## 3 Results
### Force-Displacement Curves  
The force-displacement curves are significantly different among the three breakage modes. Figure 11a shows the typical peeling force-displacement curves. The normal force increases linearly with displacement, during which the local crushing happens and then fluctuates in a certain range due to a few small fragments peeled off. The larger reduction in the normal force is attributed to the bigger fragment. For particle peeling-splitting, the normal force experiences a more visible plummet after a linearly increasing and floating process, mainly caused by more severe local crushing and peeling (see Figure 11b). In this stage, the particle crushes into two pieces. After that, the normal force increases gently because the broken pieces could continue to support the load. The force-displacement curve of the particle splitting (see Figure 11c) is similar to that of spherical particle splitting [28]. The normal force increases linearly, accompanied by the occurrence of local crushing, and then drops steeply to almost zero since the particle breaks into pieces.  
After the normal force reaches its first peak, a relatively large displacement is generated for particle peeling or peeling-splitting, while a large normal force reduction occurs for particle splitting. All the consequences might induce serious accidents in practical engineering. Therefore, the first peak value \(F_{\text{cr}}\) of the normal force is of great significance to particle breakage. The critical normal force \(F_{\text{cr}}\) is closely related to contact diameter \(d\). The relationship between the critical normal force and contact diameter is shown in Figure 12. It can be seen that for the same cone angle (here, 90\({}^{\circ}\)), the 30 and 50 mm exhibit an almost identical linearly increasing relationship between \(F_{\text{cr}}\) and \(d\), whereas, for a larger sphere diameter, the effect of the contact diameter \(d\) in \(F_{\text{cr}}\) is more prominent, evident by the highly accelerated increase of \(F_{\text{cr}}\) with an increase of \(d\). The explanation for the prominent effect of contact diameter \(d\) in \(F_{\text{cr}}\) for larger sphere diameter \(D\) is that \(F_{\text{cr}}\) is related to the crushing surface of the particle. For larger sphere diameter \(D\) and contact diameter \(d\), the crushing surface is much larger than smaller ones. The relationship between critical normal force \(F_{\text{cr}}\) and cone angle \(\theta\) can also be linearly-fitted (see Figure 13). For the same cone angle and contact diameter, it seems that the critical normal force \(F_{\text{cr}}\) has no significant  
Figure 10: Breakage modes change with the increase in sphere diameter \(D\) when cone angle \(\theta\) is 90\({}^{\circ}\) and contact diameter is 8 mm: (**a**) \(D\) = 30 mm; (**b**) \(D\) = 50 mm; (**c**) \(D\) = 70 mm.  
Figure 9: Breakage modes change with the increase in cone angle \(\theta\) when the sphere diameter \(D\) is 50 mm and contact diameter is 7 mm: (**a**) \(\theta\) = 60\({}^{\circ}\); (**b**) \(\theta\) = 90\({}^{\circ}\); (**c**) \(\theta\) =120\({}^{\circ}\).  
relationship with sphere diameter \(D\). Figure 14a shows no direct correlation between \(F_{\text{cr}}\) and \(D\) for particles of contact diameter of 8 mm and 10 mm when the cone angle is \(90^{\circ}\). The reason may lie in that the breakage modes affect \(F_{\text{cr}}\) heavily. Nevertheless, for the particles that have the same breakage mode, the critical normal force \(F_{\text{cr}}\) increases linearly with increasing sphere diameter \(D\), as shown in Figure 14b. Thus, the relationship between \(F_{\text{cr}}\) and \(D\) heavily depends on particle breakage mode.  
Figure 11: The typical force–displacement curves corresponding to three particle breakage modes: (**a**) peeling; (**b**) peeling–splitting; (**c**) splitting.  
Figure 12: Relationship between critical normal force \(F_{\text{cr}}\) and contact diameter \(d\).