Renormalized Energy for Dislocations in Quasi-Crystals

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 $\mathrm{\Gamma }$-convergence. Also, the interaction between dislocations and the Peach-K"{o}hler force at each dislocation are analyzed.