Tire road frictional interaction

Physical approach

B.G.B. Hunnekens

DCT 2008.088

TU/e Bachelor Final ProjectJune, 2008

Supervised by:dr.ir. I. Lopez Arteaga (TU/e)ir. R. van der Steen (TU/e)Eindhoven University of TechnologyDepartment of Mechanical EngineeringDynamics and Control Group

Abstract

The friction of rubber plays an important role in practical situations. Friction between the road andthe tire is such a practical application. Models, developed by B.N.J. Persson, predict the kinematicfriction coefficient as a function of the slip velocity of the rubber. The basic theory of the models includeviscoelastic energy dissipation on many different length scales.

The first model that is implemented does not include the local heating of the rubber. Results areshown for different power spectral densities of road surfaces and for different ranges of length scales. Theobtained µ(v) curves are validated with the results published by Persson.

The viscoelastic modulus of rubber strongly depends on temperature. Therefore, the second modeladdresses the local heating of the rubber. An additional heat diffusion equation is solved to account forthis raise in temperature. The results obtained are in qualitative agreement with the results presentedby Persson. An increase in temperature softens the rubber and, in practical tire-road situations, leadsto less energy dissipation of the rubber. Therefore the friction coefficient will drop compared to thetemperature-independent case. Quantitative validation of the temperature-dependent model was notpossible: Persson uses measurement data in his calculations that is not published.

The power spectral density of the road surface plays a significant role in the calculation of the frictioncoefficient. Therefore, measurements on a real road surface (surface 1) are performed with an opticalimaging profiler. The data is used to calculate the psd and the resulting friction coefficient. This is alsodone for a fitted psd and it is concluded that a fitted psd will lead to similar results. Therefore a fittedspectrum can be used to calculate the µ(v) curve. Additionally, road surface data was available fromanother project (surface 2). The spectra and the resulting friction coefficients are compared. Althoughthe two surfaces are quite different, the two µ(v) curves are comparable.

It is widely known that rain decreases the friction between the tire and the road. The measured roadprofile is ’flutted’ numerically which leads to a new ’corrected’ surface. This leads to a different psd ofthe road surface and results in a different µ(v) curve. As observed in practice, the friction coefficientdecreases a significant amount when the road is wet.

Samenvatting

De wrijving van rubber speelt een belangrijke rol in praktische situaties. De wrijving tussen het wegdeken de band is zo’n praktische toepassing. Modellen, ontwikkeld door B.N.J. Persson, voorspellen dewrijvingscoefficient als functie van de slip snelheid van het rubber. De basis van de modellen bestaat uitvisco-elastische energie dissipatie op vele verschillende lengteschalen.

Het eerste model dat is geımplementeerd bevat niet de locale opwarming van het rubber. Resul-taten worden gegeven voor verschillende power spectral densities van wegdekken en voor verschillenderanges van lengteschalen. De berekende µ(v) curves zijn gevalideerd met de resultaten die Persson heeftgepubliceerd.

De visco-elastische modulus van rubber hangt sterk af van de temperatuur. Daarom behandeldhet tweede model de locale opwarming van het rubber. Een extra warmte diffusie vergelijking wordtopgelost om de verhoging van de temperatuur in rekening te brengen. De resultaten komen kwalitatiefovereen met de resultaten van Persson. Een verhoging van de temperatuur versoepelt het rubber en,in praktische band-weg situaties, leidt tot minder energie dissipatie van het rubber. Om deze reden zalde wrijvingscoefficient afnemen in vergelijking met het temperatuur-onafhankelijke geval. Kwantitatievevalidatie van het temperatuur-afhankelijke model was niet mogelijk: Persson gebruikt gemeten data inzijn berekeningen die hij niet heeft gepubliceerd.

De power spectral density van het wegdek speelt een belangrijke rol in de berekening van de wrijv-ingscoefficient. Om deze reden zijn metingen verricht op een echt wegdek (wegdek 1) met een ’opticalimaging profiler’. De data is gebruikt om de psd en de resulterende wrijvingscoefficient te berekenen. Ditis ook gedaan voor een gefit psd en er kan geconcludeerd worden dat een gefit psd leidt gelijke resultaten.Daarom kan een gefit spectrum gebruikt worden voor het berekenen van de µ(v) curve. Ook was extrameetdata van een wegdek beschikbaar van een ander project (wegdek 2). De spectra en de resulterendewrijvingscoefficienten zijn vergeleken. Hoewel de twee oppervlakken van elkaar verschillen, zijn de µ(v)curves erg vergelijkbaar.

Het is bekend dat regen de wrijving verminderd tussen band en wegdek. Het gemeten wegdek isnumeriek ’vol laten lopen’ met water wat leidt tot een nieuw ’gecorrigeerd’ wegdek. Dit leidt tot eenander psd van het wegdek en resulteert in een andere µ(v) curve. Net als in de praktijk neemt dewrijvingscoefficient significant af als het wegdek nat is.

Acknowledgements

I would like to express my gratitude to my supervisor, dr. ir. Ines Lopez Arteaga, for the supportthroughout the project. I value the weekly meetings with room for questions, discussions and tips. Iam also grateful for the support of ir. Rene van der Steen working on the CCAR project about FEMTire modelling. He helped me a lot during the project and was always available to answer my questions.Furthermore I would like to thank Marc van Maris for measuring the road surface in such a short amountof time and providing me with excellent data.

