Homogeneous nucleation of dislocations as bifurcations in a periodized discrete elasticity model

Abstract

A novel analysis of homogeneous nucleation of dislocations in sheared twodimensional crystals described by periodized discrete elasticity models is presented. When the crystal is sheared beyond a critical strain F = Fc, the strained dislocation-free state becomes unstable via a subcritical pitchfork bifurcation. Selecting a fixed final applied strain F f > Fc, different simultaneously stable stationary configurations containing two or four edge dislocations may be reached by setting F = F f t/tr during different time intervals tr. At a characteristic time after tr, one or two dipoles are nucleated, split, and the resulting two edge dislocations move in opposite directions to the sample boundary. Numerical continuation shows how configurations with different numbers of edge dislocation pairs emerge as bifurcations from the dislocation-free state.

On the critical nature of plastic flow: One and two dimensional models

“…The ensuing discrete model with continuous dynamics can be viewed as a parallel set of coupled overdamped FK chains [113,114,115,116,117]. It has been shown before that this model allows for generation and annihilation of dislocations and that it describes adequately their long range elastic interactions [118,119]. Most importantly, both the propagation and the nucleation/anihillation mechanisms in this model are associated with reaching the same strain thresholds.…”

Theory of defect dynamics in graphene: defect groupings and their stability

“…In simple geometries, we can avoid updating by introducing a periodic function of differences in the primitive directions that automatically describes link breakup and union associated with defect motion. Besides greatly reducing computational cost, the resulting periodized discrete elasticity models allow analytical studies of defect depinning [19,14], motion and nucleation [20,21]. Another advantage of periodized discrete elasticity is that boundary conditions can be controlled efficiently to avoid spurious numerical reflections at boundaries.…”

Periodized discrete elasticity models for defects in graphene

“…Static edge dislocations can be generated using the overdamped version of (16)-(17) periodized by means of (19) - (21) and the elastic field of edge dislocations for (10)- (11). To find the stationary edge dislocation at zero stress, we first write the corresponding stationary edge dislocation of isotropic continuum elasticity.…”

“…Recently, we have developed periodized discrete elasticity models of dislocations in cubic crystals that describe their motion and interaction 19,20,21 . These models appear to provide the simplest correction to the equations of elasticity allowing nucleation and motion of defects and have two main ingredients.…”