tire modeling

Tyre/road friction modelingLiterature survey

R. van der Steen

DCT 2007.072

Coaches: I. LopezB. de BruijnA.J.C. Schmeitz

Supervisor: H. Nijmeijer

Eindhoven University of TechnologyDepartment of Mechanical EngineeringDynamics and Control group

Eindhoven, May, 2007

Contents

List of symbols 5

1 Introduction 7

2 Materials 11

2.1 Elastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Visco-elastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Friction models and surface roughness 15

3.1 Amontons-Coulomb friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Friction of viscoelastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Finite Element Method 21

4.1 Contact and friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Literature related to FEM and tyres . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Validation and modeling methods 31

5.1 LAT1OO and Tekscan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Eplexor and DIK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3 Flat plank and tyre test trailer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.4 Tyre modeling in ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Conclusions 35

6.1 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.2 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A 32th Tire Mechanics Short Course 39

A.1 Strength, wear and friction of rubber. . . . . . . . . . . . . . . . . . . . . . . . . . 39

A.2 The tire as a vehicle component. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

A.3 Tire materials and manufacturing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.4 Rules and regulations governing tires. . . . . . . . . . . . . . . . . . . . . . . . . . 44

A.5 Advanced tire modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A.6 Tire wear, traction and force generation. . . . . . . . . . . . . . . . . . . . . . . . . 47

A.7 Tire stresses and deformation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 50

Bibliography 53

List of symbols

Symbol Unit Description

α [rad] Slip angleγcr [m] Elastic slipγcr [m/s] Elastic slip rateδ [rad] Loss angleθ [ ] Field variableκ [-] Longitudinal slipλ [m] Length scaleν [-] Poisson’s ratioµ [-] Coefficient of frictionµk [-] Kinetic friction coefficientµf [-] Maximum friction coefficientµr [-] Rolling friction coefficientµs [-] Static friction coefficientφT [Hz] Segmental jump rate at temperature Tω [rad/s] Spinning velocity

aT [-] WLF shift factorci [ ] Constantd [m] Mean plane distancef [Hz] FrequencygN [m] Normal gap distancegT [m] Relative tangential displacementgT [m/s] Relative tangential velocityh [-] Fit parameterk [-] Material constantl [m] Characteristic surface lengthpN [Pa] Normal pressuretT [Pa] Shear stressz [m] Distance to wheel center

Symbol Unit Description

A [m2] Contact areaE [Pa] Young’s modulusE∗ [Pa] Complex Young’s modulusFn [N] Normal forceFx [N] Longitudinal forceFt [N] Tangential forceFr [N] Rolling friction forceG [Pa] Shear modulusG′ [Pa] Dynamic shear modulusG′′ [Pa] Dynamic loss modulusG∗ [Pa] Complex shear modulusK [Pa] Bulk modulusM [N/m] TorqueMz [N/m] Self aligning momentR [m] RadiusT [K] TemperatureTg [K] Glass temperatureVs [m/s] Sliding velocityW [N] Normal load

B [-] Magic formula coefficientC [-] Magic formula coefficientD [-] Magic formula coefficientE [-] Magic formula coefficientFf [-] Slip toleranceG [Pa/m] Penalty coefficient

Chapter 1

Introduction

This literature report has been written in the scope of the CCAR project FEM Tyre Modelling.This project aims to realize a thorough base in modeling of a tyre using a finite element method.An import aspect is the development of a robust friction model to include tyre/road interaction.The report has been written to obtain a basic understanding in the field of (rubber) friction andthe finite element method. An overview is presented with respect to friction models and surfaceroughness, contact and friction in the finite element method, and tyre analyses using the finiteelement method. It can therefore be used as a reference guide for future steps within the CCARproject.

Since the patent on a pneumatic tyre, by John Boyd Dunlop in 1888, the development of pneu-matic tyres started and is still going on today. The worldwide tyre production is now greater than1 billion units per year with a market value exceeding US $100 billion. Around 73% are passengercartyres and 70% of these tyres are replacement tyres. There are basically two types of tyres, thebias ply and, since 1948, radial ply tyres. Nowadays most of the passenger car tyres are radialtyres. The main difference is the orientation of the body plies, see figure 1.1.

(a) Bias tyre. (b) Radial tyre.

Figure 1.1: Difference between bias and radial tyres (picture taken from Gent (2007)).

The tyre construction, such as aspect ratio, belt construction and tread compound, dependson the size and target market (speed rating). This information is printed on every tyre, e.g.,215/55 R 16 97 V. The first number (215) is the nominal section width in mm, the second num-ber (55) is the percentage of the height/width ratio of the cross-section. The R (radial) standsfor the tyre construction code. Then the rim diameter (16) is given in inches and 97 is the loadcapacity index, which indicates the maximum load capacity. The last symbol (V) is the speed

8

symbol, which stands for the maximum speed. On winter tyres an additional logo is printed to-gether with the mud and snow (M&S) symbol. See table 1.1 for an overview of load indices andthe speed symbols for cars.

Table 1.1: Load indices and speed symbols for car tyres.

Load index Load (kg) index kg index kg index kg Symbol Speed (km/h)65 290 79 437 93 650 107 975 K 11066 300 80 450 94 670 108 1000 L 12067 307 81 462 95 690 109 1030 M 13068 315 82 475 96 710 110 1060 N 14069 325 83 487 97 730 111 1090 P 15070 335 84 500 98 750 112 1120 Q 16071 345 85 515 99 775 113 1150 R 17072 355 86 530 100 800 114 1180 S 18073 365 87 545 101 825 115 1215 T 19074 375 88 560 102 850 116 1250 U 20075 387 89 580 103 875 117 1285 H 21076 400 90 600 104 900 118 1320 V 24077 412 91 615 105 925 119 1360 W 27078 425 92 630 106 950 120 1400 Y 300

Basically a tyre should carry load and transmit forces. However the design of a tyre is a complextrade-off problem, because a tyre needs to perform under varying conditions. In figure 1.2 themain components of a radial tyre can be seen. All these different components provide specificproperties.For example the tread should have good traction for all weather conditions, low rolling resistanceand low wear. But low rolling resistance and a long lifetime require a hard compound, which hasless traction. For good wet traction groves in the tread are required to drain off water, but thismeans less dry traction since the contact area is decreased. And softer compounds, which providemore traction than harder compounds, also wear faster.Besides the engineering aspects, economical factors, (limitations of) the manufacturing process(Walter, 2007a) and government regulations (Walter, 2007b) have to be taken into account.

To model a tyre and the interaction with the environment different modeling approaches can beused: analytical models, empirical models or physical models. In the following the focus is onphysical models to be used in a Finite Element Analysis (FEA). Information of the used materialsand geometry of the tyre and an interaction with the road is required for the Finite ElementMethod (FEM).The basic materials used in tyres and material models are described in chapter 2. In chapter 3an overview of different friction models is given and the relation with surface roughness is shown.Chapter 4 covers the basics of FEM and numerical methods related to contact. Some aspects ofcontact in ABAQUS are mentioned and an overview of different types of tyre problems is given.Validation possibilities are given in chapter 5 and finally conclusions are presented in chapter 6.An additional appendix, which is a short summary of the 32th tire mechanics short course is alsoincluded.

1. INTRODUCTION 9

Figure 1.2: Components of a radial tyre (picture taken from Gent (2007)).

Chapter 2

Materials

Different materials are used in tyres. The rubber compounds are the most complex parts of thetyre. The principal tyre elastomers are natural rubber (NR), polyisoprene (IR, this is syntheticNR), styrene butadiene rubber (SBR), butyl rubber (IIR) and butadiene (BR). A compound canbe defined as the formulation of a mixture of rubber and additives which meets the needs of thetyre component application. Usually a compound consists of one or more polymers, vulcanizingagents, accelerators, reinforcing fillers, antidegradents, plasticizers, softeners and tackifiers (usedto bond pieces together). This explains why the properties of a compound can deviate from a pureelastomer. Bead wires are usually made of steel. Belt ply cords consist of steel or rayon, whilebody plies cords can be made of nylon or polyester. For a FE model the geometry of the parts,the applied loading and boundary conditions and the material behavior of each of the differentmaterials is required. In this chapter the basics of elastic and visco-elastic material behavior arementioned.

2.1 Elastic

2.1.1 Small strains

For small strains the material properties can be defined by the shear modulus G and the modulusof bulk compression K, which are related to the tensile or Young’s modulus E and Poisson’s ratioν as follows:

E = 2(1 + ν)G, (2.1)

ν =12

3K − 2G

3K + G. (2.2)

When a material behaves elastically, there is no permanent deformation and dissipation. Thismeans that if the stress is released the strain will become zero. Furthermore there is no time orpath history, thus the stress can be calculated if the strain is known.The strains in the belt and body plies cords, under normal conditions, are in the linear elasticrange, which means that they can be described by either G and K or E and ν.

2.1.2 Large strains

Rubber is an elastic, soft and virtually incompressible material, but it can usually be stretchedmore than 500%. Its molecular structure consists of long, linear flexible molecules forming randomcoils. The molecular segments are mobile, so-called Brownian motion (rate of segmental jumps,

12 2.1. ELASTIC

Figure 2.1: Polymer properties as function of temperature (picture taken from Gent (2007)).

which depends strongly on temperature), and the molecules are interlinked into a 3D network.Chemical crosslinks are usually made by sulfur linkages also known as vulcanization. The rubberystate of a polymer is determined by the so-called glass transition temperature Tg. If the tempera-ture is above Tg the polymer is rubbery, below Tg the polymer is glassy, see figure 2.1. For rubbersthe Tg is below room temperature.

For rubber typically K G and it follows that E = 3G and ν = 0.5. If a rubber block withν = 0.5 is compressed, it is sheared outwards and generates an internal pressure. This effectcauses a compression modulus, which is generally greater than E. To model this behavior otherconstitutive models have to be used. Hyperelastic material models are based on energy functions,which define the strain energy stored in the material per unit of reference volume, instead of theYoung’s modulus (E) and poisson ratio (ν) for isotropic linear elastic materials. In ABAQUSseveral hyperelastic material models are available, an overview can be seen in table 2.1. A specificmaterial model is usually chosen based on available test data, however it is also possible to enterspecific material constants directly. It is possible to fit material models on experimental data, suchas uniaxial, biaxial, planar and volumetric test data. For more details see the ABAQUS Analysisuser’s manual, chapter 17.

2.1.3 Filled rubbers

In tyres rubbers are usually filled with particles like carbon black or silica. This creates twoadditional effects. The so-called Payne effect and Mullins effect, see e.g. Diani et al. (2006), whichare both softening effects. The Payne effect is a softening effect for small strains (0.1%), attributedto breaking apart aggregates of filler particles. The Mullins effect is a substantial softening effectat higher strains, attributed to progressive detachment of rubber molecules from filler particles.The Mullins effect can be modeled in ABAQUS, but it cannot be used together with visco-elasticityor hysteresis options. In Bergstrom (2005) a new constitutive material model is presented, which iscompared with the standard ABAQUS material models. In Becker et al. (1998) a material model

2. MATERIALS 13

Table 2.1: Available material models in ABAQUS.

Material model:Neo-HookeanMooney-RivlinOgdenMarlowArruda-BoyceYeohVan der WaalsPolynomialReduced polynomialUser defined

for filled rubbers is presented, which is implemented into an in-house FE code and compared toexperiments.

2.2 Visco-elastic

The constitutive laws for large strains cannot be used to fully describe the stress-strain relation,since rubbers do not follow reversible stress-strain relations. When rubber is dynamically stretchedand released the returned energy is less than the energy that is put into the rubber. This visco-elastic effect cannot be described by the perfect elastic shear modulus G, but a dynamic shearmodulus G′ and a dynamic shear loss modulus G′′ is introduced to describe this hysteresis. Anotherterm that is frequently used is the loss angle, which is defined as tan δ = G′′

G′ . For an elastic solidthe strain is in phase with the stress, while for a visco-elastic solid the strain lags behind the stresswith a delay of δ

ω , where ω is the frequency (rad/sec).The universal WLF relation, named after Williams, Landel and Ferry, (Williams et al., 1955)describes the dependence of the segmental jump rate on temperature,

logφT

φTg

=17.5(T − Tg)52 + T − Tg

, (2.3)

where φTgis defined as 0.1/sec. Some polymers, such as butyl rubber, behave according to a

different relation

logφT

φTg

=17.5(T − Tg)100 + T − Tg

. (2.4)

This is possibly due to unusually long moving segments. The WLF relation can by used to predictresults at other temperatures, e.g. the frequency f at temperature T is equivalent to a frequencyf∗ at T ∗, using the following relation

f∗

f=

φT∗

φT= aT , (2.5)

where aT is known as the shift factor, this equivalence also holds for strain rates or velocities.To create master curves of material properties of most (pure) polymers this transformation canbe used. Normally experiments can only be done for a limited range of frequencies. If the sameexperiments are conducted at different temperatures, a master curve can be made for a chosenreference temperature by shifting the measurements using (2.5). An example is shown in figure2.2.Details of a FE implementation of 3D visco-elastic behavior for small and finite strains can befound in an article of Kaliske and Rothert (1997).

