Shear-horizontal waves in periodic layered nanostructure with both nonlocal and interface effects

The propagation of shear-horizontal (SH) waves in the periodic layered nanocomposite is investigated by using both the nonlocal integral model and the nonlocal differential model with the interface effect. Based on the transfer matrix method and the Bloch theory, the band structures for SH waves with both vertical and oblique incidences to the structure are obtained. It is found that by choosing appropriate interface parameters, the dispersion curves predicted by the nonlocal differential model with the interface effect can be tuned to be the same as those based on the nonlocal integral model. Thus, by propagating the SH waves vertically and obliquely to the periodic layered nanostructure, we could invert, respectively, the interface mass density and the interface shear modulus, by matching the dispersion curves. Examples are further shown on how to determine the interface mass density and the interface shear modulus in theory.     

Wave transmission characteristics in periodic media of finite length: multilayers and fiber arrays

Wave transmission characteristics in elastic media that have periodic microstructure over a finite spatial length are examined theoretically as well as numerically. Two classes of such media are demonstrated, namely, one-dimensional multilayered media with finite-length periodicity and two-dimensional composite media with square arrays of aligned fibers within a finite length. From these one-dimensional and two-dimensional analyses, the influence of the finite-length periodicity on the wave transmission characteristics is discussed. In these media, there are frequency bands (stop bands) where the energy transmission coefficient appears to vanish or takes very low values, while in pass bands it oscillates with the frequency due to the finite-length periodicity. It is theoretically demonstrated in the one-dimensional case of multilayers how the frequency intervals of the oscillation in the transmission spectrum depend on the repeating number of the periodic cells as well as other acoustic and geometrical parameters. The results of the two-dimensional fiber arrays, which are obtained numerically by solving the equations of the SH wave multiple scattering, are shown to fit well in the one-dimensional framework of multilayered structures up to a certain frequency encompassing the first stop band. This similarity between two classes of problems is demonstrated by appropriately identifying the one-dimensional reduced transfer matrix for a single cell that is representative of the periodic fiber array.     

成层饱和介质平面波斜入射问题的一维化时域方法

地震波斜入射下自由场的输入是大型结构抗震分析中亟待解决的问题之一,尤其是成层饱和多孔介质自由场问题,由于问题的复杂性,目前研究甚少. 本文基于Biot提出的饱和多孔介质动力方程,建立了一种新的求解平面波斜入射下基岩上覆饱和多孔介质成层场地自由场分析的一维化时域计算方法. 该方法首先根据Snell定律将饱和多孔介质二维空间问题转化为一维时域问题,通过对深度方向的有限元离散,得到饱和多孔介质波动问题的一维化有限元方程,然后采用单相弹性介质精确人工边界条件模拟基岩半空间的波动辐射和输入特征,通过考虑基岩与饱和多孔介质间透水或不透水边界条件以及不同饱和多孔介质交界面边界条件,形成基岩上覆成层饱和介质系统的整体有限元方程,最后采用中心差分法与Newmark平均加速度近似格式相结合的方法对时间进行离散,得到节点的动力时程的显式表达. 典型场地的地震反应分析表明,本文方法的计算结果与传递矩阵法结合傅里叶变换的计算结果完全吻合,证明了其有效性.      

Shear horizontal wave in multilayered piezoelectric structures: Effect of frequency,incidence angle and constructive parameters

In this paper, the global matrix method is applied to find the behavior of transmission coefficients for shear horizontal (SH) wave propagation with oblique incidence in piezocomposite layered systems. The elemental periodic cells of these piezocomposites are formed by two types of components: a piezoelectric PZT material and an Araldite polymer. The behavior of the transmission coefficient with respect to the frequency, incidence angle and piezoelectric volume fraction is studied. Some numerical calculations for different configurations of the composite are shown. The influence of the bonding films made of polymeric material is also studied. These results could be used to versatile transducers design.     

