# An Integrated Cellular Automata Model Improves the Accuracy of Secondary Fragmentation Prediction
## 3 Results
### Fragmentation Model with Shear Effect  
Given the results observed in Section 3.1 and to minimize differences, especially in the estimation of the smallest material size (d\({}_{10}\) and d\({}_{20}\)), \(Ri\) results were evaluated in both numerical simulations and physical experiments. In this way, the error was quantified, which made it possible to identify the discrepancies and propose adjustments to incorporate the effect of the shear. Table 4 shows the errors as \"\(\varepsilon\)\", which corresponds to the difference between the experimental and simulated \(Ri\), as obtained from the previous estimates for each characteristic size. As shown in the table, errors decrease with greater confinement pressures. Fragmentation related to shear has been identified in caving mining [6; 35; 36], and based on what was observed in Section 3.1, this error was hypothesized to occur because the shear strain effect was not considered in the fragmentation model. In this way, the reduction ratio (Equation (8)) can be expressed as follows:  
\[R_{i}=\alpha_{1}\Big{(}\frac{\sigma_{v}}{UCS}\Big{)}^{\alpha_{2}}+\varepsilon, \tag{8}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\sigma_{v}\) [MPa]} & \multicolumn{4}{c}{\(\varepsilon\)} \\ \cline{2-5}  & **R\({}_{10}\)** & **R\({}_{20}\)** & **R\({}_{50}\)** & **R\({}_{80}\)** \\ \hline
0.8 & 0.540 & 0.916 & 0.345 & 0.437 \\
3 & 0.218 & 0.111 & 0.140 & 0.067 \\
5 & 0.096 & 0.054 & 0.079 & 0.066 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Errors obtained between experimental and numerically simulated reduction ratios, Ri. Figure 8: Original numerical model vs physical model results [12]. where  
\[\varepsilon=\left(\frac{\gamma*\delta}{\sigma_{v}}\right)^{\alpha_{3}}. \tag{9}\]  
In Equation (9), \(\gamma\) is shear strain, a dimensionless parameter related to the interaction height and the width of the shear band. The vertical stress is represented by \(\sigma_{v}\), while \(\alpha_{3}\) is a calibration constant related to abrasion effects. The parameter \(\delta\) is linked to the friction of the system, a calibrated and dimensionless value. The reduction ratio results are presented in Table 5 for different fragment sizes. Thus, considering the results in Figure 9, the values of the \(\delta\) and \(\alpha_{3}\) parameters are determined through the minimization of the mean square error. After integrating this new factor (Equation (9)) incorporating shear fragmentation into the fragmentation model, a series of simulations were executed, following the same procedure used previously to ensure the consistency and comparability of the results obtained. The impact of the shear strain effect on the behavior of the model was analyzed for each of the pass-through percentages and the three confinement pressures considered in this study. The results derived from these simulations are presented in Figure 9. The characteristic sizes obtained by the numerical models (original and proposed) correspond to the average value of the five simulations carried out for each vertical stress. The new model shows a good fit for the prediction of fine fragmentation at low confinement stresses, achieving an error of 6.8% in the simulation with the new integrated model compared to the experimental results for the d\({}_{10}\) fragment size. As the stress increased, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Reduction Ratios** & \(\alpha_{3}\) & \(\delta\) \\ \hline R\({}_{10}\) & 0.79 & 1.03 \\ R\({}_{20}\) & 1.58 & 2.10 \\ R\({}_{50}\) & 0.74 & 0.53 \\ R\({}_{80}\) & 1.25 & 1.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Calibrated values for \(\alpha_{3}\) and \(\delta\). Parameters obtained after minimizing the mean square error between the experimental and numerical values. Figure 9: Results of the proposed numerical model for the pass-through percentages studied. errors decreased to 6.5% for 3 MPa and 3.8% for 5 MPa, evidencing an improvement in model accuracy (Figure 9, square marker).