Geology Reference
In-Depth Information
6.5.2 Dislocation Density and Multiplication
During plastic deformation, very marked increases can occur in the dislocation
density q, specified as the average total length of dislocation line per unit volume.
For example, commencing with a dislocation density of around 10
10
m
-2
or so,
such as might typically be found in annealed or as-grown metallic and ionic
crystals of ordinary quality, the density can increase by several orders of magni-
tude with 10 % strain. Such marked growth in density is generally thought to arise
from the presence of specific, persistent dislocation ''sources'' that can produce a
succession of new dislocation segments which augment the total population.
Dislocation growth or multiplication results from the differential movement of
parts of an existing dislocation line. A persistent source is formed when the parent
dislocation line is pinned at a certain point by an obstacle or by the deviation of the
line into another plane in which it is immobile. The adjacent freely moving seg-
ment of the dislocation line will tend to sweep circularly around the pinning point,
forming an ever-lengthening spiral of dislocation line which is added to with each
additional revolution at the source (Fig.
6.16
a). Various versions of this basic
multiplication process have been proposed, including climb sources.
Thus, if the mobile segment is bounded by a second pinning point at which
spiraling in the opposite sense occurs, the two spirals annihilate where they meet
and so outwardly expanding loops can be formed (Fig.
6.16
b). Such a ''two-
armed'' source is known as a Frank-Read source and is often invoked for dislo-
cation multiplication. However, the presence of a second suitable pinning point
within a field of view that is not obscured by additional substructure may well be
generally rather fortuitous, as is indicated by the observation of Caillard and
Martin (
1983
) that in aluminum ''one-armed'' sources can be identified but not
Frank-Read sources. Other types of dislocation source are discussed by Bilby
(
1955
).
A spontaneous generation of a dislocation loop in its slip plane with the aid of
thermal fluctuations is highly improbable (Cottrell
1953
, p. 53). However, in the
presence of high internal stresses in the neighborhood of precipitates, it is evi-
dently possible to generate a prismatic dislocation loop in climb in the absence of
pre-existing
dislocation,
as
shown
by
loops
generated
at
bubbles
in
quartz
(McLaren et al.
1983
).
If the number of dislocation sources were assumed to be proportional to the
length of dislocation line already present, then, in the absence of annihilation
processes, we would expect the dislocation density to increase at a rate propor-
tional to the existing density and to the rate of straining, that is, dq
/
qdc
;
leading
to the relation
q
¼
q
0
exp
ð
c
=
c
e
Þ
ð
6
:
23c
Þ
between the dislocation density q and the shear strain c, where q
0
is the dislocation
density at c
¼
0 and c
e
is the strain needed for an e-fold increase in q (cf. Haasen
1978
, p. 269). If, alternatively, the density of sources were proportional to the
