Analytical study of fundamental nanoindentation test relations for indenters ofnon-ideal shapes

Abstract

Nanoindentation techniques provide a unique opportunity to obtain mechanicalproperties of materials of very small volumes. The load–displacement andload–area curves are the basis for nanoindentation tests, and their interpretationis usually based on the main assumptions of the Hertz contact theory andformulae obtained for ideally shaped indenters. However, real indenters have somedeviation from their nominal shapes leading researchers to develop empirical ‘areafunctions’ to relate the apparent contact area to depth. We argue that for bothaxisymmetric and three-dimensional cases, the indenter shape near the tip can bewell approximated by monomial functions of radius. In this case problemsobey the self-similar laws. Using Borodich’s similarity considerations ofthree-dimensional contact problems and the corresponding formulae, fundamentalrelations are derived for depth of indentation, size of the contact region,load, hardness, and contact area, which are valid for both elastic andnon-elastic, isotropic and anisotropic materials. For loading the formulaedepend on the material hardening exponent and the degree of the monomialfunction of the shape. These formulae are especially important for shallowindentation (usually less than 100 nm) where the tip bluntness is of thesame order as the indentation depth. Uncertainties in nanoindentationmeasurements that arise from geometric deviation of the indenter tipfrom its nominal geometry are explained and quantitatively described.