Dynamics of screw dislocations: A generalised minimising-movements scheme approach

The gradient flow structure of the model introduced in Cermelli & Gurtin (1999, The motion of screw dislocations in crystalline materials undergoing antiplane shear: glide, cross-slip, fine cross-slip. Arch. Rational Mech. Anal. 148(1), 3–52) for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the “maximal dissipation criterion” that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in Cermelli & Gurtin is connected to a time-discrete evolution scheme which explicitly confines dislocations to move at each time step along a single glide direction. It is proved that the time-continuous model in Cermelli & Gurtin is the limit of these time-discrete minimising-movement schemes when the time step converges to 0. The study presented here is a first step towards a generalisation of standard gradient flow theory that allows for dissipations which cannot be described by a metric.