Love waves in a layered functionally graded piezoelectric structure under initial stress

Abstract

To study the effect of initial stress on the propagation behavior of Love waves in a layered functionally graded piezoelectric structure, a mathematical model is established. The piezoelectric layer is taken as exponentially graded material where as half-space is taken as simply elastic substratum. The coupled electromechanical field equations are solved analytically to obtain the mechanical displacements and electrical potential functions for the piezoelectric layer and elastic substrate. The dispersion relations are obtained for electrically open and short cases. The higher mode Love wave propagation has been considered. For numerical interpretation of the results, four sets of piezoelectric layer and elastic substrate have been taken into consideration. Graphical representation reveals about the effect of initial stress and the effect of inhomogeneity parameter on the phase velocity against wave number for electrically open and electrically short cases, respectively.     

Love waves in functionally graded piezoelectric materials by stiffness matrix method

A numerical matrix method relative to the propagation of ultrasonic guided waves in functionally graded piezoelectric heterostructure is given in order to make a comparative study with the respective performances of analytical methods proposed in literature. The preliminary obtained results show a good agreement, however numerical approach has the advantage of conceptual simplicity and flexibility brought about by the stiffness matrix method. The propagation behaviour of Love waves in a functionally graded piezoelectric material (FGPM) is investigated in this article. It involves a thin FGPM layer bonded perfectly to an elastic substrate. The inhomogeneous FGPM heterostructure has been stratified along the depth direction, hence each state can be considered as homogeneous and the ordinary differential equation method is applied. The obtained solutions are used to study the effect of an exponential gradient applied to physical properties. Such numerical approach allows applying different gradient variation for mechanical and electrical properties. For this case, the obtained results reveal opposite effects. The dispersive curves and phase velocities of the Love wave propagation in the layered piezoelectric film are obtained for electrical open and short cases on the free surface, respectively. The effect of gradient coefficients on coupled electromechanical factor, on the stress fields, the electrical potential and the mechanical displacement are discussed, respectively. Illustration is achieved on the well known heterostructure PZT-5H/SiO₂, the obtained results are especially useful in the design of high-performance acoustic surface devices and accurately prediction of the Love wave propagation behaviour.     

The effect of initial stress on the propagation behavior of SH waves in piezoelectric coupled plates

This study analytically investigates the propagation of shear waves (SH waves) in a coupled plate consisting of a piezoelectric layer and an elastic layer with initial stress. The piezoelectric material is polarized in z-axis direction and perfectly bonded to an elastic layer. The mechanical displacement and electrical potential function are derived for the piezoelectric coupled plates by solving the electromechanical field equations. The effects of the thickness ratio and the initial stress on the dispersion relations and the phase and group velocities are obtained for electrically open and mechanically free situations. The numerical examples are provided to illustrate graphically the variations of the phase and group velocities versus the wave number for the different layers comparatively. It is seen that the phase velocity of SH waves decreases with the increase of the magnitude of the initial compression stress, while it increases with the increase of the magnitude of the initial tensile stress. The initial stress has a great effect on the propagation of SH waves with the decrease of the thickness ratio. This research is theoretically useful for the design of surface acoustic wave (SAW) devices with high performance.     

Effect of periodic corrugation,reinforcement, heterogeneity and initial stress on Love wave propagation

This paper investigates the impact of corrugated boundary surfaces, reinforcement on the propagation of Love-type wave in prestressed corrugated heterogeneous fiber-reinforced layer resting over a void pores half-space. The heterogeneity in the upper corrugated layer is caused due to exponential variation in the elastic constants with respect to the space variable pointing positively downwards. The dispersion equation in the complex form has been derived using method of separation of variables. The real and imaginary parts of the complex dispersion equation were separated and found in well agreement with the classical Love wave equation. Also, the attenuation of the Love waves has been discussed. The study reveals that such a medium transmits two fronts of Love waves. The first front depends upon the change in volume fraction of the pores and the second front depends upon the modulus of rigidity of the elastic matrix of the medium. The substantial influence of corrugation parameters, reinforcement, undulatory parameter, initial stress, heterogeneity parameter and position parameter on the phase velocity, and attenuation of Love-type wave have been observed and depicted by means of graph. It has been observed that the phase velocity decreases with the increase in initial stress parameters, heterogeneity, and reinforcement in upper layer.     

