Geoscience Reference
In-Depth Information
where I chose grain-size and stress as independent
parameters for a particular set of temperature
and pressure. Similar diagrams have been con-
structed for other minerals including plagioclase
(anorthite) (Rybacki & Dresen, 2000), (Mg,Fe)O
and perovskite (Karato, 1998b) and ringwoodite
(Karato
et al
., 1998). For most of mantle miner-
als, the dominant deformation mechanism in the
typical hot mantle is diffusion creep or power-law
dislocation creep. However, in a cold subducted
lithosphere, the Peierls mechanism also plays an
important role (Karato
et al
., 2001).
more deformation will have a higher temperature
and the increased temperature enhances defor-
mation. If this positive feedback is more efficient
than the work-hardening (strain-hardening) that
will stabilize the system, then thermal runaway
will occur. The magnitude of positive feedback
is proportional to the rate of heat generation by
mechanical work and inversely proportional to
the rate of heat diffusion that is sensitive to the
space scale of deformation. Consequently, this in-
stability occurs when the rate of heating, i.e., the
energy dissipation rate, exceeds a certain value
(e.g., Argon, 1973),
4.2.7 Shear localization
π
2
hC
p
κRT
2
H
∗
L
2
σ
˙
ε>
(
σ ε
)
c
=
(4.15)
The discussions so far are all for ''steady-state''
deformation. This is a convenient assumption
that makes the formulation of flow laws easy.
However, the validity of steady-state deformation
is questionable in some cases. Particularly impor-
tant is deformation of the lithosphere. Under low
temperature conditions, nonsteady deformation
likely occurs leading to strain localization that
reduces the strength of the lithosphere substan-
tially.
The essence of the conditions for shear local-
ization is the presence of a process of positive
feedback, i.e., a process wherein the increase in
strain (or strain-rate) leads to the reduction in
the creep strength. In these cases, regions that
are deformed more become easier to deform so
that the runaway instability will occur. Such
a positive feedback is, however, not common.
In most cases, materials show work-hardening
(strain-hardening), and the resistance to defor-
mation increases with strain-rate, leading to a
negative feedback that stabilizes deformation.
However, there are several mechanisms that
lead to a positive feedback. Two processes of
such instability are well documented. One is the
thermal runaway instability where deformation-
induced heating leads to runaway instability, and
another is the instability caused by grain-size
reduction. In both cases, instability will occur
under limited conditions.
Consider thermal runaway instability. Defor-
mation produces heat and therefore a region of
∂
log
σ
∂
log
ε
where
h
1) is the coefficient of work-
hardening,
C
P
is the specific heat,
κ
is the thermal
diffusivity,
H
∗
is the activation enthalpy, and
L
is the length scale. Note that the conditions for
instability depend strongly on the space scale,
L
. Because the energy dissipation rate per unit
volume is given by
σ ε
=
η
(
T
,
P
)
=
(
∼
ε
2
, this insta-
bility occurs when viscosity exceeds a certain
value that depends on the space scale. For a typ-
ical strain-rate of
10
−
15
s
−
1
, and a space scale
∼
10
23
Pa s.
Consequently, we conclude that this mechanism
of shear localization occurs at low temperatures
where viscosity is high.
Grain-size reduction could also lead to shear
localization but again only at low temperatures.
Let us consider the grain-size reduction caused
by deformation (dynamic recrystallization). Dy-
namic recrystallization occurs when a material
deforms by dislocation creep. Small grains are
formed along the pre-existing grain-boundaries
(Figure 4.5a). The size of recrystallized grains is
inversely proportional to stress,
L
b
of
∼
100 km, the critical viscosity is
∼
A
r
μ
−
a
(
L
r
:
the size of recrystallized grains,
A
r
and
a
:con-
stants) (Karato, 2008; Poirier, 1985; Derby, 1991).
If the size of these grains is small enough, grain-
size sensitive creep such as diffusion creep will
operate there and these regions will be softer than
the initial material. Then the load is transferred to
coarse-grained regions (cores in Figure 4.5a) that
=
