Wave propagation in acoustic metamaterials with locally resonant-type microstructures was investigated. Because of their unusual forms of microstructures, these metamaterials, if represented by classical elastic continuous solids, would exhibit unusual material properties such as negative mass/mass density in certain frequency range. It was found that the range of frequencies that yield negative mass densities actually correspond to a band gap in which no harmonic wave can propagate in the meta-material without attenuation in amplitude. Moreover, the band gap can be moved by altering the local resonance frequency of the microstructure. This metamaterial can give rise to a significant wave attenuation effect near the local resonance frequency, and therefore can be used to block waves from passing the metamaterial. In two-dimensional metamaterials, it was shown that the representative classical elastic solid has an anisotropic effective mass density and that the effective mass density assumes the form of a second order tensor. Thus, the propagation directions of energy and phase are different and the longitudinal wave and shear wave are coupled in general. This unusual frequency-dependent anisotropic mass density characteristic was studied by examining harmonic wave propagations in arbitrary directions in a two-dimensional acoustic metamaterial. It was found that, for example, a pressure wave impinging on an acoustic metamaterial may be stopped directly by designing the gap frequency. Or alternatively, the impinging pressure wave can be converted to a strong shear-dominated wave mode accompanied by a weak extension-dominated wave. Since the shear-dominated wave motion can hardly be transmitted into a fluid-like material and, thus, the fluid-like material behind the metamaterial can remain mostly undisturbed. A specific metamaterial in the form of particulate composites was also considered. In this metamaterial, the resonators were embedded in a continuous elastic matrix. An efficient approach was proposed to derive the effective mass density directly from the properties of the actual matrix and microstructure. One great advantage of this method is that the dynamic behavior of the metacomposite can be fairly simply and accurately predicted by using a classical continuum model without performing any wave propagation analysis of the original metamaterial and only static analyses are needed. In addition to the use of the classical continuum model to represent metamaterials, in this study, a non-classical continuum model, a multi-displacement continuum model or microstructure continuum model, was employed to represent the metamaterial. It was found that the characteristic dynamic behavior of the metamaterial could be described without resorting to the use of negative mass/mass density.
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