14 2.2. VISCO-ELASTIC

Figure 2.2: Creation of master curve for E∗ using WLF shifts (picture taken from Gent (2007)).

Chapter 3

Friction models and surfaceroughness

In this chapter several friction models are discussed. Friction is present in all mechanical systemsand originates at an interface between bodies, which are in contact with each other. This contactenables solids to transmit forces through the surfaces. The resultant force can be resolved intoa normal and a tangential force. This is referred to as the normal force and the friction forcerespectively. Friction can now formulated as the ratio between the tangential and normal contactforce. Forces can only be transmitted if there is contact, therefore the contact area plays animport role in describing friction aspects. Several theories have been developed to describe surfaceroughness and the relation to the real contact area.It is well-known that contaminations have a strong effect on friction, e.g. lubricants. One of theclassical works on friction and lubrication is the book of Bowden and Tabor (2001).

3.1 Amontons-Coulomb friction

This is the most used friction model, usually referred to as the Coulomb friction model. It is basedon the two basic laws of friction (Bowden and Tabor, 2001). When a block is pressed upon a planewith a normal force and is pulled in the horizontal direction, there is a threshold value such thatthe block starts to move if the tangential force is bigger than this value. This threshold value isproportional to the normal force Ft = µFn, with µ the so-called coefficient of friction.The friction force always works in opposite direction of the moving velocity.Coulomb also showed that kinetic friction could be significantly lower than static friction, whichis sometimes referred to as the third friction law.

3.1.1 Deviations from Coulomb friction

Experiments often show deviations from the basic Coulomb friction model. Variants of the basismodel are the difference between a static µs and a kinetic µk friction coefficient, see figure 3.1 forsome examples. This transition can be discontinuous or continuous, using an exponential decayingfunction between the static and kinetic coefficient. More general, friction varies with slidingvelocity. An example is the so-called Stribeck curve, first the coefficient decreases to a minimumvalue and after that it increases with higher sliding velocities. The review paper of Olsson et al.(1998) describes several static and dynamic friction models. The static models include Coulombfriction, viscous friction and the models of Karnopp, where a zero velocity interval (deadzone) isintroduced, and Armstrong, where stiction and sliding are decoupled in two separate equations.The dynamic models include the reset integrator model, the Dahl model and two variants of this

16 3.2. FRICTION OF VISCOELASTIC MATERIALS

µ µ µ

µs µs µsµk

µkµk

Vs Vs Vs

µ

µs

µk

Vs

Figure 3.1: Coulomb friction model and some variations.

model, the Lugre model and the model of Bliman and Sorine.More details on the transition between sliding and sticking can be found in an article of Rabinowicz(1951). All of these models are often used in control systems, in such a way that they can be usedfor friction compensation.

3.1.2 Magic Formula

In the book of Pacejka (2006) the empirical magic formula tyre model is explained. This model iswidely used for vehicle handling simulations and based on the ‘magic’ formula

F (x) = Dsin(C arctan((1−E)Bx + E arctan(Bx))), (3.1)

where x is an input, e.g. α or κ and F an output such as Fx or Mz. The coefficients B, C, Dand E are fitted using experimental data. This model can accurately capture the global force andmoment characteristics of a tyre.Another frequently used model is the brush model, more details can be found in the sheets ofBesselink (2003) and the book of Pacejka (2006). This model is useful for a qualitative analysisof tyre behavior when limited measurement data is available.

3.2 Friction of viscoelastic materials

Friction of polymers is closely related to their viscoelastic behavior. Generally speaking, thecoefficient of friction increases with sliding velocity until a maximum value is reached at a certainspeed, followed by a decrease of the friction coefficient. This is caused by the flexibility of thepolymer chains.

3.2.1 Grosch

The article of Grosch (1963) is one of the early reports, which describes a relation between thedependence of the friction coefficient due to sliding velocity and temperature and the viscoelasticdeformation on strain rate and temperature. In this work experimental results of different vul-canized rubbers are presented. Experiments are conducted at different temperatures and shiftedto a master curve using the WLF equation. This gives a relation between the sliding velocity andtemperature in a similar way as the rate and temperature dependence of viscoelastic polymersabove their glass temperature, see figure 3.2. Grosch indicated that friction is due to energy dissi-pated when rubber is compressed and released by asperities. For dry sliding on a smooth surfacethe curve for µ vs. sliding speed resembles again the curve for tan δ versus frequency f . But nowit is displaced to much lower speeds than before. Friction in dry sliding on a smooth surface isdue to energy dissipated as rubber sticks and slips on a molecular scale. In the case of slidingon a rough dry surface the two dissipative processes appear at different speeds, corresponding to

3. FRICTION MODELS AND SURFACE ROUGHNESS 17

different length scales: asperities and molecules, see figure 3.3. Several contributions to µ canarise at the same sliding speed from stick-slip processes occurring at different length scales. Anearly theory to describe the experiments of Grosch has been developed by Schallamach (1963).An unified approach to the entire subject of friction and wear in rubbers and tyres can be foundin an article of Moore (1980).

Figure 3.2: Grosch interpretation (picture taken from Gent (2007)).

Figure 3.3: Combined effect of sliding on a rough and dry surface (picture taken from Gent (2007)).

3.2.2 Savkoor

Savkoor proposed a model, based on results of Grosch, where he described the shape of an isother-mal master curve with an empirical relation (Lemaitre, 2001, chapter 8),

µk(Vs) = µs + (µm − µs)exp

(−

(h2

2

)log2

(|Vs|Vmax

))(3.2)

where µs is the static coefficient of friction and µm the maximal coefficient for |Vs| = Vmax. h isa dimensionless parameter reflecting the width of the speed range in which friction varies signifi-cantly. More details on dry adhesive friction can be found in (Savkoor, 1965, 1966, 1986) and inthe thesis of Savkoor (1987). The advantage of (3.2) is that it can be evaluated easily for a givensliding velocity.

18 3.3. SURFACE ROUGHNESS

3.2.3 Rolling friction

Rolling friction Fr can be seen as MR , where M is a torque and R the radius of the rolling body.

For an elastic wheel the reaction force acts through the wheel center and therefore the momentM = 0. For a visco-elastic wheel the reaction force moves towards the loading side, becausestresses are lower in unloading, M = Wz where W is the normal load and z the distance from thewheel center, see figure 3.4.Displacement of the center of pressure, and hence the coefficient of friction µr, depend on speedand temperature in the same way as the loss factor tan δ. A formulation for µr as proposed byGreenwood et al. (1961),

µr = 0.33(

W

G∗R2

) 13

tan δ, (3.3)

where G∗ is the complex shear modulus. Rolling friction is mainly due to energy dissipated asrubber is compressed and released in the contact patch.

Figure 3.4: Rolling friction (picture taken from Gent (2007)).

3.2.4 Persson

Persson has published many papers (Persson, 1993, 1995, 1997, 1998, 1999, 2001a,b, 2002a,b;Persson and Tosatti, 2000; Persson and Volokitin, 2000, 2002, 2006; Persson et al., 2002, 2003,2005) on the subject of rubber friction and the role of the surface, which is in contact with therubber. These theories are a continuation of the early studies of Grosch, taking into account thatsliding friction of rubber has the same temperature dependence as that of the complex elasticmodulus. He states that the friction force (under normal circumstances) is related to the internalfriction of the rubber, which is a bulk property. The hysteretic friction component is determinedby sliding of the rubber over asperities of a rough surface. These oscillating forces lead to energydissipation. These contributions can be described with a fractal description of the surface. Everylength scale λ, up to the largest particles of asphalt, can be related to a frequency: f ∼ Vs

λ .As a result of energy dissipation heat is generated, in a recent article (Persson, 2006a) he also takesthe local heating of rubber into account, since the viscoelastic properties are strongly temperaturedependent. At very low sliding velocities the temperature increases are negligible because ofheat diffusion, but already for velocities in the order of 0.01 m/s local heating plays a role. Heshows that in a typical case the temperature increase results in a decrease in rubber friction withincreasing sliding velocity, if the sliding velocity is above 0.01 m/s.

3.3 Surface roughness

There is a long history on this subject, since it is well-known that almost all surfaces are rough ata certain length scale. This has great influence on the real contact area and therefore it is closelyrelated to friction. The real contact area is in most cases different from the apparent contact area

3. FRICTION MODELS AND SURFACE ROUGHNESS 19

and high local contact pressures can give rise to plastic deformation. The two often used methodsto describe surface roughness are statistical and fractal methods.In Archard (1957) it is shown that in order to obey the Coulomb friction law for elastic deformationa rough surface is actually needed. In elastic contact the area A is related to the load W by

A ∝ kW23 , (3.4)

where k is a constant. It is supposed that the friction force is proportional to the contact area,therefore it can not hold that the friction force is proportional to the load. For increasing loadthe friction coefficient has to decrease. To show this, Archard used a flat surface in contact with aspherical surface with large radius. However if instead of one spherical contact, several sphericalprotuberances on this initial sphere are modeled and thus multiple contact areas are created, thefollowing relation is derived

A ∝ W4445 . (3.5)

He showed that µ is now constant for an increasing load until all the local protuberances are flatand than µ drops again if the load is still increased further. This explains that for soft rubbersliding on a smooth surface (perfect contact) the frictional force is more or less constant, inde-pendent of the load W . Thus the coefficient of friction decreases continuously as the pressureincreases. However there is also a coefficient of friction, independent of pressure, for harder rubbercompounds, sliding on a rough surface. Apparently in this case contact is incomplete. An increasein pressure creates a larger true area of contact and hence a larger frictional force.

The Greenwood and Williamson (GW) contact model (Greenwood and Williamson, 1966) waspublished in 1966 and since then many revisions to this model have been made by several re-searchers. It is now known that some of the assumptions of the original model were incorrect(Greenwood and Wu, 2001).The GW model is based on contact between a rough surface and a flat rigid plane. It is a statisticaldescription of a rough surface, where the surface is covered with asperities. All these asperitieshave a spherical shape with the same radius of curvature. Further the summit surface density,the surface roughness and the surface height need to be specified. The surface height is supposedto follow a Gaussian or exponential distribution. Relations for the number of contacts, the realcontact area and the applied load are obtained.The model of Bush, Gibson and Thomas (BGT) (Bush et al., 1975) uses elliptical paraboloidsasperities with random aspect ratio and orientation, a comparison between these models can befound in an article of McCool (1986).

Fractal descriptions to describe a rough surface first appeared in the papers of Majumdar andBhushan (1990, 1991) and Majumdar and Tien (1990). If a surface exhibits geometrical self-similarity, also referred to as self-affinity, the surface appears the same under different degrees ofmagnification.This can be described by the Weierstrass-Mandelbrot function, which satisfies continuity, non-differentiability and self-affinity. The advantage of this approach is that the parameters, whichdetermine the isotropic rough surface, are independent of the roughness length scale and theresolution of the measuring instruments. This is in contrast to statistical parameters where theheight, slope and curvature of asperities depend on the the resolution of the measuring instrument.Fractal models are now widely used and compared by a great number of authors, e.g. Kluppeland Heinrich (2000); Gal et al. (2005); Morrow (2003); Jackson and Streator (2006); Ciavarellaet al. (2006a,b); Zavarise et al. (2007).Besides his work on rubber friction, Persson has also made a large contribution to the field ofcontact mechanics (Persson, 2002a; Persson et al., 2002; Peressadko et al., 2005; Persson, 2006b;Yanga et al., 2006; Persson, 2006c). An extensive overview of contact mechanics and rubberfriction can be found in a review paper of Persson (2005). A comparison of experiments and thepredictive capabilities of current physical theories is presented in Westermann et al. (2004). Thisdemonstrates the close interaction between rubber friction and surface roughness.

