Tuning of band gaps for a two-dimensional piezoelectric phononic crystal with a rectangular lattice

In this paper, the elastic wave propagation in a two-dimensional piezoelectric phononic crystal is studied by considering the mechanic-electric coupling. The generalized eigenvalue equation is obtained by the relation of the mechanic and electric fields as well as the Bloch-Floquet theorem. The band structures of both the in-plane and anti-plane modes are calculated for a rectangular lattice by the planewave expansion method. The effects of the lattice constant ratio and the piezoelectricity with different filling fractions are analyzed. The results show that the largest gap width is not always obtained for a square lattice. In some situations, a rectangular lattice may generate larger gaps. The band gap characteristics are influenced obviously by the piezoelectricity with the larger lattice constant ratios and the filling fractions.     

Local resonance phononic band gaps in modified two-dimensional lattice materials

In this paper,modified two-dimensional periodic lattice materials with local resonance phononic bandgaps are designed and investigated.The design concept isto introduce some auxiliary structures into conventional periodic lattice materials.Elastic wave propagation in this kindof modified two-dimensional lattice materials is studied using a combination of Bloch’s theorem with finite elementmethod.The calculated frequency band structures of illustrative modified square lattice materials reveal the existenceof frequency band gaps in the low frequency region due tothe introduction of the auxiliary structures.The mechanismunderlying the occurrence of these frequency band gaps isthoroughly discussed and natural resonances of the auxiliarystructures are validated to be the origin.The effect of geometric parameters of the auxiliary structures on the width ofthe local resonance phononic band gaps is explored.Finally,a conceptual broadband vibration-insulating structure basedon the modified lattice materials is designed and its capability is demonstrated.The present work is anticipated to beuseful in designing structures which can insulate mechanicalvibrations within desired frequency ranges.     

Band gaps of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress

In this paper, the stop band properties of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress are studied taking the mechanical and electrical coupling into account. The band gap characteristics for three kinds of lattice arrangements (i.e. sc, bcc and fcc) are investigated by the plane wave expansion (PWE) method. Regarding the variables of mechanical and electrical fields as the elements of the generalized state vector, the expression of the generalized eigenvalue equation for three-dimensional piezoelectric periodic structures is derived. Numerical calculations are performed for the PZT-2/polymer and ZnO/polymer phononic crystals. It can be observed from the results that the fcc lattice is more favorable to create the stop band than the sc and bcc lattices for the piezoelectric phononic crystals, which has also been proved for the pure elastic periodic structures. Compared with the PZT-2/polymer systems, the band gap of the sc lattice for the ZnO/polymer structures is narrower. However, the widths of the bcc and fcc lattices for the ZnO/polymer phononic crystals are much larger than those for the PZT-2/polymer structures. The lattice arrangements and the piezoelectricity have remarkable influences on the stop band behaviors.     

Propagation behavior of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with line defects

Using a stiffness matrix method, we in- vestigate the propagation behaviors of elastic waves in one-dimensional （1D） piezoelectric/piezomagnetic （PE/PM） phononic crystals （PCs） with line defects by calculating energy reflection/transmittion coefficients of quasi-pressure and quasi-shear waves. Line defects are created by the re- placement of PE or PM constituent layer. The defect modes existing in the first gap are considered and the influences on defect modes of the material properties and volume fraction of the defect layers, the type of incident waves, the location of defect layer and the number of structural layers are discussed in detail. Numerical results indicate that defect modes are the most obvious when the defect layers are inserted in the middle of the perfect PCs; the types of incidence wave and material properties of the defect layers have important effects on the numbers, the location of frequencies and the peaks of defect modes, and the defect modes are strongly de- pendent on volume fraction of the defect layers. We hope this paper will be found useful for the design of PE/PM acoustic filters or acoustic transducer with PCs structures.     

