7.3 Nano-Mechanical Properties of Solid Surfaces Obtained by Nano- …
195
GNDs in nano-indentation needs to be modified because a significant number
of dislocations for very small indentations spread out from the hemispherical
zone [27, 31].
Huang et al. [33] developed the nano-indentation model based on a maximum
allowable GND density, which is somewhat related to the effective storage volume
of GNDs. The strain gradient plasticity (SGP) theory based on the Taylor dislocation
model was modified to incorporate the maximum allowable GND density. In the
Taylor model, the shear flow stress τ is related to the dislocation density ρ [36]:
τ = αμb
√ ρ,
(7.14)
where μ is the shear modulus, b is the magnitude of the Burgers vector, and α is
an empirical coefficient of 0.3–0.5 depending on the tested materials [28, 33]. The
dislocation density ρ is the sum of the density ρ G for GNDs and of the density ρ S
for statistically stored dislocations (SSDs) [37]:
ρ = ρ G + ρ S .
(7.15)
The GNDs are required for the permanent shape change at the surface due to indentation, while the SSDs accumulate by random mutual trapping in the sample and
contribute to the deformation resistance. In indentation, the equivalent flow stress σ
is related to the shear stress τ by von Mises yield criterion [17, 33]:
σ =
√
3τ.
(7.16)
Furthermore, H is related to σ by the Tabor factor of 3:
H = 3σ.
(7.17)
Therefore, H can be finally expressed by
H = 3
√
3αμb
√ ρ G + ρ S .
(7.18)
Figure 7.14 shows a schematic diagram of GNDs underneath the indenter [28,
33]. In Fig. 7.14, the SSDs are not drawn to avoid the complexity of the diagram.
The calculation of ρ G is simplified by assuming that the indenter is conical, and the
indentation is accommodated by circular loops of GNDs with the Burgers vector
normal to the plane of the surface. As shown in Fig. 7.14, the angle between the
surfaces of the plane and indenter, the contact radius, and the indentation depth are
denoted by θ , a, and h, respectively. The following relationship holds between θ , a,
and h:
195
GNDs in nano-indentation needs to be modified because a significant number
of dislocations for very small indentations spread out from the hemispherical
zone [27, 31].
Huang et al. [33] developed the nano-indentation model based on a maximum
allowable GND density, which is somewhat related to the effective storage volume
of GNDs. The strain gradient plasticity (SGP) theory based on the Taylor dislocation
model was modified to incorporate the maximum allowable GND density. In the
Taylor model, the shear flow stress τ is related to the dislocation density ρ [36]:
τ = αμb
√ ρ,
(7.14)
where μ is the shear modulus, b is the magnitude of the Burgers vector, and α is
an empirical coefficient of 0.3–0.5 depending on the tested materials [28, 33]. The
dislocation density ρ is the sum of the density ρ G for GNDs and of the density ρ S
for statistically stored dislocations (SSDs) [37]:
ρ = ρ G + ρ S .
(7.15)
The GNDs are required for the permanent shape change at the surface due to indentation, while the SSDs accumulate by random mutual trapping in the sample and
contribute to the deformation resistance. In indentation, the equivalent flow stress σ
is related to the shear stress τ by von Mises yield criterion [17, 33]:
σ =
√
3τ.
(7.16)
Furthermore, H is related to σ by the Tabor factor of 3:
H = 3σ.
(7.17)
Therefore, H can be finally expressed by
H = 3
√
3αμb
√ ρ G + ρ S .
(7.18)
Figure 7.14 shows a schematic diagram of GNDs underneath the indenter [28,
33]. In Fig. 7.14, the SSDs are not drawn to avoid the complexity of the diagram.
The calculation of ρ G is simplified by assuming that the indenter is conical, and the
indentation is accommodated by circular loops of GNDs with the Burgers vector
normal to the plane of the surface. As shown in Fig. 7.14, the angle between the
surfaces of the plane and indenter, the contact radius, and the indentation depth are
denoted by θ , a, and h, respectively. The following relationship holds between θ , a,
and h: