Zero Group Velocity Resonance Research Articles

Acoustoelastic characterization of aluminum plates using zero group velocity Lamb modes

Acoustoelasticity, a characteristic of material anharmonicity, gives rise to a link between wave propagation velocity and the stress state in materials. Ultrasonic techniques to monitor this coupling, particularly with high sensitivity and in a noncontact manner, can have widespread application both in the quantification of applied and residual stress and in the characterization of nonlinear material behavior through measurement of higher order elastic constants. Here, we use a laser ultrasonic technique to excite and detect zero group velocity (ZGV) Lamb wave resonances in aluminum plates under uniaxial loading. A laser line source is used to excite these resonances at different orientations with respect to the applied load and the signals are detected using an interferometer. The effects of stress and source orientation on ZGV resonance frequencies are validated using the theory of acoustoelastic Lamb wave propagation. In addition, a model-based inversion technique is used to extract Murnaghan’s third-order elastic constants from measurements of the stress dependence of the first two ZGV modes generated parallel and perpendicular to the applied load. Laser generation and detection of ZGV resonances is shown to be an effective and powerful approach for the noncontact and nondestructive acoustoelastic characterization of elastic waveguides.

Acoustoelastic characterization of plates using zero group velocity Lamb modes

Acoustoelasticity, a characteristic of material anharmonicity, gives rise to a link between wave propagation velocity and the stress state in materials. Ultrasonic techniques to monitor this coupling, particularly with high sensitivity and in a noncontact manner, can have widespread application both in the quantification of applied and residual stress and in the characterization of nonlinear material behavior through measurement of higher order elastic constants. Here, we use a laser ultrasonic technique to excite and detect zero group velocity (ZGV) Lamb wave resonances in aluminum plates under uniaxial loading. A laser line source is used to excite these resonances at different orientations with respect to the applied load, and the signals are detected using an interferometer. The effects of stress and source orientation on ZGV resonance frequencies are validated using the theory of acoustoelastic Lamb wave propagation. In addition, a model-based inversion technique is used to extract Murnaghan's third-order elastic constants from measurements of the stress dependence of the first two ZGV modes generated parallel and perpendicular to the applied load. Laser generation and detection of ZGV resonances is shown to be an effective and powerful approach for the noncontact and nondestructive acoustoelastic characterization of elastic waveguides.

Computing zero-group-velocity points in anisotropic elastic waveguides: Globally and locally convergent methods.

Dispersion curves of elastic waveguides exhibit points where the group velocity vanishes while the wavenumber remains finite. These are the so-called zero-group-velocity (ZGV) points. As the elastodynamic energy at these points remains confined close to the source, they are of practical interest for nondestructive testing and quantitative characterization of structures. These applications rely on the correct prediction of the ZGV points. In this contribution, we first model the ZGV resonances in anisotropic plates based on the appearance of an additional modal solution. The resulting governing equation is interpreted as a two-parameter eigenvalue problem. Subsequently, we present three complementary numerical procedures capable of computing ZGV points in arbitrary nondissipative elastic waveguides in the conventional sense that their axial power flux vanishes. The first method is globally convergent and guarantees to find all ZGV points but can only be used for small problems. The second procedure is a very fast, generally-applicable, Newton-type iteration that is locally convergent and requires initial guesses. The third method combines both kinds of approaches and yields a procedure that is applicable to large problems, does not require initial guesses and is likely to find all ZGV points. The algorithms are implemented in GEW ZGV computation (doi: 10.5281/zenodo.7537442).

Characterizing bulk liquids with zero-group-velocity Lamb modes

In this work we study the coupled response of zero-group-velocity (ZGV) modes and test liquids. In particular, we assess the viability of exploiting the spatiotemporal constraint of ZGV modes to derive techniques for non-contact, non-invasive liquid characterization to support process optimization. The results demonstrate that the ZGV resonance behaves like a baffled piston source that couples into standing waves in the test liquid. Moreover the particle displacement associated with the liquid resonance extends over macroscopic length scales (∼cm). The resonance frequencies are governed by the acoustic properties and depth, h, of the test liquid. Therefore, the test liquid sound speed, cL , can be determined by analysing the spectral content of the system response. While the technique provides an accurate measure of sound speed, the morphology of immiscible mixtures (e.g. buoyancy separated or emulsion) affects the variation in sound speed with composition. In this case, it is necessary to know the mixture morphology in order to utilize the sound speed as a proxy for composition. The technology is applicable to any thin-walled (mm–cm) structure (pipe, tank, etc) commonly utilized for liquid handling. As a result, the measurement is broadly applicable to many industrial settings that require convenient and accurate methods to determine liquid composition in support of process evaluation and optimization.

