A Finite Element Material Modeling Technique for the Hysteretic Behaviour of Reinforced Rubber

DEGREE PROJECT IN VEHICLE ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2017

A finite element material modeling technique for the hysteretic behaviour of reinforced rubber

PETTER LIND

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES i

A ﬁnite element material modeling technique for the hysteretic behaviour of reinforced rubber

by Petter LIND Degree project in Solid Mechanics Second level, 30.0 HEC Stockholm, Sweden 2017

Abstract

Reinforced rubber is thanks to its elastic and dissipative properties found in industrial applications such as isolators, flexible joints and tires. Its dissipative propertied comes from material related losses which have the effect that energy invested when deform- ing the material is not retained when returning it back to its initial state. The material losses are in turn caused by interactions in the material on a level below the micro scale. These interaction forms a macro stress strain response that is dependent on both strain amplitude, strain rate and temperature. It is thus a challenge to accurately model components made of reinforced rubber and and features of interest related to them, such as the rolling resistance for a tire. It is also difficult to device general design guide lines for such components due to rubbers many and complex dependencies and a simple accurate phenomenological model for modeling these properties are highly sought for in industry today. This thesis presents a method for modeling the strain amplitude and strain rate behavior for cyclically loaded rubber along with a method of choosing its material parameters. The proposed modeling technique results in a model with the same frequency dependency over all strain rates. An approximation which is shown to be valid over a few decades of strain amplitudes and rates and is believed useful for many industrial applications. The material model presented can in addition be implemented in commercial FE- softwares by using only pre-defined material models. This was achieved by implementation of the overlay method. The thesis also presents a method for how to implement the modeling technique in simulations with purpose to determine the rolling resistance of a truck tyre. ii

En materialmodell för att modellera hysteresberoendet för förstärkt gummi

Petter LIND Examensarbete i Hållfasthetslära Avancerad nivå, 30 hp Stockholm, Sverige 2017

Sammanfattning

Förstärkt gummi används, tack vare sina elastiska och dissipativa egenskaper, i industriella komponenter som exempelvis bussningar i drivlinan, däck och flexibla gummi- kopplingar. Dissipationen orsakas av materialförluster som i sin tur orsakas av interaktioner på längdskalor kortare än micro-nivå i materialet. Dessa Interaktioner resulterar i ett material som mekaniskt kan klassificeras som ett ickelinjärt material beroende av töjningsamplitud, töjningshastighet och temperatur. Det är därför en utmaning att göra modeller som på ett korrekt sätt förutsäger be- teendet för gummikomponenter och egenskaper relaterade till dessa, som exempelvis rullmotståndet i ett däck. Det är även svårt att ge generella design riktlinjer för dessa komponenter på grund gummits många materialberoenden och enkla användvändbara fenomenologiska modeller som kan underlätta vid sådana processer är därför högt efter- frågade av industrin idag. I denna rapport presenteras en materialmodell för att modellera töjningsamplitud- och töjningshastighetsberoendet för gummi under cyklisk last samt en metod för att välja dess materialparametrar. Den föreslagna materialmoddeleringstekniken resulterar i en modell med samma töjningshastighetsberoende för alla töjningsamplituder. En approximation som är användbar inom ett antal decader av töjningsamplituder och töjning- shastigheter vilket borde vara tillräckligt för de flesta industriella tillämpningar idag. Den föreslagna materialmodellen kan dessutom implementeras i kommersiella FE- programvaror genom att endast använda i programmet inbyggda materialmodeller. Detta sker genom tillämpning av overlay-metoden. I rapporten presenteras även en metod för hur modelleringstekniken kan implementeras genom en tillämpning i simuleringar med syfte att bestämma rullmotståndet för ett lastbilsdäck. iii

Acknowledgements

I would ﬁrst like to thank my excellent supervisor at Scania, Rickard Österlöf, for his guidance throughout this whole project. It was always possible to ask questions and get some helping advices whenever I ran into a trouble spot.

I would also like to thank all co-workers at RTCC and at the neighboring group RTLC. You have all been warm, kind and supportive. You helped me with softwares, scripts, installations and even ﬁxing my bike! I am very grateful.

An extra thank you goes to my fellow students and friends Jonas Barrskog and Aron Ingi Ingvason for making my work so much easier.

Finally, I would like to thank Carl Dahlberg at KTH. Carl provided excellent support on how to structure the text in this thesis. I am also very grateful for his valuable remarks regarding the text during the last phase of this project.

Petter Lind Stockholm, 7th June 2017 iv

Contents

Abstract i

Sammanfattning ii

Acknowledgements iii

1 Introduction 1 1.1 Background ...... 1 1.1.1 Purpose ...... 3 1.1.2 Previous work ...... 3 1.2 Outline of thesis ...... 5

2 On tyres and rubber 6 2.1 The tyre ...... 6 2.1.1 Construction and terminology ...... 6 2.1.2 Rolling resistance ...... 7 2.2 Reinforced rubber ...... 9 2.2.1 Hysteresis ...... 10 2.2.2 Storage- and loss modulus ...... 10

3 Rheological model 13 3.1 Hyperelastic material definition ...... 14 3.2 Viscoelastic material definition ...... 15 3.2.1 Logarithmic distribution of time constants over the frequency axis 16 3.2.2 Power-function distribution of time constants over the frequency axis 17 3.3 Plastic material definition ...... 19

4 Finite element modeling 21 4.1 The overlay method ...... 21 4.2 Virtual material test ...... 22 4.3 Tyre modeling ...... 22 4.3.1 preprocessing ...... 22 4.4 Simulation steps ...... 24 4.5 Simpliﬁcations ...... 25 4.6 Postproccesing ...... 26

5 Results 27 5.1 Virtual material test ...... 27 5.1.1 Log distribution method ...... 28 5.1.2 Power distribution method ...... 30 5.2 Rolling resistance ...... 32 5.2.1 Mesh study ...... 33 5.2.2 Contribution from different parts ...... 33 5.2.3 Solution accuracy ...... 34 v

6 Discussion and conclusions 36 6.1 Discussion ...... 36 6.2 Conclusions ...... 36 6.3 Further work ...... 37 6.4 Best practice for tyre simulation ...... 37

A Derivations 39 A.1 Ziegler hardening law (1D) ...... 39 A.2 Stress response in one maxwell element during cyclic pertubation ..... 41

B Details 43 B.1 Test matrix ...... 43 B.2 Python code ...... 44 B.3 Simulation data ...... 45

References 47 vi

List of Figures

2.1 tyre structure ...... 6 2.2 Free body diagram of a steady state rolling tyre ...... 8 2.3 Stress-strain response for reinforced rubber [9] ...... 10 2.4 Schematic representation of a stress strain response for illustrating parameters for calculation of the storage- and loss modulus ...... 11

3.1 Rheological representation of the material model ...... 13 3.2 Stress strain response for the a generalized maxwell element, plastic and combined material model respectively...... 14 3.3 Storage modulus for ten different maxwell elements using the log distribution method ...... 16 3.4 Storage modulus comparison using the log distribution method ...... 17 3.5 Loss modulus comparison using the log distribution method ...... 17 3.6 Storage modulus for ten different maxwell elements using the power distribution method ...... 18 3.7 Storage modulus comparison using the power distribution method .... 18 3.8 Loss modulus comparison using the power distribution method ...... 19 3.9 One dimensional schematic solutions for different parameters in the Ziegler hardening law. The orange line shows the expected behavior. The Green line corresponds to C and blue line to C 0 ...... 20 !1 ! 4.1 Principle of the overlay method ...... 21 4.2 Virtual FE-specimen in (a) undeformed and (b) deformed shape ...... 22 4.3 Comparison between the mesh and a real tyre cross section ...... 23 4.4 FE-mesh for the tyre ...... 23 4.5 Simulation setup ...... 25

5.1 Stress strain response from all virtual test using the log distribution method 28 5.2 Material response for different load frequencies using the log distribution method ...... 29 5.3 Storage modulus for material using the log distribution method ...... 29 5.4 Loss modulus for material using the the log distribution method ...... 30 5.5 Stress strain response from all virtual test using the power-distribution method ...... 30 5.6 Storage modulus from material using the power-distribution method ... 31 5.7 Loss modulus from material using the power-distribution method ..... 31 5.8 Loss modulus shape for cycles with stress amplitude levels close to the yield stress ...... 32 5.9 Rolling resistance as function of velocity using the log distribution material modeling method ...... 33 5.10 Comparison of the result when computing the rolling resistance by different methods ...... 34 5.11 The result of the energy balance of the different runs ...... 35 vii

5.12 Rolling resistance comparison with shorter ...... 35

A.1 Stress vs strain for one maxwell element ...... 42

B.1 Rolling resistance result from the 10 km/h- simulation ...... 45 B.2 Velocity proﬁle used in the 10 km/h- simulation ...... 45 B.3 Rolling resistance with different mesh sizes ...... 46 B.4 Rolling resistance with hysteretic material active in only one part at the time 46 viii

List of Tables

2.1 tyre structure ...... 7 2.2 Forces and geometry from the free body diagram ...... 8

4.1 Data describing the FE-model ...... 24

5.1 Parameters used in the material model ...... 27 5.2 Range of strain amplitude and rates...... 28 5.3 Impact of mesh size ...... 33 5.4 Contribution to the total rolling resistance ...... 34

A.1 Maxwell element parameters ...... 42

B.1 Design of experiment for virtual material test ...... 43 B.1 Design of experiment for virtual material test ...... 44 1

Chapter 1

Introduction

This introduction starts of with setting the rubber material modeling technique and the value of understanding it in a larger scenery and also brieﬂy explains how it is related to the rolling resistance. It also presents the purpose of the work performed, a section of previous works and a short outline of the thesis.

