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Introduction to Viscoelasticity in Polymers and Its Impact on Rolling Resistance in Pneumatic Tyres L

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Introduction to Viscoelasticity in Polymers and Its Impact on Rolling Resistance in Pneumatic Tyres L International Journal of Squiggly and Wobbly Materials, v23(1) February 2019 Introduction to Viscoelasticity in Polymers 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 biomaterials 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 Hysteresis, 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 stress/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] 1 International Journal of Squiggly and Wobbly Materials, v23(1) February 2019 2. Viscoelasticity If a material exhibits Hookean elasticity, 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 Viscosity is defined as the state of being thick or sticky due to internal resistance2. In viscous fluids 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 liquids). 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. 2 International Journal of Squiggly and Wobbly Materials, v23(1) February 2019 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 polymer 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. 3 International Journal of Squiggly and Wobbly Materials, v23(1) February 2019 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 fluid. 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: 4 International Journal of Squiggly and Wobbly Materials, v23(1) February 2019 • 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 creep. 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 5 International Journal of Squiggly and Wobbly Materials, v23(1) February 2019 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 Solid 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.
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