## 4 Initial Optimization Design of Extrusion Roller
### Analysis of Results of Initial Optimization  
The results of the seven groups of simulation tests are presented in Table 2. \(R\) increased with the increase of \(L\), and the maximum stress value now decreased with the increase of \(L\). With \(L\) increasing from 25 mm to 85 mm and \(R\) increasing from 25 mm to 157 mm, the maximum equivalent stress decreased from 579.87 MPa to 477.12 MPa, which is a decrease of 112.75 MPa. The simulation results where \(L\) = 85 mm and \(R\) = 157 mm are shown in Figure 14. The optimization results were as follows: The contact stress value after optimization was reduced from 345.61 MPa to 289.52 MPa, which is a reduction of 16%. The contact sliding distance decreased from 1.315 mm to 0.256 mm, which is a decrease of 81%. The maximum equivalent stress decreased from 651.03 MPa to 477.12 MPa, a decrease of 26%. The maximum deformation was reduced from 1.379 mm to 1.102 mm, which is a reduction of 20%. The yield limit of the roller sleeve material was 835 MPa, and the optimized stress met the strength requirements of the extrusion roller. The optimized contact stress and maximum equivalent stress became smaller, and the stress concentration still occurred in the transition arc area. However, with the increase of the area of stress concentration area, the arc transition was smoother, which made the roller sleeve less prone to cracking. It can be seen that the predetermined optimization scheme was effective.  
Figure 11: The section of optimization model of roller shaft.  
Figure 12: The section of optimization model of roller sleeve.  
Figure 13: Parameters of transition arc.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Group** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \(L\) (mm) & 25 & 35 & 45 & 55 & 65 & 75 & 85 \\ \(R\) (mm) & 25 & 37 & 53 & 73 & 97 & 125 & 157 \\ \(\sigma_{\max}\) (MPa) & 579.87 & 585.32 & 589.87 & 565.32 & 518.71 & 490.59 & 477.12 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Stress values of different transition arc parameters.  
Figure 14: Simulation results of the improved structure: (**a**) contact stress nephogram; (**b**) sliding distance nephogram; (**c**) equivalent stress nephogram of roller sleeve; (**d**) deformation nephogram of roller sleeve; (**e**) equivalent stress nephogram of roller shaft; (**f**) deformation nephogram of roller shaft.  
The optimization scheme increased the contact area between the roller shaft and the roller sleeve by setting the transition arc so as to increase the strength of the extrusion roller and reduce the deformation. With the increase of the transition arc, the change in the trend of the contact surface between the roller shaft and the roller sleeve in the width direction of the extrusion roller decreased slowly. The stress reduction trend was also slower. The simulation test results showed that the optimization scheme was feasible. However, due to the large interval between the length and radius of the transition arc, the maximum stress was in a continual downward trend. It was therefore necessary to further refine the optimization scheme of the extrusion roller to find the optimal structural parameter values.