Harnessing Buckling to Design Tunable Locally Resonant Acoustic Metamaterials

We report a new class of tunable and switchable acoustic metamaterials comprising resonating units dispersed into an elastic matrix. Each resonator consists of a metallic core connected to the elastomeric matrix through elastic beams, whose buckling is intentionally exploited as a novel and effective approach to control the propagation of elastic waves. We first use numerical analysis to show the evolution of the locally resonant band gap, fully accounting for the effect of nonlinear pre-deformation. Then, we experimentally measure the transmission of vibrations as a function of the applied loading in a finite-size sample and find excellent agreement with our numerical predictions. The proposed concept expands the ability of existing acoustic metamaterials by enabling tunability over a wide range of frequencies. Furthermore, we demonstrate that in our system the deformation can be exploited to turn on or off the band gap, opening avenues for the design of adaptive switches.

Figure 1

Tunable acoustic metamaterial: (a) The undeformed configuration comprises resonating units dispersed into an elastomeric matrix. Each resonator consists of a metallic mass connected to the matrix through elastic beams, which form a structural coating. The black regions in the picture indicate voids in the structure. The unit cell size is $A_0=50.0\mathrm{mm}$. (b) When a compressive strain $ϵ=-0.10$ is applied in the vertical direction, buckling of the beams significantly alters the effective stiffness of the structural coating, which in turn changes the band gap frequency.

Figure 2

Static response: (a) Distribution of the normalized Von Mises stress, ${\sigma }_{\mathrm{VM}}/{\mu }_0$ at different levels of applied strain $ϵ$. (b) Effect of the applied deformation on the reaction force transmitted by the vertical beams to the matrix. The normalized reaction forces acting both in axial (normal component), $S_{\text{normal}}=R_{\text{normal}}/(t_0{\mu }_0)$, and tangential (shear component), $S_{\text{shear}}=R_{\text{shear}}/(t_0{\mu }_0)$, are reported, $R_{\text{normal}}$ and $R_{\text{shear}}$ denoting the total reaction force measured at the end of the beam in the normal and tangential directions, respectively. (c) Effect of the applied deformation on the force transmitted by the horizontal beams to the matrix.

Figure 3

Effect of the applied deformation on the band structure: Dispersion relations from Bloch-wave analysis for the infinite metamaterial in (a) the undeformed configuration and under uniaxial compressive strain (c) $ϵ=-0.065$ and (e) $ϵ=-0.10$. Shear dominated bands are colored in blue, pressure dominated bands in red and locally rotational bands in black. The grey region in (a) and (c) highlights the band gap induced by local resonance. The Bloch modes of the lowest four bands at high-symmetry points of the Brillouin zone ($X$, $M$, and $Y$) [29] are shown in (b), (d), and (f) for $ϵ=0.0,-0.065$, and $-0.10$, respectively. The distribution of the magnitude of the modal displacement field is shown.

Figure 4

Effect of the applied deformation on the band gap frequency: (a) Experimentally measured transmittance in a sample with $6\times 3$ unit cells (see Fig. 1) at different levels of applied prestrains. (b) Evolution of the resonant band gap frequency as a function of the applied strain, $ϵ$. The markers and error bars represent the lowest transmittance frequency and the $\pm 3\mathrm{dB}$ band gap width measured experimentally. The dashed line indicates the lower edge of the band gap (i.e., frequency of resonance) predicted by the FE simulations for the corresponding infinite system.