1.1 Background

The transport sector is today responsible for more than a quarter of the global energy- related green house gas emissions, a number forecasted to increase towards the year 2030 [2]. This is a relevant issue since the level of atmospheric CO2 is one of the nine planetary boundaries and one of two which already has been exceeded [7]. It is here wrong to conclude that the emissions per vehicle is increasing. The development in the sector is on the contrary keeping high pace. A modern 40 tonne truck only pollutes 5 % and emits 75 % of the CO2-levels compared to a few decades ago [11]. IPCC even forecasts that the average transport will have a 30-50 % higher efﬁcency in year 2030 with a 15-40% reduced CO2 emission compared to todays levels[2]. The development pace is despite recent progress therefore not deemed to slow down. Possible thanks to tougher legal demands, manifested by for example the new European emission standard, Euro 6 [16]. Nevertheless, legal demands and planetary boundaries are not the only challenge for the automotive industry. New vehicles still have to be better performing than the predecessor, available for an attractive price and developed within a reasonable amount of time. The development of new better perfoming tyres are an important piece when it comes to meeting the various challanges of the heavy automotive industry. Tyres are the force coupling between the road and the truck. Ideally they should provide unlimited traction and perfect isolation without any energy losses. This is of course not the case, or even possible. Modern tyres are a compromise between several factors such as rolling resistance, tyre wear, sound level, handling, wet grip etc. Measurements show that the tyres on a passenger car are responsible for up to 20 % of the energy loss due to resistive forces during motorway driving where the biggest contributor is wind resistance, responsible for up to 70% [11]. Tyres are therefore an important factor to consider when it comes to decreasing the CO2 - levels but also important when it comes to for example noise reduction in cities and the levels of micro particles which in itself is an increasing health hazard caused by tyre and road wear [10]. Needless to say, tyres have also developed. A modern car tyre has a third of the rolling resistance compared to a few decades ago, much thanks to new material compounds and tyre structure [11]. 2 Chapter 1. Introduction

Reinforced rubber components are a big part of the automotive industry today. Mainly thanks to reinforced rubbers unique properties such as high extensibility, fatique resistance and high material damping [9]. In fact, many reinforced rubber materials can en- dure strain amplitudes in the magnitude of 100 % and return back to its original con- ﬁguration with just a small set of permanent deformation. These properties have made reinforced rubber the material of choice for many industrial applications such as ﬂexible joints, shock absorbers, bushings and tyres. The mechanical behavior of rubber in combination with its main application areas often makes it suitable to model as a non-linear material, with elastic and dissipative properties dependent on both the deformation amplitude but also on the deformation rate. The mechanical behavior makes simpler linearized models, for example Hooke’s law, to be inadequate when used over wider ranges of strain- amplitudes. Rubber also has a temperature dependency which adds even more to the complexity of describing its mechanical behavior since the deformation itself causes the material to heat up which in turn alters its material properties. The rolling resistance is connected to the material behaviour and therefore also de- penent on for example both deformation rate and temperature. This makes it both troublesome to quantify the rolling resistance in a meaningful way and maybe even more troublesome when it comes to the question on how to practically test it. The industry standard in Europe is to use a standarized test, ISO 9948, for determing the value of the rolling resistance for a truck tyre [11]. The test is performed by pressing the tyre against a large drum. The drum (and tyre) is then accelerated up to a rotational speed that would correspond to a transversal velocity of 80 km/h. After which the speed is held constant in order to reach a stable temperature in the tyre. The rolling resistance can from the test setup be calculated by monitoring the wattage needed to drive the drum. Although, care needs to be taken in order to ensure that other factors, such as frictional forces between the drum and its hub does not get confused for rolling resistance. Ideally a test for the rolling resistance should give a result for the rolling resistance over a wider range of speeds, temperatures and axle loads. The major downsides with the current ISO-test is that;

• A lot of information about the rolling resistance gets lost by only measuring the rolling resistance at one boundary temperature at one speed with one level of axle load.

• The boundary conditions differ a lot from a real driving situation. The tyre is for example only rotating and not translating which reduces the level of airdrag around the tyre. This might have an impact on the the temperature gradient in the tyre which in turn affects the rolling resistance. Experiments conducted shows that a raise of the temperature will lower the amount of hysteresis in the rubber material [17]. A raise of the temperature in the tyre will therefore lower the material related losses and therefore also the rolling resistance.

• The measurements are conducted after the tyre reaches a stable temperature dis- trubution which occurs after approximately three hours in a running test rig [11]. This procedure simpliﬁes the complexity of measuring the rolling resistance since it only yields one value of the rolling resistance throughout the whole measurement time. A relative long warm up phase if compared to the 4.5 hours maximum driving time for a truck-driver in the EU [20]. The measurements are in this perspective, at the best, correct for approximately half of the driving time, given that the truck is running at a constant speed of 80 km/h. 1.1. Background 3

Simple phenomenological material modeling techniques for rubber would allow for more accurate simulations of for example the rolling resistance and is therefore of high interest in industry today. Accurate simulation methods would in turn allow for better understanding of the rolling resistance and could even open up for the possibility to adapt the tyre for a speciﬁc end user. Again, a tyre that has a low rolling resistance at a constant speed at 80 km/h is not necessarily the tyre with the lowest rolling resistance at a lower or higher speed due to the strain rate and temperature dependencies.

1.1.1 Purpose The aim of this thesis is to develop a material modeling technique that, with sufﬁcient precision, can model the strain amplitude and strain rate dependency for rubber in industrial application within a reasonable range of amplitudes and rates. The material modeling technique is presented in the thesis by implementation in simulations for de- termining the rolling resistance of a steady state rolling tyre.

1.1.2 Previous work There already exist several different suggestions on how to model rubber. For example the incompressible hyper elastic approach by Mooney [13]

n i j W = Cij (I 3) (I 3) (1.1) 1 2 i+Xj=1 where W is the strain-energy function, I1 and I2 are strain invariants and Cij are material constants. A hyper elastic model such as the Mooney-model can, in general, accurately model the stiffness behavior for a material given a deformation rate, but lacks the ability to accurately model the material related losses and the strain rate dependency. A common way to include the dissipative behavior in the hyper elastic models is to use so called pseudo-elasticity. The idea behind is that the unloading and loading can be described as separate hyper elastic models, as proposed by for example Fung [3]. It does look promising in 1D but raises a lot of problems when generalized to 3D, for example; when is a load changing from loading to unloading? And what is the effect under cyclic loading? 3D generalizations for these type of models have however been done and a possible development from the pseudo-hyper elastic models is to use a material model that is a composite of a hyper elastic and a elasto-plastic model with a damage parameter depending on the total deformation rate as suggested by for example M.M Safadi and M.B Rubi [12]. The downside with these more advanced hyper elastic models are also the lack of the strain rate dependency and it would in addition also be cumbersome to implement such a model in a FE-program since the user would have to write their own material subroutine. A simple common approach in industry today is to model the material behaviour as a viscoelastic generalized Maxwell model [11]. The equation for which can be found in course littrature in material mechanics. For example in the book by Gudmundson [5] where the elastic modulus in 1D, E, is expressed as function of time 4 Chapter 1. Introduction

N tEi /⌘i E(t) =E 1 ↵i 1 e , 0 i= X1 (1.2) * N ⇣ ⌘+ E0 =Ee ,+ Ei. - Xi=1

Where Ei and ⌘i are the stiffness and damping constant for each Maxwell element respectively and E0 is the instantaneous stiffness. The prony constant ↵i and the time constant ⌧i are for each maxwell chain deﬁned as

