Two-dimensional Phononic Crystals Research Articles

Robustly Tuning Bandgaps in Two-Dimensional Soft Phononic Crystals with Criss-Crossed Elliptical Holes

Tuning band gaps in soft materials by post-buckling deformation is becoming an appealing means to manipulate elastic waves. As one of the most favorable topologies, two-dimensional soft structures with circular holes have been extensively studied. Based on the contrarian thinking, this paper starts from the two-dimensional soft structures with criss-crossed elliptical holes, which is close to the post-buckling configuration of soft structures with circular holes, and then proposes to tune the band gaps through elongating or stretching rather than compressing. Influences of the loading magnitude and loading pattern (i.e., uniaxial and biaxial elongations) on the band gaps are studied via the nonlinear finite element simulations. Effects of the geometric parameters (the major-to-minor half-axis ratio and the porosity of the structure) are also discussed. It is shown that, compared with the traditional circular hole case, the band gaps of the unloaded structure with criss-crossed elliptical holes are much richer, and they could be reversely and continuously tuned by tensile loadings. In particular, the deformation is very robust and is insensitive to small geometric imperfections, which is always necessary for triggering the post-buckling deformations. The present work provides a useful reference to the manipulation of elastic waves in periodic structures as well as the design of soft phononic crystals/acoustic devices.

Noise isolation with phononic crystals to enhance fatigue crack growth detection using acoustic emission

The background noise imposes difficulties in monitoring crack activities in real-time monitoring of structures using the acoustic emission (AE) method. Here, a two-dimensional phononic crystal (PC) is designed to enhance fatigue crack growth detection capability of acoustic emission technique in plate-like structures, such that a single AE sensor is sufficient to monitor the crack activity. PCs, also known as acoustic metamaterials, are artificial composite, man-made periodic systems creating new responses through physical constraints in the constituent materials such that certain frequencies are prohibited by local the resonance of the periodic system. The integration of PC design with AE method can be utilized to block unwanted noise signals and enhance the crack detection ability of the AE method. The proposed PC structure to block certain frequencies is numerically studied using COMSOL Multiphysics software through applying two approaches: the unit cell analysis and the transmission loss. It is concluded that the primary reason of the band gap formation through the designed PC structure is the local resonance phenomenon. The final geometry is selected to provide a band gap near 150 kHz, which is a common frequency to monitor crack growth in metallic structures. The stubs forming the PC structure are placed on the compact specimen periodically, except along the crack growth path and sensor position, to block the friction emissions coming from the grip locations. The performance of the designed PC structure is demonstrated experimentally through the fatigue testing of aluminum standard compact tension specimens. The cumulative numbers of hits and AE energy obtained from the specimen integrated with the PC structure demonstrate the successful implementation of noise blocking in the AE testing of detecting active cracks.

Wave Propagations in Metamaterial Based 2D Phononic Crystal: Finite Element Analysis

In the present work, the acoustic band structure of a two-dimensional (2D) phononic crystal (PnC) containing composite material were investigated by the finite element method. Two-dimensional PC with triangular and honeycomb lattices composed of composite cylindrical rods are in the air and liquid matrix. The existence of stop bands are investigated for the waves of certain frequency ranges. This phononic band gap - forbidden frequency range - allows sound to be controlled in many useful ways in structures. These structures can be used as sonic filters, waveguides or resonant cavities. Phononic band diagrams ω=ω(k) for a 2D PnC were plotted versus the wavevector k along the Γ-X-M- Γ path in the first Brillouin zone. The calculated phonon dispersion results indicate the existence of full acoustic modes in the proposed structure along the high symmetry points.

The significance of temperature dependence on the piezoelectric energy harvesting by using a phononic crystal

In this study, an acoustic energy harvester based on a two-dimensional phononic crystal has been constructed. The present structure consists of silicon cylinders in the air background with a polyvinylidene fluoride cylinder as a defect to confine the acoustic energy. The presented energy harvester depends on the piezoelectric effect (using the piezoelectric material polyvinylidene fluoride) that converts the confined acoustic energy to electric energy. The maximum output voltage obtained equals 170 mV. Moreover, the results revealed that the output voltage can be increased with increasing temperature. In addition, the effects of the load resistance and the geometry of the piezoelectric material on the output voltage have been studied theoretically. Based on these results, all previous studies about energy harvesting in phononic structures must take temperature effects into account.

