Geometrical Modelling and Numerical Analysis of Dislocaion Mechanics

TL;DR

This paper develops a geometric and numerical framework to analyze dislocation mechanics, revealing detailed plastic and stress fields around dislocations while addressing classical singularities.

Contribution

It introduces a novel differential geometric model and numerical approach for dislocation analysis, including boundary conditions and non-singular stress field computation.

Findings

Distribution of plastic deformation around screw and edge dislocations

Stress fields match classical theory at a distance but are non-singular near dislocations

Surface effects influence deformation differently for plastic and elastic fields

Abstract

This study undertakes the mathematical modelling and numerical analysis of dislocations within the framework of differential geometry. The fundamental configurations, i.e. reference, intermediate and current configurations, are expressed as the Riemann-Cartan manifold, which equips the Riemannian metric and Weitzenb\"ock connection. The torsion 2-form on the intermediate configuration is obtained through the Hodge duality of the dislocation density and the corresponding bundle isomorphism is subjected to the Helmholtz decomposition. This analysis introduces the boundary condition for plastic deformation. Cartan first structure equation and stress equilibrium equation are solved numerically using weak form variational expressions and isogeometric analysis. The numerical analysis carried out for this study reveals the distribution of plastic deformation fields around screw and edge dislocations for the first time. It also demonstrates stress fields around dislocations of which the distant fields show full agreement with the classical Volterra theory, while at the same time eliminating the singularity otherwise introduced at the dislocation by classical methods. The stress fields include several characteristic features due to the geometrical nonlinearity included therein. We also demonstrate that free surfaces affect both plastic and elastic deformation, but in different ways. The mathematical framework of this study is applicable to an arbitrary configuration of dislocations.