Geology Reference
In-Depth Information
climb velocities can also be important, especially as rate-determining factors when
obstacles to glide in the primary slip plane have to be bypassed. We shall therefore
consider briefly all three aspects of dislocation velocity.
6.4.1 Glide Velocity
In the absence of energy dissipation, an isolated dislocation would tend to
accelerate indefinitely under applied stress were it not for a relativistic limitation to
the velocity of sound. A direct extension of the elastic theory of dislocations
predicts that, as the velocity of the dislocation increases, not only does a kinetic
energy term appear in its total energy but the self energy of the dislocation also
increases in a way that can be described roughly as relativistic (Hirth and Lothe
1982
, Chap. 7). The self energy approaches infinity and the stress field undergoes a
relativistic contraction in the direction of motion as the dislocation velocity
approaches the velocity of sound (the velocity for transverse or longitudinal
waves, v
s
or v
p
;
respectively, applies depending on whether screw or edge dislo-
cations are concerned). However, in practice, there are many dissipative processes
that generally limit the velocity to the sub-relativistic domain (Hirth and Lothe
1982
, Chaps. 7, 15, 16; Weertman and Weertman
1980
).
Some of the dissipative processes are intrinsic effects, applying to an isolated
dislocation in an otherwise perfect crystal. One such process is the scattering of
phonons associated with the thermal vibrations in the crystal, which becomes
important at high velocities, perhaps generally in excess of about 10
-3
v
s
or
1ms
-1
, and which gives rise to a viscous drag on the dislocations (Granato
1984
,
and other references therein; Hirth and Lothe
1982
, p. 211; Kocks et al.
1975
,
p. 85). However, when there is a significant Peierls potential, this gives rise to a
more important viscous drag on dislocations, effective at much lower velocities
and treatable in terms of kink mobility (Hirth and Lothe
1982
, Chap. 15). Hirth and
Lothe (p. 545) deduce a relationship which, in the approximation that the spacing
of the Peierls valleys and the jump distance for kink migration are each equated to
the Burgers vector b, can be written as
v
2sb
4
m
kT
exp
ð
E
kn
þ
E
km
Þ=
kT
f
g
ð
6
:
12
Þ
where s is the resolved shear stress, m the attempt frequency (of the order of the
atomic vibration or Debye frequency, say, *10
13
s
-1
), and E
kn
;
E
km
the activation
energies for kink nucleation and migration, respectively. The term sb
4
m
=
kT can be
written as bm
0
where m
0
has the character of a vibration frequency of the dislocation
line under stress s (in effect, sb
3
¼
hm
0
and kT
¼
hv
;
where h is the Planck con-
stant). In addition to the linear stress dependence introduced in this term, there is
also a stress dependence in the activation energy. Thus, the kink nucleation energy
