Dispersion relations and modes of wave propagation in inclusion-reinforced composite plates

This paper is intended to examine the effect of inclusion shapes, inclusion contents, inclusion elastic constants, and plate thickness on the dispersion relations and modes of wave propagation in inclusion-reinforced composite plates. The shape of inclusion is modeled as spheroid that enables the composite reinforcement geometrical configurations ranging from sphere to short and continuous fiber. Mori–Tanaka mean-field theory is used to predict the effective elastic moduli of the composite plate explicitly. The effective elastic moduli are able to elucidate the effect of inclusion’s shape, stiffness, and volume fraction on the composite’s anisotropic elastic behavior. The resulting moduli are then used to determine the dispersion relations and the modal patterns of Lamb waves using the dynamic stiffness matrix method. The types (symmetric or antisymmetric) of Lamb waves in an isotropic plate can be classified according to the wave motions are symmetrical or antisymmetric about the midplane of the plate. Classifying the wave type in an anisotropic plate is not as simple as that in an isotropic plate, and has not received proper attention in the literature. The wave types and orders are identified by analyzing the dispersion curves and inspecting the calculated modal patterns, and the results indicate that the Lamb waves in an orthotropic composite plate can also be classified as either symmetric or antisymmetric waves. It is also found that the inclusion contents, aspect ratios and plate thickness affect propagation velocities, higher-order mode cutoff frequencies, and modal patterns. Propagation speed is generally increased with the aspect ratio, e.g., using longer fibers generally results in a higher propagation speed.     

Long wavelength approximation for Lamb wave characterization of composite laminates

A simple method for measuring Lamb wave phase velocities is used to obtain data for the lowest symmetric Lamb mode (S ₀) and the lowest antisymmetric Lamb mode (A ₀) for composite laminates. The experimental data are compared with the results from an approximate theory for the lowest Lamb modes in the low frequency, long wavelength region for a unidirectional laminate, a symmetric cross-ply laminate, a symmetric quasi-isotropic laminate and an aluminum plate. There is good correlation between the data and the results from the approximate theory, which suggests that the approximate theory works well in the low frequency, long wavelength region in these cases. Also, this experimental procedure of measuring phase velocities of the lowest symmetric and antisymmetric modes can be used to characterize laminated composite plates with and without damage since each material and stacking sequence gives distinct lowest symmetric and antisymmetric curves.     

全局矩阵法及其在层状各向异性复合板中Lamb波传播特性研究中的应用

纤维增强复合材料中兰姆波的传播因其在无损评价中的应用而成为广泛研究的课题。一般采用传递矩阵法对各向异性多层媒质进行建模。虽然这种方法较为方便,但在大频厚积时会出现数值不稳定。本文作者对Nafeh的传递矩阵方法进行改进,所得到的全局矩阵方法具有较好的稳定性,其程序能有效地产生兰姆波的频散曲线。纤维增强复合材料中兰姆波的传播受多种因素的影响,层数的影响是其中之一。给出了不同层数复合板中兰姆波的相速度频散曲线以及两个基本模式a₀和s₀沿板厚方向的应力分布。这些数值示例为层状各向异性复合材料板的无损评价提供了理论依据。      

Group velocity and characteristic wave curves of Lamb waves in composites: Modeling and experiments

The propagation characteristics of Lamb waves in composites, with emphasis on group velocity and characteristic wave curves, are investigated theoretically and experimentally. In particular, the experimental study focuses on the existence of multiple higher-order Lamb wave modes that can be observed from piezoelectric sensors by the excitation of ultrasonic frequencies. Using three-dimensional (3-D) elasticity theory, the exact dispersion relations governed by transcendental equations are numerically solved for an infinite number of possible wave modes. For symmetric laminates, a robust method by imposing boundary conditions on the mid-plane and top surface is proposed to separate symmetric and anti-symmetric wave modes. A new semi-exact method is developed to calculate group velocities of Lamb waves in composites. Meanwhile, three characteristic wave curves: velocity, slowness, and wave curves are adopted to analyze the angular dependency of Lamb wave propagation. The dispersive and anisotropic behavior of Lamb waves in a two different types of symmetric laminates is studied in detail theoretically. In the experimental study, two surface-mounted piezoelectric actuators are excited either symmetric or anti-symmetric wave modes with narrowband signals, and a Gabor wavelet transform is used to extract group velocities from arrival times of Lamb wave received by a piezoelectric sensor. In comparison with the results from the theory and experiment, it is confirmed that multiple higher-order Lamb waves can be excited from piezoelectric actuators and the measured group velocities agree well with those from 3-D elasticity theory.     

