Friction and wear of multiple steel wires in a wire rope

The fretting wear among the steel wires aggravates the wire rope’s fatigue damage, affects the service performance of the wire ropes, and threatens mine hoisting safety. In this paper, the practical friction behavior and wear mechanism among the wires in the wire rope are investigated. A series of tests were carried out on multiple steel wires in helical contact and tension-torsion coupling under different fretting parameters, twisting parameters, and lubrication conditions by self-made friction and wear testing machine. The results show that the coefficient of friction (COF) among the steel wires decreases slightly with increasing lateral loads and tension, and the effect of twist angle on the COF has opposite results under different lubrication conditions. Lateral loads, tension of the steel wires, twist angle, and lubrication condition all affect the fretting morphology among the steel wires. Fretting wear with larger twist angle structure leads to more energy loss. The energy loss of fretting is directly related to the fretting morphology among the contact surfaces, and the dissipated energy is lower in the two forms of complete slip and sticking. The wear depth and width increase with the increase of lateral loads, steel wire tension, and twist angle. And the wear width and depth under dry friction conditions are higher than those under oil lubrication conditions. In addition, the wear mechanism under dry friction conditions is mainly abrasive wear, adhesive wear, and fatigue wear. And the wear mechanism under oil lubrication conditions is mainly abrasive wear and fatigue wear.


Introduction
Due to the advantages of safety, stability, and high reliability, steel wire ropes are widely used in cranes, lifts, ropeways, ski lifts, mine hoists, and bridge braces [1]. The wire rope, which is the main component of the mine hoisting system ( Fig. 1(a)), plays a vital role in mine production. Its endurance limit and fatigue life have a significant effect on mining production and the life safety of miners [2,3]. Steel wire rope as a complex space geometry is composed of spiral wire group, the contact form among wires is very complex, and different twisting parameters (twist distance and twist angle) lead to different inter-wire contact forms within the wire rope ( Fig. 1(b)). During the mine hoisting process, there will be tiny relative movement and relative torsion among the steel wires inside the wire rope due to its tension difference (tension or release load) and additional axial load and cyclic bending load on the guide wheel [4]. And fretting wear occurs under the action of different contact loads and contact stresses, which leads to a decrease in the cross-sectional area of the steel wire, resulting in stress concentration and even failure of the steel wire. In addition, the service conditions of wire ropes in mine hoisting are more complicated, so different lubrication conditions among the wires in the wire rope will also lead to great changes in the friction form and degree of wear [5].
In recent years, some scholars have studied the friction and wear characteristics of steel wire ropes. Some of these scholars have carried out the corresponding research for different contact conditions among wire ropes. Peng et al. [6,7] investigated the effects of different contact loads, crossover angles, and displacement amplitudes as well as impact loads on the friction and wear characteristics of wire ropes. They found that the contact load mainly affects the wear depth, the crossover angle affects the wear shape, and the coefficient of friction (COF) under impact load decreases significantly. On this basis, Chang et al. [8,9] investigated the effect of wear characteristic parameters of wire rope on its load-carrying capacity and fracture failure behavior. Meanwhile, Zhang et al. [10] investigated the effect of longitudinal vibration of different frequencies and amplitudes on the friction and of multilayer coiled wire ropes for the vibration wear of wire ropes. In order to investigate the influence of external environmental conditions on the friction and wear of wire rope, some other scholars have conducted experimental investigations and analyses on the influence of external conditions such as temperature rise [11], low temperature [12], lubrication status [5,13], corrosion [14], etc., on the friction and wear characteristics of wire rope.
In addition, some scholars have carried out theoretical and experimental studies on the inter-wire contact characteristics and fretting friction and wear behavior inside the wire rope. They have explored the friction and wear problem between wires from an elastic cylindrical model under a constant normal load [15], a finite element model of seven filaments with local constraints, and a wear evolution solution model [16][17][18]. Several authors [19][20][21][22][23][24][25][26][27][28][29][30][31] carried out experimental studies on the fretting friction and wear behavior between 2-3 wires under different fretting conditions of strain rate, cross angle, contact force, fretting amplitude, number of cycles, and fretting form. It was found that the decrease of strain ratio makes the fretting state change, the change of cross angle causes the changes of contact force and COF while the wear amount between the wires mainly depends on the contact load, and the number of cycles has almost no effect on the wear rate. In order to be closer to the contact form among the wires in the wire rope, only Xu et al. [32] used concave-convex contact instead of spiral contact form inside the wire rope, studied the friction and behavior of five individual wires under the action of tensile-torsional coupling, and found that the wear amount of convex contact is greater than that of concave contact. Considering the complex working conditions in mine hoisting, some scholars have further investigated the friction and wear behavior between steel wires under different lubrication conditions [33] [34][35][36][37]. It was found that increasing grease can effectively reduce wear. The COF between the wires was significantly reduced in the presence of acidic media, while the wear marks were smoother. Compared with that of dry friction, the wear mechanism between wires in corrosive environment changed from layered wear, abrasive wear, and oxidative wear to abrasive wear and electrochemical corrosion wear, and the COF between wires decreased as the concentration of mineral particles and coal dust increased.
