Pdf this work first introduces explicit expressions of displacements and stresses of free rayleigh waves and lamb waves propagating in an. Furthermore, the assumed displacement function introduces additional constraints which increase the stiffness of the system. Fundamentals and analysis of lamb waves springerlink. Propagation of rayleighlamb waves in multilayered plates. Guided waves play an important role in nondestructive health monitoring. Rayleigh waves, or guided modes such as lamb waves in plates. Horace lamb also established the theoretical rudiments of such waves. Substantial work using rayleigh wave has also been. Keywords dispersion equation, solid acoustic waveguide, the phase velocity, package matlab i. Surface rayleigh waves have an elliptical particle motion and travel across the surface of a material. Features of lamb waves used for nde and shm are timeofflight tof, mode conversiongeneration, change in amplitudeattenuation, velocity, etc. Rayleigh waves are distinct from other types of surface or guided acoustic waves such as love waves or lamb waves, both being types of guided waves supported by a layer, or longitudinal and shear waves, that travel in the bulk.
This research presents a 3d numerical finite element fe model which, previously developed, precisely simulates noncontact, aircoupled measurements of nonlinear rayleigh wave propagation. Crosssectional view of a meander coil emat for exciting obliquely propagating l or sv waves, rayleigh waves, or guided modes such as lamb waves in plates. In reading this paper one is struck by the fact that, except for minor changes in notation, the formulation established by rayleigh is still followed today in the theory of. We used a threelayer model similar to the one studied by ryden et al. Another such wave in platelike structure is the shearhorizontal sh wave, commonly studied for laminated composite plates 21. Their velocity is approximately 90% of the shear wave velocity of the material and their depth of penetration is approximately equal to one wavelength.
Our group has been investigating the feasibility of using a shear wave dispersion ultrasound vibrometry sduv method to excite lamb waves in organs with platelike geometry to estimate the viscoelasticity of the medium of interest. Y raylpdfx,b computes the rayleigh pdf at each of the values in x using the corresponding scale parameter, b. Interaction of lamb waves with geometric discontinuities. The distance between the monitoring position p and the center of the simulated laser source on the plate, which is referred to as the wave propagation distance, is 90 mm. May 14, 2002 laser generated rayleigh and lamb waves are studied in aluminum samples. A scalar input for x or b is expanded to a constant array with the same dimensions as the other input the rayleigh pdf is. The nondestructive testing of materials can be conducted by various techniques. The complex behavior of the lamb wave modes at different frequency ranges was explained. On lamb and rayleigh wave convergence in viscoelastic tissues. The response of rectangular piezoelectric sensors to rayleigh. The method thus reduces the dynamic system to a singledegreeoffreedom system.
Rayleigh and lamb waves have the advantage that they prop. It was lord rayleigh in 1889 who first explained wave propagation along a guided surface 1, and the waves are known as rayleigh waves today. I have just started to look into this subject area which i am completely unfamiliar with. If a and b are the parameters of the weibull distribution, then the rayleigh distribution with parameter b is equivalent to the weibull distribution with parameters a 2 b and b 2 if the component velocities of a particle in the x and y directions are two independent normal random variables with zero means. Wavelet transform is used to analyze received signals and compute the arrival times of lamb. It is shown how aperturing the laser beam produces high thermal gradients resulting in large. By solving the rayleighlamb equation, the highest group velocities of s 0 and a 0 modes are 5433 ms and 3149 ms respectively in the aluminum plate of thickness 1. A fundamental understanding of the response of piezoelectric transducer patches to ultrasonic waves is of increasing interest to the field of structural health monitoring. In practice, especially in fea you have to work with waves having wavelength waves. They can be produced in materials in many ways, such as by a localized impact or by piezoelectric transduction, and are frequently used in nondestructive testing for detecting defects. These waves were excited by an 11sec pulse, projected onto the cylindrical surface at an excitation angle given by sin 1 c o c s, where c o is the velocity of sound in water and c s is the velocity of the surface wave. The commercial fesolver abaqus is used to perform the simulations.
In 1917, the english mathematician horace lamb published his classic analysis and description of acoustic waves of this type. Preface classroom training handbook ultrasonic testing 5330. Prediction of attenuated guided waves propagation in carbon. Development of a 3d finite element model evaluating air. With lamb waves, a number of modes of particle vibration are possible. Prediction of attenuated guided waves propagation in. Laser generated rayleigh and lamb waves are studied in aluminum samples. Rayleigh probability density function matlab raylpdf. A discussion about the capability to simulate, with multiphysics finite element method commercial software, guided wave in composite material using the rayleigh damping is developed. In general, elastic waves in solid materials are guided by the boundaries of the media in which they propagate. Following rayleighs work, horace lamb, a british applied mathematician, reported the waves discovered in plates in one of his historic publications, on waves in an elastic plate, in 1917 2, and the waves were named after him as lamb waves.
According to him, only some frequencyvelocity pairs can propagate through it. Modelling of attenuation of lamb waves using rayleigh. Buy this book softcover 103,99 price for spain gross buy softcover isbn 9781489956835. Rayleighs method requires an assumed displacement function.
