Hertz elastic contact in spherical nanoindentation considering infinitesimal deformation of indenter

Abstract

The nanoindentation technique has made it possible to measure deformations at extremely low forces and displacements. Many studies have been performed to identify and analyze unusual nano-scale phenomena. The violation of Hertz elastic contact between a spherical nanoindenter and metallic materials has been discussed in previous studies. When a sharp indenter is used and elasto-plastic contact occurs, the elastic modulus is well predicted by elastic contact theory. However, since nanoindentation is widely used to measure elastic moduli of nano-size samples, unexpected results using a spherical indenter have raised doubt about elastic contact in nanoindentation. We performed fully elastic loading and unloading nanoindentation on fused silica. To characterize the actual geometry of the spherical indenter we measured it directly using an atomic-force microscope. We then confirmed the actual indenter radius in experiments by comparison to indenter radius measured from residual impression size above 200 nm indentation depth. The Hertz equation was found to underestimate the indentation depth. To understand this phenomenon, we reconsidered the frame compliance, which in general nannoindentation testing is taken as constant. The infinitesimal deformation of the spherical indenter was calculated by summing the partial compliances of the infinite cylinder of the indenter. We found that indenter compliance depends on indentation depth on a logarithmic scale. We adopted an indentation-depth-dependent frame compliance to evaluate accurate force and depth data for indentation depths less than 100 nm. The recalibrated curve is found to be identical to the Hertz equation.