Trajectories in a space with a spherically symmetric dislocation

TL;DR

This paper introduces a new type of defect called a spherically symmetric dislocation in linear elasticity, deriving its geometric properties and analyzing test particle trajectories around it.

Contribution

It develops a geometric framework for modeling spherically symmetric dislocations and predicts particle trajectories, extending the geometric theory of defects.

Findings

Test particle trajectories are curved around the defect.

Trajectories cannot form circular orbits.

The induced metric is discontinuous, yet physical predictions remain well-defined.

Abstract

We consider a new type of defect in the scope of linear elasticity theory, using geometrical methods. This defect is produced by a spherically symmetric dislocation, or ball dislocation. We derive the induced metric as well as the affine connections and curvature tensors. Since the induced metric is discontinuous, one can expect ambiguity coming from these quantities, due to products between delta functions or its derivatives, plaguing a description of ball dislocations based on the Geometric Theory of Defects. However, exactly as in the previous case of cylindric defect, one can obtain some well-defined physical predictions of the induced geometry. In particular, we explore some properties of test particle trajectories around the defect and show that these trajectories are curved but can not be circular orbits.