# Three-Dimensional Probabilistic Semi-Explicit Cracking Model for Concrete Structures
## 1 Introduction  
The accurate modeling of crack initiation and propagation holds significant importance in the numerical analysis of concrete structures, leading to the development of various constitutive models within this domain. Among them, the following approaches can be mentioned: damaged models [1; 2; 3; 4]; the smeared crack model [5; 6; 7]; plasticity [8]; the cohesive crack model or fictitious crack model [9; 10]; and lattice models [11]. These modeling approaches mainly follow deterministic principles. However, there exist also probabilistic models that utilize statistical techniques to directly address material heterogeneity, offering promising alternatives for numerically simulating the nonlinear behavior of concrete structures [12; 13; 14; 15; 16]. Among the probabilistic models, two distinct formulations have emerged, differing in their approach to cracking analysis (whether it focuses on crack initiation, localization, or propagation) and their treatment of kinematic discontinuities associated with cracks, namely, explicit and semi-explicit models. The first formulation [12] adopts a highly localized approach. This method finely models cracking by employing interface elements to simulate, with a lot of precision, all cracks opening, including very small cracks. In contrast, the second model employs linear volume elements to simulate only macrocrack propagation in large concrete structures. The utilization of interface elements in the first approach can lead to significant computation times, limiting the applicability of the explicit method. In contrast, the second model, employing volume elements for macrocrack modeling, ensures reasonable computation times, rendering it more suitable for analyzing large structures. The concept behind the semi-explicit probabilistic cracking numerical model has been developed in recent years [14; 15]. In this model, the macrocrack propagation is governed by two criteria: (a) the macrocrack initiation is associated with the uniaxial tensile strength (\(f_{t}\)) and (b) the macrocrack propagation is linked to the mode I critical fracture energy (\(G_{IC}\)) used in the well-known linear elastic fracture mechanics (LEFM) theory. The propagation of a macrocrack occurs when all \(G_{IC}\) is fully dissipated, leading to the representation of a macrocrack as a succession of fully damaged elements converging to form the macrocrack structure. The originality of the proposed approach lies in its characterization as a type of LEFM model, wherein volume elements are employed instead of singular elements usually used in this type of numerical model. This modification of finite element type results in smoothed stresses concentration at the macrocracks tip. Therefore, the model presented in this paper is not a classical damage model or smeared crack model [1; 2; 6; 17; 18; 19; 20; 21]. In these conventional models, damaged zones are treated as physically realistic entities, and crack openings are deduced from situations involving these zones of damage. On the contrary, in the proposed model, physically existing damaged zones are not considered. The introduction of a damage parameter in the theoretical approach is arbitrary and serves solely as a method to dissipate \(G_{IC}\). The mechanical properties \(f_{t}\) and \(G_{IC}\) are considered as random parameters depending on the size of the mesh elements employed. In the case of \(f_{t}\), previous experimental and numerical investigations [15; 22] have determined and validated its probabilistic properties in terms of mean and standard deviation. Estimating these values relies on factors such as the size of finite elements (FE) and the maximum aggregate size (further described in Section 2.3). Concerning \(G_{IC}\), its mean value represents an intrinsic characteristic of concrete that remains unaffected by the size of the mesh elements. However, its standard deviation exhibits size-dependent variation, as observed for \(f_{t}\). This assumption is a key aspect of the proposed model.