Effective transient behaviour of inclusions in diffusion problems

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.

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

“…ellipsoids) could easily be considered. Strategies for identifying the chemical creep function for arbitrary inclusion shapes have been proposed in Brassart and Stainier (2018).…”

“…Future work could include the extension to nonlinear diffusion behaviour due to concentrationdependent mobility coefficients and chemical moduli. A first, heuristic approach at the single inclusion level is provided in (Brassart and Stainier, 2018), however more sophisticated techniques could certainly be proposed. Extending the mean-field model to coupled chemo-mechanical problems would also be important to address problems such as diffusion in battery electrodes.…”

“…Alternatively, a computational approach for example based on modal analysis could be adopted, see e.g. Brassart and Stainier (2018). At the composite level, a suitable mean-field assumption would also need to be introduced in order to partition the strain between the phases, see e.g.…”

“…Our model relies on the exact solution for the transient diffusion problem in a single inclusion subject to a uniform, time-varying chemical potential at its boundaries developed in (Brassart and Stainier, 2018). Consider an inclusion with chemical modulus K, molecular conductivity k and diffusivity D " Kk.…”

“…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.…”