considered as a major disadvantage, but it may in fact be used for shaping by plastic
deformation via the application of relatively small forces. The third point, clearly
visible in Figure 11.20, is the increase in Young’s modulus with decreasing
grain size.
Based on the content of Figures 11.14 and 11.15, the result shown in Figure 11.20
was to be expected. Finally, many experimental indications have suggested that there
exists a grain size where strength is maximal and that beyond this grain size the yield
stress is reduced with decreasing grain size. This phenomenon is one of the clear-cut
results of these calculations. The yield stress of nanocrystalline copper as a function
of the grain size and deformation rate is displayed in Figure 11.21, where these
results are from the model calculations performed by Kim et al. [16]. In this graph,
because of the broad range of grain sizes covered in the calculations, a logarithmic
scale was selected as the abscissa and the results confirmed, perfectly, the tendency
of the experimental results as depicted in Figure 11.14.
The calculations leading to Figure 11.21 clarify that a maximum strength is found
with larger grain sizes, when the deformation rate is reduced. Figure 11.21 also
shows that the inverse Hall–Petch effect is limited to a very narrow range of grain
sizes; when the grain size reaches a lower limit, any further reduction of the yield
stress is negligible. This point is of major importance for the near-net-shape forming
of ceramic parts by plastic deformation.
Calculations that led to the graphs depicted in Figures 11.20 and 11.21 also
provided information on the contributions of different mechanisms for deformation; these contributions as a function of grain size at a strain rate of 10
À5 s
À1 are
shown in Figure 11.22, which also indicates the contributions of dislocation and
grain boundary sliding processes. The contribution of lattice diffusion processes is
so small that it was not plotted. It is important to realize that, at a strain rate of
10
À5 s
À1 , below a grain size of approximately 35 nm practically the whole
Figure 11.21 Results of model calculations for the yield stress of copper as a function of grain
size and deformation rate. The grain size where the maximum strength is observed depends on
the deformation rate [16].
11.2 Bulk Metallic and Ceramic Materials j315
deformation via the application of relatively small forces. The third point, clearly
visible in Figure 11.20, is the increase in Young’s modulus with decreasing
grain size.
Based on the content of Figures 11.14 and 11.15, the result shown in Figure 11.20
was to be expected. Finally, many experimental indications have suggested that there
exists a grain size where strength is maximal and that beyond this grain size the yield
stress is reduced with decreasing grain size. This phenomenon is one of the clear-cut
results of these calculations. The yield stress of nanocrystalline copper as a function
of the grain size and deformation rate is displayed in Figure 11.21, where these
results are from the model calculations performed by Kim et al. [16]. In this graph,
because of the broad range of grain sizes covered in the calculations, a logarithmic
scale was selected as the abscissa and the results confirmed, perfectly, the tendency
of the experimental results as depicted in Figure 11.14.
The calculations leading to Figure 11.21 clarify that a maximum strength is found
with larger grain sizes, when the deformation rate is reduced. Figure 11.21 also
shows that the inverse Hall–Petch effect is limited to a very narrow range of grain
sizes; when the grain size reaches a lower limit, any further reduction of the yield
stress is negligible. This point is of major importance for the near-net-shape forming
of ceramic parts by plastic deformation.
Calculations that led to the graphs depicted in Figures 11.20 and 11.21 also
provided information on the contributions of different mechanisms for deformation; these contributions as a function of grain size at a strain rate of 10
À5 s
À1 are
shown in Figure 11.22, which also indicates the contributions of dislocation and
grain boundary sliding processes. The contribution of lattice diffusion processes is
so small that it was not plotted. It is important to realize that, at a strain rate of
10
À5 s
À1 , below a grain size of approximately 35 nm practically the whole
Figure 11.21 Results of model calculations for the yield stress of copper as a function of grain
size and deformation rate. The grain size where the maximum strength is observed depends on
the deformation rate [16].
11.2 Bulk Metallic and Ceramic Materials j315