The paper addresses the problem of correlation within an array of parallel dislocations in a crystalline solid. The first two of a hierarchy of equations for the multi-point distribution functions were derived by treating the random dislocation distributions and the corresponding stress fields in an ensemble average framework. Asymptotic reasoning, applicable when dislocations were separated by small distances, provided equations that were independent of any specific kinetic law relating the velocity of a dislocation to the force acting on it. The only assumption made was that the force acting on any dislocation remains finite. The hierarchy was closed by making a standard closure approximation. For the particular case of a population of parallel screw dislocations of the same sign moving on parallel slip planes the solution for the pair distribution function was found analytically. For the dislocations having opposite signs the system of equations suggests that in ensemble average only geometrically necessary dislocations correlate, while balanced positive and negative dislocations would create dipoles or annihilate. Direct numerical simulations supported this conclusion. In addition, the relationship of the dislocation correlation to strain gradient theories and size effect was shown.

The Pair Distribution Function for an Array of Screw Dislocations. V.Vinogradov, J.R.Willis: International Journal of Solids and Structures, 2008, 45[13], 3726-38