Viscoelasticity and F1 tyres

Formula 1 aerodynamics is all about utilising a viscous fluid, to maximize the forces generated by a viscoelastic solid.

As Mark Hughes explained in last week's Autosport, an F1 tyre generates grip by two different mechanisms: physical grip and chemical adhesion. The latter is only triggered when the tyre reaches a certain critical temperature, and to reach that temperature, the physical grip must be used to repeatedly load and unload the tyre, the deformation cycle thereby heating up the carcass of the tyre.

However, if a rubber tyre is represented as an elastic solid, there is a puzzle here, for the theory of elasticity says that an elastic solid is non-dissipative; in other words, no heat is generated as a result of loading and unloading an elastic material. There is no net work done on a perfectly elastic substance during a load cycle. If rubber was genuinely elastic, it would deform, and then return to its initial configuration, at its initial temperature. Tyre friction between the tyre and the road surface would certainly generate heat, even if the tyre were an elastic solid, but if rubber were genuinely elastic, the carcass of the tyre would not heat up as a result of going through repeated load cycles.

To put the puzzle into context, and to understand the answer, we need to consider three types of material defined within continuum mechanics: elastic solids, viscous fluids, and viscoelastic solids.

An elastic solid undergoes elastic deformation up to a yield point, and thereafter undergoes a degree of irreversible plastic deformation, (until it finally fractures). Under plastic deformation, net work will be performed on the solid, and that net work will go into heating it up. In some respects, a solid undergoing plastic deformation behaves like an incompressible viscous fluid.

A viscous fluid, conversely, can be thought of as a solid with no elasticity and no yield point; it is something which flows under the action of an applied force, and which possesses internal resistance to shear forces.

A viscoelastic solid responds to an applied stress by undergoing simultaneous elastic deformation and viscous flow. There is a viscous flow for all stress levels, unlike plastic deformation, which only occurs in elastic solids after a yield point has been attained. The viscous flow produces heat. When the stress is removed, the material will return to its initial configuration via a different curve on a stress-strain graph. This phenomenon is referred to as hysteresis.

To reiterate, perfectly elastic materials do not dissipate energy as heat when a stress is applied and removed; there is zero net work done on an elastic material over a load cycle. The work done deforming the elastic solid will be stored as strain energy, and when the external stress is removed, the strain energy will be used to do work on the environment, resulting in zero net work being done on the elastic solid over a load cycle.

Now, rubber is slightly unusual in that it will heat up when deformed. In most elastic materials, the strain energy will be stored in the electrostatic bonds between molecules. In rubber, whilst some of the strain energy is stored in such bonds, a component of the strain energy is stored in the form of thermal energy, and when the external stress is removed, and the rubber returns to its original configuration, it will adiabatically cool.

If the unloading curve on a stress-strain graph followed the same path as the load curve, then the material would return to its initial temperature at the end of a load cycle. In contrast, a viscoelastic material dissipates a quantity of heat energy equal to the area enclosed by the curves on the stress-strain graph, and this is equal to the net work done on the material. Rubber is a viscoelastic solid. (This is also nicely explained in Pat Symonds's trilogy of articles on the science of F1 tyres, featured in the April/May/June issues of RaceTech Magazine).

Formula 1 aerodynamicists attempt to use the flow of a viscous substance over the car to maximize the downforce on the viscoelastic contact surfaces; this enables the drivers to corner at the speeds necessary to load and unload the viscoelastic contact surfaces to the point at which they dissipate sufficient heat that the chemical adhesion of the contact surfaces is then maximized.

It's all about resisting the flow.