Computing leaky Lamb waves for waveguides between elastic half-spaces using spectral collocation

TL;DR

The paper addresses the challenge of computing leaky Lamb waves in a waveguide sandwiched between elastic half-spaces, where energy leaks into surrounding media and the solutions require complex wavenumbers such as $k_x$. It introduces a spectral collocation method based on Chebyshev differentiation matrices and domain mappings to complex paths, which suppresses the exponential growth of the exterior fields while preserving physics. This approach leads to a polynomial eigenvalue problem in $k_x$ that yields the full dispersion and attenuation spectra for both shear- and fully-leaky modes, without prior knowledge of the modes. Validation against DISPERSE and finite-element simulations demonstrates high accuracy and robustness, with potential applicability to a broader class of loading configurations including solid-solid and solid-fluid interfaces.

Abstract

In non-destructive evaluation guided wave inspections, the elastic structure to be inspected is often embedded within other elastic media and the ensuing leaky waves are complex and non-trivial to compute; we consider the canonical example of an elastic waveguide surrounded by other elastic materials that demonstrates the fundamental issues with calculating the leaky waves in such systems. Due to the complex wavenumber solutions required to represent them, leaky waves pose significant challenges to existing numerical methods, with methods that spatially discretise the field to retrieve them suffering from the exponential growth of their amplitude far into the surrounding media. We present a spectral collocation method yielding an accurate and efficient identification of these modes, leaking into elastic half-spaces. We discretise the elastic domains and, depending on the exterior bulk wavespeeds, select appropriate mappings of the discretised domain to complex paths, in which the numerical solution decays and the physics of the problem are preserved. By iterating through all possible radiation cases, the full set of dispersion and attenuation curves are successfully retrieved and validated, where possible, against the commercially available software DISPERSE. As an independent validation, dispersion curves are obtained from finite element simulations of time-dependent waves using Fourier analysis.

Figures (7)

• Figure 1: Schematic of an elastic waveguide, between two elastic half-spaces, of thickness $2d$ and infinite extent in the $x$ and $z$ directions. The densities of the elastic media of the setup are $\rho$, $\rho_{1}$ and $\rho_{2}$, their longitudinal wavespeeds are $c_l$, $c_{l_1}$ and $c_{l_2}$ and their transverse wavespeeds are $c_t$, $c_{t_1}$ and $c_{t_2}$ respectively.
• Figure 2: Illustration of the different radiation cases with an increase in phase velocity for the example case of $c_{l_1}\geq c_{l_2} > c_{t_1} \geq c_{t_2}$.
• Figure 3: (Colour online) A schematic of a FE domain used to model elastic wave propagation and retrieve the dispersion curves of guided modes in a waveguide adjacent to two elastic half-spaces. In the figure, source nodes are shown in red and monitor nodes in black. Absorbing layers around the domain are shown in green while different colours denote the different materials within the domain.
• Figure 4: (Colour online) Dispersion curves of an epoxy waveguide in contact with an aluminium half-space on each side, obtained by our SCM (denoted by $\circ$), by DISPERSE (denoted by --) and by FE modelling (normalised intensity in colour).
• Figure 5: (Colour online) Comparison of attenuation values obtained by our SCM (denoted by $\circ$) and by DISPERSE (denoted by --) for waves in an epoxy waveguide in contact with an aluminium half-space on each side \ref{['Attenuation plot']}. A detail of the region with attenuation of up to 1 Np/mm \ref{['Attenuation plot']}.