Dynamic Contact of Tires with Road Tracks

Transcription

1 Paper No. 49 Dynamic Contact of Tires with Road Tracks By M. Klüppel*, A. Müller, A. Le Gal Deutsches Institut für Kautschuktechnologie e. V., Eupener Straße 33, D Hannover, FRG and G. Heinrich Continental AG, Strategic Technology, P.O. 169, D Hannover, FRG Present Address: Institute of Polymer Research, Hohe Strasse 6, D Dresden, FRG Presented at a meeting of the Rubber Division, American Chemical Society San Francisco, CA April 8-30, 003 * Speaker

2 Abstract: The frictional physics behind the interaction of tire treads with road tracks during braking is discussed on the basis of two recent theoretical approaches of dynamic contact of elastomers with rough, self-affine interfaces, considered in Refs. 1-3 and 4-6, respectively. In both models, which are briefly reviewed, the frictional force is described via the dissipated energy, resulting from stochastic excitations of the sliding rubber by surface asperities on various length scales. Fractal surface descriptors obtained from stylus measurements consider the effect of surface roughness. The hysteresis response of the rubber enters through viscoelastic master curves of the complex modulus up to high frequencies in the glassy zone. For both models, stationary friction curves are estimated numerically on a broad velocity scale, in dependence of surface roughness, load, temperature and viscoelastic properties of the rubbers. The predicted friction curves are compared to experimental friction data estimated for filled model systems sliding over well characterized hard surfaces (corundum). Furthermore, numerical predictions of the true contact area and the interval of contact (frequency regime of excitation) are presented. The obtained results emphasize the role of hysteresis and adhesion under different contact conditions, e. g. in the presence of lubricants. The investigations are shown to be useful for a quantitative understanding of the traction and wear behavior of tires under service conditions. In particular, they provide a deeper insight into the contact mechanics of tires on dry and wet road tracks during ABS-braking. 1. Introduction In recent years, much progress has been obtained in modeling the friction behavior of elastomer at rough, self-affine interfaces. 1-6 This provides a fundamental physical background for understanding the dynamic contact of tires with road tracks during cornering and breaking, especially under ABS-conditions A deeper insight into the traction mechanism of tires can serve as a valuable tool in the development of tread compounds for specific applications, e. g. for dry-, wet- or ice traction. Furthermore, it may offer useful hints in understanding the various wear mechanisms of tire treads under different service conditions, since these depend strongly on the amount of sliding. 11,1 The prediction of the traction and wear mechanisms of tires requires detailed knowledge about the contact mechanics of tire treads that operate under slip conditions, considering e. g. the tread element deformation during breaking or the decomposition of tire slip in a deformation and sliding part So far, this is not fully understood on a theoretical level, though simple analytical models, evaluating the transition point between stick- and slip area of the tread pattern under the action of lateral forces, can be formulated. 13 Nevertheless, a physically realistic description of the sliding characteristics in the contact area between tire treads and road surfaces must be formulated on the basis of a model of rubber friction with generally rough road tracks, which allows for predictions of the frictional force in dependence of contact load, temperature and sliding velocity.

