Enhanced friction modeling for steady-state rolling tires

Enhanced friction modeling for steady-state rolling tires

Citation for published version (APA):

Steen, van der, R. (2010). Enhanced friction modeling for steady-state rolling tires. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR692262

DOI:

10.6100/IR692262

Document status and date: Published: 01/01/2010 Document Version:

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Enhanced friction modeling for

steady-state rolling tires

Apollo Vredestein B.V., Enschede, the Netherlands.

René van der Steen (2010). Enhanced friction modeling for steady-state rolling tires. Ph.D. thesis, Eindhoven University of Technology, Eindhoven, the Netherlands. A catalogue record is available from the Eindhoven University of Technology Library. ISBN: 978-90-386-2390-0

Cover design: Oranje Vormgevers, Eindhoven, the Netherlands. Reproduction: Ipskamp Drukkers B.V., Enschede, the Netherlands. Copyright c _{2010 by René van der Steen. All rights reserved.}

Enhanced friction modeling for

steady-state rolling tires

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 9 december 2010 om 16.00 uur

door

René van der Steen

prof.dr. H. Nijmeijer

Copromotor: dr.ir. I. Lopez

Summary

Enhanced friction modeling for steady-state rolling tires

Tire modeling is nowadays a necessary tool in the tire industry. Car manufacturers, gov-ernments and consumers demand better traction under all circumstances, less wear and more recently less noise and a lower rolling resistance. Therefore finite element analysis is adopted in the design process of new tires to cope with these, often conflicting, de-mands. Finite element tire modeling can increase the insight on specific properties of a tire, decrease the development time and reduce development costs of new tires. However in practice most finite element models are still not able to match outdoor experiments. Both the static deformation and the dynamic response of the tire rolling on the road should be accurately predicted. The cornering, braking and traction of a tire depend on the generated friction forces. Friction depends not only on the tread properties of the tire, but also on the road surface and environmental conditions. The main goal of this thesis is to develop a robust and numerically efficient friction model for finite element tire simulations and to create a framework for the identification and implementation of friction related parameters.

The numerical modeling of a tire in combination with its environment is a challenging task, since different physical phenomena play a role. Typically the mechanical, thermal and fluid domains contribute to the tire response. This research is restricted to the me-chanical domain, where a numerical modeling framework for steady-state rolling tire simulations is defined. In future developments of the model other effects can be in-cluded using this framework as a base. One of the objectives of this thesis is to develop and validate a tire friction model for finite element analysis, which captures observed ef-fects of dry friction on the handling characteristics of rolling tires. Friction by itself is a highly complex interaction phenomenon between contacting materials and can be mod-eled on many different length scales, applying different numerical techniques. This can however lead to an enormous computational burden and as a result it can be impractical

for an industrial application. To provide a numerically feasible and relatively fast solution a phenomenological friction model is chosen, where the parameters are identified using a two step experimental / numerical approach.

Firstly, friction experiments are performed on a laboratory abrasion and skid tester to investigate the influence of contact pressure on the frictional force. In this experimental setup a small solid tire, with adjustable side slip angle, is pressed on an abrasive disk. The friction present between the abrasive disk and solid tire drives the tire and the resulting forces are measured with a force sensor. Several experiments under different normal loads and side slip angles of the tire are conducted. These measurements, under low rolling velocity, are used to identify contact pressure dependent friction parameters. The relevant parts of this setup are modeled in the commercial finite element packageABAQUS

and the steady-state performance of the small tire under different slip angles is evaluated and compared with experiments. It is shown that the present turn slip, which has great impact on the slip velocity field at the trailing edge of the contact area, is captured well with the model. Furthermore, the calculated cornering stiffness is in good agreement with the experiments.

Secondly, outdoor braking experiments at different velocities with a full scale tire are conducted to obtain a velocity dependent parameter set for the tire friction model. The derived friction model is then coupled to a finite element model of this full scale tire, which is also constructed in the software packageABAQUS. The finite element model is

validated statically using measurements of the contact pressure distribution, contact area and of the radial and axial stiffness of the tire. The steady-state transport approach in

ABAQUSis used to efficiently compute steady-state solutions at different forward velocities

as used in the outdoor experiments.

Finally, the predictive capability of the finite element tire model in combination with the proposed friction model is assessed. The basic handling characteristics, such as pure braking, pure cornering, and combined slip under different loads, inflation pressures and velocities are evaluated and validated with experiments. Based on this comparison, it can be concluded that all three basic handling characteristics are adequately predicted.

Contents

Summary v

Nomenclature xi

1 Introduction 1

1.1 General introduction . . . 1

1.2 Tire performance and modeling . . . 2

1.3 Challenges in FE tire modeling . . . 4

1.4 Objectives and research approach . . . 6

1.5 Contributions . . . 8

1.6 Outline of this thesis . . . 8

2 Modeling framework for steady-state rolling tires with friction 9 2.1 Introduction . . . 9

2.2 Overview of friction models . . . 10

2.2.1 Amontons-Coulomb . . . 10

2.2.2 Rubber friction . . . 11

2.3 Tire research using FE Models . . . 17

2.3.1 Implicit analysis . . . 17

2.3.2 Explicit analysis . . . 19

2.4 Modeling framework . . . 20

2.4.1 Numerical method . . . 20

2.4.2 Friction model . . . 21

2.5 Steady-state transport analysis . . . 22

2.5.1 Contact conditions for steady-state rolling . . . 25

2.5.2 Frictional stress for steady-state rolling . . . 26

2.6 Implementation of a friction law in steady-state rolling . . . 28

2.7 Validation of the implemented friction law . . . 30

2.7.1 Cylindrical model . . . 30

2.7.2 Comparison with standard Coulomb friction law . . . 30

2.7.3 Validation of friction law for varying friction coefficient . . . 32

2.8 Steady-state free-rolling . . . 33

2.9 Conclusions . . . 34

3 Simulation procedure to compute handling characteristics with an FE model 37 3.1 Introduction . . . 37

3.2 Approach to identify parameters of the friction model . . . 39

3.3 Design of the test tire used for experimental validation of the FE model . . 40

3.4 FE tire model . . . 41

3.4.1 2D tire cross section . . . 41

3.4.2 Rim mounting and inflation . . . 43

3.4.3 3D tire model . . . 43

3.4.4 Mesh effect on lateral force in 3D model . . . 45

3.4.5 Static loading of the tire . . . 47

3.5 Simulation procedure to compute the braking characteristic of a rolling tire 47 3.5.1 Mesh effect on the force equilibrium in vertical direction . . . 48