圆管型局域共振声子晶体三维构型振动带隙研究

采用多重多级子结构方法计算具有一定刚度的圆管型局域共振声子晶体三维构型振动带隙特性。考察包裹方向对带隙特性的影响,并对第一带隙上下边界点的单胞振动形式进行分析。结果表明,两种包裹形式都可以得到较低较宽的第一带隙,并且带隙特性相似,因而其周期结构都可以大幅减弱带隙范围内弹性波的传播。但两种构型带隙上下边界点对应振动形式不同,此外带隙特性还受单胞尺寸的影响。通过给定评价指标得到相关材料参数与带隙特性关系的相图,由此分析包裹层材料属性对带隙特性的影响。     

Propagation and localization of two-dimensional in-plane elastic waves in randomly disordered layered piezoelectric phononic crystals

Two-dimensional in-plane wave propagation and localization in the disordered layered piezoelectric phononic crystals with material 6 mm are investigated taking the electromechanical coupling into account. The electric field is approximated as quasi-static. The analytical solutions of elastic waves are obtained. The 6 × 6 transfer matrix between two consecutive unit cells is obtained by means of the mechanical and electrical continuity conditions. The expressions of the localization factor and localization length in the disordered periodic structures are presented by regarding the variables of the mechanical and electrical fields as the elements of the state vector. The numerical results of the localization factors and localization lengths are presented for two kinds of disordered piezoelectric phononic crystals, i.e. ZnO–PZT–5H and PVDF–PZT–5H piezocomposites. It is seen from the results that the incident angle of elastic waves and the thickness of the piezoelectric ceramics have significant effects on the wave localization characteristics. For different piezoelectric phononic crystals, the effects of the incident angle are very different. Moreover, with the increase of the disorder degree, the localization phenomenon is strengthened.     

Interfacial effects in electromagnetic coupling within piezoelectric phononic crystals

In this paper, we discuss waves in piezoelectric periodic composite, with the emphasis on the connection between the electromechanical coupling and the effects of dispersion of Bloch-Floquet waves. A particular attention is given to structures containing interfaces between dissimilar media and localization of the electrical fields near such interfaces.     

Mechanically tunable phononic band gaps in three-dimensional periodic elastomeric structures

Three-dimensional periodic structures have many applications in acoustics and their properties are strongly related to structural details. Here we demonstrate through simulations the ability to tune the phononic band gaps of 3D periodic elastomeric structures using deformation. The elastomeric nature of the material makes the transformation of the band gaps a reversible and repeatable process, providing avenues for the design of tunable 3D phononic crystals such as sonic switches.     

Solution of two-dimensional scattering problem in piezoelectric/piezomagnetic media using a polarization method

Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-dimensional piezoelectric/piezo-magnetic "comparison body" is formulated. For simple harmonic motion, kernel of the polarization method reduces to a 2-D time-harmonic Green's function, which is ob-tained using the Radon transform. The expression is further simplified under condi-tions of low frequency of the incident wave and small diameter of the inclusion. Some analytical expressions are obtained. The analytical solutions for generalized piezoelec-tric/piezomagnetic anisotropic composites are given followed by simplified results for piezoelectric composites. Based on the latter results, two numerical results are provided for an elliptical cylindrical inclusion in a PZT-5H-matrix, showing the effect of different factors including size, shape, material properties, and piezoelectricity on the scattering cross-section.     

Crack tip field in piezoelectric/piezomagnetic media

This paper considers the magnetoelectroelastic problem of a crack in a medium possessing coupled piezoelectric, piezomagnetic and magnetoelectric effects. Based on the extended Stroh formalism, the general two-dimensional solutions to the magnetoelectroelastic problem are obtained, involving five analytic functions of different variables. The magnetoelectroelastic field around the crack tip is given. It contains five modes of square root singularities. Expressions of the stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived and the field intensity factors are provided. The path-independent conservative integral is derived. The energy release rate is written in terms of those field intensity factors. The explicit algebraic results are given for a special case of an anti-plane crack in a magnetoelectroelastic medium.     

