Illustrated Physics
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Mechanics

The contact that slips before it slides

Static friction is supposed to hold perfectly until the load reaches μ times the normal force and then give way. Two elastic bodies pressed together do something else. Push a ball sideways on a flat with any force at all and a ring at the edge of their contact is already slipping, while the centre stays stuck. As the push grows the ring spreads inward, and gross sliding is only the moment the stuck centre shrinks to nothing. Shake the push back and forth and the ring slips each way on every cycle. The contact then loses energy though it never slides, as the cube of how hard it is shaken.
16 min read 5 figures Approximations that lieThe shape decides

Assumes: The grip that needs a little slipping · The grip that is not a coefficient

The grip that needs a little slipping found that a tyre transmitting a force is never simply rolling. Part of its contact patch is stuck to the road and part is sliding, and the force it delivers measures how much has given up. That was a rolling contact, in which fresh material keeps entering the patch at the front, and the tyre’s rubber was assumed to press on the road with a known distribution of pressure.

This essay takes away the rolling and computes the pressure. Two elastic bodies are pressed together and then pushed sideways, without turning, by a force smaller than the one that would slide them. The textbook picture of static friction says nothing happens: the force takes what it needs up to μ\mu times the normal load, and the surfaces do not move. Cattaneo in 1938 and Mindlin in 1949 worked out what actually happens when the bodies are elastic, and the answer is that the textbook picture holds nowhere in the contact. Some of the contact is slipping from the first newton of sideways force. What the picture calls “static” is a mixture of stuck and slipping regions whose proportions change continuously, and gross sliding is only the end of a process that has been going on all along.

Where the shear exceeds the grip

The starting point is Hertz’s solution for a sphere pressed on a flat. The contact is a circle of radius aa, and the pressure across it is not uniform. It rises from zero at the rim to a peak p0p_0 at the centre, as p01−r2/a2p_0\sqrt{1 - r^2/a^2}, a dome whose total is the load PP. Amontons’s law, applied locally, says each point of the contact can carry a shear stress up to μp\mu p, so the friction limit across the contact is the same dome scaled by μ\mu.

Now push the sphere sideways by QQ. If every point of the contact stuck, the sphere and the flat would move as one across the whole circle, and elasticity then fixes how the shear must be distributed to produce a uniform displacement. The answer is a shear that rises towards the rim as 1/1−r2/a21/\sqrt{1 - r^2/a^2} and becomes infinite at the edge, exactly where the pressure, and therefore the friction limit, falls to zero. So the rim cannot stick. Whatever sideways force is applied, however small, the shear near the rim exceeds the local limit, and the surfaces there slip.

The shear a pushed contact carries, and where it gives. Across a Hertzian contact of radius a, the normal pressure as a fraction of its peak, which is also the friction limit μp in units of μp₀ (dashed), and the shear traction for a sideways push of 0.3, 0.6, 0.9 times the load that would slide it, μP. Where the shear reaches the dashed ceiling the surfaces slip; inside, they stick. At 0.3 μP the stuck disc has a radius of 0.89a, so 21 per cent of the contact area is slipping; at 0.6 μP the stuck disc has a radius of 0.74a, so 46 per cent of the contact area is slipping; at 0.9 μP the stuck disc has a radius of 0.46a, so 78 per cent of the contact area is slipping. The shear is highest just inside the slipping ring and, in an elastic contact that simply stuck everywhere, would be infinite at the rim: slip at the edge is what removes that singularity.
Fig. 1 Across a Hertzian contact, the friction limit μp\mu p (dashed) and the shear traction for sideways pushes of 0.3, 0.6 and 0.9 times the sliding load μP\mu P. Where the shear touches the ceiling the surfaces slip. The stuck discs (dots at their edges) have radii 0.89, 0.74 and 0.46 of the contact, leaving 21, 46 and 78 per cent of the area slipping.

The figure draws Cattaneo and Mindlin’s resolution. In an outer ring the shear sits exactly at the friction limit, μp\mu p, and the surfaces slide against each other there. Inside a disc of radius cc they stick, and the shear is the ceiling minus a smaller dome that keeps it below the limit. The two regions join smoothly at the dots. The stuck radius is

c=a(1−QμP)1/3.c = a\left(1 - \frac{Q}{\mu P}\right)^{1/3}.

At three-tenths of the sliding load the stuck disc has shrunk to 0.89 of the contact radius and 21 per cent of the area is slipping. At six-tenths it is 0.74 and 46 per cent. At nine-tenths only a disc of 0.46 of the radius holds, and 78 per cent of the contact is sliding. The whole contact slides when the stuck disc shrinks to a point, at Q=μPQ = \mu P exactly — so Amontons’s law survives as a statement about the total, while failing everywhere as a statement about any piece of the contact.

The shear peaks just inside the slipping ring, where it runs along the ceiling before the ceiling itself turns down towards the rim. There is no singularity. The infinite shear that a fully stuck contact would need is exactly what the slip removes, and it removes it by giving up in the one place where the grip was weakest.

