Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 180

Obtaining a surface integral representation for the mass flow rates h q i is slightly

more complicated. To start, consider the expression and recall the

discussion of the special form of the divergence theorem following ( 7.9 ) in which
it was indicated that

rð q x Þ

rð q x Þ would imply that the divergence operator would be

applied to q , not to x , thus

q . Then if the second rank
tensor in the divergence theorem in the form (A184) is set equal to x

rð q x Þ ¼ ðr q Þ x þ

q ; the

divergence of x

q is then equal to x ( r q ) þ q , and (A184) yields

þ þ

q þ x ðr q Þ dv ¼

V ð

q n Þ x dv :

(7.16)

V @

A second integral formula involving q is obtained by setting r in the divergence
theorem in the form (A183) equal to q , thus
þ
V r

þ

q dv ¼ q n dv : (7.17)

@V

Now, if it is assumed that there are no sources or sinks in the volume V and that
there is no net flow across the surfaces V , both of the integrals in ( 7.17 ) are zero.

Then, employing the argument that is used to go from (3.4) to (3.5), it follows that
D ∙

∂

q ¼ 0 in the region and using ( 7.1 ) and ( 7.16 ) one may conclude that

þ
@V ð

1
V RVE

h q i¼ n q Þ x ds : (7.18)

Using the representations ( 7.18 ) and ( 7.15 ) for the volume averages of the mass
flow rates hqi and the pressure gradient hr p i , respectively, the effective anisotropic

permeability constants H eff

are defined by the relation

H eff h q i hr p i: (7.19)

This formula provides the tool for the evaluation of the effective permeability of
a porous material in terms of the porous architecture of the solid phase and the
properties of the fluid in those pores. In the section after next the result ( 7.19 )is
used to evaluate the effective permeability in a simple uniaxial model with multiple
aligned cylindrical channels.
Although it is frequently not stated, all continuum theories employ local effec-
tive constitutive relations such as those defined by ( 7.13 ) and ( 7.19 ). This is
necessarily the case because it is always necessary to replace the real material by
a continuum model that does not contain the small-scale holes and inhomogeneities
the real material contains, but which are not relevant to the concerns of the modeler.
In the presentations of many continuum theories the substance of this modeling
procedure is incorporated in a shorthand statement to the effect that a continuum
model is (or will be) employed.