The problem of spatial correlation within an array of dislocations in a two-dimensional crystalline solid was addressed. A system of equations for joint probability densities was derived based on the assumption that the force on each dislocation remained finite. For arrays of screw dislocations moving on several slip planes the equations were consistent with balanced positive and negative dislocations forming dipoles or mutually cancelling, leaving geometrically necessary dislocations to interact and correlate. The resulting pair distribution function for the geometrically necessary screw dislocations was found, and used to demonstrate the strain gradient correction emerging in the case of micro-scale plasticity.

The Pair Distribution Function for an Array of Screw Dislocations - Implications for Gradient Plasticity. V.Vinogradov, J.R.Willis: Mathematics and Mechanics of Solids, 2009, 14[1-2], 161-78