Homogenization of acoustic waves in strongly heterogeneous porous structures

We consider acoustic waves in fluid-saturated periodic media with dual porosity. At the mesoscopic level, the fluid motion is governed by the Darcy flow model extended by inertia terms and by the mass conservation equation. In this study, assuming the porous skeleton is rigid, the aim is to distinguish the effects of the strong heterogeneity in the permeability coefficients. Using the asymptotic homogenization method we derive macroscopic equations and obtain the dispersion relationship for harmonic waves. The double porosity gives rise to an extra homogenized coefficient of dynamic compressibility which is not obtained in the upscaled single porosity model. Both the single and double porosity models are compared using an example illustrating wave propagation in layered media.     

ON FREE WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA

The method of reverberation-ray matrix （MRRM） is extended and modified for the analysis of free wave propagation in anisotropic layered elastic media. A general, numerically stable formulation is established within the state space framework. The compatibility of physical variables in local dual coordinates gives the phase relation, from which exponentially growing functions are excluded. The interface and boundary conditions lead to the scattering relation, which avoids matrix inversion operation. Numerical examples are given to show the high accuracy of the present MRRM.     

非线性弹性介质中冲击波斜反射的研究

本文讨论了各向同性非线性超弹性介质在平面小应变下的冲击波斜反射问题。给出了本构关系、简单波解和冲击波解,并作为例子求解了入射冲击波在自由面的斜反射问题。     

层状介质中的双Griffith交界裂纹的SH波散射(反平面运动)——近场解

本文研究层状介质中的双Griffith交界裂纹的SH波散射。文中采用积分变换法将双裂纹的弹性波散射化为对偶积分方程,进而将其转化为第一类奇异积分方程组,借助于第一类Chebyshev多项式,给出了奇异积分方程组的解答,并得到了双裂纹的动态应力强度因子的计算公式     

Primary wave transmission in systems of elastic rods with granular interfaces

We study analytically and numerically primary pulse transmission in one dimensional systems of identical linearly elastic non-dispersive rods separated by identical homogeneous granular layers composed of n beads. The beads interact elastically through a strongly (essentially) nonlinear Hertzian contact law. The main challenge in studying pulse transmission in such strongly nonlinear media is to analyze the ‘basic problem’, namely, the dynamical response of a single intermediate granular layer, confined from both ends by barely touching linear elastic rods subject to impulsive excitation of the left rod. The analysis of the basic problem is carried out under two basic assumptions; namely, of sufficiently small duration of the shock excitation applied to the first layer of the system, and of sufficiently small mass of each bead in the granular interface compared to the mass of each rod. In fact, the smallness of the mass of the bead defines the small parameter in the asymptotic analysis of this problem. Both assumptions are reasonable from the point of view of practical applications. In the analysis we focus only in primary pulse propagation, by neglecting secondary pulse reflections caused by wave scattering at each granular interface and considering only the transmission of the main (primary) pulse across the interface to the neighboring elastic rod. Two types of shock excitations are considered. The first corresponds to fixed time duration (but still much smaller compared to the characteristic time of pulse propagation through the length of each rod), whereas the second type corresponds to a pulse duration that depends on the small parameter of the problem. The influence of the number of beads of the granular interface on the primary wave transmission is studied, and it is shown that at granular interfaces with a relatively low number of beads fast time scale oscillations are excited with increasing amplitudes with increasing number of beads. For a larger number of beads, primary pulse transmission is by means of solitary wave trains resulting from the dispersion of the original shock pulse; in that case fast oscillations result due to interference phenomena caused by the scattering of the main pulse at the boundary of the interface. Considering a periodic system of rods we demonstrate significant reduction of the primary pulse when transmitted through a sequence of granular interfaces. This result highlights the efficacy of applying granular interfaces for passive shock mitigation in layered elastic media.     

Intrinsic transfer matrix method and split quaternion formalism for multilayer media

The intrinsic transfer matrix method for 1D longitudinal elastic wave propagation through multilayer media is used to obtain an equivalent split quaternion formalism. Periodic media are analysed in this framework and the presence of a defect is considered. A simple one layer defect and an inversion defect are analysed. A commutative type of split quaternion is identified, which corresponds to defects of the periodic structure that can be placed in any position, the overall acoustic properties of the medium being conserved. Also, a nonperiodic medium composed of such commutative elements has a behaviour independent of the order of elements. Several possible applications in sound wave measurement and processing are outlined. The proposed split quaternion formalism is compact and can make analytical and numerical computations easier.     

