## 4 Conclusions  
In this study, the analysis of strain rate tensors and particle velocity fields are utilized to distinguish crack types and study the mechanical behaviors in the region between pre-existing flaws under uniaxial compression. The following conclusions have been drawn.  
1. By defining a variable \(R\) to quantify the crack failure mechanism, the types and mechanical behaviors of the secondary cracks between the flaws can be effectively distinguished. The initiation mechanism of secondary cracks between flaws is most complicated, and the types and mechanical behaviors of newly generated cracks are distinctive in different stress loading stages. According to the value of variable \(R\), we can directly understand the types mechanical behaviors of secondary cracks.
2. According to the velocity and relative motion trend of the particles on both sides of the crack, the velocity field of the particles can be obviously divided into four types. The type of particle velocity field on both sides of the newly generated cracks determines the type of crack in the measurement region. At different stress loading stages, the velocity field types of the particles on both sides of the crack are constantly evolving and complicated.
3. Combined with the particle velocity field analysis and the value of the variable \(R\), it can be seen that the macro tensile crack contains partial shear cracks, and the macro shear crack contains partial tensile cracks, indicating that the type of macro crack is not totally determined by the type of partial cracks. The secondary cracks contain tensile micro-cracks, shear micro-cracks, and compressive-shear micro-cracks, and the number of tensile micro-cracks is the largest. However, when the axial stress is reached to 35.12 MPa, the ratio (\(R_{1}=22.52\)) indicates that the newly generated cracks are shear cracks in essence. Therefore, we can see that the type of macro-crack does not depend on the dominant type of micro-cracks.
4. At the beginning stage of secondary crack initiation, SC-1 and SC-2 appear as shear cracks between pre-existing parallel flaws, then extend up and down toward the tip of the flaws, and finally connect the tips of T2 and T4, as well as T1 and T3, in the form of arcs. The particles between pre-existing parallel flaws form a confined compressive member under uniaxial compression. Under the confinement of lateral particles, the contacts between particles are broken owing to the combined compressive and shear actions, and eventually the shear cracks are successively formed.  
Conceptualization, Y.L. and X.L.; Methodology, W.C.; Software, W.C.; Validation, Y.L. and Q.Z.; Formal analysis; Writing--review and editing, W.Z. and Y.L.; Supervision, S.W. and W.C.  
This research was funded by National Natural Science Foundation of China (Grant Nos. 51879149, 51779134 and 51579142), Taishan Scholars Project of Shandong Province and Open Research Fund Program of State Key Laboratory of Water Resources and Hydropower Engineering Science (Grant No. 2018 SGG01).  
The authors would like to thank the anonymous reviewers for their constructive suggestions to improve the quality of the paper.  
The authors declare no conflict of interest.