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Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering http://pid.sagepub.com/ A physical−analytical model for a real-time local grip estimation of tyre rubber in sliding contact with road asperities Flavio Farroni, Michele Russo, Riccardo Russo and Francesco Timpone Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering published online 14 February 2014 DOI: 10.1177/0954407014521402 The online version of this article can be found at: http://pid.sagepub.com/content/early/2014/02/12/0954407014521402

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Original Article

A physical-analytical model for a real-time local grip estimation of tyre rubber in sliding contact with road asperities

Proc IMechE Part D: J Automobile Engineering 1–15 Ó IMechE 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954407014521402 pid.sagepub.com

Flavio Farroni, Michele Russo, Riccardo Russo and Francesco Timpone

Abstract This paper deals with the frictional behaviour of a tyre tread elementary volume in sliding contact with road asperities. Friction is assumed to be composed of two main components: adhesion and deforming hysteresis. The target, which was fixed in collaboration with a motorsport racing team and with a tyre-manufacturing company, is to provide an estimation of local grip for online analyses and real-time simulations and to evaluate and predict adhesive and hysteretic frictional contributions arising at the interface between the tyre tread and the road. A way to approximate the asperities, based on rugosimetric analyses on a macroscale and a microscale, was introduced. The adhesive component of friction was estimated by means of a new approach based on two different models found in the literature, whose parameters were identified thanks to a wide experimental investigation previously carried out. The hysteretic component of friction was estimated by means of an energy balance taking into account the viscoelastic behaviour of rubber (which was characterized by means of appropriate dynamic mechanical analysis tests) and the internal stress–strain distribution (which was due to indentations of the road). The model results are finally shown and discussed, and the validation experimental procedure is described. The correct reproduction of the friction phenomenology and the model prediction capabilities are highlighted, making particular reference to the grip variability due to changes in the working conditions.

Keywords Tyre–road local grip evaluation, adhesive friction, hysteretic friction, sliding contact, road asperities modelling, physical– analytical model

Date received: 11 June 2013; accepted: 6 January 2014

Introduction The frictional behaviour of rubber in tyre–road interactions is one of the main topics in a wide range of research fields. Knowledge about phenomena related to the adherence is a key factor in the development of the braking–traction systems and the stability control systems adopted in the automotive industry,1 such as in the study of innovative tyre structures and compounds which are able to minimize braking distances, to preserve the vehicle stability in panic situations and to guarantee optimal road holding on wet or icy surfaces.2 Moreover, the fact that drivers continuously seek the optimal grip for each different driving condition makes the development of a physical grip model an essential instrument for a top-ranking racing team, in particular because of the definitely lower amount of resources needed by simulations than by experimental tests carried out in order to acquire information about

the behaviour of tyres. Rubber–asphalt friction, in fact, is influenced by a great number of variables and parameters, which are often hard to control and measure;3 the macroroughness and microroughness of the bodies in contact, the pressure arising at their interface, the materials stiffness characteristics and their frequency and temperature dependences, the relative motion direction and the speed are only a few of the factors that take part in the phenomena involving contact mechanics, thermodynamics, polymers chemistry and, from a wider point of view, vehicle dynamics. Department of Industrial Engineering, University of Naples Federico II, Naples, Italy Corresponding author: Flavio Farroni, Department of Industrial Engineering, University of Naples Federico II, Via Claudio 21, Naples 80125, Italy. Email: [email protected]


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Proc IMechE Part D: J Automobile Engineering

Figure 1. The Kummer model for rubber friction mechanisms.

Since ancient times, the frictional force has been investigated, often with the aim of reducing and avoiding it, because of its tendency to impede the motion and of the consequential phenomena such as heat production, power dissipation and wear, which are considered to be undesirable. In the automotive field, and in particular in the race field, the possibility of maximizing the grip when driving in the same environmental conditions as competitors can represent a key factor in configuring an optimal set-up of the vehicle, which is designed in order to obtain the best performances under the expected loads and wheel angles, to choose the most suitable compound for each road and weather condition and to plan a proper driving strategy that is able to make the tyre work under the desired conditions predicted by means of a physical model. After the studies on the wheel invention carried out by Temistio (390–320 BC), the first attempt to formalize the relationship between the frictional force and the main variables on which it depends dates back to the time of Leonardo da Vinci, who proposed in his unpublished handbooks a linear relationship between the contact force and the vertical load; after the experimental research studies by Amontons,4 Coulomb5 theorized that, for metals, the frictional force was independent of the contact area and directly proportional to the applied normal load by means of a coefficient, expressing for the first time the well-known law m=

