Mechanics of metadamping in flexural dissipative metamaterials: Analysis and design in frequency and time domains

Locally resonant metamaterials are engineered structures with intriguing acoustic mitigation properties which stem from an interplay between incident elastic waves and a set of internal resonances. Augmented with damping, such properties are shown to vary in unique and non-trivial ways which can amplify the wave attenuation capacity in certain frequency regions, as well as curtail it at times. Wave dispersion mechanics of dissipative metamaterials in literature are typically investigated in the context of wave propagation where the wavenumber is directly involved in band structure evaluations. In this paper, we show that it is possible to conduct such investigation in the context of structural vibrations and demonstrate how modal analysis of a given finite periodic structure is sufficient to assess the accuracy of the predictions obtained from a representative unit cell. The free wave approach is known to be associated with complex frequency solutions where attenuation due to dissipation takes place only temporally and as a consequence, frequency and damping ratio band structures can be constructed. In an attempt to establish direct correlations between infinite and finite medium predictions, it is shown that a combination of frequency and damping ratio band structures (i.e. frequency and damping ratio vs. wavenumber relations) for a given unit cell leads to a new band structure directly relating the damping ratio to wave frequency, which is capable of replicating the findings obtained from the corresponding finite structure. The different phenomena encountered at the fundamental unit cell level are validated via a series of numerical simulations of metamaterial beams with a damped host medium, damped resonators, as well as combinations thereof. The results reveal that the overall damping level as well as the damping source play key roles in dictating the metadamping emergence in locally resonant metamaterials as they compare to baseline designs including phononic and homogeneous structures of equal size and shape. Finally, the notion of positive and negative metadamping is introduced and phase diagrams are constructed to guide and inform the design of such metamaterials in the presence of viscous and structural losses.