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. Liu,1 M. P. F. Sutcliffe,2 and W. R. Graham2

odeling of Tread Block Contactechanics Using Linear Viscoelastic

heory3

REFERENCE: Liu, F., Sutcliffe, M. P. F., and Graham, W. R., ‘‘Modeling of Tread BlockContact Mechanics Using Linear Viscoelastic Theory,’’ Tire Science and Technology,TSTCA, Vol. 36, No. 3, July – September 2008, pp. 211-226.

ABSTRACT: In an effort to understand the dynamic hub forces on road vehicles, an advancedfree-rolling tire-model is being developed in which the tread blocks and tire belt are modeledseparately. This paper presents the interim results for the tread block modeling. The finiteelement code ABAQUS/Explicit is used to predict the contact forces on the tread blocks basedon a linear viscoelastic material model. Special attention is paid to investigating the forces onthe tread blocks during the impact and release motions. A pressure and slip-rate-dependentfrictional law is applied in the analysis. A simplified numerical model is also proposed wherethe tread blocks are discretized into linear viscoelastic spring elements. The results from bothmodels are validated via experiments in a high-speed rolling test rig and found to be in goodagreement.

KEY WORDS: tread block, contact force, linear viscoelasticity, viscoelastic spring

ntroduction

Dynamic hub forces induced by tire/road interaction have been a topic ofnterest for automobile manufacturers for two main reasons. First, the forces atower frequencies up to 20 Hz are important in defining the quality of vehicleide and handling. Second, the forces at higher frequencies up to 1 kHz, al-hough at lower amplitudes, contribute significantly to the generation of cabinoise �1�. Since the forcing on vehicle hubs is a direct result of the dynamicehaviors of tires, it is thus essential for automobile manufacturers to have andvanced tire-model in order to predict the dynamic hub forces.

Tire belt vibration has been studied extensively, and in recent years manyodels have been developed to investigate tire behavior in the high-frequency

ange. Pinnington et al. �2� have applied one-dimensional Timoshenko beamheory to predict the frequency response of a tire belt. Larsson et al. �3� haveroposed a model based on a two-layer beam structure in which the tire tread isodeled as a separate layer. Following this approach, O’Boy and Dowling �4�

Corresponding author. Department of Engineering, University of Cambridge, Trumpington Street,Cambridge CB2 1PZ, UK. E-mail: fl247@cam.ac.ukDepartment of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ,UK.This paper was presented at the twenty-sixth annual meeting and conference of the Tire Society,

Akron, Ohio, September 25–26, 2007.

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212 TIRE SCIENCE AND TECHNOLOGY

ave presented a multilayer model that describes the tire structure more fullyith various material properties for different layers. A bending plate model has

lso been formulated by Graf et al. �5�. These models together with Pinning-on’s recent studies �6,7� are capable of predicting tire belt responses for fre-uencies up to 5 kHz.

In contrast to the prolific tire belt vibration work, contact forces betweenread blocks and roads, which are the input stimuli to the vibration system, areess well understood. In the early works of Kropp �8� and Vinesse et al. �9�, theread layer is treated as smooth and hence no effect of tread pattern or roadoughness is accounted for. Andersson et al. �10� have modeled both smoothnd patterned tires with grooved treads in either the circumferential or lateralirection. However, the validation for their models has been carried out using aynamic shaker on freely suspended tires and hence no contact mechanics haseen addressed in their study to analyze the interaction between the roads andhe grooved treads. Although it is clear from Fujikawa et al.’s �11� work that theoad roughness plays an important role in the generation of tire vibration noise,ew models are available that present a deeper understanding of contact me-hanics for tire/road interaction. Fujikawa et al. �11� have modeled the tread assmooth elastic layer and applied Hertz contact theory to calculate the contact

orces. Wullens and Kropp �12� have developed a three-dimensional contactodel using elastic half-space theory. However, they have only found good

greement for tires with circumferentially grooved tread. The lack of a suitableontact mechanics model has compromised the effectiveness of highly devel-ped tire belt vibration models, especially in the high-frequency range.

