  A three-dimensional continuum dislocation theory for single crystals
containing curved dislocations is proposed. A set of governing equations and
boundary conditions is derived for the true placement, plastic slips, and loop
functions in equilibrium that minimize the free energy of crystal among all
admissible functions, provided the resistance to the dislocation motion is
negligible. For the non-vanishing resistance to dislocation motion the
governing equations are derived from the variational equation that includes the
dissipation function. A simplified theory for small strains is also provided.
An asymptotic solution is found for the two-dimensional problem of a single
crystal beam deforming in single slip and simple shear.
