Modelling the viscoplastic properties of carbon black ...917398/FULLTEXT01.pdf · ency of ﬁlled elastomers utilizing a modiﬁed Boundary Surface Model. Inter-national Journal of

Modelling the viscoplastic properties ofcarbon black filled rubber

A finite strain material model suitable forFinite Element Analysis

Rickard Österlöf

Doctoral ThesisStockholm, Sweden

2016

Academic thesis with permission by KTH Royal Institute of Technology,Stockholm, to be submitted for public examination for the degree ofTeknologie Doctor in Vehicle and Maritime Engineering, Friday the 29th

of April, 2016 at 10:15, in D2, Lindstedtsvägen 5, 10044 Stockholm, KTHRoyal Institute of Technology, Stockholm, Sweden.

TRITA-AVE 2016:12ISSN 1651-7660ISBN 978-91-7595-902-3

c© Rickard Österlöf, 2016

Postal address: Visiting address: Contact:KTH, AVE Teknikringen 8 [email protected] for ECO2 Vehicle Design StockholmSE-100 44 Stockholm

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Abstract

An increased environmental awareness, legal demands and the largepart of total costs attributable to fuel cost are all incentives for the auto-motive industry to reduce fuel consumption. The optimal driveline toenable this reduction depends on the operational conditions and theavailable infrastructure. Examples of implementable solutions are lowerrevolutions at cruising speed, biofuels and hybrid engines running onelectrified roads. However, these drivelines have different mass, inertiaand dynamic properties. Therefore, special care is needed when devel-oping the driveline isolators, since the demands on noise, vibration andharshness (NVH) are the same regardless of driveline. Without isolat-ors that fulfil the demands of customers, both regarding reliability andNVH, the introduction of these alternative drivelines will be delayed.

Currently, a common method for developing isolators is to build pro-totypes and measure the response in a vehicle. Unfortunately, there willnever be enough resources to measure all possible combinations of driv-elines and vehicle configurations. Instead, computer aided calculationscan be used in order to evaluate a large number of configurations. How-ever, these calculations are only, at best, as good as the material modelsemployed. In the foreseeable future, rubber with reinforcing fillers willbe used in vibration isolators in order to obtain the desired properties ofthese components. This is due to the unique properties of this class ofmaterial such as a high extensibility at break, a high fatigue resistanceand damping. In order to acquire the mechanical characteristics of acomponent from calculations, the material model must be accurate overa sufficiently large range of strain amplitudes, frequencies and temper-atures. However, the stiffness and damping of rubber with reinforcingfillers are highly non-linear functions of these variables, and the avail-able material models in commercial software and in the literature areoften insufficient.

Therefore, a finite strain viscoplastic material model is derived inthe time domain and implemented as a user defined material modelin Abaqus Explicit. The model captures the strain amplitude and fre-quency dependency of the storage and loss modulus for a carbon blackfilled natural rubber. The model is accurate over a wide range of shearstrain amplitudes and frequencies, 0.2-50 % and 0.5-20 Hz, respectively,using only 5 material parameters. In addition, the model correctly cap-tures the response from bimodal excitations. This is an important prop-erty since operational conditions rarely consist of a single sinusoidal

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excitation. For rubber with reinforcing fillers, the sum of the responsefrom two sinusoidal excitations will not equal the response from whenthe two sinusoidal excitations are performed simultaneously. Since thesuperposition principle is invalid for strain amplitudes larger than 0.01-0.1 %, material models in the frequency domain are inherently complic-ated, motivating the implementation in the time domain. Moreover, themodel is capable of capturing the storage and loss modulus over a widerange of temperatures, 0 to 85 ◦C. The implementation in Abaqus Expli-cit enables component characteristics to be evaluated early in the devel-poment phase, with material parameters derived from simple test spe-cimens. Thereby, the number of costly prototypes with long lead timescan be reduced.

Even though many of the examples in this thesis are from heavytruck applications, rubber with reinforcing fillers are widely used inmany engineering fields in bushings, isolators and tires. The improvedaccuracy of simulations of these components can aid engineers developmore optimized solutions faster than with conventional methods.

Keywords: Reinforcing fillers, rubber, finite strain, viscoplastic

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Sammanfattning

En ökad miljömedvetenhet, lagkrav och att bränslekostnaden är en stordel av den totala kostnaden är alla incitament för bilindustrin att minskabränsleförbrukningen. Den optimala drivlinan för att möjliggöra dennaminskning beror på driftförhållanden och den tillgängliga infrastruktu-ren. Exempel på genomförbara lösningar är lägre motorvarvtal i marsch-fart, biobränslen och hybridmotorer som körs på elektrifierade vägar.Däremot har dessa drivlinor olika massa, tröghet och dynamiska egen-skaper. Stor aktsamhet krävs därför vid utvecklingen av drivlineisolato-rer, eftersom kraven på buller och vibrationer (NVH) är desamma oav-sett drivlina. Utan isolatorer som uppfyller kundernas krav, både närdet gäller tillförlitlighet och NVH, finns det en risk att införandet av al-ternativa drivlinor kommer att försenas.

För närvarande är en vanlig metod vid utveckling av isolatorer attbygga prototyper och mäta responsen i ett fordon. Däremot kommer detaldrig att finnas tillräckligt med resurser för att mäta alla möjliga kombi-nationer av drivlinor och fordonskonfigurationer. Istället kan datorstöd-da beräkningar användas för att utvärdera ett stort antal konfiguratio-ner. Dessa beräkningar är, i bästa fall, enbart så bra som de materialmo-deller som används. Inom en överskådlig framtid, kommer gummi medförstärkande fyllmedel användas i vibrationsisolatorer för att erhålla deönskade egenskaperna hos dessa komponenter. Detta beror på de unikaegenskaperna hos dessa material, såsom en hög töjbarhet vid brott, enhög utmattningshållfasthet och dämpning. För att erhålla de mekaniskaegenskaperna hos en komponent från beräkningar måste materialmo-dellen vara korrekt över ett tillräckligt stort område av töjningsamplitu-der, frekvenser och temperaturer. Styvheten och dämpningen av gummimed förstärkande fyllmedel är kraftigt icke-linjära funktioner av des-sa variabler, och de tillgängliga materialmodellerna i kommersiella pro-gramvaror och i litteraturen är ofta otillräckliga.

Därför är en viskoplastisk materialmodell för finita töjningar framta-gen och implementerad i den kommersiella programvaran Abaqus Ex-plicit. Modellen är härledd i tidsdomänen och fångar töjningsamplituds-och frekvensberoendet av styvhets- och förlustmodulen för naturgummimed kimrök som förstärkande fyllmedel. Modellen är tillförlitlig överett brett område av töjningsamplituder och frekvenser, 0.2-50 % respek-tive 0.5-20 Hz, med enbart 5 materialparametrar. Dessutom återskaparmodellen responsen från bimodala excitationer. Detta är en viktig egen-skap, eftersom driftsförhållanden sällan består av en ren sinusformad

v

excitation. För gummi med förstärkande fyllmedel kommer summanav responsen från två sinusexcitationer inte vara lika med responsenfrån när de två sinusexciteringarna utförs samtidigt. Eftersom super-positionsprincipen är ogiltig för töjningsamplituder större än 0.01-0.1%, är materialmodeller i frekvensdomänen komplicerade, vilket moti-verar implementeringen i tidsdomänen. Dessutom kan materialmodel-len återskapa styvhets- och förlustmodulen över ett brett temperaturin-tervall, 0-85 ◦C. Implementeringen i Abaqus möjliggör att komponente-genskaper kan utvärderas tidigt i konstruktionsfasen med materialpara-metrar som härrör från prov på enkla provkroppar. Därmed kan antaletkostsamma prototyper med långa ledtider minskas vid utveckling avnya komponenter.

Även om många av exemplen i denna avhandling är från tunga last-bilar används gummi med förstärkande fyllmedel i stor utsträckning imånga tekniska fält, som bussningar, isolatorer och däck. Den förbätt-rade noggrannheten av simuleringar av dessa komponenter kan hjälpaingenjörer utveckla mer optimerade lösningar snabbare än med konven-tionella metoder.

Nyckelord: Förstärkande fyllmedel, gummi, finita töjningar, viskoplas-tisk

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Acknowledgements

This work is supported by Scania CV AB and has been performed withinthe Centre for ECO2 Vehicle design at KTH Royal Institute of Techno-logy from January 2012 to April 2016.

I would like to thank my supervisors Leif Kari, Henrik Wentzel andJenny Jerrelind for their guidance, help and discussions during this work.

I would also like to direct my gratitude to Prof. Alexander Lion, Dipl.-Ing Nico Diercks and Dipl. -Ing Daniel Wollscheid for their help andsupport during my stay in Munich during the measurements leading toPaper A.

In addition, to all of my fellow PhD students at KTH and colleagues atScania, thank you for making sure that it is fun to go to work, even whenwork itself is tough.

Also, a big thank you to Mumford & Sons and Peter Hollens for beingthe soundtrack during the preparation of this thesis.

Last but not least, to my wife Malin, my family and friends- Thank you for being there for me in times of joy and doubt!

Rickard Österlöf

Stockholm, 29th April 2016

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Dissertation

This thesis consists of two parts: The first part gives a more thoroughbackground to the properties of filled elastomers and an overview ofthe research performed. The second part contains the following researchpapers (A-D):

Paper A

Rickard Österlöf, Henrik Wentzel, Leif Kari, Nico Diercks and DanielWollscheid. Constitutive modelling of the amplitude and frequency depend-ency of filled elastomers utilizing a modified Boundary Surface Model. Inter-national Journal of Solids and Structures 51:3431-3438, 2014.

