Modelling of the Fletcher-Gent Effect and Obtaining Hyperelastic Parameters for ﬁlled Elastomers

Modelling of the Fletcher-Gent effect and obtaining hyperelastic parameters for ﬁlled elastomers

Rickard Österlöf

Licentiate Thesis Stockholm, Sweden 2014 Academic thesis with permission by KTH Royal Institute of Technology, Stockholm, to be submitted for public examination for the degree of Li- centiate in Vehicle and Maritime Engineering, Wednesday the 8th of Oc- tober, 2014 at 09.00, in Munin, Teknikringen 8, KTH - Royal Institute of Technology, Stockholm, Sweden. TRITA-AVE 2014:47 ISSN 1651-7660 ISBN 978-91-7595-273-4 c Rickard Österlöf, 2014

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

The strain amplitude dependency , i.e. the Fletcher-Gent effect and Payne effect, and the strain rate dependency of rubber with reinforcing fillers is modelled using a modified boundary surface model and implemented uniaxially. In this thesis, a split of strain instead of stress is utilized, and the storage and loss modulus are captured over two decades of both strain amplitudes and frequencies. In addition, experimental results from bimodal excitation are replicated well, even though material parameters were obtained solely from harmonic excitation. These results are encouraging since the superposition principle is not valid for filled rubber, and real-life operational conditions in general contain several harmonics. This means that formulating constitutive equations in the frequency domain is a cumbersome task, and therefore the derived model is implemented in the time domain. Filled rubber is used irre- placeable in several engineering solutions, such as tires, bushings, vibrations isolators, seals and tread belts, to name just a few. In certain applications, it is sufficient to model the elastic properties of a component during finite strains. However, Hooke’s law is inadequate for this task. Instead, hyperelastic material models are used. Finally, the thesis presents a methodology for obtaining the required material parameters utilizing experiments in pure shear, uniaxial tension and the inflation of a rubber membrane. It is argued that the unloading curve rather than the loading curve is more suitable for obtaining these parameters, even at very low strain rates.

Keywords: Filled elastomers, natural rubber, carbon black, Fletcher- Gent effect, Payne-effect, ﬁnite strain, hyperelastic

iii Sammanfattning

Töjningsamplitudsberoendet, Fletcher-Gent effekten och Payne effekten, samt det svaga töjningshastighetsberoendet för gummin med för- stärkande fyllmedel är modellerat genom att modifiera en plasticitets- model som tillämpar begränsningsytor. Modellen har implementerats för enaxliga belastningar och använder en uppdelning av töjning istället för spänning. Den fångar styvhets- och förlustmodulen för två dekader av töjningsamplituder och frekvenser. Utöver detta återskapar modellen experimentella mätningar med bimodala exciteringar väl, trots att dessa mätresultat inte användes vid bestämningen av materialparametrarna. Dessa resultat är särskilt uppmuntrande eftersom superpositionsprinci- pen inte är giltig för gummin med förstärkande fyllmedel och att verkli- ga belastningar nästan alltid är en kombination av flera frekvenser. Detta medför en stor svårighet att formulera materialmodeller i frekvenspla- net och den presenterade modellen är därför implementerad i tidsdo- mänen. Kimröksfyllda gummin används oersättligt i ett flertal teknis- ka lösningar, såsom exempelvis i däck, bussningar, isolatorer, tätningar, slitbaneband, för att nämna ett fåtal. I somliga tillämpningar är det till- räckligt att modellera enbart de elastiska egenskaperna hos komponen- ten. Däremot är Hookes lag inte tillämpbar för att karaktärisera dessa material under finita töjningar. Istället används hyperelastiska materialmodeller och denna avhandling beskriver en metod för att bestämma dessa materialparametrar med hjälp av provning i plan töjning, enaxligt drag och uppblåsning av ett gummimembran. Vidare anses avlastnings- snarare än pålastningskurvan lämpligast att använda för att bestämma de elastiska materialparametrarna.

Nyckelord: Förstärkta elastomerer, naturgummi, kimrök, Fletcher-Gent effekt, Payne-effekt, ﬁnita töjningar, hyperelasticitet

iv Acknowledgements

This work is supported by Scania CV AB and has been performed within the Centre for ECO2 Vehicle design at KTH- Royal Institute of Techno- logy from January 2012 to October 2014.

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

I would also like to thank Prof. Alexander Lion, Dipl. -Ing Nico Diercks and Dipl. -Ing Daniel Wollscheid for their help and support during my stay in Munich during the measurements leading to Paper A. In addition, I want to thank all of my fellow PhD students at KTH and colleagues at Scania for making sure that it is fun to go to work, even when work itself is tough.

Also, a big thank you to Mumford & Sons for being the soundtrack during the preparation of this thesis.

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

Rickard Österlöf Stockholm, 3rd September 2014

Dissertation

This thesis consists of two parts: The ﬁrst part gives a more thorough background to the properties of ﬁlled elastomers and an overview of the research performed. The second part contains the following research papers (A-B):

Paper A

Rickard Österlöf, Henrik Wentzel, Leif Kari, Nico Diercks and Daniel Wollscheid. Constitutive modelling of the amplitude and frequency dependency 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 for obtaining the hyperelastic properties of filled elastomers in finite strain applications. Polymer Testing, submitted August 2014

Division of work between authors

Österlöf initiated and performed the measurements, the coding and produced the appended papers. Kari and Wentzel supervised the work, discussed ideas regarding the constitutive modelling and reviewed the written papers. Diercks and Wollscheid supervised and assisted during the measurements, reviewed the paper and provided comments and discussions on the results.

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 Model and fractional derivatives, Proceedings of RubberCon14, Manchester, United Kingdom (2014).

vii

Contents

I OVERVIEW 1

1 Introduction 3 1.1 Background ...... 3 1.2 Objective ...... 4 1.3 Scope ...... 5 1.4 Long term goal ...... 5

2 Polymers 7 2.1 Natural and synthetic rubber ...... 7 2.2 Vulcanization ...... 8 2.3 Physical properties of polymer chains ...... 8 2.4 Temperature dependency of elastomers ...... 11 2.5 Fillers ...... 12

3 Dynamic Mechanical Analysis 17 3.1 Classical deﬁnitions of storage and loss modulus . . . . . 17 3.2 Deﬁnition of storage and loss modulus used in thesis . . . 19

4 Modelling the Fletcher-Gent effect 23 4.1 Modiﬁed Boundary Surface Model ...... 25 4.2 Results from model ...... 25

5 Finite strain elasticity framework 29 5.1 Basics of continuum mechanics ...... 29 5.2 Hyperelastic materials ...... 30

ix CONTENTS

6 Obtaining hyperelastic material parameters 35 6.1 Unloading curves for obtaining material parameters . . . . 36 6.2 Uniaxial tension and pure shear experiments ...... 36 6.3 Bubble inﬂation and digital image correlation ...... 38

7 Summary and conclusions 45 7.1 Conclusions ...... 45 7.2 Future work ...... 46

8 Summary of appended papers 47

Bibliography 49

II APPENDED PAPERS A-B 55

x Part I

OVERVIEW

1Introduction

This chapter provides the background and incentive for the conducted work. The motivation for this thesis is sprung from real needs in industrial applications. Nevertheless, the research presented is general, and independent of application.

1.1 Background

The transportation of goods and people has always been crucial in humanity’s strife of improved resource efﬁciency and increased prosperity. During the end of the 1990s and beginning of the 21st century the media and society at large became aware of the unwanted effects that accom- pany the use of fossil fuels, hazardous materials and our consumptive way of living. Locally, there is the concentration of emissions from sources such as transports in a small region with apparent effects on hu- mans and the environment, as for instance in Beijing, Los Angeles and Mexico City. On a global scale, Rockstrom¨ et al. [1] show that humanity has already transgressed three out of the nine planetary boundaries within which humanity can operate safely, namely: climate change; rate of biodiversity loss and changes to the global nitrogen cycle. Of these three, climate change is closely linked to the emissions from the transport sector. The Swedish government has set the goal that by 2030 the CO2- emissions from the transport sector should be reduced by 80% from 2004 levels, and that it shall be fossil free by 2050 [2]. Even though the major- ity of emissions comes from passenger cars [3], every potential has to be tapped if Sweden is going to be able to reach its ambitious goals. Fur- thermore, the European Union has set goals to reduce greenhouse gas

3 CHAPTER 1. INTRODUCTION emissions from the transport sector with 20% by 2030, and with at least 70% by 2050, compared to 2008 years levels [2]. Enabling new technology in order to reduce contributions from the transport sector is therefore crucial. In order to minimize the emissions from heavy goods transports, alternative drive lines can be used with for instance stop-and-go technology, hybrid drive lines, lower revolutions at cruising speed and alternative fuels. However, the most cost efficient and environmentally friendly solution depends on the operational conditions. Moreover, reducing emissions is beneficial not only for the environment. There is an economical incentive for haulage companies, since a reduction in fuel consumption also lowers costs. However, vibration isolators and rubber bushings have to be designed to cope with the increased variance of dynamic load, since different drive lines have different dynamic be- haviours. This is especially important since the customer demands on noise, vibration and harshness are ever increasing. Vibration isolators and bushings are almost exclusively made out of rubber, due to the unique properties of high extensibility, damping and the introduction of a significant change in impedance in the mechanical system. This makes the material perfect for isolating and attenuating vibrations from the engine and uneven roads. Due to the increased fatigue resistance of natural rubber compared to other elastomers, it is predicted that this is the material of choice in industrial applications in the foreseeable future. Furthermore, in industrial applications, a reinforcing filler, the most common being carbon black, is often used in order to achieve sufficient tear strength, stiffness and to increase fatigue resistance [4, 5]. Unfortu- nately, the mechanical behaviour of filled rubber is complicated, with a strong temperature and strain amplitude dependency and a weak frequency dependency.

