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The elastic energy functional of a system of discrete dislocation lines iswell known from dislocation theory. In this paper we demonstrate how thediscrete functional can be used to systematically derive approximations whichexpress the elastic energy in terms of dislocation density-like variables whichaverage over the discrete dislocation configurations and represent thedislocation system on scales above the spacing of the individual dislocationlines. We study the simple case of two-dimensional systems of straightdislocation lines before we proceed to derive energy functionals for systems ofthree-dimensionally curved dislocation lines pertaining to a single, as well asto multiple slip systems. We then illustrate several applications of the theoryincluding Debye screening of dislocations in two and three dimensions, and thederivation of back stress and friction stress terms entering the stress balancefrom the free energy functionals.

Local density approximation for the energy functional of three-dimensional dislocation systems