Frequency-thickness Product Research Articles

Visualization of the interaction of guided acoustic waves with water by light refractive vibrometry

Guided acoustic waves, such as Lamb waves, are widely applied for material characterization, sensing of liquids and the generation of streaming in liquids. There are numerical simulation tools for the prediction of their propagation near a solid-liquid boundary but a demand for complementary measurement techniques for the validation of the simulation results remains. In this contribution it is demonstrated that light refractive vibrometry is a suitable approach for the visualization of the interaction of guided acoustic waves with liquids. For this purpose Lamb waves were excited by piezoelectric transducers on copper plates partially immersed in water. There the fundamental symmetric and antisymmetric modes are converted to compressional waves and quasi-Scholte plate waves below a frequency-thickness product of 1 MHz mm. From the vibrometry scans the wavelengths, radiation angles and pressure amplitudes of the involved modes could be determined and thus theoretical predictions of the attenuation of the Lamb modes and the energy distribution of quasi-Scholte plate waves between the solid substrate and the liquid environment could be confirmed.

EMAT design for minimum remnant thickness gauging using high order shear horizontal modes

Detection and sizing of corrosion are critical issues across many industries such as for the oil and gas industry or the petrochemical industry. Inspections may become difficult and time-consuming when the structures under inspection are only partially accessible such as for pipes under insulation or at pipe supports. It has been demonstrated in the literature that the cutoff frequency-thickness product of high order ultrasonic guided wave modes can be used in medium to long-range thickness gauging. As the thickness varies along an inspection line, the thickness variation acts as a low-pass filter for the high order ultrasonic guided wave modes. As the thickness drops below the cutoff frequency-thickness product of a given mode, this mode is filtered out of the propagating wave packet. The effectiveness of this technique depends on the number of excited modes and the width of the ultrasonic beam along the inspection line. Both of these parameters can easily be controlled using electromagnetic acoustic transducers (EMAT) for the excitation. Analytical and multiphysics finite element simulations were performed to optimize an EMAT that can excite enough modes to allow the measurement of the remnant thickness based on the number of modes propagating through a corroded area. The results were validated experimentally, and a thickness resolution of 2 mm was achieved in a 10 mm aluminum plate.

The nonlinear S0 Lamb mode in a plate with a linearly-varying thickness

The aim of this paper is to investigate propagation characteristics and the generation mechanism of the nonlinear lowest-order symmetric Lamb mode (S0) which propagates downslope in free elastic plates with slowly linearly varying-thickness. From theoretical analyses, in a low frequency-thickness product (fd) range, the S0 mode is slightly dispersive, it is easy to generate, and it approximately satisfies the principle of the phase velocity matching. Therefore, if a S0 mode is excited at a proper frequency in the low fd range, the amplitude of the second harmonic wave is linearly increasing in a certain propagating-distance, which is valuable for the practical NDE application of the second harmonic wave. Moreover, numerical simulations and experiments have been carried out to validate theoretical results. Our investigation of properties of the second harmonic wave can be applied to characterize and evaluate micro-structural damages in varying-thickness waveguides.

Experimental validation of an analytical method to predict lamb wave scattering from a discontinuity

This paper presents an experimental validation of an analytical method called complex mode expansion with vector projection (CMEP), which is used to calculate the scattering coefficients (amplitude of the out-of-plane velocity) of Lamb wave modes from geometric discontinuities. For a test case, a plate with a thickness step change type geometric discontinuity is considered in this paper. The scattered wave fields from the discontinuity are expanded in terms of complex Lamb wave modes with unknown scatter coefficients. These unknown coefficients are obtained by projecting the stress or displacement boundary conditions on the displacement or stress boundary conditions utilizing the power expression. In the analytical analysis, complex-valued scatter coefficients are calculated with frequency-thickness product from 50 to 1500 kHz mm for A0 incident wave. A parametric study was conducted using CMEP to find the optimized step depth ratio for the experiment. For incident A0 mode at step depth ratio of 0.6, the scattering coefficients of reflected and transmitted S0 modes are maximum. A plate of thickness 4.86 mm with a step depth ratio of 0.6 was chosen for experimental study. Long piezoelectric wafer active sensors (PWAS) were used to create straight crested Lamb wave modes. Antisymmetric Lamb wave mode selectively excited by using two PWAS in out of phase on opposite sides of the plate. Scanning laser Doppler vibrometer was used to measure the out-of-plane velocity of scattered Lamb wave fields on the plate. Scatter coefficients were calculated from Fourier transform of the time domain signal. The obtained experimental results agree well with the CMEP analytical predictions.

