Waves In Elastic Plates Research Articles

Best reconstruction frequency for time-reversal process of Lamb waves and its determination from a single measurement

This article presents a theoretical framework for the concept of the best reconstruction frequency (BRF) of Lamb waves in elastic plates and its significance in the context of structural health monitoring (SHM). The time-reversal process (TRP) of Lamb waves has been widely explored for baseline-free damage detection in thin-walled structures. Previous studies have indicated that the accuracy of damage detection can be significantly improved by exciting Lamb waves at a frequency where the time-reversibility in the undamaged state is maximum within a desired frequency range. This frequency, termed the BRF, has been determined phenomenologically in prior research but lacked a rigorous theoretical foundation. In this study, the BRF is derived as the frequency at which the amplitude dispersion of the main mode of the reconstructed signal after the TRP is minimized. Accordingly, a method is put forth to determine this frequency from a single measurement using a broadband excitation eliminating the need to compute the similarity between the reconstructed and input waveforms at different excitation frequencies. The developed theory is validated with experiments conducted on an aluminium plate with Lamb waves generated and sensed by surface-bonded piezoelectric patch transducers. Notably, the study establishes that the BRF is a system property specific to a given transducer-plate-adhesive system and is independent of the distance between transducers. This characteristic makes it suitable for identifying damage locations when employing multiple sensing paths. The findings constitute an important milestone in understanding the TRP of Lamb waves and developing robust SHM technologies based on it.

Experimental validation of zero-group-velocity feature guided waves in a welded joint utilizing the pitch-catch measurement technique with air-coupled ultrasonic transducers

Zero-Group-Velocity (ZGV) Lamb waves in elastic plates had been conducted extensive theoretical and experimental researches in the field of ultrasonic nondestructive testing. The ZGV modes in complex structures had been studied theoretically, but less attention had been paid to their experimental investigation. This paper reports the experimental observation of Zero-Group-Velocity Feature Guided Waves (ZGV-FGWs) in a welded joint using the pitch-catch measurement technique with air-coupled ultrasonic transducers. Firstly, for the elastic plate, it is verified that the received time-domain signal using the pitch-catch measurement method with air-coupled ultrasonic transducers is indeed ZGV Lamb waves. Subsequently, we applied the same pitch-catch measurement method with air-coupled ultrasonic transducers to receive time-domain signals at different excitation frequencies in the welded joint. It is observed that the received time-domain signals in the welded joint oscillate for extended periods of time. By performing short-time Fourier transforms on the received time-domain signals, we analyze the frequency content of the received time-domain signals at different excitation carrier frequencies. By analyzing the spectral amplitude variations of these signals at different excitation carrier frequencies, it can be demonstrated that the spectral amplitude corresponding to the resonance frequency is the largest. These findings collectively affirm that the received time-domain signals in the welded joint exhibit ZGV characteristics, identified as ZGV-FGWs. Consequently, from an experimental perspective, the presence of ZGV-FGWs in the welded joint is verified. Moreover, the experimentally determined resonance frequency of ZGV-FGWs concurs with the results obtained through simulation. This study confirms the feasibility of using the pitch-catch measurement method with air-coupled ultrasonic transducers to excite ZGV-FGWs in a welded joint and provides a reference for future experimental investigations of ZGV-FGWs in complex structures.

Validation of zero-group-velocity feature guided waves in a welded joint

Extensive research has been conducted on zero-group-velocity (ZGV) Lamb waves in elastic plates, demonstrating significant progress in the field of nondestructive testing. However, there is a scarcity of studies focusing on ZGV modes in complex structures. In this paper, we present our research investigating the presence of ZGV feature guided waves (FGWs) in a welded joint. Our approach follows a similar methodology used to study ZGV Lamb waves in elastic plates. By employing two-dimensional (2D) finite element (FE) modeling, we analyze the response spectra of the welded joint when subjected to a force source, revealing the occurrence of resonance in the response spectra. To investigate resonance modes in the welded joint, we employ the three-dimensional (3D) time-step FE method. By applying spatial 2D and short-time Fourier transforms to the received time-domain signals, we analyze the frequency content and spatial distribution of the signals. This analysis allows us to verify the existence of non-propagation and propagation modes in the welded joint. The non-propagation mode refers to the presence of signals with a zero wavenumber, indicating that they do not propagate or travel along the welded joint. These signals are typically associated with local resonances or vibrations within the welded joint itself. On the other hand, the propagation mode corresponds to signals with nonzero wavenumbers, suggesting that they propagate or travel along the welded joint. Furthermore, by further analyzing the propagation mode in the welded joint, similar to the analysis of ZGV modes in solid plates, we have observed that it also exhibits ZGV characteristics based on the wavenumber-frequency spectra. To further analyze acoustic field distributions at resonance frequencies, we utilize the semi-analytical finite element method in conjunction with the perfectly matched layer method. The results obtained from this analysis are consistent with those obtained from the 2D FE method and 3D time-step FE method, thereby confirming that propagation modes with ZGV characteristics at resonance frequencies correspond to FGWs, which we refer to as ZGV-FGWs. Through this step-by-step analysis, we ultimately establish the existence of ZGV-FGWs in the welded joint. This study introduces fresh ideas and serves as a point of reference for future research on ZGV-FGWs in complex structures.

