkovsky published an article in the Dec. 1, 1995, issue of Science magazine that for the first time explains what different kinds of crumpled materials have in common.

"Before this article, if you wanted to describe quantitatively how a material crumpled, you would have to describe it separately for paper, Mylar, sheet metal, et cetera," Witten said, "because the way each one crumples might depend crucially on the type of material.

"But we found that a sheet of one material crumples like a sheet of any other material, provided the sheets are large and thin. This was very much a surprise to us."

Lobkovsky added, "We had thought the problem was too complex to be able to say anything general about it."

If you look closely at a crumpled sheet of paper, you see that it forms sharp points connected by ridges. To simplify the crumpling problem theoretically, Lobkovsky and Witten decided to model it by studying the simplest case of a ridge between two points. They modeled the material on a computer and calculated the energy at different places.

"If you thought about it naively," said Lobkovsky, "you might think that the energy would be uniformly distributed throughout the sheet. But we found that it's concentrated in the narrow ridge between these two points."

The energy distribution is independent of the type of material. "As long as the material is thin -- relative to the distance between the two points -- one material looks just like another," Lobkovsky said.

Witten and Lobkovsky were also surprised by another finding. Increasing the size of the material before it's crumpled has an unexpectedly small effect on the total energy. Crumpling a small sheet of paper, and then creating an identical ball -- same ridges and points -- with a sheet eight times larger only increases the energy by twice as much.

"This was something we didn't expect. You might have thought the energy would be 64 times greater, since the area and volume are 64 times greater," said Witten. "But because the energy is concentrated in the ridges, the area over which the energy is spread is a small fraction of the total area of the sheet." This is the first time that anyone has been able to quantify this relationship.

"If we want to understand how crumpling works, we will have to understand how ridges form. Our work is an essential ingredient in understanding these phenomena," said Witten. "It's one ingredient, but by no means the only one."

But, he suggested, if materials could be designed to control where ridges form, engineers might be able to more effectively manipulate how materials crumple -- the body of a car in a head-on crash, for example.

Witten and Lobkovsky are members of the University's Materials Research Science & Engineering Center, funded by the National Science Foundation. Other collaborators on the work include undergraduate Sharon Gentges; Hao Li, a former Research Scientist here, now at the NEC Research Institute; and David Morse, of the Institute for Theoretical Physics at the University of California, Santa Barbara.

For more information about this and other research at the Materials Research Science & Engineering Center, visit the center's World Wide Web site, http://arnold.uchicago.edu/MRSEC.

-- Diana Steele