1.2 Objective

The strain amplitude dependency, also called the Fletcher-Gent effect, is most pronounced in the range of 0.1-50% strain [5–11]. This coin- cides with the range of strain amplitudes commonly experienced in industrial applications. Therefore, to model this phenomena is of utmost importance in order to enable realistic dynamic simulations of systems suspended with rubber mounts. The primary goal of this project is to

4 1.3. SCOPE develop new material models and dynamic modelling techniques for vibration isolators and rubber bushings made out of carbon black filled natural rubber. These methods should allow the response in mechanical systems to be calculated before prototypes of the isolators are available. This means that the parameters in the material model should be obtained from simple experiments with test specimens that are easy to manufacture early in a design process. For some applications, finite strains need to be taken into account, which necessitates the use of hyperelastic material models that can capture the non-linear elastic properties of polymer chains [12]. However, the methods to determine these material parameters often requires specialized testing equipment for biaxial extension [13], which is rarely found in industry. Therefore, a methodology is developed that utilizes standard tests in uniaxial tension and pure shear, together with the inflation of a rubber membrane.

1.3 Scope

In this thesis, a model is presented that captures the Fletcher-Gent effect for strain amplitudes and frequencies in the range of 0.2-50% shear strain and 0.2-20 Hz, respectively. It is implemented uniaxially with the motivation that many bushings are primarily loaded in one direction. Moreover, even though the independent rheological elements in the model have three-dimensional equivalents, the generalization to three dimensions is not trivial, and is the focus of future research. A condensed introduction to finite strain continuum mechanics is presented. The mathematical framework of hyperelasticity is used to model the finite strain elastic properties of carbon black filled elastomers.

1.4 Long term goal

It is expected that to reduce the environmental impact of commercial vehicles, different drive lines are needed for different operating conditions. In this thesis, tools are developed to aid engineers in designing rubber bushings and isolators, by enabling realistic simulations early in the design process. Furthermore, rubber is expected to be the material of choice in vibration isolation applications in the foreseeable future, in many engineering disciplines. This means that the conducted research falls perfectly in line with the objective of ECO2 to be multi-disciplinary

5 CHAPTER 1. INTRODUCTION and to develop methods that enable the transition to a more sustainable transport system.

6 2Polymers

A polymer is a very long macromolecule, composed of many smaller subunits, or monomers. The properties of the polymer chain depend on the speciﬁc monomer, the branching and the degree of crosslinking. One of the most important properties is the glass transition temperature, Tg, which states when the material starts to transit from a brittle and glassy solid to a rubbery, viscous phase [14]. The term elastomer is used to classify polymers that are used above their Tg, meaning that they have a low shear modulus, in the order of 1 MPa, and allow a re- markably high elastic strain before failure. Rubber is in turn a collective term for vulcanized elastomers. This chapter offers a basic understand- ing of the molecular structure of vulcanized ﬁlled elastomers. It also gives a plausible physical explanation from literature to the strain amplitude dependency in this class of materials, commonly referred to as the Fletcher-Gent effect and Payne effect [7, 10].

2.1 Natural and synthetic rubber

Rubber can broadly be separated into natural and synthetic rubber. The major source of natural rubber is the latex of the Pará rubber tree (Heavea Brasiliensis), with roughly 75% of the annual production originating from Thailand, Malaysia and Indonesia, since the tree only grows in tropical and subtropical climates. Even though the tree originates from South America, it is not cultivated there due to the fungal disease known as the South American leaf blight. In arid and temperate climate zones, there are other potential sources for high quality rubber, such as for instance Guayule and the Russian dandelion, respectively. Unfortunately, the cost is still too high for these alternatives to be viable in industrial

7 CHAPTER 2. POLYMERS applications [15]. Synthetic rubber is, as the name suggests, manufactured elastomers, produced mainly from petroleum products by polymerizing different monomers. These elastomers can be tailored for speciﬁc purposes and are usually superior to natural rubber when subjected to high temperatures, oils and abrasion. However, natural rubber has a fatigue resistance which currently is not possible to reproduce in its synthetic counterpart. For this reason, natural rubber is the material of choice in many engineering applications, in the foreseeable future. Furthermore, it stems from a truly renewable source, which is an advantage over petroleum based synthetic rubber.

2.2 Vulcanization

The long polymer chains in natural and synthetic rubber are initially only entangled. Under strain, the polymer chains can move relative to each other, similar to snakes in a basket. This process was named reptation by the Nobel laureate Pierre-Gilles de Gennes, from the Latin word reptare (to creep), to describe the mobility of long macromolecules [16]. In 1839, Charles Goodyear discovered that if natural rubber was mixed with sulphur and heated, it drastically changed the properties of the material [14]. From being gooey when hot, and brittle when cold, the new material was rubbery over a large range of temperatures. In addition, and most importantly, it retrieved its original shape after an applied strain was removed. The reason for this is that the sulphur creates chemical cross-links between polymer chains which inhibits them to move relative to each other during deformation. Fig. 2.1 shows an illustration of the effect of vulcanization. There are other vulcanization agents than sulphur, such as perox- ides, urethane cross-linkers and metallic oxide. However, vulcanization is a complicated chemical process which is not discussed in further de- tail. The interested reader is referred to, for instance, Barlow [17] and references therein.

2.3 Physical properties of polymer chains

The polymer chain is long and ﬂexible in its rubbery state and in constant Brownian motion. It can take on a wide range of different conformations, or shapes, for a ﬁxed distance between the endpoints. This

8 2.3. PHYSICAL PROPERTIES OF POLYMER CHAINS

Polymer melt Vulcanized rubber

Force Force

Figure 2.1: Left: Individual polymer chains can untangle from the polymer melt via reptation. Right: Vulcanization causes permanent cross-links between chains, resulting in a restoration of the rubber’s initial shape when an applied force is removed. is demonstrated in Fig. 2.2, where the endpoints of a polymer chain are ﬁxed at a distance of r1, and two out of many possible conformations are shown. The cross-link sites in a vulcanized rubber network can be viewed as endpoints for each segment of the polymer chain [18]. When the distance between the endpoints increases, the number of possible conformations is reduced. In the limit, when the distance between the end points equal the length of the chain there is only one possible con- formation. An increased endpoint distance thereby causes an increased order in the system, or correspondingly that the entropy of the rubber chain is reduced. In the statistical theory of rubber elasticity, the distance between the endpoints of the polymer chains in a network are assumed to be ran- domly distributed, often expressed with a Gaussian distribution function [14, 18, 19]. During macroscopic deformation, the polymer chains are stretched and the entropy of the system is thereby reduced. When removing the outer load, the chains strive towards increasing their entropy by recoiling to the unstretched state. By comparison, the elasticity in hard materials, such as metals and ceramics, is caused by changing the shape and size of the atomic lattice. When the load is removed, the structure reverts back to its original shape in order to minimize

9 CHAPTER 2. POLYMERS

Figure 2.2: Two conformations of a polymer chain.

the internal energy. The difference between entropic elasticity and en- ergetic elasticity results in a significantly different thermo-mechanical behaviour. One example is that a metal spring cools during stretching, whereas a rubber band heats up. Another example is that a metal spring under constant tensile loading expands during heating and shrinks during cooling, due to the thermal expansion of the material. A rubber band under the same condition shrinks when heated, and expands when cooled. Like all materials, rubber has a positive thermal expansion coef- ficient, but the stiffness is a function of temperature, and this effect is often larger than the thermal expansion. In conclusion, the mechanical behaviour of vulcanized rubber is governed by physics that are not present in hard solids. Therefore, it should come as no surprise that the material models applicable for elasticity in metals, i.e. Hooke’s law, are not sufficient for describing the elastic response in elastomers.