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.     

Minimizing influence of multi-modes and dispersion of electromagnetic ultrasonic lamb waves

Electromagnetic ultrasonic (EMU) Lamb waves excited by electromagnetic acoustic transducers (EMATs) possess many advantages in NDT. However, their characteristic multi-modes and dispersion are disadvantageous for inspection and restrict further improvements in their real applications. By deducing the excitation equation of EMU Lamb waves, the primary design parameters of EMATs and the characteristic equation of Lamb waves are combined, and excitation curves based on the excitation equation are plotted to aid the design of EMATs. The excitation characteristic of EMU Lamb waves on different thickness of plates is analyzed according to the excitation curves. The influence of multi-modes of EMU Lamb waves is minimized by choosing reasonable operating points and operating zones to excite a single-mode Lamb wave or multi-mode Lamb waves with identical or approximate propagation velocities. The influence of dispersion is minimized by searching corresponding points whose slope of group velocity tends to zero. The validity of the proposed method is verified by experiments.     

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。     

Calculations of Lamb wave band gaps and dispersions for piezoelectric phononic plates using mindlin's theory-based plane wave expansion method

Based on Mindlin's piezoelectric plate theory and the plane wave expansion method, a formulation is proposed to study the frequency band gaps and dispersion relations of the lower-order Lamb waves in two-dimensional piezoelectric phononic plates. The method is applied to analyze the phononic plates composed of solid-solid and airsolid constituents with square and triangular lattices, respectively. Factors that influence the opening and width of the complete Lamb wave gaps are identified and discussed. For solid/solid phononic plates, it is suggested that the filling material be chosen with larger mass density, proper stiffness, and weak anisotropic factor embedded in a soft matrix in order to obtain wider complete band gaps of the lower-order Lamb waves. By comparing to the calculated results without considering the piezoelectricity, the influences of piezoelectric effect on Lamb waves are analyzed as well. On the other hand, for air/solid phononic plates, a background material itself with proper anisotropy and a high filling fraction of air may favor the opening of the complete Lamb wave gaps.     

Application of 1-D transient elastography for the shear modulus assessment of thin-layered soft tissue: comparison with supersonic shear imaging technique

Elasticity estimation of thin-layered soft tissues has gained increasing interest propelled by medical applications like skin, corneal, or arterial wall shear modulus assessment. In this work, the authors propose one-dimensional transient elastography (1DTE) for the shear modulus assessment of thin-layered soft tissue. Experiments on three phantoms with different elasticities and plate thicknesses were performed. First, using 1DTE, the shear wave speed dispersion curve inside the plate was obtained and validated with finite difference simulation. No dispersive effects were observed and the shear wave speed was directly retrieved from time-of-flight measurements. Second, the supersonic shear imaging (SSI) technique (considered to be a gold standard) was performed. For the SSI technique, the propagating wave inside the plate is guided as a Lamb wave. Experimental SSI dispersion curves were compared with finite difference simulation and fitted using a generalized Lamb model to retrieve the plate bulk shear wave speed. Although they are based on totally different mechanical sources and induce completely different diffraction patterns for the shear wave propagation, the 1DTE and SSI techniques resulted in similar shear wave speed estimations. The main advantage of the 1DTE technique is that bulk shear wave speed can be directly retrieved without requiring a dispersion model.     

Lamb wave excitation by Hertzian contacts with applications in NDE

Excitation of Lamb waves in solid plates by point-like Hertzian contacts for material characterization and nondestructive testing is investigated. A 2 dimensional model using normal mode theory is used to predict the relative excitation efficiency of the lowest order Lamb waves in anisotropic solid plates. Hertzian contact transducers with PZT-5H piezoelectric material and quartz buffer rods are realized to operate in the 200 to 500 kHz range for experimental verification. Single mode operation with the lowest order antisymmetric Lamb wave (A ₀) mode is achieved in various plates at in agreement with theoretical predictions. The technique is applied for material characterization on single crystal silicon samples and defect detection in composite plates. The phase velocity anisotropy of the A₀ mode is measured with signal-to-noise levels exceeding 65 dB. In (111) cut silicon plates the absolute phase velocity is measured with ±0.05% accuracy. The phase velocity anisotropy and effects of delamination in layered composite plates are calculated using the surface impedance approach. The experiments on graphite/epoxy composite plates agree with these calculations and show the potential of the method for defect detection with high resolution     

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.     

