# A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture
## 1 Introduction
This greatly simplifies the damage model formulation and lends itself to the aSDG method; the aSDG method directly discretizes spacetime by elements that satisfy the causality constraint of the underlying hyperbolic problem being solved. The nonlocal damage models violate this causality constraint, whereas the majority of gradient-based damage models are not hyperbolic. In contrast, the time-delay damage model maintains the hyperbolicity of the elastodynamic problem. Besides, the ODE form of the governing equation greatly simplifies the application of initial and boundary conditions for the coupled problem. The distribution of material defects at microstructure can have a great effect on macroscopic fracture response, particularly for quasi-brittle materials. Some examples are high variability in fracture pattern for samples with the same loading and geometry [35], high sensitivity of macroscopic strength and fracture toughness to microstructural variations [36], and the so-called size effect [37; 38; 39], i.e., the decrease of the mean and variations of fracture strength for larger samples. Weibull model [40; 41] is one of the popular approaches for modeling the effect of defects in quasi-brittle fracture, particularly the size effect. We have used the Weibull model in the context of an interfacial damage model to capture statistical fracture response of rock, in hydraulic fracturing [42], fracture under dynamic compressive loading [43], and in fragmentation studies [44; 45]. However, these models are computationally expensive due to the use of a sharp interfacial damage model. In this manuscript, we propose a stochastic bulk damage model for rock fracture. There are two main differences to the damage model presented in [31]. First, in damage mechanics often only the spectral positive part of either strain or elastic stress tensor is used to drive damage accumulation. Moreover, upon full damage, only the negative part of the stress tensor is maintained in forming the effective stress. While these choices are appropriate for tensile-dominant fracture, they have some shortcomings for rock fracture under compressive loading. Specifically, using these models damage does not accumulate under compressive loading; even if it could, it would not have modeled the failure process as the effective stress remains the same as the elastic stress of the intact rock. Herein, we propose a new damage model based on the Mohr-Coulomb failure criterion and an effective stress that correctly represents rock failure in compression. Second, we employ a stochastic damage model wherein rock cohesion is treated as a random field. This aspect is important for the uniaxial compression examples considered, as due to the lack of macroscopic stress concentration points highly unrealistic fracture patterns will be obtained by using a homogeneous rock mass model. We note that the use of a bulk damage model makes the proposed approach significantly more efficient than the stochastic fracture problems [42; 43; 44; 45] studies by the authors using an interfacial damage model. The outline of the manuscript is as follows. The formulation of the stochastic damage model, its coupling to elastodynamic problem, and the aSDG method are discussed in Section 2. We use a dynamic uniaxial compressive example to demonstrate the effect of material inhomogeneity on fracture response in Section 3. Final conclusions are drawn in Section 4.