# Mesoscale Modelling of the Mechanical Behavior of Metaconcretes
## 4 Results
### Effect of the Resonator Ratio  
This subsection analyzes the effect of varying the resonator ratio, \(\rho\), within the concrete matrix. This research study addressed different resonator contents ranging from 1.5% to 31.8%. For each case, five FEMs were created, each with a distinct spatial distribution of the resonators, and the corresponding stress-strain curves were obtained. From these curves, [PERSON]'s modulus and the compressive strength were determined.  
Table 4 presents the mean values, SDs of [PERSON]'s modulus and compressive strength, and the number of resonators considered for each resonator ratio. As can be observed, there is little dispersion in the results due to the resonators' spatial distribution, as evidenced by the low standard deviation (SD) values below 3% for both [PERSON]'s modulus and compressive strength.  
Figure 5 shows the mean normalized values of [PERSON]'s modulus, compressive strength, and crack patterns as a function of the resonator content. The [PERSON]'s modulus and compressive strength of the concrete normalized these mean values without resonators (\(E_{0}\) = 25.42 GPa and \(f_{c0}\) = 67.49 MPa, respectively). In this figure, it can be observed that, as expected, both the compressive strength and [PERSON]'s modulus decrease with an increasing resonator ratio. Increasing the resonator ratio leads to a marked reduction in [PERSON]'s modulus, which is consistent with the predictions of a mixing law [54; 55]. Moreover, incorporating a single resonator leads to a reduction of more than 25% in compressive  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**\# Resonators** & \(\rho\) **(\%)** & \(E\) **(GPa)** & \(f_{c}\) **(MPa)** \\ \hline
0 & 0.00 & 25.42 & 67.49 \\
1 & 1.51 & 23.58 \(\pm\) 0.05 & 47.10 \(\pm\) 0.68 \\
3 & 4.54 & 21.59 \(\pm\) 0.10 & 39.13 \(\pm\) 1.11 \\
7 & 10.60 & 18.27 \(\pm\) 0.16 & 30.14 \(\pm\) 1.59 \\
11 & 16.66 & 14.62 \(\pm\) 0.31 & 22.10 \(\pm\) 2.03 \\
14 & 21.20 & 12.66 \(\pm\) 0.28 & 18.01 \(\pm\) 0.95 \\
18 & 27.26 & 10.61 \(\pm\) 0.31 & 15.55 \(\pm\) 1.26 \\
21 & 31.80 & 8.89 \(\pm\) 0.24 & 14.58 \(\pm\) 0.74 \\ \hline \hline \end{tabular} \({}^{\dagger}\)Concerning the mean and standard deviation of [PERSON]’s modulus (\(E\)) and compressive strength (\(f_{c}\)). Source: own elaboration.  
\end{table}
Table 4: Effect of resonator ratios (\(\rho\)) on mechanical properties \({}^{\dagger}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Experimental Data [36]** & **Numerical Results** \\ \hline \(E_{0}\) [GPa] & 15.34 \(\pm\) 1.91 & 18.27 \(\pm\) 0.16 \\ \(f_{c0}\) [MPa] & 29.98 \(\pm\) 1.81 & 30.14 \(\pm\) 1.59 \\ \hline \hline \end{tabular} \({}^{\dagger}\)Assuming concrete specimens with a 10.6% resonator ratio, contrasting the experimental data with the proposed numerical model. Source: own elaboration.  
\end{table}
Table 3: Comparison of [PERSON]’s modulus (\(E\)), and the compressive strength (\(f_{c}\)) \({}^{\dagger}\).  
strength. Due to the addition of further resonators, the compressive strength approaches an asymptotic value, reaching approximately 20% of the compressive strength of concrete without resonators. This trend is also evident in rubberized concrete [56], where the inclusion of a small proportion of rubber results in a significant decrease in compressive strength; however, further increases in rubber content lead to an asymptotic increase in compressive strength. Observation of the cracking patterns reveals that cracks initiate at the resonators, preferentially at their lower and/or upper extremities. These cracks follow predominantly vertical trajectories, consistent with the experimental observations reported in [36]. The existence of connections among resonators suggests that the crack paths become more erratic as the resonator ratio increases.