## 5 Conclusions  
Based on the existing laboratory experiments, a numerical model with non-parallel overlapping flaws was established in this study. It was verified that the numerical model could more truly reflect the crack propagation process. Besides the influence of flaw inclination angle on mechanical properties, crack initiation characteristics and energy evolution was studied.  
(1) The results show that for the sample model containing non-parallel overlapping flaws, the UCS and elastic modulus increase monotonously with the increase of flaw inclination angle, and the UCS is more sensitive to the change of flaw inclination angle. The larger the dip angle, the greater the initiation stress, and the lower the crack concentration generated near the flaw.  
(2) The boundary energy, strain energy and dissipated energy under peak stress increase with the increase of flaw inclination angle. The dissipated energy increases the most, followed by the boundary energy and finally the strain energy. Combined with the statistics of the number of micro-cracks, it shows that the larger the flaw inclination angle, the more serious the internal damage and the higher the damage degree.  
(3) In the early stage of crack initiation, the range of tensile stress zone on the upper and lower surfaces of the flaw decreases with the increase of the flaw inclination angle, and gradually transfers to the flaw tip, while the compressive stress area gradually expands  
Figure 14: The lateral displacement field of different numerical models under crack initiation stress (Unit: m). (**a**) Intact model; (**b**) \(\beta=15^{\circ}\); (**c**) \(\beta=30^{\circ}\); (**d**) \(\beta=45^{\circ}\); (**e**) \(\beta=60^{\circ}\); (**f**) \(\beta=75^{\circ}\).  
from the flaw tip to the middle of the flaw. Due to the stress-shielding effect, a low-stress area appears in the flaw overlapping area, and with the increase of flaw inclination angle, this area gradually decreases until it disappears. The distribution pattern of \(\sigma_{1}\) and \(\sigma_{3}\) around the flaw gradually changes from flat to proximal circular with the increase of the flaw inclination angle, the stress concentration effect around the flaw weakens, and the maximum tensile stress under the crack initiation stress decreases gradually. This is also the reason for the difficult crack initiation at the flaw tip and the gradually dispersed distribution of cracks under the large flaw inclination angle.  
(4) In the crack initiation stage, the effect of the flaw inclination angle on the lateral displacement field is mainly located at the middle of a specimen. When the flaw inclination angle is from 15\({}^{\circ}\) to 60\({}^{\circ}\), the lateral displacement field near the flaw is affected significantly. Meanwhile, the lateral displacement of particles in the small area near the flaw tip changes greatly, which makes it easy to induce micro-cracks. When the flaw inclination angle is 75\({}^{\circ}\), the lateral displacement field near the flaw is less affected than other flaw inclination angles.  
Conceptualization, P.W. and Y.C.; methodology, Q.L.; validation, Y.C.; formal analysis, P.W.; investigation, P.W.; resources, L.C. and Z.Z.; data curation, M.L.; writing--original draft prep-aration, P.W.; writing--review and editing, P.W.; visualization, Q.L.; supervision, X.M.; project administration, L.Z.; funding acquisition, Y.C. All authors have read and agreed to the published version of the manuscript.  
This work was financially supported by the National Key R&D Program of China (2022 YFC2903902) and National Natural Science Foundation of China (51974295).  
Not applicable.  
All data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.  
The authors declare no conflict of interest.