A linear-elastic-nonlinear-swelling theory for hydrogels: constitutive relations and dynamics

Hydrophilic polymers can form hydrogels with polymer volume fractions of less than 1% when brought into contact with water, and the resulting gels find numerous uses in healthcare, agriculture and industry. To describe the behaviour of such super-absorbent gels, we allow for arbitrarily large isotropic strain whilst linearising around deviatoric strains that are assumed to be small, treating the hydrogel as an instantaneously incompressible linear-elastic material. Introducing nonlinearities only in the swelling strain allows us to formulate an analytically-tractable model built only on three macroscopically-measurable material parameters, each dependent on the polymer fraction: an osmotic pressure, a shear modulus and a permeability. We detail a conceptual rheometer to measure the quantitative relationships for all of these properties based on mechanical responses of a hydrogel under compression. Given these constitutive relations, we describe various swelling and drying processes – driven by the flow of water through the polymer matrix – to illustrate the utility of the modelling approach and the role of elastic stresses in determining boundary conditions and the effective diffusivity of a gel.