## 2 Analytical Method
### Springback Prediction
#### 2.4.2 Nonlinear Unloading  
Generally, elastic unloading is applied to find the \(\Delta\)(1/R). However, it was shown that the unloading is not entirely elastic, and there are small scale plasticity effects in springback [19; 20]. In order to consider the effects of nonlinearity in unloading and springback, several methods were suggested by various researchers. [PERSON] and [PERSON] [21] introduced a quasi-plastic-elastic (QPE) strain component in addition to plastic and elastic strain components. The QPE strain allows an additional strain recovery, which cannot be captured by elastic unloading. They used some aspects of two-yield surface approaches. Another recently proposed method is a piecewise-linear approach that adopts the concept of multiple yield surfaces [22]. Although these methods are successful in capturing the nonlinearity in unloading, they have the complexity of applying the multiple yield surface approaches which make them suitable for finite element analysis. The most common approach is to define a variable elastic modulus for the unloading [19; 23; 24]. In this method, the elastic modulus is the function of the accumulated plastic strain. This method is called the chord model, since the coefficients of the proposed function are determined by fitting the function to the measured chord moduli. Chord modulus is the slope of a chord or a straight line drawn between the unloading start point and the end point (\(\sigma=0\)) of the unloading stress-strain curve. In this study, to describe the nonlinear elastic unloading [PERSON], the chord method was adopted. According to [PERSON] et al. [25], the variation of the effective elastic modulus with accumulated plastic strain can be expressed in terms of an exponential equation as described in Equation (30)  
\[E_{eff}=E_{0}-(E_{0}-E_{a})(1-exp(-\xi\overline{e}^{\prime})) \tag{30}\]  
where \(E_{0}\) and \(E_{a}\) are the initial and saturated elastic moduli, \(\overline{e}^{\prime}\) is the equivalent plastic strain and \(\xi\) is a material parameter. As it can be seen in Figure 3, Equation (29) was fitted to the experimental data achieved from experimental stress-strain curves in the loading-unloading with different pre-strain values for SUS304 TD specimens from [PERSON] et al. (2009a) [6]. The applied coefficients for the chord method are listed in Table 2, where T and C denote tension and compression, respectively. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Material** & \(\mathbf{E_{a}^{T}}\) **(GPa)** & \(\mathbf{E_{a}^{C}}\) **(GPa)** & \(\mathbf{\xi_{T}}\) & \(\mathbf{\xi_{C}}\) \\ \hline SUS304 & 148 & 133.8 & 60.76 & 75.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured coefficients of variable elastic modulus. Figure 3: Variation of effective elastic modulus versus plastic strain. The equation for the unloading moment with the variable elastic modulus \(M_{U}^{\rm VE}\) and the assumption of no neutral surface change during unloading, is given as  
\[ \begin{split} M_{u}^{\rm ve}&=\int_{y_{c}}^{y_{t}}E^{ \prime}(\frac{y}{\Delta r})ydy+\int_{y_{t}}^{y_{c}}\left(E^{\prime}-(E^{\prime}- E_{a}^{T^{\prime}})(1-exp(-\xi_{T}\overline{\varepsilon}_{T}^{p}))\right)(\frac{y}{ \Delta r})ydy\\ &\qquad\qquad\qquad\qquad\qquad+\int_{-\frac{y_{c}}{2}}^{y_{c}} \left(E^{\prime}-(E^{\prime}-E_{a}^{C^{\prime}})\left(1-exp(-\xi_{c}\overline{ \varepsilon}_{c}^{p})\right)\right)(\frac{y}{\Delta r})ydy\end{split}  \tag{31}\]  
where \(E_{a}^{T}\), and \(E_{a}^{C}\) are saturated elastic moduli in tension and compression, \(\xi_{T}\) and \(\xi_{T}\) are the material parameters in tension and compression and \(\overline{\varepsilon}_{T}^{p}\) and \(\overline{\varepsilon}_{c}^{p}\) are equivalent plastic strains in tension and compression, respectively. Although in the above equation, the nonlinear relationship between elastic modulus and plastic strain is applied, the relationship between curvature change and unloading moment is linear.