Two-dimensional Phononic Crystals Research Articles

Topological insulator based locally resonant phononic crystals: Wave propagation and acoustic band gaps

ABSTRACTIn the present work the acoustic band structure of a two-dimensional phononic crystal (PC) containing an organic ferroelectric (PVDF- polyvinylidene fluoride) and topological insulator (Bi2Te3) were investigated by the plane-wave-expansion (PWE) method. Two-dimensional PC with square lattices composed of Bi2Te3 cylindrical rods embedded in the PVDF matrix are studied to find the existence of stop bands for the waves of certain energy. Phononic band diagram ω = ω(k) for a 2D PC along the Г-X-M-Г path in the square Brillouin zone show four stop bands in the frequency range 0.01–8.0 kHz.

Phononic band gap and wave propagation on polyvinylidene fluoride-based acoustic metamaterials

In the present work, the acoustic band structure of a two-dimensional phononic crystal (PC) containing an organic ferroelectric (PVDF-polyvinylidene fluoride) and topological insulator (SnTe) was investigated by the plane-wave-expansion (PWE) method. Two-dimensional PC with square lattices composed of SnTe cylindrical rods embedded in the PVDF matrix is studied to find the allowed and stop bands for the waves of certain energy. Phononic band diagram ω = ω(k) for a 2D PC, in which non-dimensional frequencies ωa/2πc (c-velocity of wave) were plotted vs. the wavevector k along the Г–X–M–Г path in the square Brillouin zone shows five stop bands in the frequency range between 10 and 110 kHz. The ferroelectric properties of PVDF and the unusual properties of SnTe as a topological material give us the ability to control the wave propagation through the PC over a wide frequency range of 103–106 Hz. SnTe is a discrete component that allows conducting electricity on its surface but shows insulator properties through its bulk volume. Tin telluride is considered as an acoustic topological insulator as the extension of topological insulators into the field of “topological phononics”.

Boundary element method for calculation of elastic wave transmission in two-dimensional phononic crystals

A boundary element method (BEM) is presented to compute the transmission spectra of two-dimensional (2-D) phononic crystals of a square lattice which are finite along the x-direction and infinite along the y-direction. The cross sections of the scatterers may be circular or square. For a periodic cell, the boundary integral equations of the matrix and the scatterers are formulated. Substituting the periodic boundary conditions and the interface continuity conditions, a linear equation set is formed, from which the elastic wave transmission can be obtained. From the transmission spectra, the band gaps can be identified, which are compared with the band structures of the corresponding infinite systems. It is shown that generally the transmission spectra completely correspond to the band structures. In addition, the accuracy and the efficiency of the boundary element method are analyzed and discussed.

Acoustic band gaps with diffraction gratings in a two-dimensional phononic crystal with a square lattice in water

The present work reports a combined experimental and theoretical study on the acoustic band gaps in a two-dimensional (2D) phononic crystal (PC) consisting of periodic square arrays of stainless-steel cylinders with diameters of 1.0 mm and a lattice constant of 1.5 mm in water. The theoretical band structure of the 2D PC was calculated along the ΓX direction of the first Brillouin zone. The transmission and the reflection coefficients were obtained both experimentally and theoretically along the ΓX direction of the 2D PC. The 2D PC exhibited 5 band gaps at frequencies below 2.0 MHz, with the first Bragg gap being around a frequency of 0.5 MHz. To understand the band gaps in the 2D PC, we calculated the acoustic pressure fields at specific frequencies of interest for normal incidence, and we explained them from the perspective of acoustic diffraction gratings.

Research on the tunability of point defect modes in a two-dimensional magneto-elastic phononic crystal

Point defect modes in a 2D phononic crystal with giant magnetostrictive material tuned by a magnetic field and compressive stress are investigated theoretically in this study. The 3D magnetostrictive constitutive model proposed by Liu and Zheng (2005 Acta Mech. Sin. 21 278–85) is adopted to develop effective elastic, piezomagnetic, and magnetic permeability constants. The finite element method, in combination with a supercell technique, is then applied to obtain the band structures and transmission spectra of the point defect modes in a 2D phononic crystal composed of Terfenol-D rods of circular cross section embedded in a polymethyl methacrylate matrix with a square lattice. The magnetic field not only enlarges the first band gap (FBG) but also opens up a new band gap of XY modes. New point defect modes are simultaneously trapped in the band gaps. The width of the FBG and the frequencies of the point defects of the Z mode decrease as the magnetic field increases.

