The line-tension approximation as the dilute limit of linear-elastic dislocations

We prove that the classical line-tension approximation for dislocations in crystals, i.e., the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the Γ-limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length ψ(b, t), which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the selfenergy per unit length ψ0(b, t) obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation the classical energy down to its H1-elliptic envelope.