On continuous distributions of dislocations in nonlocal elasticity

Abstract

A linear, nonlocal continuum theory of dislocations is developed. The field equations are given for the dislocation density and the stress field due to continuous distribution of dislocations. Green's functions are obtained for two- and three-dimensional media and an integral formula is given for line distribution of dislocations generalizing Peach-Koehler formula of the classical (local) theory. Unlike the classical theory, no stress singularities occur so that self-stress and energies of dislocation loops can be calculated involving no divergences. Exact solutions given for the line and circular distributions of dislocations verify these expectations.