Continuum approximation of dislocation correlations for systems of curved dislocations

Crystal plasticity primarily arises from the motion and interactions of dislocations. Consequently, one may expect a continuum theory of plasticity to emerge as a statistical continuum theory of interacting 'particles' - where these particles are, in fact, dislocations, i.e., extended and flexible moving curves. Due to the long range elastic interactions between dislocations, pair correlations play an important role for the emerging plastic characteristics of crystals. For these systems correlations arise not only from the relative positions of different dislocations (pair correlations) but also from non-trivial self correlations that reflect the geometry of individual dislocations as connected lines. The self correlations reflect interactions between segments of the same dislocation, which, for example, yield a tendency for dislocations to be straight. These self correlations need to be understood because they usually dominate numerically obtained correlation data from dislocation distributions. In this paper, we derive analytical expressions for self correlations in various hypothetical dislocation configurations which can serve as approximations for more general dislocation systems. The most general configurations we analyse are uncorrelated dislocations that form circular loops with a given distribution of radii. Additionally, we also derive pair correlations in various dipole-like configurations of dislocation pairs. Furthermore, we examine the radius distribution obtained from discrete dislocation dynamics simulations and show that these match radius distributions obtained from a maximum information entropy ansatz considering dislocation curvature. Our results suggest that to obtain a physically meaningful radius distribution within this maximum information entropy framework, an additional curvature-related variable must be introduced - one that has not yet been considered in continuum dislocation dynamics.