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This paper presents a homogenisation-based constitutive model to describe theeffective tran- sient diffusion behaviour in heterogeneous media in which thereis a large contrast between the phase diffusivities. In this case mobilespecies can diffuse over long distances through the fast phase in the timescale of diffusion in the slow phase. At macroscopic scale, contrasted phasediffusivities lead to a memory effect that cannot be properly described byclassical Fick's second law. Here we obtain effective governing equationsthrough a two-scale approach for composite materials consisting of a fastmatrix and slow inclusions. The micro-macro transition is similar tofirst-order computational homogenisation, and involves the solution of atransient diffusion boundary-value problem in a Representative Volume Elementof the microstructure. Different from computational homogenisation, we proposea semi-analytical mean-field estimate of the composite response based on theexact solution for a single inclusion developed in our previous work [Brassart,L., Stainier, L., 2018. Effective transient behaviour of inclusions indiffusion problems. Z. Angew Math. Mech. 98, 981-998]. A key outcome of themodel is that the macroscopic concentration is not one-to-one related to themacroscopic chemical potential, but obeys a local kinetic equation associatedwith diffusion in the slow phase. The history-dependent macroscopic responseadmits a representation based on internal variables, enabling efficient timeintegration. We show that the local chemical kinetics can result in non-Fickianbehaviour in macroscale boundary-value problems.

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