Homogenization of Thermoelasticity Systems Describing Phase Transformations

Abstract

This thesis is concerned with the mathematical homogenization of thermoelasticity models with moving boundary describing solid-solid phase transformations occurring in highly heterogeneous, two-phase media. In the first part of this thesis, existence and uniqueness of weak solutions are established under the assumption that the changes in the geometry, which are due to the moving boundary, are given a priori. This is achieved after a transformation of coordinates to a fixed referential geometry. In addition, uniform a priori estimates are provided. Via an argument utilizing the concept of two-scale convergence, a corresponding homogenized model with distributed time and space dependent microstructures is derived. Quantitative error estimates measuring the accuracy and efficacy of the homogenized model are investigated. While such estimates seem not to be obtainable in the fully coupled setting, optimal convergence rates are proven for some special scenarios where the coupling mechanisms between the mechanical part and the heat part are simplified. In the second part, a more general scenario, in which the geometric changes are not assumed to be prescribed at the outset, is considered. Starting with the normal velocity of the interface separating the competing phases, a specific transformation of coordinates, the so-called Hanzawa transformation, is constructed. This is achieved by (i) solving a non-linear system of ODEs characterizing the motion of the interface and (ii) using the Implicit Function Theorem to arrive at the height function parametrizing this motion. Based on uniform estimates for the functions related to the transformation of coordinates, the strong two-scale convergence of these functions is shown. Finally, these results are used to establish the corresponding homogenized model.