Propagation of Iamb waves in adhesively bonded multilayered media

The effect of introducing attenuation on Lamb wave dispersion curves is studied in this paper. Attenuation is introduced to a three-layered composite plate by an adhesive bond layer with viscous behavior. No changes are required to the transfer matrix formulation for the propagation of elastic waves. By introduction of a complex wavenumber, the model can be used to the propagation of attenuative Lamb waves. Numerical examples for a three-layered aluminium-epoxy-aluminium plate show that attenuation values of each mode in plates are related not only to attenuation, but also to the thickness of the bonded layer, which is in agreement with practical situations.     

An alternative approach of acousto-ultrasonic technique formonitoring material anisotropy of fiber-reinforced composites

An alternative acousto-ultrasonic (AU) technique has been developed for nondestructive evaluation (NDE) of fiber-reinforced composites. The technique measures the time of flight (TOF) of AU waves, instead of the stress wave factor, by two low-frequency (0.5 MHz) transducers and relates TOF to material properties and fiber orientation. As the transducer separation increases, the measured time-domain AU signals clearly separate into two groups, since the excitation is under the first critical frequency, which correspond to the first two fundamental modes of the Lamb waves. One is an antisymmetric mode with slower propagation velocity and is highly dispersive, while the other is a symmetric mode with faster propagation velocity, which is very close to that of the longitudinal bulk wave, and is nearly nondispersive. The phase velocity in the composites can be accurately determined from the slopes of the TOF curves, and depends strongly on the azimuthal angle, frequency, and plate thickness. If the wave propagates away from the fiber direction, a slower but more attenuated wave is observed. Phase-velocity curves in azimuthal angles were obtained for E-glass/polyester, S-2-glass/epoxy, and Kevlar 49 composites. The theoretical solutions, for the longitudinal bulk wave and Lamb wave, are obtained by solving an eigenproblem once the material mechanical properties are defined. Good agreement is obtained between the measurements and the theoretical calculations     

套管井中泄漏弯曲型Lamb波声场的模拟与分析

基于套管中传播的最低阶泄漏弯曲型Lamb波(以下简称弯曲型Lamb波)对套管后介质尤其是低阻抗慢速水泥声学参数的敏感性,通过建立多层介质模型并利用实轴积分方法,数值模拟了有限尺寸定向辐射探头激发的声场和定点接收的全波波列。结果表明,在声源辐射到套管内壁上的入射角度约为33°时,在套管中主要激发弯曲型Lamb波。声场快照显示弯曲型Lamb波在沿着套管传播时,还会向与套管耦合的泥浆或水泥中泄漏比它速度低的波。若套管后介质是纵横波速度均低于弯曲型Lamb波相速度的慢速水泥,则套管中弯曲型Lamb波的衰减较大。利用弯曲型Lamb波对套管后耦合不同参数介质时的敏感性,可以有效地区分声阻抗接近的固体和液体,提高低阻抗水泥固井质量评价的效果。     

Guided waves in functionally graded viscoelastic plates

In this paper, a dynamic solution for the propagating viscoelastic waves in functionally graded material (FGM) plates subjected to stress-free conditions is presented in the context of the Kelvin–Voigt viscoelastic theory. The FGM plate is composed of two orthotropic materials. The material properties are assumed to vary in the thickness direction according to a known variation law. The three obtained wave equations are divided into two groups, which control viscoelastic Lamb-like wave and viscoelastic SH wave, respectively. They are solved respectively by the Legendre orthogonal polynomial series approach. The validity of the method is confirmed through a comparison with the Lamb wave solution of a pure elastic FGM plate and a comparison with the SH wave solution of a viscoelastic homogeneous plate. The dispersion curves and attenuation curves for the graded and homogeneous viscoelastic plates are calculated to highlight their differences. The viscous effect on dispersion curves is shown. The influences of gradient variations are illustrated.     

