Abstract
Bragg band gaps of phononic crystals generally, but not always, open at Brillouin zone boundaries. The commonly accepted explanation stems from the empty lattice model: assuming a small material contrast between the constituents of the unit cell, avoided crossings in the phononic band structure appear at frequencies and wavenumbers corresponding to band intersections; for scalar waves the lowest intersections coincide with boundaries of the first Brillouin zone. However, if a phononic crystal contains elastically anisotropic materials, its overall symmetry is not dictated solely by the lattice symmetry. We construct an empty lattice model for phononic crystals made of isotropic and anisotropic materials, based on their slowness curves. We find that, in the anisotropic case, avoided crossings generally do not appear at the boundaries of traditionally defined Brillouin zones. Furthermore, the Bragg “planes” which give rise to phononic band gaps, are generally not flat planes but curved surfaces. The same is found to be the case for avoided crossings between shear (transverse) and longitudinal bands in the isotropic case.
Highlights
Phononic crystals are periodic functional composites made of two or more materials with different mass densities and elastic constants [1]
A characteristic feature of the phononic crystal band structure is the existence of band gaps, i.e., frequency ranges within which the propagation of acoustic waves is prohibited, either in a specific propagation direction, or in any direction
We have constructed an empty lattice model for elastically anisotropic phononic crystals based on the slowness curves of matrix material, and have predicted the positions of band intersections and Bragg band gaps opening in reciprocal space
Summary
Phononic crystals are periodic functional composites made of two or more materials with different mass densities and elastic constants [1]. For scalar waves in an isotropic medium, the intersections of dispersion bands form planes (referred to as Bragg planes) which bisect reciprocal lattice vectors at right angles. Anisotropic materials (typically natural crystals) are widely used, and phononic crystals made of anisotropic materials have been investigated in a number of studies [1,18,20,21] While these studies were concerned with the analysis of the numerically calculated band structure, it would be desirable to understand the band gap formation in the limit of small periodicity without resorting to numerical computations, on par with the isotropic case. We numerically calculate band structures of phononic crystals with a finite periodic perturbation to demonstrate the formation of band gaps at locations predicted by the empty lattice model
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