3 In the present paper we will present new results obtained with our recently proposed model of dynamic contact and friction of elastomers at self-affine interfaces 1-3 and compare them with the predictions of the more complex theoretical approach of rubber friction introduced by Persson et al. 4-6 In the first part of the paper, basic assumptions and evaluated parameters of both models are reviewed and the main differences are discussed. In the second part, simulation results, considering variations of the most interesting model parameters, are presented. Thereby, viscoelastic master curves of unfilled and filled rubber formulations and the profile of model surfaces as well as road tracks are applied. Finally, the results are compared to experimental friction data obtained under laboratory conditions on corundum tracks.. Theory of dynamic contact and sliding friction at self-affine surfaces In the present approach of dynamic contact and sliding friction of elastomers with rough surfaces, the roughness is described by two closely related correlation functions, the height-difference correlation function C z (λ) = <(z(x+λ)-z(x)) > and the auto-correlation function Γ z (λ) = <z(x+λ)z(x)>-<z(x)>, respectively, where the average <...> is taken over all realizations of the profile z(x) of the surface. For selfaffine surfaces, characteristic scaling assumptions can be applied for the correlation functions. In particular, C z (λ) follows a power law on small length scales: C z 6 D λ ( λ ) = ξ for λ < ξll (1) ξll and approaches a constant value ξ for λ > ξ ll. The cross-over length scales ξ ll and ξ describe the maximum roughness parallel and perpendicular to the profile and the exponent is governed by the surface fractal dimension D (<D<3). A similar behavior can be derived for the Fourier transform of the auto-correlation function, i. e. the spectral power density S(f). In the case of small spatial frequencies, S(f) is constant while for large spatial frequencies one finds the scaling form: D 7 f S ( f) = k for f > f () f min min Here, f min = ξ ll -1 and k = (3-D) ξ ξ ll. The three surface descriptors of self-affine surfaces, the fractal dimension D and the two cut-off lengths ξ and ξ ll, can be obtained from an analysis of the profile of the surface via stylus- or laser measurements. An example of the height-difference correlation of a corundum surface, obtained from ten different profile measurements, is shown in Fig. 1.

4 -1 - ξ log C z ( λ ) (mm ) m=6-d log λ (mm) ξ ll Fig. 1: Height-difference correlation function of a rough corundum track (results from ten different stylus measurements). The fractal surface descriptors D, ξ and ξ ll are indicated (D =.1; ξ = 0.10 mm and ξ ll.= 0.1 mm)..1 The Greenwood-Williamson approach of elastic contact For modeling the dynamic contact of elastomers with self-affine interfaces, we refer to the Greenwood-Williamson (GW) approach of elastic contact with spheres of constant radius R and height distribution φ(z). 14 These authors calculate the load F N as a sum of N distinct contact forces F n,i of Hertzian type in dependence of the distance d between the rubber surface and the mean height <z> of the surface formed by the spherical summits: N i = 1 = 16 1/ ( ) 3 / n i N E *( ) R z d, φ( 9 d F = F ω z) dz (3) N Here, E*(ω) is the norm of the frequency dependent complex modulus that provides an extension of the model to dynamic contact conditions. The true contact area A is found as the sum of individual contact patches πa i : N i = 1 i A = π a = π N R ( z d) φ( z) dz (4) Accordingly, the averaged true contact stress σ F N /A in this approach depends on the ratio between the integrals in Equs. (3) and (4). However, the apparent stress σ o F N /A o varies linear with the integral in Equ. (3), where A o is the apparent area of contact. If the friction force F R is assumed to increase linear with the true contact area (F R ~ A), then it follows for the friction coefficient µ: d

5 ( z d) φ( z) dz FR d µ ~ (5) FN 3 / φ( z) dz ( z d ) d This relation characterizes the load dependency of µ via the mean surface distance d. In particular, as shown by Greenwood and Williamson, if the height distribution φ(z) of the spherical summits is considered to be a Gaussian distribution, then µ is found to be almost independent of load. 14 However, in general the friction coefficient may increase or decrease with load, dependent on the height distribution realized in the particular case.. Extension of the GW-concept to self-affine surfaces For an extension of the GW-concept to self-affine surfaces we consider an approximation of the rough surface by an arrangement of spherical summits, representing the asperities of the rough track on the largest length scale. The spherical summits are assumed to have a fixed curvature radius R given by the second derivative of the modulation of the surface profile on the largest roughness scale, which yields R = ξ ll /(4π ξ ). The height distribution φ s (z) of the spherical summits can be obtained via a statistical parameter α, by evaluating the different momenta of the height distribution φ(z) of the roughness profile.,15,16 Thereby, the shape of the distribution is found to be unaffected, but the variances and the mean surface heights are different. This is illustrated in Fig.. z φ S (z) φ(z) Rubber d <z S > <z> Fig. : Schematic view of the two height distributions φ(z) of the surface profile and φ s (z) of the spherical summits on the largest roughness scale. The mean profile height <z> and the summit height <z s > as well as the distance d between <z> and the rubber surface are indicated. In the present paper we use a somewhat different approach for the evaluation of the height distribution φ s (z) of the spherical summits, representing the largest asperities of the rough surface. It is obtained by an affine transformation of the height distribution φ(z) with a fixed maximum profile height z max, which leaves the shape of the distribution as well as the fractal dimension unaltered. Then, the

