Geology Reference
In-Depth Information
where
a
1
¼
1
2
dG
dp
þ
7
G
K
þ
GDV
G
ð
6
:
68b
Þ
6
kT
The first two terms in a
1
, representing the effects of pressure on the shear
modulus and on the specimen dimensions, respectively, are each of the order of
unity. They are the only terms taken into account in considerations such as those of
Seeger and Haasen (
1958
) and Jung (
1981
). However, in the case of viscous drag
control, it is possible that they may be less important than the third term, repre-
senting the effect of the dilatancy associated with the activation event; for
example,
put DV
G
= 0.01 nm
3
,
if
we
the
order
of
an
atomic
volume,
and
G = 50 GPa, then GDV
G
/kT = 30 at T = 1,200 K
In this case (
6.69
) becomes
pDV
G
kT
s
s
0
exp
ð
6
:
69
Þ
When the dislocation density is high enough that the mutual dislocation
interactions cannot be neglected, then the applied stress s is given by
s
¼
s
eff
þ
s
i
where s
eff
is given by the expression (
6.66
) and s
i
¼
aGbq
2
. In this case
ds
dp
¼
ds
eff
dp
þ
ds
i
ð
6
:
70
Þ
dp
which, with Eqs. (
6.63
) and (
6.67
), leads to the parameter a
1
in (
6.68a
) becoming
a
1
¼
3s
i
s
2s
dG
dp
þ
s
s
i
7
6
G
K
þ
GDV
G
ð
6
:
71
Þ
s
kT
The two limiting cases correspond to the simple cases already discussed:
1. s
i
¼
0 : the pure viscous drag model with low dislocation density, discussed in
the previous paragraph.
2. s
i
¼
s : the pure athermal case, discussed in the previous subsection, for which
a
1
¼
dG
=
dp.
It is thus to be expected that the pressure sensitivity of the flow stress will be
much greater when the flow is controlled mainly by viscous drag on the dislocation
motion (Peierls force, etc.) rather than by mutual dislocation interaction, provided
the activation volume DV
G
in the former case is at least of the order of an atomic
volume.
It is to be reiterated that the pressure sensitivities represented by Eqs. (
6.69
) and
(
6.71
) apply only under the condition that the dislocation content of the crystal
does not change, that is, that qb
2
is constant, elastic volume changes being taken
