  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.
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.