10 2.4. TEMPERATURE DEPENDENCY OF ELASTOMERS

2.4 Temperature dependency of elastomers

In their rubbery state, elastomers are ﬂexible and can move easily relative to one another. The underlying physical reason for the entropic elasticity is the possibility for elements in the polymer chain to rotate around single bonds in the chain. This rotation is restricted by atoms in the nearby vicinity, either in the same or neighbouring molecules. The possibility that the thermal energy of the element overcomes this re- striction is governed by a Boltzmann factor, exp(−e/kT), where k is the Boltzmann constant, T is the temperature in Kelvin and e is a potential barrier to be overcome. At a sufﬁciently low temperature, the mobility of the chains is severally reduced with a high modulus of elasticity and a low loss factor as a result. This is called the glassy state. In between these regions there is a transition region, where the loss factor of the material exhibits a maximum. For natural rubber the Young’s modulus falls from approximately 1 GPa in the glassy region to 1 MPa in the rubbery region. The transition point is generally 30oC above the glass transition temperature [14]. A schematic drawing of the modulus of elasticity and the loss factor is shown in Fig. 2.3.

Glassy region

Transition point Transition region

Rubbery region Transition point

LossFactor

Shearmodulus Temperature Temperature

Figure 2.3: Schematic temperature dependency of rubberlike materials.

The glass temperature of a speciﬁc polymer depends, among other things, on the ﬂexibility of the backbone of the polymer. For natural rubber it is approximately -70 oC. For neoprene, used in for example o wetsuits, it is -50 C whereas silicone rubber has a Tg of approximately -125 oC. In compounds, the use of plasticizers can increase the distance between polymer chains which in turn lowers the glass transition temperature.

11 CHAPTER 2. POLYMERS

2.5 Fillers

The mechanical properties of vulcanized rubber are in general insufficient in industrial applications. Therefore, a reinforcing filler is often used, the most common of which is carbon black. The word “filler” is an unfortunate choice, since it gives the impression that it is an additive used merely for filling a gap. Instead, they are equally important to obtain the desired mechanical properties, and “elastomer and reinforcing filler should be considered as two inseparable parts of equal merit in the composite” [6]. The smallest component of carbon black is called a primary particle, with sizes varying between roughly 20 - 300 nm [20]. However, in practice these particles form aggregates of several particles, which in turn can interact with other aggregates and form agglomerates. Fig. 2.4 shows an illustration of these different entities.

Figure 2.4: A. Filler primary particle. B. Filler aggregate. C. Filler agglomerate, with physical bonds between aggregates highlighted.

Even though carbon blacks have been used as a reinforcement in rubber for decades, the exact morphological characterization and how these inﬂuence the reinforcement of the compound is not fully understood. One of the reasons for this is that typical aggregates are of the dimension 50-500 nm [20], which generally make them hard to resolve in a micro- scope. Furthermore, since the smallest object in an actual compound is

12 2.5. FILLERS a branched aggregate or weakly bonded agglomerates, with an inhomogeneous dispersion, the actual strain amplification in the rubber matrix is also inhomogeneous. In agglomerates there is even the possibility of rubber being “trapped”, and thereby not subjected to strain, effectively amplifying the strain in the remaining rubber matrix even further. Generally, a smaller primary particle size increases the reinforcement, due to the increased surface area between elastomer and filler. How- ever, the smaller the primary particle, the harder it is for the mixing process to breakdown agglomerates, leading to a larger object size in the compound than expected. It is also possible that the high shear strains during mixing manages to break apart aggregates, leading to a smaller object size than expected. To summarize, even though the carbon black is characterized before mixing, the final object size in the compound is often unknown. The exact relationship of how the size and shape of carbon black in- fluences the reinforcement in a rubber compound is not completely understood. However, there is a wide consensus that the Fletcher-Gent effect is due mainly to the breakdown of filler-filler network [6]. One explanation of this process is offered by Donnet and Custodero [6]. The carbon black particles have a high surface energy which leads to the adsorption of elastomer chains on its surface. Furthermore, due to the short distance between filler particles, it is possible that the same elastomeric chain is adsorbed to two nearby particles. For a low strain amplitude, the chains are stretched and the entropy is reduced, stage 1 ⇒ 2, see Fig. 2.5. At load reversal the macroscopic deformation is therefore largely recoverable, and the dissipated energy is thereby low. When the

Figure 2.5: Filler-particle interaction at low macroscopic strain amplitudes. strain amplitude increases further, stage 2 ⇒ 3, the stored energy in the chains exceeds the adsorption energy, and the chains desorb from the surface. In addition, part of the elastically stored energy in the desorbed

13 CHAPTER 2. POLYMERS chains is mechanically lost, thereby increasing the dissipation of energy. This is schematically shown in Fig. 2.6. Due to the inhomogeneous dispersion and structure of the ﬁller aggregates, and since there is a broad distribution of elastomeric chain lengths between particles in the compound, this occurs progressively for increasing strain amplitudes. This is deemed to be the cause of the smoothness of the Fletcher-Gent effect, on a macroscopic length scale. During large cyclic deformation the

Figure 2.6: At higher macroscopic strain amplitudes, the chains desorb from the surface of the carbon black ﬁllers. compound will not immediately revert back to stage 1, but to a pseudo- equilibrium state, stage 3 ⇔ 3a, as visualized in Fig. 2.7. However, the process is reversible, and the polymer chains will adsorb on the surface once more if the macroscopic strains are low. However, due to the three- dimensional entangled polymer chains, it is very likely that the same part of the polymer chain is not adsorbed to the same active site on the carbon black surface.

Figure 2.7: At higher macroscopic strain amplitudes, the chains desorb from the surface of the carbon black ﬁllers.

The elasticity and dissipation of a reinforced rubber is therefore strain amplitude dependent, as shown schematically in Fig. 2.8, here given as

14 2.5. FILLERS the storage and loss modulus, respectively. These concepts are deﬁned in the following chapter. The pronounced reduction in storage modulus is the previously mentioned Fletcher-Gent effect, sometimes referred to as the Payne effect [7, 10]. The strongest variation is commonly in the macroscopic strain amplitude range of 0.1-50 % [5–11], which is an important strain range in industrial applications [21]. Therefore, to accurately capture and model this phenomena is of utmost importance in order to enable computer aided simulations early in the design process. This is especially important for dynamic simulations of systems suspended with rubber mounts, where the dissipation of energy is crucial for the simulated response.

E´

Modulus [Mpa] Modulus E´´

Log(Strain amplitude)

Figure 2.8: Illustration of the strain amplitude dependency of the storage (E0) and loss (E00) modulus of a reinforced rubber.

Dynamic Mechanical 3Analysis

One way to characterize the mechanical properties of rubber is with a Dynamic Mechanical Analysis (DMA). A test specimen is subjected to a sinusoidal strain and the resulting stress is measured. If a material is linear and viscoelastic, the stress is also sinusoidal. However, this is commonly not the case for filled elastomers. Nevertheless, the classical definition of storage and loss modulus is presented, in order to intro- duce concepts and to enable a comparison between the response of a linear viscoelastic material and actual carbon black filled elastomers.

3.1 Classical deﬁnitions of storage and loss modulus

For a linear viscoelastic material, the applied strain and resulting normalized stress can schematically be drawn as in Fig. 3.1. With a sinusoidal strain, the resulting stress is also sinusoidal. However, the strain and stress will be out of phase. The phase lag δ, also denoted loss angle, between stress and strain leads to a dissipation of energy for each cycle. The amount of energy lost is equal to the enclosed area when the stress is plotted as a function of strain, as in Fig. 3.2. Note that the shape of the hysteresis is elliptical. The applied strain and measured stress can be written with phasor ∗ ∗ notation as e = e0exp(iωt) and σ = σ0exp(iωt + δ), respectively. Here, 2 e0 and σ0 are the strain and stress amplitude, respectively, i = −1 and ω is the angular frequency and t is time. The complex modulus of the

17 CHAPTER 3. DYNAMIC MECHANICAL ANALYSIS

δ Strain Stress

Time

Figure 3.1: Normalized stress response of a viscoelastic material when subjected to a sinusoidal strain. material is then given by [5] ∗ ∗ σ σ0 iδ 0 00 E = ∗ = e = E + iE , (3.1) e0 e0 where E0 is the storage modulus and E00 is the loss modulus. It is obvious that E00 tan δ = . (3.2) E0 The loss angle can also be calculated from the dissipated energy, D, during one cycle. This dissipated energy is given by D=H σde. With e0(t) = e0 cos (ωt) and σ(t) = σ cos (ωt + δ), this gives

I Z de D = σde = σ cos (ωt + δ) dt = 0 dt Z − σ0e0ω cos (ωt + δ) sin ωtdt = sin δσ0e0π (3.3)

18 3.2. DEFINITION OF STORAGE AND LOSS MODULUS USED IN THESIS

Hysteresis Stress

Strain

Figure 3.2: Hysteresis in a linear, viscoelastic material. or D sin δ = , (3.4) e0σ0π where the integration is understood to be carried out over one cycle.

