Calculation of leaky Lamb waves with a semi-analytical finite element method

A semi-analytical finite element method (SAFE) has been widely used for calculating dispersion curves and mode shapes of guided waves as well as transient waves in a bar like structures. Although guided wave inspection is often conducted for water-loaded plates and pipes, most of the SAFE techniques have not been extended to a plate with leaky media. This study describes leaky Lamb wave calculation with the SAFE. We formulated a new solution using a feature that a single Lamb wave mode generates a harmonic plane wave in leaky media. Dispersion curves obtained with the SAFE agreed well with the previous theoretical studies, which represents that the SAFE calculation was conducted with sufficient accuracy. Moreover, we discussed dispersion curves, attenuation curves, and displacement distributions for total transmission modes and leaky plate modes in a single side and both two side water-loaded plate.     

Double-layer PVDF transducer and V(z) measurement system for measuring leaky Lamb waves in a piezoelectric plate

This paper presents a new experimental measurement method for leaky Lamb waves propagating in a piezoelectric plate immersed in a conductive fluid. The measurement system is a low-frequency version of lens-less acoustic microscopy which has been developed based on a line-focus double-layer PVDF transducer. The transducer and its defocusing measurement system can perform V(z) measurements on a sample plate immersed in a fluid, and therefore can obtain the leaky Lamb wave velocities with high accuracy. An X-cut LiNbO₃ plate is investigated with this experimental measurement system to find out its fluid-loading effects, especially the conductive loading effects by water of various conductivities. Angular dependence of this conductive loading effect along different propagating directions on the X-cut LiNbO₃ plate is measured. It is found out the conductive loading effects are strongly dependent on the piezoelectric coupling factor. Theoretical calculations based on partial wave theory have also been carried out and compared with experimental data. Good agreements have been observed.     

Numerical simulation and experimental detection of leaky Lamb waves induced by pulse laser at air-solid interfaces

Based on the thermoelastic mechanism of laser ultrasonic, the problems of the thermal conduction and the coupling between the motion of solid and fluid are solved by using the finite element method. And then the transient waveforms of leaky Lamb waves induced by pulse laser action on the air-aluminum interface are obtained. Experimental signals of laser-induced leaky Lamb waves at the air-aluminum interface are detected by applying an our-developed detector, based on the light deflection principle. The dispersion and attenuation properties of leaky Lamb waves are analyzed through the phase spectral analysis.     

Laser-induced thermoelastic Leaky Lamb waves at the fluid–solid interface

A model based on the theory of fluid–structure interaction is developed to simulate the laser thermoelastic generation and propagation of Leaky Lamb waves at the water–aluminum interface. Each component of displacement, stress, and temperature are derived in transform domain by the photothermoelastic transfer matrix method. The time domain solutions are obtained by numerically inverting the transforms while the dispersion curves and attenuation curves for the leaky waves are also calculated. Then the propagation characteristics of different modes are analyzed. The model establishes a quantitative relation between the laser parameters, the material parameters, the corresponding waveforms, and the dispersion curves, which provides a useful tool for the Leaky Lamb waves applied to nondestructive evaluation.     

利用激光多普勒测振仪的空气耦合超声声场测量*

进行了空气耦合超声声场测量的实验研究。使用激光多普勒测振仪测量由声波引起的激光路径上的介质折射率变化,进而得到空气和固体材料内部的时域瞬态声压。通过直接测量空耦换能器的辐射声功率,给出对单个空耦换能器灵敏度的直接评价方法。用空耦换能器激励K9玻璃板的漏兰姆波,观测到空气中的直达波和反射波、固体板内和空气中的漏兰姆波,实现了空气和固体中微弱声波的非侵入式实验测量,为空耦换能器的特性评估和空耦检测系统的声场测量提供了实验方法。     

