Exploring Geometric Frustration in Self Assembly of Mechanical Metamaterial Using a Generalized Elasticity Theory

Mechanical metamaterials have been studied extensively to explore their unusual properties owing to instabilities arising from their micro-structures. Our simple continuum framework for the mechanics of a metamaterial thin sheet shows that in the Föppl–von Kármán limit, the elasticity of the sheet screens the curvature induced stress through soft deformation modes. We use this approach to explore the characteristic shapes of a geometrically-frustrated metamaterial with an incompatibility between in-plane and and out-of-plane strain. We uncover new dimensionless numbers which, in concert with the Foppl-von Karman number dictate how the soft modes affect this competition and give rise to anomalous equilibrium shapes.