A numerical study of the interaction of laser-generated circumferential wave with defect on hollow cylinder

A finite element method is used to investigate the laser-induced circumferential waves on a hollow cylinder with a surface defect. After a control numerical experiment for correctness verification, the method is used to simulate the circumferential waves on hollow cylinder having a defect with different source–defect angles, depths, and widths. The results show that the existence of the defect will generate a bipolar waveform between the S₀ mode and the A₀ mode in the circumferential waves, and its emerging time and amplitude linearly depend on the source–defect angle and the depth of defect, respectively. Moreover, we also find that the width of defect has obvious influence on the circumferential waves.     

激光在管道中激发周向导波的有限元模拟

以超声的热弹激发机理为基础，建立了模拟激光在管道中激发周向导波的有限元模型.为了验证模型的正确性，根据Gao等人报道的实验条件［J. Appl. Phys. 91, 6114 (2002)］进行了模拟，理论模拟结果与Gao等人的实验结果符合很好，说明本数值模型的正确性.在此基础上，模拟激光在不同厚度、不同曲率半径的管道内激发的周向导波波形，同时分析了激光在管道中激发的周向导波波形与平板中激发Lamb波的差异，以及管道厚度和曲率半径对激光激发周向导波的影响.     

Wave propagation in non-homogeneous magneto-electro-elastic hollow cylinders

A dynamic solution is presented for the propagation of harmonic waves in imhomogeneous (functionally graded) magneto-electro-elastic hollow cylinders composed of piezoelectric BaTiO₃ and magnetostrictive CoFe₂O₄. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The Legendre orthogonal polynomial series expansion approach is employed to determine the wave propagating characteristics in the hollow cylinders. The dispersion curves of the imhomogeneous piezoelectric-piezomagnetic hollow cylinder and the corresponding non-piezoelectric and non-piezomagnetic hollow cylinders are calculated to show the influence of the piezoelectricity and piezomagnetism. Electric potential and magnetic potential distributions are obtained to illustrate the different influences of the piezoelectricity and piezomagnetism and the different influences of the piezoelectric effect and piezomagnetic effect on longitudinal modes and torsional modes. For the radial polarizing piezoelectric-piezomagnetic hollow cylinder, the piezoelectric effect and piezomagnetic effect take mostly on the longitudinal mode. Finally, a hollow cylinder at different ratio of radius to thickness is calculated to show the influence of the ratio on the piezoelectric effect and piezomagnetic effect.     

Natural frequencies of free finite-length circular cylinders

Sets of tables are given for the natural frequencies of the first five symmetric and first five antisymmetric modes of a hollow or solid cylinder for circumferential wave numbers n = 0, 1, 2. Contour graphs are given for the lowest frequencies as functions of (length)/(mean radius) and (thickness)/(mean radius).     

Circumferential waves in a composite circular cylinder

Circumferential waves propagating in a layered, circular cylinder are studied. The cylinder consists of an elastic circular core encased in a hollow, circular cylinder of distinctly different elastic properties. Both smooth and bonded contact are considered. The effect of curvature and layer thickness on the phase velocity of the lowest mode(s) is investigated for both an acoustically softer and an acoustically stiffer layer. When the outer radius of the composite cylinder is unbounded, Stoneley waves are a limiting case as the ratio of the radius of the core to the wavelength increases beyond bounds. When the outer radius is finite, waves in a layer and a half-space result for this limit. Some attention is also directed to the limiting case of small layer thickness to the wavelength ratio. In the limit as this ratio vanishes, the motion of the core reduces to that of Rayleigh waves on the curved surface. For smooth contact, the motion of the core becomes uncoupled from that of the layer for this limiting case, and two distinct modes are seen to exist.     

