Effect of Material Nonlinearity on Rubber Friction a Thesis Presented

Effect of Material Nonlinearity on Rubber Friction

A thesis presented to

the faculty of the Russ College of Engineering and Technology of Ohio University

In partial fulfillment

of the requirements for the degree

Master of Science

Tejas N. Bhave

December 2016

This thesis titled

Effect of Material Nonlinearity on Rubber Friction

TEJAS N. BHAVE

has been approved for

the Department of Mechanical Engineering and the Russ College of Engineering and Technology by

Alireza Sarvestani

Assistant Professor of Mechanical Engineering

Dennis Irwin

Dean, Russ College of Engineering and Technology 3

ABSTRACT

BHAVE, TEJAS N., M.S., December 2016, Mechanical Engineering

Effect of Material Nonlinearity on Rubber Friction

Director of thesis: Alireza Sarvestani

With the increase in the importance of vehicular transportation, the study of contact patch parameters including the contact patch forces and the tire-road friction has become essential from the perspective of improving vehicle safety as well as vehicle performance.

The current work aims at analyzing the effect of the nonlinear elastic and nonlinear viscoelastic nature of tire tread rubber by modifying two commonly used rubber friction models (Gim’s analytical model and Heinrich-Klüppel (HK) friction model) in order to implement the nonlinear elasticity and the nonlinear viscoelasticity of rubber. Gim’s analytical model is modified by changing the linear elastic constitutive equation used in the original model to a nonlinear elastic equation based on the strain energy density of rubber. Results are obtained for a test simulation using this modified model and an experimental method is proposed to validate the modified model’s force predictions.

The HK friction model computes the hysteretic sliding friction coefficient of rubber

based on the viscoelastic modulus. It however, does not consider the (experimentally

proven) dependence of viscoelastic modulus of rubber on the applied strain amplitude and

temperature. The current work thus aims at implementing this dependence and modifying

the classical HK friction model. A test simulation run for a rubber block sliding on a rough

surface using the modified HK friction model yielded friction results that are sensitive to

the input strain amplitude and temperature. 4

DEDICATION

Dedicated to my family

ACKNOWLEDGMENTS

I am indebted to my thesis advisor Dr. Alireza Sarvestani for his able guidance, sound advice (both technical and worldly) and unwavering support during this two-year journey that was my thesis research. It was the balance that he struck between allowing me to work independently and monitoring and helping me whenever I was stranded, that helped me develop not only as a researcher, but also as an engineering professional. I was able to achieve more in my academic life here at Ohio University than I could have ever imagined thanks to his support and encouragement.

I would also like to thank my thesis committee members Dr. John Cotton, Dr. Munir

Nazzal and Dr. Ardalan Vahidi for their valuable advice regarding my research work. Their technical expertise, help, advice and suggestions has helped me shape my research work.

Furthermore, I would like to thank my lab colleagues, Mohammad Jafari Tehrani and Mohammad Hossein Moshaei for their support, encouragement and help during the course of my research.

Thanks are due to Mayur, Manish, Shantanu, Pratik, Amit, Ajinkya, Aditya,

Aniruddha, Cody, John, Brian McCoy and the Petitt family for all their help and support.

I would also like to thank everybody, who in any way, has offered their help, support, guidance or advice throughout this process.

Most importantly, I would like to thank my family and especially my parents. The daily conversations with them helped me tide over some difficult times. If it were not for their endless affection and support, at all levels, I am sure I would not have progressed to where I stand on this day. 6

TABLE OF CONTENTS

Page

Abstract ...... 3 Dedication ...... 4 Acknowledgments...... 5 List of Tables ...... 8 List of Figures ...... 9 1 Research Background ...... 11 2 Introduction ...... 14 2.1 The Tire ...... 14 2.2 Introduction to Tread Rubber...... 15 2.3 Properties of Tread Rubber ...... 16 2.4 Contact Patch in Tires ...... 19 3 Tire Road Interaction Models ...... 22 3.1 Introduction and Classification ...... 22 3.2 Brush Model Method ...... 24 3.3 Improvements in the Classical Brush Model ...... 26 3.3.1 Combined Slips ...... 26 3.3.2 Pressure Distribution in the Contact Patch ...... 28 3.3.3 Velocity-Dependent Coefficient of Friction ...... 30 3.3.4 Pressure Dependent Coefficient of Friction ...... 32 3.3.5 The TreadSim Model ...... 33 3.3.6 Applications of the Brush Model Method ...... 34 3.4 Thesis Objective...... 35 4 Non Linear Elasticity of Rubber ...... 37 4.1 Expression for Material Nonlinearity of Rubber ...... 37 4.2 Friction Force Predictions Using Modified Brush Model Method ...... 42 4.2.1 Longitudinal Force Prediction Using the Modified Brush Model Method ...... 42 4.2.2 Implementing Nonlinearity into Lateral Force Calculations ...... 48 4.3 Proposed Method to Validate Modified Brush Model ...... 53 5 Effect of Tread Rubber Viscoelasticity on Tire Friction ...... 55 5.1 Adhesive and Hysteretic Friction...... 55 7

5.2 Introduction to Viscoelasticity ...... 56 5.3 Rubber Viscoelasticity and Hysteresis Friction ...... 59 5.4 Classical HK Friction Model ...... 61 5.5 The Payne Effect ...... 63 5.6 Modified HK Model with Payne Effect ...... 65 5.7 Temperature Dependent Rubber Viscoelasticity ...... 70 5.8 Strain Amplitude and Flash Temperature Dependent Modified HK Model ...... 73 6 Summary and Conclusions ...... 76 References ...... 79 Appendix: Supplemental Files ...... 85

LIST OF TABLES

Page

Table 1: Input Parameters for Prediction of Longitudinal Force Using the Modified Brush Model Method ...... 44 Table 2: Input Parameters for Calculating the Lateral Force Due To Slip Angle Using the Modified Brush Model Method ...... 50 Table 3: Input Parameters for Calculating the Lateral Force Due To Camber Angle Using the Modified Brush Model Method ...... 51

LIST OF FIGURES

Page

Figure 2.1 Nonlinear Stress-Strain Curve for Rubber ...... 17 Figure 2.2 Stress Strain Curve for Viscoelastic Material ...... 18 Figure 2.3 Tire Contact Patch ...... 19 Figure 2.4 Contact Patch Forces and Moments ...... 20 Figure 3.1: The Brush Model Method ...... 24 Figure 3.2: Variation of the Longitudinal Force with the Longitudinal Slip...... 26 Figure 3.3 Parabolic Pressure Distribution in the Contact Patch ...... 29 Figure 3.4: Variation of the Normalized Force with the Slip Ratio...... 31

Figure 4.1: Representation of x and L0 from Eq. (4.2) ...... 39 Figure 4.2: Determination of Fitting Parameters for Eq. (4.4) ...... 41 Figure 4.3: Variation of the Breakaway Point in the Contact Patch with the Slip Ratio for Different Tread Thicknesses ...... 45 Figure 4.4: Variation of the Longitudinal Stress in the Contact Patch ...... 46

Figure 4.5: Normalized Longitudinal Force vs Slip for Different Values of ‘L0’ ...... 47 Figure 4.6: The Tire Slip and Camber Angles ...... 48 Figure 4.7: Normalized Lateral Force Due To Slip Angle Vs Slip Angle for Different Values of ‘L0’ ...... 50

Figure 4.8: Normalized Lateral Force Vs Camber Angle for Different Values of ‘L0’ .... 52 Figure 4.9: Schematic of the Proposed Experimental Setup ...... 53 Figure 5.1: Adhesion and Hysteresis Friction in Elastomers...... 55 Figure 5.2: Stress Strain Curve for a Viscoelastic Material ...... 57 Figure 5.3: Carbon Black ...... 59 Figure 5.4: Variation of the Coefficient Of Friction with the Sliding Velocity ...... 60 Figure 5.5: Rubber Sliding on a Sinusoidal Surface...... 60 Figure 5.6: Friction Calculation using the Classical HK Model ...... 61 Figure 5.7: Variation of the Storage Modulus and Loss Tangent of Styrene Butadiene Rubber with Strain Amplitude ...... 63 Figure 5.8: Fractional Standard Linear Solid Viscoelastic Model...... 64 Figure 5.9: Rubber Block Sliding on a Sinusoidal Surface...... 66 Figure 5.10: Curve Fitting to Obtain Fitting Parameters for Viscoelastic Modulus At 8% Strain Amplitude ...... 67 10

Figure 5.11: Variation of Storage and Loss Moduli with the Strain Amplitude...... 68 Figure 5.12: Storage and Loss Moduli Using the Obtained Fitting Parameters for Applied Strain Amplitudes Of 8% And 1%...... 69 Figure 5.13: Variation of the Hysteretic Coefficient Of Friction with Sliding Velocity for 8% And 1% Strain Amplitude ...... 70 Figure 5.14: Infrared Image of the Tire Tread Exiting The Contact Patch...... 71 Figure 5.15: Coefficients of Friction with and without Flash Temperature For 8% And 1% Applied Strain Amplitude ...... 74 Figure 5.16: Friction Predictions of the Modified HK Model Compared with Experimental Data ...... 75

1 RESEARCH BACKGROUND

The United States system of roadways are the most widely used means of passenger transport within the country [1]. The system of highways handled up to 87 percent of the passenger miles [1] in the year 2012, nearly 7.5 times the passenger miles handled by airways which is the second most common means of passenger transport. Of the vehicles that use highways for passenger transport (including light vehicles, motorcycles, buses and trucks), light vehicles (personal cars) accounted for about 86% of the passenger miles [1].

