## 2 The True Stress-Strain Relationship of Superalloy GH1140  
The premise of the plastic forming problem is described with the plastic condition and constitutive equations. The functional relationship of these two equations is related to the shape and material properties of the workpiece, but is independent of the stress state [19]. The true stress-strain curve of the superalloy GH1140 is established by the uniaxial uniform tensile test method.  
According to GB/T 228.1-2010, the superalloy GH1140 sheet with thickness of 2 mm should be cut to get the sample [20]. The sample structure before and after the experiment is shown in Figure 1. Five groups of tensile tests with the conditions of 3 mm/min moving speed and 20 \({}^{\circ}\)C normal temperature were carried out on the MTS600 tensile sample machine (Jinan MTS Testing Technology Co., Ltd, Jinan, China). The nominal stress-strain curve is obtained, as shown in Figure 2.  
Figure 1: Tensile test sample of the GH1140 superalloy: (**a**) the tensile test sample structure; and (**b**) the stretched test sample.  
The nominal stress-strain curve reflects that the stress of deformation continues to increase with the increase of displacement, before the metal enters plastic deformation. The strengthening effect of the material is dominant. After entering plastic deformation, the cross-sectional area decreases, the stress concentration appears and the stress value decreases. Therefore, the nominal stress-strain curve does not apply to the tensile process that reflects the changing cross-section of the sample. It is necessary to further derive the true stress-true strain curve.  
\[\sigma_{T}=\sigma(1+\varepsilon) \tag{1}\]  
\[\varepsilon_{T}=\mathit{In}(1+\varepsilon) \tag{2}\]  
Based on the above formulas, the true stress-strain curve of the sample can be obtained by integrating the five samples' nominal stress-strain curves, as shown in Figure 3.  
The elastoplastic equations of metal materials have nonlinear properties. The strength coefficient \(K\) and strain hardening coefficient \(n\) are fitted according to elastoplastic models, which describe the plastic flow behavior of the material. However, the applicability of the model to the obtained tensile test curves is different, so the model needs to be chosen according to the fitting results [4]. As shown in Equations (3)-(6), four elastoplastic models were proposed [21; 22; 23; 24]. The corresponding fitting results are shown in Table 1.  
Figure 3: True stress–strain curve of the GH1140 superalloy.  
Figure 2: Nominal stress–strain curve of the GH1140 superalloy.  
[PERSON]:  
\[\sigma=K_{H}\varepsilon_{t}{}^{nH} \tag{3}\]  
[PERSON]:  
\[\sigma=\sigma_{\sigma}+K_{L}\varepsilon_{t}{}^{nL} \tag{4}\]  
[PERSON]:  
\[\sigma=\sigma_{\mathrm{s}}-K_{\mathrm{g}}e^{\mu_{\mathrm{g}}^{\varepsilon_{t} }\varepsilon_{t}} \tag{5}\]  
[PERSON]:  
\[\sigma=K\varepsilon_{t}^{n}+e^{(K_{1}+\mu_{1}\varepsilon_{t})} \tag{6}\]  
Among them, SSE is the sum of squares due to error. It refers to the total deviation of the sample data from the regression equation. RMSE is root mean squared error. DFE is the degrees of freedom error. The coefficient of determination R indicates the fitting degree of the regression equation corresponding to the sample data, which is used to judge the accuracy of the fitting equation. The closer the R value is to 1, the better the fitting effect. Therefore, the true stress-strain equation fitted by the Ludwik model is selected, as shown in Equation (7).  
\[\sigma_{t}=1819(\varepsilon_{t}+0.1957)^{1.099}, \tag{7}\]  
where \(\sigma_{t}\) expresses the true stress during uniaxial tension and \(\varepsilon_{t}\) expresses the true strain during uniaxial tension.  
The comparison between the fitting curve and the true stress-strain curve is shown in Figure 4. The fitting accuracy is high, which verifies the correctness of the model in this paper.