Band structure analysis of phononic crystals with imperfect interface layers by the BEM
Introduction
Phononic crystals [1], [2], [3] are typical periodic structures which are composed of one or more elastic or viscoelastic inclusions periodically embedded in a matrix material. Due to their special characteristics of elastic/acoustic band-gaps, they can be used in elastic/acoustic filters, transducers, silent blocks, sound shields, noise barriers, and so on [4], [5], [6]. Early studies on phononic crystals are mostly focused on the perfectly bonded interfaces. But in practical applications, some imperfect interfaces [7, 8] caused by various defects and damages at interfaces may appear. These imperfect interface conditions may have significant effects on the mechanical behaviors of the composite materials and the properties of the interfaces will affect the wave propagation characteristics in the phononic crystals. Hence, the investigation on the interface properties of the phononic crystals is necessary and important.
Generally speaking, the interface can be considered as a special transition layer (called interface or interphase layer) between the matrix and the inclusion, which allows a transmission of the stresses and deformations from one component material to another [9]. Its properties have a great influence on the macroscopic mechanical behavior of the composite materials. When the elastic waves propagate and meet the interfaces, they will be reflected and scattered, which is a complex mutual interaction process. It may not only change the wave propagation direction, but may also cause interface debonding or contact slipping. A lot of studies on composite materials with imperfect interfaces or interphases have been carried out over the past years. Datta et al. [10] studied the imperfect interface with both tangential and normal displacement discontinuities. Paskaramoorthy et al. [11] analyzed the effects of the interface layers on scattering of elastic waves. Hashin [12] used a Taylor expansion method to derive the imperfect interface conditions with the jumps of the interface displacements and tractions and analyzed the plane and cylindrical interfaces as special cases. Huang et al. [13] considered the shear waves propagating along the imperfectly bonded interface of a magnetoelectric composite consisting of piezoelectric (PE) and piezomagnetic (PM) phases. Li and Lee [14] studied the effects of the imperfect interface on the SH waves propagating in a cylindrical piezoelectric sensor. Chaudhary [15] adopted an analytical method to investigate the SH waves propagating in pre-stressed and rotating piezo-composite structure with imperfect interfaces. All these investigations show that the interaction of elastic waves and interfaces or interphases can change the wave propagation characteristics in a composite material evidently. Therefore, the interface or interphase properties in composite materials play an important role.
In the past years, many studies on the band structures and the transmission spectra of the phononic crystals with perfect interfaces have been carried out [16], [17], [18], [19], [20]. However, only few research works about the effects of the interface or interphase properties on the phononic band structures have been performed so far. Xu [21] used the finite difference time-domain (FDTD) method to study the effects of different contact interface modes (weld contact, radial contact, and tangential contact interfaces) on the scalar waves in phononic crystals. By comparisons, it seems that the tangential contact mode would greatly affect the dispersion relations. Lan et al. [22, 23] discussed the influences of the imperfect interface on the dispersion curves and the band-gaps of the periodic laminated piezoelectric composites based on the transfer matrix method. Guo et al. [24] used the transfer matrix method to study the two-dimensional (2D) tessellated piezoelectric phononic crystal, formed by homogeneous piezoelectric and inhomogeneous functionally graded rectangular columns. Liu et al. [25] developed the Mie scattering matrix and computed the band structures considering the interface/surface effects. They found that the phononic crystals with larger scatterers would be more affected by the interface/surface effects. Zhen et al. [26] used the Dirichlet-to-Neumann (DtN) map method to calculate the band structures in nano-scale phononic crystals. They found that the interface/surface effects may help to lower the initial forbidden frequency and widen the band-gaps. Yan et al. [27] developed a meshless radial basis function (RBF) collocation method to calculate the phononic band structures taking account of different interface models and analyzed the effects of various parameter changes on the band structures of the spring-interface model and the three-phase model. Li et al. [28] applied the boundary element method (BEM) to compute the band structures of phononic crystals with sliding interface conditions. Chen et al. [29] studied the band structures of elastic waves in nanoscale periodic piezoelectric/piezomagnetic laminates and analyzed the nanoscale size-effect on the wave propagation in phononic crystals. Zheng et al. [30] computed the band structures of nanoscale phononic crystals by using a meshfree local RBF collocation method (LRBFCM). Yao et al. [31, 32] developed the modified smoothed finite element method (M-SFEM) to analyze the band structures of the two-dimensional (2D) and three-dimensional (3D) phononic crystals. Later on, they applied the method to the analysis of 3D acoustic cavities with impedance boundary conditions and to the band structure calculations of 2D fluid-solid phononic crystals by considering the fluid-solid interactions [33, 34]. Therefore, many different methods have been developed so far for calculating the band structures of 2D and 3D phononic crystals. Among the above mentioned previous investigations, several research works have demonstrated that the imperfect interfaces or interphases may have significant influences on the wave propagation characteristics and the band structures of the phononic crystals, and the interaction between the elastic waves and the imperfect interfaces or interphases in phononic crystals is attracting more and more attention in the scientific community and engineering applications.
