Incompatibility-governed singularities in linear elasticity with dislocations

The purpose of this paper is to prove the relation inc ε = Curl κ relating the elastic strain ε and the contortion tensor κ , related to the density tensor of mesoscopic dislocations. Here, the dislocations are given by a finite family of closed Lipschitz curves in Ω ⊂ ℝ 3 . 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.