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

Abstract

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. * bg374@cam.ac.uk

The Hall–Petch and inverse Hall–Petch relations and the hardness of nanocrystalline metals

“…One important problem at present with MD simulations is that since the cost of computation time needed to track the motion of atoms in a solid is very high [35], the timescale that can be investigated is Figure 1 The relationship between lower yield point (r LYP ) and grain size, d, in mild steel. The yield stress of the single crystal was obtained from Ref.…”

Mechanisms-Based Transitional Viscoplasticity

“…In consequence, plastic strain is merely a byproduct of the requirement that stress must adhere to the yield surface. An interesting technique suitable for generating random dislocation arrangements is proposed by Gurrutxanga-Lerma [23]. The main concern of the study is focused on how a given population of dislocation structures influences the dominant features of the stress field.…”

The multipolar elastic fields of ellipsoidal and polytopal plastic inclusions

“…In the context of linear elasticity, multipolar field expansions have long been employed to model point defects [24,25], particularly because the trace of the dipolar moment tensor is the relaxation volume of the defect [26], which enables the easy modelling of the dipolar moment tensor if the relaxation volume of the point defect can be calculated from first principles [26,27] or deduced from X-ray diffraction data [28]. Generalized formalisms of the multipolar fields have been offered in the context of reconstruction of seismic sources from estimated low order multipolar moments [29,30], and applied over particular cases to circular voids in plane stress [31], prismatic loops [32], dislocation loops and cracks [33] or to stochastic ensembles of dislocations [34]. They have also been applied to ensembles of dislocations in the context of discrete dislocation dynamics and the fast multipole method [35,36], and in the context of homogenization theory, with the aim of development of effective elastic moduli and constitutive behaviours in composite materials with spherical or ellipsoidal inclusions [37][38][39].…”