Line-Tension Model for Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy

Source abstract

In this paper we rigorously derive a line-tension model for plasticity as the $\Gamma$-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off radius. The $\Gamma$-limit we obtain as the length of the Burgers vector tends to zero has the same form as the $\Gamma$-limit obtained by starting from a linear, semidiscrete dislocation energy. The nonlinearity, however, creates several mathematical difficulties, which we tackled by proving suitable versions of the rigidity estimate in non-simply-connected domains and by performing a rigorous two-scale linearization of the energy around an equilibrium configuration.