The use of light vehicles is widespread in the United States, as can be shown by the fact that about 76.5 percent of the population prefer commuting using personal vehicles as compared to carpooling or other means of transport [1]. Another fact that demonstrates the popularity of the light vehicle is that more than 7500,000 new cars were purchased / registered in 2015 [2] and the total number of cars on the road in the USA numbered more than 253 million [3].

The tire, and more specifically a part of the tire called the contact patch, is the contact point between the vehicle and the road. The main purpose of the tire is to facilitate the movement of the vehicle on the road surface by controlling friction, while also acting as shock absorbers by partially damping out the vibrations arising due to surface irregularities [4]. The interaction between the tire and the road in the contact patch affects the steering and maneuverability of the car.

Since tires play an important role in vehicle operation as described above, malfunctioning tires may lead to vehicle accidents. Among the main causes for tire-related vehicle accidents are vehicle skidding (loss of friction between the tire and the road), and 12 a sudden loss in inflation pressure while the vehicle is in operation (tire blowout) [5]. In a study on traffic accidents in Japan, it was found that about 20% of all vehicle accidents were due to loss of control over the vehicle, more than half of which happened due to the vehicle skidding [6]. Loss of tire-road friction, either due to the wheel slipping, or due to sudden change in the road surface condition, are among the major causes of vehicle skidding [6]. The severity of the loss of friction problem can be quantified by the fact that wet road surfaces (leading to a reduction in tire-road friction) are the reason for 13.5% of all fatal vehicle crashes [7].

An improvement in the tire design and operation would contribute significantly to reducing the number of tire related vehicle accidents. Since tire rubber is a mixture of different constituent materials like natural and synthetic rubber, carbon black and other filler materials, adjusting the relative quantities of these materials to get properties of a tire specific to the intended application would aid in proper tire operation [8]. In addition, a recent trend has been to use the tire itself as a sensor to monitor the tire-road interaction and get feedback to aid in vehicle operation. An example of this method is the tire pressure monitoring system (TPMS) which monitors the inflation pressure of the tire in real time, helping to avoid inflation pressure related accidents [9].

The future of the ‘tire as a sensor’ concept is using smart/intelligent tires that measures and reports a variety of parameters related to tire road interaction which makes vehicle driving easier and safer. The new spherical concept tire by Goodyear is a prime example of the direction in which tire related research is progressing [10]. Thus, as vehicles and highways become indispensable parts of the daily commute, tires, which form the only 13 points of contact between the vehicle and the ground, need to be the focus of intensive research and development.

2 INTRODUCTION

This chapter aims at giving an introduction to vehicle tires, including the development of tires, tire materials and tire tread properties. Two rubber properties, nonlinear elasticity and viscoelasticity, which form the focus of the current work are described in detail. Following this, the tire contact patch and its relationship to vehicle maneuvering and road surface wear in terms of the contact patch forces is discussed. The chapter concludes with a short discussion on the need for modelling the forces in the contact patch.

2.1 The Tire

Even though the wheel was probably invented around 3500 BC [11,12], the development of the pneumatic tire is a comparatively recent phenomenon [11,12]. Olden day wheels were made from metals or wood and although sturdy and durable, did not make for a comfortable ride [11]. Although using rubber to cover these solid wheels seemed a suitable option to increase ride comfort, the use of rubber had problems associated with it since rubber became sticky in warm weather and grew stiff in cold weather. This problem was solved when Charles Goodyear discovered the process of vulcanization of rubber

[11,12]. The modern pneumatic tire was developed by Robert Thomson and then later reinvented by John Dunlop [11]. The pneumatic tire has found widespread application and is the subject of continuous improvement [11].

Among the latest trends in tire research and development is the concept of smart/intelligent tires. Smart tires make use of the tire as a sensor to measure different variables including, but not limited to, tire pressure, tread temperature and the 15 instantaneous friction and tractive force between the tire and the road [13]. This information can be made use of to analyze the tire-road interaction at any given instant.

The smart tire would be an important part of the smart vehicle, which represents the future of automobile based transportation [14,15].

2.2 Introduction to Tread Rubber

The outer surface of the tire, called the tire tread, is made of rubber. The typical tire tread rubber consists of elastomers like natural or synthetic rubber and filler materials like carbon black or silica [16], which are dispersed in the elastomer matrix [17,18].

Additionally, it also contains extending oils, sulfur and zinc oxide from vulcanization, softeners, and other additives [16,18]. The fine carbon black filler particles are among the important constituents of tread rubber since they act like additional crosslinks in the rubber, thereby increasing its modulus [18]. Carbon black particles also affect the viscoelastic response of the tread rubber since they offer resistance to the movement of the polymer strands under an applied load [18]. The relative quantities of the raw materials (elastomers and additives) used in the tire tread rubber can be adjusted to achieve the desired tread properties [16].

An important advantage of studying tire tread rubber properties is the ability to customize tires to specific applications. The friction between the tire and the road is dependent, among other factors, on the properties of the tire tread rubber. Adjusting the construction [8,19,20] and the properties of the tire tread thus enables the tire designers to adjust the amount of friction between the tire and the road. An example of tire properties being adjusted to have optimum tire performance in different operating conditions is the 16 tread on summer and all weather tires [8]. Thus, studying tire material properties helps manufacture tires that are optimized for the anticipated tire operating conditions. The properties of tire tread rubber being focused on in the scope of the current work are the nonlinear elasticity and viscous elasticity. These properties are described in detail in the following section.

2.3 Properties of Tread Rubber

Rubber, by nature, is a nonlinear viscoelastic material. The mechanism for elasticity in rubber is different from that in materials like metals. Metal elasticity is due to the forces of intermolecular attraction, whereas rubber elasticity is entropic in nature [18]. Another important point to be noted here is that rubber can be approximated to be linear elastic only for small strains [21], whereas most metals can be considered to be linearly elastic until their yield point. When subjected to large deformations, the material behavior of rubber is found to be nonlinear. Eq. (2.1), which is a constitutive equation relating the stress and strain in rubber in uniaxial tension [18], shows this material nonlinearity.

σ = G (λ – λ-2) (2.1)

In Eq. (2.1), σ is the stress, G is the modulus of rigidity and λ is the stretch.

Fig.2.1 shows the typical stress-strain curve for rubber. It can be seen that that at

small values of strain, the stress-strain relationship can be approximated to be linear, but at

higher strains, it is nonlinear. 17

Figure 2.1 Nonlinear Stress-Strain Curve for Rubber (Reproduced From [22])

Viscoelasticity is an important phenomenon to consider while using rubber in tire treads since it affects the tire properties [18]. The viscous component of rubber elasticity manifests itself in the form of the finite amount of time that it takes a rubber specimen to return to its pre-loaded state after the applied load has been removed [18,23]. This is in contrast to metals or ceramics, which regain their original shape and size almost instantaneously after the applied load has been removed, as long as there is no permanent

(plastic) deformation. The concept of viscoelasticity can be explained by Fig.2.2 which shows the stress strain curve for a viscoelastic material. 18

Figure 2.2 Stress Strain Curve for Viscoelastic Material (Reproduced with modifications from [24])

For a viscoelastic material, the stress increases with an increase in the strain. When the loading is removed, the material regains its original state, but unlike a linear elastic material, it does not follow the same path for loading and unloading. Thus, for viscoelastic materials, there is a net loss of energy during the loading-unloading process which is the amount of energy dissipated due to the material viscosity. The study of the material viscoelasticity is important from the point of view of calculating the friction between the tire tread and the road surface, since the viscoelasticity affects the dynamic modulus of the material, which in turn, affects the friction.

2.4 Contact Patch in Tires

Figure 2.3 Tire Contact Patch (Reproduced from [25])

Fig.2.3 is a representation of the tire contact patch, which is the part of the tire that contacts the road surface at any given instant. The study of the tire-road interaction in the contact patch is essential for determining parameters such as tire-road traction forces and tire wear [26]. An analysis of the tire-road interaction in the contact patch and calculations of the relevant parameters may be applied back into designing tires with improved properties [27]. Among the different parameters that need to be determined to help improve tire properties are the forces and moments occurring due to the motion of the tire tread on the road surface. These forces and moments may be represented in terms of a co-ordinate system called the tire axis system [28]. Fig.2.4 shows the forces and the moments experienced by a tire in motion, defined in the tire axis system. 20

Figure 2.4 Contact Patch Forces and Moments (Reproduced from [28])

Since the force in the contact patch arising from the tire-road interaction affect the steering and maneuverability of the vehicle, it is necessary to estimate the values of these forces. In addition to affecting vehicle steering, the tire road contact forces also influence tire and pavement wear. Tires with softer tread rubber that are designed to provide greater grip wear out faster. The friction between the tire and the road also affects the road surface.

Winter conditions causing snow / sleet may lead to a lack of friction between the tire tread and the road surface causing accidents. To avoid this, traction sand is spread over the road surface in order to enhance the friction between the tire and the road surface. Alternatively, the tires themselves may be equipped with spikes that dig into the road for increased traction. Both these methods lead to increase in the wear of the road surface [29].

