## 1 Introduction  
High-silicon steel, which usually contains 3.0-3.5 wt% silicon, is produced by hot rolling and cold rolling. High-silicon steel sheets produced by this process serve as an important material for improving the efficiency of transformers and motors because they have a highly efficient magnetic field when charged with electricity [1]. Hereinafter, a high-silicon steel sheet (with width in the range of 1000-1200 mm) is referred to as a \"strip\" for convenience. In the hot (1100-900 \({}^{\circ}\)C) rolling process, the strip edges lose more heat than the strip center by radiation and convection into the atmosphere. Therefore, the hot ductility at the edges was significantly lower than that of the strip center and a lack of dynamic recrystallization due to temperature drop lowers the formability of the edge region [2]. Hence, microcracks always occur at the edge of the strip and develop into macro cracks (referred to in this paper simply as cracks). Therefore, a certain width on both sides of the strip is cut lengthwise after hot rolling. This cut is referred to as trimming. However, excessive trimming leads to a decrease in production and sales profits. During cold rolling, roll bending inevitably occurs because a work roll barrel is typically longer than 1680 mm. The strips have width to thickness ratios greater than several hundred times when rolled in cold rolling mills [1]. Therefore, roll bending, the elastic deformation of the work roll in the direction of the strip width during rolling, is inevitable and is significantly affected by the thickness reduction ratio per pass [3] and the strip width. For positive roll bending (Figure 1a), the strip center surface is wrinkled (center buckle) and edge cracking occurs in the strip. Edge cracking implies the onset of cracks on both sides of the strip and their propagation during rolling. Meanwhile, in the case of negative roll bending (Figure 1b), a lack of flatness occurs at the strip edges (edge waviness) and cracks grow at the strip center. An attempt to reduce the edge cracking may be considered by analyzing the roll bending conditions measured during rolling in actual cold rolling mills; however, this approach has not yielded tangible outcomes because the changes in roll bending condition in actual cold rolling mills are extremely limited. Therefore, to address this challenge, finite element (FE) simulation or pilot cold rolling test should be carried out to find a roll bending condition that reduces the edge cracking. [PERSON] and [PERSON] [4] performed a pilot cold rolling test with computer numerical control (CNC)-machined work roll surface profiles to experimentally examine the effects of roll bending on the crack growth in the 25 mm wide strip. The CNC-machined work roll surface outlines its axis as concave and convex to imitate negative and positive roll bending, respectively. [PERSON] et al. [5] conducted cold rolling testing and FE analysis coupled with the Gurson-Tvergaard-Needleman (GTN) damage model [6] to examine the effects of multiple rolling parameters (tension applied to the 2.1 wt% silicon strip, reduction ratio for one pass, and the friction coefficient between strip and work rolls) on edge cracking. They also performed cold rolling tests with strip specimens (length 1000 mm and width 100 mm). [PERSON] et al. [7] performed a scanning electron microscope (SEM) analysis of fracture surfaces that developed at the edges of a cold-rolled silicon steel strip. They reported not only the formation and growth of voids, but also the shear deformation of voids. They argued that the stress triaxiality could have a value of zero when the strip fractured during cold rolling. Recently, [PERSON] et al. [8] conducted FE analysis coupled with the Johnson-Cook ductile fracture (DF) criterion [9] to predict the length of crack growth and crack growth direction in the strip during cold rolling.