20 3.3. SURFACE ROUGHNESS

Chapter 4

Finite Element Method

The development of non-linear Finite Element Method (FEM) started about forty years ago to-gether with the development of the digital computer, with one of the first text books of Zienkiewiczand Cheung (1967). Although there are many books related to FEM nowadays, this book or latereditions are still cited in almost every article related to FEM.

There are two distinctive methods to solve FE problems, implicit and explicit. The implicit methodsolves for a statical or dynamical equilibrium, while the explicit method solves transient dynamicresponse problems using an explicit direct-integration procedure.

Besides the geometrical, material and interaction issues additional computational problems arise.Since the start of non-linear FEM many different types of elements and solution algorithms havebeen developed and implemented in specialized software such as ABAQUS, MARC, ANSYS, LS-DYNA or NASTRAN. This raises the question why one should study non-linear FEM if it isalready available in many commercial packages? The answer to this is that a commercial packageis basically a black box program providing an interface for simulations. The user has to makeseveral choices in the model formulation, i.e., the discretization of the mesh, the element types,the solution procedure, the solver tolerances etc. These selections (may) have a tremendous effecton the final result and can even lead to non-physical, thus meaningless, results (see Kouznetsova,2006).

There are four different sources of non-linearity in solid mechanics, geometrical, material, forceboundary conditions and displacement boundary conditions, see also figure 4.1.Unfortunately all four play a role in tyre modeling. One example is incompressible material be-havior of rubber. When the material response is incompressible, the solution to such a problemcannot be obtained in terms of the displacement history only, since a purely hydrostatic pressurecan be added without changing the displacements.

In ABAQUS this is solved by removing this singular behavior in the system by treating the pres-sure stress as an independently interpolated basic solution variable, coupled to the displacementsolution through the constitutive theory and the compatibility condition, with this coupling im-plemented by a Lagrange multiplier. This independent interpolation of pressure stress is the basisof the so-called hybrid elements. More precisely, they are ‘mixed formulation’ elements, using amixture of displacement and stress variables with an augmented variational principle to approx-imate the equilibrium equations and compatibility conditions. The hybrid elements also remedythe problem of volume strain ‘locking’. Volume strain locking occurs if the finite element meshcannot properly represent incompressible deformations. Volume strain locking can be avoidedin regular displacement elements by fully or selectively reduced integration (ABAQUS Analysis

22 4.1. CONTACT AND FRICTION

user’s manual).

Figure 4.1: Schematic view of non-linearities in solid mechanics (picture taken from Kouznetsova(2006)).

Two aspects are considered here, the mathematical formulations of contact and friction and im-plementation issues in ABAQUS, and computational models for tyres found in literature to givean overview of current possibilities.

4.1 Contact and friction

Starting points for this subject are the article of Doghri et al. (1998), the book of Laursen (2002)and especially the book of Wriggers (2006) on computational contact mechanics. Contact is splitinto normal contact and tangential contact.

4.1.1 Normal contact

pN

gN

Figure 4.2: Schematic view of Signorini conditions.

Normal contact can be formulated as a geometrical constraint, which describes the non-penetrationcondition of two bodies in contact, also known as ‘low contact precision’. This is the so-calledSignorini contact description, which models the non-penetrability of the bodies in contact withthe following relations. For contact it holds that gN = 0 and pN < 0, where gN is the gap distanceand pN is the normal pressure. If there is a gap between the bodies, than the pressure is equal tozero, see figure 4.2. This can be formulated as

gN ≥ 0, pN ≤ 0, pNgN = 0. (4.1)

4. FINITE ELEMENT METHOD 23

This formulation states that only compressive forces can be transmitted if there is contact (noadhesion) and forms the basis of frictionless contact problems.If for a contact problem knowledge of the micromechanical surface is essential to describe the phys-ical phenomena (‘high contact precision’) an interface law is needed to capture this phenomena.Instead of a geometrical constraint a constitutive equation is used, where the contact pressure isgiven by a general non-linear function of the mean plane distance

pN = f(d). (4.2)

An example, taken from Wriggers (2006), of this general non-linear function:

pN =c1(1617646.152 c3

c4)c2

5.5891+0.0.0711c2exp

[−1 + 0.0.0711c2

(1.363c3)2d2

], (4.3)

where c1 and c2 are mechanical constants which express the nonlinear distribution of the surfacehardness and c3 and c4 are statistical parameters of the surface profile.

4.1.2 Tangential contact

Usually friction is present between two interacting bodies and this leads to tangential forces.Tangential contact is described by constitutive laws. The response in tangential direction can bedivided into two cases. The first case is no tangential displacement in the contact zone (stick) andin the second case there is relative displacement (slip).Stick corresponds to a situation where the relative velocity (gT ) is zero and thus a zero relativedisplacement

gT = 0 ⇔ gT = 0, (4.4)

which poses a constraint on the motion in the contact interface.If the tangential forces reach a certain limit the contacting bodies will no longer stick, but slidingwill occur. In the case of Coulomb friction this can be formulated as

tT = −µ|pN |gT

||gT ||, if ||tT || > µ|pN |, (4.5)

where tT is the shear stress.

Since the friction coefficient usually depends on more parameters, a constitutive law can be writtenas a non-linear function of several parameters,

µ = µ(gT , pN , θ, . . .), (4.6)

where θ is defined as a field variable, e.g. temperature.

An example from Wriggers (2006) of a velocity and pressure dependent constitutive relation forrubber friction is the following

µ(vT , pN ) = c5

[pN

c6

]c7

+ c8 lnVs

c9− c10 ln

Vs

c11. (4.7)

Besides the Coulomb law (4.5), it is possible to formulate tangential constitutive equations byregularization of the stick-slip behavior. This is to avoid the non-differentiability of Coulomb’slaw at zero velocity, which can prevent numerical problems, some examples of this are squareroot, hyperbolic tangent of piecewise polynomial regularizations. Other formulations are in theframework of elasto-plasticity, where the key idea is to split the tangential slip in an elastic (stick)

24 4.1. CONTACT AND FRICTION

part and a plastic (slip) part.

Articles of Huemer et al. (2001a); Liu et al. (2000) and Huemer et al. (2001b) are related tofrictional behavior of rubber, where a phenomenological friction law for rubber tread blocks onsnow and concrete is presented for use in a macroscopic model. The coefficient of friction de-pends on normal pressure, sliding velocity and temperature. The friction coefficient itself is onlya function of normal pressure and sliding velocity. Temperature effects are incorporated using theWLF transformation, i.e., if the current temperature is different than the reference temperature,an equivalent new sliding velocity for the reference temperature is calculated.This research has been continued by Hofstetter et al. (2003), where a thermo-mechanical couplinghas been introduced. Energy dissipation during sliding is converted into heat and this heat fluxcauses a temperature rise of the rubber and road. In (Hofstetter et al., 2006) also simulations ofabrasion of the tread block are added.Dorsch et al. (2002) derived a phenomenological friction law using data obtained from experimentson the LAT100.The modeling and calculation of the transient rolling contact of tyres on road tracks is the subjectof a large multidisciplinary research project (FOR492) at the University of Hannover. One ofthe projects focusses on computational homogenization procedures, to develop a friction law at amacroscopic level, based on effects on a microscopic level. More details on this subject can alsobe found in the book of Wriggers (2006).

4.1.3 Constraint handling in ABAQUS

The aforementioned constraints, imposed by contact and friction, can be handled by Lagrangemultipliers, which satisfies the constraint exactly, or with penalty methods, which penalize vio-lation of constraints. All the constraint and interaction methods of ABAQUS can be found inthe ABAQUS Analysis user’s manual, chapters 28-32. One can find here that normal contact ishandled by Lagrange multipliers, penalty methods or an augmented Lagrange method. Friction,in ABAQUS/standard, is normally handled by the penalty method.While the surfaces are sticking, a certain elastic slip γcr is allowed and the magnitude of sliding islimited to this elastic slip. The allowable elastic slip is given as

γcr = Ff l, (4.8)

where Ff is an user-defined slip tolerance and l is the characteristic surface length. The slope G,which is a penalty stiffness, is adapted depending on the normal pressure, see figure 4.3. One cansee that the smaller γcr, the steeper the slope of G. Exact sticking, G = ∞, can be obtained byenforcing a Lagrange multiplier.However there is one important exception: If a so-called Steady-State Transport Analysis (whichis commonly used in tyre simulations) is performed, the penalty constraint is based on a maximumallowable slip rate

γcr = Ff2ωR, (4.9)

where ω is the spinning velocity and R the radius of the object. This introduces discontinuities if aSteady-State Transport Analysis is combined with other ABAQUS standard analyses and shouldtherefore be avoided.

4.1.4 Contact possibilities in ABAQUS

Several contact models are available in ABAQUS. In this section the import features are given.There are two contact discretizations, node-to-surface and surface-to-surface contact and twotracking approaches, small sliding and finite sliding.


γγcr

G1

G2

τ2cr = µp2

τ

τ1cr = µp1

τcr = µp

γcr = Ff l

G = τcr

γcr

Figure 4.3: Adapting penalty stiffness, based on a maximal allowable slip.

Some of the mechanical contact property models available in ABAQUS/Standard include: Softenedcontact, friction and user-defined constitutive models for surface interaction. Surface interactionin thermal or coupled thermal-mechanical contact simulations can include heat exchange by con-duction and radiation as well as the generation of frictional heat in coupled simulations.

During an increment in which the contact status has changed, ABAQUS/Standard will use thedefault hard contact criterion to determine whether the change should be reversed. In other words,if the contact status changes from open to closed during an increment, the contact pressure mustremain positive for the changed status to persist. In subsequent increments the contact point canagain sustain tensile pressures up to a specified value before the surfaces separate. This contactpressure-overclosure relationship is useful for cases where negative pressure values (such as surfacecohesion) may be allowed physically. It can also be useful in overcoming numerical problems indifficult contact simulations and in obtaining solutions without excessive iteration. Another pos-sibility to oppose the relative motion between the interacting surfaces is to specify damping.

The friction models available in ABAQUS:

• Include the classical isotropic Coulomb friction model, which in ABAQUS: in its general formallows the friction coefficient to be defined in terms of slip rate, contact pressure, averagesurface temperature at the contact point, and field variables; and provides the option todefine a static and a kinetic friction coefficient with a smooth transition zone defined by anexponential curve.

• Allow the introduction of a shear stress limit, which is the maximum value of shear stressthat can be carried by the interface before the surfaces begin to slide.

• Include an anisotropic extension of the basic Coulomb friction model in ABAQUS/Standard;

• Include a model that eliminates frictional slip when surfaces are in contact.

• Include a softened interface model for sticking friction in ABAQUS/Explicit in which theshear stress is a function of elastic slip.

• Can be implemented with a stiffness (penalty) method, a kinematic method (in ABAQUS/Explicit),or a Lagrange multiplier method (in ABAQUS/Standard), depending on the contact algo-rithm used.

26 4.2. LITERATURE RELATED TO FEM AND TYRES

• Can be defined in user subroutine FRIC (in ABAQUS/Standard) or VFRIC (in ABAQUS/Explicitfor the contact pair algorithm only), which allows modeling of very general frictional surfaceconditions.

Besides the friction models it is also possible to make use of user-defined interfacial constitutivebehavior. This is provided so that any constitutive behavior across an interface can be added tothe library of existing models. To use this the constitutive model (or a library of models) for theinterface has to be programmed in user subroutine UINTER(ABAQUS/Standard) or VUINTER.It is available only for a surface-based contact definition involved in stress/displacement, coupledtemperature-displacement, or heat transfer analysis. According to the manual the feature is verygeneral and powerful, but it requires considerable effort and expertise and is intended for advancedusers.

4.2 Literature related to FEM and tyres

Together with the development of non-linear FEM the use of FEM in tyre development started.There is a lot of literature, somehow related to tyres, on computational developments in the frame-work of FE and quite some work is also incorporated into ABAQUS. Examples are the works ofOden on friction and rolling contact (Oden and Pires, 1983, 1984; Oden and Martins, 1985; Odenand Lin, 1986; Oden et al., 1988; Faria et al., 1989), Laursen and Simo in the field of contactproblems with friction (Simo and Laursen, 1992; Laursen and Simo, 1993a,b) and Padovan onrolling visco-elastic cylinders (Zeid and Padovan, 1981; Padovan, 1987; Kennedy and Padovan,1987; Nakajima and Padovan, 1987; Padovan et al., 1992). A special issue (Vol 177, nr 3/4) of thejournal Computer Methods in Applied Mechanics and Engineering is devoted to computationalmodeling of contact and friction. More recent are the works of Wriggers (Wriggers et al., 1990;Zavarise et al., 1992; Haraldsson and Wriggers, 2000; Bandeira et al., 2004; Wriggers, 2006) onconstitutive interface laws with friction.