Although the experimental determination of contact patch forces would give the most accurate data, it might not always be feasible to arrange and conduct experiments for all the situations for which data is desired. Thus, creating models to simulate the tire-road 21 interaction and calculate the values of the necessary forces and moments is a convenient alternative. Tire models generally accept the longitudinal slip, the slip angle, the camber angle and the normal pressure acting on the tire to give an estimate of the longitudinal and lateral forces and moments [30]. These forces, defined as per the tire axis system, form the focus of the current work.

3 TIRE ROAD INTERACTION MODELS

The purpose of the current chapter is to introduce, classify and discuss the different modelling techniques used to simulate the interaction between the tire and the road.

Initially, different tire-road interaction models and their classification depending on different criteria are discussed. The succeeding section introduces a particular tire road interaction model called the Brush Model [31]. The methodology involved in calculating the forces in the contact patch is described in detail. Lastly, different modifications proposed and implemented in the classical brush model are discussed. The shortcoming of the classical brush model is highlighted and the first research objective is stated.

3.1 Introduction and Classification

There are many different tire models available to simulate the interactions in the contact patch [32]. These models can also be classified based on a variety of criteria. For instance, Pacejka [33] classifies tire models based upon their mathematical complexity and the accuracy of the results. Based on this method of classification [33], tire models can be separated into experimental, semi-empirical models (as defined in [34]), analytical/mathematical models and finite element models. Experimental models are formulated based on curve fitting of measured experimental data, while semi-empirical models extrapolate the available experimental data and apply it to situations for which experimentation / measurements have not been made. In contrast to the experimental and the semi empirical models which depend, to some extent, on experimental data, analytical models rely on mathematical modeling of the contact between the tire and the road to propose equations for the forces and the moments. The finite element models of tires are 23 similar to the analytical models, but they are used when detailed analyses are required and a large amount of computational work needs to be performed [33]

A different way of classifying tire models is the classification based on the intended area of application of the model. Using this method of classification, Li et al. [35] classify tire models as models for ride comfort analysis [35], road loads analysis [35] and handling and stability analysis [35]. The main aim of the ride comfort [35] models is to accurately simulate all the parameters that would influence the vehicle occupant’s decision about the amount of comfort in the ride. The road load analysis [35] models are applied to assess the durability of the vehicle when loaded on account of being driven over the road surface.

Finally, the handling and stability analysis [35] simulates the contact between the tire and the road and calculates the lateral and longitudinal force and the moments for both the time dependent and time independent cases [35]. The brush model method was among the earliest modeling methods to be used in the handling and stability analysis [35]. Among the current applications of the brush model method in this domain are in smart tires, as described in the following sections.

The brush model method is an analytical method for tire modeling (e.g. see [31,33]) and has served as the base for a multitude of tire models [35]. Some examples include

Gim’s analytical model [36–39] and the semi-empirical tire model proposed by Svendenius et al. [34,40,41]. A description of the brush model method is given in the following section.

3.2 Brush Model Method

The classical brush model method uses a set of mathematical equations to simulate the tire-road interactions in the contact patch in terms of the lateral and longitudinal forces and moments. This method assumes the tire tread in the contact patch to be divided into a large number of brush elements, or bristles [42]. Fig.3.1 shows a representation of the contact patch of the tire as modeled using the brush model method.

Figure 3.1: The Brush Model Method (Adapted from [43])

According to the brush modelling methodology, the contact patch is assumed to be divided into adhesion and sliding regions, with the length of the adhesion region being , and the total length of the contact patch being . The vehicle velocity, corresponding to the tire rotation shown in Fig.3.1 is denoted by vsx. The distortion of the tire tread in the contact patch gives rise to a difference in the rotational speed of the tire and the velocity of the vehicle, called as the slip. For the scope of the current work, the slip is as defined as [36] 25

(3.1) where s is the slip ratio, Vx is the longitudinal vehicle velocity and Vc is the circumferential velocity of the tire tread.

The tire tread is modelled as a series of one dimensional brush elements or bristles

(Fig.3.1). The brush elements deform as they move through the contact patch. This deformation is assumed to be a function of the slip and the position of the individual element in the contact patch. During free rolling, the brush elements are assumed to not have a component of deformation parallel to the road surface, and the effects of rolling resistance are not considered in the model [31]. Upon entering the contact patch, the brush elements undergo deformation due to the adhesion and friction forces, generated by the rolling of the tire and the vertical pressure in the contact patch. As long as the force acting on the bristle is less than the force required to overcome the static friction between the bristle end and the road surface, the tip of the brush element continues to adhere to the road surface. Once the force on the bristle overcomes the friction, the bristle starts sliding on the road surface, i.e. it is assumed to move into the sliding region. While the force in the adhesion region is considered to be a function of the bristle deformations, the force in the sliding region is dependent only on the sliding friction is independent of the brush element deformations. The force in the adhesion and the sliding regions as predicted by the brush model can then be used to calculate the coefficient of friction and the tractive force existing between the tire and the road. Fig.3.2 shows the variation of the longitudinal force in the contact patch with the longitudinal slip ratio as predicted by Gim’s analytical model. 26

Figure 3.2: Variation of the Longitudinal Force with the Longitudinal Slip. Adapted with modifications from [36]

The description mentioned in the above section briefly summarizes the brush model method. Different assumptions of the brush model method have been subject to research, and improvements have been proposed in the classical brush model, some of which are described in the following section.

3.3 Improvements in the Classical Brush Model

The previous section describes the working of the brush model. There have been different improvements that have been proposed and implemented in the classical brush model, as described below.

3.3.1 Combined Slips

The brush model method for calculating the lateral and longitudinal forces in the contact patch assumes that the brush deformation in the lateral and longitudinal directions are independent of each other [34]. This makes the longitudinal force independent of the lateral brush deformation and vice versa. In reality, however, this is not the case and the 27 forces and the moments in the contact patch are simultaneously dependent on both the lateral and the longitudinal bristle deformations [37]. Gim et al. [36–39] in their analytical model based on the brush model method, proposed a method to calculate the forces and moments for combined slips and camber angle cases. The authors [36–39] initially formulated an expression for calculating the lateral and longitudinal forces and moments for pure (independent) slip cases. The results of the independent slip cases were then combined appropriately to calculate the results for the case where the forces and the moments are simultaneously dependent on both lateral and longitudinal slips [37]. The results obtained from the analytical model [36–39] for the combined slip cases were verified experimentally [39]. Similar to the work done by Gim et al. [36–39], Svendenius et al. [34,40,41] proposed and validated a semi-empirical [40] model to incorporate the results of the pure (independent) slip cases to calculate the results for the case of forces being simultaneously dependent on the longitudinal and lateral slips and the camber angle.

Similar to the model proposed by Svendenius et al. [34,40,41], Bruzelius et al. [44] proposed a simplified model with only a few parameters. This simplified model simulated the simultaneous dependence of the contact patch forces on the slips and the camber angle, and though it [44] had certain limitations, it was proved (analytically and experimentally) to be suitable for its intended application, which was estimation the friction in the contact patch. Mavros et al. [45] proposed two models for the contact patch forces depending simultaneously on the slips and camber angle, while additionally incorporating the viscoelastic response of rubber in the form of a Kelvin element [45]. Thus, from the literature described in the current section, it can be seen that simulating the contact patch 28 forces depending simultaneously on both the lateral and longitudinal bristle deformations is an extensively researched improvement in the brush model.

3.3.2 Pressure Distribution in the Contact Patch

Another aspect of the brush model method that has received attention is the

distribution of the normal pressure along the length of the contact patch. The normal

pressure is the resultant of the vehicle weight acting on the tire and the tire inflation

pressure. Livingston et al. [46] investigated the distribution of normal pressure in the

contact patch. The authors [46] tested different pressure variations including uniform

pressure variation, elliptic pressure variation, and parabolic pressure variation by

formulating mathematical expressions for the contact patch parameters using each of the

pressure variations. Experiments were then carried out and the data was compared with the

data from the mathematical modeling. It was found that the experimental results matched

the analytical results for the parabolic pressure distribution. The brush model method too,

uses a parabolic pressure distribution as shown in Fig.3.2

Figure 3.3 Parabolic Pressure Distribution in the Contact Patch (Reproduced from [47])

Fig.3.3 shows the pressure variation along the length ‘l’ of the contact patch. The pressure distribution is assumed to be averaged along the width of the contact patch [43].

As seen in Fig.3.2, the value of the pressure is zero at either end of the contact patch, while the maximum value, denoted by ‘pm’, occurs near the center of the contact patch.

Mathematically, the parabolic pressure distribution can be represented as:

p(ξ) = 4 µ pm 1 (3.1)

where µ is the coefficient of friction in the contact patch and ξ is the contact patch length

variable such that 0 ≤ ξ ≤ l.