6 variance of the summit distribution an affine scaling factor s 1: ~ σ s is related to the variance of the profile ~ σ via ~ ~s σ = σ / s (6) The distance d s between the mean summit height <z s > and the rubber surface is given by: 1 ds d ( < zs > < z > ) = d 1 ( z max < z > ) (7) s where d is the distance between the center line of the roughness profile <z> and the rubber surface (compare Fig. ). The affine scaling factor s is a free parameter of the model. An example of the height distribution φ(z) of a road profile and the evaluated distributions φ s (z) of the largest asperities is depicted in Fig. 3. Frequency (%) profile (s=1) s=1.1 s=1. s= Height (mm) Fig. 3: Height distribution φ(z) of the profile of an asphalt road track (solid line) and evaluated height distributions φ s (z) of the largest asperities for three different values of the affine scaling factor s. The load F N or the apparent stress σ o = F N /A o can now be evaluated similar to Equ. (3), if the summit distribution φ s (z) and the corresponding distance d s are inserted: 0.53 ξ E * ( ω ) min ds σ o F 3 / 3 π s ξ ~ II σ s (8) Here, we have used the relation ~σ = 1/ ξ and the abbreviation: t ( t) = ( z t) 3 / φ ( z)dz F3 / (9) s

7 It's important to note that the above approximation used for evaluating the apparent stress insures that adjacent asperities act largely independently on the rubber. This is fulfilled, due to the restriction on contact or pressure ranges formed by the largest asperities, which can considered to be sufficiently separated from each other. This is a necessary condition for applying the Greenwood-Williamson concept Equ. (3) that calculates the total apparent stress as a sum of individual, nonintersecting stress fields of the rubber..3 Energy condition of dynamic contact and true contact area In the following we will consider the intervals of contact and the true contact area of rubbery materials at self-affine interfaces. This is done by referring to an energy condition of elastic contact on different length scales λ. Accordingly, the sum of elastic deformation work and adhesion energy upon contact should be larger than the elastically stored energy in the stress field of the rubber: σ + ~ (10) 3 ( λ) λ h ( λ) γ λ h ( λ) > E * ( λ) h ( λ) Here, a cylindrical shape of the asperities and cavities of the surface roughness is assumed, i. e. they are considered to have a diameter λ and a height or depth h(λ) (C z (λ)) 1/. E*(λ) is the norm of the frequency dependent complex modulus at frequency ω = π v/λ, where v is the sliding velocity. γ is the change of the interfacial energy upon contact and σ(λ) is the averaged true contact stress. Equ. (10) implies two regimes of contact, which are governed by the interfacial energy (adhesion) and the applied load, respectively. In the first case, when the surface energy is dominant (σ(λ) λ << γ), contact is realized on the length scale interval [λ c, λ max ] with λ max given by the condition: γ λ ( λ ) h ( ) max E * max λmax (11) The lower cut-off length of the interval of contact λ c represents the smallest possible length scale of roughness, i. e. the atomic length scale λ c m. Using the identity h(λ) (C z (λ)) 1/ together with Equ. (1), one obtains the following scaling form for λ max : λ max 1/(5 D) γ ξll ξll E *( λmax ) ξ (1) This is an implicit equation for λ max that can be solved if the explicit form of E*(λ max ) is known (e.g. for a Zener-slider or a Rouse-model). A similar relation as Equ. (1) has been derived by Persson and Tosatti. 6 Note that the interval of contact related to the surface energy vanishes and λ max becomes zero, if the surface fractal dimension D becomes larger than.5. For such extremely rough surfaces adhesion effects are not possible. For smaller values of D, the adhesive interval of contact [λ c, λ max ] is typically located in the nanoscopic length scale regime. In the second case, when the load is dominant (σ(λ) λ >> γ), contact is realized on the length scale interval [λ min, ξ ll ] with λ min given by the condition:

