Geometrically driven wrinkling observed in free plastic sheets and leaves

We have measured the multiscale wrinkling that occurs along the edge of torn plastic sheets. The plastic deformations produced by tearing define a new metric on the sheet, which then relaxes elastically. The resultant patterns of wrinkles correspond to a superposition of waves of different wavelengths. Measurements of the variation of the pattern as a function of the distance from the edge reveal a set of transitions, each of which adds a new mode to the cascade. The wavelengths lambda in the cascade depend on both a geometrical length scale L-geo given by the metric near the sheet's edge, and the sheet thickness t: lambda proportional to t(0.3)L(geo)(0.7). This scaling implies vanishingly short wavelengths in the limit t -> 0. A possible geometrical origin of this behavior is discussed. Finally, we show that our measurement and analysis techniques are applicable to the study of some wavy patterns of leaves. These measurements reveal that the intrinsic geometry of a wavy leaf resembles that of the torn plastic sheets. This supports the possibility that some leaves form waves through a spontaneous wrinkling, rather than through an explicit three-dimensional construction.