Variational modeling of dislocations in crystals in the line-tension limit

Abstract

Dislocations are line singularities in crystals which are crucial for the plastic deformation of materials. Mathematically they can be modeled as measures supported on rectifiable curves, or as vector-valued one-currents. They have a lattice-valued multiplicity, which is a conserved quantity, in the sense that the divergence of the measure (or the boundary of the current) vanishes. Dislocations are necessarily accompanied by large elastic strains, and indeed their energetics can be understood starting from the theory of elasticity, in an appropriate scaling regime. We discuss recent progress in the rigorous derivation of dislocation models in the line-tension regime from linear elasticity, and their application to specific problems in metals. We present numerical simulations on dislocations in bcc molybdenum which show how our line-tension model provides a simple and efficient description of dislocation structures.