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It is common for dispersion curves of damped periodic materials to be basedon real frequencies versus complex wavenumbers or, conversely, real wavenumbersversus complex frequencies. The former condition corresponds to harmonic wavemotion where a driving frequency is prescribed and where attenuation due todissipation takes place only in space alongside spatial attenuation due toBragg scattering. The latter condition, on the other hand, relates to free wavemotion admitting attenuation due to energy loss only in time while spatialattenuation due to Bragg scattering also takes place. Here, we develop analgorithm for 1D systems that provides dispersion curves for damped free wavemotion based on frequencies and wavenumbers that are permitted to besimultaneously complex. This represents a generalized application of Bloch'stheorem and produces a dispersion band structure that fully describes allattenuation mechanisms, in space and in time. The algorithm is applied to aviscously damped mass-in-mass metamaterial exhibiting local resonance. Afrequency-dependent effective mass for this damped infinite chain is alsoobtained.

comptes rendus physique

Generalized Bloch's theorem for viscous metamaterials: Dispersion and  effective properties based on frequencies and wavenumbers that are  simultaneously complex