## 3 Results
### Breakage Mode Discrimination  
The peeling-splitting mode can be seen as a transition between peeling and splitting as it has the characteristics of both peeling and splitting. According to the breakage modes discussed above, the relationship between contact diameter \(d\), cone angle \(\theta\) and sphere diameter \(D\) in different modes can be deduced. In order to distinguish different breakage modes, the largest contact diameter \(d\) when the particle is crushed as peeling is defined as the lower critical contact diameter \(d_{\text{crl}}\), and the smallest contact diameter \(d\) when the particle crushed as splitting is called the upper critical contact diameter \(d_{\text{cru}}\). With increasing cone angle \(\theta\), the critical contact diameters \(d_{\text{crl}}\) and \(d_{\text{cru}}\) decrease, as shown in Figure 15. It indicates that for larger cone angle \(\theta\), the particle is more susceptible to crush as splitting. For particles of sphere diameter of 50 mm, the relationship between cone angle \(\theta\) and the critical contact diameters \(d_{\text{crl}}\) and \(d_{\text{cru}}\) could be described as:  
\[\left\{ \begin{array}{l}d_{\text{crl1}}=-0.0833\theta+12.167\\ d_{\text{crl1}}=-0.0833\theta+17.167\end{array} \right. \tag{1}\]  
where \(d_{\text{crl1}}\) is the lower critical contact diameter of cone-hemispherical particles of sphere diameter of 50 mm; \(d_{\text{crl1}}\) is the upper critical contact diameter of cone-hemispherical particles of sphere diameter of 50 mm. \(\theta\) is the cone angle of cone-hemispherical particles of 50 mm. Figure 14: Relationship between critical normal force \(F_{\text{cr}}\) and sphere diameter \(D\): (**a**) contact diameters of 8 mm and 10 mm; (**b**) breakage modes of peeling–splitting and splitting. Figure 13: Relationships between critical normal force \(F_{\text{cr}}\) and \(\theta\) for particles of sphere diameter of 50 mm. For the same cone angle, particle shape is the same for the same ratio of contact diameter \(d\) to sphere diameter \(D\). As the breakage mode is related to particle shape instead of particle scale [33], ratios of the critical contact diameters \(d_{\text{crl}}\) and \(d_{\text{cru}}\) to sphere diameter \(D\) for particles of the same cone angle should be constants. For the particles of cone angle of 90\({}^{\circ}\), when sphere diameters are 30 mm, 50 mm and 70 mm, the ratios of the lower critical contact diameter \(d_{\text{crl}}\) to sphere diameter \(D\) are 0.10, 0.10 and 0.11, the ratios of the upper critical contact diameter \(d_{\text{cru}}\) to sphere diameter \(D\) are 0.23, 0.20 and 0.23, respectively. Then the relationship between critical contact diameters \(d_{\text{crl}}\) and \(d_{\text{cru}}\) and sphere diameter \(D\) can be described as:  
\[\left\{ \begin{array}{ll}d_{\text{crl2}}=0.10D\\ d_{\text{crn2}}=0.23D\end{array} \right. \tag{2}\]  
where \(d_{\text{crl2}}\) is the lower critical contact diameter of the cone-hemispherical particles of cone angle of 90\({}^{\circ}\); \(d_{\text{cru2}}\) is the upper critical contact diameter of the cone-hemispherical particles of cone angle of 90\({}^{\circ}\). \(D\) is the sphere diameter. As discussed above, the critical contact diameter \(d_{\text{cr}}\) has a linear relationship with sphere diameter \(D\) and cone angle \(\theta\). Therefore, these three variables can be fitted by plane, as shown in Figure 16. The critical contact diameter \(d_{\text{cr}}\) can be described as:  
\[\left\{ \begin{array}{ll}d_{\text{crl}}=0.125D-0.0833\theta+6.25\\ d_{\text{cru}}=0.225D-0.0833\theta+6.65\end{array} \right. \tag{3}\]  
where \(d_{\text{crl}}\) is the lower critical contact diameter of the cone-hemispherical particle; \(d_{\text{cru}}\) is the upper critical contact diameter of the cone-hemispherical particle; \(D\) is the sphere diameter, and \(\theta\) is the cone angle. Figure 16: Relationship between critical normal force \(F_{\text{cr}}\) and sphere diameter \(D\).