## 4 Analysis of Grain Shape Effects
### Effects of Grain Shape on Rock Failure Mode  
The macroscopic failure of rock is a gradual evolutionary process of internal microcracks, and the microcracking behavior has a great influence on the ultimate failure mode of rock. Figure 11 presents the spatial distributions of microcracks for numerical models with different grain shapes when the models reached the postpeak stage and the axial stress was 80% of the peak strength. The generated microcracks can be divided into two types--tensile and shear cracks--represented by black and red segments, respectively. A large number of previous studies [41; 42; 43] have shown that tensile failure plays a dominant role in rock failure mechanisms. In this study, the microcracks were dominated by tensile cracks (Table 2), accounting for more than 80% of the total number of cracks. It can also be seen from Table 2 that the percentage of tensile cracks was maximal in the model with square grains, followed by the triangular grain and strip-shaped grain models.  
Figure 12 shows the simulated stress-strain curves of the models with triangular grains (taking the models with shape numbers T3, T4, and T6 as examples). It can be seen that the stress dropped after the peak strength gradually became slower from T3 to T6. This can be explained by the fact that from T3 to T6, the surface roughness of the grain increased, and the irregularity of the grain shape increased. In the pre-peak stage, the mineral grains were cemented together. The bond damage increased after the rock reached the peak strength, then the number of microcracks increased rapidly and formed macroracks. The mineral grains on both sides of the crack began to move around each other along the grain surface and produced significant displacement. In mineral grains with a high surface roughness, the interlocking effect and friction between grains were stronger, which made samples have a higher residual strength and resulted in slower stress drop. Figure 13 shows the stress-strain curves of the models with three grain shapes (taking the models with shape numbers S6, T6, and R6 as examples). As can be seen, from the strip-shaped grain to the square grain, the surface roughness increased, and the postpeak stress drop of the three models also became slower, which further confirmed the influence of grain shape on rock characteristics after peak strength.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Ratio of Tensile Crack (\%)**} & \multicolumn{6}{c}{**Shape Number**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Strip-shaped grain & 88 & 87 & 83 & 85 & 79 & 83 \\ Triangular grain & 88 & 92 & 90 & 92 & 92 & 90 \\ Square grain & 88 & 92 & 95 & 93 & 93 & 91 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Percentage of tensile cracks of rock with three particle shapes in postpeak stage \(\sigma\) = 0.8 \(\sigma_{c}\).  
Figure 10: Effects of grain shape on rock cohesion.  
Figure 11: Failure mode of rock with three grain shapes in the postpeak stage when \(\sigma=0.8\)\(\sigma_{\rm c}\) (black segments represent tensile cracks and red segments represent shear cracks).  
Figure 12: Stress–strain curves of different triangular grain models (case study of models T3, T4, and T6).  
According to the above analysis, the PFC model was composed of grains by cementation, and the failure of the model was essentially the break of the intergrain bond. Hence, the peak strength of the model was macroscopically controlled by cohesion. After the axial stress reached the peak strength of the model, obvious displacement started to appear along the failure surface. The interlocking between grains played a major role, the relative sliding between the grains became difficult with the increasing surface roughness, and the macroresponse was that the stress drop became slower.