3.2 Deﬁnition of storage and loss modulus used in thesis

Unfortunately, the shape of the hysteresis for a carbon black ﬁlled natural rubber is in general not elliptical, as seen in Fig. 3.3. Furthermore, the strain rate dependency on the dissipated energy is weak in the frequency range considered, which is demonstrated by the fact that the response is nearly identical even though the strain rate has increased by a decade. The test specimen and experimental setup are described in Paper A.

19 CHAPTER 3. DYNAMIC MECHANICAL ANALYSIS

0.4

0.2

0 0.01 Hz 0.1 Hz Stress [MPa] −0.2

−0.4 −60 −40 −20 0 20 40 60 Strain [%]

Figure 3.3: Hysteresis in a real material, a carbon black ﬁlled natural rubber.

This means that the classical definitions of storage and loss modulus are inapplicable when studying real materials subjected to finite strains. Therefore, the following definitions are used,

σ E0 = 0 . (3.5) e0 and D tan δeq = (3.6) e0σ0π for the storage modulus and tangent of the equivalent loss angle, δeq, respectively. Here, the dissipated energy during one cycle is also given by D=H σde, and 00 0 E = tan δeqE . (3.7) The storage modulus is now defined as the secant modulus, and it can be shown that for small equivalent loss angles, cos δeq ≈ 1, Eq. 3.5-3.7 are good approximations to the classical definitions in Eq. 3.1-3.2. With these new definitions, the storage and loss modulus are calculated for a series of shear strain amplitudes and frequencies, in the range of 0.2- 50% shear strain and 0.5-20 Hz, as presented in Paper A. The results

20 3.2. DEFINITION OF STORAGE AND LOSS MODULUS USED IN THESIS are shown in Fig. 3.4 - 3.5. As seen in Fig. 3.5, the storage modulus increases almost linearly on a logarithmic frequency scale. However, the linearity is a function of the strain amplitude, with a higher increase for smaller strain amplitudes. At the largest strain amplitudes, there is almost no increase in stiffness for an increase in frequency. Similar results in this frequency range are observed in literature for other rubber compounds [11, 21, 22].

0.2 0.5Hz 1.6 0.8Hz 0.18 1.4Hz 2.3Hz 1.4 0.16 3.9Hz 6.5Hz 0.14 10.7Hz 1.2 17.9Hz

[MPa] 0.12 [MPa] ′ ′′ G 1 G 0.1

0.08 0.8 0.06

0.6 −1 0 1 2 0.04 −1 0 1 2 10 10 10 10 10 10 10 10 Strain [%] Strain [%]

Figure 3.4: Left: Storage modulus as a function of strain amplitude. Right: Loss modulus as a function of strain amplitude.

0.2 0.2% 1.6 0.5% 0.18 1% 5% 1.4 0.16 10% 20% 0.14 50% 1.2

[MPa] 0.12 [MPa] ′ ′′ G 1 G 0.1

0.08 0.8 0.06

0.6 0 1 0.04 0 1 10 10 10 10 Frequency [Hz] Frequency [Hz]

Figure 3.5: Left: Storage modulus as a function of frequency. Right: Loss modulus as a function of frequency.

Modelling the 4Fletcher-Gent effect

As stated previously, there is a general consensus that the Fletcher-Gent effect is mainly caused by the breakdown of the filler-filler network [6]. However, the details of this reinforcing structure is an active area of research [5, 6, 9, 22–26]. The breakdown introduces a strain amplitude dependency to the rubber compound, as visualized in Fig. 3.4. A strain amplitude dependency is generally present in all elastomers with reinforcing fillers. Moreover, a higher filler volume fraction and smaller filler particles generally results in a more pronounced dependency. Since the modelling of the Fletcher-Gent effect has been an area of research since the 1950s, many measurements of the storage and loss modulus of filled rubber compounds are available in the literature [5, 6, 10, 11, 23–25]. By studying these measurements, several observations are made that influence the approach chosen for capturing this effect. For strain amplitudes below a certain threshold, usually in the range of 0.01 − 0.1%, the stiffness of the material reaches a plateau, and is not increased further upon an additional reduction of strain amplitude. On the other hand, when strain amplitudes exceed this threshold the material exhibits a substantial hysteresis even at very low strain rates. Addi- tionally, the superposition principle is not valid, which means that the response from a filled rubber subjected to two simultaneous sinusoidal excitations does not equal the sum of the individual responses [27, 28]. Therefore, it is a cumbersome task to model the Fletcher-Gent effect in the frequency domain, and it is beneficial to formulate the constitutive equations in the time domain. The conventional modelling technique for filled rubber is to divide the stress into two parts acting in parallel, one elastic and one inelastic stress [11, 29–32]. Moreover, the inelastic stress can subsequently be

23 CHAPTER 4. MODELLING THE FLETCHER-GENT EFFECT comprised of strain rate independent and rate dependent components. Rubber is viscoelastic in nature, meaning that there is a strain rate dependent phase lag between the strain and an applied stress. However, researchers have shown that a single relaxation time is insufficient to describe the properties of a given material over a wide range of frequencies [33]. Instead, a broad distribution is needed to capture the strain rate dependency for a given strain amplitude [33,34]. Therefore, several linear viscoelastic elements are required if the response in a wide frequency range is to be reproduced. This increases the required number of material parameters. An alternative solution is to utilize fractional derivatives [25, 33, 35–37]. This effectively reduces the required number of material parameters. Unfortunately, state variables for all previous time steps need to be saved and utilized in the calculation of each subsequent increment, which instead increases computational time. Moreover, the main disadvantage of both modelling techniques is that they are not able to describe the amplitude dependency of filled rubber, without modific- ations.

Under certain periodical loading conditions, a suitable method for capturing the amplitude dependency of rubber components is the equivalent viscoelastic approach [38]. It is an iterative method where a spe- ciﬁc viscoelastic material model is assigned to each element depend- ing on the expected strain amplitude in that element. With non-linear viscoelastic models, a strain amplitude dependency can be introduced [11, 22]. However, it is difﬁcult to describe both the storage and loss modulus with the same set of parameters. This raises the question of whether or not these models are capable of capturing the response from real life operational conditions where the excitations very seldom are purely sinusoidal.

The mathematical framework of plasticity has been used to describe the strain amplitude dependency of filled rubber [31,39–45], where fric- tional elements are seen as one-dimensional implementations of a plastic material model. In order to include a strain rate dependency, viscoelastic elements are commonly added in parallel to the plastic element [31, 39–41, 44, 45]. In three dimensions, this can efficiently be achieved by utilizing the overlay method [46, 47]. However, the strain rate dependency for filled rubber is generally not additive, shown in that the increase in stiffness in absolute values is larger for smaller strain amplitudes [11, 24, 25, 48].

24 4.1. MODIFIED BOUNDARY SURFACE MODEL

4.1 Modiﬁed Boundary Surface Model

The final observation in the previous section is a clear indication that an additive split of stress between elastic and inelastic contributions is generally unable to capture both the Fletcher-Gent effect and strain rate dependency of filled rubber over a wide range of strain amplitudes and frequencies. Instead, it is proposed to utilize an additive split of deformation between a rheological element utilizing plasticity and a generalized Maxwell chain (GMC). This material model is presented in Paper A. The idea is that the plastic element reduces the strain rate dependency at large strain amplitudes, and that the GMC bounds the stiffness for infinitesimal deformations. A rheological model is presented in Fig. 4.1. The total strain, γ, is the sum of the strain over the plastic element, w, and the GMC q, as γ = w + q.

τ, γ

S q yield Hp0

G1 GN G˳ c1 cN

Figure 4.1: Rheological model of carbon black ﬁlled natural rubber [Paper A].

4.2 Results from model

From Paper A, the measured and simulated storage and loss modulus is presented in Fig. 4.2 and 4.3. In Fig. 4.4, a comparison between simulation and measurements of a bimodal excitation is presented. As seen, the storage and loss modulus are well replicated over a wide range of frequencies and strain amplitudes. Additionally, bimodal excitations are

25 CHAPTER 4. MODELLING THE FLETCHER-GENT EFFECT reproduced even though they were not included when obtaining the material parameters. This is encouraging since it could enable real-life operating conditions to be simulated before an actual prototype is available.

Meas.

1.6 0.2% 0.5% 1.4 1% 1.2 5%

[MPa] 10% ′ 1 G 20% 0.8 50%

0.6

Sim. 1.6 1.4 1.2 [MPa] ′ 1 G 0.8

0.6 0 1 10 10 Frequency [Hz]

Figure 4.2: Measured and simulated storage modulus. Results from Paper A.