层状横观各向同性饱和土的非轴对称动力响应

通过方位角的Fourier变换,将圆柱坐标系下横观各向同性饱和土的Biot非轴对称波动方 程转化为一组一阶常微分方程组. 然后基于径向Hankel变换,建立问题的状态方程;求解状态方程后,得到传递矩阵. 进而利用传递矩阵,结合饱和层状地基的边界条件、排水条件及层间接触和连续条件,求解 了任意震源力作用下层状横观各向同性饱和地基频域动力响应问题. 时域解可通过频率的Fourier积分得到.     

Reflection from a semi-infinite stack of layers using homogenization

A canonical scattering problem is that of a plane wave incident upon a periodic layered medium. Our aim here is to replace the periodic medium by a homogenized counterpart and then to investigate whether this captures the reflection and transmission behaviour accurately at potentially high frequencies.We develop a model based upon high frequency homogenization and compare the reflection coefficients and full fields with the exact solution. For some material properties it is shown that the asymptotic behaviour of the dispersion curves are locally linear near critical frequencies and that low frequency behaviour is replicated at these critical, high, frequencies. The homogenization approach accurately replaces the periodic medium and the precise manner in which this is achieved then opens the way to future numerical implementation of this technique to scattering problems.     

关于传递矩阵法分析饱和成层介质响应问题的讨论

水平成层土体的地震响应分析(自由场分析)是地震工程领域地震波散射问题的前提基础,由于饱和多孔方程的复杂性,以往的研究大多集中于干土情形,对于饱和土情形的研究相对较少.而实际工程中,地下水位以下,土体孔隙中充满流体,应考虑饱和多孔介质模型.基于Biot多孔介质模型,考虑饱和土中固液相对运动引起的衰减,采用Thomson--Haskell传递矩阵方法得到了饱和成层土体在地震波入射情形时的稳态反应,经傅里叶反变换,可得到时域暂态反应.通过SV波从基岩入射至上覆饱和土层的数值算例,验证了该方法的有效性.发现和初步阐明了计算中出现的两类违背因果律(即响应先于输入)的现象:(1)当SV波入射角度大于导致基岩中反射P波为非均匀波的临界角时,会使得计算结果违背因果律.因此,当入射角超过临界角时,非均匀波的表示尚需进一步完善;(2)由于P2波的衰减,当与稳态波衰减有关的渗透率、土层厚度、入射波频率等参数导致衰减系数超过计算机表示精度时,会出现结果违背因果律现象,并据此得到了满足因果律的参数范围,该范围可作为实际计算时的一个上界.该工作为采用传递矩阵法分析水平饱和土层自由场响应提供了指导依据,且地下水位以上可采用干土模型,水位以下采用饱和土模型,更符合实际情形.     

A method of interface inversion in inhomogeneous media

First, an approximate solution of time domain interface scattering field is derived by extending the classical Born approximation in the problem of interface scattering. In accordance with the solution form, a projection density compensation (PDC) inversion method is developed according to the projection slice theorem, which is valid for the cases of inhomogeneous media and wave mode transformation. Finally, in the model of layered media, the calculation algorithm and the simulation inversion comparison results of point defect, crack, and crack on an interface, as well as the experiment method and results in the condition of acoustic wave, are given. The project supported by the National Natural Science Foundation of China (No.19232031) and Chinese Education Commission Science Foundation     

Study on wave localization in disordered periodic layered piezoelectric composite structures