As part of a research project that aims to predict vehicle hub forces, thisaper presents our recent study of contact mechanics for tread blocks with aocus on the impact and release mechanisms. A finite element �FE� model basedn linear viscoelastic theory and a simplified numerical model using discreteiscoelastic spring elements have been developed. The interim results for aectangular tread block from both models are compared with the experimentalata obtained using a high-speed rolling contact test rig. The good agreementound between theory and experiments opens the way to applying these modelso tread blocks with more complicated shapes in contact with road surfaces ofarious textures.

inite Element Model

It has been widely accepted that the impact and release motions of treadlocks are the main sources of perturbation to the tire belt �13–15�. Therefore,he investigation of contact mechanics in this work has focused on the impact

nd release mechanisms of tread blocks. A FE model has been developed using

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LIU ET AL. ON MODELING OF TREAD BLOCK 213

he commercial FE software package ABAQUS/Explicit to give insight into theontact problem. Due to the dynamic nature of the tread block contact problem,he use of the explicit solver in ABAQUS enables the investigation of viscoelas-icity and the inclusion of mass effect of tread blocks at a much lower compu-ational cost than using the implicit solver.

aterial ModelTread rubber can be modeled as a viscoelastic material in ABAQUS/

xplicit by specifying its long-term and relaxation moduli. Boltzmann superpo-ition theory �16� is used to define the relaxation modulus, which is imple-ented in its numerical form in ABAQUS/Explicit as an N-term Prony series

xpansion given by �17�

E�t� = E� + �i=1

N−1

Eie−�t/�i�, N = 1,2, . . . �1�

here E� is the long-term modulus, and Ei and �i are material constants. Bypplying a Fourier transform, the relaxation modulus E�t� can be expressed inhe frequency domain as a storage modulus E���� and loss modulus E����,espectively, given by

E���� = E� + �i=1

N−1

Ei

�2�i2

1 + �2�i2

�2�

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he storage and loss moduli can be measured via standard dynamic mechanicalnalysis �DMA�, based on which all the material constants can be obtained byerforming a curve-fitting operation.

Due to the carbon black filler inside the tread compounds, a tread rubberormally also exhibits some nonlinearity with increasing strain level and hencestrain-dependent modulus. However, this nonlinearity is negligible in our

nalysis due to the fact that the tread blocks on a rolling tire undergo a high-requency loading process in each revolution in which the stiffness of the treadlock is high enough to limit the strain to a low level.

Another material property to define is the volumetric modulus. Althoughhis is difficult to measure, tread rubbers are nearly incompressible and nor-ally have a Poisson’s ratio close to 0.5, varying little with frequency �18�.herefore, it is reasonable to assume a fixed Poisson’s ratio over the entire

requency range. This renders the measurement of volumetric modulus unnec-

ssary.

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read Sample CharacterizationDMA tests for the tread sample supplied by our industrial partner were

onducted using a TA Instruments Q800 DMA machine. The acquired storagend loss moduli data were smoothed using the technique described in Ref. �19�nd referred to 22 °C by performing a WLF �Williams-Landel-Ferry� �20�ransform with standard coefficients. A 24-term Prony series expansion wastted to the data using the sign control method proposed by Bradshaw et al. �21�nd found to be an accurate approximation as shown in Fig. 1. The coefficientsf the Prony series expansion were then used as inputs to the FE code togetherith a Poisson’s ratio of 0.47. Due to the fact that hybrid elements are not

upported in ABAQUS/Explicit, this Poisson’s ratio was selected as the highestsafe” value to avoid pressure locking caused by the incompressibility of theread material �17�.

