prediction of contact area and frictional behaviour of

PREDICTION OF CONTACT AREA AND FRICTIONAL BEHAVIOUR OF RUBBER ON RIGID ROUGH SURFACES

Hagen Lind Institute of Dynamics and Vibration Research

Leibniz Universität Hannover Hannover, Germany

Email: [email protected]

Matthias Wangenheim Institute of Dynamics and Vibration Research

Leibniz Universität Hannover Hannover, Germany

ABSTRACT In the tire-road contact friction depends on several

influencing variables (e.g. surface texture, real contact area, sliding velocity, normal contact pressure, temperature, tread block geometry, compound and on the existence of a lubrication film). A multi-scale model for prediction of contact area and frictional behaviour of rubber on rigid rough surfaces at different length scales is presented. Within this publication the multi-scale approach is checked regarding convergence. By means of the model influencing parameters like sliding velocity, compound and surface texture on friction and contact area will be investigated.

NOMENCLATURE A0 Nominal contact area Ai Local contact Area Ar Real contact area Cz Height difference correlation D Fractal dimension E’ Storage modulus E’’ Loss modulus H Hurst exponent pi Pressure sA Magnification factor of amplitude sλ Magnification factor of wavelength v Sliding velocity x, y, z Cartesian coordinates ηi Sine wave density λi Wavelength µi Local friction coefficient µr Global friction coefficient ξ Magnification factor

||ξ Horizontal correlation length

⊥ξ Vertical correlation length

ω Frequency Index i, 1, 2, 3, … No. of different actions

INTRODUCTION A demand in tire development is that a vehicle gets the

shortest possible braking distance during a full brake application. This is achieved by a high friction force within the tire-road contact. The design and development of tires is strongly based on vehicle test results. In the modern tire development processes numerical simulations have become an important tool as well as the facilitation of experimental investigation of tires and tire components in the lab. By this means it is possible to reduce the time of development cycles and development costs.

The friction force of rubber is made up of four components [1]: • Adhesion: Intermolecular bindings between the rubber and

pavement surface. • Hysteresis: Energy dissipation due to deformation of

rubber caused by rough surface. • Viscous friction: Shearing of a liquid film between both

contact partners. • Cohesion: Formation of new surface (e.g. by wear). Adhesion and hysteresis dominate the dry friction of tire tread blocks. During braking, sliding friction occurs within the footprint, especially at the trailing edge of the contact zone. The friction coefficient depends on several influencing variables (e.g. surface texture, real contact area, sliding velocity, normal

1 Copyright © 2014 by ASME

Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014

November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-37998

contact pressure, temperature, tread block geometry, compound and on the existence of a lubrication film).

A multi-scale model for predicting the contact area and frictional behaviour of non-linear-elastic and non-linear-visco-elastic bodies on a rigid rough surface at different length scales is presented. The model is based on the finite element model of [2] and [3]. Within this publication the following chapters explain the multi-scale approach in detail. Starting with the scheme of approach and computation steps follows the surface characterization and approximation for the approach. Afterwards numerical examples respecting convergence and influencing parameters on the approach will be presented.

MULTI-SCALE APPROACH The multi-scale model for the estimation of real contact

area and frictional behaviour is based on [2] and [3]. The scheme of the approach is shown in Fig. 1. The visco-elastic rubber material behaviour is characterized by the frequency depended loss modulus E’’ (ω) and storage modulus E’(ω), cf. Fig. 2. The surface characteristic is build up by an approximation of the height difference correlation by using sine waves. The scales are coupled through pressure transmission.

FIGURE 1. SCHEME OF MULTI-SCALE APPROACH

Starting on the first (largest) scale the contact area Ai () and

the friction coefficient µi is computed. Note that the model is two-dimensional. The contact area has the unit mm. On the next lower scale the pressure pi+1 is given by:

,1i

iii A

pp

⋅=+

λ (1)

where λi denotes the wavelength of the sinus of the previous scale. The real contact area Ar is calculated according to [4]:

.max

1∏

=

⋅=i

iiir AA η (2)

Here ηi is the sine wave density related to the previous contact area Ai-1. It specifies how often the wavelength λi is included in the contact area Ar. The global friction coefficient µr can be calculated by:

.max

1∑

=

=i

iir µµ (3)

FIGURE 2. STORAGE MODULUS AND LOSS MODULUS OF TWO COMPOUNDS

SURFACE CHARACTERIZATION Rigid rough surfaces like asphalt are considered as self-

affine [5], [6], [7] and [8]. The roughness of macro scale occurs with a magnification factor ξ on the micro scale. The statistical properties of a surface profile z(x) are invariant under the scaling transformation:

,, zzxx Hξξ →→ (4)

with the Hurst exponent H. The Hurst exponent can take values between 0 and 1 and is related to the fractal dimension D by H = 3 – D. For a single surface profile z(x) applies 1 < D < 2. Here, D = 1 represents a smooth curve, D = 2 a smooth area.

