## 3 Validation
### Probabilistic Numerical Model  
The three-dimensional semi-explicit probabilistic model, extensively described in [12], is developed in the finite element method (FEM) context and integrates heterogeneity and volume effects using a probabilistic approach. The code is written in FORTRAN language. The model belongs to the fracture mechanics family of models and primarily deals with the propagation of mode I macroracks. However, it does not take into account mode II fracture propagation in its current version. Although it shares similarities with linear elastic fracture and non-linear fracture models, it distinguishes itself from damage or smeared crack models by not attempting to simulate the microcracking process. The model utilizes three-dimensional (3D) linear tetrahedral elements to simulate macrorack propagation. It employs a criterion based on the mode I critical fracture energy, \(G_{IC}\). For each volume element, the dissipation of the cracking energy, following the linear elastic behavior, was modeled through a softening behavior that initiated when the tensile strength, \(f_{t}\), was reached. This softening behavior, exhibiting a descending branch, was depicted by a linear relationship between the principal tensile stress and strain. The governing principle behind this linear relation is a classical isotropic damage law, uniquely characterized by the random assignment of \(G_{IC}\) and \(f_{t}\) to the mesh elements due to the the model's probabilistic nature. The basic steps of the FEM code can be seen in Algorithm 1. Once the dissipative energy associated with this softening behavior reached the value of \(G_{IC}\), the stiffness matrix of the element reduced to zero. As a result, macrorack propagation was modeled through a sequence of fully damaged elements, rather than the opening of interface elements as traditionally performed in fracture mechanics models. This characteristic defines the model as a non-explicit cracking model, as opposed to an explicit cracking model. It is important to note that in this numerical model, the utilization of a simplistic damage approach was solely aimed at dissipating the energy related to the softening behavior until reaching the \(G_{IC}\) value, and it did not hold any physical significance. The objective of the present model did not involve explicitly modeling the microcracking process. It is worth emphasizing that in the model, all mechanical criteria were assessed at the centroid of the linear volume elements. Besides, the rationale behind considering \(f_{t}\) and \(G_{IC}\) as probabilistic lay in accounting for the material's inherent heterogeneity and integrating scale effects, directly linked to this material's characteristic. As illustrated in [21], the variation in tensile strength values stemmed from this phenomenon. Thus, the  
Figure 1: Detail of the geometry and of the loading conditions related to the DCB specimen. intensity of the scale effect diminished with a higher material quality (higher \(f_{c}\) values) and reduced heterogeneity (measured as the ratio of the specimen's volume to the volume of the maximum aggregate). ```
1:Variables initialization
2:Read input data
3:Distribute random tensile strength ;// according to Weibull distribution
4:Distribute random critical fracture energy ;// according to lognormal distribution
5:istep = 0 ;// load step counter
6:while the number of load steps is not achieved do
7:istep = istep + 1
8:i = 0 ;// iteration counter
9:while the balance between external and internal forces is not achieved do
10:i = i + 1
11:Evaluate the linear solution of displacements
12:Update the total displacements
13:Update the damage of the elements
14:Update the stiffness matrix
15:Update the vector of residual forces
16:endwhile
17:endwhile
``` ```  
**Algorithm 1**Basic steps of the FEM program  
Furthermore, given the probabilistic nature of the numerical model, a Monte Carlo (MC) technique was employed to ensure statistically robust results.