Average separation between a rough surface and a rubber block: Comparison between theories and experiments

Wear 268 (2010) 984–990

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Average separation between a rough surface and a rubber block: Comparison between theories and experiments B. Lorenz a,∗ , G. Carbone b , C. Schulze c a b c

IFF, FZ-Jülich, D-52425 Jülich, Germany DIMeG-Politecnico di Bari, V.le Japigia 182, 70126 Bari, Italy ISAC, RWTH Aachen University, D-52074 Aachen, Germany

a r t i c l e

i n f o

Article history: Received 17 April 2009 Received in revised form 4 December 2009 Accepted 14 December 2009 Available online 24 December 2009 Keywords: Interfacial separation Rough surfaces Contact mechanics Persson Greenwood–Williamson Experiment

a b s t r a c t We briefly review the most important contact mechanic theories regarding the average separation between an elastic solid with a nominal flat surface and a hard solid with a randomly rough surface, as a function of the squeezing pressure. We present experimental results for a silicon rubber (PDMS) block, squeezed against two different road surfaces. Finally we compare the theoretical predictions of the different theories with the experimental data. © 2009 Elsevier B.V. All rights reserved.

1. Introduction What happens at the atomic and molecular level when surfaces come into contact with each other? And how do these events relate to macroscopic properties and observations? When two elastic solids with rough surfaces are squeezed together, one observes that the solids will not in general make contact everywhere in the apparent contact area, but only at a distribution of asperity contact spots. The separation u(x) between the surfaces vary in a nearly random way with the lateral coordinates x = (x, y) in the apparent contact area. Increasing the applied squeezing pressure, will consequently lead to a decrease of the average surface separation u = u(x) (the symbol  stands for the ensemble average). Describing these interactions upon contact between two solids plays a major role in a large number of physical phenomena and engineering applications, e.g. structural adhesives, protective coating, friction of tires, lubrication, wear and seals. The two most important physical quantities in the field of contact mechanics are the area of real contact and the interfacial separation between two solid surfaces as a function of the applied load.

Notwithstanding the research activity that has been carried out in the last decades to find reasonable and accurate models to describe this problem [1–5], a full comprehension of contact between randomly rough surfaces has not yet been achieved. In the present state of the art, there still exist different kinds of approaches to this problem and numerous theories are dealing with this topic, while the scientific community is still debating about which of them gives the most accurate results. There are some numerical investigations [1–3,6–8] aiming to understand what is the most accurate between the two most important approaches available in literature: (i) the multiasperity contact models, all based on the original idea by Greenwood and Williamson [4], and (ii) the theoretical approach by Persson [9]. However, till now no definitive answer to this question has been found. In this paper we extend the analysis presented by one of the authors in Ref. [10] to test these two different approaches and compare their theoretical predictions with our experimental data. This study has been focused on the interfacial separation between an elastic solid squeezed against two different randomly rough rigid surfaces. 2. Theory

∗ Corresponding author. Tel.: +49 2461 61 1689; fax: +49 2461 61 2850. E-mail address: [email protected] (B. Lorenz). 0043-1648/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2009.12.029

As we have mentioned before, there are mainly two different approaches to describe the mechanical contact between two rough solids: the multiasperity contact models, a generalization of

B. Lorenz et al. / Wear 268 (2010) 984–990

the Hertz contact theory, and the theoretical approach by Persson [9]. The very first idea of multiasperity contacts was proposed by Archard [11], however a profound refinement of this idea was due to Greenwood and Williamson (GW) [4], who modeled the roughness of the surface as an ensemble of identical spherical asperities with randomly distributed heights to take account of the surface statistics. The ultimate development of this idea was due to Bush, Gibson and Thomas (BGT) [12] in 1975 (see also Refs. [15,16]), who, moving from Longuet-Higgins [13] and Nayak [14] statistical theory of isotropic randomly rough surfaces, modeled the asperities as paraboloids with two different radii of curvature. The statistics of asperity height, curvatures, etc. were completely taken into account. Of course, multiasperity contact theories break down as the contact moves towards full contact conditions, i.e. these theories are believed to hold true only for small loads and contact areas. The second approach by Persson was proposed in 2001 [9]. This approach instead gives exact solution for full contact conditions. The theory, indeed, assumes that, at any given magnification, the power spectral density (PSD) of the deformed surface of the elastic half-space is well approximated by that of the underlying rigid substrate multiplied by the apparent contact area observed at the given magnification. This assumption of course is correct in full contact conditions but less accurate as we move towards small contact areas and small loads.

