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.     

Lamb waves in a thin isotropic layer between two anisotropic layers

Attenuative Lamb wave propagation in adhesively bonded anisotropic composite plates is introduced. The isotropic adhesive exhibits viscous behavior to stimulate the poor curing of the middle layer. Viscosity is assumed to vary linearly with frequency, implying that attenuation per wavelength is constant. Attenuation can be implemented in the analysis through modification of elastic properties of isotropic adhesive. The new properties become complex, but cause no further complications in the analysis. The characteristic equation is the same as that used for the elastic plate case, except that both real and imaginary parts of the wave number (i.e., the attenuation) must be computed. Based on the Lowe's solution in finding the complex roots of characteristic equation, the effect of longitudinal and shear attenuation coefficients of the middle adhesive layer on phase velocity dispersion curves and attenuation dispersion curves of Lamb waves propagating in bonded anisotropic composites is visualized numerically.     

基于谱方法分析有阻尼负载圆柱壳频散特性

以Chebyshev多项式系为基函数,采用谱方法离散弹性理论的波动方程,建立对应的广义特征值问题。依据壳体结构波运动、内部流体及外部阻尼材料在界面处的位移、应力连续条件,构造此复杂圆柱壳系统广义特征值方程。通过数值求解特征值获得对应频率下波数,进而获得圆柱壳结构的频散曲线。分别讨论充水与否、有阻尼负载圆柱壳的频散曲线,获得有价值结论。     

Elastodynamic Behavior of Waves in Thermo-Microstretch Elastic Plate Bordered with Layers of Inviscid Liquid

The elastodynamic behavior of waves in a thermo-microstretch elastic homogeneous isotropic plate bordered with layers of inviscid liquid on both sides subjected to stress-free thermally insulated and isothermal conditions is investigated in the context of Lord and Shulman and Green and Lindsay theories of thermoelasticity. The mathematical model has been simplified by using the Helmholtz decomposition technique, and the frequency equations for different mechanical situations are obtained and discussed. The special cases such as short wavelength waves and regions of the secular equations are also discussed. Finally, the numerical solution is carried out for a magnesium crystal composite material plate bordered with water. The dispersion curves, attenuation coefficients, amplitudes of dilatation, microrotation, microstretch, and temperature distribution for the symmetric and skew-symmetric wave modes are presented graphically.     

Magneto-thermo-elastic plane waves in rotating media

A study is made of the propagation of plane harmonic waves in an infinite conducting thermo-elastic solid permeated by a primary uniform magnetic field when the entire elastic medium is rotating with a uniform angular velocity. A more general dispersion relation is obtained to determine the effects of rotation, relaxation time and the external magnetic field on the phase velocity of the waves. This result indicates that if the primary magnetic field has a transverse component, then the longitudinal and transverse components of the displacement field are linked together. For the case of low frequency (Gc ? 1 where κ is the ratio of the wave frequency to the characteristic frequency), the rotation and the thermal relaxation time are found to have no influence on the phase velocity, and the attenuation factor for both finite and infinite electrical conductivity. In the case of high frequency (κ ? 1), no effect of rotation on the phase speed is observed to the first order of ($\text{1}\text{κ}$), while the relaxation time affects both the phase velocity and the specific energy loss. However, the effects of rotation on the phase speed, the attenuation factor, and the specific energy loss are found only to the second order of ($\text{1}\text{κ}$) in the case of high frequency. Several limiting cases of interest are discussed.     

碳纤维增强复合材料层合板Lamb波衰减特性研究

为提取适用于碳纤维增强复合材料层合板声发射故障诊断的模态信号,利用三维弹性理论及传递矩阵法获得Lamb波的频散曲线。以碳纤维增强复合材料层合板为研究对象搭建实验平台,改变断铅激励位置从而获得不同声发射信号。对采集的声发射信号进行小波尺度谱分析,结合频散曲线分离出不同模式的Lamb波,分别研究其不同频率的幅度及能量衰减特性。实验结果表明,较其它信号,低频率S0波幅度信号衰减速度较低,对碳纤维增强复合材料层合板的声发射故障诊断研究具有较大优势。     

Modelling of nonlinear dilatation response of fluids containing columns plastic and shear relaxation considered

Summary The non-linear wave equation governing the propagation and scatter of dilatation waves in discontinuous media is presented and its Eulerian numerical analogue is used to study scatter of acoustic dilatation waves by colums and ducts in elastic and visco-elastic fluids. Plastic and visco-elastic relaxation mechanisms are considered. The spectral form of the wave equation is developed and used to discuss dispersion. Selected applications of the numerical analogue to simulation of scatter, propagation and echo attenuation presented.     

