# Numerical Study of Concrete: A Mesoscopic Scale Simulation Methodology
## 2 Materials and Methods
### Stage 1: Geometrical Model Generation  
To carry out the first stage of modeling, coarse aggregates were spherical in shape, and three phases were established: mortar, coarse aggregates, and the ITZ. The effect of aggregate shape has been studied by other authors [15; 16] using spherical, ellipsoidal, and polyhedral shapes while keeping the rest of the parameters fixed, reaching the conclusion that the type of crack is independent of the shape of the aggregate. It was also concluded that the load capacity of concrete containing spherical and ellipsoidal aggregates is between 1% and 3% higher than that which includes polyhedral aggregates, perhaps since non-smooth aggregates have more aggregate interface elements and because the smooth edges of spherical and ellipsoidal aggregates have a more benign stress distribution that delays the fracture process and increases tensile strength [15]. Thus, in this stage, a spherical aggregate shape is adopted. In principle, the ITZ was generated by subtracting the volume of two previously defined concentric spheres: the inner sphere corresponds to the equivalent diameter of the coarse aggregate, and the outer sphere is designed with a diameter equal to that of the coarse aggregate plus twice the thickness of the ITZ. The shell, obtained by the difference between the larger and smaller spherical volumes, would precisely be this weaker zone (ITZ) of the concrete that is generated around the coarse aggregate particles. The programming in MATLAB is carried out using the following input parameters:  
* The dimensions of the Representative Volume Element (RVE), which correspond to the dimensions of the specimens tested in the laboratory. * The diameters of the coarse aggregate particles are represented as spheres. * The mass percentage of each of the sizes of the coarse aggregate particles is considered in relation to the total mass. * The coarse aggregate-cement ratio. * The volume fractions of cement, aggregates, and water. * The density of concrete. The RVE generated in MATLAB can vary in size and shape, including options such as cylinder, prism, or cube. In the simulations conducted in this study, coarse aggregate particles with diameters of 4, 6, and 8 mm are utilized, comprising 8%, 15%, and 7% of the total mass, respectively [17]. Particles smaller than 4 mm are considered mortar. These volumes correspond to those outlined in the Fuller parabola for the same particle sizes. The MATLAB programming enables the generation of a random particle distribution within the RVE, aiming to statistically mimic that of real concrete. The positioning of each coarse aggregate particle within the RVE is achieved randomly, employing the take-and-place method [9; 18], which organizes the spheres (particles) based on their size, following the mass percentages established in the initial data. The number of particles of each diameter i that is introduced in the RVE is calculated by the expression:  
\[np_{i}=\frac{V\cdot p_{i}}{v_{i}}\bigg{[}1-c\bigg{(}\frac{1}{\rho_{c}}-\frac{r _{uv}}{\rho_{w}}\bigg{)}\bigg{]}, \tag{1}\]  
where \(V\) is the volume in m\({}^{3}\) of the RVE, \(p_{i}\) is the mass fraction of the coarse aggregate of diameter \(i\) with respect to the total mass, \(v_{i}\) is the aggregate volume in m\({}^{3}\), \(c\) is the amount of cement per m\({}^{3}\) of concrete, \(r_{uv}\) is the water-cement ratio, \(\rho_{c}\) is the density of cement (3150 kg/ m\({}^{3}\)), and \(\rho_{w}\) is the density of water. The value obtained from the previous expression (1) is rounded to ensure that the number of particles is an integer. Following this, a surplus of particles from each class is generated, exceeding the quantity needed to fill the RVE, ensuring it complies with the predetermined mass percentage. Subsequently, the RVE is populated. A Cartesian coordinate system is employed in the concrete sample, where the placement of a coarse aggregate particle within the specified concrete volume is determined by the coordinates of its center.