Acoustic metamaterials exhibit unconventional and unusual approaches to control the sub-wavelength elastic/acoustic waves. Locally resonant acoustic metamaterials (LRAMs) attract great attention due to their potential applications as vibration isolation, low-frequency noise insulators, and wave filtering. It offers spectral gaps in a frequency range of two orders of magnitude lower than ones resulting from the Bragg scattering mechanism. Low-frequency band gaps associated with resonant frequencies of structural units were discovered with solid crystals consisting of soft coated hard inclusion embedded in a hard matrix. In this multiphase LRAM, bandgap originates due to negative effective density resulting from dipolar resonances. The presence of single-mode resonance in a multiphase LRAM essentially restricts the broadening of its low-frequency bandgap. Recently proposed single-phase metamaterials that are made of primarily cellular type structure have opened an avenue towards wide-bandgap metamaterials. It is found that the presence of multiple modes of local resonances such as dipolar, monopolar, and quadrupolar resonances has enabled a significant bandgap. However, the emergence of such resonances with respect to their geometric configuration and constituent materials remains majorly unexplored.
In this work, we propose a single-phase acoustic metamaterial having periodically arranged cellular lattice type of divergent shaped star structure and perform a numerical simulation to obtain the bandgaps. We develop the computational framework to simulate the bandgap of the unit cell. Further, utilizing Bloch-Floquet wave analysis for the periodic unit cell and finite element based numerical scheme an eigenvalue problem is formulated to predict the dispersion response within the irreducible Brillouin zone (IBZ). Moreover, we present a parametric study on the bandgaps with various geometrical parameters.  The vibration mode shapes are extracted to find the presence of multiple resonances associated with negative effective mass and modulus. We primarily explore the interplay of dipolar, monopolar, and quadrupolar resonances towards the evolution of low-frequency bandgap. Thus, present findings provide key understanding towards the widening of low-frequency bandgaps for single-phase metamaterials. 

Interplay between pull-in instability and a pitchfork bifurcation in soft dielectric films

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