## 4 Discussion
### Lateral Displacement Characteristics under Crack Initiation Stress  
The displacement field of rock-like material samples changes with the evolution of micro-crack distribution. Under the condition of uniaxial compression, the lateral displacement of rock-like materials occurs. This kind of material is more sensitive to lateral displacement in the process of deformation and failure due to its weak tensile resistance [43]. Therefore, the lateral displacement field of rock-like materials with non-parallel overlapping flaws is analyzed in this section.  
Figure 12: The principal stress distribution around flaw under the crack initiation stress in different models (Unit: MPa). (**a**) \(\beta=15^{\circ}\); (**b**) \(\beta=30^{\circ}\); (**c**) \(\beta=45^{\circ}\); (**d**) \(\beta=60^{\circ}\); (**e**) \(\beta=75^{\circ}\).  
Figure 14 shows the contour map of the lateral displacement of different numerical models under the crack initiation stress, and different colors represent different lateral displacements. The contour map of the intact sample model generally presents uniform strip distribution, as shown in Figure 14a. However, the numerical model with different flaw inclination angles shows different characteristics. The contour map presents non-uniform and symmetrical distribution. The contour of lateral deformation deflects near the flaw and concentrates on the flaw surface. The contour is more concentrated near the flaw tip, and the number of the contour increases with the increase of flaw inclination angle. The maximum lateral displacement near the flaw tip increases with the increase of flaw inclination angle from 15\({}^{\circ}\) to 60\({}^{\circ}\). When the flaw inclination angle \(\beta=45^{\circ}\) and \(\beta=60^{\circ}\), the maximum lateral displacement near the flaw tip is close to the maximum lateral displacement of the model, such as in Figure 14d,e. It indicates that the lateral displacement field under crack initiation stress is greatly affected. In Figure 14f, the maximum lateral displacement around the flaw and the concentration degree of contour around the flaw at \(\beta=75^{\circ}\) are lower than those at \(\beta=15^{\circ}\). Therefore, the influence on the lateral displacement field is small at \(\beta=75^{\circ}\). Due to the existence of flaws, the area of the lower lateral displacement in the overlapping zone between flaws is generated, and the contour converges to the middle of the upper and lower flaws, presenting a shape of a multi-layer arc; the average curvature radius of the contour increases with the increase in flaw inclination angle.  
It can be seen that the influence of non-parallel overlapping flaws on the lateral displacement of the sample surface is mainly concentrated near the flaw. When the flaw inclination angle \(\beta\leq 60^{\circ}\), the lateral displacement field near the flaw is obviously affected. In addition, the lateral displacement near the flaw tip has a maximum value and a large variation range, for which it is easy to induce micro-cracks. When the flaw inclination angle \(\beta=75^{\circ}\), the lateral displacement field near the flaw is less affected, the variation range of the lateral displacement near the flaw tip is relatively small, and micro-cracks are difficult to be produced. This may also explain that only a few cracks appear at the flaw tip at \(\beta=75^{\circ}\).  
Figure 13: Variation of the maximum tensile stress around the flaw and the crack initiation stress with the flaw inclination angle.