Evanescent Lamb Wave in Laminated Composites Elastic Plates

Propagation of Lamb waves in multilayered elastic anisotropic plates is studied in the framework of combination of the six-dimensional Cauchy formalism and the transfer matrix method. The closed form secular equations for dispersion curves of Lamb waves propagating in multilayered plates with arbitrary elastic anisotropy are obtained.

Underwater target acoustic scattering is an important aspect of underwater acoustic research. Research on the elastic scattering wave is related to the wave dispersion curves and the exit angle, but the excited condition of a Lamb wave is rarely discussed in the context of elastic plates. The energy distribution for the reflected Lamb waves of several modes when Lamb waves of other anti-symmetric modes are formed by reflection at the end face of an elastic plate is also rarely discussed. However, the research concentrating on the excitation condition of Lamb waves and mode conversion at the end face of a submerged elastic plate can be the basis for target recognition. Based on R.D. Mindlin Hamilton's principle and using an orthogonal complete feature, the frequency dispersion equation of a Lamb wave is derived for an elastic plate in a vacuum; and water, phase velocity, attenuation coefficients and group velocity dispersion curves are obtained. Analysis of the energy ratio between each of two anti-symmetric mode backward Lamb waves resulting from mode conversion when an anti-symmetric Lamb wave is incident at the end face of the plate in single mode is derived. The scattering wave of a submerged elastic plate is measured experimentally, which shows that only the incident anti-symmetric Lamb wave at the critical angle of one mode can directly excite a Lamb wave of this mode in the elastic plate, Lamb waves of other anti-symmetric modes are formed by reflection at the end face of the elastic plate. There is also no energy change between a symmetric Lamb wave and an anti-symmetric Lamb wave, even with mode conversion, if the waves are reflected at the end of the elastic plate. Combined with radiation efficiency of submerged elastic plate energy ratio between each two anti-symmetric mode backward lamb waves resulted from mode conversion when anti-symmetric lamb wave incident at the end face of plate in single mode confirmed validity of the analysis results.

Solitons, or by the original terminology waves of translation, were for the first time observed and described by Scott Russel (1845) as a special kind of the hydrodynamic waves that can arise and propagate in narrow channels. Solitons are: (i) solitary waves, resembling propagation of the wave front of shock waves; (ii) these waves can propagate without considerable attenuation, or (iii) change of form; or (iv) diminution of their speed; see, Craik (2004). It was shown later on, that motion of these waves can be described by a non-linear KdV differential equation; see the work by the originators of the KdV-equation Korteweg and de Vries (1885) and the subsequent works by Lax (1968), Miles (1981), and Zwillinger (1997), where some of the analytical solutions are presented and the main properties of the KdV equation are analyzed. Herein, we analyze the long-wave limits of Lamb waves propagating in multilayered elastic anisotropic plates at vanishing frequency 0   , or in terms of the wave number r , at 0 r  . These vanishing frequency Lamb waves satisfy conditions (i) – (iv), and thus, resemble the solitons. But, in contrast to the genuine solitons in hydrodynamics or their nonlinear analogues propagating in elastic solids; see, Eckl et al. (2004), Kawahara (1972), Kliakhandler, Porubov, and Verlande (2000), Planat and Hoummady (1989), Porubov et al. (1998), Samsonov (2001), our soliton-like waves are described by linear vectorial differential equations, known as the Christoffel equations for Lamb waves. Studies of Lamb waves, as solutions of linear equations of motion for the infinite plates, and the corresponding soliton-like linear waves traveling with the finite phase speed at vanishing frequency have quite a long history. Presumably, the first asymptotic analysis of the waves propagating at vanishing frequency in an isotropic plate with the traction-free outer planes was performed by Gogoladze (1947). He obtained an analytical expression for the phase speed of such a wave by asymptotic analysis of the approximate equation of motion related to the theory of plates based on the Bernoulli – Euler hypotheses. Later on, the similar approach and a more elaborate one allowing to consider plates with different boundary conditions at outer planes, but still based on the approximated theories of plates, were exploited by Mindlin (1951a, b, 1958, 1960), Mindlin and Medick (1959), Mindlin and Onoe (1957), Onoe (1955), and Tolstoy and Usdin (1953). The latter authors reported highly intricate behavior of the disperse curves in the vicinity of the zero frequency. See also a more recent work by Pagneux and Maurel (2001), where the dispersion relations in the complex