8 σ ( λ ) λ E ( λ ) h ( ) (13) min min * min λmin From this equation a similar scaling form as Equ. (1) for the minimum cut-off length λ min can be derived, provided the scaling behavior of the averaged true stress σ(λ min ) F N /A c (λ min ) is known. This requires an evaluation of the true contact area A c (λ min ), which is specified in detail in Ref.. According to this approach, the true contact area consists of contact patches of size λ c with nanoscopic asperities on the smallest roughness scale and contacts with cavities on all length scales within the interval of contact [λ min, ξ ll ]. This yields for the true stress: D 1 3 (D ) σ o λmin σ ( λmin) = ( 4) ( / ~ (14) D Fo d σ ) ξ ll where F o (d/σ ~ ) is the probability that a nanoscopic asperity is in contact with the rubber (compare Fig. ): t F ( t) = φ ( z) dz (15) o The front factor of Equ. (14) describes the contact of the rubber with nanoscopic asperities, while the power law term considers the additional contact area with cavities of the rough track within the interval of contact [λ min, ξ ll ]. If Equ. (14) is inserted into Equ. (13) and the identity h(λ) (C z (λ)) 1/ is used together with Equ. (8), one finds a scaling law for the cut-off length λ min : λ ξ min II = 1 ~ D π s ξ II 3 / ξ ( D 4) E * ( λmin) Fo ( d σ ) ( D ) E * ( ξ ) F ( d ~ σ ) II 3 s s (16) with the notation E*(ξ ll ) = E*(ω min ) and ω min = πv/ξ ll. Again, this is an implicit equation for λ min, which can be solved if the functional form of E*(λ min ) is known. Finally, we get a scaling form for the true contact area A c (λ min ) F N /σ(λ min ) from a combination of Equs. (8), (14) and (16): A c ( λ ) min = A 0 ~ ( D 4) ξii Fo ( d / σ ) F3 / ( ds / σ s ) E * ( ξll ) 3 / 808π s ( D ) ξ E * ( λ ) ~ min 1 3 (17) Note that the load dependency of the quantities evaluated by Equs. (8), (14), (16) and (17) enters through the distances d or d s between the rubber surface and the center lines <z> or <z s >, respectively (compare Fig. ). In particular for a given load, F 3/ (d s / ~ σ ) can be obtained in dependence of E*(ω min ) by inverting Equ. (8). s

