How useful are formal hierarchies? A case study on averaging dislocation dynamics to define meso-macro plasticity

TL;DR

This paper critically examines the usefulness of formal hierarchies of evolution equations in modeling dislocation dynamics, highlighting their complexity and advocating for early termination and closure relations based on microscopic dynamics.

Contribution

It derives formal evolution hierarchies for dislocation averages and emphasizes the need for closure relations, challenging the practicality of such hierarchies for meso-macro plasticity modeling.

Findings

Hierarchies are complex and non-closed, limiting their direct applicability.

Early termination of hierarchies is recommended for practical modeling.

Closure relations should incorporate full-stress-coupled microscopic dynamics.

Abstract

A formal hierarchy of exact evolution equations are derived for physically relevant space-time averages of state functions of microscopic dislocation dynamics. While such hierarchies are undoubtedly of some value, a primary goal here is to expose the intractable complexity of such systems of nonlinear partial differential equations that, furthermore, remain `non-closed' and therefore subject to phenomenological assumptions to be useful. It is instead suggested that such hierarchies be terminated at the earliest stage possible and effort be expended to derive closure relations for the `non-closed' terms that arise from the formal averaging by taking into account the full-stress-coupled microscopic dislocation dynamics, a matter on which these formal hierarchies, whether of kinetic theory type or as pursued here, are silent.