Gradual Deterioration Behavior of the Load-Bearing Strength of Main Cable Wires in a Suspension Bridge

Abstract

The main cable is the primary load-bearing component of a long-span multi-tower suspension bridge. The interaction between a dead load, vehicle load, wind load, and the corrosion environment leads the main cable wire to exhibit tribo-corrosion-fatigue behaviors. This behavior causes wire wear and deterioration, as well as a reduction in the effective cross-sectional area. This leads to the gradual deterioration of the wire’s load-bearing strength and seriously affects the load-bearing safety of the main cable. In order to ensure the safety of suspension bridges, it is critical to investigate the gradual deterioration behavior of the main cable wire’s load-bearing strength. A wire tribo-corrosion-fatigue test rig was established to test the wire under different friction pairs (saddle groove or parallel wires). The cross-sectional failure area of the wire with different pairs was obtained by super-depth electron microscopy and calculation. The damage degree evolution model and the deterioration model of the wire load-bearing strength were established by combining the theory of damage mechanics and the finite element method. The results show that, as contact and fatigue loads increase, so does the cross-sectional failure area of the fatigue steel wire. The fatigue wire’s damage degree has a good quadratic function relationship with fatigue cycles. The damage degree of the wire increases and the load-bearing strength decreases with increasing contact load and fatigue load. The load-bearing strength of the wire changes little at the beginning and decreases with increasing fatigue cycles. The results have fundamental significance for the life prediction of the main cable wires of suspension bridges.

1. Introduction

Suspension bridges represent a bridge structure with the main cable under tension as the main load-bearing component. In the operation process, under the coupling effect of a dead load (such as a stiffening beam, main cable, and sling), live load (such as automobiles and railway trains), and wind load [1,2], the main cable will generate time-varying dynamic loads on both sides of the main saddle, resulting in a force imbalance in the main cable on both sides of the middle tower [2,3]. With the continuous action of the unbalanced force, dynamic contact and “layered-slipping” phenomena occur between the main cable wires in the main saddle [4], resulting in friction, wear, and fatigue between the steel wire and the saddle material. At the same time, due to the intrusion and retention of rain and water vapor, electrochemical corrosion will occur on the steel wire inside the main cable. Corrosion-fatigue damage will occur on the steel wire of the main cable when the corrosion accumulates to a certain extent [2,3]. The main cable wires are subjected to both tribo- and corrosion-fatigue, resulting in tribo-corrosion-fatigue damage behaviors. The development of cumulative damage causes a deterioration in the wire wear and a decrease in the effective cross-sectional area, which leads to the gradual deterioration of the wire’s load-bearing strength. A decrease in the wire load-bearing capacity seriously affects the load-bearing safety of the main cable [4,5,6,7]. Therefore, it is of great significance to study the tribo-corrosion-fatigue deterioration behaviors of main cable wires to ensure the load-bearing safety of the main cable in suspension bridges.

