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predicted from first principles. All models of the flow stress and the strain hard-
ening so far proposed have therefore been in some degree phenomenological in
that they are based on assumptions about the basic parameters characterizing the
dislocation population and are therefore in that degree descriptive rather than
predictive. Although there is a large literature on such models, we can here only
indicate their broadest aspects.
The dimensionless quantity R in ( 6.32 ) is essentially an expression of the ratio
of dislocation storage to dislocation travel. This ratio is widely represented in the
various models as a function of the ratio of two parameters observable or definable
in terms of the microstructure. Thus in the often-quoted long-range model of
Seeger for stage II hardening, R is expressed, apart from a numerical factor of
order unity, as nb = LD ð Þ 1 = 2 where n is the number of dislocations released by a
given source, as determined from the slip step height nb, at the surface, L is the
mean free path of dislocations, as measured by the length of the slip traces on
the surface, and Dc is the resolved shear strain in excess of a reference strain of the
order of the strain at the onset of stage II; see brief treatments and references in
Haasen ( 1978 , p. 274) and in Weertman and Weertman ( 1983a , p. 1285). We-
ertman and Weertman ( 1983a , p. 1284) further describe a model for stage I
hardening in which R is written as d = L where d is the active slip plane spacing and
L is as before; however, they also quote the view of Hirsch that no really satis-
factory model has yet been advanced for stage I hardening.
In a survey of the effects of mutual dislocation interaction in relation to the flow
stress and the strain hardening, Kocks ( 1985a ) has emphasized that the deforma-
tion should be viewed in terms of the two-dimensional motion of dislocations in
the slip plane as a percolation process, rather than one-dimensionally as in the
previous paragraph. As relevant microstructural parameters in what is essentially a
forest-cutting model he has used the quantities l, the average spacing of the
obstacles to dislocation motion (of order q 1 = 2 ), and k, the average spacing of ''hard
spots'' in the slip plane which are not penetrated during the ''percolation'' of the
dislocation line across the slip plane as it surmounts the penetrable obstacles (''soft
spots''); k is in some way related to the scale of the cell structure ( Sect. 6.5.1 ) that
becomes obvious in stage III. Kocks then derives an expression for the strain
hardening which is equivalent to that given by putting R equal to ð l = k Þ 2
in ( 6.32 ).
The observed hardening rate in stage II is then obtained if k 10 l.
Although it is generally agreed that, in stage III, dynamical recovery effects are
causing the strain hardening to fall below the maximum rate reached in stage II
(sometimes called the athermal hardening rate), the phenomenology of stage III is
not fully understood and satisfactory models are still lacking. As already indicated,
it is widely considered that cross-slip is an important factor in the dynamical
recovery, but this view has been disputed (for example, Kuhlmann-Wilsdorf 1985 )
and there may be other factors involved. Thus, Kocks ( 1985a ) has pointed to a
possible role of the breaking of attractive junctions involved in the forest effects, a
process that would also introduce a sensitivity to stacking fault energy and a
greater temperature sensitivity, such as is observed. The formation of a more
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