7.2 Fundamentals of Nano-Indentation
185
for the calculation of h c from Eq. (7.4). The area function can be fitted to a fifth-order
polynomial form:
A = c 0 h
2
c + c 1 h c + c 2 h
1
2
c + c 3 h
1
4
c + c 4 h
1
8
c + c 5 h
1
16
c ,
(7.7)
where c i (i = 0, 1, 2, 3, 4, 5) are constants, independent of h c , and the fixed values
of c 0 = 24.5 and 2.98 are used for the Berkovich and cube-corner indenters, respectively. The values of c i in Eq. (7.7) except for c 0 are determined by curve fitting
with the A vs h c curves shown in Fig. 7.5. Once the values of c i are determined, the
projected contact area A at any h c is obtained from Eq. (7.7), and thus, the hardness
H can be determined as a function of h c from Eq. (7.1). Equation (7.5) is rewritten
by
S
−1
=
√ π
2E r
A
−
1
2 .
(7.8)
Equation (7.8) means that the relationship between S
−1 and A
−
1
2 is linear if Young’s
modulus of the sample is constant, independent of the indentation depth. Young’s
modulus E s of the sample can be determined by using Eqs. (7.6) and (7.8) from the
load-depth curves measured at various maximum loads after the calibration of the
indenter shape. First, S
−1 is plotted versus A
−
1
2 , and then the reduced elastic modulus
E r is obtained from the slope
√ π
2E r
of the linear relationship between S
−1 and A
−
1
2
(see Fig. 7.11 for a Nb-doped TiO 2 (001) crystal). Consequently, Young’s modulus
of the sample can be determined from Eq. (7.6) if Poisson’s ratio of the sample is
known.
Alternatively, the reduced elastic modulus E r can be obtained by analyzing the
initial (elastic) portion of the load-depth curve based on the Hertzian theory [5],
which predicts the elastic behavior of an indenter tip with a hemispherical end for a
flat sample in the absence of frictional and adhesive interactions by using continuum
elasticity. The Hertzian behavior is expressed by [6, 7]:
L =
4
3
E r R
1
2 h
3
2 ,
(7.9)
where R is the radius of an indenter tip, L is the load, and h is the indentation depth.
Figure 7.6 shows schematically a load-depth curve (solid line) and the Hertzian
behavior (dotted line). The dotted line in Fig. 7.6 represents Eq. (7.9). The deviation
point of the load-depth curve from the Hertzian behavior is a plastic threshold at
which the plastic deformation is initiated. If R is known, E r can be determined by
fitting the elastic portion of the load-depth curve with the Hertzian behavior. If R is
unknown, at first, the load-depth curve for a standard sample with known E s has to
be measured and then R is obtained by fitting the elastic portion of the load-depth
curve with the Hertzian behavior.
185
for the calculation of h c from Eq. (7.4). The area function can be fitted to a fifth-order
polynomial form:
A = c 0 h
2
c + c 1 h c + c 2 h
1
2
c + c 3 h
1
4
c + c 4 h
1
8
c + c 5 h
1
16
c ,
(7.7)
where c i (i = 0, 1, 2, 3, 4, 5) are constants, independent of h c , and the fixed values
of c 0 = 24.5 and 2.98 are used for the Berkovich and cube-corner indenters, respectively. The values of c i in Eq. (7.7) except for c 0 are determined by curve fitting
with the A vs h c curves shown in Fig. 7.5. Once the values of c i are determined, the
projected contact area A at any h c is obtained from Eq. (7.7), and thus, the hardness
H can be determined as a function of h c from Eq. (7.1). Equation (7.5) is rewritten
by
S
−1
=
√ π
2E r
A
−
1
2 .
(7.8)
Equation (7.8) means that the relationship between S
−1 and A
−
1
2 is linear if Young’s
modulus of the sample is constant, independent of the indentation depth. Young’s
modulus E s of the sample can be determined by using Eqs. (7.6) and (7.8) from the
load-depth curves measured at various maximum loads after the calibration of the
indenter shape. First, S
−1 is plotted versus A
−
1
2 , and then the reduced elastic modulus
E r is obtained from the slope
√ π
2E r
of the linear relationship between S
−1 and A
−
1
2
(see Fig. 7.11 for a Nb-doped TiO 2 (001) crystal). Consequently, Young’s modulus
of the sample can be determined from Eq. (7.6) if Poisson’s ratio of the sample is
known.
Alternatively, the reduced elastic modulus E r can be obtained by analyzing the
initial (elastic) portion of the load-depth curve based on the Hertzian theory [5],
which predicts the elastic behavior of an indenter tip with a hemispherical end for a
flat sample in the absence of frictional and adhesive interactions by using continuum
elasticity. The Hertzian behavior is expressed by [6, 7]:
L =
4
3
E r R
1
2 h
3
2 ,
(7.9)
where R is the radius of an indenter tip, L is the load, and h is the indentation depth.
Figure 7.6 shows schematically a load-depth curve (solid line) and the Hertzian
behavior (dotted line). The dotted line in Fig. 7.6 represents Eq. (7.9). The deviation
point of the load-depth curve from the Hertzian behavior is a plastic threshold at
which the plastic deformation is initiated. If R is known, E r can be determined by
fitting the elastic portion of the load-depth curve with the Hertzian behavior. If R is
unknown, at first, the load-depth curve for a standard sample with known E s has to
be measured and then R is obtained by fitting the elastic portion of the load-depth
curve with the Hertzian behavior.