Designing Ground States and Degeneracies of Complex Triangular Mechanical Metamaterials

Mechanical metamaterials are artificial structures exhibiting unusual mechanical properties that stem mainly from their geometrical structure rather than from the materials they are made of. We design complex responses by utilizing geometric frustration - the inability of all mechanical elements to simultaneously deform in harmony. We consider a metamaterial made of a triangular lattice of elastic beams of different widths. The three beams constituting each triangle cannot simultaneously bend to their first buckling mode and also maintain the angles at their contacts. For beams of a single width, this frustration is relieved by long-range elastic interactions, resulting in two possible ordered deformation patterns. We extend an existing mapping to an Ising spin model to theoretically show that varying the different beam widths leads to four possible deformation patterns, which we verify experimentally. Two of these phases exhibit extensively degenerate energy-minimizing states, which implies that the metamaterial deforms in a non-periodic manner, with multiple independent degrees of freedom.