frictional force between two metal surfaces applied normal load

ð1Þ

The dependence on the sliding velocity was not taken into account, but a first distinction between the static

friction coefficient and the dynamic friction coefficient was proposed and analysed. Bowden and Tabor6,7 and Rabinowicz8 introduced the theme of adhesion in the contact of polymers, investigating the frictional behaviour of rubber and highlighting its strong dependence on the load, the temperature and the relative speed. A generalized friction model was proposed by Kummer9 during his research on tyre–road interactions; his model (Figure 1) considered for the first time the resistant force as composed of three components: adhesion, deforming hysteresis and wear. In dry conditions, the frictional forces of the tyres take the form FT = FADH + FHb + FC

ð2Þ

where FT is the total frictional resistance developed between a sliding tyre and the road, FADH is the frictional contribution due to the van der Waals adhesion bonds between the two surfaces, FHb is the frictional contribution from bulk deformation hysteresis in the rubber and FC is the cohesion contribution linked with rubber wear. Kummer postulated that FADH and FHb are not independent because adhesion is able to increase the extension of the contact area and consequently the zone in which the hysteretic deformations occur. The third contribution to the resistant forces is achieved because of the removal of rubber material by the road asperities, but the contribution to friction due to this phenomenon is estimated to be around 2% on rough surfaces.10 Because of the work of Kummer9 and Savkoor,11 Moore12 hypothesized that the different components were predominant on different scales; the macroroughness affects the deformations related to hysteresis, and


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Figure 2. Influence of the roughness scales on the different friction mechanisms.

the microroughness affects the intermolecular bonds characterizing adhesion. For this reason, the two aspects may conceptually be split and treated by applying a sort of superposition principle (Figure 2). In more recent years, the cited theories concerning with friction modelling have been confirmed by further developments in the fields of tyre–road hysteretic and adhesive interaction analysis,13–15 of contact mechanics between rubber and rough surfaces16,17 and of local heat transfer in frictional phenomena.18,19 Typical methods to determine the tyre grip are based on observers and identification procedures.20–22 In this paper, a tyre/road friction physical model, named Gr.E.T.A. (Grip Estimation for Tyre Analysis) Model, will be presented. The model, which was developed in collaboration with a motorsport racing team and a tyre manufacturing company, is based on the previous considerations, providing an effective calculation of the power dissipated by the road asperities which indent the tyre tread and taking into account the phenomena involved with adhesive friction which are expressed by means of an original formulation (synthesizing some adherence models available in literature) the parameters of which are identified by dedicated experimental tests.

Table 1. Dimensions of the road texture scales.24 Range size

Megatexture Macrotexture Microtexture

Wavelength

Ra

50–500 mm 0.5–50 mm 0–0.5 mm

0.1–50 mm 0.1–20 mm 1–500 mm

Model definition and basic hypotheses In order to model the complex interactions between the tyres and the asphalt at a microscopic level, it was necessary to focus initially on the behaviour of an elementary volume of rubber in sliding contact with a limited portion of the road. Modelling asphalt, as commonly found in the literature,23 as the sum of sinusoidal waves distributed in the space characterizing the different roughness scales (Table 1), the elementary volume of the tread was defined as a square-based parallelepiped. Its height is equal to the thickness of the tyre tread and the base side to the wavelength lMACRO of the macroroughness of the road (Figure 3). The wavelengths l and the roughness indices25 Ra characterizing the soil profile were estimated by means of appropriate algorithms employed to analyse the data acquired experimentally by a laser scan on different dry tracks and to reproduce the best-fitting sinusoidal

Figure 3. Elementary tread volume and coordinate system.

waves corresponding to the macroprofiles and the microprofiles (Figure 4). The chosen Cartesian reference system, as shown in Figure 3, has its origin in the centre of the upper parallelepiped face; the x axis is in the tread surface plane and is oriented in the sliding direction of the indenter, the z axis is oriented in the direction of the tread width, and the y axis is oriented in order to obtain a righthanded coordinate system.


