Fundamental Lamb Mode Research Articles

Propagation of the Fundamental Lamb Modes in Strain Stiffened Hard-Magnetic Soft Plates

Abstract This study focuses on investigating hard-magnetic soft materials, characterized by magnetoactive polymers containing magnetically polarized particles as fillers. The research utilizes the Gent model of hyperelasticity to analyze the propagation of Lamb waves in a magnetically induced deformed compressible plate. In this investigation, we explore both finite deformations and incremental wave propagation in nonlinear hard-magnetic soft materials. The main objective is to formulate the elastic tensor and relevant wave equations within the framework of Lagrangian space. To assess the dispersion characteristics of the guided wave, the study introduces and discusses an extension of the semi-analytical finite element (SAFE) method. Using this numerical approach, the research further examines the effects of magnetic flux densities and its orientation with respect to wave propagation direction on the dispersion characteristics of the fundamental Lamb modes. The study starts by examining the limiting case of the neo-Hookean material model to explain such inherent dependencies. These dependencies are then further emphasized by including the strain-stiffening effect that the Gent material model describes. The research findings reveal the presence of a threshold applied magnetic flux, beyond which the Gent-type material may undergo a snap-through instability, resulting in changes in the dispersion characteristics of the fundamental symmetric Lamb mode.

Ultrasonic Battery Management System for Lamb wave mode tracking on Lithium-ion pouch cells

During charging and discharging, lithium-ion secondary batteries (LIBs) change their mechanical properties due to intercalation and deintercalation of lithium-ions in the electrodes. In this work, a novel ultrasonic battery management system (UBMS) with dedicated FPGA hard- and software has been developed that can track these mechanical changes by using ultrasonic sensors. In particular, the group velocity of the fundamental Lamb mode A0 is tracked and compared with the proposed simulation model. Previous studies have shown that the Young’s modulus of the anode coating changes significantly during cycling. By incorporating this information into the simulation, a change in this value will drastically affect the group velocity of the propagating wave mode A0, which can also be seen in the measurements carried out. To represent a realistic use case, the behaviour of the Lamb mode A0 is measured at different temperatures, which have a big influence on the group velocity of the propagating wave mode. The collected ultrasonic BMS data was further used to develop a state-of-charge (SOC) and temperature estimator. The accuracy of this novel method is discussed. With the addition of the ultrasonic sensors, the BMS can have a larger and better database that can be used to provide a more accurate estimate of the mechanical properties of LIBs and their SOC and temperature.

Measurement on a thin plate of stepped thickness using zero-group velocity effect of an S1 mode Lamb wave induced by a moving laser source.

The desired narrowband mode of a Lamb wave can be generated efficiently, as long as the laser's moving speed matches the mode's phase velocity, which is a typical characteristic of the moving continuous wave (CW) laser excitation method. In this paper, S 1 mode zero group velocity (ZGV) waves can be generated efficiently when the laser's moving speed matches the phase velocity of the S 1 mode at the ZGV point, while the two fundamental Lamb modes, i.e.,A 0 and S 0 waves, can be avoided. Meanwhile, measurements on thin plates of stepped thickness were carried out by using the zero-group velocity effect of the S 1 mode. Those simulation results demonstrated that the S 1-Z G V resonance can be used to measure the thickness of thin plates by tracking the resonance peak as the sample was scanned, and good accuracy can be achieved, since the measurement was reduced to a simple frequency gauging of sample resonances in the ultrasonic domain, which provides a reliable method to measure a thin plate of varying thickness.

Guided Waves in Ship Structural Health Monitoring – A Feasibility Study

Abstract Ships and offshore structures operate in a severe corrosion degradation environment and face difficulty in providing long-lasting corrosion protection. The Classification Societies recommend regular thickness measurements leading to structural component replacements, to ensure structural integrity during service life. The measurements are usually performed using ultrasonic thickness gauges and such an approach requires multiple measurements of the corroded structural components. Otherwise, the collected data are insufficient to precisely assess the corrosion degradation level. This study aims to perform numerical and experimental analyses to verify the use of guided ultrasonic waves in defining the corrosion degradation level of the corroded structural components of a ship. The study incorporates the fundamental antisymmetric Lamb mode, excited by piezoelectric transducers attached at the pre-selected points on stiffened panels, representing typical structural ship components. The specimens are exposed to accelerated marine corrosion degradation, the influence of the degree of degradation on the wave time of flight being analysed. The study indicates that guided waves are a promising approach for diagnosing corroded structural components. The signals characterised by a high signal-to-noise ratio have been captured, even for relatively long distances between the transducers. This proves that the proposed approach can be suitable for monitoring more extensive areas of ship structures by employing a single measurement.