Expansion of lower-frequency locally resonant band gaps using a double-sided stubbed composite phononic crystals plate with composite stubs

We studied the expansion of locally resonant complete band gaps in two-dimensional phononic crystals (PCs) using a double-sided stubbed composite PC plate with composite stubs. Results show that the introduction of the proposed structure gives rise to a significant expansion of the relative bandwidth by a factor of 1.5 and decreases the opening location of the first complete band gap by a factor of 3 compared to the classic double-sided stubbed PC plate with composite stubs. Furthermore, more band gaps appear in the lower-frequency range (0.006). These phenomena can be attributed to the strong coupling between the “analogous rigid mode” of the stub and the anti-symmetric Lamb modes of the plate. The “analogous rigid mode” of the stub is produced by strengthening the localized resonance effect of the composite plates through the double-sided stubs, and is further strengthened through the introduction of composite stubs. The “analogous rigid mode” of the stubs expands the out-of-plane band gap, which overlaps with in-plane band gap in the lower-frequency range. As a result, the complete band gap is expanded and more complete band gaps appear.

Low-frequency bandgaps of two-dimensional phononic crystal plate composed of asymmetric double-sided cylinder stubs

In this paper, we theoretically investigate the propagation characteristics of Lamb wave in a two-dimensional (2D) asymmetric phononic crystal (PC) plate composed of cylinder stubs of different radius deposited on both sides of a thin homogeneous plate. The dispersion relations, transmission spectra and displacement fields of the eigenmodes are calculated by using the finite element method (FEM). Two complete bandgaps (BGs) can be found in low-frequency range and the transmission spectra coincide with the band structures. We investigate the evolution of dispersion relations with the decrease of the upper stub radius. The physical mechanism of the upper stub radius effect is also studied with the displacement fields of the unit cell. Numerical results show that the symmetry of the stub radius can remarkably influence the band structures and the asymmetric double-sided plate exhibits a new bandgap (BG) in lower frequency range due to the coupling between the lower stub’s resonant mode and the plate’s Lamb mode becomes weak and the adjacent bands separate. Moreover, we further investigate the effect of the stub height on the dispersion relations and find that the BGs shift to lower frequency regions with the increase of the stub height. In addition, the BGs’ sensitivity to the upper stub radius and the stub height is discussed. The low-frequency BGs in the proposed PC plate can potentially be used to control and insulate vibration in low frequency range.

Dirac-like point at the high symmetric M point in a square phononic crystal

Abstract Using the accidental degeneracy of a doubly degenerate state and a single state, a new Dirac-like point was constructed at the high symmetric M point in a two-dimensional phononic crystal (PnC) that consists of a square array of square rods in water. When a plane wave at a frequency near the Dirac-like point impinges on the PnC slab from the left, the spatial phase experiences a minor change in the regions located near the incident interface, but this phase remains uniform in the far field. We also demonstrate two important properties that are correlated to these special field patterns: acoustic cloaking and wavefront reshaping.

Frequency characteristics of defect states in a two-dimensional phononic crystal with slit structure

In this paper, the defect state and band gap characteristics in a two-dimensional slit structure phononic crystal, consisting of slotted steel tubes embedded in an air matrix, are investigated theoretically and experimentally. Using the finite element method and supercell technique, the dispersion relationships and power transmission spectra of the slit structures are calculated. The vibration modes of the band gap edges are analyzed to clarify the mechanism of the generation of the band gaps. Additionally, the influence of the slit width on the band gaps in slit structure is investigated. The slit width was found to influence the band gaps; this is critical to understand for practical applications. Based on this finding, a method to form defect scatterers by changing the slit width of a single central scatterer, or one row of scatterers, in the perfect PC was developed. Defect bands can be induced by creating defects inside the original complete band gaps. The frequency can then be tuned by changing the slit width of defect scatterers. Meanwhile, the relationship between point defect and line defect is investigated. Finally, we verify the results of theoretical research by experiments. These results will help in fabricating devices such as acoustic filters and waveguides whose band frequency can be modulated.

