Mechanics of generically creased disks

Folded structures are often idealized as a series of rigid faces connected by creases acting as revolute hinges. However, real folded structures can deform between creases. An example of particular interest is a disk decorated by multiple radial creases. Such disks are bistable, snapping between a "natural" and "inverted" shape. We investigate the mechanical behavior of these creased disks and propose a new analytical approach to describe their mechanics. Detailed experiments are performed which show that, when indented at the center, a localized dimple forms, precluding the conical shape assumed in previous studies. As the indentation depth increases this dimple expands radially until reaching the disk edge when it snaps to the inverted shape, which has a conical form. We develop an analytical model which approximates each face as a series of rigid facets connected by hinges that can both rotate and stretch. Energy expressions are derived relating hinge rotation and stretching to compatible shell deformations of the facets and equilibrium enforced by minimizing the total strain energy. By increasing the number of facets, the mechanics of the continuum shell is approached asymptotically. The analysis shows that membrane stretching of the faces is required when a conical form of deformation is enforced. However, in the limit of zero thickness, the forming and propagation of a localized dimple is inextensional. This new approach relates the kinematic analysis of rigid origami to the mechanics of thin shells, offering an efficient method to predict the behavior of folded structures.