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 $\eta$. The mean curvature was reported to vanish at the rim where the d-cone is supported. 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 toward 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 ${(R/h)}^{5/2}F/\left(Y{R}^2\right)$, where $F$ is the pushing force and $Y$ is the Young's modulus. Scaling arguments and analytical solutions confirm the numerical results.

• Received 6 May 2011

DOI:https://doi.org/10.1103/PhysRevE.84.066603