Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load

A network of polymers can imbibe a large quantity of a solvent and swell, resulting in a gel. The swelling process can be markedly influenced by a mechanical load and geometric constraint. When the network, solvent, and mechanical load equilibrate, inside the gel the chemical potential of the solvent is homogeneous, but the concentration of the solvent and the deformation of the network can be inhomogeneous. We use the chemical potential of the solvent and the deformation gradient of the network as the independent variables of the free-energy function, and show that the boundary value problem of the swollen gel is equivalent to that of a hyperelastic solid. We implement this approach in the finite-element package, ABAQUS, and analyze examples of swelling-induced deformation, contact, and bifurcation. Because commercial software like ABAQUS is widely available, this work may provide a powerful tool to study complex phenomena in gels.