# Damage Analysis of Rolling Stock Automatic Coupler under Cyclic Loads
## 3 Results
### Damage Summation under Stationary Load  
Low-cycle failure is caused by the accumulation of fatigue damage characterized by an elastic-plastic hysteresis loop and the quasi-static damage characterized by the accumulation of unilateral plastic deformation. In the automatic coupler of a rolling stock, the forces are restricted, resulting in stress-controlled low-cycle loading where strain is unrestrained, develops freely, and quasi-static and fatigue damage accumulates. When the amplitude stresses of the cycle are close to the limit of proportionality of the material, fatigue failure occurs without visible signs of plastic deformation. At moderate amplitude stresses, the so-called intermediate or transient failure occurs, where the failure also produces a small neck and clearly visible fatigue cracks. In an automatic coupler, the stresses vary during operation over a wide range, depending on the weight, velocity of the rolling stock, and the track relief. They range from a few tenths of the limit of proportionality of the material to the stresses exceeding the limit of proportionality and yield strength of the material. In addition, the stresses vary irregularly, i.e., in a non-stationary manner. Their variation characteristics can be determined on a case-by-case basis by directly measuring the forces acting on the automatic coupler. With such a wide range of variations, the material of the automatic coupler is subject to both low and high-cycle fatigue damage. Since the tests have shown that, under stress-controlled low-cycle loading, steel 20 GL accumulates unilateral plastic strain; hence, low-cycle, quasi-static fatigue damage will accumulate where the stresses are above the limit of proportionality, and high-cycle fatigue damage will accumulate where the stresses are below the limit of proportionality. Therefore, the overall damage in the automatic coupler material can be determined by dependency [22; 23]:  
\[d_{total}=d_{Q}+d_{F} \tag{6}\]  
Fatigue damage from cyclic plastic strain:  
\[d_{F}=\frac{\sum\limits_{1}^{k}\overline{\delta}_{k}}{\frac{k_{F}}{\sum\limits _{1}^{k_{F}}\overline{\delta}_{k}}} \tag{7}\]  
where \(\sum\limits_{1}^{k}\overline{\delta}_{k}\) is the fatigue damage accumulated over \(k\) half-cycles of loading, and \(\sum\limits_{1}^{k_{F}}\overline{\delta}_{k}\) is the fatigue damage accumulated to failure when the quasi-static damage \(d_{Q}=0\), and \(\overline{\delta}_{k}=A_{1,2}\Big{(}\overline{\epsilon}_{0}-\frac{\overline{ \epsilon}_{T}}{2}\Big{)}k^{\alpha}\). Quasi-static damage:  
\[d_{Q}=\sum\limits\overline{\epsilon}_{pk}/\overline{\epsilon}_{u} \tag{8}\]  
where \(\overline{\epsilon}_{pk}\) is the accumulated unilateral plastic strain over \(k\) half-cycles of loading and \(\overline{\epsilon}_{u}\) is the continuous strain under static loading:\[\vec{e}_{u}=\ln\frac{1}{1-\psi_{u}} \tag{9}\]  
Intermediate failure covers a relatively large range of cycles with significant fatigue and quasi-static damage. In this case, Equations (7)-(9) can be used to express the failure condition as follows:  
\[ \begin{pmatrix}\frac{\sum\limits_{f}^{k_{f}}\overline{\delta}_{k}}{\sum \limits_{k_{r}}\overline{\sum\limits_{\xi}}}\\ \frac{\sum\limits_{f}^{k_{r}}\overline{\sum\limits_{\xi}}}\end{pmatrix} ^{q}+ \left(\frac{\vec{e}_{pk_{f}}}{\vec{e}_{u}}\right)^{q}=1 \tag{10}\]  
To calculate the fatigue damage \(d_{F}\) under stress-controlled loading, theoretical low-cycle fatigue curves are required, which only consider fatigue damage. To determine these curves, the stress-controlled loading is considered a non-stationary strain-controlled loading, with the accumulated damage per half-cycle expressed by the equation:  
\[d_{F}=\frac{\overline{\delta}_{k}}{k_{f}} \tag{11}\] \[\frac{\sum\limits_{1}^{k_{f}}\overline{\delta}_{k}}{1}\]  
When the material accumulates the unilateral plastic strain, difficulties arise in the determination of low-cycle fatigue damage since, in this case, it is necessary to calculate the theoretical fatigue curve corresponding only to fatigue damage.