# Three-Dimensional Probabilistic Semi-Explicit Cracking Model for Concrete Structures
## 2 Development and Implementation of the 3D Model
### Estimation of the Model's Parameters  
The estimation of Weibull distribution parameters relies on the correlation of equations for \(\mu_{w}\) and \(\sigma_{w}\) with the analytical scale law proposed by [PERSON] et al. (2018). This scale law, derived from extensive experimental investigations that correlate concrete heterogeneity and scale effects, plays a significant role in validating probabilistic models. The expressions for these correlations are detailed in Equations (13)-(16). Through these equations, the mean \((\mu_{f_{t}})\) and coefficient of variation of tensile strength \((\sigma_{f_{t}}/\mu_{f_{t}})\) for a given volume of concrete can be estimated, taking into account parameters such as specimen volume \((V_{s})\), maximum aggregate volume \((V_{a})\), and material compressive strength \((f_{c})\).  
\[\mu_{f_{t}}=a\left(\frac{V_{s}}{V_{a}}\right)^{-\gamma} \tag{13}\]  
\[\frac{\sigma_{f_{t}}}{\mu_{f_{t}}}=A\left(\frac{V_{s}}{V_{a}}\right)^{-B} \tag{14}\]  
where \(a=6.5\) MPa, \(\gamma\) is given by Equation (15), \(A=0.35\), and \(B\) is given by Equation (16). Moreover, the constant \(c_{1}=1\) MPa.  
\[\gamma=0.25-3.6\times 10^{-3}\left(\frac{f_{c}}{c_{1}}\right)+1.3\times 10^{- 5}\left(\frac{f_{c}}{c_{1}}\right)^{2} \tag{15}\]  
\[B=4.5\times 10^{-2}+4.5\times 10^{-3}\left(\frac{f_{c}}{c_{1}}\right)-1.8 \times 10^{-5}\left(\frac{f_{c}}{c_{1}}\right)^{2} \tag{16}\]  
These analytical expressions are applicable to various concrete types with compressive strengths ranging from 35 to 130 MPa, excluding fiber-reinforced and lightweight concretes. They are specifically utilized in the model at the elemental level, i.e., considering the finite element volume \((V_{e})\). Consequently, the combination of Equations (7), (8), (13) and (14) results in a nonlinear system of equations.  
To solve this system, an iterative numerical procedure is employed. Through this iterative process, a pair of \((b,c)\) values is determined for each finite element based on its \(V_{e}\), as well as the concrete's \(V_{a}\) and \(f_{c}\). Detailed implementation of this iterative procedure can be found in [PERSON] (2018).  
The lognormal distribution characterizes the heterogeneity of fracture energy, typically governed by mean and standard deviation parameters for estimation. However, experimental findings by [PERSON] (2018) suggest that fracture energy can be regarded as an intrinsic material property, whereby its mean value \(\mu(G_{IC})\) remains constant and aligns with experimentally obtained values. Consequently, in this context, the sole parameter necessitating determination is the standard deviation of the distribution, denoted as \(\sigma_{L}(G_{IC})\).  
To estimate \(\sigma_{L}(G_{IC})\), an inverse analysis procedure is adopted. This method involves conducting multiple simulations of a macrocrack propagation test on a large double cantilever beam specimen (DCB). Utilizing the outcomes from this procedure, a function relating \(\sigma_{L}(G_{IC})\) to the heterogeneity degree of mesh elements is proposed. A comprehensive description of this inverse analysis procedure is detailed in Section 3.