9 .4 Hysteresis friction The most elementary approach to sliding friction of elastomers at self-affine surfaces starts from an one-dimensional deformation of a cylindrical rubber block due to oscillating forces transmitted by the rough surface. The frictional force resulting from hysteresis effects of the rubber can be estimated via the dissipated energy: ~ E V T = 3 diss d x dt σ & ε (18) 0 0 where V is excitation volume, T is friction time and ε& is time derivative of strain. This equation can also be formulated in Fourier space via the transformations: 1 iωt σ ( t) = dωσˆ ( ω ) e π 1 ( ) ˆ iωt ε t = dω ε * ( ω ) e π (19) and 1 i ( ) ( ω ω ')t δ ω ω' = dt e π for Dirac s δ-distribution. By introducing the complex modulus: ( ω) ( ω ) I II ˆ σ E * ( ω) E ( ω ) + ie ( ω ) = (0) ˆ ε and averaging over all realizations of roughness profiles one obtains : ω ~ V max II E diss < E diss > = T dω ω ( ω ) S( ω ) π E (1) ( ) Here, according to the famous Wiener-Khinchin theorem, the correlation function in Fourier space has been expressed by the spectral power density S(ω), representing the Fourier transform of the auto-correlation of the roughness induced strain during sliding of rubber over stationary surfaces (compare Equ. ()): ω min 1 < ˆ ε ( ω ) ˆ* ε ( ω) > = S ( ω ) δ ( ω ω' ) () Since the energy dissipation per unit time (frictional power) equals the product of friction force and sliding velocity ( < Ediss > / T = FH v ), the hysteresis friction coefficient results as: µ H F F H N = 1 ( π ) ω < δ > max II dω ω σ ov E ω min ( ω) S( ω ) (3)

10 where <δ> V/A o is the layer thickness of the excitation volume. This layer thickness is assumed to increase linear with the mean penetration depth <z p > of the rubber (<δ> = b <z p >), which leaves a free parameter b. The mean penetration depth is given by: < >= ~ d z p σ F1 ~ (4) σ with t ( t) = ( z t) φ ( z)dz F1 (5) The two boundary frequencies in the integral of Equ. (3) indicate the frequency range of excitation of the rubber. They correspond to characteristic cut-off lengths of the interval of contact, e. g. ω min = πv/ξ ll and ω max = π v/λ min. In this interval of contact the rubber fills out the cavities of the rough track and excitations are generated. Indeed, a rough surface characterized by a cavity on length scale λ in contact with an elastomer solid, gives rise to oscillating forces on the rubber with a frequency ω = πv/λ. This leads to energy dissipation through hysteresis or internal friction of the material. It has been shown that Equ. (3) provides reasonable friction curves for a number of different analytical rubber models, e. g. standard linear solids (Zener slider) or Rouse like materials. 3.5 Perssons approach to sliding friction Recently, Persson 4-6 proposed a more complex model of dynamic contact and hysteresis friction of elastomers at self-affine interfaces that considers the deformation behavior of the rubber in two dimensions. Similar to the presentation in the last section, the rough surface is considered to generate fluctuating forces on a sliding rubber block within a length scale interval. In this approach, the surface power spectra is evaluated by a double Fourier transform: 1 v v S( q) = d x < h( x) h(0) > e (π ) vv iqx (6) where q=π/λ represents the norm of the wave vector at length scale λ, z = h( x v ) is the surface height relative to a reference plane <h> = 0, <...> standing for ensemble average. As in many real applications, rough surfaces like road tracks are assumed to be self-affine between a lower and upper cut-off length ξ ll and the roughness power spectra can in this case be formulated as following: ( H + 1) q S( q) = κ for q>q o (7) q o where q o = π/ξ ll represents the lowest wave number related to the size of the largest asperities of the surface, κ = H(ξ ξ ll ) /(π) 3 is the pre-factor describing the surface topography and H = 3-D is the Hurst exponent. By considering a rough

