Crumpling a sheet of paper seems simple enough and certainly doesn't require much effort, but explaining why the resulting crinkled ball behaves the way it does is another matter entirely. Once scrunched, a paper ball is more than 75 percent air yet displays surprising strength and resists further compression, a fact that has confounded physicists. A report in the February 18 issue of Physical Review Letters, though, describes one aspect of the behavior of crumpled sheets: how their size changes in relation to the force they withstand.

A crushed thin sheet is essentially a mass of conical points connected by curved ridges, which store energy. When the sheet is further compressed, these ridges collapse and smaller ones form, increasing the amount of stored energy within the wad. Sidney Nagel and colleagues of the University of Chicago modeled how the force required to compress the ball relates to its size. After crumpling a sheet of thin aluminized Mylar, the researchers placed it inside a cylinder equipped with a piston to crush the crumpled sheet. Instead of collapsing to a final fixed size as expected, the team writes, the height of the crushed ball continued to decrease, even three weeks after the weight was applied.

The scientists further found that the crumpled ball displayed a phenomenon known as hysteresis, in which the effect of forces acting upon an object lags behind its cause. For instance, the behavior of the crumpled sheet under a given weight depended in part on the amount of weight previously applied to it. Once they compensated for this response, the team deciphered the crumpled sheet's behavior and found that the force required for compression increases in proportion to the size of the scrunched sheet raised to a negative power. That is, the larger the ball, the less force is required to compress it further. The exponent calculated by the researchers differed from their prediction but finding such a power-law relationship at all may be key to understanding the peculiarities of crumpling. Because the calculations did not account for either friction or plastic flow within the creased material, however, the scientists say more research is needed.