3.5.2 Effect of penalty parameter in the friction model on the longitudi-nal force . . . 49

3.6 Simulation procedure to compute the cornering and combined slip char-acteristics . . . 50

3.7 Conclusions . . . 53

4 Friction parameter identification using a Laboratory Abrasion and skid Tester 55 4.1 Introduction . . . 55

4.2 Identification using lab scale experiments . . . 56

4.3 Experimental setup . . . 57

4.4 FE model of the setup . . . 58

4.4.1 Material model . . . 58

4.4.2 2D model . . . 62

4.4.3 3D model . . . 63

4.5 Parameter identification . . . 66

4.6 Comparison of numerical and experimental results . . . 67

4.7 Discussion . . . 72

4.8 Conclusions . . . 75

5 Friction parameter identification using longitudinal slip characteristics 77 5.1 Introduction . . . 77

5.2 Identification using full scale experiments . . . 78

5.3 Tire force and moment measurements . . . 79

CONTENTS ix

5.5 Free-rolling rotational velocity: Comparison of FEM prediction and

exper-iments . . . 85

5.6 Comparison between FEM and MF predictions . . . 86

5.7 Effect of inflation pressure on the longitudinal force . . . 88

5.8 Discussion . . . 91

5.9 Conclusions . . . 92

6 Predictive capability of the FE tire model 93 6.1 Introduction . . . 93

6.2 Prediction of the handling characteristics . . . 94

6.2.1 Pure cornering characteristic . . . 94

6.2.2 Combined slip characteristic . . . 100

6.2.3 Friction power distribution in the footprint . . . 105

6.3 Force and moment measurements and Magic Formula . . . 107

6.4 Comparison of the FE model and the Magic Formula . . . 111

6.4.1 Pure cornering . . . 111

6.4.2 Combined slip . . . 115

6.5 Conclusions . . . 119

7 Conclusions and recommendations 121 7.1 Conclusions . . . 121

7.2 Recommendations . . . 123

References 125 A Friction model implementation 133 A.1 Stick . . . 133

A.2 Slip . . . 135

B Mesh effect on the force equilibrium in vertical direction 137

C Computation of slip velocity field for the LAT 100 setup 141

Samenvatting 143

Dankwoord 145

Nomenclature

Acronyms and abbreviations

acronym description acronym description

ALE arbitrary Lagrangian-Eulerian MF Magic Formula

DoE design of experiments NASA national aeronautics and space FE(M) finite element (method) administration

FEA finite element analysis R radial

ISO international organisation for TNO Netherlands organization for applied

standardization scientific research

LAT laboratory abrasion and skid tester WLF Williams-Landel-Ferry equation L.I. load (capacity) index

Operations and notation

symbol description symbol description

a, A scalar ˆx, ˆX skew-symmetric matrix,

a, A vector or matrix associated with x, X

| · | absolute value _× outer product

X Lagrangian coordinate · inner product

x Eulerian coordinate xT_{, A}T _{vector or matrix transpose} ˙x, (¨x) (double) time derivative _{|| · ||} magnitude

∂ partial derivative

Roman symbols and letters

symbol description unit

A0 cross sectional area m2

AExp experimental contact area m2

AMod calculated contact area m2

a line length m

tuning parameter

a Eulerian acceleration vector m/s2

aT shift factor

b line length m

tuning parameter

C10 hyperelastic material coefficient Pa

CF α cornering stiffness N/deg

CM z aligning stiffness Nm/deg

c line length m

tuning parameter d tuning parameter

D1 hyperelastic material parameter Pa

E Young’s modulus Pa

E0 _{elastic storage modulus Pa}

E00 elastic loss modulus Pa

errorrel relative error

Ff ric frictional force N

Fx longitudinal force N

Fy lateral force N

Fz normal (vertical) force (load) N

f frequency rad/s

G0 _{shear storage modulus Pa}

G00 shear loss modulus Pa

h tuning parameter friction model

penetration m

I identity matrix J Jacobian matrix

k tuning parameter friction model

ks slope

L1,2,3 length m

L initial length m

l current length m

Mx overturning moment Nm

My moment or driving torque Nm

Mz self-aligning moment Nm

NOMENCLATURE xiii

symbol description unit

n cornering axis in normal direction

p contact pressure Pa

p0 tuning parameter friction model Pa p1,2 tuning parameter

plimit lower bound on contact pressure Pa Rα side slip rotation matrix

Rβ rotation matrix

Rc cornering rotation matrix Rs spinning rotation matrix

R radius m

parameter friction model

Rinner inner tire radius m

Router outer tire radius m

r radius m

re effective rolling radius m

rl loaded rolling radius m

ru unloaded rolling radius m

T rigid axle position

T torque Nm

t time s

pneumatic trail m

ti _{orthogonal unit vector, i ∈ {1, 2}}

uc longitudinal carcass deflection m

V velocity m/s

Vmax tuning parameter friction model m/s

Vsx longitudinal slip velocity m/s

Vsy lateral slip velocity m/s

Vx longitudinal (forward) velocity m/s

Vy lateral velocity m/s

v (longitudinal) slip velocity m/s

vc lateral carcass deflection m

vo lateral resultant force offset m

vs slip velocity m/s

v Eulerian velocity vector m/s

Width tire width m

X Lagrangian coordinate x Eulerian coordinate x x-position m Y Lagrangian coordinate y Eulerian coordinate y y-position m

Greek symbols and letters

symbol description unit

α side slip angle deg

β angle deg

χ deformation map

∆ increment in position m

δ phase lag rad

e nominal strain

˙γ, ˙γ slip velocity (vector) m/s

˙γcrit critical slip velocity m/s

κ longitudinal slip

κmax tuning parameter friction model

λi wavelength i m

µ friction coefficient

µk kinetic friction coefficient µlock tuning parameter friction model

µM F Magic Formula dependent friction coefficient µm tuning parameter friction model

µs tuning parameter friction model static friction coefficient

Ω rotational velocity rad/s

ω rotational velocity rad/s

ωα free-rolling rotational velocity for nonzero side slip angles

rad/s ωf ree free-rolling rotational velocity rad/s

ρ density kg/m3

σe nominal stress Pa

τ, τ frictional stress (vector) Pa

τcrit critical frictional stress Pa

Subscripts and indices

symbol description symbol description

0 center max maximum

c cornering n discrete time step

D deformable body p contact pressure dependent

disk abrasion disk R rigid foundation

eqv equivalent r reference frame

i index, i ∈ {1, 2} slip slip

j node j v slip velocity dependent

max maximum wheel wheel

NOMENCLATURE xv

ISO sign conventions

Tire force and moment

V

α

F

F

M

F

F

M

V

α

F

F

M

Top view tire

_{Rear view tire}

Tire velocity and slip velocity

V

_α

V

V

V

V

V

tan α =

κ =

₋

Top view tire

longitudinal slip

C

HAPTER ONE

Introduction

Abstract /In this chapter a general introduction of tire modeling is presented. The current chal-lenges in tire modeling using finite element methods are discussed, which form the motivation of the research objectives. Based on these objectives, a research approach is presented and the main contributions of this thesis are stated. The chapter ends with the outline of the thesis.

1.1 General introduction

The development of pneumatic tires started with the patent by John Boyd Dunlop in 1888 (Dunlop, 2010)1_{and is still going on today. The first pneumatic tires had small cross}

sec-tions and high inflation pressures, mainly for bicycle applicasec-tions. From the 1920s, larger tires were introduced for the upcoming vehicle industry. Two major evolutions took place in the 1960s, the tubeless tire was introduced and bias ply tires were replaced with radial ply tires, which improved the wear and handling properties significantly. The main dif-ference between the bias and radial ply tire is the orientation of the plies. In bias ply tires, the body ply cords are laid at angles substantially less than 90◦ _{to the tread centerline,}

extending from bead to bead. In radial tires, the body ply cords are laid radially from bead to bead, at 90◦ _{to the centerline of the tread. Two or more belts are laid diagonally}

in the tread region to obtain the required strength and stability (Gent and Walter, 2005, chapter 1). In Figure 1.1, a typical layout of a radial tire, which is now the standard for passenger car tires, is shown.

