In order to verify the accuracy and advantages of the model1 of metal wire with curvature, the operating condition parameters in Table 1 were substituted into the formula of the wear evolution models of metal wire wear in Section 4. As a result, the fretting wear degree of metal wire with curvature and without curvature under three working conditions of angle, amplitude, and the load was obtained. To ensure the simplification of variates, the amplitude and load control parameters in the contact angle working condition group are 40um and 4N, respectively. In the condition group of amplitude, the angle and load control parameters are 90° and 4N, respectively. Finally, in the condition group of load, the amplitude and angle control parameters are 40um and 90°, respectively. In addition, in the wear evolution theory of metal wire with curvature, the radius of curvature of the metal wire is 15mm, which is consistent with the curvature of the circular arc boss of the fixture in the fretting wear test, and the number of wear cycles is 10000. The comparison results are as follows:
In the condition group of amplitude, the predicted value of the model1 of the metal wire with curvature is consistent with the experimental value, as shown in Fig. 9 (a-c). The wear depth, area, and volume of metal wire with curvature were linearly and positively correlated with the fretting amplitude, obviously related to the work done by friction, which was consistent with the wear features of metal wire without curvature. Table 4 shows the prediction value and prediction Error (Eq. (16)) of the wear evolution model of metal wire with curvature or without curvature under various amplitude conditions. The model1's prediction errors are less than 10%, and the average wear depth, area, and volume errors are 8.80%, 8.01%, and 5.47%, respectively. The maximum error of the model2 is 19.49%, and the average wear depth, area, and volume error is 11.39%, 10.34%, and 14.16%, respectively. The prediction errors of model2 are generally significant, which is not suitable for predicting the fretting wear of metal wires with curvature.
$$\begin{gathered} Erro{r_{}}=\left| {\frac{{{\text{Valu}}{{\text{e}}_{warp}} - {\text{Valu}}{{\text{e}}_{Exp.}}}}{{{\text{Valu}}{{\text{e}}_{Exp.}}}}} \right| \times 100\% \hfill \\ or \hfill \\ Erro{r_{}}=\left| {\frac{{{\text{Valu}}{{\text{e}}_{str.}} - {\text{Valu}}{{\text{e}}_{Exp.}}}}{{{\text{Valu}}{{\text{e}}_{Exp.}}}}} \right| \times 100\% \hfill \\ \end{gathered}$$
16
Table 4
Wear degree and error of amplitude group
| depth(um) | Error | area(e-3·mm2) | Error | volume(e-5·mm3) | Error |
90°-4N-20um | Experimental | 6.73 | | 7.02 | | 1.95 | |
Warp-model | 6.19 | 7.99% | 6.51 | 7.26% | 1.85 | 5.01% |
Straight-model | 5.70 | 15.30% | 6.69 | 4.70% | 1.57 | 19.49% |
90°-4N-40um | Experimental | 8.40 | | 8.37 | | 3.80 | |
Warp-model | 9.16 | 9.05% | 9.12 | 8.96% | 4.02 | 5.76% |
Straight-model | 8.87 | 5.60% | 9.78 | 14.42% | 3.39 | 10.79% |
90°-4N-60um | Experimental | 10.51 | | 10.22 | | 6.13 | |
Warp-model | 11.55 | 9.94% | 11.14 | 9.01% | 6.47 | 5.59% |
Straight-model | 11.12 | 5.80% | 11.94 | 16.83% | 5.46 | 10.93% |
90°-4N-80um | Experimental | 14.87 | | 13.99 | | 9.14 | |
Warp-model | 13.04 | 8.34% | 12.84 | 8.22% | 8.86 | 3.04% |
Straight-model | 12.64 | 15.00% | 13.16 | 5.93% | 7.63 | 16.52% |
90°-4N-100um | Experimental | 16.86 | | 15.36 | | 11.18 | |
Warp-model | 15.40 | 8.69% | 14.34 | 6.64% | 10.29 | 7.96% |
Straight-model | 14.29 | 15.24% | 16.87 | 9.83% | 9.72 | 13.06% |
In the condition group of contact angle, the predicted value of the model1 of the metal wire with curvature is consistent with the experimental value, as shown in Table 5. The prediction errors of the wear evolution model1 are less than 10% at all angles. The average wear depth, area, and volume errors were 7.94%, 7.86%, and 8.86%, respectively. However, under various contact angles, the model2's prediction errors are significant. The average error of wear depth, area and volume of the model was 26.80%, 33.83% and 12.56%, respectively.
