Abstract
We study wave dispersion in a one-dimensional nonlinear elastic metamaterial consisting of a thin rod with periodically attached local resonators. Our model is based on an exact finite-strain dispersion relation for a homogeneous solid, utilized in conjunction with the standard transfer matrix method for a periodic medium. The nonlinearity considered stems from large elastic deformation in the thin rod, whereas the metamaterial behavior is associated with the dynamics of the local resonators. We derive an approximate dispersion relation for this system and provide an analytical prediction of band-gap characteristics. The results demonstrate the effect of the nonlinearity on the characteristics of the band structure, including the size, location, and character of the band gaps. For example, large deformation alone may cause a pair of isolated Bragg-scattering and local-resonance band gaps to coalesce. We show that for a wave amplitude on the order of one-eighth of the unit cell size, the effect of the nonlinearity in the structure considered is no longer negligible when the unit-cell size is one-fourteenth of the wavelength or larger.
Highlights
The nonlinearity considered stems from large elastic deformation in the thin rod, whereas the metamaterial behavior is associated with the dynamics of the local resonators
The results demonstrate the effect of the nonlinearity on the characteristics of the band structure, including the size, location, and character of the band gaps
This limitation may be overcome by using locally resonant elastic metamaterials (MMs), introduced by Liu et al.,[4] in which band gaps may open up in the subwavelength regime and do not require the unit cell size to be on the order of the wavelength
Summary
Phononic crystals (PCs) are periodic materials with a spatial modulation of inertial and/or elastic properties. For a given choice of unit-cell geometry and/or type and distribution of constituent materials, PCs can produce absolute band gaps due to Bragg scattering where acoustic/elastic waves are forbidden to propagate.[1,2,3] in order to open band gaps in the low frequency range of up to a few hundred Hertz, the dimension of a periodic structure tends to be too large for a wide range of practical applications This limitation may be overcome by using locally resonant elastic metamaterials (MMs), introduced by Liu et al.,[4] in which band gaps may open up in the subwavelength regime and do not require the unit cell size to be on the order of the wavelength. The conditions for transition between the Bragg scattering and the local resonance hybridization regimes have been investigated in depth.[26,27,28,29]
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