# Three-Dimensional Probabilistic Semi-Explicit Cracking Model for Concrete Structures
## 4 Numerical Results
### Analysis of the Results  
At the conclusion of the second step of the strategy, it was observed that \(\sigma_{L}(G_{IC})\) increases with the mesh refinement. This outcome aligns with the evolution of the concrete tensile strength standard deviation correlated to the heterogeneity degree [15; 22]. Utilizing the optimal parameters (Section 4.3) determined for each \(r_{e}^{mean}\) value (Table 3), a function has been formulated to estimate \(\sigma_{L}(G_{IC})\) corresponding to the heterogeneity degree of each finite element. This function, described in Equation (17), has been extrapolated to encompass all elements included in the three mesh refinements. \[\sigma_{L}(G_{IC})=(A\ln(r_{e})+B)\times\mu_{L}(G_{IC}) \tag{17}\]  
where \(A=-8.538\), \(B=70.88\), and \(\mu_{L}(G_{IC})\) is the mean value of \(G_{IC}\). Figure 12 illustrates the behavior of the estimated function concerning the coefficient of variation of \(G_{IC}\), represented as \(\sigma_{L}(G_{IC})/\mu_{L}(G_{IC})\). A focused view of the initial section of the function domain (encompassing the heterogeneity degree of all elements at the central part of the DCB specimen, \(r_{e}\in[3,350]\)) is presented in Figure 12a, while Figure 12b showcases the complete domain of the proposed function. Analyzing the extrapolated function from Equation (17) raises caution due to the inherent risks associated with extrapolating from only three data points. Therefore, it is imperative to approach this function with caution, emphasizing the need for future validation. To initiate the validation process using available experimental information, the following pertinent points merit examination:  
Figure 11: Comparison between the macrocrack pattern observed on both surfaces and in a cut view normal to axis Y (showing inside the specimen)—Mesh 3. 1. If Equation (17) is taken into account, it evident that \(\sigma_{L}(G_{IC})\) equals zero at a value of \(r_{e}\approx 4030\). 2. In the work developed by [PERSON] [28], acoustic emission techniques were employed to assess the dimensions of the process zone at the macrocrack tip. The findings revealed a process zone volume approximately 3600 cm\({}^{3}\) (10 cm high, 10 cm wide, 30 cm long). Considering the maximum aggregate volume of 1.13 cm\({}^{3}\) (diameter of 12 mm, Table 1), the ratio of the process zone volume (\(V_{FPZ}\)) to the larger aggregate volume (\(V_{a}\)) is \(V_{FPZ}/V_{a}\approx 3185\). 3. Additionally, in the study of [PERSON] [28], during the macrocrack propagation (excluding 2 m in length), it was evaluated that \(\sigma(G_{IC})=0.073\times 10^{-4}\) MN\(/\)m for \(\mu(G_{IC})=1.25\times 10^{-4}\) MN\(/\)m.  
Hence, the deduction that the critical fracture energy's standard deviation reaches zero when the finite element heterogeneity degree equals 4030 appears to be logical. Therefore, it could be reasonable to assume that \(\sigma_{L}(G_{IC})\) continues to remain at zero for a mesh refinement related to a value of \(r_{e}>4300\). Additionally, it is important to highlight that Equation (17) is specifically linked to the proposed numerical model and is not an inherent property of the concrete investigated in this study. Consequently, if different formulations of finite elements or alternative softening curves in tension are employed, this relationship might undergo alterations. A comprehensive review of Figures 7-9 yields the following insights and comments: (a) The experimental global response consistently resides within the bundle of numerical responses concerning a notch opening displacement (NOD) around 0.4 mm. This implies that within the context of the DCB study, the numerical model becomes pertinent from this particular NOD value. (b) It is crucial to note that this NOD value is not equivalent to the crack opening. The observed value is higher due to a certain level of flexibility inherent in the DCB specimen.