# The Generalized Mohr-Coulomb Failure Criterion
## 3 The Generalized Mohr-Coulomb Criterion  
The M-C criterion assumes that the effect of the intermediate principal stress \(\sigma_{2}\) on the shear strength is negligible. However, it has been repeatedly confirmed that this is not always the case. In order to reflect the influence of \(\sigma_{2}\) on shear strength, we assume the M-C criterion still holds in form, but cohesion \(c\) and friction factor \(f\) are related to the coefficient \(b\) of the intermediate principal stress by  
\[c_{b}=c(b)\]  
and  
\[f_{b}=f(b)\]  
respectively, with \(b\) being the coefficient of the intermediate principal stress, defined as  
\[b=\frac{(\sigma_{2}-\sigma_{3})}{(\sigma_{1}-\sigma_{3})} \tag{10}\]  
Since \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\) is specified, we have \(0\leq b\leq 1\). As a result, the generalized Mohr-Coulomb criterion (GMC) takes form in the Mohr stress space. \[\tau_{f}=c(b)+f(b)\sigma_{n} \tag{11}\]  
From the perspective of differential geometry, the strength envelope defined by Equation (11) is a ruled surface [34] in the generalized Mohr stress space of \((\sigma,\tau;b)\), with the line  
\[\left\{ \begin{array}{l}\sigma=0\\ \tau=c(b)\end{array} \right. \tag{12}\]  
being a direction of the surface; and  
\[\left\{ \begin{array}{l}\tau=c(b)\\ b=\text{const}\end{array} \right. \tag{13}\]  
being one of the rulings. Figure 2 illustrates a GMC surface in the generalized Mohr stress space of \((\sigma,\tau;b)\). Similarly, we have the failure locus of the GMC in the principal stress space  
\[(\sigma_{1}-\sigma_{3})-(\sigma_{1}+\sigma_{3})\sin\phi(b)-2c(b)\cos\phi(b)=0 \tag{14}\]  
with \(\phi(b)=\text{tg}^{-1}f(b)\). The simplest way to obtain functions \(c(b)\) and \(\phi(b)\) is to apply the Lagrange interpolation to a series of data, \((c_{i},f_{i};b_{i})\), \(i=1,\cdots\), with \(f_{i}=\text{tg}\phi_{i}\), which are based on conventional or true triaxial tests and calculated using Equations (8) and (9). Particularly, we can interpolate the shear strength parameters of \((c_{0},f_{0})\) measured from the conventional triaxial compression test (\(b\) = 0), and \((c_{0},f_{0})\) from the conventional triaxial extension (\(b\) = 1) to obtain \(c(b)\) and \(\phi(b)\), reading  
\[c(b)=c_{0}L_{0}(b)+c_{1}L_{1}(b) \tag{15}\]  
and  
\[f(b)=f_{0}L_{0}(b)+f_{1}L_{1}(b) \tag{16}\]  
where \(L_{0}(b)\) and \(L_{1}(b)\) are the linear Lagrange interpolation functions, i.e.,  
\[L_{0}(b)=1-b,\;L_{1}(b)=b \tag{17}\]  
In this way, only a piece of conventional triaxial test apparatus is adequate to develop a three-dimensional failure criterion that considers the effect of \(\sigma_{2}\). Even so, we will see shortly the GMC performs better than those celebrated failure criteria, and denote by GMC-L the GMC corresponding to \((c,f)\) obtained by the liner Lagrange interpolation. Similarly, the GMC corresponding to the quadric Lagrange interpolation to \(c\) and \(f\) is represented by GMC-Q, namely,  
\[c(b)=\sum_{k=1}^{3}c_{k}L_{k}(b) \tag{18}\]  
and  
\[f(b)=\sum_{k=1}^{3}f_{k}L_{k}(b) \tag{19}\]  
where \((c_{k},f_{k}),k\) = 1, 2, 3, are shear strength parameters corresponding to \(b_{k}\), and \(L_{k}\) quadric Lagrange interpolation functions, defined by  
\(L_{1}(b)=\frac{(b-b_{2})(b-b_{3})}{(b_{1}-b_{2})(b_{1}-b_{3})}\), \(L_{2}(b)=\frac{(b-b_{3})(b-b_{1})}{(b_{2}-b_{3})(b_{2}-b_{1})}\), and \(L_{3}(b)=\frac{(b-b_{1})(b-b_{2})}{(b_{3}-b_{1})(b_{3}-b_{2})}\). Here, \(b_{1}\) = 0, \(0<b_{2}<1\), and \(b_{3}\) = 1 are the most selected for \(b_{k}\). Figure 3 shows the surfaces of the M-C and the GMC on the \(\pi\) plane corresponding to some mean pressure \(p(=l_{1}/3)\), respectively, suggesting that the GMC surface contains the M-C surface, but coincides with the M-C surface at vertex \(A\) (\(b\) = 0).