Literature specifically related to tyres usually does not provide detailed information. The mainreason is that most tyre manufacturers use own (in-house) finite element codes and specific meth-ods are kept confidential. Even in the publications where ABAQUS is used not much backgroundinformation can be found. Besides this, most research at universities is done in cooperation witha tyre manufacturer and therefore still not much model details are given, more information canbe obtained from articles where specific subjects related to tyres are treated. However the foundliterature provides insight in the trends and developments of tyre modeling throughout the pastdecades.

According to Padovan (2007), the following list of tyre problems are successfully handled by virtualCAE modeling methodologies

• Static rim mounting, inflation and axle load/deflection analysis

– States of stress/deflection/strain in all tyre components

– Force and moment, on center feel

– Definition of footprint shape as a function of axle load state and pressure

• Rolling/handling models both quasi-static and dynamic including braking and corneringeffects

• Thermal analysis/rolling resistance under rolling conditions

• Model cure press, and other manufacturing steps


• Fatigue life prediction models under full duty cycle

• Natural frequencies, handling and critical speed analysis

• Hydroplaning models, snow traction, in mud performance

• Geometry, and cord layout optimization

• Tread lug design

• Acoustical analysis

• Suspension models incorporating tyres

In one of his presentations it is stated that the overall tyre performance is usually measured byglobal actions, such as force & moment, ride comfort or dynamic response and therefore gross FEAmodels can be employed to establish such characteristics. In contrast to this some aspects, such astyre failure, require very localized models. In this sense a form of multi length scale methodologycan be used, i.e., a global model is used to drive a local model.A summary according to Padovan of advantages and disadvantages for global and local models:An one scale model has one contiguous mesh and a refined mesh is placed in locations of interest,transition meshes are needed to link refined and coarse zones. It is in general difficult to build amesh with reliable local refinements. The models tend to be very large and time consuming andare not very adoptable making mesh modifications. The transition areas tend to yield unreliableresults.Multiple scales suffer the so-called localization syndrome: localization always requires further lo-calization, so a localization cut-off criteria is needed. Besides that stress and strain take on arandom character as localization brings on the probabilistic nature of such scale levels. Whileprobabilistic variances can be very significant, generally the mean is close to the nominal designvalue and since stiffness of cord or metallic/plastic subcomponents are orders of magnitude higherthan rubber, local actions of such subcomponents dominate; rubber is hence strictly a geometricplace holder. Also every new level should be correlated with experiments to assess accuracy andsensitivity to meshing.

Most of the mentioned tyre problems require a specific type of modeling approach, in this sensethere is no universal tyre model. This totally depends on the kind of problem one wants to solve.An overview is given according to some points of the above list, although most articles fit in morethan one category. Besides the journal of Tire Science and Technology, a good starting point arethe two overview papers of Mackerle (1998, 2004) about rubber and rubber-like materials, finiteelement analyses and simulations, where an extensive reference list can be found.

4.2.1 Static analysis

One of the first overview papers, where FE models and the contact problem for tyres are described,is the paper of Noor and Tanner (1985). In this article, for a NASA research program for thespace shuttle tyres, the current status and developments of computational models for tyres aresummarized. This review has been made again by Danielson et al. (1996). Basically all early tyremodels are axi-symmetric, with no tread pattern or only circumferential groves due to the limitsof computational resources.Another popular way to reduce the number of degrees of freedom is the application of the global-local analysis. In this approach, an analysis of the complete structure is first performed with acoarse mesh. After that a part of the structure is meshed finer and interpolated displacements areapplied at the boundaries of this region. Such an approach can be used to compute the forces inthe contact area of a deflected tyre (see Gall et al., 1995 and Meschke et al., 1997). Although a


local model is able to give detailed numerical results corresponding to a tread block, the accuracyis strongly influenced by the simple global model.With the ever increasingly computational power it is now possible to mesh a part of even thewhole tread with a detailed pattern (e.g. Cho et al., 2004) and perform all kinds of static analysissuch as footprint shapes as function of axle load state and pressure.

4.2.2 Force & Moment

In an article of Kabe and Koishi (2000) a comparison between ABAQUS standard, explicit andexperiments for steady state cornering tyres is made. Although the results of both implicit and ex-plicit are closer to each other than to the experiments, the implicit method is significantly faster, 6hours and 8 days respectively. The prediction of tyre cornering forces is given in a paper of Tonukand Unlusoy (2001), where MARC is used. An comparison with experiments is also presented.The application of a new type of non-linear 3D finite element tyre model for simulating tyre spindleforce and moment response during side slip is described by Darnell et al. (2002). The simulationmodel is composed of shell elements, which model the tread deformation, coupled to special pur-pose finite elements that model the deformation of the sidewall and contact between the treadand the ground. Despite the (seemingly) simple model the results corresponds quite well withexperiments.A FE simulation with ABAQUS, in which the effect of tyre design parameters on lateral forcesand moments is studied is presented in a paper of Olatunbosun and Bolarinwa (2004). Parametricstudies are performed on a simplified tyre (no tread, negligible rim compliance and viscoelasticproperties, shear forces modeled with Coulomb friction with constant µ).Explicit simulations to predict tyre cornering forces, using PAM-SHOCK, are presented by Koishiet al. (1998). Besides a comparison with experiments, parametric studies on the effect of inflationpressure, belt angle and rubber modulus are performed.Rao et al. (2003) simulated the dynamic behavior of a pneumatic tyre using ABAQUS/Explicit.A study on a passenger car radial tyre to simulate cornering behavior, braking behavior, and com-bined cornering & braking behavior is presented. Further the effect of camber angle and groovedtread on tyre cornering behavior is studied. To reduce computational load two tyre models areused, one treadless tyre and a tyre with five circumferential groves.A cleat test with a half rotational response treadless tyre model can be found in a paper of Olatun-bosun and Burke (2002), where NASTRAN has been used. An comparison with experiments showsdeviations as the traverse speed increases above 20 kph. The prediction of the radial force seemsbetter than the longitudinal force. In an article of Cho et al. (2005) the dynamic response of afully patterned tyre rolling over a cleat is presented, using ABAQUS/Explicit. To decrease thecomputational load the fiber-reinforced rubber is modeled with composite shell elements, insteadof rebar elements.The effect of plysteer for steady-state rolling tyres is investigated by Mundl et al. (2005), where aglobal-local analysis is used, and by Ohishi et al. (2002), where a detailed tread pattern is used inthe contact area.

4.2.3 Thermo-mechanical approaches

More recently thermo-mechanical models for prediction of the temperature inside tyres have beenpresented (Allen et al., 1996a,b; Park et al., 1997; Futamura and Goldstein, 2004; Lin and Hwang,2004). Besides the prediction of steady state temperatures for rolling tyres a small literatureoverview is given by Rao et al. (2006). Experimental results of the temperature rise in an actualcontact area between the pavement and the tread of a cornering tyre can be found in an article ofFujikawa et al. (1994). It is shown that a significant temperature rise occurs for severe slip angles.


4.2.4 Hydroplaning

Another phenomenon is hydroplaning, where the interaction with fluid (water) is taken into ac-count by coupling FEM and Computational Fluid Dynamics (CFD). For examples see Seta et al.(2000), Nakijima et al. (2000) and Cho et al. (2006).

4.2.5 Component modeling

Besides complete tyre models a lot of work focusses on the specific components of a tyre, e.g. thetyre cord reinforcements, which can be modeled using so-called rebar elements (Huh and Kwak,1990; Helnwein et al., 1993; Mesehke and Helwein, 1994). This is also incorporated in ABAQUS.Asperity modeling to simulate rubber sliding on rough surfaces is considered by Bui and Ponthot(2002) for the 2D case and by Faulkner and Arnell (2000) for two 3D hemispherical asperities.Recently a comparison between experimental and explicit FEM has been published (Kuwajimaet al., 2006) to examine the interfacial phenomena between rubber and abrasive papers, such as thereal contact length, partial slip, and apparent friction coefficient under vertical load and tangentialforce. Other examples are optimization of the carcass (Cho et al., 2002) and the optimum shapeof the tread to avoid lateral slippage (Ahmed et al., 2005).

Chapter 5

Validation and modeling methods

5.1 LAT1OO and Tekscan

The LAT100 (VMI) is a test machine originally developed by Dr. Grosch. It is designed to rel-atively rank tread compounds with respect to dry traction, wet traction, ice traction, abrasionand rolling resistance. Under certain test conditions it should be possible to relate these indoorexperiments with outdoor tyre tests. On this machine a small rubber wheel, see figure 5.1(b), isplaced on a spinning disk which represents a road surface. The temperature, speed, surface of thedisk, the slip angle and normal load on the sample can be varied. More details of the LAT100 canbe found in the documentation system of Vredestein (Doclib).

(a) LAT100 machine. (b) Sample wheel.

Figure 5.1: The Laboratory Abrasion and skid Tester.

The Tekscan machine can be used to obtain the normal contact pressure distribution for a staticallydeformed or slow rolling tyre under zero slip angle. More details on the principle of Tekscan andpossibilities of using small LAT100 samples on the Tekscan can be found in the report of Broeze(2007).

5.2 Eplexor and DIK

Material test data of rubber compounds can be acquired from the Deutsches Institut fur Kautschuk-technologie (DIK). Visco-elastic material properties can be measured using the Eplexor availableat Vredestein or at the university in cooperation with the section of Materials Technology.

32 5.3. FLAT PLANK AND TYRE TEST TRAILER

5.3 Flat plank and tyre test trailer

The global tyre deformation characteristics and cleat experiments can be measured on the flatplank machine in the automotive lab at the TU/e. The global force and moments can be measuredwith the tyre test trailer of TNO. More details are presented in the report of Cremers (2005).

5.4 Tyre modeling in ABAQUS

Guidelines for tyre analyses in ABAQUS can be found in the ABAQUS Example problems manual,chapter 3.1 and in the course notes on Tire modeling using ABAQUS.The way to obtain the global force and moment characteristics of a tyre under different drivingconditions, such as free rolling or slip and chamber angles, in ABAQUS/Standard is the following:

• Start with a 2D cross-section of the tyre.

• Perform a static analysis in which the inflation pressure (and rim mounting) is simulated.

• Revolve the axi-symmetric cross-section about the axis of revolution, with a nonuniformdiscretization along the circumference, to obtain a 3D model.

• Apply a vertical load, which represents the weight of a vehicle.

• Perform a static full 3D footprint analysis of the tyre in contact with a flat road.

• Transfer these results to perform steady-state rolling analysis under different driving condi-tions.

The great advantage of the ‘steady-state transport analysis’ is that it uses a moving referenceframe in which rigid body rotation is described in an Eulerian manner and the deformation is de-scribed in a Lagrangian manner (the so-called ALE-formulation), see also articles of Nackenhorst(2004); Ziefle and Nackenhorst (2005); Laursen and Stanciulescu (2006). This kinematic descrip-tion converts the steady moving contact problem into a pure spatially dependent simulation. Thusthe mesh need to be refined only in the contact region, the steady motion transports the materialthrough the mesh. Frictional effects, inertia effects, and history effects in the material can all beaccounted for.The limitation of a steady-state transport simulation is that the circumference of the tyre must becontinuous (in the current version). As a result only circumferential groves can be used, insteadof a detailed tread pattern.If an uniform discretization along the circumference is used, it is possible to transfer steady-staterolling results to ABAQUS/Explicit and perform dynamic/impact simulations. This is computa-tionally expensive and usually a coarse mesh and a simplified tyre model is used to handle thesekind of simulations.

Contact must be divided into normal and tangential contact as mentioned in chapter 4. Bothnormal and tangential contact or friction can accounted for by using the available algorithms orby own user subroutines.Temperature effects can be accounted for in different ways. A similar procedure for steady-state solutions can be followed as in Huemer et al. (2001b), where a reference temperature isused and WLF transformations are used to calculate equivalent sliding velocities at the referencetemperature. In an implicit thermo-mechanical approach it is possible to convert energy dissipationdue to friction into heat. If this option is used, one should be aware that material properties canbe affected by temperature changes and ideally this should be taken into account as well. Thisalso holds for the friction law, if temperature effects are taken into account the friction law shouldbe an explicit function of temperature.

