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)].

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.

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.

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.

Laser induced zero-group velocity resonances in transversely isotropic cylinder.

The transient response of an elastic cylinder to a laser impact is studied. When the laser source is a line perpendicular to the cylinder axis, modes guided along the cylinder are generated. For a millimetric steel cylinder up to ten narrow resonances can be locally detected by a laser interferometer below 8 MHz. These resonances correspond to Zero-Group Velocity guided modes or to circumferential modes. The authors observe that the theory describing the propagation of elastic waves in an isotropic cylinder is not sufficient to precisely predict the resonance spectrum. In fact, the texture of such elongated structure manifests as elastic anisotropy. Thus, a transverse isotropic (TI) model is used to calculate the dispersion curves and compare them with the measured ones, obtained by moving the source along the cylinder. The five elastic constants of a TI cylinder are adjusted, leading to a good agreement between measured and theoretical dispersion curves. Then, all the resonance frequencies are satisfactorily identified.

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.

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.

On the elasticity of transverse isotropic soft tissues (L).

Quantitative elastography techniques have recently been developed to estimate the shear modulus µ of soft tissues in vivo. In the case of isotropic and quasi-incompressible media, the Young's modulus E is close to 3 µ, which is not true in transverse anisotropic tissues such as muscles. In this letter, the transverse isotropic model established for hexagonal crystals is revisited in the case of soft solids. Relationships between elastic constants and Young's moduli are derived and validated on experimental data found in the literature. It is shown that 3 µ(⊥) ≤ E(⊥) ≤ 4 µ(⊥) and that E(//) cannot only be determined from the measurements of µ(//) and µ(⊥).

Laser ultrasonic inspection of plates using zero-group velocity lamb modes.

A noncontact laser-based ultrasonic technique is proposed for detecting small plate thickness variations caused by corrosion and adhesive disbond between two plates. The method exploits the resonance at the minimum frequency of the S(1) Lamb mode dispersion curve. At this minimum frequency, the group velocity vanishes, whereas the phase velocity remains finite. The energy deposited by the laser pulse generates a local resonance of the plate. This vibration is detected at the same point by an optical interferometer. First experiments show the ability to image a 1.5-microm deep corroded area on the back side of a 0.5-mm-thick duralumin plate. Because of the finite wavelength of the S(1)- zero group velocity (ZGV) mode, the spatial resolution is limited to approximately twice the plate thickness. With the same technique we investigate the state of adhesive bonds between duralumin and glass plates. The S(1)-Lamb mode resonance is strongly attenuated when plates are rigidly bonded. In the case of thin adhesive layers, we observed other resonances, associated with ZGV modes of the multi-layer structure, whose frequencies and amplitudes vary with adhesive thickness. Experiments were carried out on real automotive adhesively bonded structures and the results were compared with images obtained by X-ray radiography.

Excitation and focusing of Lamb waves in a multilayered anisotropic plate.

The radiation of Lamb waves by an axisymmetric source on the surface of an anisotropic plate is investigated. An analytical expression of the Green's function, valid in the far field domain, is derived. This approximation shows that the anisotropy of the propagation medium induces a focusing of Lamb modes in some directions, which correspond to minima of the slowness. Numerical simulations and experiments demonstrate that for the fundamental A(0) and S(0) modes, this phenomenon, analog to the phonon focusing effect, can be very strong in materials such as composite fiber-reinforced polymers. This effect due to the plate anisotropy must be correctly taken into account, for example, in order to develop systems for in situ structural health monitoring. The choice of the most appropriate Lamb mode, the excitation frequency, and the design of the array of piezoelectric disks used as transmitters and receivers depends on such considerations.

Influence of the anisotropy on zero-group velocity Lamb modes.

Guided waves in a free isotropic plate (symmetric S(n) and antisymmetric A(n) Lamb modes) exhibit a resonant behavior at frequencies where their group velocity vanishes while their phase velocity remains finite. Previous studies of this phenomenon were limited to isotropic materials. In this paper, the optical generation and detection of these zero-group velocity (ZGV) Lamb modes in an anisotropic plate is investigated. With a circular laser source, multiple local resonances were observed on a silicon wafer. Experiments performed with a line source demonstrated that the frequency and the amplitude of these resonances depend on the line orientation. A comparison between experimental and theoretical dispersion curves for waves propagating along the [100] and [110] directions of the silicon crystal verified that these resonances occur at the minimum frequency of the S(1) and A(2) Lamb modes. Simulations indicated that it is possible to deduce the three elastic constants of the plate material with good accuracy from these measurements.

Lamb mode spectra versus the poisson ratio in a free isotropic elastic plate.

The variation, with material parameters, of Lamb modes is investigated. Vibration spectra of traction-free elastic plates are generally presented, for a given isotropic material, as a set of dispersion curves corresponding to the various Lamb mode branches. Here, the spectrum variations, with the Poisson ratio nu, are plotted in a dimensionless co-ordinate system in the form of a bundle of curves for each Lamb mode. Except for the fundamental anti-symmetric mode A(0), this representation highlights the same behavior for all Lamb modes. V(T) denoting the shear wave velocity, the (omega,k) plane can be divided into two angular sectors separated by the line of slope V(T) [square root]2. In the upper one, corresponding to a phase velocity V=omega/k larger than V(T)[square root]2, dispersion curves are very sensitive to the plate material parameters. In the lower sector (V

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.

