Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 230

Bowen ( 1982 ) showed that it was possible to recover the Biot constitutive
equations from the mixture theory approach. In particular Bowen derived equations
(his 8.23 and 8.24) that have the same form as equations (7) and (8) of Rice and
Cleary ( 1976 ); these equations are (3) and (5) in the text below. (Actually the
second summand on the right hand side of Bowen's equation ( 8.24 ) is missing a
factor of 3; this is probably a typo or an algebraic slip.) Earlier Mow and Lai ( 1980 ,
footnote page 291) indicated how the mixture theory-based (incompressible)
biphasic theory for articular cartilage could be linearized to the incompressible
Biot theory.
In Chap. 10 the difference between the RVE approach of Biot, represented by
Fig. 8.9a and the mixture theory approach represented by Fig. 8.9b is narrowed
by minor changes in the mixture theory approach.
The third major approach to the development of the poroelastic equations is due
to Burridge and Keller ( 1981 ). These authors rederived the dynamic form of the
same basic set of equations using a two-space method of homogenization. This
method provides a systematic method for deriving macroscopic equations that
govern the behavior of the medium on the microscale. Thus, at the continuum
point, the three rigorous theoretical developments lead to the same set of equations,
and the difference between Biot (RVE), the mixture-theory-based, and the homog-
enization derivations is the method of averaging. The nature of the equations is
better understood because there are three approaches. The Biot approach provides
better insight into the nature of the parameters associated with the solid phase, the
mixture theory approach provides the mechanism for averaging over different fluid
phases, and the homogenization approach illuminates the dynamical (wave propa-
gation) characteristics of the theory.
The basic equations of quasistatic poroelasticity developed here are extended to
include a dependence the fabric tensor (introduced at the end of Chap. 7 ) in Cowin
( 2004 ).
8.12 Relevant Literature
There is only one text on poroelasticity (Coussy 1995 ), but there is much related
material in Bear ( 1972 ). There is also a topic on the very interesting historical
development of the theory, de Boer ( 2000 ), which also presents a noteworthy
presentation of the theory. The presentation of poroelasticity in this chapter was
taken from the following papers: Biot ( 1941 , 1955 , 1956a, b, 1962a, b), Rice and
Cleary ( 1976 ), Rudnicki ( 1985 ), Thompson and Willis ( 1991 ), Cowin ( 2003 ),
Cowin and Mehrabadi ( 2007 ), Cowin and Cardoso (2009) and the excellent sum-
mary of Detournay and Cheng ( 1993 ). In these papers the proofs omitted in this
chapter, as well as some technical restrictions or assumptions underlying those
proofs, are documented. The exception to the inclusion of proofs occurred at
the end of Sect. 8.2 . The proof included was the derivation of the formula ( 8.3 )