The height difference correlation is a method to characterize rough surfaces. The height difference correlation Cz(λ) specifies the mean square height difference with respect to the horizontal length λ:

( ) ( ) ( )( ) ,2xzxzCz −+= λλ (5)

where ... stands for averaging over the surface profile z(x). In

case of self-affine surfaces the height difference correlation is given by [7]:


( ) ||

26

||

2 ξλξλξλ <

=

−

⊥ forC

D

z (6)

The variables ||ξ and ⊥ξ are correlation lengths and describe

the cut-off point (Fig. 3b). The slope for ||ξλ < is

characterized by the fractal dimension D. For ||ξλ > the mean

square height difference does not increase further, due to the

maximum finite height of the surface ( )( )2⊥= ξλzC .

FIGURE 3. (a) ASPHALT SURFACE WITH SINGLE SURFACE PROFILE, (b) HEIGHT DIFFERENCE CORRELATION OF SURFACE PROFILE

SURFACE APPROXIMATION The surface is described by an approximation of the height

difference correlation by using sine waves. The superposition of the sine waves results in a comparable height difference correlation as the measured one. In Figure 4 is shown that superpositions of sine waves with a magnification factor ξ does not lead to good approximation. In micro length scale range the approximation gets worse. This means the structure of macro length scale of the asphalt surface cannot be found by magnification in micro length scale. Investigations to identify characteristic sine waves of an asphalt surface according [9] show different ratios of amplitude to wavelength regarding

macro and micro scale. Factors higher than ten are possible between the ratios. For this reason, magnification factors for wavelength sλ and amplitude sA are introduced. Thus, the differences of the amplitude wavelength ratios in macro and micro scale are taken into account and a good approximation of the height difference correlation by sine waves can be found. The result of the approximation is also shown in Fig. 4. Two magnification factors are in contradiction with the definition of self-affinity and approaches from [5-8]. In these approaches fractal descriptors or single magnification factors are used. In this work an approximation of the height difference correlation of the surface is used in contrast to the approaches above. Both, measurement and approximation have the same characteristic. The approximation by sine waves can be used independently of a fractal description.

FIGURE 4 COMPARISON OF MEASURED HEIGHT DIFFERENCE CORRELATION AND APPROXIMATION WITH SUPERPOSED SINE WAVES WITH MAGNI-FICATION FACTOR ξ AND WITH sλ AND sA

NUMERICAL EXAMPLES An example for the contact area estimation in static and

dynamic case is shown in Fig. 5. In the static case a non-linear-elastic rubber block is loaded on a rough surface with a pressure of p = 3 bar. The relative contact area is depicted as a function of wavelength. Different magnification factor combinations for the surface approximation are used to check the approach regarding the convergence. In all cases the values of amplitude and wavelength of the first scale are the same. All magnification factor combinations show qualitatively the same progression. On the highest scale respectively largest wavelength occurs the same relative contact area. With decreasing wavelength the contact areas decrease. Furthermore a convergence is apparent. The magnification factors sλ = 3.5 and sA = 2.4 lead to an optimum of the height difference correlation approximation (The difference between approximated and measured height difference correlation is minimal). It can be seen that all contact area estimations with other magnification factor combinations


converge to the contact area trend of the best height difference correlation fit. Only in the macro range, there are still differences between the different magnification factor combinations. This is due to the fact that small magnification factor combinations appear more often (cf. approximations with sλ = 3, sA = 2.2 and sλ = 6, sA = 3). The macro roughness is better approximated with smaller magnification factors. Only with increasing number of scales the contact areas converge.

FIGURE 5 RELATIVE CONTACT AREA AS FUNCTION OF WAVELENGTH AND DIFFERENT MAGNIFICATION FACTORS (a) STATIC CASE p = 3 bar, (b) DYNAMIC CASE p = 3 bar AND v = 300 mm/s

Also in the dynamic case the convergence of contact area regarding different magnification factor combinations can be seen (cf. Fig. 5b). Here a non-linear-visco-elastic rubber block is loaded with a pressure of p = 3 bar on the approximation of a rough surface and displaced with a constant sliding velocity of v = 300 mm/s. As in the static case the convergence depends mainly on the approximation quality of the surface. An optimum is at sλ = 3.5 and sA = 2.4. In contrast to the static case the storage modulus E’(ω) influences the results. A material stiffening effect can be observed, due to frequency dependent material behaviour (cf. Fig. 2). The higher the excited frequency the higher the storage modulus. The contact area gets smaller

than in static case. Not only the gradient is higher, but also absolute values are smaller.

