The Pair Distribution Function for an Array of Screw Dislocations: Implications for Gradient Plasticity

The problem of spatial correlation within an array of dislocations in a two-dimensional crystalline solid is addressed. A system of equations for joint probability densities is derived based on the assumption that the force on each dislocation remains finite. For arrays of screw dislocations moving on several slip planes the equations are 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 is found, and used to demonstrate the strain gradient correction emerging in the case of micro-scale plasticity.