A micro-macroscopic constitutive model for spatial confinement hydrogels incorporating directional chain slippage and crosslinking distance evolution

Spatial confinement hydrogels (SCHs) exhibit excellent mechanical properties, including enhanced toughness, self-healing capacity, and swelling-induced degradation, all of which stem from their unique network structures. In these structures, tangential and normal displacements at crosslinking points govern chain slippage and crosslinking distance evolution, yielding variable Kuhn segment numbers and non-zero crosslinking distances overlooked in classical models. This study proposes a micro-macroscopic constitutive model to capture the response of SCHs under large deformation. We first derive an entropic free energy function for individual chains to account for the evolution of the Kuhn segment number. By incorporating finite crosslinking distances into a micro-macroscopic transition model, we develop a macroscopic continuum model that features non-affine deformation mapping and evolution equations for both the Kuhn segment vector and the crosslinking distance. Our analysis reveals that directional chain slippage and crosslinking distance evolution collectively mitigate stress concentrations in SCHs under large strains. Furthermore, the model characterizes the strain rate-dependent behavior of SCHs, specifically the transition from a hyperelastic-dominated to a viscoelastic-dominated regime. The proposed constitutive model provides a valid description of SCH mechanical behavior and may advance engineering applications for hydrogels.