Geology Reference
In-Depth Information
envisages the line lying in a minimum energy configuration determined by the
potential fields around diffuse obstacles.
The Friedel-Fleischer theory applies for relatively strong obstacles in dilute
concentration and predicts a solute hardening component of the flow stress of
1
2
x
2
F
2
xF
3
G
b
2
ð
2T
Þ
2
1
s
f
¼
ð
6
:
25
Þ
b
3
where x is the mole fraction of solute, F the maximum interaction force per solute
atom or molecule (for estimates, see Kocks
1985b
), T the line tension or energy
per unit length of the dislocations (2T
Gb
2
from
Sect. 6.2.2
), G the shear
modulus and b the Burgess vector. The Mott-Labusch theory applies to weak,
diffuse obstacles in higher concentration and predicts a solute hardening compo-
nent of the flow stress of
3
x
3
F
3
w
3
x
2
F
4
w
4aG
2a
3
b
2
ð
Tb
Þ
3
1
s
f
¼
ð
6
:
26
Þ
b
3
where w is the ''obstacle width'' or interaction range for the solute and a is a
numerical factor of order unity deriving from the form of the interaction potential.
Both square-root and two-thirds power (or stronger) dependence on solute con-
centration have been observed for the flow stress at small strains in metallic solid
solution alloys; for details and references, see Haasen (
1983
) who also reviews the
effect of solutes on the form of the stress-strain curve for various classes of materials.
In spite of the agreement with observed concentration dependence, just noted,
reviews have emphasized that other observations indicate that, in interaction with
dislocations, the solute atoms can seldom be regarded as discrete entities or point
obstacles as in the Friedel-Fleischer theory, except possibly at very low concen-
trations (Kocks
1985b
; Nabarro
1985
). Rather, the effect of the solute atoms is
generally more smeared out, providing a series of extended potential troughs in the
path of the dislocation. This effect is recognized in some degree in the Mott-
Labusch approach but is emphasized more strongly in terms of a ''trough model''
by Kocks (
1985b
) who also recognizes a limited amount of solute mobility already
within the plateau hardening regime.
Particle hardening. The effect of precipitates or second phase particles depends
on the size and physical characteristics of the particles as well as on their spacing.
According to the nature of the interaction with the dislocations (
Sect. 6.3.2
), there
are two types of situation, distinguished by whether the dislocation cuts through
the particle (by common convention, covered by the term precipitation hardening)
or bypasses it (dispersion hardening):
1. In the particle-cutting case, the theory of the flow stress involves the same sort
of statistical considerations as in the case of solute hardening effects, the dif-
ference lying in the obstacle now being larger and possibly stronger. So long as
the spacing of the particles is sufficiently large relative to their effective
