Elasticity and Fracture of Polymer-like Entangled Network

Traditional hydrogels are often brittle and prone to mechanical failure under repeated stress. Recently, a new class of hydrogels incorporating dense polymer entanglements has emerged, significantly enhancing toughness from ∼10 J/m² to ∼3000 J/m² with negligible hysteresis. Despite these promising advances, the fundamental mechanisms that govern the deformation of entangled networks remain poorly understood. Notably, there currently exists no mathematical model linking the topological structure of entangled networks to their mechanical properties.

In this work, we develop a mathematical framework that models entangled networks as graphs, capturing the topological constraints of entanglements. We begin with a two-chain unit, comparing a spring network with a fixed node to an entangled network with a slidable node. Our analysis shows that the energy required to stretch the spring network is greater than that of the entangled network. Furthermore, we prove that entanglements reduce the system energy by enabling uniform tension along chains crossing entanglements and by redistributing stress through sliding. To validate these findings, we use hydrogel fabrics and their photoelastic properties in a custom-built setup to visualize stress distribution during deformation. Building on insights from the two-chain toy model, we construct discrete entangled lattice network models to study macroscopic mechanical behavior in elasticity and fracture. For elasticity, the strength of the network exhibits a non-monotonic trend as the number of slidable nodes (entangled nodes) increases. The strength initially decreases due to the heterogeneity introduced by entanglements and then increases once sufficient entanglements enable homogeneous stress redistribution. In fracture, entanglements markedly enhance the intrinsic fracture toughness of the network by mitigating stress concentration at crack tips. The mathematical frame and these physical findings provide a theoretical foundation for designing tough materials and a guidance to design metamaterials.