# A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture
## 2 Formulation
### Realization of Stochastic Damage Model Parameters  
As discussed in Section 1, incorporating material inhomogeneity is quite important to capture realistic failure response of quasi-brittle materials. The inhomogeneity is both in elastic and fracture properties. If an isotropic material model is assumed at the mesoscale, often only the elastic modulus is deemed to be a random field as in [54]. However, in general the entire elasticity tensor should be considered as a tensorial random field. However, often due to the higher effect that fracture properties have on macroscopic failure response, only they are considered to be random and inhomogeneous. For a general fracture model, strength, energy, and initial damage state are the main model parameters. For the MC model, the friction angle \(\phi\) (or friction coefficient \(k\)) and cohesion \(\underline{\varepsilon}\) are the model parameters used to determine \(c\) and \(D_{f}\) from (2) and (5), respectively. Cohesion is the parameter that is associated with fracture strength. The relaxation time \(\tau_{c}\) in (6) and brittleness factor \(\beta\) determine the area under the strain-stress curve for different loading rates in Figure 4. That is, they determine the fracture energy of the damage model. Finally, the initial condition for damage parameter, \(D(\mathbf{x},t=0)\), corresponds to the initial state of material. In the present work, among strength, energy, and initial damage parameters, we consider inhomogeneity only in the strength property. This is in accord with a majority of similar studies in the literature such as [55; 56; 57; 58; 59; 60]. Accordingly, the only random field in the present study is cohesion \(\underline{\varepsilon}\). For a macroscopically homogeneous material, the point-wise and two-point statistics of the random field are spatially uniform. For the point-wise statistics, the mean and standard deviation of the random field are the main parameters. For the two-point statistics the form of the correlation function and the correlation length, i.e., the length scale at which the field spatially varies are the main parameters. In [54], where the elastic modulus is considered to a random field, standard deviation and correlation length of the random field are considered as the main parameters that impact fracture response. The realization of random fields for fracture strength and the subsequent fracture analysis becomes more expensive as the correlation length tends to zero. In [54] it is shown that certain macroscopic fracture statistics converge as the correlation length tends to zero. That is, by maintaining sufficient level of material inhomogeneity through using a small enough correlation length, accurate representation of macroscopic fracture response can be obtained. We treat cohesion as a stationary random field with certain standard deviation \(\varsigma_{c}\) and correlation length \(l_{c}\). The statistics of this random field can be systematically obtained by using _Statistical Volume Elements_ (SVEs), as shown in [60; 61]. However, for simplicity and better control on the effect of these parameters, we artificially manufacture random fields with certain \(\varsigma_{c}\) and \(l_{c}\). The distribution of \(\underline{\varepsilon}\) is assumed to follow a \(\mathrm{Lognormal}(\mu_{c},\varsigma_{c})\) probability structure where \(\mu_{c}\) and \(\varsigma_{c}\) are the mean and standard deviation of the normal field. The corresponding mean and standard deviation of the log normal field for \(\underline{\varepsilon}\) are \(M_{c}=\exp\left(\mu_{c}+\varsigma_{c}^{2}/2\right)\) and \(\Sigma_{c}=\exp(\mu_{c}+\varsigma_{c}^{2}/2)\sqrt{\exp(\varsigma_{c}^{2})-1}\). Once the underlying correlation function form and length, and point-wise _Probability Distribution Function_ (PDF) are specified, there are a number of statistical methods to realize consistent random fields.