Biomedical Engineering Reference
In-Depth Information
be viewed as existing within a reservoir that is capable of supplying more mass of
any constituent or being capable of resorbing some of the mass of any constituent.
This was the approach taken in Cowin and Hegedus ( 1976 ) in the development of a
growth model for bone tissue.
18.4.2 The Biot RVE for Poroelasticity and the Mixture Theory
Approach
A key difference between the Biot effective parameter approach and the Eulerian
point approach to mixture poroelastic models is the averaging process employed.
The effective parameter approach illustrated in Fig. 18.3 is a schematic version of
the viewpoint described in Biot ( 1941 ). He wrote, 'Consider a small cubic element
of soil, its sides being parallel with the coordinate axes. This element is taken to be
large enough compared to the size of the pores so that it may be treated as homoge-
neous, and at the same time small enough, compared to the scale of the macroscopic
phenomena in which we are interested, so that it may be considered as infinitesimal
in the mathematical treatment.' This prose written by Biot appears to be the first
statement of what later came to be called the representative volume element (RVE)
concept. In Biot's proposal a small but finite volume of the porous medium is used
as a model for a continuum point in the development of constitutive equations for
the fluid-infiltrated porous solid. These constitutive equations are then assumed to
be valid at a point in the continuum. 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 struc-
ture over which the material microstructure is averaged or 'homogenized' in the
process of forming a continuum model. The homogenization approach is illustrated
in Fig. 18.3 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 effec-
tive solid mechanical parameters like effective solid moduli than does the mixture
theory approach.
The averaging process for the mixture approach is illustrated in Fig. 18.2 .This
is an Eulerian approach in that the flux of the various species toward and away
from a spatial point is considered. The spatial point is shown in Fig. 18.2 and the
vectors represent the velocities of various species passing through the referenced
spatial point. In neither approach is a length scale specified, but an averaging length
is implied in the Lagrangian material, Biot-effective modulus, approach because
a finite material 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
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