Contents

1 Introduction 1

2 Rubber friction without temperature dependence 32.1 Background of the friction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The friction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Implementation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Friction coefficient for different slip velocities . . . . . . . . . . . . . . . . . . . . . 72.3.3 Friction coefficient for different road profiles . . . . . . . . . . . . . . . . . . . . . . 7

3 Rubber friction with temperature dependence 113.1 Background of the friction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 The friction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3.1 Implementation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 Friction coefficient and temperature for different slip velocities . . . . . . . . . . . 13

4 Results for real road surfaces 174.1 The power spectral density of a real asphalt road surface . . . . . . . . . . . . . . . . . . . 174.2 Friction coefficient for a real asphalt road surface . . . . . . . . . . . . . . . . . . . . . . . 194.3 Comparison of two real road surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Influence of rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Conclusion and recommendations 275.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A Self-affine fractal surfaces 29

B The rheological model 31

C Used parameter values 33

Symbols

A area, m2

a lattice spacing, ma1 ratio moduli rheological modelaT shift factor WLF-equationC power spectral density road, m4

Cv heat capacity, J/kg·KD heat diffusivityE viscoelastic modulus, PaF0 nominal force, NFf shear force, NH Hurst exponenth height, mL length, mP apparent area of contactq wave vector, 1/mq0 lower cut-offq1 upper cut-offQ heat production per unit volume and unit time, W/m3

Rasp radius asperity region, mT temperature, K, ◦Ct time, st0 half time a asperity region is in contact with the roadT0 background temperature, K, ◦CTg glass transition temperature, K, ◦CTq temperature on length scale q, K, ◦Cv slip velocity, m/sx, y, z rectangular coordinates, m

Greek Lettersλ length scale, mλth thermal conductivity, W/m·Kµ kinematic friction coefficientν Poisson ratioω frequency of oscillation, rad/sφ angle between sliding direction and wave vector q, radρ density, kg/m3

σ0 nominal stress, Paσf shear stress, Paζ magnificationζmax maximum magnification

Chapter 1

Introduction

Rubber friction plays an important role in many practical situations. One practical application is forinstance the rubber friction between a tire and the road. It is important to know how the kinematicfriction coefficient depends on the slip velocity of the tire, in order to make good models. The friction ofrubber on a hard rough substrate differs form most other solids due to the very low elastic modulus andthe high internal friction exhibited by the rubber over a wide region of frequencies. The rubber exhibitsviscoelastic behavior and will therefore dissipate energy. The energy dissipation can be related to thefriction coefficient. When slipping, the temperature of the tire increases. The temperature increase hasa significant effect on the rubber friction. From a physical basis, models have been developed by Perssonthat predict the friction coefficient as a function of the slip velocity [4], [5].

The goal of this project can be split into three parts:

• implement a temperature-independent friction model that calculates the kinematic friction coeffi-cient. Investigate the effect of different parameters on the friction coefficient.

• implement a temperature-dependent friction model and compare the results with the temperature-independent model. Investigate the effect of heating of the rubber.

• investigate the effect of a real measured road surface on the friction coefficient.

In the first chapter the temperature-independent model will be discussed. In the subsequent chapterthe temperature-dependent model will be addressed. A comparison between the two models will be madein this chapter. The effect of a real, measured road surface on the model outcome will be discussed inchapter three. A comparison of two real road surfaces will be made. Also, the influence of rain on thefriction coefficient will be addressed in this part.

1

2

Chapter 2

Rubber friction withouttemperature dependence

2.1 Background of the friction model

Rubber friction has got two fundamental contributions: hysteric and adhesion. The hysteric contributionresults from internal friction of the rubber. The road surface exerts oscillating forces on the rubber. Dueto internal damping of the rubber this leads to viscoelastic energy dissipation. The adhesion componentof the rubber friction is only important for very clean rubber surfaces. The adhesion force is a resultfrom the van der Waals interaction between the interfaces. In [3] it is shown that adhesion on a roughsurface becomes important for velocities in order of magnitude v < 10−8 m/s. This is due to the factthat the actual contact area is only ∼ 1% of the nominal contact area. The internal or hysteric friction isfor that reason dominant in all practical cases. The adhesion component is only relevant for very cleansurfaces and is therefore not included in the here presented model.

The theory that Persson developed [4] starts with the energy dissipation ∆E of the rubber due tothe oscillating forces exerted by the road surface. This energy dissipation can be related to the frictionalshear stress σf exerted on the rubber. The kinematic friction coefficient µ can be related to the frictionalshear stress by dividing it with the nominal stress σ0. This is illustrated in figure 2.1.

Figure 2.1: Definition of the kinematic friction coefficient as the ratio between shear stress and nominalstress

3

The oscillating forces occur on many different length scales. This is due to the many irregularitiespresent on a road surface, which is illustrated in figure 2.2. The length scales that are present in theroad are defined as λ and the corresponding wave vector is:

q ≡ 2πλ

On a road surface sample of length L the longest possible length scale is L. The smallest possiblelength scale could be determined by road contamination (or a thin layer of thermally degraded rubber)and is defined as λ1.

Figure 2.2: Simplified road surface with length scales λ. The longest length scale possible is L

The length scales that are considered in the analysis are λ1 < λ < L. In terms of wave vectors thiscan be defined as:

2πL

= q0 < q < q1 =2πλ1,

where q0 is called the lower distance cutoff and q1 the upper distance cutoff.A magnification when looking at a length scale λ is defined as ζ ≡ q

q0. The maximum magnification

corresponds to a length scale λ1 and is defined as ζ = ζmax = q1q0

.To describe the wavelengths that are present in a road surface, the power spectral density C(q) can

be used. A special kind of surface is a self-affine fractal surface. From experiments it is known that atypical road surface can be approximated as being self-affine fractal. The power spectral density canthen be described by:

C(q) ∼ q−2(H+1),

with H the Hurst exponent. More information about self-affine fractal surfaces can be found in Ap-pendix A.

The road surface exerts oscillating forces on the rubber with certain frequencies. This frequency canbe related to the sliding velocity v and wavelength λ:

ω =v

λ

4

The viscoelastic modulus of the rubber material depends on this frequency. The energy dissipationin the material is maximal if the loss tangent is maximal. This frequency is located in the transitionregion between the low frequency rubbery region and the high frequency glassy region. A broader rangeof length scales will lead to more energy dissipation and hence, this will broaden the µ(v) curve and willincrease the peak maximum.

