# Meso-Scale Simulation of Concrete Based on Fracture and Interaction Behavior
## 3 Calibration of Numerical Simulation  
Modeling concrete, which contains multiple materials and interfaces at the meso-scale, requires a large number of parameters. Few experiments can provide the corresponding parameters that can one-to-one match with those used in the numerical simulation presented in this paper. Moreover, the cohesive zone model has a mesh sensitivity problem, so calibration is necessary for obtaining a suitable set of parameters and mesh precision. In the calibration, it is found that the specimens with the same grading cumulative distribution but different aggregate distributions present different compressive properties, especially during the softening stage. Therefore, it is inadequate to include a single specimen to study the entire concrete performances. In order to conduct universal studies, ten specimens with different aggregate distributions are modeled randomly according to Equation (2) and (3). The number and proportion of aggregates included in the specimens are summarized in Table 1.  
To check the mesh dependence and computing efficiency, three models with the same aggregate distribution and different mesh precisions are tested. The approximate element sizes of the models are 1.5 mm, 1.0 mm, 0.5 mm (Figure 5). The number of bulk elements and cohesive elements are listed in Table 2. The time consumption of the models with element size of 1.0 mm and 0.5 mm are 2.5 and 11.6 times that of the model with 1.5 mm element size, respectively. \begin{table}
\begin{tabular}{c c c} \hline
**Aggregate Particle Diameter D (mm)** & **Proportion** & **Particle Number** \\ \hline \(12<D\leq 15\) & 5.36\% & 8–11 \\ \(9<D\leq 12\) & 8.83\% & 20–25 \\ \(6<D\leq 9\) & 11.74\% & 65–72 \\ \(2.5<D\leq 6\) & 19.18\% & 291–308 \\ \(D\leq 2.5\) & 35.69\% & / \\ \hline \end{tabular}
\end{table} 
Table 1: Aggregate gradation and particle number. \begin{table}
\begin{tabular}{c c c} \hline
**Approximate Elements Size (mm)** & **Bulk Element** & **Cohesive Element** \\ \hline
1.5 & 22,076 & 19,877 \\
1.0 & 50,890 & 44,357 \\
0.5 & 199,038 & 166,548 \\ \hline \end{tabular}
\end{table} 
Table 2: The bulk and cohesive elements number of the model with different mesh precision. Figure 4: Illustration of quadratic nominal stress criterion and mixed-mode fracture criterion. Figure 6 shows the stress-strain curves of the specimens with different element sizes. Comparing the tension and compression simulation results, it is shown that the compressive performance is more sensitive to mesh precision. It is indicated that the damage and fracture of cohesive elements lead to stress redistribution. With the increase of plastic deformation ability in the local area, the stress distribution becomes more uniform, which can make the particles stay at a higher average stress level. Hence, the increase of cohesive element number can contribute to higher ultimate compressive strength and higher residual strength in the softening stage. The mesh size of 1.0 mm is chosen for further experiments due to its good simulation accuracy and acceptable time consumption. Based on the outcomes of previous research and the simulation technique discussed in Section 2, the models are calibrated by considering interaction relations, material and interface parameters. Through the calibration, the frictional angle of 54.4\({}^{\circ}\) and the material parameters listed in Table 3 can make the models achieve satisfactory results, which are in close agreement with experimental results in both mechanical response and failure mode. In the following, the standard models used for analysis and parametric study are the models with ten different aggregate distributions and the calibrated parameters. Figure 5: The meshes with different approximate element sizes: (**a**) boundary constraint condition marked by the blue triangles and the zoom area marked by the red square; (**b**) approximate size of 1.5 mm; (**c**) approximate size of 1.0 mm; (**d**) approximate size of 0.5 mm.