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International Journal of Squiggly and Wobbly , v23(1) February 2019

Introduction to Viscoelasticity in and its Impact on Rolling Resistance in Pneumatic Tyres L. Dunn1 The Hollyfield School and Sixth Form College, Surbiton Hill Road, Surbiton, KT6 4TU ARTICLE INFO ABSTRACT

Article history: The characteristics of viscoelasticity in a number of Received: 24 February 2019 and polymers are discussed. It is shown how Accepted: 26 February 2019 the molecular structure of rubber leads to viscoelastic behaviour. The equations for the Maxwell model are derived Keywords: and two other, more complex models are discussed. It is Rubber, Viscoelasticity, Biomaterials, shown how viscoelastic behaviour in tyres leads to rolling , Rolling Resistance resistance.

1. Introduction

The aim of this paper is to discuss viscoelasticity, materials which exhibit it, and its practical consequences in engineering, particularly rolling resistance in pneumatic tyres. Elastic, viscous and viscoelastic materials and their properties are discussed and compared, and their force/extension curves examined. The polymeric molecular structure of rubber is discussed and the changes which occur to it under tension, which lead to viscoelastic behaviour. Biomaterials exhibiting viscoelastic properties are briefly discussed, but the main focus is on rubber. The modelling of viscoelastic materials is discussed, with three common models being the focus, and the equation for the extension predicted by the Maxwell model is derived from first principles.

It is shown how viscoelastic behaviour leads to hysteresis and energy loss, and this is demonstrated using the corresponding /strain curves of certain materials.

Finally, the paper discusses rolling resistance in pneumatic tyres, and shows how viscoelastic behaviour is the principle cause of this energy loss, its effect on fuel consumption, and suggests ways in which it may be reduced.

1 Corresponding author. Email address: [email protected]

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2. Viscoelasticity If a exhibits Hookean , it is said to obey Hooke’s law, which states that the stress applied to an elastic material will be proportional to the strain produced in it (equation (1), up to a certain limit. No material is entirely elastic, but many can be modelled as such, especially if the strain is small. (Georgia State University, 1998) A key implication is that, in a perfectly elastic material, all energy used to extend it is stored internally and released with total Figure 1 A typical stress/strain efficiency when it returns to its original length. graph for an elastic material

휎 = 퐸 ∙ 휀 (1) Where σ is stress (Pa), E is the Young’s modulus of the material (Pa) and ε is strain is defined as the state of being thick or sticky due to internal resistance2. In viscous or materials, this means that some of the energy used in manipulating them will be transferred into heat energy due to these internal losses. In viscous materials, force is proportional to the rate of change of extension (equation (2).

푑휀 휎 = 휂 (2) 푑푡 Where t is time (s) and η is the viscosity (Pa∙s)

It is established that some materials can be approximated to having solely elastic properties, while others are purely viscous (namely ). However, we define viscoelasticity as the property of a material which exhibits both viscous and elastic behaviours.

2 The word viscosity comes from the Latin viscum, meaning mistletoe, from which a sticky substance was extracted and used to ensnare birds.

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3. Materials That Exhibit Viscoelasticity All polymers exhibit viscoelastic properties, due to the internal structure produced by interlinking monomers. This will be discussed in more detail in section 4. Any materials, both synthetic and natural, are viscoelastic. In nature, it can be observed in materials such as:

• Spider’s silk – Spiders produce several types of silk for different purposes. One such silk type is viscid silk, used to ensnare prey. It is desirable that this silk exhibit viscoelastic properties as to absorb energy from incoming insects as opposed to being elastic enough to bounce them away. (Univestiy of Cambridge , 2006)

Figure 2 Stress/strain graph for viscid spider's silk

• Human hair – Human hair is a formed of keratin, as with all polymers, it is viscoelastic. This is due to hydrogen bonds within the hair breaking and eventually reforming once the stress is removed, given its elastic limit was not exceeded by the stress applied. (Univestiy of Cambridge , 2006) Figure 3 Stress/strain graph for human hair

In addition, many synthetic polymers are engineered to have specific viscoelastic properties for different applications. Perhaps the most common of which being rubber.