The tire construction, such as aspect ratio and belt construction, depends on the size of the tire and the target market. This information is printed on the sidewall of every tire, e.g. 215/55 R16 97 H. The first number (215) is the nominal section width in mm, the

1_{The idea of a pneumatic tire was already patented by Thomson (1847). Dunlop’s patent was later}

declared invalid on the basis of this patent, but is generally considered as the first practical pneumatic tire. 1

Figure 1.1 /Cross section of a radial tire with the main components, Ghoreishy (2008)

second number (55) is the percentage of the height/width ratio of the cross section. The R (radial) stands for the tire construction code. The rim diameter (16) is given in inches and 97 is the load capacity index (L.I.), which is a reference to the maximum load capac-ity. The last symbol (H) is the speed symbol, which corresponds to a maximum allowable speed (European Tyre and Rim Technical Organisation, 2010).

More recent developments in the tire industry are the run-flat technology, which en-ables the vehicle to continue at reduced speeds after deflation of a tire, and the so-called ultralow-aspect tires, which have very short sidewalls (Rodgers, 2001).

1.2 Tire performance and modeling

All tires must meet the following fundamental set of performance factors (Rodgers, 2001):

• Provide load carrying capacity.

• Provide cushioning, damping and minimum noise and vibration. • Transmit forces and moments.

• Resist abrasion.

• Have a low rolling resistance.

• Be durable and safe throughout the expected lifespan.

The different components of the tire determine the tire overall characteristics in response to the application of load, torque or steering input, resulting in the generation of forces

1.2 TIRE PERFORMANCE AND MODELING 3

and deflection of the tire. The mechanical properties are often interrelated, which means that a design change for one of the performance factors, can affect the other factors both positively or negatively (Rodgers, 2001). Besides the engineering aspects, also economi-cal factors, (limitations of) the manufacturing process (Walter, 2007b) and government regulations (Walter, 2007a) have to be taken into account. All these aspects are captured in specific performance criteria, which are visualized in a performance chart in Figure 1.2. In this figure, a new tire design is compared with an existing reference tire for six perfor-mance criteria. It should be noted that these criteria are not defined unambiguously in literature, e.g. (Gent and Walter, 2005, chapter 1).

noise and vibration

dry traction wear resistance

rolling resistance wet traction 95% 100% 110% 105%

ice and snow traction

Figure 1.2 /Example of a new tire design (dashed line) related to a reference tire for sev-eral performance functions. Based on the tread performance chart (Mundl et al., 2008).

Dry, wet, and, ice and snow traction are directly related to the handling properties of the tire and as such to safety issues, e.g. the braking distance at 100 km/h. Obviously, wear resistance is related to the tire’s lifespan, while rolling resistance has direct influence on vehicle fuel consumption. A tire rolling over a road generates undesired noise both at the surroundings (exterior noise) and inside the vehicle itself (interior noise).

An accurate model of the tire behavior enables the engineer to optimize the overall per-formance, while taking into account the different performance criteria. For this purpose, several modeling techniques have been developed during the last decades.

In the case of handling models, a distinction can be made between empirical models and (simple) physical models. Each type is developed for a specific purpose (Pacejka, 2006). The (semi)-empirical models are based on experimental data. These models have a spe-cific structure and the parameters are usually obtained by regression techniques. One of the most well-known and widely used tire handling models is the Magic Formula model of Pacejka (Bakker et al., 1987). This model is very useful to reproduce and interpolate

tire properties. The great advantage is of this model is the low computational cost. This is a very important requirement for tire models that are used in vehicle dynamics, where the tire is just one part of the total vehicle model. An overview of specific tire simulation challenges in vehicle dynamics is presented by Rauh and Mössner-Beigel (2008). The simple physical models use analytical expressions to describe the forces and moments and can produce realistic results, e.g. the stretched string model (Pacejka, 2006), if the parameters are assigned appropriate values. Usually the application field of these kind of models is limited.

The main drawback of these two types of models is that the parameters are experimen-tally determined from full scale tire tests and as such these models can not be used to predict the influence of tire construction design changes. For detailed analysis of a tire itself, the Finite Element Method (FEM) can be used. Such finite element models are complex, but allow to investigate the effects of tire design parameters on the generated forces and moments. FE models are nowadays a standard tool in the tire industry, and their use opens the possibility of tire virtual prototyping.

1.3 Challenges in FE tire modeling

Finite element tire modeling can increase the insight on the relative influence of specific tire properties on the tire behavior, decrease the development time and eventually reduce development costs of new tires. However in practice most finite element models are still not able to accurately match outdoor experiments.

INTERACTION MODEL OUTPUTS INPUTS FE Model environment

contact model

Figure 1.3 /Schematic overview of the necessary components for every Finite Element model of a rolling tire interacting with the environment.

1.3 CHALLENGES INFETIRE MODELING 5

The numerical modeling of a tire in combination with its environment, shown in Figure 1.3, is a difficult task, since different physical phenomena play a role. Typically the me-chanical, thermal and fluid domains contribute to the tire response. These interrelated domains and the wide range of operating conditions provide several challenges in the modeling process:

• Material modeling. A correct description of the different rubber and cord-rubber plies is required. Large deformations and large strains, as well as incompressibility of rubber compounds should be accounted for.

• Contact modeling. Contact models in normal and tangential direction need to be defined. Especially, friction models for tangential contact depend on environmental conditions.

• Geometric modeling. Modeling of geometric shapes is nowadays not an issue any-more, but in many cases the creation of valid elements for detailed tread patterns is difficult.

• Temperature modeling. The temperature of the tire changes during operation, which affects the mechanical properties of the tire and hence its behavior.

• Steady-steady versus transient modeling. Different numerical algorithms are re-quired for steady-state rolling tires and the modeling of transient effects e.g. when impacts with obstacles occur.

• Measurements for validation purposes. This is not directly related to FE tire mod-eling, however necessary to validate FE models. Obtaining good measurements of the forces and moments or other system quantities acting on rolling tires is a difficult task.

The latest overview of the current state of the art of FE modeling of rolling tires is given in the paper of Ghoreishy (2008). In the summary, it is stated that despite the substantial progress, achieved during the last decades, the analysis of the complicated tire structure is still a formidable task. Sofar, none of the current published works is capable of giving a full analysis of the tire under different loading conditions. Instead, each work tried to focus on some critical aspect of the tire and investigated this specific feature as deep as possible.

Furthermore, the majority of tire related research in literature does not provide detailed information about the used models and parameter values are often lacking, which makes interpretation of the results difficult. Finally, a validation of the used FE models for rolling situations with full scale tire experiments is often missing.

1.4 Objectives and research approach

This thesis focuses on the FE prediction of the steady-state handling characteristics of rolling tires on dry roads. Although handling on wet roads is also important, the pres-ence of water or snow add significant complexity to the tire/road interface. Additionally, obtaining consistent measurements under controlled wet conditions is very challenging in practice. Therefore, wet roads are not considered in this research.