When rubber is squeezed onto a hard solid surface of different length scales it will fill out part of thecavities. Such a situation is sketched in figure 2.3. It is clear that the rubber pressure is high enough tomake the rubber ’follow’ the large length scale roughness of the hard solid. The small cavities howeverare not completely filled. Especially at the bottom of a cavity it is hard for the rubber to follow the solidsurface. This is due to the smaller pressure at the bottom of the cavities. The small-scale roughnessdoes contribute to the friction force but only at the top of the big asperities. This is taken into accountin this friction model. Due to the above described phenomenon the apparent area of contact is smallerthan the nominal area of contact. This apparent area of contact at magnification ζ (length scale λ) isdefined as:

P (ζ) =A(λA(L)

Figure 2.3: Only part of the cavities of the hard solid are filled with rubber. This is due to the smallerpressure at the bottom of the cavities

2.2 The friction model

This paragraph contains the equations that make up the temperature-independent model. The frictioncoefficient depends on many different parameters, the most important are µ = µ(v, P (q), C(q), E(ω)).The resulting equations are the following:

µ =12

∫ q1

q0

q3C(q)P (q)dq∫ 2π

0

cos(φ)Im(E(qv cos(φ))(1− ν2)σ0

)dφ (2.1)

P (q) = erf(

12√G

)(2.2)

G(q) =18

∫ q

q0

q3C(q)dq∫ 2π

0

∣∣∣∣E(qv cos(φ))(1− ν2)σ0

∣∣∣∣ dφ (2.3)

where ν is the Poisson ratio. The frequency ω has been written in terms of the slip velocity:

ω =v

λ= qv cos(φ),

where the factor cos(φ) accounts for the fact that the sliding direction may differ from the direction ofthe wave vector q.

5

2.3 Numerical results

2.3.1 Implementation of the model

Equations 2.1-2.3 for the friction model and equation A.1 for the road surface are implemented in Matlab.The rheological model used for the viscoelastic modulus can be found in Appendix B. The calculationscan be performed in many different ways. Especially the integrals can be determined with differentmethods. It is possible to calculate the integrals numerically with a trapezoid integration method.Another possibility is calculating the integrals explicitly as an analytical expression. This could lead tomore accurate results and less calculation time. The calculation time is especially important becausethe model that includes temperature dependence (chapter 3) needs to be solved iteratively. This leadsto much longer calculation times so it is important that this ’simple’ model is calculated as efficiently aspossible.

The friction model is implemented in three different ways. The first method uses numerical integration(trapezoid method) to evaluate the integrals. The second method performs the integration by analyticalintegration. After this, the corresponding values are substituted in the symbolic expression for theintegral (using subs(...) command in Matlab). At a certain point it is not possible to calculate everyintegral analytically because of the complexity of the expression. From that point on, one has to relyon numerical integration. The advantage of this second method is that the integrals are calculatedmore accurately. The disadvantage of this method lies in the roots of the programming language. Thesubstitution is performed not by Matlab itself but by Maple. Calling Maple functions requires a lot oftime and is therefore inefficient. The last method that is implemented does not differ very much from thesecond method: the substitution is no longer performed by Maple. Special function files are used thatcontain the analytical expressions of the integrals. When the functions are called they only calculate thenumerical value of the integral. This way calling Maple is avoided.

The implemented models calculate the kinematic friction coefficient for a certain range of slip ve-locities. As mentioned above, some integrals are necessarily evaluated numerically with the trapezoidmethod. A certain number of points has to be chosen to calculate the integrals. Table 2.1. shows thetime required to calculate the friction coefficient for the different methods. The number of velocitiessolved for is 50 and the number of points used for numerical integration is 100. It is clear that the thirdmethod reduces calculation time significantly and that this method is therefore used.

Table 2.1: Calculation time for different methods of integration, 100 integration points, 50 velocitiesMethod Calculation time [s]1st method: all numerical integration 722nd method: analytical + numerical integration with substitution 1083rd method: analytical + numerical integration without substitution 1.6

The calculation of the friction coefficient with method 3 can be divided into the following steps:

• for one certain velocity the following calculations are performed

• calculation of q3C(q)∫ 2π

0

∣∣∣E(qv cos(φ))(1−ν2)σ0

∣∣∣ dφ analytically

• calculation of G(q) = 18

∫ qqLq3C(q)dq

∫ 2π

0

∣∣∣E(qv cos(φ))(1−ν2)σ0

∣∣∣ dφ by means of the trapezoid integrationmethod

• calculation of P (q) = erf(

12√G

)• calculation of

∫ 2π

0cos(φ)Im

(E(qv cos(φ))(1−ν2)σ0

)dφ analytically

• calculation of µ = 12

∫ q1q0q3C(q)P (q)dq

∫ 2π

0cos(φ)Im

(E(qv cos(φ))(1−ν2)σ0

)dφ by means of the trapezoid

integration method

• these steps are repeated for every velocity v

6

2.3.2 Friction coefficient for different slip velocities

With the implemented model the kinematic friction coefficient can be calculated for different ranges oflength scales considered. The range of length scales is determined by the upper and lower distance cutoff:

q0 < q < q1 = ζq0

The friction coefficient curve is calculated for different values of the maximum magnification ζmax.These results are shown in figure 2.4. As discussed in paragraph 2.1, a broader range of length scales(larger ζmax) leads to a broadening of the curve and a raise of the peak maximum. The values for theparameters used in the calculations are listed in Appendix C. These results are validated with the resultsobtained by Persson in [4].

10−4

10−2

100

102

104

0

1

2

3

4

5

6

7

8

9

slip velocity [m/s]

kine

mat

ic fr

ictio

n co

effic

ient

Kinematic friction model without flash temperature

ζmax

= 10

ζmax

= 100

ζmax

= 1000

Figure 2.4: Kinematic friction coefficient for different maximum magnifications

In paragraph 2.1 the influence of the apparent contact area has been discussed. When shorter lengthscales are considered the apparent area of contact decreases. In other words, the value of P (q), whichdescribes the ratio between apparent area of contact and nominal contact area, decreases with increasingmagnification. This leads to a strong reduction of the contribution to the friction force from the small-scale roughness of the road. The reduction in apparent area of contact is visualized in figure 2.5 withP (ζ) for different slip velocities. These curves are qualitative in agreement with the curves publishedby Persson in [4]. However, the heights of the curves differ with those of Persson. Probably, the curvespublished by Persson have been produced with other parameter values.