11 profile with a characteristic length ξ ll observed at a smaller length scale λ, a scaling factor ζ = ξ ll /λ can be defined that describes the magnification level under which the rubber deformation arising from the asperities is investigated. The real area of contact is of considerable importance as it determines the amount of cavities effectively filled by the rubber, e.g. the number of local contact spots involved in the friction phenomenon. It is also well-known that this value strongly decrease when analyzing the contact on smaller roughness scale. An analytical formulation of the real area of contact as a function of the scaling factor ζ is presented in the frame of this model and described by two functions: and G sin x P( ζ ) = dx exp[ x π x ζ qo π 1 3 ( ζ ) = ( ) 8 dqq S q q 0 o 0 G( ζ )] E( qv cosφ ) dφ (1 ν ) σ o (8) (9) The angle φ introduced in the second integral describes the orientation of the surface corrugations, e. g. takes in account the contribution of the asperities over the whole sliding profile. The dependency on the rubber elastic modulus E, the Poisson ratio ν and the apparent stress σ o reveal the rubber response under sliding conditions. An expression for the stationary friction coefficient of a rubber block sliding on a hard rough substrate is then derived within a length scale interval. Using the scaling factor ζ = q/q o, one obtains for the hysteresis friction coefficient: µ H ( q h ) o o 4π H ζ max H + 1 dζ ζ P( qoζ ) dφ cosφ Im (1 1 π 0 E( ζ qov cosφ ) ν ) σ o (30) where h o ξ is the horizontal cut-off length. It follows from numerical arguments that the condition σ o << E(0), with E(0) being the low frequency modulus, implies friction values independent of the normal load. Simulations performed in the frame of this theory are quantitatively strongly influenced by the free parameter ζ as this model does not provide an explicit formulation of the lower cut-off length. However, it has been shown from physical considerations that the friction phenomenon is reasonably characterized under a length scale interval of three decades. 3. Experimental The rubbers used for laboratory measurements were a commercial silicabased tread compound and solution-styrene-butadiene-rubbers with 5 wt.% styrene and a vinyl content of 5 and 50 wt.%, respectively (BUNA VSL 55-0 M; BUNA VSL505-0 HM). Unfilled and filled (50 phr carbon black N0) testspecimens ( mm-slides) were used. The unfilled systems were mixed on a roller mill (mixing time: 5 min.) whereas the filled systems were first mixed in an internal mixer (3:30 6 min) and afterwards on the roller mill (5 min). The samples were cross-

12 linked with phr sulphur and 1.5 phr CBS. The test-specimens were cured at 160 C for 15 min. (filled) and 30 min. (unfilled) respectively. To obtain the viscoelastic master curves, a Rheometrics Dynamic Analyzer (RDA) was used to measure the temperature- and frequency-dependent modulus (temperature-frequency-sweeps). According to the time-temperature-superpositionprinciple, the single modulus-curves at the distinct temperatures were shifted on the frequency-axis in such a way, that the tan(δ)-curves are smooth (G and G are shifted by the same amount). 4. Results and discussion The ability of the presented models of dynamic contact and sliding friction of elastomers at self-affine interfaces for predicting tire traction on rough road tracks is limited by various factors. In particular, stationary sliding conditions are not always realized and the tread deformation during cornering or braking plays an important role. Nevertheless, theoretical predictions can serve as a guideline for a better understanding of tire traction. A test of the models by comparing them to experimental friction data is restricted on the one side by the open model parameters and on the other side by reliable friction data. For obtaining simulation results we refer to viscoelastic master curves that are obtained by applying the temperature-time superposition principle. This yields viscoelastic data on a broad frequency scale. An example of the real and imaginary part of the complex modulus for a filled (50phr carbon black N0) S-SBR sample is depicted in Fig log G' (Pa) log G'' (Pa) G' G'' log frequency (Hz) Fig. 4: Master curves of the complex shear modulus vs. frequency for a filled (50phr N0) S-SBR sample (5% Vinyl) at reference temperature T ref = 0 C (0.% strain).

