A distributional approach to the geometry of 2D dislocations at the continuum scale

Abstract

This paper develops a geometrical model of dislocations and disclinations in single crystals at the continuum scale, with use of the distribution theory 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 locally countably many parallel defect lines, under global 2D strain assumptions relying on the geometric measure theory. Our approach provides a new understanding of the continuum theory of defects and introduces the appropriate objective internal variables and the required mathematical framework for a rigorous homogenization of all relevant fields from mesoscopic to macroscopic scale.