Two-scale, non-local diffusion in homogenised heterogeneous media

Ramírez-Torres, A. , Penta, R. and Grillo, A. (2022) Two-scale, non-local diffusion in homogenised heterogeneous media. Archive of Applied Mechanics, 92(2), pp. 559-595. (doi: 10.1007/s00419-020-01880-3)

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We study how and to what extent the existence of non-local diffusion affects the transport of chemical species in a composite medium. For our purposes, we prescribe the mass flux to obey a two-scale, non-local constitutive law featuring derivatives of fractional order, and we employ the asymptotic homogenisation technique to obtain an overall description of the species’ evolution. As a result, the non-local effects at the micro-scale are ciphered in the effective diffusivity, while at the macro-scale the homogenised problem features an integro-differential equation of fractional type. In particular, we prove that in the limit case in which the non-local interactions are neglected, classical results of asymptotic homogenisation theory are re-obtained. Finally, we perform numerical simulations to show the impact of the fractional approach on the overall diffusion of species in a composite medium. To this end, we consider two simplified benchmark problems, and report some details of the numerical schemes based on finite element methods.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Penta, Dr Raimondo and Ramirez Torres, Dr Ariel
Authors: Ramírez-Torres, A., Penta, R., and Grillo, A.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Archive of Applied Mechanics
ISSN (Online):1432-0681
Published Online:09 March 2021
Copyright Holders:Copyright © 2021 The Authors
First Published:First published in Archive of Applied Mechanics 92(2): 559-595
Publisher Policy:Reproduced under a Creative Commons licence

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
303232EPSRC Centre for Multiscale soft tissue mechanics with MIT and POLIMI (SofTMech-MP)Xiaoyu LuoEngineering and Physical Sciences Research Council (EPSRC)EP/S030875/1M&S - Mathematics
308255The SofTMech Statistical Emulation and Translation HubDirk HusmeierEngineering and Physical Sciences Research Council (EPSRC)EP/T017899/1M&S - Statistics