Lamb Wave Dispersion Curves Research Articles

Dispersion curve estimation via a spatial covariance method with ultrasonic wavefield imaging

Numerous Lamb wave dispersion curve estimation methods have been developed to support damage detection and localization strategies in non-destructive evaluation/structural health monitoring (NDE/SHM) applications. In this paper, the covariance matrix is used to extract features from an ultrasonic wavefield imaging (UWI) scan in order to estimate the phase and group velocities of S0 and A0 modes. A laser ultrasonic interrogation method based on a Q-switched laser scanning system was used to interrogate full-field ultrasonic signals in a 2-mm aluminum plate at five different frequencies. These full-field ultrasonic signals were processed in three-dimensional space-time domain. Then, the time-dependent covariance matrices of the UWI were obtained based on the vector variables in Cartesian and polar coordinate spaces for all time samples. A spatial covariance map was constructed to show spatial correlations within the full wavefield. It was observed that the variances may be used as a feature for S0 and A0 mode properties. The phase velocity and the group velocity were found using a variance map and an enveloped variance map, respectively, at five different frequencies. This facilitated the estimation of Lamb wave dispersion curves. The estimated dispersion curves of the S0 and A0 modes showed good agreement with the theoretical dispersion curves.

Detection and localization of closely distributed damages via lamb wave sparse reconstruction

Ultrasonic Lamb wave is a promising tool for structural health monitoring and nondestructive evaluation of plate-like structures. Using an array with several piezoelectric discs for damage imaging (i.e. visual detection and localization) is of interest. Commonly used delay-and-sum method is limited for overlapped signals when several damages are closely distributed in the structure. To overcome this limitation, modal-based sparse reconstruction imaging method is applied for adjacent damages in this study. Firstly, Lamb wave dispersion curve is obtained by solving the Rayleigh-Lamb equations. Subsequently, propagation modal of the damage-reflected signal is constructed based on the solved dispersion curve. Finally, the modal is used for damage imaging via sparse reconstruction and basis pursuit de-noising. Experimental data measured in an aluminum plate is considered, and the result demonstrates that the sparse reconstruction imaging method is effective to detect and localize closely distributed damages in the presence of signal overlapping.

Reconstruction of Lamb wave dispersion curves by sparse representation with continuity constraints.

Ultrasonic Lamb waves are a widely used research tool for nondestructive structural health monitoring. They travel long distances with little attenuation, enabling the interrogation of large areas. To analyze Lamb wave propagation data, it is often important to know precisely how they propagate. Yet, since wave propagation is influenced by many factors, including material properties, temperature, and other varying conditions, acquiring this knowledge is a significant challenge. In prior work, this information has been recovered by reconstructing Lamb wave dispersion curves with sparse wavenumber analysis. While effective, sparse wavenumber analysis requires a large number of sensors and is sensitive to noise in the data. In this paper, it extended and significantly improved by constraining the reconstructed dispersion curves to be continuous across frequencies. To enforce this constraint, it is included explicitly in a sparse optimization formulation, and by including in the reconstruction an edge detection step to remove outliers, and by using variational Bayesian Gaussian mixture models to predict missing values. The method is validated with simulation and experimental data. Significant improved performance is demonstrated over the original sparse wavenumber analysis approach in reconstructing the dispersion curves, in synthesizing noise-removed signals, in reducing the number of measurements, and in localizing damage.

Derivation of acoustoelastic Lamb wave dispersion curves in anisotropic plates at the initial and natural frames of reference.

The propagation speed of ultrasonic waves in pre-stressed media can be evaluated either at the natural or initial frames of reference. In this paper general equations that can be applied to the partial wave technique are presented in order to obtain the dispersion spectra of acoustoelastic Lamb waves in anisotropic plates in either frame of reference. Employing these equations, dispersion curves for the fundamental modes in a pre-stressed transversely isotropic aluminum plate were numerically obtained in both reference frames under longitudinal and transverse loading with the material transverse axis along each of the Cartesian directions, as well as the propagation along a non-principal direction. Results confirm that due to the material natural anisotropy, the speed variation depends not only on the pre-stress direction but also on the material orientation as well as on the polarization of the propagating mode. Similar to bulk waves, the relationship between the speed at the natural and initial frames is a function of the load direction.

The Simple Lamb Wave Analysis to Characterize Concrete Wide Beams by the Practical MASW Test.

