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.     

Transverse surface waves in an FGM layered structure

The existence and propagation behavior of Love waves in a functionally graded material (FGM) layered structure are theoretically investigated in this paper based on the three-dimensional equations of linear electricity. The Wentzel–Kramers–Brillouin (WKB) method is applied to obtain the analytical solutions in the FGM layer. The dispersion equation for a Love surface wave in this kind of structure is obtained in a simple mathematical form, where the material property variation of the FGM layer is arbitrary. First, the solution is used to study the effect of the gradient coefficients on the dispersion curves and the phase velocities of Love waves. Then, the influence of gradient coefficients on the stress and displacement fields is discussed in detail. The reported results are important in the design of surface acoustic wave (SAW) devices with high performance.     

Effect of gradient dielectric coefficient in a functionally graded material (FGM) substrate on the propagation behavior of love waves in an FGM-piezoelectric layered structure

The propagation behavior of Love waves in a layered structure that includes a functionally graded material (FGM) substrate carrying a piezoelectric thin film is investigated. Analytical solutions are obtained for both constant and gradient dielectric coefficients in the FGM substrate. Numerical results show that the gradient dielectric coefficient decreases phase velocity in any mode, and the electromechanical coupling factor significantly increases in the first- and secondorder modes. In some modes, the difference in Love waves' phase velocity between these two types of structure might be more than 1%, resulting in significant differences in frequency of the surface acoustic wave devices.     

覆盖层为功能梯度材料弹性半平面中的Love波

对均匀各向同性弹性半平面上覆盖一层功能梯度材料中存在的Love波的频散问题进行了研究,给出了Love波频散方程的一般形式。利用WKBJ近似理论,给出了功能梯度材料层的位移、应力近似解析解,导出了Love波WKBJ近似频散方程的一般形式。该文以功能梯度材料层的剪切弹性模量和质量密度沿厚度方向均为指数函数变化为例,进行了实例计算和分析,给出了频散曲线,讨论了Love波在功能梯度材料覆盖层弹性半平面中传播的一般性质。这些结论对无损检测和反问题分析方法的改进提供理论依据。     

Love waves on a half-space with a gradient piezoelectric layer by the geometric integration method

The propagation of Love waves on an elastic homogeneous half-space with a piezoelectric gradient covering layer is studied by the geometric integration method in this article. First, the state transfer equation of a Love wave is derived from the governing equations and constitutive relations. Then, the transfer matrix of the state vector is obtained by solving the state transfer equation of a Love wave and then the stiffness matrix is obtained. By combining transfer matrices and the stiffness matrices of the gradient covering layer and the homogeneous half-space, the total surface stiffness matrix of a Love wave is obtained. Lastly, the application of the electrically open circuit and short circuit conditions and mechanically traction-free conditions gives the frequency dispersive relation of a Love wave. For the gradient covering layer, the material constants at the bottom of the covering layer may be greater or smaller than that at the top of the covering layer. The two situations and three kinds of gradient profiles for each of these two situations are investigated. The numerical results show that the Love wave speed is sensitive to not only the material constants at the bottom and the top of the covering layer, but also the gradient profiles of the covering layer.     

Love waves propagation in a piezoelectric layered structure with initial stresses

Summary. The propagation behavior of Love waves in a piezoelectric layered structure with inhomogeneous initial stress is studied. Solutions of the mechanical displacement and electrical potential function are obtained for the isotropic elastic layer and transversely isotropic piezoelectric substrate, respectively, by solving the coupled electromechanical field equations. Firstly, effects of the inhomogeneous initial stress on the dispersion relations and phase velocity of Love wave propagation are discussed. Then the influence of the initial stress gradient coefficient on the stress, mechanical displacement and electrical potential distribution in the layer and the substrate is investigated in detail. The results reported in this paper are not only meaningful for the design of surface acoustic wave (SAW) devices with high performance, but also effective for evaluating the residual stress distribution in the layered structures.     

