## 1 Introduction  
Concrete, commonly treated as homogeneous in macroscopic numerical models for simplicity, is inherently heterogeneous. Inner defects arise from cement paste hydration and restrained shrinkages, causing cracks even before external loads are applied. Given the inherent formation of cracks, modeling their initiation and propagation presents a critical challenge in predicting concrete behavior. The importance of crack formation has instigated a variety of studies, resulting in diverse constitutive models. Techniques for simulating the cracking process in concrete structures fall into two broad approaches, implicitly or explicitly addressing kinematic discontinuity, resulting in continuum or discrete models. In continuum models, cracks are implicitly represented, and the failure process is considered by the degradation of material stiffness, altering its constitutive equation. Some models in this field are damaged models [1, 2], the smeared crack model [3, 4], and the plasticity model [5]. Conversely, in discrete cracking models, cracks are explicitly treated as geometrical entities, manifesting as discontinuities of displacement at interfaces between finite elements or integrated into the finite element formulation. Some discrete models are the cohesive crack model or fictitious crack model [6, 7], extended finite element method (XFEM) [8], embedded finite element method (EFEM) [9] and lattice models [10]. Additionally, probabilistic models address the significant scale effect in concrete structure cracking by employing random distribution functions of material properties to explicitly consider concrete heterogeneity. In this work, a semi-explicit probabilistic cracking numerical model, based on the finite element approach, developed and validated in previous studies [11; 12], is employed. This model specifically focuses on the macrocrack propagation problem, incorporating several crucial characteristics. One significant feature of the model is its representation of macrocrack propagation using volume elements, which offers a realistic portrayal of the phenomenon. The model employs two criteria governing macrocrack propagation, as follows: (1) macrocrack initiation is linked to the uniaxial tensile strength, \(f_{t}\) and (2) macrocrack propagation is influenced by post-cracking dissipation energy in tension. The complete propagation of the macrocrack occurs when all post-cracking dissipation energy has been consumed. The evolution of post-cracking dissipation is governed by a simple damage approach. A distinctive aspect of the model's damage approach is that the post-cracking dissipation energy is derived from linear elastic fracture mechanics (LEFM), specifically utilizing the mode I critical fracture energy, referred to as \(G_{IC}\). Both \(f_{t}\) and \(G_{IC}\) are probabilistic mechanical characteristics that depend on the size of the mesh elements. While the mean value of \(G_{IC}\) is considered an intrinsic characteristic of concrete independent of the mesh element size, its standard deviation is influenced by the size effects. This probabilistic semi-explicit cracking model differs significantly from classical smeared crack or damage models [1; 2; 4; 13; 14; 15; 16; 17; 18; 19; 20]. Indeed, physically speaking, these diffused crack models, which are deterministic, consider that, for each loading level, an equivalence exists between a microcracking evolution (represented by a certain dissipation of energy) and a localized crack creation. Previous experimental and numerical studies [21; 22; 23; 24] have successfully determined and validated the probabilistic properties of \(f_{t}\) (mean and standard deviation values) for concretes with a compressive strength of up to 130 MPa, depending on the size of the mesh elements. However, acquiring equivalent information regarding \(G_{IC}\) has proven to be a challenge.