A distributional approach to the geometry of 2D dislocations at the mesoscale Part A: General theory and Volterra dislocations Part B: The case of a countable family of dislocations

TL;DR

This paper introduces a geometric and distributional framework for modeling 2D dislocations and disclinations in crystals at the mesoscale, enabling rigorous homogenization to macroscopic descriptions.

Contribution

It develops a distributional geometric model for dislocations and disclinations, extending previous work to countably many parallel defect lines with rigorous mathematical identities.

Findings

Established fundamental identities relating incompatibility tensor to defect densities.

Proved the model's applicability to countably many parallel defect lines.

Provided a mathematical framework for homogenization from mesoscale to macroscale.

Abstract

This paper develops a geometrical model of dislocations and disclinations in single crystals at the mesoscopic scale. In the continuation of previous work the distribution theory is used to represent concentrated effects in the defect lines which in turn form the branching lines of the multiple-valued elastic displacement and rotation fields. Fundamental identities relating the incompatibility tensor to the dislocation and disclination densities are proved in the case of countably many parallel defect lines, under global 2D strain assumptions relying on the geometric measure theory. Our theory provides the appropriate objective internal variables and the required mathematical framework for a rigorous homogenization from mesoscopic to macroscopic scale.