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Article Dans Une Revue Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik Année : 2018

Effective transient behaviour of inclusions in diffusion problems

Résumé

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.
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Dates et versions

hal-02159721 , version 1 (19-06-2019)

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Laurence Brassart, Laurent Stainier. Effective transient behaviour of inclusions in diffusion problems. Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2018, 98 (6), pp.981-998. ⟨10.1002/zamm.201700187⟩. ⟨hal-02159721⟩
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