# Three-Dimensional Probabilistic Semi-Explicit Cracking Model for Concrete Structures
## 2 Development and Implementation of the 3D Model  
### Probabilistic Model Formulation  
The three-dimensional semi-explicit probabilistic model focuses on crack propagation, specifically addressing failure in mode I due to tensile stresses within concrete structures. This model is developed within the framework of the finite element method (FEM), incorporating considerations of heterogeneity and volume effects through a probabilistic approach. Implemented within a platform continually refined by researchers from the Civil Engineering Department at COPPE/UFRJ, the code is written in FORTRAN language. This platform has served as a foundational tool for numerous studies across various applications over the years [16; 23; 24]. A summary of the model's formulation along with its primary features is depicted in Figure 1. This modeling approach utilizes volume elements, specifically tetrahedral solid elements employing linear interpolation functions. To account for material heterogeneity, each finite element represents a volume of heterogeneous material (Figure 1a), quantified by a heterogeneity degree (\(r_{e}\)). This degree is evaluated as the ratio of the finite element's volume (\(V_{e}\)) to that of the maximum aggregate (\(V_{a}\)) (Figure 1b). The model's probabilistic nature is established through the random distribution of tensile strength and fracture energy among the mesh elements. These properties follow Weibull and lognormal distributions, respectively. Notably, the randomness of these mechanical properties directly correlates with the material's heterogeneity degree. The selection of the Weibull distribution [25; 26] for the tensile strength was supported by experimental evidence as demonstrated by [PERSON] et al. [22]. The rationale behind this choice was rooted in the observation that concrete exhibits a highly heterogeneous nature and displays a brittle behavior under uniaxial tension. Hence, the adoption of the Weibull theory aligns well with these characteristics. On the other hand, the preference for a lognormal  
Figure 1: General aspects of the 3D probabilistic macroscopic model for “semi-explicit” cracking of concrete. (**a**) Illustrates the representation of material heterogeneity; (**b**) Showcases the correlation between the heterogeneity degree, volume effects, and the use of random mechanical properties; (**c**) Depicts the random distribution of the mechanical properties and the elementary behavior of energy dissipation during damage evolution; (**d**) Presents an example of the overall behavior acquired with the use of the Monte Carlo method. distribution to represent fracture energy is somewhat more arbitrary but remains a classical choice in such contexts. At the elementary level, specifically in each finite element volume, the cracking process induces energy dissipation, governed by an isotropic damage law. The progression of damage is characterized by the \((\sigma\times\varepsilon)\) diagram (Figure 1c). This constitutive law accounts for both the tensile strength and volumetric density of dissipated energy (\(g_{IC}\)). The value of \(g_{IC}\) is determined using an energetic regularization technique proposed by [PERSON] and [PERSON] and [PERSON] (2018). It relates to the material fracture energy as follows: \(g_{IC}=G_{IC}/l_{e}\), where \(l_{e}\) denotes the elementary characteristic length evaluated as: \(l_{e}=(V_{e})^{1/3}\). Therefore, given the random distribution of \(f_{t}\) and \(g_{IC}\) over the finite element mesh, each finite element follows distinct laws characterizing its own dissipation process. The model's global structural responses are obtained using a Monte Carlo approach (Figure 1d). This methodology involves conducting numerous simulations of the structural problem, each employing different spatial distributions of mechanical material properties, to achieve statistically significant results.