## 4 Results and Discussion
### Residual Stress Distribution Around Actual Inclusions  
When inclusions are assumed as spherical, the shape details are actually neglected. Nevertheless, as shown in Figure 3, the inclusions of different types usually possess different geometry shapes, which will lead to important stress concentration around interfaces between the inclusions and the steel matrix. In this Section, four types of inclusions in Figure 3 were inserted in the square steel area, and the residual stress distributions were simulated. The simulation results are shown in Figure 6. Generally, the residual stress also decreases when the distance to inclusion surface increases, which is consistent with the case of the perfect spherical inclusion, while the maximum residual stresses for these four types of inclusions all increased compared to the maximum stress of spherical inclusions. Besides, the residual stress on the surface between the inclusion and the steel matrix changes with the shape of inclusions. To quantitatively analyze the effect of the inclusion shape on residual stress changes, two parameters were introduced: the edge angle \(\theta_{\text{sv},i}\) and the local residual stress \(\sigma_{\text{sv},i}\). These two parameters represent the smooth of the inclusion edge and the corresponding local residual stress for each \(\theta_{\text{sv},i}\). The calculation method of \(\theta_{\text{sv},i}\) is shown in Equation (4) and the schematic is shown in Figure 7.  
\[\theta_{\text{sv},i}=\theta_{i+1}-\theta_{i} \tag{4}\]  
where \(i\) is the identification number of digital point on the inclusion edge. All the digital point were sequenced continually in an anticlockwise direction. \(\theta_{i}\) is the angle between the \(x\)-axis and the straight line through the \((i\)-\(1)^{th}\) and the \(i^{th}\) digital point on the inclusion edge, which can be calculated with Equation (5). \[\theta_{i}=\arctan\left[(y_{i}-y_{i-1})/(x_{i}-x_{i-1})\right](0\leq\theta_{i} \leq 180^{\circ}) \tag{5}\]  
where \(x_{i}\) and \(y_{i}\) are the coordinates of the digital \(i^{th}\) point on the inclusion edge. When \(\theta_{i}\) is a positive value, the local part of inclusion is convex; when \(\theta_{i}\) is a negative value, the local part of inclusion is concave. The smaller the absolute value of \(\theta_{\text{sv},i}\) is, the smoother the inclusion edge is. When \(\theta_{\text{sv},i}\) is zero, the local shape of the inclusion is a straight line. For a right angle, \(\theta_{\text{sv},i}\) is \(90^{\circ}\). It is to be noted that, the distance between each neighboring point will affect the result of \(\theta_{\text{sv},i}\). Therefore, the distance  
Figure 6: Residual stress distributions around the inclusions with their actual shapes: (**a**) Mg-Al-O; (**b**) Al-Ca-O-S; (**c**)TiN; (**d**) MnS. between each neighboring point is kept consistent in this study which offers a value of 3.6\({}^{\circ}\) for \(\theta_{\text{sv},i}\) in a perfect circle. These two parameters of each digital point on inclusion edges of Mg-Al-O, Al-Ca-O-S, TiN and MnS shown in Figure 6 were extracted and calculated. The relation between these two parameters is shown in Figure 8. To avoid the interaction of neighboring points, the data with \(\theta_{\text{sv},i}\) lower than 5\({}^{\circ}\) were neglected. As shown in the figure, the local residual stress \(\sigma_{\text{sv},i}\) increases with the absolute value of edge angle \(\theta_{\text{sv},i}\) for all four types of inclusions, which indicates that no matter the local shape of the inclusion is convex or concave, the sharp edge will lead to stress concentration. When the local shape is convex, the stress concentration is in the steel matrix side; when the local shape is concave, the stress concentration is in the inclusion side (see Figure 6). Figure 8: Relation between edge angle \(\theta_{\text{sv},i}\) and local residual stress \(\sigma_{\text{sv},i}\) for different inclusions. Figure 7: Schematic of the edge angle \(\theta_{\text{sv},i}\) for inclusions.