Periodic image effects in dislocation modelling

The use of periodic boundary conditions for modelling crystal dislocations is predicated on one's ability to handle the inevitable image effects. This communication deals with an often overlooked mathematical subtlety involved in dealing with the periodic dislocation arrays, that is conditional convergence of the lattice sums of image fields. By analysing the origin of conditional convergence and the numerical artefacts associated with it, we establish a mathematically consistent and numerically efficient procedure for regularization of the lattice sums and the corresponding image fields. The regularized solutions are free from the artefacts caused by conditional convergence and regain periodicity and translational invariance of the periodic supercells. Unlike the other existing methods, our approach is applicable to general anisotropic elasticity and arbitrary dislocation arrangements. The capabilities of this general methodology are demonstrated by application to a variety of situations encountered in atomistic and continuum modelling of crystal dislocations. The applications include introduction of dislocations in the periodic supercell for subsequent atomistic simulations, atomistic calculations of the core energies and the Peierls stress and continuum dislocation dynamics simulations in three dimensions.