Effect of prestress on phononic band gaps induced by inertial amplification

Abstract

Phononic crystals and elastic metamaterials have recently received significant attention due to their potential for unconventional wave control. Despite this interest, one outstanding issue is that their band diagram is typically fixed, once the structure designed. To overcome this limitation, periodic structures with adaptive elastic properties have recently been proposed for Bragg-and local resonancedriven structures.In this context, we report about the effect of an applied external mechanical load on a periodic structure exhibiting band gaps induced by inertial amplification mechanism. If compared to the cases of Bragg scattering and ordinary local resonant metamaterials, we observe here a more remarkable curve shift, modulated through large but fully reversible compression(stretch) of the unit cell, eventually triggering significant (up to two times) enlargement(reduction) of the width of a specific band gap.An important up(down)-shift of some dispersion branches over specific wavenumber values is also observed, showing that this selective variation may lead to negative group velocities over larger(smaller) wavenumber ranges. In addition, the possibility for a non-monotone trend of the lower limit of the first BG under the same type of external applied prestrain is found and explained through an analytical model, which unequivocally proves that this behaviour derives from the different unit cell effective mass and stiffness variations as the prestrain level increases. These peculiarities derive from the hinge-like behaviour of some regions of the unit cell, which is typical of structures exhibiting the inertial amplification mechanism.The effect of the prestress on the dispersion diagram is investigated through the development of a 2-step calculation method: first, an Updated Lagrangian scheme, including a static geometrically nonlinear analysis of a representative unit cell undergoing the action of an applied external load is derived, and then the Floquet-Bloch decomposition is applied to the linearized equations of the acousto-elasticity for the unit cell in the deformed configuration.Finally, the most evident consequence on the dispersion curves of the application of an external prestress, i.e. the band gap shift with respect to the unloaded structure, is demonstrated through nonlinear transient numerical simulations, clearly proving the capability of the structure to switch from a pass-to a stop-band behaviour over the same frequency range.The results presented herein provide insights in the behaviour of band gaps induced by inertial amplification, and suggest new opportunities for real-time tunable wave manipulation.

Inducing Dispersion Curves With Negative Group Velocity in Inertially Amplified Phononic Crystals Through the Application of an External State of Prestress

“…The band diagrams of the phononic structure can be computed from the eigenvalue problem (the mathematical expressions of the mapping operator implementing the Dirichlet boundary condition, K 3 , K 2 , K 1 and M are not reported in the present paper for the sake of brevity and can be found in Refs. [21,22]):…”

“…This implies poor versatility and adaptability to external variations, often essential for various practical applications. In this context, periodic systems with adaptive elastic properties have been proposed tuning the dispersion diagrams through the piezoelectric effect [12,13], inducing temperature variation [14], or exployting magneto-and light-based approaches [15,17], as well as applying external mechanical loads [18][19][20][21][22]. The majority of the aforementioned investigations focused their attention on the bandgap shift / widening / reduction, whereas here, we report about the possibility of inducing negative group velocity in an isolated branch of the dispersion diagram of an inertially amplified phononic crystals.…”

“…A phononic crystal exhibiting inertial amplification mechanism has been chosen because it has been shown that the large inertial forces generated by amplifying the motion of a mass increases the inertia of the overall system and lowers its resonance frequency, allowing thus for sub-wavelength phenomena, while keeping the structure lightweight [25,26]. As a consequence of this, the inertial amplification allows for a more remarkable curve shifts, if compared to the cases of Bragg scattering [21,22] and / or ordinary local resonant metamaterials. We consider the static deformation induced by the prestress to be in the linear elastic regime so to have a complete reversibility of the phenomena (tunability).…”

“…We here recall the main steps for the band diagram calculation adopting a Bloch-Floquet finite element method which takes into account the effect of the prestress in a formulation called 2-step Updated Lagrangian scheme. Refer to [21,22] for the full description of the approach. Synthesizing, the numerical procedure includes two principal computational steps: 1) a nonlinear static analysis involving large strains and displacements;…”

Broadband subwavelength tunable valley edge states induced by fluid filling acoustic metastructure

“…Mazzotti et al [21] applied compression or tension to the unit cell to lead to continuous shift in the bandgap. Miniaci et al [40] realized the continuously tunable dispersion by employing an external prestress.…”