Multiple zero-group velocity resonances in elastic layered structures

To date, the phenomenon of zero group velocity (ZGV) Lamb wave in a stress-free elastic plate has been well studied and used in non-destructive material characterization, electroacoustic devices and some other applications. This phenomenon is associated with the backward mode appearing in the range limited by the ZGV and cutoff frequencies. The modal frequency response of the structure, primarily the spectrum of the corresponding guided wave (GW) excited in the plate by a force source, is featured by a sharp peak at the ZGV frequency, allowing its experimental detection. Other peaks appearing at the cutoff frequencies indicate thickness resonances. In a layered half-space of infinite thickness, the backward-mode bend of the dispersion curve turns into a double bend that gives rise to three GWs associated with the same branch (one backward and two forward modes). Moreover, several adjacent branches with double bends, and, thus, several peaks, can appear in a narrow range as the number of layers increases, indicating multiple ZGV resonances. Such dispersion curves with double ZGV points were also found in waveguides of finite thickness with anisotropic or contrast material properties; however, their resonance manifestation is poorly studied. In this article, we consider and discuss several kinds of multi-peak ZGV responses using examples of layered samples of various materials. Along with the expected increase in the number of peaks in multilayer structures with contrast layers, it was noted that some ZGV points may or may not appear in peaks, depending on the type of load.

Nondestructive evaluation of structural adhesive bonding using the attenuation of zero-group-velocity Lamb modes

A laser ultrasonic method is proposed for the nondestructive evaluation (NDE) of structural adhesive bonding. Zero-group-velocity (ZGV) resonances were generated and detected in five trilayer assemblies composed of two asymmetric aluminum alloy plates bonded with an epoxy adhesive. Cohesive and adhesive defects were introduced to degrade the practical adhesion. The attenuation of the temporal signal of ZGV resonances was found to provide sufficient information to discriminate between strong and weak bonding. Two metrics characterizing the attenuation were identified, which allow us to quantitatively evaluate the differences between the manufactured samples. A 2D scan of a trilayer assembly with different bond defects demonstrates the imaging capability of this all-optical NDE method.

In-plane backward and zero group velocity guided modes in rigid and soft strips.

Elastic waves guided along bars of rectangular cross sections exhibit complex dispersion. This paper studies in-plane modes propagating at low frequencies in thin isotropic rectangular waveguides through experiments and numerical simulations. These modes result from the coupling at the edge between the first order shear horizontal mode SH0 of phase velocity equal to the shear velocity VT and the first order symmetrical Lamb mode S0 of phase velocity equal to the plate velocity VP. In the low frequency domain, the dispersion curves of these modes are close to those of Lamb modes propagating in plates of bulk wave velocities VP and VT. The dispersion curves of backward modes and the associated zero group velocity (ZGV) resonances are measured in a metal tape using noncontact laser ultrasonic techniques. Numerical calculations of in-plane modes in a soft ribbon of Poisson's ratio ν≈0.5 confirm that, due to very low shear velocity, backward waves and ZGV modes exist at frequencies that are hundreds of times lower than ZGV resonances in metal tapes of the same geometry. The results are compared to theoretical dispersion curves calculated using the method provided in Krushynska and Meleshko [J. Acoust. Soc. Am. 129, 1324-1335 (2011)].

