# Profile Design of the Grooved Die and Rolling Force Prediction in the Cold Pilger Rolling Process
## 1 Introduction
(1) The shape in which the radius of the surface semicircle between two angles is linear (linear profile); (2) the shape that has a cosine curve with small cross-section reduction in the inlet and outlet parts, while sudden cross-section reduction occurs in the middle part (cosine profile); and (3) the shape that has a quadratic curve with large cross-section reduction in the inlet and small cross-section reduction in the outlet (quadratic profile). For these three types, variable rolls and a mandrel were generally designed. \begin{table}
\begin{tabular}{c c} \hline \hline
**Material** & **Applications** \\ \hline Stainless steel & Chemical \\  & Medical \\  & Military \\  & Pure gas \\ \hline  & Boiler construction \\  & Chemical engineering \\ \hline  & Aeronautical engineering \\  & Medical \\ \hline  & Zircaloy \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Applications of cold pilger rolling process. Figure 1: Application of cold pilgered products. Figure 2 depicts a comparison of the distribution of the radius and the movement distance of the rolling roll for the die shape. Unlike other shapes, when the pilger die shape is linear (Figure 2a), it has a linearly increased distribution of deformation from the inlet to the outlet without being subjected to sudden deformation in a certain section. In this case, the pipe slowly reaches a steady state. Because of the large forming load at the outlet, there is a risk of mandrel breakage and low shape precision. When the pilger die shape has a cosine profile (Figure 2b), the forming load change rate is low at the inlet and outlet; however, the forming load suddenly increases in the middle part. The time to reach the steady state is relatively longer than that when the pilger die shape is linear. Nevertheless, the load can be predicted to be lower than that of the linear profile because it is formed at almost the same rate of forming load change from the middle part to the outlet. When the pilger die shape has a quadratic profile (Figure 2c), the load increases sharply at the inlet part and then slowly decreases from the middle part. To summarize the results of the three shapes presented above, the shape of the quadratic profile shows the best outcome when considering the forming load distribution, the time required to reach a steady state, and the shape change in shape in the pilger rolling process [8]. Along with die groove shape design, another important aspect of the cold pilger rolling process is the prediction of the rolling force according to the groove shape. Numerical analysis and finite element methods have been used to predict the rolling force in this process. However, in the conventional numerical method, the rolling force prediction range is constrained because the force is predicted for a limited die shape [9; 10]. To resolve this problem, a predictive model was developed using the finite element method for precise prediction of the rolling force over a wider range. However, because the analysis time is too long in this approach, it is difficult to predict various shapes [11; 12; 13; 14]. Therefore, for cold pilger rolling process design, a die shape design technique that includes various process variables, as well as rolling force prediction techniques that consider these process variables, should be simultaneously investigated. In this study, we, therefore, developed a die groove shape design method, along with a rolling force prediction model according to die groove shape, to design the cold pilger rolling process of a SUS304 seamless pipe. A rolling force prediction model is herein proposed that considers the main variables (cross-section reduction rate, principal deformation zone, roll die radius, and die groove angle) that affect the rolling force. The die groove shape was designed based on the dimensions of the initial tube and the final product, as well as on the working zone and roller specifications. A pilger rolling experiment was performed to validate the proposed die groove shape design method and the rolling force prediction model.