Refraction characteristics of phononic crystals

Abstract

Some of the most interesting refraction properties of phononic crystals are revealed by examining the anti-plane shear waves in doubly periodic elastic composites with unit cells containing rectangular and/or elliptical multi-inclusions. The corresponding band-structure, group velocity, and energy-flux vector are calculated using a powerful mixed variational method which accurately and efficiently yields all the field quantities over multiple frequency pass-bands. The background matrix and the inclusions can be anisotropic, each having distinct elastic moduli and mass densities. Equifrequency contours and energy-flux vectors are readily calculated as functions of the wave-vector components. By superimposing the energy-flux vectors on equifrequency contours in the plane of the wave-vector components, and supplementing this with a threedimensional graph of the corresponding frequency surface, a wealth of information is extracted essentially at a glance. This way it is shown that a composite with even a simple square unit cell containing a central circular inclusion can display negative or positive energy and phase-velocity refractions, or simply performs a harmonic vibration (standing wave), depending on the frequency and the wave-vector. Moreover that the same composite when interfaced with a suitable homogeneous solid can display: arXiv:1412.4019v1 [cond-mat.mtrl-sci] 10 Dec 2014 Sia Nemat-Nasser 2 MatLab codes with comments will be provided.

3-D phononic crystals with ultra-wide band gaps

“…The sensitivity of the objective function can now be calculated by differentiating (10). Finally we note that the Heaviside Projection Method (HPM) 52 is used within these formulations to control the minimum length scale of designed features.…”

“…The phononic band-structure 2 results from the periodic modulation of stress waves, and as such has deep similarities with areas like electronic band theory 3 and photonics 4 . These periodic modulations provide for very rich wave-physics and for the potential novel applications such as wave guiding 5 , ultrasound tunneling 6 , acoustic rectification 7 , sound focusing 8 , thermal property tuning 9 , and novel wave refraction applications [10][11][12] (See 13 for a comprehensive review). The definitive characteristic of a phononic crystal which distinguishes it from a homogeneous or randomly heterogeneous media is the existence of a frequency region where wave propagation is prohibited.…”

Anti-plane shear waves in periodic elastic composites: band structure and anomalous wave refraction

“…As can be seen, our method converges very rapidly, yielding essentially the final results for N = 3 (49 terms), whereas the plane-wave expansion has not yet converged even for several hundred terms. As our final illustration, figure 10 shows the equifrequency contours of the first and third pass-bands together with the superimposed energy-flux vectors, which reveal a wealth of physics for this two-dimensional phononic crystal; for a thorough exposition, discussions and several examples, see [43].…”

Maximizing bandgap width and in-plane stiffness of porous phononic plates for tailoring flexural guided waves: Topology optimization and experimental validation

“…Extraordinary features of heterogeneous lattice structures in controlling acoustic and elastodynamic waves have attracted a great deal of research and motivated development of design optimization methods. The destructive interaction of waves within these structures when the wavelength is comparable to the lattice periodicity, enables self-collimation, negative refraction and even total reflection of particular frequencies (Deymier 2011;Lin 2012;Nemat-Nasser 2015a;Park, Ma & Kim 2015). Frequency ranges over which successive in-phase (Bragg) reflection of waves at the interface of periodic heterogeneities causes exponential decay of the wave amplitude are called phononic bandgaps.…”