Latex flowing out of a rubber tree [1]
Why Does A Tyre Grip The Road? Part 1 - Rubber
By Trav Mays
In today’s article we will begin a series tackling the deceptively simple question of Why Does A Tyre Grip The Road? As you probably know, a tyre is made up of a number of components, but for today, the area of interest is the outside rubber layer. It, working with the road surface, creates grip through two main mechanisms, the road roughness effect (hysteresis) and molecular adhesion. But before we dive into those two, we need to take a step back and look at rubber more generally.
Rubber
Rubber is classified as a visco-elastic material, meaning under certain conditions it acts like a perfect spring (elastic), whilst under others it acts like a piston moving through a viscous liquid; a suspension system with a spring and damper is a useful analogy. Like a spring and damper, if we apply a force to rubber and remove it, the rubber does not spring back, it gradually moves back to its relaxed state, the viscous properties slows it down. If we plot this on a stress strain curve, we can see that depending on whether the force is being applied or removed, the amount of stress (force) required to deform (strain) the rubber is different, this difference results in energy loss or hysteresis, violet area in graph below. As we shall see in the next post, hysteresis is important in getting grip.
Figure 1. Stress strain curve of spring (Blue) vs Rubber (Purple), the violet area between the two purple curves represents the energy loss.
Click here to learn more about stress strain curves
Stress strain curves are created in the process of measuring a materials characteristics by inducing stress. One of the more frequent tests is called the tensile test, the material is placed in between to clamps and pulled apart, as this is happening the amount of force being applied and the distance the clamps have moved is recorded. As the material is pulled apart it typically goes through three phases, the elastic region, where if the force was removed the specimen would relax with no deformation. This area of the curve is linear and from which we can get the young's modulus (the ratio of the stress to strain ϑ/𝜺). The next point recorded is the yield strength (the point after which further stress will cause the material to be permanently deformed), we can see this on the graph below where the curve stops being linear. As we continue to apply stress the material begins to permanently deform, if we were to remove the force now, the material would be compromised and any part that has experienced this level of force needs to be replaced. Continuing to apply the force results in the stress building to the ultimate tensile strength, here we experience necking with the cross sectional area reducing up until the material breaks at the fracture point.
Above I have explained the tensile test, but similar plots can be made using compression testing as well. Tensile is typically used to measure a material underdevelopment, compression is more used for finished products, but they could both do either.
Stress is defined as the ratio of the applied force to the cross-sectional area of the material it is applied to
Where ϑ = Stress,
F = Force, and
A = Area
A = Area
Strain is defined as the ratio of the change in length to the initial length of the material
Where 𝜺 = strain,
l = new length, andl₀ = original length
Typical stress strain curve of mild steel [2]
Another way to visualise the energy loss is by overlaying a graph showing an extension-compressive force placed on the rubber with the rubber’s displacement over time. The left graph in Figure 2 shows this type of graph for a spring whilst the right one displays a rubber’s response. As you can see the displacement for the spring is in perfect sync with the applied force, this is because a spring follows Hooke’s Law, whereas the viscous properties of the rubber cause the displacement to be out of phase with the force. This difference in the phase is the energy lost in the system, which for a tyre, mostly gets turned into heat and is dissipated throughout the tyre and the road.
Figure 2 A extension-compressive force vs deformation for a spring (left) and rubber (right) [4]
Click here to learn more about tan (δ)
The value of tan (δ) = Viscous stress / Elastic stress, or put into words is the ratio of the rubber’s viscous response to that of its elastic response, it therefore is a measure of the dampening of the rubber.
The Stress line for rubber in Figure 2 is a combination of the viscous and elastic stresses. Using the measured total stress, phase angle (δ), and some simple trigonometry we can calculate each.
Graphing the total (measured) stress and the phase angle (δ) we can calculate the viscous and elastic stresses. [3]
Click here to learn more about Hooke’s Law
Hooke’s Law, named after Robert Hooke, is the law which governs all elastic materials (assuming the force doesn’t exceed the material’s elastic limit). It states that there is a linear relationship between the force required to compress (extend) an elastic material and the amount of distance it is compressed (extended). Under low enough forces, rubber responds as a pure elastic material and therefore follows Hooke’s law. Mathematically:
Where F = force,
x = the distance, and
k = the springs constant, determined by the springs stiffness.
Internal Friction
The reason for the energy loss is internal friction created by what rubber is made of, polymer strands. Each individual polymer strand can be thought of as a tiny spring, which when in the presence of others naturally entwines. As the wheel rotates the rubber is deformed, causing the strands to be stretched and compressed. As this is occurring the strands are also rubbing up against themselves and each other, it’s the friction caused by this rubbing that creates hysteresis.
