## 5 Fracture Initiation Model
### 3D Fracture Locus  
Stress triaxiality is a major factor that typically affects fracture [26; 27; 28; 38]. In particular, under high stress triaxiality conditions such as tension tests, this has a full effect on fracture. In the tension test, various stress triaxiality conditions are realized by varying the specimen shape and the notch shape, and the corresponding strain-to-fracture initiation and physical properties are then evaluated. [PERSON] and [PERSON] [39; 40] evaluated the yield strength and ultimate tensile strength of structural steels for each tension specimen shape and examined their dependence on stress triaxiality. [PERSON] et al. [41] calibrated a two-parameter ductile fracture model through tension test and FE analysis for stress triaxiality due to various notch changes in tensile specimens. Unlike the tension test, since the range of stress triaxiality in rolling is \(-1.2\) to \(1.2\)[8], the effect on fracture is affected not only by stress triaxiality but also by the Lode parameter [42; 43]. Therefore, a ductile fracture criterion that reflects the Lode parameter is required, and related models [10; 29; 34; 44] are the maximum shear model [44], the modified Mohr-Coulomb (MMC) fracture criterion [10] and so on. In this study, the MMC model, which can predict fracture well not only at high triaxiality but also at negative and low triaxiality, was adopted. [PERSON] and [PERSON] [10] proposed the modified Mohr-Coulomb (MMC) criterion (model) to predict the ductile fracture of 2024-T531 aluminum and TRIP690 materials during the cold forming process. Using the MMC criterion, [PERSON] et al. [45] constructed the fracture locus of AA6061-T6 and predicted the fracture occurring during the forming of multi-pass rolls through FE analysis. [PERSON] et al. [46] predicted fracture behavior during the installation and operation of an offshore pipeline made of X80 steel subjected to tension and torsion using the MMC criterion. \(\overline{\varepsilon}_{f}\big{(}\eta,\overline{\theta}\big{)}\) in the MMC criterion is expressed as follows. \[\overline{\varepsilon}_{f}\big{(}\eta,\overline{\theta}\big{)}\ =\left\{ \begin{array}{c}\frac{K}{C_{2}}\big{[}1-C_{ \eta}(\eta-\eta_{0})\big{]}\times\Big{[}C_{\theta}^{s}+\frac{\sqrt{3}}{2- \sqrt{3}}\big{(}C_{\theta}^{ax}-C_{\theta}^{s}\big{)}\Big{(}\sec\Big{(}\frac{ \pi\overline{\theta}}{6}\Big{)}-1\Big{)}\Big{]}\\ \times\bigg{[}\sqrt{\frac{1+C_{1}^{2}}{3}}\cos\Big{(}\frac{\pi\overline{ \theta}}{6}\Big{)}+C_{1}\Big{(}\eta+\frac{1}{3}\sin\Big{(}\frac{\pi\overline{ \theta}}{6}\Big{)}-1\Big{)}\bigg{]}\end{array} \right\}^{-\frac{1}{\theta}} \tag{8}\]  
Equation (8) was adopted in this study to simulate the edge cracking of a strip during cold rolling. The eight parameters (\(K\), \(n\), \(C_{\eta}\), \(\eta_{0}\), \(C_{\theta}^{s}\), \(C_{\theta}^{ax}\), \(C_{1}\), and \(C_{2}\)) must be calibrated for the construction of the fracture envelope from the MMC criterion. \(K\) and \(n\) are the same material constants adjusted for the power law (\(\overline{\sigma}=K\overline{\sigma}^{n}\)) [17]. \(C_{\eta}\) and \(\eta_{0}\) are the parameters that determine the yield criterion. The dependence of the Lode angle parameter is controlled by \(C_{\theta}^{s}\), and the asymmetry of the fracture locus is influenced by the parameter \(C_{\theta}^{ax}\). The constants \(C_{1}\) and \(C_{2}\) must be determined through fracture tests. Because the von Mises yield function was used in this study, \(C_{\eta}\) = 0 and \(C_{\theta}^{s}=C_{\theta}^{ax}\) = 1 [34].