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)

Abstract

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
• Status: Published
• Refereed: Yes
• 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
• Publisher: Springer
• ISSN: 0939-1533
• 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

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
303232		EPSRC Centre for Multiscale soft tissue mechanics with MIT and POLIMI (SofTMech-MP)	Xiaoyu Luo	Engineering and Physical Sciences Research Council (EPSRC)	EP/S030875/1	M&S - Mathematics
308255		The SofTMech Statistical Emulation and Translation Hub	Dirk Husmeier	Engineering and Physical Sciences Research Council (EPSRC)	EP/T017899/1	M&S - Statistics