Normal mode expansion method for laser-generated ultrasonic Lamb waves in orthotropic thin plates

A quantitative theory for modeling the laser-generated transient ultrasonic Lamb waves, which propagates along arbitrary directions in orthotropic plates, is presented by employing an expansion method of generalized Lamb wave modes. The displacement field is expressed by a summation of the symmetric and antisymmetric modes in the surface stress-free orthotropic plate, and therefore the theory is particularly appropriate for waveform analyses of Lamb waves in thin plates because one needs only to evaluate several lower modes. The transient waveforms excited by the thermoelastic expansion and the oil-coating evaporation are analyzed for a transversely isotropic thin plate. The results show that the theory provides a quantitative analysis to characterize anisotropic elastic stiffness properties of orthotropic plates by laser-generated Lamb wave detection.     

Ultrasonic, hardness and x-ray densitometric analysis of wood

Ultrasonic, hardness and x-ray densitometric methods were used to predict the wood quality of Douglas fir sapwood from pruned and control trees. Measurements were performed on logs and on increment cores.

On logs, hardness was tested by a pilodyn instrument. Longitudinal waves (80 kHz) were used to measure the velocity in the radial anisotropic direction of the wood. Surface waves of the same frequency were employed to measure the velocity on the circumference of the log in the longitudinal anisotropic direction of the wood.

On increment cores, 1 MHz waves, longitudinal and transverse, were used to measure the velocity of ultrasonic pulses along three anisotropic axes of the wood, using the through-transmission technique.

The predictive power of the pilodyn test and of the surface wave velocity method for logs was judged in relation to several independent variables on cores given by the densitometric method and by ultrasonic velocities and elastic constants of the wood.     

Elastic constants measurement of anisotropic Olivier wood plates using air-coupled transducers generated Lamb wave and ultrasonic bulk wave

A hybrid elastic wave method is applied to determine the anisotropic constants of Olive wood specimen considered as an orthotropic solid. The method is based on the measurements of the Lamb wave velocities as well as the bulk ultrasonic wave velocities. Electrostatic, air-coupled, ultrasonic transducers are used to generate and receive Lamb waves which are sensitive to material properties. The variation of phase velocity with frequency is measured for several modes propagating parallel and normal to the fiber direction along a thin Olivier wood plates. A numerical model based mainly on an optimization method is developed; it permits to recover seven out of nine elastic constants with an uncertainty of about 15%. The remaining two elastic constants are then obtained from bulk wave measurements. The experimental Lamb phase velocities are in good agreement with the calculated dispersion curves. The evaluation of Olive wood elastic properties has been performed in the low frequency range where the Lamb length wave is large in comparison with the heterogeneity extent. Within the interval errors, the obtained elastic tensor doesn’t reveal a large deviation from a uniaxial symmetry.     

A stationary phase argument for the modal wave beam deviation in the time-space domain for anisotropic multilayered media

The aim of the paper is to describe the physical phenomenon of the excitation of modal waves, such as Lamb waves, in anisotropic multilayered media by a monochromatic incident beam and then by a time depending signal. A modal beam is generated in the structure and, due to the anisotropy of the media constituting the structure, is deviated with respect to the sagittal plane of the incident bounded beam. Using a stationary phase approach, it is possible to determine the deviation direction of the modal beam in the far field at a given frequency. This direction is normal to the modal curve, at the point corresponding to the main modal wave vector. Using Lagrange multipliers, it is possible to obtain the equation of an oblique plane in which the modal beam reradiates in the external fluid. As the modal waves are dispersive, the group velocity and the direction of propagation of the principal modal wave vary with the frequency. So, in the far field, for a time depending signal, the different monochromatic components of the main modal wave are found in different directions. In general, the main crest line of this modal wave packet is not a straight line.     

Research on relationships between Lamb wave velocity and static stress in metal plate

On the fact that an isotropic metal solid produces anisotropic property in the state of static stress, based on the theory of the nonlinear acoustoelasticity, the equivalent second order elastic constants are calculated for metal plate with static stress. For the case of thin metal plate with stress, the two kinds of dispersion equation for Lamb waves propagating parallel and vertical to the direction of static stress are derived. Using the equations, the relationships between Lamb wave velocity and static stress in a metal plate are discussed.     

Elastic constants of Ultrasonic Additive Manufactured Al 3003-H18

Ultrasonic Additive Manufacturing (UAM), also known as Ultrasonic Consolidation (UC), is a layered manufacturing process in which thin metal foils are ultrasonically bonded to a previously bonded foil substrate to create a net part. Optimization of process variables (amplitude, normal load and velocity) is done to minimize voids along the bonded interfaces. This work pertains to the evaluation of bonds in UAM builds through ultrasonic testing of a build’s elastic constants. Results from ultrasonic testing on UAM parts indicate orthotropic material symmetry and a reduction of up to 48% in elastic constant values compared to a control sample. The reduction in elastic constant values is attributed to interfacial voids. In addition, the elastic constants in the plane of the Al foils have nearly the same value, while the constants normal to the foil direction have much different values. In contrast, measurements from builds made with Very High Power Ultrasonic Additive Manufacturing (VHP UAM) show a drastic improvement in elastic properties, approaching values similar to that of bulk aluminum.     

