Biomedical Engineering Reference
In-Depth Information
Fig. 18.3
The foundation to
which the house is moved
represents the representative
volume element (RVE)
employed in Biot's (
1941
)
development of poroelasticity
ences the flow of the fluid and vice versa. The theory was proposed and developed
by Biot (
1941
,
1956a
,
b
,
1962a
,
b
) and Biot and Willis (
1957
) as a theoretical exten-
sion of soil consolidation models for calculating the settlement of structures placed
on fluid-saturated porous soils. The theory has been widely applied to geotechnical
problems beyond soil consolidation, most notably problems in rock mechanics and
wave propagation in porous media. There are thousands of papers, and a singular,
but notable, topic on the subject is that of Coussy (
2004
). The governing equations
for anisotropic poroelasticity for quasi-static and dynamic poroelasticity were de-
veloped and extended to include the dependence of the constitutive relations upon a
pore structure fabric tensor
F
as well as the porosity (Cowin
1985
,
2004a
; Cardoso
and Cowin,
2011
; Cowin and Cardoso,
2011
).
18.4 The Alternative Formulation of Mixture Theory-Based
Poroelasticity
In this alternative formulation of mixture theory-based poroelasticity, the Eulerian
point used in mixture theory as a model of the continuum point (Fig.
18.2
)isre-
placed by a larger RVE introduced by Biot as the model of the continuum point
(Fig.
18.3
). Further, Biot's concept of the RVE level representation of the fluid ve-
locity as a function of the pore fluid velocities in the sub RVE pores is employed.
Finally, the mixture theory concept of the mean velocity of the solid and fluid con-
stituents is replaced by reference to the velocity of the solid and the diffusion veloc-
ities relative to a solid constituent.
This formulation of mixture theory is directed toward the modeling of biological
growth, that is to say changing mass and changing density of an organism. Growth
is slow, accelerationless from a mechanics viewpoint and therefore, although formu-
las for the acceleration are obtained (Cowin and Cardoso,
2012
), accelerations will
be neglected in the applications. The formulas for acceleration are obtained so that
what is neglected is specially specified. The presentation of the theory of mixtures in
Cowin and Cardoso (
2012
) is further restricted to the situation in which all the mix-
ture constituents are compressible, immiscible, reacting (chemical reactions) and all
are at the same temperature. It is assumed that terms proportional to the square of
diffusion velocities will be negligible. Bowen (
1976
), see page 27 therein, considers
