When a dry polymer network is immersed in a solvent, the solvent molecules enter the polymer network and result in swelling. The chemical potential of the solvent molecules is homogeneous in the gel and is set by the external solution. Hydrogel (the solvent is water) reaches equilibrium with the external solution when the chemical potential of the solvent equals to that in the external solution. The problem formulation and numerical method follow the one developped by Suo's group (eg.,reference [1] where further details and additional references are given). With regard to hydrogels, the free energy density function due to Flory and Rehner (1943) is expressed as a function of the deformation gradient and the nominal concentration of the solvent. It consists of two parts: stretching of the polymer network and mixing the polymer and the solvent. The kinematic constraint of incompressibility of both polymer network and solvent molecules is taken into account through the introduction of a Lagrange multiplier. The nominal stress within the gel is defined as the derivative of the free energy density function with respect to the deformation gradient. It encompasses two components, namely, the elastic part and the osmotic pressures of the solvent. This thorough approach has been implemented into the software Abaqus by coding the whole governing equations via the use of a UHYPER subroutine. The chemical potential is mimicked by a temperature-like variable, which is uniform in the gel, and is incremented as a loading parameter [1]. 

In this investigation, an attempt is made towards obtaining an estimation of the effective behaviour of porous hydrogels where microvoids are already present. To this end, a two-level representation of the material at hand is considered. The microscopic scale is treated through a representative volume element (RVE) composed of two phases: a homogeneous matrix (sound gel) and spherical void, for the sake of simplicity. The behaviour of the RVE is then appropriately averaged to provide the co-called macroscopic behaviour of the material considered as homogeneous. The preliminary computational study addresses the numerical simulations of a cylindrical representative cell containing an initially spherical void and subjected to axial and lateral overall stresses under conditions of constant prescribed overall stress triaxiality. The loading is displacement controlled in such a way that the stress triaxiality ratio retains a constant prescribed value during the whole process of deformation of the unit cell. The Riks's arc-length method in Abaqus is used in order to handle the inevitable instability of the RVE and to proceed with further calculations. The overall stress-strain behaviour of the RVE, the influence of the initial porosity, the Flory-Huggins parameter, and the nominal concentration of the solvent on the RVE behaviour have been highlighted.

[1] Hong W., Liu Z.S., and Suo Z. (2009). Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical Load, Int. J. Solids and Structures, 46, 3282-3289.