## 7 Results and Discussion
### Numerical Analysis of Edge Cracking in the Strip  
Figure 9a,b show the difference in edge cracking when 2D and 3D fracture loci are used. The thickness reduction ratio was 25%, the initial notch length was 2 mm, and the secondary radius of curvature of the work roll surface profile (positive roll bending) was 206.4 mm. The tip of the notch was enlarged to observe edge cracking in detail. The length of crack growth in the transverse direction and the length of crack growth in the rolling direction are indicated by \"LCGT\" and \"LCGR\", respectively. The direction in which the crack grew is indicated by \"CGD\". Figure 9a shows the edge cracking behavior when the 2D fracture locus (J-C criterion) was used. LCGT and LCGR are 2.09 mm and 1.73 mm, respectively. CGD is 43.68\({}^{\circ}\). The crack grows in the rolling direction and then advances in the reverse rolling direction (refer to the ellipse indicated by the yellow dashed line in Figure 9a). This is because \(\overline{e}_{f}\) at the advancing crack tip was computed as a function of only the stress triaxiality. Figure 9b shows the edge cracking behavior when the 3D fracture locus (MMC criterion) was used. The LCGT and LCGR were 1.67 mm and 1.53 mm, respectively. CGD was 45.9\({}^{\circ}\). The 2D fracture locus overestimates LCGT and LCGR by 20.1% and 11.6% more than the 3D fracture locus, respectively. This is because the \(\overline{e}_{f}\) in the 3D fracture locus in high stress triaxialities is also influenced by variations in the Lode angle parameters. Figure 9c shows the history of stress triaxialities (solid squares) from the point at which deformation begins at the notch tip of the strip until the onset of fracture during rolling. The Lode angle parameter \(\overline{\theta}\) varied from \(-\)0.81 to 0.72. The dashed line represents the plane stress condition (\(\sigma_{3}\) = 0). Because the stress triaxialities (solid squares) are above the plane stress condition (dashed line), the edge cracks initiate faster with the 2D locus than with the 3D locus. 1 indicates the point at which damage begins to be accumulated. 2 specifies the point at which fracture starts at the tip of the notch. 2 indicates the changes in the stress states at the notch tip when the strip is deformed by cold rolling. Figure 10 shows the effect of the secondary roll-bending ratio (L\({}_{2}\)/R\({}_{2}\)) on the edge cracking behavior when the thickness reduction ratio of a strip for one pass was 25%. L\({}_{2}\) is the secondary length of the work roll barrel, and \(R_{2}\) is the secondary radius of curvature of the convex profile of the work roll barrel (Figure 6), which emulates positive roll bending. Figure 10a shows that LCGT and LCGR are 1.67 mm and 1.53 mm, respectively, when L\({}_{2}\)/R\({}_{2}\) is 0.18. Figure 10b illustrates that LCGT and LCGR are 0.48 mm and 0.73 mm, respectively, when L\({}_{2}\)/R\({}_{2}\) is 0.09. Comparing the two cases, LCGT decreased by 71.2% and LCGR by 52.3%. This is because the tensile stress applied to the edge of the strip significantly reduced as R\({}_{2}\) doubled. Figure 10c shows no edge cracking when (L\({}_{1}\)/R\({}_{1}\)) and (L\({}_{2}\)/R\({}_{2}\)) are 0, which corresponds to flat rolling and is a special case. Because the profile of the work roll barrel is perfectly flat, tensile stress is scarcely applied to the strip edge, which suppresses crack initiation at the notch tip. Note that flat rolling is a condition in which actual cold rolling is almost impossible. Figure 11 shows variations in the length of crack grown in the transverse (LCGT) direction of the strip when the R\({}_{2}\) and initial notch length changes at the thickness reduction ratio of 25% for a pass. LCGT reduces nonlinearly as the secondary roll-bending ratio (L\({}_{2}\)/R\({}_{2}\)) decreases. The shorter the initial notch length, the smaller the crack growth. For example, when the ratio L\({}_{2}\)/R\({}_{2}\) is 0.15, the LCGT corresponding to the initial notch lengths of 2 mm, 1 mm, and 0.5 mm are 1.23 mm, 1.17 mm, and 0.8 mm, respectively.