Emergence and role of dipolar dislocation patterns in discrete and continuum formulations of plasticity

The plasticity transition, at the yield strength of a crystal, typically signifies the tendency of dislocation defects towards relatively unrestricted motion. An isolated dislocation moves in the slip plane with velocity proportional to the shear stress, while dislocation ensembles move towards satisfying emergent collective elastoplastic modes through the long-range interactions. Collective dislocation motions are discussed in terms of the elusively defined back stress. In this paper, we present a stochastic continuum model that is based on a two-dimensional continuum dislocation dynamics theory that clarifies the role of back stress and demonstrates precise agreement with the collective behavior of its discrete counterpart as a function of applied load and with only three essential free parameters. The main ingredients of the continuum theory are the evolution equations of statistically stored and geometrically necessary dislocation densities, which are driven by the long-range internal stress; a stochastic yield stress; and, finally, two local ``diffusion''-like terms. The agreement is shown primarily in terms of the patterning characteristics that include the formation of dipolar dislocation walls.