The handling characteristics considered in this thesis are pure braking, pure cornering and combined slip (Pacejka, 2006) and are defined as follows. In case of pure braking, the tire is braked from free-rolling up to wheel lock. For pure cornering, the free-rolling tire is steered up to ±12◦ _{side slip angle and in case of combined slip, the tire is braked}

up to wheel lock when rolling at a constant slip angle. These conditions cover driving situations from normal driving to extreme manoeuvres.

To evaluate the handling performance of a tire, both the static deformation and the dynamic response of the tire rolling on the road should be accurately predicted. The cornering, braking and traction of a tire depend on the generated friction forces. Friction depends not only on the tread properties of the tire, but also on the road surface and en-vironmental conditions. Frictional behavior for model parameterization can be acquired by extensively testing tires under different conditions. Experimental characterization of frictional properties of rubber compounds is cumbersome, since environmental conditions influence these measurements. As a result, Coulomb’s friction law, with a constant friction coefficient, is still often used in finite element simulations to predict handling characteristics. It is however clear from experiments with elastomers that rubber friction depends on various parameters like contact pressure, sliding velocity, temperature and surface roughness. Because of these dependencies Coulomb’s law is not sufficient to accurately predict the handling characteristics over the desired range. Furthermore, numerical problems occur during steering at large side slip angles when Coulomb’s law with realistic friction coefficients is used in finite element simulations. To overcome these limitations a different strategy is needed to capture observed effects of dry friction on the handling characteristics of rolling tires. Therefore the essential goal of this thesis is:

The development of a robust and numerically efficient friction model for finite element tire simulations and to create a framework for the identification and implementation of friction related parameters.

Furthermore, this friction model should capture observed effects of dry friction and it should be compatible with commercial FE codes. As mentioned above, rubber friction depends on several parameters. This thesis focuses on developing a friction model,

1.4 OBJECTIVES AND RESEARCH APPROACH 7

which includes the influence of contact pressure and sliding velocity.

To achieve the above objective, the currently available state-of-the-art numerical methods in the commercial finite element packageABAQUS are used. Emphasis is placed here on

the modeling of frictional contact of the tire with the road, the FE model of the tire itself is provided by tire manufacturer Vredestein.

Lab scale experiments

3D Forces Experiments

3D Forces Simulations Full scale experiments

Inputs

Friction

model

FE Tire

Virtual prototyping Lab scale setup

FEM 3D Forces Simulations 3D Forces Experiments Inputs

Figure 1.4 /Schematic overview of the two step experimental / numerical approach to obtain friction information using both small scale and full scale experiments.

For the parameter identification of the friction model several measurements have been carried out on two experimental setups on different scales, as shown in Figure 1.4. First, friction experiments are performed on a commercially available small scale lab setup (LAT 100). On this setup controlled experiments on a small solid tire, under low rolling velocities, are performed. These measurements are used to identify the parameters of the friction model, which are contact pressure dependent. The friction model is coupled to an FE model of this lab setup to simulate the hub forces. These forces are compared to the experimentally found hub forces and used to validate the parameters of the friction model. With the small scale setup it is only possible to conduct experiments at low sliding velocities, since excessive wear of the small tire occurs at higher velocities. Therefore outdoor braking experiments with a full scale tire at different velocities are conducted to obtain a slip velocity dependent parameter set for the friction model. These experiments are carried out with the TNO Tyre Test Trailer. Finally, the completely identified friction model is validated by comparing simulated and measured cornering and combined slip characteristics.

1.5 Contributions

The main contributions presented in this thesis are:

• The definition and implementation of a complex friction model in the commercial packageABAQUS, which is suitable for tire handling simulations in the full operating

range.

• A procedure for the identification of the parameters of the friction model using a two step experimental / numerical approach, combining small scale lab experi-ments with full scale outdoor experiexperi-ments.

• Validation of the predicted handling characteristics with full scale outdoor experi-ments, covering the entire operating range.

1.6 Outline of this thesis

In Chapter 2, the modeling framework to obtain the handling characteristics of rolling tires, with friction, is described. The choice for the numerical approach and the friction model, based on a literature overview, is made. The background of the numerical method is reviewed and the implementation of the chosen friction model is described.

After that, the strategy for identification of the parameters of the chosen friction model is explained in more detail in Chapter 3. Furthermore, the design of the used test tire and the corresponding layout of the FE model are presented. The different simulation steps required to compute the pure braking, pure cornering and combined slip characteristics are discussed.

In Chapter 4, the identification of the contact pressure dependent part of the tire friction model is given. Friction data is obtained using a commercial laboratory abrasion and skid tester. From this data the contact pressure related parameters are derived and imple-mented in an FE model for the tire/disk contact.

Next, the identification of the slip velocity dependent part, using measured axle forces, is presented in Chapter 5. The complete identified friction model is then coupled to the FE model of the tire under testing. The computed steady-state longitudinal slip characteris-tics are compared with the full scale outdoor experiments and a discussion of the results is given.

In chapter 6, the fully identified friction model is used to compute the pure cornering and combined slip characteristics. The predicted characteristics are compared with ex-periments and it is shown that the handling performance of the tire can be adequately predicted with the identified friction model.

Finally, in Chapter 7, the main conclusions are summarized and recommendations for future work are given.

C

HAPTER TWO

Modeling framework for steady-state

rolling tires with friction

Abstract /A numerical modeling framework to obtain the handling characteristics of rolling tires, with friction, is defined in this chapter. Based on a literature overview of friction models and tire simulation methods the choice for the numerical approach and the friction model is motivated. The background of the numerical method is presented together with the implementation of the chosen friction model. The validation of this implementation is presented at the end of the chap-ter.

2.1 Introduction

In this chapter the numerical approach to obtain the handling characteristics of rolling tires including friction is described. First a short literature review of friction is presented in Section 2.2, which shows the difference of rubber friction compared to other solids and in Section 2.3 Finite Element Analysis (FEA) related to tires is discussed. Based on these findings a numerical method is chosen, which is able to efficiently compute the handling characteristics of steady-state rolling tires. Furthermore, the choice of the friction model is motivated in Section 2.4. The background of the numerical method is reviewed in Sec-tion 2.5 and the implementaSec-tion of the fricSec-tion model is discussed in SecSec-tion 2.6. This implementation is validated using different test models, which are presented in Section 2.7. In Section 2.8 a description of an algorithm to obtain the solution, which corre-sponds to a so-called free-rolling tire is discussed. The chapter ends with conclusions in Section 2.9.

2.2 Overview of friction models

2.2.1 Amontons-Coulomb

Although Leonardo da Vinci (1452-1519) is generally credited as being the first to develop the basic concepts of friction (Bowden and Tabor, 1964), Amontons formulated the first two laws of friction and Coulomb the last two. These classic laws are summarized by Moore (1972) as follows:

• The friction force is proportional to load.

• The coefficient of friction is independent of the apparent contact area. • The static coefficient is greater than the kinetic coefficient.

