Effect of ridge-ridge interactions in crumpled thin sheets.

We study whether and how the energy scaling based on the single-ridge approximation is revised in an actual crumpled sheet, namely, in the presence of ridge-ridge interactions. Molecular dynamics simulation is employed for this purpose. In order to improve the data quality, modifications are introduced to the common protocol. As crumpling proceeds, we find that the average storing energy changes from being proportional to one-third of the ridge length to a linear relation, while the ratio of bending and stretching energies decreases from 5 to 2. The discrepancy between previous simulations and experiments on the material-dependence for the power-law exponent is resolved. We further determine the average ridge length to scale as 1/D(1/3), the ridge number as D(2/3), and the average storing energy per unit ridge length as D(0.881) where D denotes the volume density of the crumpled ball. These results are accompanied by experimental proofs and are consistent with mean-field predictions. Finally, we extend the existent simulations to the high-pressure region and verify the existence of a scaling relation that is more general than the familiar power law at covering the whole density range.