In this paper, we combine Bloch theory with familiar techniques of structural dynamics to study the effects of energy dissipation (i.e., damping) in an acoustic metamaterial. The formulation we present has the novel feature of incorporating a temporal component to wave attenuation in addition to the standard spatial component. The frequency band structure reflects the metamaterial response to the damping intensity. In the context of a lumped parameter nested mass model, increasing the magnitude of damping is shown to cause the band structure to descend the frequency range and reveal an intriguing phenomenon: branch overtaking. This effect occurs as dissipation causes the optical branch to descend below the acoustical branch. The resulting decrease in the width of the band gap would impact vibration minimization and isolation. We also examine the effective properties of the metamaterial, specifically, the effective mass and effective stiffness, and the conditions for these quantities to become negative. Finally, the aforementioned material results are shown to be related to their finite counterpart. For ease of exposition, we consider a special form of Rayleigh damping in which the damping is proportional to the stiffness. The intrinsic presence of dissipation in acoustic metamaterials and the limited scientific literature addressing damped wave propagation in periodic media in general motivates our present study.

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