# Failure in Confined Brazilian Tests on Sandstone
## 3 Results
### Experimental Results
Both figures confirm the role of the intermediate stress on the rock strength. The Mogi-Coulomb failure criterion has been used to describe strength of materials where the intermediate principal stress has been shown to be important [31]. The Mogi-Coulomb criterion is a more general case of the Mohr-Coulomb criterion which ignores the intermediate principal stress. The Mogi-Coulomb criterion is defined in terms of the octahedral shear stress and the mean normal stress:  
\[\tau_{\text{oct}}=\frac{1}{3}\sqrt{\left(\sigma_{1}-\sigma_{2}\right)^{2}+ \left(\sigma_{2}-\sigma_{3}\right)^{2}+\left(\sigma_{1}-\sigma_{3}\right)^{2}} \tag{4}\]  
\[\sigma_{\text{m,2}}=\ \frac{1}{2}(\sigma_{1}+\sigma_{3}) \tag{5}\]  
Figure 5: Minimum and maximum principal stresses at failure for a series of confined, flattened Brazilian disks on sandstone. Average values at each confining stress, including unconfined, are represented by an open point. Uniaxial compression strength (UCS) on the same sandstone is also shown. The dashed line represents the best fit failure curve. The Mogi-Coulomb criterion is typically represented as a straight line in terms of the octahedral shear stress and mean normal stress. The Mogi-Coulomb criterion is typically applied to failure under compressive stresses, and represents strength derived from frictional (shear) processes. The strength data from the confined Brazilian tests are given in terms of these stresses in Figure 8, and are found to fit to the straight line:  
\[\tau_{\text{oct}}=13.354+0.571\sigma_{\text{m},2} \tag{6}\]  
where R\({}^{2}\) is 0.9902. This good fit to the Mogi-Coulomb criterion suggests frictional processes are largely controlling strength in much of the confined tension region as suggested by [PERSON] (2017). In addition, the unconfined compressive strength data are fit well by this straight line, suggesting that accounting for the intermediate principal stress reconciles strengths determined under different stress paths and stress conditions. The unconfined  
Figure 6: Intermediate and maximum principal stresses at failure for a series of confined, flattened Brazilian disks on sandstone. Average values at each confining stress, including unconfined, are represented by an open point. The dashed line represents the best fit failure curve. Figure 7: Intermediate and minimum principal stresses at failure for a series of confined, flattened Brazilian disks on sandstone. Average values at each confining stress, including unconfined, are represented by an open point. The dashed line represents the best fit failure curve. Brazilian strength data fall below the extrapolation of the straight line, consistent with the strength in this configuration having a greater dependence on tensile strength and less on frictional processes during crack initiation due to more open micro-flaws in the absence of confinement. Additionally, included in the figure is the range of strengths expected from a direct tensile test. The direct tensile strength (DTS) is expected to be related to the unconfined Brazilian tensile strength (BTS) by:  
\[\text{DTS}=\text{A}\times\text{BTS} \tag{7}\]  
where A is a constant ranging from 0.6 to 0.8. From the analysis of many datasets, [PERSON] and [PERSON] [7], found A for sedimentary rocks as 0.69. [PERSON] and [PERSON] [32] reported an average value of A to be 0.77 for all rock types. A line has been drawn that connects the DTS, BTS and straight-line portion of the failure surface defined by the Mogi-Coulomb criterion. Along this line, the failure mode transitions from pure tensile to progressively more frictional (shear).