Branched flows of flexural elastic waves in non-uniform cylindrical shells.

Propagation of elastic waves along the axis of cylindrical shells is of great current interest due to their ubiquitous presence and technological importance. Geometric imperfections and spatial variations of properties are inevitable in such structures. Here we report the existence of branched flows of flexural waves in such waveguides. The location of high amplitude motion, away from the launch location, scales as a power law with respect to the variance, and linearly with respect to the correlation length of the spatial variation in the bending stiffness. These scaling laws are then theoretically derived from the ray equations. Numerical integration of the ray equations also exhibit this behaviour-consistent with finite element numerical simulations as well as the theoretically derived scaling. There appears to be a universality for the exponents in the scaling with respect to similar observations in the past for waves in other physical contexts, as well as dispersive flexural waves in elastic plates.

Localized waves in elastic plates with perturbed honeycomb arrays of constraints.

In this paper, we study wave propagation in elastic plates incorporating honeycomb arrays of rigid pins. In particular, we demonstrate that topologically non-trivial band-gaps are obtained by perturbing the honeycomb arrays of pins such that the ratio between the lattice spacing and the distance of pins is less than 3; conversely, a larger ratio would lead to the appearance of trivial stop-bands. For this purpose, we investigate band inversion of modes and calculate the valley Chern numbers associated with the dispersion surfaces near the band opening, since the present problem has analogies with the quantum valley Hall effect. In addition, we determine localized eigenmodes in strips, repeating periodically in one direction, that are subdivided into a topological and a trivial section. Finally, the outcomes of the dispersion analysis are corroborated by numerical simulations, where a time-harmonic point source is applied to a plate with finite arrays of rigid pins to create localized waves immune to backscattering.This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.

Influence of Dimensions and Shape of Process Cutouts on the Accuracy of Locating Acoustic Emission Sources

Experiments have been carried out to study the effect of the size and shape of cutouts in steel plates on the difference in the arrival time of acoustic emission pulses at receiving transducers. The data obtained were compared with the results of a numerical simulation of the propagation of elastic waves in plates with various strip and circular cutouts. The research results indicate that the shape of the cutout has a much lesser effect on the time of recording pulses by the transducers of an antenna array than the size of the cutout and the location of the receiving transducer relative to the shading zone—the edge of the cut-out. Based on the results of studies in a 40-mm–thick steel plate with a central 100 mm hole, the accuracy of locating an acoustic emission source near the edge of the hole was estimated. Studies have shown that, in this case, to reduce the measurement error to less than 10% of the antenna-array base size, the location group must include at least four transducers. Numerical simulation of the propagation of acoustic emission pulses in plates with strip and circular cutouts made it possible to significantly reduce the volume of experimental studies, while increasing their information content.

The SH0 wave manipulation in graded stubbed plates and its application to wave focusing and frequency separation

In this study, a methodology to artificially control the propagation of fundamental shear horizontal (SH0) waves in elastic plates is proposed using stubs with spatially graded height, and two novel phenomena, wave focusing and low-pass wave filtering, are realized. As a basis for this, the band structure analysis is carried out for an aluminum plate with hexagonally arranged stubs, and the relationships between the wave velocity, the band gap and the stub height are established. Based on these results, omni-directional SH0 wave-based Luneburg and Maxwell fish-eye lenses, as well as low-pass wave filter, are designed artificially using multiple stubs with spatially graded heights. It is demonstrated that both of the lenses exhibit good focusing ability, with the focusing size smaller than the wavelength. Additionally, they can work with a finite bandwidth around the design frequency. It has also been revealed both in time and frequency domain simulations that the SH0 wave with a lower frequency can travel over a longer distance when it enters the low-pass wave filter, which demonstrates its frequency separation capability.

