## 4 Second Application: Three-Point Bending Test and Comparison to Experimental Data
### Model Description  
The beam geometry and the sensor positions are shown in Figure 10a, where \(\zeta=0.0\) and 0.4 defines the fissure's position in each case. A three-point-bending test was performed by controlling the displacement velocity in the fissure's mouth (CMOD). The boundary  
Figure 9: AE magnitudes for sensor \(S_{1}\) and the corresponding increments of kinetic and dissipated energies.  
conditions are shown in Figure 10b, which also illustrates how the main fissure was modeled in the numerical model: The bars' constitutive laws were modified, reducing their corresponding strength values.  
The respective LDEM model was built with \(73\times 27\times 12\) cubic modules, with sides of 2.25 mm, that were excited through a constant-velocity-prescribed displacement. The model parameters were based on those in [41] and presented in Table 2. Two pre-fissure configurations were considered with \(s_{t}=0.5\zeta s_{a}\), where \(\zeta=0\) and \(\zeta=0.40\).  
Important considerations regarding the model calibration were as follows:  
* The Young's modulus for Berea Sandstone, as presented in [41], was in the range from 12 GPa to 15 GPa. In this study, it was considered 15 GPa to better adjust the curve force vs. global vertical displacement, as shown in Figure 11 (red line).
* \(G_{f}\) was considered a random field with a Weibull distribution, with a mean value computed by considering \(P_{crit}=736.8\) N, as shown in Figure 11, and using the classical fracture-intensity factor expression for the three-point-bending test, as provided in Equations (13) and (14) [61], yielding \(K_{Ic}=345,\)860 Nm\({}^{-3/2}\), and \(G_{f}=K_{Ic}{}^{2}/E=7.9\) N m\({}^{-1}\). \[K_{I}=\frac{P_{y}}{e\sqrt{h}}f\Big{(}\frac{a}{h}\Big{)},\] (13) where \[f\Big{(}\frac{a}{h}\Big{)}=\frac{3\frac{S_{a}}{h}\sqrt{\frac{a}{h}}}{2\Big{(} 1+2\frac{a}{h}\Big{)}\Big{(}1-\frac{a}{h}\Big{)}^{3/2}}\bigg{[}1.99-\frac{a}{ h}\Big{(}1-\frac{a}{h}\Big{)}\bigg{(}2.15-3.93\frac{a}{h}+2.7\frac{a^{2}}{h^{2}} \Big{)}\bigg{]}.\] (14)
* The material porosity used by [41] was in the [0.1 mm-0.8 mm] interval, which was lower than the discretization level adopted in this study. For this reason, we assumed \(l_{0}=l_{c}\), meaning that the random generation of each bar would be statistically independent. However, in the same reference, the variations in the tensile stress test were about 6% (\(\sigma_{u}=\) [3.4 MPa-3.6 MPa]), whereas it was around 10% in [62] for sandstone specimens with the same dimensions. Previous studies using LDEM on tensile specimens with similar sizes showed that to obtain a \(CV(\sigma_{u})\) close to 10%, the bars' \(CV(G_{f})\) had to be about 65%. The links between the toughness random field properties and the global parameter variations were discussed in more detail in [50; 63].  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(\mu(G_{f})\) & \(CV(G_{f})\) & \(E\) & \(d_{eq}\) & \(\rho\) & \(\
u\) & \(L_{n}\) & \(l_{c}\) \\ \hline
7.3 N m\({}^{-1}\) & 65\% & 15 GPa & 0.012 m & 2800 kg m\({}^{-3}\) & 0.25 & 2.25 mm & 2.25 mm \\ \hline \end{tabular}
\end{table}
Table 2: Input parameters used in the LDEM simulation.  
Figure 10: (**a**) Three-point-bending test. (**b**) LDEM model details.  
* The material's characteristic length \(d_{eq}\) was based on the Carpinteri number, assuming \(\sigma_{p}\) = 3 MPa. This parameter was slightly lower than the range used in [41] [3.4 MPa-3.6 MPa], and slightly higher than the values presented by [62]. The corresponding value for \(s\) was computed using Equation (8): \[s=\frac{\sqrt{\frac{G_{c}E}{d}}}{\sigma_{p}}=\frac{\sqrt{\frac{7.3\,\text{N}\, \text{m}^{-1}15\,\text{GPa}}{0.06\,\text{m}}}}{3\,\text{MPa}}=0.45,\] (15) Then, based on \(s\) and using Equation (8), we obtained \(d_{eq}=s^{2}d=0.45^{2}\times 0.06\,\text{m}=0.012\,\text{m}\). For a detailed discussion between \(d_{eq}\) and \(s\), see [53].