Propagation of fundamental Lamb modes along the non-principal axes of strain-stiffened soft compressible plates: A numerical investigation.

This paper numerically investigates the propagation of elastic plate waves along the non-principal directions in a prestretched compressible material described by the Gent model of hyperelasticity. We formulate the elastic tensor and the underlying wave equations in the Lagrangian space by employing the theory of nonlinear elasticity together with the linearized incremental equations. An extension of the Semi-Analytical Finite Element (SAFE) method is discussed for computing the dispersion characteristics of the two fundamental guided wave modes. The predictive capabilities of the numerical framework are established using the previously published data for a weakly nonlinear as well as hyperelastic material models. Using the numerical framework, we then bring out the effects of applied prestretch, orientation of the propagation direction, and material parameters on the dispersion characteristics of the fundamental Lamb modes. A limiting case of the neo-Hookean material model is first considered for elucidating such implicit dependencies, which are further highlighted by considering the strain-stiffening effect captured through the Gent material model. Our results indicate the existence of a threshold prestretch for which the Gent-type material can encounter a snap-through instability; leading to the change in the dispersion characteristics of the fundamental symmetric Lamb mode.

S0 Lamb Mode Scattering Studies in Laminated Composite Plate Structures with Surface Breaking Cracks: Insights into Crack Opening Behavior

Ultrasonic-guided waves are attractive for rapid inspection of laminated composite structures, where cracks developed transverse to the loading direction are the severe type of damage. This paper presents studies of the interaction of fundamental symmetric S0 Lamb mode with vertical surface-breaking cracks in laminated composite plate structures. Finite element simulations and experimental investigations are used to study the effect of crack depth on S0 wave reflection behavior. Results show a monotonic rise of the reflection coefficient for different crack depths in a manner that is strongly dependent on the orientation of the plies and transverse ply location in the vicinity of the crack. Scattered wave packets in the reflection regime are captured using an in-plane laser. The S0 Lamb mode's sensitivity is numerically presented for the different crack depths in the long wavelength limit. We also observed that the reflected wave mode depicts the information of the corresponding broken interfaces. An attempt was made to show that this behavior relates to the crack-opening behavior in response to in-plane excitation. The reflection coefficient as a characteristic polynomial is proposed for various orientations. It was observed that the dispersion at receiver nodes makes the analysis challenging for distinguishing the signal from crack faces due to the smaller dimension. The study outcomes show its prospect as a promising NDE tool for crack damage detection in thin laminated plate structures.

Scattering of the Fundamental Lamb Modes in Bent Metallic Plates

Abstract Guided elastic waves, propagating through curved waveguides, have attracted significant attention in the recent past, both from the perspectives of assessment of structural integrity and generating novel designs of acoustic waveguides. This article presents investigation of the interaction of the fundamental Lamb modes with a cylindrical bend in thin metallic plates. A hybrid numerical method is exposited, which combines the computational efficacy of the semi-analytical finite element method in modeling the long straight portions of the plate and the versatility of the conventional finite element method in modeling the bent portion. The predictive capabilities of the proposed method are validated using transient finite element simulations. Appropriate modifications to the hybrid method, needed for simulating the multimodal incidence resulting from the point-force actuation, are discussed. Using the hybrid method, the scattering and mode-conversion behavior, imparted by the cylindrical bend, is studied when the two fundamental Lamb modes are used as the incident interrogation signals. The extents of equi-modal and cross-modal contributions in both the reflected as well as transmitted waveforms are quantified in terms of the respective modal energy ratios. Explicable contour charts are presented for comprehending the scattering behavior over a wide range of frequencies and bend angles that span from 0 deg to 180 deg. For two representative cases, the modal displacement patterns inside the bend region are presented and discussed. The present investigation can find its potential use in the analysis of geometrically irregular structures leading to the design of novel acoustic waveguides.

