Energetic solutions to rate-independent large-strain elasto-plastic evolutions driven by discrete dislocation flow

TL;DR

This paper develops a rigorous mathematical framework for large-strain elasto-plastic evolution in single crystals driven by discrete dislocation flow, incorporating nonlinear elasticity, internal variables, and path-dependent plastic flow.

Contribution

It introduces a novel energetic solution concept that accounts for dislocation movement and plastic flow in a complex nonlinear setting with polyconvex energy.

Findings

Constructed an energetic solution for rate-independent flow rule.

Accounted for dislocation trajectories as integral currents in space-time.

Proved existence using a-priori estimates and rescaling techniques.

Abstract

This work rigorously implements a recent model of large-strain elasto-plastic evolution in single crystals where the plastic flow is driven by the movement of discrete dislocation lines. The model is geometrically and elastically nonlinear, that is, the total deformation gradient splits multiplicatively into elastic and plastic parts, and the elastic energy density is polyconvex. There are two internal variables: The system of all dislocations is modeled via $1$-dimensional boundaryless integral currents, whereas the history of plastic flow is encoded in a plastic distortion matrix-field. As our main result we construct an energetic solution in the case of a rate-independent flow rule. Besides the classical stability and energy balance conditions, our notion of solution also accounts for the movement of dislocations and the resulting plastic flow. Because of the path-dependence of plastic flow, a central role is played by so-called ``slip trajectories'', that is, the surfaces traced out by moving dislocations, which we represent as integral $2$-currents in space-time. The proof of our main existence result further crucially rests on careful a-priori estimates via a nonlinear Gronwall-type lemma and a rescaling of time. In particular, we have to account for the fact that the plastic flow may cause the coercivity of the elastic energy functional to decay along the evolution, and hence the solution may blow up in finite time.