The above study simply uses the fretting behavior between 2 or 3 wires to explore the friction and wear characteristics among wires inside the wire rope, ignoring the complex spiral contact form among wires. It cannot truly reflect the friction and wear characteristics among wires inside the wire rope. The friction and wear characteristics derived from it have little significance in correlation with the real internal friction and of wire ropes. Therefore, this paper innovatively designs a multi-wire helical friction test rig with seven wires independently controlled. The aim is to investigate the tension-torsion coupling fretting behavior of multiple wires twisted into strands under different twisting parameters and a helical contact form. Based on different twisting parameters, lubrication conditions, tension, and lateral loads, fretting friction and wear tests were carried out on a self-developed test rig. The effects of various test parameters on COF, relative slip, dissipated energy, wear width, wear depth, and wear mechanism were studied. The research results help us grasp the actual friction and wear behavior among the wires in the wire rope and provide essential data support for extending the service life of the wire rope.

Specimens
In each group of experiments, 6 steel wires with a diameter of 1 mm were used as the outer winding steel wires, and 1 steel wire with a diameter of 1.1 mm was used as the core wire. The chemical compositions and mechanical properties of the test steel wire are shown in Tables 1 and 2, respectively. Before the start of the test, the surface of the steel wire was repeatedly cleaned with alcohol.

Fretting test rig and test procedure
The self-made fretting friction test rig can twist the loaded seven steel wires into strands and perform tribological experiments on this basis, as shown in Fig. 2.
The test steps are as follows: (1) Load the center wire. (2) The central steel wire is stretched and twisted by a cylinder and a stepping motor. A circumferential load is added to simulate the pressure exerted by the outer strands of the steel rope. The electric cylinder tensions the center wire. (3) Load six-side wires and tighten them by the cylinder. (4) Rotate the turntable, twist the seven wires into strands, and adjust the tension of each wire. (5) Add lateral load as required, and use weights to control the size of the applied lateral load.
The steel wire twisting process is shown in Fig. 3. The steel wire rope strand is made by twisting seven independent steel wires with certain parameters. Figure 4(a) is the schematic diagram of the test. After a plurality of steel wires are twisted into rope strands with certain parameters, the two ends are fixed. The twisting area is in the middle area. The real twisting form of multiple steel wires is shown in Fig. 4(b). One end of the strands is fixed, and the wires at the other end are stretched and connected to the tension sensor. A circumferential load is added to the periphery of the strands to simulate the extrusion among the strands in the wire rope. The form of the rope strand subjected to external load is shown in Fig. 4(c). In order to understand the friction and wear behavior among the wires in the wire rope strand under different twisting   www.Springer.com/journal/40544 | Friction parameters (twist angle and twist length), tension and lateral load conditions, the values of side wire tension and lateral loads, and their values were determined before the test. Among them, the twist angle is the angle between the center line of the wire and the center line of the rope strand when twisting the strand, and the twist length is the straight line distance between the starting and ending points of the wire rotating around the core of the strand when twisting the strand (360°). Based on this, a total of 4 groups of tests were set up, of which the first and second groups were friction and wear tests with different tension and lateral loads, respectively. The third group is the friction and wear test among steel wires with different twisting parameters (twist angle and twist length) under dry friction conditions, and the fourth group is the friction and wear test among the steel wires in the wire rope strands with different twisting parameters under oil lubrication conditions. According to the actual working conditions and the material characteristics of the wire, the appropriate twisting parameters and loading load variation range are selected, where the twist angles of the rope strand are 4.2°, 5.4°, and 6.6°, single side wire tension variation is in the range of 120-180 N, and single side lateral load variation is in the range of 20-80 N. Core wire stretching end amplitude is 3 mm in the first two groups of tests and 6 mm in the third and fourth groups of test amplitude. When carrying out the oil lubrication test, apply the lubricating oil evenly on the surface of the strands after twisting, so that the lubricating oil naturally penetrates into the inside of the wire rope strands and infiltrates the surface of each contact wire. The lubricant used is specially used to lift the wire rope (ISIS-550A), and its specific performance parameters are shown in Table 3. In addition, in order to prevent accidental error and improve the accuracy of the test, each group test repeats three times, and the wear scars of the central wire obtained from the first test were chosen for observation. The specific test parameters are shown in Table 4.