26 4.2. RESULTS FROM MODEL

Meas.

0.2 0.5Hz 0.8Hz 0.15 1.4Hz 2.3Hz

[MPa] 3.9Hz ′′ 0.1 G 6.5Hz 10.7Hz 0.05 17.9Hz

Sim. 0.2

0.15 [MPa] ′′ 0.1 G

0.05 −1 0 1 2 10 10 10 10 Strain [%]

Figure 4.3: Measured and simulated loss modulus. Results from Paper A.

27 CHAPTER 4. MODELLING THE FLETCHER-GENT EFFECT

20%/1Hz + 2%/5Hz 20%/1Hz + 2%/10Hz 0.2 0.2

0.15 0.15

0.1 0.1

0.05 0.05

0 0

Stress [MPa] −0.05 Stress [MPa] −0.05 Sim. Sim. −0.1 Meas. −0.1 Meas.

−0.15 −0.15

−0.2 −0.2 −40 −20 0 20 40 −40 −20 0 20 40 Strain [%] Strain [%]

Figure 4.4: Measured and simulated response from two different bimodal excitations. Right graph from Paper A.

28 Finite strain elasticity 5framework

To capture the Fletcher-Gent effect is crucial to accurately model the dynamics of a system mounted on or connected with rubber components. For some applications though, it is sufficient to describe only the elasticity of a rubber compound under finite strains. As mentioned in Chapter 2, Hooke’s law is inadequate for this task. Instead, hyperelastic material models are employed. In Paper B, a methodology to obtain these material parameters is included. In this chapter, basic continuum mechanical concepts and especially the particular class of hyperelastic materials is briefly summarized. For a more thorough explanation of continuum mechanics, the reader is referred to a textbook such as the one by Holza- pfel [18].

5.1 Basics of continuum mechanics

Consider a deformable body, in its unloaded state as in Fig. 5.1. After a motion χ, a particle at point X is moved to point x, in reference to a fixed origo. The unloaded and deformed states are called the reference and current configuration, respectively. The position of the particle in the reference and current configuration is described be the vectors X and x, respectively. Consider now a material curve, through the point X. The line segment at point X is dX, which during the motion χ, is rotated and stretched to dx, as schematically visualized in Fig. 5.1. There exists a second order tensor F(X, t), which maps the original vector dX to dx, and this is the well known deformation gradient, i.e. dx=F(X, t) dX and dX=F−1(X, t) dx. It is a function of both position and time, hereafter referred to as F, to simplify notation.

29 CHAPTER 5. FINITE STRAIN ELASTICITY FRAMEWORK

Figure 5.1: Basic concepts of continuum mechanics. A line segment in a deformable body is for a particular motion mapped explicitly with the deformation gradient, F.

Uniaxially, the stretch is deﬁned as λ = L/L0, where L is the length after deformation and L0 is the original length, as in Fig. 5.2.

Undeformed Deformed

L0 L

Stretch ratio λ = L/L0

Figure 5.2: The stretch ratio λ is deﬁned as the current length divided by the original length.

In three dimensions, the symmetric right Cauchy-Green tensor, C = FTF, gives a measure of the stretch of the line segments initially located at a point X. This tensor is one suitable choice to use in constitutive theories for ﬁnite strain, hyperelastic materials.

5.2 Hyperelastic materials

Hyperelastic materials postulates the existence of a Helmholtz free energy function W, deﬁned in a reference volume. If W is solely a function

30 5.2. HYPERELASTIC MATERIALS of the deformation gradient F, i.e. W = W(F), or a suitable strain tensor such as for instance C, this free energy function is called a strain energy density function. In industrial applications, isolators commonly have a large unbounded surface. For this case, the assumption that rubber is incompressible is valid, and it follows that det(F) = 1, where det means the determinant of the tensor. Furthermore, if the material is isotropic, it is possible to postulate a strain energy density function which is based on the principal stretches λi, i = 1, 2 and 3, or the strain invariants of C,

2 2 2 I1 = TrC = λ1 + λ2 + λ3, (5.1)

1 h i I = (TrC)2 − Tr(C2) = λ2λ2 + λ2λ2 + λ2λ2, (5.2) 2 2 1 2 1 3 2 3 and

2 2 2 I3 = detC = λ1λ2λ3 = 1, (5.3) where Tr is the mathematical operator trace. An interesting geometrical interpretation of the three strain invariants is found when studying the continuum cube with principal stretches λi, i = 1, 2 and 3, shown in Fig. 5.3. As seen, I1 is the sum of the square of the stretch ratio of the line elements, I2 is the sum of the square of the relative increase in area elements and I3 is the square of the change of volume of the continuum cube.

Figure 5.3: The geometrical interpretation of the three strain invariants I1, I2 and I3 as the sum of the square of relative change of line elements, area elements and the volume, respectively.

In this thesis, following Rivlin [49], the general polynomial form of the strain energy density function is used,

31 CHAPTER 5. FINITE STRAIN ELASTICITY FRAMEWORK

n m i j W = ∑ ∑ Cij(I1 − 3) (I2 − 3) , (5.4) i=0 j=0 where Cij are material constants, C00 equals zero and I1 and I2 are the ﬁrst and second strain invariant, respectively. Other strain invariant based material models such as the Neo-Hookean, Mooney-Rivlin, Yeoh and Biderman model are all special cases of the general polynomial form. From Eq. 5.4, it is apparent that the strain energy density function depends on the deformation mode. For the three loading conditions, uniaxial tension (UT), equibiaxial tension (ET) and pure shear (PS), with deformations as visualized in Fig. 5.4, the ﬁrst and second strain invariants are calculated as in Table 5.1.

Figure 5.4: Uniaxial tension (UT), equibiaxial tension (ET) and pure shear (PS).

Table 5.1: First and second strain invariant of the right Cauchy-Green tensor for three different deformations in an incompressible material.

Strain invariant UT ET PS 2 −1 2 −4 2 −2 I1 λ + 2λ 2λ + λ λ + λ + 1 −2 4 −2 2 −2 I2 2λ + λ λ + 2λ λ + λ + 1 I3 1 1 1

With Table 5.1, the ﬁrst and second strain invariant can be visualized for a given stretch ratio for UT, ET and PS as in Fig. 5.5. For incompress-

32 5.2. HYPERELASTIC MATERIALS ible materials, the possible deformation modes are bounded by UT and ET [50].

λ = 1.6 λ = 2.4 7

6 λ = 2.6 2 I

4 Uniaxial tension Equibiaxial tension Pure shear 3 3 4 5 6 7 I 1

Figure 5.5: I1 and I2 for different deformation modes in an incompressible material. For each deformation mode, the maximum stretch ratio is given, as well as equidistant mark- ers 0.2 in stretch ratio apart [Paper B].

The Cauchy stress for the three different deformation modes are

2 −1 −1 σ1 = 2(λ − λ )(∂W/∂I1 + λ ∂W/∂I2), σ2 = σ3 = 0, (5.5) 2 −4 2 σ1 = σ2 = 2(λ − λ )(∂W/∂I1 + λ ∂W/∂I2), σ3 = 0, (5.6) and

2 −2 σ1 = 2(λ − λ )(∂W/∂I1 + ∂W/∂I2), −2 2 σ2 = 2(1 − λ )(∂W/∂I1 + λ ∂W/∂I2), σ3 = 0 (5.7) for UT, ET and PS, respectively [18]. Measurements in all of the deformation modes listed above are use- ful when obtaining material parameters. Unfortunately, equibiaxial testing equipment is uncommon in industry. It would nevertheless be valu- able to receive measurement data from a deformation mode with mainly biaxial tension. One such procedure is the bubble inﬂation technique, which is utilized in Paper B.

Obtaining hyperelastic material 6parameters

For certain applications it is sufficient to employ a material model capturing only the elastic properties of the material. However, since there are several hyperelastic material models available and implemented in commercial FE-software, a design engineer is faced with the question of which one to choose. In order to make this choice, experimental data from different deformation modes needs to be available, preferably from uniaxial tension (UT), pure shear (PS) and equibiaxial tension (ET), as discussed in Chapter 5. Unfortunately, biaxial testing equipment is rare in industry, whereas UT and PS tests can be performed in a common servo hydraulic rig. This chapter summarizes the methodology to obtain hyperelastic material parameters for filled elastomers from UT and PS experiments, presented in Paper B. Unloading curves are used when obtaining the elastic part of the material model, and measurements on an inflated rubber membrane is used as a reference to verify the obtained parameters. The reason for utilizing measurements in UT, PS and ET is that analytical functions, Eq. 5.5-5.7, are available to compare measurement and simulation results. Other methods to obtain hyperelastic material parameters exist in the literature, such as the modified hardness test [51] and impact testing [52]. Unfortunately, since they only utilize one deformation mode it is difficult to determine the dependencies on the first and second strain invariant in the strain energy density function. A solution is to use a material model that only utilizes the first strain invariant. However, both theoretical and experimental results show that a de-

35 CHAPTER 6. OBTAINING HYPERELASTIC MATERIAL PARAMETERS pendency on the second strain invariant is important to quantitatively capture the response from rubber in all deformation modes [21, 49, 53].

