# Fracture Threshold Analysis and Parameter Matching of Cut-Out Induced Bolts for Subway Couplers
## 2 Experiment and Simulation Analysis
### Simulation Analysis
#### 2.2.2 Material Parameter  
1. Buffer and expansion tube  
Since the complete coupler force-displacement characteristics were not obtained in the full-scale vehicle test, ideal coupler force-displacement characteristics were used as a substitute in the simulation modeling of the expansion tube. Figure 12 exhibits the force-displacement curves of the expansion tube and buffer. According to the design energy absorption parameters of the coupler, the effective stroke of the expansion tube is 245 mm, with an average crush force of 1200 kN. The quasi-static compression force-displacement characteristics of the EFG3 rubber friction buffer were obtained through quasi-static compression experiments [31]. Figure 11: The model of the test conditions. Figure 12: The force–displacement curve of the expansion tube and buffer: (**a**) expansion tube; (**b**) buffer. Bols and Other Components of Coupler  
The bolt material is 40 Cr and the stress-strain relationship of the 40 Cr material obtained by [PERSON] [29] on the universal fatigue testing machine is shown in Figure 13. The blue curve represents the engineering stress-strain relationship of the 40 Cr material, which is mainly based on the assumption that the cross-sectional area of the bolt does not change during the tensile process, and is obtained through Equations (1)-(3). \[\left\{ \begin{array}{l}\sigma\ =\ \frac{F}{A}\\ \varepsilon\ =\ \frac{l-l_{0}}{l_{0}}\end{array} \right. \tag{1}\]  
\[\sigma_{t}\ =\ \sigma(1+\varepsilon) \tag{2}\]  
\[\varepsilon_{t}\ =\ \ln(1+\varepsilon) \tag{3}\]  
In the equations, \(\sigma\) represents the engineering stress; \(\varepsilon\) represents the engineering strain; \(F\) represents the tensile load; \(A\) represents the original cross-sectional area of the sample standard distance section; \(l_{0}\) represents the original length of the sample standard distance section; \(l\) represents the length of the sample standard distance section after extension; \(\sigma_{t}\) represents the true stress; \(\varepsilon_{t}\) represents the true strain. However, in the actual tensile process, the cross-sectional area of the specimen will decrease. To further obtain the true stress-strain relationship of the material, it is necessary to make corresponding corrections to the engineering stress-strain curve. [PERSON] assumed that after the specimen undergoes necking, the cross-section at the necking position appears as a circular arc shape as shown in the figure, and then it is possible to obtain the following true stress-strain correction formulas (Equations (4)-(7)) based on the principle of volume conservation during the tensile process. \[\sigma_{t}\ =\ \frac{F}{A}\ =\ \frac{F}{\pi D_{i}^{2}} \tag{4}\]  
\[e_{t}\ =\ \ln\biggl{(}\frac{A}{A_{i}}\biggr{)}\ =\ \ln(\frac{D}{D_{i}}) \tag{5}\]  
\[L_{0}\ =\ \frac{\Delta L_{i}(2D_{0}^{2}+D_{i}^{2})}{D_{0}^{2}-D_{i}^{2}} \tag{6}\]  
Figure 13: Stress–strain curve of bolt material modified by the formula [29]. \[D_{i}\,=\,\sqrt{\frac{3D_{0}^{2}L_{0}}{L_{0}+\Delta L_{i}}-2D_{0}^{2}} \tag{7}\]  
In the equations, \(A\) represents the original cross-sectional area, \(A_{i}\) represents the cross-sectional area at time \(i\), \(D\) represents the original diameter of the necking section, and \(D_{i}\) represents the minimum diameter of the necking section at each time \(i\) after necking occurs. In the equations, \(L_{0}\) represents the length before necking, \(L_{i}\) represents the length after necking, \(D_{0}\) represents the diameter before necking, and \(D_{i}\) represents the outer arc radius after necking. \(L_{0}\) can be obtained by substituting the minimum diameter of the necking section at fracture and the final deformation into Equation (6). \(D_{0}\) can be obtained before necking occurs based on the principle of volume conservation. By substituting the minimum cross-sectional diameter \(D_{i}\) of the necking section into Equations (4) and (5) at any time, the true stress-strain values of the necking section can be obtained.