Properties of Love waves in layered piezoelectric structures

Love waves propagating in a layered structure with an elastic layer deposited on a piezoelectric substrate are analytically investigated. We present a general dispersion equation that describes the properties of Love waves in the structure. A detailed discussion regarding the dispersion equation is presented, and the parameters for Love-mode sensors are also introduced. The properties of Love waves are illustrated by means of sample results for a layered structure with an SiO₂ layer sputtered on an ST-cut 90°X-propagating quartz substrate. Interestingly, we found that a threshold-normalized layer thickness existed for the fundamental Love mode in such a structure.     

Dispersion in oceanic crust during earthquake preparation

Dispersion of Stoneley waves is studied in a sedimentary layer of ocean bottom resting over basaltic solid half space. Sedimentary layer is assumed a transversely isotropic poroelastic medium. Lower-most solid half-space is assumed to be embedded with vertically aligned saturated micro-cracks and behaves transversely isotropic to wave propagation.Frequency equation is obtained in the form of determinantal equation. Role of phase angle is eliminated by expressing slowness of waves in terms of phase velocity and elastic constants. Numerical solutions for phase velocity and group velocity are obtained for a particular model. Calculations are made for different depths of ocean and sediments. Effect of thickness and density of cracks on these velocities are observed.Special cases are discussed which represent the absence of ocean and sediments, in the model considered. Changes in dispersion are discussed during the stress accumulation in an earthquake preparation region.     

Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.     

Propagation of SH waves in an irregular monoclinic crustal layer

The present paper discusses the dispersion equation for SH waves in a monoclinic layer over a semi-infinite elastic medium with an irregularity. In the absence of the irregularity, the dispersion equation reduces to standard dispersion equation for SH waves in a monoclinic layer over an isotropic semi-infinite medium. The dispersion curves for different size of the irregularity are computed and compared for the half-space without any irregularity. It can be seen that the phase velocity is strongly influenced by the wave number and the depth of the irregularity.     

功能梯度材料板中Lamb波传播特性研究

对材料性能参数沿厚度连续变化的横观各向同性热应力缓和型功能梯度材料板中Lamb波的传播问题,采用幂级数法,求得其相速度方程.借助数值算例,分析了参数梯度变化对Lamb波频散曲线的影响,并与相应陶瓷板和金属板中的频散曲线进行了对比.进一步研究了参数梯度变化对波结构的影响,揭示了Lamb波在这种非均质板中的传播行为,所得结果可以为功能梯度材料及结构的超声表征与检测提供理论依据.     

梯度半空间梯度覆层中的Love波

对功能梯度弹性半空间上覆盖一层功能梯度材料中的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. 对功能梯度弹性半空间和功能梯度覆层的反平面剪切波的运 动控制方程进行了求解，给出了半空间和覆盖层的位移、应力解析解，给出了Love波在该解析 解下的频散方程. 以覆盖层的剪切弹性模量和质量密度均呈指数函数变化，半空间的剪切弹 性模量和质量密度均呈抛物线变化为例，利用迭代方法对频散方程进行了求解，给出了频散 曲线. 结果显示：在最低阶振型频散曲线中出现截止频率.     

Rayleigh lamb waves in micropolar isotropic elastic plate

The propagation of waves in a homogeneous isotropic micropolar elastic cylindrical plate subjected to stress free conditions is investigated. The secular equations for symmetric and skew symmetric wave mode propagation are derived. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduces to Rayleigh surface wave frequency equation. Thin plate results are also obtained. The amplitudes of displacements and microrotation components are obtained and depicted graphically. Some special cases are also deduced from the present investigations. The secular equations for symmetric and skew symmetric modes are also presented graphically.     

Love waves in functionally graded piezoelectric materials

To investigate the features of Love waves in a layered functionally graded piezoelectric structure, the mathematical model is established on the basis of the elastic wave theory, and the WKB method is applied to solve the coupled electromechanical field differential equation. The solutions of the mechanical displacement and electrical potential function are obtained for the piezoelectric layer and elastic substrate. The dispersion relations of Love waves are deduced for electric open and short cases on the free surface respectively. The actual piezoelectric layer–elastic substrate systems are taken into account, and some corresponding numerical examples are proposed comparatively. Thus, the effects of the gradient variation about material constants on the phase velocity, the group velocity, the coupled electromechanical factor and the cutoff frequency are discussed in detail. So the propagation behaviors of Love waves in inhomogeneous medium is revealed, and the dispersion and the anti-dispersion are analyzed. The conclusions are significant both theoretically and practically for the surface acoustic wave devices.     

