Physically based elastic theory of hydrogel with application to shear accounting for the effect of hydrostatic modulus

In this paper, the chemically coupled elastic theory is proposed for hydrogel, based on statistical mechanics foundations, with particular emphasis on the critical yet overlooked role of hydrostatic modulus. We propose a comprehensive free energy density formulation that systematically integrates chemical interactions with both linear and nonlinear elastic moduli. Through analysis of isotropic swelling of hydrogel, a calibration protocol is developed for the chemically coupled elastic moduli, and the explicit relationship is derived between elastic moduli and environmental parameters. These analytical expressions facilitate direct determination and dynamic monitoring of instantaneous elastic moduli, including the hydrostatic modulus, linear constants, and second-order coefficients, under varying environmental conditions through measurable material properties. Notably, our theoretical framework reveals the significant influence of hydrostatic modulus on hydrogel shear response, a previously unrecognized mechanism. As a demonstration, shear of a rectangular hydrogel block is investigated with the statistically-based phenomenological elastic theory, elucidating the impact of hydrostatic modulus and nonlinear properties through both linear and second-order nonlinear simulations. Under linear approximation, our model recovers the classical infinitesimal deformation theory, while second-order instantaneous elastic moduli prove essential for capturing finite deformation effects such as the negative Poynting effect, wherein shear induces axial contraction. Furthermore, the direct connection is established between internal micro-physical parameters and macroscopic deformation. The effects of chemical potential, Flory parameter, crosslinking degree, and related factors on shear deformation are analytically investigated and quantified for their contributions to material response. Through systematic analysis, this work advances hydrogel mechanics understanding through a unified energy formulation that bridges statistical physics with continuum mechanics. And the results obtained here may provide a comprehensive guide for analysis of complex phenomena and design of soft materials.