Saddle-point and minimization principles for diffusion in solids : phase separation, swelling and fracture

Abstract

Liquid diffusion in solids plays a major role in countless biomedical, geotechnical and everyday applications. Modern continuum mechanics delivers suitable modeling approaches for both macroscopically observable material behavior and microstructural arrangements: multi-field formulations allow for different length scales and coupled physical effects. This work contributes to the theory and finite element discretization of diffusion phenomena in solids. Foundations of large strain kinematics, Newtonian balances and constitutive theory are outlined in conjunction with solute transport and essential non-equilibrium thermodynamics. This forms the basis for three applications. First, spinodal decomposition in rigid bodies is formulated in terms of an incremental variational formulation allowing for an efficient exploitation of its saddle-point structure. Then, a new minimization formulation for Fickian diffusion in hydrogels is shown to be the counterpart of the classical saddle-point principle and implemented with non-standard FE schemes. This model is extended by a phase-field approach to fracture to account for diffusion-induced material failure.