Dipole formation and yielding in a two-dimensional continuum dislocation model
While a Taylor-type yield stress, proportional to the square-root of the dislocation density, may appear at a macroscopic scale, it can be shown with discrete dislocation simulations that it does not accurately describe dislocation motion on the scale of individual dislocations. In this article, we first demonstrate that the Taylor term fails to capture a number of features of dislocation dynamics by comparing the results of a continuum formulation using this yield stress term to discrete dislocation dynamics simulations. We then present an alternate model, based on a mean free path formulation, and demonstrate that this model effectively reproduces the results of the discrete simulation. This mean free path model is proposed as an extension to an existing continuum theory, making use of the key fact that the velocity of dislocations in a realistic system is not single valued, but distributed over several values. The velocity distribution may also change with time. It is demonstrated that this formulation predicts features of both pileups and homogenous distributions which are in agreement with discrete dislocation simulations, but are not reproducible by traditional statistical continuum theories. Finally, possible extensions to this model are discussed, which may enhance the ability to reproduce key features of dislocation yielding.