Dispersion of circumferential waves in cylindrically anisotropic layered pipes in plane strain

Dispersion spectra of circumferential waves along the periphery of circular pipes made of layered anisotropic materials do not seem to be available in literature. This note attempts to partially fill this gap by providing the dispersion spectra in two and three layered cylindrically anisotropic pipes in plane strain motion. The spectra for pipes executing time harmonic vibrations in plane strain condition are obtained as roots of a numerical characteristic equation derived extending a weighted residual method of solution of the governing equations for a single layer pipe [Towfighi et al., J. Appl. Mech. 69, 283-291 (2002)] to a general N layered pipe. The anisotropic elastic coefficients are considered to be independent of position coordinates and the bond condition at interfaces of the layers is assumed to be perfect. Numerical illustrations are presented for two and three layered pipes with anisotropy directions differing in adjacent layers. Increase in curvature of the pipe and inclination of the fiber orientation in the outermost layers to propagation direction are factors that seem to influence the mode number and pattern within the limited examples worked out.     

A study of time harmonic guided Lamb waves and their caustics in composite plates

Spatial steady-state Lamb wave propagation in an anisotropic composite plate excited by harmonic surface sources is modeled using a Green’s matrix representation in a frequency-wavenumber domain. An approach based on a residue integration technique for two dimensional wavenumber integrals for the computation of displacements outside an excitation source is presented in this paper. In the far-field zone of the excitation source, the method of stationary phase is used, which gives an asymptotic expansion of the displacement vector as a sum of cylindrical waves. Near caustic directions, a far-field solution is computed in terms of Airy functions. The results obtained applying residue integration technique and asymptotic expansion are found to be coinciding with the results of the computation by using the adaptive quadratures. Moreover, these approaches agree well with experimental data. Then, the advantages and disadvantages of the various methods applied for modeling of Lamb wave propagation are discussed in this paper. Focussing and other properties of Lamb waves are studied using numerical examples.     

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.     

Ultrasonic Lamb wave diffraction tomography.

Ultrasonic guided waves, Lamb waves, allow large sections of aircraft structures to be rapidly inspected. Unlike conventional ultrasonic C-scan imaging that requires access to the whole inspected area, tomographic algorithms work with data collected over the perimeter. Because the velocity of Lamb waves depends on thickness the travel times of the fundamental modes can be converted into a thickness map of inspected region. Lamb waves cannot penetrate through holes and other strongly scattering defects and the assumption of straight wave paths, essential for many tomographic algorithms, fails. Diffraction tomography is a way to incorporate scattering effects into tomographic algorithms in order to improve image quality and resolution. This work describes the iterative reconstruction procedure developed for Lamb wave tomography and allowing for ray bending correction for imaging of moderately scattering objects.     

利用Taylor展开法研究Lamb波在功能梯度材料中的传播特性

超声技术可用于对功能梯度材料(FGMs)的性质进行评估. 由于FGMs性质的非均匀性,采用分布函数来描述FGMs弹性常数和密度沿厚度方向的变化趋势,并提出利用Taylor展开的方法来解决分布函数为任意函数时的FGMs中Lamb波的传播问题. 利用本征函数展开法得到了铁基氧化铝FGMs中Lamb波的相速度色散曲线,讨论了材料性质分布对铁基氧化铝FGMs中Lamb波传播特性的影响. 为FGMs性质(沿板厚方向变化)的反演提供了理论依据. 关键词： 功能梯度材料 Lamb波 Legendre多项式 分布函数     

Elastic surface waves guided by the edge of a slit

Surface waves propagating along the free surface of a homogeneous, isotropic, linearly elastic half-space, are shown to have the property that the normal displacement component at the free surface is governed by a reduced wave equation. This suggests a “membrane analogy”, and a corresponding family of surface waves. Of particular interest is a three-dimensional surface wave, whose displacement components in the sagittal plane vary linearly with the co-ordinate normal to that plane, while the displacement component in the direction normal to the sagittal plane is uniform in that direction. This new wave arises when surface waves propagate along the free surfaces of a semi-infinite slit, parallel to the edge of the slit, with the classical Rayleigh wave velocity. It is also shown that a semi-infinite slit cannot support surface waves which decay with the distance from the edge of the slit.     