Numerous engineering practices and experimental studies have demonstrated that friction and corrosion interact in terms of the tribo-corrosion fatigue of steel wire [8,9]. When fatigue and corrosion damage coexist, their effects are not simply superimposed. A steel wire’s strength and toughness will deteriorate significantly as tribo-corrosion-fatigue behaviors worsen [10]. Li [11] studied the degradation of the mechanical properties of corroded wires and their relationship with different corrosion parameters through tensile tests. By introducing the damage variable d, the damage constitutive model of the corroded wire was established. Wang [12] proposed a numerical simulation method to study the damage evolution and failure process of a high-strength bridge wire with precorrosion defects under fatigue loading. Based on the theory of continuous damage mechanics, a fatigue damage model for a steel wire with precorrosion defects was established. Sun [13] established a continuous damage model and simulation algorithm to simulate the corrosion-fatigue process of high-strength bridge cable wires. Chen [14] established a multiscale corrosion-fatigue damage model based on Faraday’s law and the initiation and propagation rates of microcracks. The evolution of damage at the stages of pit growth and microcrack propagation was described. Xue [15] chose the high-strength wires of cables or slings from actual bridges as the research object to explore corrosion-fatigue failure. The fatigue tests of the steel wire under different corrosion conditions were carried out, the S-N curves of the steel wire with different corrosion degrees were given, and a service life prediction model was put forward. Cui [16] proposed an improved continuum damage mechanics model to predict the corrosion-fatigue life of high-strength steel wires by considering the influence of the corrosion environment and external forces. Many experimental studies were conducted using the replacement of old cable wire and salt fog acceleration corrosion or electrochemical corrosion of a steel wire to investigate the effect of corrosion on the fatigue and mechanical performance of the cable. The influence of corrosion on the development of the wire’s surface morphology and fracture, the mechanical properties of the corroded wire, and the corrosion-failure process of the wire were explored [17,18,19]. At present, for the deterioration study of cable steel wires, previous scholars have established a deterioration model for cable steel wires based on theory and tests and obtained the deterioration law of cable load-bearing strength [20]. In addition, the methods of numerical simulation and experiment were used to analyze metal surface defects [21,22]. However, most of these studies focus on the corrosion degradation of suspension bridge wire and cable wire in cable-stayed bridges [11,12]. Few studies have been conducted on the wear degradation of main cable wires in suspension bridges due to the combined action of tribo-corrosion fatigue. A damage degree model of steel wire and a gradual deterioration model of load-bearing strength during the tribo-corrosion-fatigue processes are barely reported. Therefore, based on the tribo-corrosion-fatigue tests between the parallel steel wires and between the steel wire and saddle groove, a quantitative representation model of wire damage degree and a deterioration model of load-bearing strength in the tribo-corrosion-fatigue process are established in this paper.

2. Experimental Details

2.1. Experimental Materials and Parameters

The experimental steel wire was a high-strength, hot-dip galvanized steel wire with a diameter of 1.4 mm and a length of 500 mm. The mechanical parameters of the wire are an elastic modulus of 166 GPa, a tensile strength of 1650 Mpa, a nominal fracture true strain of 0.482 Mpa, a yield strength of 1601 Mpa, a section shrinkage of 38.3%, and a minimum cross-sectional area (at neck reduction) of 0.9503 mm². According to references [1,4], the saddle groove material is ZG 275–485 with a yield strength of 275 Mpa, an elastic modulus of 274 Gpa, and a Poisson ratio of 0.28. The contact length between the saddle groove and the fatigue wire is 10 mm, compared to 5 mm between the loading wire and fatigue wire.

Contact loads of 60 N, 80 N, and 100 N and fatigue load ranges of 750–1000 N, 750–1100 N, and 750–1200 N were chosen for the tribo-corrosion–fatigue test. The relative displacement was 300 μm for the test between steel wire and saddle groove, compared to 160 μm for the test between parallel wires. The used corrosion solution medium was a self-prepared 3.5% NaCl solution with a constant temperature of 30 °C. A frequency of 7 Hz was set for the test. When the effect of the contact loads on the failure area of steel wires was explored, the fatigue load range was 750–1000 N, whereas, when the influence of the fatigue loads was investigated, the contact load was 60 N.

2.2. Test Rig and Method

Before the test, the wire simples were polished with the 2000-grit metallographic sandpaper and washed with industrial alcohol. As shown in Figure 1, first, an initial pull force was applied to the fatigue wire in the corrosion tank by adjusting the length of the electric cylinder pusher extension. Then, the fatigue wire was in close contact with the loading wire or saddle groove by loading weights on both sides of the suspension. The contact load was recorded as F_n. Next, the programmable controller was used to write a program to make the electric cylinder push rod reciprocate, ensuring that the fatigue wire obtained the alternating load required for the experiment. The fatigue wire slid relative to the loading wire or the saddle groove. During the test, an alternating tensile force was applied at the nonfixed end of the fatigue wire. A constant positive pressure was applied at both ends of the loading wire or saddle groove. The tension sensor and the laser displacement sensor were connected by the acquisition card to obtain the tension and displacement signals, respectively. After the test, the width of the wire wear marks was measured using a super-depth electron microscope.

