Complex-Eigenfrequency Band Structure of Viscoelastic Phononic Crystals

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