Multipole expansion of continuum dislocations dynamics in terms of alignment tensors

TL;DR

This paper introduces a multipole expansion method using alignment tensors to represent dislocation kinematics in continuum plasticity models, enabling simplified yet accurate descriptions of dislocation distributions.

Contribution

It develops a novel multipole expansion framework for dislocation density using alignment tensors, addressing the challenge of representing dislocation kinematics with few variables.

Findings

Derived a hierarchy of evolution equations for alignment tensors.

Proposed low order closure approximations for continuum dislocation dynamics.

Discussed potential for more refined theories and current modeling challenges.

Abstract

Dislocation based modeling of plasticity is one of the central challenges at the crossover of materials science and continuum mechanics. Developing a continuum theory of dislocations requires the solution of two long standing problems: (i) to find a faithful representation of dislocation kinematics with a reasonable number of variables and (ii) to derive averaged descriptions of the dislocation dynamics (i.e. material laws) in terms of these variables. In the current paper we solve the first problem. This is achieved through a multipole expansion of the dislocation density in terms of so-called alignment tensors containing the directional distribution of dislocation density and dislocation curvature. A hierarchy of evolution equations of these tensors is derived from a higher dimensional dislocation density theory. Low order closure approximations of this hierarchy lead to continuum dislocation dynamics models with only few internal variables. Perspectives for more refined theories and current challenges in dislocation density modeling are discussed.