Effect of normal load on friction coefficient for sliding

Tribology International 92 (2015) 335–343

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Effect of normal load on friction coefficient for sliding contact between rough rubber surface and rigid smooth plane Satoru Maegawa n, Fumihiro Itoigawa, Takashi Nakamura Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho Showa-ku, Nagoya, Aichi 466-8555, Japan

art ic l e i nf o

a b s t r a c t

Article history: Received 31 January 2015 Received in revised form 27 June 2015 Accepted 15 July 2015 Available online 22 July 2015

This study focused on the normal load dependence of the friction coefficient for the sliding friction of a rubber material with a rough surface. A developed friction tester was used to visualize the real contact regions distributed within the transparent contact interface between poly-dimethyl siloxane (PDMS) and glass surfaces. Based on experimental results, an adhesion friction model was developed to explain the normal load dependence of the friction coefficient. This model provides a simple technique that can roughly but easily estimate the real contact area and shear stress without in situ observation of the contact interface. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Friction coefficient Normal load Rubber Real contact area

1. Introduction The friction model is an important subject in the design stage of mechanical systems involving contacting surfaces. An accurate friction model contributes to the development of numerical simulations; it can promote the primary performance of intended systems and avoid problems such as the occurrence of frictioninduced vibrations. However, modeling the friction is not easy because the friction force depends on a number of system parameters such as the sliding speed, normal load, surface roughness, and contact configuration [1,2]. Understanding the contact mechanism is imperative to establish an accurate model of sliding systems. For conventional metallic materials, a number of theories considering the elastic, plastic, and elasto-plastic deformations of contacting asperities have been established to explain the contact mechanism by theoretically predicting the total area of real contact regions [3–8]. Some of these theories describe the total area of real contact regions Areal as linearly increasing with the normal load W, i.e., Areal pW. Therefore, assuming that the friction force F can be derived as the product of the total area of contact Areal and the shear strength τ, the independence of the normal load FZ with regard to the friction coefficient μ can be derived: μ¼F/W¼τAreal/Wpτ. The contact mechanism for rubber friction is significantly different from that for other solid materials. A number of small contacting asperities within an apparent contact region can easily

n

Corresponding author. Tel./fax: þ 81 52 735 5429. E-mail address: [email protected] (S. Maegawa).

http://dx.doi.org/10.1016/j.triboint.2015.07.014 0301-679X/& 2015 Elsevier Ltd. All rights reserved.

deform because of their low elasticity. When a sufficiently large normal pressure is applied, completely flattened contact, i.e., overall contact (full contact), can be formed. The contact theories based on the assumption of linear elasticity cannot be used because rubber materials have nonlinear elasticity [9]. Therefore, it is difficult to establish an accurate model of sliding systems involving rubbersliding surfaces. For example, predicting the theoretical normal load dependence of the real contact area is not easy. In situ observation of contact interfaces is known to be a powerful tool for understanding the friction mechanism. In particular, the use of optically transparent materials allows the direct visualization of real contact regions and provides useful information for the study of friction. Several optical systems have been used in many experiments to visualize the dynamics of real contact regions [10–17]; they have helped validate theoretical predictions and find unpredictable phenomena [18]. However, these optical methods require the use of optically transparent materials. Special constructions such as light surfaces, light detectors, and other optical components are also needed. Therefore, it is not easy to utilize in situ techniques for many practical situations. In this study, we observed the real contact area for the contact interface between surfaces made of transparent polydimethyl siloxane (PDMS) and conventional optical glass (BK7) in situ. The changes in the contact interface under increasing normal load were visualized to quantify the relationship between the total area of real contact and the normal pressure. Based on the experimental results, a simple adhesion model was developed to explain the dependence of the friction coefficient on the normal load. The model provides a simple method for estimating the total contact area and shear strength using the experimental results of friction

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tests; with this approach, these values can be roughly but easily obtained without in situ observation of the contact interfaces.

