## 3 Results and Discussion
### Effects of Fretting and Torsional Parameters on Dissipation Energy Characteristics
#### 3.2.2 Hysteresis Loops of Torque versus Torsion Angle  
(1) Effect of loading cycles  
Figure 8a shows elliptical hysteresis loops at different loading cycles, which indicates that the fretting contact surface is mainly regulated by coupled elastic and plastic deformations [29]. Increases of loading cycles induce increases in the openness of hysteresis loop, relative slip range and plastic deformation. Figure 8b shows the increase in the area of hysteresis loop with increasing loading cycles, which reveals increases in the dissipation energy due to cyclic torsional loading with fretting contact and the fretting contact surface damage. (2) Effect of contact load  
Figure 9 shows elliptical hysteresis loops of torque versus torsion angle in cases of different contact loads. At each contact load, increases of loading cycles cause increased openness of hysteresis loop. At the same loading cycles, the openness of hysteresis loop increases with increasing contact load. At loading cycles 20,000, dissipation energies due to cyclic torsional loading with fretting contact in cases of contact loads of 30 N, 40 N, 50 N and 60 N are \(9.14\times 10^{-3}\) J, \(9.97\times 10^{-3}\) J, \(11.9\times 10^{-3}\) J and \(16.2\times 10^{-3}\) J, respectively. Therefore, the dissipation energy increases with increasing contact load and thereby the fretting contact surface damage of cyclic loaded wire increases. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Crossing Angle, \(\beta\) (\({}^{\circ}\))** & **90** & **70** & **45** & **26** \\ \hline Maximum tangential force (N) & 16.2 & 19.0 & 19.4 & 20.1 \\ Relative slip (\(\mu\)m) & 58.5 & 48.6 & 59.1 & 61.2 \\ Dissipation energy (\(\times 10^{-3}\) J) & 1.46 & 1.52 & 1.83 & 2.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Tangential forces and relative slips between steel wires during tension–torsion fretting fatigue and dissipation energies of tangential force versus displacement amplitude at fatigue cycles 20,000 at different crossing angles. Figure 8: Evolutions and dissipation energy of torque versus torsion angle. (**a**) Evolution of the hysteresis loop. (**b**) Dissipation energy at loading cycles 20,000. Effect of torsion angle  
Figure 10a-d shows elliptical hysteresis loops of torque versus torsion angle in cases of different torsion angles. At each torsion angle, the openness of hysteresis loop increases with increasing loading cycles, which indicates that the cyclic torsional loading with fretting contact always runs at the partial slip state. At the same loading cycles, the openness of hysteresis loop increases with increasing torsion angle, which reveals increases in the relative slip and fretting contact surface damage. The nonlinear increase of torque with increasing torsion angle may be attributed to the elastic-plastic torsional deformation according to Material Mechanics and complex elastic-plastic fretting contact role [30]. At loading cycles of 20,000, dissipation energies due to cyclic torsional loading with fretting contact are \(4.11\times 10^{-3}\) J, \(6.73\times 10^{-3}\) J, \(9.97\times 10^{-3}\) J and \(17.0\times 10^{-3}\) J, respectively. Therefore, the dissipation energy and fretting contact surface damage of cyclic loaded wire both increase with increasing torsion angle. Figure 9: Evolutions of torque versus torsion angle during tension–torsion cyclic loading at different contact loads. (**a**) \(F_{n}\) = 30 N. (**b**) \(F_{n}\) = 40 N. (**c**) \(F_{n}\) = 50 N. (**d**) \(F_{n}\) = 60 N.  
(4) Effect of crossing angle  
Figure 11a-d shows elliptical hysteresis loops of torque versus torsion angle during the test in each crossing angle case. In cases of crossing angles of \(90^{\circ}\), \(70^{\circ}\) and \(45^{\circ}\), increases of loading cycles cause the increase in the openness of hysteresis loop. In the case of crossing angle of \(26^{\circ}\), the large openness of hysteresis loop does not vary obviously, which indicates the large relative slip during the whole test.