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.     

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.     

Ultrasonic guided wave scattering in a plate overlap

Guided wave scattering in a plate overlap is investigated by numerical calculations and experimental measurements of transmission and reflection factors from the overlap region. In the numerical study, a hybrid boundary element-finite element method is used to calculate the guided wave scattered field from the overlap region. Transmission and reflection factors are calculated for incident A0 and S0 Lamb and n0 shear horizontal waves, including higher modes generated through mode conversion phenomena. In addition, parametric studies of transmission and reflection factors in this problem are performed numerically over various incident modes, frequencies, and overlap lengths. For verification and comparison with numerical results, experiments were conducted to measure the transmission and reflection factors for incident Lamb and shear horizontal waves in steel plates with two different overlap areas. The experimental results agree well with the numerical calculations. The numerical and experimental results show that it is highly feasible to carry out efficient Lamb wave nondestructive evaluation (NDE) in overlapped plates and in multilayer structures with various lap joints by selecting various modes and tuning frequency.     

The excitation and detection of lamb waves with planar coil electromagnetic acoustic transducers

Planar coil electromagnetic acoustic transducers (EMATs) are investigated for the excitation and detection of Lamb waves in nonferromagnetic metallic wave-guides. Such EMATs are attractive for certain applications due to their omni-directional sensitivity to wave modes with predominantly in-plane surface displacement, such as the So Lamb wave mode. A model is developed that enables the modal content of the radiated Lamb wave field from a transmitting EMAT to be calculated, and the output voltage from a receiving EMAT to be predicted when a Lamb wave mode is incident on it. The predictions from this model are compared with experimental data obtained from 12 different EMATs tested on a 5-mm thick aluminum plate, and good agreement is obtained. The model then is used to analyze the different effects that contribute to the overall Lamb wave modal sensitivity of an EMAT. The relationship between coil geometry and wavelength is examined.     

薄板钢结构超声检测Lamb波激励与模态分析

运用超声Lamb波对薄板钢结构进行检测,采用斜入射方式实施Lamb波的激励,并对检测信号进行模态分析。经实验发现选择合适的频率/入射角组合可实现对Lamb波模态的选择和控制,实现对检测对象的检测策略,并具有一定的工程应用价值。     

Circular crested waves in anisotropic thermoelastic plates bordered with inviscid liquid

The propagation of circularly crested 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 conventional coupled thermoelasticity, Lord-Shulman and Green-Lindsay theories of thermoelasticity. Secular equations for circular homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. The special cases such as short wavelength waves, thin plate waves and leaky Lamb waves of the secular equation are also deduced and discussed. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic circular plate of cobalt material bordered with water. The dispersion curves for symmetric and antisymmetric 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 analytical and numerical results are found to be in close agreement.     

A lamb wave source based on the resonant cavity of phononic-crystal plates

In this paper, we propose a Lamb wave source that is based on the resonant cavity of a phononic-crystal plate. The phononic-crystal plate is composed of tungsten cylinders that form square lattices in a silicon plate, and the resonant cavity is created by arranging defects inside the periodic structure. The dispersion, transmission, and displacement of Lamb waves are analyzed by the finite-difference time-domain (FDTD) method. The eigenmodes inside the cavities of the phononic-crystal plate are identified as resonant modes. The fundamental and higher order resonant modes, which vary with the length of cavities, are calculated. By exciting the specific resonant mode in an asymmetric cavity, the 232.40 MHz flexural Lamb wave has a magnified amplitude of 78 times larger than the normal one. Thus, the cavity on the tungsten/ silicon phononic-crystal plate may serve as a source element in a microscale acoustic wave device.     

应用Poisson板理论求解S0兰姆波在通孔缺陷的散射

为降低3D精确理论散射场求解的复杂性和计算成本,应用Poisson板理论描述面内波模式,给出了S0波在各向同性板中受通孔散射的近似模型。该理论利用波函数展开技术以及孔边缘的边界条件,得到了S0和SH0模式的散射远场。将Poisson板理论与3D精确理论进行比较,两者在低频情况下具有较好的一致性。考察了两个基本导波模式在不同尺寸孔的散射特性,发现它们的散射波幅值在某些散射方向随孔径呈线性变换,该结果为兰姆波无损评价提供了依据。     

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.     