The two-dimensional wave propagation and localization in disordered periodic layered 2-2 piezoelectric composite structures are studied by considering the mechanic-electric coupling. The transfer matrix between two consecutive sub-layers is obtained based on the continuity conditions. Regarding the variables of mechanical and electrical fields as the elements of the state vector, the expression of the localization factors in disordered periodic layered piezoelectric composite structures is derived. Numerical results are presented for two cases—disorder of the thickness of the polymers and disorder of the piezoelectric and elastic constants of the piezoelectric ceramics. The results show that due to the piezoelectric effects, the characteristics of the wave localization in disordered periodic layered piezoelectric composite structures are different from those in disordered periodic layered purely elastic ones. The wave localization is strengthened due to the piezoelectricity. And the larger the piezoelectric constant is, the larger the wave localization factors are. It is found that slight disorder in the piezoelectric or elastic constants of the piezoelectric ceramics can lead to more prominent localization phenomenon.     

Surface electro-elastic shear horizontal waves in a layered structure with a piezoelectric substrate and a hard dielectric layer

The existence and behaviour of surface electro-elastic shear horizontal waves in a layered structure consisting of a piezoelectric substrate of crystal class 6, 4, 6mm, or 4mm mechanically bonded at its upper surface to an elastic dielectric layer and bounded by an adjacent dielectric medium is considered when the shear bulk wave velocity in the elastic layer is greater than or equal to that in the substrate. The dispersion equation for the existence of the surface electro-elastic SH waves with respect to the phase velocity is obtained which includes all the above crystal classes i.e. the surface wave problems related to all these classes are presented in a single mathematical model. The investigation of the solutions of the dispersion equation is carried out and all the possible cases of the behaviour of the surface electro-elastic SH wave depending on the electro-mechanical coefficients of the layered structure are revealed.     

ELASTIC SH WAVE PROPAGATION IN PERIODIC LAYERED COMPOSITES WITH A PERIODIC ARRAY OF INTERFACE CRACKS

The interaction of anti-plane elastic SH waves with a periodic array of interface cracks in a multi-layered periodic medium is analyzed in this paper. A perfect periodic structure without interface cracks is first studied and the transmission displacement coefficient is obtained based on the transfer matrix method in conjunction with the Bloch-Floquet theorem. This is then generalized to a single and periodic distribution of cracks at the center interface and the result is compared with that of perfect periodic cases without interface cracks. The dependence of the transmission displacement coefficient on the frequency of the incident wave, the influences of material combination, crack configuration and incident angle are discussed in detail. Compared with the corresponding perfect periodic structure without interface cracks, a new phenomenon is found in the periodic layered system with a single and periodic array of interface cracks.     

非均匀损伤介质中波传播的数值解

对弹性波在非均匀损伤介质中的传播理论进行了研究。通过将非均匀损伤区域离散成分层均匀的区域,结合相邻区域交界面处的连续条件,推导出了以右行波、左行波为状态向量的波动方程和传递矩阵。对几种非均匀损伤介质中波的传播进行了实例数值计算,并和其解析解的结果进行了比较,讨论了弹性波在非均匀损伤介质中传播的一般性质。     

PROPAGATION BEHAVIORS OF SHEAR HORIZONTAL WAVES IN PIEZOELECTRIC-PIEZOMAGNETIC PERIODICALLY LAYERED STRUCTURES

This paper investigates shear horizontal （SH） waves propagating in a periodically layered structure that consists of piezoelectric （PE） layers perfectly bonded with piezomagnetic （PM） layers alternately. The explicit dispersion relations are derived for the two cases when the propagation directions of SH waves are normal to the interface and parallel to the interface, respectively. The asymptotic expressions for dispersion relations are also given when the wave number is extremely small. Numerical results for stop band effect and phase velocity are presented for a periodic system of alternating BaTiO3 and Terfenol-D layers. The influence of volume fraction on stop band effect and dispersion behaviors is discussed and revealed.     

Scattering of a plane harmonic SH wave by multiple layered inclusions

Scattering of a plane harmonic SH wave by an arbitrary number of layered inclusions in a half-space is investigated by using a direct boundary integral equation method. The inclusions of arbitrary shape and placement are embedded within an elastic half-space. The effects of multiple scattering, the geometry, and the impedance contrast of the materials for layered inclusions and pipes are considered in detail.