Description of the nanoindentation unloading curves with a universal function: Theoretical consideration and applications to brittle materials

With the traditional Oliver-Pharr (OP) method, only the contact stiffness at peak load can be obtained from one load-displacement (P-h) curve and the area function used for mechanical property calculation should be precalibrated by indenting an isotropic reference material of known modulus. In the present study, a universal function was proposed to describe the nanoindentation unloading data based on a brief theoretical consideration. Analyses of the experimental data measured on several brittle materials (soda-lime glass, Y2O3 and ZrO2) confirmed that, with this universal function, the contact stiffness can be determined continuously as a function of contact depth using the data recorded during the loading segment of a nanoindentation test. Further, the obtained contact stiffness versus contact depth relation enables a self-calibration of area function with the measured P-h curve, circumventing some problems associated with pre-calibration of area function. Using the determined contact stiffness and the self-calibrated area function, Young's modulus of the test samples can be obtained and the results were shown to be comparable with the outputs of the traditional OP method. In addition, the universal function proposed in the present study was also found to have potential applications in evaluating the effect of microstructural inhomogeneity on indentation responses of materials.