Experimental Investigation of Guided Wave Imaging in Thin Soft Media under Various Coupling Conditions

Guided wave imaging for the artery remains in its infancy in clinical practice mainly because of complex arterial microstructure, hemodynamics and boundary conditions. Despite the theoretically known potential effect of the surrounding medium on guided wave propagation in thin media in non-destructive testing, experimental evidence pertaining to thin soft materials, such as the artery, is relatively scarce in the relevant literature. Therefore, this study first evaluated the propagating guided wave generated by acoustic radiation force in polyvinyl alcohol-based hydrogel plates differing in thickness and stiffness under various material coupling conditions (water and polyvinyl alcohol bulk). A thin-walled polyvinyl alcohol hollow cylindrical phantom coupled by softer gelatin–agar phantoms and an excised porcine aorta surrounded by water and pork belly were further examined. Guided waves in the thin structure and shear waves in the bulk media were captured by ultrafast ultrasound imaging, and guided wave dispersion as a function of the frequency–thickness product was analyzed using the zero-order anti-symmetric Lamb wave model to estimate the shear modulus of each thin medium studied. Results confirmed the deviated shear modulus estimates from the ground truth for thin plates, the thin-walled hollow cylindrical phantom and the porcine aorta bounded by stiffness-unmatched bulk medium. The findings indicated the need for (i) careful interpretation of estimated shear moduli of thin structure bounded by bulk media and (ii) a generalized guided wave model that takes into account the effect of coupling medium.

Quantitative Evaluation of Delamination in Composites Using Lamb Waves

Ultrasonic guided wave monitoring has become very popular in the area of structural health monitoring (SHM) of aerospace structures. Any possible type of damage must be reliably assessed. The paper deals with delamination length determination in DCB specimens using Lamb waves. An analytical equation based on the velocity dependence on variable thickness is utilized. The group velocity of the fundamental antisymmetric A0 mode rapidly changes in a particular range of the frequency-thickness product. Using the same actuation frequency the propagation velocity is different for delaminated structure. Lamb wave based delamination lengths were compared to the visually determined lengths. The method of the wave velocity determination proved to be essential. More accurate results were achieved by tracking the maximum amplitude of A0 mode than the first signal arrival. These findings are considered as the basis for the damage evaluation of complex structures.

Plane Wave SH0 Piezoceramic Transduction Optimized Using Geometrical Parameters

Structural health monitoring is a prominent alternative to the scheduled maintenance of safety-critical components. The nondispersive nature as well as the through-thickness mode shape of the fundamental shear horizontal guided wave mode (SH) make it a particularly attractive candidate for ultrasonic guided wave structural health monitoring. However, plane wave excitation of SH at a high level of purity remains challenging because of the existence of the fundamental Lamb modes (A and S) below the cutoff frequency thickness product of high-order modes. This paper presents a piezoelectric transducer concept optimized for plane SH wave transduction based on the transducer geometry. The transducer parameter exploration was initially performed using a simple analytical model. A 3D multiphysics finite element model was then used to refine the transducer design. Finally, an experimental validation was conducted with a 3D laser Doppler vibrometer system. The analytical model, the finite element model, and the experimental measurement showed excellent agreement. The modal selectivity of SH within a 20 beam opening angle at the design frequency of 425 kHz in a 1.59 mm aluminum plate was 23 dB, and the angle of the 6 dB wavefront was 86.

