# Three-Dimensional Probabilistic Semi-Explicit Cracking Model for Concrete Structures
## 3 Estimation of the Coefficients of Variation of Fracture Energy
### Inverse Analysis Methodology  
In accordance with the model's hypotheses, the parameters need to be determined relative to the finite element volume. Hence, the estimation is conducted by considering three different finite element mesh refinements to determine the relationship between the \begin{table}
\begin{tabular}{l c} \hline \hline
**Constituent** & **Quantity** \\ \hline Aggregate (4–12 mm) & 1105 (kg/m\({}^{3}\)) \\ Sand (0–5 mm) & 700 (kg/m\({}^{3}\)) \\ Cement & 400 (kg/m\({}^{3}\)) \\ Water & 190 (L/m\({}^{3}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mix design of the plain concrete. Figure 3: Experimental load–notch opening curve obtained for the DCB test [28; 38]. standard deviation of \(G_{IC}\) and the ratio \(V_{e}/V_{a}\), where \(V_{e}\) represents the volume of the finite elements and \(V_{a}\) corresponds to the volume of the larger aggregate in the concrete composition. Specifically, three meshes, referred to as Mesh 1, Mesh 2, and Mesh 3, comprising linear tetrahedral elements, are taken into account to perform this inverse analysis. Their respective three-dimensional representations are illustrated in Figure 4. The finite element meshes were produced using the software Coreform Cubit [41]. It is noticeable that the elements comprising the meshes exhibit considerable variations in size. As the focus lies on assessing the behavior of macrocrack propagation, predominantly occurring within the central part of the specimen, an average volume of the elements situated in this area will be considered. Hence, the mean heterogeneity degree for each mesh refinement, denoted as \(r_{e}^{mean}=V_{e}^{mean}/V_{a}\), is established. This assumption seems rational as it concentrates solely on the elements in this specific region, undergoing the damage process and being subject to random variations in \(G_{IC}\). Additional details regarding the three meshes are presented in Table 3, indicating: the total number of equations (Neq) and nodes (Nnode) encompassing the entire specimen; the count of elements (Numel); and the mean heterogeneity degree at the center of the specimen. This includes the standard deviation (\(r_{e}^{sd}\)), as well as its minimum (\(r_{e}^{min}\)) and maximum (\(r_{e}^{max}\)) values. The refinement of Mesh 3 was specifically designed to accounts for elements with volumes exceeding that of the maximum aggregate in the concrete composition, ensuring that the elements represent a heterogeneity degree \(r_{e}=V_{e}/V_{a}\geq 1\). In Figure 5, the frontal view of Mesh 3 illustrates the locations where the prestressing forces and imposed displacements are applied. The boundary conditions employed to accurately represent the DCB test behavior are as follows: restriction of displacements along the \(X\)-axis within the YZ plane, along the \(Y\)-axis within the XZ plane, and along the \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & **Neq** & **Nnode** & \multicolumn{3}{c}{**Numel**} & \multicolumn{3}{c}{**Heterogeneity Degree**} \\ \cline{2-9}  & **Total** & **Total** & **Total** & **Center** & \(r_{e}^{mean}\) & \(r_{e}^{sd}\) & \(r_{e}^{min}\) & \(r_{e}^{max}\) \\ \hline Mesh 1 & 3786 & 1370 & 5222 & 3953 & 114.7 & 50.7 & 34.1 & 332.3 \\ Mesh 2 & 11,848 & 4135 & 19,564 & 15,143 & 29.3 & 8.1 & 8.2 & 78.4 \\ Mesh 3 & 23,273 & 8038 & 39,600 & 34,009 & 10.8 & 2.8 & 3.1 & 31.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Description of the meshes. Figure 4: Three DCB meshes—3D view. \(Z\)-axis at the central nodes in the XZ plane. Additionally, \(Z\)-axis displacements at nodes with prescribed displacements are restricted. Prestressing force is simulated by applying prescribed compression forces in the Y-direction on the elements situated on the specimen's bottom surface at the initial load step. Each Monte Carlo simulation involves the execution of one hundred independent Finite Element Method (FEM) analyses, which proves to be a sufficient number for ensuring consistent results.