铝板中Lamb波检测的实验研究

Lamb波在传播过程中具有频散及多模态特性,若相关参数选择不当,会导致在实际应用中信号相互叠加而无法识别。本文基于Lamb波的频散曲线是其频散方程实数解分布的特点,采用二分法绘制了铝板中Lamb波的频散曲线、波结构曲线和入射角曲线。根据曲线选择S0模态的Lamb波对1mm厚铝板中不同类型的缺陷进行检测。实验结果表明,S0模态的Lamb波对裂纹型缺陷和贯穿型缺陷十分敏感,但对于裂纹型缺陷,其幅值变化并不与缺陷大小成线性关系,并且S0模态Lamb波的声场指向性十分集中,在偏离声束轴线时无法检测到缺陷。     

The band gaps of Lamb waves in a ribbed plate: A semi-analytical calculation approach

In this paper a semi-analytical method is proposed to investigate the transmission band gaps of Lamb waves through repetitive structures in a waveguide. For a unit cell the scattering matrix is obtained by the Lamb mode matching technique at each artificially sliced interface and then substituted into the Bloch theorem to solve the eigenproblem incurred by cell repetition. The method is implemented on a ribbed plate. The band gaps in the dispersion curves are confirmed by the dips of Lamb wave transmission coefficients, and attributed to the coupled vibrations of Lamb modes in different sliced sections of the unit cell.     

轨道板中超声兰姆波传播的有限元计算和实验*

该文针对我国高速铁路轨道板缺陷的非接触动态检测问题,研究了空气耦合超声兰姆波在轨道板中的传播规律。首先,给出了轨道板中超声兰姆波的相速度和群速度频散曲线,结果表明:随着频厚积的增加,频散现象越明显,并且A0相速度收敛于Rayleigh波的波速。然后,建立轨道板中波传播的有限元模型,计算得到兰姆波传播的群速度为2220 m/s,且二维傅里叶变换系数的较大值沿Rayleigh波的频散曲线分布。最后,在沪杭高铁嘉兴南站进行了现场测试,以8.8°倾斜角向轨道板激励产生超声兰姆波,激发产生的兰姆波模态群速度为2325 m/s,且二维傅里叶变换分析其系数的较大值沿Rayleigh波的频散曲线分布。有限元计算结果和实验结果均与理论计算结果一致。该研究为后续轨道板缺陷的非接触动态检测提供了理论依据和实验方法。     

Lamb waves in anisotropic plates (review)

Propagation of Lamb waves in elastic anisotropic plates is studied in the framework of the six-dimensional Cauchy formalism. Closed-form secular equations for dispersion curves for Lamb waves propagating in a plate with arbitrary elastic anisotropy are obtained.     

Determination of ultrasonic wave velocities and phase velocity dispersion curves of an Inconel 600 plate using resonant ultrasound spectroscopy and leaky Lamb waves

A plate of Inconel 600 was interrogated using the resonant ultrasound spectroscopy (RUS) and the reflected leaky Lamb waves (LLW). It was found that the plate used in the present work has anisotropy in its material properties by the RUS. The longitudinal and the transverse wave velocities of the Inconel 600 plate were determined by the RUS, ultrasonic pulse-echo method and cut-off frequencies of the LLWs. The wave velocities in the direction of thickness determined by the RUS under the assumption of the orthotropic symmetry were quite similar to those obtained by other methods, the pulse-echo method and from cut-off frequencies. The reflected LLW from the plate was measured with varying the incident angle. The dispersion curves obtained from the reflected LLWs show good agreement with the theoretical calculation in general. The mismatches may be caused by anisotropy of the plate.     

Localization of notches with Lamb waves

A time-frequency representation (TFR) is used to analyze the interaction of a multimode and dispersive Lamb wave with a notch, and then serves as the basis for a correlation technique to locate the notch. The experimental procedure uses a laser source and a dual-probe laser interferometer to generate and detect Lamb waves in a notched plate. The high fidelity, broad-bandwidth, point-like and noncontact nature of laser ultrasonics are critical to the success of this study, making it possible to experimentally measure transient Lamb waves without any frequency biases. A specific TFR, the reassigned spectrogram, is used to resolve the dispersion curves of the individual modes of the plate, and then the slowness-frequency representation (SFR) of the plate is calculated from this reassigned spectrogram. By considering the notch to be an additional (second) source, the reflected and transmitted contributions of each Lamb mode are automatically identified using the SFRs. These results are then used to develop a quantitative understanding of the interaction of an incident Lamb wave with a notch, helping to identify mode conversion. Finally, two complementary, automated localization techniques are developed based on this understanding of scattering of Lamb waves.     