3. Cross-Sectional Failure Area of the Fatigue Wire

When a pair of cylinders with parallel axes make contact and are subjected to pressure, the line contact changes to surface contact, and its contact surface is a narrow rectangle [23]. In the test between the wire and the groove in this paper, the groove can be regarded as a cylinder with an infinite radius. Therefore, the wear marks of the fatigue wire are theoretically narrow rectangles. As shown in Figure 2, the cross-section of the wear zone is approximately arched; the wear surface is a narrow rectangle. The wear depth and the cross-sectional failure area of the wear zone are calculated directly from the geometric relationship between the wear mark width and the wire diameter (Figure 2c). The shaded area is the failure area of the worn wire. L = AB, which is the width of the wear mark, r = OA = OB, which is the wire radius; θ is the angle between OB and OC. From the geometric relations, $\theta =\arcsin \frac{L}{2r}$, failure area $A^{″}=r{\theta }^2-\frac{1}{2}L\mathrm{r}\cos \theta$. As shown in

Table 1. Cross-sectional failure area of fatigue wire (pair: groove) under different contact and fatigue loads.

Parameters		N = 1 × 105	N = 2 × 105	N = 3 × 105	N = 4 × 105	N = 5 × 105
Contact load (N)	60	0.0023	0.0037	0.006	0.0086	0.0124
	80	0.0027	0.0045	0.0068	0.0101	0.0142
	100	0.0033	0.0053	0.0085	0.0127	0.0173
Fatigue load peak (N)	1100	0.0023	0.0037	0.006	0.0086	0.0124
	1200	0.0025	0.0044	0.0074	0.012	0.0182
	1300	0.0027	0.0052	0.0088	0.0139	0.0198

and

Table 2. Cross-sectional failure area of fatigue wire (pair: wire) under different contact and fatigue loads.

Parameters		N = 1 × 105	N = 2 × 105	N = 3 × 105	N = 4 × 105	N = 5 × 105
Contact load (N)	60	0.0035	0.0059	0.0086	0.0118	0.0165
	80	0.0049	0.0087	0.0136	0.0192	0.0283
	100	0.0074	0.0114	0.0167	0.0233	0.0337
Fatigue load peak (N)	1100	0.0035	0.0059	0.0086	0.0118	0.0165
	1200	0.0063	0.0105	0.0151	0.0205	0.0299
	1300	0.0076	0.0132	0.0199	0.0285	0.0407

, as the number of fatigues increases, so does the fatigue wire wear failure area. Furthermore, as the contact load and fatigue load peak increase, so does the wear failure area. This is due to the fact that as the contact load increases, so does the contact stress between the contact surfaces. The axial stress of the fatigue wire increases with increasing fatigue load. The plastic deformation of the material becomes more severe. Meanwhile, the dissolution of the material is aggravated by the action of the corrosion solution. Therefore, the cross-sectional failure area of the fatigue wire increases, and the wear is intensified.

4. Wire Damage Model

4.1. Damage Evolution Theory

As the damage to the material accumulates, its mechanical properties decrease. Kachanov [24] put forward the concept of continuous damage mechanics when studying metal creep. The main reason for the decline in material properties is the decrease in the actual bearing area caused by internal defects in the material. The concept of “continuity” is proposed to describe the damage state of materials containing defects. As shown in Figure 3, a typical volume unit is selected from the interior of the material, assuming that the total sectional area in the n direction is A. The actual bearing area A′ decreases due to the internal defects of the material. The continuity can be defined as $X=\frac{A^{\prime }}{A}$. Based on the continuity, a damage degree D (Equation (1)) that can directly describe the material damage state is introduced [25]. When the material is intact, the actual bearing area does not decrease (A′ = A), D = 0. When the material fails, the actual bearing area is 0 (A′ = 0), at which point D = 1. The effective bearing area of the material $A^{\prime }=(1-D)A$. As shown in Equation (2), the effective stress σ′ of the damaged material can be expressed as the ratio of the actual external force F to the actual bearing area A′.

