Adiabatic Lamb modes in 3D tapered waveguides: Cut-off effects and ZGV resonances

TL;DR

This work investigates adiabatic Lamb modes in 3D tapered plates with linearly varying thickness, focusing on cut-off and Zero-Group Velocity (ZGV) phenomena. The authors combine laser-based broadband excitation with 3D FFT dispersion mapping to extract local $A_1$ cut-off thicknesses and $S_1S_2$-ZGV resonances in a linearly varying Al plate, including anisotropy effects. They demonstrate two thickness-reconstruction strategies: an $A_1$-cutoff method and a $S_1S_2$-ZGV method, with the latter achieving an average relative error of $1.43\\%$ compared to $2.6\%$ for the former. The work suggests broad applicability to inhomogeneous media and AM structures, with potential extensions to elasticity and temperature gradients.

Abstract

This paper aims to enhance our understanding of the physical behavior of adiabatic modes in inhomogeneous elastic plates, particularly their remarkable capacity to adapt to gradual perturbations. The study investigates the propagation characteristics of higher-order adiabatic Lamb modes in waveguides with linearly varying thickness, with a focus on the influence of critical thicknesses on their propagation. This is achieved by leveraging the broadband excitation capabilities of a pulsed laser generating higher order Lamb modes to reveal various critical thicknesses, such as the cut-off and Zero-Group Velocity (ZGV) thicknesses. Remarkably, ZGV resonances can be induced at locations well beyond the laser source. Moreover, the mode's behavior is strongly influenced by thickness variations in all directions, imparting the plate an anisotropic-like behavior. Additionally, based on the observed effects, our experimental approach enables precise reconstruction of elastic waveguide profiles in additively manufactured aluminum plates with such thickness variations. The reconstructed profiles show a strong correlation with reference measurements across the scanned area.

Figures (9)

• Figure 1: Experimental setup for the laser generation and probing of Lamb modes on an elastic AM-plate with a slow variation of the thickness in all directions.This experimental setup is based on the principle of reciprocity: the pulsed laser is in motion and both longitudinal transducers is in the probing position to obtain a better signal-to-noise ratio.
• Figure 2: Experimental dispersion curves, measured with transducer #1, for the column at $x = 60$ mm are compared with theoretical curves calculated for the following thicknesses: $h_1$ = 4.2 mm (), $h_m$ = 3.5 mm (), and $h_2$ = 2.5 mm ().
• Figure 3: Theoretical parametric dispersion curves for the $A_1$-mode: (a) for a constant thickness $h_1$ (red) and for observed thicknesses at the edge of the scanned zone $h_e$ (green); projection of the dispersion curves presented in three different $(f,k_x,k_y)$ planes for a better overview: (b) $P_1 \to k_x=0$, (c) $P_2 \to k_y=0$, (d) and $P_3 \to f=0.5$ MHz.
• Figure 4: (a) Experimental wavefield at 35 µs propagation time; (b) 2D spatial Fourier plane at 0.5 MHz ; (c) 2D filtered spatial Fourier plane at 0.5 MHz ; Wavefield (real part) of incident $A_1$-mode at 0.5 MHz (d) and 0.6 MHz (e) obtained after spatial IFFT.
• Figure 5: (a) Measured thickness map with the laser rangefinder, (b) reconstructed thickness map, and (c) relative error between thickness maps.