Modeling to evaluate the contact areas of hard materials during the nano-indentation tests

A new mechanical model is developed in the present study for hard materials to investigate the behavior arising during the loading/unloading process of an indentation test. Two governing differential equations are derived for the depth solution of the indenter tip and the depth solution formed at the separation point expressed in a power form. The exponent value in either the loading process or unloading process is considered to be a variable as a function of the indentation depth in the governing differential equation. All coefficients shown in these governing differential equations associated with the spring and damping behavior are determined by the real-coded genetic algorithm. Quartz and silicon were used as the examples of hard materials, and the contact projected area predicted by the present model is quite close to the solution predicted by the area function of Oliver and Pharr [W.C. Oliver, G.M. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7 (4) (1992) 1564–1583]. The phase lagging behavior demonstrated in the indentation test at two different loading/unloading rates was investigated, and it is enhanced by increasing the loading/unloading rate. No restriction for hard materials in the loading rate is needed in the indentation test if the present model is employed. The contact area is decreased by increasing the loading rate.