Topological Mechanics of Entangled Networks

Entangled networks are ubiquitous in tissues, polymers, and fabrics. However, their mechanics remain insufficiently understood due to the complexity of the topological constraints at the network level. Here, we develop a mathematical framework that models entangled networks as graphs, capturing topological constraints of entanglements. We prove that entanglements reduce system energy by enabling uniform tension along chains crossing entanglements and by redistributing stress through sliding. Under this framework, we study elasticity and fracture, validated by experiments on entangled fabrics and hydrogels. For elasticity, entanglements increase strength by enabling stress homogeneity in the network. For fracture, entanglements enhance toughness by mitigating stress concentration around crack tips. We discover counterintuitive physical laws governing crack-tip stretch during crack opening: stress deconcentration at small deformation, constitutive-law independence at intermediate deformation, and linear scaling at large deformation. This framework establishes fundamental principles of linking topology to mechanics of entangled networks and offers a foundational tool for designing reconfigurable materials.