13 Figures. 5a and 5b show simulated friction curves according to the model of Klüppel/Heinrich and Persson, respectively, for two S-SBRs on a corundum track. The curves are qualitatively similar, however with the used parameters the calculated friction coefficients differ by one order of magnitude. A better convergence is reached when using a much smaller parameter ζ within the Persson approach (corresponding there a larger lower cut-off length ~ µm) or a much larger ratio of penetrated rubber volume to dynamically excited volume within the KH approach. Figures 6a and 6b demonstrate that the load dependency of the simulated friction curves (Equ. (3)) can be quite different, dependent on the roughness characteristic of the road track. On a blunt asphalt track the friction coefficient is found to increase with load, while on the sharp track it decreases with increasing load. A similar behavior has been reported by Grosch, 17 who measured the friction behavior of different tread compounds on sharp and blunt corundum surfaces and on blunt glass surfaces using the so-called Grosch-Wheel-Tester. The similar trend like in Fig. 6b could also be found with a Linear Friction Tester (LFT) 18, where rubber samples slide on an asphalt surface within a velocity range of v= mm/s. This is in contrast to Perssons model (Equ. (30)) that predicts a vanishing load dependency for typical values corresponding to tire applications (0.1-1 MPa). 5,6 Figures 7a and 7b show that the two models are quite different sensitive to slight variations in the lower cut-off length. The friction coefficient according to Equ. (3) is nearly unchanged for variations in the lower cut-off length if it is at least two decades smaller then the upper cut-off length. Figure 8 demonstrates the reasonable results for the evaluated lower cut-off length (Equ. (16)) in dependence on temperature and sliding velocity (the higher the temperature and the lower the velocity, the smaller structures can contribute to friction). The opposite effect at high velocities is due to the fact that the cut-off length is strongly influenced by the ratio of the moduli at highest and lowest excitation frequencies; and in the high-velocity range even the lower cut-off frequency is located within the rubber-glass transition region. Figures 9a and 9b show the computed areas of real contact as fraction of the nominal area of contact acc. to Equ. (17) and Equ (8), respectively. In Fig. 9a this fraction is found to decrease with decreasing temperature and increasing velocity over the whole range as it could be expected. According to Equ.8 the area of real contact has more complex dependencies on temperature and velocity, and leads partially to an inversion.

14 S-SBR (5% Vinyl) S-SBR (50% Vinyl) 1.0 µ H log v (m/s) Fig. 5a:Evaluated friction curves (Equ. (3)) for two unfilled S-SBR systems with 5 and 50 wt.% vinyl groups. The simulations are obtained for a corundum track (D =.1; ξ = 0.10 mm and ξ ll.= 0.1 mm) at reference temperature T ref = 0 C, σ o = 0 kpa, s = 1.1 and <δ>/<z p > = S-SBR (50% Vinyl) S-SBR (5% Vinyl) µ H log v (m/s) Fig. 5b:Evaluated friction curves with Perssons model Equ. (30) for the same systems and conditions as in Fig. 5a and with ζ = 50.

15 MPa 0.3 MPa 0.6 MPa µ H log v (m/s) Fig. 6a: Effect of load variation(σ o as indicated) on friction curves (Equ. (3)) for a silica based tread compound on a blunt asphalt track (D =.11; ξ = 1.1 mm and ξ ll.= 3.59 mm). The simulations are obtained for a reference temperature T ref = 0 C, s = 1.1 and <δ>/<z p > = MPa 0.3 MPa 0.6 MPa 0.5 µ H log v (m/s) 3 Fig. 6a: Effect of load variation(σ o as indicated) on friction curves (Equ. (3)) for a silica based tread compound on a high-my, sharp asphalt track (D =.14; ξ = 0.81 mm and ξ ll.= 3.87 mm). The simulations are obtained for a reference temperature T ref = 0 C, s = 1.1 and <δ>/<z p > = 50.

16 λ min =1.3 µm λ min =3.9 µm λ min =7.8 µm λ min =1.9 µm 0.0 µ log v (m/s) Fig. 7a: Effect of lower cut-off length on friction curves (Equ. (3)) for a silica based tread compound on a high-my, sharp asphalt track (D =.14; ξ = 0.81 mm and ξ ll.= 3.87 mm). The simulations are obtained for a reference temperature T ref = 0 C, σ o = 0.3 MPa, s = 1.1 and <δ>/<z p > = λ min = 1,9 µm λ min = 7,8 µm λ min = 3,9 µm λ min = 1,3 µm µ H log v (m/s) Fig. 7b:Effect of lower cut-off length in Perssons model (Equ. (30)) for a silica based tread compound on a high-my, sharp asphalt track (D =.14; ξ = 0.81 mm and ξ ll.= 3.87 mm). The simulations are obtained for a reference temperature T ref = 0 C, σ o = 0.3 MPa