Viscoelastic rheological models for guided acoustic waves: Single mode at high frequency and second harmonic generation of Lamb and SH modes

The previous published work by Othmani et al. invites a further useful discussion on how the viscoelastic rheological model affects the second harmonic and single modes of guided waves. However, this paper has two main objectives. The first objective is to establish a simple relationship between the Rayleigh waves velocity (VR), fundamental Lamb modes (A0 and S0), and the shear wave velocity (VS) in unidirectional viscoelastic carbon–epoxy at high frequency. The second objective is to study the second harmonic Lamb modes. Numerical codes based on Legendre polynomial and Newton–Raphson methods are developed and implemented to predict dispersion and attenuation curves. The second harmonic generation was investigated with the hysteretic and Kelvin–Voigt viscoelastic models. Results showed that rheological models have a much larger effect on the attenuation curves than the ones on the dispersion trajectories. Additionally, the optimal angles of Lamb and SH modes in unidirectional carbon–epoxy under Snell‐Descartes law was investigated. The present numerical results may render as a useful message for the rheological community.

Localization of a Breathing Delamination Using Nonlinear Lamb Wave Mixing

Abstract A guided wave-based method for localization of breathing delamination is presented in this investigation. The proposed technique utilizes one-way mixing of a dual-frequency fundamental antisymmetric Lamb modes with judiciously selected central frequencies. The dual-frequency interrogation signal, upon interacting with a breathing delamination, leads to additional frequency sidebands in the frequency response spectrum, strength of which is quantified in terms of the combination tone index. The numerical predictions of these sidebands are validated using an in-house experimentation. It is further exposited that the combination tone index depends strongly on the extent of the temporal overlap that the two constituent wave envelopes have as they propagate through the breathing delamination. Accordingly, for a synchronous passage (with 100% temporal overlap), the combination tone index is maximum while it reduces with the decreasing temporal overlap. By utilizing the dispersive nature of the chosen Lamb mode, a relation is then developed correlating the temporal separation of the wave envelopes at the location of the actuator, the group speeds, and the distance between the actuator and the delamination. Based on these inferences, a technique for localizing a breathing delamination is proposed, which involves interrogating the component by systematically altering the temporal overlap in the input waveform and monitoring the combination tone index for its maxima. The efficacy of the localization technique (close to 90%) is demonstrated through an illustrative case analyzed numerically as well as experimentally.

Nonlinear generation of a zero group velocity mode in an elastic plate by non-collinear mixing

Zero Group Velocity (ZGV) modes are peculiar guided waves that can exist in elastic plates or cylinders, and have proved to be very sensitive tools in characterizing materials or thickness variations with sub-percent accuracy at space resolutions of about the plate thickness. In this article we show theoretically and experimentally how such a mode can be generated as the sum-frequency interaction of two high amplitude primary waves, and then serve as a local probe of material non-linearity. The solutions to the phase matching condition, i.e. condition for a constructive non-linear effect, are obtained numerically in the mark of classical, quadratic non-linearity. The coupling coefficients that measure the transfer rate of energy as a function of time from primary to secondary modes are derived. Experiments are conducted on an aluminum plate using piezo-electric transducers and a laser interferometer, and explore the interaction for incident symmetric and anti-symmetric fundamental Lamb modes. In an experiment operated without voltage amplifier we demonstrate that the resonant nature of these ZGV modes can be leveraged to accumulate energy from long excitations and produce detectable effects at extraordinarily low input power even in such weakly non-linear material.

Radiation Characteristics of Cranial Leaky Lamb Waves.

We numerically and experimentally investigate the dispersion properties of leaky Lamb waves in the cranial bone. Cranial Lamb waves leak energy from the skull into the brain when propagating at speeds higher than the speed of sound in the surrounding fluid. The understanding of their radiation mechanism is significantly complicated by the geometric and mechanical characteristics of the cortical tables and the trabecular bone (diploë). Toward such understanding, we here analyze the sub-1.0 MHz radiation angle dispersion spectrum of porous bone phantoms and parietal bone geometries obtained from μ CT scans. Our numerical results show that, when diploic pores are physically modeled, leakage angles computed from time transient finite-element analyses correspond to those predicted by an equivalent three-layered fluid-loaded waveguide model. For the bone geometries analyzed, two main leaky branches are observed in the near-field dispersion spectrum: a fast wave radiated at small angles, which is related to the fastest fundamental Lamb mode supported by the cranial bone, and a slower wave radiated at larger angles. This observation is also confirmed by experimental tests carried out on an immersed parietal bone.

