Geoscience Reference
In-Depth Information
where
E
j
and
V
j
are the formation energy and
volume of a jog respectively. Formation of a step
such as a jog requires the distortion of a disloca-
tion line and consequently, the formation energy
and the density of these steps are anisotropic,
providing a source for large plastic anisotropy.
At high stresses, the activation enthalpy may
become stress dependent. This is the case when
the rate-controlling process is thermally activated
motion of dislocation glide over the Peierls poten-
tial (potential energy of a dislocation in a crystal).
Dislocation motion over the Peierls potential in-
volves the formation of a kink pair and their
migration. Figure 4.3b shows a saddle point con-
figuration of formation of a pair of kinks where
A
(
σ
) is an area swept by a dislocation. Because
the force exerted by the external stress on the
unit length of a dislocation is
σb
(e.g. Poirier,
1985), the stress does an extra work,
et al
., 1975). This mechanism of deformation is
often referred to as the Peierls mechanism. Note
that this relation implies that deformation at fi-
nite strain-rate is possible even at T
=
0 K if stress
approaches the Peierls stress,
σ
P
.
When the flow law described by Equation (4.12)
operates (high stress and/or low temperatures),
the temperature dependence of creep strength
comes mainly from the stress dependence of acti-
vation enthalpy and
⎡
s
⎤
⎦
q
T
T
o
σ
σ
P
≈
⎣
1
−
(4.13)
where
T
o
is a reference temperature that de-
pends on strain-rate and activation enthalpy
at zero stress. The temperature dependence of
creep strength corresponding to this mecha-
nism is weaker than that of power-law creep
(
σ
·
σb
.
Therefore, the activation enthalpy for dislocation
glide is
−
A
(
σ
)
). The Peierls stress corrected
for temperature is equivalent to the concept of
''yield stress'' often used in geodynamic modeling
(e.g., Tackley, 1998; Richards
et al
., 2001).
exp
E
pl
+
PV
pl
nRT
∝
H
glide
=
H
o
−
A
(
σ
)
·
b
·
σ
(4.11)
and hence activation energy depends strongly on
stress. Also the activation area
A
(
σ
) itself de-
pends on the shape of a dislocation at the saddle
point that depends on the stress, and this leads to
various formulae of the activation enthalpy. The
flow law for such a mechanism of dislocation
motion is given by
4.2.3 Plastic anisotropy and the relation
between single crystal and polycrystal
deformation
σ
μ
2
Flow laws such as those given by Equations (4.1),
(4.9) or (4.12) apply to deformation by each slip
system. In a given crystal, there are several slip
systems and a crystal has plastic anisotropy, i.e.,
the resistance for deformation depends on the
orientation of a crystal with respect to the ap-
plied stress. In the homogeneous deformation of
a polycrystalline aggregate, each grain needs to
be deformed to arbitrary geometry. This requires
five independent slip systems to be present (the
von Mises criterion). Consequently, the rate of
deformation of a polycrystalline aggregate is, in
most cases, controlled by the rate of the most
difficult slip system (Kocks, 1970). This is in con-
trast to the case of lattice-preferred orientation
where the easiest slip system makes the most
important contribution.
ε
glide
=
A
glide
·
exp
1
q
s
σ
σ
P
H
glide
RT
·
−
−
(4.12)
where
A
glide
is a constant with the dimension of
s
−
1
,
H
glide
is the activation enthalpy for disloca-
tion glide at zero stress,
σ
P
is the Peierls stress
(a critical stress for dislocation glide at T
=
0K),
2)
2
(Kocks
q
and
s
are constants (0
<
q
≤
1, 1
≤
s
≤
2
Strictly speaking, the rate of reverse motion of a
dislocation needs to be added to Equation (4.12) (see
Karato, 2008), but this term is not important at high
stresses. The reserve motion term is important when
this equation is used at low stresses.