Determination of COF
Friction is an important parameter of fretting friction and wear among steel wires in a wire rope. Therefore, this paper collects and analyzes friction and friction factor ( f ). It should be noted that what we collect is the comprehensive friction of the six wires on the side and the center wire. The COFs stated in this paper where F n is the contact load, P is the applied load, F av is the average friction force of the center wire under a single stroke of the electric cylinder, F f is the friction force collected each time, k is the number of points collected during the stretching stroke or the return stroke, F avmax and F avmin are the average friction forces of two adjacent ones, and their signs are opposite, f av is the average COF of a single cycle, and F lmax is the friction force of the center wire in the stable phase under the pulling force of the side wire and the lateral force.

Fig. 5
Input and output data curves. (a) Center wire tension and torque curves for any three cycles, (b) friction curves for any three cycles. F max and F min are the maximum force and minimum force provided by the electric cylinder. θ max and θ min are the maximum and minimum torsion angles provided by the stepper motor, and they are in opposite directions. F tmax is the peak friction force of a single cycle.
www.Springer.com/journal/40544 | Friction the tangential force is the friction force generated by the mutual contact between the outer steel wire and the core wire collected in the test, and the displacement parameter is the average slip value of the wear section. The displacement parameters are converted from the pulling force of the electric cylinder, the elastic modulus and cross-sectional area of the steel wire, and the ratio of the twisting position of the steel wire to the total length of the core wire. The specific conversion relationship is shown in Eqs. (4) and (5): where Δl is the elongation of the tensile end of the core wire, ΔF is the core wire tension variation range, A is the cross-sectional area of the core wire, E is the modulus of elasticity, and ΔX is the average relative slip value of the wear section. The data points of the hysteresis curve are graphically approximated as a quadrilateral symmetric about the x-axis, as shown in Fig. 6. From the hysteresis curve, we can easily extract two travel values, where the distance from the leftmost end of the quadrilateral to the rightmost end is the maximum displacement value (ΔX p-p ), and the distance between the two intersections of the quadrilateral and the horizontal axis is the actual displacement value (ΔX r ) in this cycle.
The dissipated energy of each cycle is defined as the closed area of the hysteresis curve, that is, the energy required to remove surface material in each cycle [26]. At the same time, we can obtain the dissipation energy of a single cycle by multiplying the friction force of each cycle by twice of the relative slip. The cumulative dissipated energy (ΣE d ) is the accumulation of the dissipated energy (E d ) of multiple cycles, which can be obtained by Eqs. (6) and (7): where F av is the average friction of a single cycle, ΔX r is the actual average slip in a single cycle, and n is the number of cycles.

Determination of wear characteristic
After the test, the worn section of the core wire was intercepted, and the surface morphology was observed and analyzed. As shown in Fig. 7(a), the scanning electron microscope (SEM; Zeiss Sigma 300, ZEISS, Germany) image was used to observe the size of the wear scars on the surface of the steel wire, and the width of the abrasion was measured by taking the pictures and the scale. The average width of the abrasion was obtained by taking the average of 10 measurements at equal intervals to reduce the error, and the average width of the abrasion of a single sample was the average width of the six wear scars. The wear morphology and wear mechanism of the wire surface were also analyzed by the pictures taken by the SEM. In addition, as shown in Fig. 7(b), the surface morphology of the worn section of the wire was photographed and measured by the digital microscope (DSX1000, Olympus, Japan), and the three-dimensional morphology of the worn surface, the profile curve, and the maximum wear depth could be obtained. All the experiments in the first and second groups have a similar change feature: As the number of sliding friction cycles increases, the COF increases rapidly, and then passes through a slowly rising stage, and finally stabilizes at a constant value. Therefore, the whole fretting friction and wear process can be divided into three stages, namely the break-in stage, the transition stage, and the steady-state stage. There are also significant differences between each group of tests due to the variation of the test parameters. Generally speaking, the COF of the first running-in stage increases rapidly, which is caused by the damage of the smooth protective film on the surface of the steel wire. After the protective film is damaged, it enters a transitional stage. The base material of the wire begins to come   There is also a moderating effect of the wear debris, which causes the COF to rise slowly. Finally, the protective film outside the steel wire is completely worn out. The wear debris has become finer abrasive particles, and the COF has entered a relatively stable steady-state stage. In addition, from Figs. 8(b) and 8(d), we can see that the average COFs in the steady-state stage decrease with the increase of lateral loads and tension. With increasing the lateral load from 80 to 320 N, the COF decreased from 0.73 to 0.66, and as the tension of a single steel wire increased from 120 to 180 N, the COF decreased from 0.69 to 0.66. The lateral force and the tension on the steel wire are the main two factors affecting the contact force among the steel wires, which is consistent with the conclusion drawn by Xu et al. [32].

