Complex-Eigenfrequency Band Structure of Viscoelastic Phononic Crystals

Abstract

The consideration of material losses in phononic crystals leads naturally to the introductionof complex valued eigenwavevectors or eigenfrequencies representing the attenuation of elastic wavesin space or in time, respectively. Here, we propose a new technique to obtain phononic band structureswith complex eigenfrequencies but real wavevectors, in the case of viscoelastic materials, wheneverelastic losses are proportional to frequency. Complex-eigenfrequency band structures are obtainedfor a sonic crystal in air, and steel/epoxy and silicon/void phononic crystals, with realistic viscouslosses taken into account. It is further found that the imaginary part of eigenfrequencies are wellpredicted by perturbation theory and are mostly independent of periodicity, i.e., they do not accountfor propagation losses but for temporal damping of Bloch waves.