Developable cones in annular rings*

Thin, confined sheets may distribute strain energy smoothly (e.g. the wrinkled edges of the lotus leaf), or focus it into small, sharp, sacrificial regions (e.g. the ridges and vertices, or d-cones, in discarded wrapping paper). The d-cone emerges when a circular sheet is forced, via indentation, through a ring of smaller radius. Its radial directors are unstrained, and the bulk shape follows that of an elastic ring.

Here, we perturb the d-cone structure by constraining its central region to remain flat. In experiments and molecular dynamics (MD) simulations, sheets are clamped between circular plates and quasi-statically drawn through a ring. This leads to sequential buckling of truncated cones, which arrange circumferentially around the free annulus. The size of cones is sensitive to confinement geometry. Regions of concentrated strain occur near the clamp, stiffening the system. This effect is analogous to a Winkler foundation, informing a 1D model that modifies the d-cone solution to reveal how the characteristic cone size depends on the geometric parameters. Additionally, we observe transient, small-amplitude wrinkling in the outer region of the sheet before cone formation. This diffuse-to-focused transition is abrupt, and our model captures the associated critical force.