5. VALIDATION AND MODELING METHODS 33

To incorporate transient dynamical effects, such as local temperature rise for severe slip angles,ABAQUS explicit can be used, although this probably takes a very long time to simulate and norecords of this has been found in literature sofar. Guidelines for heat-transfer problems can befound in the ABAQUS Example problems manual, chapter 5.

34 5.4. TYRE MODELING IN ABAQUS

Chapter 6

Conclusions

To model a tyre and interaction with the environment using the finite element method not onlyknowledge of materials, geometry and interaction is required, but also a thorough knowledge ofthe computational mechanics behind the finite element method is required.However the most important conclusion is that the often used Coulomb friction model with aconstant coefficient of friction is in general not realistic in the case of rubber friction.

6.1 Friction

Friction of polymers is closely related to their viscoelastic behavior. Generally speaking, the co-efficient of friction increases with sliding velocity until a maximum value is reached at a certainspeed, followed by a decrease of the friction coefficient. The viscoelastic properties of polymersdepend strongly on temperature. Grosch showed the relation between the dependence of the fric-tion coefficient due to sliding velocity and temperature and the viscoelastic deformation on strainrate and temperature.The friction force for soft rubber sliding on a smooth surface (perfect contact) is more or lessconstant, independent of the load. Thus the coefficient of friction decreases continuously as thepressure increases. However there is also a coefficient of friction, independent of pressure, forharder rubber compounds, sliding on a rough surface. Apparently in this case contact is incom-plete. An increase in pressure creates a larger true area of contact and hence a larger frictionalforce. This shows the influence of the temperature, sliding velocity and real contact area.

The current theory of rubber friction and surface roughness, which capture all these properties canbe found in the publications of Persson, (see review paper Persson, 2005). The surface roughnessis characterized by a fractal description, which exhibits geometrical self-similarity. These resultsare based on the early studies of Grosch, taking into account that sliding friction of rubber hasthe same temperature dependence as that of the complex elastic modulus. He states that thefriction force is related to the internal friction of the rubber, which is a bulk property. Thehysteretic friction component is determined by sliding of the rubber over asperities of a roughsurface. These oscillating forces lead to energy dissipation. Every length scale up to the largestparticles of asphalt, can be related to an excitation frequency. As a result of energy dissipationheat is generated, in a recent article this local heating of the rubber is also taken into account.Although this theory provides knowledge of the physical origin of rubber friction it has somedrawbacks. Exact knowledge of the contribution of each length scale is needed. Therefore it has,at the moment, limited practical use, since variable amounts of unknown foreign materials inthe interfacial region makes it almost impossible to derive quantitative estimates of the absolutemagnitude of friction from physical properties of the rubber and surface. So assumptions muststill be made.

36 6.2. FEM

6.2 FEM

There are two distinctive methods to solve FE problems, implicit and explicit. The implicit methodsolves for a statical or dynamical equilibrium, while the explicit method solves transient dynamicresponse problems using an explicit direct-integration procedure.Although a lot of procedures are implemented in ABAQUS, the user still has to make severalchoices in the model formulation, i.e., the discretization of the mesh, the element types, the solu-tion procedure, the solver tolerances etc. These selections may have a tremendous effect on thefinal result, so care must be taken.

Contact problems in FEM are divided into normal and tangential contact and interfacial laws mustbe chosen to solve the contact problem. If the tangential forces reach a certain limit bodies will nolonger stick, but start sliding. Normally a local Coulomb law is used to calculate the shear stressin this case, but it is also possible to formulate tangential constitutive equations by regularizationof the stick-slip behavior, e.g. to avoid the non-differentiability of Coulomb’s law at zero velocity.Besides a fixed friction coefficient in ABAQUS some build-in constitutive laws can used to makethe coefficient of friction dependent on pressure, sliding velocity or other field variables. ABAQUSalso provides the possibility to use own interfacial laws.

With the ‘standard options’ in ABAQUS it is nowadays possible to include a detailed tread patternon a (part of a) tyre global model, instead of the global-local approach, and perform analyses tocapture footprints as a function of axle load and inflation pressure. The stress and strain in alltyre components and the force and moment on a center feel can be calculated.The steady state option in ABAQUS makes it possible to efficiently calculate steady state solutionsfor different driving conditions, since the mesh need to be refined only in the contact region. Withthis option frictional effects, inertia effects, and history effects in the material can all be accountedfor. This makes it a powerful approach to obtain steady state solutions and determine the globalforces and moments for several variables, such as inflation pressure, normal load, velocity, slip andchamber angle.

A limitation of this so-called steady state transport simulation is that the circumference of thetyre and the road must be continuous. As a result only circumferential groves can be used in thesekind of simulations. This also implies that a macroscopic friction law is required, in which surfaceroughness is embedded in the friction model.A phenomenological approach could be used to derive a macroscopic friction law, which depend onnormal pressure, sliding velocity, temperature and surface roughness. The articles of Huemer et al.(2001a,b) are related to this context, a friction law for rubber tread blocks on snow and concrete ispresented for use in a macroscopic model. The coefficient of friction depends on normal pressure,sliding velocity and temperature. The friction coefficient itself is only a function of normal pres-sure and sliding velocity. Temperature effects are incorporated using the WLF transformation.Hofstetter et al. (2003) introduced a thermo-mechanical coupling where energy dissipation duringsliding is converted into heat and this heat flux causes a temperature rise of the rubber and roadand this influences the friction coefficient.Dorsch et al. (2002) derived a phenomenological friction law using data obtained from experimentson the LAT100.

Despite the still increasingly computational power there are still limitations in the kind of sim-ulations one can perform. Especially explicit simulations, where transient dynamic effects arecaptured, require a lot of time and as a result coarse meshes are often used to solve these prob-lems.A way to perform explicit simulations is to use an uniform discretization along the circumference.It is than possible to transfer steady state rolling results to ABAQUS/Explicit and perform dy-

6. CONCLUSIONS 37

namic/impact simulations such as rolling over a cleat.To incorporate transient local temperature effects at severe slip angles, ABAQUS explicit can beused as well in theory, although this probably takes a very long time to simulate and no recordsof this has been found in literature sofar. However experimental results of the temperature in anactual contact area between the pavement and the tread of a cornering tyre show a significanttemperature rise (Fujikawa et al., 1994).

38 6.2. FEM

Appendix A

32th Tire Mechanics Short Course

A.1 Strength, wear and friction of rubber.

The presentation of Gent (2007) deals with the mechanical properties of rubber. Four subjectsare addressed: Elasticity and visco-elasticity, strength of rubber components, rolling and slidingfriction, and abrasion of rubber and wear of tyres.

A.1.1 Elasticity and visco-elasticity

Elastic

Rubber is an elastic, soft and virtually incompressible material. Its molecular structure consists oflong, linear flexible molecules forming random coils. The molecular segments are mobile (so-calledBrownian motion) and the molecules are interlinked into a 3D network. Chemical crosslinks areusually made by sulfur linkages also known as vulcanization. The rubbery state of a polymer isdetermined by the so-called glass transition temperature Tg. If the temperature is above Tg thepolymer is rubbery, below Tg the polymer is glassy. For rubbers the Tg is below room temperature.The Brownian motion (rate of segmental jumps) depends strongly on temperature (sheet 6).

Small strains

For small strains the material properties can be defined by the shear modulus G and the modulusof bulk compression K, which are related to the tensile or Young’s modulus E and Poisson’s ratioν as follows:

E = 2(1 + ν)G, (A.1)

ν =12

3K − 2G

3K + G. (A.2)

For rubber typically K G and it follows that E = 3G and ν = 0.5. If a rubber block withν = 0.5 is compressed, it is sheared outwards and generates an internal pressure. This effect causesa compression modulus, which is generally greater than E. An example can be found on sheet 10.

Large strains

For large strains the deformation is usually described by stretch ratios λ1, λ2, λ3 in the principle di-rections. Instead of a modulus a strain energy density function is used. Several forms (constitutivemodels) are available.

40 A.1. STRENGTH, WEAR AND FRICTION OF RUBBER.

Filled rubbers

In tyres rubbers are usually filled with particles like carbon black or silica. This creates two addi-tional effects. The so-called Payne effect and Mullins effect, which are both softening effects. ThePayne effect is a softening effect for small strains (0.1%), attributed to breaking apart aggregatesof filler particles. The Mullins effect is a substantial softening effect at higher strains, attributedto progressive detachment of rubber molecules from filler particles.The constitutive laws for large strains cannot be used to fully describe the stress-strain relation,because rubbers do not follow reversible stress-strain relations.

A.1.2 Visco-elastic

When rubber is dynamically stretched and released the returned energy is less than the energythat is put into the rubber, see sheet 21. This visco-elastic effect cannot be described by the perfectelastic shear modulus G, but a dynamic shear modulus G′ and a dynamic shear loss modulus G′′

is introduced to describe this hysteresis. Another term that is frequently used is the loss angle,which is defined as tan δ = G′′

G′ . For an elastic solid the strain is in phase with the stress, while fora visco-elastic solid the strain lags behind the stress with a delay of δ

ω , where ω is the frequency(rad/sec).The dependence of segmental jump rate on temperature is described by the universal WLF relation(Williams et al., 1955),

logφT

φTg

=17.5(T − Tg)52 + T − Tg

, (A.3)

where φTg = 0.1/sec. Some polymers such as butyl rubber behave according a different relation

logφT

φTg

=17.5(T − Tg)100 + T − Tg

. (A.4)

This is possibly due to unusually long moving segments. The WLF relation can by used to predictresults at other temperatures, the frequency f at temperature T is equivalent to a frequency f∗

at T ∗, wheref∗

f=

φT∗

φT= aT (A.5)

aT is known as the shift factor, this also holds for strain rates or velocities. An example is givenon sheets 27-30. However the following remark should be taken into account: These shifts areonly valid for pure rate (viscous) processes.

Strength of rubber components

In this section the tear strength and fracture energy are discussed. Also the visco-elastic contri-bution to strength effects of rate and temperature is treated. However this is not so important atthis moment.

A.1.3 Friction

The coefficient of friction is defined as:µ =

F

N, (A.6)

where N is the normal force and F the tangential force. For soft rubber sliding on a smoothsurface (perfect contact) the frictional force F is more or less constant, independent of the loadN . Thus the ‘coefficient’ of friction decreases continuously as the pressure increases, see figureA.1. However, there is also a coefficient of friction, independent of pressure, for harder rubbercompounds, sliding on a rough surface. Apparently in this case contact is incomplete. An increasein pressure creates a larger true area of contact and hence a larger frictional force.

A. 32TH TIRE MECHANICS SHORT COURSE 41

(a) Friction definition. (b) Friction law.

Figure A.1: Friction on smooth surface.

A.1.4 Rolling friction

Rolling friction Fr can be seen as MR , where M is a torque and R the radius. For an elastic

wheel the reaction force acts through the wheel center and therefore the moment M = 0. For avisco-elastic wheel the reaction force moves towards the loading side, because stresses are lowerin unloading, M = Nz where N is the normal load and z the distance from the wheel center, seefigure 3.4.Displacement of the center of pressure, and hence the coefficient of friction µr, depend on speedand temperature in the same way as the loss factor tan δ. A formulation for µr as proposed byGreenwood et al. (1961),

µr = 0.33(

N

G∗R2

) 13

tan δ, (A.7)

where G∗ is the complex shear modulus. Rolling friction is mainly due to energy dissipated asrubber is compressed and released in the contact patch.

Figure A.2: Rolling friction.

Sliding friction

Grosch (1963) noted that the curve for µ vs. sliding speed on a lubricated rough surface resemblesthe curve for tan δ vs. frequency f (and they both obey WLF temperature shifts), see figureA.3. He claims that friction is due to energy dissipated as rubber is compressed and released byasperities. For dry sliding on a smooth surface the curve for µ vs. sliding speed Vs resembles againthe curve for tan δ vs. frequency f . But now it is displaced to much lower speeds than before.Friction in dry sliding on a smooth surface is due to energy dissipation as rubber sticks and slips ona molecular scale. In the case of sliding on a rough dry surface the two dissipative processes appear

42 A.2. THE TIRE AS A VEHICLE COMPONENT.

Figure A.3: Grosch interpretation.

at different speeds, corresponding to different length scales: asperities and molecules, see figureA.4. Conclusion: Several contributions to µ can arise at the same sliding speed from stick-slipprocesses occurring at different length scales.

Figure A.4: Combined effect of sliding on a rough and dry surface.

Abrasion

Some general remarks can be found on sheets 64-65. This is not relevant at this moment.

