Dislocation mechanism based model for Portevin-Le Chatelier like instability in microindentation of dilute alloys

摘要

Although the topic of intermittent plastic flow manifesting as load fluctuations or displacement jumps in nanoindentation (depths less than 100 nm) has attracted considerable attention, the existence of steps on load-indentation (F-z) curves reported in microindentation (depths of several microns) of samples of dilute alloys of varying concentrations and load dates, has received little attention from a modeling point of view. Following our earlier approaches to nanoindentation instabilities and indentation size effect, we develop a minimal dislocation mechanism based model that predicts all the generic experimental features by setting up time-evolution equations for the mobile, the forest, dislocations with solute atmosphere, and the geometrically necessary dislocation densities. The model includes basic dislocation mechanisms common to most plastic deformations, such as dislocation multiplication, storage, and recovery mechanisms. We model the indentation instability as a variant of the standard Portevin-Le Chatelier (PLC) effect seen in the constant strain rate condition by including collective pinning and unpinning of dislocations from solute atmosphere. The instability mechanism is further generalized to include concentration-dependent dislocation-solute interaction to capture both concentration dependence of the indentation instability and strengthening of alloy samples. Based on recent experimental observations that show small misorientation at small depths suggesting limited geometrically necessary dislocation density, we model the growth of the geometrically necessary dislocation density by the number of loops that can be activated under the contact area and the mean strain gradient. The equations are then coupled to the load rate equation. The model predicts all the generic experimental features, such as (a) the stepped nature of the F-z curves, (b) the existence of a critical load and critical indentation depth for the onset of the instability, (c) the decreasing dependence of the maximum indentation depth with concentration of the alloying element, (d) the mean critical indentation depth z* for the onset of the instability increasing with decreasing concentration with a concomitant increase in levels of fluctuations, (e) the decreasing power-law dependence of the critical indentation depth with concentration, (f) the manifestation of intermittent stepped response in a window of load rates, and (g) the magnitude of the load steps scaling linearly with the load. In essence, the basic physical mechanisms responsible for predicting all the experimental results (a)-(g) are the generalization of pinning and unpinning mechanism (of dislocations from solute atmosphere) to include concentration-dependent dislocation-solute interaction and solution hardening of alloy samples with concentration together with the inherent rate-dependent nature of the PLC instability.