The location of the maximum pressure, as seen in Fig.3.2, is in the center of the contact patch. However, when the tire rolls, the location of the maximum pressure shifts to either the front or the rear of the contact patch [43]. Taking into account this movement of the pressure peak in the contact patch results in better simulations using the brush model method [43]. 30

To simulate the variation in the location of the pressure peak in the contact patch,

Svendenius et al. [42,43,48] proposed the introduction of a calibration factor into the equation for the parabolic pressure distribution. The value of this calibration factor could be modified to change the location of the maximum pressure along the length of the contact patch. The analytical data generated from the brush model using this calibration factor was compared to the results obtained from the Magic Formula [49], and the two sets of results were found to be in agreement with each other. Thus, although the classical brush model correctly assumes the pressure distribution along the length of the contact patch to be parabolic, it does not take into account the movement of the pressure peak in the contact patch. This is yet another modification needed in the classical brush model method in order to achieve close agreement between experimental and analytical results.

3.3.3 Velocity-Dependent Coefficient of Friction

The brush model method assumes the coefficient of friction used in the estimation of the lateral and longitudinal forces in the contact patch to be constant [33]. However, the coefficient of friction between a sliding body and the ground has been shown to be dependent on the velocity of sliding [27]. This dependence of the coefficient of friction on the sliding velocity was taken into account by Dugoff et al. [50] in their analytical model.

This particular model [50] estimated the longitudinal and lateral tire forces in order to observe the effects of the variation in tire characteristics on the handling and braking performance of a vehicle. Bickerstaff et al. [51] modified the relation between the coefficient of friction and the sliding velocity proposed by Dugoff et al. [50] to include a sensitivity factor. Gim et al. [36–39], in their analytical model, noted the existence of a 31 relation between the coefficient of friction and the sliding velocity. Svendenius et al.

[42,43], in their proposed modifications to the brush model method, considered different cases of velocity dependent friction. In addition to the linear and quadratic relations available in the literature [31,36,50,51], the authors also considered the coefficient of friction to be an exponential function of the sliding velocity.

Figure 3.4: Variation of the Normalized Force with the Slip Ratio. Adapted with modifications from [43]

Fig.3.4 shows the variation of the normalized brake force with the slip ratio for velocity dependent friction approximated using a linear expression [43]. The solid line shows the predictions of the magic formula, based on experimental data, while the dashed lines show the predictions of the linearly velocity dependent friction model (for different fitting parameters) proposed by the authors [43]. Apart from considering the linear, 32 quadratic and exponential dependencies of the friction on the sliding velocity, the authors also assumed the coefficients of static and kinetic friction to be dependent alternately and simultaneously on the sliding velocity. For the case of exponential velocity dependency,

Svendenius et al. [43] found that the results from the modified brush model described certain phenomena which were observed in experiments, but not described by the Magic

Formula [49].

In conclusion, a velocity dependent coefficient of friction is an important modification in the brush model which may improve the agreement between results obtained from the brush model with those obtained from experiments.

3.3.4 Pressure Dependent Coefficient of Friction

As mentioned in the previous section, the coefficient of friction in the brush model method was assumed to be constant. However, attempts have been made to incorporate the dependency of the coefficient of friction on different contact patch parameters, one of which is the normal pressure distribution. Heinrich et al. [47] considered the coefficient of friction to be a function of the pressure distribution across the contact patch as well as the maximum pressure encountered in the contact patch. The authors calculated an (closed form) expression for the normalized longitudinal force in the contact patch with the pressure dependent coefficient of friction. They found that the normalized longitudinal force for the pressure dependent case was monotonically increasing. This is different from the pressure independent (constant) friction case in which the normalized longitudinal force increases monotonically until a certain value of slip and then remains constant [43]. The 33 authors concluded that for the case of load dependent friction coefficient, some part of the contact patch is always in the adhesion region, i.e., pure sliding never takes place.

3.3.5 The TreadSim Model

The TreadSim Model [31] is an analytical model, based on the brush model method, used to simulate the tire-road interactions in the contact patch. The model [31] improves upon the brush model method by introducing the force and moments in the contact patch as functions of certain parameters that are not taken into consideration by the classical brush model. As noted previously, the brush model method considers the coefficient of friction in the contact patch to be constant [33]. However, as seen in sections 3.3.3 and

3.3.4, this coefficient of friction is dependent on both the sliding velocity and the pressure.

This dependency is taken into account by the TreadSim model, which implements a pressure and velocity dependent coefficient of friction in the modeling methodology.

Another important advantage of the TreadSim model over the brush model method is the implementation of the combined slip cases in the model. The brush model method considers the longitudinal and lateral slips to be independent, which is not the case, as described in section 3.3.1. The TreadSim model, on the other hand, calculates the forces and the moments in the contact patch by considering the combined slip cases. The

TreadSim model also allows the usage of a non-parabolic pressure distribution in the contact patch, as well as compliances of the carcass, and the belt, which are not considered in the brush model method [31]. Thus, it can be observed that the TreadSim model incorporates into the brush model method all the improvements discussed in the previous sections. 34

3.3.6 Applications of the Brush Model Method

While the previous sections describe the working of the brush model method and the different improvements made to it, the current section is devoted to detailing its applications. One important application of the brush model method is in the smart tire domain, which makes use of the tire as a sensor in order to measure and transmit information to an onboard control system, which is then used to estimate the tire-road interactions [13,52,53]. The brush model method has also been implemented in a commercial software package as will be described later.

Smart tires measure and provide feedback about, among other parameters, the tire tread deformation as a way of estimating the tire-road interactions [13]. Applying this concept, Yi [54] proposed the development of a system that sensed the deformation of tread rubber and reported it back to the control system mounted on the vehicle in order to estimate the friction between the tire and the road surface. The author [54] make use of the brush model method in order to calculate the stresses in the adhesion and the sliding region of the contact patch. The results obtained from the analytical model were checked against a skid steered [54] robot. The preliminary results obtained by the authors indicated that the proposed method seemed to be feasible for estimating the tire-road interactions. Similarly,

Svendenius [34] used the brush model method to estimate the friction between the tire and the road surface with the purpose of analyzing the time lag in, and dependability of, the process of friction estimation. The author [34] used tire mounted sensors, optical instruments, and the brush model as three independent methods for friction estimation for a tire subjected to longitudinal forces. Omark [55] proposed a method based on the brush 35 model method to calculate the friction coefficient between the tire and the road surface.

According to the author [55], the coefficient of friction calculated using the brush model and validated using a friction estimation device would provide a good estimation of the coefficient of friction since it would be independent of the tire geometry parameters [55]

This, is advantageous since the coefficient of friction should ideally depend only on the materials of the tire tread and the road surface [55]. Like Omark [55], Singh et al. [56] also used the brush model method for friction estimation. The authors used the brush model method in combination with an intelligent tire based friction estimator to estimate the coefficient of friction. Using such an integrated approach provides a reliable friction estimation in advance [56], which is useful to active safety systems. Thus, the brush model method finds applications in estimating the tire-road interactions in smart tires.

Apart from its applications in the smart tire domain, the brush model method has also been applied in a commercial software package by Lacombe [57], who developed an analytical model based on the brush model method, in order to calculate the forces in the contact patch due to the tire-road interactions. The author [57] made use of the brush model method in order to compute the resultant force in the longitudinal direction. The developed analytical model was implemented into the commercial software package Dynamic

Analysis Design System (DADS) [57].

3.4 Thesis Objective

From the literature presented above, it can be concluded that the many improvements have been proposed and made in the classical brush model. The current work is focused on Gim’s analytical model [36–39]. An important shortcoming of the 36

Gim’s analytical model lies in the brush element stiffness term encountered in the expressions for the contact patch forces and moments. The brush element stiffness term does not seem to have a clear physical significance and its value is determined by calibrating the brush model method to experimental data. Additionally, the classical brush model assumes tire tread rubber to be a linear elastic material, whereas tread rubber, in general is nonlinear viscoelastic. The first part of the current work proposes to address these shortcomings by formulating a (physically meaningful) nonlinear constitutive equation for tire tread rubber. The specific objective of the first part of the current work is to modify the Gim’s analytical model [36–39] using a nonlinear constitutive equation relating rubber stress-deformation to be used in the calculation of the longitudinal and lateral forces. The nonlinear constitutive equation formulated in the current work replaces the expressions contacting the brush element stiffness in the calculation of the tire-road forces. The modified model thus obtained is, for the purpose of the current document, referred to as the modified brush model.

4 NON LINEAR ELASTICITY OF RUBBER

The purpose of the current chapter is to implement a nonlinear constitutive equation relating the stress-deformation of rubber in Gim’s analytical model [36–39] to give the modified brush model. This is achieved by using a nonlinear expression obtained based on the strain energy density function to relate the deformation of a tread block to the stress, as opposed to the linear expression used in Gim’s analytical model [36–39], which is based on the classical brush model. The procedure for obtaining this expression from the chosen formulation for the strain energy density function is explained in the first section of the current chapter. This is followed by the implementation of the nonlinear expression into

Gim’s analytical model [36–39] and relevant calculations. The chapter concludes with a method proposed to validate the modified brush model results. Detailed calculations are presented in the appendices.

4.1 Expression for Material Nonlinearity of Rubber

Gim’s analytical model [36–39], based on the classical brush model, has the following expression relating the deformation experienced by a tire tread block to the stress induced in it.

(4.1)

where, is the stress in the tire tread block, is the longitudinal stiffness rate per unit

area [36] and is the absolute value of the longitudinal deformation [36]. The longitudinal stiffness rate per unit area is calculated using the following expression [36].

