Incompatibility-governed singularities in linear elasticity with dislocations

Abstract

The purpose of this paper is to prove the relation relating the elastic strain and the contortion tensor , related to the density tensor of mesoscopic dislocations. Here, the dislocations are given by a finite family of closed Lipschitz curves in . Moreover the fields are singular at the dislocations, and, in particular, the strain is non square integrable. Moreover, the displacement fields show a constant jump around each isolated dislocation loop. This relation is called after E. Kröner who first derived the same formula for smooth fields at the macroscale.