Zero Group Velocity Lamb Modes Research Articles

Multiple zero group velocity Lamb modes in an anisotropic plate: propagation along different crystallographic axes

This paper studies the propagation of symmetric and antisymmetric Lamb waves along a 1 mm thick iodic acid plate (HIO3) in the 1–50 MHz frequency range. The Lamb mode propagation along three crystallographic planes was theoretically investigated, for two mutually orthogonal propagation directions. Several frequencies were found that correspond to Lamb modes with zero group velocity (ZGV) and non-null phase velocity values. The first symmetric Lamb mode, S1, was found to possess only one ZGV point, regardless of the propagation direction; higher order symmetric and antisymmetric modes with up to four ZGV points were found, depending on the propagation plane. The dependence of the ZGV frequencies on each elastic constant (c11, c13, c33, c55) of the HIO3 plate material was also investigated by changing the constant values by 5% and 10%. It was found that c33 and c55 affect the number of the ZGV points, while c11 and c13 affect the frequency of the ZGV points. The existence and frequencies of the ZGV points are strongly dependent on the proximity of standing longitudinal and transverse waves at nearby cutoff frequencies.

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.

Toward real-time assessment of material properties using elastic guided waves

Elastic guided waves are one of the most promising quantitative ultrasound techniques for the assessment of elastic properties in plate-like structures. Measurements of guided waves, associated with suitable waveguide modeling, can yield accurate estimates of waveguide properties like thickness and stiffnesses. Such a model-based approach requires solving a multi-parametric inverse problem to match experimental data with guided modes. Sensitivity studies suggest that isolated areas of the dispersion curves have predominant influence on specific model parameters. In particular, guided waves in a free plate exhibit a resonant behavior at frequencies where their group velocity vanishes while their phase velocity remains finite (so-called zero-group velocity Lamb modes). In this study, we investigate the feasibility of exploiting targeted data in the vicinity of these particular points, along with data associated with the cut-off frequencies. To this end, guided waves measurements are performed on a series of materials (anisotropic plates and tri-layer structures) using a linear multi-element transducer array and dedicated signal processing. A genetic algorithm-based inverse problem is then solved to recover the waveguide properties. Preliminary results indicate that, owing to the measurements versatility in terms of acquisition speed, this approach has the potential for inferring reliable structural and material properties in real-time.

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.

Beam shaping to enhance zero group velocity Lamb mode generation in a composite plate and nondestructive testing application

Zero group velocity (ZGV) Lamb modes have already shown their potential in nondestructive testing applications as they are sensitive to the sample structural characteristics. In this paper, we first consider an aluminum sample to validate a method based on the beam shaping of the generation laser. This method is proven to enhance ZGV Lamb modes in aluminum, and then advantageously applied to a composite material plate. Finally, based on the proposed method, scanning the sample over healthy and flawed zones demonstrates the ability to detect subsurface flaws.

Effect of refracted light distribution on the photoelastic generation of zero-group velocity Lamb modes in optically low-absorbing plates.

Zero-group velocity (ZGV) Lamb modes are associated with sharp local acoustic resonances and allow, among other features, local measurement of Poisson's ratio. While the thermoelastic generation of Lamb waves in metal plates has been widely studied, the case of materials of low-optical absorption remains unexplored. In materials such as glasses, the generation of bulk elastic waves has been demonstrated to be sensitive to the refracted light distribution. In this paper, a detailed analysis of the effect of light refraction on the laser-based generation of ZGV Lamb modes is presented. Experiments are performed on a bare glass plate without the need for an additional layer for light absorption or reflection. Using an appropriate tilted volume source, it is shown that the laser-ultrasonic technique allows non-contact measurement of the Poisson's ratio.

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.

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.

Air-coupled sensing of leaky rayleigh waves and ZGV modes in concrete slabs

Rayleigh waves and zero-group velocity (ZGV) Lamb modes (the impact-echo test) are commonly used for non-destructive evaluation (NDE) of concrete structures to extract information about material property and interior delaminations. Traditional methods in civil engineering employ point impactors to generate the waves while accelerometers measure the resulting out-of-plane motion. Though well-established, this methodology does not lend itself to efficient scanning of large areas, which is critical for monitoring the safety of infrastructure. The ability to detect these two wave types through in-air sensing of Leaky Rayleigh or ZGV Lamb waves can greatly accelerate NDE for large structures. Unfortunately, air-coupled sensing suffers from significantly decreased signal amplitude. This paper presents theoretical analysis and experimental validation efforts to amplify in-air signals resulting from Leaky Rayleigh waves and ZGV Lamb modes using a parabolic focusing mirror. A time-domain impulse response to the Ki...