(4.2) 38 where w is the width of the contact patch, l is the length of the contact patch and is the

longitudinal stiffness, defined as the slope of the longitudinal force – slip plot at zero slip

[36].

at 0 (4.3)

From Eq. (4.1) through Eq. (4.3), it can be seen that in order to estimate the value

of the longitudinal force using Gim’s analytical model [36–39], the value of the

longitudinal stiffness has to be initially estimated using experimental force-slip data.

Once is known, the longitudinal stiffness rate per unit area can be calculated and the

longitudinal stress can be estimated. Thus, once the model has been calibrated, it can be

used for force estimations. From Eq. (4.1), it can also be observed that the stress is a linear

function of the deformation. Thus, Gim’s analytical model [36–39] displays both the

shortcomings of the classical brush model discussed in section 3.4. In order to address these

shortcomings, a nonlinear stress deformation relation independent of has to be

formulated. For the purpose of primary calculations, the stress deformation relation was

assumed to be a polynomial expression as follows

(4.4)

where, is the stress in the tread block, which is a function of the position of the block

in the contact patch (ξ), x is the deformation of the tread element, L0 is the initial

undeformed length of the tread element, and b1, b2, and b3 are weight factors. Fig. 4.1

shows a representation of the undeformed length of the tread element with the

deformation that it undergoes in the contact patch. 39

Figure 4.1: Representation of x and L0 from Eq. (4.2)

Knowing the tread deformation (x) at each point in the contact patch at a particular

instant, Eq. (4.4) was used to calculate the stresses in the tread block. The stresses in turn,

were used to calculate the tire-road coefficient of friction and the tractive forces, using a

procedure similar to the one described in Appendices B through D. The results have a

qualitative agreement with the results of Gim’s analytical model [36–39]. The drawback

of using this polynomial formulation in Eq. (4.4) is that the stress deformation expression

does not have any physical interpretation. Thus, the polynomial expression in Eq. (4.4) is discarded and a different expression relating the stress deformation of the tire tread rubber, based on the strain energy density is formulated.

The constitutive equation for rubber, relating the stress to the deformation, can be

obtained through two different approaches, the micromechanics approach and the

phenomenological approach [58]. The micromechanics approach, as the name suggests,

models the microstructure of the material in order to relate the induced stress to the applied

deformation. Although more accurate than the phenomenological approach, the

micromechanics approach is often used for rubber subjected to small strains [59]. The

phenomenological approach on the other hand fits mathematical equations to experimental

stress deformation data. Once known, these expressions can be used to calculate the stress 40 for a particular deformation without the need of repetitive (and expensive) experimentation. Since the phenomenological approach formulates constitutive equations by fitting with experimental data, it can be used over a larger range of strains as compared to the micromechanical approach [59]. The drawback of the phenomenological approach is that it cannot directly relate the stress-deformation to the material microstructure [58,59].

An important parameter to be considered while formulating the constitutive equation for rubber is the strain energy density function, which gives the amount of energy stored in the material due to the deformation. An exhaustive classification and bibliography of the different strain energy density functions that can potentially be used in the constitutive modelling of rubber is given in [59]. For the purpose of the current work, the strain energy density function formulation proposed by Lopez-Paimes [60] is selected. This formulation tries to combine the advantages of the micromechanics and the phenomenological approach in that it tries to find a physical explanation for the variables in the constitutive equation while making the equation applicable for a large range of deformations [60]. The expression for the strain energy density function used by Lopez-

Paimes [60] is as follows

(4.5) where W is the strain energy density, I1 is the first invariant of the deformation tensor. μ1,

μ2, α1 and α1 are fitting parameters to be determined by comparing with experimental data.

This strain energy density function is used to calculate the shear strain and the shear

stress. (e.g. see [61]). Appendix A details the calculations involved in obtaining the shear 41 stress-shear strain relation. Since the constitutive equation involves fitting parameters arising from the expression for the strain energy density (Eq. (4.5)), calculations are initially made for the case of uniaxial tension. This enables the determination of the fitting parameters on comparison with stress strain data for uniaxial tensile test on styrene butadiene rubber [62]. The stress-strain relation for uniaxial tension is

(4.6) where λ is the stretch of the specimen subjected to uniaxial tensile loading. The stress is plotted for different values of stretch using the above parameters and compared with the experimental data as shown in Fig.4.2. Once the fitting parameters are obtained, the expression relating the shear stress to the shear deformation can be obtained using a similar method.

Figure 4.2: Determination of Fitting Parameters for Eq. (4.4) Comparing with Experimental Data [62] 42

Fig.4.2 shows the comparison of the stress-stretch plot obtained using Eq. (4.4) compared with the experimental data [62]. The fitting parameters used to fit Eq. (4.4) are inset in Fig.4.2. Once the fitting parameters have been determined, a similar procedure is used to obtain the shear stress-shear deformation relation, which is as follows.

(4.7)

Eq. (4.7) can now be used to calculate the stress instead of Eq. (4.1) used in Gim’s analytical model in order to make traction force predictions. The succeeding sections detail the implementation of Eq. (4.7) in Gim’s analytical model to give the modified brush model method and the resultant force calculations.

4.2 Friction Force Predictions Using Modified Brush Model Method

The current section presents in detail the modifications proposed in the analytical model in order to implement the material nonlinearity of rubber and the resultant force predictions. The first part of the section deals with the calculation of the longitudinal force in the contact patch using the modified brush model method, followed by the results obtained showing the variation of the normalized longitudinal force with the slip. Once the longitudinal force calculations are completed, the slip angle and the camber angle contributions to the lateral force are calculated and are plotted against the slip and camber angles respectively.

4.2.1 Longitudinal Force Prediction Using the Modified Brush Model Method

The procedure to obtain the expression to calculate the longitudinal force in the contact patch using the stress-deformation relation (Eq. (4.7)) is given in Appendix B. The 43 stress in the adhesion region of the contact patch is due to the deformation of the tread rubber [34], while the stress in the sliding region is dependent on the coefficient of friction and the pressure in the contact patch. The expressions for the stresses in the adhesion and sliding regions, based on Eq. (4.7) and the fitting parameters in Fig.4.2 are as follows

Stress in the adhesion region:

(4.8)

Stress in the sliding region:

(4.9)

At the breakaway point (transition between the adhesion and sliding regions), the stress continuity condition holds true and the expressions for the stresses in the adhesion and sliding regions can be equated to find the breakaway point [43]. Thus,

(4.10)

On equating the stresses in the adhesion and sliding regions, the location of the breakaway point in the contact patch can be solved for. The location of the breakaway point gives the lengths of the adhesion and sliding regions. Once this information is obtained, the forces in the adhesion and sliding regions can be calculated and summed to obtain the value of the total longitudinal force in the contact patch.

(4.11) 44

Eq. (4.8) through Eq. (4.11) describe the method used to calculate the longitudinal force in the contact patch using the proposed formulation. The equations are coded in

MATLAB [63] and test simulations are run using values obtained from literature in order to analyze the force predictions using the modified brush model method. The input parameters used for the simulation are given in Table 1.

Table 1: Input Parameters for Prediction of Longitudinal Force Using the Modified Brush Model Method Parameter Value and Units

Tread Thickness () values Varied between 8mm to 24 mm

Slip ratio (s) values Varied between 0 and 0.15

Coefficient of friction [39] 1.241

Normal load () [39] 200 kg = 1962 N

Width of the contact patch (w) 160 mm

Length of the contact patch (L) 100 mm

For the above defined values, the stress continuity condition at the breakaway point

can be evaluated to determine the position of the breakaway point in the contact patch at

different values of slip ratio and different tread thicknesses, as shown in Fig.4.3. 45

Figure 4.3: Variation of the Breakaway Point in the Contact Patch with the Slip Ratio for Different Tread Thicknesses

As seen in Fig.4.3, the length of the adhesion region in the contact patch tends to decrease as the value of the slip ratio increases. This trend is observed for all the values of tread thickness simulated, though the length of the adhesion region at maximum slip ratio of 0.15 goes on increasing with an increase in the tread thickness. The value of the breakaway point in the contact patch directly influences the longitudinal force in the contact patch, as seen from Eq. (4.11). Once the value of the breakaway point has been obtained and the lengths of the adhesion and sliding regions determined, the stresses in the adhesion and the sliding regions can be determined using Eq. (4.8) and Eq. (4.9) respectively. Fig. 4.4 shows the variation of the stresses in the adhesion and sliding regions along the length of the contact patch, calculated for a value of slip ratio equal to 0.12. 46

Figure 4.4: Variation of the Longitudinal Stress in the Contact Patch

Fig. 4.4 shows the longitudinal stresses in the adhesion and sliding regions in the contact patch for different values of the undeformed tread thickness (). The stress transition from the adhesion to the sliding region at the breakaway point in the contact patch can also be observed from Fig. 4.4. Once the stresses in the adhesion and the sliding regions have been calculated, the longitudinal force in the contact patch can be calculated by integrating the stresses within the lengths of the corresponding regions using Eq. (4.11).

The variation of the longitudinal force is seen in Fig. 4.5.

