The grip that is not a coefficient

Amontons’ law has two halves, and both of them are strange. The friction force is proportional to the load, and it does not depend on the apparent area of contact. A brick pushed on its narrow face and on its broad one takes the same force to move — and whether it slides or tips over is a separate question with a separate answer, and doubling its weight doubles that force exactly.

The rung below this one gives the standard account: the surfaces touch only at asperities, those asperities flow plastically until the area they present can carry the load, so the real area is W/H whatever the apparent area is, and the friction — the reaction that supplies exactly what is asked of it, which is the shear strength of the junctions times their area — comes out as sW/H. The coefficient is s/H, a ratio of two properties of one material, with the load and the size divided out.

That argument is correct and it proves less than it appears to.

Why roughness is what does the work #

The Greenwood–Williamson calculation is worth following, because the reason for the exact linearity is not obvious and it is not an approximation.

Take a surface covered in asperities of radius β whose summits are distributed in height with density φ(z). Bring a flat down to a separation d. Every asperity taller than d is compressed by ω = z − d, and Hertz gives its contact area as πβω and the load it carries as $\frac{4}{3}E^*\beta^{1/2}\omega^{3/2}$. Summing over the distribution,

$$A_r \propto \int_d^\infty (z-d)\,\varphi(z)\,\mathrm{d}z, \qquad W \propto \int_d^\infty (z-d)^{3/2}\,\varphi(z)\,\mathrm{d}z.$$

For an exponential distribution, $\varphi \propto e^{-z/\sigma}$, both integrals carry the same factor $e^{-d/\sigma}$ and everything else is a constant. Their ratio does not depend on d, so the real area is exactly proportional to the load — and the friction coefficient is exactly constant.

Why the exponential is the right tail to try. Nothing about a surface makes the whole height distribution exponential — the summits of a randomly rough surface are Gaussian, which is what a sum of many independent machining and wear processes gives. What matters is only the part of the distribution above the separation, and that is the far upper tail, where a Gaussian falls off in a way that a decaying exponential matches closely over the two or three standard deviations that carry any contact. So the exponential is not a model of a surface; it is a model of the tail of a surface, and it is a good one exactly where the physics happens.

The mechanism in a sentence. Pressing harder on a rough surface mostly brings new asperities into contact rather than flattening the ones already there, and the number of new contacts grows in the same proportion as the load they carry. On a single smooth bump there are no new contacts to recruit, so all the extra load goes into enlarging one patch, and the patch grows more slowly than the load.

Which distribution, and does it matter. An exponential tail is not arbitrary: the summits of a randomly rough surface have a Gaussian distribution, and the upper tail of a Gaussian is very nearly exponential over the few standard deviations that carry any contact. The dotted curve on the first figure repeats the integration with a Gaussian and gives 0.9695 rather than exactly one — close enough that no experiment separates them, and different enough to show that the exact linearity is a property of the exponential and the approximate linearity is a property of any distribution with a rapidly falling tail.

That is the honest form of the result. Amontons’ law is not exact and is not derived from a deep principle; it is what a broad class of rough surfaces gives to within a per cent or so, and the class is broad because the shape of the tail hardly matters.

Which mechanism a given pair actually uses #

Saying that plasticity is sufficient and not necessary leaves an obvious question: for a real pair of surfaces, which is it? Greenwood and Williamson answered that in the same paper, with one dimensionless group.

An asperity yields when the pressure Hertz puts under it reaches the hardness, and Hertz’s pressure grows with how far the asperity is squashed. So whether the tallest asperities yield before enough of them are in contact to carry the load is a competition between how soft the material is and how sharp the roughness is:

$$\psi = \frac{E^*}{H}\sqrt{\frac{\sigma}{\beta}},$$

with $\sigma$ the spread of summit heights and $\beta$ their radius. Below about 0.6 the contacts stay elastic at any load worth applying; above about one they are plastic from the first touch, and the range in between is mixed.

The two figures in this essay sit on opposite sides of it, which is the useful thing about the number. The steel-like pair — a modulus of 110 GPa against a hardness of 2 GPa, with a micrometre of roughness on asperities of fifty micrometres’ radius — gives $\psi$ of about eight, so its contacts really are plastic and the rough elastic curve drawn beside them is a counterfactual. The softer, blunter pair at the end — 3 GPa against 800 MPa, four micrometres of roughness on two-hundred-micrometre summits — gives about 0.5, so its contacts are elastic and stay elastic. Both give a slope of one.