2. Adhesion friction models First, we describe two types of adhesion friction models that explain the dependence of the friction coefficient on the normal load under two extreme conditions. The friction force of rubber materials is known to mainly comprise two different components: adhesion friction and hysteresis friction [19,20]. The former results from the intermolecular interaction between contacting surfaces, and the latter results from the energy loss due to bulk deformation processes during sliding motions. When a rubber object slides on a smooth rigid plane at a low sliding speed, the hysteresis components of the total friction force are drastically reduced; thus, the adhesion friction term is the dominant factor that determines the value of the friction force. Therefore, the friction force F is described by the products of the shear stress τ and total area of contact Areal: ð1Þ

F ¼ τAreal

Fig. 1 illustrates two types of friction models that predict the friction force acting on point contacts between an elastic hemisphere and smooth rigid plane. For model I in Fig. 1(a), the elastic hemisphere has a smooth surface. In contrast, for model II in Fig. 1(b), the surface roughness of the hemisphere is significantly high. In model I, the contact can be regarded as a huge single asperity contact because no microscopic asperities form within the apparent contact region; thus, the total area of real contact Areal is equal to the apparent contact area Aapp. Using the Hertz contact theory [4], Areal is calculated as Areal ¼ Aapp ¼ π

!2=3
 3 1 ν2 WR ; 4 E

ð2Þ

where W, R, E and ν are the normal load, radius of the hemisphere, Young's modulus of the hemisphere, and Poisson's ratio of the hemisphere, respectively. If Eq. (2) is substituted into Eq. (1), the normal load dependence of friction coefficient μ under a constant shear stress τ is given by μ¼

!2=3
 τAreal 3 1  ν2 WR ¼ πτ W  1 p W  1=3 4 E W

ð3Þ

The friction coefficient was found to have a small dependence on the normal load W. The dependences of Areal and μ on the normal load are schematically summarized in the lower graphs of Fig. 1. The similar small dependence of μ on the normal load had been found by Schallamach [21]. In model II, the real contact area is regarded as the sum of the area of multiple small contact regions. For the contact between rough surfaces, one effective approach is to use the Greenwood– Williamson theory [5], which analyzes the contact mechanism under the assumption that small contacting asperities are localized far away from each other so there is no interaction between them. Assuming that the asperity heights are distributed normally with a standard deviation but that these asperities have a constant radius of curvature, the total area of the real contact regions Areal is proportional to the normal load W. However, it was found that the GW theory is not an accurate model for rough surfaces unless roughness occurs on a single length scale [22–24]. On the other hand, Persson recently developed a new contact theory, which includes the effect of the elastic interactions between deforming asperities [8,25]. In Persson's theory, as in the case of the GW theory, the contact area depends linearly on the normal load under a small loading condition. Thus Areal p W

ð4Þ

As in the case for model I, the friction coefficient μ is given by μ¼

τAreal W 1 p 1 pW0 W W

ð5Þ

In this case, the friction coefficient does not depend on the normal load W. The normal load dependences of Areal and μ are also illustrated in the lower graphs of Fig. 1. No dependence of μ on the normal load in the case of model II corresponds to the wellknown Amonton–Coulomb friction model. Persson also discussed the changes in the total contact area during the transition from small contact (low load) to full contact (high load) [25]. In highly loading conditions, the real contact regions formed with a small amount of separation; some of them merged with each other. In this case, the normal load dependence of the real contact area significantly differs from the above prediction, in which the real contact area linearly increases with normal load.

3. Experimental details 3.1. Apparatus

Fig. 1. Schematic of adhesion friction models: (a) model I and (b) model II.