Ei ↵i = , E 0 (1.3) ⌘i ⌧i = . Ei Viscoelastic generalized maxwell material models can, for a wide range of frequencies, accurately model the stiffness behavior. This approach also combines well with the previously mentioned hyper-elastic model, equation 1.1. It is thus a well suited model for modeling the stiffness behavior of the material and the rate dependency over a limited range of strain rates. A major drawback by only using a hyper- viscoelastic model is that the material dissipation only depends on the strain rate which makes it troublesome to capture the strain amplitude dependency. Altough, some improvements can be done as shown by for example Rendek and Lion [9], where the generalized maxwell chain is modiﬁed so that the viscosities depends on additional internal state variables. In summary, a promising approach but it would, as the previous mentioned pseudo-elasticity models, be cumbersome to implement in a commercial FE-program. The material modeling technique implemented in this thesis is based upon a parallel rheological model with a non-linear plastic element in parallel with a hyper-elastic generalized maxwell element. This setup enables the possibility to model the material behaviour in a range of several decades of strain and strain rates given the approximation that the material can be modelled using the same, additative, frequecy dependency across all amplitude levels. Such an approximation could be resonable for some applications over wide ranges of strain as seen in measurements done by for example Kari et al [18]. A beneﬁt of the parallel model technique used is that the material model can be implemented in FE-simulation using built-in material models. The implementation is achieved by using a so called overlay method which connects several identical element nets over the same sets of nodes. The overlay method is not new and has already been used several times in similar models with hyper-viscoelastic elements in parallel with elastic perfect- plastic elements [15][8]. The novelty in this work lies in implementing a non-linear kinematic hardening in the plasticity model which results in a material model with few material parameters and a smooth hysteretic behavior. The thesis also provides a method for how the material modeling technique can be calibrated and implemented in FE-simulation used for simulating the rolling resistance of a tyre. 1.2. Outline of thesis 5

1.2 Outline of thesis

Chapter 2 describes characteristics of rubber and highlights some of the proposed explanations of the underlaying mechanisms that causes characteristic rubber like behavior. Chapter 3 presents a 1D rheological model with a hyper elastic generalized maxwell chain in parallel with a non-linear kinematic- plasticity element. Each elements purpose is also presented along with how its material parameters affects the material mechanical behavior. Chapter 4 presents the FE-modeling principle for the overlay model and also the FE- model of a truck tyre implementing a more general material model based upon the model presented in chapter 3. Finally, chapter 5 concludes and discusses the results. It also presents a best practice for how to simulate tyres and lastly gives proposals for future work. 6

Chapter 2

On tyres and rubber

Before a material model and its application can be formulated in later chapters it is important to start with some fundamentals about tyre and rubber. It is also worth pointing out that the model of the truck tyre and material presented in this thesis is aimed at being representative for tyres and rubber used in industry today but are not chosen in order to represent a speciﬁc material or tyre. Their purpose is rather to provide a foundation for discussion and illustration of the material modeling technique and its possibilities.

2.1 The tyre

Tyres are the connection between the vehicle and the road with the task to transfer forces between the two of them. The purpose of the tyre as a mechanical structure is to damp out the variations in the normal forces and at the same time provide high traction against the road with as few losses as possible. The forces involved can be categorized into two categories; normal forces and frictional forces. The losses will be manifested as rolling resistance, heat and tyre wear. A regular truck tyre will, without the rim, weigh around 50-60 kg with a radius around 0.5-0.6 m.

2.1.1 Construction and terminology Tyres are complex engineering constructions. Their exact geometrical structure and com- position varies with the application, manufacturer and model. However, most tyres sold on the market today have common characteristics. A proposal on how to categorize the structure of a tyre is shown below in Figure [2.1] along with a short overview of parts and purposes in Table 2.1.

FIGURE 2.1: tyre structure 2.1. The tyre 7

TABLE 2.1: tyre structure

Nr. Part Description 1 Tread The outermost thick rubber section of the tyre. The tread is the only part of the tyre that has contact with the road. Its purpose is to provide traction and and protect the tyre from wear. 2 Bead A completely embedded rubber section in the tyre with main purpose to acts as a spaceﬁller and keep the reinforcements in place. 3 Steel belts The steel belts in the tyre consists of steel wires organized into thin sheets. The sheet layers can vary signiﬁcantly in both number, thickness and orientation between different tyre models. 4 Carcass A layers of steal wires with consistent orientation around the whole tyre. The carcass forms the framework for the tyre and is in that sense the most important part of the load bearing structure. The carcass is often radially oriented and anchored in the bead wire (see below). 5 Inner Liner An air proof rubber seal preventing air leakage through the tyre structure 6 Side A rubber section enclosing the bead wire and the carcass with purpose to keep the bead wire and carcass in place and provide stiffness and damping to the structure 7 Bead wire A thick metal wire cycled around the rim several laps. It prevents the edge of the tyre to de- tach from the rim and also acts as an anchoring point for the carcass.

It is common to categorize tyres depending on how the reinforcement layers are oriented in the tyre and how the direction is oriented within the different layers of reinforcement. The far most common structure used in 98% of cars is the radial ply tyre which is characterizes by a radially oriented carcass, meaning that the steel wires in the carcass travels from bead wire to bead wire radially from the tyre center [19]. The radial ply tyre is also the tyre that will be used for the truck tyre modeled in this thesis.

2.1.2 Rolling resistance Rolling resistance is a measurement of the relative energy loss caused by the tyre when rolling. The rolling resistance has contributions from the tyre-road contact but also resistive forces from the tyre-air contact. The main contributor is the assymetric contact pressure between the tyre and the road which is responsible for up to 95 % of the total rolling resistance [11]. The rolling resistance will therefore from now on be idealized as only caused by this contact pressure. A free body diagram with all relevant forces involved is shown in Figure 2.2 below, along with explanations in table 2.2. 8 Chapter 2. On tyres and rubber

FIGURE 2.2: Free body diagram of a steady state rolling tyre

TABLE 2.2: Forces and geometry from the free body diagram

Variable Description N Normal force from the vehicle F Resultant of the contact pressure. FRR Rolling resistance force. Fµ Resultant of frictional forces between the tyre and ground. R Radius of the tyre RL Shortest radius of the deformed tyre ↵ Distance between the center of the wheel and the resultant of the normal pressure. v translational velocity for the center of the tyre.

The rolling resistance can be though of as a resistive torque. This torque is found in the ﬁgure as the resultant of the normal force, F, times its distance from the wheel center, ↵. The direction of the torque is opposed the rotation direction of the traveling tyre. The tyre thus has a traction force, Fµ, corresponding to the magnitude of the resisting torque positioned a distance RL from the wheel center. The magnitude of the pulling force, FRR, needed in order to move the tyre at constant speed can thus be calculated using Newton’s third law

Fµ RL =F ↵, (2.1) · · Fµ FRR = 0, (2.2) which, solved for FRR, result in N ↵ FRR = · . (2.3) RL 2.2. Reinforced rubber 9

An intuitive approach is to think about the rolling resistance force as the force needed in order to move the tyre at a constant speed given a certain normal load. The dimen- sionless rolling resistance coefﬁcient, CRR, can now be deﬁned as the ratio between the rolling resistance force and the normal load

FRR CRR = . (2.4) N The rolling resistance for tyres are today around 8 ‰[11].

2.2 Reinforced rubber

Rubber is a collective term that covers materials with very varying applications and properties. From tough industrial applications such as bushings or tyres in the vehicle industry to rubber used in ordinary objects, such as cookware. The word rubber will in this thesis refer to vulcanized reinforced rubber. This type of rubber can undergo deformation up to several 100% strain and return elastically to the original conﬁguration with only a small set of permanent deformation. Its mechanical properties are also dependent on temperature, strain level and strain rate. which, for most applications, makes it suitable to model with non-linear models. The dependencies of rubber can be further expanded to include for example aging or swelling but will not be discussed further in this thesis. Rubber materials used in industrial applications are often reinforced, meaning that small reinforcement particles has been added to the rubber matrix. The reinforcement has many wanted effects. It allow for a higher level of stress before fracturing, which also occur at a higher level of strain compared to a non-reinforced material [6]. It also makes the material more fatique- and wear resistant, as well as it increases the damping in the material [6]. These effects differ from the effect of similar reinforcement in other reinforced materials. Plastic, for example, increases its critical stress when reinforced, but not its critical strain [6]. A characteristic cyclic stress strain curve for reinforced rubber is shown in Figure 2.3 below. This curve illustrates the stress-strain response for rubber during cyclic loading for a stepwise increasing amplitude with a constant load rate and was conducted by M. Rendek and A. Lion [9]. 10 Chapter 2. On tyres and rubber

FIGURE 2.3: Stress-strain response for reinforced rubber [9]

The stress-strain response in cyclic loading forms so called hysteresis loops. Hystere- sis is a history dependent property which causes the unloading to follow a different load path than the loading of the material resulting in a loss of elasticity and therefore a dissipation of elastic energy during each load cycle. These so called hysteresis loops are a characteristic and important property of reinforced rubber. The width of the hysteresis loop is dependent on the strain amplitude, but also on the load rate and temperature.

2.2.1 Hysteresis The research in the field have so far agreed upon that the reinforcement particles in combination with the long polymer chain in the rubber matrix causes the hysteretic behavior. Simply since non-reinforced rubber lacks the hysteresis behaviour [9]. The interactions between the filler and the polymer chains at a small scale is however still up for discussion. A recently proposed mechanism published by J-B. Donnet and E. Custodero [6] proposes that the hysteresis is due to adsorption and desorption cycles between the polymer chains and the reinforcing filler’s surfaces. While other, for example G. Heinrich, and M. Kluppel, proposes that breaking and reforming of bonds in the polymer network is the main contributor to this behavior [4].

