Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock

In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy $E_gamma$ modelling the interactions$-$at a typical length-scale of $1/gamma$$-$of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_gamma$ in powers of $1/gamma$ by $Gamma$-convergence, in the limit $gammatoinfty$. While the zero-order term in the expansion, the $Gamma$-limit of $E_gamma$, captures the `bulk profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a `boundary-layer energy that captures the profile of the density in the proximity of the lock. This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.

Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity

In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1-currents with multiplicity in a lattice.