Biomedical Engineering Reference
In-Depth Information
Fig. 18.1
A cartoon that illustrates the thesis of this contribution; the cartoon is a paradigm for
the change in conceptual basis suggested for mixture theory based poroelasticity. The house being
transported represents a significant structure of mixture theory being moved to a representative
volume element viewpoint described in Biot (
1941
)
The most important way in which the mixture model is more general than the Biot
poroelastic model is that the mixture model admits the possibility of following many
solid and fluid constituents and it admits the possibility of having chemical reactions
occurring. Thus some constituents might vanish and others might be created. The
contrast with Biot theory is that Biot theory considers the single solid and fluid com-
ponents to be chemically inert. In several important ways the poroelastic model of
Biot (
1941
,
1956a
,
b
,
1962a
,
b
) and Biot and Willis (
1957
) offers better conceptual
mechanisms for relating the elements of the physical situation to their mathematical
representations, a principal example being a hierarchical structure that permits the
distinction between the matrix, the drained and the undrained elastic constants. It
is the objective of this contribution to transfer the selected Biot conceptual mecha-
nisms to a mixture theory formulation of poroelasticity, thus combining the advan-
tages of Biot's ideas with mixture theory.
The cartoon in Fig.
18.1
illustrates the thesis of this contribution; the cartoon is
a paradigm for the change in conceptual basis suggested for mixture-theory based
poroelasticity. The house being transported represents the significant structure of
mixture theory. The foundation from which the house was removed represents the
Eulerian point considered in the analysis of mixtures of fluids (Fig.
18.2
). The foun-
dation to which the house is moved represents the representative volume element
(REV) employed in Biot's (
1941
) development of poroelasticity (Fig.
18.3
). When
a house is moved from one location to another the plumbing must be disengaged
at the first location and reengaged at the new location. That is indeed the case for
this change in conceptual basis suggested for mixture-theory based poroelasticity.
However this change of plumbing for the theory is dealt with in a relatively simple
and simplifying way as will be described.
