## 2 Theoretical Model of the Bimetallic Tube Bending Spring-Back
### Uploading Moment of the Bimetallic Tube  
The stress-strain relationship of the two layers of metal materials can be described by the following power hardening model because of its practicality and accuracy. Different parameters of the inner layer and outer layer are distinguished by subscripts. The symbol with subscript 1 indicates the parameters of the outer layer while subscript 2 indicates inner correspondingly.  
\[\sigma_{j}=\left\{\begin{array}{l}E_{j}\varepsilon,0<\varepsilon\leq\frac{ \sigma_{Sj}}{E_{j}}\\ K_{j}\varepsilon^{n_{j}},\frac{\sigma_{Sj}}{E_{j}}<\varepsilon\end{array}\right.,j=1,2 \tag{8}\]  
where \(K_{j}\) denotes strength coefficient, \(n_{j}\) denotes hardening index, and \(\sigma_{Sj}\) denotes yield strength.  
Figure 3: Bi-layered straight tube under axial elastic tension or compression.  
The tangential stress along the bending path is the main factor effecting the plastic tension and compression deformation and the tangential strain is the main characterization of elastic recovery. To simplify the analytical process, the non-uniform deformation in bending has been ignored. It is also assumed that the tangential stress-strain relationship along the bending path is consistent under tension and compression, which conforms to the power hardening material model. Considering a certain state with a fixed bending radius and angle, the stress distribution on the circular section approximately exhibits lamellar layers parallel to the stress neutral layer. The stress value gradually increases from the neutral layer along the radial direction of the circular section. So, it can be assumed that only elastic deformation occurs in the volume layer near the neutral layer. As the section radius increases, plastic deformation happens to occur in the volume layer far from the neutral layer. For multi-layer composite bending tubes, each layer of material on the circular section conforms to this feature. According to this, the circular section deformation area of the bimetallic tube in plane CNC bending can be divided into elastic deformation region and plastic deformation region, as shown in Figure 4.  
There are interfaces between the elastic deformation region and the plastic deformation region of each layer. The position of the interfaces is determined by the yield strength \(\sigma_{S}\) of materials respectively and symmetry about the neutral layer. To simplify the calculation, the wall thinning and the flattening phenomena have been neglected. The stress neutral layer and the geometry intermediate are regarded as coincident when deriving the uploading moment. By setting the distance between interfaces and the strain neutral layer as \(h_{j}(j=1,2)\), the position of interfaces can be expressed as follows:  
\[\sigma_{Sj}=E_{j}\cdot\frac{h_{j}}{R_{0}} \tag{9}\]  
\(M_{e}\) and \(M_{p}\) denote the moments of the elastic deformation region and the plastic deformation region. By the integration of the differential layer with z distance from the strain neutral layer, the banding moment under a certain state is  
\[M=M_{e}+M_{p} \tag{10}\]  
\[\begin{array}{l}M_{e}=4\int_{0}^{h_{1}}z\cdot E_{1}\cdot\frac{z}{R_{0}}\cdot (\sqrt{r^{2}-z^{2}}-\sqrt{\left(r-t_{1}\right)^{2}-z^{2}})dz\\ +4\int_{0}^{h_{2}}z\cdot E_{2}\cdot\frac{z}{R_{0}}\cdot(\sqrt{\left(r-t_{1} \right)^{2}-z^{2}}-\sqrt{\left(r-t_{1}-t_{2}\right)^{2}-z^{2}})dz\end{array} \tag{11}\]  
Figure 4: Section deformation area of the bimetallic bending tube.  
\[\begin{split}& M_{p}=4\int_{h_{1}}^{r}z\cdot K_{1}\cdot(\frac{\tilde{z}}{ R_{0}})^{n_{1}}\cdot(\sqrt{r^{2}-z^{2}}-\sqrt{\left(r-t_{1}\right)^{2}-z^{2}})dz\\ &+4\int_{h_{2}}^{r-t_{1}}z\cdot K_{2}\cdot(\frac{\tilde{z}}{R_{0} })^{n_{2}}\cdot(\sqrt{\left(r-t_{1}\right)^{2}-z^{2}}-\sqrt{\left(r-t_{1}-t_{2 }\right)^{2}-z^{2}})dz\end{split} \tag{12}\]