*5.1. Laboratory Beams*

A diagram of the beams is shown in Figure 18. Their dimensions were 10 cm × 10 cm × 50 cm and they were made of ready market mixture of concrete with a mean compression strength of *f*cm,cube = 48 MPa after 28 days. The maximum aggregate diameter was *d*a max = 8 mm. The longitudinal lower reinforcement consisted of two bars of 8 mm in diameter, made of steel with characteristic yield point declared by the manufacturer of *f*yk = 400 MPa. The studies consisted in the measurement of the time of propagation of the longitudinal ultrasound wave using a Pundit-Lab tester and transreceiver heads with a frequency of 250 kHz. The coupling of the heads and element was provided by special gel for ultrasonic tests. The adopted system of transmitting/receiving points distant from each other by Δ<sup>p</sup> = 10 cm is shown in Figure 18. Tomographic rays were assumed between the opposite points and lying diagonally at an angle of 45◦ and 135◦ in relation to the beam axis (Figure 18). The longitudinal vertical section of the beams for tomographic imaging was located in the middle between the reinforcement bars. For comparison, the ultrasound tests were carried out in two stages: before and after the first load-unload cycle in static three-point bending (Figure 19). The first load stage was finished at the moment when the first cracks appeared, controlling the registered load (*P*) and deflections at the centre of the span (*u*), i.e., until the slope change occurs in the function *P* − *u*. After the second stage of ultrasonic testing, the beams were bent until their load capacity was exhausted. The obtained values of the cracking (*P*cr) and maximum (*P*max) loads are summarized in Table 7.

**Figure 19.** (**a**) Beam loading scheme. (**b**) Illustrative pictures of beam No. 3 before and after an action of cracking load.

**Table 7.** Cracking and maximal loads of the lab-beams.


Taking into account the measured longitudinal wave propagation times and basic frequency of ultrasonic pulses, the average wavelengths for beam No. 1, 2, and 3 were, respectively, ~1.7 cm, ~2.0 cm, and ~2.1 cm before the loading and ~1.7 cm, ~1.8 cm, and ~2.0 cm after the loading. At the same time, it allowed satisfying the basic requirement described in ASTM D2845-08 regarding the selection of frequency so that measurable longitudinal ultrasonic waves could be generated in the samples, i.e.,: dominant wavelength ≥ 3× the average grain size equal to ~4 mm.

The first visible crack appeared in each case in the middle of the beam span as perpendicular to the beam axis (Figure 19). Measured and interpolated longitudinal wave propagation times *t*path int,*<sup>i</sup>* are shown in Figure 20 where interpolated times are determined using a cubic Hermite spline. The diagrams also show *t*ray approx,*<sup>i</sup>* determined according to relation (28) with β = βopt according to (24), (25). In Figure 20, by comparing wave propagation times before and after the loading, the evolution of cracks can be clearly observed and their location initially made in the sections where these times have increased the most. It can also be seen that not taking into account the increase in the propagation time of ultrasonic pulses along the straight rays compared to the times measured for the fastest paths would lead to their underestimation of approximately 5–8% in the most damaged areas of the beams. Figure 21 shows an example of a recorded signal by the receiving head together with a reading the time of longitudinal wave propagation. In Figure 21, the signal caused by longitudinal wave propagation is visible first, and, a moment later, the signal connected with the propagation of transverse and Rayleigh waves of much higher amplitude can be noticed, but without the possibility of precise distinction of the initial moment of their registration. This was also the main reason why the authors decided to use longitudinal waves in their studies, taking into account the capabilities of their research equipment.

**Figure 20.** Propagation times *t*path int,*<sup>i</sup>* and *t*ray approx,*<sup>i</sup>* between the transmitting/receiving points in the beams before and after an action of cracking load: (**a**) No. 1, (**b**) No. 2, and (**c**) No. 3. The results are presented as a function of the position of the transmitting points: *t*path int,*<sup>i</sup>* a grey line with nodes marked with circles, and *t*ray approx,*<sup>i</sup>* a black, dashed line. The diagrams on the left refer to the paths connecting the opposite points and those on the right to the points lying diagonally at an angle of 45◦ and 135◦ to each other in relation to the axis *x*.

**Figure 21.** Example of signal recorded by the receiving head (transmitting point No. 4 and receiving point No. 3 in beam No. 3): (**a**) before load; (**b**) after an action of cracking load.