$$D=1-X=\frac{A-A^{\prime }}{A}$$

(1)

$${\sigma }^{\prime }=\frac{F}{A^{\prime }}=\frac{F}{(1-D)A}=\frac{\sigma }{1-D}$$

(2)

4.2. Wire-Saddle Groove

Figure 4 shows the fitting curves of the damage degree, D, of the fatigue wire between wire and saddle groove under different contact loads and different fatigue loads. The corresponding equations of the fitting curve are shown in

Table 3. Fitting equation for damage degree of the fatigue wire (pair: groove).

Parameters		Fitting Equation	Goodness of Fit (R2)
Contact load (N)	60	D = 4.73 × 10 14 N2 − 4.21 × 10−9 N + 0.0021	0.9987
	80	D = 5.20 × 10−14 N2 + 5.98 × 10−9 N + 0.0024	0.9996
	100	D = 5.76 × 10−14 N2 + 11.47 × 10−8 N + 0.0025	0.9990
Fatigue load peak (N)	1000	D = 4.73 × 10−14 N2 + 4.21 × 10−9 N + 0.0021	0.9987
	1100	D = 8.64 × 10−14 N2 + 5.57 × 10−10 N + 0.0022	0.9994
	1200	D = 9.01 × 10−14 N2 + 1.39 × 10−9 N + 0.0025	0.9991

. The goodness of fit (R²) is above 0.99, which indicates that the damage degree of the steel wire has a good quadratic function relationship with the fatigue cycles. Under the same contact load, the damage degree, D, of the fatigue wire increases continuously, and the increase rate becomes faster with increasing fatigue cycles, indicating that the wire damage is aggravated. When the contact load, F_n, is 60 N, 80 N, and 100 N, the maximum damage degree of the wire is 0.0161, 0.0185, and 0.0225, respectively. It demonstrates that an increase in contact load can clearly promote steel wire damage. When the fatigue load peak, F_max, is 1000 N, 1100 N, and 1200 N, the maximum damage degree of the steel wire is 0.0161, 0.0237, and 0.0257, respectively. It shows that increasing fatigue load promotes the damage of steel wire. By comparing the influence of different contact loads and fatigue loads on the damage degree of steel wire, it can be found that an increasing contact load is more damaging to steel wire than an increasing fatigue load.

4.3. Parallel Wire–Wire

Figure 5 shows the fitting curves for wire damage degree, D, between the parallel wires under different contact loads and different fatigue loads. The corresponding equations of the fitting curve are shown in

Table 4. Fitting equation for damage degree of the fatigue wire (pair: wire).

Parameters		Fitting Equation	Goodness of Fit (R2)
Contact load (N)	60	D = 2.28 × 10−14 N2 − 6.88 × 10−9 N + 0.9985	0.9970
	80	D = 5.36 × 10−14 N2 − 5.16 × 10−9 N + 0.9977	0.9965
	100	D = 6.43 × 10−14 N2 − 3.43 × 10−9 N + 0.9961	0.9971
Fatigue load peak (N)	1000	D = 2.3 × 10−14 N2 + 6.52 × 10−9 N + 0.0015	0.9971
	1100	D = 5.20 × 10−14 N2 + 5.98 × 10−10 N + 0.0031	0.9958
	1200	D = 7.08 × 10−14 N2 + 10.91 × 10−9 N + 0.0033	0.9982

. The variation law of fatigue wire damage degree with fatigue cycle is the same as the wire–groove variation law. When the contact load, F_n, is 60 N, 80 N, and 100 N, the maximum damage degree of the wire is 0.0107, 0.0184, and 0.0219, respectively. When the fatigue load peak, F_max, is 1000 N, 1100 N, and 1200 N, the maximum damage degree of the wire is 0.0107, 0.0194, and 0.0265, respectively. These show that an increase in fatigue load and the increasing contact load both promote the damage degree of the wire. It is worth noting that under the same contact load, the damage degree of the fatigue wire in the contact between the wire and the groove is greater than that in the contact between wire–wire. However, the rule is reversed under the same fatigue load. The main reason is that the material of the friction pair is different.

