# Minimising movements for the motion of discrete screw dislocations along glide directions

## TL;DR

This paper rigorously proves the convergence of a discrete scheme modeling screw dislocation motion to a continuous limit, confirming the predicted glide-direction dynamics using advanced mathematical techniques.

## Contribution

It provides the first rigorous proof of the discrete-to-continuous transition for screw dislocation motion along glide directions, validating previous formal results.

## Key findings

- Discrete scheme converges to continuous dynamics
- Motion occurs along crystal glide directions
- Rigorous mathematical proof using $ ext{Gamma}$-convergence

## Abstract

In [3] a simple discrete scheme for the motion of screw dislocations toward low energy configurations has been proposed. There, a formal limit of such a scheme, as the lattice spacing and the time step tend to zero, has been described. The limiting dynamics agrees with the maximal dissipation criterion introduced in [8] and predicts motion along the glide directions of the crystal. In this paper, we provide rigorous proofs of the results in [3], and in particular of the passage from the discrete to the continuous dynamics. The proofs are based on $\Gamma$-convergence techniques.