Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 228

8.11 Three Approaches to Poroelasticity
There are three different major approaches to the development of the same basic
equations for the theory of poroelasticity. Each approach is rigorous to its own
hypotheses and stems from well-established mathematical and/or physical models
for averaging material properties. The averaging processes are the difference
between the three approaches.
The first approach, the effective medium approach, originates in the solid
mechanics tradition and the averaging process involved is the determination of
effective material parameters from a RVE, as discussed in the work of Hashin and
Shtrikman ( 1961 ), Hill ( 1963 ) and others. Standard contemporary references are the
topics of Christensen ( 1979 ) and Nemat-Nasser and Hori ( 1993 ). The effective
medium (parameter) approach appears in a primitive form in the early Biot work
( 1941 ) and grows in sophistication with time through the work of Nur and Byerlee
( 1971 ), Rice and Cleary ( 1976 ), Carroll ( 1979 ), Rudnicki ( 1985 ) and Thompson
and Willis ( 1991 ). The development of effective moduli/parameter theory over the
last 50 years occurred almost in parallel with the increasing sophistication and
refinement of the original Biot formulation.
The second approach is called the mixture theory approach; mixture theory is
developed in Chap. 10 . Mixture theory is based on diffusion models and has a very
different philosophy and a longer history than the RVE approach. It stems from a
fluid mechanics and thermodynamical tradition and goes back to the last century.
Fick and Stefan suggested (Truesdell and Toupin 1960 , section 158) that each place
in a fixed spatial frame of reference might be occupied by several different particles,
one for each constituent. This is a Eulerian approach in that the flux of the various
species toward and away from a fixed spatial point is considered. Truesdell ( 1957 )
assigned to each constituent of a mixture in motion a density, a body force density, a
partial stress, a partial internal energy density, a partial heat flux, and a partial heat
supply density. He postulated equations of balance of mass, momentum, and energy
for each constituent and derived the necessary and sufficient conditions that the
balance of mass, momentum, and energy for the mixture be satisfied. Bowen ( 1967 )
summarized the formative years of this subject. For subsequent developments see
Bowen ( 1976 , 1980 , 1982 ) andM¨ ller ( 1968 , 1985 ). An advantage of mixture theory
approach over the other approaches appears when a number of different fluid species
are present and in relative motion. This advantage is attractively illustrated in the
theory for the swelling and deformation behavior of articular cartilage by Lai et al .
( 1991 ), Gu et al . ( 1993 ) and Huyghe and Janssen ( 1997 ).
The key difference between the effective parameter approach and the mixture
approach to poroelastic models is the averaging process employed. The effective
parameter approach illustrated in Fig. 8.9a is a schematic version of the viewpoint
described in Biot ( 1941 ). A small but finite volume of the porous medium is used
for the development of constitutive equations for the fluid-infiltrated solid. These
constitutive equations are then assumed to be valid at a point in the continuum.
This is an early form of the RVE approach used in composite material theory today.