Fig. 2 shows schematic views of the experimental setups used in this study. In order to apply a wide range of the normal load W, two different apparatuses were developed. The first apparatus is shown in Fig. 2(a) and employs the contact between a rough rubber plate made of cross-linked PDMS and a glass hemisphere lens made of BK7. The thickness of the PDMS plate and radius of the glass hemisphere are 3 and 5 mm, respectively. The PDMS plate is connected to a double-cantilever spring that is mounted on the X-directional motorized stage via a lever arm. The second apparatus is shown in Fig. 2(b) and employs a different contact configuration that comprises a smooth glass (BK7) plate and rough PDMS hemisphere with a radius of 18 mm. The PDMS hemisphere and glass plate are fixed to a threedirectional dynamometer and the X- and Y-directional stages, respectively. In both systems, an objective lens and white LED light source are placed above and under the contact interface. A CCD camera is attached to the lens system with the objective lens.

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Fig. 2. Schematic views of experimental setup: (a) test I and (b) test II. Insets in the figures show the power spectrum of surface roughness. The slopes of the solid lines in the insets correspond to the Hurst exponents (a) H¼0.92 and (b) H¼ 0.87.

Fig. 3 schematically illustrates the method for visualizing the distribution of the real contact area within the apparent contact region. The white LED light source illuminates the contact interface, and the transmitted intensity of the incident light is captured by the CCD camera. Example incident and transmitted intensity profiles are inserted in the figure. In the real contact regions, the light goes straight through with low refraction because the difference in the refractive indices of PDMS and BK7 is small. In contrast, in the noncontact regions, the air layer separates the contact between the PDMS and BK7 surfaces. Thus, the light is bent and scattered by surface irregularities and the large refractive index difference between the contacting surface materials (PDMS and BK7) and air. Therefore, the transmitted intensity profiles can be measured to visualize the space distribution of the real contact area. 3.2. Specimens Cross-linked PDMS (Dow Corning's SYLPOT 184) hemispheres and plates were prepared as rubber specimens. They were formed by a molding process: the mixture comprised a PDMS melt and cross-linker agent with a compounding ratio of 10:1 that was poured into a hemispherical steel mold with a different surface roughness. The surface was roughened by sandblasting and polishing processes. In test I, Ra ¼ 0.64 μm; in test II, Ra ¼0.93 μm. The roughness of the mold surface was transferred onto the contacting surface of the PDMS plate. Note that different methods to make the surface roughness, i.e., sandblast and polishing, were used to validate this developed model under a wide range of conditions. To obtain Young's modulus of the PDMS specimens, a JKR test [4,26] was performed between the smooth rear face and a smooth glass hemisphere. Details on the method for obtaining Young's modulus are given in [11]. Young's modulus of the PDMS plate was found to be 1.0 MPa. The Poisson ratio of the PDMS elastomers was close to 0.5 [27]. 3.3. Procedure In test I, normal loads W of 0.03, 0.06, 0.09, 0.12, and 0.15 N were applied using a lever mechanism with five different weights. After a wait of 300 s, the intensity distribution of the transmitted light was imaged using the CCD camera. A certain waiting time was necessary to eliminate the effects of the changes in real contact area with contact time [10,28]. The PDMS plate was then

Fig. 3. Method for visualizing real contact region.

driven in the X direction at a constant speed of 0.1 mm/s. During the sliding motion, the time changes in the deflection of the double-cantilever spring δ and images of the contact interface were simultaneously recorded using an eddy current displacement sensor and the CCD camera, respectively. The change in the tangential load FX was calculated as FX ¼ kδ, where k is the stiffness of the double-cantilever spring: k ¼855 N/m. In test II, the normal loads FZ of 1.1, 3.1, 5.8, 9.2 and 13.1 N were similarly applied through changes to the position of the glass plate. As is the case for test I, after a wait of 300 s, the intensity distribution of the transmitted light was imaged using the CCD camera. Subsequently, the glass plate was driven in the X direction at a constant speed of 0.1 mm/s. The changes in FX and images of the contact interface were recorded with the three-directional dynamometer and CCD camera, respectively. The resolutions of the optical system in tests I and II were 1.13 and 27.0 mm, respectively. As pointed out by Ref. [29], the measured quantity of the real contact area strongly depends on the

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Fig. 4. Captured images of contact interface in test I under stationary conditions: (a) W ¼ 0.03 N, (b) W ¼0.06 N, (c) W ¼0.09 N, (d) W ¼ 0.12 N, and (e) W ¼ 0.15 N. Color bar: transmitted light intensity.

resolution (or magnification) of used optical systems: the real contact area decreases as the magnification increases. Therefore, we cannot simply compare the real contact area measured by the optical systems with different magnifications. All experiments were performed in an air-conditioned room, where the temperature and relative humidity were approximately 25 1C and 30%, respectively.