Paper B

Rickard Österlöf, Henrik Wentzel and Leif Kari. An efficient method forobtaining the hyperelastic properties of filled elastomers in finite strain applic-ations. Polymer Testing 41:44-54, 2015.

Paper C

Rickard Österlöf, Henrik Wentzel and Leif Kari. A finite strain viscoplasticconstitutive model for rubber with reinforcing fillers. Submitted to Interna-tional Journal of Plasticity, January 2016.

Paper D

Rickard Österlöf, Henrik Wentzel and Leif Kari. Temperature dependencyof a viscoplastic material model derived for rubber with reinforcing fillers. Tobe submitted to International Journal of Solids and Structures.

Division of work between authors

Österlöf initiated and performed the measurements, the coding and pro-duced the appended papers. Kari and Wentzel supervised the work,discussed ideas regarding the constitutive modelling and reviewed thewritten papers. Diercks and Wollscheid supervised and assisted dur-ing the measurements leading up to Paper A, provided comments anddiscussions on the results and reviewed the finished paper.

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Publications not included in this thesis

Conference paper: Rickard Österlöf, Leif Kari and Henrik Wentzel. Mod-elling of carbon black filled elastomers using a modified Boundary Surface Modeland fractional derivatives, Proceedings of RubberCon14, Manchester, UnitedKingdom (2014).

Conference paper: Rickard Österlöf, Leif Kari and Henrik Wentzel. Afinite strain viscoplastic constitutive model for filled rubber, Proceedings ofICSV 22, Florens, Italy (2015).

Conference paper: Rickard Österlöf, Leif Kari and Henrik Wentzel. Tem-perature dependency of a viscoplastic constitutive model for rubber with rein-forcing fillers, Proceedings of ECCMR IX, Prague, Czech Republic (2015).

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Contents

I OVERVIEW 1

1 Introduction 31.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Long term goal . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Rubber with reinforcing fillers 72.1 Rubber, entropic elasticity and incompressibility . . . . . . 82.2 Carbon black . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Mullins effect . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Dynamic Mechanical Thermal Analysis 133.1 Temperature dependency . . . . . . . . . . . . . . . . . . . 15

4 Finite strain elasticity 194.1 Hyperelastic material models . . . . . . . . . . . . . . . . . 234.2 Unloading curves for fitting hyperelastic parameters . . . . 25

5 Modelling the Fletcher-Gent effect 275.1 Finite strain viscoplastic model . . . . . . . . . . . . . . . . 29

5.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 305.1.2 Constitutive equations . . . . . . . . . . . . . . . . . 31

5.2 Comparison to measurements . . . . . . . . . . . . . . . . . 365.2.1 Sensitivity analysis of parameters . . . . . . . . . . 365.2.2 Bimodal excitation . . . . . . . . . . . . . . . . . . . 38

5.3 FE-implementation . . . . . . . . . . . . . . . . . . . . . . . 39

xi

CONTENTS

6 Summary and conclusions 436.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2 Possible future work . . . . . . . . . . . . . . . . . . . . . . 44

7 Summary of appended papers 45

Bibliography 48

II APPENDED PAPERS 57

8 ERRATA LIST 59

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Part I

OVERVIEW

1 Introduction

The transport sector is responsible for nearly one fifth of the total greenhouse gas (GHG) emissions in the European Union (EU) [1]. In the USAthe share is over one quarter [2], and in both regions road transport isthe main contributor. Road transport is therefore closely linked to atmo-spheric CO2 concentration, one of the planetary boundaries which hasalready been exceeded [3]. In response, the EU have stated a desire toreduce the GHG emissions from 1990 levels by 80-95 % until 2050 [1],in order to limit the global increase of temperature to 2 ◦C comparedto the pre-industrial level. Reducing emissions from the transport sec-tor will be a crucial part in achieving this goal. One proposal to reducethese emissions is by globally shifting some of the freight from road torail. However, road transports will always be present due in part totheir versatility compared to trains. Therefore, enabling new technolo-gies in order to reduce emissions from road transport is crucial in orderto achieve a sustainable transport sector.

The driveline with the least amount of GHG emissions depends onthe operational conditions and the available infrastructure, and could bebio-fuels, lower revolutions at cruise speed, introducing start-stop func-tionality and hybrid engines running on electrified roadways. These al-ternative drivelines can lead to significant reductions in emissions, whileat the same time reducing the fuel consumption, and thereby fuel cost,for haulage companies. Fuel cost comprises roughly 35 % of the totalcosts for a typical long haulage company in Europe, which is a businesssector with small margins. Therefore, any reduction of fuel consumptionis highly sought for. Consequently, the demand for alternative drive-lines is expected to increase both due to legislative pressure to reduceclimate impact and customer demands to reduce operational costs.

3

CHAPTER 1. INTRODUCTION

A driveline connects to the chassis via vibration isolators, and thiscomponent is crucial in order to obtain the desired NVH characteristicsof a vehicle. When developing a vibration isolator, designing the stiff-ness of the component is a trade off. A too stiff component will result inhigher loads and more vibrations transferred to the surrounding struc-ture, however, the reliability of the isolator will generally be high. A toocompliant component will increase the travel of the driveline, leading tolarger strains in the isolator, and therefore a lower reliability. What ‘toostiff’ and ‘too compliant’ entales depends on the dynamic properties andmass of the driveline, as well as the dynamic loading conditions fromroad induced vibrations. Currently, building prototypes and measuringthe response in a vehicle is a commonly used method when developingthese components in heavy truck applications. Unfortunately, measur-ing all possible combinations of vehicle configurations and drivelineswill in practice be impossible, due to the limited amount of resourcesin development projects. An alternative method would instead be toperform computer aided calculations, which enables many more con-figurations to be studied than what is possible with measurements onprototypes. Nevertheless, these simulations are, at best, only as accur-ate as the material models employed, and herein lies the problem.

Rubber with reinforcing fillers have unique properties such as highextensibility, fatigue resistance and damping which makes it the mater-ial of choice for tires, vibration isolators and bushings [4], now and inthe foreseeable future. However, the stiffness and damping are stronglynon-linear and depends on the strain amplitude, strain rate and temper-ature. In order to obtain the mechanical properties of the component,the material model needs to capture the stiffness and damping over asufficiently large range of strain amplitudes, frequencies and temperat-ures. The models available in literature and in commercial software donot have this capability.

1.1 Objective

The main objective of this thesis is to derive a material model for rub-ber with reinforcing fillers which accurately captures the stiffness anddamping of the material under finite strains. With such a model, the in-fluence from material and geometry on a components properties can beseparated, and thereby several concepts can be evaluated, early in thedesign phase, requiring only experimental data from simple test spe-

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1.2. LONG TERM GOAL

cimens to characterize the material. Furthermore, the need for costlyprototypes is reduced, and a much larger amount of calculations can beperformed during the same time as performing a single measurementon a complete vehicle.

1.2 Long term goal

In operational conditions of vehicles, vibrations occur which are annoy-ing or even potentially harmful for the driver or passengers. Therefore,vibration isolators are used to negate this problem. This is for exampletrue for passenger cars, commercial heavy trucks, trains and in mari-time applications. Even though many of the presented examples in thisthesis are from heavy truck applications, the conducted research is mul-tidisciplinary in the sense that is applicable in many modes of transport,and that an accurate isolator model is a prerequisite for optimizing acomplete vehicle when several engineering disciplines are included inthe optimization.

A novel material model for rubber with reinforcing fillers will notin itself lower any emissions in the transport sector. However, the im-proved accuracy of simulations can help engineers to develop more op-timized solutions and deliver them to the market faster than what wouldbe possible with conventional methods. This can have a true impact onlowering emissions, while at the same time reducing costs for the cus-tomers. The presented research can thereby be said to align with theobjective of the Centre for ECO2 Vehicle Design, to focus on researchaiming towards ‘environmentally friendly and economically competit-ive rail and road vehicles in a sustainable transport system’.

1.3 Outline of thesis

The second chapter of this thesis contains a general description of rub-ber with reinforcing fillers. In Chapter 3, experiments performed duringthis thesis are summarized, and Chapter 4 gives a brief introduction tofinite strain hyperelastic material models, together with a suitable ap-proach to obtain these material parameters. In Chapter 5, a materialmodel for finite strains is derived which captures the viscoplastic prop-erties of rubber with reinforcing fillers. Moreover, the constitutive equa-tions are implemented as a user defined material model, VUMAT, in thecommercial software Abaqus Explicit and results from Finite Element

5

CHAPTER 1. INTRODUCTION

(FE) calculations are presented. Chapter 6 contains a summary of theresults and main conclusions as well as indications of future work. Fi-nally, Chapter 7 consists of the abstracts of the appended papers.

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2Rubber withreinforcing fillers

In order to understand the unique mechanical characteristics of rubber,this section will give a brief introduction to the molecular structure ofthe material and introduce some nomenclature. Rubber is a polymer,which simple means that each molecule contains many repeating sub-units, called monomers. The molecular structure of the monomer, andthe number of repetitions determines the characteristics of the polymer,which can vary immensely.