In recent years, the Lamb wave analysis by the multi-channel analysis of surface waves (MASW) for concrete structures has been an effective nondestructive evaluation, such as the condition assessment and dimension identification by the elastic wave velocities and their reflections from boundaries. This study proposes an effective Lamb wave analysis by the practical application of MASW to concrete wide beams in an easy and simple manner in order to identify the dimension and elastic wave velocity (R-wave) for the condition assessment (e.g., the estimation of elastic properties). This is done by identifying the zero-order antisymmetric (A0) and first-order symmetric (S1) modes among multimodal Lamb waves. The MASW data were collected on eight concrete wide beams and compared to the actual depth and to the pressure (P-) wave velocities collected for the same specimen. Information is extracted from multimodal Lamb wave dispersion curves to obtain the elastic stiffness parameters and the thickness of the concrete structures. Due to the simple and cost-effective procedure associated with the MASW processing technique, the characteristics of several fundamental modes in the experimental Lamb wave dispersion curves could be measured. Available reference data are in good agreement with the parameters that were determined by our analysis scheme.

Lamb waves dispersion curves for diamond based piezoelectric layered structure

The presence of spurious peaks in the amplitude-frequency response of diamond based piezoelectric layered structure was shown. Excitation of such peaks results in deterioration of an useful acoustical signal. It was shown that such spurious peaks should be associated with Lamb waves in a layered structure. By means of FEM analysis, the propagation of acoustic waves of different types in the piezoelectric layered structure “Al/AlN/Mo/(100) diamond” has been investigated in detail. By analyzing the elastic displacement patterns at frequencies from 0 up to 250 MHz, a set of all the possible acoustic waves, especially Lamb modes, have been studied, and dispersive curves of phase velocity have been plotted. A revised classification of Lamb modes has been introduced.

Detecting the thickness mode frequency in a concrete plate using backward wave propagation.

Material stiffness and plate thickness are the two key parameters when performing quality assurance/quality control on pavement structures. In order to estimate the plate thickness non-destructively, the Impact Echo (IE) method can be utilized to extract the thickness resonance frequency. An alternative to IE for estimating the thickness resonance frequency of a concrete plate, and to subsequently enable thickness determination, is presented in this paper. The thickness resonance is often revealed as a sharp peak in the frequency spectrum when contact receivers are used in seismic testing. Due to a low signal-to-noise ratio, IE is not ideal when using non-contact microphone receivers. In studying the complex Lamb wave dispersion curves at a frequency infinitesimally higher than the thickness frequency, it is seen that two counter-directed waves occur at the same frequency but with phase velocities in opposite directions. Results show that it is possible to detect the wave traveling with a negative phase velocity using both accelerometers and air-coupled microphones as receivers. This alternative technique can possibly be used in non-contact scanning measurements based on air-coupled microphones.

A study of adhesion between dental cement and dentin using non-destructive ultrasonic testing

The retention of indirect dental restoration is reliant on the degree of the adhesion between dental materials and dentin. In this work, we introduce a new approach to characterise the cement-dentin interface using a Brekhovskikh model. This model is used to study the propagation of acoustic waves in a multilayered medium. The reflection coefficient of plane waves has been calculated. Then, the Lamb waves dispersion curves for the considered media have been produced. The difference between the acoustic properties of good and bad adhesion has been found. The amplitude of signal reflection from the cement-dentine interface is higher in the case of the bad bonding than in the case of good adhesion. The analysis of dispersion curves shows that the behaviour of some modes strongly depends on the quality of bonding. The obtained results demonstrate that ultrasound method is a valuable method for analysis of cement-dentin interface.

Bayesian parameter identification of orthotropic composite materials using Lamb waves dispersion curves measurement

This paper deals with the problem of elastic constant identification in thin plates made of orthotropic composite materials. The approach is based on the analysis of Lamb wave propagation and the related dispersion curves to find the underlying material elastic constants. In the proposed implementation a scanning laser Doppler vibrometer is used to measure Lamb wave dispersion curves. The Local Interaction Simulation Approach is used simultaneously to find a solution to a high-frequency wave propagation problem. The experimental and simulated data are combined in a Bayesian framework for parameter identification which is robust in condition of parameter, modeling and measurement uncertainty. The results are discussed and compared with the results from a deterministic optimization.

Comparison of laminate stiffness as measured by three experimental methods

Three independent stiffness measurement techniques are applied to characterize virgin glass fiber laminates. Natural frequency based measurements of laminate stiffness are compared to results from simple quasi-static testing and Lamb wave dispersion curves. Genetic algorithm (GA) and a modified two-stage Simplex optimization are employed to solve parameter identification inverse problems. The two-stage Simplex method is shown to converge to similar solutions as global genetic algorithms for Lamb wave optimization. The identified elastic and shear moduli show very good agreement, confirming the feasibility of these methods for material characterization. Poisson's ratios on the other hand were not reliably obtained from these two indirect methods due to low sensitivity.

