Theory of acoustic band structure of periodic elastic composites

We study an elastic composite described by the position-dependent mass density \ensuremath{\rho}(r), the longitudinal speed of sound ${\mathit{c}}_{\mathit{l}}$(r), and the transverse speed of sound ${\mathit{c}}_{\mathit{t}}$(r). For a spatially periodic composite---a ``phononic crystal''---we derive the eigenvalue equation for the frequencies ${\mathrm{\ensuremath{\omega}}}_{\mathit{n}}$(K), where n is the serial number of the band and K is the Bloch wave vector. This is applied to the special case of a binary composite and, further, to the case of infinite cylinders that form a two-dimensional lattice. For this configuration (and no wave-vector component parallel to the cylinders) there are two independent modes of vibration. The elastic displacement u(r) is parallel to the cylinders for one of them---the transverse polarization mode. The other one is a mixed (longitudinal-transverse) polarization mode with u(r) perpendicular to the cylinders. Specifically we consider circular cylinders that form a square lattice. We compute the band structures for the transverse modes of nickel alloy cylinders in an aluminum alloy host, and vice versa. In both situations we find band gaps which extend throughout the Brillouin zone. Within these gaps the transverse vibrations, sound, and phonons are forbidden. We also investigate the dependence of the band gap on the filling fraction and on the material parameters.