Geology Reference
In-Depth Information
Integrating and putting
1
=ðÞ
db
=
dp
ð
Þ¼
1
=
3K
;
where K is the bulk modulus,
we obtain a steady-state strain rate of
p
G
c
s
¼
c
s0
exp
a
1
ð
6
:
75
Þ
and
a
1
¼
3
2
dG
dp
þ
7
G
K
þ
GDV
D
ð
6
:
76
Þ
6
kT
Alternatively, if the pressure effect is expressed as 1
=ðÞ
ds
=
d
ð Þ
c
for a constant
strain-rate test, a similar argument leads to a steady-state flow stress of
a
1
3
p
G
s
s
¼
s
s0
exp
ð
6
:
77
Þ
where s
s0
is the steady-state flow stress at zero pressure and a
1
;
is again given by
(
6.77
).
When DV
D
is of the order of an atomic volume or more, the third term in (
6.76
)
tends to be large compared with the other two terms, leading to the commonly used
expression
c
s
¼
c
s0
exp
pDV
kT
ð
6
:
78
Þ
The quantity DV in this expression corresponds to an experimentally deter-
expected to be approximately, but not exactly, equal to the activation volume for
diffusion DV
D
.
When effects such as viscous drag and cross-slip of dislocations are introduced
in more realistic thermal models of flow, the interpretation of the experimentally
determined activation volume for steady-state flow may be more complex. Only
when such effects are relatively minor compared with the diffusion-controlled
recovery effects can the empirical activation volume be expected to approximate
the diffusion activation volume DV
D
.
Even if a steady state has been attained at a given pressure, there will, of course,
tend to be transient effects when a step change is made in the pressure (Fig.
6.23
).
The relationships in Eqs. (
6.75
) and (
6.77
) can only be applied to the strain rate
and the stress measured after the transient effects have given way to a new steady
state. For the simple recovery-controlled model underlying (
6.73
), an instanta-
neous change in stress given by (
6.63
) could be expected after a step change in
pressure in a constant strain-rate experiment, but this stress jump may be difficult
to resolve if the recovery rate is high. When the steady-state flow stress includes a
viscous drag component, a larger instantaneous effect is to be expected, corre-
sponding to (
6.71
) replacing (
6.63
).
Relatively few measurements have been made of pressure effects in either creep
or diffusion in minerals or related materials. See, for example, Ross et al. (
1979
),
