# A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture
## 3 Numerical Results
### Heterogeneous Material
#### 3.2.1 Low Amplitude Load  
In this section, we study the effect of heterogeneity on damage response for the low amplitude condition. Figure 11 shows the damage response for \(l_{c}=40\) mm at various times. From the cohesion map in Figure 10a, we observe that \(\underline{c}\) varies very slowly in space. It takes the highest values near the top boundary and the lowest ones at three spots close to the left and right boundaries; weak zones are colored by blue. In Figure 8a, the initial damage zone begins when the stress waves collide in the middle of the domain. The particular form of this realization for \(\underline{c}\) actually favors damage accumulation in the center, given that a higher strength zone is near the top boundary in Figure 10a. As shown in Figure 11, damage initiates and accumulates both in this center location and in the three aforementioned weak sites close to the boundaries. Thus, the form of the failure pattern follows both the weak points in the material and locations with higher stress values in general. By comparison  
Figure 10: Random field realizations for cohesion with different correlation lengths, \(l_{c}\), equal to: (**a**) 40 mm, (**b**) 20 mm, (**c**) 10 mm, and (**d**) 5 mm.  
Figure 9: Damage evolution for the medium mesh and high amplitude load at times: (**a**) 13.9 \(\upmu\)s, (**b**) 21.1 \(\upmu\)s, (**c**) 33.2 \(\upmu\)s, and (**d**) 37.4 \(\upmu\)s. The results are shown on the deformed meshes with a magnification factor of 300.  
of Figures 8 and 11, we also observe that the earlier initiation of damage in weaker sites results in a response with more concentrated damage zones. Finally, the damage initiation time is almost the same as that for the homogeneous rock, and in both cases it is right after the collision of the waves at \(t_{\text{collision}}\approx 24\).  
Figures 12-14 show the damage evolution for heterogeneous cohesion fields with correlation lengths equal to 20 mm, 10 mm, and 5 mm, respectively. According to Figures 12a-14a, the time for damage initiation decreases as the correlation length gets smaller, i.e., when the heterogeneity is increasing.  
It is well accepted in the literature that one of the main reasons for localization and softening behavior in brittle materials is their heterogeneous structure at microscale [68; 69; 70]; the weaker points in material begin to fail earlier. This results in an increased stress concentration in the damaging zones and the shielding of the surrounding areas. That is, the inhomogeneity in material properties promotes inhomogeneity and localization in the stress field. Unlike ductile materials, there are not much energy dissipative reserves, for example from plasticity, to balance the stress field. Figures 11d-14d reveal a crucial impact of the correlation length on failure mechanism; this is a transition from diffusive damage propagation to a more localized response as the correlation length gets smaller. This agrees with the preceding discussion on the promotion of damage localization by material inhomogeneity. In fact, for the solutions with the lowest correlation length, even the mode and propagation of failure is significantly different than that of a homogenous material; in Figures 8d and 14d, the effect of the weakest point of the material is high to an extent that damage initiates and accumulates in a more distributed sense, as opposed to the damage accumulation in the central zone in Figure 8.  
Figure 11: The evolution of damage field for the low amplitude load and cohesion realization with \(l_{c}=40\) mm at times: (**a**) 25 \(\mu\)s, (**b**) 34 \(\mu\)s, (**c**) 42 \(\mu\)s, and (**d**) 53 \(\mu\)s.  
Figure 12: The evolution of damage field for the low amplitude load and cohesion realization with \(l_{c}=20\) mm at times: (**a**) 21 \(\mu\)s, (**b**) 31 \(\mu\)s, (**c**) 41 \(\mu\)s, and (**d**) 51 \(\mu\)s.