## 3 Material Failure Theory Model  
The stress state is usually represented by the stress tensor. In addition, uncoupled damage mechanics models typically adopt a stress invariant to represent the stress state. In fracture mechanics, most fracture criteria are characterized by the failure strain, which is functionally related to the stress invariant. Consequently, the stress state is related to the damage state by the failure model to describe the accumulation of material damage and the occurrence of the ultimate fracture. The stress triaxiality \(\eta\) can be defined in terms of the hydrostatic stress \(\sigma_{m}\) and the equivalent stress \(\overline{\sigma}\), represented by the following equation:  
\[\eta=\frac{\sigma_{m}}{\overline{\sigma}} \tag{1}\]  
where, \(\sigma_{m}=\frac{1}{3}(\sigma_{1}+\sigma_{2}+\sigma_{3})\); \(\overline{\sigma}=\sqrt{\frac{1}{2}\Big{[}\big{(}\sigma_{1}-\sigma_{2}\big{)}^ {2}+\big{(}\sigma_{2}-\sigma_{3}\big{)}^{2}+\big{(}\sigma_{3}-\sigma_{1}\big{)} ^{2}\Big{]}}\); \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\) represent the three principal stresses with the relationship \(\sigma_{1}>\sigma_{2}>\sigma_{3}\). The Lode angle \(\theta\) is a significant measure of the stress state and can be defined in terms of stress invariants as follows:  
\[\theta=\frac{1}{3}\left(\arccos\left(\frac{3\sqrt{3}}{2}\frac{J_{3}}{J_{2}^{ \frac{3}{2}}}\right)\right) \tag{2}\]  
Figure 2: Geometric dimensions of the specimen with dimensions in millimeters. where \(J_{2}\) and \(J_{3}\) are the second and third deviatoric stress invariants, respectively. the Lode angle can be normalized to the Lode angle parameter as:  
\[\overline{\theta}=1-\frac{6\theta}{\pi} \tag{3}\]  
The value of the Lode angle is taken within a specified range, denoted as (0, \(\overline{\hat{s}}\)). Therefore, the value range of the Lode angle parameter is (\(-1,1\)). [PERSON] and [PERSON] [23] established the MMC fracture model for the prediction of metal fracture based on the Mohr-Coulomb damage criterion. This model enables the calculation of the equivalent plastic strain at failure through the incorporation of stress triaxiality and the Lode angle parameter. \[\overline{e}_{p}^{f}=\left\{\frac{A}{c_{2}}\times\left[c_{\theta}^{s}+\frac{ \sqrt{3}}{2-\sqrt{3}}(1-c_{\theta}^{s})\left(\sec\left(\frac{\overline{\theta }\pi}{6}\right)-1\right)\right]\times\left[\sqrt{\frac{1+c_{1}^{2}}{3}}\cos \left(\frac{\overline{\theta}\pi}{6}\right)+c_{1}\left(\eta+\frac{1}{3}\sin \left(\frac{\overline{\theta}\pi}{6}\right)\right)\right]\right\}^{-\frac{1}{n}} \tag{4}\]  
where \(\overline{e}_{p}^{f}\) is the equivalent plastic strain at failure, \(A\) is an equation constant, and \(n\) is the hardening exponent. In addition, \(c_{1}\), \(c_{2}\) and \(c_{\theta}^{s}\) are material constants that need to be obtained by fitting the experimental results. The damage evolution model matched with the MMC criterion is the GISSMO model. In the GISSMO model [27; 28], the damage condition of material is characterized by the nonlinear damage accumulation factor \(D\) (\(0<D\leq 1\)):  
\[\Delta D=\frac{k}{\overline{e}_{p}^{f}}D^{\frac{k-1}{k}}\Delta\varepsilon_{p} \tag{5}\]  
where \(k\) is the damage accumulation exponent; \(\Delta\varepsilon_{p}\) is the equivalent plastic strain increment. Furthermore, the GISSMO model introduces a parameter \(F\) (\(0<F\leq 1\)) to quantify material instability, which reflects the effect of instability on material softening. In the GISSMO model, parameter \(F\) is associated with the equivalent plastic strain through a nonlinear equation similar to Equation (5):  
\[\Delta F=\frac{k}{\varepsilon_{crit}(\eta)}F^{(1-\frac{1}{k})}\Delta \varepsilon_{p} \tag{6}\]  
where \(\varepsilon_{crit}\) is the critical equivalent plastic strain at instability. When the magnitude of the instability parameter \(F\) reaches 1, the current damage accumulation factor \(D\) is stored as the critical damage parameter \(D_{crit}\).