Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities

TL;DR

This paper develops a homogenisation-based model to describe the complex transient diffusion in heterogeneous media with high contrast in phase diffusivities, capturing memory effects and non-Fickian behaviour at the macroscale.

Contribution

It introduces a semi-analytical mean-field approach for effective transient diffusion, incorporating local kinetic equations and internal variables for heterogeneous media with large diffusivity contrast.

Findings

The model captures memory effects due to phase contrast.

Macroscopic concentration relates to chemical potential via local kinetics.

The approach predicts non-Fickian diffusion behaviour.

Abstract

This paper presents a homogenisation-based constitutive model to describe the effective tran- sient diffusion behaviour in heterogeneous media in which there is a large contrast between the phase diffusivities. In this case mobile species can diffuse over long distances through the fast phase in the time scale of diffusion in the slow phase. At macroscopic scale, contrasted phase diffusivities lead to a memory effect that cannot be properly described by classical Fick's second law. Here we obtain effective governing equations through a two-scale approach for composite materials consisting of a fast matrix and slow inclusions. The micro-macro transition is similar to first-order computational homogenisation, and involves the solution of a transient diffusion boundary-value problem in a Representative Volume Element of the microstructure. Different from computational homogenisation, we propose a semi-analytical mean-field estimate of the composite response based on the exact solution for a single inclusion developed in our previous work [Brassart, L., Stainier, L., 2018. Effective transient behaviour of inclusions in diffusion problems. Z. Angew Math. Mech. 98, 981-998]. A key outcome of the model is that the macroscopic concentration is not one-to-one related to the macroscopic chemical potential, but obeys a local kinetic equation associated with diffusion in the slow phase. The history-dependent macroscopic response admits a representation based on internal variables, enabling efficient time integration. We show that the local chemical kinetics can result in non-Fickian behaviour in macroscale boundary-value problems.