# Mesoscale Modelling of the Mechanical Behavior of Metaconcretes
## 3 Validation of the Proposed Model  
This section is devoted to validating the proposed model based on the experimental results reported in [36]. Simple compression tests were conducted on cylindrical high-strength concrete specimens, each measuring 20 cm in height and 5 cm in diameter. Two configurations were considered: the first consisted of a plain concrete specimen, whereas the second involved a concrete specimen containing 10.6% resonators. These resonators comprise a spherical steel core with a diameter of 15.4 mm, covered by a 2.3 mm-thick layer of rubber. The mechanical properties of the concrete were obtained from the work published by [PERSON] et al. [36]. According to that study, the concrete exhibits a Young's modulus of 25.42 GPa, a Poisson's ratio of 0.173, a density of 2183 kg/m\({}^{3}\), a compressive strength of 67.49 MPa, and a failure strain of 0.38%. Regarding the characterization of the ITZ's mechanical behavior, the same values used for the concrete were adopted, except for the strength, which was set to 50% of that of the concrete (33.75 MPa) [48]. An ITZ thickness of 0.1 mm was also considered [40; 50; 51; 52]. The properties of steel and rubber materials were assumed to behave linearly when modeling the resonators. A Young's modulus of 210 GPa, a Poisson's ratio of 0.3, and a 7800 kg/m\({}^{3}\) density were adopted for steel. A Young's modulus of 50 MPa, a Poisson's ratio of 0.5, and a 1300 kg/m\({}^{3}\) density were considered for the rubber. Four-node plane-stress elements with reduced integration (CPS4R) from the Abaqus/Explicit library were employed [49]. Figure 2 shows the stress-strain curves obtained from compression tests on concrete specimens, including the three specimens tested in earlier experiments [36], as well as the curve obtained from the implemented FEM. Consequently, the constitutive model parameters employed to delineate the concrete's mechanical behavior exhibit high congruence with the experimental outcomes. To quantitatively compare the numerical results with the experimental data, Table 2 provides a comparison of the mean values and standard deviations of [PERSON]'s modulus (\(E\)), and the compressive strength (\(f_{c}\)). [PERSON]'s modulus was calculated from the stress-strain curves through the following expression [53]:  
\[E=\frac{\sigma_{2}-\sigma_{1}}{\epsilon_{2}-0.00005}, \tag{1}\]  
where \(\sigma_{1}\) corresponds to the stress at a strain of 0.005%, and \(\sigma_{2}\) and \(\epsilon_{2}\) are the stress and strain at 40% of the failure stress. The results in Table 2 indicate that the CDP model, \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{5}{c}{**Geometrical Parameters and Mesh Configuration**} \\ \hline Feature & \multicolumn{3}{c}{Value/Description} \\ \hline Specim size & \multicolumn{3}{c}{200 mm (height) \(\times\) 100 mm (width)} \\ Inclusion geometry & \multicolumn{3}{c}{Circular resonators, randomly distributed} \\ Element size & \multicolumn{3}{c}{1 mm} \\ \hline \multicolumn{5}{c}{**Mechanical properties of the materials**} \\ \hline Material & \(E\) [GPa] & \(f_{c}\) [MPa] & \(\rho\) [kg/m\({}^{3}\)] & \(\
u\) [\(-\)] \\ \hline Concrete & 25.42 & 67.49 & 2183 & 0.173 \\ ITZ & 25.42 & 33.75 & 2183 & 0.173 \\ Steel core & 210 & - & 7800 & 0.3 \\ Rubber coating & 0.05 & - & 1300 & 0.5 \\ \hline \end{tabular} \({}^{1}\)Source: own elaboration based on data from [36].  
\end{table} 
Table 1: Geometrical parameters and mechanical properties of the materials \({}^{1}\). utilizing the established concrete parameters, exhibits a high degree of compatibility with the experimental data. Thus, the proposed model exhibits strong alignment with [PERSON]'s modulus and the strength parameters, with deviations of less than 4%. Three FEMs, each showing a distinct distribution of resonators within the concrete matrix, were created for the specimen with resonant inclusions. Following the execution of a mesh-size sensitivity study, an element size of 1.0 mm was selected.