Bloch theorem with revised boundary conditions applied to glide, screw and rotational symmetric structures

Abstract

Wave propagation in complex periodic systems is often addressed with the Bloch theorem, and consists in applying periodic boundary conditions to a discretized unit cell. While this method has been developed for structures periodic by translation, in a recent work, for quasi-one-dimensional wave propagation, it has been shown that screw (translation plus rotation) and glide (translation plus reflection) periodicities can be accounted for as well, keeping the Cartesian coordinate system but revisiting the periodic boundary conditions. The goal of the present paper is to generalize this concept to quasi-two-dimensional wave propagation (two dimensional waves propagating in three dimensional structures). Dispersion relations for a set of reduced problems are then compared to results from the classical method, when available. By considering a smaller periodicity, the computational cost is decreased and the number of folding curves and non-interacting intersecting curves is reduced, improving their interpretability. While the size of a unit cell is divided by a factor two when glide symmetries are considered, this ratio is significantly increased for screw or rotational symmetries. Moreover, the proposed revisited Bloch method is applicable to screw symmetric structures that do not possess purely translational symmetries, and for which the classical method cannot be used (e.g. chiral nanotubes, longitudinally wrinkled helicoids).