# Mesoscale Modelling of the Mechanical Behavior of Metaconcretes
## 4 Results
### Effect of Resonator Dimensions  
#### 4.2.1 Combined Effect of Resonator Diameter and Coating Thickness  
Resonators are typically designed to yield a specific natural frequency through their dimensions. This target frequency can be estimated using the following equation [17,18]:  
\[f(\mathrm{Hz})=\frac{1}{2\pi}\sqrt{\frac{3}{2}\frac{E_{r}}{\rho_{c}tR_{c}}}, \tag{2}\]  
where \(\rho_{c}\) is the density of the core material, \(R_{c}\) is the radius of the core, \(t\) is the thickness of the coating, and \(E_{r}\) is the Young's modulus of the rubber. Equation (2) provides an estimated resonant frequency that serves as a reference for examining wave transmission behavior. For the resonators considered in this study (see Section 3), the target resonant frequency is 3696 Hz. In this section, we investigate the effect of resonator size on the mechanical properties of the concrete. To this end, a series of resonator sizes were evaluated, all designed to achieve a resonant frequency of 3696 Hz (calculated from Equation (2)) while maintaining a constant resonator ratio of 10.6%. Table 5 summarizes the rubber coating thickness, steel core diameter, overall resonator diameter, and the number of resonators within the concrete matrix that were considered in this analysis. Figure 5: Mean normalized values of [PERSON]’s modulus (\(E/E_{0}\)) and compressive strength (\(f_{c}/f_{c0}\)), along with the corresponding cracking patterns, as a function of the resonator ratio (\(\rho\)). Source: own elaboration from numerical calculations. Figure 6 displays the meaning of normalized values of Young's modulus and compressive strength as functions of the resonator diameter (\(D_{a}\)) and the coating thickness (\(t\)). As in previous cases, these values are normalized for specimens without resonators. The results indicate that the variation in [PERSON]'s modulus is below 10%, while the compressive strength varies by approximately 15-20%. The data suggest that employing larger resonators with thinner coatings is more advantageous. Additionally, the figure includes representative cracking patterns. The observed patterns indicate that cracks initiate at the interface between the concrete and the resonator and tend to propagate along pathways that connect adjacent resonators. As the number of resonators increases, these preferential paths multiply, reducing the effective load-bearing area and thereby explaining the observed decrease in compressive strength. Table 6 presents the mean values and SDs of [PERSON]'s modulus and compressive strength for various resonator diameters. This means that both [PERSON]'s modulus and compressive strength decrease as the resonator diameter decreases. Moreover, compressive strength exhibits a higher standard deviation--approximately 10%--compared to about 2% for [PERSON]'s modulus. This suggests that the spatial distribution of resonators has a more significant influence on compressive strength than on [PERSON]'s modulus. Figure 6: Mean normalized values of [PERSON]’s modulus (\(E/E_{0}\)) and compressive strength (\(f_{c}/f_{c0}\)), along with the corresponding cracking patterns, as a function of the resonator ratio (\(\rho\)). Source: own elaboration. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**\# Resonators** & \(D_{c}\) **(mm)** & \(t\) **(mm)** & \(D_{a}\) **(mm)** \\ \hline
2 & 35.4 & 1.0 & 37.4 \\
3 & 28.0 & 1.3 & 30.5 \\
4 & 23.4 & 1.5 & 26.5 \\
5 & 20.2 & 1.8 & 23.7 \\
6 & 17.6 & 2.0 & 21.6 \\
7 & 15.4 & 2.3 & 20.0 \\
8 & 13.4 & 2.6 & 18.7 \\
10 & 8.4 & 4.2 & 16.8 \\ \hline \hline \end{tabular} \({}^{\dagger}\) The table lists the steel core diameter (\(D_{c}\)), the coating thickness (\(t\)), the overall resonator diameter (\(D_{a}\)), and the number of resonators incorporated in the concrete matrix. Source: own calculation.  
\end{table} 
Table 5: Resonator Dimensions and Quantities for a Fixed Resonant Frequency (3696 Hz) \({}^{\dagger}\).