Thermoelasticity of multilayered plates with imperfect interfaces

MITC technique extended to variable kinematic multilayered plate elements

This article compares different air-coupled ultrasonic testing methods to characterize impact-type defects in a pultruded quasi-isotropic glass fiber-reinforced plastic (GFRP) composite plate. Using the air-coupled transducers, comparisons among three methods were performed, namely, bulk-wave through transmission, single-side access using guided waves, and ultrasonic-guided wave tomography. The air coupled through transmission technique can determine the size and shape of impact-type defects with a higher resolution, but with the consequence of time consumption and, more importantly, the necessity of access to both sides of the sample. The guided wave technique on the other hand, allows a single-side inspection and is relatively fast. It can be used to determine the size of the defect using ultrasonic B-scan, but the exact shape of the defect will be compromised. Thus, in this article, to determine the shape of the defect, application of the parallel beam tomographic reconstruction technique using guided Lamb waves is demonstrated. Furthermore, a numerical finite element simulation was performed to study the effects of guided wave propagation in the composite sample and interaction with the internal defect. Lastly, the results from the experiments of different techniques were compared according to possibilities of defect sizing and determination of its shape.

A non-destructive testing method using ultrasonic waves for composite materials has been studied. In this paper, a computer program of three dimensional image processing was developed to evaluate the defects in composite materials. The developed computer program has two particular functions, which are smoothing of scan data and decision of defect size. The size of defect was obtained by a gradient of echo level. As the examples, this method was applied to GFRP and CFRP laminates with artificial defects, and the effect of processing on the size of defects was investigated. As a result, the shape, size and location of defects in composite materials were obtained by the three dimensional image processing. It is also revealed that the detectability of defects is improved by this method. From these results, it can be shown that the method proposed in this paper is very useful to evaluate defects in composite materials.

Bridge superstructures, such as slabs and box girders, behave as plate-like structures where Lamb waves propagate. Lamb wave characteristics are highly sensitive to structural properties and damage. Variations in Lamb wave characteristics can thus serve as indicators of structural changes. However, conventional methods struggle to capture their wavefields due to the high cost of deploying dense sensor networks. This study presents the successful measurement of wavefields of low-frequency (1-70 Hz) flexural-mode Lamb waves in bridge box girders using distributed acoustic sensing (DAS). DAS turns pre-existing telecommunication fiber-optic cables into ultra-dense dynamic-strain-sensing arrays, enabling continuous, high-resolution vibration and seismic monitoring. Through ambient noise interferometry, we reconstruct Virtual Shot Gathers from traffic-induced vibrations, eliminating the need for active seismic sources and minimizing disruptions to regular transportation. This method captures both standing waves (resonance vibrations) and propagating flexural waves, corresponding to the fundamental antisymmetric Lamb wave mode. To validate these observations, we model Lamb wave dispersion by approximating box girders as elastic isotropic plates and Bernoulli-Euler beams, incorporating material properties from design drawings. The strong agreement between modeled and measured dispersion curves confirms the reliability of DAS-recorded flexural waves in box girder bridges. These findings demonstrate the potential of DAS for capturing low-frequency Lamb waves, providing a scalable, real-time solution for bridge condition assessment.

