Corrector Estimates for the Homogenization of a Two-Scale Thermoelasticity Problem With a Priori Known Phase Transformations

TL;DR

This paper derives corrector estimates for a two-scale thermoelasticity model in heterogeneous media with phase transformations, showing optimal convergence rates in simplified scenarios using energy and operator estimates.

Contribution

It provides the first rigorous derivation of corrector estimates for a coupled thermoelasticity problem with phase transformations in a two-scale setting.

Findings

Optimal convergence rates proven for simplified scenarios

Energy estimates and operator bounds used in proofs

Additional regularity results are established

Abstract

We investigate corrector estimates for the solutions of a thermoelasticity problem posed in a highly heterogeneous two-phase medium and its corresponding two-scale thermoelasticity model which was derived in an earlier paper by two-scale convergence arguments. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing a priori known phase transformations. While such estimates seem not to be obtainable in the fully coupled setting, we show that for some simplified scenarios optimal convergence rates can be proven rigorously. The main technique for the proofs are energy estimates using special reconstructions of two-scale functions and particular operator estimates for periodic functions with zero average. Here, additional regularity results for the involved functions are necessary.