Incompatibility-governed singularities in linear elasticity with dislocations

Abstract

The purpose of this paper is to prove the relation [Formula: see text] relating the elastic strain [Formula: see text] and the contortion tensor [Formula: see text], related to the density tensor of mesoscopic dislocations. Here, the dislocations are given by a finite family of closed Lipschitz curves in [Formula: see text]. 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.