Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 261

10.2 The Present Mixture Theory-Based
Poroelasticity Approach
Since mixture theory was first presented by Truesdell ( 1957 ) its relationship to the
previously established Biot's poroelasticity theory ( 1941 ) has been a subject of
discussion. In this Chapter the overlap in the two theories is increased. In several
important ways the mixture model of saturated porous media is more general than
the Biot ( 1941 ) model of poroelasticity; Bowen ( 1980 , 1982 ) recovered the model
of Biot ( 1941 ) from the mixture theory approach. 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 components to be
chemically inert. In several important ways the poroelastic model of Biot ( 1941 ,
1956a , b , 1962 ) offers better conceptual mechanisms for relating the elements of the
physical situation to their mathematical representations, a principal example being
in 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 mechanisms to a mixture theory formulation of poroelasticity, thus
combining the advantages of Biot's ideas with mixture theory.
The mixture theory-based poroelasticity presented in this chapter is augmented
from the usual presentation by the addition of two poroelastic concepts developed
by Biot and described in the two previous chapters. The first of these is the use of
the larger RVE in Fig. 8.9a rather that the Eulerian point often employed when the
mixture consist only of fluids and solutes, Fig. 8.9b. The second is the subRVE-
RVE velocity average tensor J , which Biot called the micro-macro velocity
average tensor and which is related to pore structure fabric by (9.35). These two
poroelastic concepts are developed in Cowin and Cardoso ( 2012 ). Traditional
mixture theory allows for the possibility constituents to be open systems, but the
entire mixture is a closed system. In this development the mixture is also considered
to be an open system. The velocity of a solid constituent is employed as the main
reference velocity in preference to the mean velocity ( 10.25 ) concept in the early
formulations of mixture theory. The mean velocity concept is avoided in the
mixture theory-based poroelasticity because the averaging of the solid velocity
and the fluid velocity is seldom a quantity of physical interest as it is in mixture in
which all constituents are fluids. The standard development of statements of the
conservation principles and entropy inequality employed in mixture theory are
modified to account for these kinematic changes and to allow for supplies of
mass, momentum, and energy to each constituent and to the mixture as a whole.
This presentation of the theory of mixtures is restricted to the situation in which
all the mixture constituents are incompressible, immiscible and all are at the same
temperature y . It is assumed that terms proportional to the square of diffusion
velocities will be negligible. Bowen ( 1976 , p. 27) considers the case where they