Geoscience Reference
In-Depth Information
where
B
is the mobility (
B
o
is a constant) and
E
disl
is the activation energy and
V
disl
is the activation
volume for dislocation motion and Equation (4.6)
becomes
H
∗
H
∗
°
=
-
s Ω
climb plane
jog
σ
μ
3
exp
.
(4.8)
atomic diffusion
E
disl
+
PV
disl
RT
ε
disl
≈
B
o
·
μ
·
·
−
glide plane
(a)
ε
∝
σ
n
(
n
Such a power-law dependence, i.e.,
3),
is often seen in laboratory studies at modest stress
levels,
=
μ
<
10
−
3
(e.g., Weertman, 1975; Karato,
2008), but more generally
σ
glide plane
kink
exp
σ
μ
n
E
pl
+
PV
pl
RT
Δ
A
b
H
∗
H
°
-
ε
pl
=
A
pl
·
−
·
(4.9)
=
(
)
b
ΔΑ
s
s
(b)
with
A
pl
=
3-5 (the suffix
pl
means
''power-law'') (Weertman, 1975, 1999; Karato,
2008; Poirier, 1985).
For simple materials such as metals, the acti-
vation energy and volume (
E
pl
,
V
pl
) agree with
those of diffusion (
E
∗
D
,
V
D
) and this is interpreted
by a model where the rate of high-temperature
creep is controlled by diffusion-controlled dis-
location climb (Weertman, 1968, 1975, 1999,
Karato, 2008, Kohlstedt, 2006) (see Figure 4.3a).
However, in oxides and silicates the activation
energy and volume often include some extra-term
caused by the high energy of dislocations in these
crystals (Karato, 2008).
1
The extra-term in dislo-
cation climb is the concentration of jogs (steps on
a dislocation line that help dislocation climb, see
two steps on a dislocation line in Figure 4.3a; e.g.,
Poirier, 1985; Karato, 2008). In oxides or silicates,
chemical bonding is strong and the unit cell tends
to be large. Consequently, the dislocation energy
is large and hence the formation of these steps
(jogs in case of dislocation climb) is difficult. In
these cases, the formation of these steps requires
thermal activation. Therefore, Equation (4.9)
B
o
μ
and
n
=
Fig. 4.3
Schematic diagram showing the thermally
activated processes in dislocation creep. (a) diffusion-
controlled dislocation climb and in (b) dislocation glide
over the Peierls potential. Dislocation climb requires
diffusion of atoms from or to a jog of the dislocation
line. When the density of jog is high, all portions of a
dislocation line act as sources or sinks for diffusion,
whereas when the density of jog is small, then
dislocation climb requires the creation of jogs. The rate
of diffusion by thermal activation assisted by stress
will be proportional to exp
H
o
−
σ
RT
−
−
exp
exp
RT
(
σ
RT
H
o
+
σ
RT
H
o
σ
RT
·
1).
Dislocation glide over the Peierls potential involves
the formation and migration of a pair of kinks. This
figure shows a saddle point configuration for the
formation of a pair of kinks.
A
is the area swept by a
dislocation to form the saddle point configuration.
Because the force per unit length of a dislocation by the
external stress is
σb
, the extra work done by the stress
is -
A
(
σ
)
−
≈
2
·
−
·
σ
.Thisterm(
A
(
σ
)
·
b
·
σ
RT
) is large at high
stress, and should be included explicitly leading to the
high sensitivity of strain-rate on stress.
·
b
is modified to
E
∗
D
+
E
j
⎛
⎝
−
⎞
⎠
1
Kohlstedt (2006) argued that dislocation creep in
olivine is directly controlled by diffusion-controlled
dislocation climb similar to deformation of metals.
However, this model is inconsistent with the presence
of large plastic anisotropy in olivine as discussed by
(Karato, 2010a).
P
(
V
D
+
V
j
)
+
A
pl
·
ε
pl
=
exp
RT
σ
μ
n
·
(4.10)
