A variational formulation of constrained dislocation dynamics coupled with heat and vacancy diffusion

We present a formulation of the discrete Dislocation Dynamics (DD) method based on Onsager׳s variational principle. The motion of discrete dislocations is treated as a generalized irreversible flux associated with conjugate thermodynamic forces causing internal production of entropy. Intrinsic in the variational principle is the role of physical constraints that limit the choice of generalized fluxes. We leverage the concept of constrained maximization to introduce the requirement that dislocation climb must be sustained by the flux of vacancies into the dislocation core. The constrained variational approach results naturally in the coupling between plastic deformation induced by discrete dislocations, vacancy diffusion, and heat propagation in solid crystals. In particular, this coupling requires that dislocation velocity and chemical potential of vacancies at the dislocation core be found simultaneously. A new numerical formulation of DD that accounts for generalized constraints imposed on dislocations is presented, based on a network discretization of the dislocation configuration. Applications illustrate the significance of constrained motion of dislocations confined in channels and pillars, and the attainment of heterogeneous dislocation structures.