## 4 Results for a Medium-to-Strong Black Box Rock
### Failure Envelopes for the Three Basic Types of Stress Paths
So, along the stress path or the 45\({}^{\circ}\) line, the failure occurred about 21% earlier, i.e., the distance between (0.0, 79.4) MPa and (8.2, 71.2) MPa vs. the distance between (0.0, 79.4) MPa and (39.7, 39.7) MPa. The conclusion for these three simulations is that the stress path had a significant effect on the failure load. Figure 5: Failure envelopes for the three basic types of stress paths (i.e., uniaxial loading (S1(loa),S3(=)), uniaxial unloading (S1(=),S3(unl)), and simultaneous loading and unloading (S1(loa),S3(unl)). The start of each stress path (i.e., an isotropic stress state) and failed stress states are indicated by crosses. The other simulations (Figure 5, Table 1) showed the same trend. The unloading of the minor principal stress only, i.e., S1(=),S3(unl), resulted in the weakest failure criterium. By simultaneously increasing the major principal stress and decreasing the minor principal stress, i.e., S1(loa),S3(unl), its failure criterium was situated between the two other criteria. The three failure criteria widened for larger stress levels. Or, in other words, they were not parallel. However, the relative effect remained similar. For example, for a confinement of 10 MPa, the failure for uniaxial loading was reached at 137.6 MPa. Failure for a uniaxial unloading from an initial isotropic stress level of 137.6 MPa was reached for a minor principal stress of 63.6 MPa instead of 10 MPa. In other words, the failure occurred about 39% earlier (53.6 MPa vs. 137.6 MPa). This percentage was 38% for a zero confinement (See Table 1 for other cases). Figure 6 shows the circles of Mohr for all simulations in a shear stress-normal stress diagram. Per type of stress path, a linear failure criterium was fitted by hand. These fits were acceptable, but the uniaxial loading would benefit most from a non-linear failure criterium. The cohesion (and friction angle) values for the three fitted criteria were about 22 MPa (38\({}^{\circ}\)), 10 MPa (15\({}^{\circ}\)), and 20 MPa (22\({}^{\circ}\)) for S1(loa),S3(=), S1(=),S3(unl), and S1(loa),S3(unl), respectively. As for the presentation in a major-minor principal stress diagram, the three failure criteria are separated clearly. \begin{table}
\begin{tabular}{l l l} \hline \hline
**(a)** & \\ \hline Initial Stress State & Stress State at Failure & \\ \hline Isotropic Stresses, & Minor Principal & Major Principal \\ MPa & Stress (S3), MPa & Stress (S1), MPa \\ \hline
0.0 & 0.0 & 79.4 \\
5.0 & 5.0 & 104.2 \\
10.0 & 10.0 & 137.6 \\
15.0 & 15.0 & 157.3 \\ \hline
**(b)** & \\ \hline Initial stress state & Stress state at failure & Strength reduction for S3, \\ \hline Isotropic stresses, & Minor principal & Major principal \\ MPa & stress (S3), MPa & stress (S1), MPa \\ \hline
79.4 & 30.1 & 79.4 \\
104.2 & 46.6 & 104.2 \\
137.6 & 63.6 & 137.6 \\
157.3 & 75.6 & 157.3 \\ \hline
**(c)** & \\ \hline Initial stress state & Stress state at failure & Strength reduction for S3 and \\ \hline Isotropic stresses, & Minor principal & Major principal \\ MPa & stress (S3), MPa & stress (S1), MPa \\ \hline
39.7 & 8.2 & 71.2 \\
54.6 & 15.4 & 93.8 \\
73.8 & 28.6 & 119.0 \\
86.2 & 35.5 & 136.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Strength values for the three types of stress paths. (a) Uniaxial loading (S1(loa),S3(=)), i.e., reference for other stress paths. (b) Uniaxial unloading (S1(=),S3(unl)) and reduction along the stress path in comparison to uniaxial loading. (c) Simultaneous loading and unloading (S1(=),S3(=)) and reduction along the stress path in comparison to uniaxial loading.