## 2 Plasticity Formulation
### Return Mapping Algorithm
#### 2.2.1 Return-Mapping Scheme for the 1D Model
\tag{27}\]

The consistency condition in this case is given by:

\[f_{1}(\sigma\,,c)=\sigma_{1}-\sigma_{3}+(\sigma_{1}+\sigma_{3})\sin\phi-2c \cos\phi=0 \tag{28}\]

By substituting Equation (27) in Equation (28) and after a few transformations, the analytical form of \(\Delta\gamma\) is obtained by:

\[\Delta\gamma=\frac{f_{1}^{trial}\Big{(}\sigma^{trial},c\Big{)}}{a} \tag{29}\]

with

\[a=4\bigg{[}G+\bigg{(}\frac{G}{3}+K\bigg{)}\sin\psi\sin\phi\bigg{]}\]

In the case where _the material properties are non-linear_, we adopt a fixed-point algorithm to complete the resolution. The fixed point algorithm applied to the return mapping of the main plane mechanism (Algorithm 3) is based on the constitutive law of the activated mechanism (Equation (25)), with the initial conditions taken in the elastic domain \(\Big{(}\sigma_{1}^{trial},\sigma_{2}^{trial},\sigma_{3}^{trial},0,\,\bar{z}_{n }^{p}=0\Big{)}\). In this algorithm, the plastic multiplier \(\Delta\gamma\) is obtained by substituting the principal trial stresses \(\sigma_{1}^{trial},\sigma_{2}^{trial},\sigma_{3}^{trial}\) in the consistency equation for this activated mechanism Equation (28). Moreover, from Equation (28) we deduce \(\sigma_{1}\). _Return mapping to the right edge \((\sigma_{1}>\sigma_{2}=\sigma_{3})\)_

Similarly, for two mechanisms activated on the right edge, the principal updated stresses are:

\[\left\{ \begin{array}{ccc}\sigma_{1}=\sigma_{1}^{trial}-\Big{[}2G\Big{(}1+\frac{ 1}{3}\sin\psi\Big{)}+2K\sin\psi\Big{]}\Big{(}\Delta\gamma^{a1}+\Delta\gamma^{b1 }\Big{)}&(a)\\ \sigma_{2}=\sigma_{2}^{trial}+\Big{(}\frac{4}{3}G-2K\Big{)}\Delta\gamma^{a1}\, \sin\psi+\Big{[}2G\Big{(}1-\frac{1}{3}\sin\psi\Big{)}-2K\sin\psi\Big{]}\Delta \gamma^{b1}&(b)\\ \sigma_{3}=\sigma_{3}^{trial}+\Big{[}2G\Big{(}1-\frac{1}{3}\sin\psi\Big{)}-2K \sin\psi\Big{]}\Delta\gamma^{a1}+\Big{(}\frac{4}{3}G-2K\Big{)}\Delta\gamma^{b1 }\,\sin\psi&(c)\end{array} \right. \tag{30}\]

with the accumulated plastic strain given by:

\[\bar{\varepsilon}_{n+1}^{p}=\bar{\varepsilon}_{n}^{p}+2\Big{(}\Delta\gamma^{ a1}+\Delta\gamma^{b1}\Big{)}\cos\phi \tag{31}\]

The combination of equations Equation (30,c) gives:

\[\left\{ \begin{array}{ccc}\sigma_{1}-\sigma_{3}=\sigma_{1}^{trial}-\sigma_{3 }^{trial}-4G\Delta\gamma^{a1}-2G(1+\sin\psi)\Delta\gamma^{b1}\\ \sigma_{1}+\sigma_{3}=\sigma_{1}^{trial}+\sigma_{3}^{trial}-4\Big{(}\frac{ \sigma}{3}+K\Big{)}\Delta\gamma^{a1}\sin\psi-\\ 2\Big{[}G\Big{(}1-\frac{1}{3}\sin\psi\Big{)}+2K\sin\psi\Big{]}\Delta\gamma^{a1} \end{array} \right. \tag{32}\]

By substitution of Equation (32) into Equation (28) and after some transformations, we obtain:

\[a\Delta\gamma^{a1}+b_{1}\Delta\gamma^{b1}=f_{1}^{trial}\Big{(}\sigma^{trial},c \Big{)} \tag{33}\]

with

\[b_{1}=2G\Big{(}1+\sin\psi+\sin\phi-\frac{1}{3}\sin\psi\sin\phi\Big{)}+4K\sin \psi\sin\phi \tag{34}\]

For the second mechanism, the consistency condition is given by:

\[f_{6}(\sigma\,,c)=\sigma_{1}-\sigma_{2}+(\sigma_{1}+\sigma_{2})\sin\phi-2c \cos\phi=0 \tag{35}\]The combination of equations Equation (30a,b) gives:

\[\left\{ \begin{array}{c}\sigma_{1}-\sigma_{2}=\sigma_{1}^{trial}-\sigma_{2}^{ trial}-4G\Delta\gamma^{b1}-2G(1+\sin\psi)\Delta\gamma^{a1}\\ \sigma_{1}+\sigma_{2}=\sigma_{1}^{trial}+\sigma_{2}^{trial}-4\Big{(}\frac{G}{3 }+K\Big{)}\Delta\gamma^{b1}\sin\psi-\\ 2\Big{[}G\Big{(}1-\frac{1}{3}\sin\psi\Big{)}+2K\sin\psi\Big{]}\Delta\gamma^{a1} \end{array} \right. \tag{36}\]

By substitution of Equation (36) into Equation (35) and after some transformations, we obtain:

\[a\Delta\gamma^{b1}+b_{1}\Delta\gamma^{a1}=f_{6}^{trial}\Big{(}\sigma^{trial}, c\Big{)} \tag{37}\]

From Equations (33) and (37), we can deduce the scalar form of \(\Delta\gamma^{a1}\) and \(\Delta\gamma^{b1}\), defined by:

\[\left\{ \begin{array}{c}\Delta\gamma^{a1}=\frac{\sigma_{1}^{trial}-b_{1} \sigma_{6}^{trial}}{Det_{1}}\\ \Delta\gamma^{b1}=\frac{\sigma_{6}^{trial}-b_{1}\sigma_{1}^{trial}}{Det_{1}} \end{array} \right.\]

or

\[\left\{