After the convergence concerning the contact area of the multi-scale approach has been demonstrated some investigations regarding contact area influencing parameters will be presented. The first parameter is the sliding velocity v (cf. Fig. 6a). With increasing sliding velocity the contact area decreases. The higher the sliding velocity the higher the material stiffness (ω = v/λ, cf. Fig. 2). This is especially with consideration of the micro roughness considerable. A smaller contact area leads to a higher material stiffing. This is because the rubber is not in full contact with the sine waves on all considered scales. From this follows that the rubber is not only excited by the fundamental frequency of each wavelength. Besides the fundamental frequency a multiple of the fundamental frequency excite the rubber. Higher frequencies due to a smaller contact area lead to higher material stiffness and thus again to a smaller contact area. The apparent contact areas in dependence of sliding velocities can be distinguished in macroscopic but above all in microscopic case.

FIGURE 6 RELATIVE CONTACT AREA AS FUNCTION OF WAVELENGTH AND (a) SLIDING VELOCITY (p = 3 bar), (b) COMPOUND (p = 3 bar, v = 3000 mm/s)


Besides the sliding velocity the influence of different rubber compounds regarding the contact area can be determined (cf. Fig. 6b). Compound A with a higher storage modulus E’(ω) than Compound B shows the expected behaviour. The macroscopic as well as the microscopic contact area of Compound A is smaller than of Compound B. In micro wavelength range the differences increase. Compound A stiffens more than Compound B due to the higher excited frequencies by smaller dynamic contact area.

FIGURE 7 (a) HEIGHT DIFFERENCE, (b) RELATIVE CONTACT AREA AS FUNCTION OF WAVELENGTH AND SURFACE (p = 3 bar, v = 300 mm/s)

Furthermore the contact areas of a compound on different

surfaces can be distinguished. In Fig. 7a Surface A represents an asphalt pavement and Surface B porous asphalt. The largest wavelengths and amplitudes differ as well as the magnification factors sλ and sA and thus the characteristic wavelengths and amplitudes (cf. Fig. 7b). It is evident that the contact area on asperities in the macro wavelengths range of Surface A gets higher than of surface B. By consideration of macro and micro roughness the differences of both contact areas decrease.

In Addition to contact area the frictional behaviour of rubber on rough surfaces can be estimated with this multi-scale approach. Statements on relevant excitation wavelengths are

possible. As with the estimation of the real contact area, the friction determination approach is checked regarding convergence by variation of magnification factor combinations of the height difference correlation, cf. Fig 8a.

FIGURE 8 FRICTION COEFFICIENT AS FUNCTION OF WAVELENGTH AND (a) DIFFERENT MAGNIFICATION FACTORS (p = 3 bar, v = 300 mm/s), (b) SLIDING VELOCITY (p = 3 bar)

It is becoming evident that for smaller length scales the friction coefficient increases, independent of magnification factor combinations. The rubber is excited on all length scales with frequencies which are smaller than the frequency ωE’’,max, where the loss modulus E’’ (ω= ωE’’,max) gets a maximum. The friction coefficients converge for larger wavelengths. For smaller wavelengths the friction coefficients diverge, especially with the worst height difference correlation approximation (sλ = 6 and sA = 3). This is because the rubber is not in full contact with the sine waves on all considered scales. From this follows that the rubber is not only excited by the fundamental frequency of each wavelength. Besides the fundamental frequency a multiple of the fundamental frequency excites the rubber. The worst height difference correlation approximation leads to much higher excited frequencies and thus higher friction values. The height difference correlation fits with


similar magnification factor combinations lead to nearly the same friction curves. In the further course of the work the best height difference correlation approximation is used (sλ = 3.5 and sA = 2.4).

The velocity dependency of a sliding rubber block on a rough surface is shown in Fig. 8b. Here, the influence of wavelength for different sliding velocities is apparent. For all sliding velocities an increasing characteristic can be seen for decreasing wavelengths. The higher the velocity the sooner a maximum of friction potential is reached and with further decreasing wavelength the friction coefficient decreases. A rubber block with low sliding velocity shows the lowest friction coefficients in the macro length scale range, because the corresponding excited frequencies are smaller than for higher velocities. Thus the friction coefficient continues to rise with decreasing wavelengths up to ω = ωE’’,max.

