Continuum Theory and Deformation Control in Origami Metamaterials

The careful addition of creases to a thin sheet according to a mechanism, such as the Miura fold pattern, unlocks a special pathway of motion that allows the macroscopic sheet to access nonlinear shape changes at very low energy cost. This special mechanism behavior makes these origami metamaterials ideal candidates for controllable soft robotics. However, even rudimentary mechanical probing of these folded sheets reveals a broad variety of soft response which does not resemble the mechanism. Recently, some of us have revealed a new principle in which a geometric compatibility relation guarantees a space of soft motions affiliated with any planar mechanism. Here, we assemble these previous works in the context of differential geometry to reveal a continuum theory governing the generic soft response of origami sheets. Despite the presence of additional compatibility requirements and local modes of deformation, known colloquially as twist and bend, these exotic soft motions remain subextensive and are encoded on the sheet boundary. Our approach, which we confirm in a variety of loading simulations, closes a gap between current theory and tangible origami sheet behavior, while also revealing rich new possibilities for the precise control of deformation via boundary actuation.