Effective dislocation lines in continuously dislocated crystals. I. Material anholonomity

TL;DR

This paper introduces a geometric framework for modeling continuously dislocated crystals, capturing the effects of dislocations and point defects through an anholonomic triad and Riemannian geometry.

Contribution

It proposes a novel continuous geometric description of dislocated crystals using an anholonomic triad and Riemannian space, extending prior discrete models.

Findings

Modeling of dislocation distributions via anholonomic triad.

Representation of point defects within a Riemannian material space.

Discussion of implications for crystal defect analysis.

Abstract

A continuous geometric description of Bravais monocrystals with many dislocations and secondary point defects created by the distribution of these dislocations is proposed. Namely, it is distinguished, basing oneself on Kondo and Kroners Gedanken Experiments for dislocated bodies, an anholonomic triad of linearly independent vector fields. The triad defines local crystallographic directions of the defective crystal as well as a continuous counterpart of the Burgers vector for single dislocations. Next, the influence of secondary point defects on the distribution of many dislocations is modeled by treating these local crystallographic directions as well as Burgers circuits as those located in such a Riemannian material space that becomes an Euclidean 3-manifold when dislocations are absent. Some consequences of this approach are discussed.