Estimation of ultrasonic guided wave mode conversion in a plate with thickness variation

The hybrid boundary element method aimed at analyzing Lamb wave scattering from defects can provide us with an excellent numerical tool for tackling complicated mode conversion phenomena under waveguide thickness variation. In this paper, utilization of hybrid boundary element modeling for specific Lamb wave mode incidence situations with special energy distributions along the structural cross section is proposed for estimating reflection and transmission from various scatterers, such as a step discontinuity and tapered parts of a waveguide, etc. Interaction of individual Lamb wave modes with scatterers that represent arbitrary thickness variation along the direction of guided wave propagation is investigated by calculating the scattered fields for varying incident modes, frequency, and scatterer shape. The mode conversion phenomena through step discontinuity in a plate are also experimentally explored. The theoretical predictions of reflection and transmission by boundary element methods and the utility of dispersion curves are compared with experiments for specific modes. Results in this paper can be used to improve inspection sensitivity and penetration power for a variety of practical NDE applications, notably those in which thickness variation is found. In addition, the feasibility of inspecting sections located behind a waveguide thickness variation region and subsequent mode control will also be discussed.     

A mode expansion technique for rigorously calculating the scattering from 3D subwavelength structures in optical recording

Abstract

We apply a mode expansion technique to the three-dimensional vectorial diffraction problem of an electromagnetic field that is incident on a perfectly conducting, metallic disc containing a pattern of rectangular pits. The mode expansion technique reduces the three-dimensional diffraction problem to a two-dimensional numerical problem. Furthermore, by choosing a particular numerical discretization, the a priori unknown amplitudes of the propagating and evanescent scattered plane waves in the half space above the metallic plate can be eliminated from the system of equations. The relatively small remaining system of equations for the amplitudes of the propagating and evanescent modes inside the pits, can be solved very rapidly. Some first results are presented. Furthermore, the application of the scanning of an optical beam by a pit structure on a metallic optical disc is discussed.     

板状粘接结构中对称和反对称纵波与界面的相互作用

为了激发粘接结构中的导波或界面波,通常需要将声波从两半无限介质同相位或反相位地同时入射多层系统。针对此问题,基于矩阵方法,推导了界面处于理想连接的情况下,对称或反对称纵波从上下半无限空间入射时,三层板状粘接结构中纵、横波的反射与透射系数表达式。分析了入射角度、粘接层厚度以及基体材料等对声波反射(或透射)特性的影响。结果表明,对称或反对称纵波垂直入射时不发生波型转换。粘接结构中声波的反射(或透射)特性与入射角度、频率以及粘接层厚度等参数密切相关。在相同的粘接层厚度(或频率)范围内,随着声波频率(或粘接层厚度)的增加,谐振频率曲线向低频漂移。该方法可作为粘接结构中体波或导波传播特性研究的重要理论基础。     

Diffraction of elastic waves by an elliptic crack

The integral equation method is used to obtain the scattered field of a normally incident plane wave from an elliptic crack embedded in an isotropic elastic medium. It is shown that the determination of the diffracted field depends on the solution of integro-differential equation. A formal power series solution, in the low frequency limit, is obtained. Expressions are derived for the scattered amplitudes and the dynamic stress intensity factor.     

An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate

Summary This paper concerns an asymptotic analysis of the dispersion relation for wave propagation in a pre-stressed incompressible elastic plate. Asymptotic expansions for the wave speed as a function of wavenumber and pre-stress are obtained. These expansions have important potenatial applications to many dynamic problems such as impact problems. It is shown that in the large wavenumber limit the wave speed of the fundamental modes of both symmetric and anti-symmetric motions tends to the associated Rayleigh surface wave speed, on the other hand, the wave speeds of all the harmonics tend to a single limit which is the corresponding body wave speed. It is also shown that, whereas the fundamental modes are very sensitive to changes in the underlying pre-stress, the harmonics are little affected by such changes, espcially in the small and large wavenumber limits.     

Scattering of SH-waves by an interface cavity

Summary. The scattering of the SH-wave and dynamic stress concentrations near an arbitrary cavity situated at the planar interface separating two different elastic media are investigated. The total wave field can be obtained by superposition of the free field and the scattered field. The free field is composed of the incident, reflected and refracted waves. The scattered wave fields in adjacent media are expressed respectively, and the method of wave functions expansion is applied to obtain the solutions for these fields. The scattered wave functions can be expanded into Hankel-Fourier series with unknown coefficients. In solving for the unknown coefficients according to the boundary conditions for the total wave field at the interface and at the cavity wall, the non-orthogonality makes the system of equations for the unknown coefficients infinite and coupling each other. Another key point is to extend each scattered wave field from its own half-plane domain into the full plane domain by a certain way keeping the total wave field unchanged for the non-orthogonal Fourier integrals around the cavity. Finally, the scattering of the SH wave by an interface ellipse with different ratios between long and short axis is considered, and the distributions of dynamic stress concentration factors at the cavity wall are presented.     

