Geology Reference
In-Depth Information
The last effect will tend to limit severely the amount of strain that can be
produced at the initial rate given by (
5.15
), making the dislocation climb creep a
transient effect. However, if a certain threshold stress is exceeded the dislocation
segments between the pinning points can act as Bardeen-Herring climb sources for
dislocation multiplication (Bardeen and Herring
1952
) and unlimited amounts of
climb can be produced. The dislocation spacing will still be limited by the need for
the externally applied stress to balance the internal stress arising from mutual
repulsion of like dislocations. These internal stresses are proportional to the
spacing between the dislocations and hence inversely proportional to q
2
;
leading to
the dislocation density being of the order of r
1
r
3
Þ
2
=
G
ð
2
:
Using this rela-
ð
tionship in (
5.15
) leads to
Þ
3
e
¼
C
N
V
m
D
V
RT
ð
r
1
r
3
ð
5
:
16a
Þ
G
ð
2
or, with V
m
Lb
3
;
bD
V
G
2
RT
e
¼
C
0
N
Þ
3
ð
r
1
r
3
ð
5
:
16b
Þ
which is the formula of Nabarro (
1967
); G is the shear modulus, b the Burgers
vector, L the Avagadro constant and C
N
;
C
0
N
numerical constants of order 0.01 and
0.01/L respectively.
If the transfer process is pipe diffusion along the dislocation cores, we have
A
t
b
2
instead of q
1
in (
5.7
), where b is the Burgers vector, and hence instead of
(
5.15
) we have the initial strain rate
e
¼
C
DC
=
P
V
m
D
P
b
2
q
2
RT
ð
r
1
r
3
Þ
ð
5
:
17
Þ
where D
P
is the pipe diffusion coefficient and C
DC
=
P
is a numerical constant, again
probably of the order of magnitude of unity to ten. Dislocation multiplication
leading to a steady state dislocation density of the order of r
1
r
ð Þ
2
=
G
ð
2
would correspondingly yield the further formula of Nabarro (
1967
) for the case of
pipe diffusion,
bD
P
G
4
RT
Þ
5
e
¼
C
0
N
=
P
ð
r
1
r
3
ð
5
:
18
Þ
again with C
0
N
=
P
of the order 0.01.
If we use the parameters C
N
¼
C
N
=
P
¼
0
:
01
;
V
m
¼
10
4
m
3
;
T
¼
1
;
200
K, G
¼
50 GPa and b
¼
0
:
5 mm, and we consider a stress r
1
r
3
¼
10 MPa, the
Nabarro formula (
5.16b
) gives a strain rate of the order of 10
-10
s
-1
for dislocation
climb creep sustained by Bardeen-Herring dislocation multiplication if we assume
material transfer by volume diffusion with D
V
¼
10
18
m
2
s
1
:
Alternatively, (
5.18
)
gives a strain rate of the order of 10
-11
s
-1
if we assume transfer by dislocation core
