7.3 Nano-Mechanical Properties of Solid Surfaces Obtained by Nano- …
197
slope of the solid line in Fig. 7.13. An included angle of 142.3° for the Berkovich
indenter used experiments in Fig. 7.13 is nearly equal to a cone angle of conical
indenter (140.6°) employed as an ideal indenter for the derivation of Eq. (7.25).
Equation (7.22) indicates that ρ G increases with decreasing indentation depth h in
the nanometer range.
However, ρ G cannot increase infinitely because of strong repulsive forces between
GNDs which impose dislocations to spread beyond the hemisphere at very small
indentation depth [27, 31]. A maximum allowable GND density ρ G, max [33] was
introduced to explain the nano-indentation hardness for which ρ G can never exceed
ρ G, max :
ρ G ≤ ρ G, max ,
(7.26)
where ρ G, max is determined from experiments, and it is on the order of 10
16 m
−2 [33].
Furthermore, the finite element method for strain gradient plasticity (SGP) theory [33,
38–40] based on the Taylor dislocation model was employed to examine the effect
of maximum allowable GND density in micro- and nano-indentation. In the finite
element analysis, it is not necessary to assume GND distributions underneath the
indenter, nor the storage volume for GNDs since they are automatically determined
in the computations. The result obtained by the finite element analysis for MgO
is represented by the dot-dash line in Fig. 7.13. The dot-dash line with a value of
ρ G, max = 1.28 × 10
16 m
−2 coincides well with the experimental data except for the
intermediate zone between
1
h
= 0.03 and 0.08 nm
−1 . The dotted line in Fig. 7.13
represents the result without consideration for ρ G, max , which coincides well with
the micro-indentation data
1
h
< 0.05 nm
−1
but not with the nano-indentation data.
In addition, the micro- and nano-indentation data for iridium as well as MgO were
explained by taking the maximum allowable GND density into account [33]. The
finite element analysis was also achieved for the effect of the indenter tip radius in
the micro- and nano-hardness of MgO, leading to the result that the effect of the tip
radius effect alone cannot explain the nano-indentation size effect without taking the
maximum allowable GND density into account.
Chicot [17] proposed a new indentation parameter named “ hardness length scale
factor (H LSF )” and indicated that the H
2 versus h
−1 curve for MgO is composed
of two separate linear lines covering micro- and nano-indentation hardness data as
shown in Fig. 7.15. Equation (7.13) can be rewritten by
H
2
= H
2
0 +
H
2
0 h
∗
h
.
(7.27)
The hardness length scale factor H LSF is defined by
H LSF = H 0
√
h ∗ ,
(7.28)
197
slope of the solid line in Fig. 7.13. An included angle of 142.3° for the Berkovich
indenter used experiments in Fig. 7.13 is nearly equal to a cone angle of conical
indenter (140.6°) employed as an ideal indenter for the derivation of Eq. (7.25).
Equation (7.22) indicates that ρ G increases with decreasing indentation depth h in
the nanometer range.
However, ρ G cannot increase infinitely because of strong repulsive forces between
GNDs which impose dislocations to spread beyond the hemisphere at very small
indentation depth [27, 31]. A maximum allowable GND density ρ G, max [33] was
introduced to explain the nano-indentation hardness for which ρ G can never exceed
ρ G, max :
ρ G ≤ ρ G, max ,
(7.26)
where ρ G, max is determined from experiments, and it is on the order of 10
16 m
−2 [33].
Furthermore, the finite element method for strain gradient plasticity (SGP) theory [33,
38–40] based on the Taylor dislocation model was employed to examine the effect
of maximum allowable GND density in micro- and nano-indentation. In the finite
element analysis, it is not necessary to assume GND distributions underneath the
indenter, nor the storage volume for GNDs since they are automatically determined
in the computations. The result obtained by the finite element analysis for MgO
is represented by the dot-dash line in Fig. 7.13. The dot-dash line with a value of
ρ G, max = 1.28 × 10
16 m
−2 coincides well with the experimental data except for the
intermediate zone between
1
h
= 0.03 and 0.08 nm
−1 . The dotted line in Fig. 7.13
represents the result without consideration for ρ G, max , which coincides well with
the micro-indentation data
1
h
< 0.05 nm
−1
but not with the nano-indentation data.
In addition, the micro- and nano-indentation data for iridium as well as MgO were
explained by taking the maximum allowable GND density into account [33]. The
finite element analysis was also achieved for the effect of the indenter tip radius in
the micro- and nano-hardness of MgO, leading to the result that the effect of the tip
radius effect alone cannot explain the nano-indentation size effect without taking the
maximum allowable GND density into account.
Chicot [17] proposed a new indentation parameter named “ hardness length scale
factor (H LSF )” and indicated that the H
2 versus h
−1 curve for MgO is composed
of two separate linear lines covering micro- and nano-indentation hardness data as
shown in Fig. 7.15. Equation (7.13) can be rewritten by
H
2
= H
2
0 +
H
2
0 h
∗
h
.
(7.27)
The hardness length scale factor H LSF is defined by
H LSF = H 0
√
h ∗ ,
(7.28)