The grip that needs a little slipping

A wheel rolling freely has, at the point of contact, no relative motion between tyre and road. That is what rolling means, and it is the reason a wheel is a good idea. It is also the reason a freely rolling wheel transmits no force along the road at all.

Every force a wheel delivers — accelerating, braking, cornering — arrives with slip attached. Not as a failure or a limit, but as the mechanism: the tread must move relative to the road for the road to push back, and the amount it moves sets the amount of push. A car cruising at a steady speed on a flat road is running each driven tyre at something like one or two per cent slip, and about a quarter of its footprint is sliding.

The bristles, and what they do #

The model that gets this right is crude enough to describe in a sentence. Treat the tread as a row of independent elastic bristles standing off the carcass. Each is laid down on the road at the leading edge of the contact patch, undeflected. The road holds it where it landed while the carcass carries its root along at the wheel’s own speed, so the bristle stretches, and the longer it has been down the more it is stretched. When its elastic pull exceeds what friction can hold at that point, it lets go and slides.

Two consequences follow immediately and neither is obvious beforehand.

The sliding starts at the back. Each bristle’s deflection grows with the distance it has travelled since landing, so the most stretched bristles are the ones nearest the trailing edge, and they are the first to break away. The patch therefore divides into a stuck region at the front and a sliding region at the back — never the other way round, and never uniformly.

The friction bound falls to zero at both edges. Contact pressure between two elastic bodies is not uniform; it is roughly parabolic, largest in the middle and vanishing at the edges. So the bristles near the trailing edge are being asked to hold the most while being pressed down the least, which is why the sliding region eats forward so readily.

At two per cent slip the patch is stuck over most of its length and sliding over the rest. At twelve per cent almost nothing is stuck. The transition is gradual and there is no threshold anywhere in it: full grip and full slip are the two ends of one continuum, and every useful state of a tyre is somewhere in the middle.

The slope that has no coefficient in it #

Integrating the bristles gives the curve in the first figure, and the most useful thing about it is the part nearest the origin.

At vanishing slip, nothing is sliding. Every bristle is stuck, each is stretched in proportion to how far it has travelled, and the total force is simply the tread’s elasticity multiplied by the slip. The coefficient of friction cannot appear, because friction has not yet been asked to do anything. The figure checks this rather than asserting it: the slope at the origin, measured from the integration, is exactly the bristle bed’s own stiffness.

That is a strong statement about what “grip” means, and it cuts against the way the word is used. Two tyres with identical friction coefficients and different tread stiffness deliver different forces at every slip a driver actually uses, respond differently to a steering input, and feel completely different — while agreeing exactly about the one number a textbook quotes. The coefficient decides the ceiling; the stiffness decides everything on the way up, which is where the driving happens.

The same distinction runs through the grip that is not a coefficient, which reaches it from the other end: a coefficient is a summary of a contact whose real behaviour depends on load, speed, temperature and time. Here the summary fails for a more basic reason — the force at small slip is elastic rather than frictional, and no summary of friction can describe it.

The numbers, so the curve has a scale #

The figures are drawn for a patch twelve centimetres long carrying four thousand newtons — one corner of an ordinary car — with a tread stiffness chosen so the initial slope is about eighty kilonewtons per unit slip. Those are the three numbers that fix everything else.

They put full sliding at fifteen per cent slip, which means the whole interesting range of the curve is compressed into the first tenth of it. At one per cent slip the tyre delivers around eight hundred newtons, a fifth of what it can; at five per cent, most of it. A wheel on a car doing 100 km/h is turning about fourteen times a second, and a slip of two per cent means its tread surface is moving half a metre a second faster than the road — walking pace, at the contact, under a car that feels perfectly planted.

That comparison is the one to keep. The relative motion at a contact people describe as “not slipping” is of the same order as the speed of a slow walk, sustained continuously, over an area the size of a hand. It is not a small correction to rolling; it is the mechanism by which a wheel does anything at all, and the reason a tyre wears out.

Where the maximum comes from #

The brush model with a fixed coefficient rises and then flattens. It has no maximum. Every measured tyre curve does, and the maximum matters more than anything else on the plot, so it is worth knowing exactly where it comes from.

Rubber’s coefficient falls with sliding speed over the range that matters, for reasons about polymer relaxation that belong with the liquid that remembers rather than with contact mechanics. The sliding speed in the patch is proportional to the slip, so as the slip grows the sliding part of the patch gets both larger and less effective, and past a certain point the second beats the first.

The branch beyond the peak is unstable, and that is the real difficulty. On the rising side, a wheel that slips a little more delivers a little more force, which slows the wheel less relative to the car, which reduces the slip: the system corrects itself. On the falling side every one of those signs reverses. A wheel pushed past the maximum slips more, delivers less, slows further, and locks — in a fraction of a second, with no intermediate state worth naming. This is the same structure as the negative slope that makes the chatter a stiffer holder removes squeal: a force falling with speed is an instability wherever it appears.

What it costs, in metres #

The peak is at ten per cent slip or so, which is neither rolling nor sliding, and holding a wheel there is the whole job of an anti-lock system.

Eleven metres is two and a half car lengths, and it is the difference between stopping and not. The asymmetry of the curve is the design constraint: overshooting the peak costs much more than undershooting it, so a controller that cannot measure the peak directly — and none can, since the peak depends on the road surface, which changes without warning — errs low deliberately.

And the distance understates the case badly. A locked wheel is sliding straight, so its friction opposes the direction it is already going and nothing else. It delivers no sideways force whatever, which means a car braking on locked wheels cannot be steered at all. The reason anti-lock braking became compulsory is not primarily the eleven metres; it is that a driver who can steer during an emergency stop has options that a driver sliding in a straight line does not.