6.1 Unloading curves for obtaining material parameters

As shown in Chapter 3, filled rubber can exhibit a hysteresis at very low strain rates. Furthermore, the elastic-plastic material model presented in Chapter 4 is capable of capturing the Fletcher-Gent effect over several decades of strain amplitude. Additionally, several measurements in literature show that the hysteresis can grow with increasing strain amplitudes [40, 41, 54]. This leads to the conclusion that it is important to distinguish between the loading and unloading curve in experimental data. Specifically, it is assumed that the loading curve is comprised of both elastic and inelastic contributions. Therefore, the preferred method for obtaining elastic properties is to use the unloading curve. Nevertheless, it is important to report both the loading and unloading curve from experiments together with the response from simulations with the fitted material model [55]. The difference between loading and unloading can be substantial, and the design engineer must decide if a purely elastic material model is sufficient, or if a more advanced model is required.

6.2 Uniaxial tension and pure shear experiments

Uniaxial tension and pure shear experiments are readily performed in a common servo hydraulic rig, with test specimens as in Fig. 6.1. Note that the uniaxial test specimen can be cut from the one used for pure shear experiments, ensuring that the same material is studied in both deformation modes. Due to boundary conditions, the strain state in UT and PS test specimens are inhomogeneous. Nevertheless, the inﬂu- ence of this deviation can be studied with FE-software and the analytical solutions in Eq. 5.5-5.7, as shown in Paper B. Generally, if the geometries of the test specimens are chosen appropriately this deviation can be neglected. In the following, all test specimens are assumed to be preconditioned with ﬁve cycles of the largest expected strain amplitude. This is to en-

36 6.2. UNIAXIAL TENSION AND PURE SHEAR EXPERIMENTS sure that the initial stress softening known as the Mullins effect [56] is reduced, and that a sufﬁciently steady-state condition is reached. Moreover, the Mullins effect is directional [57]. Therefore, the preconditioning is performed in all deformation modes prior to collecting measurement data.

Figure 6.1: Test specimens for uniaxial tension and pure shear experiments [Paper B].

The unloading curves from the UT and PS measurements are shifted to end at origo, to enable a comparison with analytical elastic stresses. The offset is due in part to the inelastic properties of the material. Mater- ial parameters in Eq. 5.4 are chosen to minimize the stress error function

φ = ∑ [σm,PS(λm,PS) − σa,PS(λm,PS)]+ ∑ [σm,UT(λm,UT) − σa,UT(λm,UT)] (6.1) where the subscripts m, a, PS and UT stand for the measured and analytical values in pure shear and uniaxial tension, respectively, and the summation is made over all measured stretch ratios. Any number of material parameters in Eq. 5.4 can be chosen, and more parameters will result in a better ﬁt to the available measurement data. However, there is a risk that FE-simulation in a deformation mode not included when obtaining the parameters is inaccurate [14]. There-

37 CHAPTER 6. OBTAINING HYPERELASTIC MATERIAL PARAMETERS fore, the amount of parameters should be kept to a minimum. Never- theless, the inclusion of an extra parameter dependency on the second strain invariant enables a much better ﬁt between measurement and simulation results, as seen in 6.2. Here, a comparison is made between the Yeoh and Biderman model, with strain energy density functions

2 3 W = C10(I1 − 3) + C20(I1 − 3) + C30(I1 − 3) (6.2) and 2 3 W = C10(I1 − 3) + C20(I1 − 3) + C30(I1 − 3) + C01(I2 − 3), (6.3) respectively. Moreover, if only UT measurement data is utilized when obtaining the parameters for the Yeoh model, the capability to predict the response in another deformation mode is poor.

3 3 3 PS, exp PS, exp PS, exp UT, exp UT, exp UT, exp PS, Biderman PS, Yeoh PS, Yeoh, only UT 2.5 2.5 2.5 UT, Biderman UT, Yeoh UT, Yeoh, only UT

2 2 2

1.5 1.5 1.5

True stress [MPa] 1 1 1

0.5 0.5 0.5

0 0 0 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 Strain [%] Strain [%] Strain [%]

Figure 6.2: Measurement and simulation results. Left: Biderman. Middle: Yeoh-Gent, using both UT and PS data. Right: Yeoh-Gent, using only UT data [Paper B].

6.3 Bubble inﬂation and digital image correlation

As mentioned previously, experimental data from a deformation mode in equibiaxial tension is important to obtain hyperelastic material parameters. In Paper B, a methodology is developed that utilizes the bubble

38 6.3. BUBBLE INFLATION AND DIGITAL IMAGE CORRELATION inflation technique together with standard UT and PS experiments to determine these parameters. The general idea of the bubble inflation method is to clamp a rubber membrane between two flanges and ap- ply pressure to one side. The technique has been used previously in literature to obtain biaxial deformation [58, 59]. The advantages with the presented methodology is that a single sheet of rubber is used in all experiments, thereby reducing the variance on the material parameters originating from the mixing and vulcanization process. Therefore, the variance on the elastic properties of a compound from different batches is possible to study. In the 1940s, researchers manually drew and measured the vertical and horizontal projection of lines during the inflation of the membrane, in order to calculate the principal stretches [59]. Today, with the aid of modern digital image correlation systems, the same measurements are easy and quick to perform. For a detailed discussion the mathematical theories in digital image correlation techniques, the reader is referred to, for instance, the textbook by Sutton et al. [60]. In Paper B, the GOM ARAMIS 5M system is used, with two 50 mm focal lenses. A schematic layout of the measurement system is presented in Fig. 6.3, the stochastic pattern on the rubber membrane in Fig. 6.4, and Fig. 6.5 shows the rubber membrane during inflation. By measuring the inflation pressure simultaneously as the strain field, experimental data is collected allowing a comparison with FE-simulations. The inflation pressure and the measured strain fields at the four pressure peaks are shown in Fig. 6.6 and 6.7. The acquired inflation pressure and strain field enables a comparison between measurement and simulation results in a deformation mode not included when obtaining the parameters, as in Fig. 6.9. The material parameters for the Biderman model that enabled a good fit between measurement and simulation results in UT and PS results in a too stiff response for simulation in biaxial extension. Therefore, a constraint on the ratio between the parameters C10/C01 is imposed. Results for different ratios as well as for the Yeoh model is shown in Fig. 6.9. In order to reduce the error in all deformation modes, a ratio of 10 is chosen for the Biderman model in Paper B. A comparison between the measured and simulated reaction force in the UT and PS experiments is shown in Fig. 6.10.

39 CHAPTER 6. OBTAINING HYPERELASTIC MATERIAL PARAMETERS

Figure 6.3: A schematic overview of the experimental setup. The temperature and pressure are measured simultaneously as the strain and height of the inﬂated bubble [Paper B].

40 6.3. BUBBLE INFLATION AND DIGITAL IMAGE CORRELATION

Figure 6.4: Left: Stochastic pattern on the rubber membrane used during measurements. Right: Zoomed area of Speckle pattern. [Paper B].

Figure 6.5: Inﬂated rubber membrane [Paper B].

41 CHAPTER 6. OBTAINING HYPERELASTIC MATERIAL PARAMETERS

0.8

0.6

0.4 Pressure [bar]

0.2

0 0 50 100 150 Time [s]

Figure 6.6: Pressure as a function of time in the bubble experiment. Instants of strain recording are marked with red circles [Paper B].

Figure 6.7: Measured major strain component for the four pressure peaks in Fig. 6.6 [Paper B].

42 6.3. BUBBLE INFLATION AND DIGITAL IMAGE CORRELATION

0.8

0.6

0.4 Pressure [bar]

0.2

0 0 10 20 30 40 50 60 70 80 Biaxial strain at pole [%]

Figure 6.8: Inﬂation pressure as a function of the measured biaxial strain at the pole [Paper B].

1 Measurements C /C no cstr. 0.8 10 01 C /C =3 10 01 C /C =5 0.6 10 01 C /C =7 10 01 C /C =10 0.4 10 01

Pressure [bar] C /C =20 10 01 0.2 Yeoh Yeoh, only UT

0 0 20 40 60 80 Biaxial strain at pole [%]

Figure 6.9: Measurement and simulation results for different ratios of C10/C01 [Paper B].

