Mesoscale harmonic analysis of homogenous dislocation nucleation

We perform atomistic computer simulations to study the mechanism of homogeneous dislocation nucleation in two dimensional (2D) hexagonal crystalline films during indentation with a circular nanoindenter. The nucleation process is governed by the vanishing of the energy associated with a single normal mode. This critical mode is largely confined to a single plane of adjacent atoms. For fixed film thickness, L, the spatial extent, \xi, of the critical mode grows with indenter radius, R. For fixed R/L, the spatial extent \xi, grows roughly as \xi ~ L^0.4. We, furthermore, perform a mesoscale analysis to determine the lowest energy normal mode for mesoscale regions of varying radius, r_{meso}, centered on the critical mode's core. The energy, \lambda_{meso}, of the lowest normal mode in the meso-region decays very rapidly with r_{meso} and \lambda_{meso} ~= 0 for r_{meso} >~ \xi. The lowest normal mode shows a spatial extent, \xi_{meso}, which has a sublinear power-law increase with r_{meso} for r_{meso} <~ \xi, and saturates at r_{meso} \gtrsim 1.5\xi. We demonstrate a universal relationship between \xi_{meso}/ \xi versus r_{meso}/ \xi: independent of film thickness or indenter radius. The scenario that emerges is one where the analysis of small regions, r_{meso} <~ \xi, in the material \emph{can} reveal the presence of incipient instability even when the region being probed is much smaller than the spatial extent of the critical mode. However, the mesoscale analysis gives good estimates for the energy and spatial extent of the critical mode \emph{only} for r_{meso} >~ 1.5 \xi. In this sense homogeneous dislocation nucleation should be understood as a quasi-local phenomenon.