Chapter 2. Nanoindentation Techniques
2.2 Force-displacement curves
2.2.2 Extraction of elastic modulus and hardness from the P-δ curves
2.2.2.1 Unloading curve method
Modelling the material response of a loading curve is much more complex, as it may consist of both plastic and elastic behaviours. Therefore, the unloading curve is normally used to obtain elastic modulus and hardness by assuming there is only elastic behaviour involved in the unloading. Based on the analysis for the elastic unloading curve with a flat punch indenter (Sneddon, 1965), Doerner and Nix proposed a linear relationship in the first one third of the unloading curve for nanoindentation by a flat punch indenter (Doerner and Nix, 1986). Soon, Oliver and Pharr extended this method to the cases of various indenters by proposing a power law relationship in the initial part of the unloading curve. The validity of Sneddon analysis for various indenters was verified by them as well (Oliver and Pharr, 1992). In the Sneddon analysis, the elastic modulus is given by (Sneddon, 1965),
𝑆 = 𝑑𝑃
𝑑ℎ = 2
√𝜋 𝐸
𝑟√𝐴
𝑐(2.1)
where 𝐸𝑟 is the reduced modulus that combines the modulus of the indenter and the test specimen, which can be described as,
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𝐸
𝑟= 1 − 𝑣
𝑠2𝐸
𝑠+ 1 − 𝑣
𝑖2𝐸
𝑖(2.2)
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where 𝐸𝑠 and 𝑣𝑠 are the Young’s modulus and the Poisson’s ratio for the specimen, respectively. 𝐸𝑖 and 𝑣𝑖 are the Young’s modulus and the Poisson’s ratio for the indenter, respectively. For a commercially used diamond tip, the Young’s modulus is 1141 GPa and the Poisson’s ratio is 0.07.
In order to obtain the contact stiffness from the P-δ curve, Oliver and Pharr proposed a power law relationship for the initial part of the unloading curve, which is expressed as,
𝑃 = 𝐵(𝛿 − 𝛿
𝑟𝑒𝑠)
𝑚(2.3)
where 𝐵, 𝑚 and
𝛿
𝑟𝑒𝑠 are the fitting constants determined by the least squares fitting procedure. By differentiating Equation 2.3 with respect to the depth, the unloading slope at the peak depth can be mathematically expressed as,𝑆 = 𝑑𝑃
𝑑𝛿 |
𝛿=𝛿𝑚𝑎𝑥
= 𝑚𝐵(𝛿
𝑚𝑎𝑥− 𝛿
𝑟𝑒𝑠)
𝑚−1(2.4)
According to the Oliver and Pharr method, the contact area is calculated from the contact depth rather than the optical measurement. In order to unveil the relationship between different dimensions, a cross section through an indentation was provided by Oliver and Pharr, as shown in Figure 2.2.
Figure 2.2 Schematic of a cross section of an indentation with the assumption that piling-up and sinking-in are negligible. Various dimensions that used in the
analysis are indicated (Oliver and Pharr, 1992).
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At any time during the loading, total depth, 𝛿, can be written as,
𝛿 = 𝛿
𝑐+ 𝛿
𝑠(2.5)
where 𝛿𝑐 is the contact depth between the contact circle and the apex of the indenter. 𝛿𝑠 is the elastic deflection of the surface which equals the depth between the contact circle and the sample free surface. In the Sneddon analysis, it is given by,
𝛿
𝑠= 𝜀 𝑃
𝑚𝑎𝑥𝑆 (2.6)
where 𝜀 is a geometric constant. For a flat punch indenter, 𝜀 = 1. For a spherical (paraboloid of revolution) indenter, 𝜀 = 0.75. For a Berkovich (pyramidal) indenter and conical indenter, 𝜀 should be theoretically equal to 2𝜋(𝜋 − 2), but Oliver and Pharr have indicated that 𝜀 = 0.75 will better represent experimental results (Oliver and Pharr, 1992).
Whereupon, by substituting Equation 2.6 into Equation 2.5, the contact depth can be expressed as,
𝛿
𝑐= 𝛿
𝑚𝑎𝑥− 𝜀 𝑃
𝑚𝑎𝑥𝑆 (2.7)
Once contact depth is calculated, the contact area can be presented as a function of contact depth with the known geometry of the indenter (Pethica et al., 1983). For a perfect Berkovich indenter,
𝐴(𝛿
𝑐) = 𝐶
0𝛿
𝑐2= 24.5𝛿
𝑐2(2.8)
In practice, more terms are added to Equation 2.8 to account for the inaccuracies of the area brought by a non-perfect indenter,
𝐴(𝛿
𝑐) = 𝐶
0𝛿
𝑐2+ 𝐶
1𝛿
𝑐+ 𝐶
2𝛿
𝑐1/2+ 𝐶
3𝛿
𝑐1/4+ ⋯ + 𝐶
8𝛿
𝑐1/128(2.9)
where 𝐶0 is the geometric constant which represents the projected area to depth ratio of an ideal indenter, and 𝐶0 = 24.5 for a Berkovich indenter. 𝐶1 to 𝐶8 are the fitting parameters obtained by calibration tests on a reference material with known
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mechanical properties. Actually, elastically isotropic materials, such as fused quartz and aluminium, are widely used as the reference materials due to their elastic moduli are independent of displacement (Oliver and Pharr, 1992). Besides, the surface quality of the reference material should be carefully controlled, as this may affect the independency of its elastic modulus (Zheng et al., 2007).
The hardness of test sample is given by,
𝐻 = 𝑃
𝐴(𝛿
𝑐) (2.10)
It is very important to note that one of the simplified assumptions for the Sneddon equation (in the case of a flat punch indenter) is that the sides of the shape after fully unloaded are straight. Since most commercial indenters are not flat-ended punches, a correction for the contact stiffness in the Sneddon equation is required to account for the corresponding inward deformation of the surface (as shown in Figure 2.3) after removal of the indenter. With the extensive use of the finite element modelling (FEM) method, Hay et al. indicated the discrepancy relates to both the geometry of the indenter and the test sample, and introduced a correction factor 𝛾. For a pyramidal or conical indenter with 𝜃>60˚, 𝛾 is given by (Hay et al., 1999; Hay and Wolff, 2001; Malzbender, 2002),
𝛾 = 𝜋
For a pyramidal or conical indenter with 𝜃≤60˚, this factor is described as,
𝛾 = 1 + 2(1 − 2𝑣)
4(1 − 𝑣) tan 𝜃 (2.12)
For a spherical indenter this factor is given by,
𝛾 = 1 + 2(1 − 2𝑣)𝑎
𝑐3𝜋(1 − 𝑣)𝑅 (2.13)
where 𝜃 is the half included angle of the indenter, 𝜈 is the Poisson’s ratio of the test material, 𝑎𝑐 is the contact radius and 𝑅 is the radius of the spherical indenter.
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For a pyramidal indenter, 𝜃 should be its effective cone angle, namely, 𝜃=70.3˚
for a Berkovich indenter.
Figure 2.3 Schematic of the radial displacement of the deformed surface after load removal (Hay et al., 1999).
For the symmetry of the indenter, King introduced an extra correction factor 𝛽 to account for the deviation of data from the non-axially symmetric indenter (King, 1987). Though many researchers reported different values of 𝛽, a value of 1.034 has actually been widely adopted. Taking these corrections into consideration, the reduced modulus of specimen can be obtained by rewriting Equation 2.1,