Oliver-Pharr model

It should be also reminded that plastic deformation is not considered in any models mentioned above. For a punch that includes both the elastic and plastic deformation, the Oliver-Pharr model is widely known [29]. However, it does not take adhesion into account, and therefore it is often inadequate to analyze AFM experimental data. In many cases, it is preferable to operate at possibly light load to avoid damage on both the probe and sample surfaces. 

Fig. 7. (a) Load-displacement curve of a typical elastoplastic material and (b) the schematic of the indentation model of Oliver and Pharr [40]. S—contact stiffness he— contact depth /imax—indenter displacement at peak load hf—plastic deformation after load removal hs—displacement of the surface at the perimeter of the contact. 

It is worth to mention that, during such measurements, the combined responses of the coating and the substrate are measured. Several relationships have been proposed for modeling this effect (see Pharr, 1992 Fischer-Cripps, 2000 Malzbender et al., 2002 Oliver, 2004). The contribution of the substrate becomes important when the indentation depth exceeds 10-25% ofthe film thickness and, therefore, may become critical when very thin films are measured. 

The mentioned heterogeneities were also visible in the mechanical analysis of the nano indentation experiments [63, 75, 77]. We calculated the hardness H and the elastic modulus E from the indentation curves following the model by Oliver and Pharr [80, 81]. In the case of the dense amorphous structure, the mechanical modulus of the sample was homogeneous over the tested area. In contrast, strong scattering of the data was observed in the case of the polycrystalline sample. The latter result reflects the intrinsic structural heterogeneity of the sample. 

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