205
needed to break through the obstacle. There comes a point, however,
at which N falls to 1 and no pileup is possible. This occurs at a grain
size d*, which, from Equations 7.3 and 7.6, is given by
d
b
E
Ck
*
*
<
2
(7.8)
Figure 7.7, top, shows measurements of the hardness of electrodeposited nickel as a function of the size of the grains. As with copper,
the hardness (and strength) rises by a factor of 7 as the grains size
d decreases from 10 µm to 10 nm. The data are replotted in the way
suggested by Equation 7.6 in the lower figure, with d
−1/2 on the x
axis. The measurements are consistent with the equation, with a
value of k* of 5 GPa. Inserting this value into Equation 7.8 predicts a critical grain size below which the simple Hall-Petch relation
breaks down. Its value, 10 nm, is plotted in the figure.
Figure 7.7
The increase in strength of nickel as the grain
size is reduced to nanodimension (top); the same
data plotted to reveal the Hall-Petch relationship
(bottom). (Data from Weertman and Averbach,
1996.)
0
2
4
6
8
10
0
2000
4000
6000
8000
10000
Grain size d (nm)
Hardness H (GPa)
Nickel
electroplates
0
2
4
6
8
10
0.0
0.1
0.2
0.3
0.4
(Grain size d) -1/2 (nm -1/2 )
Hardness H (GPa)
Nickel
electroplates
Breakdown of
Hall-Petch
equation
Mechanical Properties
needed to break through the obstacle. There comes a point, however,
at which N falls to 1 and no pileup is possible. This occurs at a grain
size d*, which, from Equations 7.3 and 7.6, is given by
d
b
E
Ck
*
*
<
2
(7.8)
Figure 7.7, top, shows measurements of the hardness of electrodeposited nickel as a function of the size of the grains. As with copper,
the hardness (and strength) rises by a factor of 7 as the grains size
d decreases from 10 µm to 10 nm. The data are replotted in the way
suggested by Equation 7.6 in the lower figure, with d
−1/2 on the x
axis. The measurements are consistent with the equation, with a
value of k* of 5 GPa. Inserting this value into Equation 7.8 predicts a critical grain size below which the simple Hall-Petch relation
breaks down. Its value, 10 nm, is plotted in the figure.
Figure 7.7
The increase in strength of nickel as the grain
size is reduced to nanodimension (top); the same
data plotted to reveal the Hall-Petch relationship
(bottom). (Data from Weertman and Averbach,
1996.)
0
2
4
6
8
10
0
2000
4000
6000
8000
10000
Grain size d (nm)
Hardness H (GPa)
Nickel
electroplates
0
2
4
6
8
10
0.0
0.1
0.2
0.3
0.4
(Grain size d) -1/2 (nm -1/2 )
Hardness H (GPa)
Nickel
electroplates
Breakdown of
Hall-Petch
equation
Mechanical Properties