2.2.2 Storage- and loss modulus A pure viscoelastic material subjected to a cyclic sinusoidally varying load will have a cyclic varying stress and strain response, only shifted in phase and magnitude. The phase difference between the responses is commonly referred to in the literature as the phase angle. The stiffness and dissapated energy are for every frequency quantiﬁed by implementation of two scalar values. They are the storage and loss modulus and can for small loss angles be deﬁned as [4] 2.2. Reinforced rubber 11

⌧amp G0 = , (2.5) amp

G00 =G0 tan eqv , (2.6) ⇣ ⌘ where W tan eqv = . (2.7) ⇡⌧ampamp ⇣ ⌘ The variables ⌧amp and amp refers to the amplitude of stress and strain respectively and W is the enclosed area, encircled in the stress-strain plot by the steady state cyclic response, W = v (✏) d✏. All of which are drawn into a schematical stress- strain response in Figure 2.4 below. H

FIGURE 2.4: Schematic representation of a stress strain response for illustrating parameters for calculation of the storage- and loss modulus

The storage- and loss modulus is commonly presented as a function of frequency. The frequency dependency is however a simpliﬁcation of the underlaying strain rate behavior. The material properties will change when the frequency of a sinusoidal load is altered, but only because the strain rate changes with a change of the frequency. It is however powerful to present the material behavior as a function of altering load frequency when performing material test with altering sinusoidal load frequencies since it connects the material behavior with the choice of loading rate. Nevertheless, not all rubber components are used in applications with a sinusoidally varying load which makes the frequency analogy of less interest in for example bump 12 Chapter 2. On tyres and rubber stops or ﬂexible joints. The strain rate dependency will thus not be referred to as a frequency dependency in this thesis. Later chapters will however present the result from material tests as a function of frequency and strain level thanks to its strong connections to the test settings, as discussed above. 13

Chapter 3

Rheological model

A rubber material model used in tyres should be able to accurately capture the strain amplitude and strain rate dependency of the material within a reasonable range of amplitudes and rates. It should also be straight forward to generalize the model to 3D and implement it in a FE-software. The model presented in this chapter has therefore several well known material models in parallel which makes it straight forward to implement in FE-programs, as will be shown in Chapter 4. The idea is to split the material response into two parts; one part capturing the non-reinforced rubber elastic behavior and one modeling the dissipative. The global material response is obtained as sum of the stress in each sub-material model

i=n = i, (3.1) Xi=1 where represents the total stress response of the material and i represents the contribution from each sub-model respectively. This was achieved in 1D by arranging material models in parallel. The material models chosen were a generalized maxwell element with hyper elastic stiffness deﬁnition and a non-linear elastic-plastic element. As shown in the rheological model presented in Figure 3.1 below.

FIGURE 3.1: Rheological representation of the material model 14 Chapter 3. Rheological model

The characteristic dynamic stress-strain response for the plastic and hyper-viscoelastic model respectively as well as an additive, total, response is shown in Figure 3.2. The stress strain result shown was conducted using a sinusoidally varying load with constant frequency and amplitude. The result shown differ from a quasi-static simulation since the losses from the viscoelastic model in that case would be zero witch would reduce the cyclic shape of the viscoelastic response to a line.

FIGURE 3.2: Stress strain response for the a generalized maxwell element, plastic and combined material model respectively.

The material model will only have the sought for behavior over a predetermined range of strain amplitudes and strain rates, as will be further discussed for each material sub-model separately later in this chapter. The relevant strain- amplitude and rate interval for the model is dependent of application. It will for the tyre simulation be determined from a simulation of a steady state rolling tyre with only elastic material properties from which the values of the maximum and minimum strain amplitude and rates can be measured. The difference of the deformed shape for the tyre when switching material model will be of less importance, since the reinforcement layers and not the rubber are the main load carrying structure, as previously discussed in Chapter 2.

3.1 Hyperelastic material deﬁnition

The hyper elastic model was modeled using a the Mooney Rivelin model, equation 1.1. This model was reduced to two material constants C10 and C01 by summing all combinations up to n = 1 and modiﬁed, with an additional term added, in order to include compressible materials

1 2 W = C10 (I1 3) + C01 (I2 3) + (I3 1) , (3.2) D1 where Ii are the stretch invariants and D1 is a function of the bulk modulus, K

D1 = 2/K. (3.3)

The value for D1 was expressed as function of the shear modulus, G, and Poisson’s ratio ⌫ 3.2. Viscoelastic material deﬁnition 15

3(1 2⌫) D = . (3.4) 1 G(1 + ⌫)

The value for the shear modulus, G, was in turn determined from the parameters C10 and C01 from Equation 3.2.

G = 2(C10 + C01), (3.5) The hyper elastic model was then reduced to only two material constant by using the standard ratio between the hyperelastic parameters for natural rubber [1] C 10 = 7. (3.6) C01 Rubber is often referred to as an incompressible material. It is here worth pointing out that experiments tells us that the bulk modulus for rubber is several order of magnitudes higher than its shear modulus. It is also the reason why the incompressible approximation often is suitable in rubber simulations. However, the bulk modulus for rubber is still smaller than the bulk modulus of for example most metals. The conclusion is that rubber is more compliant to purely hydrostatic loads than for example construction steel, but referred to as incompressible due to the high ratio between its shear and bulk modulus. It is therefore reasonable to set the bulk modulus according to Equation 3.4 both for preventing convergence issues in later FE-simulations and also since rubber is not ideally incompressible.

3.2 Viscoelastic material deﬁnition

The purpose of the viscoelastic elements are to add a frequency dependency to the material model. This approach comes with a limitation caused by the parallel structure of the material model which causes the viscoelastic properties to have the same effect over all strain amplitudes. An increase in strain rate thus has the same stiffening effect over the whole amplitude range. This is not a characteristic of reinforced rubber and a limitation of the model [18]. The model is therefore limited to model the response over strain amplitude levels where the rate dependency can be approximated as constant. The generalized maxwell element was defined by a prony series, in accordance with equation 1.2-1.3. Meaning that the stiffness of each element in the maxwell chain are expressed as fractions of the total stiffness where the total instantaneous stiffness is set to the stiffness of the hyperelastic material defined in equation 3.2-3.5. The parameters for the material model can be fitted against some experimental data by implementation of an optimization algorithm. This is in general a good approach, commonly used when selecting parameters for the overlay model [15][8]. However, a best-fit optimization of the material parameters adapts the material parameters to the test(s) performed. The behavior of such a model is uncertain if the material is subjected to other deformation modes than it was originally fitted against. In extension leading to the conclusion that different material models can model the same material accurately in different deformation modes. Rubber in tyres are loaded multi ax- ially which makes it tough to construct tests that accurately reflects the material response in the real application. It is thus a strength of the modeling technique if the behavior of the model is predictable even if the material is loaded in another deformation mode than what was tested or with significantly higher or lower magnitudes of strain rates and amplitudes then what was expected. 16 Chapter 3. Rheological model

Two different suggestions on how to calibrate the model without the need of optimization are described in the following Chapters 3.2.1 -3.2.2. Both methods uses the same stiffness over all maxwell elements in the chain. Implemented by setting the prony constants, ↵i, to the same fraction of the total instantaneous stiffness. The first method results in a exponentially increasing frequency behavior for the storage modulus with a constant loss modulus. The second results in a increase for both the storage and loss modulus. Both methods could be useful depending on the feature of interest where the first method offers a slightly simpler model and the second a possibility for adapting a rate dependent loss modulus. All figures following in this chapter referring to the frequency distribution assumes an interval of frequencies from 0.1 to 1000 Hz, corresponding to a strain rate interval in the same order of magnitude. This interval was chosen only for illustration purpose.

3.2.1 Logarithmic distribution of time constants over the frequency axis The rate-stiffening of rubber can, from measurements, over small rate intervalls be approximated as exponentially increasing with an increase of strain rate [9][18]. This behavior is achieved in the model by choosing values on the damper constants, ⌘i, so that the time constants, Ti, becomes logarithmically distributed. This method of selecting the time constants will in this report be referred to as the log distribution method. The principle of this method is demonstrated using a few maxwell elements. The equations describing the cyclic stress strain response for one maxwell chain are derived in Appendix A. Figure 3.3 below illustrates the distribution of the time constants and the rapid transition of the storage modulus in each maxwell element in a region around its characteristic frequency.

FIGURE 3.3: Storage modulus for ten different maxwell elements using the log distribution method

A comparison of the superimposed storage modulus using 3 and 10 elements are shown in Figure 3.4. Both of which has the same total stiffness for all element and the same total span between its time constant along the frequency axis. The only difference between the curves are the number of maxwell elements covering the frequency span. 3.2. Viscoelastic material deﬁnition 17

FIGURE 3.4: Storage modulus comparison using the log distribution method

The same superposition analogy can be applied for calculation of the loss modulus for which the results are shown in Figure 3.5 below.

