*4.3. Determining the Elastic Modulus of Concrete Plates*

To compare the numerical results with theoretical predictions, the elastic modulus was estimated in two ways. First, the shape of dispersion curves was reconstructed based on numerical results. Secondly, it was calculated based on known proportions between aggregate particles and mortar matrix.

The elastic modulus identification procedure was developed in a MATLAB environment using function fminsearch. The procedure of curve shape reconstruction can be summarized in four steps. In the first step, the data obtained from the FEM analysis were transformed in the wavenumber–frequency domain using a 2D-FFT-based algorithm. Further considerations were conducted for the limited representative area of data (Figure 10a). In our case, due to the use of excitation in the form of wave packets with central frequencies of 25, 50, 100, and 150 kHz, the frequency range, which was taken into account, was 0–250 kHz and the corresponding wavenumber range was 0–1000 m−<sup>1</sup> . To optimize the calculation process, the 2D-FFT results were normalized with respect to the maximum amplitude value for each frequency:

$$\hat{Y}(k\_{\dot{\nu}}\omega\_{\dot{l}}) = \frac{Y(k\_{\dot{\nu}}\omega\_{\dot{l}})}{\max\{Y(\omega\_{\dot{l}})\}}\tag{20}$$

The normalization process excluded the unequal influence of the individual frequencies. Moreover, there was no need to use weight functions. Examples of normalized data are shown in Figure 11b. Normalization was the last stage of data preparation. In the next step, the procedure for determining the elastic modulus was initiated. In the first step, the dependence for the first antisymmetric Lamb mode was calculated for the pre-established Young modulus. The determined values creating a dispersion curve were considered as the sets of *n* pairs of numbers:

$$K = \left[ (\Delta\omega, k(\Delta\omega)), (\Delta\Delta\omega, k(\Delta\Delta\omega)), \dots, (n\Delta\omega, k(n\Delta\omega)) \right] = \left[ (\Delta\omega, k\_1), (\Delta\Delta\omega, k\_2), \dots, (n\Delta\omega, k\_n) \right] \tag{21}$$

**Figure 11.** The Young modulus determining algorithm: (**a**) The 2D-FFT map and the data area selected for further analysis; (**b**) results of the normalization; (**c**) visualization of 2D-FFT interpolation along the dispersion curve; (**d**) comparison of interpolation results for a pre-selected and finally appointed Young modulus values.

 

Next, the standardized 2D-FFT values for particular angular frequencies were interpolated, forming a second set of data (Figure 10c,d):

$$\hat{Y}\_{\text{int}} = \langle \hat{Y}(\Delta\omega, k\_1), \hat{Y}(2\Delta\omega, k\_2), \dots, \hat{Y}(n\Delta\omega, k\_n) \rangle = \langle \hat{Y}\_1, \hat{Y}\_2, \dots, \hat{Y}\_{\text{int}} \rangle \tag{22}$$

Finally, the value of the following function was calculated:

$$\hat{F} = \frac{1}{n\sum\_{i}^{n} \hat{Y}\_{i}^{2}} \tag{23}$$

The procedure was repeated for different values of elastic moduli until the function *F*ˆ reaches a minimum value. The minimum value of the function *F*ˆ indicated that the dispersion curve that analytically determined the best coincides with the dispersion curve visible in the map. This procedure was tested first for the homogeneous material with parameters *E* = 26 GPa, *v* = 0.2, and ρ = 2100 kg/m<sup>3</sup> . The elastic modulus determined based on numerical results was 25.92 GPa and the percentage error was 0.309%, which is a satisfactory consistency of results. The analytical dispersion curves for the finally determined Young modulus were imposed on the numerical maps by red dashed lines (Figure 10), while the corresponding values of Young's modulus denoted as *E DC <sup>c</sup>* are summarized in Table 3.


**Table 3.** Parameters of numerical models.

The second stage of the analysis involved the theoretical calculation of Young's modulus. There are several theoretical models, which allow the elastic modulus of two-phase heterogeneous materials to be estimated. Two most common approaches were proposed by Voigt and Reuss. The use of either of these two models requires knowledge of the modulus of elasticity of mortar and aggregate and the volume of aggregates. According to the Reuss model, the elastic modulus is calculated in the following way:

$$E\_{\mathbb{C}}^{R} = \frac{E\_m}{1 + \left(\frac{E\_m}{E\_a} - 1\right)V\_a} \tag{24}$$

The overall elastic modulus of concrete by Voigt is:

$$E\_{\mathbb{C}}^V = E\_m \left( 1 + \left( \frac{E\_a}{E\_m} - 1 \right) V\_a \right) \tag{25}$$

where *E<sup>m</sup>* and *E<sup>a</sup>* are elastic moduli of mortar and aggregate, respectively, and *V<sup>a</sup>* is the volume fraction of aggregate. The elastic modulus calculated for all nine models is summarized in Table 3. Additionally, the result for the homogeneous concrete model was added for comparison. It can be seen that if *E<sup>a</sup>* > *Em*, the Voigt model always predicts higher values of the elastic modulus than the Reuss model. The experimental study presented in previously published papers showed that these two models usually define the upper and lower bound of the concrete elastic modulus and the exact value usually lies between their predictions [27]. Indeed, the modulus value predicted on the basis of dispersion

curves *E DC c* lies between theoretically determined boundary values. The differences between particular results, as well as the percentage errors, are reported in Table 4.


**Table 4.** Differences between theoretical and numerical results.

It can be seen that Young's modulus calculated according to both theoretical models clearly increased with the volume fraction of aggregate particles. The results obtained using the dispersion curves do not show the same increasing tendency. As expected, the elastic modulus was found to be the highest for models A3, B3, and C3, but the value of *E DC <sup>c</sup>* was higher for model A1 than for A2, and that for model C1 was higher than that for C2. This means that the aggregate presence affected wave propagation velocity. Moreover, the wave-aggregate interactions could affect the signal characteristics, which, in turn, resulted in a change in the shape of the dispersion curves.

Comparing the values of percentage errors reported in Table 4, it is clearly visible that the theoretical Reuss models are better suited to numerical results obtained by dispersion curves' reconstruction. The average percentage error for the Reuss model is 5.714%, while for the Voigt model, it is 9.633%. The greatest differences were reported for models with an aggregate ratio of 40%: 8.614% for A3, 9.636% for B3, and 13.943% for C3. One can conclude that the discrepancy between theoretical and experimental results clearly increases with the number of scatterers, but also with their size. ห െ ห ቚ ି ቚ⋅% ห െ ห <sup>ቚ</sup> ି ቚ⋅%

The presented results indicate that the heterogeneity of concrete influences wave propagation characteristics. The elastic modulus estimated using most common theoretical models differ from the modulus estimated based on dispersion curves. Meanwhile, the difference in modulus values leads to discrepancies in wave velocity estimation, which is particularly important if the wave velocity is used as an indicative parameter in the diagnostic process. To illustrate the differences in wave velocity, the Lamb dispersion equations were solved for model C3, for which the highest error was noted. Figure 12 contains a comparison of the first symmetric and antisymmetric modes. Disregarding the impact of the concrete mesostructure may lead to incorrect velocity determination—for some frequencies, the discrepancy may reach over 400 m/s.

**Figure 12.** Comparison of (**a**) antisymmetric and (**b**) symmetric modes traced for various values of Young's modulus.