Valley-based splitting of topologically protected helical waves in elastic plates

Topological protection offers unprecedented opportunities for wave manipulation and energy transport in various fields of physics, including elasticity, acoustics, quantum mechanics and electromagnetism. Distinct classes of topological waves have been investigated by establishing analogues with the quantum, spin and valley Hall effects. We here propose and experimentally demonstrate the possibility of supporting multiple classes of topological modes within a single platform. Starting from a patterned elastic plate featuring a double Dirac cone, we create distinct topological interfaces by lifting such degeneracy through selective breaking of symmetries across the thickness and in the plane of the plate. We observe the propagation of a new class of heterogeneous helical-valley edge waves capable of isolating modes on the basis of their distinct polarization. Our results show the onset of wave splitting resulting from the interaction of multiple topological equal-frequency wave modes, which may have significance in applications involving elastic beam-splitters, switches, and filters.

Conical dispersion of Lamb waves in elastic plates

Guided elastic waves in homogeneous isotropic plates exhibit conical dispersion at zero wave number for discrete values of Poisson's ratio where accidental degeneracy between longitudinal and transverse thickness resonances occur. Waves excited at the coincidence frequency have the infinite phase velocity associated with thickness mode resonances, but the group velocity remains finite. This leads to propagating waves that oscillate in time but are spatially uniform over the plate surface. Here, accidental degeneracy is induced in an aluminum plate by cooling the plate to tune the Poisson's ratio through the degenerate point. We measure linear dispersion in the transition from forward to backward propagating waves near zero wave number. In addition, we demonstrate that waves generated near the coincidence frequency exhibit spatially uniform phase over the plate surface, and show angle independent mode conversion upon encountering the free edge of the plate. We propose that elastic plates offer a simple system to explore the physics of conical dispersion at zero wave number, and that this phenomenon may find application in acoustic devices and nondestructive testing.

Time reversal focusing of elastic waves in plates for an educational demonstration.

The purpose of this research is to develop a visual demonstration of time reversal focusing of vibrations in a thin plate. Various plate materials are tested to provide optimal conditions for time reversal focusing. Specifically, the reverberation time in each plate and the vibration coupling efficiency from a shaker to the plate are quantified to illustrate why a given plate provides the best spatially confined focus as well as the highest focal amplitude possible. A single vibration speaker and a scanning laser Doppler vibrometer (SLDV) are used to provide the time reversal focusing. Table salt is sprinkled onto the plate surface to allow visualization of the high amplitude, spatially localized time reversal focus; the salt is thrown upward only at the focal position. Spatial mapping of the vibration focusing on the plate using the SLDV is correlated to the visual salt jumping demonstration. The time reversal focusing is also used to knock over an object when the object is placed at the focal position; some discussion of optimal objects to use for this demonstration are given.

English

In this article, gauge condition in elastodynamics is explored more to revive its potential capability of simplifying wave propagation problems in elastic medium. The inception of gauge condition in elastodynamics happens from the Navier-Lame equations upon application of Helmholtz theorem. In order to solve the elastic wave problems by potential function approach, the gauge condition provides the necessary conditions for the potential functions. The gauge condition may be considered as the superposition of the separate gauge conditions of Lamb waves and shear horizontal (SH) guided waves respectively, and thus, it may be resolved into corresponding gauges of Lamb waves and SH waves. The manipulation and proper choice of the gauge condition does not violate the classical solutions of elastic waves in plates; rather, it simplifies the problems. The gauge condition allows to obtain the analytical solution of complicated problems in a simplified manner.

Broad-angle negative reflection and focusing of elastic waves from a plate edge

Guided elastic waves in plates, or Lamb waves, generally undergo reflection and mode conversion upon encountering a free edge. In the case where a backward-propagating Lamb wave is mode-converted to a forward-propagating wave or vice versa, the mode-converted wave is reflected on the same side of the surface normal as the incident wave. In this paper, we study such negative reflection and show that this effect can be achieved over a broad angular range at a simple plate edge. We demonstrate, through both numerical and experimental approaches, that a plate edge can act as a lens and focus a mode-converted Lamb wave field. Furthermore, we show that as the wave vectors of the incident and mode-converted Lamb waves approach each other, the mode-converted field nearly retraces the incident field. We propose that broad-angle negative reflection may find application in the nondestructive testing of structures supporting guided waves and in the development of new acoustic devices including resonators, lenses, and filters.

Gradient Index Devices for the Full Control of Elastic Waves in Plates.

In this work, we present a method for the design of gradient index devices for elastic waves in plates. The method allows the design of devices to control the three fundamental modes, despite the fact that their dispersion relation is managed by different elastic constants. It is shown that by means of complex graded phononic crystals and thickness variations it is possible to independently design the three refractive indexes of these waves, allowing therefore their simultaneous control. The effective medium theory required for this purpose is presented, and the method is applied to the design of the Luneburg and Maxwell lenses as well as to the design of a flat gradient index lens. Finally, numerical simulations are used to demonstrate the performance of the method in a broadband frequency region.

