Dynamical approach to displacement jumps in nanoindentation experiments

The load-controlled mode is routinely used in nanoindentation experiments. Yet there are no simulations or models that predict the generic features of force-displacement F-z curves, in particular, the existence of several displacement jumps of decreasing magnitude. Here, we show that the recently developed dislocation dynamical model predicts all the generic features when the model is appropriately coupled to an equation defining the load rate. Since jumps in the indentation depth result from the plastic deformation occurring inside the sample, we devise a method for calculating this contribution by setting up a system of coupled nonlinear time evolution equations for the mobile and forest dislocation densities. The equations are then coupled to the force rate equation. We include nucleation, multiplication, and propagation threshold mechanisms for the mobile dislocations apart from other well known dislocation transformation mechanisms between the mobile and forest dislocations. The commonly used Berkovitch indenter is considered. The ability of the approach is illustrated by adopting experimental parameters such as the indentation rate, the geometrical quantities defining the Berkovitch indenter including the nominal tip radius, and other parameters. We identify specific dislocation mechanisms contributing to different regions of the F-z curve as a first step for obtaining a good fit to a given experimental F-z curve. This is done by studying the influence of the parameters on the model F-z curves. In addition, the study demonstrates that the model predicts all the generic features of nanoindentation such as the existence of an initial elastic branch followed by several displacement jumps of decreasing magnitude, and residual plasticity after unloading for a range of model parameter values. Further, an optimized set of parameter values can be easily determined that gives a good fit to the experimental force-displacement curve for Al single crystals of (110) orientation. The stress corresponding to the maximum force on the elastic branch is close to the theoretical yield stress. We also elucidate the ambiguity in defining the hardness at nanometer scales where the displacement jumps dominate. For larger indentation depths where displacement jumps disappear, the indentation size effect is predicted. The approach also provides insights into several open questions