A Numerical Investigation of the Pinning Phenomenon in Quasi-Periodic Frenkel Kontrova Model Under an External Force

We perform a numerical study on Frenkel–Kontrova (FK) models in quasi-periodic media subject to a constant external force. The quasi-periodic FK models appear in several physical problems such as deposition, spin waves in 1-D, planar dislocations in 3-D either in a quasi-crystal or on a cleaved surface with irrational slopes of a periodic crystal. The phenomenon that we study is pinning. When there is no force, the very homogeneous equilibria are abundant, but when there is a constant force, the uniformly distributed equilibrium states just slide away and the physically relevant equilibria are configurations with frequencies (inverse of densities in some interpretation) that are resonant with the frequencies of the medium. We use a dynamical interpretation of the equilibrium equation and perform a systematic Monte-Carlo exploration of the equilibria. We study the geometry and quantitative properties of the equilibria after the application of a constant external force. This leads to several quantitative regularities on the abundance and geometric properties of the pinned states in the models we consider. This complements recent qualitative studies.

Duke Scholars

Author Hossein Salahshoor Civil and Environmental Engineering