2.3.3 Friction coefficient for different road profiles

The characterization of a road profile is discussed in Appendix A. The power spectral density C(q) canbe used to describe the road surface. For self-affine fractal surfaces the Hurst exponent H determinesthe slope of C(q), it tells us the presence of different length scales found in the road surface. A low Hurstexponent means that all length scales (in the considered domain) are all present in a same amount. Thiswill lead to more energy dissipation and hence to a higher friction coefficient. The kinematic frictioncoefficient µ has been calculated for different values of the Hurst exponent. The corresponding power

7

0 20 40 60 80 10010

−5

10−4

10−3

10−2

10−1

100

Apparent contact area

ζ [−]

P [−

]

v = 0.00005 m/sv = 0.013 m/sv = 0.316 m/sv = 7.85 m/sv = 187 m/s

Figure 2.5: Ratio of apparent area of contact to nominal contact area for different slip velocities

spectral densities are shown in figure 2.6. The results are visualized in figure 2.7, where ζmax = 100 wasused for the maximum magnification. The other parameter values are those tabulated in Appendix C.These results are again identical to the results published in [4].

This model does not include the effect of the raise of temperature of the rubber due to the energydissipation of the rubber. In reality the rubber will heat up and the characteristics of the rubber willchange. This effect of local warming of the rubber, referred to as the flash temperature, will be consideredin the next chapter.

8

103

104

105

106

10−22

10−21

10−20

10−19

10−18

10−17

10−16

10−15

10−14

Wavevector q [1/m]

C [m

4 ]

Power Spectral Density of asphalt road surfaces

H = 0.2H = 0.4H = 0.6H = 0.8

Figure 2.6: Power spectral densities of road surfaces with different Hurst exponents

10−4

10−2

100

102

104

0

5

10

15

20

25

slip velocity v [m/s]

kine

mat

ic fr

ictio

n co

effic

ient

µ

Kinematic friction model without flash temperature

H = 0.2H = 0.4H = 0.6H = 0.8

Figure 2.7: Kinematic friction coefficient for different Hurst exponents

9

10

Chapter 3

Rubber friction with temperaturedependence

3.1 Background of the friction model

In this chapter the friction model with temperature dependence will be discussed. In practical situationsthe friction of the tire with the road will raise the temperature of the tire. Especially for rubbers, thishas an effect on the energy dissipation due to internal friction, because the viscoelastic modulus stronglydepends on the temperature and therefore the modulus can be written as:

E = E(ω, T )

Just below the surface of the rubber tire (small length scale λ, i.e. a large wave vector q) the tem-perature will be higher than deep inside the tire. Therefore, the local temperature (flash temperature)has to be calculated for different length scales. Persson developed a theory in [5] that provides the basisof the model discussed in this chapter.

The basis for the theory starts from the heat diffusion equation, based on stationary sliding:

−D∇2T =Q

ρCv,

with D = λth/ρCv the heat diffusivity, ρ the mass density, λth the thermal conductivity and Cv theheat capacity of the rubber. Q is the heat production per unit volume and unit time due to the internalfriction of the rubber.

3.2 The friction model

This paragraph contains the equations that make up the temperature dependent model. In [5] Perssonshows that the heat diffusion equation can be reduced to the following equations that determine thetemperature for a velocity v and length scale λ = 2π/q:

Tq = T0 +∫ ∞

0

g(q, q′)f(q′)dq′ (3.1)

g(q, q′) =1π

∫ ∞0

1Dk2

(1− e−Dk

2t0) 4q′

k2 + 4q′24q2

k2 + 4q2(3.2)

f(q′) =vq′4

ρCvC(q′)

P (q′)P (qm)

∫ q1

q0

cos(φ)Im(E(qv cos(φ)), Tq

(1− ν2)

)dφ (3.3)

where T0 is the background temperature and t0 is roughly half the time a rubber patch is in contact witha so called macro-asperity contact region. When a rubber is squeezed against a hard surface it looks as

11

if complete contact occurs at many regions. These regions are called marco-asperity contact regions andthese can be observed at a magnification ζm = qm/q0. Typically this magnification appears to be in therange 2 < ζm < 5. If the magnification is increased the macro-asperity contact regions break up intomuch smaller and closely separated micro-asperity contact regions. The separation of the macro-regionsis relatively large and thus the interactions between these regions can be neglected.

The half-time a rubber patch is in contact with a macro-asperity contact area can be approximatedby:

t0 =Raspv

,

with:

Rasp =a+ bP (qm)c

qm,

where a, b and c are constants.

The friction coefficient can be determined in a way similar to the temperature-independent model.However, the visco-elastic modulus now depends on the flash temperature of the rubber:

µ =12

∫ q1

q0

q3C(q)P (q)dq∫ 2π

0

cos(φ)Im(E(qv cos(φ), Tq)

(1− ν2)σ0

)dφ (3.4)

P (q) = erf(

12√G

)(3.5)

G(q) =18

∫ q

q0

q3C(q)dq∫ 2π

0

∣∣∣∣E(qv cos(φ), Tq)(1− ν2)σ0

∣∣∣∣ dφ (3.6)

The relation between the visco-elastic modulus and the temperature can be approximated by thetemperature-frequency shift described by the Williams-Landel-Ferry equation (WLF-equation) [2]:

E(ω, Tq) = E(ωaT , T0) (3.7)

where the shift factor aT is given by:

log10(aT ) ≈ −8.86Tq − Tg − 50

51.5 + Tq − Tg,

here Tg is the glass transition temperature of the rubber. The WLF-equation states that at a certaintemperature Tq and frequency ω the modulus is equal to the modulus considered at a temperature T0

and frequency ωaT .

