## 2 Plasticity Formulation
### Return Mapping Algorithm
#### 2.2.1 Return-Mapping Scheme for the 1D Model
\begin{array}{ccc}\frac{ \partial\mathbf{\varepsilon}_{xx}}{\partial\mathbf{\varepsilon}_{xx}}&\frac{\partial \mathbf{\varepsilon}_{xx}}{\partial\mathbf{\varepsilon}_{xx}}&\frac{\partial\mathbf{ \varepsilon}_{xx}}{\partial\mathbf{\varepsilon}_{xx}}\\ \frac{\partial\mathbf{\varepsilon}_{yy}}{\partial\mathbf{\varepsilon}_{xx}^{\text{ }T}}&\frac{\partial\mathbf{\varepsilon}_{yy}}{\partial\mathbf{\varepsilon}_{yy}^{ \text{ }T}}&\frac{\partial\mathbf{\varepsilon}_{yy}}{\partial\mathbf{\varepsilon}_{yy}^{ \text{ }T}}\\ \frac{\partial\mathbf{\varepsilon}_{yy}}{\partial\mathbf{\varepsilon}_{xx}^{\text{ }T}}&\frac{\partial\mathbf{\varepsilon}_{yy}}{\partial\mathbf{\varepsilon}_{yy}^{ \text{ }T}}&\frac{\partial\mathbf{\varepsilon}_{yy}}{\partial\mathbf{\varepsilon}_{yy}^{ \text{ }T}}\\ \end{array} \right)_{(n+1)}^{(k)} \tag{61}\]

Each component of the strain vector is thus perturbed according to the same principle stated in Section 2.3.1, and so allows an approximation of the elastoplastic tangent matrix to be calculated for each activated mechanism. ## 3 Numerical Study

In this section, we show the efficiency of the proposed numerical methods. For this purpose, two examples are considered. The first example concerns the solution of a bar (under a traction load) problem proposed by [PERSON] [14]. This is a classical benchmark test for which there exists an analytical solution. Therefore, any numerical algorithm in plasticity must pass it. The second example deals with a symmetrical rockfill dam modeled in plane strains according to Mohr-Coulomb's law. ### Example 1: Nonlinear Bar Problem

The bar considered is uniform with a cross-sectional area \(A=1\times 10^{-4}\ m\) and a unit length \(L=1\ m\). A traction force of magnitude \(F=50\ kN\) is applied at the end of the bar (Figure 5). The yield strength is \(\sigma_{Y}=400\ MPa\). In the elastic region, the bar's Young's modulus is \(E=200\ GPa\) and in the plastic region, its plastic modulus \(K\) is zero. Its yield function obeys Equation (7). Figure 6 shows the stress-strain for a single element of the 1D bar. Ten equal load levels are applied to the bar. In the legend, Ana-NR represents the results from the analytical calculation of \(D^{elp}\), Num-NR represents the numerical calculation of \(D^{elp}\), and (Kim) represents the exact solution [14]. The results obtained for both models and for the exact solution (Ana-NR, Num-NR and Kim) are virtually identical (Figure 6). For loads below 40 kN, all models converge (Table 1) in a single iteration (elastic domain). Above 40 kN (the plastic domain), the Ana-NR and Num-NR models converge (i.e., meet the convergence criterion which is \(\frac{\|F_{ext}-F_{int}\|}{\|F_{ext}\|}\leq 10^{-6}\) where \(F_{ext}\) represents the external force, \(F_{int}\) the internal force and \(\|\cdot\|\) the Euclidean norm) after at most two iterations. All the results show that the proposed numerical model for the calculation of the tangent module (Num-NR) is sufficiently accurate and efficient. In summary, the results prove that the exact solution is obtained no matter how many

Figure 5: Nonlinear bar problem [14]. Figure 6: Stress-strain for a single element considered for \(\omega=10^{-6}\) and \(tol=10^{-6}\). elements are used. The case of 10 elements is to show that the code behaves correctly for any mesh. Figure 6 also shows that the yield strength is reached at 400 MPa, and beyond this value, the bar becomes in plastic mode. To better observe these areas, we divided the bar into ten identical elements of two nodes each. Ten successive loads in 5 kN steps were applied. The evolution of plasticity in the bar during these loads for the Num-NR model is presented in Figure 7.

### Example 2: Rockfill Dam

To test the efficiency of the algorithm in the case of a more complex behavior, we consider an example of a rockfill dam. This rockfill dam is assumed to be dry, homogeneous and symmetrical, and subject to its own weight.