Geology Reference
In-Depth Information
where C
0
¼
C
1
C
2
is a dimensionless constant. In order to bring out the physical
content of (
5.6
) it is simplest to consider the two limiting cases.
a. Diffusion-controlled case. When the rate coefficients k
s
are sufficiently large
relative to the diffusion coefficient D that the bracketed term in the
denominator sum can be neglected, the diffusion in the transfer path is rate
limiting and the creep rate is
D
lV
=
A
t
V
m
r
1
r
3
ð
Þ
e
¼
C
0
V
m
c
ð
5
:
7
Þ
RT
Expressing l
;
V and A
t
in terms of grain and grain boundary dimensions leads to
Nabarro-Herring, Coble and fluid transfer diffusion creep laws, as will be shown
later.
b. Reaction-controlled case. When the diffusion coefficient is sufficiently large
relative to the rate coefficients so that the bracketed term in the denominator
sum of (
5.6
) is much greater than unity, the reactions are rate controlling and
the creep rate is
V
m
r
1
r
3
ð
Þ
bk
V
=
A
s
e
¼
C
0
ð
5
:
8
Þ
RT
where we have, for illustration, assumed that the reaction at one of the source-sink
pair is much slower than at the other and we have written k
¼
a
s
k
s
;
an effective
rate coefficient with the dimensions of frequency.
2. If the Dl
s
have independently fixed values, (
5.4
) and (
5.5
) must be indepen-
dently substituted in (
5.1
) to calculate virtual strain rates, the minimum of
which will be the realizable strain rate. It immediately follows that a positive
strain rate can only be obtained if
C
2
V
m
r
1
r
3
ð
Þ
[
Dl
so
ð
Þþ
Dl
si
ð
Þ
that is, that there will be a threshold stress for creep since
Dl
s
ð Þ
and
Dl
s
ð Þ
must both be positive for source and sink to function as such (cf. Ashby
1969
).
At stresses immediately above the threshold stress the diffusion will be rate
controlling because of the relatively small value of
Dl
D
but at stresses above
a certain higher level the reaction rate at source or sink will become rate
controlling and the strain rate will become independent of further increase in
stress.
Before proceeding to consider more specific models of atomic transfer flow,
attention should be drawn to some general aspects of this class of deformation
mechanisms:
1. In the case of ionic substances or multicomponent bodies the individual ions or
components may possibly travel along different transfer paths and tend to have
