Constraint reaction and the Peach–Koehler forcefor dislocation networks

In the presence of dislocations, the elastic deformation tensor F is not a gradient but satisfies the condition Curl F = Lambda(T)(L) (with the dislocation density 3 L a tensor-valued measure concentrated in the dislocation L). Then F is an element of L-p with 1 <= p < 2. This peculiarity is at the origin of the mathematical difficulties encountered by dislocations at the mesoscopic scale, which are here modeled by integral 1-currents 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 PeachKoehler force. Moreover, it is shown that the Peach-Koehler 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.