• The coefficient of friction is independent of the sliding velocity. This can be formulated as

Ff ric = µFz, (2.1)

where Ff ric is the frictional force, µ the constant friction coefficient and Fz the applied

load and (2.1) is usually referred to as the Coulomb friction model. Deviations from Coulomb friction

Experiments often show deviations from the basic Coulomb friction model. Variations of the basic model are the difference between a static µsand a kinetic µkfriction coefficient,

see Figure 2.1 for some examples. This transition can be discontinuous or continuous, e.g. using an exponential decaying function between the static and kinetic coefficient. In general, the friction coefficient varies for increasing sliding velocity especially in the

µ

µ

µ

µ

µ

µ

µ

µ

µ

v

v

v

µ

µ

µ

v

2.2 OVERVIEW OF FRICTION MODELS 11

presence of lubricants, e.g. the Stribeck curve (Stribeck, 1902). First the coefficient de-creases to a minimum value and after that it inde-creases with higher sliding velocity. The review paper of Olsson et al. (1998) describes several static friction models, which focus only on sliding velocity, and dynamic friction models. The dynamic models also describe the stick-slip transitions around zero sliding velocity. Examples are the Dahl model and two variants of this model, the LuGre model and the model of Bliman and Sorine. These models are often used in control systems, in such a way that they can be used for friction compensation in mechanical systems.

2.2.2 Rubber friction

Rubber friction differs from friction of most other solids. According to Moore (1972) the second friction law appears to be valid only for materials possessing a definite yield point and it is does not apply to elastic or viscoelastic materials. The third law is not obeyed by any viscoelastic material and the fourth law is not valid for any material (Moore, 1972). It is also clear from experiments with elastomers that rubber friction depends on various parameters like contact pressure, sliding velocity, temperature and surface roughness. Because of these dependencies Coulomb’s law is not sufficient to model the frictional response of an elastomer.

The friction force between rubber and a rough surface has two contributions commonly described as the adhesion and hysteretic components (Moore, 1972). The hysteretic com-ponent results from internal friction of the rubber; during sliding asperities of a rough surface exert oscillating forces on the rubber surface. This leads to cyclic deformations of the rubber and to energy dissipation caused by the internal damping of the rubber (Pers-son, 2001). The adhesion component is caused by the intermolecular attractive forces between the contacting bodies (Wriggers and Reinelt, 2009).

The article of Grosch (1963) is one of the early reports, which describes the analogy be-tween the friction coefficient as function of sliding velocity v and the energy dissipated per cycle (tan δ) as function of frequency f, see Figure 2.2. The phase angle is given by tan δ = E00_/E0_{, with δ the phase lag between stress and strain; E}0 _{and E}00 _{are the}

storage and loss modulus respectively. In this work experimental results of different vul-canized rubbers are presented. The experiments are conducted at different temperatures and shifted, with shift factor aT, to a master curve using the WLF equation, developed by

Williams et al. (1955).

Grosch indicates that friction is due to energy dissipated when rubber is compressed and released by asperities. Friction for dry sliding on a smooth surface is due to energy dissipated as rubber sticks and slips on a molecular scale. In the case of sliding on a rough dry surface the dissipative process appears at different speeds, corresponding to different length scales: asperities (hysteretic component) and molecules (adhesion component).

Several contributions to the friction coefficient µ can arise at the same sliding speed from stick-slip processes occurring at different length scales, see Figure 2.3.

An overview of models to describe the frictional interaction between tire and road is given next.

Figure 2.2 /Grosch’s interpretation of the equivalence between master curves of rubber friction versus sliding velocity and tan δ versus frequency f, Gent (2007).

Figure 2.3 /Combined effect of sliding on a rough and dry surface, Gent (2007).

Savkoor proposed a model, based on results of Grosch, where he described the shape of an isothermal master curve with an empirical relation (Savkoor, 1966, 1987)

µ(v) = µs+ (µm− µs) exp  −h2_log2  v Vmax  , (2.2)

where µsis a static coefficient of friction, µmthe peak value of the function (which occurs

at |v| = Vmax) and h is a dimensionless parameter reflecting the width of the speed range

in which friction varies, as shown in Figure 2.4. According to Savkoor (1966), the values of Vmax and µm depend on the viscoelastic properties of the rubber. At higher

tempera-2.2 OVERVIEW OF FRICTION MODELS 13

tures Vmaxincreases and the friction curve is shifted significantly towards higher speeds.

The advantage of this function is that the parameters are related directly to the shape of the friction curve, which makes a physical interpretation possible. The disadvantage of this model is that only the influence of sliding velocity on the friction coefficient is con-sidered, at constant temperature, and that the influence of contact pressure is not taken into account. µ v h µs µm Vmax

Figure 2.4 /Model proposed by Savkoor to describe an isothermal master curve.

Based on the original Schallamach 2D model (Schallamach, 1971) a generalized theory that predicts the buckling effects, known as Schallamach waves (Schallamach, 1952; Bar-quins, 1992) which occur at very smooth surfaces, is presented by Berger and Heinrich (2000). In contrast to the original Schallamach model, where Coulomb friction is used, this generalized theory considers a friction coefficient that depends on the local normal pressure over the contact line,

µ(p) = µ0

p E

n

(2.3) with E the linear elastic modulus and µ0 and n fitted parameters. A similar pressure

dependent model is used in the work of Trinko (2007), where µ0 and E are replaced with

a single constant.

The phenomenological models proposed by Dorsch et al. (2002) are fully empirically, where a velocity v and pressure dependency p for the friction coefficient is assumed.

µ(v, p) = c1pc2vc3 (2.4)

µ(v, p) = c1p + c2p2+ c3v + c4v2+ c5pv (2.5)

The friction coefficient is described by a power law or a linear approximation, in Dorsch et al. (2002) a so-called full quadratic model is presented. Although these are very simple models, several experiments are required to identify the coefficients ci. Design of

sliding velocity values needed (Montgomery and Runger, 1999).

A velocity and pressure dependent model (Wriggers, 2006), which is proposed by Nack-enhorst (2000) is given by µ(v, p) = c1  p c2 c3 + c4ln v c5 − c 6ln v c7 , (2.6)

where all seven parameters must be determined using experiments. In this model the influence of contact pressure and sliding velocity is decoupled and summed to obtain the total friction coefficient, which indicates that this model also originates from statistical techniques. All these three models are able to accurately fit experimental data, but the obtained parameters of these models have no straightforward physical interpretation. The model proposed by Huemer et al. (2001a) is developed for sliding rubber blocks on ice and concrete. The phenomenological friction law is given by

µ(v, p) = α|p| n−1_{+ β} a + _|v|1/mb + c |v|2/m (2.7) and is developed for a macroscopic model. In this approach the coefficient of friction de-pends on normal pressure p, sliding velocity v and temperature. The friction coefficient itself (2.7) is only a function of normal pressure and sliding velocity. Temperature effects are incorporated using the WLF transformation, i.e., if the current temperature is differ-ent from a reference temperature, an equivaldiffer-ent new sliding velocity for the reference temperature is calculated.

The friction law in (2.7) requires seven parameters (a, b, c, n, m, α, β), which must be identified using experimental data. They account for the dependence of the friction coef-ficient on friction surface, rubber compound, and dimensions and geometric shape of the rubber block. The identification is based on a least square error method and performed iteratively. First the coefficients a, b and c related to the sliding velocity are fitted and then αand β related to the contact pressure, this process is repeated until a defined error cri-terium is reached. The whole process is done for every value of n and m. Furthermore, in the identification procedure the contact pressure is replaced with the averaged pressure on the contact surface. An evaluation of the model is shown in Figure 2.5.