Experimental and numerical analysis of Lamb wave transmission through the coronal suture of a dry human skull

Cranial Lamb waves have recently gained attention as potential means for ultrasound imaging and diagnostics of the skull. The inherent capability of these waves to travel for long distances across the cranial vault requires the understanding of transmission mechanisms across cranial sutures, which remain mostly unexplored. This work investigates such mechanism in the coronal suture of a dry human skull. The transmission analysis is performed experimentally by recording velocity arrays induced by short tone-bursts along curvilinear paths that intercept the suture at different locations. For each path, the experimental frequency-wavenumber spectrum of the incident wave indicates the presence of fundamental and higher order Lamb modes, with transmission being mainly associated with the fundamental antisymmetric Lamb mode. By calculating the corresponding transmission coefficients and comparing them with those obtained from a semi-analytical multilayered model of the cranial bone, we demonstrate that the coronal suture can be described by a simplified constitutive law defined by distributed shear and compressional stiffness coefficients. The experimental results and the estimated suture constitutive parameters provide the basis for a comparative analysis of the suture mechanical properties at different spatial locations, which may be employed for suture characterization and imaging.

Abnormal dispersion of fundamental Lamb modes in FG plates: II—symmetric versus asymmetric variation

A closed-form dispersion equation for Lamb waves propagating in functionally graded (FG) plates with arbitrary varying anisotropic physical properties in transverse direction is constructed by applying Cauchy sextic formalism. Comparative analyses of the dispersion curves related to the fundamental modes of homogeneous and FG plates reveal their substantial and visually detected differences even at sufficiently small transverse inhomogeneity. The observed discrepancy in dispersion curves may serve for developing acoustic nondestructive methods for determination of possible transverse inhomogeneity.

Modulating optical properties of Lithium Niobate through acoustic stress

Optical index modulation in Z-X Lithium Niobate slab through acoustic stress is theoretically studied. To generate acoustic stress, fundamental symmetric Lamb mode is excited in the slab. A periodic disturbance of period 858 nm in 500 nm thick slab is noticed when 5 GHz frequency Lamb wave mode is used. The displacement field pattern caused by 5 GHz acoustic disturbance is then estimated using potential techniques and applied to photo-elastic relations. Theoretical calculation reveals that the disturbance modulates optical index sinusoidally, with alternate decrement and increment in optical index around the unperturbed value. The maximum observed variation was found along the direction of acoustic wave propagation is of order of 1.5 percent.

Scattering of Lamb Waves Near Lamé Point at an Opening Crack

Abstract The scattered fields of Lamb waves resulting from damage have been extensively studied because they carry rich structural information that allow us to identify and evaluate damage. This paper investigates the interaction of the fundamental symmetric Lamb mode (S0) near the first Lamé point with opening cracks of various heights under the plane strain condition. The scattering coefficients obtained by the semi-analytical method indicate a significant non-mode-conversion point, where the mode conversion is very weak. The frequency of this non-mode-conversion point originates in the first Lamé point and increases linearly with the crack height when the normalized height is less than 25%. The finite element simulations validate the calculation results. In addition, similar phenomena are also observed for the scattering at a symmetric internal crack. These interesting scattering characteristics are related to the special mode shapes near the Lamé point, which may provide a candidate to evaluate the height of an opening crack.

Monitoring internal corrosion in steel pipelines: a two-step helical guided wave approach for localization and quantification

This article presents a two-step approach for the assessment of internal corrosion in cylindrical structures using helical guided ultrasonic waves. The approach consists of two steps such as (1) localization and (2) estimation of the size of the corroded area. Localization is performed with the algebraic reconstruction technique where the energy ratio of the two fundamental Lamb modes S0 and A0 is used as input damage coefficient. Using the output of the localization step, the size of the corroded area is estimated by iteratively solving the eikonal equation using a finite-difference approach. The proposed approach is validated by an accelerated corrosion test. Furthermore, numerical simulations are carried out to study the interaction of the energy ratio and the phase velocity travel time with various thickness profiles acquired from the experiment. The proposed approach is validated against the experimental data up to approximately 50% thickness loss.

Influence of a piezoelectric ZnO intermediate layer on Rayleigh waves propagating in Sc43%AlN57%/ZnO/diamond hetero-structures subjected to uniaxial stress

