Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 260

based on diffusion models that originated in a fluid mechanics and thermodynamics
tradition and was formulated in the century before last.
The modeling of pore fluid flow in porous materials whose fluid and matrix
constituents are solid components, solvents and solutes, and which may be modeled
as incompressible, is the main subject of this chapter. The modeling of pore fluid
flow in geological and biological materials whose components are compressible
was dealt with in Chap. 8 on quasistatic poroelasticity. In Chap. 8 it was shown how
the incompressible case may be approached from the compressible and thus
demonstrating that it is possible to treat the compressible and incompressible
cases jointly, but it is easier to address them separately because the interaction of
the interstitial fluid flow and the solid matrix in these two tissue types is signifi-
cantly different. Chap. 9 concerned wave propagation in poroelasticity and neces-
sarily deal with the case when all constituents were compressible.
The porous medium behavior of hard biological tissues is similar to the behavior
of saturated porous rocks, marble and granite, while the porous medium behavior of
soft tissues is similar to the behavior of saturated soils, the sort of geological
deposits one might call “swampy soils.” Although both are saturated porous
media, their detailed modeling and physical behavior are quite different. The
term unsaturated porous media generally refers to cases when the matrix pores
are filled with a fluid and a gas as, for example, the soil near the roots of a plant may
contain water, air, and soil solids. In the case of hard tissues and saturated porous
rocks, the fact that the bulk stiffness of the matrix material is large compared to the
bulk stiffness of water means that (1) only a fraction of the hydrostatic stress in
the matrix material is transferred to the pore fluid, and (2) the strains levels in many
practical problems of interest are small. In the case of soft tissues and the saturated
porous soils, (1) the strains can be large (however, only small strains are considered
in this chapter), and (2) the bulk stiffness of the matrix material is about the same as
the bulk stiffness of water which means that almost all of the hydrostatic stress in
the matrix material is transferred to the pore fluid. The effective Skempton param-
eter defined by (8.49) or (8.45) is a measure of the fraction of the hydrostatic stress
in the matrix material that is transferred to the pore fluid. For soft tissues and
saturated soils the Skempton parameter approaches one, indicating that almost
100% of the hydrostatic stress in the matrix material is transferred to the pore
fluid. As a consequence of this fact that the response to volumetric deformation of
the fluid and the solid matrix in soft tissue is much stiffer than the deviatoric
response, it is reasonable to assume that the soft tissues and the contained fluid
phase are incompressible. Thus soft tissues are “hard” with respect to hydrostatic
deformations and soft with respect to shearing or deviatoric deformations. This fact
is the justification for the assumption of incompressibility of both the matrix
material and the pore fluid made in the development of porous media models for
soft tissues. The assumption of incompressibility is not correct for marble, granite,
and hard tissue. For marble and granite the Skempton parameter is between 0.5 and
0.6 and for the lacunar canalicular porosity of bone it is between 0.4 and 0.5. This
means that only about 50% of the hydrostatic stress in the matrix material is
transferred to the pore fluid in these materials with a stiff bulk modulus.