# Damage Analysis of Rolling Stock Automatic Coupler under Cyclic Loads
## 2 Materials and Methods
### Static and Low-Cycle Results for the Automatic Coupler Steel
The fatigue curves (Figure 4) show that there were three failure zones. The quasi-static failure zone is characteristic of high stress levels. The quasi-static failure zone extends up to \(k\approx 30\), where an intense accumulation of unilateral plastic strain takes place and the specimen failure occurs at the neck. The location of the zones is seen more clearly by looking at the variation of reduction of area \(\psi\). The fatigue fracture zone starts with \(k\approx 600\), since the accumulation of unilateral plastic strain is lower in this zone. The transient zone extends from \(k\approx 30\) to \(k\approx 600\). Here, the failure is due both to the formation of the neck and to fatigue. Figure 5 shows that the material tested is cyclically softening, i.e., the loop width is increasing under stress-controlled loading. This is typical for low-carbon alloy steel. The dependence was used to analytically describe the cyclic strain diagrams [22; 23; 24]:  
\[\overline{\epsilon}_{k}=\overline{S}_{k}+A_{1,2}\Bigg{(}\overline{\epsilon}_{0 }-\frac{\overline{S}_{T}}{2}\Bigg{)}k^{\alpha} \tag{4}\]  
The cyclic material characteristics are considered in relative coordinates, which are obtained by dividing the stresses by \(\sigma_{pr}\) and strain by \(e_{pr}\). Figure 4: Stress-controlled loading low-cycle fatigue curves for steel 20 GL (1\(-\overline{\sigma}_{0}\), symmetric cycle; 2\(-\overline{\sigma}_{0}\), asymmetric cycle; 3\(-\psi\), symmetric cycle). The investigations have shown that the cyclic limit of proportionality \(\overline{S}_{T}=1.83\) and the cyclic strain parameters, characterizing the widths of the plastic strain hysteresis loops of the first and second half-cycles are \(A_{1}=0.35\) and \(A_{1}=0.3785\), respectively, and the coefficient of cyclic softening, showing the variation of the hysteresis loop depending on the number of loading half-cycles, \(k\), is \(\alpha=0.123\). The material accumulates residual plastic strain in the direction of initial loading (tension) (Figure 6) because the hysteresis loop width of the paired half-cycles is greater than the loop width of the adjacent unpaired half-cycles. The theoretical accumulated unilateral plastic strain can then be presented as follows:  
\[\overline{e}_{pk}=\overline{e}_{0}-\overline{e}_{0}+\sum_{1}^{k}(-1)^{k} \overline{\delta}_{k} \tag{5}\]  
Figure 6 shows the calculated curves and the experimental values of the accumulated unilateral plastic strain \(\overline{e}_{pk}\). The low-cycle strain test data shown in Figures 4-8 suggest that the material of the automatic coupler is cyclically softening and accumulates unilateral plastic strain. Therefore, it is clearly undergoing an accumulation of cyclic and quasi-static damage. For this reason, the investigation of durability under strain-controlled loading s with the strain restrictions applied has also been carried out (Figures 6 and 7). Figure 8 shows the calculated curves for the asymmetric cycle and the experimental \(\overline{e}_{pk}\) values. Investigations have shown that the cyclic strain parameters of the asymmetric cycle characterizing the hysteresis loop widths of the plastic strain of the first and second half-cycles are \(A_{1}=0.032\) and \(A_{2}=0.034\). In the case of strain-controlled loading, quasi-static  
Figure 5: Dependence of hysteresis loop width on the number of half-cycles and load level (dashed lines for corresponding levels of even half-cycles \(\overline{\delta}_{k}\), continuous lines for corresponding levels of odd half-cycles \(\overline{\delta}_{k}\)). Figure 6: Accumulated unilateral plastic strain during the symmetric cycle. damage is eliminated. The shares of damage caused by fatigue and quasi-static damage can be determined by using the strain-controlled and stress-controlled loading durability curves. This is very important for calculating the total low-cycle damage [22]. The specimens were made from the body of an automatic coupler and were therefore limited in number.