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.     

Active control of flexural waves in a phononic crystal beam with staggered periodic properties

The band gaps of a phononic crystal beam with staggered periodic structure are investigated. The periodic system consists of a pure elastic (i.e. PMMA) matrix beam and some piezoelectric (i.e. PZT) patches with coupling between the mechanical–electrical components. The PZT patches connected by negative capacitance circuits are applied to function as the active control system. Based on the condition at the interface between adjacent unit cells, the transfer matrix and localization factor are derived. The influence of the degree of interlacing and negative capacitance circuits are discussed. The numerical results show that another band gap can be generated by the staggered periodic structure of PZT patches. The widths and locations of the band gaps can be changed by the degree of interlacing.     

INFLUENCES OF ANISOTROPY ON BAND GAPS OF 2D PHONONIC CRYSTAL

Band gaps of 2D phononic crystal with orthotropic cylindrical fillers embedded in the isotropic host are studied in this paper. Two kinds of periodic structures, namely, the square lattice and the triangle lattice, are considered. For anisotropic phononic crystal, band gaps not only depend on the periodic lattice but also the angle between the symmetry axis of orthotropic material and that of the periodic structure. Rotating these cylindrical fillers makes the angle changing continuously; as a result, pass bands and forbidden bands of the phononic crystal are changed. The plane wave expansion method is used to reduce the band gap problem to an eigenvalue problem. The numerical example is given for YBCO/Epoxy composites. The location and the width of band gaps are estimated for different rotating angles. The influence of anisotropy on band gaps is discussed based on numerical results.     

一维磁电弹准周期结构带隙特性的研究

采用传递矩阵方法,研究了横波(SV波)垂直入射时压电/（弹性/压磁）和（压电/弹性）/压磁两种Fibonacci准周期结构的频带特性,通过计算局部化因子和位移透射系数,数值揭示了此两种Fibonacci准周期结构频带特性的差异以及与相应理想周期结构频带特性的不同,而且表明（压电/弹性）/压磁Fibonacci准周期结构的频带特性与纯弹性材料Fibonacci准周期结构的频带特性是相似的。     

Novel mechanical behavior of periodic structure with the pattern transformation

In periodic cellular structures, novel pattern transformations are triggered by a reversible elastic instability under the axial compression. Based on the deformation-triggered new pattern, periodic cellular structures can achieve special mechanical properties. In this paper, the designed architecture materials which include elastomer matrixes containing empty holes or filled holes with hydrogel material are modeled and simulated to investigate the mechanical property of the periodic materials. By analyzing the relationship between nominal stress and nominal strain of periodic material, and the corresponding deformed patterns, the influence of geometry and shapes of the holes on the mechanical property of architecture material is studied in more details. We hope this study can provide future perspectives for the deformation-triggered periodic structures.     

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.     

低频弹性波超材料的若干进展

超材料是一类新兴的具有超常物理性质的人造周期/拟周期材料, 能够改变电磁波、声波以及弹性波等在介质中的传播特性. 因在航天、国防以及民用科学等方面的巨大应用潜力, 超材料自被提出后便受到极大的关注并引发研究热潮. 弹性波超材料是超材料的一种, 能够基于弹性波与超材料结构的相互耦合作用实现对弹性波的操控. 带隙是评估弹性波超材料实现弹性波操控的重要工具, 其性质与超材料的材料参数、晶格常数以及局域振子的固有频率相关. 受制于超材料的承载能力、外观尺寸以及局域振子结构等因素, 利用传统超材料开启低频(约100 Hz)弹性波带隙依然存在较大困难. 文章首先简要介绍超材料开启弹性波带隙的基本原理, 然后从低频弹性波超材料基本结构与低频带隙实现方法、低频带隙优化与调控策略、低频带隙潜在应用等三个方面详细总结低频弹性波超材料的研究工作. 其中, 低频带隙超材料的基本结构主要包括布拉格散射型超材料、传统局域共振型超材料以及准零刚度局域共振超材料. 文章通过总结低频弹性波超材料的研究进展, 分析了目前研究中的不足并对未来低频弹性波的研究方向进行了展望.      