This work has been continued by Hofstetter et al. (2003), where a thermo-mechanical coupling has been introduced. Energy dissipation during sliding is converted into heat and this heat flux causes a temperature rise of the rubber and road. Simulations of abrasion of the rubber block are considered in Hofstetter et al. (2006), where the obtained numerical results are also compared with experimental data. Material loss occurs only at the front edge and is captured qualitatively with the proposed friction and abrasion models. However the model also predicts material loss in the middle of the

2.2 OVERVIEW OF FRICTION MODELS 15 0 0.5 1 1.5 2 0.65 0.7 0.75 0.8 0.85 0.9 log(v) [mm/s] µ [−]

Figure 2.5 /Friction coefficient as function of sliding velocity using (2.7), the parameters are extracted from Hofstetter et al. (2006) and a chosen contact pressure of 0.5 MPa.

bottom surface, but this is not observed experimentally.

A different approach is used by Persson, who has published many papers (Persson, 1993, 1995, 1998, 1999, 2002; Persson and Tosatti, 2000; Persson and Volokitin, 2000, 2002; Persson et al., 2002) on the subject of rubber friction and the role of the surface, which is in contact with the rubber. This theory is a continuation of the early studies of Grosch. Persson states that the friction force is related to the internal friction of the rubber, which is a bulk property of the material. The hysteretic friction component is determined by sliding of the rubber over asperities of a rough surface. These oscillating forces lead to energy dissipation. The contribution of a every asperity size can be described with a fractal description of the rough surface. Every length scale λ, up to the largest particles of asphalt, can be related to a frequency: f ∼ v/λ. The friction coefficient is based on analytical expressions, which limits the rubber models to linear elastic theory.

A similar method is described by Klüppel and Heinrich (2000), but this theory is based on a cylindrical rubber block undergoing only a one-dimensional deformation during sliding contact with a rough surface. A comparison of experiments and the predictive capabilities of the physical theories of Klüppel and Heinrich (2000) and Persson (2001) is presented by Westermann et al. (2004). It is concluded that each theory has one open parameter, which needs to be fitted on experimental data to obtain a good agreement between theory and data. However this match only holds for a sliding speed interval up to a few cm/s. Towards higher sliding speeds, systematic deviations appear, which are related to flash temperature effects in the contact patch.

he also takes the local heating of the rubber into account. At very low sliding velocities the temperature increases are negligible because of heat diffusion, but already for velocities in the order of 0.01 m/s local heating plays a role. He shows that in a typical case the temperature increase results in a decrease in rubber friction with increasing sliding velocity, if the sliding velocity is above 0.01 m/s. The advanced model, with flash temperature effects included, is able to predict the friction coefficient for slightly higher sliding speeds compared to the model without flash temperature. However validated models for the range of sliding velocities that appear in tire/road contact are not available yet.

An FE multi-scale approach for frictional contact is proposed by Wriggers and Reinelt (2009). The proposed model is based on the analytical models of Klüppel and Heinrich (2000) and Persson (2001), however it is well-known that large deformations occur when a tire is in contact with a rough road surface and therefore the numerical simulations are based on finite deformation models instead of linear elastic theory. The numerical calculation of a rough surface demands a very detailed model to include all length scales and a model of a tire tread block also requires a very fine mesh to accurately describe the contact surface. This is not possible with current computer resources and therefore the rough surface is modeled with only a few superimposed harmonic functions and the tread is replaced with a small rubber block. On each length scale several simulations for varying sliding velocity are carried out, where the following function is fitted though the obtained data points

µ(v, p) =  2vv v2_{+ v}2 c µmax_{, (2.8)}

in which v denotes the sliding velocity, where the maximum point of the friction curve µmax_{is reached. The dependency of the applied normal pressure is included by the}

func-tions

v = ap, (2.9)

µmax = b

parctan(dp), (2.10)

which describe the effect that for increasing global pressure the maximum friction value decreases and is shifted to larger velocities. The friction law on micro-scale requires a fit of four parameters a, b, c and d, which are obtained by a nonlinear least-square method. The obtained homogenized friction law is then applied within each so-called representative contact element at the next larger scale. This method is able to predict the qualitative frictional behavior, but is not suitable for complete tires due to computational limitations.

2.3 TIRE RESEARCH USINGFE MODELS 17

2.3 Tire research using FE Models

The use of FEM in tire development started together with the development of nonlinear FEM. All four types of nonlinearities (Kouznetsova, 2006) in solid mechanics are present in pneumatic tires. Geometric nonlinearity occurs due to the large displacements and rotations involved, while material nonlinearity is present for the almost incompressible (visco)elastic rubbers. Force boundary conditions nonlinearities arise as pressure load inside the tire and displacement boundary conditions nonlinearities are present due to contact with a foundation. This creates changes in the boundary conditions during a simulation.

There is a vast literature, sometimes related to tires, on computational developments for contact problems in FE and quite some work is also incorporated intoABAQUS. Examples

are the works of Oden on friction and rolling contact (Oden and Pires, 1983, 1984; Oden and Martins, 1985; Oden and Lin, 1986; Oden et al., 1988; Faria et al., 1989), Laursen and Simo in the field of contact problems with friction (Simo and Laursen, 1992; Laursen and Simo, 1993b,a) and Padovan on rolling viscoelastic cylinders (Zeid and Padovan, 1981; Padovan, 1987; Kennedy and Padovan, 1987; Nakajima and Padovan, 1987; Padovan et al., 1992). 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) on constitutive interface laws with friction.

Literature specifically related to tires does usually not provide detailed information. The main reason is that most tire manufacturers use own (in-house) finite element codes and specific implementations are kept confidential. However the literature provides insight in the trends and developments of tire modeling throughout the past decades. Besides the journal of Tire Science and Technology the reader is referred to the two overview papers of Mackerle (1998, 2004) about rubber and rubber-like materials, finite element analyses and simulations for an extensive reference list.

The following overview focuses on methods to obtain the cornering and braking forces acting on a rolling tire with friction. A distinction is made between implicit and explicit methods.

2.3.1 Implicit analysis

Static and quasi-static analysis

Static analyses are used when inertia effects can be neglected and time-dependent mate-rial effects are not included. In this case the time increments are then simply fractions of the total period of the step, which are used as increments in the analysis. If time-dependent material effects are taken into account, such as viscoelastic materials, the ap-proach is called quasi-static. An implicit analysis is solved using an incremental-iterative

procedure, which requires matrix inversion (SIMULIA, 2009c).

One of the first overview papers, where FE models and the contact problem for tires are described, is the paper of Noor and Tanner (1985). In this article, for a NASA research program for the space shuttle tires, the current status and developments of computational models for tires are summarized. This review has been made again by Danielson et al. (1996). Basically all the tire models are axisymmetric, with no tread pattern or only cir-cumferential groves due to the limits of computational resources. The effect of friction, using the Coulomb model, is investigated on the vertical load versus vertical deflection curve.

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 a coarse mesh. After that a part of the structure is meshed finer and in-terpolated displacements are applied at the boundaries of this region. Such an approach can be used to compute the forces in the contact area of a deflected tire (Gall et al., 1995; Meschke et al., 1997). Although a local model is able to give detailed numerical results corresponding to a tread block, the accuracy is strongly influenced by the simple global model.

With the ever increasingly computational power it is nowadays possible to mesh a part or even the whole tread of the tire with a detailed pattern, e.g. Cho et al. (2004), and perform very detailed static analyses including footprint shapes as function of axle load and inflation pressure.