Band structures of surface phononic crystals with one-dimensional periodicity are investigated in this paper. Real and complex dispersion relations of surface acoustic waves are calculated by a semi-analytical finite element method in the frequency domain. The model applies 2D nine-noded elements in the periodic direction to discretize a unit cell with finite depth. Propagation perpendicular to the periodic direction is modeled by analytic functions. Two eigenproblems are obtained from the method which lead to real or to complex dispersion relations of the one-dimensional phononic crystal. The model results in a Lamb wave band structure where the surface modes are identified by their displacement and elastic energy distribution. The complex band structure includes imaginary and complex modes in addition to the real dispersion relation describing also standing and evanescent modes. Such evanescent waves not only include the folded surface waves, but it is also shown that an evanescent mode is present within the stop band.

Within the second-order perturbation approximation, this paper investigates the physical process of generation of the time-domain second harmonic by a primary Lamb wave waveform in an elastic plate. The present work is performed based on the preconditions that the phase velocity matching is satisfied and that the transfer of energy from the primary Lamb wave to the double frequency Lamb wave is not zero. It investigates the influences of the difference between the group velocities of the primary Lamb wave and the double frequency Lamb wave, the propagation distance and the duration of the primary Lamb wave waveform on the envelope shape of the time-domain second harmonic. It finds that the maximum magnitude of the envelope of the second-harmonic waveform can grow within some propagation distance even if the condition of group velocity matching is not satisfied. Our analyses also indicate that the maximum magnitude of the envelope of the second-harmonic waveform is kept constant beyond a specific propagation distance. Furthermore, it concludes that the integration amplitude of the time-domain second-harmonic waveform always grows with propagation distance within the second-order perturbation. The present research yields new physical insight not previously available into the effect of generation of the time-domain second harmonic by propagation of a primary Lamb wave waveform.

The interaction of acoustic waves generated by a point source with an elastic plate immersed in acoustic fluid is considered. To estimate the range of applicability of various theoretical approaches, a hierarchy of mathematical and computer models has been developed. The models differ in the governing equations for the sounded elastic laminate structure (from a simplified Kirchhoff-Love plate to a multi-layered anisotropic composite plate obeying the full set of 3D Navier-Lame equations in each sublayer). Explicit integral representations for generated and scattered wave fields have been derived using the Fourier transform technique. In the far-field, the acoustic and guided waves are described by asymptotic representations obtained from those integrals using the stationary phase method and the residual technique. Among others, the derived solutions enable quantitative analysis of time-averaged wave energy fluxes in the considered source-structure system. Analytic representations for the total source energy and its portions carried by reflected, transmitted and guided waves have also been obtained.

Exact analysis of the propagation of acoustic waves in multilayered piezoelectric plates is performed using the transfer matrix method. A general technique for analyzing layered piezoelectric resonators under thickness and lateral field excitation is presented and is applied to the study of zinc oxide on silicon thin film resonators. Both one and two-dimensional analysis with general material anisotropy is performed, and a simplified method for incorporating thin conducting electrodes on the plate's free surfaces is presented. The general methodology described is summarized into efficient algorithms to aid in the implementation of the procedures and some computational aspects are discussed. Results are presented for cutoff behavior as well as general dispersion characteristics for two and three layered plates.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A solution has been found to the problem of calculating the stress and displacement fields caused by a rectilinear dislocation in an anisotropic elastic plate. Special cases of anisotropy have been found with solutions represented by elementary functions. Certain problems in describing crystal plastic deformation phenomena make it vital to know the fields of the elastic stresses and displacements caused by an individual dislocation in a bounded crystal. It is interesting to study the effect of crystal boundaries on these fields with a simple model which approximates fairly closely to experimental conditions. The model selected is shown in Fig, 1. A dislocation with a Burgers vector β (b1, b2, b3) is situated in an infinite elastic anisotropic plate of thickness 2h. The dislocation line is parallel to the plate boundaries. The following restriction is introduced in relation to the plate's elastic properties: the medium has a plane of elastic symmetry perpendicular to the dislocation line. The selection of the coordinate system and position of the dislocation are shown in Fig. 1. The requirement is to find the stresses and displacements at an arbitrary point in the plate. One limited special form of this problem has been solved by Kroupa [1]. The limitations which he introduced are as follows: the medium is isotropic, the dislocation is at the precise center of the band and the Burgers vector has only one component b2 differing from zero (the same coordinates were chosen in [1] as in Fig. 1). Thus Kroupa's results can be obtained from the results of the present work as a special case. Other special cases arising from this problem are those concerning the elastic stress and displacement fields caused by a dislocation in anisotropic semi-bounded [2] and bounded [3] media. It is immediately apparent that the problem is a plane one, in the sense that the fields to be found do not depend on coordinate z. Since the medium has a plane of elastic symmetry perpendicular to the dislocation line, it is clear from [4] that the system of stresses and strains in such a medium can be divided into two independent subsystems. The first of these is plane deformation with stress components σxx, σyy and σxy differing from zero and displacement vector components ux and uy, the second is “antiplane” deformation with stress components σxz and σyz differing from zero and the displacement vector component uz. In the case under examination, the plane deformation is caused by the Burgers vector edge components bx and by and the antiplane deformation by the screw component bz. The solution is therefore divided into two stages, corresponding to edge and screw dislocations.