A.2 The tire as a vehicle component.

The presentation of Potts (2007) consists of vehicle dynamics and can be compared with the bookof Pacejka (2006) or the lecture notes on vehicle dynamics (Besselink, 2003). The presentationactually starts more or less at sheet 44 with the definition of a tyre: A Compliant roller bear-ing with tractive properties. Further load support and the so-called carpet plots are explained.The importance of vehicle understeer and the contribution of the tyre is treated in sheets 67-79.Testing equipment and results of forces and moments measurements as well as eigenfrequenciesare presented in the remainder of the presentation. The tyre models used for vehicle handling are(semi-)analytical dynamic models.


A.3 Tire materials and manufacturing.

The materials and manufacturing processes used for pneumatic tyres are discussed in this pre-sentation (Walter, 2007a). The worldwide tyre production is greater than 1 billion units with amarket value exceeding US $100 billion. Around 73% of this is contributed by passenger tyres and70% of these tyres are replacement tyres.The presentation is divided in four subjects: the tyre components, tyre elastomers, the cord rein-forcements and factory processing.

A.3.1 Tyre components

There are basically two types of tyres: bias and radial tyres, however the focus in this report willbe on radial tyres, since these are manufactured at Vredestein. The main difference is the orien-tation of the plies, see figure A.5. Every rubber component in a tyre requires specific properties.

(a) Bias tyre. (b) Radial tyre.

Figure A.5: Difference between bias and radial tyres.

An overview of these components can be found on sheet 10. Further the tyre construction, such asaspect ratio, belt construction and tread compound, depends on the target market (speed rating).

The principal tyre elastomers are natural rubber (NR), polyisoprene (IR, this is synthetic NR),styrene butadiene rubber (SBR), butyl rubber (IIR) and butadiene (BR). The reinforcement ma-terials are: a) The fillers, such as carbon black and silica. b) The organic textile plies, such aspolyester or rayon bodies, nylon cap or kevlar. And c) Steel, which is used in the belt cord andthe bead wires. Some details and definitions are given on sheets 15–23.The definition of a compound (sheet 24) is the following: the formulation of a mixture of rubberand additives which meets the needs of the tyre component application. Usually a compound con-sists of one or more polymers, vulcanizing agents, accelerators, reinforcing fillers, antidegradents,plasticizers, softeners and tackifiers (used to bond pieces together). This explains why the prop-erties of a compound can deviate from a pure elastomer.

44 A.4. RULES AND REGULATIONS GOVERNING TIRES.

A.3.2 Tyre elastomers

In this part (sheet 29–64) the aforementioned rubbers are described in detail and the effect offillers is discussed. Some sheets deal with the dynamic mechanical properties, this overlaps withGent et al. (2007).

A.3.3 Cord reinforcements and Factory processing

In the last part the different cord reinforcements, textile and steel, are discussed. Different types ofsteel cord constructions are explained and finally the manufacturing processes is described. Thiswill not be discussed here, however is should be noted that (limitations of) the manufacturingprocess greatly affects the final tyre. So it is worthwhile to scroll through this slides.

A.4 Rules and regulations governing tires.

This is an interesting presentation of Walter (2007b) about all the rules, which again affect tyre de-sign. Although this presentation is focussed on the US federal government regulations, it providesa background on the fair amount of rules, which tyre manufactures have to cope with.

A.5 Advanced tire modeling.

The presentation of Padovan (2007) introduces the course goals of finite element analysis in tyreanalysis. There are in total 12 presentations with over 600 sheets, so in the remainder of thischapter references to a presentation are made using the number (#) shown in table A.1. In thisoverview I’ll give some remarks to highlight interesting parts. In the introduction presentation

Table A.1: Presentations of Padovan.

Number Title1 Advanced tire modeling2 Intro to FEA modeling3 Multilength scale FEA model of tire4 Dynamic elastomer properties5 Critical speed6 Rolling resistance7 FEA modeling of tires stochastic effects8 Optimization of cord spacing9 Fracture mechanics10 Modeling stochastic effects on durability11 Overal fatigue analysis procedure12 Hydroplaning analysis

there is, besides the outline of the other presentations, one sheet giving an overview of tyreproblems, which are successfully handled by virtual CAE modeling methodologies (1):

• Static rim mounting, inflation and axle load/deflection analysis

– States of stress/deflection/strain in all tyre components

– Force and moment, on center feel


– Definition of footprint shape as a function of axle load state and pressure

• Rolling/handling models both quasi-static and dynamic including braking and corneringeffects

• Thermal analysis/rolling resistance under rolling conditions

• Model cure press, and other manufacturing steps

• Fatigue life prediction models under full duty cycle

• Natural frequencies, handling and critical speed analysis

• Hydroplaning models, snow traction, in mud performance

• Geometry, and cord layout optimization

• Tread lug design

• Acoustical analysis

• Suspension models incorporating tyres

The second presentation is the introduction to finite elements, this presentation is more elaboratethan the other presentations. These sketch the general problem and show the FE solution.It is important to determine the levels of deformation/strain to use appropriate stress-strain (linearvs. nonlinear) relations (2). In general tyre problems are large deformation/strain problems:

• Small deformationDeformation <<<<< Characteristic length

• Small strainStrain = Deformation/Characteristic length <<<<< 1.

• Large deformationDeformation = O( Characteristic length)

• Large strainStrain = Deformation/Characteristic length = O(1.)

In the third presentation it is stated that the overall tyre performance is usual measured by globalactions, such as force & moment, ride comfort or dynamic response and therefore gross FEA mod-els can be employed to establish such characteristics. In contrast to this some aspects, such as tyrefailure, require very localized models. In this sense a multi length scale methodology can be used,i.e., a global model is used to drive a local model. A summary of advantages and disadvantagesfor global and local model is given in this presentation and also stated here.A one scale model has one contiguous mesh and a refined mesh is placed in locations of interest,transition meshes are needed to link refined and coarse zones. It is in general difficult to build amesh with reliable local refinements. The models tend to be very large and time consuming andare not very adoptable making mesh modifications. The transition areas tend to yield unreliableresults.Multiple scales suffer the so-called localization syndrome: localization always requires further lo-calization, so a localization cut-off criteria is needed. Besides that stress and strain take on arandom character as localization brings on the probabilistic nature of such scale levels. Whileprobabilistic variances can be very significant, generally the mean is close to the nominal designvalue and since stiffness of cord or metallic/plastic subcomponents are orders of magnitude higherthan rubber, local actions of such subcomponents dominate; rubber is hence strictly a geometricplace holder. Also every new level should be correlated with experiments to assess accuracy and

46 A.5. ADVANCED TIRE MODELING.

sensitivity to meshing.Sheets 30-36 give an example of a global-local approach to investigate local cord-rubber interaction.It is however stated here that with a global model the usual performance characteristics are usable.

Presentation 4 deals with material properties of elastomers, such as stress-strain and creep, andfriction. The friction depends on interfacial pressure and dynamic effects, see figures A.6 and A.7.

µ

Interfacial pressure

1

1000 psi

Figure A.6: Quasi static.

µ

Percentage slip

1

100%

O(10− 20% slip)

dry

icy/wet

Figure A.7: Dynamic effect for ‘average roughness road’.

Presentation 5 is about critical speeds of tyres and describes four methodes to analyze this, butit is not relevant at the moment.Rolling resistance is the subject of presentation 6 and is primary caused (95%) by mechanical workstored in rubber that is partially converted to heat and secondary cause (5%) is the mechanicalwork stored in cords that is converted to heat and friction work induced in the footprint areaduring rolling, which generates heat in addition to wear. Two different approaches are given todetermine the rolling resistance.The other presentations go into specific subjects and are not relevant at the moment.


A.6 Tire wear, traction and force generation.

The footprint is the subject of the presentation by Pottinger (2007). A footprint is a complexobject, since the tyre is a doubly curved surface and is pneumatically pre-stressed. And frictionaffects the deformation of the tyre.Contact leads to a stress distribution across the interface, with normal component σZ , and shearcomponents at each point, σX and σY . The nature of this distribution has a complex dependenceon conditions of use. Two distinctive measurement techniques can be identified: Measure σZ atever finer resolutions or examine the entire stress field at somewhat coarser resolutions. A methodfor measuring σZ is pressure sensitive film (like the Fuji-film) for a static case. Electronic pressuremats (like Tekscan) can be used for static and slow rolling cases. These cannot be used for largeshears, which occur in cornering, braking or accelerating. The dramatic difference between staticand slow rolling is shown in figure A.8, this shows why static footprints can be misleading. Sheets18-30 give some examples of machines and concepts for measuring footprints.

Figure A.8: Static and slow rolling footprint.

Temperature plays a great role in the footprint. The surface temperature, in particular the changethrough contact, is related to wear. The bulk temperature determines the stiffness of the treadcompound, see also chapter A.1. Both wear resistance and handling depend on tyre temperaturein the footprint because it influences abradability, stiffness and also the coefficient of friction. Infigure A.9 the effect of temperature rise on different surfaces is shown. It can be seen that theconductivity of the surface has great influence on the tread surface temperature, note that the slipangle is 6. This slip angle is a very severe condition for wear and will cause gumming in racingtyres. Some years ago Michelin published data showing that over 99% of all driving in the UnitedStates occurred at less than 6 slip angle. The low slip angles involved in actual operation arewhy tyres have a reasonable wear life.

Figure A.10 shows basic passenger tyre footprint longitudinal and lateral stress characteristics.These figures are for zero slip and chamber angle. It can be seen that σX is positive at the frontand negative at the back. The shoulders have a larger effective radius than the crown in this figure.The lateral stresses are pointed outwards. The effect of slip angle can be seen in figure A.11. A

48 A.6. TIRE WEAR, TRACTION AND FORCE GENERATION.

Figure A.9: Temperature effects.

slip angle reshapes the footprint into a rough trapezoid. One of the shoulders becomes longer, forexample, the right shoulder, when turning left, and the other grows shorter (the left shoulder inthis case). The normal stress increases on the long shoulder and falls on the short shoulder.


(a) Longitudinal stress. (b) Lateral stress.

Figure A.10: Longitudinal and lateral stress.

Figure A.11: Effect of slip angle.

50 A.7. TIRE STRESSES AND DEFORMATION ANALYSIS.

The tread pattern on tyres exist only as traction aids for contaminated surfaces, such as water orsnow. These grooves cause uneven wear and noise. A remarkable citation from sheet 61: Industrialdesign of the pattern is artistic icing on an engineering necessity.For good dry traction one should put as much rubber on the road as possible. The followingtraction summary shows the trade-off in tread design:

• Dry

– Maximize contact area– Keep σZ low– Use soft compound

• Wet

– Medium-hard compound– Maximize groove volume fraction– Minimize element size– Long narrow footprint

• Ice

– Low TG rubber– Lots of element edges

• Snow

– Large void projection times void width product perpendicular to direction of requiredforce

– Flexible tread elements to insure cleaning, so void doesn’t clog

Sheets 73-97 shows different aspects of wear. A relation between wear and shear energy intensityis presented. Some guidelines for outdoor and indoor testing are given. The last part (98-116) ofthe presentation focusses on the limits of force generation for several conditions, mostly based onexperimental data, such as pavement texture, hydroplaning, compounds and snow traction.This outlines again that a lot of environmental issues influence the force generation, and as a resultfriction and wear.

A.7 Tire stresses and deformation analysis.

This last presentation given by Trinko (2007) gives an overview of the stresses and strains in tyres.It consists of three parts: Force topics, composite properties and stresses and strains.

A.7.1 Force topics

This section (sheets 3–49) is divided into 5 topics: Load transfer, cord tension, bead calculation,Purdy equations and air diffusion. Interesting part is air diffusion, which is equivalent to a heattransfer problem.

A.7.2 Composite properties

In this part of the presentation (sheets 51-86) the laminate theory is explained. This theory canprobably be used to do some basic calculations to rubber/cord laminates and compare that to theso-called rebar elements in ABAQUS.


A.7.3 Stress and strain

The last part gives some FE results about slip velocity and friction. Although this is anotherprogram than ABAQUS, it shows that a rolling tyre with friction and different slip angles ispossible. In figure A.12 the slip and friction for 0 and 1 slip angle are shown. There are two

(a) Slip 0. (b) Friction 0.

(c) Slip 1. (d) Friction 1.

