Page 268 - Materials Science and Engineering An Introduction

240 • Chapter 7 / Dislocations and Strengthening Mechanisms

From Figures 7.19a and 7.19c, a yield strength of 410 MPa (60,000 psi) and a ductility of 8%EL
are attained from this deformation. According to the stipulated criteria, the yield strength is
satisfactory; however, the ductility is too low.
Another processing alternative is a partial diameter reduction, followed by a recrystal-
lization heat treatment in which the effects of the cold work are nullified. The required yield
strength, ductility, and diameter are achieved through a second drawing step.
Again, reference to Figure 7.19a indicates that 20%CW is required to give a yield strength
of 345 MPa. However, from Figure 7.19c, ductilities greater than 20%EL are possible only for
deformations of 23%CW or less. Thus during the final drawing operation, deformation must
be between 20%CW and 23%CW. Let’s take the average of these extremes, 21.5%CW, and
then calculate the final diameter for the first drawing d 0 , which becomes the original diameter
for the second drawing. Again, using Equation 7.8,
2 2
d 0 5.1 mm
a b p - a b p
2 2
21.5%CW = * 100
2
d 0
a b p
2
Now, solving for d 0 from the preceding expression gives
d 0 = 5.8 mm (0.226 in.)

7.13 GRAIN GROWTH
After recrystallization is complete, the strain-free grains will continue to grow if the
metal specimen is left at the elevated temperature (Figures 7.21d to 7.21f ); this phenom-
grain growth enon is called grain growth. Grain growth does not need to be preceded by recovery and
recrystallization; it may occur in all polycrystalline materials, metals and ceramics alike.
An energy is associated with grain boundaries, as explained in Section 4.6. As grains
increase in size, the total boundary area decreases, yielding an attendant reduction in
the total energy; this is the driving force for grain growth.
Grain growth occurs by the migration of grain boundaries. Obviously, not all grains
can enlarge, but large ones grow at the expense of small ones that shrink. Thus, the
average grain size increases with time, and at any particular instant there exists a range
of grain sizes. Boundary motion is just the short-range diffusion of atoms from one side
of the boundary to the other. The directions of boundary movement and atomic motion
are opposite to each other, as shown in Figure 7.24.
For many polycrystalline materials, the grain diameter d varies with time t accord-
ing to the relationship
n
n
For grain growth, d - d 0 = Kt (7.9)
dependence of grain
size on time
where d 0 is the initial grain diameter at t 0, and K and n are time-independent con-
stants; the value of n is generally equal to or greater than 2.
The dependence of grain size on time and temperature is demonstrated in Figure 7.25,
a plot of the logarithm of grain size as a function of the logarithm of time for a brass
alloy at several temperatures. At lower temperatures the curves are linear. Furthermore,
grain growth proceeds more rapidly as temperature increases—that is, the curves are
displaced upward to larger grain sizes. This is explained by the enhancement of diffusion
rate with rising temperature.
The mechanical properties at room temperature of a fine-grained metal are usually
superior (i.e., higher strength and toughness) to those of coarse-grained ones. If the