Rim curvature anomaly in thin conical sheets revisited

摘要

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 containerof 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 reportedto 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 thetwo principal curvatures versus sheet thickness $h$ over a wider dynamic rangethan was used previously, holding $ R $ and $ \eta $ fixed. Instead of tendingtowards 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 previouslyclaimed. Moreover, we find that the normalized rim profile of radial curvaturein a d-cone is identical to that in a "c-cone" which is made by pushing aregular cone into a circular container. In both c-cones and d-cones, the ratioof 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. Scalingarguments and analytical solutions confirm the numerical results.