# A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture
## 2 Formulation
### Bulk Damage Problem Description  
#### 2.1.1 Damage Driving Force  
As will be discussed in Section 2.2, the damage parameter \(D\in[0,\ 1]\) gradually reduces the elasticity stiffness in the process of material degradation. Damage evolution if generally driven by the strain field \(\epsilon\). For the remainder of the manuscript, we assume that the spatial dimension is two. The symmetric _elastic stress tensor_\(\sigma\) is defined as,  
\[\sigma=\mathbf{C}\epsilon,\qquad\text{where}\quad\sigma= \begin{bmatrix}\sigma_ {xx}&\sigma_{xy}\\ \sigma_{yx}&\sigma_{yy}\end{bmatrix} \quad\text{and}\quad\epsilon= \begin{bmatrix} \epsilon_{xx}&\epsilon_{xy}\\ \epsilon_{yx}&\epsilon_{yy}\end{bmatrix}  \tag{1}\]  
are the expressions of stress and strain tensors in global coordinate system \((x,y)\) and \(\mathbf{C}\) is the elasticity tensor. Instead of \(\epsilon\), damage evolution can be expressed in terms of \(\sigma\). This is more suitable for rock fracture given that many known _failure criteria_ such as _Mohr-Coulomb_ (MC) or Hoek-Brown [46] are expressed in terms of the stress tensor. Figure 1 shows the Mohr-Coulomb failure criterion in terms of normal \(\sigma\) and shear \(\tau\) traction components on a fracture surface. We employ the tensile positive convention for \(\sigma\). The failure criterion is determined by the _colhesion_\(\underline{c}\) and _friction angle_\(\phi=\tan^{-1}(k)\), where \(k\) is the friction coefficient. In the figure, the Mohr circle for a stress tensor A (red semi-circle) corresponding to principal stresses \(\sigma_{2}<\sigma_{1}\) is shown. Since only isotropic rocks are considered herein, \(\underline{c}\) and \(\phi\) are assumed to be constant with respect to the orientation of principal stresses (with respect to the global coordinate system axes). We define the _scalar stress_ as,  
\[c(\sigma,\phi):=\frac{R}{\cos\phi}+\sigma_{ave}\tan\phi \tag{2}\]  
where as shown in Figure 1, \(c\) is the ordinate of the tangent line on the Mohr-circle with angle \(\phi\), and the radius \(R\) and average normal stress \(\sigma_{ave}\) are given by,\[R :=\frac{\sigma_{1}-\sigma_{2}}{2}=\sqrt{\frac{(\sigma_{xx}-\sigma_{yy })^{2}}{4}+\sigma_{xy}{}^{2}}, \tag{3a}\] \[\sigma_{ave} :=\frac{\sigma_{1}+\sigma_{2}}{2}=\frac{\sigma_{xx}+\sigma_{yy}}{ 2}, \tag{3b}\]  
Figure 1 shows two stress states. For the stress state B, the entire Mohr circle is below the failure criterion curve, thus no degradation is expected. For the stress state A, the Mohr circle expands beyond the failure criterion curve; in a binary intact and failed classification, this stress state would be considered failed. These stages correspond to \(c(\sigma,\phi)<\underline{c}\) and \(c(\sigma,\phi)\geq\underline{c}\), respectively. Some specific strengths corresponding to the MC criterion \(c(\sigma,\phi)=\underline{c}\) are shown in Figure 2 and are given by,  
\[\underline{s}_{\rm ht} =\frac{\underline{c}}{\tan\phi}\] Hydrostatic tensile strength (4a) \[\underline{s}_{\rm at} =\frac{2\underline{c}\cos\phi}{1+\sin\phi}\] Uniaxial tensile strength (4b) \[\underline{s}_{\rm s} =\underline{c}\cos\phi\] Shear strength (4c) \[\underline{s}_{\rm ac} =\frac{2\underline{c}\cos\phi}{1-\sin\phi}\] Uniaxial compressive strength (4d)  
Figure 1: Mohr-Coulomb failure criterion and scalar stress \(c\) for a given stress state. As will become clear later, the damage model, regularizes the process of failure. Otherwise, failure for a stress state occurs instantaneously once MC criterion \(c(\sigma,\phi)=\underline{c}\) is satisfied; for example, when \(\sigma_{xx}=\sigma_{yy}>0\) reaches \(\mathrm{s}_{\mathrm{ht}}\) (\(\sigma_{xy}=0\)). To facilitate this, the _damage force_ is defined as,  
\[D_{f}(c,\underline{c},\overline{c}):= \begin{cases}0&c\leq\underline{c}\\ \frac{c-\underline{c}}{\overline{c}-\underline{c}}&\underline{c}<c<\overline {c}\\ 1&\overline{c}\leq c\end{cases}