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Figure 4. (a) Road acquired profile: (b) analysis of a a two-dimensional (2D) section of it.

The tread’s rubber and the road are considered as isotropic and homogeneous materials; moreover, the road is modelled as perfectly rigid.

Material characterization The rubber employed in tyre treads is a hyperelastic, soft and virtually incompressible material (Poisson’s ratio n = 0.5), but it can usually be stretched by more than 500%. Its molecular structure consists of long linear flexible molecules interlinked into a threedimensional (3D) network. Chemical cross-links are usually made by sulphur linkages which appear after a technological process, known as vulcanization. The rubbery state of a polymer is determined by the so-called glassy transition temperature Tg. When the working temperature is above Tg, the polymer shows a rubbery behaviour. Otherwise, it has a glassy behaviour. The analysis of the response to deformation for a viscoelastic solid can be conveniently conducted by referring to sinusoidal loads. When an elastomeric material is subjected to a harmonic deformation e1 = e01 sinðvtÞ

ð3Þ

where e01 is the amplitude of the applied deformation and v is the angular frequency, the induced stress s1 is harmonic too as given by s1 = s01 sinðvt + DÞ

ð4Þ

with the same frequency but out of phase with respect to the deformation. The stress s1 can be expressed as the sum of two contributions, one in phase with the imposed deformation and a second in quadrature phase, according to s1 = e01 ½E0 sinðvtÞ + E00 cosðvtÞ

ð5Þ

where E0 , the said storage modulus, is the part of the elastic modulus relative to the in-phase response of the material and E00 , called the loss modulus, represents the elastic modulus of the part in quadrature phase. A very common index used to describe the dissipative attitude of a compound is the loss angle tangent, defined as tanðdÞ =

E00 E0

ð6Þ

The stiffness parameter adopted for viscoelastic materials in place of Young’s modulus is the complex dynamic modulus pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E = E0 2 + E00 2 ð7Þ The E0 and E00 values and tan(d) are strongly dependent on the temperature and on the frequency at which the rubber is stressed, as schematically represented in Figure 5. As regards the influence of the temperature, the typical behaviour of polymers is characterized by a decrease in the dynamic modulus with increasing temperature, while the phase angle increases until it reaches a maximum before decreasing again. Experimental tests were initially carried out on common passenger-tyre rubber materials with the aim of acquiring data useful to model properly the behaviour of styrene–butadiene rubber (SBR) copolymers constituting the tread; rubber specimens, which had been appropriately cut and prepared, were dynamically tested following dynamic mechanical analysis (DMA) procedures26,27 in a three-point bending proof, in order to acquire data on the storage modulus and tan(d). Tests were carried out at a fixed frequency (1 Hz) and a fixed displacement (1%), making the temperature increase at 1 °C/min from –50 °C to 100 °C. The results of the E0 characterization are shown in Figure 6; as expected for the weakly cross-linked


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Figure 5. Relationships of E0 and tan(d) with the frequency and the temperature.

Figure 6. (a) Passenger-tyre E0 thermal characterization data (frequency, 1 Hz); (b) detail of the 0–100 °C zone.

polymers employed in the manufacture of tyres, it was possible to identify three areas: a glassy region (low temperature and high storage modulus), a transition region and a rubbery region (high temperature and low storage modulus). Figure 7 highlights, despite some irregularities, the expected tan(d) trend, with an absolute maximum localized in the thermal transition zone. For common passenger tyres, the glass transition temperature is often below 0 °C, so that the usual working conditions are localized in the rubbery zone in order to provide optimal frictional performances. Sport and high-performance tyres, which are characterized by the employment of different rubber compounds and fillers, exhibit lower values of the dynamic modulus; this results in a softer and highly wearable tread, a more hysteretic attitude, a definitely higher Tg and, consequently, a higher thermal optimal working range. When both the frequency and the temperature vary, it is possible to make use of the property whereby an appropriate shift operation is capable of combining

their effects; the main element on which the temperature–frequency equivalence principle is based is that the values of the complex modulus components at any reference frequency and temperature (f1, T1) are identical with those observable at any other frequency f2 at an appropriately shifted value of temperature a(T1) according to Eðf1 , T1 Þ = E½f2 , aðT1 Þ

ð8Þ

The relationship most widely used to describe the equivalence principle is the Williams–Landel–Ferry28 (WLF) transform. For passenger-tyre rubber it can be employed in a simplified way in order to determine the unknown equivalent temperature T = a(T1) from f2 T T1 ð9Þ log = f1 DT in which a common DT value, identifiable by means of DMA tests at different frequencies, is about 8 °C.


