Emergent subharmonic band gaps in nonlinear locally resonant metamaterials induced by autoparametric resonance

Abstract

With the aim of developing high-performance locally resonant metamaterials, the effect of nonlinear hyperelastic interactions between a rubberlike elastomeric local resonator and the host matrix is investigated. The results reveal a new emergent physical phenomenon not previously reported within the framework of elastoacoustic metamaterials: The appearance of a half subharmonic attenuation zone complementing the local resonance band gap around the fundamental frequency. Evidence of the emergent attenuation zone is provided by direct numerical simulations as well as semianalytical developments via the method of multiple scales. The analyses demonstrate that, in the considered nonlinear locally resonant metamaterial, the combined effects of autoparametric and local resonance induce saturation of the primary wave at certain conditions and, subsequently, promote energy exchange from a primary propagating wave to an evanescent subharmonic wave, giving rise to an additional attenuation zone. This opens new possibilities for the design of passive filtering devices for elastoacoustic waves.

Experimental proof of emergent subharmonic attenuation zones in a nonlinear locally resonant metamaterial

“…As explained in Silva et al 32 , the harmonic excitation at one end of the metamaterial induces wave propagation with the frequency of the harmonic excitation, but also the near-field (evanescent) waves. For locally resonant metamaterials, the resonance of the local resonator opens up a band gap and an associated evanescent wave.…”

“…1 of Supplementary Information), which quantifies the pre-deflection of the beams, it is possible to control the relative magnitude of the quadratic term compared to the cubic one and optimize the geometry accordingly. In particular, as shown in Silva et al 32 , to achieve a half-subharmonic attenuation in the metamaterial under study, the quadratic contribution needs to be dominant over the cubic one.…”

“…To exploit nonlinear phenomena, such as those leading to the appearance of a subharmonic attenuation zone, the resonator springs must be nonlinear. In particular, in Silva et al 32 , it is shown that a quadratic nonlinearity in the relation between the force and the displacement of the resonator mass is required to promote the phenomenon. In practice, this can be achieved through different approaches, e.g.…”

“…For locally resonant metamaterials, the resonance of the local resonator opens up a band gap and an associated evanescent wave. In Silva et al 32 , it was demonstrated that when the local resonance frequency is around half the excitation frequency and a quadratic nonlinear interaction exists between the host and the local resonator, the propagating wave and the half-subharmonic evanescent wave couple strongly with each other for high amplitudes of excitation, leading to parametric excitation of the evanescent wave, and energy transfer from the propagating to the evanescent wave in a similar fashion as it occurs for autoparametric resonance in nonlinear dynamical systems. For the metastructure under concern, energy that was initially imputed as a propagative wave with frequency of 700 Hz is converted to the evanescent wave with a frequency around the half-subharmonic frequency of 350 Hz, which is within the local resonance band gap.…”

“…To this aim, a locally resonant unit cell has been designed exhibiting predominantly quadratic nonlinear coupling between the host and the resonator, which, according to Silva et al 32 , is the prerequisite for promoting the phenomenon under investigation. This has been achieved by exploiting the quadratic geometric nonlinearity arising in arched beams.…”