Abstract
We have proposed an “exact” strain gradient (SG) continuum model to properly predict the dispersive characteristics of diatomic lattice metamaterials with local and nonlocal interactions. The key enhancement is proposing a wavelength-dependent Taylor expansion to obtain a satisfactory accuracy when the wavelength gets close to the lattice spacing. Such a wavelength-dependent Taylor expansion is applied to the displacement field of the diatomic lattice, resulting in a novel SG model. For various kinds of diatomic lattices, the dispersion diagrams given by the proposed SG model always agree well with those given by the discrete model throughout the first Brillouin zone, manifesting the robustness of the present model. Based on this SG model, we have conducted the following discussions. (I) Both mass and stiffness ratios affect the band gap structures of diatomic lattice metamaterials, which is very helpful for the design of metamaterials. (II) The increase in the SG order can enhance the model performance if the modified Taylor expansion is adopted. Without doing so, the higher-order continuum model can suffer from a stronger instability issue and does not necessarily have a better accuracy. The proposed SG continuum model with the eighth-order truncation is found to be enough to capture the dispersion behaviors all over the first Brillouin zone. (III) The effects of the nonlocal interactions are analyzed. The nonlocal interactions reduce the workable range of the well-known long-wave approximation, causing more local extrema in the dispersive diagrams. The present model can serve as a satisfactory continuum theory when the wavelength gets close to the lattice spacing, i.e., when the long-wave approximation is no longer valid. For the convenience of band gap designs, we have also provided the design space from which one can easily obtain the proper mass and stiffness ratios corresponding to a requested band gap width.
Highlights
During the last few decades, researchers have been exploiting various potential applications of metamaterials[1,2]
With the Taylor expansion series of displacement modified, an exact strain gradient (SG) model has been established for diatomic lattice metamaterials
Some fundamental topics related to the present model have been studied, that is, the influence of different parameter ratios, appropriate truncation of SG orders, and local/nonlocal interactions. (I) The band gap can be designed by adjusting the parameters, and at this moment, the SG continuum model can still effectively predict the band gap characteristics and dispersion behavior of diatomic lattice metamaterials
Summary
During the last few decades, researchers have been exploiting various potential applications of metamaterials[1,2]. The dispersion of one-dimensional (1D) periodic mass-spring system has been investigated as the discrete lattice model, and as the continuum model with microstructures considered. Dispersions of the 1D periodic lattice to be studied here have been a benchmark problem and attracted extensive research interest Even though it can be solved by the discrete model, a robust continuum model capable of giving correct results throughout the whole first Brillouin zone is still lacking[12,30]. We attempt to construct an exact SG continuum model to predict dispersive characteristics of diatomic lattice metamaterials with local and nonlocal interactions throughout the first Brillouin zone.
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