Band structures and localization properties of aperiodic layered phononic crystals

The band structures and localization properties of in-plane elastic waves with coupling of longitudinal and transverse modes oblique propagating in aperiodic phononic crystals based on Thue-Morse and Rudin-Shapiro sequences are studied. Using transfer matrix method, the concept of the localization factor is introduced and the correctness is testified through the Rytov dispersion relation. For comparison, the perfect periodic structure and the quasi-periodic Fibonacci system are also considered. In addition, the influences of the random disorder, local resonance, translational and/or mirror symmetries on the band structures of the aperiodic phononic crystals are analyzed in this paper.     

Wave localization in two-dimensional porous phononic crystals with one-dimensional aperiodicity

The localization properties of in-plane elastic waves propagating in two-dimensional porous phononic crystals with one-dimensional aperiodicity are initially analyzed by introducing the concept of the localization factor that is calculated by the plane-wave-based transfer-matrix method in this paper. The band structures characterized by using localization factors are calculated for different phononic crystals by altering matrix material properties and geometric structure parameters. Numerical results show that the effect of matrix material properties on wave localization can be ignored, while the effect of geometric structure parameters is obvious. For comparison, the periodic porous system and Fibonacci system with rigid inclusion are also analyzed. It is found that the band gaps are easily formed in aperiodic porous system, but hard for periodic porous system. Moreover, compared with aperiodic system with rigid inclusion, the wider low-frequency band gaps appear in the aperiodic porous system.     

Elastic wave localization in two-dimensional phononic crystals with one-dimensional random disorder and aperiodicity

The band structures of in-plane elastic waves propagating in two-dimensional phononic crystals with one-dimensional random disorder and aperiodicity are analyzed in this paper. The localization of wave propagation is discussed by introducing the concept of the localization factor, which is calculated by the plane-wave-based transfer-matrix method. By treating the random disorder and aperiodicity as the deviation from the periodicity in a special way, three kinds of aperiodic phononic crystals that have normally distributed random disorder, Thue-Morse and Rudin-Shapiro sequence in one direction and translational symmetry in the other direction are considered and the band structures are characterized using localization factors. Besides, as a special case, we analyze the band gap properties of a periodic planar layered composite containing a periodic array of square inclusions. The transmission coefficients based on eigen-mode matching theory are also calculated and the results show the same behaviors as the localization factor does. In the case of random disorders, the localization degree of the normally distributed random disorder is larger than that of the uniformly distributed random disorder although the eigenstates are both localized no matter what types of random disorders, whereas, for the case of Thue-Morse and Rudin-Shapiro structures, the band structures of Thue-Morse sequence exhibit similarities with the quasi-periodic (Fibonacci) sequence not present in the results of the Rudin-Shapiro sequence.     

Localisation of elastic waves in two-dimensional randomly disordered solid phononic crystals

Combined with the supercell technique, the plane wave expansion method is used to calculate the band structures of the two-dimensional solid–solid phononic crystals with the random disorders in either radius or location of the scatterers. Phononic systems with plumbum scatterers embedded in an epoxy matrix are calculated in detail. The influences of the disorder degree on the band structures for both anti-plane and in-plane wave modes are investigated. It is found that, with increase of the disorder degree, the band gaps become narrower with more flat bands appearing in the gaps. Both displacement distribution and response spectra show that at the flat bands, elastic waves are localised due to the presence of the disorder. Wave localisation is more pronounced at the flat bands near the lower/upper edge for the radius/location disorder. Wave propagation and localisation in a randomly disordered system with a point defect is also studied. The influence of the disorder on the point-defect state is discussed. The results show that the disorder can tune the frequencies of the defect states. It is particularly noticed that the double degenerate mode appearing within the gap of the mixed in-plane waves is split up into two separated ones when the random disorder is introduced into the system. Generally, the influence of the disorder is more pronounced for the mixed in-plane modes than the anti-plane modes. The analysis of this paper is relevant to the assessment of the influences of manufacture errors on wave behaviours in phononic crystals as well as the possible control of wave propagation by intentionally introducing disorders into periodic systems.     

Absolute band gaps and waveguiding in free standing and supported phononic crystal slabs

Using the finite element method (FEM), we investigate the existence of absolute band gaps and localized modes associated with a guide in thin films of phononic crystals. Two different structures based on two-dimensional (2D) phononic crystals are considered, namely a free standing plate and a plate deposited on a silicon substrate. The 2D phononic crystal is constituted by a square array of cylindrical holes drilled in an active piezoelectric PZT5A matrix. We demonstrate the existence of absolute band gap in the band structure of the phononic crystal plate and, then, the possibility of guided modes inside a linear defect created by removing one row of air holes. In the case of the supported plate, we show the existence of an absolute forbidden band in the plate modes when the thickness of the substrate significantly exceeds the plate thickness.     

