## 2 Materials and Methods
### Numerical Models
#### 2.3.1 Concrete Damage Plasticity Model (CDP)  
The CDP was formulated to model the brittle behavior of concrete, even though it proved to be suitable to catch the response of all brittle materials, such as masonry. It is an isotropic plasticity model presented for the first time by [PERSON] [33] and subsequently implemented by [PERSON] and [PERSON] [34]. The first formulation introduced the concept of distinct responses in tension and compression [33]. However, the improvement [34] was considered thanks to the contribution of the stiffness recovery, following the cracks' closing under cycle loads. The damage is defined by the effective configuration (damaged). To accomplish this, the equivalence between tension stress in undamaged (\(\sigma_{0}\)) and damaged (\(\sigma\)) configurations must be imposed, taking the damage parameter (\(d\)) into account:  
\[\sigma=d\cdot\sigma_{0}, \tag{1}\]  
written under the hypothesis of strain equivalence:  
\[\varepsilon=\varepsilon_{0}. \tag{2}\]  
In the plasticity theory, the strain tensor can be expressed as the sum of its elastic and plastic components:  
\[\varepsilon=\varepsilon^{e}+\varepsilon^{p} \tag{3}\]  
where the elastic stress can be written as a function of the elastic damage stiffness matrix \(E\) using the Hooke law:  
\[\sigma=E(\varepsilon-\varepsilon^{p}) \tag{4}\]  
To consider the damage irreversibility, the degraded stiffness matrix is expressed as:  
\[E=(1-d)E_{0}, \tag{5}\]  
where \(E_{0}\) identifies the undamaged elastic stiffness matrix. When the material is intact, the \(d\) value is zero, whereas, when it is completely broken, it assumes a value of one. This consideration affects the relationship between stress and strain, which becomes:  
\[\sigma=(1-d)E_{0}(\varepsilon-\varepsilon^{p}). \tag{6}\]To account for the higher stiffness degradation in tension (\(d_{t}\)) compared to that in compression (\(d_{c}\)), under mono axial stress (Figure 4a,b), two distinct damaged variable laws must be defined:  
\[\sigma_{t}=(1-d_{t})E_{0}\Big{(}\varepsilon_{t}-\varepsilon_{t}^{p}\Big{)}, \tag{7}\]  
\[\sigma_{c}=(1-d_{c})E_{0}\Big{(}\varepsilon_{c}-\varepsilon_{c}^{p}\Big{)}. \tag{8}\]  
Equations (7) and (8) assume increasing values as the elastic strain increases. Instead, under cyclic loads, the crack closure produces a partial elastic stiffness recovery, especially passing from tension to compression status. In this case, it is assumed that:  
\[(1-d)=(1-s_{t}d_{c})(1-s_{c}d_{t}). \tag{9}\]  
The scalars \(s_{t}\) and \(s_{c}\) hold the stiffness recovery during the load inversion. They are defined as functions of weight factors as:  
\[s_{t}=1-w_{t}r^{*}(\underline{\sigma}), \tag{10}\]  
\[\varepsilon=\varepsilon_{0}. \tag{2}\]0  
where \(r^{*}(\underline{\tilde{\sigma}})\) is equal to zero if the principal stresses are all positive and, when the primary stresses are all negative, it takes unity. \(w_{t}\) and \(w_{c}\) are the scalar considering the stiffness recovery: assuming zero, the stiffness recovery is not considered; if they are equal to one, a complete recovery is allowed. To define the boundary of the elastic domain, a yield strength function based on the Drucker-Prager is employed (Figure 4c):  
\[\varepsilon=\varepsilon_{0}. \tag{2}\]1  
with:  
\(I_{1}\) the first invariant of effective stress;  
\(I_{2}\) the second invariant of effective stress;  
\(\underline{\tilde{\sigma}}\) maximum principal effective stress. Although, \(\alpha\) and \(\beta\) are two constants defined using the initial uniaxial compressive yield stress \(f_{b0}\) and the initial equiaxial compressive yield stress as:  
\[\varepsilon=\varepsilon_{0}. \tag{2}\]2  
\[\varepsilon=\varepsilon_{0}. \tag{2}\]3  
and \(c_{c}(\kappa)\) and \(c_{t}(\kappa)\) are the effective cohesion in compression and tension, respectively. Finally, the constant \(\gamma\) is calculated as:  
\[\varepsilon=\varepsilon_{0}. \tag{2}\]4  
where \(K_{c}(0.5\leq K_{c}<1)\) is a parameter that defines the yield surface shape. It can assume a value between 0.5 and 1.