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Figure 7. Passenger-tyre tan(d) thermal characterization data (frequency, 1 Hz).

The physical meaning of the law is that rubber stressed at a high frequency behaves as if the stress is applied at a lower frequency but, at the same time, at a colder working temperature. The high frequency acts to reduce the time between two consecutive stresses and does not allow complete relaxation of the rubber in the same way as a low working temperature would do. A further aspect of the behaviour of viscoelastic materials that has to be taken into account in contact mechanics with rigid indenters is the increasing trend that the compression force shows with respect to the displacement in the first phase of indentation. Stressing a cylindrical specimen in a monoaxial compression test allowed us to investigate this effect, which is attributed to an increase in the dynamic modulus of rubber and to shape modifications of the contact area, both of which are responsible for the decrease in friction with decreasing contact pressure. In Figure 8, the global dynamic modulus E, adimensionalized with respect to its value at a compression rate of 0.01, is plotted as a function of the compression rate itself, expressed as e=

t0 t t0

ð10Þ

in which t0 and t are the initial thickness and the actual thickness respectively of the specimen during the test. The experimental results also highlight the fact that, for indentation levels mainly smaller than those able to induce the typical hardening phenomenon29 due to compaction of the polymeric chains, the penetration of an asphalt asperity into a rubber layer has an influence on the characteristics of the rubber layer, modifying the interaction between the two bodies; hypothesizing the penetration of an asperity 1 mm high into a 10 mm tyre tread, it is possible to observe a dynamic modulus value about 8.5 times higher than in unloaded conditions.

Figure 8. Passenger-tyre tread hardening phenomenon observed by means of a monoaxial compression test.

Adhesion model Adhesive friction, which is regarded as being the primary contributor when a rubber block slides over a smooth unlubricated surface, is usually pictured as being due to molecular bonds between the rubber chains and the molecules of the track. For this reason a satisfactory modelling of such a friction mechanism cannot exclude knowledge of the complex phenomena concerning the chemistry of polymers and molecular physics. With the aim of reproducing the functionalities between the adhesive friction and the main variables influencing it (i.e. the sliding velocity Vs, the contact pressure p and the temperature T), a model which takes into account both the approach of Le Gal and Klu¨ppel30 and that of Momozono and Nakamura31 was adopted, giving FADH FN t s AC = p A0

mADH =


ð11Þ

Farroni et al.

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(f)

(g)

Figure 9. The pin-on-disc tribometer.

in which, from the Le Gal–Klu¨ppel model, E‘ =E0 t s = t s, 0 1 + ð1 + vc =Vs Þn

ð12Þ

and, from the Momozono–Nakamura model, the ratio of the real actual contact area Ac to the apparent contact area A0 is approximated by Ac 2 FN = pffiffiffiffiffiffiffiffiffi m2 p A0 E A0 2 p = pffiffiffiffiffiffiffiffiffi m2 p E

ð13Þ

Because of a wide experimental investigation performed by the present authors using a pin-on-disc machine,32,33 it has been possible to identify the parameters of the Le Gal–Klu¨ppel model which are the most difficult to define, i.e. the interfacial shear strength ts,0, the critical velocity vc and the viscoelastic dissipation parameter n. A pin-on-disc tribometer (Figure 9) is often employed to measure the friction and sliding wear properties of dry or lubricated surfaces of a variety of bulk materials and coatings. The elements of the machine are as follows: (a) an electric motor, driven by an inverter; (b) a metal disc, moved by the motor through a belt, which can be covered with other different materials; (c) an arm on which a tyre rubber specimen is housed; (d) a load cell, interposed between the specimen and the arm, which allows measurement of the tangential force;

an incremental encoder, installed on the disc axis in order to measure its angular position and velocity; an optical pyrometer pointed on the disc surface in the proximity of the contact exit edge, which provides an estimation of the temperature at the interface; a thermocouple located in the neighbourhood of the specimen, which is used to measure the ambient temperature.

