  The elastic energy functional of a system of discrete dislocation lines is
well known from dislocation theory. In this paper we demonstrate how the
discrete functional can be used to systematically derive approximations which
express the elastic energy in terms of dislocation density-like variables which
average over the discrete dislocation configurations and represent the
dislocation system on scales above the spacing of the individual dislocation
lines. We study the simple case of two-dimensional systems of straight
dislocation lines before we proceed to derive energy functionals for systems of
three-dimensionally curved dislocation lines pertaining to a single, as well as
to multiple slip systems. We then illustrate several applications of the theory
including Debye screening of dislocations in two and three dimensions, and the
derivation of back stress and friction stress terms entering the stress balance
from the free energy functionals.
