Geometric Dependence of Curvature-Induced Rigidity

Gauss’ Theorema Egregium is colloquially known as the ‘pizza theorem’ due to the rigidity conferred along a slice’s length when a curvature or fold is imposed on the crust. But there are physical limitations to this curvature-induced rigidity, as any pizza enthusiast would appreciate. These limits depend critically on the geometry of the material. We investigate how shape affects the deflection of a curved structure under its weight using experiments, finite-element simulations, and a geometrically nonlinear plate theory. Different settings in which the curvature-induced rigidity is more pronounced for rectangular or triangular elastic sheets are explored. Furthermore, we uncover and analyze a hysteretic region of bistability that exists for only certain sheet geometries.