Asymmetric propagation of the first order antisymmetric lamb wave in a tapered plate based on time domain analysis

The asymmetric propagation of the first order antisymmetric (A₁) Lamb wave in a tapered plate respectively carved with sharp bottom corner and round bottom corner is theoretically investigated. Through numerical simulation of A₁ Lamb wave in time domain, we find that when the thickness of the waveguide abruptly decreases to below the cut-off thickness, about half of the A₁ mode is converted into the fundamental symmetrical S₀ and antisymmetrical A₀ modes to pass through the defected region. Furthermore, the transmitted modes A₀ and S₀ are completely apart from each other and can be quantitatively evaluated. Conversely, when the thickness change is very smooth, most of the energy of A₁ Lamb wave is reflected back. It is the unique mode conversion behavior that leads to great transmission difference value of A₁ Lamb wave along the opposite directions. Finally, the influence of geometrical parameters on the transmission coefficient is also studied. The higher efficiency and proper working frequency range can be realized by adjusting the slope angle θ, height h ₁ and h ₂. The simple asymmetric systems will be potentially significant in applications of ultrasound diagnosis and therapy.     

无限流体中孔隙介质圆柱周向导波的传播特性

为研究无限大流体约束的孔隙圆柱中周向导波的传播规律,分析孔隙参数对导波传播特性的影响,建立了无限流体中孔隙介质圆柱的理论模型,利用孔隙介质弹性波动理论,建立了周向导波频散方程,通过数值模拟计算得到无限流体中孔隙介质圆柱的频散曲线,探讨了圆柱半径和孔隙参数对导波传播特性的影响,并对导波的衰减特性进行了分析;通过数值计算,得到了周向导波的时域波形,讨论了孔隙参数对波形的影响.结果表明,孔隙介质圆柱半径的改变影响圆柱尺度,孔隙度的改变影响孔隙介质中体声波的波速,都对周向导波频散曲线产生一定的影响,所得到的频散曲线特征及衰减曲线与时域波形吻合.研究结果对开展无限流体中孔隙介质圆柱的超声无损评价提供了一定的理论参考.     

Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection

Recent improvements in tomographic reconstruction techniques generated a renewed interest in short-range ultrasonic guided wave inspection for real-time monitoring of internal corrosion and erosion in pipes and other plate-like structures. Emerging evidence suggests that in most cases the fundamental asymmetric A₀ mode holds a distinct advantage over the earlier market leader fundamental symmetric S₀ mode. Most existing A₀ mode inspections operate at relatively low inspection frequencies where the mode is highly dispersive therefore very sensitive to variations in wall thickness. This paper examines the potential advantages of increasing the inspection frequency to the so-called constant group velocity (CGV) point where the group velocity remains essentially constant over a wide range of wall thickness variation, but the phase velocity is still dispersive enough to allow accurate wall thickness assessment from phase angle measurements. This paper shows that in the CGV region the crucial issue of temperature correction becomes especially simple, which is particularly beneficial when higher-order helical modes are also exploited for tomography. One disadvantage of working at such relatively high inspection frequency is that, as the slower A₀ mode becomes faster and less dispersive, the competing faster S₀ mode becomes slower and more dispersive. At higher inspection frequencies these modes cannot be separated any longer based on their vibration polarization only, which is mostly tangential for the S₀ mode while mostly normal for the A₀ at low frequencies, as the two modes become more similar as the frequency increases. Therefore, we propose a novel method for suppressing the unwanted S₀ mode based on the Poisson effect of the material by optimizing the angle of inclination of the equivalent transduction force of the Electromagnetic Acoustic Transducers (EMATs) used for generation and detection purposes.     

Time reversal technique for health monitoring of metallic structure using Lamb waves

Time reversal active sensing using Lamb waves is investigated for health monitoring of a metallic structure. Experiments were conducted on an aluminum plate to study the time reversal behavior of A₀ and S₀ Lamb wave modes under narrow band and broad band pulse excitation. Damage in the form of a notch was introduced in the plate to study the changes in the characteristics of the time reversed Lamb wave modes experimentally. Time–frequency analysis of the time reversed signal was carried out to extract the damage information. A measure of damage based on wavelet transform was derived to quantify the hidden damage information in the time reversed signal. It has been shown that time reversal can be used to achieve temporal recompression of Lamb waves under broadband signal excitation. Further, the broad band excitation can also improve the resolution of the technique in detecting closely located defects. This is demonstrated by picking up the reflection of waves from the edge of the plate, from a defect close to the edge of the plate and from defects located near to each other. This study shows the effectiveness of Lamb wave time reversal for temporal recompression of dispersive Lamb waves for damage detection in health monitoring applications.     

