Geology Reference
In-Depth Information
(
5.11
), except that now d becomes the mean thickness of the intergranular film
(or d
=
d is replaced by the volume fraction / of fluid) and D
GB
is replaced by
V
m
cD
F
where c and D
F
are the molar concentration and diffusion coefficient,
respectively, of the material in the fluid (V
m
c is the volume fraction of the fluid
phase occupied by the diffusing material, here V
m
is, strictly, the molar volume of
the material in solution rather than in the solid phase but in the present approxi-
mation we shall ignore the distinction). Thus for diffusion-limited fluid transfer
creep we have the relation
e
¼
C
FT
V
m
cD
F
d
RT
r
1
r
3
d
3
ð
5
:
12
Þ
where the value of the numerical constant C
FT
is similar to that for Coble creep.
Relations of this form or equivalent to it have been derived by Weyl (
1959
),
Stocker and Ashby (
1973
), Rutter (
1976
,
1983
), Elliott (
1973
), Raj and Chung
(
1981
), Raj (
1982
).
Two limitations on the applicability of (
5.12
) can be foreseen. First, if the fluid
itself is moving through the porous body, the material transfer rate may be
modified in a way that depends on the direction and rate of the fluid flow relative to
the principal stress directions. Second, because of the more rapid diffusion in
liquids or, particularly, the more rapid transport in a moving fluid, the transfer
kinetics are less likely to be rate controlling in fluid transfer creep than are the
interface kinetics at source and sink. We now consider the situation where the
latter are rate controlling.
5.6 Reaction-Controlled Creep
Each of the specific models in the previous three sections has been treated under
the assumption that the diffusion from source to sink is the rate-controlling step.
We now consider the parallel cases in which the creep rate is controlled by the rate
of reaction involved in the release and/or absorption the diffusing species at the
source and/or sink, respectively, as first discussed by Ashby (
1969
) and Green-
wood (
1970
) and subsequently by Raj and Chyung (
1981
) and others. If the
chemical potential difference driving the reaction is determined by the flux from
which the strain rate derives, then the general expression (
5.8
) can be applied. The
diffusion path no longer enters into consideration; only the source-sink geometry
and the reaction rate are involved.
Taking the sources and sinks to be grain boundaries in all three cases so that
V
d
3
and A
s
d
2
, then V
=
A
s
d in (
5.8
) and we obtain the creep equation
e
¼
C
R
V
m
bk
RT
r
1
r
3
d
ð
5
:
13
Þ
