## 2 Plasticity Formulation
### Return Mapping Algorithm
#### 2.2.1 Return-Mapping Scheme for the 1D Model
If the final stress is located on the left edge of the cone relative to the sextant plan, then the point is singular and also two mechanisms are activated, \(m=2\). If the final stress is not located inside the load surface or on an edge, then it is projected to the top of the cone, the point is singular and six mechanisms are activated, \(m=6\). Figure 2 illustrates the plastic flow rule of Mohr-Coulomb's law. Figure 2a shows how, according to the condition \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\) linking the principal stresses, the main plane and the two possible configurations for two mechanisms are activated simultaneously: the right-edge mechanism or the left-edge mechanism and the apex (Figure 2b). For a single activated mechanism, the plastic flow rule is written:

\[\dot{\epsilon}^{p}=\dot{\gamma}\ N^{a} \tag{19}\]

with \(N^{a}\) the flow vector normal to the corresponding plane defined by \(f_{1}=0\), given by:

\[N^{a}=\frac{\partial g_{1}}{\partial\sigma}=(1+\sin\psi)e_{1}\otimes e_{1}-( 1-\sin\psi)e_{3}\otimes e_{3} \tag{20}\]

For two activated mechanisms, the plastic flow rule is written:

\[\dot{\epsilon}^{p}=\dot{\gamma}^{a}N^{a}+\dot{\gamma}^{b}N^{b} \tag{21}\]

with \(N^{b}\) the normal at the intersection of the two planes \(f_{1}=0\) and \(f_{6}=0\) or \(f_{1}=0\) and \(f_{2}=0\), given by:

\[N^{b}=\left\{ \begin{array}{l}N^{6}=(1+\sin\psi)e_{1}\otimes e_{1}-(1-\sin \psi)e_{2}\otimes e_{2}\\ N^{2}=(1+\sin\psi)e_{2}\otimes e_{2}-(1-\sin\psi)e_{3}\otimes e_{3}\end{array} \right. \tag{22}\]

For six activated mechanisms, the plastic flow rule is written

\[\dot{\epsilon}^{p}=\sum_{i=1}^{6}\dot{\gamma}_{i}N^{i} \tag{23}\]

Figure 2: The Mohr–Coulomb flow rule; (**a**) faces and edges, and (**b**) apex [1]. The updated stress tensor in the return mapping algorithm is generally given by equation Equation (24). For an isotropic model, and taking into account the elastic rigidity matrix linking the principal stresses and strains given to Equation (5), [PERSON]'s constitutive law written in principal stresses is ([1,19]):

\[\sigma_{j}=\sigma_{j}^{trial}-\sum_{i=1}^{6}\Delta\gamma^{i}\bigg{(}2G\Big{[}N _{d}^{i}\Big{]}_{j}+K\,N_{v}^{i}\bigg{)},\,\,\,j=1,2,3 \tag{24}\]

where \(N_{v}^{i}\equiv\text{tr}\Big{[}N^{i}\Big{]}\) represents the \(i\)-th volumetric component of the updated plastic flow vector, \(\text{tr}\big{[}\,\big{]}\) the trace of the second-order tensor and \(\Big{[}N_{d}^{i}\Big{]}_{j}\) the \(j\)-th eigenvalue of its deviatoric projection. From Equation (24), four return mappings of Mohr-Coulomb's law according to the active mechanisms follow: to the smooth portion, to the right edge, to the left edge, and to the apex. (a) _Return mapping to the smooth portion_\((\sigma_{1}>\sigma_{2}>\sigma_{3})\)

For a single activated mechanism and taking into account its flow rule, the principal updated stresses give:

\[\left\{ \begin{array}{cc}\sigma_{1}=\sigma_{1}^{trial}-\Big{[}2G\Big{(}1+ \frac{1}{3}\sin\psi\Big{)}+2K\sin\psi\Big{]}\Delta\gamma&(a)\\ \sigma_{2}=\sigma_{2}^{trial}+\Big{(}\frac{4}{3}G-2K\Big{)}\Delta\gamma\,\, \sin\psi&(b)\\ \sigma_{3}=\sigma_{3}^{trial}+\Big{[}2G\Big{(}1-\frac{1}{3}\sin\psi\Big{)}-2K \sin\psi\Big{]}\Delta\gamma&(c)\end{array} \right. \tag{25}\]

with the accumulated plastic strain:

\[\bar{z}_{n+1}^{p}=\bar{z}_{n}^{p}+2\Delta\gamma\cos\phi \tag{26}\]

By combining Equation (25a,c) we have:

\[\left\{ \begin{array}{cc}\sigma_{1}-\sigma_{3}=\sigma_{1}^{trial}-\sigma_{3 }^{trial}-4G\Delta\gamma\\ \sigma_{1}+\sigma_{3}=\sigma_{1}^{trial}+\sigma_{3}^{trial}-4\Big{(}\frac{G}{ 3}+K\Big{)}\Delta\gamma\sin\psi\end{array} \right.