Scattering and active acoustic control from a submerged piezoelectric-coupled orthotropic hollow cylinder

A general exact analysis for three-dimensional scattering of a time-harmonic plane-progressive sound wave obliquely incident upon an arbitrarily thick bilaminated circular hollow cylinder of infinite extent, which is composed of a cylindrically orthotropic axially polarized piezoelectric inner layer perfectly bonded to an orthotropic outer layer, is presented. An approximate laminate model in the context of the so-called state space formulation along with the classical T-matrix solution technique involving a system global transfer matrix is employed to solve for the unknown modal scattering and transmission coefficients. Numerical example is given for an air-filled and water-submerged two-layered elasto-piezoelectric hybrid (steel/PZT4) hollow cylinder insonified by an obliquely incident unit-amplitude plane sound wave. Following the acoustic resonance scattering theory (RST), the total form function amplitude together with the associated global scattering, the far-field inherent background, and the resonance scattering coefficients of the nth normal mode are computed as a function of dimensionless frequency for selected angles of incidence, piezoelectric layer thickness parameters, and electrical boundary conditions (i.e., open/closed circuit or active). Also, the electrical voltage coefficients required for partial or complete cancellation of the reflected sound field are calculated. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.     

Circumferential waves for a cylindrical shell in smooth contact with a continuum

Dispersion relations are determined for circumferential waves propagating in a layered, circular cylinder by using shell equations to approximate the behavior of the outer layer. These equations include the effects of transverse shear deformation and rotatory inertia. The cylinder consists of an elastic core in smooth contact with a hollow, circular cylinder of distinctly different elastic properties. Two distinct modes exist as the shell thickness reduces to zero. One mode is recognized to be surface waves on the convex cylindrical surface of the core; the second mode is associated with long longitudinal waves in the shell. The approximate dispersion curves for these modes are compared with curves obtained by employing elasticity equations for the layer. As the curvature increases, the agreement of the two theories becomes progressively poorer whether or not any disagreement exists for the case of no curvature. The agreement of the two theories is better when the layer is relatively stiff than when the layer is relatively soft. The shell equations simplify the calculations necessary to produce the dispersion curves.     

An experimental study on the behavior of Ao mode transduced using misaligned air-coupled transducers in composite laminates

Air-coupled transducers can be employed for generation and reception of Lamb waves in composite laminates. To generate and receive a particular Lamb mode, the transducers are oriented in a particular angle. The orientations of transducers also determine amplitude of a particular Lamb mode of interest. Any deviation (misalignment) from the right orientation results in a reduction in the amplitude. Therefore, an attempt has been made to establish variation in amplitude of the fundamental anti-symmetric Lamb mode (A_o) when misalignments were purposefully introduced in the air-coupled probes. In the present work, three different misalignments – linear, orientation and synchronized orientation were envisaged. Experiments were performed on two laminates – unidirectional ([0₃]_s) and cross-ply ([0/90₂/0]_s) of GFRP (Glass Fiber Reinforced Plastic) material. Curves were fitted on experimental data to characterize variation in amplitude of A_o mode in each misalignment. Moreover, wave splitting and merging was observed when the probe(s) was (were) rotated in a particular direction in orientation and synchronized orientation misalignments.     

Wave propagation in axially polarized piezoelectric hollow cylinders of sector cross section

In this paper, the problem of electroelastic waves propagating in piezoelectric hollow cylinders of sector cross section is studied for the case when the boundary surfaces of sector cut are covered by non-extensible membranes. The three-dimensional linear equations of motion for the piezoelectric cylinder are analytically integrated and different boundary conditions on the cylindrical surfaces yield frequency equations, which relate the frequencies of elastic waves to their wavenumbers. Numerical results for waveguides with various boundary conditions are presented to illustrate the approach. Analysis of the dispersion spectra is carried out, and cutoff frequencies are obtained and characterized; mode asymptotic behavior and amplitude distributions of wave characteristics are analyzed. The main effects of their transformation by variation of the angular measure and the ratio of inner and outer radii are discussed. The results obtained are in good agreement with the results for the special case of a hollow semicircular cylinder.     