功能梯度压电层/压电半空间结构中Love波的传播:一种改进的拉盖尔多项式方法

使用传统的拉盖尔多项式方法求解层状半空间结构时,存在因层间材料差异所造成的应力、电位移不连续的现象。为了克服此方法的不足,提出了一种改进的拉盖尔多项式方法,研究了功能梯度压电层状半空间中Love波的传播特性。与文献中应用WKB法得到的结果进行对比,验证了该方法的正确性。计算和分析了相应的频散曲线、应力和电位移分布曲线。结果表明:该方法能够避免因层间材料差异所造成的应力、电位移不连续现象的出现;高频Love波的应力和电位移主要分布在功能梯度压电层中速度较低的一侧。该研究为基于Love波传感器的设计与优化奠定了一定的理论基础。     

On dispersion relations of Rayleigh waves in a functionally graded piezoelectric material (FGPM) half-space

For propagation of Rayleigh surface waves in a transversely isotropic graded piezoelectric half-space with material properties varying continuously along depth direction, the Wentzel–Kramers–Brillouin (WKB) technique is employed for the asymptotic analytical derivations. The phase velocity equations for both the electrically open and shorted cases at the free surface are obtained. Influences of piezoelectric material parameters graded variations on Rayleigh wave dispersion relations, particles’ displacements magnitude and corresponding decay properties are discussed. Results obtained indicate that coupled Rayleigh waves can propagate at the surface of the graded piezoelectric half-space, and their dispersion relations and the particles displacements ellipticity at the free surface are dependent upon the graded variation tendency of the material parameters. By the Rayleigh surface waves phase velocities relative changing values combined with the relationship between the wave number and the material graded coefficient, a theoretical foundation can be provided for the graded material characterization by experimental measurement.     

Rayleigh surface waves problem in linear thermoviscoelasticity with voids

In this paper, we study the propagation of the Rayleigh surface waves in a half-space filled by an exponentially functionally graded thermoviscoelastic material with voids. We take into consideration the dissipative character of the porous thermoviscoelastic models upon the propagation waves and study the damped in time wave solutions. The propagation condition is established in the form of an algebraic equation of tenth degree whose coefficients are complex numbers. The eigensolutions of the dynamical system are explicitly obtained in terms of the characteristic solutions. The concerned solution of the Rayleigh surface wave problem is expressed as a linear combination of the five analytical solutions, while the secular equation is established in an implicit form. The explicit secular equation is obtained for an isotropic and homogeneous thermoviscoelastic porous half-space, and some numerical simulations are given for a specific material.     

Guided Waves in Functionally Graded Rods with Rectangular Cross-Section under Initial Stress

The characteristics of the guided waves propagation in functionally graded rods with rectangular cross-section (finite width and height) under initial stress are investigated in this paper based on Biot’s theory of incremental deformation. An extended orthogonal polynomial approach is present to solve the coupled wave equations with variable coefficients. By comparisons with the available results of a rectangular aluminum rod, the validity of the present approach is illustrated. The dispersion curves and displacement profiles of various rectangular functionally graded rods are calculated to reveal the wave characteristics, and the effects of different width to height ratios and initial stress and gradient functions on the guided waves are discussed in detail.     

Dynamic stress from a cylindrical inclusion buried in a functionally graded piezoelectric material layer under electro-elastic waves

This paper presents a theoretical method to investigate the multiple scattering of electro-elastic waves and dynamic stress around a subsurface cylindrical inclusion in a functionally graded piezoelectric material layer bonded to homogeneous piezoelectric materials. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions around the inclusion. The image method is used to satisfy the mechanical and electrically short conditions at the free surface of the structure. Through the numerical solutions of dynamic stress concentration factors around the inclusion, it is found that when the cylindrical inclusion possesses higher rigidity and greater piezoelectric constant than the two phases of functionally graded materials, the dynamic stress around the inclusion increases greatly. When the distance between the surface of the structure and the inclusion is smaller, the effect of the properties of the inclusion becomes greater. When the cylindrical inclusion possesses lower rigidity and smaller piezoelectric constant than the two phases of functionally graded materials, the maximum dynamic stress shows little difference; however, the variation of the distribution of the dynamic stress around the inclusion is greater. The effect of the properties of the inclusion on the dynamic stress around the inclusion is greater than that on the electric field. The effects of wave frequency on the dynamic stress and electric field are also examined.     