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。     

A traction‐based equilibrium finite element free from spurious kinematic modes for linear elasticity problems

This paper presents equilibrium elements for dual analysis. A traction‐based equilibrium element is proposed in which tractions of an element instead of stresses are chosen as DOFs, and therefore, the interelement continuity and the Neumann boundary balance are directly satisfied. To be solvable, equilibrated tractions with respect to the space of rigid body motion are required for each element. As a result, spurious kinematic modes that may inflict troubles on stress‐based equilibrium elements do not appear in the element because only equilibrium constraints on tractions are required. An admissible stress field is eventually constructed in terms of the equilibrated tractions for the element, and hence, equilibrium finite element procedures can proceed. The element is also generalized to accommodate non‐zero body forces, nonlinear boundary tractions and curved Neumann boundaries. Numerical tests including a single equilibrium element, error estimation of a cantilever beam and an infinite plate with a circular hole are conducted, displaying excellent convergence and effectiveness of the element for error estimation. Copyright © 2014 John Wiley & Sons, Ltd.     

Quantitative flaw reconstruction from ultrasonic surface wavefields measured by electronic speckle pattern interferometry

A new method for imaging flaws in plate and shell structures is presented. The method employs two-dimensional ultrasonic surface wave data obtained by optical electronic speckle pattern interferometry (ESPI) techniques. In the imaging method, the measured out-of-plane displacement field associated with an externally excited ultrasonic Lamb wave is processed to obtain the spatial frequency domain spectrum of the wavefield. A free space Green's function is then deconvolved from the wavefield to obtain quantitative images of effective scattering sources. Because the strength of these effective sources is directly dependent on local variations in sample thickness and material properties, these images provide a direct map of internal inhomogeneities. Simulation results show that the method accurately images flaws for a wide range of sizes and material contrast ratios. These results also demonstrate that flaw features much smaller than an acoustic wavelength can be imaged, consistent with the theoretical capability of the imaging method to employ scattered evanescent waves. Reconstructions are also obtained from ultrasonic Lamb wave displacement fields recorded by ESPI in a flawed aluminum plate. These reconstructions indicate that the present method has potential for imaging flaws in complex structures for which ESPI wavefield measurements cannot be straightforwardly interpreted     

Rayleigh-Lamb waves in magneto-thermoelastic homogeneous isotropic plate

The propagation of magnetic-thermoelastic plane wave in an initially unstressed, homogeneous isotropic, conducting plate under uniform static magnetic field has been investigated. The generalized theory of thermoelasticity is employed, by assuming electrical behaviour as quasi-static and the mechanical behaviour as dynamic, to study the problem. The secular equations for both symmetric and skew-symmetric waves have been obtained. The magneto-elastic shear horizontal (SH) mode of wave propagation gets decoupled from rest of the motion and it is not influenced by thermal variations and thermal relaxation times. At short wavelength limits, the secular equations for symmetric and skew-symmetric modes reduce to Rayleigh surface wave frequency equation, because a finite thickness plate in such a situation behaves like a semi-infinite medium. Thin plate results are also deduced at the end. Dispersion curves are represented graphically for various modes of wave propagation in different theories of thermoelasticity. The amplitudes of displacement, perturbed magnetic field and temperature change are also obtained analytically and computed numerically. The result in case of elastokinetics, magneto-elasticity and coupled magneto-elasticity has also been deduced as special cases at appropriate stages of this work.     

Phononic plate waves

In the past two decades, phononic crystals (PCs) which consist of periodically arranged media have attracted considerable interest because of the existence of complete frequency band gaps and maneuverable band structures. Recently, Lamb waves in thin plates with PC structures have started to receive increasing attention for their potential applications in filters, resonators, and waveguides. This paper presents a review of recent works related to phononic plate waves which have recently been published by the authors and coworkers. Theoretical and experimental studies of Lamb waves in 2-D PC plate structures are covered. On the theoretical side, analyses of Lamb waves in 2-D PC plates using the plane wave expansion (PWE) method, finite-difference time-domain (FDTD) method, and finite-element (FE) method are addressed. These methods were applied to study the complete band gaps of Lamb waves, characteristics of the propagating and localized wave modes, and behavior of anomalous refraction, called negative refraction, in the PC plates. The theoretical analyses demonstrated the effects of PC-based negative refraction, lens, waveguides, and resonant cavities. We also discuss the influences of geometrical parameters on the guiding and resonance efficiency and on the frequencies of waveguide and cavity modes. On the experimental side, the design and fabrication of a silicon-based Lamb wave resonator which utilizes PC plates as reflective gratings to form the resonant cavity are discussed. The measured results showed significant improvement of the insertion losses and quality factors of the resonators when the PCs were applied.