# Two-Dimensional Mesoscale Finite Element Modeling of Concrete Damage and Failure
## 3 Automatic Generation of Cohesive Elements
Figure 7 shows the mean stress-strain curve obtained for each mesoscale specimen. The mean stress is calculated by dividing the total reaction forces of the left edge nodes by the specimen length at each step frame. As can be seen from Figure 7, the dependence of the obtained stress-strain curve is negligible for specimens with discretization steps of 1 mm and smaller than 1 mm. Figure 8 shows the final crack pattern for specimens with different mesh sizes. The propagation of macrocracks is illustrated by visualizing the so-called scaler damage variable (SDEG) of cohesive interface elements in Abaqus/CAE with a value greater than 0.9. It is worth noting that the numerical counterpart of the variable D in Equation (20) is the SDEG variable in Abaqus/CAE, so an SDEG value of zero for any CIE represents no damage and when it reaches 1, complete failure has occurred. As can be seen, the final macrocrack patterns for mesh sizes of 1 mm and smaller than 1 mm (see Figure 8c-f) are quite similar despite small differences. Considering the good agreement of the mean stress-strain curve and macrocrack pattern of the specimen with a mesh size of 1 mm with those for specimens with lower discretization levels, and taking into account the computational cost, a mesh size of 1 mm is chosen for the following finite element simulations. The convergence study in this paper highlights the critical role of fracture paths in determining the optimal mesh size, alongside stress-strain curves. While larger mesh sizes may yield comparable stress-strain curves, they can produce significantly different fracture paths. Mesh sizes below a certain value yielded consistent fracture paths and stress-strain curves, but further reduction significantly increased computational costs. Figure 7: Mean stress–strain curve for a concrete sample shown in Figure 8 with different levels of discretization. All mesoscale samples are subjected to a uniform displacement of 0.15 mm in the x-direction. Strain is determined by dividing the displacement at each step frame by the initial length of the specimen. ### Uniaxial Tension Simulation

After calibrating the input data parameters of the TSL and performing a mesh sensitivity analysis, mesoscale concrete simulations are conducted and validated against experimental results obtained from [52]. The simulations involve ten mesoscale samples with various spatial distributions and polyhedral aggregate particles, accounting for a 50% volume fraction. The samples are subjected to tensile test simulations, and the final stress-strain curves are obtained by averaging the results from each simulation. Figure 9 shows the stress-strain curve obtained from the uniaxial tensile simulations, along with the experimental data. It can be seen that the peak stress and softening behavior are close for the two curves. It should be noted, however, that a quantitative comparison is not possible here because these curves represent samples with completely different spatial distributions of aggregate particles. Figure 8: Cohesive fracture paths for a unique concrete sample with different discretization levels. Figure 9: Mean stress–strain curve under uniaxial tension. All mesoscale samples are subjected to a uniform displacement of 0.15 mm in the x-direction. Strain is determined by dividing the displacement at each step frame by the initial length of the specimen. ### Propagation of Microcracks and Formation of Macroracks

To illustrate the formation and propagation of microcracks leading to the formation of macroracks, the mesoscale concrete specimen with smoothed polyhedral aggregates with a total volume fraction of 50% without voids is utilized. As shown in Figure 10, the formation of macroracks in eight continuous phases is illustrated. The corresponding value of each stage within the mean stress-strain curve is depicted by marked points in Figure 9. To better visualize the generated microcracks, the displacement values are enlarged by a factor of 50.