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

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.