Constraint reaction and the Peach–Koehler force for dislocation networks

Abstract

In the presence of dislocations, the strain field F is not a gradient, but satisfies the condition Curl F = Λ T L (with Λ L a measure concentrated on the dislocation L). Then F ∈ L p with 1 ≤ p < 2. This peculiarity is at the origin of the mathematical difficulties encountered with dislocations at the mesoscopic scale, which are here modeled by integral 1-currents which are free to form complex geometries in the bulk. In this paper, we first consider an energy minimization problem among the couples (F, L) of strains and dislocations, and then we exhibit a constraint reaction field arising at minimality, due to the satisfaction of the condition on the deformation curl, hence providing explicit expressions of the Piola-Kirchhoff stress and Peach-Kohler force. Moreover, it is shown that the Peach-Kohler force is balanced by a defect-induced configurational force, a sort of line tension. The functional spaces needed to mathematically represent dislocations and strains are also analyzed and described in a preliminary part of the paper.