2.1 Friction coefficient
Figure 1 shows the friction coefficient (µ) without/with glycerol plotted against the sliding distance, d. The different colors indicate the results for seven types of rubber sheets where D is within the range of 0.231–7.80 mm. Regardless of the lubricant conditions, µ increased with d at d = 0.0–2.0 mm with a constant value at d = 2.0–20.0 mm. Using the results at d = 10.0 mm as the steady state, the µ of the unlubricated condition (mean µ = 0.940, standard deviation = 0.102) was 30.1 times that of the lubricated condition (mean µ = 0.0306, standard deviation = 0.0043). Considering that the loss tangent of the silicone rubber was so minimal (~ 0.05), the Fhys was sufficiently small and the difference in µ corresponded to the change in the Fad. Thus, owing to the high viscosity of glycerol compared with that of air, it was considered that the real contact area between the rubber and steel sphere drastically decreased because of the lubricant invasion.
2.2 Major and minor principal strains during rubber friction against steel sphere
Figure 2 shows the vectors of the major and minor principal strains at d = 0.0, 0.7, and 10.0 mm and D = 0.231, 1.48, and 7.80 mm under unlubricated conditions. The red and blue vectors indicate tension and compression, respectively. The cross points of the vectors are the center of each subset. The changes in the position of the subsets were emphasized three times in the x- and y-directions. Regarding the results at d = 0.0 mm, which corresponded to the phase of after contacting but before sliding, it was confirmed that, in certain subsets, the major and minor principal strains were positive. At D = 0.231 and 1.48 mm, the major and minor principal strains were relatively large at one spot. This spot would be the position under the contact point between the rubber sheet and steel sphere because compression in the z-direction could have caused the increase in the major and minor principal strains. Interestingly, the major and minor principal strains in this spot at d = 1.48 mm exceeded those at d = 0.231 mm. The phase at d = 0.7 mm corresponded to the phase when sliding was about to occur. Although the strain distribution at D = 7.80 mm did not drastically change, the strain distribution at D = 0.231 and 1.48 mm exhibited asymmetry in the x-direction. This dependency was more prominent at d = 10.0 mm. In front of the contact point, the subsets were compressed and extended in the x- and y-directions, and the positions of the subsets shifted backward. However, the subsets at the rear of the contact point extended in the x-direction, with a small strain in the y-direction. At this point, the subsets were about to return to their initial position. Owing to a high µ for the unlubricated condition, it was considered that the changes in the strain and subset position at D = 0.231 and 1.48 mm were caused by F. In addition, it can be said that the influence of F on the strain distribution and subset position was relatively low at D = 7.80 mm.
Figure 3 shows the position of the subsets and vectors of the major and minor principal strains in rubber bulk during friction against the steel sphere at d = 0.0, 0.7, and 10.0 mm and D = 0.231, 1.48, and 7.80 mm under glycerol-lubricated conditions. Similar to Fig. 2, the change in the subset position is emphasized three times, and the red and blue vectors indicate tension and compression, respectively. The changes in the subset position and strain distribution under glycerol-lubricated conditions were smaller than those under unlubricated conditions. The distribution of the subset position and strain was symmetric during friction compared with those under unlubricated conditions. µ was considered too small to have affected the distribution.
2.3 Position changes of points in rubber bulk during friction
Under unlubricated conditions, the experimental results showed that each subset shifted during friction, at least horizontally. To discuss the position change, this section describes the position trajectory. The strain distribution analysis was applied at d = 0.0–10.0 mm; therefore, the position trajectory of each subset was practically within 10 mm in the x-direction. However, the stress field in the x-direction can exceed the analysis range. Thus, the position trajectory was calculated based on the superposition of the position change of five subsets in the same x-position.
Figure 4(a) shows the position trajectory of points beside and including the contact point center (center line, x-axis) at D = 0.231–0.780 mm under unlubricated conditions. The subset, 6.0 mm, beside (corresponding to five subsets) the x-axis was collected as a result of points beside the center line. Figure 4(a) confirms that the points under the center line shifted backward to the sliding direction and returned to the initial position and that the shifting distance decreased with D. For the points beside the center line, the position change in the x-direction was smaller but similar to that of the points under the center line. However, the points beside the center line shifted to the outer side and returned to the initial position, resulting in a semicircle. An asperity contacting on a rubber surface has been reported to induce vertical and cyclic deformation during friction 12. However, the present study findings experimentally showed that horizontal and cyclic deformation also occurred during rubber friction when contacting a steel sphere.
Figure 4(b) shows the position trajectory of points under and beside the center line at D = 0.231–0.780 mm under glycerol-lubricated conditions. Compared with the results obtained under unlubricated conditions, the position trajectory for glycerol-lubricated conditions was symmetric in the x- and y-directions, and the shifting distance was smaller. Focusing on the displacement in the x-direction, each position shifted backward, forward, and then backward to the sliding direction, resulting in a circle for the points beside the center line. Interestingly, the rotational direction was opposite that under lubricated conditions.
2.4 Strain changes of points in rubber bulk during friction
Apart from the position trajectory, the time dependency of the strain in the x- and y-directions and the shear strain in the xy plane was calculated based on the superposition of the position change of five subsets on the same x-axis. It was technically difficult to collect the subsets exactly beneath the center line because the subsets were set at every 1.2 mm. Thus, this section describes the strain change in the rubber bulk beside the contact line. Figure 5 shows the time dependency of strain in the x-direction, εx, on the points beside the center line at D = 0.231–7.80 mm under unlubricated and glycerol-lubricated conditions. Time t = 0.0 s corresponded to when the steel sphere passed the subset position. The εx decreased and increased under unlubricated conditions, indicating that the rubber bulk was compressed and extended by F during friction. The change range in εx was observed to decrease with D. Under glycerol-lubricated conditions, the changes in εx were symmetric and the εx peak decreased with D. The rubber bulk was slightly compressed before and after passing beside the contact point, whereas the rubber bulk was extended in passing beside the contact point. These results suggested that the strain under glycerol-lubricated conditions was mainly caused by the normal load owing to a small F (µ = 0.0306).
Figure 6 shows the time dependency of strain in the y-direction, εy, on the points beside the center line at D = 0.231–7.80 mm under unlubricated and glycerol-lubricated conditions. The changes in εy at D = 0.231–3.27 mm (unlubricated) and D = 0.231–2.14 (glycerol) were complex, resulting in a semicircle or circle. Even if the strain distribution in the z-direction was not measured here, the vertical strain could have also caused this behavior. Under the other aforementioned conditions, the rubber bulk was slightly extended in passing beside the contact point, which could have been caused by F.
Figure 7 shows the time dependency of shear strain in the xy plane, γxy, on the points beside the center line at D = 0.231–7.80 mm under unlubricated and glycerol-lubricated conditions. In passing beside the contact point, γxy increased, and the maximum values of γxy decreased with D under unlubricated conditions. These results were considered to have been caused by F. Conversely, under glycerol-lubricated conditions, the γxy was positive and negative before and after passing beside the contact point, respectively. These results could have been caused by the squeezing out of the rubber bulk under the contact point.
Owing to a low loss tangent (0.0398–0.0717), the Fhys was small, and the Fad was dominant in the rubber friction in the present study. The experimental results indicated that the increase in Fad caused an increase in the strain range, resulting in a semicircle-shaped displacement. If the loss tangent was sufficiently large, the Fhys would increase with the strain range, which can be enlarged by Fad. Because the loss tangent of rubber for outer shoe soles and vehicle tire treads can be > 0.5, the relationship between friction behavior and strain distribution for rubbers with high loss tangents should be investigated.