During the test the arm vertically approaches the rotating disc surface and, through the application of calibrated weights, the normal force between specimen and disc can be varied. The variations in the adhesion with the sliding velocity are explicit in the model, while the thermal effect is modelled by means of the variation in the dynamic modulus E of rubber induced by the temperature. The modulus EN of the glassy region and the modulus E0 of the rubbery region were identified by means of viscoelastic characterization tests; m2 is the second profile moment of the probability distribution function of the track (equivalent to the r.m.s. slope) and was computed utilizing road data in accordance with ASME B46.1. It is relevant to emphasize that the effect of the frequency on the rubber for the sliding phenomenon in the field of adhesive friction estimation is taken into account considering only the microroughness profile, as a consequence of the superposition principle previously described. The expression for the adhesion is actually able to model the frictional interactions arising at the microroughness level; the values of the identified parameters take into account the microhysteretic effects and the indentation phenomena highlighted in the above-cited experimental tests.

Hysteresis model The modelling of the hysteresis starts from the expression of the power dissipated by a rubber block which slides with a speed Vs under a vertical load Fz over a generic macrorough surface. Because of the complexity of the real track surface, each elementary volume of the deformed compound block is subjected to a local stress–strain field (which is variable with the time), resulting in a dissipated power due to the viscoelastic behaviour of the polymers. In general, considering the volume VTOT of the elementary tread element, it is possible to express the dissipated power WDISS at time t as ð WDISS ðtÞ = wðx, y, z, tÞ dV VTOT

ð

=

de1 ðx, y, z, tÞ dV s1 ðx, y, z, tÞ dt

VTOT


ð14Þ

8


in which w(x, y, z, t) represents the dissipated power at each point of the deformed elementary volume at time t. Hypothesizing that Vs is constant in the sliding over a single asperity, each stress–strain cycle can be considered as performed in a period equal to T0 =

lMACRO Vs

ð15Þ

Figure 10. Detail of the contact between the sinusoidal asperity apex and the tyre tread.

The average value of the dissipated power must thus be evaluated over this time period according to 1 wðx, y, zÞ = T0

T ð0

wðx, y, z, tÞ dt

ð16Þ

0

In the same time period, considering Vs and FN as constants, the global power dissipated in the elementary volume, can be expressed as WDISS = FT Vs ð17Þ

= mFN Vs = mpA0 Vs

R=

in which p represents the average contact pressure in the nominal elementary area A0, equal to (lMACRO)2. In this way, the balance between the global dissipated power and the local dissipated power is ð mpA0 Vs =

2

41 T0

VTOT

T ð0

3 de1 ðx, y, z, tÞ dt5 dV s1 ðx, y, z, tÞ dt

0

ð18Þ

From the work by Etienne and David,34 we can write 1 T0

T ð0

s1 ðx, y, z, tÞ

multi-dimensional approach was not able to satisfy the real-time requirements and the low computational loads that the applications for which the model has been developed need. Because of the studies by Kuznetsov and Gorokhovsky,37–39 it is possible to calculate the stress state induced in a rubber elastic body by a periodic sinusoidal, perfectly rigid indenter in sliding contact with it. Once the radius of curvature R of the road sinusoidal indenter at the apex given by

de1 1 ðx, y, z, tÞ dt = ve21 E tanðdÞ 2 dt

0

1 1 s21 2p tanðdÞ 2 T0 E Vs s21 =p tanðdÞ lMACRO E =

ð19Þ

allowing us to formulate the final hysteretic friction expression Ð 2 tanðdÞ VTOT s1 dV m=p ð20Þ lMACRO E pA0 In order to estimate this friction coefficient, knowing the polymer characteristics, the road wavelength and the input variables, it is necessary to provide the stress s21 at each point of the discretized elementary tread volume and, in particular, of the stress components along the x, y and z directions. Although many researchers have proposed formulations able to estimate the stress distributions and contact area extension,35,36 a