43 CHAPTER 6. OBTAINING HYPERELASTIC MATERIAL PARAMETERS

600 25 PS, exp. UT, exp. Biderman Biderman 500 Yeoh 20 Yeoh Yeoh, only UT Yeoh, only UT 400 15 300

Force [N] Force [N] 10 200

5 100

0 20 40 60 80 0 20 40 60 80 Displacement [mm] Displacement [mm]

Figure 6.10: Comparison between simulations and experiments in pure shear and uniaxial tension, with the set of material parameters obtained for C10/C01 = 10 [Paper B].

44 Summary and 7conclusions

The main contributions of this work are:

- A material model that captures the Fletcher-Gent effect and strain rate dependency for carbon black ﬁlled natural rubber over a wide range of frequencies and strain amplitudes with a limited number of material parameters.

- A methodology to determine material parameters for ﬁlled rubber when the hyperelastic properties are sufﬁcient.

7.1 Conclusions

By utilizing a split of strain rather than stress, the Fletcher-Gent effect is efficiently described with only a few material parameters. The pro- ficiency of the model is demonstrated in that both the storage and loss modulus are captured over a wide range of frequencies and strain amplitudes. Moreover, bimodal excitations are well replicated, even though they were not included when obtaining material parameters. The long term goal of modelling filled rubber is to reproduce stresses and strains in a component during complex real-life operating conditions. Since these very seldom are purely sinusoidal, the results from the bimodal experiments are truly encouraging. When designing any new component much is gained if an accurate material model is available early in the product development phase. For many applications, the hyperelastic material parameters of a filled rubber is sufficient at this stage. The presented methodology for obtaining these parameters is efficient and only requires a single sheet of

45 CHAPTER 7. SUMMARY AND CONCLUSIONS rubber. The implementation of an accurate material model enables more concepts and geometries to be investigated at an early stage, before an actual component prototype is available. This reduces the risk of costly changes late in a project.

7.2 Future work

Many isolators and bushings are loaded mainly in one direction, which motivates the uniaxial implementation of the derived material model. Nevertheless, for some components there is a need to generalize the material model to three dimensions. Even though there are three-dimensional equivalents to the individual rheological elements in the model, the task is not trivial. This is the main focus of future research.

46 Summary of 8appended papers

Paper A - Constitutive modelling of the amplitude and frequency dependency of ﬁlled elastomers utilizing a modiﬁed Boundary Surface Model

Rickard Österlöf, Henrik Wentzel, Leif Kari, Nico Diercks and Daniel Wollscheid. International Journal of Solids and Structures, 51:3431-3438, 2014. A phenomenological uniaxial model is derived for implementation in the time domain, which captures the amplitude and frequency dependency of filled elastomers. Motivated by the experimental observation that the frequency dependency is stronger for smaller strain amplitudes than for large ones, a novel material model is presented. It utilizes a split of deformation between a generalized Maxwell chain in series with a bounding surface plasticity model with a vanishing elastic region. Many attempts to capture the behaviour of filled elastomers are found in the literature, which often utilize an additive split between an elastic and a history dependent element, in parallel. Even though some models capture the storage and loss modulus during sinusoidal excitations, they often fail to do so for more complex load histories. Simulations with the derived model are compared to measurements in simple shear on a compound of carbon black filled natural rubber used in driveline isolators in the heavy truck industry. The storage and loss modulus from simulations agree very well with measurements, using only 7 material parameters to capture 2 decades of strain (0.5-50% shear strain) and frequency (0.2-20 Hz). More importantly, with material parameters ex- tracted from the measured storage and loss modulus, measurements of a dual sine excitation are well replicated. This enables realistic operat-

47 CHAPTER 8. SUMMARY OF APPENDED PAPERS ing conditions to be simulated early in the development process, before an actual prototype is available for testing, since the loads in real life operating conditions frequently are a combination of many harmonics.

Paper B - An efficient method for obtaining the hyperelastic properties of filled elastomers in finite strain applications

Rickard Österlöf, Leif Kari and Henrik Wentzel. Polymer Testing submitted Aug 2014

An efficient methodology for obtaining hyperelastic material parameters for filled elastomers is presented utilizing unloading curves in uniaxial tension, pure shear and the inflation of a rubber membrane. Experimental results from biaxial extension are crucial when fitting hyperelastic material parameters and the bubble inflation technique is an excellent method of obtaining this data when specialized testing equipment is unavailable. Moreover, filled elastomers have a considerable hysteresis, and the hysteresis grows with increasing strain amplitudes. Therefore, the loading curve is in general comprised of both elastic and inelastic contributions, even at very low strain rates. Consequently, it is deemed more accurate to use experimental data from the unloading curve to describe the elastic behavior of the material. The presented methodology enables obtainment of parameters related with both the first and second strain invariant, which is required for a good fit between measurement and simulation results. Finally, that a chosen material model is accurate in all deformation modes is essential when designing components subjected to a complex, multi-axial load history. An accurate material model enables more concepts and geometries of a component to be studied before a physical prototype is available.

48 Bibliography

[1] J. Rockström, W. Steffen, K. Noone, Å. Persson, F. S. Chapin, III, E. Lambin, T. M. Lenton, M. Scheffer, C. Folke, H. J. Schellnhuber, B. Nykvist, C. A. de Wit, T. Hughes, S. van der Leeuw, H. Rodhe, S. Sörlin, P. K. Snyder, R. Costanza, U. Svedin, M. Falkenmark, L. Karlberg, R. W. Corell, V. J. Fabry, J. Hansen, B. Walker, D. Liver- man, K. Richardson, P. Crutzen, and J. Foley. Planetary boundaries: Exploring the safe operating space for humanity. Ecology and Soci- ety, 14(2):32, 2009.

[2] Traﬁkverket. Målbild för ett transportsystem som uppfyller klimat- mål och vägen dit. Publikationsnummer: 2012:105 ISBN: 978-91- 7467-303-6, September 2012.

[3] Traﬁkverket. Nationell plan för transportsystem 2010-2021. Pub- likationsnummer: 2011:067 ISBN: 978-91-7467-132-2, March 2011.

[4] Gert Heinrich, Manfred Klüppel, and Thomas A. Vilgis. Reinforce- ment of elastomers. Current Opinion in Solid State and Materials Sci- ence, 6:195–203, 2002.

[5] Gert Heinrich and Manfred Klüppel. Recent advances in the theory of ﬁller networking in elastomers. Advances in Polymer Science, 160:1–44, 2002.

[6] Jean-Baptiste Donnet and Emmanuel Custodero. The Science and Technology of Rubber, chapter Reinforcement of Elastomers by Par- ticulate Fillers, pages 383–416. Academic press, The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK, fourth edition, 2013.

49 BIBLIOGRAPHY

[7] W. P. Fletcher and A. N. Gent. Nonlinearity in the dynamic properties of vulcanized rubber compounds. Transactions of the Institution of the Rubber Industry, 29(5):266–280, October 1953.

[8] J. Fröhlich, W. Niedermeier, and H.-D. Luginsland. The effect of filler-filler and filler-elastomer interaction on rubber reinforcement. Composites: Part A, 36:449–460, 2005.

[9] Gerard Kraus. Mechanical losses in carbon-black-ﬁlled rubbers. Journal of Applied Polymer Science: Applied Polymer Symposium, 39:75–92, 1984.

[10] A. R. Payne. The dynamic properties of carbon black-loaded natural rubber vulcanizates. part I. Journal of Applied Polymer Science, VI(19):57–63, 1962.

[11] Martin Rendek and Alexander Lion. Amplitude dependence of ﬁller-reinforced rubber: Experiments, constitutive modelling and FEM - implementation. International Journal of Solids and Structures, 47:2918–2936, 2010.

[12] M. Mooney. A theory of large elastic deformation. Journal of Applied Physics, 11:582–592, 1940.

[13] R. W. Ogden, G. Saccomandi, and I. Sgura. Fitting hyperelastic models to experimental data. Computational Mechanics, 34(6):484– 502, 2004.

[14] L. R. G. Treloar. The Physics of Rubber Elasticity. Oxford University Press, UK, 2005.

[15] Jan B. van Bellen and Yves Poirier. Guayule and russian dandelion as alternative sources of natural rubber. Critical Reviews in Biotech- nology, 27:217–231, 2007.

[16] P. G. de Gennes. Reptation of a polymer chain in the presence of ﬁxed obstacles. The Journal of Chemical Physics, 55:572–579, 1971.

[17] Fred W. Barlow. Rubber Compounding: Principles: Materials, and Tech- niques. Taylor & Francis, 2nd edition, 1993.

[18] Gerhard A. Holzapfel. Nonlinear Solid Mechanics: A Continuum Ap- proach for Engineering. John Wiley & Sons Ltd, 2006.

