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

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.