Wrinkling and developable cones in centrally confined sheets

TL;DR

This study investigates how thin sheets respond to central confinement, revealing the formation of developable cones and wrinkles through experiments and simulations, and how these patterns depend on geometric parameters.

Contribution

It provides a detailed analysis of stress focusing and pattern formation in confined sheets, expanding understanding of the d-cone phenomenon in realistic conditions.

Findings

Wrinkles develop then transition to cones in confined sheets.

The number and size of cones depend on geometry.

A coarse-grained model describes geometric dependence.

Abstract

Thin sheets respond to confinement by smoothly wrinkling, or by focusing stress into small, sharp regions. From engineering to biology, geology, textiles, and art, thin sheets are packed and confined in a wide variety of ways, and yet fundamental questions remain about how stresses focus and patterns form in these structures. Using experiments and molecular dynamics (MD) simulations, we probe the confinement response of circular sheets, flattened in their central region and quasi-statically drawn through a ring. Wrinkles develop in the outer, free region, then are replaced by a truncated cone, which forms in an abrupt transition to stress focusing. We explore how the force associated with this event, and the number of wrinkles, depend on geometry. Additional cones sequentially pattern the sheet, until axisymmetry is recovered in most geometries. The cone size is sensitive to in-plane geometry. We uncover a coarse-grained description of this geometric dependence, which diverges depending on the proximity to the asymptotic d-cone limit, where the clamp size approaches zero. This work contributes to the characterization of general confinement of thin sheets, while broadening the understanding of the d-cone, a fundamental element of stress focusing, as it appears in realistic settings.