There are three possibilities to obtain the cornering and braking forces acting on a rolling tire, which follow on a (quasi)static analysis. The first possibility is an implicit dynam-ical analysis, the second one is the arbitrary Lagrangian-Eulerian method and the last possibility is an explicit analysis.

Dynamic analysis

An implicit dynamical analysis is not often used for rolling tires, since it is well-known that this type of analysis is not efficient in solving changing contact conditions. The nonlinear equation solving process is expensive due to the Newton iterations, and if the equations are very nonlinear, as in the case of changing contact, it may be difficult to obtain a solution (SIMULIA, 2009c).

Arbitrary Lagrangian-Eulerian method

The arbitrary Lagrangian-Eulerian (ALE) method is developed for numerical analysis of rolling contact problems, see the articles of Nackenhorst (2004); Ziefle and Nackenhorst (2005) and Laursen and Stanciulescu (2006). This method converts the steady state mov-ing contact problem into a pure spatially dependent simulation, where the mesh is fixed

2.3 TIRE RESEARCH USINGFE MODELS 19

in space and the material flows trough the mesh. Thus the mesh needs to be refined only in the contact region, which leads to a computational time reduction.

This type of modeling, for simulating tire spindle force and moment response under side slip angles is described by Darnell et al. (2002). The simulation model is composed of shell elements, which model the tread deformation, coupled to special purpose finite ele-ments that model the deformation of the sidewall and contact between the tread and the ground, where Coulomb friction is used. Despite the seemingly simple model the results correspond quite well with experiments.

FE simulations with ABAQUS, in which the effect of tire design parameters on lateral

forces and moments is studied, are presented in a paper of Olatunbosun and Bolarinwa (2004). Parametric studies are performed on a simplified tire (no tread, negligible rim compliance and viscoelastic properties, shear forces modeled with Coulomb friction with constant µ) and a comparison with literature is made to show the computational time re-duction of the method compared to explicit analyses. The effects of variations in stiffness and geometry on the nonuniformity of tires is investigated by Jeong et al. (2007) using

ABAQUS, where also a Coulomb friction model is used for the tire-road interaction.

2.3.2 Explicit analysis

In an explicit analysis the dynamic response problems are solved using an explicit direct integration procedure. The displacements and velocities are calculated in terms of quan-tities that are known at the beginning of an increment and no iterations and no tangent stiffness matrix are required, which is an advantage compared to the implicit method. However due to the explicit time integration a very small time-step, which depends on the highest frequency present in the model, is usually required. Therefore this approach is ideal to simulate transient behavior in a short time span, such as impact of a tire with a cleat. It can also be used to compute handling characteristics, but longer time spans are required to reach the steady-state situation. Furthermore for these longer time spans the risk of error accumulation is present as shown by Tönük and Ünlüsoy (2001).

Explicit simulations to predict tire cornering forces with a maximum side slip angle of three degrees, using the package PAM-SHOCK, are presented by Koishi et al. (1998).

Be-sides a comparison with experiments, parametric studies on the effect of inflation pres-sure, belt angle and rubber modulus are performed. Koishi et al. (1998) use the Coulomb friction model, with coefficient equal to one.

A prediction of tire cornering forces on a drum is given in a paper of Tönük and Ünlü-soy (2001), where the FE packageMARC is used. A comparison with experiments is also

presented. With the model a maximum side slip angle of five degrees is obtained, higher angles created problems caused by the used Coulomb friction model and error accumu-lation dominates the model results before a steady-state is reached, which occurs even earlier with higher normal loads. In the region below five degrees, when the cornering

force is almost linear for increasing side slip angle, the results show a good comparison with experimental data.

A cleat test, with a treadless tire model, can be found in a paper of Olatunbosun and

Burke (2002), where the package NASTRAN has been used. A comparison with

experi-ments shows deviations as the traverse speed increases above 20 km/h.

Rao et al. (2003) simulated the dynamic behavior of a pneumatic tire using

ABAQUS/EXPLICIT to replace the extensive measurement program test to fit Magic

For-mula parameters. A study on a passenger car radial tire to siFor-mulate cornering behavior, braking behavior, and combined cornering & braking behavior is presented. Furthermore the effect of camber angle and grooved tread on tire cornering behavior is studied. To re-duce computational load two tire models are used, one treadless tire and a tire with five circumferential groves. Again the Coulomb friction model with coefficient equal to one is used. The simulations are however unstable for side slip angles above five degrees and longitudinal slip larger than 12%, which is explained due to the use of an inadequate friction model.

In an article of Cho et al. (2005) the dynamic response of a fully patterned tire rolling over a cleat is presented, using ABAQUS/EXPLICIT. A constant friction coefficient of one

is used for the tire-cleat contact. To decrease the computational load the fiber-reinforced rubber is modeled with composite shell elements and mass lumping is used to decrease computation time (SIMULIA, 2009c). Kerchman (2008) conducted an analysis for ride and harshness analysis with a detailed tire-wheel model coupled with a suspension and attached to a simplified vehicle model, again with the Coulomb friction model.

2.4 Modeling framework

2.4.1 Numerical method

Based on the literature overview it is clear that there are only two methods suitable to ob-tain the cornering and braking forces of a rolling tire: the ALE method or the dynamical explicit approach. The explicit method does not require the inversion of the global mass and stiffness matrices, which is an advantage. However due to the explicit time integra-tion a very small time-step is required to obtain a stable soluintegra-tion. Therefore it is ideal to simulate transient behavior in a very short time span, such as impact of a tire with a cleat. It is however not ideal to compute handling characteristics, since longer time spans are required and this results in unacceptable computation times. To overcome this the ALE method can be used, which is also available in ABAQUS, and is developed to

com-pute the steady-state response of rolling structures. This is an effective method to obtain the global force and moment characteristics of a tire under different driving conditions, such as braking or side slip and camber angles (SIMULIA, 2000). This so-called

steady-2.4 MODELING FRAMEWORK 21

state transport analysis (SIMULIA, 2009a, chapter 6.4) uses a moving reference frame in which rigid body rotation is described in an Eulerian manner and the deformation is described in a Lagrangian manner (the ALE-formulation). Furthermore frictional effects, inertia effects, and history effects in the material can all be accounted for.

In an article of Kabe and Koishi (2000) a comparison between the steady-state transport method and ABAQUS/EXPLICIT and experiments for steady-state cornering tires is made.

Although the results of both methods are closer to each other than to the experiments, the steady-state transport method is significantly faster, 6 hours and 8 days respectively. This, together with the possibility to reduce the number of elements outside the contact area, clearly shows the benefits of the steady-state transport analysis and therefore this method is chosen to compute the steady-state characteristics of a rolling tire.

2.4.2 Friction model

From the literature review it follows that Coulomb friction is still often used in numeri-cal simulations. Deviations from the experiments are often attributed to this model and this indicates that a different model should be used to improve the results. One of the restrictions of the steady-state transport approach is that the underlying surface must be (rigid) flat, convex or concave (SIMULIA, 2009a, chapter 6.4). This means that it is not possible to incorporate a rough road surface as used by Wriggers and Reinelt (2009). So the possible surface effects should be captured in the parameters of a friction model, since a flat surface can not exert oscillating forces on the rubber surface. Although the models of Klüppel and Heinrich (2000) and Persson (2001) incorporate the effect of sur-face roughness directly in their model description, the models are not suitable for high sliding velocities yet, which occur when tires are tested at high velocities.