COF
After the contact force increases, the average COF in the steady-state stage will decrease. The effect of twisting parameters on the COF is shown in Fig. 9, which shows the variations of the COF with the number of cycles under dry friction and oil lubrication conditions and the average COFs in the steady-state stage. The COF of the steel wires in the oil-lubricated state is much smaller than that in the dry friction state. Similarly, the change process of COF is also divided into three stages, namely the running-in stage, the transition stage, and the steady-state stage. The increase of COF under dry friction conditions shows an almost linear increase and finally tends to be stable. In contrast, the increase of COF under oil lubrication condition enters a slowgrowth stage after an initial rapid increase and stabilizes at last. As shown in Figs. 9(a) and 9(b), the COFs under dry friction conditions are about 0.58-0.75, and the COFs of the two curves with a twist angle of 6.6° and a lateral load of 80 and 160 N have been greatly reduced. Mainly due to the phenomenon of static friction in the friction process, the relative sliding among the wires tends to zero, and the resulting COF is the coefficient of static friction. From the tests of the other two groups, it is clear that a larger lateral load causes a slight decrease in the COF, while the twist angle has a greater effect on the COF compared to the lateral load. Under dry friction conditions, the COF decreases as the twist angle increases. Due to the existence of the wire surface film, the initial COF among the wires is small; with the intensification of wear, a large number of abrasive debris generates between the two contact surfaces, the twist angle is small rope strands of the core and side wire abrasion direction, and the direction of movement is more consistent, so the generated abrasive debris is more difficult to discharge, impeding the relative movement among the wires. Twist angle increases after the two contact surfaces between the abrasive debris will be discharged with the reciprocal movement of the contact area, so the COF will also be reduced. As shown in Figs. 9(c) and 9(d), the COF under oil lubrication is about 0.16-0.22. Among them, a larger lateral load will lead to a slight decrease in the COF, while different twist angles lead to a large difference in the steady-state value of the COF. When the twist angle is 4.2°, the COF is about 0.17, and when the twist angle is increased to 6.6°, the COF increases to about 0.22. Unlike dry friction, the COF increases with the increase of the twist angle under oil lubrication conditions. This is because the wear debris generated by wear is mixed in the lubricating oil, which plays the role of rolling friction between the contact surfaces and reduces the friction resistance. The wear debris particles in the lubricating oil are more likely to form rolling than the wear debris in dry friction. Friction, rather than sticking and blocking on the material surface, is also evidenced by the lubricating oil changing from an initial translucent light-yellow liquid to a viscous black liquid after the tests. In addition, due to the change of the twist angle, the number of particles existing between the contact surfaces will also change.
The larger the twist angle, the fewer particles rolling between the contact surfaces, and the COF will also increase. Here, the COFs for dry friction and oil lubrication have diametrically opposite responses to the twist angle.

Relative slip
The hysteresis curve can be obtained from the tangential force (friction force) change with the displacement amplitude. From the hysteresis curve, we can recognize the fretting state of the fretting region. According to different fretting states, the shape of the closed region of the hysteresis curve can be roughly divided into three categories: a parallelogram representing slip, a quasi-parallelogram representing partial slip, and a straight-line representing adhesion [28,38,39]. Figure 10 shows the evolution of the hysteresis curves with the number of cycles for different combinations of tension and lateral loads. From Fig. 10, we can observe that the initial shape of the hysteresis curve is a standard parallelogram, and the fretting state among the wires is a complete slip. As the number of cycles increases, the friction force increases, and the relative slip value decreases continuously. The shape of the hysteresis curve becomes a quasiparallelogram representing a partial slip state. Finally, the friction force tends to be stable, and the shape of the hysteresis curve is stable at partial slip or linear viscous state. This is due to a protective film on the wire surface at the beginning of the test cycle, which makes the friction at a low level, thus making the fretting among the wires completely slippery. With the destruction of the surface film, the contact surface becomes rough. The formed wear debris hinders the relative sliding, resulting in the continuous increase of the friction force and the continuous reduction of the relative sliding. With the destruction of the surface film, the hysteresis curve tends to be stable, thus entering the stable wear stage. Obviously, the whole fretting process is a typical mixed fretting state. The evolution process of the hysteresis curve is different under different conditions of tensile force and lateral load. Larger tensile and lateral loads will accelerate the evolution of the fretting state, and the final stable fretting state tends to be more linear.
