Page 258 - Materials Science and Engineering An Introduction

230 • Chapter 7 / Dislocations and Strengthening Mechanisms

Figure 7.14 The motion of a dislocation
as it encounters a grain boundary, illustrating Grain boundary
how the boundary acts as a barrier to
continued slip. Slip planes are discontinuous
and change directions across the boundary.
(From L. H. Van Vlack, A Textbook of Materials
Technology, Addison-Wesley Publishing Co., 1973.
Reproduced with the permission of the Estate of Slip plane
Lawrence H. Van Vlack.)

Grain A Grain B

“pile up” (or back up) at grain boundaries. These pile-ups introduce stress concentra-
tions ahead of their slip planes, which generate new dislocations in adjacent grains.
A fine-grained material (one that has small grains) is harder and stronger than one
that is coarse grained because the former has a greater total grain boundary area to
impede dislocation motion. For many materials, the yield strength s y varies with grain
size according to
Hall–Petch
equation— -1/2
dependence of yield s y = s 0 + k y d (7.7)
strength on grain size
In this expression, termed the Hall–Petch equation, d is the average grain diameter, and
s 0 and k y are constants for a particular material. Note that Equation 7.7 is not valid for
both very large (i.e., coarse) grain and extremely fine grain polycrystalline materials.
Figure 7.15 demonstrates the yield strength dependence on grain size for a brass alloy.
Grain size may be regulated by the rate of solidification from the liquid phase, and
also by plastic deformation followed by an appropriate heat treatment, as discussed in
Section 7.13.
It should also be mentioned that grain size reduction improves not only the strength,
but also the toughness of many alloys.

Figure 7.15 The influence of Grain size, d (mm)
grain size on the yield strength of a –1 –2 –3
70 Cu–30 Zn brass alloy. Note that 10 10 5 × 10 30
the grain diameter increases from 200
right to left and is not linear.
(Adapted from H. Suzuki, “The
Relation between the Structure and 150
Mechanical Properties of 20
Metals,” Vol. II, National
Physical Laboratory, Symposium No. Yield strength (MPa) Yield strength (ksi)
15, 1963, p. 524.) 100

50 10

0 0
4 8 12 16
d –1/2 (mm –1/2 )