Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation

A thermodynamics-based variational method is developed to establish the equations of motion for three-dimensional (3D) interacting dislocation loops. The approach is appropriate for investigations of plastic deformation at the mesoscopic scale by direct numerical simulations. A fast sum technique for determination of elastic field variables of dislocation ensembles is utilized to calculate forces acting on generalized coordinates of arbitrarily curved loop segments. Each dislocation segment is represented by a parametric space curve of specified shape functions and associated degrees of freedom. Kinetic equations for the time evolution of generalized coordinates are derived for general 3D climb/glide motion of curved dislocation loops. It is shown that the evolution equations for the position $\left(\mathbf{P}\right),$ tangent $\left(\mathbf{T}\right),$ and normal $\left(\mathbf{N}\right)$ vectors at segment nodes are sufficient to describe general 3D dislocation motion. When crystal structure constraints are invoked, only two degrees of freedom per node are adequate for constrained glide motion. A selected number of applications are given for: (1) adaptive node generation on interacting segments, (2) variable time-step determination for integration of the equations of motion, (3) dislocation generation by the Frank-Read mechanism in fcc, bcc, and dc crystals, (4) loop-loop deformation and interaction, and (5) formation of dislocation junctions.