5. Deterioration Model for Wire Load-Bearing Strength

5.1. Theory of Strength Deterioration

Under the continuous action of cyclic loading, the damage to the material will continue to accumulate. The mechanical properties of the material deteriorate, which manifests as a decline in the bearing capacity, meaning the deterioration of the residual strength R(n) [26]. In the early stage of material damage, the damage caused by cyclic loading has little influence on the residual strength, while during the late stage of the damage, it increases the failure area of the material. The residual strength of the material drops sharply and fails suddenly. R(0) = σ_b; that is, the initial value of the residual strength is equal to the tensile strength of the material. R(N) = σ_e, which means that the residual strength is equal to the peak fatigue load at the fracture time. R(n) is monotonically decreasing, and the intensity degrades slowly at the initial time and has the characteristic of “sudden death” when approaching the fracture time [27]. They are the boundary conditions of the residual strength degradation model. Residual strength is associated with the load level and the cycles (Equation (3)).

$$R(n)=f(n,{\sigma }_{\max },S)$$

(3)

where R(n) is the residual strength (MPa), n is the number of cycles, and S is the cyclic stress ratio ().

5.2. Finite Element Model for Wire Tensile Testing

A large general-purpose finite element software called ABAQUS is used to establish the tensile finite element simulation model of the wire. Firstly, the finite element model of the unworn wire is established (Figure 6a), and the analysis results of the finite element model of the unworn wire are compared with the tensile test results to verify the correctness of the finite element model. Then, based on the results of the test, the actual wear mark size of the worn wire is regarded as the damage gap of the three-dimensional model (Figure 6e). The worn wire’s tensile finite element model is established (Figure 6c).

The tensile fracture of the wire in the worn zone is the main concern of this paper. In order to improve the calculation efficiency of the tensile simulation model, the total length of the wire was selected as 25 mm after a lot of pretests. The material characteristics of the wire are shown in

Table 5. Plastic material data of the steel wire.

$\mathit{\sigma }\left(\mathbf{MPa}\right)$	Plastic Strain	$\mathit{\sigma }\left(\mathbf{MPa}\right)$	Plastic Strain	$\mathit{\sigma }\left(\mathbf{MPa}\right)$	Plastic Strain
1601.00	0	1636.81	0.00193	1666.89	0.00226
1603.19	0.00169	1640.82	0.00196	1668.25	0.00229
1606.31	0.00171	1643.49	0.00198	1669.19	0.00233
1609.20	0.00173	1645.96	0.002	1670.35	0.00236
1612.53	0.00175	1648.65	0.00203	1670.85	0.0024
1615.43	0.00177	1650.45	0.00206	1671.35	0.00243
1618.55	0.00179	1653.14	0.00208	1671.61	0.00246
1622.97	0.00182	1656.49	0.00211	1671.68	0.0025
1625.87	0.00184	1658.52	0.00214	1671.97	0.00254
1628.56	0.00186	1660.53	0.00216	1672.28	0.0026
1631.46	0.00188	1662.56	0.00219	1672.41	0.00263
1634.12	0.00191	1665.06	0.00223

. The eight-node linear element (C3D8R) is used for meshes, and the mesh of the worn area is encrypted. In the simulation model of the unworn wire, there are 56,000 finite element mesh elements and 61,997 nodes. The mesh number and node number of the finite element model of the worn wire were 95,400 and 104,526, respectively. As shown in Figure 6b,d, two reference points, RP1 and RP2, were established in the interaction module. Then, the reference point was coupled with the corresponding surface, and the constraint mode was in the motion-coupling mode to ensure that the established reference point was consistent with the corresponding surface displacement. In the finite element calculation model, three translational and three rotational degrees of freedom are constrained at the reference point RP1 at the bottom of the steel wire. All degrees of freedom, except in the Y direction at the reference point RP2 at the top of the wire, are constrained by applying a Y direction velocity of 5 mm/s. The stress–strain curve and tensile strength of the steel wire were obtained by extracting the force and displacement information at the coupling reference point RP2.

