## 3 Numerical Simulation
### Numerical Simulation Analysis
As the dynamic crack propagation continues, the free surface of the dynamic crack collaborates with the boundary of two empty holes, reshapes stress wave distribution, and results in transforming the curvature of ellipse shown in Figure 10e,f, which, in response, causes the direction of particles moving between two holes to change with time and leads the compressive stress field to have different effects on the running crack. For instance, when t = 335.7 \(\upmu\)s, the compressive stress field has a strong impact on the arresting running crack, during which the running crack may stop, as HT-2 in Figure 5b shows. However, when t = 391.8 \(\upmu\)s, this stress field might facilitate crack propagation, enhancing the velocity of the running crack, with Figure 9a providing evidence that the crack propagation speed increases after the running crack goes across the two holes' zone. Also, the compressive stress contour is extracted from numerical simulation and plotted in Figure 11 with the dynamic crack path at the corresponding time. Figure 10: HT-1 specimen; (**a–c**) simulation results of the particle velocity vector contour plot calculated by AUTODYN; (**d–f**) principle diagram of the holes’ deformation. Figure 11 illustrates directly that the compressive stress field exists between two holes and the intensity of the compressive stress field near the holes is higher than in other zones. Meanwhile, this compressive stress field presents most obviously when the running crack tip is in front of the two hole zone. If the crack pierces the two hole zone, the compressive stress field would be influenced by the tensile stress field around the main crack tip, as shown in Figure 11. When t = 335.7 \(\upmu\)s and t = 391.8 \(\upmu\)s, the two stress fields overlap with each other, reconstruct the whole stress field and dominate the crack propagation. The secondary cracks initiate on the back of the surfaces of the holes when t = 335.7 \(\upmu\)s, while they appear on the front of surfaces when t = 391.8 \(\upmu\)s. As the main crack has gone through the centre of the two holes, the secondary cracks on the back surface of the holes generate more actively as the tensile stress field become stronger than before. Thus, with the influence of the compressive stress field between the two holes and the secondary cracks, which share the energy required for the main crack propagation, the main crack propagates with very low speed. In order to quantitatively illustrate the effect of the compressive stress field and its relationship with loading rate, the maximum compressive stress of twenty-one gauge points is detected from numerical simulation results, and the curves of the maximum compressive stress of three typical loading rates are plotted in Figure 12.  
Figure 11: Numerical simulation results of crack propagation paths and contour plot of the compressive stress calculated by AUTODYN. (**a**) The compressive stress contour plot when t = 294.2 \(\upmu\)s. (**b**) The compressive stress contour plot when t = 335.7 \(\upmu\)s. (**c**) The compressive stress contour plot when t = 391.8 \(\upmu\)s.  
According to the compressive stress curves in Figure 12, it can be concluded that the intensity of compressive stress in the two hole zone is much greater than the rest of the gauge points during the stress wave's progress in the specimen, and the maximum compressive stress appears at the center of the holes. Particularly, the maximum compressive stress monitored by gauges presents an entire increase tendency with the increase in loading rate. The maximum compressive stresses under loading rate are 1000 GPa-s\({}^{-1}\), 1500 GPa-s\({}^{-1}\) and 2000 GPa-s\({}^{-1}\) are 32.52 MPa, 49.66 MPa, and 60.46 MPa, respectively. Therefore, the compressive stress field demonstrates a more powerful influence on crack propagation when the loading rate becomes higher. This can explain the different behaviors that a dynamic crack shows when it is subjected to different loading rates.