12

3.3 Numerical results

3.3.1 Implementation of the model

Equations 3.4-3.6 determine the temperature of the tire rubber for a certain velocity v and length scaleλ. It is not possible to obtain an explicit expression for the flash temperature due to the complexityof the equations. Therefore the temperature and corresponding shift factor aT has to be calculated inan iterative way. The iterative method that is applied is the bisection method. This method uses apredefined range that has to contain the answer Tq. The range is divided into two sections. The sectionthat doesn’t contain the answer is discarded and a new range is defined. This method is repeated untilthe solution is below a certain specified tolerance. The calculation of the temperature and the shift factorcan be divided into the following steps:

• for one velocity v calculate P (ζm) and the half-contact time t0

• loop over different wave vectors q and calculate g(q, q′) = 1π

∫∞0

1Dk2

(1− e−Dk2t0

)4q′

k2+4q′24q2

k2+4q2 dk

by means of the trapezoid integration method

• start the iteration loop until error < tolerance

• calculate the shift factor aT with log10(aT ) ≈ −8.86 Tq−Tg−5051.5+Tq−Tg

for the last calculated temperatureTq

• calculate G(q) analytically and calculate P (q′)

• calculate f(q′) = vq′4

ρCvC(q′) P (q′)

P (qm)

∫ q1q0

cos(φ)Im(E(qv cos(φ)),Tq

(1−ν2)

)dφ analytically

• calculate the new temperature Tq = T0 +∫∞0g(q, q′)f(q′)dq′ by means of the trapezoid integration

method

• calculate the error: error =∣∣∣Tq,new

Tq,old− 1∣∣∣

• apply the bisection method and discard one half of the range of temperatures

• end the iteration loop if error < tolerance and store the temperature Tq and shift factor aT

• perform this iteration loop for all wave vectors q and velocity v

The nature of the calculation of the friction coefficient doesn’t differ very much from the temperature-independent model. The difference lies in the fact that the shift-factor aT is included in the analysis.For each velocity a range of shift factors aT is calculated for a range of wave vectors q. These are usedto calculate the viscoelastic modulus at the calculated temperatures through the use of the temperature-frequency shift. The subsequent steps are identical to those discussed in paragraph 2.3.1.

3.3.2 Friction coefficient and temperature for different slip velocities

With the method described above, the kinematic friction coefficient can be calculated for a range ofvelocities. The temperature at magnification ζ is also calculated. The same rheological model is used inthis temperature-dependent model (see appendix B). The parameter values can be found in appendixC. These are different from the ones used in the temperature-independent model because Persson usesdifferent values in [5].

The results for the friction coefficient are depicted in figure 3.1. For comparison the results withoutflash temperature are shown too (constant background temperature). The friction coefficient lowerswhen the flash temperature is included in the analysis. This can be easily understood: when the rubberwarms up, the visco-elastic modulus shifts to higher frequencies (see WLF-equation 3.7). This leads toa more elastic, less viscous, rubber. The frequency where the loss tangent reaches its maximum is inpractical applications in tire friction always higher then the excitation frequency. Therefore, a shift tohigher frequencies will lead to less energy dissipation. This is visualized in figure 3.2.

13

The effect of a decrease of the friction coefficient is also found by Persson. However the quantitativeresults presented here are not in good agreement with the results published by Persson. He uses mea-surement data for the visco-elastic modulus and for the shift factor aT . This data is not published inthe paper and could therefore not be used to validate the model. A good description of the tire rubberwould require a broad range of relaxation times. However, the rheological model used here only uses onerelaxation time. This could explain the difference in the results obtained.

10−8

10−6

10−4

10−2

100

102

0

1

2

3

4

5

6

7

8

slip velocity [m/s]

kine

mat

ic fr

ictio

n co

effic

ient

Kinematic friction model comparison

without flash temperaturewith flash temperature

Figure 3.1: Kinematic friction coefficient for models with and without flash temperature

100

105

0

0.2

0.4

0.6

0.8

1

ω [s−1]

loss

tang

ent

100

105

0

0.2

0.4

0.6

0.8

1

ω [s−1]

loss

tang

ent

Figure 3.2: A shift to higher frequencies due to a higher temperature will lead to less energy dissipationin practical tire applications

14

The temperature at magnification ζ is also calculated for different velocities. The results are presentedin figure 3.3 for a magnification of ζ = 1 and maximum magnification ζ = ζmax = 400. At a lengthscale λ a rubber volume ∼ λ3 heats up. The length scale corresponding to ζ = 1, therefore equalsλ = 2π/q0 ≈ 4 mm. Thus, the temperature at magnification ζ = 1 corresponds to the temperature4 mm below the surface of the tire. Equivalently, a magnification of ζ = 400 corresponds to a lengthscale 2π/q0 ≈ 10 µm. The rubber just below the surface can reach temperatures of 145 ◦C. Deeperinside the rubber, the temperature is of course lower. This can also be concluded from figure 3.3: themaximum temperature a few millimeters inside the tire reaches only 70◦C. The raise in temperature leadsto a frequency shift of the visco-elastic modulus. This leads to less energy dissipation in the rubber asdiscussed above. Again, the qualitative results are the same as the results obtained by Persson. However,the quantitative agreement of the results is relatively poor. As explained above, the difference in theresults could originate from the fact that Persson uses measurement data, and here a simple rheologicalmodel is used.

10−8

10−6

10−4

10−2

100

102

50

60

70

80

90

100

110

120

130

140

150

slip velocity [m/s]

flash

tem

pera

ture

°C

Flash temperature

ζ = 1ζ = 400

Figure 3.3: Flash temperature for magnification ζ = 1 and maximum magnification ζmax = 400

15

16

Chapter 4

Results for real road surfaces

In the previous chapters the numerical data that was calculated with power spectral densities (psd) asdepicted in figure A.1. The psd is calculated through the use of formula A.1, as proposed by Persson.It could be questioned if the use of such an ’idealized’ surface will generate realistic results. Therefore,measurements have been made on a real asphalt road surface. From this data, a spectrum of theroad is calculated and compared with the spectrum used in chapter 3. This is discussed in paragraph4.1. The influence of a measured psd on the friction coefficient will be discussed in paragraph 4.2. Inthe subsequent paragraph two real road surfaces are compared since data of another road surface wasavailable from another project. Finally, the influence of rain on the spectrum and the friction coefficientwill be addressed in paragraph 4.4.