One important class of polymers that have very weak inter-molecularforces while at the same time exhibiting both viscosity and elasticity arecalled elastomers, an abbreviation of elastic polymer. Figure 2.1 showscis-1,4-polyisoprene, commonly known as latex, the main constituent ofnatural rubber, which is an important elastomer for engineering applic-ations. The number of repeating monomers of natural rubber can be ofthe order 10 000 [5, 6].

CH3

CH2 CH

2

C C

H

n

Figure 2.1: Chemical structure of the polymer chains in natural rubber, where n repres-ents the number of repeating monomers.

The molecules in an elastomer, with natural rubber as an example,are initially not connected and can therefore move relative to each other,

7

CHAPTER 2. RUBBER WITH REINFORCING FILLERS

resulting in a large permanent deformation upon the removal of an ap-plied load [5]. Therefore, it was fairly useless as a material in vibrationisolators, until Charles Goodyear in 1839 discovered the process of vul-canization [5,6]. Through this process, the long elastomer molecules arefused together with strong chemical crosslinks at some points along themolecule chains. A common method to achieve crosslinking is by theaddition of sulphur and heat. Nevertheless, other methods exist thatutilize, for instance, per-oxide based cure systems. For more informa-tion of the process of vulcanization, the interested reader is referred toCoran [6] and Barlow [7], and references therein. Figure 2.2 shows aschematic overview of three vulcanized elastomer molecules. The dens-ity of crosslinks along the elastomer, as well as the number of sulphuratoms in each crosslink greatly influence the mechanical properties ofthe vulcanized compound. Finally, modern rubber compounds containnumerous plasticizers and additives, e.g. accelerators, activators andantidegradants, in order to obtain the desired characteristics.

Figure 2.2: Illustration of crosslinking between three natural rubber molecules, shown inblack, red and blue. The yellow dots represents crosslink sites.

2.1 Rubber, entropic elasticity andincompressibility

Rubber is a collective term for a large amount of materials with similarcharacteristics, in much the same way as many materials fall under thecategory of metals. In this thesis, the word rubber is preserved for vul-canized elastomers, i.e. materials that can withstand large deformationsand return elastically to the original shape when the load is removed,

8

2.1. RUBBER, ENTROPIC ELASTICITY AND INCOMPRESSIBILITY

with only a small permanent set. The physical reason for this elastic con-traction is completely different from the physical reason for the elasticityin hard materials, such as metals and ceramics. In hard materials, theinternal energy changes rapidly when an external load is applied, dueto distance changes between atoms. In general, a substantial volumechange accompanies the elastic deformation of hard materials.

For unfilled rubber, the stress in the material is instead caused by achange in entropy. With the exception of sites with double bonds, theatoms in the backbone of the polymer chains can rotate fairly easy inrelationship to each other. Due to the Brownian motion of the atoms, aninitially straight polymer chain will curl in on itself, and this is the pre-ferred shape of an elastomer. When a curled elastomer is straightenedthe disorder, or entropy, is reduced, and this requires an external force.Upon the removal of the load, the elastomer tends towards increas-ing the disorder by curling in on itself. Since the interatomic distanceshardly change during the deformation, the internal energy does notchange with deformation, and there is almost no change in volume dur-ing a uniaxial tension test [5, 8].

However, during a compression test when the volume of a rubbersample is reduced, the interatomic distances are changed. Consequently,the bulk modulus is of the order 10 GPa, whereas the shear modulusof an ordinary rubber is of the order 1 MPa [5]. This large differencebetween shear and bulk modulus is the reason why rubber can be mod-elled as an incompressible material. As long as a rubber component hasa sufficiently large free surface so that the shape of the rubber is allowedto change, nearly identical reaction forces will be obtained in calcula-tions with either 10 GPa or an infinite bulk stiffness. When comparingthe shear and bulk modulus of steel, 80 GPa and 160 GPa, respectively,they are of the same order of magnitude. Therefore, steel cannot be mod-elled as incompressible during elastic deformations, even though it re-quires substantially less force to change the volume of rubber than tochange the volume of steel.

Moreover, the characteristics of rubber differs greatly depending onthe temperature. For engineering applications it is important that thetemperature is well above the glass transition temperature. Below thistemperature, rubber is hard and in a relatively brittle ‘glassy’ state, hencethe name. Depending on the rubber, this transition occurs at differenttemperatures, and for natural rubber it is approximately -70 ◦C [5].

Finally, due to the large deformations, finite strains need to be takeninto account, and the concept of continuum mechanics is used through-

9


out the thesis. An alternative approach is to model the interactions ofthe molecules directly, using a micromechanical approach. However,the length scale required would result in unreasonable large calculationsfor even the simplest of components.

2.2 Carbon black

Mechanical properties of unfilled rubber, such as abrasion resistance,tensile strength, elastic modulus and ultimate strength at break, are en-hanced with the addition of a reinforcing filler [9, 10]. Examples of re-inforcing fillers are precipitated silica, fumed silica, mineral fillers andcarbon black, where carbon black is the most common [9,10]. It consistsof hard carbon particles, smaller than 0.1 µm, fused together into aggreg-ates in a random cluster. These aggregates can in turn interact and formagglomerates through physical bonds [9, 11]. The breaking and reform-ing of the bonds in the filler network has been proposed as one of themain contributors to energy dissipation in filled rubber, and a well dis-persed filler is crucial for low energetic losses in the material [9, 12, 13].In recent years, the adsorption and desorption of polymer chains to thesurface of carbon black particles have been attributed a large contribu-tion to the strain amplitude dependent stiffness and dissipation [14–16].This view is one inspiration to the proposed modelling technique, aswill be shown in Chapter 5.

In Figure 2.3, the carbon particles in NR3015, a natural rubber with54 phr carbon black, are shown in a sweep electron microscope with 20000 times magnification. The rubber sample was dispensed in liquidnitrogen and subsequently cracked, in order to produce a surface withthe carbon particles clearly visible. Note that the primary particle size issmaller than 100 nm in diameter.

2.3 Mullins effect

Unfilled rubber can be accurately modelled with hyperelastic materialmodels [8, 17–20]. However, the inclusion of a reinforcing filler intro-duces a strong amplitude and temperature dependency on the material,which will be discussed in Chapter 3, as well as a strong strain historydependency. The strain history dependency, commonly known as theMullins effect, means that the material properties are a function of thepreviously maximum experienced strain amplitude [21]. To illustrate

10

2.3. MULLINS EFFECT

Figure 2.3: SEM-image of the carbon black particles in NR3015.

this, Figure 2.4 shows measurements performed during this thesis fora previously unloaded natural rubber with 54 phr carbon black in uni-axial tension. The strain amplitude is initially 10 %, after 5 cycles it isincreased to 20 %, and after 5 more cycles to 40 %. Thereafter the strainamplitude is increased in steps of 40 % until the strain amplitude is 240%. As seen in Figure 2.4 (a), the maximum stress in the first and fifthcycle is equal at 10, 20 and 40 % strain. However, at the higher strainamplitudes, there is a distinct softening of the material. Moreover, a newstationary condition is visible already after a few cycles. The fifth cyclefor all strain amplitudes is shown in Figure 2.4 (b), and by comparingthe difference in stiffness at 160 % strain for a previous maximum strainof 160 % and 240 %, it is obvious that the Mullins effect can have a sig-nificant influence at larger strain amplitudes.

At room temperatures this softening is irreversible, and moreover, ithas been shown that the stress softening is direction dependent. Thismeans that when a sample is sheared with a pulsating load in one direc-tion until a stationary condition, the material still exhibits strain soften-ing when the sample is sheared in the opposite direction [22–25].

For rubber components where the maximum strain amplitudes areexpected to be large, such as bumpstops, the Mullins effect is crucialto take into account. Either, a material model which captures this phe-nomena should be used [26, 27], or material parameters derived frommeasurements pre-cycled with the maximum expected strain amplitudeare needed. The second alternative is not trivial, since the strain amp-litudes vary throughout a component, implying that different materialparameters are needed in different parts of the component.

Fortunately, the Mullins effect is not that pronounced at moderate

11


Time [s]0 2000 4000 6000 8000 10000

Eng

. str

ess

[MPa

]

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

(a) Stress as a function of time

Eng. strain [%]0 50 100 150 200 250

Eng

. str

ess

[MPa

]

0

0.5

1

1.5

2

2.5

3

3.5

(b) Stress as a function of strain, fifth cycle ateach strain level

Figure 2.4: Characteristic example of the Mullins effect. The material response dependson the previous maximum strain amplitude, and the material softens at each new strainamplitude level.

strain amplitudes, as seen in Figure 2.4 (a), and in order to ensure anacceptable reliability vibration isolators need to be designed in such away that the maximum experienced strains are moderate. Therefore,this effect is not included in the derived material model. However, thematerial parameters are derived from test specimens pre-cycled with themaximum strain amplitude in the measurement series, as discussed inPaper A [28].

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3Dynamic MechanicalThermal Analysis

This section presents experimental results performed during this thesiswork. Three blends of natural rubber are studied, two with reinforcingcarbon black fillers. The studied rubber compounds are manufacturedby TBVC, and the composition is given in Table 3.1. The experiments areperformed at strain amplitudes, frequencies and temperatures that areexpected to occur during operational conditions for vibration isolatorsin heavy truck applications.

A common way to characterize rubber is with the aid of DynamicMechanical Thermal Analysis (DMTA). Specialized testing equipment isused and a very precise deformation is applied to a rubber compound,while the resulting reaction force is measured. Normally, a double sheartest specimen is used yet other deformation modes are possible to meas-ure.