1 ð2p=lMACRO Þ2 ðRaMACRO Þ

ð21Þ

is determined, it is possible to estimate, by means of the Kuznetsov–Gorokhovsky formula, the half-length N of the contact area (Figure 10) as a function of the radius R, of the average contact pressure p in the nominal area A0, of the rubber dynamic modulus E (calculated taking into account the working conditions acting on the examined elementary volume) and of the macroasperity wavelength according to " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# l 4pRð1 n2 Þ 1 p ð22Þ N = sin p l pE The Kuznetsov–Gorokhovsky method for planar stress calculation can be used to determine the 3D field starting from the vertical planes x–z (with sliding velocity) and y–z (without sliding velocity) shown in Figure 11 and 12, which are localized under the asperity apex. The model does not need the direction of the sliding velocity on each asperity as an input because the orientation of the elementary volume adapts automatically to it, considering the x axis to be parallel to Vs. In this way the only kinematic input is the modulus of the sliding velocity, and it is taken into account by means of the frequency effect that it has on rubber for the WLF law and on adhesion for the functionality discussed in the respective submodel. The strong relationship between the adhesive friction and the hysteretic friction is taken into account by means of the Kuznetsov–Gorokhovsky parameter K, which is assumed to be equal to the adhesive friction coefficient. In Figure 13 the effect of adhesion on the stress profile is shown; the increase in the adhesive component causes a progressive asymmetrization of the stress field in the direction of sliding.


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Figure 11. (a) s1x distribution in the x–z plane, localized as shown in (b), under a pressure of 125 kPa.

Figure 12. (a) s1y distribution in the y–z plane, localized as shown in (b), under a pressure of 125 kPa.

Applying the Kuznetsov–Gorokhovsky equations in the x–z plane, it is possible to calculate the stress components s1x and s1z generated by the sliding indenter, noting in Figure 11(a) that the compressive tangential stress is localized before the indenter and the traction state behind the indenter, owing to the presence of the adhesive contribution. Because of the self-orientation of the elementary volume, it is possible to state that, in the y–z plane, the sliding velocity components are absent; this means that, by applying the Kuznetsov– Gorokhovsky equations in this plane and imposing K equal to zero, the perfectly symmetric tangential stress thus obtained can be considered as an estimation of the stress component s1y. With the aim of extending the 2D results to the whole 3D elementary volume (Figure 14), the planar

components were scaled (reducing the stress entity at increasing distance from the asperity apex) by means of a quadratic function identified on the basis of appropriate tests carried out with a commercial finit element method solver (Figure 15). The tests, which are characterized by high computational loads, confirmed the accuracy of the stress fields calculated much more easily with the Kuznetsov–Gorokhovsky equations, which, once implemented, represent an optimal solution for the needs of a real-time physical model. Moreover, information from finite element analysis allowed us to neglect the stress components t xy, txz and tyz, because their contribution to power dissipation is about one order of magnitude lower than that relative to the components s. Finally, indentation tests confirmed the contact zone extension provided by the work


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of Kuznetsov and Gorokhovsky by means of the parameter N. Knowledge of the described stress components allows us to calculate the numerical integral extended to the elementary tread volume of equation (20). With this aim, the volume was discretized in 200 nodes along x and y and in 50 nodes along z; this number represents an optimal trade-off between the stability of the results and the computational performances.

Figure 13. s1z in the x–z plane for adhesion coefficients (Kuznetsov–Gorokhovsky parameters K) equal to (a) 0, (b) 0.5 and (c) 1 under a pressure of 125 kPa and a sliding velocity of 1 m/s directed from left to right.

Further developments will consider the possibility of expressing explicitly the microhysteretic dissipations, taking into account the indentation phenomena involving the microroughness, replicating the same procedure employed for the macroroughness.