A rapid, fully non-contact, hybrid system for generating Lamb wave dispersion curves

A rapid, fully non-contact, hybrid system which encompasses an air-coupled transducer (ACT) and a laser Doppler vibrometer (LDV) is presented for profiling A0 Lamb wave dispersion of an isotropic aluminum plate. The ACT generates ultrasonic pressure incident upon the surface of the plate. The pressure waves are partially refracted into the plate. The LDV is employed to measure the out-of-plane velocity of the excited Lamb wave mode at some distances where the Lamb waves are formed in the plate. The influence of the ACT angle of incidence on Lamb wave excitation is investigated and Snell’s law is used to directly compute Lamb wave dispersion curves including phase and group velocity dispersion curves in aluminum plates from incident angles found to generate optimal A0 Lamb wave mode. The measured curves are compared to results obtained from a two-dimensional (2-D) Fast Fourier transform (FFT), Morlet wavelet transform (MWT) and theoretical predictions. It was concluded that the experimental results obtained using Snell’s law concept are well in accordance with the theoretical solutions. The high degree of accuracy in the measured data with the theoretical results proved a high sensitivity of the air-coupled and laser ultrasound in characterizing Lamb wave dispersion in plate-like structures. The proposed non-contact hybrid system can effectively characterize the dispersive relation without knowledge of neither the materials characteristics nor the mathematical model.

Obtaining complementary Lamb wave dispersion information by two signal processing methods on an all-optical non-contact configuration

We present the experimental realisation of an all-optical non-contact tool for the extraction of the Lamb wave propagation dispersion characteristics in thin plates. We describe and assess two different signal processing approaches used to transform the acoustic impulse response optically generated and measured into a frequency response from which to extract the Lamb wave dispersion curves. We demonstrate that the two different techniques enable complementary portions of the ultrasonic transmission path to be addressed.Finally we demonstrate the sensitivity of this approach, in terms of material testing, to thermal changes in the ultrasonic transfer function and its capability to respond to damage to the sample structure.

Fully Noncontact Wave Propagation Imaging in an Immersed Metallic Plate with a Crack

This study presents a noncontact sensing technique with ultrasonic wave propagation imaging algorithm, for damage visualization of liquid-immersed structures. An aluminum plate specimen (400 mm × 400 mm × 3 mm) with a 12 mm slit was immersed in water and in glycerin. A 532 nm Q-switched continuous wave laser is used at an energy level of 1.2 mJ to scan an area of 100 mm × 100 mm. A laser Doppler vibrometer is used as a noncontact ultrasonic sensor, which measures guided wave displacement at a fixed point. The tests are performed with two different cases of specimen: without water and filled with water and with glycerin. Lamb wave dispersion curves for the respective cases are calculated, to investigate the velocity-frequency relationship of each wave mode. Experimental propagation velocities of Lamb waves for different cases are compared with the theoretical dispersion curves. This study shows that the dispersion and attenuation of the Lamb wave is affected by the surrounding liquid, and the comparative experimental results are presented to verify it. In addition, it is demonstrated that the developed fully noncontact ultrasonic propagation imaging system is capable of damage sizing in submerged structures.

A study of pressure-shear vertical wave propagation in periodically layered fluid and piezoelectric structure

In this paper, the pressure (P) and shear vertical (SV) wave propagation in the periodically layered fluid and piezoelectric structure are theoretically studied. A single piezoelectric layer is first analyzed for three scenarios: piezoelectricity-ignored, electrically open, or electrically closed. It is observed that both the Lamb wave dispersion curves and the transmission coefficients are affected by piezoelectricity and electrical boundary conditions under any incident angle. From the analyses of the periodically layered structure, it is further found that the piezoelectricity and the electrical boundary conditions have influence on band gaps for a given incident angle.

Acoustoelastic Lamb wave propagation in biaxially stressed plates

Acoustoelasticity, or the change in elastic wave speeds with stress, is a well-studied phenomenon for bulk waves. The effect of stress on Lamb waves is not as well understood, although it is clear that anisotropic stresses will produce anisotropy in the Lamb wave dispersion curves. Here the theory of acoustoelastic Lamb wave propagation is developed for isotropic media subjected to a biaxial, homogeneous stress field. It is shown that, as expected, dispersion curves change anisotropically for most stresses, modes, and frequencies. Interestingly, for some mode-frequency combinations, changes in phase velocity are isotropic even for a biaxial stress field. Theoretical predictions are compared to experimental results for several Lamb wave modes and frequencies for uniaxial loads applied to an aluminum plate, and the agreement is reasonably good.

