Geology Reference
In-Depth Information
near to unity (theoretical estimates of a vary from 1/2 to 1/3, while measurements
on copper suggest a
¼
1
=
3
;
as quoted by Haasen (
19781
, p. 25); see also Basinski
and Basinski (
1979
), The flow stress is then obtained by putting s
¼
s
i
:
In models based on the forest-cutting effect, an account is taken of the work
required for a moving dislocation to cut through the dislocations that cross the slip
plane, involving mainly short-range interactions (
Sect. 6.3.4
). Since the stress
needed will be proportional to the number of such intersections per unit distance
traveled, the flow stress can again be expected to be proportional to q
2
(at least
insofar as dislocation networks of different densities are self-similar or q is now
taken to be the density of the forest dislocations). The proportionality factor will
presumably depend on the nature of the impediment to dislocation movement
arising from the intersection, for which there are a number of models (see sum-
mary by Weertman and Weertman
1983a
, Sect. 5.5). Also, if there is significant
thermal activation of the intersection process, some temperature dependence may
be introduced in addition to that deriving from the shear modulus G.
An alternative to the forest-cutting view is to place emphasis on the ''mesh
length'' or ''link length'' of the three-dimensional network of dislocations that
builds up during activity of the various slip systems (see, for example, Burton
1982b
; Kuhlmann-Wilsdorf
1985
). Again, the average mesh length is inversely
proportional to q
2
and so the formal theoretical implications tend to be the same.
Thus, regardless of whether long-range or short-range effects are involved, a
flow stress determined by mutual dislocation interactions can be expected to have
the form
s
¼
aGbq
2
ð
6
:
31
Þ
where q is a dislocation density, which may refer to the total dislocation popu-
lation or to some significant part of it, and a is a numerical factor of the order of
unity. The particularities of specific models enter through the way in which the
quantity q is expressed in terms of resolved shear strain c in order to obtain a
stress-strain relation s
ð
c
Þ:
The central aim of the models is to predict the strain-
hardening rate
¼
aRG
2
h
¼
ds
dc
¼
aG
b
q
2
dq
dc
¼
bG
ð
6
:
32
Þ
2
where R
¼
b
=
q
1
=
2
ð
dq
=
dc
Þ
and b
¼
aR
=
2 is a constant insofar as R is a constant.
h thus reflects the value of dq
=
dc at a given value of q
ð
c
Þ
and hence involves the
problems of dislocation multiplication (
Sect. 6.5.2
). Given (
6.32
), the stress-strain
relation can then, in principle, be obtained by integration.
As pointed out by Hirth and Lothe (
1982
, Chap. 22) there has been little
progress in developing fundamental theories of the flow stress and strain hardening
on the basis of the properties of mutual dislocation interaction without introducing
ad hoc assumptions about the characteristics of the dislocation population; that is,
the form of development of the actual dislocation substructure cannot yet be
