## 3 Validation
### Numerical Simulations  
The numerical simulations conducted in this work for validation purposes focused on a high strength concrete with the following properties: \(f_{c}=105\) MPa, \(E=53.4\) GPa, maximum aggregate size = 20 mm, and \(G_{IC}=1.52\times 10^{-4}\) MN/m. These concrete parameters were obtained from [42] and were suitable for performing the numerical simulations using the present model (as discussed in Section 2). It is worth noting that the mechanical characteristics and maximum aggregate size of this particular concrete differed significantly from those used in the simulations [12] from which Equation (2) was derived. Therefore, if Equation (8) was validated for this new high-strength concrete, it could be considered valid for a wide range of typical concretes. Figure 3 displays both frontal and 3D perspectives of the finite element mesh utilized in the simulation, pinpointing the locations where the prestressing force and imposed displacements were applied. The mesh comprised 19,564 tetrahedral solid elements with linear interpolation. The simulation of the DCB test involved several boundary conditions to accurately represent its behavior. These conditions included the restriction of displacements along the X axis within the YZ plane, along the Y axis within the XZ plane, and along the Z axis at the central nodes in the XZ plane. Additionally, there were restrictions on Z axis displacements at nodes where prescribed displacements were applied. Moreover, the simulation incorporated the application of prescribed forces specifically in the Y direction, exerted on the elements situated on the specimen's bottom surface. The Monte Carlo simulation consisted of the execution of 30 independent finite element analyses. As shown in [12], this number of MC samples was sufficient to produce a consistent outcome concerning the variability of the average curve from the numerical simulations. For this level of mesh refinement, Monte Carlo simulations employing 30 or more finite element analyses did not exhibit significant variability in the average curve. The loading force versus notched opening displacement curves obtained from the Monte Carlo simulation results are presented in Figure 4. These numerical curves were then compared with the experimental data obtained from [42] for a comprehensive evaluation. Additionally, Figure 5 provides an example of the crack pattern obtained from the numerical simulations, with cracked elements represented in red and uncracked elements in blue. Upon examining the graph presented in Figure 4, several significant observations emerge. Firstly, the peak loads evident in the numerical curves consistently exhibited lower values in contrast with those observed in the experimental curve. This aligned with the findings reported in [12]. A depiction of the Monte Carlo (MC) outcome conducted for 100 samples is presented in Figure 6, further supporting this observation. Figure 4: Loading force versus notched opening displacement curves — numerical and experimental results. Figure 3: 3D Finite element mesh of the DCB specimen—frontal and 3D view. As explained in detail in [12], this difference can be attributed to the fact that the proposed model primarily focused on macrorack propagation rather than the localization process, which was responsible for the peak load. Therefore, for a valid comparison, it was important to consider the behavior of the descending branch of the curves, representing macrocrack propagation. Moreover, for simplification purposes, the numerical modeling of the notch tip was represented as a line. In contrast, the actual DCB specimen featured a notch tip thickness of 0.5 mm, as depicted in Figure 3. This discrepancy led to higher stress concentrations at the numerical front tip compared with the experimental values. Additionally, it is noteworthy that, after a small notch opening, the experimental curve aligned within the range of the numerical curves. This observation concurred with the findings detailed in [12].