Spontaneous curvature cancellation in forced thin sheets

Abstract

In this paper we report numerically observed spontaneous vanishing of mean curvature on a developable cone made by pushing a thin elastic sheet into a circular container [1]. We show that this feature is independent of thickness of the sheet, the supporting radius and the amount of deflection. Several variants of developable cone are studied to examine the necessary conditions that lead to the vanishing of mean curvature. It is found that the presence of appropriate amount of radial stress is necessary. The developable cone geometry somehow produces the right amount of radial stress to induce just enough radial curvature to cancel the conical azimuthal curvature. In addition, the circular symmetry of supporting container edge plays an important role. With an elliptical supporting edge, the radial curvature overcompensates the azimuthal curvature near the minor axis and undercompensates near the major axis. Our numerical finding is verified by a crude experiment using a reflective plastic sheet. We expect this finding to have broad importance in describing the general geometrical properties of forced crumpling of thin sheets.

Dispersion Theory in Electromagnetic Interactions

“…1 (Q 2 ) is an observable which has been measured at JLab/Hall C (124). One can then use the measured value on the lhs of the sum rule of Eq.…”

“…At the real photon point, both observables yield the Baldin sum rule value for α E1 + β M 1 (30). The data point at Q 2 = 0.3 GeV 2 (left panel) is from JLab/HallC (124). In order to completely fix the term of O(Q 4 ) in the subtraction function T N B 1 (0, Q 2 ), one needs to determine the low-energy coefficient b3,0.…”

Energy Scaling Law for the Regular Cone

“…In this reference, one can find ansatz-based discussions and numerical simulations of the regular cone. For more recent numerical simulations, see Liang and Witten (2006). In , minimizers of the free elastic energy have been described in some detail under the assumption of small deflections and rotational symmetry.…”

Compensation of Gaussian curvature in developable cones is local

“…As the thickness of the elastic sheet goes to zero, they found that within the numerical accuracy, the radial curvature C rr and the azimuthal curvature C θθ are equal and opposite at the rim, so that the mean curvature virtually vanishes there. Though it arises from mechanical equilibrium, the phenomenon is purely geometric: it does not involve material parameters [1,5]. The von Kármán equations must be able to account for the cancellation effect, but this has not been done yet.…”

“…It is nonlocal because it depends on the overall geometry of the sheet. The curvature does not vanish when one modifies this geometry by cutting the sheet radially or by replacing the flat sheet by a cone [5]. The effect also depends on the nonlocal forcing at the vertex.…”