From dislocation motion to an additive velocity gradient decomposition, and some simple models of dislocation dynamics

Amit Acharya         Xiaohan Zhang

(Chinese Annals of Mathematics, 36(B), 2015, 645-658.  Proceedings of the International Conference on Nonlinear and Multiscale Partial Differential Equations: Theory, Numerics and Applications held at Fudan University, Shanghai, September 16-20, 2013, in honor of Luc Tartar.)

A mathematical theory of time-dependent dislocation mechanics of unrestricted
geometric and material nonlinearity is reviewed. Within a “small deformation” setting, a
suite of simplified and interesting models consisting of a nonlocal Ginzburg Landau equa-
tion, a nonlocal level set equation, and a nonlocal generalized Burgers equation is derived.
In the finite deformation setting, it is shown that an additive decomposition of the total
velocity gradient into elastic and plastic parts emerges naturally from a micromechanical
starting point that involves no notion of plastic deformation but only the elastic distortion,
material velocity, dislocation density and the dislocation velocity. Moreover, a plastic spin
tensor emerges naturally as well.