Characterization of mechanical properties of thin-film Li-ion battery electrodes from laser excitation and measurements of zero group velocity resonances

The mechanical properties of thin-film Li-ion battery electrodes are controlled by the microstructure of the constituent materials. In this work, a noncontact and nondestructive measurement of the mechanical properties of electrode films is performed by measurement of zero-group velocity (ZGV) resonances. Theoretical models are used to quantify the sensitivity of the ZGV resonances to changes in mechanical properties. The ZGV Lamb modes of a solid bilayer consisting of a thin metallic layer and a thin compliant coating layer are shown to be dependent on Young’s moduli, thicknesses, densities, and Poisson’s ratios of the layers. Experimental ZGV resonances are excited using a pulsed infrared laser and detected using a laser interferometer. Commercial-grade battery films with different coating materials, densities, and thicknesses are measured. Young’s moduli of the battery electrode layers are estimated using the combination of a theoretical model and experimental results. The effect of the calendering process on the battery materials is also investigated. Results suggest that Young’s modulus of the electrode coating increases drastically after the battery films are calendered. This technique can be used to quantitatively study the mechanical properties of Li-ion battery electrodes in order to improve overall battery performance.

Optical absorption effect on laser-generated resonances in semi-transparent solid

We studied the effect of optical absorption on the Lamb wave thermoelastic generation in isotropic plates. For metallic plates, high optical absorption results in a surface source and Lamb mode amplitudes depend on the total deposited energy and laser source shape. In the case of moderately absorbing plates, the laser beam can penetrate the sample to some optical depth, producing a bulk source. Consequently, the radiation characteristics of laser-ultrasound are significantly different. The displacement amplitudes can be controlled by the incidence angle of the laser beam and depend on the material absorption. We used a laser-based ultrasonic setup for the generation and detection of guided waves in semi-transparent plate-like structure. In a free elastic plate, resonances occur at Zero-Group Velocity (ZGV) points where the phase velocity remains finite. For studying the transition from a surface source to a bulk source we performed local measurements in neutral density filters having different absorbance. ZGV resonances amplitude increase with absorption until a maximum is reached depending on the order of the mode. Furthermore, these measurements allow to discriminate the so-called thickness resonances, associated to the precursors, from ZGV resonances. Finally, the experimental results are compared with semi-analytical simulations based on Spicer model.

Laser beam shaping for enhanced Zero-Group Velocity Lamb modes generation.

Optimization of Lamb modes induced by laser can be achieved by adjusting the spatial source distribution to the mode wavelength (λ). The excitability of Zero-Group Velocity (ZGV) resonances in isotropic plates is investigated both theoretically and experimentally for axially symmetric sources. Optimal parameters and amplitude gains are derived analytically for spot and annular sources of either Gaussian or rectangular energy profiles. For a Gaussian spot source, the optimal radius is found to be λZGV/π. Annular sources increase the amplitude by at least a factor of 3 compared to the optimal Gaussian source. Rectangular energy profiles provide higher gain than Gaussian ones. These predictions are confirmed by semi-analytical simulation of the thermoelastic generation of Lamb waves, including the effect of material attenuation. Experimentally, Gaussian ring sources of controlled width and radius are produced with an axicon-lens system. Measured optimal geometric parameters obtained for Gaussian and annular beams are in good agreement with theoretical predictions. A ZGV resonance amplification factor of 2.1 is obtained with the Gaussian ring. Such source should facilitate the inspection of highly attenuating plates made of low ablation threshold materials like composites.

Backward waves with double zero-group-velocity points in a liquid-filled pipe.

Hollow cylinders often exhibit backward propagation modes whose group and phase velocities have opposite directions, and these exhibit a minimum possible frequency at which the group velocity vanishes at a nonzero wavenumber. These zero-group-velocity (ZGV) points are associated with resonant conditions in the medium. On the basis of ZGV resonances, a non-contact and laser ultrasound technique has been developed to measure elastic constants of hollow pipes. This paper provides a theoretical and numerical investigation of the influence of the contained liquid on backward waves and associated ZGV modes, in order to explore whether this ZGV technique is suitable for in-service non-destructive evaluations of liquid-filled pipes. Dispersion spectra and excitation properties have been analyzed. It is found that the presence of the liquid causes an increased number of backward modes and ZGVs which are highly excitable by a point source. In addition, several guided modes twice undergo a change of sign in the slopes of their dispersion curves, leading to two ZGV points. This phenomenon of double ZGVs in one backward wave, which is caused by strong mode repulsions, has not been found in isotropic hollow cylinders, but it can be observed in a fluid-filled thin-walled pipe.