半无界分层介质表面任意面源激发的弹性波场

研究了半无界层状介质自由表面任意形状的面源产生的弹性波场。首先,我们将层状介质中的传递矩阵理论从二维推向三维空间,在频率域研究了任意面源激发的三维弹性波理论问题。然后,深入研究了Rayleigh波和Love波的激发与传播特性,发现Rayleigh波和Love波的传播速度在与自由表面平行的平面内与传播方位角θ无关,但其激发强度却强烈地依赖于传播方位角θ。最后,我们具体研究了矩形源、无穷长条形源和圆盘激励出的弹性波场,并通过数值计算给出了Rayleigh波和Love波的位移指向性分布图。     

Love waves in layered flexoelectric structures

Abstract

In this paper, the flexoelectric effect on Love waves propagating in a structure with a nanoscale piezoelectric guiding layer deposited on an isotropic elastic substrate is analytically investigated. Transcendental complex dispersion equations are obtained and solved numerically which are corresponding to the electrically open and short conditions at the free surface. A detailed discussion about the dispersion relations of the fundamental mode is subsequently presented. The results indicate that flexoelectricity has a substantial effect on Love wave propagation. The presence of flexoelectricity leads to a complex phase velocity with a negative/positive imaginary part, which means Love waves attenuate/grow over time. In addition, the phase velocity dispersion relations depend greatly on the thickness and flexoelectric coefficients of the guiding layer. The current work is the first attempt to explore the flexoelectric effect on the propagation characteristics of surface acoustic waves (SAWs). And the results would be beneficial to achieve a better performance of SAW devices.     

Theoretical validation on the existence of two transverse surface waves in piezoelectric/elastic layered structures

In this paper, we analytically study the dispersion behavior of transverse surface waves in a piezoelectric coupled solid consisting of a transversely isotropic piezoelectric ceramic layer and an isotropic metal or dielectric substrate. This study is a revisit to the stiffened Love wave propagation done previously. Closed-form dispersion equations are obtained in a very simple mathematical form for both electrically open and shorted cases. From the viewpoint of physical situation, two transverse surface waves (i.e., the stiffened Love wave and the FDLW-type wave) are separately found in a PZT-4/steel system and a PZT-4/zinc system. All the observed dispersion curves are theoretically validated through the discussion on the limit values of phase velocity using the obtained dispersion equations. Those validation and discussion give rise to a deeper understanding on the existence of transverse surface waves in such piezoelectric coupled structures. The results can be used as a benchmark for the study of the wave propagation in the piezoelectric coupled structures and are significant in the design of wave propagation in the piezoelectric coupled structures as well.     

Propagation of Bleustein-Gulyaev waves in a prestressed layered piezoelectric structure

Based on the theories of nonlinear continuum mechanics, piezoelectricity and elastic waves in solids, theoretical analysis of Bleustein-Gulyaev surface acoustic wave propagation in a prestressed layered piezoelectric structure are described. Numerical calculations are performed for the case that the layer and the substrate are identical LiNbO(3) except that they are polarized in opposite directions. It is found that an almost linear behavior of the relative change in phase velocity versus the initial stress is obtained for both surface electrically free and shorted cases. Potential applications in the design of acoustic wave devices are suggested.     

应用于气体传感器的多波导层Love波器件的温度特性研究

为获得趋于零温度系数的多波导层Love波器件结构参数,对Love波器件的温度稳定性进行了研究。首先构建了这种多波导层条件下Love波传播特性的理论模型,即根据层状结构声波传播方程和边界条件推导出Love波的色散方程,再结合色散方程和Tomar的方法,成功提取出趋于零温度系数的Love器件结构参数,并通过实验验证了理论计算结果,实验测试得到的基于ST-90°X石英且采用SU-8与SiO₂双波导层的Love波延迟线器件的频率温度系数(tcf)仅为2.16 ppm/℃。优化后的器件具有很好的实际意义。     

Propagation of SH-waves in two anisotropic layers bonded to an isotropic half-space under gravity

This study reports a theoretical investigation of the propagation of SH-wave in a piezoelectric layer superimposed on a self-reinforced layer overlying an isotropic gravitational half-space. The expressions of the dispersion relation of SH-wave have been established for electrically open and electrically short conditions in closed form. For the purpose of numerical computation, lithium niobate piezoelectric material has been considered. The dispersion curves have been depicted graphically and the prominent impacts of piezoelectric constant, dielectric constant, reinforced parameter, width ratio, and Biot’s gravity parameter on the phase velocity of SH-wave have been unraveled for both the electrical conditions. As a special case of the problem, it is found that the obtained dispersion relation concurs with classical Love wave equation for both the electrical conditions. Moreover, some important peculiarities have also been traced out through numerical computations for both the electrical cases.     

Longitudinal wave propagation in a rod with variable cross-section

Vibration data are required for condition monitoring in machinery, and can only be collected indirectly after transferring through rods, shells, rotating shafts or other components in many engineering applications. Investigation on the transfer characteristics of vibration in these components is very helpful to guarantee the efficiency of the data collected indirectly. Here, the longitudinal wave propagation in a rod with variable cross-section is investigated. First, the equations of motion are established for the rod based upon the elementary wave theory, the Love theory and the Mindlin–Herrmann theory. Second, the transfer matrix method is employed to explore the propagation characteristics of the rod from the derived equations of motion. Finally, two kinds of rods with the cross-sections varying in the exponential and the polynomial forms are used to illustrate the analytical predictions of the propagation characteristics of the longitudinal wave, which are compared with the results from the finite element analysis (FEA) method. It is shown that Poisson's effect or the shear deformation plays a very important role in the longitudinal wave propagation in the rod and can widen the rod's stop band moderately. Moreover, the cut-off frequency of the rod is unconcerned with the variation form of the cross-section, but dependent on the area ratio between both the ends of the rod, even though Poisson's effect or shear deformation is included.     

