  In recent years, van der Waals (vdW) heterostructures and homostructures,
which consist of stacks of two-dimensional (2D) materials, have risen to
prominence due to their association with exotic quantum phenomena. Atomistic
scale relaxation effects play an extremely important role in the electronic
scale quantum physics of these systems. We investigate such structural
relaxation effects in this work using atomistic and mesoscale models, within
the context of twisted bilayer graphene -- a well-known heterostructure system
that features moire patterns arising from the lattices of the two graphene
layers. For small twist angles, atomic relaxation effects in this system are
associated with the natural emergence of interface dislocations or strain
solitons, which result from the cyclic nature of the generalized stacking fault
energy (GSFE), that measures the interface energy based on the relative
movement of the two layers. In this work, we first demonstrate using atomistic
simulations that atomic reconstruction in bilayer graphene under a large twist
also results from interface dislocations, although the Burgers vectors of such
dislocations are considerably smaller than those observed in small-twist
systems. To reveal the translational invariance of the heterointerface
responsible for the formation of such dislocations, we derive the translational
symmetry of the GSFE of a 2D heterostructure using the notions of coincident
site lattices (CSLs) and displacement shift complete lattices (DSCLs). The
workhorse for this exercise is a recently developed Smith normal form
bicrystallography framework. Next, we construct a bicrystallography-informed
and frame-invariant Frenkel-Kontorova model, which can predict the formation of
strain solitons in arbitrary 2D heterostructures, and apply it to study a
heterostrained, large-twist bilayer graphene system.
