Biomedical Engineering Reference
In-Depth Information
significantly different averaging lengths in the two approaches reflect the difference
in the averaging methods.
18.4.3 The Larger RVE Hypothesis
The larger RVE for mixture theory-based poroelasticity is considered to have a
length or size many times larger than the length scale of the pores (Fig.
18.3
). It
is assumed that the pores represent a lesser length scale that is sub RVE. The length
of the RVE is the length of the material structure over which the porous microstruc-
ture is averaged or 'homogenized' in the process of forming a continuum model.
The RVE is of sufficient size so that three sets of elastic constants (the drained and
the undrained and those of the matrix material) may be represented as well as the
porosity and the pore structure fabric tensor
F
. Pore structure fabric is a quantitative
stereological measure of the degree of structural anisotropy in the pore architecture
of a porous medium (Hilliard,
1967
; Whitehouse,
1974
; Whitehouse and Dyson,
1974
; Oda,
1976
; Cowin and Satake,
1978
; Oda et al.,
1980
,
1985
; Satake,
1982
;
Kanatani,
1983
; Harrigan and Mann,
1984
; Kanatani,
1984a
,
b
,
1985
; Odgaard
1997
,
2001
; Odgaard et al.,
1997
; Matsuura et al.,
2008
). The governing equations for
anisotropic poroelasticity were developed and extended to include the dependence
of the constitutive relations upon pore structure fabric (Cowin
1985
,
2004a
). Dy-
namic poroelasticity was extended by Cardoso and Cowin (
2011
) and Cowin and
Cardoso (
2011
) to include the pore structure fabric tensor as a variable. The pore
structure of the RVE is assumed to be characterized by porosity and a pore structure
fabric tensor
F
.
18.5 The Hypothesis for Representing Microflows at the RVE
Level
Let
v
(a/s)
denote the fluid sub RVE velocity of constituent 'a' relative to the selected
(solid) constituent 's.' This is a velocity that exists only in the small pores of the
solid matrix. The general hypothesis for representing microflows at the RVE level
is that a homogenization process over the RVE may be constructed to determine
the RVE level fluid velocity
v
(a/s)
from the fluid sub RVE velocity
v
(a/s)
.This
homogenization process will depend on the pore structure fabric tensor
F
of the
RVE since the process is accomplished over the porous structure of the RVE.
The precise homogenization process employed is likely to depend upon the par-
ticular problem being studied so no general mathematical formulation is proposed
here. For this presentation the Biot hypothesis (
1956a
) for representing sub RVE
flows at the RVE scale is adopted:
v
(a/s)
,
v
(a/s)
=
ยท
J
(18.1)