4.1 The power spectral density of a real asphalt road surface

In order to calculate the power spectral density of a road surface, the height profile of the road has to bemeasured. Because small length scales are considered in these models, down to the µm range, a detailedscan of the road surface is required.

A piece of asphalt with dimensions 56 x 61 mm2 has been scanned with an optical imaging profiler(’Sensofar’). A rectangular grid of 978 x 1088 points has been scanned with a spacing of 55.4 µm. Onlysmall areas can be scanned during one measurement. Therefore, subsequent measurements are partiallymapped with previous measurements. A plot of the measured road surface is shown in figure 4.1.

In [1] Persson describes a way to calculate the power spectral density from surface height data. First,a two-dimensional Fast Fourier Transform (FFT) is performed to calculate the spatial frequencies thatare present in the surface:

hA(q) ≈ a2

(2π)2Hm,

where a is the lattice spacing of the data points (55.4 µm) and Hm is the two-dimensional FFT of thesurface height. The FFT is performed on discrete data, which leads to leakage: the fourier transform ofthe actual frequencies present is smeared out onto neighboring frequencies. Also, the data is not periodicwhich leads to aliasing. The most common way to prevent leakage and aliasing is to make use of awindow. The height data is windowed in two dimensions by means of tukey-windows. The advantageof a tukey window is that the shape of the window can be altered. The ratio between the tapered andconstant section can be set by means of the factor α. A value of α = 0 corresponds to a rectangularwindow and a factor of α = 1 corresponds to the hamming window. The use of a hamming windowwould alter the height data to much. A rectangular window would not satisfy the periodicity of the data.A factor α = 0.25 is used in the calculation and leads to satisfactory results. The used window and thewindowed data are shown in figure 4.2.

17

Figure 4.1: Asphalt road surface measured with an optical imaging profiler

Figure 4.2: Used 2-dimensional tukey window with α = 0.25 (left) and surface data windowed (right)

18

The power spectral density can easily be calculated as:

C(q) =(2π)2

A〈|hA(q)|2〉,

where A is the surface area of the asphalt piece that is studied and 〈...〉 stands for surface averaging.

It is assumed that the psd is independent of the direction of q. This makes it possible to calculatean angular average of C(q) which means C(q) = C(q) with q = |q|.

Through the above described procedure a psd of the measured road surface is calculated. Theresulting spectrum is shown in figure 4.3. The smallest wave vector is determined by the dimensions ofthe measured surface: q0 = 2π/L = 103 m−1. The largest possible wave vector is determined by theNyquist frequency stated by the Nyquist-Shannon sampling theorem: the highest possible reconstructiblefrequency equals half the sample frequency. Therefore q1 = π/a = 5.6 · 104 m−1. It can be concludedfrom the figure that the calculated spectrum is very similar to the spectrum used for the temperature-independent model. The Hurst exponent H can be calculated from the slope of the curve. It appears tobe 0.86, only a deviation of 7% with the value that was used in the calculations of the friction coefficient inthe previous chapter (0.8). Persson also published spectra of road surfaces calculated from measurementdata. The results are in excellent agreement with the results presented here.

102

103

104

105

10−20

10−18

10−16

10−14

10−12

10−10

10−8

Power spectral density of the road

Wave vector q [m−1]

C [m

4 ]

Calculated spectrumSpectrum temperature−independent model

Figure 4.3: Calculated power spectral density of measured road surface and the psd used in chapter 3

4.2 Friction coefficient for a real asphalt road surface

With the spectrum discussed in paragraph 4.1, the friction coefficient can be calculated for different slipvelocities. All the measured length scales are considered so there will not be any extra cut-off lengthscales. A significant difference with the spectra used in the previous chapters is the range of wave vectorsq. The psd of figure 4.3 is limited to 102 < q < 5.6 · 104 m−1. The psd used in for example thetemperature-dependent model is defined for 1.5 · 103 < q < 6 · 105 m−1.

19

In figure 4.4 the calculated psd is shown together with a complete fitted spectrum. The slope of thecurve determines the Hurst exponent. The roll off of the curve is determined by q0 and the height of thecurve is determined by the rms value of the surface height h0. The resulting friction coefficient for thesetwo spectra can be found in figure 4.5. Here the flash temperature is not included in the analysis, becausethe temperature-independent implementation of the model is validated and the temperature-dependentmodel is not. As figure 4.5 indicates, the deviation between the friction coefficients is small. This makesit possible to make use of a fitted model for a road surface when calculations are performed. However,one should bear in mind that the upper cut off is now determined by the spacing of the measurementpoints. A more detailed analysis of the road surface would yield a larger cut-off wave vector. In practice,the upper cut off wave vector is determined by contamination of the tire or due to a thermally degradedlayer of tire rubber. Extrapolation of the power density spectrum could therefore lead to more realisticresults. For example, if the spectrum is extrapolated to a upper cut-off of q1 = 6 · 105 m−1, the frictioncoefficient will increase with 58%.

102

103

104

105

10−22

10−20

10−18

10−16

10−14

10−12

Comparison of calculated and fitted psd

Wave vector q [m−1]

C [m

4 ]

Calculated spectrumFitted spectrum

Figure 4.4: Calculated and complete fitted power spectral density of the measured road surface

20

10−4

10−2

100

102

104

0

0.5

1

1.5

2

2.5

slip velocity [m/s]

kine

mat

ic fr

ictio

n co

effic

ient

µ

Kinematic friction model without flash temperature

Calculated spectrumFitted spectrum

Figure 4.5: Kinematic friction coefficient for the calculated and fitted spectrum. Flash temperature isnot included

