Asymptotic Expansion of the Elastic Far-Field of a Crystalline Defect

TL;DR

This paper derives a rigorous, computable asymptotic expansion for the elastic fields of crystalline defects, combining continuum and discrete multipole terms, with a focus on the decay rates and core structure localization.

Contribution

It introduces a novel asymptotic expansion for lattice defect fields, integrating continuum correctors and discrete multipole terms with explicit decay rates.

Findings

The expansion accurately models long-range elastic fields.

Truncation yields a localized core structure with algebraic decay.

The method provides a computable approach to defect field analysis.

Abstract

Lattice defects in crystalline materials create long-range elastic fields which can be modelled on the atomistic scale using an infinite system of discrete nonlinear force balance equations. Starting with these equations, this work rigorously derives a novel far-field expansion of these fields: The expansion is computable and is expressed as a sum of continuum correctors and discrete multipole terms which decay with increasing algebraic rate as the order of the expansion increases. Truncating the expansion leaves a remainder describing the defect core structure, which is localised in the sense that it decays with an algebraic rate corresponding to the order at which the truncation occurred.