as

985

2

mp =

d(cos ) 0

p

∞

dqq1+p C(q)

(2)

0

In order to determine the area of contact and the load between the rough rigid surface and an initially flat elastic half-space, multiasperity contact theories make use of the Hertz’s theory to calculate, for a given penetration ı = 1 − u (where u is the distance of the approaching elastic half-space from the mean plane of the rough surface and 1 is the height of summit), the contact area and load upon contact between the elastic half-space and the single rigid asperity. Hertz’s theory states that, besides the elastic properties of the contacting bodies, the contact area and load depend only on the penetration ı and on the principal radii of curvature of the contacting asperities. Thus, one can calculate the fraction of area in contact Ac /A0 and the mean pressure in the nominal contact area  = F/A0 as Ac = A0 F = A0



+∞

d1 u



 





 



d1 d2 AH 1 , 1 , 2 P 1 , 1 , 2

(3)

D



+∞

d1 u

d1 d2 FH 1 , 1 , 2 P 1 , 1 , 2 D

(4)



where the domain D = {(1 , 2 ) ∈ 2 |2 ≤ 1 ≤ 0}, AH 1 , 1 , 2



2.1. Multiasperity contact theories Within the framework of multiasperity contact models, the determination of area of contact and load as a function of the distance between the two approaching bodies moves from the knowledge of the joint probability distribution P(1 , 1 , 2 ) of summits with heights 1 , and curvatures 1 and 2 . Observe that we define 1 being the maximum curvature and 2 the minimum curvature of the summit, thus the following inequality holds true 2 < 1 < 0. The condition 1 > 2 is necessary since two different summits with different orientation but with the same maximum curvature 1 and minimum curvature 2 are equivalent. Recalling the Longuet-Higgins [13] and Nayak [14] analysis of surface statistics it is possible to show that for an isotropic surface the joint probability distribution P 1 , 1 , 2 is given by [15,16] 1 P(1 , 1 , 2 ) = √ (4)2 m2 m4 m0 m4

⎡

1/2



and FH 1 , 1 , 2 are Hertzian contact area and load on each asperity in contact. Bush et al. [12] developed the most complete theory of contact mechanics within the framework of multiasperity contact models. They made use of Eqs. (3) and (4) but in a different form. Instead of focusing on the radii of curvature of the asperities, 1 and 2 , (which in agreement with the Hertz theory were treated as paraboloidal asperities), they developed calculations by referring to the semiaxes of the ellipse of contact, a1 and a2 . We refer the reader to the original paper by Bush Gibson and Thomas [12] and to Ref. [16] for a detailed description of the model. The major result of the BGT theory was, that in the limiting case of large separations, the area of true contact Ac is proportional to the applied load F, and equal to just half of the bearing area. 2.2. The Persson theory of contact mechanics

√ 27

×C1





exp ⎣−C1

×1 2 exp



−

2⎤ 3 1 + 2 1 ⎦ (1 − 2 ) + √ 1/2 1/2

m0

2 ˛ 2m 4



3 [3(1 + 2 )2 − 81 2 ] 16m4

(1)

where the quantities m0 , m2 , and m4 are the zero, second and fourth moments of the power spectral density (PSD) C (q) = (2)−1 dxh(x)h(0)e−iqx of a profile obtained by the intersection of the randomly rough rigid surface z = h(x) with a plane perpendicular to the mean plane of the surface itself, along the direction . We can write in general mp = dqC (q)qp . Because of isotropy, the PSD C (q) does not depend on the orientation of the plane intersecting the surface, therefore one can, for instance, choose the plane qy = 0 and write, C (qx ) = C=0 (qx ) = dqy C(qx , qy ), with



C(qx , qy ) = C(q) = (2)−2 d2 xh(x)h(0)e−iq·x being the surface PSD. In Eq. (1) the breadth parameter (as defined by Nayak) is ˛ = m0 m4 /m22 and C1 = ˛/(2˛ − 3). Observe that for a isotropic surface the surface PSD C(q) depends only on the modulus q = |q| of the wave-vector, therefore one can also calculate the moments mp

