## 5 Conclusions and Discussion  
In this study, our primary objective was to enhance the applicability of a semi-explicit macroscopic probabilistic model by devising strategies to estimate both the mean and standard deviation of \(G_{IC}\), which are parameters of the model. For this aim, an equation to estimate the coefficient of variation of \(G_{IC}\) in relation to \(r_{e}\), the mean and standard deviation of the tensile strength was proposed. The methodology's validation involved a simulation of an experimental DCB test using high-strength concrete with \(f_{c}=105\) MPa. The simulation confirmed the main assumption of the study, enabling an extension of the model's applicability to diverse concrete mixtures. Additionally, a strategy to estimate the mean value of \(G_{IC}\) was introduced, enabling the assessment of this value using more readily available data, such as compressive strength, \(f_{c}\), or tensile splitting strength, \(f_{ls}\). Thus, this paper offers an approach to address an ongoing issue in the literature regarding the definition of material inputs for modeling concrete cracking, considering the size effects and material heterogeneity.  
The numerical model employed in this study, based on finite element theory, is designed to analyze the propagation of macrocracks in concrete structures. It incorporates therandom distribution of material properties over the mesh, accounting for crack propagation through energy dissipation. The random mechanical properties considered in the model are the tensile strength, \(f_{t}\), and the mode I critical fracture energy, \(G_{IC}\). The model assumes that the mean value of \(G_{IC}\) remains constant regardless of scale, while its standard deviation varies based on the volume of the mesh elements.  
In conclusion, this study extends the applicability of a 3D probabilistic semi-explicit cracking numerical model to concrete mixtures with compressive strengths below 130 MPa and the largest aggregate diameter ranging between 10 mm and 20 mm. The findings contribute to describing the macrocrack propagation in concrete elements. However, for effective application of the model in real concrete structures, further advancements are necessary, particularly in modeling steel rebars and concrete/steel bond. Moreover, future research should focus on extending the model to simulate macrocrack propagation in fiber-reinforced concrete.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14010462/s1](https://www.mdpi.com/article/10.3390/app14010462/s1)]([https://www.mdpi.com/article/10.3390/app14010462/s1](https://www.mdpi.com/article/10.3390/app14010462/s1)), Section S1: Coefficient of Variation of Tensile Strength; Section S2: Coefficient of Variation of Fracture Energy; Section S3: Deduction of the analytical relation between \(\xi_{\mu}^{\xi}(G_{IC})\) and \(\xi_{\mu}^{\xi}(f_{t})\).  
Conceptualization, P.R. and M.R.R.; methodology, P.R.; software, M.R.R. and F.L.B.R.; validation, M.R.R.; formal analysis, M.R.R.; investigation, P.R.; writing--original draft preparation, P.R. and M.R.R.; writing--review and editing, P.R., M.R.R. and E.d.M.R.F; supervision, P.R., E.d.M.R.F. and F.L.B.R. All authors have read and agreed to the published version of the manuscript.  
This study was financed by Brazilian scientific agencies, namely the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq), and the Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES)--Finance Code 001.  
Not applicable.  
The data presented in this study are available in the article.  
The authors declare no conflicts of interest.