3D analysis of interaction of Lamb waves with defects in loaded steel plates

The objective of the research presented here is the investigation of the interaction of guided waves with welds, defects and other non-uniformities in steel plates loaded by liquid. The investigation has been performed using numerical simulation for 2D and 3D cases by the finite differences method, finite element method and measurement of 3D distributions of acoustic fields. Propagation of the S(0) mode in a steel plate and its interaction with non-uniformities was investigated. It was shown that using the measured leaky wave signals in the water loading of the steel plate and by application of signal processing, the 3D ultrasonic field structure inside and outside of the plate can be reconstructed. The presence of leaky wave signals over the defect caused by the mode conversion of Lamb waves has been proved using the numerical modelling and experimental investigations. The developed signal and data processing enables to visualise dynamics of ultrasonic fields over the plate, and also to estimate spatial positions of defects inside the steel plates.     

Generalized thermoelastic Lamb waves in a plate bordered with layers of inviscid liquid

The propagation of thermoelastic waves in a homogeneous isotropic, thermally conducting plate bordered with layers of inviscid liquid or half-space of inviscid liquid on both sides is investigated in the context of generalized theories of thermoelasticity. Secular equations for the plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave modes in completely separate terms are derived. The results for coupled and uncoupled theories of thermoelasticity have been obtained as particular cases. The different regions of secular equations are obtained and special cases, such as Lame modes, thin plate waves and short wavelength waves of the secular equations are also discussed. The secular equations for thermoelastic leaky Lamb waves are also obtained and deduced. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for an aluminum-epoxy composite and aluminum materials plate bordered with water. The dispersion curves for symmetric and antisymmetric thermoelastic wave modes and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement. The results have been deduced and compared with the relevant publications available in the literature at the relevant stages of the work.     

Measurement of cylindrical Rayleigh surface waves using line-focused PVDF transducers and defocusing measurement method

Line-focused PVDF transducers and defocusing measurement method are applied in this work to determine the dispersion curve of the Rayleigh-like surface waves propagating along the circumferential direction of a solid cylinder. Conventional waveform processing method has been modified to cope with the non-linear relationship between phase angle of wave interference and defocusing distance induced by a cylindrically curved surface. A cross correlation method is proposed to accurately extract the cylindrical Rayleigh wave velocity from measured data. Experiments have been carried out on one stainless steel and one glass cylinders. The experimentally obtained dispersion curves are in very good agreement with their theoretical counterparts. Variation of cylindrical Rayleigh wave velocity due to the cylindrical curvature is quantitatively verified using this new method. Other potential applications of this measurement method for cylindrical samples will be addressed.     

基于频散补偿和分数阶微分的多模式兰姆波分离

由于兰姆波的多模和频散特性,实际检测时在同一激发频率下存在多种模式的混合信号,而各模式信号有不同的频散特性,使得在时频混叠的情况下兰姆波的检测变得十分复杂.本文在频散补偿的基础上,通过时延函数建模,依靠不同模式频散趋势的差异性,将时频混叠信号的分离问题转化为部分模式混叠信号的分离问题.基于分数阶微分的理论,用信号幅值谱分数阶微分极大值和对应频率分别与微分阶次拟合多项式实现特征参数的提取并依靠特征参数重建幅值谱.结合相位谱重构时域信号以实现部分混叠信号中频散补偿后的模式的分离.最后恢复频散获得分离后的兰姆波信号.仿真和实验结果表明,本文方法不仅可以实现时频混叠多模式兰姆波信号的分离,更能保证分离精度,有助于复杂多模式频散信号的分离与处理的进一步研究.     

