Background theory

A schematic representation of a typical loading-unloading curve is presented in Figure 3-8a. The sample is loaded until it reaches the maximum depth (ht) at a corresponding

maximum load (Pt). The indenter is allowed to dwell for the specified period of time

and then unloads leaving a residual indent of depth hf. The difference between ht and hf shows that there has been some (but not total) elastic recovery. Sneddon (1965)

proposed that during the initial part of the unloading stage (where the curve is linear) the contact area between the sample material and the indenter remains constant and so exhibits elastic, ‘Hertzian’ behaviour. It is this part of the loading curve that may therefore be used to calculate the elastic modulus for the indented material. Pharr et al. (1992) showed that a power law could be fitted to the unloading curve in the form:

!

P="(h#h_f)m (3-9)

where h is a point on the unloading curve. By differentiating for the point at the start of the unloading curve (i.e. ht) it is possible to calculate the gradient, S.

To calculate the modulus and hardness it is necessary to know the contact area between the indenter and the material at the point of maximum load. An assumption of the original work by Sneddon (1965) and Pharr et al. (1992), which is also a limiting factor in the applicability of the method (Oliver and Pharr, 2004), is that sink- in occurs. Sink-in is the result of the displacement of material downwards during the indentation process and is illustrated in Figure 3-9. Sink-in results in the reduction of the area of contact between the indenter and the material, and so it is necessary to calculate the reduction of the indent depth due to sink-in (hs) in order to calculate the

contact depth (hc) and therefore contact area (A). hs may be calculated using the

gradient of the unloading curve with the following relationship:

!

h_s="1 P_t

S

(3-10)

where ε1 depends on the indenter geometry and is 0.75 for a Berkovich indenter. In

Figure 3-8b it can be seen that hc = ht - hsand so the contact depth can be calculated

!

h_c =h_t"#Pt

S

(3-11)

For a Berkovich indenter the contact area (A) can be calculated using:

A = 24.50 hc2 (3-12)

It is possible to calculate the reduced modulus (Er) using:

!

Er =

" S

2 A

(3-13)

Er is actually a function of the sample and the indenter material properties as shown in

equation (3-14) and so may be used to calculate the modulus of the sample (Es) if the

indenter modulus (Ei) and the Poisson ratio (ν) for the sample and the indenter are

known. For Al νs = 0.33 (Deiter, 1988) and for a diamond indenter νi = 0.07 and Ei =

1141 GPa (Beake et al., 2003).

! 1 Er = 1"#s 2

(

)

(

)

The hardness value (H) can be simply calculated by dividing the maximum load by the contact area calculated in equation (3-12):

!

H= Pt A

(3-15)

As has previously been mentioned, an assumption that sink-in occurs is made in the analysis of Sneddon (1965) and Pharr et al. (1992). This limits the applicability of the Pharr et al. (1992) method for calculating the modulus and hardness because this is not always the case. Tabor (1951) notes that pile-up may also occur and describes sink-in and pile-up as the displacement of metal during the indentation process; these mechanisms are illustrated in Figure 3-9. Sink-in is the displacement of the material downwards and pile-up is the displacement of the material upwards. If a material does not sink-in (as the theory predicts) then the contact depth (and so contact area) will be underestimated. Bolshakov and Pharr (1998) note that the contact area can be

underestimated by as much as 60%, which can lead to a large over-estimation of the material hardness using the Pharr et al. (1992) method.

The degree of pile-up has been shown to be dependent on the amount a material work hardens (Oliver and Pharr, 2004). Materials which work harden, for example annealed metals, are less prone to pile-up as the material on the sample surface next to the indenter hardens during deformation thus constraining the flow of the material

underneath, this inhibits pile-up. However in materials which do not work harden (for example soft metals which have been cold worked) the surface material is unable to constrain the flow from underneath. Bolshakov and Pharr (1998) also identified that in materials where elastic recovery is low (hf/ht > 0.7) pile-up may occur.

The Pharr et al. (1992) method of calculating modulus and hardness is the method recommended in the British Standard BS EN ISO 14577-1:2002. No approach is currently suggested for correcting for pile-up when using a Berkovich indenter and so instrumented hardness results need to be carefully assessed and the required caveats made when pile-up may occur.

3.9.4

Samples

Nanoindentation samples are typically of size 10x10x10 mm although they can be smaller than this. The samples were taken from the central section of the piston crown. The samples were aged at 260°C for 100 hours so that they were consistent with the mechanical testing samples. The samples were mounted in Bakelite and one face of the sample was polished to a 1µm finish using the polishing route described in Table 3-2. The opposite face of the embedded sample was ground to a 600 grit finish so that the non polished face of the sample was visible. An automatic polisher was used to ensure that the faces remained parallel. Both sides of the sample were exposed so that the only compliance not accounted for by the calibration procedure was that of the sample. The samples were mounted on an aluminium sample stub, which in turn was attached to the positioning stage during indentation. A schematic diagram of the nanoindenter is presented in Figure 3-7 and this shows the position of the sample stub during indentation.