Ultra-low thermal conductivity of two-dimensional phononic crystals in the incoherent regime

Two-dimensional silicon phononic crystals have attracted extensive research interest for thermoelectric applications due to their reproducible low thermal conductivity and sufficiently good electrical properties. For thermoelectric devices in high-temperature environment, the coherent phonon interference is strongly suppressed; therefore phonon transport in the incoherent regime is critically important for manipulating their thermal conductivity. On the basis of perturbation theory, we present herein a novel phonon scattering process from the perspective of bond order imperfections in the surface skin of nanostructures. We incorporate this strongly frequency-dependent scattering rate into the phonon Boltzmann transport equation and reproduce the ultra low thermal conductivity of holey silicon nanostructures. We reveal that the remarkable reduction of thermal conductivity originates not only from the impediment of low-frequency phonons by normal boundary scattering, but also from the severe suppression of high-frequency phonons by surface bond order imperfections scattering. Our theory not only reveals the role of the holey surface on the phonon transport, but also provide a computation tool for thermal conductivity modification in nanostructures through surface engineering.

Band gap structures for 2D phononic crystals with composite scatterer

We investigated the band gap structures in two-dimensional phononic crystals with composite scatterer. The composite scatterers are composed of two materials (Bragg scattering type) or three materials (locally resonance type). The finite element method is used to calculate the band gap structure, eigenmodes and transmission spectrum. The variation of the location and width of band gap are also investigated as a function of material ratio in the scatterer. We have found that the change trends the widest band gap of the two phononic crystals are different as the material ratio changing. In addition to this, there are three complete band gaps at most for the Bragg-scattering-type phononic crystals in the first six bands; however, the locally resonance-type phononic crystals exist only two complete band gap at most in the first six bands. The gap-tuning effect can be controlled by the material ratio in the scatterer.

Electrical detection and analysis of surface acoustic wave in line-defect two-dimensional piezoelectric phononic crystals

Line-defect piezoelectric phononic crystals (PCs) show good potential applications in surface acoustic wave (SAW) MEMS devices for RF communication systems. To analyze the SAW characteristics in line-defect two-dimensional (2D) piezoelectric PCs, optical methods are commonly used. However, the optical instruments are complex and expensive, whereas conventional electrical methods can only measure SAW transmission of the whole device and lack spatial resolution. In this paper, we propose a new electrical experimental method with multiple receiving interdigital transducers (IDTs) to detect the SAW field distribution, in which an array of receiving IDTs of equal aperture was used to receive the SAW. For this new method, SAW delay lines with perfect and line-defect 2D Al/128°YXLiNbO3 piezoelectric PCs on the transmitting path were designed and fabricated. The experimental results showed that the SAW distributed mainly in the line-defect region, which agrees with the theoretical results.

Wave bandgap formation and its evolution in two-dimensional phononic crystals composed of rubber matrix with periodic steel quarter-cylinders

The band structure of a two-dimensional phononic crystal, which is composed of four homogenous steel quarter-cylinders immersed in rubber matrix, is investigated and compared with the traditional steel/rubber crystal by the finite element method (FEM). It is revealed that the frequency can then be tuned by changing the distance between adjacent quarter-cylinders. When the distance is relatively small, the integrality of scatterers makes the inner region inside them almost motionless, so that they can be viewed as a whole at high-frequencies. In the case of relatively larger distance, the interaction between each quarter-cylinder and rubber will introduce some new bandgaps at relatively low-frequencies. Lastly, the point defect states induced by the four quarter-cylinders are revealed. These results will be helpful in fabricating devices, such as vibration insulators and acoustic/elastic filters, whose band frequencies can be manipulated artificially.

A hollow inclusion self-similarity phononic crystal with an ultra-low-frequency bandgap

FEA method is applied in order to investigate the bandgaps of two-dimensional (2D) phononic crystals which are composed with self-similarity hollow inclusions. Transmission spectra together with dispersion relations and displacement fields have been studied with FEA in detail. The simultaneous mechanisms of Bragg scattering enable multiterm bandgaps to be unfolded by the structure that can be effectively shifted by changing the lattice constant, matrix density and degree of self similarity. The smallest bandgap is located in the infrasonic wave range. Our results verify these phononic crystals with self-similar hollow inclusion structure which can change the number, location and width of bandgaps.

Pseudospins and topological edge states in elastic shear waves

We present a new framework to realize topological edge states in elastic shear waves in a two-dimensional (2D) phononic crystal (PC). The PC has a simple structure and is composed of a triangular array of core-shell cylinders embedded in an epoxy background. By tuning the geometrical parameters of the cylinders, band inversion between E1 and E2 eigenstates can be achieved at the Brillouin zone (BZ) center, which signifies a topological phase transition from a trivial PC to a nontrivial PC. An effective Hamiltonian is developed to characterize the topology of the PC around the Γ point, and spin Chern numbers are identified as the appropriate topological invariant. Helical edge states are formed at the interface between topologically distinct PCs, and these edge modes exhibit interesting one-way propagation behaviors with little backscattering. With full-wave simulations, we unambiguously demonstrate the robustness of the edge states against different types of defects, which is due to the nontrivial topology of the system. These unidirectional and robust transport phenomena of elastic shear wave thus offer people a new degree of freedom to control and manipulating elastic waves and are expected to find potential applications in diverse fields.

