Geology Reference
In-Depth Information
and thus the strain hardening. Since the introduction of a dislocation assemblage of
density q leads to a small relative volume increase, of the order of qb
2
(
Sect.
6.2.2
), there should be some increase in the strain hardening rate under pressure for
a given rate of increase of q. This effect can be estimated by equating the extra
work done by the stress, ds
dc, to the work done in relative volume increase,
d
V
¼
b
2
dq where dq is the increment in dislocation density and ds the extra
increment in stress during the strain increment dc
:
It follows from the internal
work expended per unit volume of the specimen,
dW
¼
s dc
p dV
¼
s dc
pb
2
dq
that the strain-hardening rate will be
dc
þ
pb
2
d
2
q
h
¼
ds
dc
¼
ds
0
ð
6
:
63
Þ
dc
2
where s
0
¼
dW
=
dc is the stress in the absence of a superposed hydrostatic pressure
(the pressure dependence of G is not being taken into account at this point). If, to
gain some idea of the relative values of the terms in (
6.63
), we put s
0
¼
aGbq
2
and
q
¼
q
0
exp c
=
c
e
ð
Þ
from Eqs. (
6.31
) and (
6.23a
), we obtain
h
¼
ad
2
1
þ
2d
a
p
G
ð
6
:
64
Þ
G
where d
¼
bq
2
=
c
e
:
Putting b
10
13
m
;
q
10
12
m
2
and c
e
10
2
suggests a
value of the order of 10
-2
for d, which, with a
0
:
3, would give a strain-hardening
rate consistent with that observed in stage II and early stage III of stress-strain
curves at atmospheric pressure (
Sect. 6.6.3
). Thus, (
6.64
) indicates that the extra
strain hardening associated with the dilatancy from dislocations is negligible in the
normal experimental range of pressures. It can be concluded that, in the athermal
field, the pressure effect in the stress-strain curve will derive almost entirely from
the pressure effect on the shear modulus G in the absence of effects associated with
dynamical recovery at larger strains.
With the onset of dynamical recovery, other pressure effects may arise. Thus, it
has been observed that in stage III of the stress-strain curve the effect of pressure
can be to reduce the flow stress, as, for example, in NaCl (Aladag et al.
1970
). The
reduction has been attributed to an increase in the stacking fault energy with
increase in pressure, facilitating the constriction of dissociated dislocations as
required for cross-slip, although this explanation has also to be questioned (Puls
and
So
1980).
See
Poirier
(
1985
,
pp.
155-156)
for
further
discussion
and
references.
The effects described in the previous paragraphs should apply for both single
crystals and polycrystals. However, additional pressure effects are possible in
polycrystals through intergranular interaction effects associated with anisotropic or
inhomogeneous properties of the grains and through volume changes associated
with porosity. Thus, if the linear compressibility (the relative change in linear
