Geology Reference
In-Depth Information
If we now assume a first order kinetics of recovery in which
ð
oq
=
ot
Þ
c
is propor-
tional to the dislocation density q itself and to the dislocation climb velocity v
c
;
and inversely proportional to the climb distance needed to bring about annihila-
tion, say half the dislocation spacing,
2
q
2
;
then
1
¼
a
0
qv
c
oq
ot
q
2
¼
a
0
v
c
q
2
c
where a
0
is a numerical constant. Inserting this expression in (
6.44
) and recalling
s
¼
aGbq
2
leads to
h
¼
a
0
s
2
v
c
G
2
b
c
s
¼
r
2ab
In the case of diffusion-controlled climb, we can, from (
6.23a
), put v
c
2Db
2
s
=
kT (assuming r
¼
2s) to obtain finally
3
c
s
¼
b
0
GbD
kT
s
G
ð
6
:
45
Þ
where b
0
a
0
=
ab is a numerical constant.
The form of the creep relation (
6.45
) suggests that in recovery-controlled creep
the steady-state strain rate can be expected to depend fairly strongly on the stress.
The cube stress exponent can be arrived at in a number of ways and has been
described as the ''natural'' exponent for a high-temperature creep law (Weertman
1975
). While various materials do conform approximately to (
6.45
) in their stress
dependence, many others show a stronger stress dependence in high-temperature
creep, leading Dorn to propose the semiempirical law
n
c
s
¼
A
GbD
kT
s
G
ð
6
:
46a
Þ
where A and n are empirical constants, found to be related according to
Þ
n
2
:
7
A
1025
ð
ð
6
:
46b
Þ
(Brown and Ashby
1980
; Poirier
1985
, p. 85). Various theoretical strategems
have been proposed for rationalizing values of n [ 3
;
based on more sophisti-
cated recovery models (for example Poirier
1985
, p. 110; Weertman
1975
).
It may also be noted that n
¼
3in(
6.45
) derives, in part, from an assumption of
linear strain hardening and that the alternative of a parabolic athermal strain
hardening would immediately lead to n
¼
4
:
However, it has been found to be
difficult to rationalize values of n greater than 5 and doubt has been expressed
that a power-law representation of the creep behavior is the most appropriate in
such cases (Poirier
1985
, p. 111)
In Eqs. (
6.45
) and (
6.46a
) the temperature dependence of c
s
derives mainly
from the temperature dependence of the diffusion coefficient D, which, in the case
