Viscoelastic dynamics of a soft strip subject to a large deformation.

To produce sounds, we adjust the tension of our vocal folds to shape their properties and control the pitch. This efficient mechanism offers inspiration for designing reconfigurable materials and adaptable soft robots. However, understanding how flexible structures respond to a significant static strain is not straightforward. This complexity also limits the precision of medical imaging when applied to tensioned organs like muscles, tendons, ligaments and blood vessels among others. In this article, we experimentally and theoretically explore the dynamics of a soft strip subject to a substantial static extension, up to 180%. Our observations reveal a few intriguing effects, such as the resilience of certain vibrational modes to a static deformation. These observations are supported by a model based on the incremental displacement theory. This has promising practical implications for characterizing soft materials but also for scenarios where external actions can be used to tune properties.

Beating resonance patterns and extreme power flux skewing in anisotropic elastic plates.

Elastic waves in anisotropic media can exhibit a power flux that is not collinear with the wave vector. This has notable consequences for waves guided in a plate. Through laser-ultrasonic experiments, we evidence remarkable phenomena due to slow waves in a single-crystal silicon wafer. Waves exhibiting power flux orthogonal to their wave vector are identified. A pulsed line source that excites these waves reveals a wave packet radiated parallel to the line. Furthermore, there exist precisely eight plane waves with zero power flux. These so-called zero-group-velocity modes are oriented along the crystal's principal axes. Time acts as a filter in the wave-vector domain that selects these modes. Thus, a point source leads to beating resonance patterns with moving nodal curves on the surface of the infinite plate. We observe this pattern as it emerges naturally after a pulsed excitation.

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).

Weight of single and recurrent scattering in the reflection matrix of complex media.

In a heterogeneous medium, the wave field can be decomposed as an infinite series known as the Born expansion. Each term of the Born expansion corresponds to a scattering order, it is thus theoretically possible to discriminate single and multiple scattering contribution to the field. Experimentally, what is actually measured is the total field in which all scattering orders interfere. Conventional imaging methods usually rely on the assumption that the multiple scattering contribution can be disregarded. In a back-scattering configuration, this assumption is valid for small depths, and begins to fail for depths larger than the scattering mean-free path ℓ_{s}. It is therefore a key issue to estimate the relative amount of single and multiple scattering in experimental data. To this end, a single-scattering estimator ρ[over ̂] computed from the reflection matrix has been introduced in order to assess the weight of single scattering in the backscattered wave field. In this paper, the meaning of this estimator is investigated and a particular attention is given to recurrent scattering. In a diffraction-limited experiment, a multiple scattering sequence is said to be recurrent if the first and last scattering events occur in the same resolution cell. Recurrent scattering is shown to be responsible for correlations between single scattering and higher scattering orders of the Born expansion, inducing a bias to the estimator ρ[over ̂] that should rather be termed confocal scattering ratio. Interestingly, a more robust estimator is built by projecting the reflection matrix in a focused basis. The argument is sustained by numerical simulations as well as ultrasonic data obtained around 1.5 MHz in a model medium made of nylon rods immersed in water. From a more general perspective, this work raises fundamental questions about the impact of recurrent scattering on wave imaging.

Soft elastomers: A playground for guided waves.

Mechanical waves propagating in soft materials play an important role in physiology. They can be natural, such as the cochlear wave in the inner ear of mammalians, or controlled, such as in elastography in the context of medical imaging. In a recent study, Lanoy, Lemoult, Eddi, and Prada [Proc. Natl. Acad. Sci. U.S.A. 117(48), 30186-30190 (2020)] implemented an experimental tabletop platform that allows direct observation of in-plane guided waves in a soft strip. Here, a detailed description of the setup and signal processing steps is presented as well as the theoretical framework supporting them. One motivation is to propose a tutorial experiment for visualizing the propagation of guided elastic waves. Last, the versatility of the experimental platform is exploited to illustrate experimentally original features of wave physics, such as backward modes, stationary modes, and Dirac cones.