alidation of Tread Material ModelThe effectiveness of the linear viscoelastic model in ABAQUS/Explicit has

een validated via experiments using the resonance method described in Ref.22�. As shown in Fig. 2, a simple test rig has been set up where a tread blocks fixed on a solid bench. A loading plate is bonded on top of the tread block inhich an accelerometer is embedded. Provided the mass of the loading plate isuch greater than that of the tread block, this setup can be regarded as a

ingle-degree-of-freedom system where a single mass is connected to a dampedpring. An impact hammer has been used to excite the system to its resonancerequency by hitting the center of the loading plate. Signals from both theccelerometer and impact hammer have been logged using a computer equippedith an NI-6024 data acquisition card. The resonance frequency in each test has

FIG. 1 — 24-term Prony series curve-fit for measured tread compound property.

een identified by performing a fast Fourier transform �FFT� for the logged

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LIU ET AL. ON MODELING OF TREAD BLOCK 215

cceleration data. By varying the size of the tread block and the mass of theoading plate, resonance frequencies up to 1.4 kHz have been achieved in thexperiments.

A model has been built in ABAQUS/Explicit to simulate the tread blockssed in the experiments, with their material model defined by the parametersbtained in the DMA test. The logged impact hammer forces have been used inhe model to replicate exactly the same loading conditions in the experiments.he predicted time domain results are plotted in Fig. 3 together with the experi-ental data for three resonance frequencies. It has been observed during the

alidation that the resonance frequencies predicted by the FE model match thexperimental data well up to about 1.4 kHz. The damping of the tread materialas been modeled well in the lower and medium frequency range althoughlightly underestimated in the higher frequency range above 1 kHz. The full setf results for the predicted resonance frequencies from the FE model are com-ared with the experimental results in Table 1.

ontact ModelTo investigate the tread block impact and release mechanisms via FE analy-

is, a simple approach has been taken in which the top nodes of a tread blockre tied to a rigid ring, as shown in Fig. 4. The tread block is meshed using the-node linear brick element C3D8R with reduced integration and hourglassontrol. The mesh is refined at the bottom of the tread block where contact takeslace. The ring then rotates at a prescribed speed to bring the tread block intoontact with the counter surface. The counter surface in this model is also aigid ring whose radius is three times that of the ring that the tread block is tiedo. This ring rotates at the same tangential speed as the surface nodes of theeformed tread block to create a free-rolling condition. The ratio between thewo radii matches that of the test rig described in the later sections and is alsolose to that of the full-size tire drum test rig of our industrial partner. The first

FIG. 2 — Setup of test using resonance method to validate material model.

alf of the contact motion created in this model corresponds to the impact and

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216 TIRE SCIENCE AND TECHNOLOGY

he second half corresponds to the release motion of the tread block. By varyinghe geometry of the tread block and the topography of the counter surface, thisodel enables the simulation of tread blocks of various shapes in contact with

urfaces of different roughnesses.As Persson et al. �23� have pointed out, the coefficient of friction �COF� of

IG. 3 — Validation of linear viscoelastic material model via resonance method. (a) 298 Hz test;b) 502 Hz test; (c) 1364 Hz test.

ubber is both pressure- and slip-rate-dependent. To simulate the contact pair in

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LIU ET AL. ON MODELING OF TREAD BLOCK 217

ur test rig described in the section, “Tread Block Contact Experiments,” i.e.,read rubber vs steel, the COF has been measured as a function of appliedressure and sliding velocity using a simple pin-on-plate tribometer. A diagramf the tribometers is shown in Fig. 5, where a small piece of tread rubber samples fixed on the tip of a sample holder which is pressed onto a steel plate using

spring. A linear actuator provides horizontal motion for the plate at differentpeeds. The normal and tangential forces are measured using a biaxial load cell

TABLE 1 — Linear viscoelastic material model validation results.

est number

Resonance frequency

Experimental results �Hz� Finite element results �Hz�1 96 1032 298 2763 469 4594 502 5075 883 8616 1233 12027 1364 1369

FIG. 4 — Normal stress distribution during contact.

