Scaling the Crumpled Heights

Tough guys take note: you ain't even as strong as a wimpy wad of paper.  Crumple a sheet of paper in your hand and you'll end up with a ragged little ball that resists further compression, even though it's roughly 75% air. For years physicists have wondered how a crumpled sheet can be so  extraordinarily rigid. In the 18 February print issue of PRL, a team unfolds a key piece of the mystery by determining how the size of a  crumpled wad varies with the force applied to it.

A crumpled sheet resembles a cubist's vision of a mountain range: it's a mass of conical peaks connected by a tangle of ridge lines. The peaks and ridges store energy, and as the wad is compressed, their numbers increase so that the total energy contained in the wad grows rapidly. Soon, it requires a tremendous squeeze to supply enough energy to make the wad even a tiny bit smaller.

To better understand crumpling, Sidney Nagel of the University of Chicago, his former student Kittiwit Matan (now at MIT), and their colleagues, came up with a simple model for a crumpled sheet and tested it with experiments. Based on "scaling"--the mathematics of how a system's properties change with its size--the model predicted that the force needed to compress the wad grows in proportion to its size raised to a negative exponent, a so-called power law. The theory assumes that the sheet is only loosely crumpled, that no energy is lost in friction, and that the sheet is not permanently deformed. When squeezed by a steady force, such a sheet should quickly collapse toward a final fixed size. But their experiments showed otherwise.

The researchers crumpled a circular sheet of aluminized Mylar 34 cm in diameter and placed it inside a 10-cm-wide cylinder. They squashed the Mylar into a pancake-like wad by placing a weight on the container's free-floating top, which acted like a piston. Instead of approaching some final value, the height of the pancake continued to fall, even three weeks after the weight was applied. The researchers also found that the height for a given weight depended upon the weight that had been applied before, a phenomenon known as hysteresis. For instance, a light weight would not rise up to its original height after it had been replaced briefly by a heavy weight.

Nevertheless, the team was able to decipher the crumpled sheet's behavior. To compensate for hysteresis, they first applied up to 60 kg to "train" the crumpled sheet. They then removed this weight and worked their way back up to it in increments. After adding each increment, they sidestepped the problem of creeping descent by waiting just 100 seconds to measure the new height. The researchers found that the relationship between the height of the wad and the applied mass was indeed a power law, although the exponent differed from their prediction.

But merely seeing a power law--rather than some other relationship--suggests that scaling is key to understanding the quirks of crumpling, says Jim Sethna of Cornell University in Ithaca, NY.  More realistic theories will have to account for interactions between the sheet and the walls of the container and between the sheet and itself, says Lakshminarayanan Mahadevan of Cambridge University in the UK. "This should provide a lot of impetus to theorists and, in fact, other experimentalists," he says, "because there are still many aspects of the system to be probed."