Acoustic frequency filter based on anisotropic topological phononic crystals

We present a design of acoustic frequency filter based on a two-dimensional anisotropic phononic crystal. The anisotropic band structure exhibits either a directional or a combined (global + directional) bandgap at certain frequency regions, depending on the geometry. When the time-reversal symmetry is broken, it may introduce a topologically nontrivial bandgap. The induced nontrivial bandgap and the original directional bandgap result in various interesting wave propagation behaviors, such as frequency filter. We develop a tight-binding model to characterize the effective Hamiltonian of the system, from which the contribution of anisotropy is explicitly shown. Different from the isotropic cases, the Zeeman-type splitting is not linear and the anisotropic bandgap makes it possible to achieve anisotropic propagation characteristics along different directions and at different frequencies.

Band structure analysis of phononic crystals based on the Chebyshev interval method.

Due to environmental changes and manufacturing errors, uncertainties inherently exist in phononic crystals, especially in the material properties and geometric parameters. To handle the uncertainties with limited information, two different methods are proposed for band structure analysis of phononic crystals with uncertainties. One is the Monte Carlo method (MCM). The main shortcoming of it is the high computational cost. The other is the simplex Chebyshev polynomial expansion (SPCE) method. In addition, the computational efficiency of the SPCE method is much higher. In the SPCE method, the simplex Chebyshev is employed to estimate the band structures of phononic crystals. Meanwhile, the simplified incremental sampling strategy is introduced for the simplex Chebyshev to retain calculation accuracy and improve computational efficiency simultaneously. In the Chebyshev surrogate model, the samples yielded with the MCM are used to calculate the interval ranges of the band structures in phononic crystals. Three numerical examples, including a two-dimensional (2D) Helmholtz resonator phononic crystal, a 2D solid-solid phononic crystal, and a three-dimensional phononic crystal, are introduced to verify the effectiveness and efficiency of the proposed method.

Band Structures Analysis Method of Two-Dimensional Phononic Crystals Using Wavelet-Based Elements

A wavelet-based finite element method (WFEM) is developed to calculate the elastic band structures of two-dimensional phononic crystals (2DPCs), which are composed of square lattices of solid cuboids in a solid matrix. In a unit cell, a new model of band-gap calculation of 2DPCs is constructed using plane elastomechanical elements based on a B-spline wavelet on the interval (BSWI). Substituting the periodic boundary conditions (BCs) and interface conditions, a linear eigenvalue problem dependent on the Bloch wave vector is derived. Numerical examples show that the proposed method performs well for band structure problems when compared with those calculated by traditional FEM. This study also illustrates that filling fractions, material parameters, and incline angles of a 2DPC structure can cause band-gap width and location changes.

Simultaneous multi-band valley-protected topological edge states of shear vertical wave in two-dimensional phononic crystals with veins

The introduction of the concept of valley pseudospin to phononic crystals has made a remarkable topologically protected interface transport of sound, which opens a novel research area referred to as valley Hall topological insulators. Here, we demonstrate the simultaneous multi-band edge states of shear vertical waves in two-dimensional phononic crystals with veins. The multi-band edge states are topologically valley-protected and are obtained by simultaneously gapping multiple Dirac points at K (or K′) under the inversion symmetry breaking. As the relative radius of the two adjacent steel columns varies, the band diagram undergoes a topological transition which can be characterized by topological charge distributions and opposite valley Chern numbers. Subsequently, the vortex chirality of the bulk valley modes is unveiled. With numerical simulations, simultaneous multi-band valley dependent edge states and the associated valley-protected backscattering suppression around the curved waveguide are further demonstrated. Our work could become a promising platform for applications of multi-functional topological acoustic devices.

Broadband All-angle Negative Refraction by Optimized Phononic Crystals

All-angle negative refraction (AANR) of phononic crystals and its frequency range are dependent on mechanical properties of constituent materials and their spatial distribution. So far, it is impossible to achieve the maximum operation frequency range of AANR theoretically. In this paper, we will present a numerical approach for designing a two-dimensional phononic crystal with broadband AANR without negative index. Through analyzing the mechanism of AANR, a topology optimization problem aiming at broadband AANR is established and solved by bi-directional evolutionary structural optimization method. The optimal steel/air phononic crystal exhibits a record AANR range over 20% and its refractive properties and focusing effects are further investigated. The results demonstrate the multifunctionality of a flat phononic slab including superlensing effect near upper AANR frequencies and self-collimation at lower AANR frequencies.

