## 3 Model Validation  
The nature of the bond between the concrete and the rebar is three-fold: chemical adhesion, friction, and mechanical (especially with deformed rebar with the mechanical interlock of its ribs). Of the three, the chemical adhesion is the least significant and it is broken after only some slight movement [6; 24]. Concerning the mechanical bond,  
Figure 4: Representation of the Kuhn–Tucker contact conditions. Figure 5: Final mesh of the numerical model. since the rebars ribs have not been modeled, the bond strength will mainly depend on the friction. However, corrosion deteriorates the rib's geometry of the deformed bar, leading to a significant decrease of the lug height [11]. The impact of the mechanical bond of corroded deformed bars becomes less important in comparison with non-corroded deformed bars [6, 11, 16]. Overall, three failure modes are possible in a pull-out test: the pull-out failure mode when there is a perfect slip of the rebar, the cover splitting mode characterized by the splitting of a cone of concrete, and the reinforcement failure with the rebar yielding. In our model, only the first two modes are considered, and only the first one is studied. The second one is detected in our simulations as convergence problems due to the extreme damaging of the concrete cover, since it is impossible to model the splitting of concrete. As explained previously, after applying the volume expansion on the rebar, a displacement is imposed at its free end. This allows us to determine the necessary load to pull the rebar for the various degrees of corrosion studied, as presented in Figure 6. As we can see in this same figure, the load necessary to remove the rebar decreases greatly after some slight movements. We can also notice that this same load increases for small degrees of corrosion before it drops for larger degrees. This is due to the fact that, at lower degrees of corrosion, the concrete at the interface region is still not damaged, and the volume expansion only leads to a better adherence between the reinforcement and the concrete cover. The volume expansion of the rebar increases the radial forces applied on the concrete interface which enhances the holding capacity of the concrete cover. Experimentally, this increase is also observed in [14, 16, 24]; it was explained due to the rough nature of the rust material in contact with concrete. Here, we can conclude that this increase in the pulling load of the corroded rebar in lower degrees of corrosion is more significant in reality since it takes these two behaviors into account. When the degree of corrosion starts to grow, the rust damages the concrete at the interface. Cracks start to appear due to the tensile stresses inside the concrete cover, generated by the volume expansion of the rebar, and due to the quasi-brittle nature of the concrete. It is at this moment that the pulling load starts to decrease, as we can see  
Figure 6: Load-slip pull-out curves for different corrosion degrees. in Figure 6. The evolution of the bond stress as a function of the corrosion degree is also investigated (Figure 7), using the following formula [6; 19; 25; 26]:  
\[\tau=\frac{N_{\text{max}}}{\pi h_{\text{eff}}d}, \tag{8}\]  
where \(\tau\) is the bond stress, \(N_{\text{max}}\) the maximum load of the pull-out simulation, \(d\) the diameter of the rebar, and \(h_{\text{eff}}\) the embedment depth of the rebar. To validate our model, only experimental data from the literature were used. The bond stresses reported in the literature are very scattered, and there is a clear difference in term of magnitudes [14; 24]. This can be explained with the sensitive nature of concrete, the types of reinforcement used or the conditions of the experimental program. The results of [PERSON] et al. [16] are plotted in Figure 8. They worked on the effect of corrosion on the bond stress of plain and deformed rebar.