The wire contact surfaces become rough over time, resulting in a decrease in relative slip and evolution of the fretting form from the initial complete slip to partial slip and sticky state. Figure 11 shows the relative slip evolution curves of the whole test process under different tension and lateral load conditions. With the increase in the number of cycles, the relative slip shows a decreasing trend. The evolution trend of www.Springer.com/journal/40544 | Friction relative slip can be fitted to an exponential function (y = ae x/t + b). When the tension and lateral loads increase, the values of a and t increase, and the value of b decreases. The relative slip decreased rapidly in the early stage of the test and changed more smoothly in the second half of the test. And with the increase of tension force and lateral load, the curvature of the relative slip evolution curve is larger in the first 10,000 cycles, and the curvature is smaller after 35,000 cycles. In the first 10,000 cycles, as the loading force increases, the friction also increases in the initial stage. Therefore, the wire surface film breaks down more rapidly, increasing the degree of wear. As a result, the friction force increases faster, and the relative slip decreases faster. After 35,000 cycles, the higher the loading force, the earlier the test group enters the stable wear phase, so the change in relative slip is smaller. Figures 12(a)-12(f) show the evolution of the hysteresis curve with the number of cycles for different twist angles and lateral load combinations under dry friction, corresponding to the fourth set of tests in Table 4. We can observe that in a complete test, the hysteresis curves all go through the grinding phase of complete slip first, which has less friction and more relative slip; then go through the transition phase of partial slip, which has more friction and less relative slip; and finally, into the stabilization phase, which is stable in the partial slip or sticky state and has stable friction and stable relative slip. The larger lateral load of the same twist angle causes the hysteresis curve to evolve faster and enter the steady-state phase earlier, resulting in greater friction and smaller steady relative slip values. And the hysteresis curve  shape in the final steady-state phase is closer to quasi-parallelogram and linear shape. This is due to the increased contact load between the contact surfaces caused by the greater lateral load, resulting in the increased wear between the contact surfaces. The increase in contact surface wear causes the wire surface protective film to break down faster, and thus the wear enters the steady-state phase faster. Under the same lateral load and different twist angles, the shape of the hysteresis curve is quite different. With the increase of the twist angle, the evolution speed of the hysteresis curve increases, and the shape of the stable hysteresis curve also changes from a quasiparallelogram to a straight line. This is mainly due to the fact that increasing the twist angle changes the contact form, which causes an increase in friction. Therefore, the friction force level is a key factor determining the evolution speed of the hysteresis curve. Figures 12(g)-12(l) show the evolution of the hysteresis curve with the number of cycles for different combinations of twist angle and lateral loads under oil lubrication conditions, corresponding to the fourth group of the tests in Table 4. Under oil lubrication conditions, the lateral load and twist angle have almost no effect on the evolution rate of the hysteresis curve. This is because the COF is smaller under oil lubrication, and the response of the friction to the change in contact force is smaller. Another reason is that the stable wear state between the contact surfaces is established earlier due to lubricant, so the hysteresis curve has less variation in form throughout the test. Under the same twist angle condition, the larger lateral load leads to greater friction force, which causes the hysteresis curve to develop a quasi-parallelogram shape with relative slip. And different twist angle conditions have little effect on the shape and evolution of the hysteresis curve. Figures 13 and 14 show the relative slip evolution trends for different twist angles and lateral loads under dry friction and oil lubrication conditions, respectively. The relative slip values of both dry friction and oil lubrication tended to be stable with the rapid decrease of the number of cycles. The evolution curve of relative slip with the number of cycles can be fitted to an exponential function (y = ae x/t + b). The a and t values increase with the increasing twist angle and lateral load, and the b value decreases. As shown in Fig. 13, the initial relative slips all start at 800 μm. As the cycle increases, the relative slip of the test group with a larger twist angle and larger lateral loads decreases faster, and the final stable relative slip value is smaller. It can be concluded that there is an overall difference between oil lubrication condition and dry friction condition. In oil lubrication, the  friction values are smaller, and the relative slip is larger. The evolution rate of relative slip is positively correlated with the lateral load and twist angle size. The magnitude of the stable value of relative slip is negatively correlated with the lateral load and twist angle size. Figure 15 shows