# A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture
## 3 Numerical Results
### Heterogeneous Material
#### 3.2.2 High Amplitude Load  
Figures 15-18 show the evolution of the damage field for correlation lengths \(l_{c}=40\) mm to \(l_{c}=5\) mm. We observe a very good match between damage localization sites and the locations of material weak points in Figure 10b-d. Moreover, as we decrease the correlation length, the time of damage initiation decreases; cf. Figures 15a-18a. From the final damage pattern in Figure 9d for the homogeneous domain, one observes that for high amplitude loading extensive damage is experienced almost everywhere, especially close to the top and bottom boundaries. There is little resemblance between this solution and those for high correlation random fields in Figures15d and 16d. Similarly for the low amplitude load, high differences are observed between the solutions of homogeneous, Figure 8d, and inhomogeneous domains with high correlation lengths, Figures 11d and 12d. This is due to the fact that for such large correlation lengths, the large islands of low strength greatly impact the response. In contrast, as the correlation length decreases, the overall material properties are almost the same in all areas, except the inhomogeneities that are observed at smaller length scales. Consequently, in comparison of damage patterns for the homogeneous rock in Figure 9d and rocks with small correlation length for \(\underline{c}\) in Figures 17d and 18d, a very similar overall response is observed; in all cases, damage is widespread in the domain, with the top and bottom sides experiencing the highest damage. In contrast, there is no resemblance between the damage patterns of homogeneous domain in Figure 8d and those for low correlation length fields in Figures 13d and 14d. The reason is that for this low amplitude of load, damage can only accumulate in the center of the homogeneous domain, whereas for inhomogeneous domains damage can accumulate from weak points outside of this zone; this greatly affect the final damage pattern. Figure 14: The evolution of damage field for the low amplitude load and cohesion realization with \(l_{c}=5\) mm at times: (**a**) 10 \(\upmu\)s, (**b**) 20 \(\upmu\)s, (**c**) 30 \(\upmu\)s, and (**d**) 40 \(\upmu\)s.  
Figure 13: The evolution of damage field for the low amplitude load and cohesion realization with \(l_{c}=10\) mm at times: (**a**) 16 \(\upmu\)s, (**b**) 26 \(\upmu\)s, (**c**) 36 \(\upmu\)s, and (**d**) 46 \(\upmu\)s.  
Figure 16: The evolution of damage field for the high amplitude load and cohesion realization with \(l_{c}=20\) mm at times: (**a**) 10 μs, (**b**) 20 μs, (**c**) 30 μs, and (**d**) 40 μs. The results are shown on the deformed mesh with a magnification factor of 100. Figure 17: The evolution of damage field for the high amplitude load and cohesion realization with \(l_{c}=10\) mm at times: (**a**) 8 μs, (**b**) 18 μs, (**c**) 28 μs, and (**d**) 38 μs. The results are shown on the deformed mesh with a magnification factor of 100. Figure 18: The evolution of damage field for the high amplitude load and cohesion realization with \(l_{c}=5\) mm at times: (**a**) 7 μs, (**b**) 17 μs, (**c**) 27 μs, and (**d**) 37 μs. The results are shown on the deformed mesh with a magnification factor of 100. The statistical continuum damage model enhances the accuracy of conventional continuum damage models, and its solutions are more consistent with sharp interface fracture models. The reason are as follows. First, damage initiation zones from material weak points are more concentrated and better resemble crack nucleation events. Second, damaged zones tend to propagate in crack-like features with specific inclined directions rather than the diffuse response around the initiation points. For example, in Figure 18d, many localized zones resemble cracks at 45 degree and steeper relative to the vertical direction. This features qualitatively match other numerical and experimental observations [71; 72; 73; 74; 75]. Specifically, based on the MC failure criterion, cracks are formed at angles \(\pm(45^{\circ}+\phi/2)\) with respect to the compressive loading direction.