# The Generalized Mohr-Coulomb Failure Criterion
## 5 The Justification of Non-Smoothness of Failure Surfaces of Geomaterials
### Deduction from Characteristic Theory of Stress Tensors  
Suppose the geomaterial of interest is isotropic, and accordingly, the equation of the failure surface in the stress space is still written as \(F(\sigma_{1},\sigma_{2},\sigma_{3})=0\), where \(\sigma_{k}\) is the \(k\)-th principal stress (\(k=1\), \(2\), \(3\)) and, in some subspace of the principal stress space, such as \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\), regarded as the smooth function of the Voigt stress components \(\sigma_{11}\)(=\(\sigma_{x}\)), \(\sigma_{22}\)(=\(\sigma_{y}\)), \(\sigma_{33}\)(=\(\sigma_{z}\)), \(\sigma_{12}\)(=\(\tau_{xy}\)), \(\sigma_{23}\)(=\(\tau_{yz}\)), and \(\sigma_{31}\)(=\(\tau_{xx}\)). Then, the gradient of \(F\) at the stress point \(\sigma_{ij}\) is  
\[\frac{\partial F}{\partial\sigma_{ij}}=\sum_{k=1}^{3}\frac{\partial F}{ \partial\sigma_{k}}\frac{\partial\sigma_{k}}{\partial\sigma_{ij}} \tag{24}\]  
In order to calculate \(\frac{\partial\sigma_{k}}{\partial\sigma_{ij}}\), we start from the characteristic equation of stress tensor  
\[\sigma_{k}I_{k}=SI_{k} \tag{25}\]  
Here, the index \(k\) does not refer to summation; \(I_{k}\) is the \(k\)-th principal direction corresponding to \(\sigma_{k}\), with \(\left\|I_{k}\right\|_{2}=1\); and \(SI\) is the \(3\times 3\) symmetric matrix, with \(\sigma_{ij}\) or \(\sigma_{ji}\) being the components. Premultiplying \(I_{k}^{T}\) and then differentiating with respect to \(\sigma_{ij}\) on both sides of Equation (25), we have  
\[\frac{\partial\sigma_{k}}{\partial\sigma_{ij}}=\frac{\partial I_{k}^{T}}{ \partial\sigma_{ij}}SI_{k}+I_{k}^{T}\frac{\partial SI}{\partial\sigma_{ij}}I _{k}+I_{k}^{T}S\frac{\partial I_{k}}{\partial\sigma_{ij}} \tag{26}\]  
Due to \(SI=SI^{T}\), the first item on the right-hand side equals the third one, leading to  
\[\frac{\partial\sigma_{k}}{\partial\sigma_{ij}}=2\frac{\partial I_{k}^{T}}{ \partial\sigma_{ij}}SI_{k}+I_{k}^{T}\frac{\partial SI}{\partial\sigma_{ij}}I _{k} \tag{27}\]  
The substitution of Equation (25) into Equation (27) gives rise to  
\[\frac{\partial\sigma_{k}}{\partial\sigma_{ij}}=2\sigma_{k}\frac{\partial I_{k} ^{T}}{\partial\sigma_{ij}}I_{k}+I_{k}^{T}\frac{\partial SI}{\partial\sigma_{ ij}}I_{k} \tag{28}\]  
The first item of the right-hand side of Equation (28) vanishes due to \(I_{k}^{T}I_{k}=1\), leading to  
\[\frac{\partial\sigma_{k}}{\partial\sigma_{ij}}=I_{k}^{T}\frac{\partial SI}{ \partial\sigma_{ij}}I_{k}=\big{(}2-\delta_{ij}\big{)}I_{k}^{i}I_{k}^{j} \tag{29}\]where \(\delta_{ij}\) is the Kronecker delta, and none of indices, \(i\), \(j\), or \(k\), refers to summation; \(l_{k}^{i}\) is the \(i\)-th component of the \(k\)-th principal direction \(l_{k}\). It is noted that the final result of Equation (29) is derived by deeming \(\sigma_{ij}\) and \(\sigma_{ji}\) the same variable, implying that  
\[\frac{\partial\mathcal{S}}{\partial\sigma_{22}}= \begin{bmatrix}0&0&0\\ 0&1&0\\ 0&0&0\end{bmatrix} ,\,\frac{\partial\mathcal{S}}{\partial\sigma_{23}}= \begin{bmatrix} 0&0&0\\ 0&0&1\\ 0&1&0\end{bmatrix} \]  
and so on. If \(\sigma_{1}=\sigma_{2}\), either \(I_{1}\) or \(I_{2}\) is indeterminant, and any two unit vectors perpendicular to each other can be \(I_{1}\) and \(I_{2}\), which are both perpendicular to \(I_{3}\), thereby causing the partial differentials of both \(\frac{\partial\sigma_{1}}{\partial\sigma_{ij}}\) and \(\frac{\partial\sigma_{2}}{\partial\sigma_{ij}}\) indeterminant. As a result, the normal vector \(\frac{\partial F}{\partial\sigma_{ij}}\) of function \(F\) is also indeterminant at the state of \(\sigma_{1}=\sigma_{2}\) according to Equation (24). Once again, we prove that the yield surface is not smooth at \(\sigma_{1}=\sigma_{2}\). As for those smooth yield criteria, such as the Mises criterion, the yield function \(F\) can be always written as a smooth function of the stress component \(\sigma_{ij}\) in the whole principal stress space.