Asymmetric tension–compression connectivity governs deformation delocalization in truss-based metamaterials

Failure in most material systems is characterized by strain localization, where deformation concentrates within a narrow region. Recently, a class of truss-based metamaterials has been shown to undergo severe deformation without exhibiting localization [1]. The mechanisms underlying this unusual delocalized response remain unknown. Here, we employ graph theory to elucidate the origins of this behavior. Each lattice is represented as a pair of graphs—the tension and compression networks—and their topological properties are quantified using graph-theoretic metrics. We find that the onset of localization correlates strongly with connectivity measures of these graphs. Specifically, deformation delocalization arises from an asymmetry in connectivity between these networks: when the tension graph remains more connected than the compression graph, deformation spreads throughout the structure instead of localizing. Connectivity measures such as average global efficiency[2] capture this transition quantitatively. This framework provides design principles for creating materials and metamaterials that intrinsically resist failure localization.