Geology Reference
In-Depth Information
c
¼
q
2
bDAv
0
exp
ð
DE
sbDA
Þ=
kT
½
ð
6
:
33
Þ
where q is the dislocation density and the other symbols are as used in
Sect. 6.4.1
and Fig.
6.19
. The quantity DA
is viewed as increasing progressively according to
the exhaustion or hardening effect just discussed. In the hardening case we can
write
DE
¼
Ds
d
bDA
þ
U
ð
6
:
34
Þ
where Ds
d
bDA
represents the work needed to overcome the local fluctuation in
mutual dislocation interaction that is holding up the dislocation, and U is any
additional barrier (such as a Peierls or other short-range barrier) that must also be
overcome when moving the dislocation (note that the activation area DA
in (
6.34
)
has been taken as being identical with that in (
6.33
), but it is possible that in some
cases a distinction will need to be maintained). On the view that the mutual
dislocation interactions can be described in terms of an internal stress (
Sect. 6.6.3
),
Ds
d
is the local fluctuation in internal stress s
i
to be overcome by thermal acti-
vation. We now suppose that the mutual dislocation interaction effect increases
linearly with strain, putting Ds
d
¼
hc
;
where h is the strain-hardening coefficient
and c is the creep strain. In this case, the relationship (
6.33
) can be integrated at
constant stress to give the creep law
c
tot
¼
c
inst
þ
c
0
ln
ð
1
þ
mt
Þ
ð
6
:
35a
Þ
(Friedel
1964
, p. 306; Haasen
1978
, p. 277) {Weertman 1983 #7385, whose
expression for v lacks an explicit stress dependence}, where c
tot
and c
inst
are the
total and instantaneous (elastic plus plastic) strains, respectively, and c
0
;
v are
given by
kT
hbDA
c
0
¼
ð
6
:
35b
Þ
v
¼
q
2
bDAm
0
c
0
Þ
bDA
exp
U
s
c
inst
h
ð
ð
6
:
35c
Þ
kT
The relation in (
6.35a
) leads to a creep rate
c
0
m
1
þ
vt
c
¼
ð
6
:
35d
Þ
Therefore, the initial creep rate is predicted to be c
0
v and the creep rate becomes
c
0
=
t for vt
1.
The model just described is still a phenomenological one in that the parameters
h
;
DA
and DA have not been given precise physical meaning in microstructural
terms, but it serves to illustrate the essential nature of logarithmic creep, which, in
this model, derives from the linear strain dependence of the activation energy DE
through (
6.34
).