Folding beam-type piezoelectric phononic crystal with low-frequency and broad band gap

A folding beam-type piezoelectric phononic crystal model is proposed to isolate vibration. Two piezoelectric bimorphs are joined by two masses as a folding structure to comprise each unit cell of the piezoelectric phononic crystal. Each bimorph is connected independently by a resistive-inductive resonant shunting circuit. The folding structure extends the propagation path of elastic waves, while its structure size remains quite small. Propagation of coupled extension-flexural elastic waves is studied by the classical laminated beam theory and transfer matrix method. The theoretical model is further verified with the finite element method(FEM). The effects of geometrical and circuit parameters on the band gaps are analyzed. With only 4 unit cells, the folding beam-type piezoelectric phononic crystal generates two Bragg band gaps of 369 Hz to1 687 Hz and 2 127 Hz to 4 000 Hz. In addition, between these two Bragg band gaps, a locally resonant band gap is induced by resonant shunting circuits. Appropriate circuit parameters are used to join these two Bragg band gaps by the locally resonant band gap.Thus, a low-frequency and broad band gap of 369 Hz to 4 000 Hz is obtained.     

The propagation and localization of Rayleigh waves in disordered piezoelectric phononic crystals

In this paper, the propagation and localization of Rayleigh waves in disordered piezoelectric phononic crystals with material 6 mm are studied taking the electromechanical coupling into account. The electric field is approximated as quasi-static. The analytical solutions of Rayleigh waves are obtained. The 6×6 transfer matrix between two consecutive unit cells is obtained by means of the mechanical and electrical continuity conditions. The expression of the localization factor in disordered periodic structures is presented by regarding the variables of the mechanical and electrical fields as the elements of the state vector. The numerical results for a piezoelectric phononic crystal—PVDF-PZT-2 piezocomposite—are presented and analyzed. From the results we can see that the localization is strengthened with the increase of the disorder degree. The characteristics of the passbands and stopbands are influenced by different ratios of the thickness of the polymers to that of the piezoelectric ceramics. Disorder in elastic constant c₁₁ of PZT-2 can also result in the localization phenomenon. The propagation and localization of Rayleigh waves in piezoelectric phononic crystals may be controlled by properly designing some structural parameters.     

Wave mixing in coupled phononic crystals via a variable stiffness mechanism

We investigate wave mixing effects in a phononic crystal that couples the wave dynamics of two channels – primary and control ones – via a variable stiffness mechanism. We demonstrate analytically and numerically that the wave transmission in the primary channel can be manipulated by the control channel's signal. We show that the application of control waves allows the selection of a specific mode through the primary channel. We also demonstrate that the mixing of two wave modes is possible whereby a modulation effect is observed. A detailed study of the design parameters is also carried out to optimize the switching capabilities of the proposed system. Finally, we verify that the system can fulfill both switching and amplification functionalities, potentially enabling the realization of an acoustic transistor.     

功能梯度压电压磁材料中断裂问题分析

分析了功能梯度压电/压磁材料中裂纹在反平面剪切载荷下的断裂问题. 为了便于分析，假设材料性质沿着裂纹的法线方向呈指数变化. 利用Fourier变换，问题可以转化为对未知数是裂纹表面张开位移的一对对偶积分方程的求解，此对偶积分方程采用Schmidt方法求解. 最后分析了裂纹长度及表征功能梯度材料的参数βl对应力，电位移和磁通量强度因子的影响.     

Wave localization in randomly disordered layered three-component phononic crystals with thermal effects

In this paper, the propagation and localization of elastic waves in randomly disordered layered three-component phononic crystals with thermal effects are studied. The transfer matrix is obtained by applying the continuity conditions between three consecutive sub-cells. Based on the transfer matrix method and Bloch theory, the expressions of the localization factor and dispersion relation are presented. The relation between the localization factors and dispersion curves is discussed. Numerical simulations are performed to investigate the influences of the incident angle on band structures of ordered phononic crystals. For the randomly disordered ones, disorders of structural thickness ratios and Lamé constants are considered. The incident angles, disorder degrees, thickness ratios, Lamé constants and temperature changes have prominent effects on wave localization phenomena in three-component systems. Furthermore, it can be observed that stopbands locate in very low-frequency regions. The localization factor is an effective way to investigate randomly disordered phononic crystals in which the band structure cannot be described.     