Surface acoustic waves (SAWs) are of practical importance across several fields. Particularly, a wide range of SAW devices is available, which are based on the stacking piezoelectric layered hetero-structures. As such, efficient and stable algorithm of analyzing electromechanical coupling factor (K2) should be of specific interest, inspiring a variety of piezoelectric layered structures. Motivated by the above considerations, this paper exploits the performance of Sc43% AlN57%/ZnO/diamond piezoelectric hetero-structures using matrix and polynomial methods. Consequently, dispersion curves, the electromechanical coupling factor and profiles of Rayleigh waves are calculated, in which the properties of hexagonal intermediate ZnO layer are varied. Here, $$ (0001) - [10\bar{1}0]{\text{ZnO}} $$ or $$ (11\bar{2}0) - [0001]{\text{ZnO}} $$ directions of the middle layer were integrated with Sc43% AlN57%/diamond to be a composite hetero-structure with a larger K2 of SAW. Accordingly, a commercial ZnO specimen was characterized by scanning electron microscopy (SEM) as well as Raman and XRD spectroscopy to determine its available crystallographic directions and the C-axis hexagonal character. Comparing with conventional AlN/diamond hetero-structures, the inclusion of ZnO as the intermediate layer provides with an excellent electromechanical coupling factor which significantly improves the $$ (11\bar{2}0) - [0001] $$ structure. However, comparing with the ZnO/diamond hetero-structure, a high phase velocity (Vph) was observed. Therefore, the development of a well-balanced structure that maximizes the values of Vph and K2 is systematically challenging. Moreover, it was found that initial stresses have a significant effect on the Rayleigh wave velocities. Fundamental A0 Lamb mode can be generated in piezoelectric hetero-structures when the properties of substrate remain very low relative to the film. Numerical results obtained in this paper can provide a deeper insight into the nature of the Rayleigh wave behaviors in piezoelectric hetero-structures, involving a ZnO intermediate layer.

Lamb wave propagation in anisotropic multilayered piezoelectric laminates made of PVDF-θ° with initial stresses

In this paper, the influence of an initial stresses on the Lamb waves propagation in an anisotropic multilayered piezoelectric laminates is investigated. Formulations are given for only open circuit surface; while the Legendre polynomial method is employed for calculate the wave propagating characteristics. To validate our numerical strategy, this polynomial approach was applied to simulate lamina composite structures. Relatively good results were obtained compared to reliable data available from the literature ones, with relatively significant computation effort. Convergence of the present method is discussed. Influence of initial stresses and fiber orientation angle on the behaviors of Lamb waves is analyzed. Results confirm that the behavior of Lamb modes depends not only on the initial stress but also on fiber orientation. Next, the dispersion curves of fundamental Lamb modes in sandwich plates with 0°/θ°/0 configuration are calculated. Finally, the optimal angles in PVDF-θ° with initial stresses for which the Lamb modes exist are also analyzed. These results for the composite structures may render as a useful reference for the design of ultrasonic transducers in industry for non-destructive testing especially that encounters initial stresses.

Inversion algorithm for Lamb-wave-based depth characterization of acoustic emission sources in plate-like structures

An inversion algorithm (termed AEDep) is proposed for estimating the depth of acoustic emission (AE) sources in plate-like structural components. The work is motivated by the need for characterizing early-stage fatigue crack growth in such components. The algorithm achieves depth estimation by automatically extracting the depth-dependent amplitude ratio between the fundamental Lamb modes which comprise the AE signals. A finite element model is designed to study the frequency-dependent forward problem of Lamb wave motion due to a given source, from which the relation between source depth and amplitude ratio is established. Elastodynamic theory is used to validate the model in the frequency domain, as well as to derive a sensor tuning factor which may be incorporated into the solution. The proposed algorithm was tested on two plate-like specimens: a 6061-T6 aluminum plate and a 2025-T6 aluminum aircraft fuselage panel. Validation of the algorithm was achieved by generating controlled AE sources at various depths along the edges of the specimens, in the form of Hsu-Nielsen pencil lead breaks. Good agreement was found in the aluminum plate between the true and estimated source depths. A slight decrease in accuracy was found in the fuselage panel between the true values and their estimations. However, both experimental cases demonstrated the ability to distinguish between sources originating near the mid-plane of a plate-like structure from those near the surface. Lastly, the fast computation of the inversion algorithm shows strong potential for real-time monitoring applications.

Abnormal dispersion of flexural Lamb waves in functionally graded plates

A closed-form dispersion equation for Lamb waves is obtained for analyzing dispersion in functionally graded (FG) plates with arbitrary anisotropy and arbitrary transverse inhomogeneity. The dispersion equation is derived and analyzed by a variant of the sextic formalism, previously developed for Lamb waves propagating in anisotropic layered plates with homogeneous layers. For the case of FG plates with transverse asymmetric inhomogeneity, some peculiarities in dispersion of the fundamental flexural Lamb mode are observed, revealing discrepancy in asymptotic behavior at high frequencies between homogeneous and FG isotropic plates.