loading conditions becomes clear to all researchers in crystal plasticity. One of the main guiding principles in seeking such a continuum dislocation theory has first been proposed by @Hansen1986 in form of the so-called LEDS-hypothesis: the true dislocation structure in the final state of deformation minimizes the energy of crystal among all admissible dislocation configurations. In view of numerous experimental evidences supporting this hypothesis (see, i.e., [@Hughes1997; @Kuhlmann1989; @Kuhlmann2001; @Laird1986]), its use in constructing the continuum dislocation theory seems to be quite reasonable and appealing. For the practical realization one needs to i) specify the whole set of unknown functions and state variables of the continuum dislocation theory, and ii) lay down the free energy of crystals as their functional to be minimized. Such program has been implemented by @Berdichevsky06a in the linear, and by @Le2014nonlinear in the nonlinear setting of CDT for networks of dislocations, whose lines are straight and remain so during the whole deformation process (see also [@Ortiz99; @Ortiz00]). The developed CDT has been successfully applied to various two-dimensional problems of dislocation pileups, bending, torsion, as well as formation of dislocation patterns in single crystals (see [@Berdichevsky-Le07; @Kaluza2011torsion; @Kochmann08b; @Kochmann08a; @Kochmann09; @Koster2015; @Le08a; @Le08b; @Le2009plane; @Le2012polygonization; @Le2013on]). 