Figure 4.5: Normalized Longitudinal Force vs Slip for Different Values of ‘L0’

Fig.4.5 represents a plot of the longitudinal force in the contact patch normalized with respect to the normal load plotted against the slip ratio. It can be seen from Fig.4.1 that changing the value of the tread block thickness, changes the value of the longitudinal force in the contact patch. The values of tread block thickness are selected so that they would reflect those actually used in vehicle tires. The tread block thickness of approximately 24 mm is used in trucks [64] while new passenger car tires have a tread block thickness of approximately 8 mm [64]. Fig. 4.5 also shows the variation of the normalized longitudinal force with the slip ratio predicted by the Magic Formula based on experimental data [48]. It is seen that the longitudinal force predictions of the modified brush model method are a good qualitative match with the predictions of the Magic

Formula. 48

4.2.2 Implementing Nonlinearity into Lateral Force Calculations

The lateral force in the contact patch is assumed to be the sum of the lateral force due to the slip angle (α) and the lateral force due to the camber angle (γ). The methodology used to calculate the slip angle and the camber angle contributions to the lateral force is similar to the one used to calculate the longitudinal force, as presented in the following sections

Figure 4.6: The Tire Slip and Camber Angles Reproduced with modifications from [65,66]

As shown in Fig.4.3 the tire slip angle is the angle made by the actual path traversed by the tire with the direction in which it is being steered. Gim’s analytical model [36–39] assumes the slip angle to contribute to forces in the lateral direction (Y axis in Fig.2.4).

The procedure to calculate the lateral force arising due to the slip angle is given in

Appendix C. Similar to the longitudinal force calculations presented in the previous sections, the contact patch is assumed to be divided into adhesion and sliding regions, with the stress in either region being calculated using the following expressions 49

Stress in the adhesion region:

(4.12)

Stress in the sliding region:

(4.13)

The stress continuity condition is implemented at the breakaway point and the location of the breakaway point in the contact patch is calculated for. Once the value of the breakaway point is known, the lateral force contribution of the slip angle is calculated using the expression

(4.14)

Table 2 shows the parameters used as input in the simulations performed to predict the values of the lateral force due to the slip angle

Table 2: Input Parameters for Calculating the Lateral Force Due To Slip Angle Using the Modified Brush Model Method Parameter Value and Units

Tread Thickness () values Varied between 8mm to 24 mm

Slip angle values [39] Varied between 0 and 8 degrees

Coefficient of friction [39] 1.19

Normal load () [39] 200 kg = 1962 N

Width of the contact patch (w) 160 mm

Length of the contact patch (L) 100 mm

Fig.4.7 shows the lateral force due to slip angle normalized w.r.t the normal load, plotted against the slip angle.

Figure 4.7: Normalized Lateral Force Due To Slip Angle Vs Slip Angle for Different Values of ‘L0’ 51

The tire camber angle as shown in Fig.4.6 is the inclination of the vertical tire axis with the vertical axis of the vehicle. The details of the method used to calculate the contribution of the camber angle to the lateral force in the contact patch is given in

Appendix D. The calculations for the test simulations are performed by coding the method shown in Appendix D in MATLAB [63]. The input data used for the test simulations are given in Table 3 as follows.

Table 3: Input Parameters for Calculating the Lateral Force Due To Camber Angle Using the Modified Brush Model Method Parameter Value and Units

Tread Thickness () values Varied between 8mm to 24 mm

Camber angle values [39] Varied between 0 and 6 degrees

Coefficient of friction [39] 1.00

Normal load () [39] 500 lbf = 2224.9 N

Tire Radius () [39] 16 inches = 406.4 mm

Width of the contact patch (w) 160 mm

Length of the contact patch (L) 100 mm

The expression used to calculate the force is as follows

(4.11)

Fig.4.8 shows the lateral force due to camber angle normalized w.r.t the normal load,

plotted against the camber angle. 52

Figure 4.8: Normalized Lateral Force Vs Camber Angle for Different Values of ‘L0’

The same trend can be observed in Fig.4.7 and Fig.4.8 as was evident in Fig.4.5, i.e. an increase in tread block thickness corresponds with a closer agreement to the results of the original analytical model which assumed rubber to be linear elastic.

Fig.4.5 thorough Fig.4.8 give the contact patch force predictions using the modified brush model method, which can be used to estimate the tire-road friction. From the results, it can be observed that a decrease in tire tread thickness leads to an increase in the forces in the contact patch. The forces exerted by the tire in the contact patch are important for estimation of the tire wear and fatigue. Although experimental validations of the force predictions presented above are beyond the scope of the current work, an experimental setup to estimate tire-road friction in real time is proposed in the following section, which 53 could be used to validate the modified brush model method presented in the previous sections.

4.3 Proposed Method to Validate Modified Brush Model

The previous section details the calculations of the longitudinal and the lateral forces in the contact patch using the modified brush model method. The current section provides a brief description of a proposed experimental setup that could be used to validate the modified brush model method.

Figure 4.9: Schematic of the Proposed Experimental Setup

Fig.4.6 shows a 3D model of the proposed experimental setup. A piezoelectric sensor [67] is attached to the tread block of a vehicle tire to measure the instantaneous deformation of the tread block and relay the data to an onboard data processing unit. The experimentally measured tread block deformation would be compared to the data obtained from the slip and contact patch position dependent deformation expressions used in the modified brush model method in order to check the agreement between the two sets of data. The deformation data would be then be used as an input for the modified brush model 54 method and the longitudinal and lateral forces in the contact patch could be predicted based on this deformation input using the methodology presented in the previous sections. The coefficient of friction between the tire tread could be computed analytically based on the predicted values of the longitudinal and lateral contact patch forces [68]. Experimental measurements of the friction coefficient could also be made using an appropriate device, e.g. the Dynamic friction tester [69]. Comparing the two coefficients of friction thus obtained would enable the validation of the proposed modified brush model method. The proposed experimental setup presented in the current section is a part of a research grant submitted to the Ohio University Research Committee (OURC). The submitted grant proposal is currently under review.

The current chapter presented a method to modify Gim’s analytical model [36–39] to calculate the friction forces in the contact patch. Gim’s analytical model, based on the classical brush model method divides the contact patch to be divided into adhesion and sliding regions. At low slip ratios (low velocities), the contact patch is assumed to be divided into adhesion and sliding regions, with both regions contributing to the contact patch forces [36]. At higher slip ratios, however, the contact patch force is assumed to be a function of the sliding coefficient of friction and the normal force in the contact patch

[36]. The second part of the current work thus analyzes the sliding friction of rubber using the Heinrich-Klüppel (HK) [70] friction model.

5 EFFECT OF TREAD RUBBER VISCOELASTICITY ON TIRE FRICTION

While the preceding chapters proposed a modification to the brush model [31] to consider the effect of nonlinear elasticity of rubber on friction, the current chapter will investigate the effects of the nonlinear viscoelastic nature of rubber on the sliding friction of rubber friction. Unlike elastic materials, the constitutive equation for viscoelastic materials is time dependent [23]. The previous chapters provide a method to relate tread rubber friction to the stress induced in the tire tread due to the deformation that it undergoes in the contact patch, without considering the time dependency of the stress. The focus of the current chapter thus, is to analyze the effects of tread rubber viscoelasticity on the coefficient of sliding friction. Detailed calculations using the HK [70] friction model modified to implement the effects of temperature and strain amplitude on the loss modulus are presented in Appendix E.

5.1 Adhesive and Hysteretic Friction

Figure 5.1: Adhesion and Hysteresis Friction in Elastomers. Reproduced from [94] 56

The friction of a rubber block sliding on a rough surface is assumed to have two different contributions, adhesive and hysteretic [71–74] as seen in Fig. 5.1. The adhesion between sliding rubber and the surface at low velocities is responsible for the adhesive contribution to the sliding friction [75]. The hysteretic contribution to rubber friction, dominant at larger sliding velocities [73,74], arises due to rubber viscoelasticity. As the rubber slides on a rough surface, the surface asperities cause time and surface profile dependent deformations of the rubber block. A part of the energy of deformation is dissipated due to the viscoelasticity of rubber, giving rise to hysteretic friction.

5.2 Introduction to Viscoelasticity

Viscoelasticity represents the time-dependent elastic response of a material to the applied loading. For a purely elastic material, the magnitude of stress developed in the material is dependent only on the magnitude of the deformation. On the other hand, for a viscoelastic material, the magnitude of stress induced in the material is dependent not only on the magnitude of the deformation, but also the rate at which the material deforms. This dependence of stress on strain and time for a viscoelastic material can be verified through material testing using plots similar to the one shown in Fig.5.2. 57

Figure 5.2: Stress Strain Curve for a Viscoelastic Material Adapted with modifications from [24]

Fig.5.2 shows the schematic stress-strain plot for a viscoelastic material. The stress strain paths during loading and unloading are different indicating a portion of the input energy is dissipated during deformation. Experimentally, the viscous portion of deformation of viscoelastic material can be characterized by Dynamic Mechanical

Analysis (DMA). Assume the material is subjected to small amplitude harmonic strain as

, sin (5.1)

where ω is the frequency of deformation. The corresponding stress reads

, sin (5.2)

The complex modulus is defined as

, ∗ (5.3) , 58

Here, ′ and " are the storage and the loss moduli, respectively. This complex dynamic modulus can be separated into a real part, also called the storage modulus and an imaginary part, called the loss modulus. The storage modulus represents the amount of energy stored in the material due to deformation that can be recovered elastically. The loss modulus represents that part of the internal energy that is lost through dissipation. It is this energy lost through dissipation, calculated in terms of the loss modulus that plays an important role in hysteretic friction calculations, as shown in later sections. The ratio

" tan is often referred to as the loss factor.