the cumulative dissipated energy (ΣE d ) for different tension and lateral load combinations under dry friction conditions, obtained by integrating the hysteresis curves for each cycle throughout the test. Dissipated energy per cycle is determined by relative slip and friction. Figure 15 shows that the cumulative dissipated energy also increases as the  number of cycles increases. When the tension force is 120 N, and the lateral loads are 80 and 160 N, the cumulative dissipated energy increases linearly with the number of cycles. The slopes of the straight lines (y = kx + b) obtained by the two fittings are similar, and the slopes are 3.61 and 3.76. In addition, the five fitted curves almost overlap until 20,000 cycles. The cumulative dissipation energy and the number of cycles for all the five curves during this stage are linear. It is worth mentioning that three cumulative dissipated energy curves show nonlinear growth after a specific cycle when the tension force and lateral load increase. This is due to the fact that at a low number of cycles, each test group is in a state of slippage. There is a linear relationship between the increase in friction and the decrease in relative slip to ensure a linear increase in the cumulative dissipation energy. However, with the increase of cycles, in the test group with higher tension and lateral loads, the contact surfaces of the wires produce extrusion and tangential motion under higher level of contact force, causing local sticking between the contact surfaces. The appearance of sticking caused the relative slip to decrease rapidly and even gluing in some areas. Compared with the initial stage, after the relative slip decreases to a certain level, the cumulative dissipation energy and the number of cycles are out of a linear relationship, which leads to a lower growth level of the cumulative dissipation energy in the second half of the test. Figures 16 and 17 show the fitted curves for the variation of cumulative dissipation energy with the number of cycles for different twisting angles and lateral loads under dry friction and oil lubrication conditions, respectively. Under dry friction conditions, the cumulative dissipation energy showed a linear increase in the four test groups with twist angles of 4.2° and 5.4° and lateral loads of 80 and 160 N. The cumulative dissipation energy of the test group with higher lateral load was slightly larger than that of the group with lower lateral load, and the slope of both was similar. The two test groups with a twist angle of 5.4° have the same rate of increase in the initial phase as the test group with a twist angle of 4.2°. After 3,000 cycles, the cumulative dissipated energy  increased nonlinearly, and the growth rate decreased. As the number of cycles increases, the relative slip state changes from a partial slip state to a sticking state, and the area within the hysteresis loop decreases. Therefore, the dissipation energy of a single cycle decreases, and the growth rate of the accumulated dissipation energy also decreases. When the twist angle is 6.6°, the cumulative dissipated energy maintains the same growth rate as the other test groups in the initial stage. And after 10,000 cycles, its increase rate drops rapidly, and the growth tends to stagnate. This is due to the fact that in the initial stage, the hysteresis loop is in a state of complete and partial slippage. As the degree of wear increases, the hysteresis curve tends to be linear, and the dissipation energy within a single cycle reaches a minimum value, so the growth rate of the accumulated dissipation energy tends to be minimal. As shown in Fig. 17, the cumulative dissipation energy shows a linear increase under oil lubrication conditions. The straight-line slope is different for different twist angles and lateral loads. When the twist angle is 4.2°, and the lateral loads are 80 and 160 N, the slopes of the fitted straight lines are 2.61 and 2.74, respectively. Moreover, when the twist angle is 5.4°, the slopes of the straight line increase to 3.34 and 3.47. When the twist angle increases to 6.6°, the straight-line slopes increase to 4.83 and 5.06.

Dissipated energy
In summary, we can conclude that the cumulative dissipation energy increases with the number of cycles. In addition, wire tension, lateral load, twist angle, and lubrication all affect the evolution of the fretting state, thus affecting the change of cumulative dissipation energy. The larger the wire tension, lateral load, and twist angle, the higher the rate of growth of the accumulated dissipation energy in the early stage, which is consistent. The increase rate decreases when the partial slip state changes into the sticking state. The oil lubrication makes the fretting form vary less in the same set of tests, so it can make the cumulative dissipation energy grow steadily. Under dry friction condition, the contact state between the contact surfaces is more variable, and most of them are mixed fretting states, which will experience multiple fretting morphology in the same test, so the growth rate of the accumulated dissipation energy also changes.

Average wear width and maximum wear depth
Different force loading conditions, lubrication conditions, and twisting parameters (twist angle and twist length) will affect the friction and wear among the wires. We characterize the degree of wear by observing the shape parameters of the wear scar.
