Biomedical Engineering Reference
In-Depth Information
Chapter 18
Reformulation of Mixture Theory-Based
Poroelasticity for Interstitial Tissue Growth
Stephen C. Cowin
Abstract
This contribution presents an alternative approach to mixture theory-
based poroelasticity by transferring some poroelastic concepts developed by Biot
to mixture theory. These concepts are a larger RVE and the subRVE-RVE velocity
average tensor, which Biot called the micro-macro velocity average tensor. This ve-
locity average tensor is assumed here to depend upon the pore structure fabric. The
formulation of mixture theory presented is directed toward the modeling of inter-
stitial growth, that is to say changing mass and changing density of an organism.
Growth is slow and accelerations are neglected in the applications. The velocity of
a solid constituent is employed as the main reference velocity in preference to the
mean velocity concept from the original formulation of mixture theory. The standard
development of statements of the conservation principles and entropy inequality em-
ployed in mixture theory are easily modified to account for these kinematic changes
and to allow for supplies of mass, momentum and energy to each constituent and to
the mixture as a whole. The objective is to establish a basis for the development of
constitutive equations for growth of tissues.
18.1 Introduction
The purpose of this contribution is to present an alternative approach to mixture
theory-based poroelasticity by transferring some poroelastic concepts developed by
Biot (
1935
,
1941
,
1956a
,
b
,
1962a
,
b
) and Biot and Willis (
1957
) to mixture the-
ory. The long-term objective of this study is facilitating the mixture modeling of
biological growth phenomena. Since mixture theory was first presented by Trues-
dell (
1957
) its relationship to the previously established Biot's poroelasticity theory
(Biot,
1941
) has been a subject of discussion. In this contribution the overlap in the
two theories is increased. In several important ways the mixture model of saturated
porous media is more general than the Biot (
1941
) model of poroelasticity; Bowen
(
1980
,
1982
) recovered the model of Biot (
1941
) from the mixture theory approach.
S.C. Cowin (
)
The Department of Biomedical Engineering, Grove School of Engineering of The City College
and The Graduate School, The City University of New York, New York, NY 10031, USA
e-mail:
sccowin@gmail.com