Investigation of interfacial stiffnesses of a tri-layer using Zero-Group Velocity Lamb modes.

Zero-Group Velocity (ZGV) Lamb waves are studied in a structure composed of two plates bonded by an adhesive layer. The dispersion curves are calculated for a Duralumin/epoxy/Duralumin sample, where the adhesion is modeled by a normal and a tangential spring at both interfaces. Several ZGV modes are identified and their frequency dependence on interfacial stiffnesses and on the bonding layer thickness is numerically studied. Then, experiments achieved with laser ultrasonic techniques are presented. Local resonances are measured using a superimposed source and probe. Knowing the thicknesses and elastic constants of the Duralumin and epoxy layers, the comparison between theoretical and experimental ZGV resonances leads to an evaluation of the interfacial stiffnesses. A good agreement with theoretical dispersion curves confirms the identification of the resonances and the parameter estimations. This non-contact technique is promising for the local evaluation of bonded structures.

Non contact probing of interfacial stiffnesses between two plates by zero-group velocity Lamb modes

A non contact technique using zero-group velocity (ZGV) Lamb modes is developed to probe the bonding between two solid plates coupled by a thin layer. The layer thickness is assumed to be negligible compared with the plate thickness and the acoustic wavelength. The coupling layer is modeled by a normal and a tangential spring to take into account the normal and shear interfacial stresses. Theoretical ZGV frequencies are determined for a symmetrical bi-layer structure and the effect of the interfacial stiffnesses on the cut-off and ZGV frequencies are evaluated. Experiments are conducted with two glass plates bonded by a drop of water, oil, or salol, leading to a few micrometer thick layer. An evaluation of normal and shear stiffnesses is obtained using ZGV resonances locally excited and detected with laser ultrasonic techniques.

Guided modes with multiple zero-group-velocity points in fluid-filled cylindrical pipes

It is known that guided modes in isotropic hollow cylinders exhibit backward wave propagation with negative group velocity. And interferences between backward and forward waves generate zero-group-velocity (ZGV) resonances with a finite wavenumber but vanishing group velocity. These ZGV resonances can be applied for non-destructive evaluation (NDE) of hollow pipes. In this paper, influences of a fluid-loading on ZGV resonances in pipes are studied for the possible application of integrity inspections of oil transportation pipelines. From numerically simulated frequency-wavenumber spectra of axisymmetric guided modes, in addition to the backward mode with a single ZGV point, certain branches change the sign of their slopes for twice (i.e., two ZGV points in one branch). Such multiple ZGV modes might be caused by the strong repulsion between the backward mode with a single ZGV point that is propagating in the hollow pipe and a number of longitudinal modes in the fluid cylinder. It is found that, from wave structure analyses, ZGV points correspond to relatively large displacement amplitudes at the pipe’s inner and outer interfaces. It indicates that guided modes with multiple ZGV points can be sensitive to the surface features of fluid-filled pipes, which is useful for NDE application.

Elastic characterization of nanoporous gold foams using laser based ultrasonics

A resonance based laser ultrasonics technique is explored for the characterization of low density nanoporous gold foams. Laser generated zero group velocity (ZGV) lamb waves are measured in the foams using a Michelson interferometer. The amplitude spectra obtained from the processed time-domain data are analyzed using a theoretical model from which the foam Young’s modulus and Poisson’s ratio are obtained. The technique is non-contact and nondestructive, and the ZGV resonance modes are spatially localized, allowing for spatial mapping of the bulk sample properties. The technique may be suitable for process control monitoring and mechanical characterization of low density nanoporous structures.