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218 TIRE SCIENCE AND TECHNOLOGY

nd the coefficient of friction can be derived. The linearly interpolated measure-ent results are presented as a map in Fig. 6. This COF map has been imple-ented in ABAQUS/Explicit via its VFRIC user subroutine.

iscoelastic Spring Model

Although FE analysis is a robust numerical tool to investigate tread blockorces, it is desirable to develop a simplified numerical model which can beoupled to the tire belt vibration model to achieve the ultimate goal of predict-ng dynamic hub forces. In this section we propose a tread block contact me-hanics model that is based on discrete viscoelastic spring elements.

In Winkler elastic foundation theory �24�, an elastic solid can be discretizednto uncoupled springs. The original theory only models each spring as purelylastic with no damping. In our case, damping can be introduced into the modely making the spring constant time dependent. As shown in Fig. 7, consideringhat the tread block has a surface area of A and a thickness of d, the time-ependent stiffness k�t� of the tread block is then given by

FIG. 5 — Pin-on-plate tribometers used to characterize friction coefficient of tread sample.

FIG. 6 — Pressure and slip-rate dependent friction coefficient.

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LIU ET AL. ON MODELING OF TREAD BLOCK 219

k�t� =E�t�A

d�3�

here E�t� is the relaxation modulus defined in Eq �1�. If the tread block haseen discretized into n springs, each spring will then have a time-dependent

pring stiffness k̄�t� which is expressed as

k̄�t� =k�t�n

�4�

uring contact, a discrete spring is subject to a displacement ��t�. The contactorce on this spring is then given by a convolution integral as

F�t� = �−�

t

k̄�t − ��d�

d�d� �5�

he advantage of the viscoelastic spring model is that the geometrical complex-ty of a tread block can be easily accounted for by applying a nonuniformpatial discretization, i.e., there are no springs in the areas where sipes areresent on a tread block.

The same pressure and slip-rate-dependent coefficient of friction can bencorporated in this model by tracing the force and slip rate for each spring in

FIG. 7 — Discretization of tread block.

very time step.

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read Block Contact Experiments

A high-speed rolling test rig has been developed to simulate the impact andelease motions of tread blocks. As shown in Fig. 8, the test rig consists of amall wheel of 0.2 m diameter that is mounted directly on a solid base and a bigheel of 0.6 m diameter that is connected to the base by a rocking arm. Aiaxial �normal and tangential� load cell is embedded in the small wheel tohich the tread block sample is attached via a sample holder. The small wheel

s driven by an electric motor with controlled speeds up to 600 rpm. A rubberelt is attached around the small wheel with a cutout window to accommodatehe tread block sample. This belt is used to transmit torque to the big wheel. Theurface of the tread block sample is at the same level as that of the rubber belt.nce the big wheel is accelerated to the same tangential speed as the smallheel, its inertia helps maintain its speed, with only minimal power drawn from

he small wheel to overcome the frictional loss on the shaft and the damping

(b)

(a)

FIG. 8 — Tread block rolling test rig.

oss in the contact area with the rubber belt. With the rubber belt continuously

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LIU ET AL. ON MODELING OF TREAD BLOCK 221

n contact with the big wheel, a constant tangential speed is achieved on the bigheel, even at the moments when the tread block sample impacts and loses

ontact with the big wheel. This ensures a nearly free-rolling condition for thenvestigation of contact mechanics of tread blocks. Two locking bars are used toock the shafts of the two wheels in position with a prescribed degree of inter-erence between the rubber belt and big wheel to create compression for theread block sample.

After the degree of interference is set by securing the locking bars, theheels are driven to the required speed. At the same time, the force signalsenerated by the load cell are amplified by the bridge amplifiers and transmittedhrough a slip ring unit off the rotating wheel. A thermocouple that is in contactith the tread block sample provides in situ monitoring of the operating tem-erature of the tread block during testing. The motor speed is controlled by aoeller DF51 electronic inverter control unit and all signals from the sensors

ave been logged during the experiments using the same computer described inhe section, “Finite Element Model.”

esults and Discussion

To demonstrate the robustness of both the FE and viscoelastic spring mod-ls, we present the modelling results based on a plain tread block of rectangularhape, together with the experimental results. Since the load cell in the test rigeasures the forces applied on the entire tread block, the results from the FE

nd viscoelastic spring models are produced as the summation of the forces ofll the surface nodes and all the discrete springs, respectively, in each time step.