Results and validation The connection of the presented grip model both with an interaction model40,41 (which is able to provide for each step time the contact pressure and the sliding velocity at which each tread element interacts with the corresponding asperity) and with a thermal model18 (the output of which is the tread temperature in the same step time, assumed to be uniform in the neighbourhood of the contact area with a single road asperity20) gives the possibility of estimating the friction arising at the tyre–road interface as the sum of the adhesive contribution (equation (11)) and the hysteretic contribution (equation (20)). As shown in Figures 16, 17 and 18, the model is able to give a coherent response to input variations; in particular, in Figure 16 is reported a 3D plot obtained for a passenger GT tyre, whose compound has been fully characterized, sliding at different Vs and P over a road profile, employed also for the analyses discussed in the following, described by macro and micro sinusoids having lMACRO = 9.9 mm, Ra MACRO = 0.8 mm, lMiCRO = lMACRO/100 and Ra MiCRO = Ra MACRO/100, chosen basing on a similarity conception introduced by Persson. Figure 17(a) and (b) highlights the dependence of friction on the temperature and on the storage modulus of rubber. As expected for polymers, SBR is highly sensitive to the working temperature, confirming that selection of the proper thermal range in which the compound will work is the main key factor for the maximization of the grip performance. A variation of 30 °C has a drastic effect on the friction of this kind of tyre. Different tyre models characterized by different designs and compounds might have an optimal working range

Figure 14. (a) s1x stress in the x–z plane for K = 1 under a pressure of 125 kPa and a sliding velocity of 1 m/s directed from left to right; (b) extension of s1x to the whole 3D elementary volume.


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Figure 15. s1z stress resulting from finite element method simulations along the y direction compared with the quadratic function chosen to extend the planar stress field in the 3D tread volume.

Figure 16. 3D plot showing the friction coefficient for a passenger GT tyre as a function of the sliding velocity Vs and contact pressure p, at a tread average temperature of 25 °C.

centred on other temperatures; they may be lower, for common passenger tyres, in order to ensure safety in the widest range of weather conditions, or they may be higher, for racing tyres operating under high tangential loads and able to reach high levels of friction and wear. With respect to the parameters that affect the storage modulus, the response of the model reproduces the expected physical behaviour, describing the decrease in the frictional attitude of the tyre consequently as due to an increase in the E0 value, which generates a lower indentation level and a less adhesive interaction. Figure 18 allows an analysis of the effect that a variation in the road roughness has on friction and in particular on its two components; as noticeable in the plot and as expected from the analytical modelling, the decrease in Ra (open triangles) is responsible for the increase in the adhesion (symbols with no curves through them), because a smoother microprofile allows optimization of the available contact area, in which a greater number of intermolecular bonds can be created. On the other hand, the decrease in the roughness is responsible for the lower indentation degree, for the consequential lower dissipated power and finally for the decrease in the hysteretic friction component. The differences in the rubber structure make a compound more or less capable of maximizing adhesion rather than hysteresis; the analysed tyre, because of its structure and compound, showed a high dependence of the adhesion on the roughness, with the final result that the global friction manifested an increasing trend with decreasing roughness. A different tyre characterized by different E0 and E00 moduli could have shown the opposite tendencies owing to a major degree of indentations. An experimental study provided a comparison between the model simulation results and the friction tests carried out with the described pin-on-disc tribometer (Figure 9). In the experimental investigation, which aimed to investigate the hysteretic contribution to friction, the disc was covered with 3M anti-slip tape (Ra = 27 mm)

Figure 17. (a) Friction coefficient for a passenger GT tyre as a function of the sliding velocity Vs for different tread temperatures, under a pressure of 300 kPa; (b) influence of the storage modulus at 25 °C and 300 kPa.


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and its surface was kept wet during contact by means of a thin water film. The role played by the water is to avoid adhesive interaction between the rubber and the 3M tape, removing the global friction from this contribution. In this way the road asperities, which are able to break the water film, reach the specimen’s surface and indent it, isolating the pure hysteretic contribution. The specimens employed in the tests were extracted from two slabs made up of two different compounds (C1 and C2) and characterized by means of DMA procedures to obtain E0 (Figure 19) and tan(d) (Figure 20) curves as functions of temperature. The hysteresis model outputs were compared with the experimental points, highlighting the good agreement for both

compounds; in Figure 21, some results of the experimental validation are shown. The figures show as points the mean value of the friction coefficient for the two compounds in both dry and wet conditions. The applied load is constant (10 N or 20 N) and the sliding velocity assumes the values 0.05 m/s, 0.5m/s, 1 m/s or 2 m/s. The hysteretic friction trend provided by the model under the same operating conditions (i.e. applied load, surface roughness and rubber characteristics) is plotted as a function of Vs in order to highlight the good agreement with the experimental results.32,33 As stated above, the differences between the dry and the wet experimental results can be attributed to the substantial reduction in the adhesive component of friction in wet conditions. Further experimental studies will concern the investigation of the hysteretic component of friction in the interaction with surfaces characterized by a lower roughness.

