Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezoelectric media under initial stresses

This paper is devoted to study a problem of reflection and refraction of quasi-longitudinal waves under initial stresses at an interface of two anisotropic piezoelectric media with different properties. One of the two media is aluminum nitride, which is considered the down piezoelectric medium and the above medium is chosen as PZT-5H ceramics. The two piezoelectric media welded are assumed to be anisotropic of a type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). The equations of motion and constitutive relations for the piezoelectric media have been written. Suitable boundary conditions are used to obtain the reflection and refraction coefficients. For an incidence of quasi-longitudinal plane waves, four independent-type amplitude ratios of elastic displacement components for plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves, are shown to exist. Also, it is observed that there exist four dependent amplitude ratios of electric potential, which are proportional to the previous four types. Finally, it is found that the coefficients of reflection and refraction are functions of angle of incidence, elastic constants, piezoelectric potential parameters and the initial stresses. Numerical computations and the results obtained are depicted graphically. In the end, a particular case has been reduced from the present study. This investigation is considered important because the initial stresses in such practical problems are inevitable and may result in frequency shift, a change in the velocity of surface waves and controlling the selectivity of a filter compensation of the devices.     

The influence of the viscosity and the magnetic field on reflection and transmission of waves at interface between magneto-viscoelastic materials

The influence of the viscosity on reflection and refraction of plane shear elastic waves in two magnetized semi-infinite media is investigated. The numerical results for the absolute values of the reflection and refraction coefficients and their relative changes for a particular choice of the media are presented graphically. The relative changes of these coefficients are calculated for two special orientations of the magnetic field. It is found that the absolute values of these coefficients are not only functions of the angle of incidence but they are also functions of both the large primary magnetic field and the viscosity of the media. They also vary with the orientation of the magnetic field. Finally, we show that the results of earlier works could be obtained here as particular cases. Finally, this study is regarded an attempt to accommodate magnetic field in visco-elastic media where reflection and refraction of plane shear waves are considered. Results which are obtained in this investigation useful for practical applications or for understanding some aspects of physical acoustic.     

Influence of imperfect bonding on the reflection and transmission of QP-wave at the interface of two functionally graded piezoelectric materials

This paper studies the reflection and transmission of two dimensional quasi P wave incident at an imperfect interface between two dissimilar Functionally Graded Piezoelectric Materials (FGPM) half-spaces. The imperfect bonding behavior between the two considered half-spaces is described by the interfacial imperfections. The imperfection is characterized by the normal stiffness and tangential stiffness using the linear spring model. These interface parameters (i.e normal stiffness and tangential stiffness) are dependent on the elastic properties of interphase. Secular equations have been derived analytically for both the half-spaces. Different cases of imperfect interfaces namely perfect interface, slip interface, weak bonding interface and unbounded interface have been assumed and discussed. Influence of material gradients on the reflection and transmission coefficients (RTC’s) have been inflicted graphically for all the four considered interface conditions. Further, a comparative study of the RTC’s with respect to the incident angle has been carried out for the different cases of imperfections. The obtained results may be useful for measuring imperfection at the interface and designing of SAW devices.     

垂直入射弹性波在压电体单侧接触界面上的传播特性

弹性波与压电材料接触界面的相互作用问题是工程应用中常见而复杂的问题,入射波足够强会引起界面出现滑移和分离,但滑移和分离的边界未知,边界条件具有非线性特性。通过Fourier分析,将混合边值问题的求解转化为非线性代数方程,利用软件通过迭代修正的方法进行了求解;给出3种状态边界的求解,分析入射波强度、外加应力及电场对界面状态的影响,并对高频谐波的特性进行分析,通过实例对理论推导进行验证,结果显示:入射波强度、外加荷载和电场的大小及摩擦因数均会影响到界面,通过改变这些条件可以控制界面状态,另外检测高频谐波的信号也可以反映界面状态。     

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

Reflection and transmission of quasi-plane waves at the interface of piezoelectric semiconductors with initial stresses

We examine the reflection and transmission phenomena of quasi-longitudinal plane(QP) waves in an AlN-ZnO laminated composite structure. The structure is designed under the influence of the initial stresses in which one carrier piezoelectric semiconductor(PSC) half-space is in welded contact with another PSC half-space.The secular equations in the transversely isotropic PSC material are derived from the general dynamic equation, taking the initial stresses into consideration. It is shown that the incident quasi-longitudinal wave(QP-mode) at the interface generates four types of reflected and transmitted waves, namely, QP wave, quasi-transverse(QSV) wave,electric-acoustic(EA) wave, and carrier plane(CP) wave. The algebraic equations are obtained by imposing the boundary conditions on the common interface of the laminated structure. Reflection and transmission coefficients of waves are obtained by implementing Cramer's rule. Profound impacts of the initial stresses and exterior electric biasing field on the reflection and transmission coefficients of waves are investigated and presented graphically.     

Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers

The transmission and reflection of one-component elastic, acoustic, optical waves on a stack of arbitrary number of different homogeneous layers have been intensively studied in the literature. However, all obtained formulas for the reflection and transmission coefficients are in implicit form. In this paper, we provide the explicit formulas for them. From these formulas we immediately arrive at the explicit formulas for the reflection and transmission coefficients of one-component waves through an FGM layer. Based on the obtained exact formulas, approximate formulas for the reflection and transmission coefficients are established for a stack of thin layers and for a thin FGM layer. It is numerically shown that they are good approximations. Since the obtained formulas are totally explicit, they are useful in evaluating, not only numerically but also analytically, the transmission and reflection coefficients of one-component waves.     

Reflection and transmission of elastic waves at five types of possible interfaces between two dipolar gradient elastic half-spaces

Reflection and transmission of an incident plane wave at five types of possible interfaces between two dipo-lar gradient elastic solids are studied in this paper. First, the explicit expressions of monopolar tractions and dipolar trac-tions are derived from the postulated function of strain energy density. Then, the displacements, the normal derivative of displacements, monopolar tractions, and dipolar tractions are used to create the nontraditional interface conditions. There are five types of possible interfaces based on all possible combinations of the displacements and the normal derivative of displacements. These interfacial conditions with consid-eration of microstructure effects are used to determine the amplitude ratio of the reflection and transmission waves with respect to the incident wave. Further, the energy ratios of the reflection and transmission waves with respect to the incident wave are calculated. Some numerical results of the reflection and transmission coefficients are given in terms of energy flux ratio for five types of possible interfaces. The influences of the five types of possible interfaces on the energy parti-tion between the refection waves and the transmission waves are discussed, and the concept of double channels of energy transfer is first proposed to explain the different influences of five types of interfaces.     

Reflection and transmission of plane waves at an imperfectly bonded interface between piezoelectric and piezomagnetic media

The paper analyzes the reflection and transmission of plane waves at an imperfect interface between piezoelectric (PE) and piezomagnetic (PM) media. The materials are assumed to be transversely isotropic. The linear spring model is used to describe the imperfection of bonding behavior at the interface. According to this model, the properties of the interface can be characterized by the normal and tangential interfacial stiffnesses. Numerial examples are performed for BaTiO₃/CoFe₂O₄ material combination. Four cases, a perfect, slip, normal weak bonding and unbonding interfaces for the coupled quasi-pressure (QP) wave incidence from BaTiO₃ medium are compared in detail. Numerical results of the reflection and transmission coefficients (RTCs) varying with incident angle for different interfacial stiffnesses are presented. Results show that the transmitted QP/reflected QSV waves are the strongest in the perfect/unbonded cases, respectively. The scattered waves in the slip and normal weak bonding cases are between those of the perfect and unbonded cases generally. Critial angles have noticeable effect on the RTCs and energy coefficients for the coupled scattered waves of the perfect and slip cases but have a little/no effect on those of the normal weak bonding/unbonded cases. It is found that the sum of the energy carried by the transmitted/reflected QP and QSV waves is less than unit for the imperfect bonding between BaTiO₃/CoFe₂O₄ solids. These results may provide some useful reference datum for the imperfection measurment at the PE and PM interface.     

Reflection and refraction of plane quasi-P waves at a corrugated interface between distinct triclinic elastic half spaces

The reflection and refraction pattern of elastic waves at a corrugated interface between two triclinic half-spaces is discussed. The incident wave is taken to be the cause of the interface disturbance and the reflected and refracted waves are effects. This leads to the causality requirement that the reflected and refracted waves must propagate away from the interface. Closed form expressions of reflection and transmission coefficients are derived using Rayleigh’s method of approximation. The formulae of reflection and transmission coefficients are derived in closed form for the first-order approximation of the corrugation. The analytical expressions of all the three phase velocities of qP, qSV and qSH waves have been derived. The variation of reflection and refraction coefficients with the angle of incidence and also with the corrugation parameter is shown. In this paper we have developed Graphical User Interface (GUI) Software in MATLAB which shows the variation of reflection and refraction coefficients with respect to incident angle and corrugation parameter. This software can be generalized to show the variation of reflection and refraction coefficients. Numerical computations are performed for a scientific model and the results obtained are shown graphically.     

