Bistability and equilibria of creased annular sheets and strips

Abstract

A creased thin disk is generally bistable since the crease could be pushed through to form a stable cone-like inverted state with an elastic singularity corresponding to the vertex of the conical surface. In a recent study, we found that this bistability could be destroyed by removing the singularity through cutting a hole around the vertex, depending on the size and shape of the hole. Particularly, to maintain the bistability, a circular hole normally cannot exceed approximately 20% of the disk size. This paper extends our recent work and is based on the following observations in tabletop models of creased disks with circular holes: (i) reducing the circumference of the creased disk by removing an annular sector could increase the hole size to be as large as the disk without destroying the bistability, (ii) with a single crease, the circular hole could be as large as the disk without loss of the bistability, and (iii) a family of stable inverted states can be obtained by inverting the disk almost anywhere along the crease. An inextensible strip model is implemented to investigate these phenomena. We formulate a minimal facet of the creased disk as a two-point boundary value problem with the creases modeled as nonlinear hinges, and use numerical continuation to conduct parametric studies. Specifically, we focus on geometric parameters which include an angle deficit that determines the circumference of the disk, the rest crease angle, the number of evenly distributed creases, and an eccentricity that determines the position of the hole on the crease. Our numerical results confirm the qualitative observations in (i)-(iii) and further reveal unexpected results caused by the coupling between these geometric parameters. Our results demonstrate that by varying the geometry of a simply creased disk, surprisingly rich nonlinear behaviors can be obtained, which shed new light on the mechanics and design of origami, kirigami, and morphable structures.