## 2 Plasticity Formulation
### Return Mapping Algorithm
#### 2.2.1 Return-Mapping Scheme for the 1D Model
\right.\)
5.If \(\alpha_{n+1}-\alpha_{n+1}\geq 0\)then
6.\(\sigma_{n+1}=q_{n+1}+\left[\sigma_{Y}+K\alpha_{n+1}\right]\)
7.else
8.\(\sigma_{n+1}=q_{n+1}-\left[\sigma_{Y}+K\alpha_{n+1}\right]\)
9.endif
10.\(Z=\left[\sigma_{n+1},\;\alpha_{n+1},\Delta\gamma,\;q_{n+1},\;\varepsilon_{n+1}^ {p}\right]\)
11.Update \(Z_{0}=Z\)
12.EndWhile
```

**Algorithm 2** Fixed point method applied to the return mapping algorithm. #### 2.2.2 Return-Mapping Scheme for the Mohr-Coulomb (MC) Law

The MC model generally depends on two parameters: the internal friction angle \(\phi\) of the material and the cohesion \(c\), which can also depend on the accumulated plastic strain \(\bar{\varepsilon}_{n}^{p}\) and its increment \(\Delta\bar{\varepsilon}^{p}(c\left(\bar{\varepsilon}_{n}^{p},\Delta\bar{ \varepsilon}^{p}\right))\). It also involves the shear stress \(\tau\) and normal stress \(\sigma_{n}\). For this criterion, the plastic strain begins when the shear stress \(\tau\), as well as the normal stress \(\sigma_{n}\) become critical. These two stresses are linked by the relationship:

\[\tau=c-\sigma_{n}\;\tan(\phi) \tag{13}\]

The plastic function \(f\) is expressed in terms of principal stresses by:

\[f(\mathbf{\sigma},c)=\sigma_{1}-\sigma_{3}+(\sigma_{1}+\sigma_{3})\sin\phi-2c\cos\phi \tag{14}\]with the convention \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\) where \(\sigma_{i}\) (\(j=1,2,3\)) is the \(j\)-th principal stress. The representation of its yield surface is a hexagonal pyramid whose axis is colinear to the hydrostatic pressure (Figure 1) defined by:

\[p=c\cot(\phi) \tag{15}\]

The six edges of this criterion in three-dimensional space, allow it to be expressed by six load functions given by [1]:

\[ \begin{array}{l}f_{1}(\sigma,c)=\sigma_{1}-\sigma_{3}+(\sigma_{1}+\sigma_{3 })\sin\phi-2c\cos\phi\\ f_{2}(\sigma,c)=\sigma_{2}-\sigma_{3}+(\sigma_{2}+\sigma_{3})\sin\phi-2c\cos \phi\\ f_{3}(\sigma,c)=\sigma_{2}-\sigma_{1}+(\sigma_{2}+\sigma_{1})\sin\phi-2c\cos \phi\\ f_{4}(\sigma,c)=\sigma_{3}-\sigma_{1}+(\sigma_{3}+\sigma_{1})\sin\phi-2c\cos \phi\\ f_{5}(\sigma,c)=\sigma_{3}-\sigma_{2}+(\sigma_{3}+\sigma_{2})\sin\phi-2c\cos \phi\\ f_{6}(\sigma,c)=\sigma_{1}-\sigma_{2}+(\sigma_{1}+\sigma_{2})\sin\phi-2c\cos \phi\end{array}  \tag{16}\]

The two-by-two intersections of these load surfaces and the vertex form singularities that pose some problems for numerical integration. Each of the six load surfaces has a plastic flow rule associated with it. In addition, the behavior of the soil during loading involves friction, and so non-associated plastic flow rules must be used to avoid excessive expansion of the material [9]. For this purpose, an expansion angle, noted as \(\psi\), that allows the introduction of six plastic flow potentials \(g_{i}\) (I = 1,..., 6) associated with each yield function \(f_{i}\). These plastic potentials are written in the same way as their respective yield surfaces (the relationships in (Equation (16)). For example, for the yield surface \(f_{1}(\sigma,c)\), we have:

\[g_{1}(\sigma,c)=\sigma_{1}-\sigma_{3}+(\sigma_{1}+\sigma_{3})\sin\psi-2c\cos\psi \tag{17}\]

The plastic flow rule is then written as:

\[\dot{\epsilon}^{p}=\sum_{i=1}^{m}\dot{\gamma_{i}}\frac{\partial g_{i}(\sigma, c)}{\partial\sigma} \tag{18}\]

where we recall that \(m\) is the number of active mechanisms. Thus, with the principal stresses ordered as follows: \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\), the flow rule can be formulated in the edge space located in the same plane of the principal stresses space (the

Figure 1: Mohr-Coulomb yield surface in principal stress space [1]. sertant plane) (Figure 2). There are then four distinct possibilities for defining the plastic flow rule:

If the final stress is within the load surface, then the point is regular and a mechanism is activated, \(m=1\). If the final stress is located on the right edge of the cone relative to the sextant plan, then the point is singular and two mechanisms are activated, \(m=2\).