50 BIBLIOGRAPHY

[19] S.F. Edwards and T.A. Vilgis. The tube model theory of rubber elasticity. Reports on Progress in Physics, 51:243–297, 1998. [20] G. Townson and J. Hallet. An introduction to the terminology used in the carbon black industry. In P.-E. Austrell and L. Kari, editors, Constitutive Models for Rubber IV, pages 471–476. A. A. Balkema Publishers, 2005. [21] Rickard Österlöf, Henrik Wentzel, Leif Kari, Nico Diercks, and Daniel Wollscheid. Constitutive modelling of the amplitude and frequency dependency of filled elastomers utilizing a modified boundary surface model. International Journal of Solids and Struc- tures, 51(19-20):3431–3438, 2014. [22] P. Höfer and A. Lion. Modelling of frequency- and amplitude- dependent material properties of filler-reinforced rubber. Journal of the Mechanics and Physics of Solids, 57:500–520, 2009. [23] H. R. Ahmadi and A. H. Muhr. Dynamic properties of filled rubber. Part II: Physical basis of contributions to the model. Rubber Chemistry and Technology, 84(1):24–40, 2011. [24] L. Chazeu, J. D. Brown, L. C. Yanyo, and S. S. Sternstein. Modulus recovery kinetics and other insights into the payne effect for filled elastomers. Polymer Composites, 21(2):202–222, April 2000. [25] A. Lion and C. Kardelky. The payne effect in finite viscoelasticity: constitutive modelling based on fractional derivatives and intrinsic time scales. International Journal of Plasticity, 20:1313–1345, 2004. [26] A. I. Medalia. Effect of carbon black on dynamic properties of rubber vulcanizates. Rubber Chemistry and Technology, 51(3):437–523, July 1978. [27] M. Sjöberg and L. Kari. Testing of nonlinear interaction effects of sinusoidal and noise excitation on rubber isolator stiffness. Polymer Testing, 22:343–351, 2003. [28] C. Wrana and V. Härtel. Dynamic mechanical analysis of filled elastomers. KGK - Kautschuk Gummi Kunststoffe, December:647–655, December 2008. [29] J. S. Bergström and M. C. Boyce. Constitutive modelling of the large strain time-dependant behaviour of elastomers. Journal of the Mech- anics and Physics of Solids, 46(5):931–954, 1998.

51 BIBLIOGRAPHY

[30] Alexander Lion. On the large deformation behaviour of reinforced rubber at different temperatures. Journal of the Mechanics and Physics of Solids, 45(11/12):1805–1834, 1997. [31] M. J. García Tárrago, L. Kari, J. Vinolas, and N. Gil-Negrete. Fre- quency and amplitude dependence of the axial and radial stiffness of carbon-black ﬁlled rubber bushings. Polymer Testing, 26:629–638, 2007. [32] L.A. Gracia, E. Liarte, J.L. Pelegay, and B. Calvo. Finite element simulation of the hysteretic behaviour of an industrial rubber. application to design of rubber components. Finite Elements in Analysis and Design, 46:357–368, 2010. [33] Klas Adolfsson, Mikael Enelund, and Peter Olsson. On the fractional order model of viscoelasticity. Mechanics of Time-Dependant Materials, 9:15–34, 2005. [34] G. Marin, J. J. Labaig, and Ph. Monge. Dynamic viscoelasticity of entangled polymers. Polymer, 16:223–226, March 1975. [35] Klas Adolfsson. Nonlinear fractional order viscoelasticity at large strains. Nonlinear Dynamics, 38:233–246, 2004. [36] Klas Adolfsson and Mikael Enelund. Fractional derivative viscoelasticity at large deformations. Nonlinear Dynamics, 33:301–321, 2003. [37] K. Adolfsson, M. Enelund, and S. Larsson. Adaptive discretization of fractional order viscoelasticity using sparse time history. Com- puter Methods in Applied Mechanics and Engineering, 193:4567–4590, 2004. [38] Per-Erik Austrell and Anders K. Olsson. Considering amplitude dependence during cyclic loading of elastomers using an equivalent viscoelastic approach. Polymer Testing, 31:909–915, 2012. [39] M. Sjöberg and L. Kari. Non-linear behaviour of a rubber isolator system using fractional derivatives. Vehicle System Dynamics, 37:3:217–236, 2002. [40] Christiane Netzker, Hüsnü Dal, and Michael Kaliske. An endo- chronic plasticity formulation for ﬁlled rubber. International Journal of Solids and Structures, 47:2371–2379, 2010.

52 BIBLIOGRAPHY

[41] D. Besdo and J. Ihlemann. A phenomenological constitutive model for rubbrubber materials and its numerical applications. Interna- tional Journal of Plasticity, 19:1019–1036, 2003.

[42] V. A. Coveney, D. E. Johnsson, and D. M. Turner. A triboelastic model for the cyclic mechanical behaviour of ﬁlled vulcanizates. Rubber Chemistry and Technology, 68:660–670, 1995.

[43] V. A. Coveney and D. E. Johnson. Rate-dependent modeling of a highly ﬁlled vulcanizate. Rubber Chemistry and Technology, 73:565– 577, 2000.

[44] Hoda Yarmohamadi and Viktor Berbyuk. Computational model of conventional engine mounts for commercial vehicles: validation and application. Vehicle System Dynamics, 49:5:761–787, 2010.

[45] Mats Berg. A rubber spring model for dynamic analysis of rail vehicles. Technical report, Railway Technology, Department of Vehicle Engineering, Royal Institute of Technology, 1995.

[46] Per-Erik Austrell. Modeling of Elasticity and Damping for Filled Elast- omers. PhD thesis, Structural Mechanics, Lund University, 1997.

[47] P. -E. Austrell and A. K. Olsson. A method to analyse the non-linear dynamic behaviour of carbon-black ﬁlled rubber components using standard FE codes. In D. Besdo, R. H. Schuster, and J. Ihlemann, editors, Constitutive models for Rubber II, pages 231–237. CRC Press, 2001.

[48] Wenbo Luo, Xiaoling Hua, Chuhong Wang, and Qifu Li. Frequency- and strain-amplitude-dependent dynamical mechanical properties and hysteresis loss of cb-ﬁlled vulcanized natural rubber. International Journal of Mechanical Sciences, 52:168–174, 2010.

[49] R. S. Rivlin and D. W. Saunders. Large elastic deformation of isotropic materials. VII. experiments on the deformation of rubber. Philosophical Transactions of the Royal Society of London A, 243:251– 288, 1951.

[50] T. Guélon, E. Toussaint, J.-B. Le Cam, N. Promma, and M. Grédiac. A new characterisation method for rubber. Polymer Testing, 28:715– 723, 2009.

53 BIBLIOGRAPHY

[51] P. -E. Austrell and A. Ahadi. Hyperelastic constants from a modi- ﬁed hardness test using energy balance obtained from FE-analysis. In Nere Gil-Nigrete and Asier Alonso, editors, Constitutive models for Rubber VIII, pages 225–231. CRC Press/Balkema, 2013. [52] P. -E. Austrell and A. K. Olsson. Dynamic characterization of elastomers using impact testing. In G. Heinrich, M. Kaliske, A. Lion, and S. Reese, editors, Constitutive models for Rubber VI, pages 161– 166. CRC Press, 2009. [53] Cornelius O. Horgan and Giuseppe Saccomandi. Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility. Journal of Elasticity, 56:159–170, 1999. [54] F. Laraba-Abbes, P. Ienny, and R. Piques. A new ’tailor-made’ methodology for the mechanical behaviour analysis of rubber-like materials: II. application to the hyperelastic behaviour characterization of a carbon-black ﬁlled natural rubber vulcanizate. Polymer, 44:821–840, 2003. [55] R. K. Luo, X. P. Wu, and W. J. Mortel. Rubber-unloading-behaviour evaluation using product-oriented specimen based on a resilience test. Polymer Testing, 34:49–57, 2014. [56] L. Mullins. Effect of stretching in the properties of rubber. Rubber Chemistry and Technology, 21:715–723, 1948. [57] N. Diercks and A. Lion. Modelling deformation-induced aniso- tropy using 1d-laws for mullins-effect. In Nere Gil-Nigrete and As- ier Alonso, editors, Constitutive Models for Rubber VIII, pages 419– 424. CRC Press, 2013.

[58] P. H. Mott and C. M. Roland. Strains in an inﬂated sheet. Rubber Chemistry and Technology, 73:326–333, 2003. [59] L. R. G. Treloar. Strains in an inﬂated rubber sheet, and the mech- anism of bursting. Rubber Chemistry and Technology, 17(4):957–967, 1944.

[60] M. A. Sutton. Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications. Springer Sci- ence+Business Media, 2009.

54 Part II

APPENDED PAPERS A-B