An analysis of surface acoustic wave propagation in functionally graded plates with homotopy analysis method

Wave propagation in solids of functionally graded materials is difficult to analyze due to the resulting differential equations of variable coefficients associated with the spatial variation of the material properties. In this study, the homotopy analysis method is applied to obtain the solution of surface acoustic waves in a plate of functionally graded material. The variation of material properties as a polynomial of the thickness coordinate is studied along with the convergence of the solution. It is found that the homotopy analysis method is effective in solving wave propagation problems arising from functionally graded materials with differential equations of variable coefficients in particular.     

非理想界面功能梯度压电/压磁层状半空间结构中的SH波

研究非理想界面下功能梯度压电/压磁层状半空间结构中SH波的传播特性。界面性能由“弹簧”模型表征, 假设功能梯度压电层材料性能沿层厚度方向指数变化, 其表面为电学开路。推导了频散方程, 并结合数值算例分析了界面性能、功能梯度压电层的梯度变化和厚度对相速度的影响。研究结果对功能梯度压电/压磁复合材料在声波器件中的应用提供了理论依据。     

SH-SAW propagation in layered functionally graded piezoelectric material structures loaded with viscous liquid

We investigate the properties of shear horizontal surface acoustic wave propagation in layered functionally graded piezoelectric material structures loaded with viscous liquid. The piezoelectric material is polarized in the z-direction and the material properties change gradually along the thickness of the layer. Interfacial mechanical conditions are continuity of particle velocity and stress components at the interface. We here assume that the liquid is electrically insulated and its permittivity is much less than that of the piezoelectric material. The solutions of dispersion relations are obtained for insulated liquid with electrically open or shorted conditions by means of transfer matrix method. The effects of the gradient variation of material constants on the phase velocity and attenuation are presented and discussed in detail. The analytical method and the results are useful for the design of the resonators and sensors.     

Non-destructive Detection of a Circular Cavity in a Finite Functionally Graded Material Layer Using Anti-plane Shear Waves

A semi-analytical method is proposed to investigate the non-destructive detection of a circular cavity buried in a functionally graded material layer bonded to homogeneous materials, and the multiple scattering effect of shear waves is described accurately. The image method is used to satisfy the traction free boundary condition at the edge of the functionally graded material layer. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions at the edge and around the cavity. The analytical and numerical solutions of dynamic stress concentration factors around the cavity are presented. The effects of the position of the cavity in the material layer, the incident wave number, and the properties of the two phases of materials on the dynamic stress concentration factors are analyzed. Analyses show that when the buried depth of the cavity and the thickness of the layer are relatively small, the properties of the two phases of materials have great effect on the distribution of dynamic stress around the cavity. In the region of higher frequency, the effects of the position of the cavity and the properties of the two phases of materials on the maximum dynamic stress are greater.     

