Renormalized Energy for Dislocations in Quasi-Crystals

TL;DR

This paper develops a variational approach to analyze the energy and interactions of screw dislocations in hexagonal quasi-crystals, providing precise asymptotic estimates and insights into dislocation forces.

Contribution

It introduces a core-regularized energy functional and applies higher-order -convergence to derive sharp asymptotic energy estimates for dislocations in quasi-crystals.

Findings

Asymptotic energy estimates as core radius tends to zero

Analysis of dislocation interactions and Peach-Kf6hler forces

Variational characterization of elastic equilibrium in quasi-crystals

Abstract

Anti-plane shear deformations of a hexagonal quasi-crystal with multiple screw dislocations are considered. Using a variational formulation, the elastic equilibrium is characterized via limit of minimizers of a core-regularized energy functional. A sharp estimate of the asymptotic energy when the core radius tends to zero is obtained using higher-order $\Gamma$-convergence. Also, the interaction between dislocations and the Peach-K\"{o}hler force at each dislocation are analyzed.