The Walgraef-Aifantis Dislocation Dynamics: From Discrete to Continuous Analysis

The celebrated Walgraef-Aifantis (WA) model for dislocations has motivated many studies ranging from the continuum to the discrete micro-scale level. Here, we perform a computational-assisted analysis of the (W-A) model of dislocation patterning in one dimensional finite domain. We perform linear stability analysis with respect to the size of the finite domain and we extract the critical size at which the solution disappears. We investigate symmetric properties of the non homogeneous solutions and construct the bifurcation diagram with respect to the domain size. Furthermore, by considering the Lattice Boltzmann discretization of the WA, we exploit the Equation Free method (EFM) in order to reconstruct the free energy and calculate the first time passage between metastable states related to the different material states under loading (i.e. from veins to persistent sleep bands). Finally, we study a 2D discrete dislocation model and we link the different scales of descriptions, with the use of Diffusion Maps. Within this framework, we reduce the high dimensional representation to a few macroscopic variables by passing an explicit statistical mechanics hierarchy and reduction. Possible connections between the reduced macroscopic description and the W-A model are also discussed.