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

TL;DR

This paper introduces a generalized Bloch's theorem for 1D damped periodic systems, enabling the calculation of dispersion relations with simultaneously complex frequencies and wavenumbers, capturing all attenuation mechanisms.

Contribution

It develops an algorithm for 1D systems that computes dispersion curves with complex frequencies and wavenumbers, extending Bloch's theorem to include damping effects in both space and time.

Findings

Dispersion band structure describing all attenuation mechanisms.

Application to a viscously damped mass-in-mass metamaterial.

Frequency-dependent effective mass for the damped chain.

Abstract

It is common for dispersion curves of damped periodic materials to be based on real frequencies versus complex wavenumbers or, conversely, real wavenumbers versus complex frequencies. The former condition corresponds to harmonic wave motion where a driving frequency is prescribed and where attenuation due to dissipation takes place only in space alongside spatial attenuation due to Bragg scattering. The latter condition, on the other hand, relates to free wave motion admitting attenuation due to energy loss only in time while spatial attenuation due to Bragg scattering also takes place. Here, we develop an algorithm for 1D systems that provides dispersion curves for damped free wave motion based on frequencies and wavenumbers that are permitted to be simultaneously complex. This represents a generalized application of Bloch's theorem and produces a dispersion band structure that fully describes all attenuation mechanisms, in space and in time. The algorithm is applied to a viscously damped mass-in-mass metamaterial exhibiting local resonance. A frequency-dependent effective mass for this damped infinite chain is also obtained.