Some aspects of numerical simulation for Lamb wave propagation in composite laminates

Some aspects of numerical simulation of Lamb wave propagation in composite laminates using the finite element models with explicit dynamic analysis are addressed in this study. To correctly and efficiently describe the guided-wave excited/received by piezoelectric actuators/sensors, effective models of surface-bounded flat PZT disks based on effective force, moment and displacement are developed. Different finite element models for Lamb wave excitation, collection and propagation in isotropic plate and quasi-isotropic laminated composite are evaluated using continuum elements (3-D solid element) and structural elements (3-D shell element), to elaborate the validity and versatility of the proposed actuator/sensor models.     

Detecting lamb waves with broadband acousto-ultrasonic signals in composite structures

Lamb waves can be produced and detected in ceramic matrix composites (CMC) and metal matrix composites (MMC) plates using the acousto-ultrasonic configuration employing broadband transducers. Experimental dispersion curves of lowest symmetric and lowest antisymmetric modes behave in a manner analogous to the graphite/polymer theoretical curves. In this study a basis has been established for analyzing Lamb wave velocities for characterizing composite plates. Lamb wave dispersion curves and group velocities were correlated with variations in axial stiffness and shear stiffness in MMC and CMC. For CMCs, interfacial shear strength was also correlated with the first antisymmetric Lamb mode.     

Thermoelastic Lamb waves in a transversely isotropic plate bordered with layers of inviscid liquid

Analysis for the propagation of thermoelastic waves in a homogeneous, transversely 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 coupled theory of thermoelasticity. Secular equations for homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and anti-symmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. It is shown that the purely transverse motion (SH mode), which is not affected by thermal variations, gets decoupled from rest of the motion of wave propagation and occurs along an in-plane axis of symmetry. The special cases, such as short wavelength waves and thin plate waves of the secular equations are also discussed. The secular equations for 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 transversely isotropic plate of zinc material bordered with water. The dispersion curves for symmetric and anti-symmetric wave modes, attenuation coefficient 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.     

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.     

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.     

Modelling of attenuation of Lamb waves using Rayleigh damping: Numerical and experimental studies

In this study, the Lamb mode attenuation constants were derived in terms of the attenuation coefficient, group velocity and central frequency of excitation of the Lamb mode, using the Rayleigh damping model. Attenuation of Lamb waves, both fundamental symmetric and anti-symmetric modes, propagating through viscoelastic media (cross-ply glass/epoxy laminate) was modelled using the Finite Element Method. Numerically simulated attenuation of Lamb waves using Lamb mode attenuation constants was found to be in good agreement with the assumed attenuation. Experiments were performed on a quasi-isotropic laminate, employing air-coupled ultrasonic transducers, to measure the attenuation coefficient. Lamb mode attenuation constants, computed using the attenuation coefficient, were used to model the attenuation of the Lamb mode in quasi-isotropic laminates. Numerically simulated amplitude variation was found to be in good agreement with that computed from experiments.     

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.     

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.     

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     

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.     

Thermo elastic waves with thermal relaxation in isotropic micropolar plate

In the present investigation, we have discussed about the features of waves in different modes of wave propagation in an infinitely long thermoelastic, isotropic micropolar plate, when the generalized theory of Lord–Shulman (L–S) is considered. A more general dispersion equation is obtained. The different analytic expressions in symmetric and anti-symmetric vibration for short as well as long waves are obtained in different regions of phase velocities. It is found that results agree with that of the existing results predicted by Sharma and Eringen in the context of various theories of classical as well as micropolar thermoelasticity.     

金属薄板中导波的模态识别和波速测定

按照Lamb波理论,板内导波由多阶对称波和反对称波组成。当波长远大于板厚时,利用经典平板理论对Lamb方程进行简化,此时板内主要有零阶对称波S0和反对称波A0两种模态。认识薄板内声波模态,不仅有助于分析声源性质,提取有效信号,而且可以提高声发射检测中时差定位的准确性。利用PAC公司的Fieldcal标准信号发生器和Nielsen Hsu断铅法作为信号源在0.7 mm薄钢板上进行声波传播实验。通过比较实测波速和理论波速甄别板内的声波模态。实验发现对称波S0实测波速和理论波速比较吻合,而反对称波A0的波速由于频散而较难精确测量。综合两种模态波的波速情况,声发射检测中利用对称波速度进行声源定位计算更加接近薄壁结构声波的传播状况。     

Damage assessment in layered composites using spectral analysis and Lamb wave

Piezo-ceramic transducers of the surface mounted type are commonly used for structural health monitoring (SHM) techniques. But, there is a disadvantage to use piezo-ceramic transducers of the surface mounted type in Lamb wave application. Due to the symmetric and antisymmetric Lamb wave modes generated by the surface mounted piezo-ceramic transducers simultaneously, the received signals are very complex and it is difficult to extract damage information from the signals.

