Mesoscale modelling of dislocation evolution: Physically based requirements on stochastic differential equations

The complexity of collective dislocation behaviour has motivated the use of mesoscopic models, which describe the average behaviour of dislocations. Due to the statistical nature of such models, it has been argued that a stochastic theory provides a more complete physical description. A number of stochastic differential equation (SDE)-based dislocation models have been proposed over the past three decades; however, there is no generally accepted method for transforming a deterministic model into a stochastic model. A range of approaches exist in the literature, but the choice of method is ad hoc and their physical consequences are not considered. Such transformations therefore may not preserve certain properties of the corresponding deterministic theory, namely that of non-negativity of the dislocation density. In the present work, we show that the drift and diffusion coefficients must extend to zero on the semi-closed interval [0, & INFIN;) to guarantee non-negativity of the dislocation density. We also demonstrate the breakdown of non-negativity in SDE-based dislocation models through analytical treatment of a class of mesoscale models, and propose a possible modification. We bring to attention a physically motivated method for deriving stochastic models which preserves non-negativity, and in general ensures that trajectories exist almost surely in some given set, so that it obeys the desired physical bounds on the state variables - called the stochastic invariance principle. These results can provide a better understanding of dislocation models used to describe microstructural damage accumulation in metallic structures, which can be incorporated into fatigue lifing models.