Cloaking, trapping and superlensing of lamb waves with negative refraction.

We report on experimental and numerical implementations of devices based on the negative refraction of elastic guided waves, the so-called Lamb waves. Consisting in plates of varying thickness, these devices rely on the concept of complementary media, where a particular layout of negative index media can cloak an object with its anti-object or trap waves around a negative corner. The diffraction cancellation operated by negative refraction is investigated by means of laser ultrasound experiments. However, unlike original theoretical predictions, these intriguing wave phenomena remain, nevertheless, limited to the propagating component of the wave-field. To go beyond the diffraction limit, negative refraction is combined with the concept of metalens, a device converting the evanescent components of an object into propagating waves. The transport of an evanescent wave-field is then possible from an object plane to a far-field imaging plane. Twenty years after Pendry's initial proposal, this work thus paves the way towards an elastic superlens.

Laser-excited elastic guided waves reveal the complex mechanics of nanoporous silicon.

Nanoporosity in silicon leads to completely new functionalities of this mainstream semiconductor. A difficult to assess mechanics has however significantly limited its application in fields ranging from nanofluidics and biosensorics to drug delivery, energy storage and photonics. Here, we present a study on laser-excited elastic guided waves detected contactless and non-destructively in dry and liquid-infused single-crystalline porous silicon. These experiments reveal that the self-organised formation of 100 billions of parallel nanopores per square centimetre cross section results in a nearly isotropic elasticity perpendicular to the pore axes and an 80% effective stiffness reduction, altogether leading to significant deviations from the cubic anisotropy observed in bulk silicon. Our thorough assessment of the wafer-scale mechanics of nanoporous silicon provides the base for predictive applications in robust on-chip devices and evidences that recent breakthroughs in laser ultrasonics open up entirely new frontiers for in-situ, non-destructive mechanical characterisation of dry and liquid-functionalised porous materials.

Localization of unbounded contacts on vibrating elastic plates.

Detection and localization of unbounded contacts in industrial structures are crucial for user safety. However, most structural health monitoring techniques are either invasive, power-consuming, or rely on time-varying baseline comparison. A passive acoustic method is proposed to localize unbounded contacts in plate-like structures, using the acoustic emissions by the contacts when they are excited by ambient noise. The technique consists of computing the correlation matrix of the signals measured by a set of receivers and applying to this matrix a beamforming algorithm accounting for flexural wave dispersion. To validate the technique, an experimental setup is developed in which three idealized unbounded contacts are created on a thin plate excited by a shaker. How the quality of the defect localization depends on the defect type, receiver number, and the characteristics of the noise is investigated. Finally, it is shown that the localization of unbounded contacts is possible using either an acoustic ambient noise source or a more realistic jet engine noise.

Dirac cones and chiral selection of elastic waves in a soft strip.

We study the propagation of in-plane elastic waves in a soft thin strip, a specific geometrical and mechanical hybrid framework which we expect to exhibit a Dirac-like cone. We separate the low frequencies guided modes (typically 100 Hz for a 1-cm-wide strip) and obtain experimentally the full dispersion diagram. Dirac cones are evidenced together with other remarkable wave phenomena such as negative wave velocity or pseudo-zero group velocity (ZGV). Our measurements are convincingly supported by a model (and numerical simulation) for both Neumann and Dirichlet boundary conditions. Finally, we perform one-way chiral selection by carefully setting the source position and polarization. Therefore, we show that soft materials support atypical wave-based phenomena, which is all of the more interesting as they make most of the biological tissues.

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)].

Accuracy of Green's function estimation from correlation of diffuse elastic waves on thin plates.

