# A Three-Dimensional Elastoplastic Constitutive Model for Geomaterials
## 3 Three-Dimensional Generalized Mohr-Coulomb Plastic Model  
The effect of \(\sigma_{2}\) on the shear strength is not considered by the M-C failure criterion. Alternatively, this condition might be seen as just being applicable in the case where \(\sigma_{2}=\sigma_{3}\). However, it is impossible to overlook the impact of \(\sigma_{2}\) on the geotechnical materials' shear strength [18]. For over-consolidated clays, plane strain angles, for example, are typically 10 percent higher than the comparable value obtained using traditional triaxial tests. Additionally, the peak value of \(\phi\) in plane strain for dense sands may be \(4^{\circ}\) or \(5^{\circ}\) higher [34].  
\(\sigma_{2}\) always varies in the interval of \([\sigma_{3},\sigma_{1}]\). From past experiences, the geotechnical materials' shear strength gradually increases with the increase in \(\sigma_{2}\). That is, when \(\sigma_{2}=\sigma_{3}\), the shear strength is the minimum; when \(\sigma_{2}=\sigma_{1}\), it is the maximum. [PERSON] [35] also observed similar situations.  
To reflect the influence of \(\sigma_{2}\) on the shear strength, we assume the M-C criterion still holds in form, but the cohesion \(c\) and friction factor \(f\) are related to the coefficient \(b\) of the intermediate principle stress \(\sigma_{2}\) by  
\[c_{b}=c(b)\]  
and  
\[f_{b}=f(b),\]  
respectively; with \(b\) defined as  
\[b=\frac{\sigma_{2}-\sigma_{3}}{\sigma_{1}-\sigma_{3}} \tag{5}\]  
Since \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\) is specified, \(0\leq b\leq 1\).  
Therefore, the generalized Mohr-Coulomb criteria (GMC) adopts this form in the Mohr stress space:  
\[\tau_{f}=c(b)+f(b)\sigma_{n} \tag{6}\]  
Figure 1: Haigh-Westergaard stress space: (**a**) Principal stress space; (**b**) \(\pi\) plane.  
The simplest way to obtain functions \(c(b)\) and \(f(b)\) is to apply the linear Lagrange interpolation to a series of data, \((c_{i},f_{i};b_{i})\), \(i=0,\cdots\), with \(f_{i}=tg\phi_{i}\), which are obtained from conventional or true triaxial tests.  
Particularly, we can interpolate the shear strength parameter of \((c_{0},f_{0})\) from the conventional triaxial compression test with \(b=0\), and that of \((c_{1},f_{1})\) from the conventional triaxial extension test with \(b=1\), to obtain \(c(b)\) and \(f(b)\), reading  
\[c(b)=c_{0}l_{0}(b)+c_{1}l_{1}(b) \tag{7}\]  
and  
\[f(b)=f_{0}l_{0}(b)+f_{1}l_{1}(b) \tag{8}\]  
where \(l_{0}(b)\) and \(l_{1}(b)\) are the linear Lagrange interpolation functions, i.e.,  
\[l_{0}(b)=1-b,l_{1}(b)=b \tag{9}\]  
In this way, a three-dimensional failure criterion that considers the intermediate principal stress's influence may be developed using just a piece of traditional triaxial test equipment.  
We can obtain the yield surface of the GMC-L in the principal stress space:  
\[(\sigma_{1}-\sigma_{3})-(\sigma_{1}+\sigma_{3})\sin\phi_{b}-2c_{b}\cos\phi_{b}=0 \tag{10}\]  
Each \(\phi_{b}\) corresponds to a unique \(b\) value.  
Figure 2a shows the yield image of the GMC-L criterion and the M-C failure criterion on the principal stress space. The GMC-L and M-C criteria are similar in shape and are hexagonal pyramids. Figure 2b shows the comparison of the GMC-L and M-C criteria on the \(\pi\) plane. The GMC-L surface contains the M-C surface inside but coincides with the M-C surface at vertex \(A\) (\(b=0\)).  
Figure 2: Failure envelope in principal stress space: (**a**) M-C and GMC-L criteria; (**b**) projection on \(\pi\) plane.