Rim curvature anomaly in thin conical sheets revisited

This paper revisits one of the puzzling behaviors in a developable cone (d-cone), the shape obtained by pushing a thin sheet into a circular container of radius $ R $ by a distance $ \eta $ [E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} {\bf 401}, 46 (1999)]. The mean curvature was reported to vanish at the rim where the d-cone is supported [T. Liang and T. A. Witten, {\sl Phys. Rev. E} {\bf 73}, 046604 (2006)]. We investigate the ratio of the two principal curvatures versus sheet thickness $h$ over a wider dynamic range than was used previously, holding $ R $ and $ \eta $ fixed. Instead of tending towards 1 as suggested by previous work, the ratio scales as $(h/R)^{1/3}$. Thus the mean curvature does not vanish for very thin sheets as previously claimed. Moreover, we find that the normalized rim profile of radial curvature in a d-cone is identical to that in a "c-cone" which is made by pushing a regular cone into a circular container. In both c-cones and d-cones, the ratio of the principal curvatures at the rim scales as $ (R/h)^{5/2}F/(YR^{2}) $, where $ F $ is the pushing force and $ Y $ is the Young's modulus. Scaling arguments and analytical solutions confirm the numerical results.