The dimensions of the tread block were 16�10�6.35 mm. The interfer-nce was set in the range of 0.1 to 0.3 mm. During each test, the small wheelas accelerated to 300 rpm and left to run for 1 min. Within this period of time,

he operating temperature was maintained at 22 °C with no significant risebserved. A spatial resolution of 0.5 mm was used to discretize the tread blockor the viscoelastic spring model. The normal forces predicted by both modelsre compared with the experimental data in Fig. 9.

It can be seen from the results that the normal contact force rises sharply ashe leading edge of the tread block impacts the counter surface. The normalontact force reaches its peak value as the contact patch extends towards theenter of the tread block where the maximum overall compression is achievedn the sample. As the tread block starts to be released from the contact patch,he normal force declines quickly and reaches zero as the trailing edge losesontact. The results from both the FE and viscoelastic spring models agree wellith the experimental data. At the higher compression of 0.3 mm, both models

ave predicted slightly higher forces than the measured results. Besides the

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222 TIRE SCIENCE AND TECHNOLOGY

anufacturing error in the tread block sample dimensions, the nonlinearity ofhe tread material at a higher load, i.e., above 2.5 MPa at 0.3 mm compression,ost likely has contributed to the discrepancy between the measured and pre-

IG. 9 — Comparison between modelling and experimental results. (a) 0.1 mm interference; (b).2 mm interference; (c) 0.3 mm interference.

icted results.

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LIU ET AL. ON MODELING OF TREAD BLOCK 223

The results also indicate that at 22 °C the maximum contact pressure on theread block exceeds 1.5 MPa even with only 0.2 mm interference. This value isuch higher than the actual normal pressure imparted on the tread blocks of an

rdinary passenger car tire whose maximum value is about two times the infla-ion pressure of 0.2 MPa �25�. Although the typical operating temperature ofires is between 50 and 60 °C, the DMA tests have shown that the modulus ofhe tire tread only decreases by approximately 50% at 60 °C. By applying the

LF transform and a temperature field in the FE model, further analysis hashown that given the same interference of 0.2 mm the maximum contact pres-ure decreases to 0.96 MPa at 60 °C, still significantly higher than the realisticressures on the tread blocks of a free-rolling tire. This leads to the conclusionhat on a free-rolling tire the strains on tread blocks are sufficiently small that ainear viscoelastic material model is suitable for modeling the dynamic behaviorf tread blocks.

The bi-axial load cell has also enabled the measurement of tangential forcesn the tread block samples. Figure 10 illustrates the predicted and measuredorce histories in both the normal and tangential directions for a tread blocknder testing at three speed settings �150, 300, and 600 rpm� and at the sameaximum compression of 0.2 mm. It can be seen that the FE model predicts a

inusoidal-like force history in which the positive force in the first half of theontact cancels the negative force in the second half. This corresponds to theree-rolling condition imposed on the model, in which no net force is presenturing the contact period other than that required to overcome the damping lossn the tread block. In contrast, the experimental data indicate some net forces onhe tread block which shifts from positive traction at lower speed to negativeraction at higher speed. This is due to the fact that the free-rolling condition inhe test rig is maintained mainly by the inertia of the big wheel. At lower speed,ince the inertia of the big wheel is not large enough to overcome the dampingoss in the rubber belt, some power is drawn from the tread block to acceleratehe big wheel. At higher speed, since the inertia of the big wheel is larger thanequired to overcome the loss due to a reduced contact area in the cutoutindow, the big wheel is subject to deceleration as the tread block makes

ontact with the big wheel. The magnitudes of both the normal and tangentialorces increase slightly with the applied speeds due to viscoelasticity, as pre-icted by the FE model.

