Homogenization of elasto-plastic evolutions driven by the flow of dislocations

TL;DR

This paper rigorously derives a continuum model of elasto-plasticity driven by dislocation motion from discrete dislocation models, using advanced mathematical tools to ensure stability and convergence.

Contribution

It introduces a rigorous homogenization process from discrete to continuous dislocation models in elasto-plasticity, employing space-time currents for solution concepts.

Findings

Existence of rate-independent dislocation-driven evolutions as limits of discrete models

Development of a recovery construction for space-time slip trajectories

Transfer of quasi-static stability from discrete to continuum models

Abstract

Starting from a prototypical model of elasto-plasticity in the small-strain and quasi-static setting, where the evolution of the plastic distortion is driven exclusively by the motion of discrete dislocations, this work performs a rigorous homogenization procedure to a model involving continuously-distributed dislocation fields. Our main result shows the existence of rate-independent evolutions driven by the motion of dislocation fields, obtained as limits of discrete dislocation evolutions. For all notions of solutions we employ the recent concepts of space-time integral and normal currents, which is richer than the classical approach using the Kr\"oner dislocation density tensor. The key technical challenge is to find discrete dislocation evolutions approximating a given dislocation field evolution, which requires a careful recovery construction of space-time slip trajectories and associated displacements. These methods enable one to transfer the properties, most importantly the quasi-static stability, from the discrete to the field regime.