A mathematical model was studied which described dislocation dynamics in crystals. The phase-field model was based upon the introduction of a core tensor which modified the singular field at the core of the dislocation. This model was presented for the case of the motion of a single dislocation, without cross-slip. The dynamics of a single dislocation line, moving in its slip plane, was described by using a Hamilton–Jacobi equation. Its velocity was a non-local quantity which depended upon the whole shape of the dislocation line. By introducing a level set formulation of the equation, the existence and uniqueness of a continuous viscosity solution was proved when the dislocation remained a graph in one direction. A numerical scheme was also proposed for which it was proved that the numerical solution converged to the continuous solution.

Dislocation Dynamics Described by Non-Local Hamilton–Jacobi Equations. O.Alvarez, E.Carlini, P.Hoch, Y.Le Bouar, R.Monneau: Materials Science and Engineering A, 2005, 400-401, 162-5