Although only normal force data are presented in this paper for the vis-oelastic spring model, the model is equally capable of predicting tangentialorces. The prediction of tangential force in the viscoelastic spring model ischieved by introducing tangential spring elements with appropriate time-ependent stiffness. It should also be noted that the magnitude of the tangentialorce obtained in the high-speed rolling test rig is about 1 /10 that of the normal

orce. This indicates that no slip has been present in the tread block contact in

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224 TIRE SCIENCE AND TECHNOLOGY

he test rig, given the relatively high coefficient of friction shown in Fig. 6. Onfree-rolling tire, however, the ratio between the maximum tangential and

ormal force magnitudes is normally between 1 /3 and 1 /2 �25� and some slipay be induced on the tread blocks in the impact and release motions. This is

IG. 10 — Normal and tangential forces on a tread block at 0.2 mm maximum compression (a),50 rpm, (b) 300 rpm, and (c) 600 rpm.

ue to the fact that the contact kinematics of the rig differs from that of a

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eformed tire belt. The analysis of the tangential force generation mechanismnd the prediction of tangential force using viscoelastic spring model based onhe actual geometry of a deformed tire belt will be reported at later stages of ourroject. The future work will also include the investigation of effects of shape,emperature, and counter-surface roughness for the tread blocks, through mod-ling and experiments using the high-speed rolling rig.

onclusions

A FE model has been developed to predict contact forces on a tread blockuring its impact and snap-out motions. A simplified model has been proposedhich uses discrete viscoelastic spring elements. A pressure and slip-rate-ependent coefficient of friction has been adapted to both models. The twoodels have proved capable of producing results that agree well with the ex-

erimental data obtained in a high-speed rolling test rig. The investigation hashown that a linear viscoelastic model defined by a Prony series expansion isufficient in modeling the dynamic behavior of tread blocks.

cknowledgmentsThis work is supported by the UK Engineering and Physical Sciences Re-

earch Council �grant number: EP/D011035/1�. The authors are most grateful tour industrial partners Jaguar Cars Ltd., Land Rover, and Goodyear Tire &ubber Company, for their financial and technical support and supply of testaterials. Thanks are also due to Dr. Roger Bradshaw for his advice on Prony

eries and to Mr. Robert Cornell for his help with the DMA tests.

eferences

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4� O’Boy, D. J., and Dowling, A. P., “Development of an Automotive Tyre Vibration Model foruse in a Noise Prediction Program,” in Proceedings of INTERNOISE 2004, 2004.

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7� Pinnington, R. J., “A Wave Model of a Circular Tyre. Part 2: Side-Wall and Force Transmission

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226 TIRE SCIENCE AND TECHNOLOGY

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17� ABAQUS/Explicit User’s Manual, Version 6.5, Hibbitt, Kalsson & Sorensen, Inc., 2005.18� Gnörich, W., Goodyear Tire & Rubber Company, personal communication.19� Park, S. W., and Kim, Y. R., “Fitting Prony-Series Viscoelastic Models with Power-Law Pres-

moothing,” Journal of Materials in Civil Engineering, Vol. 13, 2001, pp. 26–32.20� Williams, M. L., Landel, R. F., and Ferry, L. D., “The Temperature Dependence of Relaxation

Mechanisms in Amorphous Polymers and Other Glass-Forming Liquids,” Journal of the Ameri-can Chemical Society, Vol. 77, 1955, pp. 21–39.

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22� Oyadiji, S. O., “How to Analyse the Static and Dynamic Response of Viscoelastic Compo-nents,” NAFEMS, 2004.

23� Persson, B. N. J., and Tosatti, E., “Qualitative Theory of Rubber Friction and Wear,” Journal ofChemical Physics, Vol. 112, 2000, pp. 2021–2029.

24� Johnson, K. L., Contact Mechanics, Cambridge University Press, 1985.25� Koehne, S. H., Matute, B., and Mundl, R., “Evaluation of Tire Tread and Body Interactions in

the Contact Patch,” Tire Science and Technology, TSTCA, Vol. 31, 2003, pp. 319–338.