In a reverberant cavity, when a noise field is sufficiently diffuse, the correlation of the signal measured by two sensors provides an estimation of the Green's function (GF) between them. Here, the convergence of this passive estimation in the case of elastic waves on thin plates is studied. A statistical approach is proposed, which relates the similarity between the cross correlation and the GF to the structural properties of the plate and the number of uncorrelated sources. The analysis is sustained by experimental results obtained on an aluminum plate. This study allows us to evaluate the efficiency of passive structural health monitoring of plate-like structures based on noise correlation. Finally, a most interesting finding shows an absolute upper bound of the signal-to-noise ratio for GF quality reconstruction: 4Ns/5, independently of the plate properties.

Selective remote excitation of complex structures using time reversal in audible frequency range.

Generation of elastic waves is a major issue in nondestructive testing. Structural health monitoring of a thin element can be achieved through the analysis of its resonance spectrum. A time reversal mirror (TRM) operating in the audible frequency range (1-10 kHz) is used to remotely excite thin resonant elastic elements. The generation of elastic waves is studied with respect to the geometry of the TRM. It is observed that the quality of focusing only weakly depends on the number of loudspeakers (LS) in the TRM. When the air/plate coupling is at its maximum, the energetic efficiency of the TRM is estimated to be about 0.02%. The TRM is shown to efficiently and selectively excite a small structure embedded in a complex environment such as a hollow cylinder. Finally, the results are discussed in light of the DORT method (French acronym for "decomposition of the time reversal operator"). In particular, the optimal LS placement and emission signals in this configuration to excite individual eigenmodes of a plate is determined.

Negative reflection of elastic guided waves in chaotic and random scattering media.

The propagation of waves in complex media can be harnessed either by taming the incident wave-field impinging on the medium or by forcing waves along desired paths through its careful design. These two alternative strategies have given rise to fascinating concepts such as time reversal or negative refraction. Here, we show how these two processes are intimately linked through the negative reflection phenomenon. A negative reflecting mirror converts a wave of positive phase velocity into its negative counterpart and vice versa. In this article, we experimentally demonstrate this phenomenon with elastic waves in a 2D billiard and in a disordered plate by means of laser interferometry. Despite the complexity of such configurations, the negatively reflected wave field focuses back towards the initial source location, thereby mimicking a phase conjugation operation while being a fully passive process. The super-focusing capability of negative reflection is also highlighted in a monochromatic regime. The negative reflection phenomenon is not restricted to guided elastic waves since it can occur in zero-gap systems such as photonic crystals, chiral metamaterials or graphene. Negative reflection can thus become a tool of choice for the control of waves in all fields of wave physics.

2-D and 3-D Reconstruction Algorithms in the Fourier Domain for Plane-Wave Imaging in Nondestructive Testing.

Time-domain plane-wave imaging (PWI) has recently emerged in medical imaging and is now taking to nondestructive testing (NDT) due to its ability to provide images of good resolution and contrast with only a few steered plane waves. Insonifying a medium with plane waves is a particularly interesting approach in 3-D imaging with matrix arrays because it allows to tremendously reduce the volume of data to be stored and processed as well as the acquisition time. However, even if the data volume is reduced with plane wave emissions, the image reconstruction in the time domain with a delay-and-sum algorithm is not sufficient to achieve low computation times in 3-D due to the number of voxels. Other reconstruction algorithms take place in the wavenumber-frequency (f-k) domain and have been shown to accelerate computation times in seismic imaging and in synthetic aperture radar. In this paper, we start from time-domain PWI in 2-D and compare it to two algorithms in the f-k domain, coming from the Stolt migration in seismic imaging and the Lu theory of limited diffraction beams in medical imaging. We then extend them to immersion testing configurations where a linear array is facing a plane water-steel interface. Finally, the reconstruction algorithms are generalized to 3-D imaging with matrix arrays. A comparison dwelling on image quality and algorithmic complexities is provided, as well as a theoretical analysis of the image amplitudes and the limits of each method. We show that the reconstruction schemes in the f-k domain improve the lateral resolution and offer a theoretical and numerical computation gain of up to 36 in 3-D imaging in a realistic NDT configuration.

Towards real-time assessment of anisotropic plate properties using elastic guided waves.