功能梯度压电圆板自由振动问题的三维精确分析

本文对周边为广义刚性滑动和广义简支两种边界条件下的功能梯度压电材料圆板自由振动问题进行分析。根据轴对称横观各向同性压电材料基本方程,并利用有限Hankel变换得到了功能梯度压电材料圆板的状态空间方程。假设材料的机械和电学性质均沿板厚方向按统一的指数函数形式梯度分布,从而获得了周边为广义刚性滑动和广义弹性简支两种边界条件下功能梯度压电圆板自由振动问题的三维精确频率方程,该方程是一个关于自由振动频率的超越方程,通过求解该超越方程可得到在不同板厚以及不同的材料性质梯度变化情况下的圆板自由振动频率值,结果表明在相同的材料性质梯度变化情况下频率均随着板厚增加而增大,而在相同的板厚情况下频率则随材料性质梯度变化指数的增大而减小的结论。     

AN ACCURATE TWO-DIMENSIONAL THEORY OF VIBRATIONS OF ISOTROPIC,ELASTIC PLATES

An infinite system of two-dimensional equations of motion of isotropic elastic plates with edge and corner conditions are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions and a linear function of the thickness coordinate of the plate. The linear term in the expansion is to accommodate the in-plane displacements induced by the rotation of the plate normal in low-frequency flexural motions. A system of first-order equations of flexural motions and accompanying boundary conditions are extracted from the infinite system. It is shown that the present system of equations is equivalent to the Mindlin’s first-order equations, and the dispersion relation of straight-crested waves of the present theory is identical to that of the Mindlin’s without introducing any corrections. Reduction of present equations and boundary conditions to those of classical plate theories of flexural motions is also presented.     

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.     

The spreading of nonlinear elastic surface waves

A theory for the lateral spreading of a beam of nonlinear surface acoustic waves across the surface of an arbitrary, homogeneous, elastic half-space is developed. The resulting evolution equation generalizes that obtained for uni-directional waves by replacing an ordinary derivative by a diffusion operator of Schrödinger type. The coefficients arising in the evolution equation are related to partial derivatives of the dispersion relation for linearized surface waves on the half space. Details are given for isotropic materials and for two special cases of beams travelling along axes of high elastic symmetry.     

On coalescence of frequencies and conical points in the dispersion spectra of elastic bodies

In this paper we discuss a general procedure for determining the critical points of the dispersion spectrum at which there is a coalescence of frequencies, i.e. critical points which are roots of double multiplicity. We further show how the general behavior of the dispersion surface in the neighborhood of the critical points can be determined analytically. For the purpose of illustration, we consider (a) plane waves propagating in an infinite, elastic, isotropic plate, which corresponds to the case of a differential equation with constant coefficients, and (b) Floquet waves of the SH-type propagating in a layered, elastic medium, which corresponds to the case of a differential equation with periodic coefficients.     

Wave propagation in layer with two preferred directions

This article is concerned with overall or macroscopic properties of a composite material with no distinction made between the fibres and the matrix which they are embedded in. All the properties with dimensions larger than the fibre diameter and spacing are regarded as averaged over a volume of material. The systems of particular interest here are in the fibre reinforced composites with the fibres being very much stiffer and stronger than the matrix.Laminated plates of fibre-reinforced material are often fabricated from prepreg tapes, laid up according to some specific arrangement of fibre orientation and then bonded together. An angle-ply laminate is formed by alternating plies so that the families in adjacent laminas are inclined by angle ϕ and −ϕ to given direction alternately. The process of fabricating a multilayered plate of this material gives rise to a laminate in which the plies are separated by resin rich layer, and when this layer is thin enough that its thickness is negligible it may be regarded as plate reinforced by two families of fibres. Problems shall be considered in three dimensions, but attention shall be restricted to linear elasticity theory. The plate under consideration is reinforced by two mechanically equivalent families of fibres, but with no other preferred directions, so that it is locally orthotropic with respect to the plane of the fibres and to the two planes that orthogonally bisect the fibres.In this article linear elastic stress–strain relation is employed to derive dispersion curves for plane harmonic waves propagating in a plate of finite thickness but of infinite lateral extent. Attention is restricted to waves propagating in the plane parallel to stress free plate faces where waves travelling at any angle to one of the families of very strong fibres are examined. The dispersion equations, relating the phase velocity to the wavelength, are obtained. The fundamental modes are examined for symmetric as well as for anti-symmetric deformations. This leads to full understanding of displacement field as well as stress field.     

