Band structure of phononic crystals with general damping

In this paper, we present theoretical formalisms for the study of wave dispersion in damped elastic periodic materials. We adopt the well known structural dynamics techniques of modal analysis and state-space transformation and formulate them for the Bloch wave propagation problem. First, we consider a one-dimensional lumped parameter model of a phononic crystal consisting of two masses in the unit cell whereby the masses are connected by springs and dashpot viscous dampers. We then extend our analysis to the study of a two-dimensional phononic crystal, modeled as a dissipative elastic continuum, and consisting of a periodic arrangement of square inclusions distributed in a matrix base material. For our damping model, we consider both proportional damping and general damping. Our results demonstrate the effects of damping on the frequency band structure for various types and levels of damping. In particular, we reveal two intriguing phenomena: branch overtaking and branch cut-off. The former may result in an abrupt drop in the relative band gap size, and the latter implies an opening of full or partial wavenumber (wave vector) band gaps. Following our frequency band structure analysis, we illustrate the concept of a damping ratio band structure.