Geometric mechanics of thin curved surfaces and origami

Thin surfaces are ubiquitous in nature, from leaves to cell membranes, and in technology, from origami to corrugated containers. Structural thinness imbues them with flexibility, the ability to easily bend under light loads, even as their relatively high stretching stiffness can bear substantial stresses. When surfaces have patterns of hills and valleys, this can substantially modify their mechanical response. We show that for any such surface, there is a duality between the surface rotations of an isometric deformation and the in-plane stresses of a force-balanced configuration. This duality means that of the six possible combinations of global in-plane strain and out-of-plane bending that can be applied to a periodic thin surface, exactly three must be consistent with microscopic isometries and thus favored energetically. We show further that stressed configurations can be expressed in terms of both the applied deformation and the isometric deformation that is dual to the pattern of stress that arises. This further reveals that there are requirements rooted in symplectic geometry that constrain the set of three isometric deformations that a single surface can possess. The framework developed here can be extended to derive new fundamental limits on the mechanical response of thin periodic surfaces.