黏弹性正交各向异性空心圆柱中纵向导波的传播

基于线性三维弹性理论和Kelvin-Voigt模型,采用勒让德正交多项式展开法推导了黏弹性正交各向异性空心圆柱中纵向导波的波动方程,数值求解了波动方程并阐述了相关方程的含义。首先计算了大径厚比下黏弹性管的相速度频散曲线和衰减曲线,并与已发表文献的结果进行了对比,验证了程序的正确性,并进一步计算了低阶纵向导波的位移分布和应力分布曲线,验证了方法的可靠性。然后利用方程的解耦特性,分别求解了不同径厚比、不同黏性常数下纵向模态和扭转模态的频散和衰减曲线,研究了径厚比和黏性常数效应对两种模态的影响。最后针对扭转模态导波,研究了材料相关黏弹性常数对其频散特性和衰减特性的影响。      

Dispersion and characteristic surfaces of waves in hybrid multilayered piezoelectric circular cylinders

The frequency and group velocity dispersion behaviors, and characteristic surfaces of waves in a hybrid multilayered piezoelectric circular cylinder are investigated. The associated frequency dispersion equation is developed using an analytical-numerical method. In this method, the cylinder is modeled using the three-nodal-line layer element; the coupling between the elastic field and the electric field is considered in each element. A system of governing differential equations of each layer element is obtained following the Hamilton Principle. The phase velocity and slowness as well as the group velocity and slowness are established in terms of the Rayleigh quotient. Six characteristic wave surfaces, e.g. the phase velocity, slowness and wave surfaces as well as the group velocity, slowness and wave surfaces, are introduced to visualize the effects of anisotropy and piezoelectricity on wave propagation. A corresponding program code is developed and numerical examples are presented for hybrid multilayered piezoelectric circular cylinders with two ratios of radius to thickness.     

Wave propagation in an asymmetric fibre-reinforced laminated plate

Summary This paper examines the propagation of waves in a 2-ply laminate composed of a transversely isotropic elastic material with the fibre direction in the two plies being at right angles to each other. Attention is confined to waves propagating in the fibre direction in one of the plies. The dispersion equation, relating the wave speed to the wavelength, is derived and is solved numerically for a number of different values of the ratio of the thickness of the two layers. The long and short wavelength behaviour is examined analytically. Comparisons are made between this asymmetric composite and a symmetric 3-ply laminate composed of the same material. Results are plotted for the fundamental mode and first harmonic of the dispersion relation of the asymmetric laminate. The high frequency limit of the dispersion curve for the fundamental mode is the velocity of a Rayleigh surface wave in the cross-fibre direction, whilst that of the first harmonic is the velocity of a shear wave in the cross-fibre direction. Corresponding results are obtained for the ideal material which is inextensible in the direction of transverse isotropy, and these are compared with those for the extensible material.With 3 Figures     

非饱和地基中Love波的传播特性

基于非饱和多孔介质的波动方程,考虑了土中水,气体与土骨架之间的粘性耦合作用,建立了弹性半空间上非饱和土层中Love波的弥散方程。首先分析了饱和度与频率对非饱和孔隙介质中剪切波速的影响。然后运用数值方法得到了不同饱和度下土层中多种Love模态波的弥散特性和位移分布情况,并用图表的形式给出。数值计算结果表明,上覆非饱和土层中Love波的传播速度和衰减系数不仅具有频散性,而且与土层的饱和度有关。在不同饱和度时的高模态(n≥2)的Love波的截止频率值不同。此外,讨论了饱和度对Love波水平位移幅值的影响。     

狭缝空腔间粘弹性体中弹性波传播特性的研究

利用笛卡尔坐标下的Kelvin-Voigt线性粘弹性模型,研究了无限长粘弹性狭缝通道中波的传播和衰减。利用自适应绕数求根方法,针对具有狭缝形空腔周期性分布覆盖层的吸声问题,求得频散方程的根,并且得到无限长狭缝空腔中相速度频散曲线和衰减曲线。分析了各阶模式对无限长粘弹性狭缝结构中波传播的相速度和衰减的影响。通过引入复粘弹性模量,粘弹性波动方程具有弹性波动方程相对应的形式,计算中可用复值粘弹性模量代替相应的弹性模量。粘弹性狭缝结构中所有波传播模式均存在衰减,高阶波在某个频率以下衰减非常大,而在该频率以上逐渐减小;最低阶模式的相速度与材料无损耗的情况非常接近。粘弹性狭缝结构中波传播的衰减特性不仅与狭缝的结构参数有关,还与粘弹性材料本身的复弹性模量有关。     

Assessment of viscous and elastic properties of sub-wavelength layered soft tissues using shear wave spectroscopy: theoretical framework and in vitro experimental validation

In elastography, quantitative imaging of soft tissue elastic properties is provided by local shear wave speed estimation. Shear wave imaging in a homogeneous medium thicker than the shear wavelength is eased by a simple relationship between shear wave speed and local shear modulus. In thin layered organs, the shear wave is guided and thus undergoes dispersive effects. This case is encountered in medical applications such as elastography of skin layers, corneas, or arterial walls. In this work, we proposed and validated shear wave spectroscopy as a method for elastic modulus quantification in such layered tissues. Shear wave dispersion curves in thin layers were obtained by finite-difference simulations and numerical solving of the boundary conditions. In addition, an analytical approximation of the dispersion equation was derived from the leaky Lamb wave theory. In vitro dispersion curves obtained from phantoms were consistent with numerical studies (deviation <1.4%). The least-mean-squares fitting of the dispersion curves enables a quantitative and accurate (error < 5% of the transverse speed) assessment of the elasticity. Dispersion curves were also found to be poorly influenced by shear viscosity. This phenomenon allows independent recovery of the shear modulus and the viscosity, using, respectively, the dispersion curve and the attenuation estimation along the propagation axis.     

