## 2 STACER Theoretical Model
According to the geometric relationship between parameters, it can be found that:  
\[\mathrm{R_{0}}=\mathrm{T_{e}}+\mathrm{R_{1}}\sin\frac{\theta_{0}}{2}, \tag{5}\]  
From Equations (2)-(4), it can be calculated:  
\[\mathrm{R_{1}}+\mathrm{t}=\mathrm{T}+\mathrm{R_{d}}(1-\sin\frac{\theta_{0}}{2 })+\mathrm{R_{1}}\sin\frac{\theta_{0}}{2}, \tag{6}\]  
Solving is available:  
\[\mathrm{R_{1}}=\frac{\mathrm{T}-\mathrm{t}}{1-\sin\frac{\theta_{0}}{2}}+ \mathrm{R_{d}}, \tag{7}\]  
Bringing Equation (7) into Equation (3), we can obtain:  
\[\mathrm{R_{0}}=\frac{\mathrm{T}-\mathrm{t}\sin\frac{\theta_{0}}{2}}{1-\mathrm{ sin}\frac{\theta_{0}}{2}}+\mathrm{R_{d}}, \tag{8}\]  
The mathematical relationship between the curvature radius of the inner and outer layers of the strip with other factors is not able to be considered through Equations (6) and (7). The strip reduction during rolling is not considered. When considering strip reduction, \(\Delta\mathrm{t}\
eq 0\), the following relationship can be obtained from the assumption that the volume is constant before and after deformation of the material:  
\[\left\{ \begin{array}{l}L_{10}t_{0}=L_{1}t_{1}\\ L_{0}t_{0}=\frac{\theta}{2}(R_{0}^{2}-R_{1}^{2})=R_{m}\theta t\\ L_{20}t_{0}=L_{2}t_{2}\end{array} \right., \tag{9}\]  
In the equation, \(L_{10}\), \(L_{0}\), and \(L_{20}\) are the original lengths of the three regions before processing; \(L_{1}\) and \(L_{3}\) are the strip lengths corresponding to Zones I and III after processing; t\({}_{0}\) is the initial thickness of the strip; R\({}_{0}\) and R\({}_{1}\) is the inner and outer radii of the strip, respectively; R\({}_{m}\) is the curvature radius of the mid-section of the strip; \(\theta\) is the angle of curvature; R\({}_{n}\) is the curvature radius of the neutral layer. The following relationship can be obtained:  
\[R_{m}=\frac{1}{2}(R_{0}+R_{1}), \tag{10}\]  
\[R_{n}=\frac{L_{0}}{\theta}, \tag{11}\]  
According to the [PERSON]. [PERSON] principle, for an ideal strip, when elastic and plastic deformation occurs, the bend radius of neutral layer of a strip is:  
\[R_{n}=\sqrt{R_{0}R_{1}}, \tag{12}\]  
In summary, after the simultaneous calculation of Equation (7), Equation (9), Equation (10), and Equation (11), Equation (13) can be obtained as follows:  
\[\left(\frac{T-t}{1-\sin\frac{\theta_{0}}{2}}+R_{d}\right)^{2}+\left(\frac{T-t }{1-\sin\frac{\theta_{0}}{2}}+R_{d}\right)t=\frac{t^{4}}{4\left(t_{0}^{2}-t^{2 }\right)}, \tag{13}\]  
The geometric relationship between the strip thickness t after passing through the CSPB process and the convex filled R\({}_{d}\), the gap T, the curvature R of the STACER, and the taper angle \(\theta_{0}\) of the mold can be obtained in consideration of the strip reduction. The key geometric dimensions of the STACER include the helix angle \(\alpha\), the pitch L, and the diameter D. In the theoretical case, according to the equations above, STACERs with different sizes can be designed. The STACER discussed in this paper is a STACER prepared with a post-tension of 10 N, a punch radius of 0.5 mm, a strip thickness of 0.15 mm, a bandwidth of 20 mm, and an angle between the strip and the mold of 60\({}^{\circ}\). \(\alpha\) is 60\({}^{\circ}\), L is 127 mm, and D is 18.6 mm at the smallest diameter.