Golay code modulation in low-power laser-ultrasound

The current work presents a correlation-based detection technique with application in modulated laser-ultrasonics. In standard use of coded sequences the impulse response of a system is recovered in the time domain with improved signal to noise ratio (SNR). The presented method is an extension of this technique, where the response to a chirped waveform is restored with improved SNR; hence, the response is in a well-defined frequency range. To achieve this goal the chirped waveforms are modulated by Golay codes. It will be shown that the response to this bandlimited carrier waveform can be recovered in the time domain with improved signal to noise ratio using a cross-correlation technique. Improvement in the SNR is discussed analytically and it is shown that this improvement is proportional to the square root of the length of the applied sequences. Experimental applications in laser-ultrasound are shown using modulated laser diodes as excitation sources with an output power of ∼1W. In the experiments a plate with a thickness of 50μm is investigated using Lamb waves in the MHz range to confirm the predicted improvement in the SNR. Golay codes with three different lengths were used with 7, 9 and 11 bits resulting in 27=128, 29=512, and 211=2048 repetitions in an individual signal, respectively. The predicted improvements of 2 in the SNR between the 7 and 9 bits, and between the 9 and 11 bits waveforms, respectively, were well approximated by the experimentally obtained values of 1.83 and 2.17. As Lamb wave dispersion curves can be used for the characterization of plates or layered samples by inverse problems, it is also shown that by using multiple measurement points the recovered waveforms can be utilized in the evaluation of the dispersion relation.

Crossing characteristics of lamb wave modes in [001]c and [011]c polarized Pb(Mg1/3Nb2/3)O3-PbTiO3 crystal plates

Multiple crossings of Lamb modes found in [001]c and [011]c polarized (1−x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-xPT, x=0.33 and 0.29, respectively) plates have been studied by comparing their slowness patterns with their dispersion patterns. Multiple bandgaps and crossings of Lamb wave dispersion curves are directly related to the multivalued curves in region II, and a pair of complex conjugate b2 values in region I of their slowness curves. An expression composed of elastic and piezoelectric constants has been derived to judge whether S0 and A0 modes cross with each other. This expression can be greatly simplified for non-piezoelectric materials, and there are no crossings of Lamb modes for isotropic materials.

A homogenised vibratory model for predicting the acoustic properties of hollow brick walls

The prediction and the physical understanding of sound transmission through masonry walls made of hollow bricks remain an open question. To solve this problem a semi-analytical approach is proposed. The inhomogeneous structures of the brick wall are homogenised and a simplified analytical model is established to calculate the transmission loss of an equivalent finite and multilayered anisotropic plate. An efficient numerical homogenisation technique is derived to define the equivalent anisotropic brick. This process only needs the knowledge of the elastic tensor of the brick material that has been determined using ultrasonic measurements. The features of the simplified brick wall have been then investigated through Lamb waves dispersion curves. Finally, the model has been used to explain the transmission loss curve of a wall and a good agreement between predictions and test data is obtained.

Nondestructive Determination of Elastic Constants of Thin Isotropic Plates Based on Poly(vinylidene fluoride–trifluoroethylene) Copolymer Ultrasound Focusing Transducers and Lamb Wave Measurements

In this paper we present a nondestructive method for determining the elastic constants of thin isotropic plates using poly(vinylidene fluoride–trifluoroethylene) [P(VDF–TrFE)] copolymer ultrasound focusing transducers and Lamb wave measurements. Lensless P(VDF–TrFE) line-focus transducers are fabricated, and Lamb wave defocusing measurements are performed to measure the multimode Lamb wave dispersion curves of stainless steel and glass thin-plate samples of thicknesses ranging from 100 to 250 µm. Dispersion-curve-sensitivity analysis is carried out to determine the best method of inverse determination of the elastic properties. A search method based on the downhill simplex algorithm is applied to inversely determine the elastic constants of isotropic plate samples. The obtained results are in good agreement with the data reported by the manufacturers. Measurement accuracy and potential applications of the P(VDF–TrFE) ultrasound focusing transducers and measurements for other types of nondestructive evaluation are addressed.

Accuracy and Convergence Using a Local Interaction Simulation Approach in One, Two, and Three Dimensions

Guided waves are utilized in structural health monitoring for identifying damage in material components. Simulations can be used to examine how elastic waves propagate in components to help in selecting measurement and data analysis techniques. In this work, the influence of grid size and the frequency sample rate on the amplitude accuracy and convergence of local interaction simulation approach/sharp interface model (LISA/SIM) numerical simulations are studied as they pertain to guided wave propagation in structural materials. These issues are studied in all three dimensions, and amplitude distortion with respect to the Courant–Friedrich–Lewy criterion is explored. The LISA/SIM enables accurate and fast modeling of localized and sharp changes in material properties across interfaces associated with heterogeneities and/ or boundaries. The validity of the simulation is demonstrated by comparing simulated responses with experimentally measured data. Additionally, Lamb wave dispersion curves are extracted through the course of the convergence study using a broadband pulse and the two-dimensional fast Fourier transform method.