A discrete crystal model in three dimensions: the line-tension limit for dislocations

TL;DR

This paper introduces a comprehensive discrete lattice model for 3D dislocations in crystals, demonstrating that its energy converges to a line-tension limit consistent with semi-discrete models, capturing complex lattice effects.

Contribution

It develops a new discrete model accounting for lattice symmetry, geometry, and slip systems, and proves its energy converges to a line-tension limit via $ ext{Gamma}$-convergence, aligning with semi-discrete models.

Findings

Discrete energy converges to line-tension energy for dilute dislocations.

Line-tension limit matches semi-discrete models with core cutoff.

Line-tension energy differs from classical quadratic Burgers vector models.

Abstract

We propose a discrete lattice model of the energy of dislocations in three-dimensional crystals which properly accounts for lattice symmetry and geometry, arbitrary harmonic interatomic interactions, elastic deformations and discrete crystallographic slip on the full complement of slip systems of the crystal class. Under the assumption of diluteness, we show that the discrete energy converges, in the sense of $\Gamma$-convergence, to a line-tension energy defined on Volterra line dislocations, regarded as integral vector-valued currents supported on rectifiable curves. Remarkably, the line-tension limit is of the same form as that derived from semi-discrete models of linear elastic dislocations based on a core cutoff regularization. In particular, the line-tension energy follows from a cell relaxation and differs from the classical ansatz, which is quadratic in the Burgers vector.