Asymptotic and numerical models of waves in tunable multi-scale structures

The paper presents asymptotic models and numerical illustrations of periodic systems which possess band gaps and support standing waves at low frequencies. The structures considered here include periodic systems of defects (cracks or resonators of different types). Tuning mechanisms are described to control the position of band gaps in dispersion diagrams.     

Surface effect on band structure of magneto-elastic phononic crystal nanoplates subject to magnetic and stress loadings

This paper presents a theoretical model for the size-dependent band structure of magneto-elastic phononic crystal(PC) nanoplates according to the Kirchhoff plate theory and Gurtin-Murdoch theory, in which the surface effect and magneto-elastic coupling are considered. By introducing the nonlinear coupling constitutive relation of magnetostrictive materials, Terfenol-D/epoxy PC nanoplates are carried out as an example to investigate the dependence of the band structure on the surface effect, magn...     

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。     

蜂窝材料的弹性波传播特性

通过研究蜂窝材料的弹性波频散关系,分析了其弹性波传播特性. 采用基于小波理论的分析方法,将材料参数和位移均展开为双正交周期小波基函数的形式,利用Bloch定理将波动方程转化为特征值方程,求解该方程得到3种典型结构------正方、三角与六角排列的铝(Al)和聚丙烯(PP)蜂窝材料的频散关系,并进行了比较分析. 结果显示:结构形式的不同显著地影响其波动特性,而制作材料的不同则没有影响;3种结构形式都不存在完全带隙,但正方和三角形结构在一定的传播方向范围内存在方向带隙,而六角形结构则在任何方向都不存在方向带隙;与正方结构相比,三角结构在相同孔隙率下,在更广的传播方向内和更低的频率下,能产生较宽的方向带隙.      

Low-frequency band gaps in chains with attached non-linear oscillators

The aim of this article is to investigate the wave propagation in one-dimensional chains with attached non-linear local oscillators by using analytical and numerical models. The focus is on the influence of non-linearities on the filtering properties of the chain in the low frequency range. Periodic systems with alternating properties exhibit interesting dynamic characteristics that enable them to act as filters. Waves can propagate along them within specific bands of frequencies called pass bands, and attenuate within bands of frequencies called stop bands or band gaps. Stop bands in structures with periodic or random inclusions are located mainly in the high frequency range, as the wavelength has to be comparable with the distance between the alternating parts. Band gaps may also exist in structures with locally attached oscillators. In the linear case the gap is located around the resonant frequency of the oscillators, and thus a stop band can be created in the lower frequency range. In the case with non-linear oscillators the results show that the position of the band gap can be shifted, and the shift depends on the amplitude and the degree of non-linear behaviour.     

Transmission and band gaps of elastic SH waves in functionally graded periodic laminates

Time-harmonic plane elastic SH-waves propagating in periodically laminated composites with functionally graded (FG) interlayers are investigated in this paper. A finite stack of periodic layers between two identical elastic half-planes is considered. Two different power laws are used to describe the property variation of the FG interlayers within the unit-cell. Two different models are developed to deal with the FG interlayers, namely, the explicit FG model and the multilayer model. In conjunction with the transfer matrix method, the wave reflection and transmission coefficients, and band gaps of the FG periodic laminates are computed. Numerical results are presented and discussed to reveal the influences of the FG and homogeneous interlayers, the incidence angle of time-harmonic plane SH wave on the location and width of band gaps. The explicit FG model developed in this study is accurate and capable to simulate the full wave pattern within the periodic laminates, and it can be easily extended to periodic laminates with defects. The corresponding results presented in this paper may have important applications in optimizing and developing novel acoustic devices such as wave filters and noise insulators.     

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.     

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.     

