## 2 Plasticity Formulation
### Return Mapping Algorithm
#### 2.2.1 Return-Mapping Scheme for the 1D Model
This dam has a width \(L\) of 320 m, a height \(h\) equal to 100 m and a crest length of 9 m. The depth of the dam axis is assumed to be infinite, which makes it possible to carry out a study of in-plane strain. Vertical and horizontal movements are zero at the base. [PERSON]'s law is used in cases where the material has an elastic behavior. In the plastic case, the Mohr-Coulomb model is used in its non-associated version to avoid excessive expansion of the material [9]. The material properties correspond to those used in [21] of a rockfill dam: [PERSON]'s modulus \(E=17\times 10^{7}\) Pa, Poisson coefficient \(\
u=0.33\), density \(\rho=2370\) Kg/m\({}^{3}\), material cohesion \(c=0\), internal friction angle \(\phi=45^{\circ}\) and material dilatancy angle \(\psi=15^{\circ}\). We propose to compare the results obtained with our code developed in the Matlab environment and those obtained using the commercial software PLAXIS for the mesh sizes presented in Figure 8. The construction of the dam is simulated by adding successive layers, with 20 layers in total. The vertical and horizontal displacements, as well as the principal stresses in the first and second direction for the entire domain, are presented in Figure 9. Although the comparison of isovalues for the Matlab and PLAXIS models must be done with care, the displacements and calculated main stresses show a satisfactory correspondence. Indeed, for displacements, we note a relative difference of only 2% and 0.1% respectively for the maximum values of u\({}_{\mathrm{x}}\) and u\({}_{\mathrm{y}}\). For the maximum stresses, there is a relative difference of 4% and 11% for the principal stress 1 and principal stress 2, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Convergence: \(\frac{\|F_{\mathrm{est}}-F_{\mathrm{est}}\|}{\|F_{\mathrm{est}}\|}\)\textless{}10\({}^{-6}\)**} & \multicolumn{2}{c}{**Number of Iterations**} \\ \hline Load (kN) & Ana-NR & Num-NR & Ana-NR & Num-NR \\
5 & 0 & 0 & 1 & 1 \\
10 & 0 & \(4.6\times 10^{-11}\) & 1 & 1 \\
15 & \(1.2\times 10^{-16}\) & \(3.1\times 10^{-11}\) & 1 & 1 \\
20 & 0 & \(2.7\times 10^{-11}\) & 1 & 1 \\
25 & 0 & \(4.9\times 10^{-12}\) & 1 & 1 \\
30 & 0 & \(9.9\times 10 Figure 8: Mesh used (**a**) PLAXIS: 327 triangular elements with 15 nodes and (**b**) Matlab: 800 quadrilateral isoparametric bilinear quadrilateral elements (40 \(\times\) 20) with 4 nodes. Figure 7: Evolution of the plasticization in the bar with to the applied load. Figure 9: Comparisons of horizontal (\(\mathrm{u_{x}}\)) and vertical (\(\mathrm{u_{y}}\)) displacements and normal stresses calculated for \(\omega=10^{-6}\) and \(tol=10^{-10}\). Comparisons of the values obtained for this approach, particularly in the column and row under consideration (Figure 10), show that the vertical (u\({}_{y}\)) and horizontal (u\({}_{x}\)) displacements, as well as the normal stresses in the X and Y directions, are also in agreement with those obtained with PLAXIS (Figures 11 and 12, respectively). Figure 11: Displacements and stresses in the column considered for Matlab and PLAXIS models according to mesh size. Figure 10: Choice of the studied areas for the comparison of variables: (**a**) column considered and (**b**) row considered. However, some small differences are observed. On the one hand, these could be attributed to the mesh sizes and to the shape of the elements, as those developed for our Matlab code differ from those used for the commercial software PLAXIS. On the other hand, these differences may also be due to how the loading was applied. It should be noted, however, that the results are very encouraging, especially since we do not need to calculate the tangent matrix analytically with our numerical model for more complex constitutive laws. The code developed for Matlab also offers the possibility of studying the problem layer by layer according to the imposed load, and by refining the mesh size. These loads are represented by the different layers of the domain.