FIGURE 9 FRICTION COEFFICIENT AS FUNCTION OF WAVELENGTH AND (a) COMPOUND (p = 3 bar, v = 3000 mm/s), (b) SURFACE (p = 3 bar, v = 300 mm/s)

Furthermore the frictional behaviour of different compounds

can be analyzed (Fig. 9a). In this example the same compounds as in the contact area estimation are used. Besides the storage modulus E’(ω) they differ in their loss modulus E’’ (ω) (cf. Fig. 2). By the highest wavelength Compound A and B

show nearly the same friction coefficient. Compound B has the smaller loss modulus but larger contact area and thus larger excited volume. Furthermore the loss modulus do not differ for low frequencies. The smaller the wavelength the higher the loss modulus of Compound A concerning Compound B. The difference in friction coefficient increases up to the wavelength where both have their maximum (approximately same ωE’’,max). Afterwards the friction coefficients approach once again to lower values. Here also the loss moduli of both compounds become similar.

The surfaces can be distinguished regarding their friction characteristics. In Fig. 9b the difference between the standard asphalt Surface A and the porous Surface B is presented. Despite larger contact area of the rubber on Surface A, the friction coefficient on Surface B is higher than on Surface A. This is due to the fact that not only the contact area influences the frictional behaviour but also the ratio of amplitude to wavelength. Here Surface B has a higher ratio of amplitude to wavelength.

CONCLUSION AND OUTLOOK To improve the grip performance of a tire at breaking

conditions the knowledge of the occurring friction mechanisms is necessary. The multi-scale model allows determining the frictional behaviour of rubber compounds and their contact area on rigid rough surfaces.

The model uses an approximation of the height difference correlation to characterize rough surfaces. The approximation is made by a superposition of sine waves. It has been shown that one magnification factor is insufficient to build up the approximation. This shows that rough surfaces like asphalt are not self-affine For the approximation separate magnification factors sλ and sA for wavelength and amplitudes were introduced. First simulations regarding the convergence of different magnification factor combinations show good results. The estimated contact areas and friction values converged to the best approximation parameters for the height difference correlation.

Simulation are performed regarding the influence parameter sliding velocity, compound and surface texture. It was presented that the multi-scale model can show the effects of the different influence parameters on contact area and frictional behaviour. Statements about relevant wavelengths for the friction process are possible.

In the future the model has to be validated with measurements, carried out at the High Speed Linear Test Rig [10]. Furthermore a thermo-mechanical coupling of the model will be provided due to the temperature dependent material behaviour [11], [12].

REFERENCES [1] Kummer, H. W., 1966, “Unified theory of rubber and tire

friction”, Engineering research bulletin B-94, Pennsylvania State Univ.


[2] Dobberstein, J. H., Wangenheim, M., Schmerwitz, F., Lind, H., Wriggers, P. and Wies, B., 2011, “Simulation of the friction of rubber compounds on real road surfaces”, Proceedings 2nd International Conference on Computational Contact Mechanics (ICcCM 2011), Hannover.

[3] Wriggers, P., Reinelt, J., 2009, “Multi-scale approach for frictional contact of elastomers on rough rigid surfaces”, Comput. Methods Appl. Mech. Engrg., 198, pp.1996-2008.

[4] Jackson, R. L. and Streator, J. L., 2006, A multi-scale model for contact between rough surfaces, Wear, 261, pp. 1337-1347.

[5] Goedecke, A., Jackson, R.L., Mock, R., 2013, “A fractal expansion of a three dimensional elastic –plastic multi-scale”, Tribology International 59, pp. 230-239.

[6] Persson, B. N. J., 2001, “Theory of rubber friction and contact mechanics”, Journal of Chemical Physics, Volume 115, Number 8, pp. 3840-3861.

[7] Klüppel, M., Heinrich, G., 2000, “Rubber friction on self-affine road tracks”, Rubber Chemistry and Technology 73, pp. 578-606.

[8] Heinrich, G., Klüppel, M., 2008, “Rubber friction, tread deformation and tire traction”, Wear 265, pp. 1052-1060

[9] Lind, H., Kues, H., Linke, T., Schmerwitz, F. and Wies, B., 2013, Reduced Description of three-dimensional measured surface texture, Proceedings 5th World Tribology Congress 2013, Turin.

[10] Moldenhauer, P., Ripka, S., Kröger, M., 2008, “Tire tread block dynamics: investigating sliding friction“, Tire Technology International, The annual review of tire materials and tire manufacturing technology, pp. 96-100.

[11] Williams, M., Landel, R. F., Ferry, J. D., 1955, “The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids”, J. Am. Chem. Soc. 77, 99. 3701-3707.

[12] Persson, B. N. J., 2011, “Rubber friction and tire dynamics”, Journal of Physics, 23, 14pp.


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