*2.3. Numerical Modeling*

The numerical simulations of the guided wave propagation in the considered specimens were conducted in Abaqus/Explicit software (ver 6.14, Dassault Systemes, Vélizy-Villacoublay, France) applying the finite element method. All structural elements (steel plate, concrete beam and adhesive film) were assumed to be independent parts bonded rigidly by surface to surface tie connection, ensuring compatibility of translational degrees of freedom at contacting nodes. The materials of modelled parts were supposed to meet the requirements of a homogeneous, isotropic and linearly elastic material model. The mechanical properties of each material are presented in Table 1. The transient dynamic analysis was conducted with the use of the central difference method. Total time of calculations was assumed as 2 ms for each simulation. The propagation of wave was analyzed by considering the behavior of models under the concentrated load with varying amplitude, applied in a specific node of a discretized structure. The results of the analyses were out-of-plane (vertical) velocity signals collected in a certain number of nodes.


**Table 1.** Material parameters for each material in numerical simulations.

Two kinds of models were prepared for different purposes, plane (2-D) and spatial (3-D). Two-dimensional models (example in Figure 4a) were prepared for specimens #1 and #5 to analyze the differences between guided wave fields in a single-layer plate and a three-layer medium. Two-dimensional four-node plane strain elements with reduced integration (CPE4R) were used. The element size was constant and equal to 1 <sup>×</sup> 1 mm2 throughout the whole model. This value satisfies the requirement of the appropriate mapping of wave behavior (at least 10 nodes for the length of the shortest considered wave) presented in [48]. The calculations were conducted with a fixed time step with a value of 10−<sup>7</sup> s that meets the recommendation of at least 20 integration points per cycle of the wave with the highest frequency of interest [49]. For initial analysis, the wave was excited at one end of the plate (point R1) and its signal was collected at another end (point R5) to determine the time-of-flight (TOF) of the wave through the length of the specimen. In the same simulation, guided wave field was saved for the whole model. The excitation had the form of a concentrated force varying in time in accordance with a 5-cycle tone burst with a center frequency of 10, 23, and 100 kHz (three different simulations). The plane models were also used to determine the numerical dispersion curves, similarly as in the experimental investigations (cf. Figure 3b). The input signal was applied in points R1 and R2 (independently) as a wave packet modulated from a single-cycle sine function with a different frequency (50, 100, 150, 200, and 250 kHz), resulting with ten consecutive simulations. The guided wave responses were obtained at 101 points distributed along the line with a length of 100 mm, giving 1 mm spacing between each point. It is worth noting that the measurement path was oriented perpendicularly to the one in experimental investigations. However, the results of both approaches could be compared, as long the materials used were modeled as isotropic and only the initial parts of signals were further analyzed. Three-dimensional models (example in Figure 4b) were prepared for specimens #1–#5 to verify the results of experimental measurements. Eight-node linear brick elements with reduced integration (C3D8R) and a global size of 2 mm were applied. The integration step

was 2 <sup>×</sup> 10−<sup>7</sup> s. Both element size and time step met the requirements from [48,49], described above. The scheme of excitation and collection of signals was identical with the experimental measurements (cf. Figure 3c). The wave (Hanning windowed five-cycle sine function with a carrier frequency of 100 kHz) was excited in point R3 or R4 and collected in 2323 points covering the area of 88 <sup>×</sup> 400 mm2.

**Figure 4.** Numerical models for specimen #1: (**a**) 2-D plane model and (**b**) 3-D spatial model.
