Influence of penetration depth and mechanical properties on contact radius determination for spherical indentation

Knowledge of the relationship between the penetration depth and the contact radius is required in order to determine the mechanical properties of a material starting from an instrumented indentation test. The aim of this work is to propose a new penetration depth–contact radius relationship valid for most metals which are deformed plastically by parabolic and spherical indenters. Numerical simulation results of the indentation of an elastic–plastic half-space by a frictionless rigid paraboloïd of revolution show that the contact radius–indentation depth relationship can be represented by a power law, which depends on the reduced Young’s modulus of the contact, on the strain hardening exponent and on the yield stress of the indented material. In order to use the proposed formulation for experimental spherical indentations, adaptation of the model is performed in the case of a rigid spherical indenter. Compared to the previous formulations, the model proposed in the present study for spherical indentation has the advantage of being accurate in the plastic regime for a large range of contact radii and for materials of well-developed yield stress. Lastly, a simple criterion, depending on the material mechanical properties, is proposed in order to know when piling-up appears for the spherical indentation.