What it costs, in watts #

The sliding part of the patch is doing work against friction, and that work becomes heat in a piece of rubber the size of a hand.

A tyre working normally dissipates a few hundred watts in its footprint, which the road and the airstream carry away without difficulty. A tyre held at half slip dissipates tens of kilowatts into the same area — four domestic electric showers’ worth of power, concentrated into a hand-sized patch of a material that degrades above about 150 °C. The rubber does not survive it, which is why a locked wheel leaves a black line on the road: that line is the tyre.

Two limits arrive at nearly the same slip, and it is worth noticing that they are independent. The force peaks at around ten per cent because the coefficient falls with sliding speed. The heat becomes destructive at a few tens of per cent because the dissipation grows as force times slip. Nothing connects those two facts, and yet a tyre that could grip at thirty per cent slip would destroy itself doing so. The useful operating region is narrow from both sides.

The other dissipation, which is not this one #

A tyre also loses energy without transmitting any force, and it is worth separating the two because they are often confused.

Roll a wheel along with no torque at all. The slip is zero, nothing in the patch is sliding, and yet it takes power to keep it going: the rubber is compressed as it enters the patch and released as it leaves, and rubber does not give back everything it takes. The loss is hysteresis inside the material, exactly the loop area that the liquid that remembers measures, and it appears as a resistance of roughly one per cent of the load — forty newtons on the four-thousand-newton corner above, or about a kilowatt for a car at motorway speed.

That loss has nothing to do with the contact patch sliding. It happens at zero slip, it does not depend on the friction coefficient, and reducing it is a matter of the rubber compound rather than of the tread pattern. The two dissipations coexist: the hysteretic one is always there and roughly constant, and the frictional one is zero at zero slip and grows without limit past it. Confusing them makes the tyre’s behaviour incomprehensible, because one is a property of the material being flexed and the other a property of a surface being dragged.

What a driver is actually feeling #

The steering wheel reports the same curve sideways.

A cornering tyre does not point where it is going. It runs at a slip angle — a few degrees between the direction it is pointing and the direction it is travelling — and the sideways force it produces is the same brush-model curve with the deflection accumulating across the patch instead of along it. Every cornering car is crabbing slightly, and the front and rear slip angles differing is what makes a car understeer or oversteer.

The reason a good driver can feel the limit approaching is that the slope of the curve falls to zero before the force does. The steering effort a driver senses is roughly proportional to how the force responds to a further change in angle, so it goes light as the peak is neared — a warning that arrives before the loss of grip rather than with it. A tyre engineered for a sharp peak has more grip and gives less notice, and that trade is a design decision rather than a fact about rubber.

There is a second thing the steering reports and it belongs to the geometry rather than to the rubber. The sideways force does not act at the centre of the patch: because the sliding region is at the back, the resultant sits behind the wheel’s centre line, by a distance called the pneumatic trail. That offset multiplied by the force is a torque trying to straighten the wheel, and it is most of what a driver feels as self-centring. As the patch saturates, the sliding region eats forward and the resultant moves towards the middle, so the trail collapses before the force does — which is why the steering goes light early, and why the warning is a property of where the force acts rather than of how large it is.

None of this is available from a coefficient. It requires the shape of a curve, and the shape comes from a contact patch that is partly stuck and partly sliding.

Where the model stops #

The bristles are independent, and real tread is not. A rubber block deflected at one point pulls its neighbours, which stiffens the response and smooths the transition between the stuck and sliding regions. The correction is quantitative rather than structural, and the qualitative shape survives it.

The pressure distribution is taken as parabolic and fixed. A real patch changes shape with load, inflation pressure and speed, the tread lifts at the leading edge at high speed, and above a critical speed a standing wave forms in the sidewall that destroys the patch entirely. None of that is here.

Temperature is treated as constant. The coefficient of rubber on road depends strongly on temperature, and the previous figure shows the tyre heating itself. A serious model runs the thermal problem and the mechanical one together, and the coupling is positive: more slip makes more heat, which changes the coefficient, which changes the slip.

And the load is fixed. A braking car transfers weight forward, so the front tyres are more heavily loaded during the stop and the rear ones less. Since the peak force is not proportional to load — a doubly loaded tyre delivers appreciably less than twice the force — this matters for the total and is one of the reasons a real braking system distributes pressure between the axles rather than sharing it equally.

What the pictures cannot show #

The patch figure draws one line of bristles, and a real contact patch is two-dimensional: the sliding region is a shape rather than a segment, and it is not symmetric when the tyre is cornering and braking at once. The interaction between longitudinal and lateral slip — the fact that a tyre has one budget to spend in both directions and cannot deliver its maximum in each simultaneously — is a two-dimensional statement and does not appear anywhere here.

Nor do any of the figures show time. Every curve is a steady state, reached after the wheel has settled at a given slip, and a real emergency stop lasts three seconds during which nothing is steady. The instability past the peak is a statement about the slope of a steady curve, and the dynamics that follow from it — how fast a wheel actually locks, and what a controller has to do about it — need the wheel’s own rotational inertia, which is not in any figure here.

Where the ladder goes next #

The friction ladder began with the force that takes what it needs, where friction is whatever it has to be up to a limit, and went on through the chatter a stiffer holder removes and the grip that is not a coefficient, which take apart the two halves of the schoolbook law — the constant coefficient and the independence from area. Most recently the turn that multiplies what a hand can hold put friction to work along a curve. This rung asks what happens when the surfaces are neither stuck nor sliding but both at once.

The rung after it is the one where the two bodies are elastic in the normal direction as well as the tangential one, and the pressure distribution has to be solved for rather than assumed. The habit worth carrying is the one this rung turns on: when a law says a force is either static or kinetic, ask what the contact is doing in between, because the answer is usually that both are happening in different places and the useful regime is the mixture.