  Long-period moir\'e superlattices at the twisted interface of van der Waals
heterostructures relax into preferential-stacking domains separated by
dislocation networks. Here, we develop a mesoscale theory for dislocations in
the networks formed in twistronic bilayers with parallel (P) and antiparallel
(AP) alignment of unit cells across the twisted interface. For P bilayers we
find an exact analytical displacement field across partial dislocations and
determine analytic dependences of energy per unit length and width on
orientation and microscopic model parameters. For AP bilayers we formulate a
semi-analytical approximation for displacement fields across perfect
dislocations, establishing parametric dependences for their widths and energies
per unit length. In addition, we find regions in parametric space of crystal
thicknesses and moir\'e periods for strong and weak relaxation of moir\'e
pattern in multilayered twistronic heterostructures.

Mesoscale and Nanoscale Physics;Materials Science