Geology Reference
In-Depth Information
Appropriate expressions for the dislocation velocity v and the mobile disloca-
tion density q are derived and inserted in (
6.36
). In this way, the strain rate is
obtained directly and the stress-strain curve is obtained by integrating (
6.36
), in
both cases taking into account the dependences of q and v on the strain c as well as
on stress s, temperature T, etc. Expressions for q and v have been discussed in
general in
Sects. 6.5.3
and
6.4.1
, respectively.
The model of (Haasen
1964
) (see also Alexander and Haasen
1968
), developed
for application to silicon and germanium and similar materials, takes into account
strain hardening by assuming that the stress s acting at a dislocation, designated
the ''effective stress'' s
eff
, is equal to the applied stress s
appl
minus an internal
stress or ''back stress'', imagined to arise from the long-range stress fields of the
other dislocations or otherwise to express the effect of their interaction. Then, from
(
6.30
)or(
6.31
),
s
¼
s
eff
¼
s
appl
aGbq
2
ð
6
:
37
Þ
Haasen also uses the velocity relation in (
6.16
) and a development of (
6.23a
) for
the dislocation density in which c
e
/
vs
eff
:
The stress-strain curve or creep curve
is then obtained by numerical iteration using these three relationships together
with (
6.36
).
One of the interesting features of Haasen's model is that it reproduces the
striking yield drop that is often observed in stress-strain tests in silicon, germa-
nium and other nonmetallic materials having a low initial dislocation density. This
effect can be attributed to a softening arising from the circumstance that, as the
dislocation density increases with straining, the dislocation velocity required for
achieving the specified strain rate decreases and so the required stress falls; on
continuing to larger strains strain hardening eventually takes effect and the
required stress rises again. In a creep test the corresponding phenomenon is the
generation of a sigmoidal creep curve. This type of behavior has been observed in
quartz (Fig.
6.21
) and the application of the Haasen model to quartz has been
explored by Hobbs et al. (
1972
) and Griggs (
1974
). Such a yield drop effect is to be
distinguished from that arising from the Cottrell solute locking effect (
Sect. 6.3.1
and Fig.
6.9
), tending to give a less sharp stress drop than in the latter case.
The model of Weertman (
1957
) is developed in a somewhat similar way. An
expression for the dislocation velocity analogous to Eqs. (
6.12
)with(
6.13a
) and
E
km
¼
0 is used and the dislocation density is treated as depending on a given
density of sources and on the existence of pile-ups, for which certain parameters
are assumed (see also Poirier
1985
, p. 119). The microdynamical models can be
applied in cases of viscous drag effects such as solute drag or jog drag by using an
appropriate expression for the dislocation velocity derived from (
3.15
). Thus, if the
mobility M of a dislocation segment that is dragging a solute atom is taken as equal
to the mobility of the solute atom, that is, D
=
kT
;
where D is the diffusion coef-
ficient of the solute in the crystal, and if the force acting on the dislocation segment
is, from (
6.6a
), s
eff
bl
;
where s
eff
is the effective stress (
6.37
) and l is the length of
dislocation line per solute atom being dragged, then we have