固-液-固3层结构板中超声兰姆波的频散特性

文章采用传统矩阵方法研究固－液－固3层结构板中兰姆波的传播,数值计算结果证明了用该方法分析层状材料中的兰姆波的有效性与实用性,文章计算出的多种频散曲线对超声兰姆波的应用是有益的。     

Numerical simulation of the guided Lamb wave propagation in particle reinforced composites

This paper deals with the investigation of the Lamb wave propagation in particle reinforced composites excited by piezoelectric patch actuators. A three-dimensional finite element method (FEM) modeling approach is set up to perform parameter studies in order to better understand how the Lamb wave propagation in particle reinforced composite plates is affected by change of central frequency of excitation signal, volume fraction of particles, size of particles and stiffness to density ratio of particles. Furthermore, the influence of different arrangements is investigated. Finally, the results of simplified models using material data obtained from numerical homogenization are compared to the results of models with heterogeneous build-up. The results show that the Lamb wave propagation properties are mainly affected by the volume fraction and ratio of stiffness to density of particles, whereas the particle size does not affect the Lamb wave propagation in the considered range. As the contribution of the stiffer material increases, the group velocity and the wave length also increase while the energy transmission reduces. Simplified models based on homogenization technique enabled a tremendous drop in computational costs and show reasonable agreement in terms of group velocity and wave length.     

Measuring Lamb wave dispersion curves of a bi-layered plate and its application on material characterization of coating

This paper investigates the Lamb wave dispersion curves of a bi-layered plate and evaluates the feasibility of using the dispersion data to characterize the coating's material properties. The measurement of dispersion curves is based on a focusing PVDF transducer operating in a pulse/echo mode. An image displaying technique is used to determine the dispersion relation of Lamb waves from the measured data. Multiple dispersion curves of Lamb waves are accurately determined over a wide frequency range (4 to 20 MHz). Lamb wave dispersion curves for thin metal sheets electrodeposited with nickel coatings are measured. The elastic constants of the nickel coating are determined by comparing the experimental dispersion data with the theoretical ones calculated numerically. Potential applications of this measurement method are addressed     

Mode-selective excitation and detection of ultrasonic guided waves for delamination detection in laminated aluminum plates

Selective modes of guided Lamb waves are generated in a laminated aluminum plate for damage detection using a broadband piezoelectric transducer structured with a rigid electrode. Appropriate excitation frequencies and modes for inspection are selected from theoretical and experimental dispersion curves. Dispersion curves are obtained experimentally by short time Fourier transform of the transient signals. Sensitivity of antisymmetric and symmetric modes for delamination detection are investigated. The antisymmetric mode is found to be more reliable for delamination detection. Unlike other studies, in which the attenuation of the propagating waves is related to the extent of the internal damage, in this investigation, the changes in the time-of-flight (TOF) of guided Lamb waves are related to the damage progression. The mode conversion phenomenon of Lamb waves during progressive delamination is investigated. Close matching between the theoretical and experimentally derived dispersion curves and TOF assures the reliability of the results presented here.     

On the influence of moisture absorption on Lamb wave propagation and measurements in viscoelastic CFRP using surface applied piezoelectric sensors

The understanding of the impact of environmental influence factors on propagation and damping of Lamb waves in composite materials is a topic of great interest for both design and utilization of structural health monitoring (SHM) systems. In this work, the influence of humidity absorption on the dispersive behavior of Lamb waves propagating in viscoelastic composite materials is investigated. Using a transversely isotropic material model and DMA measurements, the changes in the viscoelastic material properties due to water absorption are characterized. By means of a higher order plate theory and those mechanical properties, the dispersion curves for unconditioned and hot/wet-conditioned UD reinforced CFRP plates are then predicted. Both the changes in Lamb wave velocity and Lamb wave damping are investigated and compared with experimental values. Additionally, the changes of the sensor response, which are related to both the changes of the material properties and that of the adhesive layer, are investigated. The large impact of moisture absorption on Lamb wave excitation and propagation and its relevance for structural health monitoring (SHM) applications is shown and discussed.