A stochastic study of the collective effect of random distributions of dislocations

The effect that random populations of dislocations have on a material is examined through stochastic integration of a random cloud of dislocations lying at some distance away from a material point. The problem is studied in one, two, and three dimensions. In 1D, the cloud consists of individual edge dislocations placed along the real line; in 2D, of edge dislocations and edge dipoles on the plane; in 3D, of dislocation loops. In all cases, the dislocation cloud is randomly distributed in space, associated to which several relevant physical parameters, including the material's slip geometry, the dislocation's sign, and its relative orientation, are also stochastically treated. A fully disordered population, i.e., one where the dislocation's signatures and orientations are entirely random, is first studied. It is shown that such disordered systems entail a strong indeterminacy in the collective stress fields, which here is solved by enforcing mass conservation locally. In 2D, this is achieved by modelling a cloud of edge dipoles instead of individual dislocations; in 3D, this is naturally guaranteed by the modelling of closed dislocation loops. The long-range fields of the dipoles in 2D and of the loops in 3D is modelled via their multipolar force expansions, which greatly simplifies the analytical treatment of the problem. The cloud's effect is then studied by performing the stochastic integration of the multipolar fields via Campbell's theorem. The local order, but not the magnitude of the dislocation density, is shown to be critical in contributing to the plastic relaxation of the material: fully disordered systems are shown to self-attenuate, leading to plastic neutrality; ordered and partially ordered systems, achieved when dislocation signatures are aligned, display a direct relationship between the dislocation density and the average stress shielding the material. We establish and generalise the conditions that a system of dislocations must fulfil to display Taylor's equation and the Hall–Petch relation, and offer adequate scaling laws related to this.