Band structure computation of in-plane elastic waves in 2D phononic crystals by a meshfree local RBF collocation method

In this paper, the band structures of in-plane elastic waves in two-dimensional (2D) phononic crystals are calculated by using a meshfree local radial basis functions (RBF) collocation method. In order to improve the stability of the local RBF collocation method, special techniques suggested in our previous work for anti-plane waves are further improved and extended for calculating the primary field quantities and their normal derivatives required by the treatment of the boundary conditions in the local RBF collocation method for computing the band structures of the in-plane elastic waves in 2D phononic crystals. The developed meshfree local RBF collocation method for the band structure calculations of in-plane elastic waves propagating in 2D phononic crystals is validated by using the corresponding numerical results obtained with the finite element method (FEM). The band structures of different material combinations or acoustic impedance ratios, different filling fractions, various lattice forms and scatterer shapes are computed numerically to show the accuracy and the efficiency of the meshfree local RBF collocation method for computing the band structures of in-plane elastic waves in 2D phononic crystals.

Complete low-frequency bandgap in a two-dimensional phononic crystal with spindle-shaped inclusions**Project supported by the China Scholarship Council.

A two-dimensional phononic crystal (PC) structure possessing a relatively low frequency range of complete bandgap is presented. The structure is composed of periodic spindle-shaped plumbum inclusions in a rubber matrix which forms a square lattice. The dispersion relation, transmission spectrum and displacement field are studied using the finite element method in conjunction with the Bloch theorem. Numerical results show that the present PC structure can achieve a large complete bandgap in a relatively low frequency range compared with two inclusions of different materials, which is useful in low-frequency noise and vibration control and can be designed as a low frequency acoustic filter and waveguides. Moreover, the transmission spectrum and effective mass are evaluated to validate the obtained band structure. It is interesting to see that within the band gap the effective mass becomes negative, resulting in an imaginary wave speed and wave exponential attenuation. Finally, sensitivity analysis of the effect of geometrical parameters of the presented PC structure on the lowest bandgap is performed to investigate the variations of the bandgap width and frequency.

P Matrix Analysis of Surface Acoustic Waves in Piezoelectric Phononic Crystals.

Large time/memory costs have constituted a significant obstacle for accurately analyzing surface acoustic waves (SAWs) in large size two-dimensional (2-D) piezoelectric phononic crystals (PnCs). To overcome this obstacle, this study introduces the unit P matrix and its associated cascading. To obtain an accurate unit P matrix, the Y parameters of the SAW delay lines were derived using a three-dimensional (3-D) finite element model (FEM) with and without 2-D piezoelectric PnCs, respectively, on the transmitting path. A time window function was adopted to extract the desired signals from the P matrix analysis. Then, unit P matrix cascading was used to obtain SAW propagation parameters for the large size piezoelectric PnCs. Using this method, the SAW in aluminum (Al) /128º-YXLiNbO3 PnCs was analyzed over 150 periods. Experiments were also conducted. To choose the appropriate size of the unit P matrix, the variance between experimental results and theoretical results, and time/memory cost were compared for different periods. The results indicate that cascading by unit P matrix of 25 PnCs periods can be appropriately adopted to accurately derive the SAW propagation parameters over 150 periods. This indicates the accuracy of the unit P matrix derived by 3-D FEM and the effectiveness of P matrix analysis.