Design of Waveguide Bars for Transmitting a Pure Shear Horizontal Wave to Monitor High Temperature Components.

Guided wave technique could be a possible method for monitoring components working in high temperature above 350 °C. However, this would require the design of an appropriate waveguide bar to transmit the wave, so that its sensing part is not influenced by the high temperature. In the present study, the shape of waveguide bars is designed based on the analysis of wave source characteristics. The critical frequency-width and frequency-thickness products of waveguide bars are analyzed theoretically and numerically to transmit the zeroth shear horizontal wave SH0* in bars. The results show that waveguide bars can cut off all the other wave modes when their frequency-thickness products are smaller than the critical value fd*, and frequency-width products are not smaller than the critical value fw*. Six waveguide bars are designed and fabricated based on the design criteria, and an experiment system is set up to check their work performance. The testing results indicate that the wave signals of the SH0* mode propagate clearly in waveguide bars, and cut off all the other modes when the frequency-thickness products and frequency-width products of the bars meet the design criteria. It is also demonstrated that the dependency of the experimental group velocity of each waveguide bar on frequency is in good agreement with the numerical result. High-temperature experiments also validate the reliability of the designed waveguide bars. Therefore, the critical frequency-thickness product and frequency-width product can be the basis for the design of the waveguide bars.

Excitation of Lamb waves over a large frequency-thickness product range for corrosion detection

For corrosion detection, it is often desirable that a Lamb wave mode is highly sensitive to surface thinning and enjoys some degree of mode purity at a particular frequency. In view of this, this paper aims to generate a variety of Lamb wave modes over broad frequency bands to ensure an abundant supply of candidates for corrosion detection, and further, establish a strategy to find appropriate operation points efficiently and effectively. Firstly, a short-duration laser pulse is applied to generate Lamb waves over a large frequency-thickness product range. The selection of symmetric modes or anti-symmetric modes is obtained by addition or subtraction of signals captured by two identical transducers which are symmetrically coupled on both sides of the plate. Subsequently, the S0 mode at a non-dispersive frequency bandwidth is employed to improve the accuracy of the transmitter–receiver distance. Based on those, three selection criteria including mode separability, amplitude ratio and corrosion sensitivity, are presented to efficiently determine the suitable operation points (i.e., mode types and frequencies). The experimental results show that the simulated corrosion could be correctly detected and accurately localized at the chosen modes and frequencies.

Propagation properties of interface waves at fluid-coated solid interface

The interface waves propagating along liquid-solid interface are widely studied and used in a lot of fields, especially in ocean acoustics, ocean engineering, and ocean geophysics. The dispersion characteristics of this kind of interface wave are closely related to the seafloor medium parameters, which is an effective means for the inversion of the seafloor sediments. However, the interface wave is difficult to use for ultrasonic nondestructive material characterization, especially for stiff and dense solid materials, owing to the mode shape or wave structure of the liquid-solid interface waves.The fraction of the total wave energy that travels in the fluid compared with the solid depends on the properties of the solid material. Usually, for a stiff and dense solid compared with the fluid, most of the energy travels in the fluid, while for a soft solid more energy travels in the solid. Therefore, it is difficult to use this kind interface wave for stiff solid material characterization. However, in the case of liquid-coated solid interface, the behaviors and properties of interface waves are quite different.In this paper, we use pulsed laser to generate the interface waves at the water-coated solid interfaces. The theoretical analysis of the laser-induced excitation of acoustic waves propagating along a plane interface between liquid and layered elastic solid is perforemd first. The general solution for the interface motion is derived. The analytic expression of the transient response is then obtained. Based on this expression, the dispersion characteristics of the interface waves, which propagate along the fluid-coated solid interface for the cases of slow coating on fast substrate and fast coating on slow substrate, are calculated and analyzed. The transient response signals are further calculated. In the case of slow coating on fast substrate, the interface wave shows an evident dispersion, in which its phase velocity is larger than its group velocity. In the case of fast coating on slow substrate, the interface wave also shows a remarkable dispersion within a smaller frequency-thickness product range, in which its phase velocity is less than its group velocity. The theoretical transient signals show the same properties.In order to verify the theoretical results, an experimental system is set up, and the interface waves are generated and measured. The experimental system mainly consists of pulsed laser, hydrophone, oscilloscope, and movable translation stage. The pulsed laser is used to excite the interface waves, and the hydrophone mounted on the movable translation stage is placed near the interface to receive the signals. Two kinds of samples of slow coating on fast substrate and fast coating on slow substrate are made and measured. The recorded testing signals are then processed and analyzed.The theoretical results and the experimental ones are in good agreement. The research results presented in this paper can provide theoretical basis for ultrasonic nondestructive characterization of coating and film material in immersion testing mode, and also for seafloor sediment parameter inversion.

