A previous deterministic model for the evolution of the dislocation density was here extended so as to take account of the stochastic nature of the underlying processes at the microstructural level. The main assumption which was made was that the dislocations were organized into a cell structure whose size scaled with the dislocation spacing. Dislocation storage and annihilation were considered to occur predominantly within the cell. Stochasticity was introduced by including fluctuations of the glide distance of dislocations in the cell. On the basis of a Langevin equation, a Fokker-Planck equation was derived for the distribution function. The corresponding distribution function for the dislocation link-length was found as a function of the strain. It was noted that the form of the distribution function suggested that the dislocation link length was not symmetrically distributed around its mean value. This led to a so-called diffusion-induced hardening term which was equivalent in form to dislocation storage at fixed obstacles such as grain boundaries or second-phase particles. A comparison with experimental data supported the theory. The stochastic model was applied to stage-IV hardening, but might easily were extended to stages II and III. In general, the model was expected to be applicable to any situation in which dislocation storage obeyed the principle of similitude.

A Stochastic Model for Dislocation Density Evolution. H.Braasch, Y.Estrin, Y.Bréchet: Scripta Materialia, 1996, 35[2], 279-84