The sound transmission through tube arrays in power boilers based on phononic crystals theory

The heat exchanger structure in power plants can be treated as two-dimensional phononic crystals. The aim of this paper is to show the sound transmission characteristics through the heat exchanger. The plane wave expansion method is presented to compute the band gaps of the tube arrays which are composed of square or rectangular lattices of solid cylinders in the air matrix. By using the Matlab and Comsol software, and setting the pitch and diameter parameters of the tube arrays and the temperature of circumstance, the sound band-gap diagram and the impact of the sound transmission are got to understand the acoustic signal processing in power station boilers. In simulation, two-dimensional pressure acoustics Application Mode (acpr) is chosen, and finite difference method is used to divide the grid. For the sound vertically incident upon the tube arrays, we can get that the experimental results can be matched with the numerical results quite well. The results obtained through experiments and calculations can tell us that the band gaps are associated with the structure and the temperature. The study can provide new idea and method on theory and application of sound propagation in power station boilers.

Acoustic band structures and time reversal of elastic waves in two- and three-dimensional phononic crystals

The present study aims to investigate experimentally the acoustic band structures of elastic waves in two-dimensional (2D) and three-dimensional (3D) phononic crystals (PCs) consisting of periodic square arrays of cylindrical air holes in a solid matrix made of cubic polyoxymethylene. In the 2D PC, the first band gap appeared at frequencies between 0.50 and 0.77 MHz, and the second pass band at frequencies between 0.77 and 1.17 MHz. On the other hand, the 3D PC exhibited a wider band gap extending from a frequency of 0.44 MHz with noise level transmission at a frequency around 0.93 MHz. The time compressions of the time-reversed waves through the 2D and the 3D PCs were also obtained by using a time reversal process. The 2D PC exhibited better time compression than the 3D PC in the time-reversed waves, consistent with the fact that the time compression achieved by time reversal is sensitive to the number of frequency information grains.

Reduction of thermal conductance by coherent phonon scattering in two-dimensional phononic crystals of different lattice types

The impact of lattice type, period, porosity and thickness of two-dimensional silicon phononic crystals on the reduction of thermal conductance by coherent modification of phonon dispersion is investigated using the theory of elasticity and finite element method. Increase in the period and porosity of the phononic crystal affects the group velocity and phonon density of states and, as a consequence, reduces the in-plane thermal conductance of the structure as compared to unpatterned membrane. This reduction does not depend significantly on the lattice type and thickness of phononic crystals. Moreover, the reduction is strongly temperature dependent and strengthens as the temperature is increased.

Semi-Dirac cone and singular features of two-dimensional three-component phononic crystals

Due to accidental degeneracy, a semi-Dirac point is realized at the center of the Brillouin zone in a two-dimensional phononic crystal (PC) consisting of a square array of core-shell-structure elliptical cylinders in water. In the vicinity of the semi-Dirac point, the dispersion is linear along the X direction, but it is quadratic along the Y direction. The semi-Dirac point is formed by the degeneracy of dipole and quadrupole modes, through accurately adjusting the radius of the cores and shells, the two modes will coincide and the dispersion relation will become linear. It is worth to be emphasised that the frequency of the semi-Dirac point is very low in our designed PC, and this is exactly the special advantage of a three-component system. Since the dispersion relation is different in the vicinity of the semi-Dirac point, some new features may be seen. Firstly, the anisotropic transmission phenomenon is demonstrated. A PC slab is placed in a rectangular waveguide where the sound hard boundary conditions are used on the upper and lower walls; a plane wave impinges on the PC slab along the X direction at the semi-Dirac point frequency, and total transmission can be achieved, so that the sound energy transmissivity is also equal to one. In the meantime, the waves experience no spatial phase changes when they are transmitting through the PC slab; this behavior indicates that the PC can be equivalent to zero index medium along the X direction. However, when the plane wave is incident along the Y direction, the transmitted field is very weak, and the sound energy transmission is nearly zero. Secondly, the properties of the semi-Dirac point can be applied to design acoustic diode. The scatterers of the PC are arranged in triangular prism shapes and placed into a straight waveguide; when the wave is incident along the X direction, it can be transmitted through the PC slab and emerge in the right area, but when the waves is incident from the opposite direction, it will be totally reflected back. Therefore, the semi-Dirac point in PC provides a way to realize the acoustic diode. Finally, the unidirectional wave-front shape effect can also be observed in our considered system. We put a square sample with 16-by-16 coating rods into water medium. When a tightly focused Gaussian beam impinges on the PC sample along the X direction at the semi-Dirac point frequency, the outgoing wave will be modulated to a plan wave. Whereas, when the incident wave along the Y direction, the Gaussian beam will be totally reflected. In conclusion, the singular features of semi-Dirac point in PC will provides an advantageous means to manipulate acoustic waves and exploit new functional materials.