In this paper, the practical method for SHM was proposed using piezo-ceramic transducers of the surface mounted type and Lamb wave. In order to overcome the difficulties in the signal processing of the simultaneous modes, the symmetric and antisymmetric modes were separated by using the two sensors bonded on the opposite surfaces at the same point. Also, spectral analyses of the separated symmetric and antisymmetric Lamb waves showed that each mode propagated with different frequency characteristics in the exciting frequency range.

By making use of these findings, the changes of power spectrum density in characteristic frequency band of symmetric and antisymmetric modes are proportional to the delamination size in quasi-isotropic Gr/Ep laminates. Therefore, this paper presents the damage assessment technique to extract damage information from the complicated PZT signals that could not be interpreted in time domain.     

识别薄板中兰姆波和体波的阶梯板方法

斜探头在某些频率下激励出的兰姆波,其群速度与体波的传播速度相近,所以通过判断传播速度不易区分出兰姆波和体波。通过数值模拟和实验,分别研究了激励频率为2 MHz的纵波和S0模态兰姆波在阶梯板上的反射特性,发现:在阶梯板上入射S0模态兰姆波时,有反射回波;而入射纵波时,无反射回波。基于这种反射特性的差别,提出了一种利用阶梯板区别薄板中兰姆波和体波的方法,该方法可用于确认探头的激励特性。     

基于三维弹性理论的碳纤维板中Lamb波建模

基于主动Lamb波的结构健康监测是目前复合材料结构损伤监测技术研究的热点之一,了解Lamb波的传播特性对进行可靠的损伤监测非常重要.本文结合经典三维弹性理论与Lamb波的运动位移方程,对碳纤维复合材料板中传播的Lamb波传播特性进行了建模研究,在此基础上推导了碳纤维板的相速度频散曲线,并讨论了Lamb波传播方向与坐标轴之间的夹角及碳纤维铺层方向对频散曲线的影响,建模结果证明了这种建模方法的正确性.     

基于Lamb波的薄壁槽状结构损伤检测研究

研究复杂工程结构的结构健康监测技术具有现实意义。使用基于Lamb波的仿真和实验方法,对“U”形截面的铝合金构件中的损伤检测问题进行了研究。建立了构件的三维有限元模型并实现了Lamb波传播过程的动态仿真;实验中使用锆钛酸铅压电晶片（PZT wafer）来激发和接收在构件中传播的Lamb波。借助于连续小波变换（CWT）和希尔伯特变换（HT）等方法对仿真和实验中采集到的Lamb波信号进行处理,从中提取了与损伤有关的时域特征,建立了损伤位置和损伤反射波包飞行时间（ToF）之间的定量关系。     

Lamb波传播特性研究

应用三维弹性理论对Lamb波频散曲线进行理论建模.在Lamb波主动监测系统中,应用Gabor小波变换理论分析Lamb波信号的时延及相位角,获得了板中Lamb波的相群速度频散曲线.比较理论曲线与实验数据,证明了三维弹性理论建模方法的有效性.     

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。      

A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators

This paper presents the development of a semi-analytical axisymmetric shell finite element model with piezoelectric layers using the 3D linear elasticity theory. The piezoelectric effect of the material could be used as sensors and/or actuators in way to control shell deformation. In the present 3D axisymmetric model, the equations of motion are expressed by expanding the displacement field using Fourier series in the circumferential direction. Thus, the 3D elasticity equations of motion are reduced to 2D equations involving circumferential harmonics. In the finite element formulation the dependent variables, electric potential and loading are expanded in truncated Fourier series. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.     

液体-压电晶片界面上电边界条件对兰姆波激发的影响

本文对液体-压电晶片结构中,不同电边界条件下叉指换能器所激发的兰姆波传播特性进行了分析研究,给出了不同电边界条件下兰姆波的相速度、叉指换能器激发兰姆波的机电耦合系数与压电晶片的归一化厚度、晶体切向之间的关系曲线,为实际设计中电边界条件的正确选择提供了理论基础。