# Numerical Simulation of Anchor Pullout and Shear Tests Using a Regularized Damage Model
## 2 Advanced Numerical Strategy for the Simulation of Anchorage Behaviors  
For the simulation of anchorage behaviors, an isotropic damage model is proposed for concrete, based on the initial work from [PERSON] [32]. As for the so-called local damage models, one of the main limits of the initial model is that it is highly dependent on the mesh due to the localization issue [37]. To minimize mesh dependence, different solutions exist, including non-local integral [38] or gradient [19] solutions, for example. These non-local solutions allow for accurate calculations with low mesh dependence, but they drastically increase the computational cost, which is not compatible with the aim of our approach. That is why energetic approaches introducing fracture energy are preferred in this contribution. The concept is thus to introduce fracture energy, which is defined by the area under the stress-displacement curve and which is a material parameter. This regularization is widely used for elements in tension [33; 34], as in most cases, concrete fails in tension before the elements in compression crack. However, a model regularized in both tension and compression [35] is proposed in this contribution. It is indeed important, as the loading conditions (shear, for example) will produce compressive stresses under which concrete may be damaged. Damage variable D characterizes the mechanical degradation and ranges between 0 and 1. It is calculated following the initial proposition from [PERSON] [32]:  
\[\sigma=(1-D)E\ \varepsilon. \tag{1}\]  
\[D=\alpha_{t}\ D_{t}+\alpha_{c}\ D_{c}. \tag{2}\]  
In Equation (2), the first term is used to calculate the damage due to tension, and the second term for damage due to compression. The coefficients \(\alpha_{t}\). and \(\alpha_{c}\) represent the proportion of the stress state in tension and compression, with the condition that \(\alpha_{t}+\alpha_{c}=1\). Compared to the initial formulation, modifications are proposed to calculate \(D_{t}\) and \(D_{c}\) by introducing the fracture energy in both tension and compression. The two damage variables depend on material parameters, including the concrete strengths, the fracture energy, and the relative strain in concrete at the peak stress in compression (\(\varepsilon_{c1}\) et \(\varepsilon_{c2}\)) [39]. \[D_{t}=1-\frac{\varepsilon_{d0}}{\varepsilon_{eq}}\exp\left(-B_{t}\big{(} \varepsilon_{eq}-\varepsilon_{d0}\big{)}\right)\ \text{with}\ B_{t}=\frac{l_{c}\times f_{t}}{G_{ft}} \tag{3}\]  
\[D_{c}=\left\{ \begin{array}{ll}1-\frac{\big{(}\varepsilon_{c2}-\varepsilon_{ c2}^{2}\big{)}}{1+(k-2)\varepsilon_{c}}\frac{f_{c}}{E_{c,eq}},\ if\ \varepsilon_{c,eq}\leq\varepsilon_{c1}\\ 1-\frac{f_{c}}{E_{c,eq}},\ if\ \varepsilon_{c1}<\varepsilon_{c,eq}\leq \varepsilon_{c2}\\ 1+\frac{k_{1}}{E}-\frac{k_{2}}{E_{c,eq}},\ if\ \varepsilon_{c2}<\varepsilon_{c,eq} \leq\varepsilon_{cu}\end{array} \right.\ \ \text{with}\ k_{1}=\frac{f_{c}}{\frac{2G_{fc}}{l_{c, fc}}+\varepsilon_{c1}-2\varepsilon_{c2}} \tag{4}\]  
\(G_{ft,c}\) in Equation (5) represents the energy required so that the crack can propagate [33], which depends only on the material, either in tension or in compression. \(l_{c}\) is the element size. However, defining the size of an element is not necessarily straightforward. In this contribution, it is assumed that the element size \(l_{c}\) is equal to the cubic root of the element'svolume. It assumes that all elements should be as cubic as possible in the simulations, so that the length of the element in each direction is equal to \(l_{c}\). \[Gf_{c,t}=\int_{upic}^{+\infty}\sigma du=l_{c}\int_{elpic}^{+\infty}\sigma d\varepsilon \tag{5}\]  
The constitutive law of concrete is shown in Figure 1 for pure tension (a) and pure compression (b) behavior. The applicability of the model in case of high confinement depends mainly on the capacity of the concrete constitutive law to represent concrete behavior in this situation. The proposed model includes a correction in biaxial and triaxial compression.