Nonlocal dynamics of dissipative phononic fluids

We describe the nonlocal effective properties of a two-dimensional dissipative phononic crystal made by periodic arrays of rigid and motionless cylinders embedded in a viscothermal fluid such as air. The description is based on a nonlocal theory of sound propagation in stationary random fluid/rigid media that was proposed by Lafarge and Nemati [Wave Motion 50, 1016 (2013)]. This scheme arises from a deep analogy with electromagnetism and a set of physics-based postulates including, particularly, the action-response procedures, whereby the effective density and bulk modulus are determined. Here, we revisit this approach, and clarify further its founding physical principles through presenting it in a unified formulation together with the two-scale asymptotic homogenization theory that is interpreted as the local limit. Strong evidence is provided to show that the validity of the principles and postulates within the nonlocal theory extends to high-frequency bands, well beyond the long-wavelength regime. In particular, we demonstrate that up to the third Brillouin zone including the Bragg scattering, the complex and dispersive phase velocity of the least-attenuated wave in the phononic crystal which is generated by our nonlocal scheme agrees exactly with that reproduced by a direct approach based on the Bloch theorem and multiple scattering method. In high frequencies, the effective wave and its associated parameters are analyzed by treating the phononic crystal as a random medium.

Transmission Calculation of Three-phase Phononic Crystals

Boundary element method (BEM) is employed to compute the transmission of two-dimensional (2D) three-phase phononic crystals. The system is in a square lattice which has finite layers along the x-direction and is infinite along y-direction. The cross sections of the inclusions are circular. In a periodic unit cell, boundary integral equations of the matrix, the coating and the inclusion are given. A linear equation set is formed by substituting the periodic boundary conditions and the interface conditions, from which the transmission response can be obtained. Then the band gaps of the phononic system can be found. The obtained results are basically consistent with the band structures from the corresponding infinite systems. It is shown that the present method can accurately compute the transmission.

Irreducible Brillouin Zone for 2D Phononic Crystals with Different Configurations

Plane-wave expansion method are employed to investigate the effect of point-group symmetry to the irreducible Brillouin zone(BZ) of two-dimensional phononic crystals(PCs). Based on the calculations of four kind configurations with different symmetry, we find that the irreducible BZ will enlarge with the lower of the point-group symmetry of PCs and some of the maximums or minimums of bands are not locate at the periphery of the irreducible BZ. Therefore, the previous calculations of band structure, considering only the periphery of the irreducible BZ, is incomplete for PCs with low point-group symmetry. We analyze this problem and give a guide for further investigations in the paper, which is important to obtain accurate band structure of PCs with low point-group symmetry.

Acoustic radiation and cold point defect in boiler tube arrays based on phononic crystals theory

The structure of heat exchangers is similar to two-dimensional phononic crystals in boilers. In this study, with the given tube array parameters of the heat exchanger, we investigate the effects of sound source location and different frequencies on acoustic radiation. Based on phononic crystals theory, the Plane Wave Expansion method is used to calculate the band gap structure, and the Finite Element method is used to study the acoustic insertion loss and radiation characteristics. In the simulations, constant boundary conditions are imposed on the boundary between the air and steel boards, the radiation boundary conditions are applied to all boundaries of the air domain cell, and the free triangular is used in the simulated domain. When there is tube leakage, a cold point defect exists in the tube arrays. We also study the sound transmission characteristics of the cold point defect at a different location. This study can provide an innovative concept and method for leakage location and sonic soot cleaning in boiler tube arrays.

Acoustic band gaps due to diffraction modes in two-dimensional phononic crystals

In this study, we experimentally and theoretically investigated acoustic band gap control with diffraction modes in two-dimensional (2D) phononic crystals (PCs) consisting of periodic arrays of stainless steel (SS) rods immersed in water. We could classify the acoustic band gaps into two types with diffraction modes in the reflection region, and control the center frequencies of the band gaps by varying the vertical lattice constants. Pressure transmission coefficients and acoustic pressure fields were calculated using the finite element method (FEM), to classify and control the acoustic band gaps. As the vertical lattice constants were varied, the center frequencies of the band gaps, where only normal reflection occurred, were almost constant while those of the band gaps, where additional reflected waves with different propagation directions occurred, decreased with increasing the vertical lattice constants. This work can be used to manipulate acoustic band gap adding, splitting, and shifting.