Materials for engineering, 3rd Edition - (Malestrom)

Metals and alloys 79
On being held up by a precipitate, a dislocation can continue in its path
across the crystal in two possible ways. If the particles are very close together,
the dislocation may cut through each particle, but if the particles are further
apart, the dislocation may loop between the particles. During the ageing
process, as the particles grow, the stress increment required to make the
dislocations cut them also rises (Fig. 3.6, curve C). The increase in shear
stress due to precipitate cutting, ∆τ c , is given by an equation of the form:
∆τ c = Af 1/2 r 1/2 [3.4]
where r is the particle radius, f the volume fraction of precipitate, and A a
material constant.
As ageing proceeds, the particles gradually increase in size and, because
they are fewer in number, the average spacing between the particles also
increases. The stress increment to cause dislocation looping ( ∆τ l ) decreases
as the inter-particle spacing increases (curve L, Fig. 3.6), depending on the
precipitate size and volume fraction according to:
∆τ 1 = BGbf 1/2 r –1 [3.5]
where B is a constant dependent upon precipitate particle shape.
As ageing continues, the measured yield stress would therefore be expected
to follow the form of the dotted curve in Fig. 3.6 and this general pattern of
C
∆τ
L
Ageing time
3.6 Showing change in yield stress (∆τ) with ageing time for a
precipitation-hardened alloy. Curve C is followed if the precipitates
are cut by dislocations, and curve L is followed if the dislocations
loop between the particles. The response is given by the dashed
curve.