Globally Frustrated Thin Sheets

It is known that geometric frustration can lead to fascinating and surprising physics in condensed matter systems. In thin elastic sheets, frustration-induced prestresses lead to degenerate ground states, topological rigidity, and stress-focusing. In these examples, the frustration arises from the failure of the so-called Gauss-Mainardi-Codazzi (GMC) equations, a set of local compatibility conditions that the prescribed rest lengths and curvatures must satisfy for the sheet to be stress-free. We show that even if the GMC equations are satisfied, so that the lengths and curvatures are locally compatible, the topology of the sheet can lead to a global frustration that induces prestresses. We characterize this new frustration by relating the GMC equations to the winding of moving frames on the sheet around closed loops. Finally, we construct globally frustrated annuli that pop into programmed shapes after slicing, opening up a possible new avenue to design deployable structures.