FIGURE 3.5: Loss modulus comparison using the log distribution method

Note that the loss modulus remains approximately constant in an interval between the lowest and highest time constant in the maxwell chain.

3.2.2 Power-function distribution of time constants over the frequency axis Another frequency dependency for the generalized maxwell chain is obtained by changing the distribution pattern over the frequency axis. The distribution chosen has the same 1 pattern over the log scale interval as the power function y(x) = x 2 has in the interval 0 to 1. This method will in this report be referred to as the power distribution method. The distribution pattern was generated using a Python code, which is appended in Ap- pendix B. The chosen distribution causes the distances between the time constants to be narrower with an increasing frequency, as shown in Figure 3.6 below. 18 Chapter 3. Rheological model

FIGURE 3.6: Storage modulus for ten different maxwell elements using the power distribution method

A comparison of the storage modulus for a generalized maxwell chain using 10 and 100 elements respectively are shown in Figure 3.7.

FIGURE 3.7: Storage modulus comparison using the power distribution method

The ﬁgure above illustrates the sought for smooth transition in the beginning of the frequency range which is achieved in addition to an overall similar stiffening shape as previously shown in Figure 3.4. A comparison of the loss modulus for a generalized maxwell chain using 10 and 100 elements respectively are shown in Figure 3.8. 3.3. Plastic material deﬁnition 19

FIGURE 3.8: Loss modulus comparison using the power distribution method

The different methods only results in a small change of the shape for the storage modulus’s frequency dependency with a more distinct difference for the result of the loss modulus. Its skew shape results in a loss modulus that is exponentially increasing with frequency over the major part of the frequency span which corresponds well with measurements found in litrature [9][18].

3.3 Plastic material deﬁnition

The purpose of the plastic elements is to add a strain amplitude dependency to the material model. The plasticity model chosen is a non-linear kinematic hardening Ziegler- model with a boundary surface [14]. The usage of this built in plasticity models has two weaknesses. First, the software (Abaqus) does not allow for setting the yield stress to zero which, as later shown, would be the sought after criteria. Second, it is not possible to set the value to an arbitrary low number either, since it creates severe non-linearities which in turn causes convergence problems for the FE-solver. A high value on the yield stress would on the other hand cause a pure elastic behavior in a wider range of strains and thus remove much of the dissipative material properties. The Ziegler hardening law develops a backstress, ↵, as formulated in the constitutive law [14]

1 pl pl ↵˙ k = Ck ( ↵) ✏¯˙ k ↵k ✏¯˙ , (3.7) 0 where Ck and k are parameters describing the non-linear hardening curve, 0, is the initial yield stress, ✏¯pl, is the effective plastic strain and ↵ is the total backstress obtained from

m ↵ = ↵k . (3.8) Xk=1 This expression can for m = 1 in 1D loading be shown to equal a spring in series with a frictional element, in detail shown in Appendix A. The sketch below, Figure 3.9, is a overview for the different hardening behavior in 1D when varying the parameters C and . 20 Chapter 3. Rheological model

FIGURE 3.9: One dimensional schematic solutions for different parameters in the Ziegler hardening law. The orange line shows the expected behavior. The Green line corresponds to C and blue line to C 0 !1 !

The maximum value for the strain amplitude dependent hysteresis in the model, max, can from the ﬁgure above be determined to

2 c max = 2 + 0 , (3.9) r3 * + where a factor 2 comes from the fact that the total amplitude is doubled during cyclic , - loading, as seen in for example Figure 3.2. It can from the ﬁgure above be concluded that the relation between the material parameters C and determines the width of the hysteresis where C affects the slope of the curve far away from the boundary surface. The contribution from the second term 0 is expected to be much smaller than the contribution from the ﬁrst term since the yield stress, ideally, should be close to zero in order for allowing amplitude related losses even for very small levels of strain amplitudes. 21

Chapter 4

Finite element modeling

This chapter describes how to implement the rheological model presented in Chapter 3 into a FE-model. It also present the setup of the tyre model used for rolling resistance calculation and walks the reader through the different steps needed for performing the rolling simulation and acquiring the results.

4.1 The overlay method

The rheological model presented in Chapter 3 can be implemented by writing a general material model (UMAT in Abaqus), and then plugging it into a FE-software. This is possible but cumbersome. An easier way is to implement a so called overlay method. The idea behind the method is to superimpose combinations of built in material models that the software otherwise would not allow for. This is practically achieved by overlaying several layers of ﬁnite element meshes with the exact same topology but with different material description. Meaning that all element meshes share the same set of nodes. The computational cost will increase with every extra set of elements due to the increased complexity of the problem even though the total degrees of freedom will remain the same, since the sets share nodes. This means that the extra computational time expected is mainly due to the complexity of the material deﬁnition in each extra layer but also to the computational cost related to the increased number of gauss points.

FIGURE 4.1: Principle of the overlay method 22 Chapter 4. Finite element modeling

4.2 Virtual material test

The mechanical behavior of the material model was investigated through a virtual material test. The purpose of which was to validate the storage- and loss modulus behavior for the model with a test similar to what can be found in litrature [9]. The FE-specimen used and its boundary condition along with its deformed shape is shown in Figure 4.2 below.

FIGURE 4.2: Virtual FE-specimen in (a) undeformed and (b) deformed shape

The bottom surface of the specimen was constrained in all directions while the top surface was displaced using a sinusoidally varying displacement

u(t) = uamp sin(!t), (4.1) where uamp is the amplitude of the displacement and ! is the displacement frequency. The deformation rate variation becomes, u˙ = uamp! cos(!t), with a maximum displacement rate of u˙max = uamp!. The stresses and strains were evaluated in the middle of the material from which the state for different levels of strain- and stress levels could be obtained. Each combination of strain amplitude and rate resulted in one value for the storage and loss modulus, calculated by insertion of the cyclic stress strain response into equations 2.5-2.7. The results of which is shown in Chapter 5. The number of elements in the FE-specimen can be reduced without signiﬁcant loss of accuracy. However, the computational cost for the virtual test was still relative low in comparison to the computational capacity available. This relative ﬁne mesh and thick specimen was therefore used throughout the test serie.

4.3 Tyre modeling

4.3.1 preprocessing A full 3D-mesh of a truck tyre was generated by revolving a 2D-mesh of a tyre cross section around the tyre center. The 2D mesh of the cross section used is shown in Figure 4.3a along with a ﬁgure of a real truck tyre cross section for comparison purpose. 4.3. Tyre modeling 23

(A) FE-mesh (B) Truck tyre

FIGURE 4.3: Comparison between the mesh and a real tyre cross section

A representative part of the 3D-mesh is shown in Figure 4.3a below along with some relevant data describing the FE-model in Table 4.1.

FIGURE 4.4: FE-mesh for the tyre 24 Chapter 4. Finite element modeling

TABLE 4.1: Data describing the FE-model

Data type Value Weight of model 55 kg Tyre radius 0.56 m Number of element 85940 Components Element type bead, tread, side, bead wire Fully integrated 8-node linear bricks with approximately 1% 6-node linear triangu- lar prisms (C3D8 and C3D6 in Abaqus) carcass, steel belts Linear shell elements with reduced integration (SFM3D4R in Abaqus) road Rigid elements (R3D4 in Abaqus)

The reinforcement layers in the model were modeled using special purpose rebar shell-elements. The rebar layers were embedded in the 3D-solids by deﬁning the nodes for every rebar-shell in between the nodes of one 3D-solid. The embedded elements were constrained to move relative their host element by a kinematic coupling which enforces the position of the host nodes and embedded elements nodes to have a mutually linear dependency. The method was chosen since it allows for an almost arbitrary placement and number of layers without increasing the number of general 3D-solids in the model.

4.4 Simulation steps

Three sequential steps were used for solving the model; inflation, footprint and rolling. The result from the inflation step were used as the initial configuration in the footprint and the result from the footprint step were then used as initial configuration for the rolling step. Figure 4.5 below provides a schematic view of the simulation setup. The inflation step was performed using a quasi-static simulation run. All nodes on the inner edges of the tyre, marked with green dashed lines in Figure 4.5, was constrained to move as if connected to a rigid rim. This was achieved by defining a kinematic coupling which enforced the nodes along the inner edges to retain their original position relative each other and the reference node. The reference node, from now on called the center node, was constrained in all degrees of freedoms. The inflation pressure was applied by defining nodal forces summing up to a pressure of 9 bar evenly distributed along the inside surface of the tyre. The footprint step moves the tyre and road into contact. It was performed with the same quasi-static settings as the previous step. The road was introduced as a 2D rigid element, with a penalty contact definition defined between the road surface and the surface of the tyre. The tyre was constrained using the same boundary conditions as in the inflation step. The road was during the step moved with a prescribed displacement from a position just outside contact to a position 4 cm closer to the wheel center along the z-direction. The last rolling step was performed using a dynamic solver setting where the center node was moved in the x-direction with a prescribed acceleration up to a target speed. After which the acceleration was kept constant for three full revolutions of the tyre. Its 4.5. Simplifications 25 rotational degree of freedom around the y-axis was for this step released, allowing for free rotation around that axis. It was in that way a simulation of a non-driven tyre.

