Geology Reference
In-Depth Information
In order to use such a relationship the rate coefficient k must be independently
known (see Tsai and Raj
1982b
for a calculation of a creep rate using an inde-
pendently determined rate coefficient). It will be noted that, provided k and the
grain size d are independent of the stress, the creep rate is linear or Newtonian in
the stress and is inversely proportional to the grain size, in contrast to the d
2
and
d
3
dependence of Nabarro-Herring and Coble creep, respectively.
The relationship (
5.13
) has been developed further by Burton (
1972
,
1983
)
under the assumption that, in the absence of fluid phases and chemical reactions,
the rate coefficient k is determined by a spiral dislocation growth mechanism
within the grain boundaries, according to which k
/
D r
1
r
ð Þ=
E
d
where D is
the diffusion coefficient, and E
d
the energy per unit length of the grain boundary
dislocations. The equation (
5.13
) then becomes
Þ
2
e
¼
C
B
V
m
bD
E
d
RT
ð
r
1
r
3
ð
5
:
14
Þ
d
where C
B
is a new dimensionless constant incorporating C
R
and the proportionality
constant in k. This equation illustrates how a non-Newtonian flow law can arise
from stress dependence in the kinetic factors.
The numerical constant C
R
might be expected to be of similar order to the
constants for the Nabarro-Herring and Coble formulae because of similar com-
patibility requirements, although the formulae of Raj and Chyung (
1981
) implies
that it is of order unity; the most appropriate value does not seem to have been
investigated closely. In Burton's model the constant C
B
incorporates a factor
specifying the density of jogs on the grain boundary dislocation, Burton's estimate
of which leads to C
B
1
=
10; Burton also estimates E
d
to be one-tenth that for
intragranular dislocations, giving E
d
10
9
Jm
1
:
Using C
R
¼
10
;
V
m
¼
10
4
m
3
mol
1
and RT
¼
10
4
Jmol
1
;
(
5.13
) becomes e
¼
10
6
s
1
at r
1
r
3
¼
10 MPa and d
¼
1 lmifk
¼
10
12
ms
1
(a measurable rate,
falling within the range of those determined in a ceramic system by Tsai and Raj
(
1982a
,
1982b
); such a creep rate is comparable to that for Nabarro-Herring creep
under these conditions. From (
5.6
) and using the assumptions underlying (
5.9
) and
(
5.13
), the cross-over from reaction control at relatively small grain sizes to diffusion
control at larger grain sizes is, in fact, to be expected at d
D
=
k, that is, at d
1 lm
when D
¼
10
18
m
2
s
1
and k
¼
10
12
ms
1
:
Alternatively, if Burton's model is
E
d
¼
10
9
Jm
1
;
applied,
with
C
B
¼
1
=
10
;
b
¼
1 nm
and
and
hence
k
¼
10
18
Þ
m
2
s
1
;
the cross-over is to be expected at d
¼
100
=
r
1
r
3
ð
r
1
r
3
ð
Þ
m
when D
¼
10
18
m
2
s
1
;
that is, at d
¼
10 lmif r
1
r
3
ð Þ ¼
10 MPa; below this
stress e
/
r
1
r
ð Þ
2
and above it e
/
r
1
r
ð Þ
until dislocation creep with high
stress exponent (
Sect. 6.6.6
)
takes over, thus giving one possible explanation of the
sigmoidal shaped log strain rate versus log stress plots sometimes observed around
the ''superplastic'' regime (Burton
1972
).
The numerical illustrations given in the previous paragraph are most likely to be
relevant to single phase systems without a fluid phase and indicate a marginal
