PSI - Issue 52

Carlos A. Galán Pinilla et al. / Procedia Structural Integrity 52 (2024) 20–27 Carlos A. Gala´n-Pinilla et al. / Structural Integrity Procedia 00 (2023) 000–000

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lies Ochoˆa et al. (2019). Guided waves can be excited in a wide range of modes depending on the material properties, waveguide geometry, and ultrasonic pulse frequency Gao et al. (2019). Due to the dispersive and multimodal nature of the guided waves, some di ffi culties emerge in the execution and interpretation of the experimental data Olisa et al. (2021). This dispersion consists of the spatial distortion in time of the wave packet as the pulses propagate with di ff erent velocities through the waveguide reducing the scanning distance and the detection sensitivity. Consequently, the localization of defects requires exact knowledge of the characteristics of the propagating wave mode, such as its dispersion, propagation velocity, and specific wave structure Groth et al. (2020),Quiroga et al. (2017). Di ff erent approaches have been studied to estimate the dispersion curves, such as analytical, based on the charac teristic Rayleigh-Lamb equations Gala´n Pinilla et al. (2023), numerical, experimental, and finally, an e ffi cient and e ff ective combination of analytical and numerical methods, named semi-analytical, the best option for waveguides of complex cross-section geometry. Based on this last approach, several algorithms have been developed and pro grammed to computationally generate dispersion curves, such as DISPERSE (Lowe.) and GUIGUW Bocchini et al. (2011). In Semi-analytical approaches such as the SAFE (Semi-Analytical Finite Element) method, the waveguide cross-section is unidimensionally discretized with several finite elements whose number depends on the thickness of the material, requiring considerable refinement and the assembly of a global sti ff ness matrix. As a result, the method demands considerable computational costs and di ffi culties in dealing with complex configurations, as with multilayer materials with viscoelastic characteristics. Despite these advances, experimental di ffi culties are reported in industrial applications of this inspection method, where reduced scanning distances are achieved when the metallic material has coatings, especially those of the vis coelastic type, such as bituminous or epoxy Zhu et al. (2018). These coatings are mainly used in pipes and components of the oil-gas sector, as well as di ffi culties in the interpretation of the data. Thus, viscoelastic coated materials present a challenge for guided wave inspection due to wave dispersion and attenuation, which generally leads to a considerable reduction in amplitude with traveling distance. Therefore, it is important to understand the e ff ect of the coating on the dispersion to predict the appropriate wave modes in terms of sensitivity and good penetration power for proper damage characterization. Lamb wave propagation in viscoelastic materials presents dispersion which is related to the real components of the wavenumbers and attenuation which is related to the imaginary components of the complex wavenumbers. For the approach of the solution we considered in the present work the consitutive viscoelastic equations with the complex formulation of the Young’s modulus of the material. The solutions are presented in terms of phase velocity curves, group velocity, and the attenuation curve which relates the absorption of the material. Finally, the wave structures present the wave propagation shape, so they are required in order to exploit the potential of the technique. This paper extends the formulation of the Scaled-Boundary Finite Element Method and the Gauss-Lobatto Legendre Spectral Collocation to estimate the dispersion and attenuation curves and the wave structures in an elastic viscoelastic (Steel-Bitumen) bilayer specimen. The results allow for comparing the e ff ect of the coating on the prop agation of the lamb waves in plates. In the implementation, the cross-section was discretized using one-dimensional spectral elements and appropriate shape functions to coincide with the location of the nodes and the points where the Gaussian quadrature function is evaluated. Our results suggest that the SBFEM method is useful in terms of perfor mance and versatility when applied to the field of ultrasonic-guided wave propagation to generate dispersion curves and wave structures. A numerical example for the S 0 mode is shown to demonstrate the relation among dispersion, attenuation, and wave structure with respect to frequency for both studied cases.

2. SBFEM for Lamb waves in plates with viscoelastic coatings

The SBFEM method is applied to describe the behavior of guided waves in a semi-infinite coated plate of constant thicknesses t 1 (elastic) and t 2 (viscoelastic). Here, the thickness of the plate is in the y direction, and the waves propagate in the z direction as shown in Fig. 1. The amplitudes of the displacements within the element u e ( y , z ) are approximated from the interpolation of their nodal displacements as