Revisiting the curvature cancellation in forced thin sheets

We revisit the numerically observed spontaneous vanishing of mean curvature [1] on a developable cone or ``d-cone'' [2] made by pushing a thin elastic sheet into a circular container. The deflection of the d-cone is the distance by which the sheet is pushed into the container. We investigate the ratio of the two principal curvatures versus sheet thickness $h$ over a wider dynamic range than was used previously, holding the deflection and radius fixed. Instead of tending towards 1 as suggested by previous work, we find that the ratio scales as $h^{1/3}$. Scaling arguments and geometric variants support this $h^{1/3}$ finding. Thus the mean curvature does not vanish for very thin sheets as previously claimed. \\[4pt] [1] T. Liang and T. A. Witten, {\sl Phys. Rev. E} {\bf 73}, 046604 (2006). \newline [2] E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} {\bf 401}, 46 (1999).