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

Abstract

Research on phononic and acoustic materials and structures emerged in the recent decade as a result of switching from theoretical physics to applications in various engineering fields. Periodicity is the main characteristic of the phononic medium stemming from periodic material phases, geometry or the boundary condition with wave propagation properties analysed through frequency band structure. To obtain these characteristics, the generalized Bloch theorem is usually applied to obtain the dispersion relations of viscously damped resonant metamaterials. Here we develop a novel analytical approach to analyse the fractionally damped model of phononic crystals and acoustic metamaterials introduced through the fractional-order Kelvin-Voigt and Maxwell damping models. In the numerical study, the results obtained using the proposed models are compared against the elastic cases of the phononic crystal and locally resonant acoustic metamaterial, where significant differences in dispersion curves are identified. We show that the fractional-order Maxwell model is more suitable for describing the dissipation effect throughout the spectrum due to the possibility of fitting both, the order of fractional derivative and the damping parameter.

Fractional damping in a monoatomic chain with cubic nonlinearity

“…Besides the classical approaches to describe energy dissipation in materials, in the literature one can find more general fractional viscoelasticity models used in mechanics and structural dynamics to describe this phenomenon for linear, Rossikhin and Shitikova (2010), as well as for nonlinear problems, Rossikhin and Shitikova (2009). Despite a number of applications of fractional damping models, only a recent work by Cajić et al (2020) provided a detailed study of fractional metadamping in linear metamaterials and phononic crystals.…”