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4. The Physical Structure of Rubber Rubber is the most notable example of a polymer which exhibits viscoelastic properties. At a molecular level, rubber is formed of many polymer chains, the arrangement of which giving rise to its useful properties. Before any tension is applied to the rubber, these chains tend to coil and tangle around each other (Figure 4). However, when a force is applied, the conformation changes and the chains uncoil and extend in the direction of the line of action of the force (Figure 5).

Figure 4 Coiled chains Figure 5 Uncoiled chains under tension

While these chains exhibit elastic properties (i.e. they store the energy used in displacing them), friction between the chains causes a net loss of energy after they return to their original positions, and causes the force applied to be proportional to the speed at which the rubber is extended. Thusly rubber is said to have viscoelastic properties. (University of Cambridge, 2006) In addition to being tangled together, under certain conditions covalent bonds can be formed between atoms in separate chains through a process known as cross linking, this reduces the mobility of the chains and therefore increases the stiffness of the rubber. This process is the basis of vulcanization, a process in which another material, commonly sulphur, is added to the rubber. The added material then forms additional cross links between polymer chains. This can be used to harden the rubber and make it more durable, which is desirable for many applications.

5. Modelling Viscoelastic Materials

Being an archetypal viscoelastic material, rubber cannot be modelled as an ideal spring, nor can it be modelled as a viscous . Instead, viscoelastic materials can be modelled using a number of different arrangements, each involving an ideal spring to represent the elastic elements, and a dashpot to represent the viscosity. (Morro, 2016) Many different models are used, depending on many aspects of the situation in question, the most prominent models are:

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• The Maxwell Model – The Maxwell model is the simplest way to model viscoelasticity, comprising of a spring and dashpot in series (Figure 6). This model is useful as it is very easy to interpret and illustrates many key properties of Figure 6 The Maxwell model viscoelastic materials, such as . However, this model is not suitable for modelling materials over long periods of time, as it places no limit upon how much the dashpot can extend (Equation(9). As components are in series, force is equal in each

휎푡 = 휎푠 = 휎푑 (3)

Total extension is sum of extension of each component

휀푡 = 휀푠 + 휀푑 (4)

Differentiating equation (4) gives

푑휀푡 푑휀푠 푑휀푑 = + (5) 푑푡 푑푡 푑푡

Differentiating equation (1) with respect to time and rearranging gives

푑휀푠 1 푑휎푠 = ∙ (6) 푑푡 퐸 푑푡

Rearranging equation (2) gives

푑휀푑 휎푑 = (7) 푑푡 휂 Substituting equations (6)) and (7)) into (5)

푑휀푡 1 푑휎푠 휎푑 = ∙ + (8) 푑푡 퐸 푑푡 휂 Integrating (8) with respect to t gives

휎푠 휎푑 휀 = + 푡 (9) 푡 퐸 휂

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As is shown by the above equations, the Maxwell model predicts that when a stress is applied, there is an instant extension from the spring, followed by an indefinite extension from the dashpot, as a function of time.

• The Kelvin-Voigt Model – This model represents the viscoelastic element of a material as a spring and dashpot in a parallel arrangement as opposed to a series one (Figure 7). Here, the extension is equal across both components, and the total force is split across the dashpot and spring. According to this model, the dashpot will only extend to the extension produced in the spring. Figure 7 The Kelvin-Voigt model

• The Standard Linear Model – The SLS model is a more complex model which combines elements of both the Maxwell and Kelvin-Voigt models. It has elements in both series and parallel arrangements and can predict and account for many phenomena associated with viscoelastic materials. However, simpler models are sometimes favoured for their ease of use. Figure 8 The Standard Linear Solid model 6. Hysteresis As seen in Figure 1, the stress/strain graph for a perfectly elastic material follows the same path when the force is loaded and unloaded. This makes sense, given that the area under the graph tells us the energy used to extend the material (or the energy returned when it returns to its original size), and that one condition for perfectly elastic materials is that they return all energy exerted in manipulating them. However, looking at graphs for viscoelastic materials such as Figure 2 or Figure 3, it can be seen that, as the area under the unloading curve is smaller than the area under the loading curve, there is a net loss of energy caused by this process. The difference between these areas (i.e. the area enclosed by the two lines) giving the energy lost. This is an example of hysteresis. In the case of viscoelastic materials, this hysteresis is due to the viscous component of their properties, when a force is applied, there is a resistance to this force, causing more energy to be exerted (larger area under loading curve) than there would be

6 International Journal of Squiggly and Wobbly Materials, v23(1) February 2019 to extend a similar purely elastic material. This resistance is then still present in unloading, now causing there to be a lesser amount of energy returned (smaller area under unloading curve).

