A multiplicative finite element algorithm for the inhomogeneous swelling of polymeric gels

Abstract The paper presents a nonlinear Finite Element (FE) algorithm to simulate the inhomogeneous steady swelling of polymeric gels. The algorithm is developed based on the multiplicative decomposition of the deformation gradient into an elastic part and a swelling part. The corresponding constitutive framework and linearization of the FE equation are elaborated, which leads to an equivalent body force driving the materials to swell. A staggered iterative procedure is designed to ensure the strongly coupled chemical and mechanical fields to reach the equilibrium state simultaneously. The constitutive equations are constructed in the framework of the volumetric–isochoric splitting of the free energy function, which is helpful both for a different treatment of the incompressible part in the FE equation and for a physical interpretation of the coupling effects between the chemical and mechanical fields. In the present work, an additive free energy function, coupling the contributions of the Flory–Huggins model and the non-Gauss statistical–mechanical model, is adopted and the corresponding consistent tangent modulus in the current configuration is also derived. To implement the FE algorithm, two two-dimensional (2D) elements and one three-dimensional (3D) element are developed, which are validated by some analytical solutions, such as free swelling of gels in 2D and 3D cases, squeezing a swollen gel with uniform pressure and the constrained swelling of a single-layer and a core–shell gel ring. The algorithm is also proved to be capable of handling the surface instabilities and some engineering applications such as hydrogel-based composite beams and thin films.