## 2 STACER Theoretical Model  
Consider in this study in the scope of materials and its processing technology, an analytic model of the forming process of the STACER cylinder is proposed. It aims to provide a general method for the design and production of a STACER from a theoretical perspective. The key size parameters of STACER include gradient diameter D, helix angle \(\alpha\), and pitch L, as shown in Figure 1[32]. In Figure 2, the helix angle \(\alpha\) is determined by the angle at which the strip is placed between the die. The gradual diameter D and the pitch L are influenced by various factors, including material and forming process parameters. STACER is produced by a compositing stretch and press bending (CSPB) process. The forming principle is shown in Figure 2. The strip enters from the left side at an angle \(\alpha\) to the mold, is plastically deformed by the mold, and is collected on the right side. This process can be divided into three areas: I is an elastic deformation zone, which is subjected to tension F1 and remains straightened without deformation. Zone II is a plastic deformation zone, which the strip is bent and deformed by the action of convex and concave molds and tension on both sides. Zone III is a complex deformation area; the strip is straightened backwards in the exit's direction [33; 34]. In order to establish a theoretical model for STACER, the theoretical calculation hypothesis is proposed:  
1. The strip used for CSPB forming is a thin strip with a thickness direction much smaller than the width direction. It is considered that the strain in the width direction during the bending process is 0. Figure 2: Geometric parameters and area division of strip CSPB forming process. The tangential tensile stress on the strip width direction is evenly distributed along the thickness direction. The tensile stress exists to reduce the thickness of the strip and lead to the curved neutral layer moving inward. 3. The strip is subjected to the Kirchhoff straight normal assumption during the bending forming process. The Bausching effect is neglected during the strip forming process. 4. The strip has a constant volume during the CSPB forming process [35; 36; 37]. Based on the above assumptions, a mathematical model of STACER is established. During the forming process of the material by the CSPB method, the effect of the convex and concave mold on the sheet material can be simplified as the effect on the three points A, B, and C on the strip. Considering the punch radius \(\mathrm{R_{d}}\), the action point on the sheet is moved downwards and the radius of curvature has been increased. The change in mold fillet can be completely equivalent to the change of mold gap. The equivalent mold clearance is:  
\[\mathrm{T_{e}}=\mathrm{T}+\mathrm{T_{d}}=\mathrm{R_{d}}(1-\sin\frac{\theta_{0 }}{2})+\mathrm{T}, \tag{1}\]  
Among them, T and \(\mathrm{T_{d}}\) are the incremental gaps between the mold gap and the mold radius. The bending angle \(\theta\) is related to the mold cone angle \(\theta_{0}\) as follows:  
\[\theta=\pi-\theta_{0}, \tag{2}\]  
From the formula above, the increase of the mold angle increases the gap of the equivalent mold, and the diameter of the obtained spring reel is increased. Assuming that the strip has a thickness t after forming, the relationship between the outer radius of curvature \(\mathrm{R_{0}}\) and the inner radius of curvature \(\mathrm{R_{1}}\) of the strip after forming is as follows:  
\[\mathrm{R_{0}}=\mathrm{R_{1}}+\mathrm{t}=\mathrm{R_{1}}+\mathrm{t_{0}}-\Delta \mathrm{t}, \tag{3}\]  
\[\mathrm{R_{m}}=\frac{1}{2}(\mathrm{R_{1}}+\mathrm{R_{0}}), \tag{4}\]  
where \(\mathrm{t_{0}}\) is the initial thickness of the strip and \(\Delta\mathrm{t}\) is the thickness increment before and after the strip is deformed. \(\Delta\mathrm{t}=\mathrm{t_{0}}-\mathrm{t}\).