As shown in Fig. 18, there are multiple spiral wear scars on the outer surface of the core wire, and the wear scars are rectangular-like concave wear scars. Under different loading methods, twisting parameters (twist angle), and lubrication conditions, the wear scars on the outer surface have different shapes and wear mechanisms. From Fig. 18, we can see two different wear scars on the surface of the steel wire. The surface of the steel wire under oil lubrication conditions is relatively clean. The width of the wear scar is small. The wear scar is more regular and smoother. On the other hand, the surface of the steel wire under the condition of dry friction is attached with a layer of dark red oxide due to the oxidation of wear debris. The width of the wear scar is large, and the inside of the wear scar is uneven.
The maximum wear depth and average wear width curves of the core wire under different tension and lateral loads are shown in Fig. 19. The average wear width and maximum wear depth increased with the increase of tension and lateral loads, the maximum wear depth increased from 10.9 to 23.1 μm, and the average wear width increased from 111.5 to 135.8 μm. The tension and lateral loads increase the contact stress, which increases the maximum wear depth and average wear width, and the larger tension and lateral loads lead to the contact deformation of the side wire and the core wire, which increases the average wear width. We can clearly see that the wear width is positively correlated with tension and lateral loads. The growth rate of the wear width showed a decreasing trend with the increase of tension and lateral loads. This is because the outer surface of the steel wire is cylindrical. As the wear depth increases, more surface material needs to be removed to increase the same wear depth and width. Therefore, the average wear width and the maximum wear depth are nonlinear with their growth and the increase of tension and lateral loads. Figure 20(a) shows the maximum wear depth values for different lateral loads and twisting parameters (twist angle and twist length) under dry friction and oil lubrication conditions. Figure 20(a) shows that the maximum wear depths under dry friction conditions are overall higher than those under oil lubrication conditions. In addition, higher lateral loads lead to larger wear depth values. The increase in twist angle leads to an increase in wear depth. This is consistent with the finding of Xu et al. [40] that large crossover angles lead to an increase in the maximum wear depth. Under dry friction conditions, the wear depth decreases when the twist angle increases from 5.4° to 6.6°. This is due to the fact that when the twist angle is 6.6°, in the second half of the test, the fretting state enters a fully sticking state, with almost static friction between the contact surfaces, and the relative sliding tends to zero. Therefore, the degree of wear is also reduced. While in the case of dry friction with twist angles of 4.2° and 5.4°, the fretting morphology is a mixture of complete and partial slip, and the wear depth increases with the twist angle. Figure 20(b) shows the average wear widths of the core wire for different lateral loads and twist angles under oil lubrication and dry friction conditions. Among them, we can observe that the average wear widths in the dry friction condition are overall higher than those in the oil lubrication condition. The average wear widths in the dry friction condition are in the range of approximately 160-240 μm. In contrast, the  Under oil lubrication conditions, the increase in lateral load and twist angle causes the average wear width to increase at a relatively constant growth rate. In contrast, the dry friction condition with a larger twist angle results in a lower maximum wear depth due to the sticky state in the second half of the run. The wear width is unchanged compared to the previous set of tests, and the reason why the maximum wear depth is not reduced is that the larger contact force makes the contact surface extruded and deformed, thus having a higher level of average wear width.

Wear mechanism
The wear scar morphology can reflect the wear behavior and wear mechanism in tension-torsion coupling and helical contact in the wire rope to a certain extent. Different force loading conditions, lubrication status, and twisting parameters affect the friction and wear behavior between the wires and thus further analyze the changes in their wear mechanisms.
Rope strands with true spiral contact form within the wire rope are subjected to fretting under tensile-torsional coupling with different contact force conditions, lubrication conditions, and twisting parameter conditions. The surface morphology of the core wire produced at the end of the test can reflect its friction and characteristics, and thus derive the wear mechanism. Figures 21-23 show the surface wear morphologies obtained by the SEM after the following tests. They are the surface wear morphology under dry friction with different force loading conditions, the surface wear morphology under dry friction with different lateral loads and twisting parameters (twist angle and twist distance), and the surface wear morphology under oil lubrication with different lateral loads and twisting parameters (twist angle and twist distance). It can be observed that the wear scars are all in the form of concave bands along the contact line among the wires, and the wear scars are spirally presented on the outer surface of the wires according to the twist angle.