Reflection and transmission coefficients of the SH0 mode in the adhesive structures with imperfect interface

Compared with body waves, ultrasonic guided waves can provide more local characteristic information about the interface in the defect detection of adhesive structures. In the paper, the expressions of the reflection and transmission coefficients of the lowest SH mode (SH0) in multilayered plate-like adhesive structure were deduced on the basis of wave propagation controlling equations and tangential stiffness coefficient KT was contained in the expressions. Then, the expressions were compared with the previous results to verify their applicability and correctness. Then, aluminum/epoxy resin/aluminum adhesive structures were used to explore the effects of the changes in incident angle, frequency-thickness product and tangential stiffness coefficient on SH wave propagation characteristics in adhesive structures with different interface quality (perfect, weak bonding, and slip/debonding interfaces). The results showed that the propagation mode of SH wave in adhesive structures was mainly determined by the incident angle, frequency, adhesive layer thickness and tangential stiffness coefficient. With the increase in the frequency-thickness product, multi-order resonance is generated in the reflection and transmission coefficient curves of SH wave under the perfect and weak bonding interfaces. If proper values of the incident angle of acoustic waves and frequency-thickness product are selected, the perfect, weak bonding, and slip/debonding interfaces can be differentiated from each other, but the slip and debonding interfaces cannot be distinguished from each other. The study provides theoretical contribution to the detection of multilayered plate-like adhesive structure by SH wave.

Ultrasonic high-frequency guided waves for the inspection of shell weld in core support structure of fast breeder reactors

This article reports an ultrasonic high-frequency guided wave based methodology for inspection of shell weld of core support structure of the main vessel in 500-MWe sodium-cooled prototype fast breeder reactor. The high-frequency guided waves are generated in 30-mm-thick plate of core support structure at 2 MHz (frequency–thickness product of ∼60 MHz-mm). These wave modes are fairly nondispersive over considerable distance of propagation through the material. In order to understand the generation and propagation characteristics of these high-frequency guided waves, a series of two-dimensional plane strain finite element models are developed using explicit scheme. The optimum transducer placement for generating efficient high-frequency guided wave is achieved by sweeping the transducer position along the curvature of the main vessel plate with respect to the centerline of the vertical shell support plate. The generated high-frequency guided waves are analyzed and identified with the help of dispersion curves and time–frequency analysis. The simulation results are found to capture the features observed in the experimental measurements rather well, and the time of flight of the reflected waves in the simulation is found to be in excellent agreement with the experimental results. The optimum transducer locations for inspection of a general T-joint configuration with different angles between the web and the base have been studied in detail using finite element simulations. With optimum transducer position, the high-frequency guided waves are also used to inspect the artificially introduced defects in the connecting weld of shell support plate. It has been demonstrated that the high-frequency guided wave could detect defects of 20% wall thickness (6 mm) in the connecting shell weld.

