Mechphononic metamaterials are novel materials artificially fabricated and designed tocontrol, direct, and manipulate mechanical shock waves which would otherwise damagestructural elements through propagation. Independent of its source, a shock wave's amplitudeis a function of frequency in the frequency domain. Locally resonant units can be introducedinto the material to stop the damaging components of the wave from propagating through asolid medium. Thus an artificial micro-structure can be designed to render the otherwisemonolithic material a mechphononic metamaterial. The stiffness and inertial specifications ofthe introduced locally resonant units can be tailored so that a given range of frequencies isfiltered out. This is equivalent to designing mechphononic metamaterials with negativeeffective mass density and/or stiffness, a situation similar to that of designing electromagneticones with negative effective inductance.The present study investigates band structure in two-dimensional (2D) anisotropicmechphononic metamaterials encompassing locally resonant mass-in-mass units buried in amassless soft matrix of a dissimilar material. The 2D lattice problem is formulated and theequations of motion are derived using Hamilton's principle. Only “resemblant anisotropy”has been considered with attenuation-free shock waves. Floquet-Bloch's principle is appliedto the irreducible Brillouin zone in the k-space to extract the band structure. It was found thatthe phenomenon of frequency filtering exists and its extent is contingent upon the valuesassigned to a finite set of parameters. Dimensionless groups have been defined so that thedegree and direction of influence of each parameter on the band structure can be appreciated.
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