Rim curvature anomaly in thin conical sheets revisited

Abstract

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 $(R/h{)}^{5/2}F/(Y{R}^2)$, where $F$ is the pushing force and $Y$ is the Young's modulus. Scaling arguments and analytical solutions confirm the numerical results.

25 pages, 12 figures. Added references. Corrected typos. Results unchanged

References in corpus (5)

• Intrinsic ripples in graphene
• Virus shapes and buckling transitions in spherical shells
• Ridge Network in Crumpled Paper
• Mechanical response of a self avoiding membrane: fold collisions and the birth of conical singularities
• The compensation of Gaussian curvature in developable cones is local

Cited by in corpus (2)

• Finite curved creases in infinite isometric sheets
• Wrinkling and developable cones in centrally confined sheets