4. Experimental results 4.1. Visualization of real contact area Figs. 4 and 5 show captured images of the contact interfaces in test I under stationary conditions. The color bars indicate the transmitted light intensity I. The white and black regions indicate real contact and non-contact regions, respectively. The apparent contact area expanded as the normal load W increased. In addition, the number density of the white regions gradually increased toward the center of the apparent contact regions. This contact area distribution was attributed to the ellipsoidal pressure distribution in the apparent contact region derived from the Hertz contact theory. The small region indicated by square in Fig. 4(a) is magnified in Fig. 5. The real contact and non-contact regions are clearly distinguished. The lower graph shows the profile of the transmitted intensity at the white dashed line drawn horizontally in the upper image. In the lower graph, the contact and non-contact regions are distinguished by the boundary value of the transmitted light intensity IB ¼104, which is denoted by a dashed horizontal line. The value of IB should be carefully determined because the estimated real contact area strongly depends on IB. In general, it is determined based on an appropriate model that considers some effects of the optical configuration and evanescent light effects. In this study, however, we roughly estimated the value of IB as described below; this value is validated by Fig. 12.

Fig. 5. Magnified image of contact region denoted by solid circle in Fig. 4(a). The lengths of each side of the image are 0.1 mm. Lower graph: intensity profile of white dashed line in upper image.

To find an appropriate value of IB, histograms of the intensity distribution within the apparent contact region were calculated, as shown in Fig. 6. The ordinate and abscissa are the normalized number of pixels and transmitted intensity, respectively. Two different peaks are clearly seen in the histogram curves: 1 and 2. Note that the histogram curves are normalized at the height of peak 1. Considering the attributes of the transmitted optical system, the positions of these peaks IP1 and IP2 indicate the feature quantities of the transmitted intensities that indicate real contact and noncontact regions. The normalized height of peak 2 increased with the normal load, but the position of IP2 stayed constant. This agreed with the experimental results, where the total area (or total number) of real contact regions increased with the normal load.

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In this study, we roughly estimated the value of IB using the following relationship: IB ¼ (IP1 þ IP2)/2¼ 104. Fig. 7 shows binarized images of Fig. 4. All pixels in the images were binarized using the threshold IB. When IZIB, the position of the pixel indicates the contact region (i.e., I¼255), and it is depicted in white. In contrast, when IoIB, the position of the pixel indicates the non-contact region (i.e., I¼0), and it is depicted in black. Fig. 8 shows the profile of the angle-averaged intensity Iave against the position in the radial direction r for the results in Fig. 7, where Iave was obtained in this manner; I was averaged azimuthally over the angle θ. Upward bell-shaped curves were observed under all loading conditions, which is similar to the elliptical distribution of the normal contact pressure used in the Hertz contact theory [4]. Thus, the total area (or total number) of the real contact regions increased with the contact pressure. Close inspection showed that the increment in the averaged intensity at the center of contact was relatively small compared to the surrounding regions. The averaged intensity at the central region was close to the maximum value of the transmitted intensity (i.e., 255). When all of the contact regions are merged (i.e., overall contact forms), the averaged intensity Iave should be 255; in this case, a white map was spread throughout the apparent contact region.

Fig. 6. Histograms of intensity distribution within apparent contact area in test I.