Shear wave propagation in periodic phononic/photonic piezoelectric medium

Coupled electro-elastic SH waves propagating oblique to the lamination of a one dimensional piezoelectric periodic structure are considered in the framework of the full system of Maxwell’s electrodynamic equations. The dispersion equation has been obtained and numerical analyses carried out for two kinds of composites both consisting of two different piezoelectric materials. The results demonstrate the significant effect of piezoelectricity on the widths of band gaps at acoustic frequencies and confirm that it does not affect the band gaps at optical frequencies.     

An interface crack with contact zones in a piezoelectric/piezomagnetic bimaterial

An interface crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric/piezomagnetic spaces under the action of a remote mechanical loading, magnetic and electric fluxes as well as concentrated forces at the crack faces is considered. Assuming that all fields are independent on the coordinate x ₂ co-directed with the crack front, the stresses, the electrical and the magnetic fluxes as well as the derivatives of the jumps of the displacements, the electrical and magnetic potentials are presented via a set of analytic functions in the (x ₁, x ₃)-plane with a cut along the crack region. Two cases of magneto-electric conditions at the crack faces are considered. The first case assumes that the crack is electrically and magnetically permeable, and in the second case the crack is assumed electrically permeable while the open part of the crack is magnetically impermeable. For both these cases due to the above-mentioned representation the combined Dirichlet–Riemann boundary value problems have been formulated and solved exactly. Stress, electric and magnetic induction intensity factors are found in a simple analytical form. Transcendental equations and a closed form analytical formula for the determination of the real contact zone length have been derived for both cases of magnetic conditions in the crack region. For a numerical illustration of the obtained results a bimaterial BaTiO₃–CoFe₂O₄ with different volume fractions of BaTiO₃ has been used, and the influence of the mechanical loading and the intensity of the magnetic flux upon the contact zone length and the associated intensity factors as well as the energy release rate has been shown.     

包裹层结构对轻质声子晶体低频振动带隙的影响

基于局域共振机理提出新型轻质声子晶体包裹层结构设计方法。借助有限元法,计算新型声子晶体能带结构、本征模态,分析包裹层总缺口度数一定时不同包裹层缺口数量及布置位置对第一完全带隙截止频率的影响;设计包裹层与散射体连接形式为线连接与点连接两种新型声子晶体模型,分别得到起始频率为37.4Hz及19.0Hz的第一完全带隙,分析第一完全带隙起始频率处散射体本征模态,揭示新型声子晶体极低第一完全带隙起始频率产生机理;进而,与传统面连接型声子晶体通过增加散射体质量降低第一完全带隙起始频率的方法对比。研究结果表明,包裹层总缺口度数一定时,采用缺口数量更多且缺口位置距离连接短板更大的包裹层布置形式能得到更宽的第一完全带隙;提出的包裹层与散射体线连接与点连接型声子晶体在获得极低第一完全带隙起始频率的同时,显著降低了声子晶体质量,突破了传统声子晶体通过增加散射体质量降低第一完全带隙起始频率的限制,为轻质声子晶体获得极低局域共振带隙起始频率的研究设计提供了参考。     

Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique

A multiple monopole （or multipole） method based on the generalized mul- tipole technique （GMT） is proposed to calculate the band structures of scalar waves in two-dimensional phononic crystals which are composed of arbitrarily shaped cylinders embedded in a host medium. In order to find the eigenvalues of the problem, besides the sources used to expand the wave field, an extra monopole source is introduced which acts as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure is obtained. Some numerical examples are presented to validate the proposed method.     

Band structure calculations of in-plane waves in two-dimensional phononic crystals based on generalized multipole technique

A numerical method, the so-called multiple monopole (MMoP) method, based on the generalized multipole technique (GMT) is proposed to calculate the band structures of in-plane waves in two-dimensional phononic crystals, which are composed of arbitrarily shaped cylinders embedded in a solid host medium. To find the eigenvalues (eigenfrequencies) of the problem, besides the sources used to expand the wave fields, an extra monopole source is introduced which acts as the external excitation. By varying the excitation frequency, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and the boundary of the irreducible first Brillouin zone (FBZ), the band structures can be obtained. Some typical numerical examples with different acoustic impedance ratios and with inclusions of various shapes are presented to validate the proposed method.