Mathematical modeling of elastic wave propagation in crystals: 3D-wave surfaces

A new method for pure modes of elastic waves search in crystals is presented. The method is based on the analysis of 3D-surfaces of phase velocities and differs from the known analogous methods by accounting for the piezoeffect. The piezoelectric corrections have an essential effect on the result of the search. The computer code is prepared to construct these surfaces and their sections by the base planes of a crystal. The code also permits to calculate the directions of pure longitudinal and pure transverse elastic wave propagation on these surfaces.     

The Character of Deformations at the Interface of Elastic Media under the Slip Condition

Fresnel coefficients obtained when solving the problem of wave propagation through the interface of two elastic media and expressions for components of the elastic-distortion tensor allow one to study the character of dynamic deformations at the interface. Deformation modes different from zero at the interface of the elastic media under the slip-contact condition have been determined. Dependences of deformation amplitudes at the interface on the wave incidence angle and parameters of the adjacent media for incident longitudinal and transverse waves have been constructed and analyzed.     

Reflection and refraction of plane waves at interface between two piezoelectric media

This paper analyses reflection and refraction of plane waves at a perfect interface between two anisotropic piezoelectric media. The equations of elastic waves, quasi-static electric field, and constitutive relationships for the piezoelectric media are derived. A solution based on the inhomogeneous wave theory is developed to address the inconsistency between the numbers of independent wave modes in the media and the numbers of interfacial boundary conditions to obtain accurate reflection and refraction coefficients in case of strong piezoelectric media, where all the elastic and electric continuity conditions across the interface are satisfied simultaneously. The study shows that there exist independent and zero energy wave modes satisfying the general Snell’s law and propagating along the interface for any incident wave angle. These waves can be treated as pseudo surface waves. It is further found that all the reflection/refraction waves including the pseudo surface waves obey the energy conservation law at the interface boundary. In addition, the analysis also reveals that the reflection and refraction elastic waves can turn into pseudo surface waves at some critical incident angles.     

Multi-modal wave propagation in smart structures with shunted piezoelectric patches

The propagation of wave modes in elastic structures with shunted piezoelectric patches is dealt with in this work. The wave finite element approach, which is based on the finite element method and periodical structure theory, is firstly developed as a prediction tool for wave propagation characteristics in beam like structures, and subsequently extended to consider shunted piezoelectric elements through the diffusion matrix model (DMM). With these numerical techniques, reflection and transmission coefficients of propagating waves in structures with shunted piezoelectric patches can be calculated. The performance of shunted piezoelectric patches on the control of wave propagation is investigated numerically with the DMM. Forced response of the smart structure can also be calculated, and based on which the time response of the structure can be obtained via an inverse discrete fourier transform approach. These general formulations can be applied to all types of slender structures. All these numerical tools can facilitate design modifications and systematic investigations of geometric and electric parameters of smart structures with shunted piezoelectric elements.     

Longitudinal and transverse waves in anisotropic elastic materials

Summary It is shown that a necessary and sufficient condition for a longitudinal wave to propagate in the direction n in an anisotropic elastic material is that the elastic stiffness C ₁₁ (n) is a stationary value (maximum, minimum or saddle point) at n. Explicit expressions of all n and the corresponding elastic stiffness C ₁₁ (n) for which a longitudinal wave can propagate are presented for orthotropic, tetragonal, trigonal, hexagonal and cubic materials. As to longitudinal waves in triclinic and monoclinic materials, only few explicit expressions are possible. We also present necessary and sufficient conditions for a transverse wave to propagate in the direction n. As an illustration, explicit expressions of all n, the polarization vector a and the wave speed c for which a transverse wave can propagate in cubic and hexagonal materials are given. The search for n in hexagonal materials confirms the known fact that a transverse wave can propagate in any direction. A longitudinal wave is necessarily accompanied by two transverse waves. However, a transverse wave can propagate without being accompanied by a longitudinal wave.     

