Elastic models of dislocations based on atomistic Kanzaki forces

This paper studies the relationship between the atomistic representation of crystalline dislocations as Kanzaki forces and the continuum representation of dislocations as Burridge-Knopoff (BK) force distributions. We first derive a complete theory of the BK force representation of dislocations in an anisotropic linear elastic continuum, showcasing a number of fundamental features found when dislocations are represented as distributions of body forces in defect-free continuum media. We then build, within the harmonic approximation, the Kanzaki force representation of dislocations in atomistic lattice models. We rigorously show that in the long-wave limit, the Kanzaki force representation converges to the continuum BK representation. We therefore justify employing the Kanzaki forces as source terms in continuum theories of dislocations. We do this by establishing a methodology to compute the Kanzaki forces of dislocations via the force constant matrix of the material's perfect lattice. We use it to study a model of a screw dislocation in bcc tungsten, where we show the existence of two distinct Kanzaki force terms: the slip Kanzaki forces, which we show directly correspond with the BK forces implied by a Volterra dislocation; and the core Kanzaki forces, which are computed from the relaxed dislocation structure, and serve to model all core effects not captured by the Volterra dislocation. We build a multipolar field expansion of both the core and the slip Kanzaki forces, showing that the dislocation core is agreeable to correction via the multipolar field expansion of the core Kanzaki forces.