Continuum descriptions of plastic deformation have found applications from the length scale of large-scale structures down to the length scale of grains in a polycrystalline material. At small length scales, it is necessary to account for the discreteness of slip systems inside individual grains, but the plastic part of deformation is still modeled in a continuum sense. With continued miniaturization, a limit to the applicability of continuum descriptions of plastic flow is reached. At sufficiently small sizes, the discrete nature of dislocations induces a length scale of the same order of magnitude as the size of the component or of the wave length of the deformation or stress field.
Discrete dislocation plasticity provides a description of plastic flow in which dislocations are treated individually, as line singularities in an elastic continuum. Determining the stress and deformation field of a body with dislocations essentially is an elasticity problem; but a very complex one, due to the singularities and the presence of boundaries. A methodology has been developed that decouples these two difficulties and allows arbitrary boundary-value problems to be solved. The physics of dislocation motion, annihilation, generation and pinning at obstacles is supplied in terms of relatively simple rules: the constitutive equations. This paper will first summarize the approach, with an emphasis on the implementation for three dimensional problems. The role of boundaries and the accuracy of satisfying the boundary values are highlighted. Next, the application of dislocations in a quantum dot is discussed, in order to address the issue of the critical size for dislocation nucleation. Also, a few two-dimensional applications are presented, including the problem of stress relaxation in a thin film.