Which is the sharpest form of the essay’s claim. The two mechanisms are not rivals to be decided between; they occupy different regions of one parameter, most engineering metals are in the plastic region, most polymers and rubbers in the elastic one, and Amontons’ law holds across the boundary without noticing it.

What actually breaks the law #

Three cases, and none of them is a small correction.

There is one more consequence of the same integral worth having. Because both the area and the load fall off exponentially with the separation, halving the load does not halve the separation — it increases it by σ ln 2, about seven tenths of a roughness. The two surfaces barely move relative to each other over decades of load, which is why the stiffness of a rough contact rises so steeply with load and why the stiffness of a joint is enormously higher when it is tight. That stiffness is the quantity a vibration engineer needs, it comes out of the same calculation as the friction, and it is where the chatter a stiffer holder removes begins.

A polished elastic contact. A steel ball on a glass flat, both optically smooth, has one contact rather than a million, and it behaves as the dashed line: the friction coefficient falls with load. This is not exotic — it is the standard configuration of a ball-on-flat tribometer, and the fact that it disobeys Amontons’ law is why measurements on smooth contacts have to quote the load.

A material that dissipates in its bulk. Rubber’s modulus is four orders of magnitude below steel’s, so the real contact area approaches the apparent one and the “asperities” picture stops applying. Worse, the dissipation is no longer at the interface: as a hard asperity ploughs through soft rubber, which remembers what was done to it, the rubber is compressed ahead of it and relaxes behind it, and because it is viscoelastic it does not give the energy back.

And adhesion, where there is no load at all. Two clean metal surfaces in vacuum weld on contact, because making a surface costs energy and losing one returns it, because there is no oxide layer to keep them apart and the junctions are as strong as the bulk. The effective coefficient is then several, and the concept of a coefficient has broken down because the friction no longer goes to zero as the load does.

Every surface costs energy to make, so two surfaces that meet and become one release it — and that release is a force pulling them together. That is what runs the adhesive cases: where the contact is clean and the materials are compatible, the junction is genuinely a weld, and separating it costs the surface energy back. Amontons’ law has no term for this, which is why it fails for clean metals in vacuum by a factor of ten.

A third case, and it is the one that seems least likely. A gecko holds to a ceiling with no normal load at all, and the mechanism is van der Waals attraction over an enormous real contact area — a foot divided into hundreds of thousands of setae, each split into hundreds of spatulae a couple of hundred nanometres across. The design is the exact inverse of everything above: instead of a rough surface making the real area a millionth of the apparent, a hierarchically split one makes the real area an appreciable fraction of it. Splitting a contact into n pieces multiplies the adhesive force by √n at fixed total area, which is why the subdivision goes on for four levels.

The general lesson is worth stating without the animal. Everything in this essay depends on the real contact area, and there are only two ways to make friction large: press hard, or make the surfaces conform. Nature reaches for the second and engineering almost always reaches for the first.

Roughness has no single scale #

There is an assumption buried in every formula above, and it is the one a modern treatment removes: that a surface has an asperity radius and a height spread, as though roughness were a layer of bumps of one size.

Measure a real surface and it is not. Look at a millimetre of it and there is structure at a hundred micrometres; magnify a hundred-micrometre patch and there is structure at a micrometre; magnify that and there is structure at a nanometre. Machined, worn and fractured surfaces are very often self-affine over three or four decades, which means there is no scale at which the bumps stop and the flats begin. The “asperity radius” that appears in the calculations above is therefore a property of how closely the surface was examined, not of the surface — and doubling the resolution of the measurement halves it.

That sounds fatal and is not, which is the interesting part. Treating the contact scale by scale — the load carried at each magnification, with the contacts at one scale resolving into smaller contacts at the next — gives the real area proportional to the load again, over the range where the contact remains a small fraction of the apparent area. No exponential tail is assumed anywhere and no single asperity size appears in the answer.

So the linearity survives being told that its ingredients do not exist. Three independent routes — plastic flow, an exponential spread of elastic asperities, and a self-affine surface with no characteristic size at all — arrive at the same proportionality, which is the strongest statement available about why Amontons’ law is so hard to break. What the routes disagree about is the value of the coefficient, and that is exactly the quantity nobody has ever been able to predict from first principles.

