25 nm are displayed. The decrease in hardness can be seen to follow more or less
distinctly the inverse Hall–Petch relationship, which, for copper, is quite well
fulfilled. However, in the case of palladium the fit was quite poor. Overall, this
indicates that the exponent 0.5 is a quite rough approximation.
This inverse Hall–Petch relationship demands other deformation mechanisms as
discussed above. In conventional materials, at high temperatures – and especially for
ceramics – plastic deformation processes via grain boundary mechanisms are active,
although a few of these processes have been described in the literature. For technical
materials, the most important are the grain boundary deformation processes
described by Nabarro–Herring or Coble. Deformation processes according to
Nabarro–Herring act via volume diffusion when, under constant stress s, the
deformation rate _
e is given by:
_
e /
D vol
d
2
ð11:7Þ
where D vol is the volume diffusion coefficient and d is the grain size. The Coble
mechanism uses grain boundary diffusion with the diffusion coefficient D GB as the
rate-controlling mechanism; hence, the deformation rate is in that case:
_
e /
D GB
d
3
ð11:8Þ
As a rule of thumb, the Nabarro–Herring mechanism acts at higher temperatures,
perhaps close to the melting point, while the Coble mechanism is active at lower
temperatures. Therefore, for nanocrystalline materials, the Nabarro–Herring mechanism can be excluded as in the temperature range where this mechanism operates
the nanocrystalline materials are no longer stable and grain growth occurs. When
considering the exponent at the grain size, it is clear that the grain boundary
diffusion mechanism according to Coble has a higher exponent describing the grain
size dependency as compared to the Nabarro–Herring mechanism. In both deformation mechanisms, the shape of the grains will be stretched during deformation;
such behavior is depicted in Figure 11.16, in an idealized manner.
In Figure 11.16 and all subsequent figures, the grains are depicted as hexagons of
equal size. Although extremely idealized; however, this is the configuration best
accessible for the theoreticians. From Eqs. (11.7) and (11.8), it is clear that grain
boundary mechanisms become more prominent for decreasing grain sizes when, as
found in nanomaterials, the contribution derived from the grain boundary processes
is increased. Ashby and Verall [13] added a grain deformation mechanism to
accommodate the grains with their new shapes in their new configuration. Also
in this case, the deformation rate is proportional to d
À3
. The principle of this
mechanism is depicted in Figure 11.17, which also shows clearly why this deformation mode is called the “grain switching” mechanism.
Chang et al. [14] developed a concise experimental proof of the validity of the
Ashby–Verall process for the deformation process of nanocrystalline intermetallics
by comparing the experimental results of hardness measurements on TiAl with
model calculations. The results of this study are shown in Figure 11.18, where the
11.2 Bulk Metallic and Ceramic Materials j311
distinctly the inverse Hall–Petch relationship, which, for copper, is quite well
fulfilled. However, in the case of palladium the fit was quite poor. Overall, this
indicates that the exponent 0.5 is a quite rough approximation.
This inverse Hall–Petch relationship demands other deformation mechanisms as
discussed above. In conventional materials, at high temperatures – and especially for
ceramics – plastic deformation processes via grain boundary mechanisms are active,
although a few of these processes have been described in the literature. For technical
materials, the most important are the grain boundary deformation processes
described by Nabarro–Herring or Coble. Deformation processes according to
Nabarro–Herring act via volume diffusion when, under constant stress s, the
deformation rate _
e is given by:
_
e /
D vol
d
2
ð11:7Þ
where D vol is the volume diffusion coefficient and d is the grain size. The Coble
mechanism uses grain boundary diffusion with the diffusion coefficient D GB as the
rate-controlling mechanism; hence, the deformation rate is in that case:
_
e /
D GB
d
3
ð11:8Þ
As a rule of thumb, the Nabarro–Herring mechanism acts at higher temperatures,
perhaps close to the melting point, while the Coble mechanism is active at lower
temperatures. Therefore, for nanocrystalline materials, the Nabarro–Herring mechanism can be excluded as in the temperature range where this mechanism operates
the nanocrystalline materials are no longer stable and grain growth occurs. When
considering the exponent at the grain size, it is clear that the grain boundary
diffusion mechanism according to Coble has a higher exponent describing the grain
size dependency as compared to the Nabarro–Herring mechanism. In both deformation mechanisms, the shape of the grains will be stretched during deformation;
such behavior is depicted in Figure 11.16, in an idealized manner.
In Figure 11.16 and all subsequent figures, the grains are depicted as hexagons of
equal size. Although extremely idealized; however, this is the configuration best
accessible for the theoreticians. From Eqs. (11.7) and (11.8), it is clear that grain
boundary mechanisms become more prominent for decreasing grain sizes when, as
found in nanomaterials, the contribution derived from the grain boundary processes
is increased. Ashby and Verall [13] added a grain deformation mechanism to
accommodate the grains with their new shapes in their new configuration. Also
in this case, the deformation rate is proportional to d
À3
. The principle of this
mechanism is depicted in Figure 11.17, which also shows clearly why this deformation mode is called the “grain switching” mechanism.
Chang et al. [14] developed a concise experimental proof of the validity of the
Ashby–Verall process for the deformation process of nanocrystalline intermetallics
by comparing the experimental results of hardness measurements on TiAl with
model calculations. The results of this study are shown in Figure 11.18, where the
11.2 Bulk Metallic and Ceramic Materials j311