339

Fig. 9 summarizes the relationship between the total area of real contact regions Areal and the normal contact pressure P. The ordinate and abscissa show the normalized averaged intensity Iave/ Imax ¼ Iave/255 and normal contact pressure P, respectively. Each plot in Fig. 9 corresponds to a given location in the contact; the normal contact pressure at the location was calculated by the Hertz contact theory with that position. Iave generally tends to increase with the normal contact pressure P. It should be noted that the surface roughness reduces the maximum contact pressure and increases the apparent contact area compared to the theoretical prediction by the Hertz contact theory [4]. In this study, the effect of the surface roughness is not so much because the bulk compression at the center of the contact is much larger than the mean height of the summits on the rough surfaces. Therefore, this study did not consider the effect of the surface roughness on the normal contact pressure.

Fig. 8. Angle-averaged intensity for increasing normal load W. The intensity profile increases with the normal load W. Embedded image shows the coordinates system with X, Y, r, and θ.

Fig. 7. Binarized images of contact interface in test I under stationary conditions: (a) W ¼ 0.03 N, (b) W ¼0.06 N, (c) W¼ 0.09 N, (d) W¼ 0.12 N, and (e) W ¼0.15 N.

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The solid line shows a fitting curve based on Persson's theory, i.e., Eq. (7). The details on the formation of this fitting curve are given in the next section. The slope of the increase in area gradually decreases as the normal pressure P increases. Iave/Imax apparently asymptotically approaches unity. Assuming that the averaged intensity indicates the real contact area, rubber contact was found to show a different tendency from the results expected with the GW theory. In GW theory, the real contact area linearly increases with the normal pressure. 4.2. Measurement of friction force Fig. 10 shows the time changes in the tangential load FX of tests I and II. In test I, each curve was found to have two distinct loading phases: an initial quasi-linearly loading phase followed by a steady loading phase. First, FX increased linearly; then, the slope gradually decreased with time. After this, FX finally reached a steady value at the kinematic friction force Fk. We found a large difference between the results of tests I and II. In test I, we can find the local maximum value of tangential load, maximum static friction force, between the initial quasi-linearly loading phase and the steady loading phase. In contrast, in test II, the maximum friction force was not observed. Thus, in this case, friction force reached a steady value without a fall in friction force. This was caused by the difference in the test configuration. In test I, the glass hemisphere lens is embedded to flat elastomer. In contrast, in test II, the elastomer lens is only flattened against the glass plate. The effect of the configuration on friction force was discussed by Gabriel [30]. As a similar result, the system dependence of the static friction coefficient is reported by David et al. [31]. They pointed out that

static friction is not a material constant; thus, the value of static friction strongly depends on the shape of slider, operating conditions, and test configuration. It indicates that it is difficult to discuss the intrinsic value of the static friction coefficient through different experiments. Fig. 11 shows the normal load dependence of the friction coefficient μ, which was calculated using the kinetic friction force Fk: μ¼Fk/W. The value of μ gradually decreased as the normal load W increased. The results tended to fall in between the two models shown in Fig. 1(a) and (b). In models I and II, the friction coefficients decreased according to the relationships μpW  1/3 and μpW0, respectively. In contrast, the rough estimation of Fig. 11 shows that μ decreased according to the relationship μpW  0.11. The value of the superscript must depend on the kind of material and surface roughness of the specimens. The difference is caused by the difference in the normal pressure dependence of the real contact areas shown in Figs. 1 and 9. To understand the normal load dependence of the friction coefficient, we developed a simple model, which is described in the next section. The solid lines in Fig. 11 indicate fitting curves based on the developed model given by Eq. (11).

5. Discussion 5.1. Adhesion friction model considering normal pressure dependence of real contact area Fig. 12 shows a double logarithmic plot for the relationship between the friction force Fk and total area of real contact Areal. This was calculated by Areal ¼ Aapp

N X

I=I max ;

ð6Þ

1

Fig. 9. Effect of normal contact pressure P on averaged intensity.

