Geology Reference
In-Depth Information
common in the literature (see discussion and references in Hirth and Lothe
1982
,
pp. 17-24). The important point to make is that, whatever the sign convention,
parallel positive and negative dislocation segments having the same Burgers vector
are mutually annihilating if brought together.
It follows from the nature of a dislocation that there is always an internal stress
field associated with it in the crystal. In the approximation that the material is
taken to be elastically isotropic, with shear modulus G and Poisson ratio v, the
stresses around a dislocation in an infinite crystal, expressed in cylindrical coor-
dinates r, h, z with z parallel to the dislocation line, are as follows for pure screw
and pure edge dislocations (Hirth and Lothe
1982
, Chap. 3):
Screw dislocation:
r
rr
¼
r
hh
¼
r
zz
¼
0
r
hz
¼
Gb
2pr
;
r
zr
¼
r
rh
¼
0
ð
6
:
3a
Þ
Edge dislocation:
Gb sin h
2p
ð
1
v
Þ
r
;
r
rr
¼
r
hh
¼
r
zz
¼
2vr
rr
r
rh
¼
Gb cos h
2p
ð
1
v
Þ
r
;
r
hz
¼
r
zr
¼
0
ð
6
:
3b
Þ
(Note that the sign convention of compressive stress being positive has been
used, leading to the signs of r
rr
;
r
hh
and r
zz
being opposite to those usually given
in textbooks; also for edge dislocations, h is taken to be zero when r coincides with
b. For the corresponding elastic displacements, see Hirth and Lothe (
1982
, Chap.
3).
The importance of dislocations in crystal plasticity lies in the slip or twinning
that is brought about when they are moved through the crystal. The direction of the
Burgers vector can then be identified with the slip direction and the plane con-
taining the Burgers vector and the dislocation line is the slip plane (corresponding
glide elements are defined in the case of twinning). If q is the total length of
mobile dislocation line per unit volume (that is, the dislocation density) and s is the
average distance moved by the dislocation line in the slip plane, then the mac-
roscopic resolved shear strain resulting from the dislocation motion is
c
¼
qbs
ð
6
:
4a
Þ
where b is the magnitude of the Burgers vector. This relationship or its kinetic
equivalent
c
¼
qbv
ð
6
:
4b
Þ
where c is the strain rate and v the dislocation velocity, is known as the Orowan
equation.
