Let us mention here the similar approaches suggested in [@Acharya2000; @Acharya2001; @Engels2012; @Gurtin2002; @Gurtin2007; @Mayeur2014; @Oztop2013] which do not use the LEDS-hypothesis explicitly but employ instead the extended principle of virtual work for the gradient plasticity. However, as experiences and experiments show, dislocation lines are in general loops that, as a rule, can change their directions and curvatures depending on the loading condition and crystal’s geometry. Therefore the extension of CDT to networks of dislocations whose lines are curves in the slip planes is inevitable. To the best of author’s knowledge, such three-dimensional continuum dislocation theory based on the LEDS-hypothesis for curved dislocations has not been developed until now. It became also clear to him that the latter’s absence was due to the missing scalar dislocation densities for the network of curved dislocations.

The first attempt at constructing a continuum theory that can predict in principle not only the dislocation densities but also the direction and curvature of the dislocation lines has been made by @Hochrainer2007 in form of the so-called continuum dislocation dynamics. Their theory starts with the definition of the dislocation density that contains also the information about the orientation and curvature of the dislocation lines. Then the 