FIGURE 4.5: Simulation setup

4.5 Simpliﬁcations

All simplifications stated here are simplifications from the already idealized axisymmet- ric geometry of a tyre. The purpose of further simplifications were only to reach a faster convergence for the FE-solver. Both for decreasing the computational time but also since a too smal time step might cause the model to be practically unsolvable. One problematic area when modeling tyres is the area close to the connection between the tyre and the rim. The connection has double purpose; to keep the tyre in place and to prevent air leakage in between the rim and tyre. The tyre-rim connection is enforced by a high inflation pressure which causes the tyre ends to lock against the rim. The shape of the surrounding tyre is constrained by the stiff bead wire. However, the stress in the rubber material in this area between the rim and the bead wire is constantly high with a relative low variation during the rolling cycle. This high stress in combination with rubber’s high ratio between its bulk- and shear modulus can be a cause for slow convergence. The reason is that small change of an elements volume have a much larger impact on the stress state than a change of its shape which in combination with the stiff boundary conditions in that area causes severe non-linearities. This rubber volume was for computational purpose replaced by an extension of the bead wire, as shown in Figure 4.3a. Information about the rubber state in this area are in this way lost, but judged to have a small impact on the rolling resistance since the stress state in this area probably has very small variations over time. The bead wire was in contrast to the other reinforcements in the tyre modeled as a solid and not a rebar. The difference being that the bead wire can carry all stress components and not just one normal stress component. The bead wire is in reality a wire, circled several times around the rim to a total thickness of over 1 cm, kept in place by a stiff rubber matrix. The bead wire will due to this provide some bending and shear stiffness to the tyre but the magnitude of which is hard to estimate. The bead wire was, based on these consideration, modeled as a solid, causing it to be stiffer than in reality. The idealization can be motivated by showing that the deformation close to the rim is small and that most of the stresses in the wire is in the normal direction. 26 Chapter 4. Finite element modeling

The carcass close to the bead area was moved to the nodes at the inner surface of the tyre. Meaning that no inner liner of rubber was transferring the load from the inﬂation pressure down to the load bearing reinforcements in this part of the tyre. The high ratio between its shear and bulk modulus could, as previously noted, cause a small change of force to have a major impact on the deformation. The approximation conducted prescribed more constraints on the rubber and reduced the non-linearity of the mechanical behavior in this area. Also, the purpose of the liner is not load carrying but, as stated in Table 2.1, to prevent air leakage through the material.

4.6 Postproccesing

The rolling resistance was from the FE-simulation result computed in two different ways. The ﬁrst method uses the contact forces from each node, Fi, multiplied with its distance from the wheel center, ↵i, in accodance to the previously derived equations 2.3-2.4

k i=1 Fi↵i Crr = , (4.2) R N P L where k is the number of nodes in contact with the road. The second method uses the reaction forces from the constrained middle node and computes the rolling resistance in accordance with equation 2.4

Rx Crr = , (4.3) Rz where Rx and Rz are the reaction forces from the center node in the x- and z-direction respectively. 27

Chapter 5

Results

The result presented for the material model and for the rolling resistance does not aim towards representing any speciﬁc tyre rubber compound or tyre. The result is presented with purpose to show how the model is intended to be used and that it produces result that is in the correct order of size with a material similar to what have been presented previous in literature [9].

5.1 Virtual material test

A serie of 50 virtual test were performed for each of the material modeling methods described in Chapter 3. The virtual test used unique combination of deformation amplitude and rate- levels. All combinations are presented in the test matrix, appended in Appendix B. The material chosen for the plastic and hyperelastic material elements are shown in Table 5.1 below.

TABLE 5.1: Parameters used in the material model

Viscoelastic Plastic Parameter value Parameter Value

C10 [kPa] 400 E [MPa] 27 C01 (= C10/7) [kPa] 57 Y [kPa] 27 D01 [1/GPa] 44 C1 [MPa] 9.8 1 [-] 80 C2 [kPa] 10 2 [-] 0

The model uses two set of parameters for defining the development of the backstress in the Ziegler hardening law. The second set is applied with purpose to speed up the convergence rate, as further discussed in Chapter 6. The parameters for the generalized maxwell chains vary depending on the method chosen as described in Chapter 3. However, the parameters are in both cases determined by setting the total stiffening of the generalized maxwell chain to 60 %. This value was only selected in order to get a significant frequency dependency for illustrative purpose and do not match any specific experiment made on rubber. The relevant interval of strain rates and frequencies over which the model should be adapted was determined from first performing a 10 km/h rolling simulation with a pure elastic material description for all its modeled rubber parts. The strain rates and amplitudes are not expected to change drastically from the values of such a simulation 28 Chapter 5. Results since the rubber is not the main load carrying structure, as discussed previous in Chapter 2. The result for the representative upper and lower values of the strain amplitude and rates as well as the intervals used for calibration of the model are presented in the Table 5.2 below.

TABLE 5.2: Range of strain amplitude and rates.

Parameter Symbol Value

Max first principal strain rate ✏˙max 10 [%/s] Min first principal strain rate ✏˙min 0.1 [%/s] Max first principal strain ✏ max 20 [%] Min first principal strain ✏ min 0.1 [%] 1 4 Model strain rate interval Tm 10 10 [%/s] Model strain amplitude interval ✏ 0.1 20 [%]

The strain rate interval over which the strain rates were distributed was selected larger than the rates expected in the model. The reason for the wider range is to ensure the predictive behavior of the viscoelastic model, as previous discussed in Chapter 2.

5.1.1 Log distribution method Below are stress strain curves from the virtual material test using the modeling method with logarithmically distributed time constants. Ten elements were used in the generalized maxwell serie, each with a prony constant of 0.06 which results in a total stiffening of 60 % over the whole frequency range. The stress-strain response of all simulations are presented ﬁrst Figure 5.1. Followed by a few selected results for illustrating the frequency stiffening behavior of the hysteresis loops, Figure 5.2.

FIGURE 5.1: Stress strain response from all virtual test using the log distribution method 5.1. Virtual material test 29

FIGURE 5.2: Material response for different load frequencies using the log distribution method

It can be noted from this ﬁgure that only the stiffness is varying with a change of frequency while the width of the loops remains constants, as what was expected. Each virtual material test results in one value for the storage and loss modulus, presented in Figure 5.3 and Figure 5.4 below.

FIGURE 5.3: Storage modulus for material using the log distribution method 30 Chapter 5. Results

FIGURE 5.4: Loss modulus for material using the the log distribution method

The ﬁgure above show that the values for the loss modulus is varying with amplitude but not with frequency which corresponds well to the expected behavior for the modeling technique derived in Chapter 3. The small differences between the curves in the ﬁgure are changing with the sampling rate of the data from the virtual material test where a higher sampling rate results in smaller variation.

5.1.2 Power distribution method The exact same virtual tests were conducted using the power-distribution modeling technique. A hundred maxwell elements were used in the generalized maxwell serie. Each with a prony constant of 6 10 3 which results in a total stiffening of 60 % over the whole · frequency range. The power distribution technique have a similar amplitude dependency as the previous shown log distribution method. However, the result of the loss modulus differs signiﬁcantly between the methods. Figure 5.5 below show an example on how both the storage modulus, inclination of the loop, and the loss modulus, area of the loop, increases with an increasing frequency when modeled with the power distribution method.

FIGURE 5.5: Stress strain response from all virtual test using the power- distribution method 5.1. Virtual material test 31

The result of the storage- and loss modulus using the power distribution method are shown in Figure 5.6 and Figure 5.7 respectively.

FIGURE 5.6: Storage modulus from material using the power-distribution method

FIGURE 5.7: Loss modulus from material using the power-distribution method

The ﬁgures above show that the storage and loss modulus varies both with strain- rate and amplitude which corresponds well to the expected behavior as discussed in Chapter 3. Both methods of selecting the viscoelastic parameters results in two distinct peaks in the loss modulus diagrams, as shown in Figure 5.4 and Figure 5.7. The peak close to 7% strain amplitude is expected and accurately captures the characteristic rubber behavior [18][9]. The peak around 0.2 % strain is non-physical and caused by limitations of the model. The reason is that the hysteresis loops with stress amplitude in magnitudes close to the yield stress forms loops that differs from the charecteristic smooth shape of the hysteresis loop for rubber. The shape for one of these amplitudes are clearly visible in Figure 5.8 below. 32 Chapter 5. Results

FIGURE 5.8: Loss modulus shape for cycles with stress amplitude levels close to the yield stress

The peak in the loss modulus ﬁgures show that the dissipated energy for loops in this low amplitude regime will be relatively bigger than for loops with larger amplitude. However, the energy dissipated in the larger loops will be greater and thus probably make this relative difference negligible, since the total amount of dissipated energy per cycle at these strain levels are very low. However, the sharp transition of behavior for both the loss- and storage modulus for cycles with stress amplitude close to the yield stress along with the constant frequency dependency over the whole amplitude range are unconditional features of the modeling technique. The model is in that way not suitable for modeling the dissipative properties of rubber at very low strain rates, which could be of interest in for example acoustic simulations.

