# A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture
## 1 Introduction  
_Interfacial_, _particle_, and _bulk_ or _continuum_ models form the majority of approaches used for failure analysis of quasi-brittle materials at continuum level. Interfacial models directly represent sharp fractures in the computational domain. Some examples are _Linear Elastic Fracture Mechanics_ (LFEM), cohesive models [1; 2], and interfacial damage models [3; 4; 5; 6; 7]. Since cracks are explicitly represented, interfacial methods are deemed accurate when crack propagation is the main mechanism of material failure. However, external criteria are needed for crack nucleation and propagation (direction and extension). Moreover, accurate representation of arbitrary crack directions can be cumbersome in computational settings. Mesh adaptive schemes [8; 9; 10], _eXtended Finite Element Methods_ (XFEMs) [11; 12; 13], and _Generalized Finite Element Methods_ (GFEMs) [14; 15] address this problem to some extend. However, for highly dynamic fracture simulations and fragmentation studies, even these methods have challenges in accurate modeling of the fracture pattern. Particle methods such as _Peridynamics_[16; 17; 18] have been successfully used to model highly complex fracture patterns that are encountered in dynamic (rock) fracture. They model continua as a collection of interacting particles. _Bulk_ or _continuum_ damage models approximate the effect of material microstructural defects and their evolution, e.g., microcrack nucleation, propagation, and coalescence, through the evolutionof a damage parameter. Due to the implicit representation of microcracks and other defects, bulk damage models are more efficient than interfacial and especially particle methods. In addition, damage pattern is obtained as a part of the solution and no external criteria are needed for crack nucleation and propagation. Finally, since damage is a smooth field interpolated within finite elements, complex fracture patterns can be easily modeled by damage models, wherein the thickness of cracks is effectively regularized by the damage field. Earlier bulk damage models, however, suffered from mesh sensitivity problem where the width of the localization and damaged region was proportional to element size; as a result, finer meshes resulted in a more brittle fracture response. This problem is related to the loss of ellipticity/hyperbolicity of the (initial) boundary value problem for the earlier formulations [19; 20], and can be resolved by the introduction of an intrinsic length scale to the damage evolution formulation. In gradient-based models, this is achieved by adding higher order derivatives of the damage or strain fields to the damage evolution equation [21; 22]. In nonlocal approaches, strain or damage field employed in a local damage formulation, is in turn computed over a neighborhood of finite size [23; 24]. Finally, time-relaxed damage formulations possess an internal time parameter which through its interaction with elastic wave speeds introduce a finite length scale for the damage model in transient settings [25; 26; 27; 28]. Related to these remedies is the _phase field_ method which closely resembles a gradient-based damage model [29]. The sharper approximation of crack width is one of the main advantages of the phase field methods to gradient-based damage models [30]. We have presented a time-delay damage model for dynamic brittle fracture in [31]. The coupled elastodynamic-damage problem is solved by the _asynchronous spacetime Discontinuous Galerkin_ (aSDG) method [32; 33]. This damage model addresses the mesh sensitivity problem of the earlier damage models by the third approach discussed above, in that, damage evolution is governed by a time-delay model. In addition, the existence of a maximum damage evolution rate results in an increase in both the maximum attainable stress and toughness as the loading rate increases. This loading rate dependency of strength and toughness is experimentally verified; see for example [24; 34]. Finally, the damage evolution law is an _Ordinary Differential Equation_ in time.