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 container of radius R by a distance eta [1]. The mean curvature was reported to vanish at the rim where the d-cone is supported [2]. 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) 5/2F/(Y R2), where F is the pushing force and Y is the Young's modulus. Scaling arguments and analytical solutions confirm the numerical results.