Geoscience Reference
In-Depth Information
Anisotropy in deformation for individual grains
leads to plastic anisotropy of a rock if the distri-
bution of crystallographic orientation of grains
is not random. In these cases, rheological prop-
erties must be treated as anisotropic properties
(effective viscosity becomes a fourth-rank tensor
same as elastic constants). Influence of plastic
anisotropy on geodynamic processes has been
studied by Saito and Abe (1984); Honda (1986);
and Lev and Hager (2008), but plastic anisotropy
of rocks has not been studied extensively. In case
of olivine, where rheological properties have been
studied extensively, a comparison on rheological
properties for tri-axial compression and simple
shear shows that plastic anisotropy is only mod-
est
3
. However, for highly anisotropic crystal such
as
hcp
metals (e.g., zinc and
ε
-iron in the in-
ner core), plastic anisotropy of an aggregate can
be very large (Frost & Ashby, 1982). This may
have an important effect on the dynamics of the
inner core.
(Karato, 2010a) discussed the influence of
anisotropic diffusion on the strength of a poly-
crystalline aggregate. For diffusion creep, it
is the diffusion along the slowest orientation
that controls the overall rate of deformation
(see also Lifshitz, 1963). For dislocation creep
controlled by dislocation climb, which is in
turn controlled by diffusion, the rate controlling
diffusion coefficient is the intermediate diffusion
coefficient. This is because the rate of dislo-
cation climb is controlled by diffusion in the
direction normal to the dislocation line, and the
rate-controlling step of deformation of a polycrys-
talline aggregate is deformation by the hardest
slip system.
4
4.2.4 The role of grain-boundary sliding
Deformation of a polycrystalline material can
occur by grain-boundary sliding (Langdon, 1975,
1994). However, grain-boundary sliding creates
gaps and overlaps of grains and therefore some
processes must operate to remove gaps or
overlaps. Processes of accommodation include
diffusion creep and dislocation creep.
Grain-boundary sliding accommodated by dif-
fusion creep is a typical example. Raj and Ashby
(1971) presented a theoretical analysis of the inter-
play between grain-boundary sliding and diffusion
creep. They showed that these processes must
operate simultaneously and therefore a more dif-
ficult process controls the overall rate of deforma-
tion. In many cases, grain-boundary sliding is eas-
ier than diffusional mass transport and the flow
law is essentially the same as the diffusion creep.
When dislocation processes are involved in the
intra-granular deformation, somewhat different
flow law may arise. When accommodation by
dislocation creep is much more difficult than
grain-boundary sliding, a flow law similar to dis-
location creep will apply (e.g., Chen and Argon,
1979). However, there is a narrow parameter space
where dislocation creep rate can be affected by the
stress concentration at grain-boundaries. In these
cases, a nonlinear, grain-size dependent flow law
will be obtained, viz.,
exp
σ
μ
n
b
L
m
E
gbs
+
PV
gbs
RT
˙
ε
gbs
=
A
gbs
·
−
·
·
(4.14)
where
A
gbs
is a constant with the dimension of
s
−
1
,
n
1-3 (e.g., Nieh
et al
., 1997),
and
E
gbs
and
V
gbs
are the activation energy and
activation volume respectively (Table 4.1). Nieh
et al
. (1997) listed a number of examples of such
a rheological behavior in metals and ceramics.
A similar rheological behavior is also reported
in geological materials (Goldsby & Kohlstedt,
2001; Hiraga
et al
., 2010; Hansen
et al
., 2011).
It is sometimes suggested that this mechanism
is crucial for shear localization (Warren & Hirth,
2006; Precigout
et al
., 2007). However, the param-
eter space in which this mechanism dominates is
=
2-3 and
m
=
3
Recently Hansen et al. (2012) reported high plastic
anisotropy of olivine aggregates. However, their formu-
lation of plastic anisotropy is mathematically incorrect
(see Chapter 3 of Karato [2008]). Therefore the validity
of their conclusion is therefore unclear.
4
One compares
2
,
D
22
+
D
3
2
(
D
ii
: diffu-
sion coefficient along the
i
-th direction), and the small-
est one controls the rate of deformation of a polycrystal.
D
11
+
D
22
,
D
11
+
D
33
2