The theoretical approach by Persson removes the assumption, which is implicit in the multiasperity contact theories, that the area of real contact is small compared to the nominal contact area. On the contrary, he moves from the limiting case of full contact conditions (where it gives the exact solution) between a rigid rough surface and an initially flat elastic half-space, and accounts for partial contact by requiring that, in case of adhesionless contact (as those discussed in this paper) the stress probability distribution vanishes when the local normal surface stress  vanishes. The theory yields very simple formulas and needs as inputs only the surface power spectral density C(q) and the elastic properties of the contacting bodies. Consider the contact between an elastic solid (elastic modulus E and Poisson ratio ) with a flat surface and a rigid, randomly rough surface with the height profile z = h(x) (see Fig. 1). When the applied squeezing force  increases, the separation between the surfaces at the interface will decrease, and one can consider  = (u) as a function of u. The elastic energy Uel (u) stored in the substrate asperity–elastic block contact regions equals the work done by the external pressure  in displacing the lower surface of the block towards the substrate. Thus, (u) = −

1 dUel , A0 du

(5)

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B. Lorenz et al. / Wear 268 (2010) 984–990

Fig. 2. A rubber block in contact with a rigid, randomly rough substrate. Left: no applied load. Right: The rubber block is squeezed against the substrate with the force F. The upper surface of the rubber block moves of a quantity s, whereas the mean plane of the deformed lower surface penetrates inside the surface roughness of a quantity w.

3. Experiment Fig. 1. An elastic block squeezed against a rigid rough substrate. The separation between the average plane of the substrate and the average plane of the lower surface of the block is denoted by u. Elastic energy is stored in the block in the vicinity of the asperity contact regions.

Eq. (5) is exact for purely elastic materials [17,18]. Theory shows that for low squeezing pressure, the area of real contact A varies linearly with the squeezing force F = A0 , and that the interfacial stress distribution and the size-distribution of contact spots are independent of the squeezing pressure [19]. That is, with increasing  existing contact areas grow and new contact areas form in such a way that in the thermodynamic limit (infinite-sized system) the quantities referred to above remain unchanged. It follows immediately that for small load the elastic energy stored in the asperity contact region will increase linearly with the load, i.e., Uel (u) = u0 A0 (u), where u0 is a characteristic length which depends on the surface roughness but is independent of the squeezing pressure . Thus, for small pressures (1) takes the form (u) = −u0

To compare the different theories presented above, we performed the experiments indicated in Fig. 2. We squeezed a rubber block with a nominal flat surface against different types of road samples with randomly rough surfaces. We controlled the displacement s of the upper surface of the rubber block and changed it in steps of 0.02 mm. For each step, we measured the restoring force F. For this experiment we used the test stand shown in Fig. 3 produced by SAUTER GmbH (Albstadt, Germany), normally used to measure spring constants. Using this test stand, we are able to mea-

d du

or [22] (u)∼e−u/u0 .

(6)

To quantitatively derive the relation (u), an analytical expression for the asperity induced elastic energy is needed. Within the contact mechanics approach of Persson one obtain [19–21] Uel ≈ A0 E

∗

2

q1

dq q2 P(q, p)C(q),

(7)

q0

where E ∗ = E/(1 − 2 ) and where P(q, p) = A( )/A0 is the relative contact area which depends on the applied pressure  when the interface is studied at the magnification = q/q0 . Substituting (7) in (5) gives for small squeezing pressures [17]:  = ˇE ∗ e−u/u0

(8)

For self-affine fractal surfaces, the length u0 and the parameter ˇ depend on the fractal dimension Df of the surface, and on the rolloff wave vector q0 and on the short distance cut-off wavevector q1 . Most surfaces that are self-affine fractals have a fractal dimension Df < 2.5. For these surfaces u0 and ˇ are nearly independent of the highest surface roughness wavevector, q1 , included in the analysis. The derivation for u0 and ˇ are given in [23]. Using (5) we finally get log

  E

 = log

4ˇ 3

 −

u u0

(9)

where  = F/A0 is the squeezing pressure. Here we have used that E ∗ /E = 1/(1 − 2 ) ≈ 4/3 since for rubber  ≈ 0.5.

