## 2 Numerical Models
### Ductile Damage Models  
Two models are considered in this work, the Bai-Wierzbicki model (BW) [28] and the Coppola-Cortese-Folgarait (CCF) [30], both of which fall in the class of empirical models. Under these models, damage is supposed to accumulate with equivalent plastic strain \(\varepsilon_{p}\), as follows:  
\[D=\int_{0}^{\varepsilon_{f}}f(T,X)d\varepsilon_{p} \tag{3}\]  
where the evolution of ductile damage \(D\) is described by the integral of the function \(f(T,X)\) between the zero plastic strain and the maximum admissible strain at failure \(\varepsilon_{f}\). The parameters \(T\), Equation (4), and \(X\), Equation (5), govern the damage evolution and are called the triaxiality and Lode parameters, respectively. They in turn depend on the stress state of the material, indicated through the first invariant of the stress tensor \(I_{1}\), and the second and third invariants of the deviatoric stress tensor, \(J_{2}\) and \(J_{3}\). \[T=\frac{1}{3}\frac{I_{1}}{\sqrt{3J_{2}}} \tag{4}\]  
\[X=\frac{27}{2}\frac{J_{3}}{\left(\sqrt{3J_{2}}\right)^{3}} \tag{5}\]Fracture onset occurs when \(D\) reaches a critical value, conventionally set at 1. Under the hypothesis of proportional loading, which implies constant \(T\) and \(X\), by posing a \(D\) equal to one and inverting Equation (3), the following expression is obtained:  
\[\epsilon_{f}=f^{-1}(T,X) \tag{6}\]  
The graphical representation of such a function is a three dimensional surface in the space \(\left(T,X,\epsilon_{p}\right)\), commonly named the \"fracture surface\"; for each couple of \((T,X)\), i.e., a specific stress state, a specific value of plastic strain is computed that coincides with the onset of failure. Generally, proportional loading conditions are not satisfied because \(T\) and \(X\) are not perfectly constant, and the deformation and necking onset may be further affected by temperature and strain rate variations occurring during the tests [39]. However, we here assume the average values of the two parameters \(T_{avg}\), Equation (7), and \(X_{avg}\), Equation (8), in order to meet the proportional or quasi proportional loading hypothesis, and use Equation (6) to derive the fracture surface. \[T_{avg}=\frac{1}{\epsilon_{f}}\int_{0}^{\epsilon_{f}}T(\epsilon)de_{p} \tag{7}\]  
\[X_{avg}=\frac{1}{\epsilon_{f}}\int_{0}^{\epsilon_{f}}X(\epsilon)de_{p} \tag{8}\]  
The models adopted in this work are characterized by a different function \(f^{-1}(T,X)\). In Equations (9) and (10), the analytical expressions of the BW and CCF models are reported [28; 30]. \[\epsilon_{f}=\left\{\frac{K}{C_{2}}\left[\sqrt{\frac{1+C_{1}^{2}}{3}}cos\ cos \left(\frac{\pi}{6}-\theta\right)+C_{1}\left(T+\frac{1}{3}sin\ sin\ \left(\frac{\pi}{6}-\theta\right)\ \right)\right]\right\}^{-\frac{1}{ \theta}},\ \text{(BW)} \tag{9}\]  
\[\epsilon_{f}=\frac{1}{c_{1}}e^{-c_{2}T}\left(\frac{cos\left[\beta\frac{\pi}{6 }-\frac{1}{3}arccos(\gamma)\right]}{cos\left[\beta\frac{\pi}{6}-\frac{1}{3} arccos(X\gamma)\right]}\right),\ \ \text{(CCF)} \tag{10}\]  
\(K\) and \(n\) are, respectively, the strength coefficient and strain hardening exponent of [PERSON]'s power law \(\sigma=K\epsilon^{n}\). The Lode angle is used in place of the Lode parameter \(X\) in the BW expression, Equation (9). The following relation holds between the two:  
\[\theta=\frac{1}{3}arccosX \tag{11}\]  
The first model is characterized by only two material constants, \(C_{1}\) and \(C_{2}\), if the [PERSON] yielding condition is adopted, while the second is characterized by four constants, \(C_{1},\ C_{2},\ \beta\) and \(\gamma\). The constants of both models can be identified for a specific material using the following calibration procedure. A suitable number of experimental tests are executed under different loading conditions, usually tensile tests (RB, RN), plane strain tensile tests (PS) or torsion tests, and the resulting global quantities, load and displacement at failure, are measured and recorded.