Effective transient behaviour of inclusions in diffusion problems

TL;DR

This paper develops a low-cost, accurate method to predict the transient diffusion response of inclusions in heterogeneous media with high contrast in diffusivities, capturing memory effects at the macroscopic scale.

Contribution

It introduces a chemical creep function and reduced relaxation mode estimates for the effective transient response of inclusions, enabling efficient homogenization.

Findings

Effective inclusion response can be approximated with few relaxation modes.

The method predicts transient behaviour under arbitrary boundary conditions.

A heuristic extension to concentration-dependent diffusion is proposed.

Abstract

This paper is concerned with the effective transport properties of heterogeneous media in which there is a high contrast between the phase diffusivities. In this case the transient response of the slow phase induces a memory effect at the macroscopic scale, which needs to be included in a macroscopic continuum description. This paper focuses on the slow phase, which we take as a dispersion of inclusions of arbitrary shape. We revisit the linear diffusion problem in such inclusions in order to identify the structure of the effective (average) inclusion response to a chemical load applied on the inclusion boundary. We identify a chemical creep function (similar to the creep function of viscoelasticity), from which we construct estimates with a reduced number of relaxation modes. The proposed estimates admit an equivalent representation based on a finite number of internal variables. These estimates allow us to predict the average inclusion response under arbitrary time-varying boundary conditions at very low computational cost. A heuristic generalisation to concentration-dependent diffusion coefficient is also presented. The proposed estimates for the effective transient response of an inclusion can serve as a building block for the formulation of multi-inclusion homogenisation schemes.