where Aapp, N, I, and Imax are the total area of the apparent contact region, number of pixels included in the apparent contact region, binarized intensity, and maximum value of the binarized intensity, respectively. When a pixel indicates the real contact region, I/Imax ¼ 255/255 ¼ 1. In contrast, when a pixel indicates a noncontact region, I/Imax ¼0/255 ¼ 0. The solid line in the figure shows the linear fitting line derived as F¼ τfitAreal, where the fitting parameter τfit has the same dimension as the shear strength at the real contact regions. The least-squares method was used to determine τfit ¼0.156 MPa, which agreed well with the values obtained in other experimental results [32,33]. All of the plots could be placed on a single fitting line, even though the contact area was very large. This indicates that the estimation method for the total area of real contact regions used in this study is appropriate. The close fit indicates that the friction force is mainly

Fig. 10. Time changes in tangential load FX: (a) test I and (b) test II.

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Fig. 11. Normal load dependence of friction coefficient μ: (a) test I and (b) test II.

Fig. 13. Calculation results: dependence of kinetic friction coefficient μ on normal contact load W. Fig. 12. Relationship between friction force Fk and real contact area Areal: (solid circles) test I and (open circles) test II.

determined by the real contact area. Thus, when rubber-like materials slide on a rigid smooth plane at a low sliding speed, the friction force can be attributed to the total area of the real contact regions. Note that, when a rubber material slides on a rough surface, the effects of the hysteresis friction should not be neglected [19]. The above results showed that the real contact area is the most important factor for determining the value of the friction force. As shown in Fig. 9, the total area of real contact regions Areal asymptotically increases with the normal pressure P and approaches the maximum value at which the overall contact forms. Thus, the required constraint condition is that the fitting curve has asymptotic behavior. For the transition from the multiasperity contact to the full contact, the changes in the real contact area with the normal contact pressure was theoretically derived by Persson [8,25] as below   Z P Areal 1 P 2 ¼ dσe  σ =4K ¼ erf ð7Þ 1=2 1=2 Aapp ðπKÞ 2K 0 where K is the characteristic normal pressure to determine the normal pressure dependence of the real contact area, and it is defined as E2 κ 2 K¼
2 ; 8 1  ν2

ð8Þ

where κ was the rms slope, which was described in Refs. [8,25,34]. Based on the fitting results with Eq. (7), we calculated the total area of real contact Areal for the overall point contact. The normal

pressure distribution within the Hertz contact region is derived by   r 2 1=2 3W P¼ 1  ; ð9Þ a 2πa2 where a is the radius of contact and r is the distance from the center of contact. Eq. (9) was substituted into Eq. (7) to derive the total real contact area Areal through the following equation: ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z a  r 2 =2K 1=2 2πrdrdr; erf P 0 1  ð10Þ Areal ¼ a 0 where P0 is the maximum contact pressure derived by the Hertz theory, i.e., P0 ¼3W/2πa2. Additionally, based on the adhesion theory, we can derive the friction coefficient μ as ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z a  r 2 τA τ μ ¼ real ¼ =2K 1=2 2πrdr; erf P 0 1  ð11Þ W 0 a W 5.2. Determination of K and τ without in situ observation of contact interfaces In many cases, in situ observation of the contact interface is difficult because the use of transparent materials and optical equipment is required. In addition, quantitatively determining the real contact area is not easy; special calibrations are needed to confirm the accuracy of the measurements. In fact, determining the physical meaning of IB derived in Fig. 6 is difficult. To avoid the above problems, we determined the values of K and τ without in situ observation of the contact interface. The solid lines in Fig. 11 are fitted curves based on Eq. (11). The experimental results obtained from direct observation of the contact interface

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Fig. 14. Captured images of contact interface in test II: (a) before sliding and (b) steady sliding.

