## 2 Theoretical Model of the Bimetallic Tube Bending Spring-Back  
### Principle of the Bimetallic Tube Spring-Back  
The stress-strain relationship in the process of tube bending can be described simply as a power hardening model. Assuming that unloading is a completely elastic deformation process, the elastic-plastic deformation of tube bending is shown in Figure 1. Point M is the dividing point between elastic deformation and plastic deformation in the loading process. Point \(N\) is the machining maximum stress point. The load gradually increases from point \(O\) to point \(N\) to meet the requirements of bending processing. Unloading from point \(N\) to point \(P\). The position of point \(P\) indicates that there are residual stress \(\sigma^{\prime}_{N}\) and residual strain \(\varepsilon^{\prime}_{N}\) in the component. Considering the fiber layer with radial distance y from the stress neutral layer at point \(N\) shown in Figure 2. The micro internal force \(\Delta\sigma dS\) in each layer of the circular section forms a group of spatial parallel force systems. Since the bending moment on the tube is zero after unloading, the stress change during unloading satisfies the following relationship:  
Figure 1: Bending loading (unloading) process and deformation area of cross section. \[\Delta\sigma=\sigma_{N}-\sigma_{N}^{\prime} \tag{1}\]  
\[\int y\cdot\Delta\sigma dS=M_{O} \tag{2}\]  
where \(M_{O}\) denotes bending moment on the tube when loaded to the maximum stress. Ignoring the residual stress and the delayed rebound after instantaneous rebound, the unloading process line can approximately extend to the point where the stress is zero along the straight line NP. Assuming that the stress neutral layer and the strain neutral layer at point \(N\) coincide, the variation of strain corresponding to the variation stress above is  
\[\Delta\varepsilon=\frac{D+y-R_{0}}{R_{0}}-\frac{y}{R_{0}^{\prime}}=\frac{ \Delta\sigma}{E} \tag{3}\]  
where \(D\) denotes curvature radius of the stress neutral layer at point \(N\), \(R_{0}\) denotes curvature radius of the central axis of the tube before spring-back, and \(R_{0}^{\prime}\) denotes curvature radius of strain neutral layer after spring-back. In the analysis of the unloading process above, the relationship between \(R_{0}^{\prime}\) and \(M_{O}\). \(M_{O}\) can also be calculated through the material model and yield strength theory during tube loading. Further, \(R_{0}^{\prime}\) can be solved. According to the volume incompressible condition, the unloading spring-back angle is  
\[\Delta\theta=\theta\cdot\frac{\theta-\theta^{\prime}}{\theta}=\theta\cdot(1- \frac{R_{0}}{R_{0}^{\prime}}) \tag{4}\]  
Compared with the integrated tube fittings, the bimetallic tube is composed of two layers of materials, each layer of which has its own elastic modulus. When the elastic modulus of the two layers is not equal, the spring-back behavior cannot perform following their own elastic modulus due to the mutual restraint between two layers. Since the effective bending requirement of the bimetallic tube is that the two layers do not separate in the whole forming process, the composite tube's spring-back behaviors can be regarded as the integral tube's elastic deformation following a certain composite elastic modulus \(E_{c}\). Figure 2: Differential of circular section and spring-back diagram of bending tube. Consider a double-layer tube with elastic tension or compression deformation under axial force \(F\). Figure 3 illustrates the situation. The strain on each part of the tube is equal, which is expressed by \(\varepsilon\). According to the fact that the axial forces on any circular section are equal, Formulas (5) and (6) can be obtained, and Formula (7) for calculating the composite elastic modulus \(E_{c}\) of the double-layer tube can be derived.