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Using the connection between depth-sensing indentation by spherical indenters and mechanics of adhesive contact, a new method for non-direct determination of adhesive and elastic properties of contacting materials is proposed. At low loads, the forcedisplacement curves reflect not only elastic properties but also adhesive properties of the contact, and therefore one can try to extract from experiments both the elastic characteristics of contacting materials (such as the reduced elastic modulus) and characteristics of molecular adhesion (such as the work of adhesion and the pull-off force) using a non-direct approach. The direct methods of estimations of the adhesive characteristics of materials currently used in experiments are rather complicated due to the instability of the experimental force-displacement diagrams for ultra-low tensile forces. The proposed method is based on the use of the stable experimental data for the elastic stage of the force-displacement curve and the mechanics of adhesive contact for spherical indenters. Since the experimental data always have some measurement errors, mathematical techniques for solving ill-posed problems are employed.

Nanoindentation techniques provide a unique opportunity to obtain mechanical properties of materials of very small volumes. The load-displacement and load-area curves are the basis for nanoindentation tests, and their interpretation is usually based on the main assumptions of the Hertz contact theory and formulae obtained for ideally shaped indenters. However, real indenters have some deviation from their nominal shapes leading researchers to develop empirical 'area functions' to relate the apparent contact area to depth. We argue that for both axisymmetric and three-dimensional cases, the indenter shape near the tip can be well approximated by monomial functions of radius. In this case problems obey the self-similar laws. Using Borodich's similarity considerations of three-dimensional contact problems and the corresponding formulae, fundamental relations are derived for depth of indentation, size of the contact region, load, hardness, and contact area, which are valid for both elastic and non-elastic, isotropic and anisotropic materials. For loading the formulae depend on the material hardening exponent and the degree of the monomial function of the shape. These formulae are especially important for shallow indentation (usually less than 100 nm) where the tip bluntness is of the same order as the indentation depth. Uncertainties in nanoindentation measurements that arise from geometric deviation of the indenter tip from its nominal geometry are explained and quantitatively described.

Statistical models of rough surfaces are widely used in tribology. These models include models based on assumption of normality of the asperity heights or similar assumptions that involve Gaussian distributions, models based solely on properties of the power spectral density of the surface heights along with models based on assumption of fractal character of roughness. It is argued that models describing surface roughness solely by its fractal dimension or its autocorrelation function (or its power spectral density) do not reflect tribological properties of surfaces. Then typical experimental data obtained for rough engineering surfaces prepared by grinding have been studied at nano and microscales. The heights of the micro-asperities were determined by a profilometre (stylus), while the data for nano/atomic scale was obtained by AFM (Atomic Force Microscopy). The assumption of the normal distribution for the roughness heights has been studied by application of various modern tests of normality. It was found that the height distribution of the surfaces under investigation were not Gaussian at both nano and microscales. Hence, the statistical models of rough surfaces under consideration cannot be used for description of the surfaces and there is a need in critical re-examination of the current statistical approaches.

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