5.2 Rolling resistance

The rolling resistance is simulated by using the log method with the same material parameters as in the virtual test, presented in Table 5.1. Figure 5.9 below shows the rolling resistances for steady state rolling at different velocities. The data from each simulation were computed using the contact force approach, equation 4.2. Example data showing the full result from 10 km/h simulation are appended in Appendix B. 5.2. Rolling resistance 33

FIGURE 5.9: Rolling resistance as function of velocity using the log distribution material modeling method

It is worth pointing out that the rolling resistance can vary with the speed of the tyre even if the loss modulus for the material used in the tyre is constant with respect to a change of strain rate, as for the case with the log distribution method. Take for example the signiﬁcant decrease of the rolling resistant between the result for 1 km/h and 2 km/h. A proposed explanation could be that the amplitude related losses decreases with a higher speed due to the rate- stiffening of the material which affect the magnitude of the deformation and thus also could decrease the level of strain amplitude related losses in the material.

5.2.1 Mesh study It is straight forward to claim that the resolution of the contact footprint would change with a change of mesh and in that way affect the rolling resistance. The effect of this change was however small even when the resolution of the mesh was changed signiﬁ- cally, as shown in Table 5.3.

TABLE 5.3: Impact of mesh size

Mesh variation Average rolling resistance Twice as ﬁne mesh 12 [‰] Reference mesh 12 [‰] A third as ﬁne mesh 11 [‰]

It can from this investigation be concluded that the result was not sensitive to a change of mesh size. The raw data used for computation of the average values shown in the table above are appended in Appendix B.

5.2.2 Contribution from different parts The inﬂuence of the rolling resistance from the different parts was investigated by only using the hysteretic material model in one part of the model at the time. The result of 34 Chapter 5. Results which is presented in Table 5.4 below. The full result from each run is appended in appendix B.

TABLE 5.4: Contribution to the total rolling resistance

Part with hysteretic model active Average rolling resistance Side 6 [‰] Thread 6 [‰] Bead 6 [‰]

The contribution from all parts are approximately of the same size. Showing that the hysteretic behavior in all parts need to be accounted for when modeling tyres and that they are in some way equally important. Although, a better approach of measuring the contributions from the different parts of the tyre would be to investigate which elements undergo largest strain changes during every revolution in the full 3D-model. The amplitude of strain should correspond well with the losses since the amplitude dependent contribution is expected to be much larger than the rate dependent dissipation.

5.2.3 Solution accuracy The result for the rolling resistance from the pressure method was then compared with the rolling resistance calculated using only the reaction forces acting on the middle node as shown in Figure 5.10 below.

FIGURE 5.10: Comparison of the result when computing the rolling resistance by different methods

Both methods should in theory produce the same result but the figure show that the average value for the computed rolling resistance are differing and that the fluctuation noted from the pressure- method are amplified. The accuracy of these simulations are connected to the accuracy of the total energy balance. That is the external work subtracted from all energy stored in the model which in theory should sum up to zero. This was not the case for the simulation method used. The 5.2. Rolling resistance 35 energy balance for simulations with different solver and simulation settings are shown in Figure 5.11 below. The different settings used for the dynamic solver in Abaqus are appended in Appendix B.

FIGURE 5.11: The result of the energy balance of the different runs

The value of the total energy is in the same magnitude, or greater, than the magnitude of the external work needed for moving a truck tyre that distance. Which should be in the magnitude of 20 kJ, assuming a rolling resistance of 1 %, an normal load of 4 104 N · and a total displacement of 50 m. The figure above shows that the error in the energy balance is changing with the value for the friction coefficient and the chosen time incrementation method. The total energy remain zero in all cases when the tyre is locked in its rotational degree of freedom. A proposed error could thus be related to the rotating motion of the tyre and the computations of kinetics related to it. The improved result when changing to a finer time incrementation method is shown in Figure 5.12 below.

FIGURE 5.12: Rolling resistance comparison with shorter

The figure shows that the fluctuation diminishes for both methods of measuring the rolling resistance compared to the previous shown result, in Figure B.2. It also shows that the different results are significantly closer to each other which is in better accordance to what is expected. 36

Chapter 6

Discussion and conclusions

6.1 Discussion

The major limitation of the modeling technique are due to the computational cost for solving the non-linearities related to the material models. Where the main contributor is the plasticity model. Its smooth softening behavior is, as described in Chapter 4, con- trolled by the parameters Ci, i and the yield stress y. The value for these parameters have huge impact on the convergence rate and therefore also to the computational time. The model converges quicker for higher Ci- and lower i values since the plasticity model in those cases moves towards a linear kinematic behavior. However, such values are a bad approximation of the smooth hysteresis loops. A better option for reducing computational cost is to determine for which levels of strain the hysteresis behavior is relevant, since a decrease of the initial yield surface quickly raises the computational cost. The simulation result of the rolling resistance, Figure 5.9, shows that the tire simulation captures the rolling resistance within the correct order of magnitude. It is however almost twice as high as the rolling resistance expected for a truck tire [11]. This is explained by the fact that the same material in the different tyre parts was never fitted against accurate test data. In reality different rubber compounds would be used in the different tyre sections each having properties adapted for their purpose as explained in Table 2.1. The plastic material model is with only one parameter quite limited to maintain the same hardening shape as shown in the report. Nevertheless, more sets of material parameters could be added to the Ziegler hardening law in order to allow a more versatile behavior. Improvements could for example be to better model the initial stiffness transition and the overall smooth characteristic shape of the hysteresis loop as shown in Figure 2.3. The main problem with adding more parameters is that it will be more cumbersome to fit the parameters to test data and the model will also become less user friendly since more advanced fitting procedures will be needed.

6.2 Conclusions

• The proposed material modeling technique can model the storage- and loss modulus over a few decades of strain rates and amplitudes for a reasonable computational cost. The proposed modeling technique with selection of the visco elastic material parameters by using the proposed distribution method results in an FE-model with a minimum set of overlaying element meshes and fewer material parameters to calibrate than similar models found in litrature [8][15].

• The rolling resistance simulation shows that the model can be implemented in FE- simulations and model rubber components with a storage- and loss modulus of the 6.3. Further work 37

correct order of magnitude. It also shows that the modeling technique presented is suitable for modeling rubbers in other applications with cyclical loading other then tires, such as bushings or ﬂexible joints. Although, some work remain on the FE-modeling method before the rolling resistance of a tyre can be determined accurately using only FE-simulations.

• The investigation of the contribution to the rolling resistance from the tyres different parts show that much effort should be put in ﬁtting the model to the different materials, and that the material used in all parts of the model have a signiﬁcant impact on the rolling resistance.

6.3 Further work

It is possible that more questions have been raised than answered during the course of the work with this thesis. The result presented has only scratched the surface of the field and much work remains to be done. Both considered the material- and FE- modeling technique. A valuable contribution to the material modeling technique would be a non-linear kinematic hardening model adapted for low yield stress. Ideally a material model allowing for plastic deformation from the first increment. It would also be valuable with a model that add the temperature dependency to the plastic behavior. Especially since the tyre, and other industrial rubber applications, will heat up during usage. Following that a contribution to the modeling technique would be to implement a thermo-mechanical coupling. In this way the temperature gradient in the tyre would be captured and more accurate simulations could be performed. The FE-modeling technique would also greatly benefit from only solving for the pe- riodic, steady state, solution. The greatest benefit of using a steady state solution would hopefully be a reduction of the computational time needed for simulating a steady state solution. However, the methods for solving for the steady state solution available in Abaqus today do not perform well with the plasticity model used. Probably because the evolution of the plastic behavior is history dependent. It should however not be impos- sible to find a repeated pattern in the plastic response even if it only repeats itself over several revolutions of the tyre. An interesting development to the simulation method that completely would reduce the kinetics of the tyre to only include the pure deformation would be to keep the tyre fixed and introduce a prescribed movement of the road, around the tyre. As if the coor- dinate system was fixed in the tyre center and rotated with the tyre. Such a simulation would greatly contribute to answer the question if the kinetics related to the large dis- placements of the rolling motion itself is the cause of the energy loss or if there are other errors in the model. In summary, it remains much work left to be done in order to fully answer the question of how to accurately and computational effective simulate the rolling resistance of a tyre.