Fig. 3. The test stand used for the interfacial separation experiment. It is fabricated by SAUTER GmbH (Albstadt, Germany). During the experiment, we have controlled the displacement in steps of 0.02 mm and then measured the restoring force.

B. Lorenz et al. / Wear 268 (2010) 984–990

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Fig. 4. Pictures of the two different surfaces used for the experimental investigation. An asphalt road surface, denoted as surface 1 (a); a concrete road surface denoted as surface 2 (b).

sure forces up to 500 N, and displacements with a resolution of 0.01 mm. When the block is squeezed against a rigid, randomly rough countersurface, the upper surface of the rubber block will move downwards by the distance s (see Fig. 2), which is the sum of a uniform compression of the rubber block, d/E, and a movement (or penetration) w of the average position of the lower surface of the rubber block into the valleys or cavities of the countersurface: s=w+

d E

(10)

In Eq. (10) the quantity d is the thickness of the elastic block (in our case d = 10 mm). We denote u as the separation between the average surface plane of the block and the average surface plane of the substrate (so that u = 0 corresponds to perfect contact with u ≥ 0). Then, we can write w = hmax − u

(11)

where we have assumed that the initial position of the lower surface of the block corresponds to the separation at which the block makes contact only with the highest substrate asperity (as in Fig. 2, left), which is located a distance hmax above the average plane of the underlying rough surface. Using (10) and (11) we get u = hmax +

d −s E

(12)

Substituting (12) in (9) gives log or log

  E

  E



= log

4ˇ 3

1 =B+ u0





−

1 u0

 s−d E



hmax − s + d

 E



 (13)

where B = log(4ˇ/3) − hmax /u0 . Since the contact between the elastic block and the rough surface is dry no slip occurs at the interface. For no-slip boundary conditions we replace the elastic modulus E by the effective elastic modulus E , using E > E (see below). Thus, in this case (13) takes the form log

 E

= B +

1 u0



s−d

 E



where B = log(4ˇE/3E ) − hmax /u0 .

(14)

The rubber block that we used for this experiment was made from a silicone elastomer (PDMS). We have used polydimethylsiloxane because of its almost purely elastic behavior. Therefore viscoelastic effects under our experimental conditions can be neglected. The PDMS samples were prepared using a two-component kit (Sylgard 184) purchased from Dow Corning (Midland, MI). This kit consists of a base (vinyl-terminated polydimethylsiloxane) and a curing agent (methylhydrosiloxane–dimethylsiloxane copolymer) with a suitable catalyst. From these two components we prepared a mixture of 10:1 (base/cross linker) in weight. The mixture was degassed to remove the trapped air induced by stirring from the mixing process and then poured into cylindric casts (diameter D = 3 cm and height d = 1 cm). The bottom of these casts were made from glass to obtain smooth surfaces (negligible roughness). The samples were cured in an oven at 80 ◦ C for over 12 h. The road surfaces used in this experiment were provided by Pirelli and by Baustoffprüfung-Aachen (Germany). The topography was measured with contact-less optical methods using a chromatic sensor with two different optics produced by Fries Research & Technology GmbH (Bergisch Gladbach, Germany). The statistical analysis of the two surfaces has been carried out with the aid of a code we have developed ad hoc. The code allows us to calculate the PSD of the two surfaces and through Eq. (2) all the moments of the PSD, the rms roughness and the rms slope of the surfaces. In addition we have also calculated the height probability density function (PDF) and the fractal dimension of both surfaces. Below we report the main data of the surfaces: (1) The asphalt road surface is shown in Fig. 4(a). We have calculated the  PSD,  the fractal dimension Df = 2.2  and the quantities m0 = h2 = 0.091 mm2 , m2 = 0.5 ∇ h2 = 2.67, m4 = 2.1 × 104 mm−2 , ˛ = 263. The highest point of the surface is located at a distance hmax = 1.37 mm from the mean plane. We show the corresponding power spectrum C(q) in Fig. 5 and the height PDF in Fig. 6. (2) The concrete road surface is shown in Fig. 4(b). For this surface, we have calculated the PSD, the fractal dimension Df = 2.3 and the quantities m0 = h2  = 0.071 mm2 , m2 = 0.5∇ h2  = 0.27, m4 = 62 mm−2 , ˛ = 60. The highest point is located at a distance hmax = 1.1 mm. The power spectrum C(q) of this second surface is shown in Fig. 7 and the height PDF in Fig. 8. We observe that

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B. Lorenz et al. / Wear 268 (2010) 984–990

Fig. 5. The surface roughness power spectrum C(q), as a function of the wavevector q (log–log scale), for the surface 1. The straight green line has the slope −3.7, corresponding to a fractal dimension Df = 2.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Fig. 8. The height probability density function of surface 2 (solid line) and the Gaussian approximation (dashed line).

