Mathematical and computational modelling of correlations in dislocation dynamics

Abstract

This work focuses on the mathematical and computational modelling of dislocation correlation in the context of dislocation dynamics modelling of mesoscale plasticity. A hierarchical system of kinetic equations describing the evolution of dislocation densities of various orders is presented to illustrate the role of correlation in dislocation dynamics, which is followed by a mathematical description of the spatial and line-orientation statistics of dislocation systems within the framework of stochastic fiber processes. In this framework, the pair correlation is related to the second moment measure of the dislocation distribution in the crystal and dislocation line-tangent spaces. The stochastic fiber process description of correlation is used in conjunction with the method of dislocation dynamics simulation to compute the pair correlation in both face-centered cubic and body-centered cubic crystals undergoing small homogeneous plastic deformation at a constant rate. An edge correction scheme is employed as part of the overall methodology for computing the pair correlation. The main characteristics of correlations, including its oscillatory behavior, anisotropy and symmetry, are studied in detail. The self-correlation and cross correlation of dislocations of various species are compared. The implications of the present results for the development of kinetic theory of dislocations are discussed.

Dislocation pair correlations from dislocation dynamics simulations

“…This dependence of the pair correlation on the type of dislocation species exhibits more features than shown in Fig. 3 for the cross correlation between dislocations of different Burgers vectors, which is discussed in detail elsewhere [15].…”

“…A generalization of the density expression (1) and the associated kinetic equation (2) to higher orders is thus required in order to obtain a self-consistent solution for the evolution of the dislocation system. This generalization is carried out elsewhere [15], which leads to a hierarchy of coupled equations, the solution of which, although is both difficult and impractical, gives a precise description of the evolution of the dislocation system. The traditional approach to follow for this type of problems is to aim at finding a solution for the lowest order dislocation density, by representing the pair density ψ (α 1 α 2 ) in terms of the pair correlation ζ (α 1 α 2 ) in the form…”

“…In the current preliminary investigation, the nearest neighbor images of the central box in all directions have been considered in computing the correlations and, therefore, the results are most trusted within one box size around the origin. A more formal edge correction procedure is considered in [15]. have a profile that resembles (1 + cos( φ)) over most of the φ axis, with an oscillatory behavior along θ (at small φ) that is nearly symmetrical about θ = π.…”

The discrete-continuum connection in dislocation dynamics: I. Time coarse graining of cross slip

“…The first is whether the dislocation dynamics simulation cube size of 5 µm used to collect cross slip data is large enough to obtain the ensemble average of cross slip rates as the volume average, and the second is the effect of time coarse graining on the cross slip rate. An earlier work by Deng and El-Azab [8], which focused on dislocation correlations, used simulation cubes with sizes of 4.9 µm, 7.35 μm and 9.8 μm to estimate the spatial dislocation correlations using the microMegas model. For all practical purposes, negligible differences were found between the correlations evaluated from these different volumes.…”

“…Earlier works [4,8] suggest that, in order to incorporate processes such as annihilation, dislocation reactions, and cross slip in density based models, it is critical to assume that the densities should be defined as temporally coarse grained. In particular, the discussion in [4] suggests that temporal coarse graining can be achieved using the method of dislocation dynamics simulations.…”