4.3 Comparison of two real road surfaces

In the previous paragraph the results for the friction coefficient for a real road surface (surface 1) hasbeen discussed. Another road surface has been measured with the same optical imaging profiler duringanother project (surface 2). The data of the two roads will be used here to compare the two roadsand the resulting friction coefficients with each other. Road surface 2 consisted of much coarser stoneparticles than road surface 1. Therefore direct measurement with the ’Sensofar’ was not possible. Aclay impression was made in order to get good quality data. The resulting road surface (negative ofthe clay impression) is shown in figure 4.6. With the data from surface 2 a power spectral density iscalculated. The two spectra are shown in figure 4.7. The psd of surface 2 is slightly larger at small wavevectors. This is due to the lager stone particles that are present in the second road surface. As can beconcluded from the figure, the Hurst exponent of road surface 2 is slightly lower (smaller slope of thecurve) and the smallest length scales (large q) are more present). Calculations have been made for thetemperature-independent model. The resulting friction coefficient is shown in figure 4.8 for different slipvelocities. The two spectra lead to different µ(v) curves, the difference can be as high as 50%. Roadsurface 2 has got a higher psd at small wave vectors, therefore the friction curve is higher for surface2 at low slip velocities (corresponding to low frequencies and low wave vectors). At high slip velocitiessurface 1 is dominant and will therefore exhibit more friction at these velocities. However, caution hasto be taken when interpreting these results: the clay impression that was made of the road surface didnot fill out the deep cavities of the road. Therefore the clay impression is not the exact negative of theroad surface. Furthermore it has not been validated that the clay ’method’ will lead to accurate results.It could be that the clay impression does not resemble the road surface very well, especially on smalllength scales. One way to check the validity of this method is to make a clay impression of road surface 1and scan that with the optical imaging profiler. If the results are identical to the original measurements(figure 4.1) it is allowed to use a clay impression in the analysis.

21

Figure 4.6: Asphalt road surface (surface 2) measured with an optical imaging profiler

22

101

102

103

104

105

10−20

10−19

10−18

10−17

10−16

10−15

10−14

10−13

10−12

Comparison of calculated and fitted psd

Wave vector q [m−1]

C [m

4 ]

Spectrum surface 1Spectrum surface 2

Figure 4.7: Comparison of the power spectral densities of two real road surfaces

10−4

10−2

100

102

104

0

0.5

1

1.5

2

2.5

slip velocity [m/s]

kine

mat

ic fr

ictio

n co

effic

ient

µ

Kinematic friction model without flash temperature

Spectrum surface 1Spectrum surface 2

Figure 4.8: Comparison of the friction coefficient for two real road surfaces

23

4.4 Influence of rain

It is widely known that the friction between the tire and road decreases when the road is wet. A possibleexplanation for the decrease in friction could be the following: rain fills up some of the cavities present inthe road surface. When the rubber is moving over these cavities it seals them off, preventing the rubberfrom filling the deep holes in the road surface. The rubber will therefore ’feel’ a different ’corrected’ roadsurface. It is possible to calculate numerically a ’flutted’ surface. The function imfill(...) in Matlabis normally used for imaging processing of pictures with missing data. However, it can be used perfectlyto calculate the ’corrected’ road surface. In figure 4.9 a detailed view of a part of the surface (surface 1)is shown with the original road and the filled road.

Figure 4.9: Detailed view of original road surface and filled road surface

The filling of the cavities will lead to a more smooth road surface. The rms-value of the surfaceheight will therefore be lower for the wet surface. The power spectral density for the dry surface andfor the wet surface is shown in figure 4.10. The psd of the wet surface is about four times lower thenthat of the dry surface. This will therefore lead to less energy dissipation in the rubber. As a result,the friction coefficient will be lower for the wet surface. A comparison of the friction coefficient for thespectra of figure 4.10 is shown in figure 4.11. The figure shows what is widely known in practice: thefriction between a tire and the road is lower for a wet surface.

24

102

103

104

105

10−22

10−20

10−18

10−16

10−14

10−12

Comparison of psd of wet road and dry road

Wave vector q [m−1]

C [m

4 ]

Wet roadDry road

Figure 4.10: Comparison of the power spectral densities for the dry and wet road

10−4

10−2

100

102

104

0

0.5

1

1.5

2

2.5

slip velocity [m/s]

kine

mat

ic fr

ictio

n co

effic

ient

µ

Kinematic friction model without flash temperature

Wet roadDry road

Figure 4.11: Comparison of the friction coefficient for the dry and wet road

25

26

Chapter 5

Conclusion and recommendations

5.1 Conclusion

The implementation of the temperature-independent model has been validated with the results presentedby Persson. The influence of different road surfaces and the effect of the range of length scales has beeninvestigated. A lower Hurst exponent H of the road surface leads to a ’flatter spectrum’ and leads tohigher peaks of the µ(v) curve. If the range of length scales is increased, more length scales will lead tomore energy dissipation in the rubber. Therefore, a larger ζmax will lead to a broadening and increaseof the µ(v) curve.

The temperature-dependent model includes the heat diffusion equation to account for the fact thatthe rubber heats up due to the energy dissipation of the rubber. The qualitative results are as expected:an increase in temperature softens the rubber and, in practical tire-road situations, leads to less energydissipation of the rubber. Therefore the friction coefficient will drop compared to the temperature-independent case. The temperature of the rubber at different length scales is in relatively good agreementwith the temperatures calculated by Persson. However, quantitative validation of the model was notpossible. Persson uses measurement data for the viscoelastic modulus E(ω) and for the shift factor aT ,but does not publish this data in his papers.

Successful measurements have been made with an optical imaging profiler on a real road surface. Withthis data, the psd of the measured road has been calculated (surface 1). The results are in very goodagreement with the psd’s published by Persson. The kinematic friction coefficient has been calculatedfor this spectrum and for a fitted one. The resulting µ(v) curves are similar and therefore it is concludedthat fitted psd’s can be used in the calculations. Comparison between two different real road surfaceshas been made with data available from another project (surface 2). The surfaces are quite different, andthis has an effect on the resulting µ(v) curves. Surface 2 has got a higher psd at low wave vectors andtherefore exhibits more friction at low slip velocities. At high slip velocities surface 1 is mostly dominantand therefore surface 1 results in a higher friction coefficient at these high sliding velocities. Caution hasto be taken when interpreting the results because measurements have been made of a clay impression ofthe road surface. Another interesting feature that is investigated is the influence of rain on the frictioncoefficient. A flutted surface makes the road more smooth and will lead to a lower psd of the road. Anumerical procedure is used to ’fill’ the surface with water. The result is what is expected from practice:the friction coefficient decreases a significant amount when the road is wet.