As shown in Figure 7, the wire tensile strength obtained by the universal testing machine is 1651.1 MPa, compared to 1648.8 MPa obtained by the simulation experiment. The relative error is 1.4%. The minimum diameter of the experimental tensile fracture is 1209 µm, compared to 1170 µm obtained by the simulation. The tensile fracture is basically the same, and the relative error of the two is 3.2%. Therefore, the simulation model established in this paper has a certain degree of correctness, so the load-bearing strength of steel wire with different amounts of wear can be obtained according to the finite element simulation model.

5.3. Analysis of the Simulation Results

Figure 8 shows the simulated stress–strain curves of the worn wire–groove under different contact loads and different fatigue loads. When the number of fatigues is 1 × 10⁵, the load-bearing strength of the steel wire decreases very little because the damage to the steel wire is small in the early stage of the tribo-corrosion-fatigue test. The load-bearing strength of the steel wire gradually decreases with increasing fatigue cycles. When the number of cycles is 5 × 10⁵, the load-bearing strength of the steel wire decreases from the initial 1650 MPa to 1622 MPa, with a small degree of reduction. When the contact load F_n is 60 N, 80 N, and 100 N, the load-bearing strength of the steel wire is 1622 MPa, 1619 MPa, and 1610 MPa, respectively. It shows that with the increase in contact load, the load-bearing strength of steel wire decreases gradually, and the increasing contact load can promote damage to the steel wire. By comparing the stress–strain curves under different fatigue loads, when the fatigue load peak F_max is 1000 N, 1100 N, and 1200 N, the load-bearing strength of the steel wire is 1622 MPa, 1617 MPa, and 1612 MPa, respectively. It shows that the load-bearing strength of the wire decreases gradually with increasing fatigue load, and the increase in fatigue load aggravates the damage to steel wire. Figure 9 shows the simulated stress–strain curves of the worn wire–wire under different contact loads and different fatigue loads. The changing trend of the curves is consistent with the change in the wire groove. The load-bearing strength of the fatigue wire decreases from the initial 1650 MPa to 1629 MPa with increasing fatigue cycles. The load-bearing strength of the wire decreases significantly with the increase in contact load and increasing fatigue load. The influence of the contact load and fatigue load on the load-bearing strength of the wire creates almost no difference.

5.4. Deterioration of the Wire Load-Bearing Strength

According to the tensile finite element analysis results of the worn wire described in Section 5.3, the fitting method was used to establish the deterioration model of the wire’s load-bearing strength. The life of the experimental steel wire was predicted.

(1)

Wire-saddle groove

As shown in Figure 10, the quadratic function is used to fit the data points obtained by tensile simulation, and the goodness of fit is above 0.98. At a contact load of 60 N, the load-bearing strength of the fatigue wire changes little at the beginning with increasing fatigue cycles. As the frequency of fatigue continues to increase, the load-bearing strength of the wire begins to decrease continuously because of the increased damage to the wire. By comparing the curves under the different contact loads, it can be seen that, with an increase in contact load, the load-bearing strength of the wire under the same number of cycles decreases. This is because the increase in contact load can promote damage to the steel wire. The larger the contact load is, the larger the failure area of the wire is and the lower the bearing strength of the wire is. From Figure 10b, the larger the fatigue load is, the lower the load-bearing strength of the wire is. The reason for this is the same as that affected by the contact load. As shown in

Table 6. Fitting equation of worn wire (pair: groove) strength degradation.

