Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 229

Continuum points
(b) The diffusion approach.
(a) The effective parameter approach.
Fig. 8.9 Illustrations of the effective medium approach and the mixture theory approach
The length or size of the RVE is assumed to be many times larger than the length
scale of the microstructure of the material, say the size of a pore. The length of the
RVE is the length of the material structure over which the material microstructure is
averaged or “homogenized” in the process of forming a continuum model. Biot's
presentation is consistent with the notion of an RVE, although the RVE terminol-
ogy did not exist when his theory was formulated. The homogenization approach is
illustrated in Fig. 8.9a by the dashed lines from the four corners of the RVE to the
continuum point. The material parameters or constants associated with the solid
phase are more numerous and difficult to evaluate compared to those associated
with the fluid phase. The Biot—effective modulus approach provides a better
understanding of the effective solid mechanical parameters like effective solid
moduli and constituent compressibility than does the mixture theory approach.
The averaging process for the mixture approach is illustrated in Fig. 8.9b . This is
a Eulerian approach in that the flux of the various species toward and away from a
fixed spatial point is considered. The fixed spatial point is shown in Fig. 8.9b and
the vectors represent the velocities of various species passing through the fixed
spatial point. It is important to note that, for mixture theory, the averaging is density
weighted on the basis of the density of each species in the mixture, instead of being
averaged over a finite volume of the porous solid as in the Biot approach. This is the
key difference between the Biot and the mixture theory approach. In neither
approach is a length scale specified, but an averaging length is implied in the
Lagrangian nor material, Biot—effective modulus, approach because a finite mate-
rial volume is employed as the domain to be averaged over. On the other hand the
mixture theory is Eulerian and considers a fixed spatial point through which
different materials pass and, as with the Biot approach, no length scale is suggested.
It is difficult to imagine a length scale for the mixture theory approach other than
one based on the mean free paths associated with the constituents. The significantly
different averaging lengths in the two approaches reflect the difference in the
averaging methods. Cowin and Cardoso ( 2012 ) suggest a method for developing
the Biot- effective modulus approach in mixture theory that requires small
modifications of mixture theory.