Asymptotic Analysis of High-Contrast Phononic Crystals and a Criterion for the Band-Gap Opening

We investigate the band-gap structure of the frequency spectrum for elastic waves in a high-contrast, two-component periodic elastic medium. We consider two-dimensional phononic crystals consisting of a background medium which is perforated by an array of holes periodic along each of the two orthogonal coordinate axes. In this paper we establish a full asymptotic formula for dispersion relations of phononic band structures as the contrast of the shear modulus and that of the density become large. The main ingredients are integral equation formulations of the solutions to the harmonic oscillatory linear elastic equation and several theorems concerning the characteristic values of meromorphic operator-valued functions in the complex plane, such as the generalized Rouché’s theorem. We establish a connection between the band structures and the Dirichlet eigenvalue problem on the elementary hole. We also provide a criterion for exhibiting gaps in the band structure which shows that smaller the density of the matrix is, the wider the band-gap is, provided that the criterion is fulfilled. This phenomenon was reported by Economou and Sigalas (J Acoust Soc Am 95:1734–1740, 1994) who observed that periodic elastic composites whose matrix has lower density and higher shear modulus compared to those of inclusions yield better open gaps. Our analysis in this paper agrees with this experimental finding.     

Complete band gaps in two-dimensional piezoelectric phononic crystals with {1–3} connectivity family

In this paper, we present results of full band structures for two-dimensional piezoelectric phononic crystals with {1–3} connectivity family. The plane-wave-expansion (PWE) method is applied to the theoretical derivation of secular equations of the two polarization modes: a transverse polarization mode and a mixed (longitudinal-transverse) polarization mode. And the band structures of the two modes for both the case of piezoelectric rods embedded in a polymer matrix and the case of polymer rods embedded in a piezoelectric matrix are calculated for two different cross-sections of the rods, i.e., circular and square, considering the practical fabrication of phononic crystals. We reveal the existence of several very large complete band gaps in a material of practical interest such as PZT rods reinforced polythene composite. The effects of shapes and filling fraction of the rods on band gaps are discussed in detail. The existence of these gaps in relation to the physical parameters of the constituent materials involved is studied. Understanding the band structures of piezoelectric phononic crystals can give some information for improvements in the design of acoustic transducers.     

桁架材料弹性波带隙特性分析

研究了弹性波在周期性桁架材料中的传播特性,并根据桁架材料的周期性特点和杆纵向振动模态,给出了基于单胞的桁架材料弹性波色散(dispersion)方程。分析了1维和2维问题的色散特性,研究了相应的弹性波带隙性质;以CAE分析软件为工具平台对桁架材料的带隙特性进行了数值仿真实验,给出了基于谐响应和特征频率变化特征的仿真实验方法。仿真实验确认了所分析的桁架材料的带隙特性,同时说明所用的仿真实验方法是可行的。     

ELASTIC WAVE LOCALIZATION IN TWO-DIMENSIONAL PHONONIC CRYSTALS WITH ONE-DIMENSIONAL QUASI-PERIODICITY AND RANDOM DISORDER

The band structures of both in-plane and anti-plane elastic waves propagating in two-dimensional ordered and disordered （in one direction） phononic crystals are studied in this paper. The localization of wave propagation due to random disorder is discussed by introducing the concept of the localization factor that is calculated by the plane-wave-based transfer-matrix method. By treating the quasi-periodicity as the deviation from the periodicity in a special way, two kinds of quasi phononic crystal that has quasi-periodicity （Fibonacci sequence） in one direction and translational symmetry in the other direction are considered and the band structures are characterized by using localization factors. The results show that the localization factor is an effective parameter in characterizing the band gaps of two-dimensional perfect, randomly disordered and quasi-periodic phononic crystals. Band structures of the phononic crystals can be tuned by different random disorder or changing quasi-periodic parameters. The quasi phononic crystals exhibit more band gaps with narrower width than the ordered and randomly disordered systems.     

利用结构基因法研究非线性周期性板结构的等效力学性能

非线性周期性板结构是一类在智能复合材料领域具有巨大应用潜力的结构,因其构成材料的非线性特性,以及结构中经常包含增强纤维、肋板和空洞等复杂微结构导致的材料几何非线性,利用常规的有限元方法进行建模和分析较为困难.本文提出了一种结构基因法,通过提取非线性周期性板结构的最小模型单元作为其结构基因,将异质周期性板结构等效为均质板结构,便捷地求解了非线性周期性板结构的微观力学性能和整体等效力学性能.算例表明,结构基因方法可用来分析复杂非线性复合材料结构问题,计算结果精度足够,为复合材料微观力学研究提供了有价值的参考.