Renormalized Energy and Peach-Köhler Forces for Screw Dislocations with Antiplane Shear

We present a variational framework for studying screw dislocations subject to antiplane shear. Using a classical model developed by P. Cermelli and M. E. Gurtin [The motion of screw dislocations in crystalline materials undergoing antiplane shear: glide, cross-slip, fine cross-slip, Arch. Ration. Mech. Anal. 148 (1999) 3--52], methods of Calculus of Variations are exploited to prove existence of solutions, and to derive a useful expression of the Peach-Köhler forces acting on a system of dislocation. This provides a setting for studying the dynamics of the dislocations, which is done in another paper of T. Blass, I. Fonseca, G. Leoni and M. Morandotti [Dynamics for systems of screw dislocations, SIAM J. Appl. Math. 75 (2015) 393--419].