Fig. 9. A rubber block between two flat and rigid solid plates. Undeformed state (a); squeezed block under no slip conditions at the rubber-plate interfaces (b); squeezed block under perfect slip conditions at the rubber-plate interfaces (c).

Fig. 6. The height probability density function of surface 1 (solid line) and the Gaussian approximation (dashed line).

the height probability density functions of both surfaces deviate from the ideal Gaussian distribution, which, instead, is the implicit assumption of both BGT and Persson’s theories. However, we note that no real surface is perfectly Gaussian, and that a theory is really useful only when it gives good results in practical applications when its hypotheses are only partially satisfied. Under this perspective and observing that for the two surfaces the deviation of the real height PDF from a Gaussian distribution is not very exaggerated (at least for surface 2), we believe that we can still employ the BGT and Persson’s theories in our case.

Fig. 7. The surface roughness power spectrum C(q), as a function of the wavevector q (log–log scale), for the surface 2. The straight green line has the slope −3.4, corresponding to a fractal dimension Df = 2.3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

To estimate the effective elastic modulus E, the PDMS sample was first squeezed against a smooth substrate in a compression test. We have both studied dry and lubricated interfaces resulting in no slip and perfect slip at the two rubber–wall interfaces (see Fig. 9). As lubricant we used polyfluoroalkylsiloxane (PFAS), a fluorinated silicone oil supplied by ABCR GmbH & Co. KG (Karlsruhe, Germany). Because of its high viscosity ( = 1000 cSt), the fluid is an excellent lubricant also under extreme pressure applications and should therefore not easily be squeezed out of the contact area. Also it does not react (or interdiffuse) with the PDMS elastomer. During the compression test in dry conditions no slip occurs at the interfaces and the PDMS sample was deformed laterally as shown in Fig. 9(b). On the other hand, the presence of the lubricant led to perfect slip conditions and the PDMS did not deform laterally during the compression test [see Fig. 9(c)]. In Fig. 10 we show the measured relation between the stress and the strain for lubricated surfaces (so that the shear stress vanish on the boundaries). If the stress is normalized with E = 2.3 MPa a nearly straight line with slope 1 is obtained so that the relation  = Es/d holds true. The elastic modulus E = 2.3 MPa is consistent with the elastic modulus reported in the literature for similar silicon rubbers [24].

Fig. 10. The stress  (in units of the elastic modulus E) as a function of the strain s/d under perfeclty slip conditions at the interfaces. We have obtained E = 2.3 MPa.

B. Lorenz et al. / Wear 268 (2010) 984–990

Fig. 11. The stress  (in units of the effetive elastic modulus E ) as a function of the strain s/d, under perfect no-slip conditions. We have calcualted an effective modulus E = 4.2 MPa. The two experimental curves corresponds to loading and unloading conditions.

989

Fig. 13. The natural logarithm of the quantity /E as a function of s − d/E for surface 1 in dry contact conditions (no-slip). The effective elastic modulus is E = 4.2 MPa and B = −7.02.

4. Results As said, we have also performed experiments for dry surfaces. In this case no (or negligible) slip occurs at the interface with the confining walls, and a visual inspection of the system showed that the rubber bulge laterally at the force-free area [see Fig. 9(b)]. We still expect a linear (or nearly linear) relation between stress and strain but the effective elastic modulus E should be larger than for lubricated interfaces. This is just what we observed: the effective elastic modulus deduced from the experimental data (see Fig. 11) E ≈ 4.2 MPa is about 80% larger than for the lubricated interface. To check the measuring system for hysteresis effects, some of the experiments were performed bidirectional. The results are shown in Fig. 11 where the strain was increased and after that slowly decreased again. Negligible hysteresis occurs, as expected because of the low glass transition temperature of the PDMS. The increase of the effective elastic modulus during compression tests, from 2.3 MPa to 4.2 MPa, when going from slip to no-slip boundary condition, is consistent with the predictions of the Lindley equation [25], which in the present case takes the form E ≈ E(1 + 1.4S 2 ) For a cylinder the shape factor S = R/2d. In the present case E = 2.3 MPa and S = 0.75 giving E = 4.1 MPa which agree very well with the measured value (4.2 MPa).

