  An atomistic model of near-crack-tip plasticity on a square lattice under
anti-plane shear kinematics is formulated and studied. The model is based upon
a new geometric and functional framework of a lattice manifold complex, which
ensures that the crack surface is fully taken into account, while preserving
the crucial notion of duality. As a result, existence of locally stable
equilibrium configurations containing both a crack opening and dislocations is
established. Notably, with the boundary in the form of a crack surface
accounted for, no minimum separation between a dislocation core and the crack
surface or the crack tip is required. The work presented here constitutes a
foundation for several further studies aiming to put the phenomenon of
near-crack-tip plasticity on a rigorous footing.
