Reinforced concrete consists of reinforcement bars and prestressing tendons in addition to the conglomerate of aggregates and pores embedded in a cement matrix. These heterogeneities influence to a high extent the ultrasonic wave propagation by causing a high attenuation due to scattering. Reflections and mode conversions at the specimen boundaries, cracks, aggregates, reinforcement bars and tendons cause a very complex wave field. Often it is not possible to detect other waves than the Pwave, that is why for the quantitative analysis only the arrival times of the Pwaves are used. More details about AE analysis of concrete can be found in various papers, e.g. [79].
For the numerical EFIT simulations the geometry of an existing prestressed concrete beam was chosen. The specimen was used for fatigue tests and was assigned for AE measurements as well. The goal of the simulations is to lead to a better understanding of the wave propagation process in the specimen and to investigate the influence of prestressing and reinforcement on wave propagation. Furthermore, synthetic data can be used to test and optimize the inversion algorithms for source location and source mechanism (moment tensor inversion).
The cross section in Fig. 4 (on the right) shows four steel reinforcement bars with diameters of 22 mm in the corners and a polyethylene tendon duct with inner diameter of 100 mm and a wall thickness of 3 mm, filled with mortar and 19 steel strands. The concrete has an Agrading curve with a maximum grain size of 16 mm. The porosity is 1 vol.% with a maximum pore size of 2 mm. The locations of various AE sources are marked by red asterisks in Fig. 4 (on the right). The black rectangles mark the positions of five sensors where the synthetic waveform data was calculated. The picture on the left shows the discrete 2D EFIT model of the specimen with thousands of grains and pores embedded in the cement matrix. Table 1 shows the material parameters that were used in the simulations.
Fig 4: Crosssection of the reinforced concrete specimen with reinforcement bars and tendon duct as used for the AE investigations. The left picture shows the 2D EFIT model. The picture on the right displays the locations of various AE sources (red asterisks) as well as the positions of the sensors (black rectangles).

Figure 4: Crosssection of the reinforced concrete specimen with reinforcement bars and tendon duct as used for the AE investigations. The left picture shows the 2D EFIT model. The picture on the right displays the locations of various AE sources (red asterisks) as well as the positions of the sensors (black rectangles).
Material

c_{L} [m/s]

c_{S} [m/s]

r [kg/m^{³}]

Cement matrix  3950  2250  2050

Gravel & sand aggregates (mean value)  4400  2500  2610

Steel reinforcement  5900  3200  7820

Polyethylene tendon duct  2300  1200  950

Table 1: Material parameters used for the EFIT calculations. 
The capability of EFIT for AE source modeling and wave propagation calculations is demonstrated by wave front pictures (Figs. 5 and 6) according to two of the four different source locations represented in Fig. 4 (on the right). One position is inside the cement matrix between the tendon duct and the right boundary of the specimen, another position is directly at the tendon/concrete interface. In order to show the effect of an adjacent boundary we used an isotropic source as described in Fig. 3 (first row). The implementation of other source mechanisms is straightforward as was demonstrated in section 3. The input pulse that was used had a maximum frequency of 250 kHz. The discretized model in Fig. 4 (on the left) consisted of 918 x 918 grid cells (440 x 440 mm^{2}, Dx = Dy = 479.8 µm). Moreover 3478 time steps were used (t_{max} = 200 µs, Dt = 57.5 ns).
Fig 5 (Slide Show):
2D EFIT simulation of elastic wave propagation caused by an isotropic AE source in the cement matrix. The wave front snapshots represent the absolute value of the particle velocity vector using a linear color scale. The wave field is shown in equidistant time intervals of 11.5 µs.

Fig. 5 shows the 2D EFIT simulation of elastic wave propagation caused by an isotropic source in the cement matrix. One can see that this type of source produces only pressure waves and that shear waves are generated later by mode conversion at internal and external boundaries. Additionally it is remarkable that the main parts of the elastic waves are diffracted around the tendon duct and only a small portion of the wave field is transmitted through the duct. This is most likely caused by the large impedance mismatch between concrete and the polyethylene wall (compare acoustic parameters in Tab. 1).
Fig 6 (Slide Show): 2D EFIT simulation of elastic wave propagation caused by an isotropic AE source at the tendon/concrete interface. The wave front snapshots represent the absolute value of the particle velocity vector using a linear color scale. The wave field is shown in equidistant time intervals of 11.5 µs.

Fig. 6 shows a numerical simulation where the isotropic source is located directly at the tendon/concrete interface. This causes the generation of shear waves in addition to the pressure waves. Moreover Lamb waves can be identified within the tendon wall. Similar to the previous case (Fig. 5) the body waves (P and S) are diffracted around the tendon and only a small portion of the waves is transmitted through the duct.
One can see from the wave front pictures that the tendon duct has a significant effect on the wave propagation and thus can influence the accuracy of source localization algorithms. By calculating the time signals of velocity or displacement at the positions of the sensors as exemplary shown in Fig. 7, the numerical simulations can systematically be used to optimize the inversion algorithms, for example by taking into account the location and geometry of the tendon ducts as well as the specific peculiarities of the wave propagation.
Fig 7: Time signals of normal velocity component at five different sensor positions as calculated by the numerical EFIT code. The corresponding wave propagation process is shown in Fig. 5.