Existence of longitudinal and transverse waves in anisotropic thermoelastic media

Four waves propagate in an anisotropic thermoelastic medium. The fastest among them is a quasi-longitudinal wave. The slowest of them is a thermal wave. The remaining two are called quasi-transverse waves. The prefix ‘quasi’ refers to their polarizations being nearly, but not exactly, parallel or perpendicular to the direction of propagation. The polarizations of these four waves are not mutually orthogonal. Hence, unlike anisotropic elastic media, the existence of a longitudinal wave may not imply the existence of a transverse wave, by default. The existence of a purely longitudinal wave in an anisotropic thermoelastic medium is ensured by the stationary characters of three expressions. These expressions involve components of phase direction with elastic (stiffness and coupling) and thermal coefficients of the thermoelastic medium. The existence of a purely transverse wave is ensured by the two equations restricting the choice of thermoelastic (stiffness and coupling) coefficients. The existence of longitudinal and transverse waves along the coordinate axes and in the coordinate planes are discussed for general anisotropy. The discussion is extended to orthotropic materials, and the existence of pure phases is explored along few specific phase directions.     

Unique Cuspidal Features in the Group-Velocity Surface of Barium Sodium Niobate and Rochelle Salt

Crystals that have sufficient elastic anisotropy can exhibit cuspidal features in the group-velocity surface so that the group velocity is double- or triple-valued. In nonpiezoelectric crystals such cuspidal features occur only in the quasi-transverse (QT) modes when the wave vectors are restricted to the symmetry planes of the crystal. Sufficient piezoelectric stiffening of the elastic constants, however, can generate cuspidal features in other modes when the wave vectors are restricted to these same symmetry planes. In barium sodium niobate (BSN) there is insufficient elastic anisotropy to generate cuspidal features when piezoelectric stiffening is neglected. The addition of piezoelectric stiffening changes only the quasi-longitudinal (QL) and QT modes in the (100) and (010) planes, and changes only the pure transverse (T) mode in the (001) plane. There is sufficient piezoelectric stiffening, however, to generate small cuspidal features in the QL mode about the [001] and axes in the (100) and (010) planes. In Rochelle salt, however, there is sufficient elastic anisotropy to generate cuspidal features in the QT modes about collinear axes along nonsymmetry directions in the (100) and (010) planes. The addition of piezoelectric stiffening changes only the T mode in the (100), (010) and (001) symmetry planes. This piezoelectric stiffening is large enough, however, to generate large cupsidal features in the T mode about new collinear axes along nonsymmetry directions in the (010) and (001) planes.     

一种由横波激发的特殊非线性声现象

目前对非线性超声的研究多集中在纵波激发的谐波性质以及对材料微观结构变化的实验检测上,横波激发的非线性声波性质少有研究。对横波激发的一维非线性声波方程入手,利用摄动法求解该方程,并改写为一阶偏微分方程,然后利用交错网格的有限差分形式进行数值求解。结果表明:采用横波激发,能产生线性横波和非线性纵波,且纵波的高次谐波内有两个信号,分别以纵波和横波两种速度传播。若采用较长的激发信号,纵波谐波能形成"拍"现象,成为一种奇特的声传播现象。     

Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material

The propagation of ultrasonic guided waves in an elastic hollow cylinder with a viscoelastic coating is studied. The principle motivation is to provide tools for performing a guided wave, nondestructive inspection of piping and tubing with viscoelastic coatings. The theoretical boundary value problem is solved that describes the guided wave propagation in these structures for the purpose of finding the guided wave modes that propagate with little or no attenuation. The model uses the global matrix technique to generate the dispersion equation for the longitudinal modes of a system of an arbitrary number of perfectly bonded hollow cylinders with traction-free outer surfaces. A numerical solution of the dispersion equation produces the phase velocity and attenuation dispersion curves that describe the nature of the guided wave propagation. The attenuation dispersion curves show some guided wave modes that propagate with little or no attenuation in the coated structures of interest. The wave structure is examined for two of the modes to verify that the boundary conditions are satisfied and to explain their attenuation behavior. Experimental results are produced using an array of transducers positioned circumferentially around the pipe to evaluate the accuracy of the numerical solution.     

