  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.