Table 3.1: Composition of the rubber compounds in parts per hundred rubber by weight,[phr], and Shore A hardness.

Compound NR Filler Plasticizer Additives Shore A[phr] [phr] [phr] [phr]

NR3233 100 31 2 22 45NR3015 100 54 13 19 50NR3301 100 0 0 12 35

Figure 3.1 shows the measured response from a 50 % shear strainat 0.5 Hz for NR3233 and NR3001, a 31 phr carbon black and unfillednatural rubber, respectively. As seen, the carbon black reinforced rubber

13

CHAPTER 3. DYNAMIC MECHANICAL THERMAL ANALYSIS

compound is substantially stiffer, and exhibits a significant hysteresiscompared to the unfilled rubber, however the shape is not elliptical. Thisis a first indication that this class of materials should not be modelledwith linear viscoelastic material models.

Eng. strain [%]-50 0 50

Eng

. str

ess

[MPa

]

-0.4

-0.2

0

0.2

0.4

(a) NR3233 with 31 phr carbon black

Eng. strain [%]-50 0 50

Eng

. str

ess

[MPa

]-0.4

-0.2

0

0.2

0.4

(b) Unfilled natural rubber, NR3001

Figure 3.1: Measurement result for a 50 % shear strain at 0.5 Hz.

Moreover, the stiffness varies as a function of the strain amplitude,as seen in Figure 3.2. This is the Fletcher-Gent effect [29, 30], whichstates that the stiffness decreases with an increase of strain amplitude,for strain amplitudes above 0.01-0.1 %. A common way to visualize thisstrain amplitude dependency is by plotting the stiffness as a function ofstrain amplitude, as seen in Figure 3.3. Furthermore, note that the strainrate dependency on the stiffness is higher at small strain amplitudesthan at large strain amplitudes, also shown in Figure 3.3. Moreover,the Fletcher-Gent effect is reversible, in comparison to the irreversibleMullins effect. The non-linear frequency dependency and the reversib-ility are two important characteristics of a material model that aims tocapture the behavior of rubber with reinforcing fillers.

In this thesis, two scalar values relating to the stiffness and dissipatedenergy during a steady state sinusoidal excitation are used in order tocompare measurements and simulations. The storage modulus, G′, andloss modulus, G′′, are hereby defined as

G′ =τamp

γamp(3.1)

14

3.1. TEMPERATURE DEPENDENCY

Shear strain [%]-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

Shea

r st

ress

[M

Pa]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Figure 3.2: Experimental data at 0.5 Hz for a double shear experiment on NR3233 at 23◦C.

and

G′′ = G′ tan δeqv, (3.2)

where

tan δeqv = W/(πτampγamp). (3.3)

Furthermore, τamp and γamp are correspondingly the amplitude of theresulting shear stress at a given shear strain amplitude. Finally, W=

∮τdγ

is the total dissipated energy during one cycle. For a linear viscoelasticmaterial with cos δ ≈ 1 it can be shown that the definitions in Eqs. 3.1-3.3 are good approximations of the classic definitions of the storage andloss modulus [9].

3.1 Temperature dependency

The temperature has a strong influence on the storage and loss modulusof rubber with reinforcing fillers. Figure 3.4 show the storage modulusfor NR3015 for amplitudes between 0.2-50 % shear strain at 0.8 Hz attemperatures of -30, 0, 23, 60 and 85 ◦C. Moreover, the hysteresis for the

15


Strain [%]10-1 100 101 102

Stor

age

mod

ulus

[M

Pa]

0.6

0.8

1

1.2

1.4

1.60.5Hz0.8Hz1.4Hz2.3Hz3.9Hz6.5Hz10.7Hz17.9Hz

Figure 3.3: Storage modulus at 23 ◦C for NR3233, for two decades of strain amplitudesand frequencies [Paper D].

largest strain amplitudes are plotted in Figure 3.5. As seen, the differ-ence at elevated temperatures and large strain amplitudes is negligible,yet at small strain amplitudes there is a significant difference. Thesegraphs show that the temperature dependency is important for rubberwith reinforcing fillers. At the very least, material parameters need tobe obtained from experiments performed at the same temperature asexpected in operational conditions. Finally, the storage modulus andloss modulus of NR3015 is plotted in Figure 3.6, for strain amplitudesbetween 0.2-50 % shear strain, 0.8-17.9 Hz and all temperatures.

Figure 3.6 clearly shows that the frequency dependency is larger atthe lower temperatures. Finally, the frequency dependency at elevatedtemperatures is clearly less pronounced than at lower temperatures, yetit is not negligible, as seen in Figure 3.7 and Figure 3.8.

16

3.1. TEMPERATURE DEPENDENCY

Stor

age

mod

ulus

[M

Pa]

0

1

2

3

4

5

6

7

Strain [%]10-1 100 101 102

-30oC0oC23oC60oC85oC

(a) Storage modulus

Los

s m

odul

us [

MPa

]

0

0.1

0.2

0.3

0.4

0.5

Strain [%]10-1 100 101 102

(b) Loss modulus

Figure 3.4: Measurement result for NR3015 at 0.8 Hz for five temperatures.

Shear strain [%]-60 -40 -20 0 20 40 60

Shea

r st

ress

[M

Pa]

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

-30oC0oC23oC60oC85oC

Figure 3.5: Hysteresis curves at 50 % shear strain for NR3015 at five temperatures.

100

Temperature [Celsius]

500

-50102Strain [%]100

5

10

0

Stor

age

mod

ulus

[M

Pa] 0.8Hz

1.4Hz2.3Hz3.9Hz6.5Hz10.7Hz17.9Hz

(a) Storage modulus

100

Temperature [Celsius]

500

-50102Strain [%]100

1

2

3

4

5

0

Los

s m

odul

us [

MPa

]

(b) Loss modulus

Figure 3.6: Results for complete measurement range for NR3015.

17


Stor

age

mod

ulus

[M

Pa]

0.5

0.7

0.9

1.1

1.3

1.5

Strain [%]10-1 100 101 102

0.8Hz1.4Hz2.3Hz3.9Hz6.5Hz10.7Hz17.9Hz

Figure 3.7: Storage modulus at 85 ◦C for NR3015 [Paper D].

Los

s m

odul

us [

MPa

]

0.06

0.08

0.1

0.12

0.14

0.16

Strain [%]10-1 100 101 102

Figure 3.8: Loss modulus at 85 ◦C for NR3015 [Paper D].

18

4Finite strain elasticity

The concept of hyperelastic material models has successfully been usedto model unfilled rubber under quasi-static loading conditions [8,17–20].This section gives a brief overview of the differences between some ofthe most common hyperelastic material models and how material para-meters can be derived.

Consider a deformable rubber like solid, and let F denote the localdeformation gradient in every point X in the reference configuration.For every point, the line segment dX in the reference configuration ismapped to the current configuration via dx = FdX. The right and leftCauchy-Green tensors, C = FTF and b = FFT, respectively, are there-fore measures of the finite deformation in each point, with •T denotingthe transpose of a tensor. It can be shown that C and b have the sameprincipal invariants, namely

I1 = tr(C) = λ21 + λ2

2 + λ23

I2 =12

[tr(C)2 − tr(C2)] = λ21λ2

2 + λ21λ2

3 + λ22λ2

3

I3 = det(C) = λ21λ2

2λ23,

(4.1)

where tr(C) denotes the trace of C and det(C) is the determinant of C,and where λi, i = 1, 2, 3, are the principal stretch ratios [8]. For an in-compressible material, the volume is constant, and therefore I3 = 1.

The three strain invariants have an interesting geometrical interpret-ation, as visualized in Figure 4.1. The first invariant, I1, relates to thesum of the squares of principal stretch ratios, whereas the second in-variant, I2, corresponds to the sum of the squares of the change in area.As stated earlier, I3, represents the change in volume, squared.

19

CHAPTER 4. FINITE STRAIN ELASTICITY

1

23DeformedUndeformed

a0

a0

a0

λ1a

0

λ3a

0λ2a0

λ1λ

2a

02

λ1λ

3a

02

λ2λ

3a

02

a0

2

a0

2

a0

2

I1= λ

1

2 + λ2

2 + λ3

2

I2= λ

1

2λ2

2 + λ1

2λ3

2 + λ2

2λ3

2

I3= λ

1

2λ2

2λ3

2

Figure 4.1: Geometrical interpretation of the three strain invariants.

Depending on the deformation mode, the ratio between I1 and I2will differ for the same principal stretch ratio. For the three deforma-tion modes uniaxial tension (UT), plane shear (PS) and equibiaxial ten-sion (ET), the invariants as a function of the stretch ratio are given inTable 4.1. Figure 4.2 illustrates how the invariants vary depending ondeformation mode. It can be shown that for incompressible materials,all possible deformations will result in a combination of strain invari-ants that are bounded by the lines for uniaxial tension and equibiaxialtension [31]. Moreover, the material response is in general a function ofboth the first and the second strain invariant, as will be shown later.

Table 4.1: First and second strain invariant of the right Cauchy-Green tensor for threedifferent deformations in an incompressible material.