Compared with other numerical methods, BEM has certain advantages in accuracy, convergence, computing time, etc., and it can appropriately treat the different interface conditions. Gao et al. [35] developed a BEM combined with the block Sakurai-Sugiura method to compute the band structures of phononic crystals. Li et al. [36, 37] implemented and applied a BEM to calculate the band structures and transmission spectra of two-dimensional (2D) phononic crystals with perfect or sliding interfaces. Zhu et al. [38] computed the band structures of the anti-plane waves in the phononic crystals with the spring-interface based on the BEM. In this paper, a BEM based on our previous works is further extended and applied to calculate the band structures of 2D phononic crystals with three different kinds of imperfect interface layers or interphases (spring-interface model, mass-interface model and spring-mass-interface model), and to analyze the effects of the imperfect interface layers on the dispersion curves and band-gaps of 2D phononic crystals.
The paper is arranged as follows. The anti-plane and in-plane elastic wave equations in the scatterer and the matrix, and three different kinds of imperfect interface models (spring-interface model, mass-interface model and spring-mass-interface model) are presented in Section 2. The boundary integral equations, their discretization and the resulting eigenvalue equations for computing the dispersion curves or band structures of 2D phononic crystals are then formulated in Section 3. Subsequently, some numerical examples are illustrated in Section 4, where the effects of the three different kinds of the imperfect-interface models are analyzed in detail, and the influences of the interface parameters and the mass density ratio are also investigated and discussed. Finally, some main concluding remarks are given in Section 5.
Section snippets
Problem statement
A 2D phononic crystal which is composed of solid scatterers periodically embedded in a solid matrix forming a square lattice is considered. The cross-sections of the scatterers may have arbitrary shapes. The lattice constant is a. It is assumed in this analysis that the interface or interphase layers between the matrix and the scatterers are sufficiently thin, so that they can be considered as idealized interfaces of zero thickness. The interface or interphase layer properties will be described
BEM and eigenvalue formulations for band structure calculations
In this paper, the BEM developed in Ref. [36] is extended to calculate the band structures of phononic crystals with different imperfect interface models. In the following, we will only give a brief description of the method for the sake of brevity.
Numerical results and discussions
Base on the different interface models and the BEM as described in the previous sections, numerical examples are presented in this section to verify the accuracy of the developed numerical method and analyze the influences of the interface imperfections on the band structures. The phononic crystal consists of a square lattice of Wu circular cylinders embedded in the aluminum (Al) matrix. The lattice constant is a = 20mm and the radius of the scatterer is r = 6mm. The parameters of the component
Conclusions
In the present paper, a BEM is implemented and applied to compute the band structures of phononic crystals with different interface models. In particular, the spring-interface, mass-interface and spring-mass-interface models are considered. Both anti-plane and in-plane elastic waves in 2D phononic crystals are investigated. The effects of the different interface models and mass density ratios on the band-gaps are analyzed and discussed. From the present study, the following main conclusions can
Appreciatory remarks in honor and commemoration of professor frank Rizzo
Professor Frank Rizzo passed away in April 2020 and the BEM community lost a pioneer and giant. For many years, Professor Frank Rizzo was an internationally leading expert on boundary integral equation method (BIEM) or boundary element method (BEM). Chuanzeng Zhang had the honor and pleasure to have many opportunities to meet and talk with him, and the last one was at the “NSF Workshop on the BEM: Bridging Education and Industrial Applications” in April 23–26, 2012, University of Minnesota,
Declaration of Competing Interest
None.
Acknowledgments
The authors gratefully acknowledge the financial support by the National Natural Science Foundation of China (Grant Nos. 11872127, 11732005), the German Research Foundation (DFG, ZH 15/27-1), the Joint Sino-German Research Project (Grant No. GZ 1355), and the China Scholarship Council (CSC, No. 201908110115) to support Feng-Lian Li as a Visiting Scholar at the Chair of Structural Mechanics, University of Siegen, Germany.
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