Dynamics of discrete screw dislocations via discrete gradient flow

We present variational approaches (developed in [3,4,11]) to the study of statics and dynamics of screw dislocations in crystals. We model the crystal as a cubic lattice and we give the asymptotic Γ-convergence expansion of the elastic energy induced by a finite family of screw dislocations as the lattice spacing goes to zero. We show that the effective energy associated to the presence of a finite system of screw dislocations coincides with the renormalized energy, studied within the Ginzburg-Landau framework and ruling the interactions between the dislocations. As a byproduct of this analysis, we show the existence of many metastable configurations of dislocations pinned by energy barries. Using the minimizing movement approach á la De Giorgi, we introduce a discrete-in-time variational dynamics, referred to as Discrete Gradient Flow, which allows to overcome these energy barriers. More precisely, we show that lettting first the lattice spacing and then the time step of minimizing movements tend to zero, dislocations move accordingly with the gradient flow of the renormalized energy. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)