Conical dispersion of Lamb waves in elastic plates

This paper provides a theoretical investigation of negative refraction and focusing of elastic guided waves in a free-standing plate with a step-like thickness change. Under certain conditions, a positive phase velocity (forward) Lamb mode can be converted into a negative phase velocity (backward) mode at such interface, giving rise to negative refraction. A semi-analytical model is developed in order to study the influence of various parameters such as the material Poisson's coefficient, the steplike thickness, the frequency and the incidence angle. To this end, all the Lamb and shear horizontal propagating modes, but also a large number of their inhomogeneous and evanescent counterpart,s are taken into account. The boundary conditions applied to the stress-displacement fields at the thickness step yields an equation system. Its inversion provides the transmission and reflection coefficients between each mode at the interface. The step-like thickness and Poisson's ratio are shown to be key parameters to optimize the negative refraction process. In terms of material, Duralumin is found to be optimal as it leads to a nearly perfect conversion between forward and backward modes over broad frequency and angular ranges. An excellent focusing ability is thus predicted for a flat lens made of two symmetric thickness steps. Theoretical results are confirmed by a numerical FDTD simulation and experimental measurements made on an optimized Duralumin flat lens by means of laser interferometry. This theoretical study paves the way towards the optimization of elastic devices based on negative refraction, in particular for cloaking or super-focusing purposes.

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“…4.c). Such case has been recently investigated by Stobbe et al 28 as it also gives rise to a Dirac cone in the dispersion curves.…”

Section: Optimization Of the Negative Refraction Phenomenonmentioning

confidence: 95%

Negative refraction of Lamb modes: A theoretical study

et al. 2018

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“…Indeed, for ν ≈ 0.5, the plate velocity V P is equal to 2V T which results in a Dirac cone at the frequency f = V T /b as it happens for Lamb modes in a plate of Poisson's ratio ν = 1/3. 5,39,40 This is illustrated in Figure 2(b) displaying the in-plane mode dispersion curves for a 2 mm thick and 20 mm wide tape of velocities V L = 1500 m/s and V T = 5 m/s (ν = 0.4998). Such low shear velocity corresponds to typical velocities measured in agar gel or soft tissues.…”

Section: In-plane Modes Of a Thin Rectangular Beammentioning

confidence: 98%

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

Laurent

Royer

Prada

2020

The Journal of the Acoustical Society of America

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Elastic waves guided along bars of rectangular cross section 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 ZGV resonances are measured in a metal tape using non-contact 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 zero group velocity 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, 2011). arXiv:1907.09813v1 [physics.class-ph]

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“…based on ZGV frequency measurements 8 . Further details on our measurement approach are available in the literature 29 . In sample 1, the transducer was placed 60 mm from a 6 mm diameter hole in a mm thick plate.…”

Section: Experimental Set-upmentioning

confidence: 99%

“…Recently, it was recognized and experimentally demonstrated that a simple, homogenous, aluminum plate shows linear dispersion through k = 0 at a particular Poisson’s ratio where accidental degeneracy occurs 28,29 . Degeneracy was induced by adjusting the elastic properties of the plate over a small range through temperature change.…”

Section: Introductionmentioning

confidence: 99%

Propagation and Scattering of Lamb Waves at Conical Points in Plates

Stobbe

Grünsteidl

Murray

2019

Sci Rep

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Lamb waves exhibit conical dispersion at zero wave number when an accidental degeneracy occurs between thickness mode longitudinal and shear resonances of the same symmetry. Here we investigate the propagation of Lamb waves generated at the conical point frequency and the interaction of these waves with defects and interfaces. The group velocity and mode shapes of Lamb waves at the conical point are found, and it is shown that as the wavenumber gets close to zero, considerable group velocity is seen only for material properties supporting a degeneracy or near-degeneracy. The unusual wave propagation and mode conversion of Lamb waves generated at the conical point are elucidated through numerical simulations. Experimental measurements of near conical point Lamb wave interaction with holes in a plate demonstrate that these waves flow around defects while maintaining a constant phase of oscillation across that plate surface.

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Conical dispersion of Lamb waves in elastic plates

Cited by 10 publications

References 31 publications

Negative refraction of Lamb modes: A theoretical study

Negative refraction of Lamb modes: A theoretical study

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

Propagation and Scattering of Lamb Waves at Conical Points in Plates

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