Identification of chemical inhibitors of protein-kinase CK2 subunit interaction.

Protein kinase CK2 is a multi-subunit complex whose dynamic assembly appears as a crucial point of regulation. The ability to interfere with specific protein-protein interactions has already provided powerful means of influencing the functions of selected proteins within the cell. CK2beta-derived cyclopeptides that target a well-defined hydrophobic pocket on CK2alpha have been previously characterized as potent inhibitors of CK2 subunit assembly [9]. As a first step toward the rational design of low molecular weight CK2 antagonists, we have in the present study screened a collection of podophyllotoxine indolo-analogues to identify chemical inhibitors of the CK2 subunit interaction. We report the identification of a podophyllotoxine indolo-analogue as a chemical ligand that binds to the CK2alpha/CK2beta interface inducing selective disruption of the CK2alpha/CK2beta assembly and concomitant inhibition of CK2alpha activity.

Stresses and displacements for some Rayleigh-type surface acoustic waves propagating on an anisotropic half space.

Specific relations between mechanical displacements and stresses for Rayleigh-type surface acoustic waves propagating on an anisotropic half space are demonstrated. For 16 symmetry configurations belonging to the orthorhombic, tetragonal, hexagonal and cubic systems, involving only two displacement and stress components, it is shown that the ratio between the shear and normal stresses inside the propagation media is equal to the ratio between the normal and in-plane displacement components at the free surface. This result generalizes the previous one obtained in the case of an isotropic solid [W. Hassan and P. B. Nagy, J. Acoust. Soc. Am. 104, 3107-3110 (1998)].

Nonlinear shear wave interaction in soft solids.

This paper describes nonlinear shear wave experiments conducted in soft solids with transient elastography technique. The nonlinear solutions that theoretically account for plane and nonplane shear wave propagation are compared with experimental results. It is observed that the cubic nonlinearity implied in high amplitude transverse waves at f(0)=100 Hz results in the generation of odd harmonics 3f(0), 5f(0). In the case of the nonlinear interaction between two transverse waves at frequencies f(1) and f(2), the resulting harmonics are f(i)+/-2f(j)(i,j=1,2). Experimental data are compared to numerical solutions of the modified Burgers equation, allowing an estimation of the nonlinear parameter relative to shear waves. The definition of this combination of elastic moduli (up to fourth order) can be obtained using an energy development adapted to soft solid. In the more complex situation of nonplane shear waves, the quadratic nonlinearity gives rise to more usual harmonics, at sum and difference frequencies, f(i)+/-f(j). All components of the field have to be taken into account.

Optical measurements of the self-demodulated displacement and its interpretation in terms of radiation pressure.

Using a sensitive optical interferometer, the low frequency displacement nonlinearly generated by an ultrasonic tone burst propagating in a liquid is studied. Close to the source, the low frequency displacement contains a quasi-static component, which is affected by diffraction effects farther from the transducer. The experimental setup provides quantitative results, which allow the determination of the nonlinearity parameter of the liquid with a good accuracy. Such measurements are carried out in water and ethanol. Finally, the pressure associated with the low frequency displacement is discussed. Introducing the temporal mean value of the displacement, as already done in lossless solids, the noncumulative part of this second order pressure is associated with the static part of the low frequency displacement. This interpretation leads to extend the definition of the Rayleigh radiation pressure usually introduced for a continuous plane wave radiated in a confined fluid.

An improved approximation for the Rayleigh wave equation.

We derive a simple expression, which gives an approximate value of the Rayleigh wave velocity in an isotropic solid. This approximation is five times better than that given by Viktorov. The velocity equation can be easily inverted in order to obtain an accurate determination of the elastic constants. This procedure can be worthwhile for elastic microanalysis of bulk materials by scanning acoustic microscopy.

Flow Thermolysis Rearrangements in the Indole Alkaloid Series: Strictamine and Akuammicine Derivatives. The Absolute Configurations of Ngouniensine and epi-Ngouniensine.

Flow thermolysis of strictamine (1) generated two of the predictable rearrangement products, resulting from [1,5]-sigmatropic shifts: akuammicine (2) and indolenine 9. Besides formation of these two compounds, a quite different pathway gave rise to a novel rearrangement leading to indole 6, with the framework of the natural alkaloid ngouniensine (19). Rearrangement to the ngouniensine skeleton became the major pathway when the akuammicine derivatives 10, 12, and 17 were submitted to thermolysis, generating compounds 11, 13 + 15, and 18, respectively. These results allowed us to assign the absolute configuration of (-)-ngouniensine (19) (3R,20R) and that of (-)-epingouniensine ((-)-21) (3R,20S).