The guided wave has been widely employed to characterize thin plates and layered media. The dispersion curves of phase and group velocities are essential for the quantitative application of guided waves. The technique using leaky Lamb wave (LLW) is one of the excellent methods to obtain dispersion curves. In the present work, a fully automated system for the measurement of backward radiation of LLW has been developed. The specimen moves in two dimensional planes as well as in angular rotation. The signals of backward radiation of LLW were measured from an elastic plate in which specific modes of Lamb wave were strongly generated. Phase velocity of the corresponding modes was determined from the incident angle. The generated Lamb waves propagated forward and backward with the leakage of energy into water. Backward radiated LLW was detected by the same transducer and its frequency components were analyzed to extract the related information to the dispersion curves. The dispersion curves of phase velocity were measured by varying the incident angle. Moving the specimen in the linear direction of LLW propagation, group velocity was determined by measuring the transit time shift in the rf waveform.

Lamb waves at ultrasonic frequencies travel with little attenuation in thin elastic plates, and we demonstrate their use in pulse-echo behavior to monitor plate integrity. We envision using a single PZT wafer-type transducer to generate waves and to receive reflections from distant flaw or boundary locations. However, Lamb waves generally have multiple modes, each of them highly dispersive, and in consequence pulse dispersion can become pronounced and can make difficult or impossible the interpretation of pulse-echo responses. We show that selective generation of the S0 wave will overcome those difficulties; therefore, selection of transducer dimensions and pulse characteristics to achieve selective generation should be considered mandatory for most intended applications. We first review the work of others identifying a basic relationship between transducer dimension and excitation frequency for selective generation of the S0 wave. We then summarize our extensive experimental studies of wafer-type transducers with particular attention to S0 and A0 mode behavior, both in transmission and reception. We next report our two-dimensional finite element simulation of the same problem performed in FEMLAB, requiring transient simulation of the coupled electromechanical problem. We simulate the piezoelectric response of the wafer-type transducer coupled to the elastic plate, both as transmitter and receiver, as well as the development of Lamb waves within the source region and their subsequent propagation along the plate. Simulations illustrate the development and separation of the S0 and A0 modes and reproduce the expected group velocities and dispersion behavior. We show good agreement between our experiments and our simulations regarding S0 mode behavior, and we summarize the results to guide a designer in choosing transducer dimensions. In particular, good selectivity between the S0 and A0 mode generation can be obtained with appropriate choice of transducer size and center frequency. We show the results of experiments on an aluminum plate in which excitation of a single PZT wafer-type transducer at 6.5 V (peak-to-peak) produces reflected signals of ample strength (tens of mV) from distant boundaries and from partial thickness flaws.