Next time you crumple a sheet of paper, reflect on the fact that no matter how hard you squeeze the ball of paper, it will still be more than 75% air. Intrigued by this surprisingly large resistance to compression, a team of physicists in the US has investigated the relationship between the size of a crumpled sheet and the force applied to it (K Matan et al 2002 Phys. Rev. Lett. 88 076101).
A crumpled sheet consists of peaks connected by curved ridges, and the ridges store most of the energy. A mathematical model of these peaks and ridges predicts that the force required to crumple the sheet increases exponentially as the size of the resulting wad decreases.
To investigate this relationship, Sidney Nagel and colleagues at the University of Chicago in the US placed circular sheets of aluminized Mylar – 34 centimetres across and 12.5 micrometres thick – inside a plastic cylinder of diameter 10.2 centimetres. A weighted piston compressed the sheets inside the cylinder, and the team monitored this compression by measuring the height of the piston above the base of the cylinder.
You might expect the piston to compress the sheet and settle at a fixed height, but the team found that the piston height continued to fall logarithmically with time for up to three weeks. The team found that vibrations in the laboratory were too weak to affect the compression, and concluded that the extra compression must be caused by energy dissipation, either due to friction or to plastic deformation of the sheet.
The crumpled sheets also exhibited hysteresis: when weights were removed from the piston it did not return to its initial height. In order to give reproducible results the sheets had to be specially prepared.
With this method the team showed that as the size of a crumpled sheet decreases, the force needed to crumple it further increases exponentially. This agreed with the scaling relation of the mathematical model, but the experimental value for the exponent differed from the prediction. The researchers believe that frictional forces and plastic deformation may account for the discrepancy.
The result raises questions that could be the basis of further study. For example, how would the relation change with sheets of different thickness and size? Is the creeping descent of the piston due to frictional effects or deformation? The team suggests that rubber sheets could be used to investigate the role of plastic deformation because the plastic effects would be minimal, so the height of the piston would reach its final value more quickly.