Separation of Leaky Lamb Modes for Ultrasonic Evaluation of Multilayer Structures

Ultrasonic guided Lamb waves are a major tool for nondestructive evaluation of cylindrically layered, fluid–loaded, elastic structures. When used in fluid–immersed steel pipes with a frequency–thickness product in the range of 1 to 3 MHz mm, fundamental pseudo–Lamb modes – the antisymmetric flexural and symmetric extensional modes –interfere and complicate the evaluation of elastic properties of an inaccessible medium outside of a pipe. We examine two techniques designed to separate the flexural mode from mixed leaky modes. Both techniques, a mode decomposition algorithm and a pursuit algorithm, are tested on synthetic and experimental data.

Investigation of Scholte and Stoneley Waves in Multi-layered Systems

Interface waves are elastic waves that can propagate at the interface between two solids (Stoneley wave) or between a solid and a liquid (Scholte wave). In this study, properties of generalized Stoneley and Scholte waves are investigated analytically in a multi-layer system with both liquid-solid and solid-solid interfaces. The interface waves are modeled using partial waves in layers with finite thicknesses to trace quasi- and non-dispersive modes. Dispersion curves of the propagating modes and corresponding particle displacement profiles are obtained using numerical solution techniques with the global matrix method. Limiting conditions of quasi-modes are evaluated analytically for thickness and material selection. Furthermore, interference of the two interface waves and plate modes are investigated for small frequency-thickness products in the multi-interface system using dispersion curves and particle displacement profiles. Preliminary sensitivity analyses are also performed for development of multi sensing physical quantities such as temperature, viscosity and density simultaneously using interface waves.

Determination of the in-plane components of motion in a Lamb wave from single-axis laser vibrometry.

A method is proposed for determining in-plane components of motion in a Lamb wave from laser vibrometer measurements of surface motion out of plane. The approach relies on a frequency domain transformation that assumes knowledge only of the plate thickness and the bulk wave speeds. An outline of the relevant theory is followed by several validation case studies that generally affirm a useful level of accuracy and robust performance across a relatively wide frequency-thickness product range. In a comparison to the two-angle vibrometry approach, the proposed method is shown to be simpler to implement and to yield estimates with a consistently higher signal to noise ratio. The approach is then used to furnish estimates of the in-plane strains in Lamb waves propagating in an aluminum plate at frequencies below the first cut-off. These estimates are compared to strain measurements obtained from an adhesively bonded fiber Bragg grating. The agreement is shown to be excellent overall with an average discrepancy of less than 6%; however, systematic errors of twice that amount were recorded in the low-frequency-thickness product regime. These low-frequency discrepancies are not consistent with known sources of experimental error and cannot be explained by shear-lag theory.

High order shear horizontal modes for minimum remnant thickness

Thickness mapping in aging structures suffering from corrosion is challenging especially when the structure is only partially accessible. In plates the high order shear horizontal guided wave modes all have a cutoff frequency thickness product below which they cannot propagate. This property is potentially attractive to estimate the minimum remnant thickness between two transducers. When using a source and a sensor array it is possible to control the number of modes being excited and the size of the region interrogated by the technique. Finite element simulations were used to show that by exciting multiple guided wave modes simultaneously and identifying which modes are received by a sensor array it is possible to estimate the minimum remaining thickness along the propagation path. Initial experimental results showed excellent agreement with the finite element simulations when the plate is uniform and with a thickness reduction between the source and the sensor arrays the minimum remnant thickness was underestimated by approximately 20%.