Similar to linear and nonlinear elasticity, viscoelasticity can also be classified as

linear or nonlinear viscoelasticity based on the relation between the stress and the strain

rate. Linear viscoelastic behavior is generally observed for viscoelastic materials subjected

to infinitesimal strains and strain rates. Mechanical models [23] of linear viscoelasticity

such as Maxwell, Kelvin-Voigt, Standard Linear Solid and the Zener use a series of springs

and dashpots to simulate viscoelastic behavior. Nonlinear viscoelasticity is generally

observed at finite deformations of viscoelastic materials.

A point to be noted here is that the current work does not consider tire tread rubber

to deform plastically. While raw rubber deforms plastically through strain induced

crystallization when subjected to load, the process of vulcanization, as undergone by tire

tread rubber, improves rubber elasticity [76]. Vulcanized rubber may still go undergo

plastic deformation if subjected to elongations of more than 400% [76], but, the current

work focusses on styrene butadiene rubber used in tire treads which does not crystallize,

and hence, does not undergo plastic deformation irrespective of the magnitude of the 59 applied loading. Thus, tire tread rubber is assumed to be viscoelastic and plastic deformations are neglected in the current work.

5.3 Rubber Viscoelasticity and Hysteresis Friction

Rubbers, and more generally polymers, are regarded as viscoelastic materials [23].

The viscoelasticity of rubber is rooted in the relaxation time of strands, the sub-chains between two consecutive crosslink points. As seen in Chapter 1, tire tread rubber is made of different constituents, primary among which are natural and synthetic rubber and filler materials like carbon black and silica. It is known that the rubber energetically interacts with the surface of filler particles in addition to the interaction of the filler particles due to van der Waals forces between particles. Thus, the rubber viscoelasticity is controlled by polymer relaxation as well as polymer/filler or filler/filler interactions [77]. Fig. 5.4 shows depicts natural rubber with carbon black added as a filler.

Figure 5.3: Carbon Black Reproduced from [78]

The hysteretic friction of a tire tread block moving over the road surface is dependent, among other parameters on the road surface profile and the sliding velocity of the tread block. Fig. 5.4 shows the variation of coefficient of friction with sliding velocity.

Figure 5.4: Variation of the Coefficient Of Friction with the Sliding Velocity Reproduced from [79]

Figure 5.5: Rubber Sliding on a Sinusoidal Surface. Adapted with modifications from [80]

Fig.5.5 shows a rubber block sliding along a rough surface. As the tread moves along the road surface, the surface asperities penetrate into the rubber causing time dependent deformations, similar to the loading applied in a DMA test. The energy dissipation, characterized by the rubber’s loss modulus contributes to the hysteretic friction of rubber. The Persson friction model [80] and the Heinrich-Klüppel (HK) friction model

[70] are the most widely used models to analyze the viscoelasticity dependent hysteretic friction of rubber. The current work focusses on the classical HK friction model [70], which has been widely used in the determination of tire-road friction [77,81] and proposes changes in the method used to calculate the hysteretic friction of rubber. A brief description of the classical HK model [70] is provided in the following section.

5.4 Classical HK Friction Model

Figure 5.6: Friction Calculation using the Classical HK Model Adapted with modifications from [80]

Heinrich and Klüppel [70] used a one dimensional viscoelastic element sliding with velocity on a randomly rough surface, as shown by Fig. 5.3. The variation of the surface roughness along its length can be visualized by the plot on the bottom half of Fig.5.3. is the variation in the height of the surface from a datum along the length . The coefficient of friction can be calculated in terms of the amount of energy dissipated due to viscoelasticity of rubber in terms of the product of the friction force and the sliding velocity.

The expression for the same is as follows

(5.4) where is the mean penetration depth which is assumed to be the penetration of the

surface asperities into the rubber surface averaged over the sliding area, σ0 is the normal

stress, is the sliding velocity, ” is the frequency dependent loss modulus of the

rubber specimen and is a quantity that measures the surface roughness.

Eq. (5.1) is the expression for the hysteretic coefficient of friction calculated using

the classical HK model [70]. As seen from the equation, the coefficient of friction is

dependent on the loss modulus of the rubber. The loss modulus itself is considered to be a

function of the frequency of the load application. However, for applying the model for tire-

road friction calculations, the dependence of the viscoelastic modulus of rubber on the

applied strain amplitude (Payne effect) needs to be considered. The following sections

briefly describe the Payne effect and a model used to implement the Payne effect in the

friction calculations of the classical HK friction model [70]. 63

5.5 The Payne Effect

The Payne effect, a characteristic of the behavior of rubbers with filler materials

[82], is the change in the viscoelastic modulus of rubber with an increase in applied strain amplitude. This dependency of the viscoelastic modulus on the applied strain amplitude is a result of the filler-filler interactions [83,84]. Fig.5.4 shows the variation of the storage and the loss moduli with the applied strain amplitude for styrene butadiene rubber [85].

Figure 5.7: Variation of the Storage Modulus and Loss Tangent of Styrene Butadiene Rubber with Strain Amplitude Adapted with modifications from [85]

From Fig.5.7, it can be seen that the storage modulus starts decreasing with an increase in the strain amplitude till it reaches a certain minimum value which is unique to the rubber and the filler being used. The loss tangent shows a maximum occurring in the 64 range of strain amplitudes where the storage modulus is decreasing. It is necessary to implement this dependency of the viscoelastic modulus on the applied strain amplitude in the friction calculations of the classical HK model [70]. The modelling of the Payne effect is still an unsolved problem in the field of solid mechanics. One of the possible solutions to this problem involves the use of fractional derivatives, which is the approach adopted in the model proposed by Lion et al. [86]. Fig.5.8 shows the fractional viscoelastic model used in the Lion et al. [86] model to formulate the expressions for storage and loss moduli.

Figure 5.8: Fractional Standard Linear Solid Viscoelastic Model. Reproduced from [86]

The expressions for the storage and loss moduli according to this model [86] are as follows

Storage Modulus

(5.5)

Loss Modulus

(5.6) with,

(5.7)

Here, ’ and ” are the storage modulus and the loss modulus respectively, and are fitting parameters with the units of material modulus (Pa), is the relaxation time, = 1s is a constant with the units of time, while , and are fitting parameters.

5.6 Modified HK Model with Payne Effect

As shown in the previous section, the variation of the viscoelastic modulus of rubber with the strain amplitude is proposed to be implemented in the classical HK friction model [70] using the model proposed by Lion et al. [86]. The current section shows friction calculations for a rubber block that is assumed to slide on a rough surface. For simplicity, we assume that the profile of the rough surface is sinusoidal with wavelength and amplitude as shown in Fig.5.5. For test simulations run using the surface shown in

Fig.5.9, the surface is assumed to have = 0.82 mm and = 0.2 mm 66

Figure 5.9: Rubber Block Sliding on a Sinusoidal Surface. Adapted with modifications from [87]

In order to calculate the strain amplitude dependent viscoelastic modulus of the sliding rubber block, the rubber is assumed to be a blend of 60 phr natural rubber and 40 phr butadiene rubber with 65 phr carbon black N220 filler. Desired experimental data is available for this rubber [88] in order to determine the fitting parameters required in the expression [86] for the viscoelastic modulus, as shown in Fig.5.10.

Figure 5.10: Curve Fitting to Obtain Fitting Parameters for Viscoelastic Modulus At 8% Strain Amplitude

Once the fitting parameters for the storage and loss moduli are available at the specified strain, the variation of the viscoelastic modulus with frequency can be calculated along with the variation of the hysteric friction coefficient with the sliding velocity. The calculations involved in this procedure are detailed in Appendix E and are coded into

MATLAB [63] to calculate the results of the test simulations. The variation of the storage and loss moduli with the strain amplitude using the above fitting parameters is presented in Fig. 5.11. 68

Figure 5.11: Variation of Storage and Loss Moduli with the Strain Amplitude

Fig.5.11 shows the decrease in the storage modulus with the increase in the strain amplitude, the Payne effect. The loss modulus can be seen to have a maxima in the strain amplitude range where the storage modulus starts decreasing. In order to observe the effect of changing strain amplitude on the frequency dependence of the viscoelastic modulus and the coefficient of friction, the strain amplitude value is changed from 0.08 (for which the fitting parameters are obtained) to 0.01, keeping all other fitting parameters the same.

Fig.5.12 shows the variation of the storage and loss moduli with the frequency, while

Fig.5.13 gives the variation of the coefficient of friction with the sliding velocity. 69

Figure 5.12: Storage and Loss Moduli Using the Obtained Fitting Parameters for Applied Strain Amplitudes Of 8% And 1%

Figure 5.13: Variation of the Hysteretic Coefficient Of Friction with Sliding Velocity for 8% And 1% Strain Amplitude

It can be observed from Fig.5.12 that the storage and the loss moduli are greater over the given frequency range for the case of 1% applied strain amplitude as compared to

8% applied strain amplitude. This trend is in agreement with experimental data [88]. The hysteretic coefficient of friction depends on the loss modulus following Eq. (5.4). The manifestation of Payne effect at larger strains has a dramatic effect on coefficient of friction and leads to the reduction of its value. Thus, this implies that rougher surfaces have lesser contribution in hysteresis friction.