Table 5
Wear degree and error of contact angle group
| depth(um) | Error | area(e-3·mm2) | Error | volume(e-5·mm3) | Error |
15°-4N-40um | Experimental | 6.65 | | 10.29 | | 2.93 | |
Warp-model | 7.30 | 9.52% | 11.29 | 9.72% | 2.74 | 6.48% |
Straight-model | 3.89 | 46.71% | 19.42 | 72.01% | 2.20 | 24.91% |
30°-4N-40um | Experimental | 8.97 | | 10.51 | | 3.22 | |
Warp-model | 8.48 | 5.41% | 9.75 | 7.23% | 3.03 | 5.90% |
Straight-model | 5.70 | 32.78% | 13.79 | 41.44% | 2.55 | 20.81% |
45°-4N-40um | Experimental | 8.17 | | 8.43 | | 3.37 | |
Warp-model | 8.93 | 9.30% | 9.18 | 8.90% | 3.23 | 4.15% |
Straight-model | 6.64 | 25.64% | 11.57 | 26.03% | 2.83 | 16.02% |
60°-4N-40um | Experimental | 9.90 | | 9.67 | | 3.41 | |
Warp-model | 9.20 | 7.02% | 9.01 | 6.93% | 3.33 | 2.35% |
Straight-model | 7.81 | 15.11% | 11.20 | 24.31% | 3.06 | 10.26% |
75°-4N-40um | Experimental | 8.58 | | 8.61 | | 3.52 | |
Warp-model | 9.31 | 8.46% | 9.16 | 6.51% | 3.32 | 5.68% |
Straight-model | 8.03 | 13.75% | 9.65 | 5.35% | 3.21 | 8.81% |
According to the wear features under different contact angles, the wear depth and volume of metal wire with curvature are positively related to the angle, and the wear area is negatively associated with the contact angle of metal wire. The changing trend slows down with the increase of angle. However, under different contact angles, there is a significant difference in the wear degree of metal wires with and without curvature. The overall features are shown as follows: under low contact angle, there is a significant difference in wear degree between metal wires with and without curvature, and the difference decreases with the increase of angle, as shown in the shaded part of Fig. 10(a-c). The main reasons for this phenomenon are: In the loaded state, when the metal wire without curvature contacts at a low angle, the two metal wires are in line contact with a large contact area; However, due to the curvature of the wire, the metal wire with curvature is approximately a point contact, and the contact area is small. When the contact angle of the wire increases, the difference between the wire with and without curvature decreases gradually, causing the prediction Error of the model2 of wire without curvature to decrease with the increase of contact angle, as shown in Table 4. Therefore, when the metal wire has curvature, the wear evolution model2 derived for the metal wire without curvature is too ideal to be applicable in the angle condition.
The wear rule of metal wire with curvature in the load condition group is similar to that in the amplitude group. The predicted values of the model1 for the metal wires with curvature are consistent with the experimental values. The model1's prediction errors under all load cases are less than 10%, and the average wear depth, area, and volume errors are 7.19%, 6.42%, and 7.41%, respectively, as shown in Table 6. The wear depth, area, and volume of metal wire with curvature are linearly and positively correlated with the load, consistent with the wear law of wire without curvature, as shown in Fig. 11(a-c). The maximum error of the model2 is 16.58%, and the prediction error of the model is generally significant, more than 10%, which is not suitable for predicting the fretting wear of metal wires with curvature. The average wear depth, area, and volume error are 7.21%, 12.77%, and 10.43%, respectively.
Table 6
Wear degree and error of contact load group
| depth(um) | Error | area(e-3·mm2) | Error | volume(e-5·mm3) | Error |
90°-2N-40um | Experimental | 6.29 | | 5.91 | | 1.96 | |
Warp-model | 6.70 | 6.53% | 6.43 | 8.80% | 2.17 | 10.60% |
Straight-model | 6.15 | 2.23% | 6.89 | 16.58% | 1.82 | 7.14% |
90°-3N-40um | Experimental | 8.72 | | 7.49 | | 3.31 | |
Warp-model | 8.14 | 6.53% | 7.88 | 5.21% | 3.19 | 3.63% |
Straight-model | 7.54 | 13.53% | 8.45 | 12.82% | 2.84 | 14.20% |
90°-4N-40um | Experimental | 10.03 | | 9.52 | | 4.39 | |
Warp-model | 9.35 | 6.80% | 9.11 | 4.31% | 4.20 | 4.33% |
Straight-model | 8.75 | 12.76% | 9.97 | 4.73% | 3.72 | 15.26% |
90°-5N-40um | Experimental | 9.51 | | 9.39 | | 4.73 | |
Warp-model | 10.40 | 9.33% | 10.15 | 8.09% | 5.19 | 9.75% |
Straight-model | 9.93 | 4.42% | 10.93 | 16.40% | 4.45 | 5.92% |
90°-6N-40um | Experimental | 10.63 | | 10.57 | | 5.69 | |
Warp-model | 11.35 | 6.75% | 11.17 | 5.68% | 6.17 | 8.40% |
Straight-model | 10.30 | 3.10% | 11.98 | 13.34% | 5.28 | 7.61% |
In summary, the wear evolution model1 of the metal wire with curvature can predict the degree of wire fretting wear well. Its prediction error is less than 10% and applies to different contact angles, amplitudes, and load conditions. When the contact angle is 90°, the prediction results of the model1 are different from those of the model2 under load and amplitude conditions. As a whole, the predicted wear depth of model1 is more significant than model2, and the wear area is the opposite. Significantly, under slight angle working conditions, the influence of curvature of bent wire on fretting wear degree cannot be ignored. Therefore, the wear evolution model established for metal wire without curvature is not applicable. Moreover, through the small-ball algorithm and the taboo search algorithm, the wire contact angle of the virtual metal rubber was counted during the loading process, which was prepared and moulded with the blank winding angle of 30 ° and 60° [24], as shown in Fig. 12. When the winding angle of MR blank is 30°, the number of contact points whose internal wire turn contact angle is less than 60° accounts for 20% − 30%. On the other hand, when the winding angle of MR blank is 60°, the number of contact points whose internal wire turn contact angle is less than 60° accounts for 70% − 85%. This indicates that many contact pairs with curvature contacting at a slight contact angle exist in MR. Therefore, the existing wear evolution model of metal wire without curvature is not suitable for predicting the fretting wear of the internal spiral contact of metal rubber, reflecting that it is crucial and practical to explore the fretting wear characteristics of metal wire with curvature and establish the evolution model of metal wire with curvature for the study of metal rubber fretting wear and overall fatigue failure.