Figure A.12: Difference between 0 and 1 slip angle.

approaches to rolling tyres. First a Lagrangian approach, where the mesh rolls. To use thisapproach equal segments around the whole tyre must be used. This allows a full tread pattern.The second one is a mixed Euler-Lagrange approach, where the material flows through the mesh.The mesh can deform, but does not roll. This can only be used for steady state solutions. Themain advantage is that a fine mesh is only needed in the contact area, however this requiresan axi-symmetric model with a smooth or circumferential grooved tread pattern. The proposedfrictional stress for steady state rolling is

F = µ(P,∆V )P, (A.8)

where ∆V is Vtyre − Vroad and P the normal pressure. Two friction laws are given, the first oneas function of pressure,

µ(P ) = CP−γ , (A.9)

where C and γ are constants and the second one as a function of ∆V , see figure A.13. Thepower law (A.9), see also figure A.6, seems reasonable, according to Trinko, however the aligningmoments are wrong compared to experiments. Trinko thinks this is due to a wrong friction law,

52 A.7. TIRE STRESSES AND DEFORMATION ANALYSIS.

but he’s not aware of improved friction laws. Another remark is that there is still no conclusivecorrelation with wear. After the end of the presentation some additional sheets (143-169) are

µ

∆V

Figure A.13: Friction law as function of ∆V for rolling tyre.

added. These describe some measurement techniques and explanations behind calculations.

Bibliography

ABAQUS Analysis user’s manual. Version 6.6. 2006.

ABAQUS Example problems manual. Version 6.6. 2006.

S. Reaz Ahmed, S.K. Deb Nath, and M. Wahhaj Uddin. Optimum shapes of tire-treads foravoiding lateral slippage between tires and roads. International Journal for numerical methodsin Engineering, 64:729–750, 2005.

J. Mc Allen, A.M. Cuitino, and V. Sernas. Numerical investigation of the deformation charac-teristics and heat generation in pneumatic aircraft tires part I. mechanical modeling. FiniteElements in Analysis and Design, 23:241–263, 1996a.

J. Mc Allen, A.M. Cuitino, and V. Sernas. Numerical investigation of the deformation characteris-tics and heat generation in pneumatic aircraft tires part II. thermal modeling. Finite Elementsin Analysis and Design, 23:265–290, 1996b.

J.F. Archard. Elastic deformation and the laws of friction. Proceedings of the Royal Society ofLondon, Series A, 243(1233):190–205, 1957.

A.A. Bandeira, P. Wriggers, and P. de Mattos Pimenta. Numerical derivation of contact mechanicsinterface laws using a finite element approach for large 3D deformation. International Journalfor numerical methods in Engineering, 59:173–195, 2004.

A. Becker, V. Dorsch, M. Kaliske, and H. Rothert. A material model for simulating the hystereticbehavior of filled rubber for rolling tires. Tire Science and Technology, 26(3):132–148, 1998.

J.S. Bergstrom. Modeling of the dynamic mechanical response of elastomers. Tire Science andTechnology, 33(2):120–134, 2005.

I. Besselink. Voertuigdynamica. Collegesheets, Eindhoven, 2003.

F.P. Bowden and D. Tabor. The friction and lubrication of solids. Oxford University Press, oxfordclassics series edition, 2001.

J. Broeze. Surface pressure on dr. Grosch tyre. In press, Eindhoven University of Technology,2007.

Q.V. Bui and J.P. Ponthot. Estimation of rubber sliding friction from asperity interaction mod-eling. Wear, 252:150–160, 2002.

A.W. Bush, R.D. Gibson, and T.R. Thomas. The elastic contact of a rough surface. Wear, 35:87–11, 1975.

J.R. Cho, H.S. Jeong, and W.S. Yoo. Multi-objective optimization of tire carcass contours using asystematic aspiration-level adjustment procedure. Computational Mechanics, 29:498–509, 2002.

J.R. Cho, K.W. Kim, W.S. Yoo, and S.I. Wong. Mesh generation considering detailed tread blocksfor reliable 3D tire analysis. Advances in Engineering Software, 35:105–113, 2004.

54 BIBLIOGRAPHY

J.R. Cho, K.W. Kim, D.H. Jeon, and W.S. Yoo. Transient dynamic response analysis of 3-Dpatterned tire rolling over cleat. European Journal of Mechanics A/Solids, 24:519–531, 2005.

J.R. Cho, H.W. Lee, J.S. Sohn, G.J. Kim, and J.S. Woo. Numerical investigation of hydroplaningcharacteristics of three-dimensional patterned tire. European Journal of Mechanics A/Solids,25:914–926, 2006.

M. Ciavarella, V. Delfine, and V. Demelio. A new 2D asperity model with interaction for studyingthe contact of multiscale rough random profiles. Wear, 261:556–567, 2006a.

M. Ciavarella, C. Murolo, and G. Demelio. On the elastic contact of rough surfaces: Numericalexperiments and comparisons with recent theories. Wear, 261:1102–1113, 2006b.

R. Cremers. Investigating dynamic tyre model behaviour. Master’s thesis, Eindhoven Universityof Technology, 2005. DCT 2005.37.

K.T. Danielson, A.K. Noor, and J.S. Green. Computational strategies for tire modeling andanalysis. Computers & Structures, 61(4):673–693, 1996.

I. Darnell, R. Mousseau, and G. Hulbert. Analysis of tire force and moment response during sideslip using an efficient finite element model. Tire Science and Technology, 30(2):66–82, 2002.

J. Diani, M. Brieu, and P. Gilormini. Observation and modeling of the anisotropic visco-hyperelastic behavior of a rubberlike material. International Journal of Solids and Structures,43:3044–3056, 2006.

Doclib. Internal documentation system Vredestein Banden B.V.

I. Doghri, A. Muller, and R.L. Taylor. A general three-dimensional contact procedure for implicitfinite element codes. Engineering Computations, 15(2):233–259, 1998.

V. Dorsch, A. Becker, and L. Vossen. Enhanced rubber friction model for finite element simulationsof rolling tyres. Plastics, Rubber and Composites, 31(10):458–464, 2002.

L.O. Faria, J.M. Bass, J.T. Oden, and E.B. Becker. A three-dimensional rolling contact model fora reinforced rubber tire. Tire Science and Technology, 17(3):217–233, 1989.

A. Faulkner and R.D. Arnell. The development of a finite element model to simulate the slidinginteraction between two, three-dimensional, elastoplastic, hemispherical asperities. Wear, 242:114–122, 2000.

FOR492, May 2007. URL http://www.ibnm.uni-hannover.de/FOR492/index.html.en.

T. Fujikawa, A. Funazaki, and S. Yamazaki. Tire tread temperatures in actual contact areas. TireScience and Technology, 22(1):19–41, 1994.

S. Futamura and A. Goldstein. A simple method of handling thermomechanical coupling fortemperature computation in a rolling tire. Tire Science and Technology, 32(2):56–68, 2004.

A. Le Gal, X. Yang, and M. Kluppel. Evaluation of sliding friction and contact mechanics ofelastomers based on dynamic-mechanical analysis. Journal of Chemical Physics, 123, 2005. doi:10.1063/1.1943410.

R. Gall, F. Tabaddor, D. Robbins, P. Majors, W. Sheperd, and S. Johnson. Some notes on thefinite element analysis of tires. Tire Science and Technology, 23(3):175–188, 1995.

A.N. Gent. Mechanical properties of rubber. Presentation, 32th Tire Mechanics Short Course,Koln, 2007.

A.N. Gent, J.B. Suh, and S.G. Kelly. Mechanics of rubber shear springs. International Journal ofNon-Linear Mechanics, in press, 2007.

http://www.ibnm.uni-hannover.de/FOR492/index.html.en

BIBLIOGRAPHY 55

J.A. Greenwood and J.B.P. Williamson. Contact of nominally flat surfaces. Proceedings of theRoyal Society of London, Series A, 295(1442):300–319, 1966.

J.A. Greenwood and J.J. Wu. Surface roughness and contact: An apology. Meccanica, 36:617–630,2001.

J.A. Greenwood, H. Minshall, and D. Tabor. Hysteresis losses in rolling and sliding friction.Proceedings of the Royal Society of London, Series A, 259(1299):480–507, 1961.

K.A. Grosch. The relation between the friction and visco-elastic properties of rubber. Proceedingsof the Royal Society of London, Series A, 274(1356):21–39, 1963.

A. Haraldsson and P. Wriggers. A strategy for numerical testing of frictional laws with applicationto contact between soil and concrete. Computer Methods in Applied Mechanics and Engineering,190:963–977, 2000.

P. Helnwein, C.H. Liu, G. Meschke, and H.A. Mang. A new 3-d finite element model for cord-reinforced rubber composites application to analysis of automobile tires. Finite Elements inAnalysis and Design, 14:1–16, 1993.

K. Hofstetter, J. Eberhardsteiner, and H.A. Mang. A thermo-mechanical formulation describingthe frictional behavior of rubber. Proceedings in Applied Mathematics and Mechanics, 2:238–239,2003.

K. Hofstetter, Ch. Grohs, J. Eberhardsteiner, and H.A. Mang. Sliding behaviour of simplified tiretread patterns investigated by means of FEM. Computers & Structures, 84:1151–1163, 2006.

T. Huemer, W.N. Liu, J. Eberhardsteiner, and H.A. Mang. A 3D finite element formulationdescribing the frictional behavior of rubber on ice and concrete surfaces. Engineering Compu-tations, 18(3/4):417–436, 2001a.

T. Huemer, W.N. Liu, J. Eberhardsteiner, H.A. Mang, and G. Meschke. Sliding behavior of rubberon snow and concrete surfaces. Kautschuk Gummi Kunststoffe, 54:458–462, 2001b.

H. Huh and K. Kwak. Finite element stress analysis of the reinforced tire contact problem.Computers & Structures, 36(5):871–881, 1990.

R.L. Jackson and J.L. Streator. A multi-scale model for contact between rough surfaces. Wear,261:1337–1347, 2006.

K. Kabe and M. Koishi. Tire cornering simulation using finite element analysis. Journal of AppliedPolymer Science, 78:1566–1572, 2000.

M. Kaliske and H. Rothert. Formulation and implementation of three-dimensional viscoelasticityat small and finite strains. Computational Mechanics, 19:228–239, 1997.

R. Kennedy and J. Padovan. Finite element analysis of steady and transiently moving/rolling non-linear viscoelastic structure-II. shell and three-dimensional simulations. Computers & Structures,27(2):259–273, 1987.

M. Kluppel and G. Heinrich. Rubber friction on self-affine road tracks. Rubber chemistry andtechnology, 73:578–606, 2000.

M. Koishi, K. Kabe, and M. Shiratori. Tire cornering simulation using an explicit finite elementanalysis code. Tire Science and Technology, 26(2):109–119, 1998.

V. Kouznetsova. Non-linear finite element method for solids. Collegesheets, Eindhoven, 2006.

M. Kuwajima, M. Koishi, and J. Sugimura. Contact analysis of tire tread rubber on flat surfacewith microscopic roughness. Tire Science and Technology, 34(4):237–255, 2006.

56 BIBLIOGRAPHY

T.A. Laursen. Computational contact and impact mechanics. Springer, 2002.

T.A. Laursen and J.C. Simo. A continuum-based finite element formulation for the implicitsolution of multibody, large deformation frictional contact problems. International Journal fornumerical methods in Engineering, 36:3451–3485, 1993a.

T.A. Laursen and J.C. Simo. Algorithmic symmetrization of coulomb frictional problems usingaugmented lagrangians. Computer Methods in Applied Mechanics and Engineering, 108:133–146,1993b.

T.A. Laursen and I. Stanciulescu. An algorithm for incorporation of frictional sliding conditionswithin a steady state rolling framework. Communications in numerical methods in engineering,22:301–318, 2006.

J. Lemaitre. Handbook of materials and behavior models, volume 2. Acedemic press, 2001.

Y.J Lin and S.J. Hwang. Temperature prediction of rolling tires by computer simulation. Mathe-matics and Computers in Simulation, 67:235–249, 2004.

W.N. Liu, G. Meschke, and H.A. Mang. On the approximations of the tangential slip in frictionalcontact analyses. Computers & Structures, 78:53–62, 2000.

J. Mackerle. Rubber and rubber-like materials, finite-element analyses and simulations, an adden-dum: a bibliography (1997-2003). Modeling and Simulation in Materials Science and Engineer-ing, 12:1031–1053, 2004.

J. Mackerle. Rubber and rubber-like materials, finite-element analyses and simulations: a bibliog-raphy (1976–1997). Modeling and Simulation in Materials Science and Engineering, 6:171–198,1998.