Strain invariant UT ET PS

I1 λ2 + 2λ−1 2λ2 + λ−4 λ2 + λ−2 + 1I2 2λ + λ−2 λ4 + 2λ−2 λ2 + λ−2 + 1I3 1 1 1

Consequently, measurements in all three deformation modes statedabove are important for determining the simplest model capable of cap-turing the response in all expected deformation modes. Uniaxial tensionand pure shear measurements are easy to perform in any uniaxial testingequipment. Unfortunately, equibiaxial tension measurements are not assimple to perform. Preferably, a biaxial testing machine should be usedwith test specimens as presented by Fujikawa et al. [32]. However, since

20

3 4 5 6 73

4

5

6

7

I1

I 2

Uniaxial tensionEquibiaxial tensionPure shear

λ = 2.6

λ = 2.4λ = 1.6

Figure 4.2: I1 and I2 for different deformation modes in an incompressible material.For each deformation mode, the maximum stretch ratio is given, as well as equidistantmarkers 0.2 in stretch ratio apart [Paper B].

this equipment is expensive and uncommon in industry, a bubble infla-tion technique, as presented in Paper B [33], could be used instead. Thedeformation state on the pole of the bubble is equibiaxial and the stretchratio as a function of inflation pressure can be measured by digital imagecorrelation techniques, resulting in images as Figure 4.3-4.5.

(a) Experimental setup (b) Zoomed area of Specklepattern

Figure 4.3: Experimental setup during bubble inflation measurements [Paper B].

21


Figure 4.4: Inflated rubber membrane [Paper B].

By measuring the inflation pressure simultaneously as the strain field,experimental data is collected allowing a comparison with FE-simulations.The measured strain fields at four pressure peaks are shown in Figure4.5.

Figure 4.5: Measured major strain component for four pressure peaks [Paper B].

Finally, in finite strain continuum mechanics many measures of stressexist, with two commonly used examples being the first Piola-Kirchoffand the Cauchy stress tensor, P and σ, respectively. They are the finite

22

4.1. HYPERELASTIC MATERIAL MODELS

strain equivalence of engineering and true stress, i.e. force per nom-inal area and force per deformed area, respectively, and are connectedthrough P = det(F)σF-T.

4.1 Hyperelastic material models

All hyperelastic material models assume the existence of a strain energydensity function, W, defined per unit reference volume. In the follow-ing, W is solely a function of F or some strain tensor. For an isotropic, in-compressible material, the strain energy density function depends onlyon the first two strain invariants, resulting in

σ = 2(

∂W∂I1

+ I1∂W∂I2

)b− 2

∂W∂I2

b2 − pI (4.2)

and

P = 2F([

∂W∂I1

+ I1∂W∂I2

I]− ∂W

∂I2C)− pF-T, (4.3)

where p is a hydrostatic pressure.Many forms of hyperelastic strain energy density functions have been

presented in the literature, and they are commonly divided into phe-nomenological and micromechanical models. The exact composition ofthe rubber compounds from a component supplier will almost alwaysbe unknown, therefore phenomenological material models are used. Thesematerial models have strain energy density functions that are functionsof for instance the strain invariants, such as the Mooney-Rivlin general-ized polynomial [34]

W =n

∑i=0

m

∑j=0

Cij(I1 − 3)i(I2 − 3)j, (4.4)

where Cij are material parameters and C00 = 0. Other models utilize theprincipal stretch ratios, as in the Ogden model [35],

W =N

∑n=1

βn

αn(λαn

1 + λαn2 + λαn

3 − 3), (4.5)

where βn and αn are material parameters. For a good fit to experimentaldata in all deformation modes, at least 6 parameters are needed in theOgden model [17].

23


Some truncations of Eq. 4.4 are worth mentioning in this context,such as the Neo-Hookean model [36],

W = C10(I1 − 3), (4.6)

the Mooney model [37],

W = C10(I1 − 3) + C01(I2 − 3), (4.7)

the Yeoh model [20],

W = C10(I1 − 3) + C20(I1 − 3)2 + C30(I1 − 3)3 (4.8)

and the Biderman model [38],

W = C10(I1 − 3) + C20(I1 − 3)2 + C30(I1 − 3)3 + C01(I2 − 3). (4.9)

The main benefit of a model is to make predictions outside of therange where measurements are available. The reason for choosing a ma-terial model with few parameters is that more parameters increases therisk of large errors outside of the deformation range in which they weredetermined. Furthermore, for unfilled rubber at moderate strains (be-low 200 %) the Neo-Hookean and Mooney models are often sufficientlyaccurate [17]. They are often good choices in engineering applications,also for rubber with reinforcing fillers. Nevertheless, they are incapableof capturing the increased stiffness at larger strain amplitudes. In ap-plications where this is important, more sophisticated material modelsare required.

It should be noted that material models that are solely functions ofthe first strain invariant yet derived for capturing the increasing stiffnessof rubber at larger strains, such as the Yeoh model, are unable to predictbiaxial extension when the material parameters are derived from uni-axial tension [17,33]. Furthermore, it can be shown that a dependency onthe second strain invariant is crucial to accurately capture both the axialforce and torsional moment on a rubber cylinder loaded in torsion [39]Therefore, the Biderman model is preferred over the Yeoh model, as dis-cussed in Paper B [33], since the extra material parameter connected tothe second strain invariant enables a better fit to measurements over alldeformation modes.

At moderate strain amplitudes, where the Mullins effect is not aspronounced, the following scheme is a good practice for obtaining hy-perelastic material parameters. The analytical engineering stress in UT,PS and ET are, respectively [8],

24

4.2. UNLOADING CURVES FOR FIT-TING HYPERELASTIC PARAMETERS

PUT =2λ

(λ2 − λ−1)(∂W/∂I1 + λ−1∂W/∂I2), (4.10)

PET =2λ

(λ2 − λ−4)(∂W/∂I1 + λ2∂W/∂I2), (4.11)

and

PPS =2λ

(λ2 − λ−2)(∂W/∂I1 + ∂W/∂I2), (4.12)

where λ is the stretch ratio in the direction of loading.With measurements in the three deformation modes UT, PS and ET,

the parameters are obtained by minimizing the error between the meas-ured engineering stress and the analytical engineering stress, at the meas-ured stretch ratios.

φ =N

∑i=1

[PPS,m(λPS,i)− PPS,a(λPS,i)]2

+N

∑j=1

[PUT,m(λUT,j)− PUT,a(λUT,j)]2

+N

∑k=1

[PET,m(λET,k)− PET,a(λET,k)]2

(4.13)

where the subscripts m, a, PS, UT and ET stand for the measured andanalytical values in pure shear, uniaxial tension and equibiaxial tension,respectively. Furthermore, the measured data should be processed sothat values are obtained at equidistant strain levels, to reduce the bias ofincreasing accuracy around a strain level with many measuring points.

4.2 Unloading curves for fitting hyperelasticparameters

As seen in Figure 2.4, rubber with reinforcing fillers have strong non-linear characteristics, especially at larger strain amplitudes. This is es-pecially true around the turning points, where the stiffness can changeabruptly with a change in loading direction. This sudden change in stiff-ness is one motivation for choosing a viscoplastic modelling approachfor capturing the Fletcher-Gent effect.

25


Furthermore, when the elastic properties of the material are suffi-cient for solving the problem at hand, it is argued in Paper B [33] thatthe unloading curve is more closely linked to the elastic behavior of thematerial compared with the loading curve. The reason for this is thatinitially, after a turning point, inelastic phenomena are present. An ex-ample is given in Figure 4.6, where the elastic behavior of NR3233 sub-jected to shear strain is best represented by the tangential stiffness faraway from the turning points. The measured data points needs to betranslated to remove the offset caused by the hysteresis in the material.

Eng. strain [%]-50 0 50

Eng

. str

ess

[MPa

]

-0.4

-0.2

0

0.2

0.4

Figure 4.6: Elastic behavior of carbon black filled rubber should be studied far away fromthe turning points.

26

5Modelling theFletcher-Gent effect

For driveline isolators in heavy truck applications, the strain amplitudescorresponding to NVH characteristics are in the region of 0.1-1 %, com-pared to the strain amplitudes from rough road induced vibrations whichare approximately 50-100 %. Moreover, 0.2-20 Hz is an important rangeof frequencies for both road induced vibrations and NVH related prob-lems. As shown in Chapter 3, the stiffness, dissipation and frequency de-pendency of the material varies significantly between these strain levels.The characteristics are also strongly influenced by temperature. An ac-curate material model for the Fletcher-Gent effect in these ranges ofstrain amplitudes and frequencies are crucial to enable reliable FE sim-ulations.

(a) Overlay method utilizing plasticity (b) Non-linear viscoelastic models

Figure 5.1: Two common methods to model rubber with reinforcing fillers, all elementscan be non-linear.

27

CHAPTER 5. MODELLING THE FLETCHER-GENT EFFECT

The modelling and understanding of the amplitude dependent Fletcher-Gent effect has been a research area since the 1950s. Figure 5.1 showsrheological models of two common methods found in the literature ofcapturing this effect, namely the overlay method utilizing plasticity [40–50] and non-linear viscoelasticity [4, 51–54]. The dashpot in the overlaymethod could be any strain rate dependent material model, such as frac-tional derivatives and a generalized Maxwell chain. In this model, theamplitude dependency is captured by the frictional element. However,the strain rate dependency will be additive. An example is given inFigure 5.2, where a Berg friction model [55] is placed in parallel with asingle Maxwell element. As depicted, the increase in stiffness is equallylarge regardless of strain amplitude, in stark contrast with the measure-ments in Chapter 3 and similar measurements in literature [4, 53, 54].Nevertheless, the overlay method has a simple implementation in fi-nite strain three dimensional FE-simulations. Three overlayed modelsutilizing the built-in hyperelastic, viscoelastic and elastoplastic materialmodels in commercial FE-software can be used in an analogous way toFigure 5.1. Therefore, for applications where the operational conditionsare limited to a narrow range of frequencies and strain amplitudes, theoverlay method can produce accurate results. Strain rate independentmodels are successful at capturing the amplitude dependency of rubberwith reinforcing fillers, however they are obviously incapable of repro-ducing the strain rate dependency of the material [56–58].