Thin layer thickness measurements by zero group velocity Lamb mode resonances

Local and non-contact measurements of the thickness of thin layers deposited on a thick plate have been performed by using zero group velocity (ZGV) Lamb modes. It was shown that the shift of the resonance frequency is proportional to the mass loading through a factor which depends on the mechanical properties of the layer and of the substrate. In the experiments, ZGV Lamb modes were generated by a Nd:YAG pulsed laser and the displacement normal to the plate surface was measured by an optical interferometer. Measurements performed at the same point that the generation on the non-coated face of the plate demonstrated that thin gold layers of a few hundred nanometers were detected through a 1.5-mm thick Duralumin plate. The shift of the resonance frequency (1.9 MHz) of the fundamental ZGV mode is proportional to the layer thickness: typically 10 kHz per μm. Taking into account the influence of the temperature, a 240-nm gold layer was measured with a ±4% uncertainty. This thickness has been verified on the coated face with an optical profiling system.

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.

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...

Local vibration of an elastic plate and zero-group velocity Lamb modes

Elastic plates or cylinders can support guided modes with zero group velocity (ZGV) at a nonzero value of the wave number. Using laser-based ultrasonic techniques, we experimentally investigate some fascinating properties of these ZGV modes: resonance and ringing effects, backward wave propagation, interference between backward and forward modes. Then, the conditions required for the existence of ZGV Lamb modes in isotropic plates are discussed. It is shown that these modes appear in a range of Poisson's ratio about the value for which the cutoff frequency curves of modes belonging to the same family intercept, i.e., for a bulk wave velocity ratio equal to a rational number. An interpretation of this phenomenon in terms of a strong repulsion between a pair of modes having a different parity in the vicinity of the cutoff frequencies is given. Experiments performed with materials of various Poisson's ratio demonstrate that the resonance spectrum of an unloaded elastic plate, locally excited by a laser pulse, is dominated by the ZGV Lamb modes.

Zero-group velocity modes and local vibrations of an elastic plate

Elastic plates or cylinders can support guided modes with zero group velocity (ZGV) at a nonzero value of the wave number. The conditions required for the existence of ZGV Lamb modes in isotropic plates are discussed. It is shown that these modes appear in a range of Poisson's ratio over the value for which the cut-off frequency curves of modes belonging to the same family intercept, i.e for a bulk wave velocity ratio equal to a rational number. An interpretation of this phenomenon in terms of a strong repulsion between a pair of modes having a different parity in the vicinity of the cutoff frequencies is given. Using laser-based ultrasonic techniques, we experimentally investigate some properties of these ZGV Lamb modes: resonance, backward wave propagation, interference between backward and forward waves. Experiments performed with materials of various Poisson's ratio demonstrate that the resonance spectrum of an unloaded elastic plate, locally excited by a laser pulse, is dominated by the ZGV Lamb modes. From these local resonance frequencies, thickness variations can be measured accurately and material properties like the Poisson's ratio, bulk acoustic wave velocities or material attenuation can be determined without any mechanical contact.

Second order dispersive effect on zero-group velocity Lamb modes

After a local and transient loading of a plate, the acoustic energy decay in the source area can be explained by three mechanisms: the energy transport phenomenon at the Lamb wave group velocity, the material damping and the second-order dispersive effect. Generally, the first mechanism dominates the other two. However, some dispersion curves of Lamb modes present a minimum frequency for a nonzero wave number. Since at these points the group velocity vanishes, no energy transport occurs, and the slower other two phenomena can be observed. Using laser-based ultrasonic techniques, we show that the mechanical response of various thin metallic plates to a laser pulse impact is dominated by the resonance of the zero group velocity S1-Lamb mode. In the first microseconds, we observed a t-1/2 decay of the local vibration amplitude, which cannot be explained by the material damping. We demonstrated that this effect is ascribed to the second-order term in the dispersion relation. After this power law decay, the mechanical displacement undergoes an exponential decay corresponding to the wave damping. Then, we show that the local attenuation of the plate material can be measured without any mechanical contact.

Power law decay of zero group velocity Lamb modes

Elastic plates or cylinders can support guided modes with zero group velocity (ZGV) for nonzero wave numbers. At these ZGV-points of the dispersion curves the acoustic energy does not propagate in the waveguide, resulting in sharp resonance effects. In this paper, using laser-based ultrasonic techniques, we investigate the time-decay of the mechanical displacement for ZGV Lamb modes excited by a pulsed laser in various thin metallic plates. In the first microseconds of the local plate vibration, we observed a t - 1 / 2 decay due to the second order term in the dispersion relation. This effect is dominant because the first order term, proportional to the group velocity, vanishes for ZGV-modes. After this power law decay, the mechanical displacement undergoes an exponential decay corresponding to the wave damping. Then, the local attenuation of the plate material can be estimated at the ZGV-resonance frequency.