Nonlinear wave propagation in locally dissipative metamaterials via Hamiltonian perturbation approach

Abstract

The cellular microstructure of periodic architected materials can be enriched by local intracellular mechanisms providing innovative distributed functionalities. Specifically, high-performing mechanical metamaterials can be realized by coupling the lowdissipative cellular microstructure with a periodic distribution of tunable damped oscillators, or resonators, vibrating at relatively high amplitudes. The benefit is the actual possibility of combining the design of wavestopping bands with enhanced energy dissipation properties. This paper investigates the nonlinear dispersion properties of an archetypal mechanical metamaterial, represented by a one-dimensional lattice model characterized by a diatomic periodic cell. The intracellular interatomic interactions feature geometric and constitutive nonlinearities, which determine cubic coupling between the lattice and the resonators. The nondissipative part of the coupling can be designed to exhibit a softening or a hardening behavior, by inde-

Free propagation of resonant waves in nonlinear dissipative metamaterials

“…According to a discrete Lagrangian description, a 2-dof model is formulated to govern the damped free dynamics of the periodic cell. The free propagation problem for the nonlinear waves propagating through the dissipative lattice is based on the Floquet-Bloch theory for periodic structures [29].…”

“…This approach aligns with conventional assumptions centred around the presence of weak nonlinearities and minimal or negligible damping effects. In a recent contribution by Fortunati et al [29], the nonlinear dispersion properties of a locally resonant metamaterial endowed with cubic stiffness and damping were described by adopting an extended Hamiltonian perturbation technique, borrowed from the field of Celestial Mechanics. Specifically, a perturbation scheme based on Lie series operators (see, e.g.…”

A multi-bandgap metamaterial with multi-frequency resonators

“…This nonlinear metamaterial concept was recently explored in Shen and Lacarbonara 27,37 where the combined effects of the nonlinearity of the hosting beam and the softening/hardening nonlinearity of the resonators on the stop bands were tackled by an asymptotic approach. The effects of dissipation in a 1D linear elastic medium with a periodic distribution of nonlinear, viscoelastically damped resonators, vibrating at relatively large amplitudes, were predicted using Lie series by Fortunati et al 38…”

“…This nonlinear metamaterial concept was recently explored in Shen and Lacarbonara 27,37 where the combined effects of the nonlinearity of the hosting beam and the softening/hardening nonlinearity of the resonators on the stop bands were tackled by an asymptotic approach. The effects of dissipation in a 1D linear elastic medium with a periodic distribution of nonlinear, viscoelastically damped resonators, vibrating at relatively large amplitudes, were predicted using Lie series by Fortunati et al 38 A novel periodic and aperiodic composite metamaterial was proposed and experimentally investigated Lim et al 39 The metamaterial resonant system, which mimics the idea of locally resonant sonic crystals, 40 consists of a polymeric casing which embeds in its skeleton spherical and cylindrical steel masses; the rigid steel masses enhance the effective mass density, and thus allow the resonant system to generate wide LF bandgaps distributed over a broad frequency range with a bandwidth gap to mid-gap ratio of 181%. Vibration attenuation of the proposed metastructure was demonstrated by performing frequency response analysis both numerically and experimentally.…”

Wave propagation in viscoelastic metamaterials via added-state formulation

“…In this respect, the spectral design of dissipative metamaterials poses serious conceptual and methodological challenges, including for instance (i) the enlargement of the parameter space, extended to embrace the dissipation properties of the dampers or harvesters, (ii) the non-standard (e.g. non-polynomial) nature of the eigenproblem governing the dispersion problem, if some common viscoelastic formulations are adopted, (iii) the complexification of the dispersion relations defining the frequency spectrum, in which the number of frequency-wavevector curves can also exceed the model dimension, due to the presence of pure attenuation branches associated to standing damped waveforms, (iv) the role played by geometrical and constitutive nonlinearities, which may become crucial if the oscillation amplitudes require to be maximized to enlarge the hysteresis cycles or improve the efficiency of the energy conversion [35]. This entire motivating background can be efficiently synthesized by recognizing that a systematic improvement in the description of the linear and nonlinear dissipation phenomena is the milestone for planning future advances in the energetically consistent modelization and spectral design of mechanical metamaterials [3,36].…”