Low frequency phononic band structures in two-dimensional arc-shaped phononic crystals

The low frequency phononic band structures of two-dimensional arc-shaped phononic crystals (APCs) were studied by the transfer matrix method in cylindrical coordinates. The results showed the first phononic band gaps (PBGs) of APCs from zero Hz with low modes. Locally resonant (LR) gaps were obtained with higher-order rotation symmetry, due to LR frequencies corresponding to the speeds of acoustic waves in the materials. These properties can be efficiently used in a structure for low frequencies that are forbidden, or in a device that permits a narrow window of frequencies.     

平面波展开法计算二维磁振子晶体带结构

磁振子晶体是光子晶体或声子晶体在磁性材料领域内的替代品,是近来的一个研究热点. 本文提出了磁振子晶体领域内的一种平面波展开法,其较传统的平面波展开法能节约一半以上的计算时间. 采用此方法,数值计算了由Fe/EuO二种铁磁材料构成的二维磁振子晶体带结构. 数值计算结果表明,在一定的体积填充率下,有自旋波带隙的出现;影响磁振子晶体带隙结构形成的主要因素是有效场中的交换作用场,其他作用场的影响相对很小. 关键词： 磁振子晶体 带隙 平面波展开法     

Band structures of two dimensional solid/air hierarchical phononic crystals

The hierarchical phononic crystals to be considered show a two-order “hierarchical” feature, which consists of square array arranged macroscopic periodic unit cells with each unit cell itself including four sub-units. Propagation of acoustic wave in such two dimensional solid/air phononic crystals is investigated by the finite element method (FEM) with the Bloch theory. Their band structure, wave filtering property, and the physical mechanism responsible for the broadened band gap are explored. The corresponding ordinary phononic crystal without hierarchical feature is used for comparison. Obtained results show that the solid/air hierarchical phononic crystals possess tunable outstanding band gap features, which are favorable for applications such as sound insulation and vibration attenuation.     

局域共振复合单元声子晶体结构的低频带隙特性研究

本文提出了一种新型局域共振复合单元声子晶体结构, 并结合有限元方法对结构的带隙机理及低频共振带隙特性进行了分析和研究. 共振带隙产生的频率位置由所对应的局域共振模态的固有频率决定, 并且带隙宽度与局域共振模态的品质因子及其与基体之间的耦合作用强度有关. 采用局域共振复合单元结构可以实现声子晶体的多重共振, 在低频范围能打开多条共振带隙, 但受到共振单元排列方式的的影响. 由于纵向和横向局域共振模态的简并, 复合单元结构能在200 Hz以下的低频范围打开超过60%宽度的共振带隙, 最低带隙频率低至18 Hz. 这为声子晶体结构获得低频、超低频带隙提供了一种有效的方法. 关键词： 局域共振 低频带隙 复合单元 声子晶体     

Band gap structures for viscoelastic phononic crystals based on numerical and experimental investigation

Many materials used as phononic crystals (PCs) are viscoelastic one. It is believed that viscosity results in damping to attenuate wave propagation, which may help to tune the defect modes or band gaps of viscoelastic phononic crystals. To investigate above phenomenon, firstly, we have extended the application of boundary element method (BEM) to the study of viscoelastic phononic crystals with and without a point defect. A new developed BEM within the framework of Bloch theory can easily deal with viscoelastic phononic crystals with arbitrary shapes of the scatterers. Experimental methods have been put forward based on the self-made viscoelastic phononic crystals. Verified by the experimental results, systematic comprehensive parametric studies on the band structure of viscoelastic phononic crystals with varying factors (final–initial value ratio, relaxation time, volume fraction of scatterers, shapes of scatterers) have been discussed by the numerical simulation. To further address the possibility to change the defect modes, the band structure of viscoelastic phononic crystals with a point defect has been studied based on the numerical and experimental methods. From present research work, it can be found that by adjusting the two viscous parameters combined with considering the effect of volume fraction and shapes, a wider and lower initial forbidden frequency or lower and higher quality factor resonant frequency can be obtained.     

Frequency-dependent localization length of SH-wave in randomly disordered piezoelectric phononic crystals

In this paper, the localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. Numerical results of the localization lengths of SH-wave are presented and discussed in ordered and disordered piezoelectric phononic crystals to identify the different effect degrees for the decay of electrical potential in the polymers and the randomness on the localization level. For the disordered case, disorder in the thickness of the polymers and disorder in the elastic constant of the piezoelectric ceramics are all considered. The results show that some parameters such as the incident angle of elastic wave, the randomness degree and the piezoelectricity of piezoelectric ceramics and so on have pronounced effects on the frequency-dependent localization length.     

