A mixture theory for the genesis of residual stresses in growing tissues I : a general formulation

McElwain, Sean & Araujo, Robyn (2005) A mixture theory for the genesis of residual stresses in growing tissues I : a general formulation. SIAM Journal on Applied Mathematics, 65(4), pp. 1261-1284.

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Abstract

In this paper a theoretical framework for the study of residual stresses in growing tissues is presented using the theory of mixtures. Such a formulation must necessarily be a solid-multiphase model, comprising at least one phase with solid characteristics, owing to the fundamental role played by the incompatibility of strains in generating residual stresses. Since biological growth involves mass exchange between cellular and extracellular phases, field equations are presented for individual phases and for the mixture as a whole which incorporate this phenomenon. Appropriate constitutive equations are then deduced from first principles, appealing to the second law of thermodynamics.

The analysis shows that the distinguishing feature of multiphase models involving mass exchange is the necessity to propose an additional constitutive postulate between the variables in the mass-balance equation in order to close the model. In particular, the defining characteristic of a solid-multiphase model which describes biological growth is a constitutive postulate which relates the process of interphase mass exchange (cell proliferation/cell death) with the expansion or contraction of the solid phase. Thus, the framework presented here represents a new class of mathematical models which extends the concepts of poroelasticity to accommodate continuous volumetric growth. A set of modelling equations is then proposed for the simplest case of a solid-multiphase model, being a biphasic mixture of a linear-elastic solid and an inviscid fluid.

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ID Code: 10258
Item Type: Journal Article
Refereed: Yes
Additional Information: The contents of this journal can be freely accessed online via the journal’s web page (see hypertext link).
Keywords: tissue growth, mixture theory, residual stresses, continuum mechanics, constitutive equations, porous media
DOI: 10.1137/040607113
ISSN: 0036-1399
Divisions: Current > QUT Faculties and Divisions > Faculty of Health
Past > QUT Faculties & Divisions > Faculty of Science and Technology
Current > Institutes > Institute of Health and Biomedical Innovation
Copyright Owner: Copyright 2005 Society for Industrial and Applied Mathematics
Copyright Statement: Reproduced in accordance with the copyright policy of the publisher.
Deposited On: 19 Oct 2007 00:00
Last Modified: 29 Feb 2012 13:13

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