多通道超声兰姆波检测板状结构中的裂纹

超声兰姆(Lamb)波在结构缺陷检测方面愈来愈受到重视,但目前Lamb波的应用局限于单信号源激励,单通道接收的方法,容易受到噪声干扰,其后续的信号分析处理也比较复杂。本文旨在采用多通道Lamb波对板状结构中的裂纹进行定量分析与诊断。在铝板的表面凿刻出不同深度的凹槽作为裂纹,通过线阵换能器采集在一定传播距离内的多通道Lamb波信号,并采用二维傅里叶变换分析在不同深度的裂纹下,Lamb波模式能量的变化规律。结果表明,相对于完好铝板中的Lamb波信号,裂纹的存在会使Lamb波发生模式转换现象,并且转换模式能量百分比随裂纹深度的增加而线性增加。其结果为Lamb波评价板状结构中的裂纹状况提供了一种可能的方法。     

Determination of the elastic constants of a composite plate using wavelet transforms and neural networks

An inverse method based on a combination of the wavelet transform and artificial neural networks is presented. The method is used to recover the elastic constants of a fiber-reinforced composite plate from experimental measurements of ultrasonic Lamb waves generated and detected with lasers. In this method, the elastic constants are not recovered from the dispersion curves but rather directly from the measured waveforms. Transient waveforms obtained by numerical simulations for different elastic constants are used as input to train the neural network. The wavelet transform is used to extract the eigenvectors from the Lamb wave signals to simplify the structure of the neutral network. The eigenvectors are then introduced into a multilayer internally recurrent neural network with a back-propagation algorithm. Finally, experimental waveforms recoded on a titanium-graphite composite plate are used as input to recover the elastic constants of the material.     

Wideband dispersion removal and mode separation of Lamb waves based on two-component laser interferometer measurement

Ultrasonic Lamb waves are considered as a sensitive and effective tool for nondestructive testing and evaluation of plate-like or pipe-like structures. The nature of multimode and dispersion causes the wave packets to spread, and the modes overlap in both time and frequency domains as they propagate through the structures. By using a two-component laser interferometer technique, in combination with a priori knowledge of the dispersion characteristics and wave structure information of Lamb wave modes, a two-component signal processing technique is presented for implementing dispersion removal and mode separation simultaneously for two modes mixture signals of Lamb waves. The proposed algorithm is first processed and verified using synthetic Lamb wave signals. Then, the two-component displacements test experiment is conducted using different aluminum plate samples. Moreover, we confirm the effectiveness and robustness of this method.     

Scattering of plane acoustic waves by a transversely isotropic cylindrical shell—application to material characterization

Scattering of obliquely incident plane acoustic wave by an air-filled, transversely isotropic cylindrical shell immersed in water is investigated theoretically and experimentally. The normal mode expansion technique is employed for analyzing the scattering field, and then resonance modes of the shell appearing in modal scattering form functions are identified performing the resonance scattering analysis. Dispersion curves for Sholte-Stoneley, SH and Lamb waves are obtained and their characteristics are interpreted. Calculated backscattering and resonance spectra as well as dispersion curves are compared with those from ultrasonic experiments for two composite samples having the same nominal composition but fabricated under different conditions. Sensitive change of the dispersion curves is observed for both normal and oblique incidences, which demonstrates the feasibility of systematic inverse evaluation of damage or elastic constants of the composite shell using data from the acoustic scattering measurements.     

Ultrasonic plate waves in paper

A theoretical and experimental study of ultrasonic plate waves in machine-made paper is described. The paper is assumed to behave as a homogeneous orthotropic plate. The dispersion equation for orthotropic plate waves in principal directions is then developed analytically. It is shown that at low frequencies the orthotropic dispersion equation depends only on in-plane parameters. Using appropriate elastic constants, a computer is used to construct the dispersion curves and describe the internal motion for plate waves in paper. Finally, a portion of the dispersion curve is checked experimentally.