One-Dimensional Transient Dynamic Response in Functionally Graded Layered Media

In this study, one-dimensional transient wave propagation in multilayered functionally graded media is investigated. The multilayered medium consists of N different layers of functionally graded materials (FGMs), i.e., it is assumed that the stiffness and the density of each layer are varying continuously in the direction perpendicular to the layering, but isotropic and homogeneous in the other two directions. The top surface of the layered medium is subjected to a uniform dynamic in-plane time-dependent normal stress; whereas, the lower surface of the layered medium is assumed free of surface tractions or fixed. Moreover, the multilayered medium is assumed to be initially at rest and its layers are assumed to be perfectly bonded to each other. The method of characteristics is employed to obtain the solutions of this initial-boundary-value problem. The numerical results are obtained and displayed in curves showing the variation of the normal stress component with time. These curves reveal clearly the scattering effects caused by the reflections and refractions of waves at the boundaries and at the interfaces of the layers. The curves also display the effects of non-homogeneity in the wave profiles. The curves further show that the numerical technique applied in this study is capable of predicting the sharp variations in the field variables in the neighborhood of the wave fronts. By suitably adjusting the material constants, solutions for the case of isotropic, homogeneous and linearly elastic multilayered media and for some special cases including two different functionally graded layers are also obtained. Furthermore, solutions for some special cases are compared with the existing solutions in the literature; very good agreement is found.     

功能梯度压电双材料板中厚度-扭曲波的传播

分析了厚度-扭曲波在无限大功能梯度压电双材料板中的传播性能,板的上表面和下表面是机械自由和电学开路的,材料常数在厚度方向按指数规律变化。首先推导了满足控制方程和边界条件的电弹场,然后利用界面条件得到了厚度-扭曲波传播所应满足的关系。通过算例表明了材料梯度变化对厚度-扭曲波传播性能的影响,结果对功能梯度压电材料在声波器件中应用有参考价值。     

Shear-horizontal surface waves on piezoelectric ceramics with depolarized surface layer

Theoretical analysis and numerical results describing the propagation of SH (shear-horizontal) surface waves on piezoelectric ceramics with a depolarized surface layer are described. SH surface waves propagating in piezoelectric ceramics with a depolarized surface layer are shown to be a mixture of the Bleustein-Gulyaev surface wave, electrical potential, and the Love surface-wave mechanical displacement. Depolarization of the surface layer in piezoelectric ceramics produces strong dispersion and a multimode structure of the SH surface wave. The penetration depth of the SH surface waves propagating on an electrically free surface of a piezoelectric ceramic with a depolarized surface layer can be significantly smaller than that of the Bleustein-Gulyaev surface waves propagating on a free piezoelectric half-space. It is concluded that piezoelectric ceramics with a depolarized surface layer can be used in hybrid piezoelectric semiconductor convolvers of reduced size.     

Transverse surface waves in a piezoelectric material carrying a gradient metal layer of finite thickness

The existence and propagation behavior of transverse surface waves in a layered structure concerning a piezoelectric substrate and a gradient metal layer are theoretically investigated in this paper. The Wentzel-Kramers-Brillouin (WKB) method is applied to obtain the analytical solutions in the gradient metal layer. The dispersion equation for transverse surface waves in such structure is obtained in a quite simple mathematical form, where the material gradient of the metal layer assumes arbitrary functions. Effects of material gradient on three types of dispersion behavior are discussed in detail based on a proper classification. Numerical results show that the material gradient in the metal layer evidently affects the fundamental mode shape of the transverse surface waves but has negligible effects on the higher order modes.     

Wave propagation and transient response of a FGM plate under a point impact load based on higher-order shear deformation theory

In this paper, the wave propagation and transient response of an infinite functionally graded plate under a point impact load are presented. The effective material properties of functionally graded materials (FGMs) for the plate are assumed to vary continuously through the plate thickness and be distributed according to a volume fraction power law along the plate thickness. Based on the higher-order shear deformation theory and considering the effect of the rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by means of integral transforms and a complete discussion of dispersion for the functionally graded plate is given. Then, using the dispersion relation and integral transforms, exact integral solutions for the functionally graded plate under a point impact load are obtained. The transient response curves of the functionally graded plates are plotted and the influence of volume fraction distributions on transient response of functionally graded plates is analyzed. Finally, the solutions of the higher-order shear deformation theory and the first-order shear deformation theory are studied.