Figure 22 shows the tomographic reconstructions of longitudinal wave velocity maps in the vertical longitudinal section of the beams. Reconstructions were determined by a randomized Kaczmarz method in accordance with the information presented in point 2, where for *c*L ref the maximum measured mean speed of the longitudinal wave over all rays before loading was adopted (4547 m/s, 5377 m/s, and 5587 m/s for beams No. 1, No. 2, and No. 3, respectively). The resolution of δ<sup>1</sup> × δ<sup>2</sup> = 2 cm × 2 cm was applied and the arrangement of rays as in Figure 18 with addition of rays between real rays connecting fictitious transmitting/receiving points at a distance of every 1 cm. The results are shown only in the middle area of the beam section, separated by a red dashed line through which all types of rays passed due to their inclination. The maps presented here are calculated on the basis of Equation (3) with propagation times *t*ray,*<sup>i</sup>* = *t*ray approx,*<sup>i</sup>* in accordance with (28) for the paths connecting the opposite points and *t*ray,*<sup>i</sup>* = *t*path int,*<sup>i</sup>* for the diagonal paths. For this purpose, the values of *t*path int,*<sup>i</sup>* are taken as shown in Figure 20. Optimal coefficients β necessary for the determination of *t*ray approx,*<sup>i</sup>* were calculated in accordance with (24) and (25) and their values are summarized in Table 8. The table also shows the values of *d*cL ray max determined from Equation (29).


**Table 8.** βopt calculated on the basis of data from Figure 20 in case of the opposite transmitting/receiving points and *d*cL ray max according to Equation (29) before and after an action of cracking load.

Figure 22 shows clearly formed elastically degraded zones caused by load *P* = *P*cr. Because of the static scheme of the bent beams, they were created in the middle of their span in the lower part of the cross-section. Based on Equation (20), the maximal tangential changes of Young's moduli, defined by the ratio of min(*E*D/*E*0) at the level 0.81, 0.50, and 0.68 in beam No. 1, No. 2, and No. 3, respectively, can be estimated for these zones. In turn, their widths in these places is within the range of 16–18 cm. They are very close to the width of the localized elastically degraded zones which were calculated theoretically in point 3. For the ratio min(*E*D/*E*0) in the 0.5–0.8 interval, this corresponds to the width of the damaged area from approximately 17 cm to 19 cm, which can be read from Figure 5a or Figure 6a. This result indirectly pre-confirms the validity of the identification method for the internal length

*l*<sup>c</sup> of concrete proposed in point 3. However, these preliminary out-comes need necessarily further intense experimental verification. In addition, it can be seen in Figure 22 that, under real conditions, the heterogeneity of the concrete itself can have a non-negligible effect on the results, as can be seen in the reconstructions of wave velocity distributions before the loading the beams. This is also evidenced by the determined values of *d*cL ray max at this stage of the study (Table 8). In turn, after the bending moment load initiating the appearance of the first cracks, the values of *d*cL ray max from Table 8 show that, in the investigated case, not taking into account deflections of the fastest propagation paths may lead to overestimation of the velocity of longitudinal waves on average by approximately 5–9%.

**Figure 22.** Tomographic reconstructions of the distribution of wave velocity according to Equation (3) in the longitudinal section of the beam before and after an action of the cracking load: (**a**) No. 1, (**b**) No. 2, and (**c**) No. 3 (red dotted lines—explanation in the text).

Another interesting issue that can be seen is that, despite the same period of care in the water of all beams, the speed of longitudinal waves in the intact configuration in beam No. 1 (stored an additional 1 week in the room conditions and isolation) was lower by about 20% if compared to the speed in beams No. 2 and 3 (which were tested right after removing from water). The explanation for this may be the fact of the phenomenon of self-drying of young concrete as a result of hydration processes [54]. On the other hand, as predicted by poromechanics [55], the initial tangent Young's modulus of the porous material and the Poisson's ratio in the state of full saturation is higher than in the dry state, which may result in a corresponding decrease in longitudinal wave velocity. In the case of cement matrix materials, such changes in Young's modulus and Poisson's ratio were measured by means of static tests among others in works [56–58]. For example, the Young's modulus of concrete at the age of 51 days with a mean compressive strength *f*cm,cube = 64.6 MPa (in the state of water saturation) varied from 47.2 GPa to 45.1 GPa and the Poisson's ratio from 0.25 to 0.15 at the transition from the saturation with water to a moisture concentration reduced by approximately 2.2% by weight [58]. Assuming proportional changes for dynamic values of this parameters and taking into account the formula expressing the speed of longitudinal waves:

$$\text{cl}\_{\text{L}} = \sqrt{\frac{E\_0(1-\nu)}{\rho(1-2\nu)(1+\nu)}}\,\text{}\,\text{}\,\text{}$$

where: ν—Poisson's ratio [-], ρ—density (kg/m3), its relative decrease with the quoted range of changes in *E*0, ν and density would be about 8%. The effect can be further enhanced by micro-cracks in concrete arising as a result of its autogenic and/or moisture shrinkage [54] (the latter in the case of absence of drying protection). The comparison of preliminary results from beams No. 1, 2, and 3 presented in this paper obviously requires further testing on a larger number of samples. Nevertheless, it can be unequivocally stated that obtaining reliable, reference longitudinal wave speed, necessary to assess the scale of damage evolution in tomographic tests, must always be determined for concrete without damage of the same composition, and which is stored in the same conditions as the assessed concrete. In practice, this may be the maximal speed determined at the time of the test on the concrete structural member in a place where there are no defects, or on a sample without defects taken from the member. It should be emphasized in the light of outcomes of the tested beams for which only self-drying concrete and extending the period before the tomographic investigation by 7 days changed the reference speed by approximately 20%.
