## 2 Plasticity Formulation
### Return Mapping Algorithm
#### 2.2.1 Return-Mapping Scheme for the 1D Model
\begin{array}{c}\Delta\gamma^{a1}=\frac{f_{1}^{trial}-b_{1}\Delta \gamma^{b1}}{Det_{1}}\\ \Delta\gamma^{b1}=\frac{f_{6}^{trial}-b_{1}\Delta\gamma^{a1}}{a}\end{array} \right. \tag{38}\]

with

\[Det_{1}=a^{2}-b_{1}^{2}\]

**Remark 1**. : _Regardless of the <<physical>> values taken by parameters K, G, \(\phi\) and \(\psi\), \(Det_{1}\
eq 0\). (see Figure 3)._

The fixed-point algorithm applied to the return mapping of the right edge mechanism (Algorithm 4) is based on the constitutive law given in Equation (30). Its initial conditions are also taken in the elastic domain (\(\sigma_{1}^{trial},\sigma_{2}^{trial},\sigma_{3}^{trial},0,0,\ \bar{\varepsilon}_{n}^{p}=0\)). Figure 3: Example of \(Det_{1}\) values for \(K=1.67\times 10^{8}\) and \(G=6.39\times 10^{5}\). _Return mapping to the left edge_ (\(\sigma_{1}=\sigma_{2}>\sigma_{3}\))

For two mechanisms activated on the left edge, principal updated stresses are given by:

\[\left\{\begin{array}{c}\sigma_{1}=\sigma_{1}^{trial}-\Big{[}2G\Big{(}1+\frac{1} {3}\sin\psi\Big{)}+2K\sin\psi\Big{]}\Delta\gamma^{\prime 2}+\Big{(}\frac{4}{3}G-2K \Big{)}\Delta\gamma^{\prime 2}\ \sin\psi\ \ \ \ \ (a)\\ \sigma_{2}=\sigma_{2}^{trial}+\Big{(}\frac{4}{3}G-2K\Big{)}\Delta\gamma^{ \prime 2}\ \sin\psi-\ \Big{[}2G\Big{(}1+\frac{



The combination of equations Equation (39a,b) gives:

\[\left\{ \begin{array}{c}\sigma_{2}-\sigma_{3}=\sigma_{2}^{trial}-\sigma_{3}^{ trial}-4G\Delta\gamma^{b2}-2G(1-\sin\psi)\Delta\gamma^{\prime 2}\\ \sigma_{2}+\sigma_{3}=\sigma_{2}^{trial}+\sigma_{3}^{trial}-4\Big{(}\frac{G}{3} +K\Big{)}\Delta\gamma^{b2}\sin\psi+\\ 2\Big{[}G\Big{(}1+\frac{1}{3}\sin\psi\Big{)}-2K\sin\psi\Big{]}\Delta\gamma^{ \prime 2}\end{array} \right. \tag{45}\]

By substituting Equation (45) into Equation (44) and making some transformations, we obtain:

\[a\Delta\gamma^{b2}+b_{2}\Delta\gamma^{\prime 2}=f_{2}^{trial}\Big{(}\sigma^{trial},c\Big{)}. \tag{46}\]

From Equations (42) and (46), the scalar forms of \(\Delta\gamma^{\prime 2}\) and \(\Delta\gamma^{b2}\) are:

\[\left\{ \begin{array}{c}\Delta\gamma^{\prime 2}=\frac{af_{1}^{trial}-b_{2}f_{2}^{ trial}}{Det_{2}}\\ \Delta\gamma^{b2}=\frac{af_{2}^{trial}-b_{2}f_{2}^{ trial}}{Det_{2}}\end{array} \right.\]

or

\[\left\{ \begin{array}{c}\Delta\gamma^{\prime 2}=\frac{f_{1}^{trial}-b_{2} \Delta\gamma^{\prime 2}}{d}\\ \Delta\gamma^{\prime 2}=\frac{f_{2}^{trial}-b_{2}\Delta\gamma^{\prime 2}}{d} \end{array} \right. \tag{47}\]

with

\[Det_{2}=a^{2}-b_{2}^{2}.\]

**Remark 2**. : _Similarly, for <<physical>> values of parameters K, G, \(\phi\) and \(\psi\), \(Det_{2}\
eq 0\) (see Figure 4)._

The fixed-point algorithm applied to the left edge mechanism (Algorithm 5) is based on the constitutive law of the activated mechanism (Equation (39)). Its initial conditions are taken in the elastic domain (\(\sigma_{1}^{trial},\sigma_{2}^{trial},\sigma_{3}^{trial},0,0,\ \sharp_{n}^{p}=0\)). Figure 4: Example of \(Det_{2}\) values for \(K\) = 1.67 \(\times\) 10\({}^{8}\) and \(G\) = 6.39 \(\times\) 10\({}^{5}\). _(d) Return mapping to the apex \((\sigma_{1}=\sigma_{2}=\sigma_{3})\)_

Finally, in the case of the Apex, the stress is located on the hydrostatic axis. The hydrostatic pressure is then written:

\[p_{n+1}=p_{n+1}^{trial}-K\,\Delta\varepsilon_{v}^{p}, \tag{48}\]

where \(\Delta\varepsilon_{v}^{p}\) represents the volumetric plastic strain increment and is given by:

\[\Delta\varepsilon_{v}^{p}=2\sin(\psi)\sum_{i=1}^{6}\Delta\gamma^{i}. \tag{49}\]

The updated stress and the accumulated plastic strain are given by [1]:

\[\left\{ \begin{array}{c}\sigma_{n+1}=p_{n+1}\,\mathbf{1}\\ \bar{\varepsilon}_{n+1}^{p}=\bar{\varepsilon}_{n}^{p}+\beta\Delta\varepsilon_{ v}^{p}.\end{array} \right. \tag{50}\]

with \(\beta\equiv\frac{\cos\phi}{\sin\psi}\). The fixed-point algorithm applied to the return mapping to the Apex mechanism (Algorithm 6) is also based on the constitutive law of the activated mechanism (Equation (48)). Its initial conditions, taken in the elastic domain, are \(p_{n+1}^{trial},0\) and \(\bar{\varepsilon}_{n}^{p}=0\).