Mathematical Framework and Numerical Methods for the Modeling of Mechanochemistry in Multi-Phase Materials

Abstract

This dissertation presents a suite of mathematical formulations and numerical methods for modeling the interactions between solid mechanics and chemistry in multi-phase materials. In all cases, the treatments rely on the free energy of the system, which potentially includes the strain energy, the chemical free energy, and the interfacial energy. Variational methods are applied to the free energy functionals to derive equilibrium conditions for mechanics and to identify constraints on kinetic laws for chemistry. The applications of this class of variational methods include evolving material configurations associated with phase changes, both diffusive (e.g. oxidation) and non-diffusive (e.g. martensitic transformations). Motivated by the need to represent multi-well, oscillatory, free energy densities, a study is presented comparing spline and polynomial forms for these functions. An alternative approach to phase-field dynamics for finding a minimum energy state is demonstrated, with Mg alloy precipitates as an example. It involves learning the free energy surface as a function of key geometric features with machine learning techniques, which are then used to predict a minimum energy state. This collection of mathematical formulations and numerical methods is aimed at explorations of the physics underlying observed phenomena in multi-phase materials, with potential use in materials' design.