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Thermo-Elasticity Problems with Evolving Microstructures

Michael Eden, Adrian Muntean

Abstract

We consider the mathematical analysis and homogenization of a moving boundary problem posed for a highly heterogeneous, periodically perforated domain. More specifically, we are looking at a one-phase thermo-elasticity system with phase transformations where small inclusions, initially periodically distributed, are growing or shrinking based on a kinetic under-cooling-type law and where surface stresses are created based on the curvature of the phase interface. This growth is assumed to be uniform in each individual cell of the the perforated domain. After transforming to the initial reference configuration (utilizing the Hanzawa transformation), we use the contraction mapping principle to show the existence of a unique solution for a possibly small but $\varespilon$-independent time interval ($\varespilon$ is here the scale of heterogeneity). In the homogenization limit, we discover a macroscopic thermo-elasticity problem which is strongly non-linearly coupled (via an internal parameter called height function) to local changes in geometry. As a direct byproduct of the mathematical analysis work, we present an alternative equivalent formulation which lends itself to an effective precomputing strategy that is very much needed as the limit problem is computationally expensive.

Thermo-Elasticity Problems with Evolving Microstructures

Abstract

We consider the mathematical analysis and homogenization of a moving boundary problem posed for a highly heterogeneous, periodically perforated domain. More specifically, we are looking at a one-phase thermo-elasticity system with phase transformations where small inclusions, initially periodically distributed, are growing or shrinking based on a kinetic under-cooling-type law and where surface stresses are created based on the curvature of the phase interface. This growth is assumed to be uniform in each individual cell of the the perforated domain. After transforming to the initial reference configuration (utilizing the Hanzawa transformation), we use the contraction mapping principle to show the existence of a unique solution for a possibly small but -independent time interval ( is here the scale of heterogeneity). In the homogenization limit, we discover a macroscopic thermo-elasticity problem which is strongly non-linearly coupled (via an internal parameter called height function) to local changes in geometry. As a direct byproduct of the mathematical analysis work, we present an alternative equivalent formulation which lends itself to an effective precomputing strategy that is very much needed as the limit problem is computationally expensive.
Paper Structure (11 sections, 21 theorems, 140 equations, 3 figures)

This paper contains 11 sections, 21 theorems, 140 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting and Preliminaries
  3. Coordinate transform
  4. Additional auxiliary results
  5. Assumptions and weak formulations
  6. Analysis of the $\varepsilon$-problem
  7. Tracking of the interface and estimates
  8. Analysis in initial domain
  9. Fixed-point argument
  10. Homogenization
  11. Discussion

Key Result

Lemma 2.2

The derivatives of $d$ and $P$ are given via ($y\in U$) where In addition, we have the implicit relation

Figures (3)

  • Figure 1: Left: The geometry of the unit cell with the inclusion $Z$. Right: Visualization of a tubular neighborhood $U_\Gamma$ (grey area) introduced in \ref{['ssec:Hanzawa']} inside which the interface is allowed to grow. Please note that these are cross sections as we are working in 3D.
  • Figure 2: Left: A two dimensional cross section of the initial periodic geometry. Right: The changed geometry of this cross section at some time $t>0$ with growth and shrinkage in different parts of the domain. Please note that these are cross sections as we are working in 3D.
  • Figure 3: Simple example demonstrating the construction of $[x]$ and $\{x\}$ in $\mathbb{R}^2$

Theorems & Definitions (46)

  • Remark 2.1: The averaging assumption
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.3.1
  • proof
  • Remark 2.4: Example: Growing and shrinking balls.
  • Lemma 2.5: Extension operators
  • proof
  • ...and 36 more