Contact problems and depth-sensing nanoindentation for frictionless and frictional boundary conditions

The Hertz type contact problems and the analytical treatment of depth-sensing nanoindentation are under consideration. Fundamental relations of nanoindentation tests are derived for various boundary conditions within the contact region. First, a frictionless contact problem for a convex punch of revolution is studied and the connection between Galin and Sneddon solutions is shown. The Bulychev-Alekhin-Shorshorov (BASh) relation that is commonly used for evaluation of elastic modulus of materials by nanoindentation, is discussed. An analogous relation is derived that is valid for adhesive (no-slip) contact. Similarly to the Pharr-Oliver-Brotzen frictionless analysis, the obtained relation is independent of the geometry of the punch. Further, we study solutions to adhesive contact for punches whose shapes are described by monomial functions and obtain exact solutions for punches of arbitrary degrees of the monoms. These formulae are similar to the formulae of the frictionless Galin solutions and coincide with them when the material is incompressible. Finally, indenters having some deviation from their nominal shapes are considered. It is argued that for shallow indentation where the tip bluntness is on the same order as the indentation depth, the indenter shapes can be well approximated by non-axisymmetric monomial functions of radius. In this case problems obey the self-similar laws. Using one of the authors' similarity approach to three-dimensional contact problems and the corresponding formulae, other 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 various boundary conditions. In particular, it is shown that independently of the boundary conditions, the current area of the contact region is a power-law function of the current depth of indentation whose exponent is equal to a half of the degree of the monomial function of the shape.