A method to recover the elastic properties, thickness, or orientation of the principal symmetry axes of anisotropic plates is presented. This method relies on the measurements of multimode guided waves, which are launched and detected in arbitrary directions along the plate using a multi-element linear transducer array driven by a programmable electronic device. A model-based inverse problem solution is proposed to optimally recover the properties of interest. The main contribution consists in defining an objective function built from the dispersion equation, which allows accounting for higher-order modes without the need to pair each experimental data point to a specific guided mode. This avoids the numerical calculation of the dispersion curves and errors in the mode identification. Compared to standard root-finding algorithms, the computational gain of the procedure is estimated to be on the order of 200. The objective function is optimized using genetic algorithms, which allow identifying from a single out-of-symmetry axis measurement the full set of anisotropic elastic coefficients and either the plate thickness or the propagation direction. The efficiency of the method is demonstrated using data measured on materials with different symmetry classes. Excellent agreement is found between the reported estimates and reference values from the literature.

Effect of microstructural elongation on backscattered field: Intensity measurement and multiple scattering estimation with a linear transducer array.

The effect of microstructural elongation on ultrasonic backscattered fields was studied. Two methods for determining the elongation direction of macrozones in titanium alloys, using the anisotropic spatial coherence of the backscattered field, are presented. Both methods use a phased array attached on a rotative holder that records the array response matrix at several angles. Two titanium alloys were investigated: TA6V and Ti17. TA6V exhibited a strong macrozone elongation, whereas Ti17 macrozones were found equiaxial. The first method is based on the measurement of backscattered intensity in function of the probe angle relative to the macrozones elongation direction. An angular dependence of backscattered intensity is observed in presence of elongated scatterers, and their elongation direction is collinear with the probe direction corresponding to a minimal intensity. This variability is linked to both piezoelectric shape and the backscattered field spatial properties. The second method is based on the measurement of the relative proportion of single to multiple scattering in a diffusive media, using a simplified version of the single scattering filter developed in Aubry and Derode (2009). It allows the measurement of the level of multiple scattering: both titanium alloys exhibited strong multiple scattering. The elongation direction was determined as the direction of minimal multiple scattering. Furthermore, these results were confirmed by the measurement of the coherent backscattering cone on both samples.

Ultrasonic Imaging in Highly Attenuating Materials With Hadamard Codes and the Decomposition of the Time Reversal Operator.

In this paper, defects in a high density polyethylene pipe are imaged with the total focusing method. The viscoelastic attenuation of this material greatly reduces the signal level and leads to a poor signal-to-noise ratio (SNR) due to electronic noise. To improve the image quality, the decomposition of the time reversal operator method is combined with the spatial Hadamard coded transmissions before calculating images in the time domain. Because the Hadamard coding is not compatible with conventional imaging systems, this paper proposes two modified coding methods based on sparse Hadamard matrices with +1/0 coefficients. The SNRs expected with the different spatial codes are demonstrated, and then validated on both simulated and experimental data. Experiments are performed with a transducer array in contact with the base material of a polyethylene pipe. In order to improve the noise filtering procedure, the singular values associated with electronic noise are expressed on the basis of the random matrix theory. This model of noise singular values allows a better identification of the defect response in noisy experimental data. Finally, the imaging method is evaluated in a more industrial inspection configuration, where an immersion array probe is used to image defects in a butt fusion weld with a complex geometry.

Causality, Nonlocality, and Negative Refraction.

The importance of spatial nonlocality in the description of negative refraction in electromagnetic materials has been put forward recently. We develop a theory of negative refraction in homogeneous and isotropic media, based on first principles, and that includes nonlocality in its full generality. The theory shows that both dissipation and spatial nonlocality are necessary conditions for the existence of negative refraction. It also provides a sufficient condition in materials with weak spatial nonlocality. These fundamental results should have broad implications in the theoretical and practical analyses of negative refraction of electromagnetic and other kinds of waves.

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.