We will now present experimental results for a rubber block squeezed against the road surfaces described before. We have performed the squeeze test, schematically shown in Fig. 2, under dry conditions, so that the effective elastic modulus is E = 4.2 MPa. Some photographs of the compression test against one of the two rough surfaced are shown in Fig. 12. The figure clearly shows that as the compressive force is increased, the number and the extension of the contact areas rapidly increase. Our measurements of load and penetration are collected in Figs. 13 and 14, where we show the natural logarithm of the squeezing pressure  (divided by the effective elastic modulus E ) as a function of the penetration w = s − d/E , where d = 1 cm is the thickness of the rubber block. We plot the experimental observations as red lines. For each experiment two lines have been represented to give an indication of the precision of the experiment. Indeed, all the other experimental curves are located in between the two red lines shown in the figures. The lower line is the minimum run obtained during the experiment while the upper line is the maximum. We show the predictions of the BGT theory as a blue line while the green line is used for Persson’s predictions. Figs. 13 and 14 show that the predictions of the Persson theory are in relative good agreement with the experiments. We observe only a slightly different slope, that disappears almost completely if one uses an effective

Fig. 12. Pictures of the asperities penetrating into the rubber sample with increasing normal pressure. (a) Zero pressure, (b) minimal pressure, (c) medium pressure, and (d) maximum pressure.

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B. Lorenz et al. / Wear 268 (2010) 984–990

However, comparing BGT predictions to our data, we do not find agreement even in this limiting case. Therefore, the predictions of multiasperity contact theories, which predicts that the squeezing 1/2

Fig. 14. The natural logarithm of the quantity /E as a function of s − d/E for surface 2, under dry conditions (no-slip). The effective elastic modulus is E = 4.2 MPa and B = −6.38.

elastic modulus equal to 4.8 MPa instead of 4.2 MPa. The diagrams also confirm the correctness of the predicted exponential law Eq. (6) between the separation u and squeezing pressure . However, we note that for small s − d/E the experimental curves drop off faster with decreasing interfacial separation than the Persson’s curves. The same effect has also been observed in molecular dynamics calculations [18] and other numerical calculations [8]. The explanation for this behavior is that in the experiments we have used a finite system, whereas Persson’s theory is for an infinite system. An infinite system has, indeed, (arbitrarily many) arbitrarily high asperities, which always allow the contact between the two solids to occur for arbitrarily large surface separations. But a finite system has asperities with height below some finite value hmax , and for u > hmax no contact can occur between the solids. As a consequence  must rapidly decrease to zero as u approaches hmax . Now let us focus on the BGT theory and compare its predictions with the experimental data. In this case we do not find a good agreement between theory and experiment, even at large separation where the BGT theory (and many other multiasperity contact theories [15,16]) predicts ∼t −1 exp(−t 2 /2) [12,15] where 1/2

t = u/h2  is the dimensionless separation. One necessarily concludes that, at least for the surfaces utilized in our experiments, the BGT theory and in general multiasperity contact models cannot be utilized to correctly predict the real separation as a function of the applied load. 5. Summary and conclusion We have carried out experimental investigations to test the theoretical predictions of BGT (and similar multiasperity contact models) and Persson’s theories of contact mechanics. A PDMS rubber block has been pressed, under displacement controlled conditions, against two different road surfaces by employing a very simple experimental apparatus. We have measured the applied load as a function of the separation between the two approaching surfaces and compared these data with the theoretical predictions. The multiasperity contact theories, of which the most representative is the BGT model, are believed to give reasonable results for very small normal pressures and therefore very large separations.