Effect of misalignment of air-coupled probes on Ao Lamb mode propagating in a metal plate

Proper alignment of air-coupled ultrasonic transducers for generation and reception of Lamb waves is vital in order to acquire a high amplitude wave group. Any misalignment with either the transmitter or the receiver or both adversely influences the amplitude of a Lamb mode. This paper reports a systematic attempt to quantify the reduction in the amplitude of the fundamental anti-symmetric Lamb mode (A_o) in a metal plate caused by misalignments in air-coupled probes. Three different types of misalignments – linear, orientation and synchronised orientation were deliberately introduced in the transducers, and experiments were performed on a 6 mm thick aluminium plate. Amplitudes of A_o mode measured at various configurations were normalised with that of A_o mode, captured in a reference configuration. Suitable curves fitted over the experimental data points revealed that Gaussian curves represent appropriately the variations in normalised amplitudes of A_o mode. Moreover, analytical expressions were derived to predict the difference in arrival times of Lamb mode(s) due to orientation and synchronised orientation misalignments.     

铌锌酸铅-钛酸铅薄层中Lamb波模式的交叉特性

以[001]_c和[011]_c极化的铌锌酸铅-钛酸铅晶体为研究对象, 利用子波理论对其无限大自由薄层中传播的Lamb波的色散及模式交叉特性进行了研究. 发现只有[001]_c极化的晶体中的对称与反对称模式Lamb波之间出现了多次交叉, 并且变化规律与铌镁酸铅-钛酸铅的情形相同. Lamb波的A₀和S₀模式的交叉是由准纵向剪切波慢度曲线的多值关系引起的, 此时其x₃方向的波数在一定范围内存在一对非纯虚数的复共轭根. 利用此结论推导出A₀和S₀模式交叉时弹性常数需要满足的条件, 为判断正交、四方对称性晶体中Lamb波的A₀和S₀模式是否交叉提供了一种直观、简便的方法.     

The use of an orthogonality relation for reducing the size of finite element models for 3D guided waves scattering problems

The scattering of guided waves by complex shaped defects in three-dimensional (3D) waveguides is considered. For such problems, analytical solutions do not exist, and modal decomposition techniques based on the establishment of the displacement and stress fields in the vicinity of the scatterer are quite heavy and complicated to perform. On the other hand, finite elements (FE)-based methods constitute a powerful way to obtain solutions, but they are known to be very memory consuming. This paper proposes a post-processing technique, based on a 3D orthogonality relation, to decompose a complex acoustic field produced by a scatterer and predicted by a 3D FE model, into plane waves, the amplitudes of which are quantified in the far field. This technique allows important reductions in the size of the FE models to be made. Two applications are presented to demonstrate the potential of this method. The first one concerns the scattering of the S₀ Lamb wave incident on a flat bottom circular hole. In this example, the amplitude of each mode is calculated via the orthogonality relation-based method, and compared to that obtained by simply monitoring the displacements at appropriate through-thickness positions. In the second application, the incident S₀ Lamb mode is converted into five modes scattered by a defect of complex geometry.     

Zeitliche Entwicklung des Reflexionskoeffizienten einer Plasmaschicht bei starker elektromagnetischer Einstrahlung

We have calculated numerically the temporal evolution of the nonlinear reflection coefficient R of an overdense plasma layer by solving the system of partial differential equations consisting of the wave equation for the slowly varying amplitude of the electric field and the hydrodynamic equations for the ion motion, including a ponderomotive force term. In dependence of the (normalized) amplitude of the incident wave u_A two regimes exist: Below a critical amplitude u_A^* ? 1 the reflection coefficient is approximately independent on the amplitude u_A and temporally constant. In the opposite case u_A>u_A^*, on the other hand, ‖R‖² decreases slowly with time down to a minimum value and after that it increases rapidly to the initial value. We think, that our results are important to interpret the anomalous reflectivity observed in some experiments when strong electromagnetic waves are incident on an overdense plasma.     

