Homogenization of a Fully Coupled Thermoelasticity Problem for a Highly Heterogeneous Medium With a Priori Known Phase Transformations

TL;DR

This paper develops a homogenized model for a complex thermoelastic system in a highly heterogeneous medium with phase transformations, using two-scale convergence to handle microstructure evolution.

Contribution

It introduces a rigorous homogenization approach for a coupled thermoelasticity problem with moving interfaces in heterogeneous media.

Findings

Proves well-posedness of the coupled model.

Derives $oldsymbol{ extit{ ext{ε}}}$-independent a priori estimates.

Shows convergence to an upscaled model with microstructure evolution.

Abstract

We investigate a linear, fully coupled thermoelasticity problem for a highly heterogeneous, two-phase medium. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing an a priori known interface movement due to phase transformations. After transforming the moving geometry to an $\varepsilon$-periodic, fixed reference domain, we establish the well-posedness of the model and derive a number of $\varepsilon$-independent a priori estimates. Via a two-scale convergence argument, we then show that the $\varepsilon$-dependent solutions converge to solutions of a corresponding upscaled model with distributed time-dependent microstructures.