Density-based crystal plasticity : from the discrete to the continuum
  Because of the enormous range of time and space scales involved in dislocation dynamics, plastic modeling at macroscale requires a continuous formulation. In this paper, we present a rigorous formulation of the transition between the discrete, where plastic flow is resolved at the scale of individual dislocations, and the continuum, where dislocations are represented by densities. First, we focus on the underlying coarse-graining procedure and show that the emerging correlation-induced stresses are scale-dependent. Each of these stresses can be expanded into the sum of two components. The first one depends on the local values of the dislocation densities and always opposes the sum of the applied stress and long-range mean field stress generated by the geometrically necessary dislocation (GND) density; this stress acts as a friction stress. The second component depends on the local gradients of the dislocation densities and is inherently associated to a translation of the elastic domain; therefore, it acts as a back-stress. We also show that these friction and back- stresses contain symmetry-breaking components that make the local stress experienced by dislocations to depend on the sign of their Burgers vector. 
P.L. Valdenaire, Y. Le Bouar, B. Appolaire and A. Finel
cond-mat.mtrl-sci