Surface waves in an elastic half-space with voids

Summary Surface waves of general type propagating in a plane-faced, homogeneous and isotropic linear elastic halfspace containing a distribution of vacuous pores (voids) are studied. Assuming that the face of the halfspace is free of external loads, it is found that the motion is not necessarily two-dimensional and that all physical variables associated with the waves are derivable from a single scalar function. The phase speed equation reveals that, unlike in the classical problem, the waves are dispersive, in general. If the frequency is very small, the motion is qualitatively similar to that in the classical problem, but the wave speed is modified due to the presence of voids.     

粘弹正交各向异性空心圆柱体周向SH波

基于Kelvin-Voight模型,推导了粘弹正交各向异性空心圆柱体周向SH波控制方程,得出了SH波特性受两个弹性常数与粘性系数的影响。通过第一类和第二类Bessel函数,求出频散方程。计算了四种不同径厚比下的相速度频散曲线和衰减曲线,分析了径厚比的变化对频散和衰减曲线的影响。为了与正交各向异性相比较,计算了粘弹各向同性空心圆柱体中周向SH波,并讨论了两个弹性常数和粘性系数对频散曲线和衰减曲线的影响。     

Love type waves in a porous layer with irregular interface

The dispersion equation is derived relating to the frequency and the phase velocity of propagation of Love waves in a nondissipative liquid filled porous solid underlain by an isotropic and homogeneous half space. The rectangular irregularity in the interface between the upper porous layer and the lower semi-infinite medium with a source in it is studied herein. The modified dispersion equation of Mal and the standard dispersion equation of Love waves are deduced as particular cases. In the present study, the frequency equation is obtained by applying the method of perturbation and the phase velocity curves have been drawn for different irregularities by using the numerical parameteric values as suggested by Biot.     

饱和土中Love波弥散特性

本文利用有限元法推导了饱和土中Ｌｏｖｅ面波的弥散特性方程，讨论了饱和土中Ｌｏｖｅ面波波速度的弥散特性及位移分布规律，并与相应弹性地基Ｌｏｖｅ面波的波速度弥散曲线进行了比较。     

Wave dispersion and attenuation in fiber composites

 This work deals with the dispersion and attenuation of elastic plane waves propagating in a single layer fiber reinforced composite, in a direction which is perpendicular to the fibers. An iterative effective medium model, based on single scattering considerations, for the quantitative estimation of wave dispersion and attenuation is proposed. The single scattering problem is solved numerically by means of a 2-D boundary element methodology. Numerical results concerning the plane velocity and the attenuation coefficient of longitudinal or transverse SH, SV waves propagating in two types of fiber reinforced composite materials, are presented. The obtained results are compared to those taken either experimentaly or numericaly by other investigators.     

An explicit numerical time domain formulation to simulate pulsedpressure waves in viscous fluids exhibiting arbitrary frequency powerlaw attenuation

An explicit time domain algorithm is developed which is capable of numerically simulating pulsed pressure waves propagating through media whose attenuation increases with frequency according to a power law dependency. Because of possible noninteger exponents in the power law formulation, standard temporal differential operators cannot be defined and traditional finite difference approximations are therefore inappropriate. We derive a method that is consistent with the fact that the complex wavenumber often must include a nonlinear phase if system causality is to be ensured. This phase is derived from the power law attenuation and, due to their interdependency, the two terms in the wave equation corresponding to attenuation and phase can be combined into a single factor. This so-called dispersion wave equation is mapped into complex discrete-time frequency. In this domain, noninteger exponents can be eliminated via a power series expansion, and the resulting equations transform naturally to discrete time operators. The algorithm is tested by comparing numerically evaluated attenuation with the exact power law form, and the issues of stability in relation to the convergence of the power series and the accuracy of the mapping are investigated     

Various numerical techniques for analysis of longitudinal wave propagation in inhomogeneous one-dimensional waveguides

Summary.  This paper presents various numerical techniques for the analysis of the one dimensional wave equation with variable coefficients, for governing materials with spatially varying thermal, inertial and elastic properties. Approximate time domain Finite Elements (FE) and frequency domain Spectral Finite Elements (SFE) are formulated, being computationally efficient, accurate and fast convergent compared to any other existing tool. The Sturm-Liouville problem corresponding to this wave equation is discussed and numerically solved using Prüfer transformation, and the effect of inhomogeneity on the natural frequencies and mode shapes is elicited. As an application, Functionally Graded Materials (FGM), where the material properties are tailored to vary in spatial directions, are chosen, and the longitudinal wave propagation in an FGM rod due to high frequency impact load is studied. As a possible application of FGM, attenuation of waves reflection at a joint of dissimilar materials is investigated. The notion of inhomogeneous waves, used in dissipative media of more than one dimension, is retained in this one-dimensional approximation. Received October 10, 2002; revised January 13, 2003 Published online: May 20, 2003