Patterns and Singularities in Elastic Shells

Abstract

We describe how thin elastic shells may be patterned with generic creases, cuts, edge curvatures, and growth patterns to alter their mechanical and geometric properties. We suggest simple scaling laws for these behaviors over a wide range of geometric parameters as the shells deform in two and three dimensions, which can be used to create robust shaping mechanisms for the design of functional materials. We detail the geometry and mechanics of a thin sheet with a single cut, then describe how the behavior of sheet with a generic pattern of cuts may be approximated with a geodesic construction and present a theorem for the flat-foldability of extended cut sheets in the inextensible limit. We then discuss force scalings for idealized creases in one and two dimensions, and a curious instability that leads to twisting in straight, finite creases. Turning to the question of patterning by simple growth laws, we study the localized singularities that form on the boundaries of flat sheets subject to curvature growth, like the cusps that form at the edges of rose petals and other natural bilayer structures, and their umbilic structure. Aside from out-of-plane curvature growth, we also present results of in-plane metric growth for small patches and stripes, which lead to curious patterns of Gauss curvature and geometric features on length scales much larger than the local patterning. We conclude with a brief introduction to convex integration, where deformations generated by in-plane growth may be parameterized over different choices of surface corrugation; this suggests an outlook on elastic shell patterning as a selection problem over corrugation parameters.