27

5.2 Recommendations

The main drawback of this report is the lack of validation of the temperature-dependent friction model.This is due to the fact that the measurement data that Persson uses in his calculations is not published.Furthermore it would be interesting to perform experiments with rubber blocks and compare the resultswith the theory presented here.

The four recommendations for future study on this subject are therefore:

• Validate the temperature dependent model

• Use measurement data for the viscoelastic modulus and for the shift factor aT . In order to usemeasurement data in the implemented model, some modifications have to be made: the analyticalintegrals that are used have to be adapted: either numerical integration should be used or themeasurement data has to fitted and integrated analytically. The second method is preferable, ifpossible, because it needs much less calculation time. The results obtained from real data will bemore realistic. The rheological model used in this report is a very simple one: only one relaxationtime is considered. A real rubber, however, can only be appropriately described by the use of abroad range of relaxation times.

• Perform experiments to obtain the friction coefficient for a range of slip velocities. This way thetheoretical model can be compared with the experimental data. If possible, temperature should beconsidered in the experiments because it has a great impact on the friction coefficient.

• Perform measurements on a clay impression of road surface 1. If the results are identical to theoriginal measurements performed directly on the road surface, the ’clay’ method can be used andthe data from road surface 2 is valid.

28

Appendix A

Self-affine fractal surfaces

In practice it has been found that many real surfaces can be described as self-affine fractal surfaces.A somewhat different surface is a self-similar surface. A self-similar surface has the property that thestatistical properties do not change if a magnified version of the surface is considered. If the equation ofthe surface is given by z = h(x, y), then it cannot be distinguished from its magnified version:

z = h(x, y) = ζh(x/ζ, y/ζ)

A self-affine surface is a bit different. The statistical properties of such a surface do not change ifthe magnification in the z-direction is chosen differently. The magnification in the x- and y-direction isζ and in the z-direction ζH . The Hurst exponent H defines the scaling in the z-direction. The Hurstexponent is related to the fractal dimension Df as H = 3 − Df . For a self-affine fractal surface thefollowing relation holds:

z = h(x, y) = ζHh(x/ζ, y/ζ)

For a self-affine fractal surface the following relation holds for the power spectral density of the road:

C(q) = k

(q

q0

)−2(H+1)

(A.1)

with k =(h0q0

)2H2π . Here h0 is the rms-value of the height of the road surface.

29

In figure A.1. the power spectral density is shown for a road surface with rms-value h0 = 0.5 mmand Hurst exponent H = 0.8 is shown.

103

104

105

106

10−22

10−21

10−20

10−19

10−18

10−17

10−16

10−15

10−14

Wavevector q [1/m]

C [m

4 ]Power Spectral Density of an asphalt road surface

q0

q1

Figure A.1: Power spectral density of an asphalt road surface. The rms surface height h0 = 0.5 mm andthe Hurst exponent H = 0.8

30

Appendix B

The rheological model

In order to use equations 2.1-2.3 to compute the friction coefficient, a rheological model has to be chosenfor the behavior of the rubber. The response of a rubber (like many polymers) depends on the excitationfrequency. In other words, rubbers exhibit viscoelastic behavior.

A simple model that can be used to describe viscoelastic behavior is the Standard Linear Solid (SLS)model. This rheological model can be visualized as in figure B.1 with a spring in series with a Maxwellelement (spring and damper parallel). The springs have spring constants E1 and E2 and the damper hasa time constant of τ . The (complex) elastic modulus of this system is described by:

E(ω) =E1(1− iωτ)1 + a1 − iωτ

,

here a1 = E(∞)/E(0) is the ratio between the elasticity modulus in the high frequency (glassy) regionE1 and the modulus in the low frequency (rubber) region E2. Figure B.2 shows the real and imaginarypart of the elastic modulus for the values shown in table B.1.

Figure B.1: SLS model

Parameter ValueE1 109 Paa1 1000τ 10−3 s

Table B.1: Parameter values used in the SLS model

31

10−2

100

102

104

106

108

100

102

104

106

108

1010

frequency omega [rad/s]

E−

mod

ulus

[Pa]

Complex E−modulus for rheological model

Re(E)Im(E)

Figure B.2: Real and imaginary part of the viscoelastic modulus

32

Appendix C

Used parameter values

The table below shows the values that are used for the parameters in the temperature-independent modeland in the temperature-dependent model.

Table C.1: Parameter values used in the friction modelsParameter Value temperature-independent model Value temperature-dependent modelH 0.85 0.8q0 2000 m−1 1500 m−1

h0 5 ·10−4 m 6.67 ·10−4 mE1 109 Pa 109 Paa1 1000 1000τ 10−3 s 10−3 sν 0.5 0.5σ0 0.2 ·106 Pa 0.4 ·106 PaTg -30 ◦CT0 60 ◦C

33

34

Bibliography

[1] U. Tartaglino A.I. Volokitin E. Tosatti B.N.J. Persson, O. Albohr. On the nature of surface roughnesswith application to contact mechanics, sealing, rubber friction and adhesion. Journal of Physics:Condensed Matter, 17:R1–R62, 2005.

[2] J.D. Ferry M.L. Williams, R.F. Landel. The temperature dependence of relaxation mechanisms inamorphous polymers and other glass-forming liquids. Journal of the American Chemical Society,77:3701, 1955.

[3] B.N.J. Persson. On the theory of rubber friction. Surface Science, 401:445–454, 1998.

[4] B.N.J. Persson. Theory of rubber friction and contact mechanics. Journal of Chemical Physics,115:3840–3861, 2001.

[5] B.N.J. Persson. Rubber friction: role of the flash temperature. Journal of Physics: CondensedMatter, 18:7789–7823, 2006.

35