A generalized continuum elasticity theory for mechanical metamaterials

Mechanical metamaterials have been studied extensively to explore the unusual mechanical properties owing to instabilities arising from their microstructures, yet challenges still remain in studying their complex deformations. We explore a simple continuum framework for the mechanics of a thin, metamaterial sheet in which a scalar field, coupled to the reference metric, captures the soft in-plane deformation of the material. In the Föppl–von Kármán limit, we show how the unusual elasticity of the sheet screens curvature-induced stress in the bulk for metamaterials with soft conformal and soft simple shear modes. We further use this framework to study the buckling of geometrically-frustrated metamaterial sheets.