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

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.

On the impact of damping on the dispersion curves of a locally resonant metamaterial: Modelling and experimental validation

“…However, as the materials used are often plastic or rubber-like, measured attenuation performances are often clearly affected by the presence of damping in these LRM constituents and are showing discrepancies with the numerical attenuation predictions. Only a limited number of works investigate the impact of damping on the LRM attenuation performance, often confined to mass-spring-damper lattices/ resonant structures [13,21,22,23,24].…”

Formation of local resonance band gaps in finite acoustic metamaterials: A closed-form transfer function model

“…The AMM consists of a chain of spring-mass unit cells shown in Figure 1. In the presented analysis, damping elements are excluded from both the base and the local structure for two important reasons: (1) to neutralize the effect of dissipation on the band gaps, an effect that has been recently investigated in a number of efforts [35][36][37][38], and (2) to ensure that any damping captured in the numerically computed poles or zeros in the lengthy expressions of the developed dynamic model of the finite AMM are merely a result of minor errors or computational precision, as will be highlighted later in the discussion. The limiting case of the presented approach as the length of the AMM chain approaches infinity matches the traditional Bloch-wave analysis and bridges the gap between the two approaches.…”

A fractional calculus approach to metadamping in phononic crystals and acoustic metamaterials

“…Besides choosing the appropriate damping model, it is important to choose whether the frequency or the wavenumber is set to be real or complex values. Based on this, Frazier and Hussein [21] defined two classes of problems dealing with damped phononic materials. If the frequencies are assumed to be real, the damping effect is manifested in the form of complex wavenumbers defining one class of problems.…”

“…If the frequencies are permitted to be complex and wavenumbers are real, the dissipation effect is represented in the form of temporal attenuation [22]. For example, this class of problems can be physically represented by a free dissipative wave motion in a medium due to impulse loading [21]. The problem considered in this study belongs to the second class of the aforementioned problems with a specified real wavenumber and complex frequency as a solution, where the real part represents the damping factor and the imaginary part is the damped frequency.…”