Acoustic band structures in two-dimensional phononic crystals consisting of periodic square arrays of stainless-steel cylinders in water

The present study aims to investigate experimentally and theoretically the acoustic band structures in two-dimensional phononic crystals (PCs) consisting of periodic square arrays of stainlesssteel solid cylinders with diameters of 1.0 mm in water. Two types of PCs with different lattice constants (a), 1.5 and 3.0 mm, were assembled to investigate the influence of structural parameters on their band gaps. The dispersion relations were calculated along the ΓX direction of the first Brillouin zone by using the finite element method. The experimental and the theoretical transmission coefficients were obtained for 2, 4, 6, 8, and 10 layers of cylinders along the ΓX direction. The first band gap of the PC with a = 1.5 mm was found around a frequency of 0.5 MHz while that of the PC with a = 3.0 mm was found around 0.25 MHz. Relatively wide band gaps were observed in the PC with a = 1.5 mm compared to that with a = 3.0 mm. The combination of the two different PCs in tandem with different structural parameters was demonstrated to lead to a superposition of their individual band gaps.

Elastic wave band gaps tuned by configuring radii of rods in two-dimensional phononic crystals with a hybrid square-like lattice

The plane wave expansion (PWE) method is used to calculate the band gaps of two-dimensional (2D) phononic crystals (PCs) with a hybrid square-like (HSL) lattice. Band structures of both XY-mode and Z-mode are calculated. Numerical results show that the band gaps between any two bands could be maximized by altering the radius ratio of the inclusions at different positions. By comparing with square lattice and bathroom lattice, the HSL lattice is more efficient in creating larger gaps.

Evanescent coupling between surface and linear-defect guided modes in phononic crystals

Evanescent coupling between surface and linear-defect waveguide modes in a two-dimensional phononic crystal of steel cylinders in air is numerically demonstrated. When the ratio of scatterer radii to the lattice constant is set to 0.47 in the square phononic crystal, the two types of modes start interacting if there is one-row separation between the surface and waveguide. Supercell band structure computations through the Finite Element Method suggest that the waveguide band is displaced significantly, whereas the surface band remains almost intact when the waveguide and surface are in close proximity. The two resultant hybrid bands are such that the coupling length, which varies between 8 and 22 periods, initially changes linearly with frequency, while a much sharper variation is observed towards the top of the lower hybrid band. Such small values facilitate the design of compact devices based on heterogeneous coupling. Finite-element simulations demonstrate bilateral coupling behaviour, where waves incident from either the surface or waveguide can efficiently couple to the other side. The coupling lengths calculated from simulation results are in agreement with the values predicted from the supercell band structure. The possible utilisation of the coupling scheme in sensing applications, especially in acoustic Doppler velocimetry, is discussed.

Resonant and nonlocal properties of phononic metasolids

We derive a general theory of effective properties in metasolids based on phononic crystals with low frequency resonances. We demonstrate that in general these structures need to be described by means of a frequency-dependent and non-local anisotropic mass density, stiffness tensor and a third- rank coupling tensor, which shows that they behave like a non-local Willis medium. The effect of non-locality and coupling tensor manifest themselves for some particular resonances whereas they become negligible for other resonances. Considering the example of a two-dimensional phononic crystal, consisting of triangular arrangements of cylindrical shells in an elastic matrix, we show that its mass density tensor is strongly resonant and anisotropic presenting both positive and negative divergent values, while becoming scalar in the quasi-static limit. Moreover, it is found that the negative value of transverse component of the mass density is induced by a dipolar resonance, while that of the vertical component is induced by a monopolar one. Finally, the dispersion relation obtained by the effective parameters of the crystal is compared with the band structure, showing a good agreement for the low-wave number region, although the non-local effects are important given the existence of some resonant values of the wave number.