A statistical microphase-field model for axial deformation localization and cyclic damage memory in double-network hydrogels

Deformation localization, such as macroscopic necking, is a fundamental structural instability in double-network (DN) hydrogels under large tensile deformation and plays a key role in their energy dissipation and toughening behavior. In this work, we develop a microphase-field framework grounded in micro-statistics to capture the experimental features of the initiation, propagation, and history-dependent evolution of localized deformation in DN hydrogels under uniaxial tension. The model combines inverse Langevin chain statistics, the Kuhn–Grün end-to-end distance distribution, and a gradient-regularized Allen–Cahn phase-field formulation to capture the transition from homogeneous deformation to localized neck propagation in a thermodynamically consistent manner. By calibrating the model and comparing its predictions with literature-reported monotonic and cyclic tensile experiments, the framework reproduces the stress plateau, strain hardening, irreversible damage accumulation, and cyclic damage memory observed in DN hydrogels. Parametric studies further reveal how the chain density and segment number of the two networks regulate localization onset, plateau stress, energy dissipation, and post-necking hardening. This study provides a fundamental basis for relating network-level structural parameters to macroscopic localization, dissipation, and cyclic damage memory in DN hydrogels.