Characterization of mechanical properties of a hollow cylinder with zero group velocity Lamb modes

Hollow cylinders used in the industry must be regularly inspected. Elastic guided waves, similar to Lamb modes in a plate, can propagate in the axial direction or around the circumference. They are sensitive to geometrical and mechanical parameters of the cylindrical shell. The objective of this paper is to show that zero group velocity (ZGV) Lamb modes can be used to bring out anisotropy and to measure elastic constants of the material. This study provides experimental and numerical investigations on a Zirconium alloy tube extensively used by the nuclear industry in reactor core components. A non-contact method, based on laser ultrasound techniques and ZGV Lamb modes, demonstrates that the difference observed between axial and circumferential guided waves cannot be explained by an isotropic model. Then, a transverse isotropic model is used for the Zircaloy tube. Four of the five elastic constants are directly extracted from ZGV resonance frequencies. The last one is deduced from the measured dispersion spectra. With this complete set of constants, a good agreement is obtained between theoretical and experimental dispersion curves for both axially and circumferentially propagating guided waves.

Thin layer Characterization by ZGV Lamb modes

Ultrasonic non-destructive testing of plates can be performed with Lamb modes guided by the structure. Non contact generation and detection of the elastic waves can be achieved with optical means such as a pulsed laser source and an interferometer. With this setup, we propose a method using zero group velocity (ZGV) Lamb modes rather than propagating modes. These ZGV modes have noteworthy properties, in particular their group velocity vanishes, whereas their phase velocity remains finite. Thus, a significant part of the energy deposited by the pulsed laser can be trapped in the source area. For example, in a homogeneous isotropic plate and at the minimum frequency of the S1-Lamb mode a very sharp resonance can be observed, the frequency of which only depends on the plate thickness, for a given material. In fact, other ZGV modes exist and the set of ZGV resonance frequencies provide a local and absolute measurement of Poisson's ratio. These non-propagating modes can also be used to characterize multi-layered structures. Experimentally, we observed that a thin (500 nm) gold layer deposited on a thick (1.5 mm) Duralumin plate induces a sensitive down-shift of the set of ZGV resonance frequencies. This shift, which is typically 5 kHz for the S1-Lamb mode at 1.924 MHz, can be approximated by a formula providing the layer thickness. Thickness down to 100 nm can be estimated by this method. Such a sensitivity with conventional ultrasound inspection by acoustic microscopy would require an operating frequency in the GHz range.

Zero group velocity Lamb modes and edge resonance in a semi‐infinite plate.

The local resonances of an elastic plate with a free edge are investigated using laser ultrasonic techniques. It is known that the long wavelength vibrations of a plate can be interpreted in terms of Lamb modes. Low‐frequency vibrations can be ascribed to the flexural A0 mode, while high‐frequency shear or stretch vibrations occur at the cutoff of higher order Lamb modes. These thickness resonances correspond to an in‐phase motion of the whole plate, associated to vanishing Lamb wave number. An elastic plate can also support local vibrations at nonzero values of the wave number for which Lamb mode dispersion curves undergo a minimum. Recently, experiments using laser ultrasonic techniques demonstrated that the local resonance of an infinite elastic plate is entirely governed by these zero group velocity (ZGV) Lamb modes. For a semi‐infinite plate, ZGV modes can be observed at a distance as close as two thicknesses from the edge. Coming toward the edge, ZGV resonances slowly vanish, whereas the edge mode resonance appears, whose frequency depends on the thickness and on the bulk velocities of the plate. Using a laser excitation on the edge, the amplitude profile of the normal displacement of the edge was determined.

Non contact thin film local thickness measurements through a thick plate using zero group velocity lamb modes.

It has been shown that a homogeneous isotropic plate can support non‐propagating Lamb modes having a zero group velocity (ZGV) for nonzero values of their wave number. Optical means such as a pulsed laser source and a heterodyne interferometer can be used for non‐contact generation and detection of these elastic waves. With this experimental setup, local and accurate measurements of material parameters such as bulk waves velocities and very small plate thinning detection were obtained from the resonance spectrum of ZGV Lamb modes. A thin layer deposited on the plate induces a significant down shift of the set of ZGV resonance frequencies. Experimentally, thin gold layers down to 100 nm were detected through a 1.5‐mm thick Duralumin plate. Optical measurements were performed at the same point as the generation on the non‐covered face of the thick plate. The frequency shift, which is typically 1 kHz for the S1‐Lamb mode at 1.9 MHz, can be approximated by a formula, which allows us to calculate the layer thi...