pressure  depends on the dimensionless separation t = u/h2  through the relation ∼t −1 exp(−t 2 /2), can not describe qualitatively the relation between the load and interfacial separation at least for the system and rough surfaces we used in our experiments. We have also compared Persson’s theory prediction with experiments and find a much better agreement. Indeed the measured  vs. u relation seems to be quite well described by an exponential function of the type ∼e−u/u0 , which is the one also predicted by Persson’s theory. We plan to extend the study to surfaces with different surface parameters to test the theory in more general cases and under more general conditions. Acknowledgments We thank O.D. Gordan (IBN, FZ-Jülich) for help with preparing the PDMS rubber blocks and S. Dirrichs (Straßen NRW, Baustoffprüfung, Aachen) for supplying us with road surface drill cores. This work, as part of the European Science Foundation EUROCORES Programme FANAS was supported from the EC Sixth Framework Programme, under contract N. ERAS-CT-2003-980409. References [1] S. Hyun, L. Pei, J.-F. Molinari, M.O. Robbins, Phys. Rev. E 70 (2004) 026117. [2] M. Borri-Brunetto, B. Chiaia, M. Ciavarella, Comput. Methods Appl. Mech. Eng. 190 (2001) 6053. ˜ M.H. Müser, Europhys. Lett. 77 (3) (2007) 38005. [3] C. Campanà, [4] J.A. Greenwood, J.B.P. Williamson, Proc. R. Soc. Lond. A 295 (1966) 300. [5] J.A. Greenwood, Wear 261 (2006) 191–200. [6] S. Hyun, L. Pei, J.F. Molinari, M.O. Robbins, Phys. Rev. E70 (2004) 026117. [7] C. Yang, U. Tartaglino, B.N.J. Persson, Eur. Phys. J.E: Soft Matter 19 (1) (2006) 47–58. [8] G. Carbone, M. Scaraggi, U. Tartaglino, Eur. Phys. J. E: Soft Matter 30 (2009) 65–74. [9] B.N.J. Persson, J. Chem. Phys. 115 (2001) 3840. [10] B. Lorenz, B.N.J. Persson, J. Phys.: Condens. Matter 21 (1) (2009) 015003. [11] J.F. Archard, Proc. R. Soc. A243 (1957) 190. [12] A.W. Bush, R.D. Gibson, T.R. Thomas, Wear 35 (1975) 87. [13] M.S. Longuet-Higgins, Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci. 249 (966) (1957) 321–387. [14] P.R. Nayak, J. Lubr. Technol. 93 (1971) 398–407. [15] G. Carbone, J. Mech. Phys. Solids 57 (7) (2009) 1093–1102. [16] G. Carbone, F. Bottiglione, J. Mech. Phys. Solids 56 (8) (2008) 2555–2572. [17] B.N.J. Persson, Phys. Rev. Lett. 99 (2007) 125502. [18] C. Yang, B.N.J. Persson, J. Phys.: Condens. Matter 20 (2008) 215214. [19] B.N.J. Persson, Surf. Sci. Rep. 61 (2006) 201. [20] B.N.J. Persson, Eur. Phys. J. E8 (2002) 385. [21] C. Campana, M.H. Müser, M.O. Robbins, J. Phys.: Condens. Matter 20 (2008) 354013. [22] We note that the result (2) differs drastically from the prediction of asperity contact mechanics theories such as those of the Bush et al. [A.W. Bush, R.D. Gibson, T.R. Thomas, Wear 35 (1975) 87] and the theory of Greenwood and Williamson [J.A. Greenwood, J.B.P. Williamson, Proc. R. Soc. Lond. Ser. A 295 (1966) 300]. [23] C. Yang, B.N.J. Persson, J. Phys.: Condens. Matter 20 (2008) 215214. [24] See, e.g., Bongaerts et al. [J.H.H. Bongaerts, K. Fourtouni, J.R. Stokes, Tribol. Int. 40 (2007) 1531], where they report the Young’s modulus E = 2.4 MPa for PDMS prepared in the way as in our case, using Sylgard 184 with a base/curing agent mass ratio 10:1. Similarly, Scheibert et al. [J. Scheibert, A. Prevost, J. Frelat, P. Rey, G. Debregeas, EPL 83 (2008) 34003] obtain the Young’s modulus 2.2 ± 0.1 MPa. [25] See, e.g., http://www.rubber-stichting.info/art2nr13.html.