Dislocation Dynamics as Gradient Descent in a Space of Currents

Recent progress in continuum dislocation dynamics (CDD) has been achieved through the construction of a local density approximation for the dislocation energy and the derivation of constitutive laws for the average dislocation velocity by means of variational methods from irreversible thermodynamics. Individual dislocations are driven by the Peach–Koehler-force which is likewise derived from a variational principle. This poses the question if we may expect that the averaged dislocation state expressed through the CDD density variables is driven by a variational gradient of the average energy, as is assumed in irreversible thermodynamics. In the current contribution we do not answer this questions, but rather present the mathematical framework within which the evolution of discrete dislocations is literally understood as a gradient descent. The suggested framework is that of de Rham currents and differential forms. We briefly sketch why we believe the results to be useful for formulating CDD theory as a gradient flow.