A linear-elastic-nonlinear-swelling theory for hydrogels. Part 1. Modelling of super-absorbent gels

Abstract

We introduce a new approach for modelling the swelling, drying and elastic behaviour of hydrogels, which leverages the tractability of classical linear-elastic theory whilst incorporating nonlinearities arising from large swelling strains. Relative to a reference state of a fully swollen gel, in which the polymer scaffold may only comprise less than$1\,\%$of the total volume, a constitutive model for the Cauchy stress tensor is presented, which linearises around small deviatoric strains corresponding to shearing deformations of the material whilst allowing for a nonlinear relation between stress and isotropic strains. Such isotropic strains are considered only to be a consequence of losses and gains of water, while the hydrogel is taken to be instantaneously incompressible. The dynamics governing swelling and drying are described by coupling the interstitial flow of the water through the porous gel with the elastic response of the gel. This approach allows for a complete description of gel behaviour using only three macroscopic polymer-fraction-dependent parameters: an osmotic modulus, a shear modulus and a permeability. It is shown how these three material parameters can, in principle, be determined experimentally using a simple rheometry experiment in which a gel is compressed between two plates surrounded by water and the total force on the top plate is measured. To illustrate our approach, we solve for the swelling of a gel under horizontal confinement and for a partially dried hydrogel bead placed in water.