Therefore, the analytically estimated arrival times. Thus, rayleighs method yields an upper limit of the true fundamental frequency. Rayleighlamb wave from eric weissteins world of physics. The response to both rayleigh waves and lamb waves is examined, starting with harmonic excitation fields and moving to broadband and narrowband excitation fields. First, frequency dependent pressure wave attenuation is investigated numerically to reconstruct the sound pressure distribution along the. Rayleigh waves are a type of surface acoustic wave that travel along the surface of solids. Rayleigh and lamb waves a basic principles wiley online library. Nondestructive examination nde technology and codes student manual volume 2 chapter 8.
Rayleigh waves are part of the seismic waves that are produced on the earth by earthquakes. Pwas and the lamb wave propagation method in structural health monitoring. This article theoretically studies the symmetry characteristics of rayleigh lamb guided waves in nonlinear, isotropic plates. Such structures being an integral part of an aircraft has resulted the popularity of lamb wave. Lamb 1917 derived the dispersion equation that can handle the transitions between these types of waves. A comparison of lamb waves generated by unfocused and linefocused laser sources is conducted. An approach to guided wave propagation, widely used in physical acoustics, is to seek sinusoidal solutions to the wave equation for linear elastic waves subject to boundary conditions representing the structural geometry.
The existence of such waves was predicted by lord rayleigh in 1885. Pdf rayleigh wave propagation for the detection of near. The derivation of the characteristic equations and wave speed dispersion curve of lamb waves are provided. There is a description of algorithm in the book ultrasonic waves in solid media by joseph l. These waves may develop after an initial transient phase, and are straightcrested waves which propagate in one of the inplane directions and are stationary in the throughthickness direction. Generation and detection of lamb waves for the characterization of. Motion characteristic of the rayleigh waves and antisymmetric lamb waves was produced by oscillating the respective sources with four cycles of sinusoidal waves in the frequency range 25600 hz with the displacements in tens of micrometers. An analytical approach compared with fem banibrata poddara and victor giurgiutiua adepartment of mechanical engineering, university of south carolina, usa abstract. Ultrasonic waves used to check the surface layer of various device components and specimens are another example of rayleigh waves. The rayleigh distribution can be used to model the lifetime of an object or a service time.
In the thermoelastic regime the rayleigh wave impulse response from a laser line source is a negative going inward displacement delta function. Both rayleigh and lamb waves are constrained by the elastic properties of the surfaces that. Characterization of the viscoelastic material properties of soft tissue has become an important area of research over the last two decades. Rayleigh wave propagation and scattering characteristics at. The basic principles in rayleigh and lamb wave propagation with respect to their use in. The rayleigh distribution is a special case of the weibull distribution. This article theoretically studies the symmetry characteristics of rayleighlamb guided waves in nonlinear, isotropic plates. Defect detection with rayleigh and lamb waves generated by. Rayleigh waves have horizontal displacement and the motion of a point in the medium is an ellipse.
X and b can be vectors, matrices, or multidimensional arrays that all have the same size, which is also the size of y. Threedimensional lamb wave propagation excited by laser pulses is modeled. Following rayleighs work, horace lamb, a british applied mathematician, reported the waves discovered in plates in one of his historic publications, on waves in an elastic plate, in. Nakagami distributions can be reduced to rayleigh distributions, but give more control over the extent of the fading. When a harmonic load or displacement is applied on a halfspace, rayleigh wave will contain the biggest portion of energy when compared to other types of body waves swaves and pwaves, and it decays at a much slower rate along the surface. Physical theory and applications ultrasonic technology softcover reprint of the original 1st ed. Broadband rayleigh waves are generated using a tem00 mode nd. Lamb waves and dispersion curves in plates and its. Lamb waves are formed by interference of multiple reflections and mode conversion of longitudinal waves p waves and shear waves s waves at the free surfaces of the plate viktorov, 1967. Features of the interpretation of the dispersion equation. Rayleighlamb wave definition of rayleighlamb wave by. For this the etching process is another alternative for the fabrication of thin membranes. Design, sensing mechanisms and applications generation of shear waves.
While analytical solutions exist on the interaction of a piezoelectric actuator with the generated lamb waves, the behavior of a piezoelectric sensor has only been examined for the limited case of a piezoactuated lamb wave. Find all the books, read about the author, and more. The first investigation will report on the scattered wave pattern and amplitude for various angles of. The response of rectangular piezoelectric sensors to. Lamb waves with particle displacement in both the x and ydirection actually represents a group. A short instruction of the shorttime fourier transformation will provided.
For what thickness of plate, rayleigh wave will become lamb wave. The reader is referred to either the book literature 17,22232425 or tutorialtype material 53 for more details. Rayleighlamb wave article about rayleighlamb wave by the. General oblique incidence of the wave on rectangular sensors is treated first. Douglas, scattering of rayleighlamb waves by a surface breaking crack in an elastic plate, j acoust soc am.
