Anisotropic Swelling in Fiber-reinforced Hydrogels: An Incremental Finite Element Method and Its Applications in Design of Bilayer Structures

Abstract

In this paper, we present a theoretical and finite element framework for the inhomogeneous swelling of fiber-reinforced anisotropic hydrogels and investigate the effects of their orientations and moduli on the swelling behaviors of hydrogel bilayers. The effects of fibers on swelling are incorporated into the constitutive equations by introducing the corresponding energy contribution, expressed as the invariants related to the deformation of fibers, into the Flory–Rehner model. Unified constitutive equations and corresponding tangent moduli for anisotropic hydrogels reinforced with two families of fibers are deduced. A nonlinear finite element procedure is developed to simulate the inhomogeneous steady anisotropic swelling, where the chemical potential is prescribed in an incremental way. The accuracy and effectiveness of the numerical procedure are demonstrated by the anisotropic swelling of hydrogel blocks with one or two families of fibers. Based on the numerical procedure, many kinds of swelling configurations for hydrogel bilayers with different fiber orientations and modulus coefficients can be obtained, ranging from pure bending, twisting or saddle-like to combinations of them, which provide systematic guidance for the design of anisotropic-hydrogel based bilayers.