Assur graphs, marginally jammed packings, and reconfigurable metamaterials

Isostatic frames are mechanical networks that are simultaneously rigid and stress-free. Isostaticity itself is a powerful concept in understanding phase transitions in soft matter and designing mechanical metamaterials. We show that the contact network of marginally jammed packings approaches not only isostaticity but minimal isostaticity in the thermodynamic limit. We define in turn this minimal isostaticity and describe how global nonlocal mechanical responses are hallmarks of minimally isostatic graphs (MIGs) known previously as Assur graphs. By using the related concept of Assur decomposition, which we generalize for periodic boundary conditions, we not only assess our claim about jammed packings but we also offer a new design principle for mechanical metamaterials in which motion and stress can propagate in reconfigurable pathways, while rigidity of the entire structure is maintained. We also briefly note an apparent relationship between fully repulsive interactions and the emergence of MIGs at the unjamming point.