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

We prove that the classical line-tension approximation for dislocations in crystals, that is, 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 self-energy 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 of the classical energy down to its H1-elliptic envelope.