Second harmonic generation of shear waves in crystals

Nonlinear self-interaction of shear waves in electro-elastic crystals is investigated based on the rotationally invariant state function. Theoretical analyses are conducted for cubic, hexagonal, and trigonal crystals. The calculations show that nonlinear self-interaction of shear waves has some characteristics distinctly different from that of longitudinal waves. First, the process of self-interaction to generate its own second harmonic wave is permitted only in some special wave propagation directions for a shear wave. Second, the geometrical nonlinearity originated from finite strain does not contribute to the second harmonic generation (SHG) of shear waves. Therefore, unlike the case of longitudinal wave, the second-order elastic constants do not involve in the nonlinear parameter of the second harmonic generation of shear waves. Third, unlike the nonlinearity parameter of the longitudinal waves, the nonlinear parameter of the shear wave exhibits strong anisotropy, which is directly related to the symmetry of the crystal. In the calculations, the electromechanical coupling nonlinearity is considered for the 6 mm and 3 m symmetry crystals. Complement to the SHG of longitudinal waves already in use, the SHG of shear waves provides more measurements for the determination of third-order elastic constants of solids. The method is applied to a Z-cut lithium niobate (LiNbO/sub 3/) crystal, and its third-order elastic constant c/sub 444/ is determined.     

声学法测量复合绝缘子弹性常数

提出了一种实用的声学法测量复合材料弹性常数的方法.依据复合材料中声波速度与材料弹性常数之间的本构关系,测量不同传播方向的少数几个声速,就可以得到其弹性常数矩阵.以复合绝缘子为测量样品,为提高测量横波的精度,根据超声波在边界上的波型转换条件,使用常规纵波探头对由掠入射纵波产生的横波进行测量.     

Love waves in a smart functionally graded piezoelectric composite structure

The propagation of Love waves in a smart functionally graded piezoelectric structure is analyzed by applying elastic wave theory. There is an additional functionally graded layer between the piezoelectric layer and the substrate in this smart structure. When the piezoelectric and dielectric constants vary individually in a functionally graded layer, the asymptotic solutions of Love waves are obtained by applying the WKB method and solving the fourth order differential equation with variable coefficients. The effects of gradient variation on the phase velocity and the coupled electromechanical factor are discussed in detail. The analysis shows that the number of vibration modes is greater than that in the non-graded layer structure, and the coupled electromechanical factor increases with the increase of piezoelectric constant graded variation. Presented results are useful for the improvement of properties of surface acoustic wave (SAW) devices.     

Shear wave passage through a vacuum gap between piezoelectric crystals with relative longitudinal displacement

The influence of relative longitudinal displacements of two piezoelectric crystals separated by a vacuum gap on the reflection and transmission of shear waves through the gap between these crystals is considered. We predict the possibility of enhancing the reflected wave under conditions of wave front reversal for the wave transmitted through the gap.     

Effects of the initial stress on the propagation and localization properties of Rayleigh waves in randomly disordered layered piezoelectric phononic crystals

In this paper, the effects of the initial stress on the propagation and localization properties of the Rayleigh surface waves in randomly disordered layered piezoelectric phononic crystals are studied. Due to different mechanical properties between the piezoelectric material and the polymer, different initial stresses in these two layers satisfying the equilibrium condition and interfacial compatibility are considered, which is more suitable for the practical cases. The transfer matrix between two consecutive piezoelectric unit cells is derived according to the continuity conditions. The expression of the localization factor is presented, and the wave localization properties are analyzed. Numerical calculations for the PVDF/PZT–2 periodic composites with the initial stress are performed. The band gap characteristics are studied taking the mechanical and electrical coupling into account. It is found that the localization degree can be influenced by the piezoelectric constants. With the increase in the piezoelectric constant, the stop band regions are enlarged for the ordered structures, and the localization properties of Rayleigh waves are strengthened for the disordered systems. The Rayleigh waves will be localized in mistuned periodic piezoelectric composites. The characteristics of band gaps and wave localization in ordered and disordered piezoelectric phononic crystals can be significantly changed by tuning the initial stress.     

A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.     

