# The Generalized Mohr-Coulomb Failure Criterion
## 6 Comparisons with Commonly Used Failure Criteria
### Comparisons of Deviatoric Stress  
In the above, we compared performances of the four failure criteria in the Mohr stress space, which represent the capacity for them to predict the shear strength of geomaterials. Here, we compare the performances of the same four failure criteria in the stress space, which represent the capacity for them to predict both the shear strength and the deformation of geomaterials because the deformation analysis is usually carried out in the stress space rather than the Mohr stress space. Let us compare the deviatoric stress \(q\) predicted by the four criteria. The square of \(q\) is defined as  
\[q^{2}=\frac{1}{2}\Big{[}(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{ 2}+(\sigma_{1}-\sigma_{3})^{2}\Big{]} \tag{39}\]  
As we know, \(q\) plays the most important role in the shear deformation of geomaterials. In calculating \(q\) in Equation (39), \(\sigma_{2}\) and \(\sigma_{3}\) always take values from experimental tests. The value of \(\sigma_{1}\) has two choices: if \(\sigma_{1}\) takes values from experimental tests, then the relevant deviatoric stress is denoted by \(q_{T}\); if, otherwise, \(\sigma_{1}\) is calculated using Equation (30) for the L-D criterion, then the relevant deviatoric stress is represented by \(q_{LD}\). Similarly, we have the deviatoric stress \(q_{SMP}\) evaluated by the SMP criterion from Equation (35) and \(q_{GMC}\) by the GMC criterion from Equation (14). If the difference \(|q_{T}-q_{LD}|\) is the minimum, say, then the L-D criterion matches best with the test result. Figure 7: Relationships between friction angle \(\phi_{b}\) and coefficient of intermediate principal stress \(b\) (**a**) Ham River sand [35]; (**b**) Shanghai fine sand [38]; (**c**) Coarse-grained materials (\(\sigma_{3}=200\) kPa) [39]; (**d**) Mizuho tracbyte (\(\sigma_{3}=75\) MPa) [40]; (**e**) Shirahama sandstone [41]; (**f**) Taiwan siltstone [45]; (**g**) Yuubari shale (\(\sigma_{3}=25\) MPa) [41]. Figure 8: Relationships between friction angle \(\phi_{b}\) and \(b\) for geotechnical materials (**a**) Santa Monica, sand [37]; (**b**) Ottawa sand [36]; (**c**) Gravel (\(\sigma_{3}\) = 100 kPa) [46]; (**d**) Gravel (\(\sigma_{3}\) = 125 kPa) [46]; (**e**) Gravel (\(\sigma_{3}\) = 150 kPa) [46]; (**f**) Coarse-grained materials (\(\sigma_{3}\) = 600 kPa) [39]. Figures 12 and 13 display the bar charts of the deviatoric stress \(q\) given by the four failure criteria for different \(\sigma_{2}\). For all the cases,  
(1) GMC-Q matches best with the tests, and GMC-L second;  
(2) \(q_{MC}\) is the minimum, suggesting that using the M-C criterion always overestimates the shear deformation because the geomaterial is easiest to reach the deviatoric stress strength and start plastic deformation under the same external load;  
(3) \(q_{LD}\) is the maximum, implying that the application of the L-D criterion always underestimates the shear deformation because the geomaterial is hardest to reach the deviatoric stress strength and start plastic deformation under the same external load. Figure 9: Relationships between friction angle \(\phi_{b}\) and \(b\) for rocks: (**a**) Mizuho trachyte (\(\sigma_{3}=45\) MPa) [40]; (**b**) Mizuho trachyte (\(\sigma_{3}=60\) MPa) [40]; (**c**) Yuubari shale (\(\sigma_{3}=50\) MPa) [41]; (**d**) Bentheim sandstone [47]; (**e**) KTB amphibolite [48]; (f) Dunham dolomite [40]. Lastly, we show in Figure 14 the comparisons in the \(\sigma_{1}\)-\(\sigma_{2}\) plane and for some fixed \(\sigma_{3}\) between experimental data of true triaxial tests and the predictions of L-D, SMP, and GMC-Q, respectively. It is observed that for different soils and rocks, GMC-Q is always in best agreement with the test results; meanwhile, \(\sigma_{1}\) varies almost linearly with \(\sigma_{3}\) by using GMC-L.  
More test results have been collected and utilized to make comparisons. All the comparisons confirm without exception that GMC matches best with the experimental observations. To avoid lengthily repeating, the presentation concludes here.