Band structure of phononic crystals with general damping

Abstract

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.

Introduction to Phononic Crystals and Acoustic Metamaterials

“…It was also shown that viscoelasticity did not only attenuate acoustic waves traversing a rubber-based phononic crystal but also modified the frequency of passing bands in the transmission spectrum [43]. A theory of damped Bloch waves [44] was employed to show that damping alters the shape of dispersion curves and reduces the size of band gaps as well as opens wave vector gaps via branch cutoff [45]. Loss has an effect on the complete complex band structure of phononic systems including the group velocity [46].…”

Harnessing structural hierarchy to design stiff and lightweight phononic crystals

“…A direct comparison between the geometric features of the regular honeycomb and hierarchical honeycombs leads us to believe that different mechanisms of band gaps formation are intrinsically dictated by the slenderness ratio and coordination number of the lattice. It should be pointed out that damping effect resulting from the viscoelastic feature of the glassy polymer may make some contribution to the wave attenuation [24]. However, recent experimental results indicate that the damping effect will not swamp the band gaps in the phononic dispersion relations [25].…”

The Three-Dimensional Phononic Crystals

“…Damping is an interesting tool in further engineering phononic structures [14] to comply with specific needs in complex applications. An interesting example is the case of an fcc crystal of close-packed rubber spheres in air (see Fig.…”