A. Majumdar and B. Bhushan. Role of fractal geometry in roughness characterization and contactmechanics of surfaces. Journal of Tribology, 112:205–216, 1990.

A. Majumdar and B. Bhushan. Fractal model of elastic-plastic contact between rough surfaces.Journal of Tribology, 113:1–11, 1991.

A. Majumdar and C.L. Tien. Fractal characterization and simulation of rough surfaces. Wear,136:313–327, 1990.

J.I. McCool. Comparison of models for the contact of rough surfaces. Wear, 107:37–60, 1986.

G. Meschke, H.J. Payer, and H.A. Mang. 3d simulations of automobile tires: Material modeling,mesh generation, and solution strategies. Tire Science and Technology, 25(3):154–176, 1997.

G. Mesehke and P. Helwein. Large-strain 3D-analysis of fibre-reinforced composites using rebarelements: hyperelastic formulations for cords. Computational Mechanics, 13:241–254, 1994.

D.F. Moore. Friction and wear in rubbers and tyres. Wear, 61:273–282, 1980.

C.A. Morrow. Adhesive Rough Surface Contact. PhD thesis, University of Pittsburgh, 2003.

R. Mundl, M. Fischer, M. Wajroch, and S.W. Lee. Simulation and validation of the ply steerresidual aligning torque induced by the tyre tread pattern. Vehicle System Dynamics, 43:434–443, 2005.

U. Nackenhorst. The ALE-formulation of bodies in rolling contact theoretical foundations andfinite element approach. Computer Methods in Applied Mechanics and Engineering, 193:4299–4322, 2004.

BIBLIOGRAPHY 57

Y. Nakajima and J. Padovan. Finite element analysis of steady and transiently moving/rollingnonlinear viscoelastic structure-III. impact/contact simulations. Computers & Structures, 27(2):275–286, 1987.

Y. Nakijima, E. Seta, T. Kamegawa, and H. Ogawa. Hydroplaning analysis by FEM and FVM:Effect of tire rolling and tire pattern on hydroplaning. International Journal of automotivetechnology, 1(1):26–34, 2000.

A.K. Noor and J.A. Tanner. Advances and trends in the development of computational modelsfor tires. Computers & Structures, 20(1/3):517–533, 1985.

J.T. Oden and T.L. Lin. On the general rolling contact problem for finite deformations of aviscoelastic cylinder. Computer Methods in Applied Mechanics and Engineering, 57:297–367,1986.

J.T. Oden and J.A.C. Martins. Models and computational methods for dynamic friction phenom-ena. Computer Methods in Applied Mechanics and Engineering, 52:527–634, 1985.

J.T. Oden and E.B. Pires. Numerical analysis of certain contact problems in elasticity with non-classical friction laws. Computers & Structures, 16(1/4):481–485, 1983.

J.T. Oden and E.B. Pires. Algorithms and numerical element approximations of results for finitecontact problems with non-classical friction laws. Computers & Structures, 19(1/2):137–147,1984.

J.T. Oden, T.L. Lin, and J.M. Bass. A finite element analysis of the general rolling contactproblem for a viscoelastic rubber cylinder. Tire Science and Technology, 16(1):18–43, 1988.

K. Ohishi, H. Suita, and K. Ishihara. The finite element approach to predict the plysteer residualcornering force of tires. Tire Science and Technology, 30(2):122–133, 2002.

O.A. Olatunbosun and O. Bolarinwa. Fe simulation of the effect of tire design parameters onlateral forces and moments. Tire Science and Technology, 32(3):146–163, 2004.

O.A. Olatunbosun and A.M. Burke. Finite element modelling of rotating tires in the time domain.Tire Science and Technology, 30(1):19–33, 2002.

H. Olsson, K.J. Astrom, C. Canudas de Wit, M. Gafvert, and P. Lischinsky. Friction models andfriction compensation. European Journal of Control, 11:16–52, 1998.

H.B. Pacejka. Tyre and vehicle dynamics. Butterworth-Heinemann, 2nd edition, 2006.

J. Padovan. Advanced tire modeling. Presentation, 32th Tire Mechanics Short Course, Koln,2007.

J. Padovan. Finite element analysis of steady and transiently moving/rolling nonlinear viscoelasticstructure-I. theory. Computers & Structures, 27(2):249–257, 1987.

J. Padovan, A. Kazempour, F. Tabaddor, and B. Brockman. Alternative formulations of rollingcontact problems. Finite Elements in Analysis and Design, 11:275–284, 1992.

H.C. Park, S.-K. Youn, T.S. Song, and N.-J. Kim. Analysis of temperature distribution in a rollingtire due to strain energy dissipation. Tire Science and Technology, 25(3):214–228, 1997.

A.G. Peressadko, N. Hosoda, and B.N.J. Persson. Influence of surface roughness on adhesionbetween elastic bodies. Physical Review Letters, 95, 2005. doi: 10.1103/PhysRevLett.95.124301.

B.N.J. Persson. Elastic instabilities at a sliding interface. Physical Review B, 63, 2001a. doi:10.1103/PhysRevB.63.104101.

58 BIBLIOGRAPHY

B.N.J. Persson. Elastoplastic contact between randomly rough surfaces. Physical Review Letters,87(11), 2001b. doi: 10.1103/PhysRevLett.87.116101.

B.N.J Persson. Adhesion between an elastic body and a randomly rough hard surface. Europeanphysical journal E, 8:385–401, 2002a.

B.N.J. Persson. Adhesion between elastic bodies with randomly rough surfaces. Physical ReviewLetters, 89(24), 2002b. doi: 10.1103/PhysRevLett.89.245502.

B.N.J. Persson. On the nature of surface roughness with application to contact mechanics, sealing,rubber friction and adhesion. Journal of Physics: Condensed matter, 17:R1–R62, 2005.

B.N.J. Persson. Rubber friction: role of the flash temperature. Journal of Physics: Condensedmatter, 2006a. doi: arXiv:cond-mat/0605273.

B.N.J. Persson. Contact mechanics for randomly rough surfaces. Surface Science Reports, 61:201–227, 2006b.

B.N.J. Persson. Contact mechanics for randomly rough surfaces. Journal of Physics: Condensedmatter, 2006c. doi: arXiv:cond-mat/0603807v1.

B.N.J. Persson. Theory and simulation of sliding friction. Physical Review Letters, 71(8):1212–1215, 1993.

B.N.J. Persson. Theory of friction: Stress domains, relaxation, and creep. Physical Review B, 51(19):568–585, 1995.

B.N.J. Persson. Sliding friction: the contribution from defects. Journal of Physics: Condensedmatter, 9:2869–2889, 1997.

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

B.N.J. Persson. Sliding friction. Surface Science Reports, 33:83–119, 1999.

B.N.J. Persson and E. Tosatti. Qualitative theory of rubber friction and wear. Journal of ChemicalPhysics, 112(4):2021–2029, 2000.

B.N.J. Persson and A.I. Volokitin. Dynamical interactions in sliding friction. Surface Science,457:345–356, 2000.

B.N.J. Persson and A.I. Volokitin. Theory of rubber friction: Nonstationary sliding. PhysicalReview B, 65, 2002. doi: 10.1103/PhysRevB.65.134106.

B.N.J. Persson and A.I. Volokitin. Rubber friction on smooth surfaces. European physical journalE, 21:69–80, 2006.

B.N.J. Persson, F. Bucher, and B. Chiaia. Elastic contact between randomly rough surfaces:Comparison of theory with numerical results. Physical Review B, 65, 2002. doi: 10.1103/PhysRevB.65.184106.

B.N.J. Persson, O. Albohr, F. Mancosuc, V. Peveric, V.N. Samoilov, and I.M. Sivebaek. On thenature of the static friction, kinetic friction and creep. Wear, 254:835–851, 2003.

B.N.J. Persson, U. Tartaglino, O. Albohr, and E. Tosatti. Rubber friction on wet and dry roadsurfaces: The sealing effect. Physical Review B, 71, 2005. doi: 10.1103/PhysRevB.71.035428.

M.G. Pottinger. Tire wear, traction and force generation. Presentation, 32th Tire Mechanics ShortCourse, Koln, 2007.

G.R. Potts. The tire as a vehicle component. Presentation, 32th Tire Mechanics Short Course,Koln, 2007.

BIBLIOGRAPHY 59

E. Rabinowicz. The nature of the static and kinetic coefficients of friction. Journal of AppliedPhysics, 22(11):1373–1379, 1951.

K. Rao, R. Kumar, and P. Bohara. Transient finite element analysis of tire dynamic behavior.Tire Science and Technology, 31(2):104–127, 2003.

K.V. Narasimha Rao, R. Krishna Kumar, P.C. Bohara, and R. Mukhopadhyay. A finite elementalgorithm for the prediction of steady-state temperatures of rolling tires. Tire Science andTechnology, 34(3):195–214, 2006.

A.R. Savkoor. Dry adhesive friction of elastomers. PhD thesis, Technische Universiteit Delft,1987.

A.R. Savkoor. On the friction of rubber. Wear, 8:222–237, 1965.

A.R. Savkoor. Some aspects of friction and wear of tyres arising from deformations, slip andstresses at the ground contact. Wear, 9:66–78, 1966.

A.R. Savkoor. Mechanics of sliding friction of elastomers. Wear, 113:37–60, 1986.

A. Schallamach. A theory of dynamic rubber friction. Wear, 6:375–382, 1963.

E. Seta, Y. Nakajima, T. Kamegawa, and H. Ogawa. Hydroplaning analysis by fem and fvm:Effect of tire rolling and tire pattern on hydroplaning. Tire Science and Technology, 28(3):140–156, 2000.

J.C. Simo and T.A. Laursen. An augmented lagrangian treatment of contact problems involvingfriction. Computers & Structures, 42(1):97–116, 1992.

Tire modeling using ABAQUS. Lecture notes, 2000.

E. Tonuk and Y.S. Unlusoy. Prediction of automobile tire cornering force characteristics by finiteelement modeling and analysis. Computers & Structures, 79:1219–1232, 2001.

M. Trinko. Tire stresses and deformation analysis. Presentation, 32th Tire Mechanics ShortCourse, Koln, 2007.

VMI, May 2007. URL http://www.vmi.nl/uploads/documents/10.pdf.

Vol 177, nr 3/4. Special issue: Computational modeling of contact and friction. Computer Methodsin Applied Mechanics and Engineering, 177(3/4):163–468, 1999.

J.D. Walter. Tire materials and manufacturing. Presentation, 32th Tire Mechanics Short Course,Koln, 2007a.

J.D. Walter. Rules and regulations governing tires. Presentation, 32th Tire Mechanics ShortCourse, Koln, 2007b.

S. Westermann, F. Petry, R. Boes, and G. Thielen. Experimental investigations into the predictivecapabilities of current physical rubber friction theories. Kautschuk Gummi Kunststoffe, 57:645–650, 2004.

M.L. Williams, R.F. Landel, and J.D. Ferry. The temperature dependence of relaxation mecha-nisms in amorphous polymers and other glass-forming liquids. Journal of the American ChemicalSociety, 77:3701–3707, 1955.

P. Wriggers. Computational contact mechanics. Springer, second edition, 2006.

P. Wriggers, T. Vu Van, and E. Stein. Finite element formulation of large deformation impact-contact problems with friction. Computers & Structures, 37(3):319–331, 1990.

http://www.vmi.nl/uploads/documents/10.pdf

60 BIBLIOGRAPHY

C. Yanga, U. Tartaglinoa, and B.N.J. Persson. A multiscale molecular approch to contact me-chanics. European physical journal E, 15(1), 2006.

G. Zavarise, P. Wriggers, E. Stein, and B.A. Schrefler. Real contact mechanisms and finite ele-ment formulation – A coupled thermomechanical approach. International Journal for numericalmethods in Engineering, 35:767–785, 1992.

G. Zavarise, M. Borri-Brunetto, and M. Paggi. On the resolution dependence of micromechanicalcontact models. Wear, 262:42–54, 2007.

I. Zeid and J. Padovan. Finite element modeling of rolling contact. Computers & Structures, 14(1/2):163–170, 1981.

M. Ziefle and U. Nackenhorst. A new update procedure for internal variables in an ale-descriptionof rolling contact. Proceedings in Applied Mathematics and Mechanics, 5:71–74, 2005.

O.C. Zienkiewicz and Y.K. Cheung. The Finite Element Method in Structural and ContinuumMechanics, volume 1-3. McGraw-Hill Publ, London, first edition, 1967.