Zur nichtlinearen Theorie der Strahl-Plasma-Wechselwirkung. II. Das asymptotische Verhalten des Systems

The asymptotic behavior of an amplifying beam-plasma-system for great distances x from the point where the beam enters the plasma is investigated on the basis of an expansion in powers of the electric field amplitude A₁. It is shown, that a necessary condition for approaching a steady state with a constant wave amplitude is that the first 2N ?1 derivations of the distribution function f₀(u) at the point u_φ vanish if the approximation is to N-th order in A₁. Further it is demonstrated, that an approximation including the third power of A₁ involves the important physical effects as mode coupling and wave amplitude saturation, and that fifth order correction terms are unimportant, if the beam parameters are chosen so that the difference between beam and phase velocities is not to small. To illustrate this point some numerical examples are considered.     

Natural beam focusing of non-axisymmetric guided waves in large-diameter pipes

The propagation of non-axisymmetric guided waves in larger diameter pipes is studied in this paper by treating the guided waves as corresponding Lamb waves in an unwrapped plate. This approximation leads to a simpler method for calculating the phase velocities of hollow cylinder guided waves, which reveals a beam focusing nature of non-axisymmetric guided waves generated by a partial source loading. The acoustic fields in a pipe generated by a partial-loading source includes axisymmetric longitudinal modes as well as non-axisymmetric flexural modes. The circumferential distribution of the total acoustic field, also referred as an angular profile, diverges circumferentially while guided waves propagate with dependence on such factors as mode, frequency, cylinder size, propagation distance, etc. Exact prediction of the angular profile of the total field can only be realized by numerical calculations. In particular cases, however, when the wall thickness is far less than the cylinder diameter and the wavelength is smaller than or comparable to the pipe wall thickness, the acoustic field can be analyzed based on the characteristics of Lamb waves that travel along a periodic unwrapped plate. Based on this assumption, a simplified model is derived to calculate the phase velocities of non-axisymmetric flexural mode guided waves. The model is then applied to discussions on some particular characteristics of guided-wave angular profiles generated by a source loading. Some features of flexural modes, such as cutoff frequency values are predicted with the simpler model. The relationship between the angular profiles and other factors such as frequency, propagation distance, and cylinder size is obtained and presented in simple equations. The angular profile rate of change with respect to propagation distance is investigated. In particular, our simplified model for non-axisymmetric guided waves predicts that the wave beam will converge to its original circumferential shape after the wave propagates for a certain distance. A concept of "natural focal point" is introduced and a simple equation is derived to compute the 1st natural focal distance of non-axisymmetric guided waves. The applicable range of the simplified equation is provided. Industrial pipes meet the requirement of wall thickness being far less than the pipe diameter. The approximate analytical algorithms presented in this paper provides a convenient method enabling quick acoustic field analysis on large-diameter industrial pipes for NDE applications.     

Experimental study of the A0 and S0 Lamb waves interaction with symmetrical notches

The aim of this work is to study the fundamental Lamb modes interaction with defects in isotropic plates. For these experimental investigations, symmetrical notches with various depths milled in aluminum plates are considered. Moreover, the incident Lamb wave of a specific mode is generated by means of two identical thin piezoceramic transducers placed at the opposite sides of the plate. The waves scattered by the notch are recorded with conventional transducers located on the plate surface in front and behind the defect. The selection of the A₀ or the S₀ modes is obtained by exciting the transducers with anti-phased or in-phased signals, respectively. Furthermore, a calibration process is investigated to correct errors caused by the presence of the receiver between the emitters and the defects. The power reflection and transmission coefficients are then obtained and the power balance is verified. Finally, these measurements are compared successfully with those obtained by a numerical method using the finite-element modeling described in a previous work.