Step by step building of a model for the Berkovich indentation cycle

The indentation cycle obtained from hardness testing of material has two singular points. The first one is the right end of the definition interval of loading/unloading curves; it corresponds to the cycle tip and poses no difficulties for mathematical analysis. The second one is the latest contact point between the material and the indenter tip in phase of withdrawal, it is located inside the interval of definition, more towards the left- or the right-hand side depending on the elasticity degree of the material. This second point is impractical for the analytical modelling of the cycle as the unloading curve loses there all mathematical properties of derivability and differentiability. This difficulty induced a tendency for empirical models built from experimental results that are, like any empirical laws, affected by some lack of precision. It also led to an intense focusing on the unloading curve to which main nanomechanical and structural properties have been connected, forgetting sometimes the loading curve and the valuable information it can provides. In this article, we work out an analytical model to represent the two curves of the indentation cycle as accurately as possible. In this step by step modeling we use functional analysis and force the modelling curves to fit the interval of definition, the concavity direction and the materials energetic properties.