Geology Reference
In-Depth Information
obstacles. In this case, the obstacles will give rise to an athermal, frictional
resistance to the dislocation motion (the term ''frictional'' is used in analogy with
ordinary sliding friction in the sense of being relatively rate insensitive, but it is
not fully analogous because of the absence of the strong normal-stress dependence
characteristic of ordinary friction). It is the objective of the models treated in this
subsection to calculate the stress component s
f
that must be applied to the crystal
in order to bring about a macroscopic plastic deformation by dislocation motion in
the presence of the more or less athermal frictional resistance from the discrete
obstacles.
In practical terms, two categories of athermal obstacle models can be usefully
distinguished, namely, those of solute hardening and of particle hardening (or
strengthening). Both are based on heterogeneities in the crystal but at different
scales, as distinguished in
Sects. 6.3.1
and
6.3.2
. For general reviews, see Brown
and Ham (
1971
), Kocks et al. (
1975
), Gerold (
1979
), Martin (
1980
), Haasen
(
1978
, Chap. 14,
1983
), Strudel (
1983
), Ardell (
1985
), Humphreys (
1985
), Kocks
(
1985b
) and Nabarro (
1985
). Haasen (
1983
) also considers the effects of long-
range ordering in solid solutions.
Solute hardening. Solid solution effects can be athermal when the temperature
is moderately low and the solute atoms are effectively immobile or only slightly
mobile. The various ways in which solute atoms can impede the motion of dis-
locations and thus increase the flow stress are listed in
Sect. 6.3.1
. We shall be
concerned here mainly with the elastic interactions since the effect of core inter-
actions will tend to be more strongly thermally activated (
Sect. 6.6.5
).
It was noted in
Sect. 6.3.1
that prior segregation of solute at pre-existing dis-
locations will tend to lock them so that a higher applied stress may be required to
initiate dislocation movement than to sustain it later, giving rise to an initial yield
drop in the stress-strain curve. We are here considering the effect of the solute on
the dislocations when they are in motion, an effect that dominates the flow stress at
small strains in many alloys of low initial dislocation density and low Peierls
stress. Although thermal activation may be important at very low temperatures, as
shown by a marked decrease in flow stress with rise in temperature, there tends to
be a temperature range, around room temperature for common metals or ionic
compounds, in which the flow stress at small strains is almost independent of
temperature, a range known as the plateau hardening regime. Because the plateau
hardening effect is relatively insensitive to strain rate, it can be referred to as a
frictional effect rather than as a viscous drag. It is to the plateau hardening regime,
therefore, that the main athermal solute hardening theories are directed.
The basic theoretical problem is to calculate, given a distribution of obstacles, the
number of them per unit length that are interacting effectively with a given dislo-
cation segment at any instant and to determine the applied stress needed to overcome
the combined effect of the interactions. Two limiting cases are commonly considered
(Haasen
1978
, p. 327,
1983
; Hirth and Lothe
1982
, p. 681). The first, Friedel-
Fleischer theory, envisages the dislocating line being forced by the applied stress
against point obstacles of given strength, while the second, Mott-Labusch theory,