Selective generation of ultrasonic Lamb waves by electromagnetic acoustic transducers

In this paper, we describe a modal expansion approach for the analysis of the selective generation of ultrasonic Lamb waves by electromagnetic acoustic transducers(EMATs). With the modal expansion approach for waveguide excitation, an analytical expression of the Lamb wave's mode expansion coefficient is deduced, which is related to the driving frequency and the geometrical parameters of the EMAT's meander coil, and lays a theoretical foundation for exactly analyzing the selective generation of Lamb waves with EMATs. The influences of the driving frequency on the mode expansion coefficient of ultrasonic Lamb waves are analyzed when the EMAT's geometrical parameters are given. The numerical simulations and experimental examinations show that the ultrasonic Lamb wave modes can be effectively regulated(strengthened or restrained) by choosing an appropriate driving frequency of EMAT, with the geometrical parameters given. This result provides a theoretical and experimental basis for selectively generating a single and pure Lamb wave mode with EMATs.     

An investigation of generalized lamb waves using ultrasonic reflectivity measurements

Ultrasonic critical angle reflectivity method is used to measure the phase velocity of generalized Lamb waves in cadmium and zinc platings. In addition to the free propagating modes, leaky-wave modes are also observed. Leaky waves are considered using principles familiar from the discussion of surface waves in anisotropic media and a relatively good agreement is obtained between the theoretical computations and measured data.     

A homogenised vibratory model for predicting the acoustic properties of hollow brick walls

The prediction and the physical understanding of sound transmission through masonry walls made of hollow bricks remain an open question. To solve this problem a semi-analytical approach is proposed. The inhomogeneous structures of the brick wall are homogenised and a simplified analytical model is established to calculate the transmission loss of an equivalent finite and multilayered anisotropic plate. An efficient numerical homogenisation technique is derived to define the equivalent anisotropic brick. This process only needs the knowledge of the elastic tensor of the brick material that has been determined using ultrasonic measurements. The features of the simplified brick wall have been then investigated through Lamb waves dispersion curves. Finally, the model has been used to explain the transmission loss curve of a wall and a good agreement between predictions and test data is obtained.     

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。     

The propagation of coupled Lamb waves in multilayered arbitrary anisotropic composite laminates

Based on linear three-dimensional elasticity theory, the wave equations of coupled Lamb waves in multilayered arbitrary anisotropic composite laminates are derived using a Legendre orthogonal polynomial approach. The elastodynamic solution for the propagation of coupled Lamb waves in composite plates is also presented to determine the characteristics of coupled Lamb waves. To verify the applicability and validity of the method, two cases of bi-layered plates formed with isotropic components and anisotropic components, respectively, are primarily manipulated for comparison with earlier known results. Next, the dispersion curves, displacements and stress distributions of Lamb waves in multilayered anisotropic laminates are calculated. The effects of coupling and fiber orientation on the characteristics of the Lamb waves are illustrated. The potential usefulness of the fundamental modes of the coupled Lamb waves is discussed in detail.     

用声波速度确定骨组织弹性参数

本文用超声方法确定密质骨的弹性特征,其基本原理是基于弹性波理论中的Christoffel方程。本文应用液浸、声波穿透技术,对牛胫骨样品中一些特殊入射方向的声波速度进行了测试,利用固体中沿任意方向的弹性波传播速度与材料的弹性系数间的关系,通过计算机编程反演出具有六方晶体对称结构的密质骨组织的5个独立弹性常数。结果与相关文献中其它方法所得结果一致。     

Anisotropy of the superconducting gap of the borocarbide superconductor YNi2B2C with ultrasonic attenuation

The ultrasonic attenuation alpha of the highly anisotropic s-wave superconductor YNi2B2C has been measured for all the symmetrically independent elastic modes to explore the location of the zero superconducting gap region on the Fermi surface. The attenuation of the longitudinal mode shows a pronounced anisotropy in the superconducting state: While alpha shows a thermally activated behavior along [110] and [001] directions, it shows T-linear dependence along [100]. These results together with those for the transverse modes demonstrate the presence of point nodes or zero-gap regions along [100] and [010] directions. This is a clear demonstration of ultrasonic attenuation as a powerful probe for the structure of the anisotropic superconducting gap.     

Stiffness matrix invariants to validate the characterization of composite materials with ultrasonic methods

This paper presents a method of testing the ultrasonic measurements of the stiffness matrix, and the identification of the anisotropic behaviour, of composite materials. Some linear combinations of elastic constants are invariants for a rotation around an axis of symmetry. If the stacking sequence is the only parameter which changes in a set of long-fibre composites made of the superimposition of plies, the composites must own these invariants. So, PEEK-carbon fibre composite samples were constructed in this way to measure their elastic properties by immersion and contact ultrasonic methods, and to compare the results with predicted invariants. By changing the stacking sequence of plies three anisotropic models are tested: orthotropic, hexagonal and quadratic. Measurements of ultrasonic velocities in various planes of propagation permit the identification of the elastic constant and invariants. From the invariance of these linear combinations, the precision of the three-dimensional effective moduli can be estimated.