6.4 Best practice for tyre simulation

This section aims at describing some empirical findings that hopefully might ease the work for anyone following with similar work. The element type that performed best for the 3D-model turned out to be fully integrated linear elements. Again, the major issue for the solver for the simulations done in 38 Chapter 6. Discussion and conclusions this work was not the number of elements but the non-linearity of the material model. Changing from fully integrated to reduced elements only had a small trade-off when it came to the computation time. Abaqus tutorial recommends hybrid element formulation when modeling rubber in tyre applications. However, hybrid elements were not chosen since they caused convergence problems in areas subjected to high pressure such as the rubber between the reinforcement layers and the road. These convergence problems are probably caused by the high ratio between the shear and bulk modulus which is enforced by the hybrid element formulation and causes severe non linear for the FE-solver, since volumetric change of the rubber elements yields far greater changes of stress than a change of the elements shape. The simplifications mentioned in Chapter 4 are highly recommended when it comes to reducing the computational cost for the simulations. Another useful trick in order to decrease the calculation time was to change settings regarding the intial step length at every iteration. The complexity of the material model sometimes made the solver to converge slowly since the low yield stress causes almost the whole model to have plastic non-linear behavior. The solver settings were set so that the step length were shorter, or did not grew as fast between steps. These settings lowered the number of iterations needed for each increment which lead to faster convergence. The calculation time was further reduced by introduction of an extra backstress, as shown in Table 5.1. The extra parameter C2 superimpose a linear hardening law on top of the existing hardening behavior. The extra parameter makes the hardening curve of the plastic element to asymptotically smoothen out to a linearly increasing curve. If the parameters are removed the stress in the plasticity elements converges towards some value with zero slope. This will cause instabilities for the solver since a small variation of stress yields a very large variation of strain. The linear curve chosen has a stiffness several orders of magnitude lower than the stiffness of the hyper elastic elements and should due to this not affect the stress strain response in the cyclical behavior. Although, a too high value for the linear slope will have a unphysical ratcheting effect in the cyclical stress- strain plot causing each cycle to have higher stress level than the previous. It is also worth discussing to which accuracy the reinforcement layers should be modeled. A rule of thumb should be that there is a small to insignificant effect of having more than one layer of reinforcement embedded in one layer of elements. At least as long linear element definitions are used since additional layers add no new constraint. They only alters the stiffness of the kinematic couplings already defined in the current element layer. Too much reinforcements through the solids will also over constrain the rubber. Causing volumes of elements entangled in the stiff rebar layers to be unable to bulge and thus increase the stiffness in an unphysical way. 39

Appendix A

Derivations

A.1 Ziegler hardening law (1D)

Below is an derivation for the initial stiffness, H0, for the ziegler hardening law with one backstress C ↵˙ = ( ↵) ✏¯˙plastic ↵✏¯˙pl. (A.1) 0 Assume that the boundary surface is far away, i.e. = 0, which reduces the above equation to C ↵˙ = ( ↵) ✏¯˙plastic. (A.2) 0

Assume also that the all stress components ij are zero except for x. Two condition are met when reaching the yield surface for the ﬁrst time. First, the effective von mises stress, eq, have to equal to the yield stress. Which means that the only stress component is the initial yield limit

x = 0. (A.3) Second, the ﬂow rule for the kinematic hardening yields that

˙ x ↵˙ = 0 (A.4) Now, by inserting Equation A.3 and Equation A.4 into Equation A.2 the following expression is obtained

plastic ˙ x C✏¯˙ = 0 (A.5) where, ✏¯˙plastic can be expressed as

plastic 2 plastic ✏¯˙ = ✏˙x . (A.6) r3 plastic The initial stiffness is now obtained by solving for ✏˙x from Equation A.5 and tot inserting it into the expression for the total strain increment, ✏˙x :

tot ˙ x plastic ✏˙x = + ✏˙ . (A.7) E x From which the result is can be expressed as 40 Appendix A. Derivations

˙ x 1 H0 = tot = , (A.8) ✏˙x 1 + 1 ! E 2 3 C ✓q ◆ which is the well recognized expression for springs in series. Lets now explore what happens when varying C. Firstly if C inf the expression ! boils down to

H0 = E, (A.9) and if C 0 the expression moves towards !

H0 = 0 (A.10) and does in such a case, as expected, reduce the ziegler hardening law to a linear kinematic hardening model. A.2. Stress response in one maxwell element during cyclic pertubation 41

A.2 Stress response in one maxwell element during cyclic pertubation

Below is a derivation of the equation describing the steady state stress-strain response for one maxwell element subjected to a cyclic pertubation

✏ = ✏0 sin(!t), (A.11) with corresponding strain rate,

✏˙ = ✏0! cos(!t), (A.12) where ! is the frequency and ✏0 is the amplitude of the sinusoidally varying load. The amplitude is from now set to ✏0 = 1 in order to reduce the number of terms in the equation. The equation for one maxwell element is obtained by expressing the total strain rate for a dashpot and spring in series subjected to a constant stress

d✏ d✏ spring d✏ dashpot 1 d = + = + , (A.13) dt dt dt E dt ⌘ where ✏ is the strain, E, is the stiffness of the spring and ⌘ is the damping constant. This linear ﬁrst order differential equation can be solved for (t), resulting in

E cos (!t) + ⌘! sin (!t) Et/⌘ (t) = E!⌘ + Ce , (A.14) E2 + (⌘!)2 where C is a integration constant. The time dependent result is obtained by setting a initial condition. For example assuming that the initial conﬁguration is stress free

E2⌘! (0) = 0 C = . (A.15) ! E2 + (⌘!)2 However, the value of the constant C is not changing the shape of the steady state cyclic response and is therefore not affecting the values for the storage or loss modulus. The transient term is therefore removed by selecting C = 0, leading to a ﬁnal version of the equation

E cos (!t) + ⌘! sin (!t) (t) = E!⌘ . (A.16) E2 + (⌘!)2 The result from this equation is shown in Figure A.1 below. 42 Appendix A. Derivations

FIGURE A.1: Stress vs strain for one maxwell element

The parameters used in the ﬁgure above are presented in Table A.1 below.

TABLE A.1: Maxwell element parameters

Parameter ! [rad/s] E [MPa] ⌘ [1/Pa s] · Value 740 0.1 770 43

Appendix B

Details

B.1 Test matrix

Below is the test matrix deﬁning all 50 virtual material tests performed.

TABLE B.1: Design of experiment for virtual material test

Nr. frequency, ! [rad/s] Amplitude, uamp, [%] 1 0.01 0.1 2 0.01 1 3 0.01 10 4 0.01 20 5 0.01 100 6 0.1 0.1 7 0.1 1 8 0.1 10 9 0.1 20 10 0.1 100 11 0.21 0.1 12 0.21 1 13 0.21 10 14 0.21 20 15 0.21 100 16 0.46 0.1 17 0.46 1 18 0.46 10 19 0.46 20 20 0.46 100 21 1 0.1 22 1 1 23 1 10 24 1 20 25 1 100 26 2.15 0.1 27 2.15 1 28 2.15 10 29 2.15 20 30 2.15 100 31 5 0.1 32 5 1 33 5 10 44 Appendix B. Details

TABLE B.1: Design of experiment for virtual material test

Nr. frequency, ! [rad/s] Amplitude, uamp, [%] 34 5 20 35 5 100 36 10 0.1 37 10 1 38 10 10 39 10 20 40 10 100 41 20 0.1 42 20 1 43 20 10 44 20 20 45 20 100 46 50 0.1 47 50 1 48 50 10 49 50 20 50 50 100

B.2 Python code

The appended code below produces the logarithmically distributed time constants, Tm, as described in Chapter 3. #!/usr/bin/env python2 # coding : utf 8 * * #WrittenbyPetterLind,201505 17 import numpy as np import matplotlib . pyplot as plt def ExpSpacing ( lb ,ub, steps , order ) :

#UsesapowerfunctionX^Nasdistribution,whereN=Order, #lb=lowerbound,ub=upperboundand #steps=Nrofstepswithinthemeasurementinterval.

span = (ub lb ) dx = 1.0 / ( steps 1) return [ub (i dx ) order span for i in range(steps)] * ** *

ExpDistribution = ExpSpacing( 3, 1, 10, 0.5) Tm = [ 1 0 ** i for i in ExpDistribution ] B.3. Simulation data 45

B.3 Simulation data

The data used for the 10 km/h average presented in Figure 5.9 is for illustrative purpose appended below along with its corresponding velocity proﬁle. Only the values after the tyre reaches the target speed were used in the average.

FIGURE B.1: Rolling resistance result from the 10 km/h- simulation

FIGURE B.2: Velocity proﬁle used in the 10 km/h- simulation

The data used for the averages in Table 5.3-5.4 are the 20 last data points from each line in Figures B.3-B.4. 46 Appendix B. Details

FIGURE B.3: Rolling resistance with different mesh sizes

FIGURE B.4: Rolling resistance with hysteretic material active in only one part at the time 47

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