A gauge-like elastic theory for conformal metamaterials

We develop a framework to understand the mechanics of a metamaterial sheet that can undergo local, biaxial dilations and rotations with no energy cost (a conformal metamaterial). We construct an elastic theory for the buckling of the sheet in the Föppl–von Kármán limit, valid for small deformations, by introducing an additional, scalar gauge-like field that couples to the internal metric of the sheet. The resulting elastic energy can describe how an ideal conformal metamaterial buckles out-of-plane. We then introduce additional terms that explicitly break the gauge symmetry and discuss possible generalizations to understand the mechanics of metamaterial sheets more broadly.