Minimization- and Saddle-Point-Based Modeling of Diffusion-Deformation-Processes in Hydrogels

Hydrogels have gained importance during the last years due to their wide range of synthetically fabricable elastic properties as well their increasing meaning in biomedical applications. Future exploitation of the vast prospects of hydrogels is however only feasible by establishing reliable material models that precisely capture their behavior in different environments. To this end, we propose a consistent variational framework for deformation-diffusion processes, offering a canonically compact approach to the chemo-mechanical coupling of hydrogels via a saddle-point as well as a new minimization formulation. The work depicts the construction of rate-type potentials for the chemo-mechanical evolution problem and their transformation into time-discrete incremental potentials. In terms of spatial discretization, the finite element method is employed, benefiting from the intrinsic symmetric structure of the variational foundation. While the saddle-point formulation yields the well-known LBB condition as a constraint for finite element interpolations, on the part of its minimizing counterpart H(Div, ℬ︁)-conforming elements have to be chosen. We illustrate appropriate solutions to both challenges, using mixed Taylor-Hood for the saddle-point and Raviart-Thomas elements for the minimization formulation and discuss advantages of the new approach. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)