17 -6 log (m) λmin log v (m/s) T=0 C T=0 C T=40 C T=60 C Fig. 8: Effect of temperature on the evaluated lower cut-off length λ min vs. velocity (Equ. (16)) for a silica based tread compound on a high-my, sharp asphalt track (D =.14; ξ = 0.81 mm and ξ ll.= 3.87 mm). The simulations are obtained for a reference temperature T ref = 0 C, σ o = 0.3 MPa and s = 1.1 (closed symbols) or s = 1. (open symbols).

18 - log A / A c 0-3 T=0 C T=0 C T=40 C T=60 C log v (m/s) Fig. 9a: Effect of temperature on the true contact area A c /A o vs. velocity (Equ. (17) for a silica based tread compound on a high-my, sharp asphalt track (D =.14; ξ = 0.81 mm; ξ ll = 3.87 mm). Simulations are obtained for σ o = 0.3 MPa and s = 1.1 (closed symbols) or s = 1. (open symbols) T=0 C T=0 C T=40 C T=60 C log A(ζ)/A o log v (m/s) Fig. 9b:Effect of temperature on the true contact area P(ζ) = A(ζ)/A o vs. velocity according to Perssons theory (Equ. (8) for a silica based tread compound on a high-my, sharp asphalt track (D =.14; ξ = 0.81 mm; ξ ll = 3.87 mm). The simulations are obtained for σ o = 0.3 MPa and ζ = 500 (closed symbols) or ζ = 100 (open symbols).

19 VSL 55-0 mit 0 phr N SC 80 SC 180 SC 80 Vaseline 1.0 SC180 Vaseline Friction coefficient µ <δ> = 3.3 <z p > v / ms -1 Fig. 10: Comparison of evaluated friction curves (lines) according to Equ. (3) with experimental friction data (symbols) for a filled S-SBR composite on two lubricated silicon carbide tracks. 7,8 5. Conclusions and Outlook In this paper two models of contact mechanics and friction of elastomers are reviewed and numerical results are presented and compared. Both models in their present form do not consider the frictional heating during rubber sliding. Therefore, a comparison with experimentally data makes sense only at slow sliding velocities and for wet surfaces. We should keep clearly in mind that the incorporation of these conditions leads to further complications within the theory. However, it is advantageous that in the wet case the friction performance actually varies with temperature in a simpler manner. (Note: Temperature is probably one of the most important basic condition. The friction coefficient between the road surface and rubber varies with the season. As a consequence, for example, the maximum lateral acceleration, when a passenger car is turning in a circle of a certain radius on a skid pad, falls as the water temperature increases. 19 ) In most studied cases both approaches to hysteresis rubber friction lead to qualitative similar results. However, with the used apparent reasonable parameters the calculated friction coefficients as a function of velocity differ by one order of magnitude. A better convergence is reached when using a much smaller parameter ζ within the Persson approach (corresponding there to a larger lower cut-off length ~ µm) or a much larger ratio of penetrated rubber volume to dynamically excited rubber volume within the KH approach.

20 The simulations support to establish a ranking between different tread compounds within the corresponding velocity range of operating tires during ABSbraking whereby the absolute value of the friction is not the aim. We finally note that a reliable prediction of friction and wet skid performance of tires can be obtained through an appropriate physical friction model in combination with an appropriate mechanical analysis and with phenomenological observations rather than from a purely physical consideration Acknowledgements The authors like to thank B. Persson (Forschungszentrum Jülich) for useful discussions and the Bayer AG and the Continental AG for support and permission to publish this paper.

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