Rubber in its natural state is too soft to be used to generate grip, the polymers are able to move too freely. To stiffen the rubber up the polymers are bound together through a process called vulcanisation, in which the rubber is heated up and sulphur is added to the mix. This sulphur connects the polymer strands together, creating what is called sulphur bridges. The number of these sulphur bridges and the length of them can have large implications on how the rubber performs. Too many or too short and it becomes too brittle, too few or too long and it is too soft. The right number and lengths is therefore a delicate balancing act, all depending on the compound of the tyre, whether it is a wet or a dry tyre, the type of racing it will be used for, and a whole heap of other considerations, very delicate stuff.
Figure 3 A ball of polymer strands that has been vulcanised with the sulphur bridges highlighted [4]
Temperature and stress frequency
The number and lengths of the sulphur bridges are not the only things that influence the behaviour of the rubber, its temperature and the frequency of stress are also important. Depending on the temperature and stress frequency the tyre is seeing, it can act in three states, a rubbery state, where it acts as a spring, the zone of maximum hysteresis, where energy loss is at a maximum, and a glassy state, where the rubber becomes brittle. The zone of maximum hysteresis is the area where we see most grip from a tyre, unfortunately its peak is very close to the glass transition temperature (Tg, the temperature at which increasing it further makes the rubber more elastic, decreasing it makes the rubber stiffer), making tyre management extremely difficult.
At low temperatures the individual polymers act like they are frozen in place, causing the rubber to be very brittle, similar to glass. As the temperature increases the polymers are able to move more freely and quickly reach the point where the perfect balance between free moving and internal friction is created, the point of maximum grip. As the tyre continues to heat up the polymers are able to move too freely, reducing internal friction to the point where the rubber acts more like a spring.
Interestingly, while increasing temps moves the rubber towards the rubbery state, increasing the stress frequency moves the rubber into the glassy state. At low stress frequencies the rubber barely deforms, so it acts more like a spring. At higher frequencies the time between stresses is reduced, meaning that the polymer strands have less time to revert back to their relaxed position and therefore hysteresis is increased. If we continue to increase the frequency the strands revert to their relaxed state less and less, creating a situation where the strands appear frozen, creating the glassy state.
Figure 4. The impact of both temperature and stress frequency on the energy loss and modulus of a given tyre [4]
Williams Landel Ferry Equation
As you may have gathered, these two forces don’t work in isolation, an increase in one affects the other. Increasing the temperature moves the tyre into the rubbery state, but if we then increase the frequency, we are back where we need to be, in the zone of maximum hysteresis. Therefore we need to balance the two forces and the best way to do that is to understand the effect they have on each other. The WLF (Williams Landel Ferry) equation, allows us to do this. Whilst not perfect, as it uses two universal constants which aren’t universal, it is a good approximation of the viscosity at or near the glass transition temperature. If experimental data can be used instead of these constants, it allows for more accurate predictions.
Where aT = WLF Shift factor,
C1 & C2 = Constants,
T = Temperature, and
Tg = Glass transition temperature
Graphically the interconnected nature of the two looks like the image below. As you can see, depending on the stress frequency, the temperature to keep the rubber within its hysteresis zone changes, and vise versa, we need to walk along the tight rope all while racing a bunch of other tight rope walkers, challenging stuff.
Figure 5. Graphical representation of results from the WLF equation [4]
Now that we have a fairly good understanding of rubber, it’s time to move onto what it is about rubber that causes it to grip the road. As I mentioned at the start of this post, this predominantly comes down to two main factors, Molecular Adhesion and the Road Roughness effect (hysteresis), read about them in the second installment of Why Does A Tyre Grip The Road?.
Don’t forget to sign up to the newsletter so you don’t miss an article and as always, if you have any comments, suggestions, noticed I made a mistake, either write it down in the comments or get in contact either via email or using the contact page, I’d love to hear it. Thanks for reading and I hope you have a great day.
By Trav Mays
Next Post
Why Does A Tyre Grip The Road Series
References:
[1] Encyclopædia Britannica, https://www.britannica.com/science/rubber-chemical-compound/Tapping-and-coagulation#/media/1/511800/109292, retrieved on 10/03/22
[2] Faridmehr, Iman, et al. "Correlation between Engineering Stress-Strain and True Stress-Strain Curve." American Journal of Civil Engineering and Architecture 2.1 (2014): 53-59, http://pubs.sciepub.com/ajcea/2/1/6/index.html, retrieved on 10/03/22
[3] Schaefer R, Chapter 33, Mechanical Properties of Rubber, Link, retrieved on 02/03/22
[4] Michelin, The Tyre Grip, Link, retrieved on 03/03/22
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