Axisymmetric free vibrations of infinite thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Axisymmetric free vibrations of infinite micropolar thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Modified equations of finite-size layered plates made of orthotropic material. Comparison of the results of numerical calculations with analytical solutions

This paper describes the modified bending equations of layered orthotropic plates in the first approximation. The approximation of the solution of the equation of the three-dimensional theory of elasticity by the Legendre polynomial segments is used to obtain differential equations of the elastic layer. For the approximation of equilibrium equations and boundary conditions of three-dimensional theory of elasticity, several approximations of each desired function (stresses and displacements) are used. The stresses at the internal points of the plate are determined from the defining equations for the orthotropic material, averaged with respect to the plate thickness. The construction of the bending equations of layered plates for each layer is carried out with the help of the elastic layer equations and the conjugation conditions on the boundaries between layers, which are conditions for the continuity of normal stresses and displacements. The numerical solution of the problem of bending of the rectangular layered plate obtained with the help of modified equations is compared with an analytical solution. It is determined that the maximum error in determining the stresses does not exceed 3 %.     

An asymptotic strain gradient Reissner-Mindlin plate model

In this paper we derive a strain gradient plate model from the three-dimensional equations of strain gradient linearized elasticity. The deduction is based on the asymptotic analysis with respect of a small real parameter being the thickness of the elastic body we consider. The body is constituted by a second gradient isotropic linearly elastic material. The obtained model is recognized as a strain gradient Reissner-Mindlin plate model. We also provide a mathematical justification of the obtained plate model by means of a variational weak convergence result.     

非理想界面弹性层/压电柱结构中SH波的传播特性

研究了各向同性弹性层与压电柱之间非理想连接时沿周向传播的SH波的频散特性.弹性层表面力学自由;弹性层与压电柱之间应力连续、位移间断.通过求解控制方程,将问题的解用Bessel函数表示,利用界面条件和边界条件得到频散方程,然后对其进行数值求解,分析了界面性态、材料常数和几何尺寸对SH波传播特性的影响.     

A physical perspective of the length scales in gradient elasticity through the prism of wave dispersion

The goal of this study is to understand the physical meaning and evaluate the intrinsic length scale parameters, featured in the theories of gradient elasticity, by deploying the analytical treatment and experimental measurements of the dispersion of elastic waves. The developments are focused on examining the propagation of longitudinal waves in an aluminum rod with periodically varying cross-section. First, the analytical solution for the dispersion relationship, based on the periodic cell analysis of a bi-layered laminate and Bloch theorem, is compared to two competing models of gradient elasticity. It is shown that the customary gradient elastic model with two length-scale parameters is able to capture the dispersion accurately up to the beginning of the first band gap. On the other hand, the gradient elastic model with an additional length scale (affiliated with the fourth-order time derivative in the field equation) is shown to capture not only the first dispersion branch before the band gap, but also the band gap itself and the preponderance of the second branch. Closed form relations between the microstructure parameters and the intrinsic length scales are obtained for both gradient elasticity models. By way of the asymptotic treatment in the limit of a weak contrast between the laminae, a clear physical meaning and scaling of the length-scale parameters was established in terms of: (i) the microstructure (given by the size of the unit cell and the contrast between the laminae), and (ii) thus induced dispersion relationship (characterized by the location and the width of the band gap). The analysis is verified through an experimental observation of wave dispersion, and wave attenuation within the band gap. A comparison between the analytical treatment, the gradient elastic model with three intrinsic length scales, and experimental measurements demonstrates a good agreement over the range of frequencies considered.