5.7 Temperature Dependent Rubber Viscoelasticity

The previous sections have addressed the amplitude and frequency dependence of the viscoelastic modulus of rubber and the calculation of the hysteretic friction coefficient using the loss modulus of rubber. Apart from being dependent on the frequency of loading and the amplitude of the load applied, the viscoelastic modulus of rubber is also sensitive to the temperature at which the measurements take place. The temperature dependency of 71 the viscoelastic modulus is accounted for using time-temperature superposition [89]. Since the hysteretic friction depends on the loss modulus of rubber as seen in previous sections, it can be concluded that the hysteretic coefficient of friction is temperature dependent.

As the tire tread block moves on the road surface, the road surface asperities deform the tread rubber surface. A part of the energy of the energy of deformation is dissipated due to the viscoelastic nature of rubber. This dissipated energy can be assumed to be converted into heat energy [90], which increases the temperature in the surroundings of the part of the tire tread in the contact patch. This phenomenon can be confirmed through thermal imaging of the tire tread as it emerges from the contact patch, as shown in the

Fig.5.14 [90].

Figure 5.14: Infrared Image of the Tire Tread Exiting The Contact Patch. Adapted with modifications from [87]

Since the viscoelastic modulus of rubber depends on the temperature, the local rise in temperature over the surroundings will change the properties of rubber in the contact patch. This, in turn will affect the friction between the tire tread and the road. An interesting point to note here is that the local rise in temperature is the result of heat dissipation which is dependent on the loss modulus of the tire tread rubber. However, the loss modulus itself, is dependent on the local temperature since the viscoelastic properties of rubber are temperature dependent. Thus, the calculation of the temperature dependent viscoelastic moduli of tire tread rubber has to be performed iteratively. To solve this problem, the current work uses the methodology used in the Persson friction model [90]. The following equation is used to calculate the temperature at a specific location in the tire tread at a specific instant in time.

(5.8)

is the heat dissipated due to rubber hysteresis per unit volume per unit time, is the heat diffusivity, is the density and is the specific heat. Eq. (5.4) is solved for the surface

shown in Fig.5.9 to obtain the expression for the temperature rose due to the heat

dissipation as

(5.9)

where and are surface parameters, ” is the loss modulus. Equation 5.5 can be

used to calculate the increase in temperature over the background temperature iteratively

until a temperature value is converged upon. This value of temperature is used to calculate

the viscoelastic modulus and the coefficient of friction (using Eq. (5.4)) from the reference 73 data using time temperature superposition. The time temperature superposition methodology used in the current work is based on the Williams-Landel-Ferry expression

[91]. It should be noted that the current work assumes the time temperature superposition method, which is generally applied for thermorheologically simple materials [92], holds true for describing the temperature dependence of the nonlinear viscoelastic behavior of tire tread rubber.

Thus, the current work proposes to implement the strain amplitude and temperature dependence of the viscoelastic modulus of rubber in the classical HK friction model [80] using the methodology described in the preceding sections. A test simulation is run using

MATLAB [63] for the and the variation of the coefficient of friction with the sliding velocity is calculated for the strain amplitude and temperature dependent HK friction model.

5.8 Strain Amplitude and Flash Temperature Dependent Modified HK Model

The current section calculates the dependence of the coefficient of friction on the flash temperature using the HK model modified to include temperature and strain amplitude dependence. As mentioned previously, the temperature and viscoelastic modulus are interdependent and need to be solved iteratively. The same friction scenario as mentioned in previous section is considered, with rubber sliding on a simple sinusoidal surface (Fig.5.9). The effects of local heating are considered and the coefficient of friction is calculated for, as shown in the Fig.5.15. 74

Figure 5.15: Coefficients of Friction with and without Flash Temperature For 8% And 1% Applied Strain Amplitude

As seen in Fig.5.15, for both 1% and 8% applied strain amplitude, the coefficient of friction with the flash temperature implemented is lower over the range of sliding velocities considered, as compared to the friction coefficient without considering the effects of flash temperature. The converged temperature was 292 K for the case of 8% strain amplitude, while it was 304 K at 1% strain amplitude. The reason for this trend is that the relaxation modulus of rubber decreases with an increase in temperature, thereby decreasing the hysteretic friction coefficient of rubber. It can also be observed that the temperature has a pronounced effect on the variation of the coefficient of friction with sliding velocity at lower strain amplitudes. Further validation of the proposed modified HK model is carried out by comparing the coefficient of friction predictions with experimental data [88], as shown in Fig. 5.16. 75

Figure 5.16: Friction Predictions of the Modified HK Model Compared with Experimental Data

The figure shows the variation of the coefficient of friction with the sliding

velocity for a rubber block sliding on a rough surface approximated using a sinusoidal profile compared with the experimentally observed variation [88]. The predictions of the proposed model show good agreement with the experimental data over the tested sliding velocity range, thereby validating the proposed model. 76

6 SUMMARY AND CONCLUSIONS

The first objective of the current work was to analyze the contact patch force predictions of the reformulated Gim’s analytical model [36–39], called the modified brush model method for the purpose of the current work. The modifications proposed in the original model included relating the stress-deformation using a nonlinear constitutive equation whose parameters could be calculated from simple uniaxial testing of tire tread rubber. Reformulating the Gim’s analytical model using this nonlinear constitutive equation also introduced the dependency of the contact patch forces on the undeformed thickness of the tire tread (L0), which was not seen in the original model.

From the predictions of the modified brush model method, it was found that the

contact patch forces increase with a decrease in the value of the undeformed tread

thickness. This was because a decrease in L0 resulted in an increase in the shear strain (Eq.

(B.1)) which caused an increase in the shear stress and thus the force in the contact patch.

The longitudinal contact patch force predictions were found to have a good qualitative agreement with the predictions of the Magic Formula, which were based on experimental data. The scope of the current work did not include experimental calculations of the contact patch forces. An experimental setup for the force calculations was proposed in the current work, and will be used in the future for validation of the method proposed in the current work.

While the classical brush model uses 1D brush elements to predict the contact patch forces, finite element modelling, using 2D and 3D elements, is a more powerful and accurate tool. Modelling the interaction of tire tread blocks and the road surface using finite 77 element models introduces additional modelling capabilities like analyzing the effect of tread sipes [93]. The advent of powerful modelling and analysis packages has made finite element modelling more convenient and thus the author recommends the use of finite element modelling of the contact patch for contact patch force and friction predictions as the future path for the current work.

The second objective was to modify the classical HK friction model [70] to implement the variation of the viscoelastic modulus with the strain amplitude (Payne effect) and the temperature. The dependency of the viscoelastic modulus on the strain amplitude was implemented using the model proposed by Lion et al [86], while the temperature dependency was implemented using a method proposed by Persson [90]. Test simulations were performed to analyze the hysteretic friction predictions of the modified model for rubber sliding on a rough surface approximated to have a sinusoidal profile characterized by the amplitude and the wavelength.

The hysteretic friction predictions showed that the coefficient of friction decreased with an increase in the magnitude of the strain applied on the sliding rubber block. The applied strain amplitude, in turn, was directly proportional to the amplitude of the rough surface. The reason for this particular observation was that the loss modulus of rubber decrease with an increase in the strain amplitude, causing a decrease in the predicted value of the coefficient of hysteretic friction.

The temperature dependency of rubber’s viscoelastic modulus was implemented to account for the variation of the viscoelastic modulus of tire tread rubber with increase in temperature due to local heating in the contact patch. The hysteretic coefficient of friction 78 was found to decrease with an increase in the temperature due to decrease in the loss modulus of rubber with an increase in temperature for a particular loading frequency. From a comparison between plots showing the variation of the coefficient of friction with the sliding velocity with and without implementing the effects of temperature, it was found that the reduction in the coefficient of friction when the effects of temperature rise were implemented was more pronounced at lower strain amplitudes.

The rough surface on which the rubber block slides was approximated using a sinusoidal profile for the scope of the current work. The HK and Persson friction models, which are amongst the most widely used friction models, approximate the rough road surface as a fractal surface with roughness on multiple scales, which affects the friction calculations. Thus, the next step in the current line of research would be to implement the strain amplitude and temperature dependency of the viscoelastic modulus in the classical

HK friction model with the sliding surface approximated as a fractal surface instead of a sinusoidal surface.

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APPENDIX: SUPPLEMENTAL FILES

The appendices A through E referred to in the document have been attached

as a supplemental file. Following is a list of the appendices.

1) Appendix A: Constitutive Equation for Rubber based on the Strain Energy Density

Function

2) Appendix B: Expressions for Calculating the Adhesion Length and the Longitudinal

Forces based on the Modified Brush Model

3) Appendix C: Expressions for the Adhesion Length and the Lateral Force due to Slip

Angle based on the Modified Brush Model

4) Appendix D: Calculating the Lateral Force due to Camber Angle using the Modified

Brush Model Method

5) Appendix E: Calculations using a Strain Amplitude and Temperature Dependent

Modified HK Model ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Thesis and Dissertation Services ! !