## 2 Theoretical Background  
In empirical damage models, the material is expected to accumulate damage (\(D\)) locally according to Equation (1), where \(\varepsilon_{f}\) is the equivalent plastic strain at failure, \(\varepsilon_{p}\) is the accumulated equivalent plastic strain, and \(f(T,X)\) is a function that includes the effect of the stress state described by the triaxiality parameter \(T\) (Equation (2)) and the deviatoric parameters \(X\) (Equation (3)); \(\sigma_{VM}\) is the von Mises equivalent stress. It is worth noting that \(X\) is related to the Lode Angle \(\theta\) through Equation (4), and that \(T\) and \(X\) are defined by the principal stresses through \(I_{1}\), \(I_{2}\), and \(I_{3}\), namely the first invariant of the stress tensor and the second and third invariants of the deviatoric stress tensor, respectively. The damage indicator \(D\) falls in the range \([0,\,1]\), and the onset condition of plastic failure occurring for \(D\) is equal to 1. \[D=\int_{0}^{\varepsilon_{f}}f(T,X)d\varepsilon_{p}\ \ \ 0\leq D\leq 1 \tag{1}\]  
\[T=\frac{1}{3}\frac{I_{1}}{\sigma_{VM}} \tag{2}\]  
\[X=\frac{27}{2}\frac{J_{3}}{\sigma_{VM}}^{3}\ \ \ -1\leq X\leq 1 \tag{3}\]  
\[\theta=\frac{1}{3}arccosX\ \ \ 0\leq\theta\leq\pi/3 \tag{4}\]  
Assuming proportional loading conditions, which implies \(T\) and \(X\) being constant with loading history, and setting \(D\) = 1, Equation (1) can be inverted leading to Equation (5), which mathematically represents a surface in the space (\(T,X,\varepsilon_{f}\)) called the fracture locus or fracture surface. This surface provides indications about the ductility of a material in terms of achievable equivalent strain to fracture for any applied stress state. Perfectly proportional loading is uncommon in actual applications; however, if \(T\) and \(X\) exhibit limited variations with plastic strain, average values \(T_{avg}\) and \(X_{avg}\) can be assumed; see Equations (6) and (7). \[\varepsilon_{f}=f^{-1}(T,X) \tag{5}\]  
\[T_{avg}=\frac{1}{\varepsilon_{f}}\int_{0}^{\varepsilon_{f}}T(\varepsilon)d \varepsilon_{p} \tag{6}\]\[X_{arg}=\frac{1}{\varepsilon_{f}}\int_{0}^{\varepsilon_{f}}X(\varepsilon)d \varepsilon_{p} \tag{7}\]  
As for Equation (5), different formulations for \(f(T,X)\) have been proposed in the literature; in this work, the Modified Mohr-Coulomb (MMC) model, devised by [PERSON] [34], the one by [PERSON] (CC) [29], and the Hosford-Coulomb (HC) by [PERSON] [30] were considered. The MMC model is defined by Equation (8) and the CC model is based on Equation (9), while the HC model is based on the Equations (10)-(12), where \(\overline{\theta}\) is the normalized Lode angle parameter; see Equation (13). \[\varepsilon_{f}=\left\{\frac{K}{c_{2}}\left[\sqrt{\frac{1+c_{1}^{2}}{3}}\text {cos}\left(\frac{\pi}{6}-\frac{1}{3}arccos(X)\right)+c_{1}\left(T+\frac{1}{3} \text{sin}\bigg{(}\frac{\pi}{6}-\frac{1}{3}arccos(X)\bigg{)}\right)\right] \right\}^{-\frac{1}{\theta}} \tag{8}\]  
\[\varepsilon_{f}=\frac{1}{c_{1}}e^{-c_{2}T}\left(\frac{\text{cos}\left[\beta \frac{\pi}{6}-\frac{1}{3}arccos(\gamma)\right]}{\text{cos}\left[\beta\frac{ \pi}{6}-\frac{1}{3}arccos(X\gamma)\right]}\right)^{\frac{1}{\theta}} \tag{9}\]  
\[\varepsilon_{f}=b\left(\frac{1+c}{g_{HC}\big{[}T,\overline{\theta}\big{]}} \right)^{\frac{1}{\theta}} \tag{10}\]  
where  
\[T=\frac{1}{3}\frac{I_{1}}{\sigma_{VM}} \tag{2}\]0  
\[T=\frac{1}{3}\frac{I_{1}}{\sigma_{VM}} \tag{2}\]1  
\[T=\frac{1}{3}\frac{I_{1}}{\sigma_{VM}} \tag{2}\]2  
The material constants which have to be tuned on the specific material are \(c_{1}\) and \(c_{2}\) for the MMC, \(c_{1}\), \(c_{2}\), \(\beta\), and \(\gamma\) for the CC, and \(a\), \(b\), and \(c\) for the HC models. The parameters \(K\) and \(n\) are the hardening coefficient and the hardening exponent of the Hollomon's power law, reported in Equation (14), which approximates the elasto-plastic constitutive behavior of the material. For the HC model only, the parameter \(n\) is recommended to be fixed to 0.1 [35; 36].