Transformation cloaking and radial approximations for flexural waves in elastic plates

It is known that design of elastic cloaks is much more challenging than that of acoustic cloaks, cloaks of electromagnetic waves or scalar problems of anti-plane shear. In this paper, we address fully the fourth-order problem and develop a model of a broadband invisibility cloak for channelling flexural waves in thin plates around finite inclusions. We also discuss an option to employ efficiently an elastic pre-stress and body forces to achieve such a result. An asymptotic derivation provides a rigorous link between the model in question and elastic wave propagation in thin solids. This is discussed in detail to show connection with non-symmetric formulations in vector elasticity studied in earlier work.

Propagation of thickness-twist waves in elastic plates with periodically varying thickness and phononic crystals

We study the propagation of thickness-twist (TT) waves in a crystal plate of AT-cut quartz with periodically varying, piecewise constant thickness. The scalar differential equation by Tiersten and Smythe is employed. The problem is found to be mathematically equivalent to the motion of an electron in a periodic potential field governed by Schrodinger’s equation. An analytical solution is obtained. Numerical results show that the eigenvalue (frequency) spectrum of the waves has a band structure with allowed and forbidden bands. Therefore, for TT waves, plates with periodically varying thickness can be considered as phononic crystals. The effects of various parameters on the frequency spectrum are examined.

Effective Models of PZT Actuators for Numerical Simulation of Elastic Wave Propagation

In this study, to accurately identify the functions of piezoelectric actuators and sensors for the generation and collection of elastic waves in typical engineering structures, several effective models of surface-bounded flat PZT disks are further developed and validated for numerical modelling of elastic wave propagations. Based on these models, a series of finite element models of elastic waves in plates are devised using both implicit and explicit dynamics analysis techniques and those numerical simulations are conducted and verified one another. The results flowed from the present research is being used to study the elastic wave propagation in pipes and develop an online structural health monitoring (SHM) system with an integrated piezoelectric actuator-sensor network.

Moulding and shielding flexural waves in elastic plates

Platonic crystals (PlCs) are the elastic plate analogue of the photonic crystals widely used in optics, and are thin structured elastic plates along which flexural waves cannot propagate within certain stop band frequency intervals. The practical importance of PlCs is twofold: These can be used either in the design of microstructured acoustic metamaterials or as an approximate model for surface elastic waves propagating in meter scale seismic metamaterials. Here, we make use of the band spectrum of PlCs created by an array of either very small or densely packed clamped circles to achieve surface wave reflectors at very large wavelengths, a flat lens, a waveguide effect, a directive antenna near the stop band frequencies. The limit in which the circles reduce to points is particularly appealing as there is an exact dispersion relation available so the origin of these phenomena can be explained and interpreted using Fourier series and high-frequency homogenization (HFH). We then enlarge the radius of clamped circles, which both makes the zero-frequency stop band up to five times wider and flattens the dispersion curves. Here, HFH notably captures the essence of localized modes, one of which appears in the zero-frequency stop band and is used in the design of a highly directive waveguide.

Dirac cone dispersion of acoustic waves in plates without phononic crystals (L).

Lamb waves in elastic plates exhibit Dirac-like cone dispersion at zero wavevector at the points of accidental degeneracy between longitudinal and transverse thickness resonances. The Lamb mode dispersion can be fine-tuned to yield a conical point by coating a plate with a layer of a different material and varying the thickness of the latter. Similarities and differences with respect to Dirac-like cone dispersion in phononic crystals and possible observable effects are discussed.

Transmission, trapping and filtering of waves in periodically constrained elastic plates

The paper discusses properties of flexural waves in elastic plates constrained periodically by rigid pins. A structured interface consists of rigid pin platonic gratings parallel to each other. Although the gratings have the same periodicity, relative shifts in horizontal and vertical directions are allowed. We develop a recurrence algorithm for constructing reflection and transmission matrices required to characterize the filtering of plane waves by the structured interface with shifted gratings. The representations of scattered fields contain both propagating and evanescent terms. Special attention is given to the analysis of trapped modes which may exist within the system of rigid pin gratings. Analytical findings are accompanied by numerical examples for systems of two and three gratings. We show geometries containing three gratings in which transmission resonances have very high quality factors (around 35 000). We also show that controlled lateral shifts of three gratings can give rise to a transmission peak with a sharp central suppression region, akin to the phenomenon of electromagnetic-induced transparency.

On the thermodynamically consistent fractional wave equation for viscoelastic solids

Since the well-known methods for the computation of harmonically induced LAMB waves in elastic plates cannot be applied directly to viscoelastic material, an enhanced material model is developed. It is based on fractional time derivatives and incorporates a fractional KELVIN–VOIGT model. A proof of the thermodynamic consistency of this model is given. On the basis of the developed material model, the fractional wave equation is derived which allows for the calculation of LAMB waves in viscoelastic solids.