chaPter 7 nanomaterials: Properties
204
pile up against them until the force on the one closest to the boundary exceeds f* (see Figure 7.6). The number N of dislocations in
such a pileup scales with the applied shear stress (τ − τ 0 ) and the
distance L between the dislocation source and the obstacle—here,
half the diameter d of a grain:
N
L
Gb
=
−
(
) −
(
)
π
ν τ τ
1
0
(7.2)
Here ν is Poisson’s ratio (approximately 0.3), and τ 0 describes the
contributions of all the other strengthening mechanisms shown in
Figure 4.41 of Chapter 4. The shear stress τ caused by a tensile or
compressive stress σ is τ ≈ σ/2. Equating (τ − τ 0 ) to (σ − σ 0 )/2 and
the shear modulus G by 3E / 8 (as it is for most materials) gives
N
Cd
Eb
=
−
(
)
σ σ 0
(7.3)
where C is a dimensionless constant with a value of about 2. The
force these exert on the obstacle is magnified by their number, so
the obstacle will be overcome when
N
b f
τ τ
−
(
) ≥
0
*
(7.4)
Replacing τ with σ / 2 as before and eliminating N from these two
equations, we get
σ σ
− =
0
1 2
1 2
2 f E
Cb
b
d
*
(7.5)
or
σ σ
− =
0
1 2
k
b
d
*
(7.6)
where
k
f E
Cb
*
*
=
2
1 2
(7.7)
The quantity k* has the dimensions of stress; it characterizes the
strength of the boundary. Its value typically lies in the range 5 to
15 GPa—it is about equal to the ideal strength. This result is known
as the Hall-Petch equation, with k* the Hall-Petch constant. The
hardness H is just three times the strength σ.
Equation 7.3 says that the smaller the grain size d, the fewer the
number of dislocations that can be packed into a pileup. The lower,
then, is its magnifying effect, and the greater is the applied stress
Figure 7.6
Pileups in a grain and a layer of a nanolayer
structure (top); the pileup in more detail (bottom).
T
T
T
T
T T T T
S
T
T
T
T
T T
T
T S
T T T T T
Grain
size, d
Bilayer
spacing, d
Slip plane
Stress σ
Stress σ
Slip planes
in next grain
Grain
boundary
Pileup
Pileup
.. ..
.. ..
.. ..
.. ..
.. ..
.. ..
.. ..
.. ..
.. ..
Force Nb(s - s 0 )
d
d
204
pile up against them until the force on the one closest to the boundary exceeds f* (see Figure 7.6). The number N of dislocations in
such a pileup scales with the applied shear stress (τ − τ 0 ) and the
distance L between the dislocation source and the obstacle—here,
half the diameter d of a grain:
N
L
Gb
=
−
(
) −
(
)
π
ν τ τ
1
0
(7.2)
Here ν is Poisson’s ratio (approximately 0.3), and τ 0 describes the
contributions of all the other strengthening mechanisms shown in
Figure 4.41 of Chapter 4. The shear stress τ caused by a tensile or
compressive stress σ is τ ≈ σ/2. Equating (τ − τ 0 ) to (σ − σ 0 )/2 and
the shear modulus G by 3E / 8 (as it is for most materials) gives
N
Cd
Eb
=
−
(
)
σ σ 0
(7.3)
where C is a dimensionless constant with a value of about 2. The
force these exert on the obstacle is magnified by their number, so
the obstacle will be overcome when
N
b f
τ τ
−
(
) ≥
0
*
(7.4)
Replacing τ with σ / 2 as before and eliminating N from these two
equations, we get
σ σ
− =
0
1 2
1 2
2 f E
Cb
b
d
*
(7.5)
or
σ σ
− =
0
1 2
k
b
d
*
(7.6)
where
k
f E
Cb
*
*
=
2
1 2
(7.7)
The quantity k* has the dimensions of stress; it characterizes the
strength of the boundary. Its value typically lies in the range 5 to
15 GPa—it is about equal to the ideal strength. This result is known
as the Hall-Petch equation, with k* the Hall-Petch constant. The
hardness H is just three times the strength σ.
Equation 7.3 says that the smaller the grain size d, the fewer the
number of dislocations that can be packed into a pileup. The lower,
then, is its magnifying effect, and the greater is the applied stress
Figure 7.6
Pileups in a grain and a layer of a nanolayer
structure (top); the pileup in more detail (bottom).
T
T
T
T
T T T T
S
T
T
T
T
T T
T
T S
T T T T T
Grain
size, d
Bilayer
spacing, d
Slip plane
Stress σ
Stress σ
Slip planes
in next grain
Grain
boundary
Pileup
Pileup
.. ..
.. ..
.. ..
.. ..
.. ..
.. ..
.. ..
.. ..
.. ..
Force Nb(s - s 0 )
d
d