Figure 3 illustrates the relationship between largescale and smallscale fading. Modeling of threedimensional lamb wave propagation excited. On the existence of antisymmetric or symmetric lamb waves. According to the restrictions of manual handling and the fabrication process, the minimum thickness is limited to around 100 m microns. Scattering of the fundamental symmetrical lamb wave mode by a small edge crack in an isotropic plate 211 symmetrical modes with a small compared to wavelength edge crack.
Multimodal lamb wave dispersion curves are measured and analyzed to obtain elastic. Lamb wave damping coefficient was modeled using the multiphysics finite element method and compared with experimental results. In practice, especially in fea you have to work with waves having wavelength oct 21, 2011 a point source on the surface of the plate was used to excite rayleigh waves. Acoustic waves propagating on the surface of solid and hollow elastic cylinders immersed in water were observed and their characteristics shown to be consistent with theory. Modeling of threedimensional lamb wave propagation. Lamb waves as we said, lamb derived the dispersion relation for different waves traveling across the plane of a free plate. A numerical solution of the dispersion equation of the. Following is an example of the inversion steps needed when dealing with lamb wave data patterns. Selffocusing of rayleigh surface waves for offaxis defect, a unfocused transmission, b unfocused reception, c superposition of unfocused signals, d transmission. I have been told to get myself a copy of rayleigh and lamb waves.
More experiments on selffocusing for rayleigh waves in the case of multiple reflectors in the steering range and selffocusing of lamb waves are discussed in 3. Nondestructive examination nde technology and codes student. Following is an example of the inversion steps needed when dealing with lambwave data patterns. Crosssectional view of a normal field emat for exciting plane polarized shear waves propagating normal to the surface.
On the existence of antisymmetric or symmetric lamb waves at. Rayleigh waves have a speed slightly less than shear waves by a factor dependent on the elastic constants of the material. Rayleigh lamb wave synonyms, rayleigh lamb wave pronunciation, rayleigh lamb wave translation, english dictionary definition of rayleigh lamb wave. The rayleigh waves r2, to be used in the next section, are related to the lamb waves. Rayleigh and nakagami distributions are used to model dense scatters, while rician distributions model fading with a stronger lineofsight. Apart from these, circumferential, axial, and radial wave propagation in cylindrical structures and pipes are also studied by several researchers. Lamb waves are generated at an incident angle in which the parallel component of the velocity of the wave in the source is equal to the velocity of the wave in the test material.
They are produced when the plate thickness 2h is much greater than the wave length and they are propagated along the free boundary and independently of the plate thickness. Features of the interpretation of the dispersion equation of. Physical theory and applications ultrasonic technology at. Crosssectional view of a periodic permanent magnet emat for exciting grazing or obliquely propagating horizontally polarized sh waves or guided sh modes in plates.
Rayleigh waves occur on the earths surface during earthquakes. These waves are employed for nondestructive evaluation nde and structural health monitoring shm applications. Ultrasonic rayleigh and lamb waves have been utilized to reconstruct tomographic images of defects. The shorter wavelength, the closer velocity and polarization of lamb waves to rayleigh. A type of seismic surface wave that moves with a rolling motion, causing rock particles to vibrate in directions both perpendicular and parallel to the.
When a harmonic load or displacement is applied on a halfspace, rayleigh wave will contain the biggest portion of energy when compared to other types of body waves s waves and p waves, and it decays at a much slower rate along the surface. Rayleighlamb theory for plate structures was obtained. Scattering of the fundamental symmetrical lamb wave mode. Lamb waves with particle displacement in both the x and ydirection actually represents a group of wave types including the bending wave, the rayleigh wave, and the quasilongitudinal wave. Rayleighlamb wave definition of rayleighlamb wave by the. Rayleigh fading channels in mobile digital communication. A scalar input for x or b is expanded to a constant array with the same dimensions as the other input. Rayleighlamb wave article about rayleighlamb wave by. Rayleigh and lamb waves physical theory and applications.
For what thickness of plate, rayleigh wave will become. Lamb waves are ultrasonic waves which propagate through thin platelike structures. Several forms of guided waves have been studied for the application to shm. One of the most common is the lamb wave which propagates through thin plates and shells bounded by stress free surfaces. Lamb waves will travel several meters in steel and so are useful to scan plate, wire, and tubes. It is now a century since lord rayleigh published his theoretical proof of the existence of elastic waves propagating along the plane surface of an elastic solid 1. Rayleighlamb waves are a type of planestrain, guided waves that exist in thin platelike structures constrained by two free surfaces graff, 1991. Lamb waves propagating in an elastic plate have a great deal of interest for. Rayleigh and lamb waves transport in random waveguides. Lamb waves in a plate with defects of different lengths are examined. Introduction elastic isotropic acoustic waveguides in the form of plates or rod form the basis for constructing a large number of technical devices. Pdf solutions of rayleigh waves and lamb waves generated by a. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the plane normal the direction perpendicular to the plate.
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