Derivation of a Line-Tension Model for Dislocations from a Nonlinear Three-Dimensional Energy: The Case of Quadratic Growth

Abstract

In this paper we derive a line tension model for dislocations in 3D starting from a geometrically nonlinear elastic energy with quadratic growth. In the asymptotic analysis, as the amplitude of the Burgers vectors (proportional to the lattice spacing) tends to zero, we show that the elastic energy linearizes and the line tension energy density, up to an overall constant rotation, is identified by the linearized cell problem formula given in [S. Conti, A. Garroni, and M. Ortiz, Arch. Ration. Mech. Anal., 218 (2015), pp. 699--755].