Strain [%]-50 0 50

Stre

ss [

MPa

]

-1

-0.5

0

0.5

11 Hz100 Hz

(a) Small to moderate strain amplitudes

Strain [%]10-1 100 101 102

Stor

age

mod

ulus

[M

Pa]

1

1.5

2

2.5

3

3.5

4

4.5

(b) Storage modulus

Figure 5.2: Simulations at 1 and 100 Hz, for a Berg friction element placed in parallelwith a Maxwell element [Paper D].

Non-linear viscoelastic material models captures the Fletcher-Genteffect through process dependent viscosities and intrinsic process de-

28

5.1. FINITE STRAIN VISCOPLASTIC MODEL

pendent time scales [4, 53, 54]. Nevertheless, it has proven difficult tocapture both the storage and loss modulus with the same set of materialparameters [53, 54].

5.1 Finite strain viscoplastic model

G2

GN

cN

c2

Hp0

Syield

G∞

q

τ, γ

Figure 5.3: A rheological model for capturing the storage and loss modulus of filledrubber [Paper C].

In this thesis, the Fletcher-Gent effect is modelled by placing a plasticelement in series with a viscoelastic network, as illustrated in Figure5.3. At low strain amplitudes, the plastic element is stiff and the re-sponse is mainly influenced by the viscoelastic network. At large strainamplitudes, the plastic element is more compliant and reduces the in-fluence of the viscoelastic network. The derived model is phenomeno-logical, nevertheless the physical inspiration for this model is given bythe explanation of the Fletcher-Gent effect offered by Donnet and Custa-dero [15]. Therein, the rubber molecules adsorb to the surface of the car-bon black particles through physical bonds. Upon stretching, some ofthese bonds break, which at a macroscopic level results in a softening ofthe material. When the strain amplitude is reduced, the polymer chainsadsorb once again to the carbon black, explaining why the Fletcher-Genteffect is reversible. In the presented material model, the viscoelastic net-work corresponds to the polymer chains and the plastic element cor-responds to the dissipation of energy associated with adsorption anddesorption of polymer chains on the reinforcing fillers.

29


A one-dimensional implementation is presented in Paper A [28], andthe generalization to three dimensions is given in Paper C [59]. Thekinematics and constitutive equations of the later model is presentedbelow.

5.1.1 Kinematics

reference

intermediateequilibrium

current

F=RUeUvUp

Up

Uv

RUe

RUev

Figure 5.4: Kinematics of the material model [Paper D].

Following Lee [60], the multiplicative split of the deformation gradi-ent is utilized,

F = RUeUvUp, (5.1)

where R is the rotation tensor, Ue is the right elastic stretch tensor, Uvis the right viscous stretch tensor and Up is the right plastic stretchtensor. These deformation tensors act on the configurations given in Fig-ure 5.4, and as is intrinsically implied, all rotations are included in theelastic deformation gradient, since the plastic and viscous rotations areassumed to be small compared to the elastic rotations. Furthermore, thisassumption simplifies the kinematics and constitutive equations, sinceotherwise the rotational part of the elastic, viscous and plastic deforma-tion gradients would be indeterminate by the multiplicative split in Eq.5.1. For a deeper discussion on the non-uniqueness of the multiplicativesplit, the interested reader is refereed to Xiao et al. [61].

Moreover,

Uev = UkeUk

v k = 1, 2, ..., N, no summation, (5.2)

30


meaning that the viscoelastic stretch tensor Uev acts on N internal sets,corresponding to the N Maxwell elements in Figure 5.3. Furthermore,the first Maxwell element has an infinite relaxation time, meaning thatfor k = 1 Uk

v = I, where I is a second order unity tensor, and the firstMaxwell element thereby corresponds to G∞ in Figure 5.3.

The spatial velocity gradient, l = FF-1, where (•) symbolizes the ma-terial time derivative, is

l = le + lv + lp, (5.3)

where lv = FelvF-1e and lp = FevlpF-1

ev, with le = FeF-1e , lv = FvF-1

v andlp = FpF-1

p .Moreover, the rate of deformation tensor is

d = (l + lT)/2 = de + dv + dp, (5.4)

where de = (le + lTe )/2, dv = (lv + l

Tv)/2 and dp = (lp + l

Tp)/2.

Finally, the rotated rate of deformation tensor is defined as

DR = RTdR = De|R + Dv|R + Dp|R, (5.5)

where De|R = RTdeR = sym(UeU-1e ), Dv|R = sym(UelvU-1

e ) and Dp|R =sym(UevlpU-1

ev), with sym(•) denoting the symmetric part of a tensor •.In this thesis, the constitutive equations are formulated as functions ofthese rates.

5.1.2 Constitutive equationsIn Paper A [28], a smooth plastic element is placed in series with a vis-coelastic network, and the storage and loss modulus is accurately cap-tured over two decades of strain amplitudes and frequencies. In PaperC [59], this idea is generalized to three dimensional finite strains, withthe assumption that the true stress for an incompressible material inthe current configuration is uniquely defined by the elastic deformationmeasured from the equilibrium configuration. Therefore, constitutiveequations for the Maxwell elements are derived for implementation ina co-rotational explicit scheme, generalized to finite strains, and imple-mented in an Euler forward technique. The Kirchoff stress in the currentconfiguration for the kth Maxwell element is [8]

τ k = pkI + 2(

∂W∂I1

+ I1∂W∂I2

)kbk

e − 2∂W∂I2

(bke)2. (5.6)

31


Since the strains are expected to be moderate, a strain energy densityfunction of Neo-Hookean type is expected to be sufficient [17]. There-fore, ∂W/∂Ik

1 = µk and ∂W/∂Ik2 = 0, where µk is a material parameter for

the kth Maxwell element. In a co-rotational framework, the co-rotateddeviatoric Kirchoff stress in the kth Maxwell element is thereby

τ kR = RTτ kR = 2µkCk

e, (5.7)

where • denotes the deviatoric part of tensor •, and Cke = Uk

eUke. Only

the deviatoric stress is of interest, since the material is incompressible.The objective, associated stress rate for every Maxwell element is [8]

τkR = 2µkC

ke = 2µk2FT

e deFe

= 4µkUkeDk

e|RUke.

(5.8)

Eq. (5.7) together with the fact that for an incompressible material,

det(Uke ) = 1, (5.9)

makes it possible to determine Uke from τ k

R.Finally, the total co-rotated, deviatoric Kirchoff stress in the current

configuration is

τR =N

∑k=1

τ kR, (5.10)

where N is the total number of Maxwell elements in the model. Inthe beginning of each time increment, for deformation driven finite ele-ment applications, the total deformation gradient F and therefore DR areknown, including all state variables from the end of the previous incre-ment. However, in Eq. (5.8), Dk

e|R is utilized. Therefore, viscous and

plastic flow rules are required in order to determine Dkv|R and Dp|R,

respectively, and obtain Dke|R through Eq. (5.5) as

Dke|R = DR − D

kv|R − Dp|R. (5.11)

Viscous flow rule

A viscous flow rule is proposed based on the following reasoning. Thematerial is incompressible, therefore the flow rule is based on the de-viatoric stresses. Furthermore, a dissipative behavior is assured with

32


the following form, D = S/η, where D is a deformation rate, S is a workconjugate stress measure to D, and η is a positive scalar, where η is a ma-terial parameter. The stress measure which is work conjugate to Dv|R isτR [8], as discussed in Paper C [59]. Therefore, the following viscousflow rule is proposed,

Dkv|R = 0, k = 1 (5.12)

and

Dkv|R =

τ kR

ηk , k = 2, ..., N. (5.13)

Plastic flow rule

The plastic flow rule is an extension of the boundary surface modelpresented by Dafalias and Popov [62] to finite strains. The elastic regionis shrunk to a vanishing size, i.e. plastic strains occur in every increment,and a bounding surface is defined in stress space such that

Φ(τ ∗,β, σy) = 0, (5.14)

where τ ∗ is the projection of τR on the surface Φ along the direction m,β is a back stress and σy is the radius of the bounding surface in stressspace, as shown in Figure 5.5. The bounding surface manifests purelykinematic hardening.

As illustrated in Figure 5.5, τ ∗ = τR + δm, where δ is a scalar valuerepresenting the distance from the current stress state to the boundingsurface. The unit vector m is given by m = τR/‖τR‖, where ‖•‖ implies

the Frobenius norm, ‖A‖ =√

tr(ATA). The distance to the boundarysurface at each increment is calculated through

Φ = ‖τ ∗ − β‖ − σy = 0. (5.15)

An associative flow rule is postulated for the back stress and plasticstrains as

β = γH0∂Φ∂τ ∗

(5.16)

and

Dp|R = γ∂Φ∂τ ∗

, (5.17)

33


τ1

τ2

β .