By observing the morphology of the abrasion scars in Fig. 21, we can find that ploughing is the main feature of the wear scars after wear. The fretting behavior produces micro-cutting debris (abrasive chips) between the wire contact surfaces, and the addition of micro-cutting debris makes the two-body wear into three-body wear. The fretting debris produces a micro-cutting effect between the contact surfaces and the fretting behavior, causing the wire contact surface to form a ploughing in the fretting gradually. Other features such as pitting, material adhesion, and microcracking can occur in the ploughing. This is due to the contact surface in many reciprocating motions, the material contact surface fatigue fracture, the formation of pitting, material adhesion and microcracking, and other features. Moreover, oxide particles and oxide clusters are left in the wear scars. This is because the wear particles in the wear process  | https://mc03.manuscriptcentral.com/friction will accumulate on both sides of the wear scars due to the squeezing pressure. A part of the wear particles remaining in the wear scars will continue to carry out the three-body wear. These wear particles will form oxides through the oxidation of air, and the oxides will remain near the wear scars and within the wear scars at the end of the test. When the tension and lateral load levels are low, the interior of the wear scar is relatively flat, and the interior features are mainly ploughing, with a small amount of pitting and material adhesion in the ploughing and a small amount of oxide accumulation on both sides of the wear scar. When the tension and lateral load level increase, the internal ploughing of the wear scars becomes smoother and flatter, and pitting and material adhesion increases. Moreover, there is a breakage of the material, and larger pieces come off. As the squeezing pressure increases, the wear debris is ground more acceptable, the ploughing is smooth, and the wear debris increases, resulting in an increased pitting and material adhesion. The increase in squeeze pressure causes the material to have a greater shear force during the reciprocating motion, causing larger pieces of material to fall off. The increased squeezing pressure leads to increased wear, which causes a large amount of wear debris to accumulate at the end of the wear scars and more wear oxide particles to appear inside the wear scars. The main wear mechanisms are abrasive wear caused by abrasive chips, fatigue wear caused by fretting, and adhesive wear caused by contact surface contact.
The differences between the two sets of morphology shown in Figs. 22 and 23 are significant, and the main reason for the differences is the different lubrication conditions. The wear scar morphologies under dry friction are mainly shallow ploughing. They are accompanied by a large number of wear particles and blocky spalling. There are dark oxides inside the wear scars. In contrast, the wear scars under oil lubrication is dominated by deep ploughing. The number of ploughing is large, and the spacing and width are small. There is a small number of wear particles in the ploughing. Due to the dry friction conditions, the two contact surfaces have significant    | https://mc03.manuscriptcentral.com/friction frictional resistance, and the wear surface will have a block of flaking. The abrasive chips in it become finer particles under the action of more considerable squeezing pressure and tangential force, thus making the ploughing formed after the abrasive particles in it produce chip action flatter. The oxidation of air causes the abrasive particles to form dark oxides inside the wear scars. Multiple sets of these morphologies in Fig. 22 differ due to different lateral loads and twist angles. In the test group with a small twist angle and small lateral load, the surface morphology is mostly flat ploughing with some oxide particles and material adhesion. In the larger twist angles and enormous lateral loads, the width of the ploughing in the surface morphology increased, and irregular surface flaking appeared. This is due to the larger tangential and extrusion forces that make the material more likely to flake off from the wear surface. The main wear mechanisms are abrasive, adhesive, and fatigue. In Fig. 23, the differences between different morphologies are slight because the addition of lubricant weakens the effects of twist angle and lateral load. The different twist angles lead to different angular trends of the wear scars, and the different lateral loads only lead to changes in the width and depth of the wear scars, while there is no difference in the internal morphology of the wear scars. The main mechanisms of wear, in this case, are fatigue wear and abrasive wear.

Conclusions
The study on the fretting friction and wear behavior of steel wire in the form of tension-torsion coupling and helically twisted multi-filament contact shows as follows: 1) The COF decreases with the increasing tension and lateral loads under different fretting conditions, and it increases with the increasing twist angle under oil lubrication conditions and vice versa under dry friction conditions.
2) Different twist structures lead to different energy losses, and larger twist angles lead to greater energy losses. The energy loss is mainly related to the fretting state among the wires, and the cumulative dissipated energy growth rate is lower in the fully slipped and sticking state.
3) When the wire rope strand has a larger twist angle and a larger load, the widths and depths of wear among the wires increase, so the smaller twist angle of the rope strand twist structure and a smaller load of the wire rope in the anti-wear performance have better performance. The wear degree of oil lubrication state among the wires is overall lower than dry friction conditions, so a good lubrication state can reduce the degree of wear among the wires within the wire rope.
4) The primary morphologies of fretting wear among wires under different twisting structures and force loading conditions are ploughing. The wear mechanism among wires under dry friction conditions is mainly abrasive wear, adhesive wear, and fatigue wear, and the wear mechanism among wires under oil lubrication conditions is primarily abrasive and fatigue wear.