Scanning electron microscopy observation of surface acoustic wavepropagation in the LiNbO3 crystals with regular domainstructures

This paper reports a scanning electron microscopy investigation of surface acoustic wave propagation in the LiNbO₃ crystals with regular domain structures. The regular domain structures in the LiNbO ₃ crystals were formed by the method of the thermo-electric treatment after growth. We investigated two modes of the interaction between the surface acoustic waves and regular domain structures in the LiNbO₃ crystals: excitation of the surface acoustic waves by the curved regular domain structure, and propagation of the surface acoustic waves along and across the regular domain structures. It is shown that the regular domain structure in the first case can be used as an interdigital transducer for excitation of the surface acoustic waves through the longitudinal piezoelectric effect. In the second mode of the interaction we observed that the regular domain structure can be used as an acoustic wave-guide in the process of the propagation of the traveling surface acoustic waves along the regular domain structure. Also, we demonstrate the reflection of the surface acoustic waves from the domain walls in the process of surface acoustic wave propagation across the regular domain structure     

Generalized thermoelastic waves in functionally graded plates without energy dissipation

In this paper, a dynamic solution of the propagating thermoelastic waves in functionally graded material (FGM) plate subjected to stress-free, isothermal boundary conditions is presented in the context of the Green–Naghdi (GN) generalized thermoelastic theory. The FGM plate is composed of two orthotropic materials. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The coupled wave equation and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of the inhomogeneous thermoelastic plate and the corresponding pure elastic plate are compared to show the characteristics of thermal modes and the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions of elastic modes and thermal modes are shown to discuss their differences. A plate with a different gradient variation is calculated to illustrate the influence of the gradient field on the wave characteristics.     

声波由粘弹性介质向流体介质入射情况下的声反射分析

在研究含腔体的结构的声学特性时,会遇到声波从粘弹性介质向流体介质入射的情况。主要研究了声波从粘弹性介质向流体介质斜入射情况下的声反射特性,分析表明：反射声波中有横波和纵波同时存在,纵波声压反射系数随入射角度的增大逐渐增大,横波的声压反射系数随入射角度的增加先增大后减小,在垂直入射时只有纵波,而没有横波存在。这些结论为进一步研究带空腔结构的声学特性打下了基础。     

LiNbO3和LiTaO3晶体电子结构、平面声波和光折射特性

本研究基于密度泛函理论的第一性原理超软赝势平面方法计算了LiNbO₃和LiTaO₃的晶格参数、电子结构和弹性常数, 并利用Christoffel方程研究了二者平面声波特征。结果表明: 两者的理论计算晶格参数和弹性常数与实验值接近, 禁带宽度分别为3.78和3.98 eV, 导带底和价带顶主要由O-2p和Nb-4d(Ta-5d)态电子贡献。化学键理论揭示Li和Nb(Ta)与O原子之间有两种成键类型。 电荷布局分析结果显示有两种相应的重叠布居数, Nb(Ta)-O键呈现强共价键作用, 并且Nb-O(Ta-O)键长小于Li-O键长。LiNbO₃和LiTaO₃晶体平面声波有两支横波和一支纵波, 纵波速度大于横波速度, 在xy平面呈现六重对称性, 在xz和yz平面各向异性程度强于xy平面, 沿[001]、晶向上两支横波振动速度相等。最后利用模守恒赝势(Norm-conserving)计算了介电常数和静态折射率, 计算表明LiNbO₃晶体的折射性能和非寻常光(e光)离散程度均强于LiTaO₃晶体。     

Acoustic wave propagation in piezoelectric fibers of hexagonalcrystal symmetry

An exact analysis is presented for acoustic wave propagation in cladded acoustic fibers having a core and an infinite thick cladding both made of piezoelectric hexagonal crystal of 6 mm point group symmetry. The crystalline Z axes of both the core and cladding coincide with the fiber axis. A general dispersion equation is derived for all the acoustic modes propagating along the fiber axis. Two simpler and independent equations which represent the dispersion relations of torsional and radial-axial modes can be separated from the general dispersion equation. It has been found that the radial-axial and general flexural modes are piezoelectrically active while the torsional modes are not. Approximate dispersion relations for pure guided modes in weakly guiding weakly piezoelectric fibers which are much simpler than the exact ones are also given. Numerical results are only presented for pure guided modes. Exact and approximate dispersion curves of several lower order pure guided flexural, radial-axial and torsional modes in a weakly guiding ZnS fiber are compared and they are in good agreement