A Splitting Scheme for Solving Reaction-Diffusion Equations Modeling Dislocation Dynamics in Materials Subjected to Cyclic Loading

Abstract

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion along with accompanying production/annihilation processes of dislocations lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result into the development of various types of dislocation micro‐structures, such as dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc., depending on the externally applied loading and the intrinsic lattice constraints. The Walgraef‐Aifantis (WA) (Walgraef and Aifanits, J. Appl. Phys., 58, 668, 1985) model is an example of a reaction‐diffusion model of coupled nonlinear equations which describe 0 formation of forest (immobile) and gliding (mobile) dislocation densities in the presence of cyclic loading. This paper discuss two versions of the WA model and focus on a finite difference, second order in time 1‐Nicolson semi‐implicit scheme, with internal iterations at each time step and a spatial splitting using the Stabilizing, Correction (Christov and Pontes, Mathematical and Computer Modelling, 35, 87, 2002) for solving the model evolution equations in two dimensions. The results of two simulations are presented. More complete results will appear in a forthcoming paper.