A non-singular theory of dislocations in anisotropic crystals

TL;DR

This paper introduces a non-singular, anisotropic gradient elasticity theory for dislocation loops that regularizes classical singularities, enabling more accurate modeling of dislocation interactions in crystals with various symmetries.

Contribution

It generalizes classical anisotropic dislocation theory by incorporating non-singular fields using gradient elasticity with multiple length scales, applicable to arbitrary crystal symmetries.

Findings

All elastic fields are non-singular and converge to classical solutions away from the core.

Interaction energy between dislocation loops is expressed as a double line integral without stress functions.

The theory applies to arbitrary anisotropic media and accounts for crystal symmetry through length scale parameters.

Abstract

We develop a non-singular theory of three-dimensional dislocation loops in a particular version of Mindlin's anisotropic gradient elasticity with up to six length scale parameters. The theory is systematically developed as a generalization of the classical anisotropic theory in the framework of linearized incompatible elasticity. The non-singular version of all key equations of anisotropic dislocation theory are derived as line integrals, including the Burgers displacement equation with isolated solid angle, the Peach-Koehler stress equation, the Mura-Willis equation for the elastic distortion, and the Peach-Koehler force. The expression for the interaction energy between two dislocation loops as a double line integral is obtained directly, without the use of a stress function. It is shown that all the elastic fields are non-singular, and that they converge to their classical counterparts a few characteristic lengths away from the dislocation core. In practice, the non-singular fields can be obtained from the classical ones by replacing the classical (singular) anisotropic Green's tensor with the non-singular anisotropic Green's tensor derived by \cite{Lazar:2015ja}. The elastic solution is valid for arbitrary anisotropic media. In addition to the classical anisotropic elastic constants, the non-singular Green's tensor depends on a second order symmetric tensor of length scale parameters modeling a weak non-locality, whose structure depends on the specific class of crystal symmetry. The anisotropic Helmholtz operator defined by such tensor admits a Green's function which is used as the spreading function for the Burgers vector density. As a consequence, the Burgers vector density spreads differently in different crystal structures.