If the sliding velocity influence in the friction models (2.2), (2.4), (2.5), (2.6) and (2.7) is compared, it follows that all models are able describe the shape as observed by Grosch. The models given by (2.4), (2.5) and (2.6) provide however no direct insight in the fric-tional properties and are therefore not preferred. The model of Huemer et al. (2001a) is developed for a rubber block with a specific geometry and dimension, which means that (2.7) describes the combined effect of contact and block geometry on the friction force. Therefore the model of Savkoor is chosen to describe the sliding velocity dependence, since the parameters in this model are directly related to the shape of the friction curve. It is shown by Lupker et al. (2004) that this model can also be used in situations with very high sliding velocities.

Furthermore the pressure dependence in most models show a decreasing friction coeffi-cient for increasing pressure, which is not included in the original model of Savkoor. To incorporate this effect (2.2) is extended with a pressure term (Lupker et al., 2004), to give

a friction law, which depends on the contact pressure and the slip velocity as follows µ(p, v) = p p0 −k µs+ (µm− µs) exp  −h2_log2  v Vmax  , (2.11)

where the parameters p0 and k are related to the contact pressure and the parameters

µs, µm, hand Vmaxare related to the sliding velocity. This model can also be found in the

work of Smith et al. (2008), where the steady-state approach ofABAQUSis used to predict

the wear of a tread profile.

2.5 Steady-state transport analysis

In this section the relevant parts of the ALE approach are presented, which are based on the documentation of SIMULIA (2009c). The kinematics of the rolling problem are described in terms of a coordinate frame that moves along with the ground motion of the body. In this moving frame the rigid body rotation is described in a spatial or Eule-rian manner and the deformation in a material of Lagrangian manner. This kinematic description converts the steady-state moving contact field problem into a purely spatially dependent simulation. In the following derivation Lagrangian coordinates are denoted in uppercase (e.g. X) and the Eulerian coordinates are denoted in lowercase (e.g. x).

A deformable body is rotating with a constant angular rolling velocity ω around a rigid axle T at X0, which in turn rotates with a constant angular velocity Ω around the fixed

cornering axis n, which is normal to the rigid surface, through point Xc, see Figure 2.6.

Hence, the motion of a particle X at time t consists of a rigid rolling rotation to position

ω

Ω

n

T X0

Xc

Figure 2.6 /Constant cornering motion in the ALE approach. Figure reproduced from (SIMULIA, 2009c).

2.5 STEADY-STATE TRANSPORT ANALYSIS 23

Y, described by

Y = Rs· (X − X0) + X0, (2.12)

where the spinning rotation matrix Rsis defined as

Rs= exp(ˆωt), (2.13)

with ˆωthe skew-symmetric matrix associated with the rotation vector ω = ωT. This rigid rolling rotation is followed by a deformation to a point x, and a subsequent cornering rotation around n to position y so that

y = Rc· (x − Xc) + Xc, (2.14)

where Rcis the cornering rotation given by

Rc= exp( ˆΩt), (2.15)

and ˆΩ is the skew-symmetric matrix associated with the rotation vector Ω = Ωn. The velocity of the particle then becomes

v = ˙y = ˙Rc· (x − Xc) + Rc· ˙x. (2.16)

To describe the deformation of the body a map χ(Y, t), shown in Figure 2.7, is intro-duced, which gives the position of a point x at time t as a function of its location Y at time t so that x = χ(Y, t). (2.17) X Rs χ Initial configuration configurationCurrent Reference configuration Y x

The time derivative of (2.17) is given by ˙x = ∂χ ∂Y · ∂Y ∂t + ∂χ ∂t, (2.18) where ∂Y ∂t = ˙Rs· (X − X0) = ωT× (Y − X0). (2.19)

Noting that ˙Rs= ˆω· Rs= ω ˆT· Rs, and introducing the circumferential direction

S = T_×Y− X0

R , (2.20)

where R = |Y − X0| is the radius of a point on the reference body, the velocity of the

reference body can now be written as ∂Y

∂t = ωRS (2.21)

and (2.18) can be written as

˙x = ωR∂Y ∂t · S + ∂χ ∂t = ωR ∂χ ∂S + ∂χ ∂t, (2.22)

where S is the distance-measuring coordinate along the streamline. The derivative of (2.15) is given by

˙

Rc= ˆΩ· Rc = Ωˆn· Rc. (2.23)

The velocity of the particle can be written as v = Ωn× (x − Xc) + ωRRc·

∂χ

∂S + Rc· ∂χ

∂t. (2.24)

The acceleration is obtained by differentiation of (2.24) a = Ω2(nn− I) · (x − Xc) + 2ωΩRn× Rc· ∂χ ∂S + 2Ωn× Rc· ∂χ ∂t (2.25) +ω2_R2_R c· ∂2_χ ∂S2 + 2ωRRc· ∂2_χ ∂S∂t+ Rc· ∂2_χ ∂t2 .

To obtain expressions for the velocity and acceleration in the reference frame tied to the body, the following transformations are used

vr= RcT · v, ar= RcT · a, (2.26) such that vr= Ωn× (x − Xc) + ωR ∂χ ∂S + ∂χ ∂t (2.27)

2.5 STEADY-STATE TRANSPORT ANALYSIS 25 and ar = Ω2(nn− I) · (x − Xc) + 2ωΩRn× ∂χ ∂S + 2Ωn× ∂χ ∂t (2.28) +ω2_R2∂2χ ∂S2 + 2ωR ∂2_χ ∂S∂t + ∂2_χ ∂t2 .

For steady-state conditions it holds that ∂χ

∂t = 0and these expressions reduce to

vr= Ωn× (x − Xc) + ωR ∂χ ∂S (2.29) and ar= Ω2(nn− I) · (x − Xc) + 2ωΩRn× ∂χ ∂S + ω 2_R2∂ 2_χ ∂S2. (2.30)

The first term of (2.30) can be seen as the acceleration that gives rise to centrifugal forces resulting from rotation about n. The second term can be identified as the acceleration that gives rise to Coriolis forces. The last term combines the acceleration that give rise to Coriolis and centrifugal forces resulting from rotation about T. When the deformation is uniform along the circumferential direction, both Coriolis effects vanishes so that the acceleration gives rise to centrifugal forces only.

The velocity of the center of the body X0is given by

v0 = Ωn× (X0− Xc) (2.31)

since the motions due to rolling and deformation vanish on the axis. In the case of straight line rolling Ω → 0 (2.29) and (2.30) reduce to

v = v0+ vr = v0+ ωR ∂χ ∂S (2.32) and a = ar = ω2R2 ∂2_χ ∂S2. (2.33)

2.5.1 Contact conditions for steady-state rolling

Given two points on the surfaces of two bodies in contact, the relative velocity can be expressed as

v = vD− vR, (2.34)

where vD is the velocity of a point on the deformable body, see (2.27), and vR the

veloc-ity of a point on the rigid foundation. This can be split into the normal and tangential components. The rate of penetration is

˙h = −n · v = n · vR− ωRn ·

∂χ ∂S − n ·

∂χ