Parameters		Fitting Equation	Goodness of Fit (R2)	Predicted Life
Contact load (N)	60	y = −5.47 × 10−11 N2 − 2.79 × 10−5 N + 1650	0.9875	4.02 × 106
	80	y = −5.87 × 10−11 N2 − 3.16 × 10−5 N + 1650	0.9878	3.86 × 106
	100	y = −7.20 × 10−11 N2 − 4.28 × 10−5 N + 1650	0.9974	3.44 × 106
Fatigue load peak (N)	1000	y = −5.47 × 10−11 N2 − 2.79 × 10−5 N + 1650	0.9875	4.02 × 106
	1100	y = −7.00 × 10−11 N2 − 3.12 × 10−5 N + 1650	0.9890	3.43 × 106
	1200	y = −7.52 × 10−11 N2 − 3.75 × 10−5 N + 1650	0.9936	3.16 × 106

, the predicted life of the fatigue wire decreases from 4.02 × 10⁶ to 3.44 × 10⁶ times with increasing contact load. The predicted life of the wire decreases from 4.02 × 10⁶ to 3.16 × 10⁶ times with an increase in fatigue load. In conclusion, the larger the contact load and the larger the fatigue load peak, the lower the predicted life of the fatigue steel wire.

(2)

Parallel wire–wire

As shown in Figure 11 and

Table 7. Fitting equation of worn wire (pair: wire) strength degradation.

Parameters		Fitting Equation	Goodness of Fit (R2)	Predicted Life
Contact load (N)	60	y = −5.34 × 10−11 N2 − 1.36 × 10−5 N + 1650	0.9835	4.20 × 106
	80	y = −6.29 × 10−11 N2 − 2.96 × 10−5 N + 1650	0.9882	3.76 × 106
	100	y = −8.42 × 10−11 N2 − 4.26 × 10−5 N + 1650	0.9617	3.20 × 106
Fatigue load peak (N)	1000	y = −5.34 × 10−11 N2 − 1.36 × 10−5 N + 1650	0.9835	4.02 × 106
	1100	y = −5.84 × 10−11 N2 − 4.11 × 10−5 N + 1650	0.9751	3.66 × 106
	1200	y = −10.37 × 10−11 N2 − 4.25 × 10−5 N + 1650	0.9879	2.70 × 106

, the quadratic function is used to fit the data points obtained by the tensile simulation. The goodness of fit under different contact loads is above 0.96, as compared to 0.97 under different fatigue loads. At a contact load of 60 N, the load-bearing strength of the fatigue wire decreases greatly at the initial stage, which is different from that between the wire and the saddle groove. This is attributed to the fact that the tribo-corrosion-fatigue test between the two wires has four wear marks, while the test between the wire and the groove has only two wear marks. The greater damage caused by the four wear marks decreases the load-bearing strength rapidly. As the fatigue frequency continues to increase, the load-bearing strength of the wire decreases further due to increasing damage to the wire. Under the same fatigue cycles, as the contact and fatigue loads increase, the load-bearing strength of the fatigue wire decreases. The reason is the same as it was with the wire and groove. Table 7 shows that the life of the fatigue wire (pair: wire) decreases faster than that of the wire (pair: groove) with increasing contact load and fatigue load. The difference in the pair of materials and the contact mode is the main reason for this phenomenon.

6. Conclusions

(1)

The cross-sectional failure area of the fatigue wire increases with increasing contact load and fatigue load;

(2)

The fitting equations with a goodness of fit above 0.99 for the damage degree, D, of the worn wire were established. The wire’s D shows a good quadratic function relationship with the fatigue cycles;

(3)

The damage degree of the worn wire increases with increasing fatigue cycles. Under the same fatigue cycle, the greater the contact and fatigue load, the greater the D of the worn wire. The influence of contact load on wire damage degree is greater than that of fatigue load;

(4)

The finite element simulation model of the wire was established, and the stress–strain curve and residual load-bearing strength of the wire under various working conditions were obtained. Increasing contact load and fatigue load decreases the load-bearing strength of the wire;

(5)

The fitting equations between the load-bearing strength and fatigue cycles of the worn wire were established, and the life of the wire was predicted.