Conclusions

Figure 18. Friction components for a passenger GT tyre as a function of the sliding velocity Vs for different roughness indices, at 25 °C and under a pressure of 300 kPa.

A model, called GrETA, aimed to evaluate the local grip in the interaction between the tyre tread and the road was described in this paper. The complex rubber viscoelastic behaviour and the randomness of the road surface texture make modelling of the grip phenomenon particularly challenging and lead usually to very onerous models from the analytical and computational point of view. The main task that the developed model had to deal with concerned the need to describe the adhesion and hysteresis frictional mechanisms adopting a physical–analytical formulation as efficiently as possible, in order to make the model employable in driving simulations and real-time analyses. With particular reference to racing tyres and to highperformance tyres, the possibility of estimating the grip

Figure 19. (a) Passenger-tyre E0 thermal characterization data (frequency, 1 Hz); (b) detail in the tyre’s thermal working range.


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in various operating conditions is a key factor in maximizing the race performances and defining the optimal tyre and vehicle set-up; on the other hand, grip optimization is a fundamental theme in the passenger-tyre field, in which the road holding during cornering and the reduction in the braking space in dry and wet conditions have to be taken into account for adequate modern vehicles because of the increasing safety standards. Road profiles, after a rugosimetric analyses stage, were modelled by means of a double sinusoidal wave; this is able to take into account the macroroughness and microroughness scales, which, by applying a sort of superposition principle, can be attributed to hysteretic submodels and adhesive submodels respectively. The SBR polymers employed in tread production were

characterized by DMA tests which provided measurements of the storage modulus E0 and tan(d). As concerns the adhesive component of friction, the work was based on models available in the literature, resulting in an original formulation which summarizes peculiar aspects of the various studies. The results of the adhesion model, with its parameters identified in order to fit experimental data provided from previous experimental tests, were self-validated; in the text a description of the pin-on-disc tribometer employed to carry out the experiments was provided. The hysteretic friction model, which was applied in innovative conditions because of the estimation of 3D stress–strain states by means of properly adapted Kuznetsov equations, was validated, and the test procedure and results were provided and discussed. The contact and indentation submodel was validated by means of finite element method simulations carried out under analogous working conditions. Final results which are consistent with the theory and in good agreement with the expected physical behaviour were reported. Further developments will concern the investigation of the wet contact phenomena and the implementation and validation of submodels which are able to describe them. Funding This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. Declaration of conflict of interest

Figure 20. Passenger-tyre tan(d) thermal characterization data (frequency, 1 Hz).

The authors declare that there is no conflict of interest.

Figure 21. Estimated hysteretic friction compared with experimental points for 3M tape at 40 °C: (a) vertical load, 10 N; (b) vertical load, 20 N.


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Appendix 1 Notation Ac A0 E E0 E00 E0 EN f FADH FC FHb FN FT m2 n N p

real actual contact area (m2) apparent contact area (m2) complex dynamic modulus of rubber (Pa) storage modulus of rubber (Pa) loss modulus of rubber (Pa) modulus of the rubbery region (Pa) dynamic modulus of the glassy region (Pa) frequency (Hz) adhesion force (N) cohesion force linked with rubber wear (N) hysteretic force (N) normal load (N) frictional force of the tyre (N) r.m.s. slope of the track (–) viscoelastic dissipation parameter (–) contact half-length (m) average contact pressure (Pa)

R Ra t T T0 vc Vs VTOT WDISS d e l m s ts,0 v

curvature radius of the asperity apex (m) roughness index (m) time (s) temperature (K) period of the stress–strain cycle (s) critical velocity (m/s) sliding velocity (m/s) total tread volume in contact with an asperity (m3) dissipated power (W) phase angle between the storage modulus and the loss modulus (rad) strain (–) wavelength of the roughness (m) friction coefficient (–) stress (Pa) interfacial shear strength (Pa) angular frequency of the stress (rad/s)