.τ .m

R τ

n

δ

2σy

*

Figure 5.5: Illustration of the stress states for a bounding surface model with a vanishingelastic region [Paper C].

where H0 is a scalar material parameter and γ is the plastic multiplier.To simplify notation, ∂Φ/∂τ ∗ = n, and the plastic hardening is thereforegiven as

τR : n = Hγ, (5.18)

with

H = H0 + hδ

δin − δ, (5.19)

where H is the plastic modulus, and h is a parameter that controls theshape of the hardening. This definition differs from the one given inPaper C [59], where h = H0. The additional parameter h is needed inorder to fit the model to measurements over a larger temperature range.

Since δ is the distance from the current stress state to the boundingsurface along direction m, it follows that δin is an internal variable whichremembers the distance to the bounding surface at the last turning point.A new turning point is defined whenever δ > δin and when m : n < 0,in which case δin takes the value of δ [62].

An elastic predictor, plastic corrector, scheme is implemented [63,64],wherein the plastic strains are zero in the predictor step and, from Eqs.

34


(5.7-5.11),

τpredictor =

N

∑k=1

4µk(

Uke(DR − Dv|R)Uk

e

). (5.20)

The elastic predictor is used to calculate m, δ and n in each time step.Thereafter, the plastic multiplier is given from

Hγ = n : τR

=N

∑k=1

n : 4µk(

Uke(DR − D

kv|R)Uk

e − γ(UkenUk

e))

,(5.21)

as

γ =n : τpredictor

(H + n : ∑Nk=1 4µk(Uk

enUke))

. (5.22)

With the plastic multiplier given from Eq. (5.21), Dp|R is calculatedthrough Eq. (5.17). Note that the same plastic strains act on all vis-

coelastic elements. Finally, with Dkv|R given from Eq. (5.12-5.13), D

ke|R is

given from Eq. (5.11). The rate of stress for every element and the rate ofthe backstress is calculated, using Eq. (5.8) and Eq. (5.16), respectively.Finally, at the end of each increment, the stress in each Maxwell elementis updated, together with the backstress.

To summarize, the material model has 3 + 2N parameters; σy, theradius of the bounding surface; H0, the plastic modulus for the backstress; h, a shape parameter for the plastic modulus; and the stiffnessand damping coefficient of each Maxwell element, µk and ηk, respect-ively. In order to reduce the number of material parameters, µk = µ,for k = 2,...,N and µ1 = µ∞. This is a simplification motivated by therelatively linear increase in stiffness for increasing frequencies on a log-arithmic frequency scale [28]. Moreover, the characteristic frequency ofthe kth Maxwell element is f k = µk/ηk, and following the same reas-oning as above, the characteristic frequencies can be determined a pri-ori, as evenly spaced on a logarithmic frequency scale. Since the fre-quency range of interest is 0.2-20 Hz, the characteristic frequencies, f k,are chosen as 0.033, 0.1, 0.33 1, 3.3, 10, 33, 100 and 330 Hz. Hence, thematerial model has 10 Maxwell elements and a total of 5 material para-meters.

35


5.2 Comparison to measurements

The commercial optimization software HEEDS MDO [65] is used to ob-tain the 5 parameters. This is done by minimizing the error between themeasured and simulated storage and loss modulus, calculated accord-ing to the definitions in Eq. 3.1-3.3. Figure 5.6 shows the relative er-ror between measured and calculated moduli for the complete measure-ment range. The graphs show the mean error over all frequencies at eachstrain amplitude, 0, 23, 60 and 85 ◦C. The error bars indicate the standarddeviation of the error over all frequencies. As seen, the presented ma-terial model is capable of capturing both the storage and loss modulusfor temperatures between 0 and 85 ◦C. At -30 ◦C, the presented mater-ial model is not capable accurately capturing the energy dissipation inthe material. Nevertheless, in heavy truck applications the self heatingof the material combined with the heat radiation from the driveline willcause the isolators operational temperature to be well above room tem-perature, even when the ambient temperature surrounding the vehicleis at -30 ◦C.

Strain amplitude0.2% 0.5% 1% 5% 11% 53%

G' m

eas/G

' sim

[%]

-6

-4

-2

0

2

4

6

(a) Storage modulus


G'' m

eas/G

'' sim

[%]

-40

-30

-20

-10

0

10

20

30

400oC23oC60oC85oC

(b) Loss modulus

Figure 5.6: Relative errors for measured compared to simulated moduli 0, 23, 60 and 85◦C [Paper D].

5.2.1 Sensitivity analysis of parameters

Figure 5.7 shows the results for the 5 parameters as a function of tem-perature, normalized at 85 ◦C. As shown, the parameters required for agood fit at each temperature vary in a non-linear fashion.

36

5.2. COMPARISON TO MEASUREMENTS

Temperature [C]0 20 40 60 80

Rel

ativ

e va

lue

0

1

2

3

4

5

6

7 µ∞

µ

σy

H0

h

Figure 5.7: Effect of temperature on the five material parameters in the model [Paper D].

In order to study the importance of acquiring parameters at the ex-act operational conditions, a new set of parameters at 23 and 60 ◦C areacquired by assuming a linear interpolation between 0 to 60 and 23 to 85◦C, respectively. Table 5.1 shows the optimal parameters at 0, 23, 60 and85 ◦C, as well as the interpolated parameters at 23 and 60 ◦C.

Table 5.1: Optimal parameters at 0, 23, 60 and 85 ◦C, as well as interpolated parameters at23 and 60 ◦C.

Temperature µ∞ µ σy H0 h[◦C] [MPa] [MPa] [MPa] [MPa] [MPa]

0 1.25 0.22 0.14 1.67 4.823 0.96 0.12 0.063 1.76 4.560 0.6 0.059 0.047 1.72 3.385 0.58 0.031 0.047 1.9 2.223intp 1.0 0.16 0.10 1.69 4.260intp 0.73 0.066 0.054 1.84 3.1

The calculated storage and loss modulus at the intermediate pointsare compared with the previously calculated moduli, and the relativeerror is summarized in Figure 5.8. The results show that the differencebetween the storage and loss modulus for the interpolated and original

37



G' in

trpl

/G' or

ig. [%

]

0

10

20

30

40

50

60

23oC60oC

(a) Storage modulus


G'' in

trpl

./G'' or

ig. [%

]

0

10

20

30

40

50

60

(b) Loss modulus

Figure 5.8: Relative errors for interpolated compared to original moduli [Paper D].

set of parameters is smaller at elevated temperatures. Currently, the ma-terial parameters at elevated temperatures cannot be extrapolated frommeasurements at room temperature. At the same time, there is a strongnon-linear temperature dependency of the material as seen in Chapter3. Therefore, for operational conditions at 85 ◦C, material parametersshould be obtained from measurements at that temperature. In the end,it is up to the design engineer to determine the level of accuracy requiredof the material model, depending on the application and operationalconditions. In some cases, for instance early in the development process,material parameters obtained at room temperature could be sufficient inorder to compare different geometries. This could be accurate enoughin order to choose the concept with highest potential.

5.2.2 Bimodal excitation

Since the superposition principle is invalid for rubber with reinforcingfillers [66, 67], a demanding test for a material model is to capture theresponse from bimodal excitations. The success of models utilizing plas-ticity to capture this kind of excitations is an additional motivation forutilizing plasticity based material models for this class of materials [58].In Figure 5.9, the measured and calculated response from three bimodalexcitations on NR3233 are shown. Most importantly, the material para-meters used are derived by minimizing the error between calculated andmeasured storage and loss modulus, i.e. at single sinusoidal excitations.The bimodal measurement data is not included when deriving the para-meters, yet the model accurately captures the response.

38

5.3. FE-IMPLEMENTATION

Strain [%]-30 -20 -10 0 10 20 30

Stre

ss [

MPa

]

-0.2

-0.1

0

0.1

0.2

Stre

ss [

MPa

]

-0.2

-0.1

0

0.1

0.2

Stre

ss [

MPa

]

-0.2

-0.1

0

0.1

0.2

Meas.Sim.

Figure 5.9: Measured and calculated response from bimodal excitations. Top: 2 %/2 Hzshear strain overlayed on 20 %/1 Hz. Middle: 2 %/5 Hz shear strain overlayed on 20%/1 Hz. Bottom: 2 %/ 10Hz shear strain overlayed on 20 %/1 Hz.

5.3 FE-implementation

A vibration isolator in a heavy truck application is analyzed at 23 and85 ◦C, in shear as well as tension and compression, with the two sets ofderived material parameters. The material model is implemented as auser defined material in Abaqus Explicit, and the FE-model is shown inFigure 5.10. As seen, the isolator consists of two rubber pads separatedby a metal plate, where the rubber pads are modelled with 840 C3D8elements and the metal plate is modelled as a rigid body.

In an explicit FE-solver, the material needs to be slightly compress-ible, otherwise the wave speed would be infinitely high and the smallest

39


(a) FE-mesh (b) Cross section of isolator

Figure 5.10: FE-model of a vibration isolator, two rubber pads separated by a metal plate.

stable time increment tends towards zero. Therefore, a simple materialmodel is incorporated for the hydrostatic stress,

p = κ log(J), (5.23)

where κ is the bulk modulus, and J is the volume ratio between thecurrent and reference configuration, i.e. the determinant of the total de-formation gradient. Herein, the bulk modulus is set to 1000 MPa, whichis in the right order of magnitude for this class of materials. Moreover,in this work, there is no temperature dependency on the bulk modulus.Since the free surface area of the isolator is fairly large compared to thebound surface, and the bulk modulus is several orders of magnitudelarger than the shear modulus, this simplification is deemed reasonable.