were not used. In two experiments, i.e., tests I and II, identical PDMS was used. Thus, the values of τ in both tests should be the same. Therefore, only three fitting parameters were used: τ, K (test I), and K (test II). From the least-squares method, these values can easily be obtained. Consequently, the values of τ, K (test I), and K (test II) were determined to be 0.149, 4.49  109, and 7.22  109 kg m  1 s  2, respectively. The estimated value of τ, which was determined without any optical measurement, agreed well with the results based on direct observation of the contact interface shown in Fig. 12. The fitting curve in Fig. 9 was based on Eq. (7), where the value of K determined by the fitting process was 4.67  109. Additionally, the shear stress determined in Fig. 12, i.e., τfit was 0.156 MPa. Fig. 13 shows typical calculation results for the normal load dependence of the friction coefficient. When K was small, the friction coefficient depended on the normal load. This corresponds to the results for model I shown in Fig. 1(a). When K was large, the friction coefficient did not depend on the normal load. This is similar to the characteristics of model II shown in Fig. 1(b). The similar small dependence of the friction coefficient on the normal load was also found in the reference by Schallamach [21]. The value of K strongly depended on the difference in specimens. As expected, the surface topography was an important factor for determining the value of the friction force. Based on the developed model, the value of τ can be explicitly obtained without direct observation and quantitative analysis of the real contact area. 5.3. Limitations of developed model Fig. 14 shows snapshots of the contact surface in test II at a normal load of 3.8 N, where a different PDMS hemisphere with a surface roughness Ra of 0.64 was used. This was the same value used in test I. However, the normal load was significantly different; therefore, the contacting asperities were easily flattened by the large normal contact pressure. The left side of the images in Fig. 14(a) shows the contact interface before sliding, and the right side in Fig. 14(b) shows the contact interface during the steady sliding. The shapes of the apparent contact region were quite different. The trailing edge of the contact interface was peeled because of the large tensile stress. This macroscopic deformation of the contact region is a known unique phenomenon for the sliding friction of rubberlike materials [35]. A large friction force due to a low surface roughness and large normal load causes macroscopic deformation of the contact interface. In this case, the use of the developed model is obviously inappropriate. As other examples of limitations, dynamic behavior of the contact interface such as the

propagation of Schallamach waves is well-known. In these situations, the developed model similarly cannot be used to determine the normal load dependence of the friction coefficient. When rubber materials slide on a substrate with very smooth surface, the energy dissipation due to the viscoelastic deformations in the bulk of the rubber is unimportant for determining the value of the friction. In this case, the friction is caused by local stick-slip events at the sliding interface [36,37]. In contrast, for a rough substrate, as well known, the energy dissipation is an important factor, i.e., hysteresis friction term [8]. Therefore, bulk deformation due to the sliding motion on rough surfaces and the viscoelastic properties of rubber should be considered for evaluating the mechanism of friction [19].

6. Conclusion The normal load dependence of the real contact area between a rough PDMS surface and smooth glass surface was directly quantified based on in situ observation of the contact interfaces. The developed transmitted optical system can measure the total area of the real contact regions without complicated calibration. Based on the experimental results, a simple adhesion friction model for considering the normal pressure dependence of the real contact area was developed. The model expresses the normal load dependence of the friction coefficient well. The model provides a simple method for estimating the real contact area and shear stress based on a fitting process using the results of friction tests. With this approach, these values can be roughly but easily estimated without in situ observation of the contact interface. References [1] Persson BNJ. Sliding friction: physical principles and applications. 2nd ed. Heidelberg: Springer; 2000. [2] Popov VL. Contact mechanics and friction: physical principals and applications. Heidelberg: Springer; 2010. [3] Bowden FP, Tabor D. Friction and lubrication of solids. New York: Wiley; 1966. [4] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 2003. [5] Greenwood JA, Williamson JBP. Contact of nominally flat surfaces. Proc R Soc A 1966;295:300–19. [6] Greenwood JA, Tripp JH. The elastic contact of rough spheres. Trans ASME: J Appl Mech 1967;34:153–9. [7] Adams GG, Nosonovsky M. Contact modeling – forces. Tribol Int 2000;33:431–42. [8] Persson BNJ. Theory of rubber friction and contact mechanics. J Chem Phys 2001;115:3840–61. [9] Boyce MC, Arruda EM. Constitutive models on rubber elasticity: a review. Rubber Chem Technol 2000;73:504–23.

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