Love wave propagation in piezoelectric layered structure with dissipation

We investigate analytically the effect of the viscous dissipation of piezoelectric material on the dispersive and attenuated characteristics of Love wave propagation in a layered structure, which involves a thin piezoelectric layer bonded perfectly to an unbounded elastic substrate. The effects of the viscous coefficient on the phase velocity of Love waves and attenuation are presented and discussed in detail. The analytical method and the results can be useful for the design of the resonators and sensors.     

Effects of initial compression stress on wave propagation in carbon nanotubes

An analytical method to investigate wave propagation in single- and double- walled carbon nanotubes under initial compression stress is presented. The nanotube structures are treated within the multilayer thin shell approximation with the elastic properties taken to be those of the graphene sheet. The governing equations are derived based on Flügge equations of motion. Frequency equations of wave propagation in single and double wall carbon nanotubes are described through the effects of initial compression stress and van der Waals force. To show the effects of Initial compression stress on the wave propagation in nanotubes, the symmetrical mode can be analyzed based on the present elastic continuum model. It is shown that the wave speed are sensitive to the compression stress especially for the lower frequencies.     

Love-type wave propagation in a pre-stressed viscoelastic medium influenced by smooth moving punch

In the present paper, a mathematical model studying the effect of smooth moving semi-infinite punch on the propagation of Love-type wave in an initially stressed viscoelastic strip is developed. The dynamic stress concentration due to the punch for the force of a constant intensity has been obtained in the closed form. Method based on Weiner–hopf technique which is indicated by Matczynski has been employed. The study manifests the significant effect of various affecting parameters viz. speed of moving punch associated with Love-type wave speed, horizontal compressive/tensile initial stress, vertical compressive/tensile initial stress, frequency parameter, and viscoelastic parameter on dynamic stress concentration due to semi-infinite punch. Moreover, some important peculiarities have been traced out and depicted by means of graphs.     

Effect of external electrical field on characteristics of a Lamb wave in a piezoelectric plate

The influence of a homogeneous electrical field E on the characteristics and propagation conditions of the Lamb wave in a piezoelectric crystalline plate is considered on the basis of the theory of acoustic wave propagation in piezocrystals under the effect of an external electrical field.     

Love-type waves in functionally graded piezoelectric material (FGPM) sandwiched between initially stressed layer and elastic substrate

This paper investigates the propagation behavior of Love-type surface waves in three-layered composite structure with initial stress. The composite structure has been taken in such a way that a functionally graded piezoelectric material (FGPM) layer is bonded between initially stressed piezoelectric upper layer and an elastic substrate. Using the method of separation of variables, frequency equation for the considered wave has been established in the form of determinant for electrical open and short cases on free surface. The bisection method iteration technique has been used to find the roots of the dispersion relations which give the modes for electrical open and short cases. The effects of gradient variation of material constant and initial stress on the phase velocity of surface waves are discussed. Dependence of thickness on each parameter of the study has been shown explicitly. Study has been also done to show the existence of cut-off frequency. Graphical representation has been done to exhibit the findings. The obtained results are significant for the investigation and characterization of Love-type waves in FGPM-layered media.     

Study of Love-type wave propagation in an isotropic tri layers elastic medium overlying a semi-infinite elastic medium structure

This paper deals with the propagation of Love-type wave in a composite isotropic structure embraced of tri layers elastic medium overlying a semi-infinite elastic medium. The heterogeneity is caused due to the variation of linear, exponential, and quadratic with respect to the depth. Modified Bessel function with Debye Asymptotic Expansion approach is used to achieve closed form of dispersion equation analytically and found to be in well agreement to the classical Love wave equation. Numerical computation has been carried out to accomplish the graphical demonstration to unravel some important peculiarities of wave number associated in presence or absence of layers medium and effect of heterogeneities on phase velocity of Love-type wave.     

Technique for determining the channel expansion rate at the stage of electrical breakdown using a grounded intercepting ring

A method is described that allows one to study the conductivity dynamics of a channel produced by explosion of a wire at the stage of electrical breakdown. Experimental data have been presented for the expansion rate of the conductivity channel in extended (up to 1.9 m long) arbitrarily shaped gapes that were produced by an exploding copper wire 90 µm in diameter. The initial stored energy and applied voltage were, respectively, 2.7–3.7 kJ and 8–10 kV. It has been shown that the expansion rate of the conductivity channel coincides with the propagation rate of a shock wave and is inversely proportional to the square root of its radius and propagation time. The radius of the shock wave is a linear function of the square root of its propagation rate. Experimental data are in satisfactory agreement with the calculated results obtained by Lin [18] in terms of the model of an intense shock wave. It has been shown that the diameter of the conductivity channel depends on the position of the trailing edge of the shock wave.