# Mesoscale Modelling of the Mechanical Behavior of Metaconcretes
## 2 Description of the Numerical Model  
A custom Python script constructs the FEM using Abaqus (2017 version), automating the entire process from geometry creation to meshing, material assignment, assembly, and load application.  
The two-dimensional specimen has a height of 200 mm and a width of 100 mm. Consequently, the study randomly embedded circular resonators within this rectangular concrete matrix until the desired area fraction was reached. Each resonator comprises three regions: an internal steel core that provides high stiffness, a compliant annular rubber coating designed to introduce flexibility, and an interfacial transition zone (ITZ), which is explicitly modeled as a thin layer (0.1 mm) surrounding each resonator [48]. The ITZ represents a weak bonding interface between the concrete and the resonator, which captures local effects such as stress concentrations and potential debonding phenomena. A random positioning algorithm prevents resonator overlaps, thus ensuring a minimum separation distance between each other and specimen boundaries. A recursive verification routine iteratively reassigns positions whenever an overlap occurs. Figure 1 shows a schematic illustration of the resonator configuration.  
Each model component is assigned specific material properties to represent its mechanical response accurately. These models are based on the concrete damaged plasticity (CDP) model, which is used to capture material fracture. This constitutive model can simulate nonlinear behavior and damage evolution under compressive loading by accounting for both tensile cracking and compressive crushing. The CDP model provides a realistic representation of failure mechanisms in concrete-like materials. The CDP model represents concrete and ITZ, while the rubber coating and steel core are expressed as linear elastic materials.  
Instantiating the concrete sample with each resonator component--including the ITZ, rubber coating, and steel core--creates the assembly. The interaction between the different phases has been modeled with perfect adhesion using a tie contact approach, which allows for an accurate representation of the system when phases are in mutual contact [49]. Adopting a tie contact enables the independent meshing of each phase, thereby avoiding the need for node matching at the interfaces and simplifying model generation, particularly when the inclusions are randomly distributed. Because the tie constraint interpolates displacements without permitting separation or sliding, the associated computational cost remains low for two-dimensional analyses.  
For the loading scenario, a smooth displacement load is applied to the top edge of the concrete sample, while the bottom edge is fully fixed to simulate support. An explicit dynamics step, with automatically controlled time incrementation, is defined to capture the quasistatic compressive response, employing explicit integration throughout the simulation. According to the data summarized in Table 1, the complete model generation has been automated via Abaqus' Python scripting interface. This script sets key simulation parameters (e.g., material properties, boundary conditions, and load amplitudes), generates specimen geometry with randomly distributed inclusions, and performs meshing operations for each part. Moreover, CDP parameters are calibrated by comparing numerical predictions with experimental data [36], ensuring that the simulated response accurately reflects the behavior of the observed material.  
Figure 1: Schematic representation of the resonant inclusion modeled in the study plotted to: (**a**) Perspective view; (**b**) Plain view. Explanatory notes: The inclusion consists of a steel core (grey) of diameter \(D_{c}\), surrounded by a compliant rubber coating (orange) with the thickness \(t\) embedded within a concrete matrix; the ITZ, depicted in blue, explicitly models the weakened bonding interface; the outer diameter of the resonator, excluding the ITZ, is denoted by \(D_{a}\). Source: the authors’ own elaboration based on research design.