Michael J. Frazier

Metadamping: An emergent phenomenon in dissipative metamaterials

Abstract Using a generalized form of Blochs theorem, we derive the dispersion relation of a viscously damped locally resonant metamaterial modeled as an infinite mass-in-mass lumped parameter chain. For comparison, we obtain the dispersion relation for a statically equivalent Bragg-scattering mass-spring chain that is also viscously damped. For the two chains, we prescribe identical damping levels in the dashpots and compare the damping ratio associated with all propagating Bloch modes. We find that the locally resonant metamaterial exhibits higher dissipation throughout the spectrum which indicates a damping emergence phenomena due to the presence of local resonance. This phenomenon, which we define as “metadamping”, provides a new paradigm for the design of material systems that display both high damping and high stiffness. We conclude our investigation by quantifying the degree of metadamping as a function of the long-wave speed of sound in the medium or the static stiffness.

Band structure of phononic crystals with general damping

In this paper, we present theoretical formalisms for the study of wave dispersion in damped elastic periodic materials. We adopt the well known structural dynamics techniques of modal analysis and state-space transformation and formulate them for the Bloch wave propagation problem. First, we consider a one-dimensional lumped parameter model of a phononic crystal consisting of two masses in the unit cell whereby the masses are connected by springs and dashpot viscous dampers. We then extend our analysis to the study of a two-dimensional phononic crystal, modeled as a dissipative elastic continuum, and consisting of a periodic arrangement of square inclusions distributed in a matrix base material. For our damping model, we consider both proportional damping and general damping. Our results demonstrate the effects of damping on the frequency band structure for various types and levels of damping. In particular, we reveal two intriguing phenomena: branch overtaking and branch cut-off. The former may result in...

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

It is common for dispersion curves of damped periodic materials to be based on real frequencies as a function of complex wavenumbers or, conversely, real wavenumbers as a function of 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 Blochs 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.

Viscous-to-viscoelastic transition in phononic crystal and metamaterial band structures

The dispersive behavior of phononic crystals and locally resonant metamaterials is influenced by the type and degree of damping in the unit cell. Dissipation arising from viscoelastic damping is influenced by the past history of motion because the elastic component of the damping mechanism adds a storage capacity. Following a state-space framework, a Bloch eigenvalue problem incorporating general viscoelastic damping based on the Zener model is constructed. In this approach, the conventional Kelvin-Voigt viscous-damping model is recovered as a special case. In a continuous fashion, the influence of the elastic component of the damping mechanism on the band structure of both a phononic crystal and a metamaterial is examined. While viscous damping generally narrows a band gap, the hereditary nature of the viscoelastic conditions reverses this behavior. In the limit of vanishing heredity, the transition between the two regimes is analyzed. The presented theory also allows increases in modal dissipation enhancement (metadamping) to be quantified as the type of damping transitions from viscoelastic to viscous. In conclusion, it is shown that engineering the dissipation allows one to control the dispersion (large versus small band gaps) and, conversely, engineering the dispersion affects the degree of dissipation (high or low metadamping).

Damped Phononic Crystals and Acoustic Metamaterials

The objective of this chapter is to introduce the topic of damping in the context of both its modeling and its effects in phononic crystals and acoustic metamaterials. First, we provide a brief discussion on the modeling of damping in structural dynamic systems in general with a focus on viscous and viscoelastic types of damping (Sect. 6.2) and follow with a non-exhaustive literature review of prior work that examined periodic phononic materials with damping (Sect. 6.3). In Sect. 6.4, we consider damped 1D diatomic phononic crystals and acoustic metamaterials as example problems (keeping our attention on 1D systems for ease of exposition as in previous chapters). We introduce the generalized form of Bloch’s theorem, which is needed to account for both temporal and spatial attenuation of the elastic waves resulting from the presence of damping. We also describe the transformation of the governing equations of motion to a state-space representation to facilitate the treatment of the damping term that arises in the emerging eigenvalue problem. Finally, the effects of dissipation (based on the two types of damping models considered) on the frequency and damping ratio band structures are demonstrated by solving the equations developed for a particular choice of material parameters.

Damping and nonlinearity in elastic metamaterials: Treatment and effects

Locally resonant acoustic/elastic metamaterials have been the focus of extensive research efforts in recent years due to their attractive dynamical characteristics, such as the possibility of exhibiting subwavelength bandgaps. In this work, we present rigorous formulations for the treatment of damping (e.g., viscous/viscoelastic) and nonlinearity (e.g., geometric/material) in the analysis of elastic wave propagation in elastic metamaterials. In the damping case, we use a generalized form of Blochs theorem to obtain the dispersion and dissipation factors for freely propagating elastic waves. In the nonlinear case, we combine the standard transfer matrix with an exact formulation we have recently developed for finite-strain elastic waves in a homogeneous medium to obtain the band structure of a 1D elastic metamaterial. Our analysis sheds light on the effects of damping and nonlinearity on the dispersive characteristics in the presence of local resonance.

Metadamping in Dissipative Metamaterials

Using a generalized form of Bloch’s theorem, we derive the dispersion relation of a viscously damped locally resonant metamaterial modeled as an infinite mass-in-mass lumped parameter chain. For comparison, we obtain the dispersion relation for a statically equivalent Bragg-scattering mass-spring chain that is also viscously damped. For the two chains, we prescribe identical damping levels in the dashpots and compare the damping ratio associated with all propagating Bloch modes. We find that the locally resonant metamaterial exhibits higher dissipation throughout the spectrum which indicates a damping emergence phenomena due to the presence of local resonance. This phenomenon, which we define as metadamping, provides a new paradigm for the design of material systems that display both high damping and high stiffness. We conclude our investigation by quantifying the degree of metadamping as a function of the long-wave speed of sound in the medium or the static stiffness.Copyright

Dissipation-triggered phenomena in periodic acoustic metamaterials

In designing a periodic acoustic metamaterial it is possible to have one or more of the constituent material phases to be damped (i.e., lossy/dissipative), for example by using a viscoelastic material such as rubber. The presence of damping results in temporal attenuation of the acoustic/elastic waves as they freely propagate through space in the periodic media. In this work we develop Bloch wave propagation models for damped periodic acoustic metamaterials and study the effects of damping on the dispersion relation. We demonstrate several intriguing phenomena that get triggered due to the presence of inherent dissipation.

Bloch-Theory-Based Analysis of Damped Phononic Materials

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.© 2011 ASME

Dispersion Relation for Generally Damped Periodic Materials

In this paper, we present a theoretical formalism for the study of damped periodic materials. First we consider a lumped parameter model consisting of two masses in the unit cell whereby the masses are connected by springs and dashpot viscous dampers. We then extend our analysis to the study of a two-dimensional phononic crystal, modeled as an elastic continuum, and consisting of a periodic arrangement of square inclusions distributed in a matrix base material. For our damping model, we consider both proportional damping and general damping. Our results demonstrate the effects that damping have on the dispersion relation and damping ratio band structure.Copyright