Geoscience Reference
In-Depth Information
D
o
μ
anelastic energy dissipation (see Chapter 3, this
volume, above). In calculating strain-rate for diffu-
sion creep, heterogeneous stress distribution and
resultant diffusional mass flux must be solved
self-consistently to obtain correct steady-state
strain-rate.
A
diff
·
with
A
diff
=
RTb
2
is a constant with a di-
mension of s
−
1
,
E
VD
(
BD
)
,
V
VD
(
BD
)
are the acti-
vation energy and volume for volume or grain-
boundary diffusion.
In a compound, the diffusion coefficient in
Equation (4.3) or (4.4) must be a combination
of diffusion coefficients of various species. For
olivine, there are at least three species (Mg (Fe), Si
and O) to consider. Usually diffusion of Si is the
slowest and diffusion coefficient of Si is used to
calculate creep rate for diffusion creep from dif-
fusion coefficients (e.g., Shimojuku
et al
., 2004;
Yamazaki
et al
., 2000). However, this assump-
tion may not be valid in some cases. First, in
many oxides, slow diffusion species in the bulk
tends to have high diffusion coefficients along
grain-boundaries but the diffusion of fast diffu-
sion species (in the bulk) is not much enhanced
along grain-boundaries (Gordon, 1973). In these
cases, the net diffusion coefficient of a species that
has the faster bulk diffusion can become slower
than the diffusion of other species. In these cases,
a species that has a fast volume diffusion coeffi-
cient could become the rate-limiting step. Second,
in (Mg, Fe)SiO
3
perovskite, Holzapfel
et al
. (2005)
showed that Mg (Fe) diffusion is the slowest in the
bulk (Xu
et al
. (2011) showed that diffusion coef-
ficients of Si and Mg are similar in perovskite).
A subtle but important point in diffusion creep
is that the rate of deformation depends on the gra-
dient in point defect concentration that depends
on the stress state at grain-boundaries. The stress
state at grain-boundary is in turn controlled by
deformation because deformation relaxes stress
concentration. This point was elegantly studied
by Raj and Ashby (1971). Upon the application of
stress to a specimen, stress concentration will
occur at grain corners (weak grain-boundaries
are assumed). High stress concentration enhances
diffusional flow, and reduces the stress concen-
tration. Consequently, stress distribution will be
modified, and at steady state, smooth distribu-
tion of stress is achieved. A faster strain-rate is
observed in the initial transient period (see also
Lifshitz & Shikin, 1965). Such a transient creep
behavior is potentially important in the analy-
sis of post-glacial rebound (Karato, 1998a) and in
4.2.2 Dislocation creep
Plastic deformation occurs also by a collective
motion of atoms such as the migration of crystal
dislocations. A dislocation is defined as a propa-
gation front line of a slip that is characterized by
the slip plane and slip direction. A combination
of a slip plane and slip direction defines a slip
system. The flow law by dislocation creep can be
described by the Orowan equation (e.g. Poirier,
1985; Karato, 2008; Orowan, 1940):
ε
disl
=
bρυ
,
(4.5)
where
b
is the length of the Burgers vector (the
unit displacement associated with a dislocation),
ρ
is the dislocation density (the total length of
dislocations per unit volume) and
υ
is the disloca-
tion velocity. Using Equation (4.2), this equation
leads to
b
−
1
σ
μ
2
ε
disl
(
σ
,
T
,
P
;
X
)
≈
·
υ
(
σ
,
T
,
P
;
X
),
(4.6)
where
X
is a set of parameters representing chem-
ical environment such as water fugacity and
oxygen fugacity. In general the dislocation ve-
locity increases with applied stress and therefore,
the rate of deformation by dislocation creep is
a nonlinear function of stress. If one defines ef-
fective viscosity by
η
eff
=
σ
˙
ε
, then the effective
viscosity is not a constant but it decreases with
stress or strain-rate.
At high temperatures (relative to the melting
temperature (T
m
), i.e., T
/
T
m
>
0.5), dislocation
motion is thermally activated, and at low stresses,
the dislocation velocity is a linear function of
stress. In these cases, the dislocation velocity can
be written as
exp
E
disl
+
PV
disl
RT
υ
=
Bbσ
=
b
·
B
o
·
−
·
σ
,
(4.7)
