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

Abstract

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).

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

“…For the ω(µ) approach, imposing freely propagating waves, thus real propagation constants µ, leads to complex frequency solutions ω corresponding to temporally decaying waves. The ω(µ) approach has been developed and applied to analyse the effects of damping on dispersion relations for PCs in [32,34,35] and for LRMs in [21,36]. The µ(ω) approach on the other hand imposes time harmonic wave motion, thus real frequencies ω, and solves the EVP for complex propagation constants µ, corresponding to spatially decaying waves.…”

“…In [21] Hussein and Frazier report a metadamping phenomenon in one-dimensional viscously damped mass-in-mass chain LRMs, leading to considerable amplification of the dissipation compared to their PC counterparts. In [36], they further investigate viscous and viscoelastic damping effects in one-dimensional PCs and LRMs, observing a narrowing of the band gaps for viscous damping, with an inverse behaviour for viscoelastic damping. Andreassen and Jensen [32] report an upward frequency shift of the dispersion curves when structural damping is increased in a two-dimensional PC.…”

Reduced Linear Constrained Elastic and Viscoelastic Homogeneous Cosserat Media as Acoustic Metamaterials

“…It is difficult to make a precise comparison with our case, since we do not consider in this work a discrete structure of metamaterial, as it is done in [46]. A similar model is investigated in [47]. It is also concluded that viscosity contracts existing band gaps, and elasticity enhances them.…”

Anisotropic dissipation in lattice metamaterials

“…For example, viscous or viscoelastic metamaterials possessing internal resonating bodies have been shown to demonstrate enhanced dissipation under certain conditions (i.e., beyond what may be deduced from the rule of mixtures according to Voigt or Reuss rules). [4][5][6] Increased damping capacity based on observed rises in the wavenumber-dependent damping ratios has also been demonstrated using negative-stiffness elements. 7 The ability to enhance dissipation without negatively impacting the overall stiffness (and the load-bearing capacity in general) is beneficial in many applications, particularly when excessive vibrations affect a structure's performance or structural integrity.…”

“…Each damped band diagram shows a small drop in frequency when compared with the undamped model, as is typical with viscous damping. 6,19 Dispersion branches are colored based on the type of displacement polarization, e.g., longitudinal or transverse, that is dominant in the Bloch mode shape. We define the following metric to measure the extent to which a mode shape shows longitudinal or transverse character in its displacement profile,…”