Wavelet-based method for computing elastic band gaps of one-dimensional phononic crystals

A wavelet-based method was developed to compute elastic band gaps of one-dimensional phononic crystals. The wave field was expanded in the wavelet basis and an equivalent eigenvalue problem was derived in a matrix form involving the adaptive computation of integrals of the wavelets. The method was then applied to a binary system. For comparison, the elastic band gaps of the same one-dimensional phononic crystals computed with the wavelet method and the well-known plane wave expansion (PWE) method are both presented in this paper. The numerical results of the two methods are in good agreement while the computation costs of the wavelet method are much lower than that of PWE method. In addition, the adaptability of wavelets makes the method possible for efficient band gap computation of more complex phononic structures. Supported by the National Natural Science Foundation of China (Grant No. 10632020)     

一维准周期声子晶体中平面波的传播和局部化

本文研究了平面波在一维准周期声子晶体中的传播,引进局部化因子的概念研究了结构的带隙特性和局部化特征,利用Wolf方法给出了局部化因子的表达式并用传递矩阵法计算了局部化因子,考查了平面波垂直入射和斜入射的情形,并与相应的周期结构及随机失谐结构进行了比较。     

Designing broad phononic band gaps for in-plane modes

Phononic crystals are known as artificial materials that can manipulate the propagation of elastic waves, and one essential feature of phononic crystals is the existence of forbidden frequency range of traveling waves called band gaps. In this paper, we have proposed an easy way to design phononic crystals with large in-plane band gaps. We demonstrated that the gap between two arbitrarily appointed bands of in-plane mode can be formed by employing a certain number of solid or hollow circular rods embedded in a matrix material. Topology optimization has been applied to find the best material distributions within the primitive unit cell with maximal band gap width. Our results reveal that the centroids of optimized rods coincide with the point positions generated by Lloyd's algorithm, which deepens our understandings on the formation mechanism of phononic in-plane band gaps.     

弹性波通过一维复合材料系统的透射性质

提出了不同结构的一维弹性波复合材料系统模型,包括一维周期结构声子晶体、标准Fibonacci准周期结构声子晶体、广义Fibonacci准周期结构声子晶体以及完全无序结构的复合材料系统. 采用模式匹配理论法,数值计算了弹性波通过一维复合材料系统的透射系数. 计算结果表明,利用特殊的准周期结构声子晶体可获得比周期结构声子晶体更宽的带隙范围,准周期结构排列的复合材料系统相当于在周期结构中引入了缺陷体一样,带隙内出现了丰富的局域模式. 对弹性波/声波在复合材料系统中局域态性质的研究有助于弹性波/声波滤波器、导波器 关键词： 弹性波复合材料 局域化     

Wave propagation in one-dimensional solid-fluid quasi-periodic and aperiodic phononic crystals

The propagation of the elastic waves in one-dimensional (1D) solid-fluid quasi-periodic phononic crystals is studied by employing the concept of the localization factor, which is calculated by the transfer matrix method. The solid-fluid interaction effect at the interfaces between the solid and the fluid components is considered. For comparison, the periodic systems and aperiodic Thue-Morse sequence are also analyzed in this paper. The splitting phenomenon of the pass bands and bandgaps are discussed for these 1D solid-fluid systems. At last the influences of the material impedance ratios on the band structures of the 1D solid-fluid quasi-periodic phononic crystals arranged as Fibonacci sequence are discussed.     

Flexural vibration band gaps in thin plates with two-dimensional binary locally resonant structures

The complete flexural vibration band gaps are studied in the thin plates with two-dimensional binary locally resonant structures, i.e. the composite plate consisting of soft rubber cylindrical inclusions periodically placed in a host material. Numerical simulations show that the low-frequency gaps of flexural wave exist in the thin plates. The width of the first gap decreases monotonically as the matrix density increases. The frequency response of the finite periodic thin plates is simulated by the finite element method, which provides attenuations of over 20dB in the frequency range of the band gaps. The findings will be significant in the application of phononic crystals.     

Hypersonic modes in nanophononic semiconductors

Frequency gaps and negative group velocities of hypersonic phonon modes in periodically arranged composite semiconductors are presented. Trends and criteria for phononic gaps are discussed using a variety of atomic-level theoretical approaches. From our calculations, the possibility of achieving semiconductor-based one-dimensional phononic structures is established. We present results of the location and size of gaps, as well as negative group velocities of phonon modes in such structures. In addition to reproducing the results of recent measurements of the locations of the band gaps in the nanosized Si/Si{0.4}Ge{0.6} superlattice, we show that such a system is a true one-dimensional hypersonic phononic crystal.     

Simultaneous complete elastic and electromagnetic band gaps in periodic structures

We study elastic and electromagnetic properties in periodic structures and present “deaf and blind” structures, i.e. materials having simultaneous complete phononic and photonic band gaps, that is, transverse electric (TE) and transverse magnetic (TM) electromagnetic waves, pure shear elastic waves, and mixed shear and dilatation elastic waves, cannot propagate within these structures. These composite materials can control the flow of light and sound at the same time. The existence of complete gaps for electromagnetic and elastic waves can lead to the simultaneous localization of light and sound, a novel phenomena that can have strong influence on photon–phonon interactions. We study the dependence of the simultaneous and complete gaps on material parameters to provide design guidelines on how to create these photonic–phononic crystals. PACS 78.66.Sq; 78.20.Bh; 63.20.-e