# The Generalized Mohr-Coulomb Failure Criterion
## 4 The Justification of GMC  
The GMC expressed in the Mohr stress space, i.e., Equation (11), is based on two facts. Firstly, the failure of geomaterials belongs to shear failure, and the shear strength comes from the cohesion and the friction strength. Secondly, according to laboratory tests, including direct shear tests and conventional triaxial tests, shear strength decreases as \(\sigma_{2}\) approaches \(\sigma_{3}\) and reaches the minimum while \(\sigma_{2}=\sigma_{3}\) or \(b=0\), while shear strength increases as \(\sigma_{2}\) exceeds \(\sigma_{3}\).  
In the case of dense sands, for example, it has been shown in Craig [43] that the peak value of \(\phi\) in plane strain (\(b>0\)) can be \(4^{\circ}\) or \(5^{\circ}\) higher than the corresponding value obtained by conventional triaxial compression (\(b=0\)). In the case of clays, for another example [2], friction angles in plane strain (\(b>0\)) are typically 10 percent higher than that in conventional triaxial compression (\(b=0\)). These facts lead us to conclude that the friction factor \(f\) is a function of \(b\), denoted by \(f\)(b).  
Just recently, [PERSON] and [PERSON] [44] did an in-depth study on mechanism that the plane-strain frictional angle developed by cohesionless soils is above the conventional triaxial compression frictional angle.  
It can be justified that the cohesion \(c\) is also a function of \(b\), written as \(c(b)\). Here is a deduction for overconsolidated clays in conventional triaxial compression.  
According to the critical state theory [8], the state (\(p,q,v\)) of an overconsolidated clay element in the peak strength is on the Hvorslev surface with the equation  
\[q=(M-m)\exp\frac{\Gamma-v}{\lambda}+mp \tag{20}\]  
Here, \(q\) is known as the deviator stress, defined by \(q=\sigma_{1}-\sigma_{3};p\) as the mean stress, by \(p=\frac{1}{3}(\sigma_{1}+2\sigma_{3})\); and \(v\) as the specific volume, by \(v=1+e\) with \(e\) the void ratio. In Equation (20), \(M\), \(m\), \(\lambda\), and \(\Gamma\) are mechanical constants of the clay.  
The M-C criterion can also be expressed in terms of (\(p,q\)) as  
\[q=\frac{6c\cos\phi}{3-\sin\phi}+\frac{6\sin\phi}{3-\sin\phi}p \tag{21}\]  
Figure 5: Relationships between shear strength \(\tau_{f}\) and \(b\) of some geomaterials with different \(\sigma_{3}\) using GMC-L (**a**) Geotechnical materials [35; 36; 37; 38]; (**b**) Coarse-grained materials [39]; (**c**) Solenhofen limestone [40]; (**d**) Duham dolomite [40]; (**e**) Mizuho trachyte [40]; (**f**) Yuubari shale [41]; (**g**) Shirahama sandstone [41]; (**h**) Westerly granite [42].  
By comparing Equations (20) and (21), it is followed that admitting the critical state theory leads us to the conclusion that the peak friction angle \(\phi\) is also a constant that is uniquely determined by \(m\), and the peak cohesion \(c\) is a monotonically decreasing function of \(v\), i.e.,  
\[c=\frac{1}{3}\Lambda\exp\frac{\Gamma-v}{\lambda} \tag{22}\]  
with  
\[\Lambda=\frac{(M-m)(3-\sin\phi)}{6\cos\phi} \tag{23}\]  
As \(\sigma_{2}\) varies from \(\sigma_{3}\) to \(\sigma_{1}\), or as \(b\) increases from zero to one, the mean stress \(p\) increases from \(\frac{1}{3}(\sigma_{1}+2\sigma_{3})\) to \(\frac{1}{3}(2\sigma_{1}+\sigma_{3})\), because \(p\) is related to \(b\) by  
\[p\equiv\frac{1}{3}(\sigma_{1}+\sigma_{2}+\sigma_{3})=\frac{1}{3}(\sigma_{1}+2 \sigma_{3})+\frac{1}{3}(\sigma_{1}-\sigma_{3})b\]  
Prior to the peak strength, \(v\) is dominantly controlled by \(p\), and hence by \(b\). Therefore, the cohesion \(c\) is a function of \(b\) according to Equation (22). This concludes the deduction of the GMC criterion.