Damped Phononic Crystals and Acoustic Metamaterials

Abstract

The objective of this chapter is to introduce the topic of damping in the context of both its modeling and its effects in phononic crystals and acoustic metamaterials. First, we provide a brief discussion on the modeling of damping in structural dynamic systems in general with a focus on viscous and viscoelastic types of damping (Sect. 6.2) and follow with a non-exhaustive literature review of prior work that examined periodic phononic materials with damping (Sect. 6.3). In Sect. 6.4, we consider damped 1D diatomic phononic crystals and acoustic metamaterials as example problems (keeping our attention on 1D systems for ease of exposition as in previous chapters). We introduce the generalized form of Bloch’s theorem, which is needed to account for both temporal and spatial attenuation of the elastic waves resulting from the presence of damping. We also describe the transformation of the governing equations of motion to a state-space representation to facilitate the treatment of the damping term that arises in the emerging eigenvalue problem. Finally, the effects of dissipation (based on the two types of damping models considered) on the frequency and damping ratio band structures are demonstrated by solving the equations developed for a particular choice of material parameters.