Geology Reference
In-Depth Information
dimensions, the Friedel-Fleischer relation in (
6.25
) can be applied; rewriting
this relation in terms of the spacing l of the obstacles in the slip plane, using
l
¼
bx
2
;
we obtain an estimate of the flow stress component due to precipi-
tation hardening as
F
2
bl
ð
2T
Þ
2
F
2
b
2
lG
2
s
f
¼
ð
6
:
27
Þ
[An analogous expression can be obtained from (
6.26
) for the case of diffuse
precipitates interacting over a range comparable to their spacing but it is probably
of more limited applicability; see discussion by Ardell (
1985
)]. The evaluation of
F depends on the nature of the dislocation-particle interaction and may involve
coherency, surface energy, ordering, stacking fault, and elastic modulus effects
(Ardell
1985
; Gerold
1979
; Martin
1980
, pp. 53-60). If F is taken as being
proportional to d
n
;
where d is the width or diameter of a particle, and we introduce
the volume fraction x
v
of particles through x
v
d
2
=
l
2
;
then (
6.27
) leads to
3n
v
l
3
2
1
s
f
/
x
ð
6
:
28
Þ
indicating that, for a given volume fraction of particles, the flow strength will
increase as the precipitate coarsens, provided n [ 2
=
3.
2. In the bypassing case, the theoretical model envisages that the dislocation line
bows around the particles, leaving residual loops (''geometrically necessary
dislocations'') as it proceeds further. This process requires a flow stress com-
ponent corresponding to the Orowan bow-out stress
s
f
¼
aGb
l
ð
6
:
29
Þ
(Embury
1985
; Haasen
1978
, p. 250; Martin
1980
, p. 62) where G
;
b
;
l are as
defined for Eqs. (
6.25
) and (
6.26
) and a is a numerical factor of the order of unity.
Such a relation is found to fit experimental observations on initial yield stress quite
well in metals with varying concentrations of dispersed hard particles, especially if
l is taken as the distance between grain surfaces rather than between their centers
(Martin
1980
, p. 63).
The opposite dependences of (
6.28
) and (
6.29
) on the spacing l (provided
n [ 2
=
3) indicate that during particle coarsening at constant volume fraction,
known as ''Ostwald ripening'', (Haasen
1978
, p. 207) a critical size and spacing of
particles will tend to be reached beyond which bypassing becomes easier than
cutting, this size corresponding to a maximum in the precipitation hardening of the
material. The decrease in strength upon ''over-aging'' of age-hardening alloys is
commonly attributed to such an effect, although this explanation is questioned by
Ardell (
1985
).
