7.3 Nano-Mechanical Properties of Solid Surfaces Obtained by Nano- …
193
16
14
12
10
8
6
4
2
0
Hardness,
H / GPa
150
100
50
0
Contact depth, h c / nm
Magnetite (100)
Fig. 7.12 Relationship between hardness H and contact depth h c obtained from the load-depth
curves for a single crystal magnetite (100) surface [25]. Reprinted from [25] by permission of Taylor
& Francis Group LLC (CRC Press)
in MgO [27]. The cleaved and epi-polished surfaces of MgO were used for the
indentation tests. The cleaved surface has facets of (100) planes with a size of about
100 µm, while the epi-polished surface has an orientation of (100)± 0.2° and a
surface roughness <1 nm. The value of H decreased from 12.5 GPa to 9.5 GPa with
increasing indentation depth h, and there was no observable difference between the
indentation data for cleaved and epi-polished samples, indicating significant ISE for
both samples. The following relationship between hardness H and indentation depth
h in the micrometer depth regime has been established by Nix and Gao [28]:
H
H 0
2
= 1 +
h
∗
h
,
(7.13)
where H 0 is the indentation hardness in the region of h h
∗ , and h
∗ is the characteristic scale length which depends on the properties of indented material and included
angle of the indenter tip. Equation (7.13) was derived by introducing the concepts of
geometrically necessary dislocations (GNDs) and of strain gradient plasticity (SGP)
based on Taylor’s dislocation theory. The above relationship coincided well with the
micro-indentation data for polycrystalline and single crystal copper surfaces [29] as
well as for single crystal silver surfaces [30].
The nano-indentation hardness data [31, 32], however, do not follow Eq. (7.13)
since the relationship between H
2 and h
−1 deviates downwards from the linearity
with decreasing h in the depth range of h < 200 nm as shown in Fig. 7.13 for MgO
[33]. The solid line in Fig. 7.13 represents the linear relationship between H
2 and h
−1
corresponding to Eq. (7.13). The values of H 0 = 9.27 GPa and h
∗
= 91.0 nm are
193
16
14
12
10
8
6
4
2
0
Hardness,
H / GPa
150
100
50
0
Contact depth, h c / nm
Magnetite (100)
Fig. 7.12 Relationship between hardness H and contact depth h c obtained from the load-depth
curves for a single crystal magnetite (100) surface [25]. Reprinted from [25] by permission of Taylor
& Francis Group LLC (CRC Press)
in MgO [27]. The cleaved and epi-polished surfaces of MgO were used for the
indentation tests. The cleaved surface has facets of (100) planes with a size of about
100 µm, while the epi-polished surface has an orientation of (100)± 0.2° and a
surface roughness <1 nm. The value of H decreased from 12.5 GPa to 9.5 GPa with
increasing indentation depth h, and there was no observable difference between the
indentation data for cleaved and epi-polished samples, indicating significant ISE for
both samples. The following relationship between hardness H and indentation depth
h in the micrometer depth regime has been established by Nix and Gao [28]:
H
H 0
2
= 1 +
h
∗
h
,
(7.13)
where H 0 is the indentation hardness in the region of h h
∗ , and h
∗ is the characteristic scale length which depends on the properties of indented material and included
angle of the indenter tip. Equation (7.13) was derived by introducing the concepts of
geometrically necessary dislocations (GNDs) and of strain gradient plasticity (SGP)
based on Taylor’s dislocation theory. The above relationship coincided well with the
micro-indentation data for polycrystalline and single crystal copper surfaces [29] as
well as for single crystal silver surfaces [30].
The nano-indentation hardness data [31, 32], however, do not follow Eq. (7.13)
since the relationship between H
2 and h
−1 deviates downwards from the linearity
with decreasing h in the depth range of h < 200 nm as shown in Fig. 7.13 for MgO
[33]. The solid line in Fig. 7.13 represents the linear relationship between H
2 and h
−1
corresponding to Eq. (7.13). The values of H 0 = 9.27 GPa and h
∗
= 91.0 nm are