Experimental studies of ignition and transition to detonation induced by the reflection and diffraction of shock waves

This paper presents results from a program of experimental studies of ignition induced by the interaction of an initially planar shock wave with an obstacle in its path. With the aid of pressure measurements, spark schlieren photography and smoked foil techniques it is shown how, given favourable initial conditions, the two-dimensional multiple shock reflection and diffraction can promote ignition and transition to detonation in reactive gaseous mixtures. Comparison of the results with those of a non-reactive gas distinguishes the gas dynamic and chemical processes involved, and experimentally determined detonation cell sizes are compared with values predicted using chemical kinetic rate data. The systems investigated were argon, air, propane-air, propane-oxygen-argon and ethylene-oxygen-argon. Received: 3 December 1998 / Accepted: 27 October 1999     

波在双孔隙与单孔隙介质界面上的反射与透射

基于Biot理论和双重孔隙介质理论研究了弹性波在双重孔隙介质与流体饱和单一孔隙介质 界面的反射和透射问题,在界面上假定裂缝孔隙流体相对于固体骨架的位移为零,推导了反 射系数和透射系数的计算公式,数值讨论了反射系数和透射系数随入射角和频率的变化关 系. 同时,讨论了双重孔隙介质中3种压缩波(P-1, P-2和P-3波)和一种剪切波(S波) 的频散和衰减特性.     

Response of loose bonding on reflection and transmission of elastic waves at interface between elastic solid and micropolar porous cubic crystal

The problem of reflection and transmission of plane periodic waves incident on the interface between the loosely bonded elastic solid and micropolar porous cubic crystal half spaces is investigated. This is done by assuming that the interface behaves like a dislocation, which preserves the continuity of traction while allowing a finite amount of slip. Amplitude ratios of various reflected and transmitted waves have been depicted graphically. Some special cases of interest have been deduced from the present investigation.     

Surface waves at the interface between an inviscid fluid and a dipolar gradient solid

This paper is about the dispersion analysis of surface waves propagating at the interface between an inviscid fluid and a higher gradient homogeneous elastic solid modelled as a dipolar gradient continuum. In order to compare the results, a second gradient model is also evaluated. The analysis is carried out by finding the roots of the secular equation, and by carefully studying their physical meaning. As it is well known, higher gradient continua are dispersive, i.e. phase and group velocities are frequency dependent. As a consequence, the existence of surface waves will indeed depend on frequency. In order to investigate the behaviour of surface waves in this specific fluid–solid configuration, a complete dispersion analysis is performed, with a particular focus on the frequency range in which the phase velocity of shear waves is lower than the speed of waves of the fluid. Surface waves of the type Leaky Rayleigh and Scholte–Stoneley are observed in this frequency range. This work extends the knowledge on surface waves in the case of higher gradient solids and applications of these results can be found in the field of non-destructive damage evaluation in micro structured materials, composites, metamaterials and biological tissues.     

气相爆轰波冲击气固界面的透反射特性

为了研究气相爆轰波冲击气固界面过程中透射波和反射波的相关特性,建立爆轰波冲击气固界面的一维理论模型,对不同初始压力条件下爆轰波到达气固界面后的界面两侧的压力和界面速度变化进行分析。利用时空守恒元求解元方法对气相爆轰波冲击气固界面过程进行数值模拟,分析气体部分反射波的压力分布和速度变化规律及透射入固体中应力波的波形和波速特征,并搭建气相爆轰波冲击活塞实验装置进行进一步验证。结果表明：气体爆轰波到达气固界面后,在固体中透射指数形式的弹性波,并在界面处向气体区反射一道激波。爆轰波后的稀疏波与反射激波相交,削弱反射激波,最终形成稳定激波回传。气固界面在稀疏波和反射稀疏波的作用下,压力和速度逐渐下降,最终也形成稳定状态。在不同混气初始压力情况下,爆轰波冲击过程中产生的最高压力和爆压的比值基本保持不变。理论模型对特征点相关物理量的计算值和实验数据符合的较好。

     

Investigation of the dynamic behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces by a new method

The dynamic behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces subjected to the harmonic waves is investigated by a new method. The cracks are parallel to the interfaces in the mid-plane of the piezoelectric layer. By using the Fourier transform, the problem can be solved with two pairs of triple integral equations. These equations are solved by using Schmidt’s method. This process is quite different from that adopted previously. Numerical examples are provided to show the effect of the geometry of cracks, the frequency of the incident wave, the thickness of the piezoelectric layer and the constants of the materials upon the dynamic stress intensity factor of cracks.     

Scattering of love waves by an interface crack between a piezoelectric layer and an elastic substrate

The scattering of Love waves by an interface crack between a piezoelectric layer and an elastic substrate is investigated by using the integral transform and singular integral equation techniques. The dynamic stress intensity factors of the left and the right crack tips are determined. It is found from numerical calculation that the dynamic response of the system depends significantly on the crack configuration, the material combination and the propagating direction of the incident wave. It is expected that specifying an appropriate material combination may retard the growth of the crack for a certain crack configuration. Project supported by the National Natural Science Foundation of China (No. 19891180), the Fundamental Research Foundation of Tsinghua University (JZ 2000.007) and the Fund of the Education Ministry of China.