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A new continuum theory for incompressible swelling materials

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Title: A new continuum theory for incompressible swelling materials
Authors: Degond, P
Ferreira, M
Merino-Aceituno, S
Nahon, M
Item Type: Journal Article
Abstract: Swelling media (e.g. gels, tumors) are usually described by mechanical constitutive laws (e.g. Hooke or Darcy laws). However, constitutive relations of real swelling media are not well-known. Here, we take an opposite route and consider a simple packing heuristics, i.e. the particles can’t overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can’t swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets while in the parallel direction, it is determined by a Laplace-Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law.
Issue Date: 4-Feb-2020
Date of Acceptance: 25-Nov-2019
URI: http://hdl.handle.net/10044/1/75398
DOI: 10.1137/18M1203158
ISSN: 1540-3459
Publisher: Society for Industrial and Applied Mathematics
Start Page: 163
End Page: 197
Journal / Book Title: SIAM: Multiscale Modeling and Simulation
Volume: 18
Issue: 1
Copyright Statement: © 2020 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license
Sponsor/Funder: The Royal Society
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: WM130048
EP/M006883/1
EP/N014529/1
Keywords: Science & Technology
Physical Sciences
Mathematics, Interdisciplinary Applications
Physics, Mathematical
Mathematics
Physics
packing
nonoverlapping constraint
minimization
bathtub principle
level sets
continuity equation
domain velocity
Laplace-Beltrami
TISSUE-GROWTH
SOLID TUMOR
MODEL
SYSTEM
SIMULATION
math.AP
math.AP
q-bio.CB
70G75, 76Z99, 74L15, 92C10
Applied Mathematics
0102 Applied Mathematics
Publication Status: Published
Online Publication Date: 2020-02-04
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



This item is licensed under a Creative Commons License Creative Commons