Junction growth, and where a coefficient near one comes from #

One more mechanism belongs here, because it explains a number the junction picture otherwise cannot: why clean metals give coefficients around one rather than the tenth or so that a shear strength divided by a hardness suggests.

A junction carrying a normal stress is at the point of yielding under that stress alone. Add a shear stress and the combination exceeds the yield criterion, so the junction yields further — which enlarges it, which lowers the normal stress, until it is once again just at yield under the pair together. The contact therefore grows while it is being sheared, and by the time it slides its area is several times what the normal load alone would have produced.

In air the growth is arrested early, because the oxide or adsorbed film on the surface shears at a much lower stress than the metal and the junction never reaches the state where it would grow. In vacuum, with the film removed, nothing stops it, and the area runs away until the two surfaces have welded. That is the whole distance between a coefficient of 0.2 for steel on steel at a bench and the seizure of the same pair in orbit, and nothing about either material changed.

Where the coefficient goes above one #

The tyre’s number is not a shear strength divided by a hardness. It is the sum of two mechanisms — adhesion at the true contact patches and hysteresis in the bulk as the road’s texture is worked through the rubber — and the second of those grows with how rough the road is, which is the opposite of what a junction model predicts. Polished ice is slippery for rubber; polished granite is not. Adding roughness to a road surface raises the grip, up to the point where the texture is coarse enough to reduce the true contact area again, and the optimum is a texture around a millimetre — which is why road surfaces are specified by their texture depth and not by what they are made of.

What the models still share, and where they part #

All three contact models in the first figure agree that the friction is proportional to the real contact area. They differ only in how that area grows with load. That is worth stating plainly, because it identifies exactly which assumption each case is breaking.

Where the real area approaches the apparent area, everything changes. All three models assume the contact spots are a small fraction of the footprint. Push hard enough, or use a soft enough material, and they merge — at which point the real area stops growing, the friction force saturates, and the coefficient falls. That is why a very heavily loaded contact departs from Amontons’ law even in a metal, and it is a completely different failure from the smooth-contact one.

The thing none of these models has is time. A stationary contact creeps, so its junctions grow while nothing is moving, and the force needed to start rises with how long it has been sitting. That is why a static coefficient is not a constant but a function of dwell time, and why a machine left overnight is harder to start than one stopped a minute ago.

Whatever the microscopic account, the macroscopic statement is one inequality: the tangential reaction cannot exceed some fraction of the normal one. That is what a free-body diagram uses and what a designer needs, and it survives every complication above — the complications change what the fraction is and whether it is constant, and none of them changes the shape of the statement.

The two-century gap #

Amontons published the law in 1699, restating what Leonardo had found and not published two hundred years earlier. Coulomb added the speed-independence of kinetic friction in 1785. The explanation — that the apparent area is irrelevant because the real area is not the apparent area — waited until Bowden and Tabor’s work in the 1930s and 1940s, and the demonstration that the real area is a tiny fraction of the apparent one was made by passing a current between the surfaces and measuring the contact resistance.

The gap is instructive. A law can be exactly right, universally applicable and completely mysterious for two hundred and fifty years, and the reason it survived that long unexplained is that its two strange features — proportionality to load, indifference to area — are exactly the features that make it useful without an explanation. An engineer needs one number per pair of materials, and that is what the law provides. It is the sort of situation in which nobody is pushed to look underneath, and the thing found underneath — that the surfaces touch over a few parts per million of themselves — is much stranger than the law it explains.

And one number the models agree on that nobody can measure directly. The real contact area is 0.025 square millimetres under a fifty-newton load, and it cannot be seen, photographed or reached. The measurements that establish it are indirect — electrical resistance across the contact, which depends on the size of the conducting spots; optical reflection at a transparent interface; the thermal resistance of a joint. All three give the same answer and none of them is a picture, which is why the figures in this essay are calculations rather than observations.

Where this ladder goes next #

Three rungs have now treated the contact as static or steadily sliding. The next asks about the layer in between: what happens when there is a film of liquid, thick enough to keep the surfaces apart and thin enough that its own molecules feel both walls. The friction then depends on the sliding speed and the viscosity rather than on the load at all, the coefficient collapses by two or three orders of magnitude, and the transition between the two regimes has a shape — the Stribeck curve — with a minimum in it that decides how every bearing ever built is designed.