Geology Reference
In-Depth Information
with heterogeneity or anisotropy, such as the closure of cracks or pores, and the
generation of internal stresses due to heterogeneity in elastic compressibility from
grain to grain or to anisotropy of linear compressibility in adjoining grains of
different orientation.
6.9.2 Pressure Effects in the Athermal Regime
In single crystals the influence of pressure on the flow stress in the athermal regime
results primarily from changes in the elastic modulus. On the basis that the flow
stress s can be expressed as aGbq
2
(
Sect. 6.6.3
) and that the term bq
2
will not vary
with pressure at a given dislocation content since it is dimensionless, it follows that
for a given dislocation density, the influence of pressure p on the flow stress will be
given by
1
s
dp
¼
1
ds
dG
dp
G
and hence
s
¼
s
0
exp
Z
p
0
1
G
dG
dp
dp
where s
0
is the flow stress at zero pressure (or other reference pressure). For
relatively small variations in G with p, we thus have
s
0
p
G
p
G
s
¼
s
0
exp a
1
1
þ
a
1
ð
6
:
62a
Þ
where
a
1
¼
dG
dp
ð
6
:
62b
Þ
In these expressions, the elastic isotropy approximation is assumed in speci-
fying the shear modulus G. The value of dG
=
dp is between 1 and 2 for most
materials, thus giving a 1-2 % increase in flow stress on raising the pressure by
500 MPa if G is 50 GPa. In the case of solute and particle hardened materials
(
Sect. 6.6.2
), since the interaction force F in expressions such as Eqs. (
6.25
) and
(
6.26
) can be expected to be proportional to G, the flow stress can again be
expressed as the product of G and a dimensionless term, so that the pressure
dependence of the flow stress is still represented by (
6.62
).
The pressure effect represented in (
6.62
) applies only for a given dislocation
configuration. It should apply to the stress at a given strain as determined in an
instantaneous pressure step test, but it does not necessarily apply to the comparison
of complete stress-strain curves determined at different pressures since the pres-
sure may also influence the evolution of the dislocation density q with straining,