Simulations are performed at 1 Hz with an amplitude of 1 and 20% strain in both shear and tension/compression, with the two sets ofparameters. The resulting hysteresis are shown in Figure 5.11. The sim-ulations clearly show the capability of the model to capture variationsin dissipated energy and stiffness as a function of temperature.

40

5.3. FE-IMPLEMENTATION

-1 -0.5 0 0.5 1

Rea

ctio

n fo

rce

[kN

]

-2

-1

0

1

2Tension/Compression

23oC

85oC

Engineering strain [%]-20 -10 0 10 20

Rea

ctio

n fo

rce

[kN

]

-30

-20

-10

0

10

-1 -0.5 0 0.5 1-0.2

-0.1

0

0.1

0.2Shear

Engineering strain [%]-20 -10 0 10 20

-2

-1

0

1

2

Figure 5.11: Resulting hysteresis from Abaqus simulations at 23 and 85 ◦C, in shear andtension/compression.

41

6Summary andconclusions

The main contributions of this work are:

• A finite strain, three dimensional material model is developed thatcaptures the frequency and strain amplitude dependency over awide range of strain amplitudes and frequencies for rubber withreinforcing fillers. Moreover, it is applicable in a large range oftemperatures.- Paper C [59] and Paper D

• An implementation of the constitutive equations as a user definedmaterial model in Abaqus Explicit.- Paper C [59]

• The response of the material model from bimodal excitations is in-vestigated and compared to measurements. Measurement resultsare accurately reproduced even though the bimodal data is not in-cluded when obtaining the material parameters for the model.- Paper A [28] and Paper C [59]

• The temperature dependency of a carbon black filled natural rub-ber is presented over a large range of strain amplitudes and fre-quencies.- Paper D

• The importance of including a dependency on the second straininvariant in hyperelastic material models for larger strain amp-litudes is discussed. Moreover, an experimental method is presen-ted for obtaining equibiaxial measurement data through the bubble

43

CHAPTER 6. SUMMARY AND CONCLUSIONS

inflation technique and digital image correlation.- Paper B [33]

6.1 Conclusions

To place a viscoelastic network in series with a plastic element enablesthe storage and loss moduli to be captured over a large range of strainamplitudes and frequencies, using only a few material parameters. Themodelling technique is valid over a large range of temperatures, includ-ing the temperature range of importance for isolator modelling in heavytruck applications. Especially the stiffness is accurately captured, whichis of great importance since the stiffness of the material determines theresulting strains from an applied load. These strains in turn are utilizedwhen estimating the reliability of a component. The separation of mater-ial and geometrical influences on the properties of a component has thepotential to save resources in product development, since accurate sim-ulations of component properties are possible early in the design phase,with material properties obtained from simple test specimens.

6.2 Possible future work

The temperature has a strong non-linear dependency on the material re-sponse, which results in the obtained material parameters having a lar-ger variation at lower temperatures than at elevated temperatures. Dueto the accurate prediction of the moduli over a large range of strain amp-litudes, frequencies and temperatures, future work could be focused to-wards including a thermo-mechanical coupling to the material model.

In many engineering problems the strains are of necessity small tomoderate in order to ensure a high reliability of the component, and forthese situations the presented model is sufficiently accurate. Neverthe-less, the presented material model does not capture the strain historydependency, commonly known as the Mullins effect. For componentswhere the operational strains are high, the addition of this modellingcapability is important.

44

7Summary ofappended papers

Paper A - Constitutive modelling of the amplitude andfrequency dependency of filled elastomers utilizing amodified Boundary Surface Model

Rickard Österlöf, Henrik Wentzel, Leif Kari, Nico Diercks and DanielWollscheid. International Journal of Solids and Structures, 51:3431-3438,2014.

A phenomenological uniaxial model is derived for implementationin the time domain, which captures the amplitude and frequency de-pendency of filled elastomers. Motivated by the experimental observa-tion that the frequency dependency is stronger for smaller strain amp-litudes than for large ones, a novel material model is presented. It util-izes a split of deformation between a generalized Maxwell chain in serieswith a bounding surface plasticity model with a vanishing elastic region.Many attempts to capture the behaviour of filled elastomers are foundin the literature, which often utilize an additive split between an elasticand a history dependent element, in parallel. Even though some mod-els capture the storage and loss modulus during sinusoidal excitations,they often fail to do so for more complex load histories. Simulations withthe derived model are compared to measurements in simple shear on acompound of carbon black filled natural rubber used in driveline isol-ators in the heavy truck industry. The storage and loss modulus fromsimulations agree very well with measurements, using only 7 mater-ial parameters to capture 2 decades of strain (0.5-50% shear strain) andfrequency (0.2-20 Hz). More importantly, with material parameters ex-tracted from the measured storage and loss modulus, measurements ofa dual sine excitation are well replicated. This enables realistic operat-

45

CHAPTER 7. SUMMARY OF APPENDED PAPERS

ing conditions to be simulated early in the development process, beforean actual prototype is available for testing, since the loads in real lifeoperating conditions frequently are a combination of many harmonics.

Paper B - An efficient method for obtaining thehyperelastic properties of filled elastomers in finite strainapplications

Rickard Österlöf, Henrik Wentzel and Leif Kari. Polymer Testing 41:44-54,2015

An efficient methodology for obtaining hyperelastic material para-meters for filled elastomers is presented utilizing unloading curves inuniaxial tension, pure shear and the inflation of a rubber membrane.Experimental results from biaxial extension are crucial when fitting hy-perelastic material parameters and the bubble inflation technique is anexcellent method of obtaining this data when specialized testing equip-ment is unavailable. Moreover, filled elastomers have a considerablehysteresis, and the hysteresis grows with increasing strain amplitudes.Therefore, the loading curve is in general comprised of both elastic andinelastic contributions, even at very low strain rates. Consequently, itis deemed more accurate to use experimental data from the unload-ing curve to describe the elastic behavior of the material. The presen-ted methodology enables obtainment of parameters related with boththe first and second strain invariant, which is required for a good fitbetween measurement and simulation results. Finally, that a chosenmaterial model is accurate in all deformation modes is essential whendesigning components subjected to a complex, multi-axial load history.An accurate material model enables more concepts and geometries of acomponent to be studied before a physical prototype is available.

Paper C - A finite strain viscoplastic constitutive modelfor rubber with reinforcing fillers

Rickard Österlöf, Henrik Wentzel and Leif Kari. International Journal ofPlasticity submitted Jan 2016

A three dimensional viscoplastic constitutive model for finite strainsin a co-rotational explicit scheme is developed and implemented usingfinite elements that captures the amplitude dependency, commonly re-ferred to as the Fletcher-Gent effect, and frequency dependency of rub-

46

ber with reinforcing fillers. The multiplicative split of the deformationgradient is utilized and the plastic flow rule stems from an extensionto finite strains of a boundary surface model with a vanishing elasticregion. The storage and loss modulus for a 50 phr carbon black fillednatural rubber are captured over a large range of strain amplitudes,0.2-50% shear strain, and frequencies, 0.2-20 Hz. In addition, bimodalexcitation is replicated accurately, even though this measurement datais not included when obtaining material parameters. This capability isessential when non-sinusoidal loading conditions are to be replicated.By separating the material and geometrical influence on the propertiesof a component, the design engineers have the capability to evaluatemore concepts early in the design phase. This also reduces the need ofcomplex prototypes for physical testing, thereby saving both time andmoney.

Paper D - Temperature dependency of a viscoplasticmaterial model derived for rubber with reinforcing fillers

Rickard Österlöf, Henrik Wentzel and Leif Kari. To be submitted to Inter-national Journal of Solids and Structures

The temperature dependency of a finite strain time domain visco-plastic material model for filled rubber is studied over the ranges 0.2-50% shear strain, 0.8-17.9 Hz and 0 to 85 degrees Celsius. The storageand loss modulus of a 54 phr carbon black filled natural rubber are ac-curately captured by the model at each temperature, thereby supportingthe notion that a viscoelastic network in series with a plastic element issuitable material model for rubber with reinforcing fillers. The variationof the moduli as a function of temperature is non-linear, with a strongerdependency at lower temperatures. Therefore, a set of measurementsare representable in a broader temperature range at higher comparedto lower temperatures. Highly filled rubber are not thermo-rheologicalsimple materials, and an elevated temperature influences the stiffnessand dissipation differently at low and high strain amplitudes. Therefore,relying merely on low strain amplitude measurements can led to erro-neous conclusions, when strain amplitudes in operational conditions aremoderate to large. The presented modelling technique enables an accur-ate evaluation between compounds, early in the development phase ofnew components in industrial applications.

47

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55

Part II

APPENDED PAPERS

8ERRATA LIST

In Paper A, Table 1 should be

NR Filler Plasticizer Additives Shore A[phr] [phr] [phr] [phr]

100 31 2 22 45

In Paper B, Table 2 should be

NR Filler Plasticizer Additives Shore A[phr] [phr] [phr] [phr]

100 31 2 22 45

In Paper C, Abstract, line 8. 50 phr should be 31 phr.

Modelling the viscoplastic properties of carbon black ...917398/FULLTEXT01.pdf · ency of ﬁlled elastomers utilizing a modiﬁed Boundary Surface Model. Inter-national Journal of

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