# A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture
## 2 Formulation
### Properties of the Damage Model
#### 2.3.2 Damage Evolution: Rate Effects and Mesh Sensitivity  
Figure 4 compares strain stress responses for three different model and loading scenarios. The loading considered can correspond to any of the strengths in (4). The nondimensional scalar elastic stress, strain, and effective stress are defined as \(\sigma^{\prime}=\sigma/\underline{\sigma}\), \(\epsilon^{\prime}=\sigma^{\prime}=C\epsilon/\underline{\sigma}\), and \(\sigma^{\prime}_{\text{eff}}=\sigma_{\text{eff}}/\underline{\sigma}\), respectively, where \(\sigma\), \(\epsilon\), and \(\sigma_{\text{eff}}\) are the scalar elastic stress, strain, and effective stress. (Note that the scalar elastic stress measure \(\sigma\) in this section is different from the normal stress component in the Mohr-Coulomb criterion; cf. Figure 1.) These scalar values, \(\underline{\sigma}\), and stiffness \(C\) correspond to a particular loading condition; for example for uniaxial tensile loading \(\sigma=\sigma_{xx}\), \(\epsilon=\epsilon_{xx}\), \(\underline{\sigma}=\underline{\sigma}_{\text{at}}\), (cf. (4b)). The corresponding stiffness is \(C=E\) and \(E/(1-\
u^{2})\), for plane stress and plane strain conditions, respectively, where \(E\) and \(\
u\) are the elastic modulus and Poisson ratio.  
As loading (\(\epsilon^{\prime}\)) increases, the scalar stress \(c\) increases until \(c=\underline{c}\) in Figure 2 for the given loading condition. This corresponds to \(\epsilon^{\prime}=1\). For the MC model, material is deemed to fail instantaneously, for \(\sigma^{\prime}_{\text{eff}}=\sigma^{\prime}=1\). This sudden failure is shown by the green circle in the figure. The damage model regularizes the MC failure criterion. For the quasi-static loading \(\dot{D}\approx 0\), thus \(D\approx D_{f}\) throughout the loading. Given the linear dependence of \(D_{f}\) on \(c\) in (5), \(\sigma_{\text{eff}}\) linearly decreases from unity to zero as \(\epsilon^{\prime}\) increases from unity to \(\overline{c}/\underline{c}=1/\beta\). As, \(\beta\to 1\) the response of the damage model tends to that of the un-regularized MC model, clarifying why \(\beta\) is called the brittleness factor. Regardless of the rate of loading for \(\epsilon^{\prime}\), \(\dot{D}\) remains bounded by \(1/\tau_{\dot{c}}\); cf. (6). This results in a delayed damage response where \(D\) falls far behind its quasi-static limit \(D_{f}\) for higher rates of loading for \(\epsilon^{\prime}\). This, in turn, increases the maximum effective stress, \(\max(\sigma_{\text{eff}})\), failure strain, \(\epsilon^{\prime}(D=1)\), and toughness, i.e., the area under the strain-stress curve. That is, the time-delay evolution law (6) can qualitatively model material's well-known stress rate effect. The dynamic solution in Figure 4 corresponds to a nondimensional strain rate of 3. For lower and higher nondimensional loading rates, the stress response gets closer to the quasi-static response and further expands, respectively.  
If the quasi-static damage model \(D=D_{f}\) where to be used, it would suffer the mesh-sensitivity problem of the early damage models. The introduction of an intrinsic length scale addresses this issue. The length scale \(l_{d}\) is either used in conjunction of added higher spatial order derivative terms in a local damage model [21, 22] or by nonlocal integration of certain fields, e.g., strain, over neighborhoods of size \(l_{d}\). However, both approaches are computationally expensive. The proposed damage model is much easier to implement, since it is simply an ODE in time. It also maintains the hyperbolicity of the elastodynamic problem which is critical for the solution of the coupled problem by the aSDG problem. Finally, the interaction of elastic wave speeds with the intrinsic time scale \(\tau_{c}\) indirectly introduces a length scale \(l_{d}\) for the damage problem. While this length scale is not relevant for very low rate loading problems [52], at moderate to high loading rates it is expected to resolve the mesh sensitivity problem of local damage models.