## 4 Results and Discussion
### Residual Stress Distribution Around Spherical Inclusions  
The residual stress distribution around inclusions was already calculated by [PERSON] and [PERSON] [33]. In their model, inclusions were simplified as perfect spherical. The radial stress in the surface between inclusion and steel matrix can be calculated with Equations (2) and (3).  
\[\sigma_{\text{R}}=\frac{(\alpha_{\text{M}}-\alpha)\Delta T}{\frac{0.5(1+\
u_{ \text{M}})+(1-2\
u_{\text{M}})d^{3}}{E_{\text{M}}(1-d^{3})}+\frac{(1-2\
u)}{E}} \tag{2}\]  
\[d=R/R_{\text{M}} \tag{3}\]  
where \(\alpha_{\text{M}}\) and \(\alpha\) are the coefficients of linear expansion of the steel matrix and the inclusion, respectively; \(v_{\text{M}}\) and \(v\) are the [PERSON]'s ratios of the steel matrix and the inclusion, respectively; \(E_{\text{M}}\) and \(E\) are the [PERSON]'s modulus of the steel matrix and inclusion, respectively; \(\Delta T\) is the difference between the holding temperature before vacuum quenching and the room temperature; \(R\) is the radius of the inclusion, and \(R_{\text{M}}\) is the radius of the steel matrix around a single inclusion, which is half of the distance between two inclusions with the same type.  
To clarify the influence of inclusion type on residual stress distribution and make a comparison of the calculation results in the present study with the model of Brooksbank and Andrews, perfect spherical Mg-Al-O, Al-Ca-O-S, TiN, and MnS inclusions were inserted in the square steel matrix. The geometry model and the calculation results of residual stress distribution are shown in Figure 4. It is obvious that the residual stress in the steel matrix around the inclusion decreases when the distance to inclusion surface increases. The position of the maximum stress is on the surface between the inclusion and steel matrix. When the inclusion is spherical, the stress on the surface between inclusion and steel matrix keeps constant around the inclusion. This constant maximum stress varies with inclusion type. The value of Mg-Al-O inclusion is the largest, followed by TiN and Al-Ca-O-S inclusions while the value of MnS inclusion is the smallest. These simulation results are also compared with the calculation results with Equations (2) and (3) in Figure 5. Except for TiN inclusions, the order of the values of the maximum residual stress on the interface between inclusions and steel matrix calculated with the two methods keeps consistent, while the results of TiN inclusions differ more greatly. It is noted that all the residual stresses around perfect spherical Mg-Al-O, Al-Ca-O-S, TiN, and MnS inclusions are compressive due to smaller \(\alpha\) of inclusions than steel matrix.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & \
\begin{tabular}{c} **Coefficient of Linear** \\ **Expansion, \(\alpha\) (\(10^{-6}\cdot^{\circ}\)C)** \\ \end{tabular} &
\begin{tabular}{c} **Young’s Modulus,** \\ **E (GPa)** \\ \end{tabular} & **Poisson’s Ratio, \(\
u\)** \\ \hline Mg-Al-O & 8.4 & 271 & 0.26 \\ Al-Ca-O-S & 5.0 & 113 & 0.23 \\ MnS & 14.8 & 103 & 0.30 \\ TiN & 9.4 & 320 & 0.19 \\ Steel matrix & 23.0 & 206 & 0.30 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mechanical properties of the steel matrix and the inclusions in the martensitic steel [33; 34; 46].  
Figure 4: (**a**) Geometric illustration of the model for the residual stress simulation during cooling; residual stress distributions around the spherical inclusion of (**b**) Mg-Al-O; (**c**) Al-Ca-O-S; (**d**) TiN; (**e**) MnS.  
Figure 5: Comparison of simulated residual stresses and the maximum residual stress on the interface between steel matrix and inclusions calculated with Equation (2): (**a**) Mg-Al-O; (**b**) Al-Ca-O-S; (**c**) TiN; (**d**) MnS.