# Mesoscale Modelling of the Mechanical Behavior of Metaconcretes
## 4 Results
### Effect of Resonator Dimensions
#### 4.2.2 Combined Effects of Core Size and Coating Thickness  
This subsection examines the combined influence of the rubber coating thickness surrounding the steel resonators and the corresponding variation in core size on the compressive mechanical behavior of MCs. Whilst analyzing this effect, numerical simulations were indeed conducted by varying the rubber thickness (\(t\)) while keeping the total metamaterial diameter constant (\(D_{a}=20\) mm) and maintaining a fixed resonator volume fraction of \(\rho=10.6\%\). This implies that the core diameter (\(D_{c}\)), defined as \(D_{c}=D_{a}-2t\), decreases as the rubber thickness increases. By modifying \(t\), the steel resonator's core size is also altered, which can significantly impact the composite material's stiffness distribution and mechanical performance. Five random spatial configurations of resonators were generated for each rubber thickness to account for variability in spatial distribution. The influence of these geometric variations was analyzed by measuring the compressive strength (\(f_{c}\)) and Young's modulus (\(E\)). Table 7 presents the mean values and SDs for different rubber thicknesses, highlighting the simultaneous changes in \(D_{c}\). As the rubber coating thickness increases (resulting in a reduction of \(D_{c}\)), a decrease in compressive strength is observed, while variations in Young's modulus remain relatively small. This suggests that the core size reduction, combined with the increase in rubber content, plays a critical role in the overall mechanical response. The SD values indicate the dispersion of measurements around the mean, providing insight into the consistency and reliability of the results. \begin{table}
\begin{tabular}{c c c} \hline \(D_{a}\) (mm) & \(E\) (GPa) & \(f_{c}\) (MPa) \\ \hline
37.4 & 19.15 \(\pm\) 0.26 & 36.87 \(\pm\) 2.78 \\
30.5 & 18.55 \(\pm\) 0.40 & 31.20 \(\pm\) 3.18 \\
26.5 & 17.39 \(\pm\) 0.30 & 30.48 \(\pm\) 0.54 \\
23.7 & 18.93 \(\pm\) 0.25 & 31.19 \(\pm\) 1.75 \\
21.6 & 18.04 \(\pm\) 0.51 & 31.58 \(\pm\) 2.23 \\
20.0 & 18.27 \(\pm\) 0.16 & 30.14 \(\pm\) 1.59 \\
18.7 & 17.66 \(\pm\) 0.32 & 28.16 \(\pm\) 0.13 \\
17.6 & 17.95 \(\pm\) 0.50 & 28.00 \(\pm\) 2.22 \\
16.8 & 17.23 \(\pm\) 0.21 & 23.28 \(\pm\) 1.40 \\ \hline \end{tabular} \({}^{\dagger}\) By the values of the mean and standard deviation of Young’s modulus (\(E\)) and compressive strength (\(f_{c}\)). Source: own calculation.  
\end{table} 
Table 6: Effect of resonator diameter (\(D_{a}\)) on mechanical properties \({}^{\ddagger}\). \begin{table}
\begin{tabular}{c c c c} \hline \(t\) (mm) & \(D_{c}\) (mm) & \(E\) (GPa) & \(f_{c}\) (MPa) \\ \hline
1.0 & 18.0 & 18.25 \(\pm\) 0.44 & 32.25 \(\pm\) 1.60 \\
2.0 & 16.0 & 17.91 \(\pm\) 0.48 & 31.63 \(\pm\) 2.00 \\
3.0 & 14.0 & 17.94 \(\pm\) 0.55 & 30.36 \(\pm\) 1.88 \\
4.0 & 12.0 & 17.49 \(\pm\) 0.40 & 29.18 \(\pm\) 2.78 \\
5.0 & 10.0 & 17.91 \(\pm\) 0.47 & 28.58 \(\pm\) 3.47 \\ \hline \end{tabular} \({}^{\dagger}\) By the values of the mean and standard deviation of [PERSON]’s modulus (\(E\)) and compressive strength (\(f_{c}\)). Source: own calculation.  
\end{table} 
Table 7: Effect of core size (\(D_{c}\)) and coating thickness (\(t\)) on mechanical properties \({}^{\ddagger}\). Figure 7 presents the normalized values of [PERSON]'s modulus and compressive strength to facilitate the interpretation of trends. The normalization was performed relative to the reference properties of plain concrete without resonators (\(E_{0}=25.42\) GPa and \(f_{c0}=67.49\) MPa, respectively). Normalized values enable a more straightforward comparison of relative changes across different rubber thicknesses while maintaining a dimensionless representation of mechanical performance. Figure 7 illustrates the relationship between rubber thickness, core size, and the normalized values of [PERSON]'s modulus and compressive strength.