High frequency guided ultrasonic waves for defect monitoring

High frequency guided ultrasonic waves offer a good compromise between area coverage and defect detection sensitivity for the health monitoring of critical sections of structures. Using standard ultrasonic transducers with single sided access to the structure, guided wave modes were generated that penetrate through the complete thickness of the structure. The wave propagation and interference of the guided wave modes depends on the frequency thickness product. Laboratory experiments were conducted using a laser interferometer and the wave propagation dependence on the wall thickness reduction verified against theoretical predictions. Measurements were conducted using accelerated corrosion in a salt water bath. From the measured signal changes due to the wave mode interference the wall thickness reduction due to corrosion was monitored. The energy transfer between the plate surfaces results in a good sensitivity for the detection of small defects on both surfaces. Experimentally the reflected wave was measured using standard pulse-echo equipment. Using a combination of evaluation in the time and frequency domain, the defect location and damaged plate side can be accurately determined. The high frequency guided waves have the potential for damage monitoring at critical and difficult to access locations from a stand-off distance.

The effect of frequency on secondary wave mode generation for ultrasonic health monitoring of fatigue damaged plate structures

ABSTRACTA computational frequency analysis of secondary wave mode generation is considered in this paper. Secondary wave mode conversion at the tip of a through‐thickness slit in a circular isotropic plate is analysed similar to a classical Sommerfeld's diffraction problem. The analysis is useful for quantitative non‐destructive evaluation of fatigue cracked plate structures that involve propagation of a single wave mode (a vertically polarised symmetric Lamb mode) and measurement of the amplitude of a new mode (a horizontally polarised shear mode). A two‐dimensional finite element model is first introduced as a baseline reference for the non‐dispersive scenario. This model is used to study the effect that P‐wave angle of incidence has on the mode converted S‐wave from the tip of the slit. A dispersive three‐dimensional finite element model of the circular plate is then introduced at a number of frequency‐thickness products within the range 0.5–3 MHz.mm−1. Within this frequency–thickness range, significant changes to the through‐thickness displacement profiles of the incident S0 Lamb mode occur. The secondary SH0 plate wave mode generated by mode conversion from the 45o incident S0 Lamb mode at the tip of a slit is analysed to establish the efficiency of the mode conversion. Visualisations of the wave fields demonstrate that at frequency–thickness products higher than the first symmetric Lamé mode (where there is no in‐plane surface displacement), the in‐plane through‐thickness displacement profile of the incident S0 Lamb mode becomes incompatible with that of the SH0 plate mode, and the mode conversion phenomenon is thus severely compromised.

Research on Elastic Scattered Wave of Submerged Elastic Plate

It is proved by theory and experiment that the arrival time of elastic scattering wave is determined by group velocity of Lamb wave in plate and the speed of elastic scattering wave in water. The frequency dispersion equation of Lamb wave is derived for submerged elastic plate, and the phase velocity and group velocity dispersion curves are obtained by numerical calculation method. It is found that the phase velocity is greater or less than the group velocity at different frequency-thickness products. The energy propagation speed of wave is group velocity, so the arrival time of elastic scattering wave is determined by group velocity of Lamb wave in plate and the speed of sound in water. Experimental results show that elastic scattering wave is ahead of or behind the edge wave in echoes of elastic steel plate. The experiment results confirm validity of the theoretical analysis results.

Generation and detection of higher-order mode clusters of guided waves (HOMC-GW) using meander-coil EMATs

This paper reports on a new means of generating higher-order mode clusters of guided waves (HOMC-GW) using a meander-coil (MC) electromagnetic acoustic transducer (EMAT) in plates at frequencies significantly higher than the lower-order plate modes. These wave modes are considerably less dispersive and they occur at much higher frequency-thickness (f x d) products. Our studies cover the f x d range of 13 to 20 MHz·mm. Experimental measurements were carried out on Al plate samples of different thicknesses using three different EMAT coil periods. To understand the generation and propagation characteristics of HOMC-GW with EMATs, several simulations were carried out using 2-D finite element models at different f x d products. These simulations captured all features observed in the experiments. The time-frequency smoothed pseudo Wigner-Ville distribution (SPWVD) was used to analyze the HOMC-GW modes. Defect detection measurements using HOMC-GW generated using EMATs were made on Al plates with machined defects.