7. Tyres and Rolling Resistance

Pneumatic tyres, used by nearly all modern cars, experience a resistive force on them known as rolling resistance. Rolling resistance is separate from any other forces resisting the car’s motion (such as aerodynamic drag) and is due to the nature of the experienced by the tyres as they travel along the road surface.

Tyres, being made from rubber (Section 4), a viscoelastic material, experience hysteresis, and it is this which is responsible for rolling resistance. As the tyre rolls, the area of the tyre in contact with the road experiences a force due to the weight of the vehicle, causing a compression in it. This compression is greatest directly beneath the centre of the wheel, and after that point the force is effectively being unloaded.

However, due to hysteresis in the tyre (Section 6), there is a lesser force exerted as the tyre returns to its original position than the force originally required to compress it (Figure 9). Meaning there is, in total, less force experienced ‘behind’ the centre of the tyre. The consequence of this being a moment produced by the net total of the compressive forces, Figure 9 Forces in rolling resistance which directly opposes the torque on the wheel, causing rolling resistance. (Andersen, 2015)

The rate of energy lost per second by a wheel to rolling resistance is given by

Volume of rubber deformed × Hysteresis losses of rubber × Frequency of rotation

In order to reduce rolling resistance losses, one or more of these variables should be limited or reduced.

The volume of rubber deformed may be reduced with thinner wheels, or a lesser contact patch, achieved through greater tyre . However, this may have adverse consequences on the tyre’s grip.

The hysteresis losses can only be minimised by employing a different polymer, which experiences lesser energy losses through its viscoelastic properties.

And frequency of rotation can be decreased by travelling at lower speeds.

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8. Summary and Conclusion

This paper set out to review the viscoelastic behaviour of materials and in doing so reviewed three well- known models based on arrangements of springs and dashpots (namely Maxwell, Kelvin-Voigt and the Standard Linear Solid). It has been shown how viscoelastic behaviour relates to the internal polymeric structure of a named polymer. It has been explained how viscoelastic behaviour results in hysteresis in the loading/unloading stress/strain curve and how this leads to energy losses. This was linked to rolling resistance in pneumatic tyres and how this can be used to maximise efficiency in vehicles.

Conflict of Interest Statement The author received no financial support for the research, authorship, or publication of this article, and declares that there is no conflict of interest.

9. Works Cited

Andersen, L. G. (2015). Rolling Resistance Modelling. PhD Thesis, Roskilde University, Denmark, IMFUFA, Department of Science, Systems, and Models, Roskilde.

Georgia State University. (1998). Elasticity, Periodic Motion. Retrieved from Hyper Physics: http://hyperphysics.phy-astr.gsu.edu/hbase/permot2.html

Morro, A. (2016, November). Modelling of viscoelastic materials and creep behaviour. Meccanica, 52(13), 3015-3021. doi:10.1007/s11012-016-0585-x

Sasaki, N. (2012). Chapter 5. Viscoelastic Properties of Biological Materials. In J. De Vicente, Viscoelasticity - From Theory to Biological Applications. IntechOpen. doi:http://dx.doi.org/10.5772/49979

University of Cambridge. (2006, August). Theory of Rubber Conformation. Retrieved from DoITPoMS: https://www.doitpoms.ac.uk/tlplib/stiffness-of-rubber/rubber-conformation.php

Univestiy of Cambridge . (2006, April). Viscoelasticity and Hysteresis. Retrieved from DoITPoMS: https://www.doitpoms.ac.uk/tlplib/bioelasticity/viscoelasticity-hysteresis.php

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