Band structure of a rotating helical phononic crystal

We consider an elastic helical medium whose tensor stiffness twirls uniformly along the helix axis. We are interested in analyzing the band structure when the whole material is externally forced to rotate around the helix axis to a fixed constant frequency. Departing from a general dynamic description of the elastic phenomena, we establish a set of equations for the displacement vector and the stress tensor. These equations allow us to calculate the band structure parametrized by the externally imposed rotating frequency. We find that the band structure strongly depends on the rotation frequency, and we show that backward and forward modes propagate differently, particularly for the longitudinal and right-polarized modes.

Figure 1

(a) Schematics of the helical triclinic structure of the crystalline solid. In the example, the pitch is right handed and the solid rotates around the helix axis with a frequency $\mathrm{\Omega }$. (b) Rotation of the lattice vectors of the helical structure around the $z$ axis as function of $z$ at a given instant. The pitch $p$ is the length along the $z$ axis required for the lattice vectors to give a complete turn. (c) Schematics of the eigenmodes propagating through the structure for low frequencies. Two of the elastic modes are mainly transversal with opposite elliptic polarizations. The third elastic mode is mainly longitudinal.

Figure 2

Eigenmodes for different values of the normalized rotating frequency $F$. Thick blue curves correspond to left circularly polarized waves, regular green curves correspond to right circularly polarized waves, and thin orange curves are associated with longitudinal waves (sound). Continuous lines indicate waves traveling backward, and the dashed lines indicate waves traveling forward.