The exponent 3/2 at pyramidal nanoindentations

The analysis of published loading curves reveals the exponent 3/2 to the depth for nanoindentations with sharp pyramidal or conical tips. This has geometric reasons, as it occurs independent on the bonding states and indentation mechanisms. Nevertheless, most mathematical deductions and finite element simulations of nanomechanical parameters in the literature continue using the experimentally not supported Hertzian exponent 2. Therefore, numerous published loading curves of various authors are plotted using the experimental exponent 3/2 to present unbiased proof for its generality with metals, oxides, semiconductors, biomaterials, polymers, and organics. Linearity is independent of equipment and valid for load controlled, or depth controlled, or continuous stiffness, or AFM force measurements. The linearity with exponent 3/2 often extends from the nano- into the microindentation ranges. The tip rounding and taper influence of the "geometrical similar" indenters are discussed. When kinks occur in such linear plots through the origin, these indicate change of the materials' mechanical properties under pressure by phase transition. These events are discussed for nanoindentations with respect to the known hydrostatic transformation pressures that are, of course, always higher than the necessary indentation mean pressure. Numerous Raman, as well as X-ray and electron diffraction results from the literature support the phase transitions that are now easily detected. Nanoporous materials first fill the pores upon indentation. Published loading curves exhibit more information than hitherto assumed.