*3.1. Wave Propagation in Single-Layer and Multi-Layer Media*

The first part of this section aimed at comparing the characteristics of guided waves propagating in a single layer plate (simulating the steel plate #5) and a two-layer plate (steel-adhesive) on a concrete half-space (three-layer medium modeling steel-adhesive-concrete composite beam #1). The materials for all the layers were assumed to be homogeneous, isotropic and linearly elastic (according to the material parameters presented in Table 1). Dispersion curves were calculated using a theoretical, numerical and experimental approach. The theoretical dispersion curves for the single-layer plate #5 were obtained in the way of solving well-known Rayleigh–Lamb equations in Matlab®. In the case of the three-layer medium, the theoretical curves were calculated using an open-source toolbox called Elastic-Matrix [59], based on the partial-wave method. Although the proposed algorithm is stated to be inaccurate for leaky cases, the appropriate solution can be obtained by selecting appropriate calculation parameters. The experimental and finite element method (FEM) numerical results were attained from the collected time-domain wave signals (91 and 101 signals, respectively) using the Matrix Pencil Method [56–58]. The pencil parameter and tolerance were chosen heuristically and equal to *p* = 16, *l* = 1.4% for the experimental analysis and *p* = 13, *l* = 2.2% for the numerical one. The determination of optimal values of these parameters requires further work; however, this aspect is not essential for the usefulness of results obtained in this research. The value of *p* was greater in the experimental calculation because the signal noise required computing more estimates, however, the tolerance needed to be more rigorous to exclude false results. The quality of results was enhanced by the authorial script deleting 'lonely' points. The complete wavenumber–frequency spectra were obtained by superimposing the dispersion curves calculated for the individual measurements with different location of the actuator and the excitation frequency.

The dispersion curves obtained from the three approaches are presented in Figure 5. The wavenumber–frequency relations for specimen #1 (Figure 5a) reveal the fact, that two flexural (A0, A1) and one longitudinal (S0) modes can propagate in the frequency range up to 400 kHz. It is essential to note, that the fundamental S0 mode does not appear in the range up to 122 kHz and the A1 mode is present for the frequencies above 260 kHz, thus only one mode can propagate for the assumed excitation frequency equal to 100 kHz. The comparison of the three approaches allowed concluding that the overall agreement between the numerical, experimental and theoretical results was very good. However, the higher mode A1 was not identified in the experimental approach, probably due to the setup limitations. It is also important to note that neither experimental nor numerical calculations detected A0 in the range up to 60 kHz, what could have been caused by the strict tolerance level. However, the increase of *l* would significantly decrease the legibility of the curves obtained. The analysis of dispersion curves for specimen #5 (Figure 5b) leads to similar conclusions. Compared with the results for beam #1, the S0 mode can propagate in the whole assumed frequency range, such as A0. The dispersion relations for both specimens for the frequencies above 150 kHz are comparable. The important differences are observable in the lower frequency range. To expose these dissimilarities, the theoretical dispersion curves for group velocity were prepared for both #1 and #5 specimens (Figure 5c). No significant difference can be seen for the A1 curve. The shapes of A0 and S0 curves are also very similar for the higher frequency range. As mentioned above, S0 mode is not present in the lower frequency range. To be noticed, it is important to analyze the shape of the A0 mode, because for the damage imaging, measurements the waves were excited perpendicularly to the plate surface and the out-of-plane velocity signals were collected. It is interesting that in the frequency range up to about 23 kHz it propagates with higher velocity in the specimen #1, whereas above this limit, the velocity of propagation is higher in specimen #5.

**Figure 5.** Dispersion curves for analyzed specimens: (**a**) wavenumber vs. frequency for three-layer plate #1; (**b**) wavenumber vs. frequency for single-layer steel plate #5; and (**c**) theoretical group velocity vs. frequency for three-layer plate #1 and single-layer plate #5.

The simple FEM numerical simulations were performed to check whether this dispersion relationship is actually observable. Figure 6 shows the time signals of propagating waves in specimens #1 and #5 obtained from the numerical models in point R5 as the results of R1 excitation (cf. Figure 4a). It is clearly visible, (especially in close-ups) that the wave packet related to the A0 mode appears earlier in the signal from specimen #1 for the excitation frequency equal to 10 kHz (Figure 6a). If the excitation frequency is equal to 23 kHz, no difference in time of flight (TOF) of the first wave packet is

observable (Figure 6b). The further increasing of the frequency value leads to the situation when the wave propagates faster in specimen #5 (Figure 6c). The TOF values for both models were calculated based on the dispersion curves (theoretical TOF) and are presented in Table 2, together with the values got from signals recorded. The relative differences between theoretical and numerical TOFs do not exceed a few percent, which indicates on the good compliance between dispersion curves and FEM simulations.

**Figure 6.** Normalized velocity signals (whole signals and zoom) registered at point R5 in numerical models of specimens #1 and #5 for different excitation frequencies: (**a**) 10 kHz; (**b**) 23 kHz; and (**c**) 100 kHz.

**Table 2.** Time of flight of the wave packet based on the dispersion curves and the wave signals.


In addition to the analysis of the time signals, the guided wave fields representing the propagation of A0 mode for specimens #1 and #5 for a specific time instance 0.17 ms are shown in Figure 7. It is visible that the incident wave with the frequency of 10 kHz propagates faster in beam #1 compared with plate #5 (Figure 7a), what stays in the agreement with the comparison of dispersion curves. What is also important, the wave propagates through the three-layer medium as a single waveguide,

although the wave was excited only in the steel plate. It is possible because the A0 wavelength is large enough (271 mm), compared to the overall sample thickness (108 mm). The length of the A0 mode in beam #1 is significantly smaller (74 mm), but much greater than the thickness of the plate (6 mm). For the frequency equal to 23 kHz (Figure 7b) the wave packets in both specimens propagate with similar velocity. The distribution of the wave field trough the thickness of specimen #1 is not uniform, the amplitude decreases with the depth because the wavelength (107 mm) is slightly smaller than the specimen thickness. The interesting situation takes place when the excitation frequency increases to 100 kHz (Figure 7c). The wave propagation velocity in specimen #1 is smaller compared with specimen #5 (what is confirmed by the dispersion curves), but the wave propagates mainly in the steel plate and adhesive film. The reason is that the wavelength is now much smaller (22 mm) than the overall specimen thickness. For the concrete part, only the Rayleigh waves can be observed propagating at the interface between concrete and adhesive and the bottom boundary. The length of the wave propagating in plate #5 is comparable with the one present in the beam #1 and equal to 20 mm, which is also clearly visible in the snapshot. The analysis of the FEM simulations illustrating in Figure 7 allows concluding that the differences in the dispersion curves for both, three-layer and single-layer media are less pronounced for the higher frequency range (above about 120 kHz), because in this case, the wave propagates mainly in the steel plate, passing by the influence of the concrete part. Thus, the damage visualization in analyzed specimens would not be successful in this case. On the other hand, lower frequencies result in poorer image resolution. As the compromise, the excitation frequency equal to 100 kHz was chosen for further investigations aimed at damage detection and imaging.

**Figure 7.** Numerical wave fields in 2-D finite element method (FEM) models of specimens #1 and #5 (*t* = 0.17 ms) for different excitation frequencies: (**a**) 10 kHz; (**b**) 23 kHz; and (**c**) 100 kHz.
