# A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture
## 4 Conclusions  
We presented a dynamic bulk damage model, based on the time-delay evolution law in [3]. The relaxation time \(\tau_{c}\) indirectly introduces an intrinsic length scale for dynamic fracture problems. This resolves the mesh sensitivity problem of early local damage models. Moreover, by limiting the maximum damage rate, the model qualitatively captures stress rate effect, in that, both strength and toughness increase when the loading rate increases. The ODE form of the evolution model greatly simplifies the implementation of the damage model and maintains the hyperbolicity of the elastodynamic problem. The coupled elastodynamic-damage problem was implemented by the aSDG method to solve a uniaxial compressive fracture problem for rock. The MC model is used to formulate a damage force model. In the process of damage accumulation, the effective stress tends from the initial elastic limit at \(D=0\) to its hydrostatic compressive value at \(D=1\). The MC model also captures rock strengthening effect as hydrostatic pressure increases. In contrast, damage models that are based on spectral positive and negative decomposition of strain (or stress) tensor, fail to model failure under compressive response. To model the effect of material inhomogeneity, cohesion was assumed to be a random field. Two different macroscopic compressive load amplitudes were used for this study. For a homogeneous material, the higher load amplitude initiates damage as the compressive wave enters the domain, whereas for the lower load damage initiates only in the center of the domain where stress doubling effect occurs upon the intersection of compressive waves. Four lognormal fields with different correlation lengths \(l_{c}\) were generated for \(\underline{c}\). It was shown that inhomogeneity could significantly alter the failure response of an otherwise homogeneous rock. For example, for the higher load amplitude, unlike the homogeneous case, damage initiates in the center of the domain. This is due to the particular form of the realized random field where a large zone of low \(\underline{c}\) is sampled in the center of the domain. Moreover, for the lower load amplitude damage can initiate everywhere in the domain as the waves travel toward the center of the domain. This is due to the weaker sampled \(\underline{c}\) at these locations, which does not require the stress wave doubling effect to initiate damage. Moreover, even the zones that eventually accumulate the highest damage can be significantly different between models with homogeneous and inhomogeneous properties, even as the correlation length tends to zero (low load amplitude example). Another problem of using a homogeneous material model is the inability or difficulty of bulk damage models to capture sharp localization zones. In contrast, as lower correlation lengths were used for inhomogeneous domains, the fracture pattern became more realistic and resembled the results that are obtained by more accurate sharp interface models [43]. In particular, the MC model predicts fractures at \(\pm(45+\phi/2)\) degree angles with respect to the compressive load direction. For the lowest correlation lengths, localized damage zones with angles roughly in the range \(\pm 45\) to \(\pm(45+\phi/2)\) are observed. These features are better resolved with the higher resolution finite element mesh, confirming that finer meshes are required for the solution of problems with more rapid variation of material properties. There are several extensions to the present work. First, the form of effective stress (8) implies that friction coefficient is zero at complete damage (\(D=1\)), whereas jointed (damaged) rock may still possess some residual friction coefficient. This will enhance the angle of localized regions in Figure 18d. Second, MC criterion is not appropriate for rock tensile fracture analysis and the damage force can be formulated by Hoek-Brown [76] and other more accurate models.