Wrinkling and developable cones in centrally confined sheets.

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 simulations, we probe the confinement response of circular sheets, flattened in their central region and quasistatically 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 paper 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.