Slipping from the first newton

The stuck radius’s cube-root dependence has a consequence that the drawing understates: slip starts immediately and spreads fast.

How much of a contact is already sliding. For a sphere pressed on a flat and pushed sideways, the radius of the stuck disc as a fraction of the contact radius, (1 − Q/μP) to the one-third, and the fraction of the contact area that is slipping, against the push as a fraction of the sliding load. The slip starts the moment any sideways load is applied and spreads fast: at a tenth of the sliding load 7 per cent of the area is slipping, at half of it 37 per cent, and half the area is slipping at 0.65 of the sliding load. Gross sliding is simply the moment the stuck disc shrinks to a point. There is no load at which the contact goes from wholly stuck to wholly sliding; it is always partly both.
Fig. 2 The stuck radius as a fraction of the contact radius (solid) and the fraction of the contact area that is slipping (dashed), against the sideways push as a fraction of the sliding load. At a tenth of the sliding load 7 per cent of the area slips; at half of it 37 per cent. Half the area is slipping at 0.65 of the sliding load (dot).

The figure plots both the stuck radius and the fraction of area that has given up against the push. At a tenth of the sliding load 7 per cent of the area is already slipping. At half the sliding load it is 37 per cent, and half the area has gone by 0.65. Near the limit the stuck radius falls steeply, as the cube root of the remaining margin, so the last few per cent of push takes it from a sizeable disc to nothing.

There is no load at which the contact goes from wholly stuck to wholly sliding. It is always partly both. That undermines the picture of static friction as a threshold, and it does so for any pair of elastic bodies with a curved contact, which is to say for every real contact. What the threshold picture gets right is that nothing macroscopic happens until the limit. What it gets wrong is that the microscopic contact is continuously and irreversibly changing from the first newton of sideways force.

The same mixture governed the tyre in the grip that needs a little slipping and the belt in the part of the wrap that is actually gripping, where a creeping arc and an idle arc divided the wrap. The three problems share a structure. A load is carried by friction along an interface whose ability to carry it varies from place to place, and the interface gives up first where its capacity is lowest relative to its demand. In the belt that was the end where the tension changed; in the ball it is the rim, where the pressure falls to zero.

The creep before the slide

A slipping ring means the sphere moves. Its displacement relative to the flat grows with the push, and the relation between them is the contact’s tangential stiffness.

The creep before the slide. The sideways push on a Hertzian contact, as a fraction of the sliding load μP, against the tangential displacement it produces, in units of 3μP/16Ga — the displacement at which gross sliding begins. The curve starts with the stiffness of a fully stuck contact, 8Ga (dashed), and softens as the slipping ring spreads, reaching zero slope just as the whole contact slides. The textbook picture of static friction, no motion at all until the limit, is the vertical axis. For a 20 mm steel ball pressed on a steel flat with 100 N and μ = 0.5, the contact is 0.37 mm across with a peak pressure of 1.37 GPa, and it creeps 2.1 μm sideways before it slides.
Fig. 3 The sideways push, as a fraction of μP\mu P, against the displacement, in units of 3μP/16G∗a3\mu P/16G^*a. The curve starts with a fully stuck contact’s stiffness (dashed), softens as the slipping ring spreads, and reaches zero slope as the whole contact slides. The textbook picture is the vertical axis. A 20 mm steel ball pressed on steel with 100 N creeps 2.1 μm before it slides.

Mindlin’s displacement is

δ=3μP16G∗a[1−(1−QμP)2/3],\delta = \frac{3\mu P}{16 G^* a}\left[1 - \left(1 - \frac{Q}{\mu P}\right)^{2/3}\right],

where G∗G^* combines the two bodies’ shear moduli and Poisson ratios. The figure plots the push against it. At the origin the slope is the stiffness of a contact that was fully stuck, 8G∗a8G^*a, drawn dashed. As the slipping ring spreads, less and less of the contact resists and the curve softens, reaching zero slope at the sliding load, where the displacement is exactly 3μP/16G∗a3\mu P/16G^*a and the sphere moves off. The textbook picture of static friction is the vertical axis: no displacement at all until the limit, then unlimited sliding.

For a steel ball 20 millimetres across pressed on a steel flat with 100 newtons, with μ=0.5\mu = 0.5, the contact is 0.37 millimetres across with a peak pressure of 1.37 gigapascals, and the ball creeps 2.1 micrometres sideways before it slides. That is small, but it is not zero, and in precision instruments it is the whole problem. A stage that must move by nanometres, held on balls or on rough surfaces, moves first through this compliant regime, where the force needed is not the sliding friction but something smaller and history-dependent. The stiction and the “stick–slip at small amplitude” that plague positioning systems at that scale are this curve, not the stick–slip a stiffer holder removes, which is a dynamic instability of full sliding.

The displacement also shows why the grip that is not a coefficient could treat friction as a property of the total load. That essay found the friction force proportional to the load because a rough surface’s real contact area is. Here, one of those microscopic contacts is examined on its own, and its sliding force is exactly μP\mu P as Amontons said. The coefficient is honest about the total, and silent about everything that happens before it is reached.

A loop traced without sliding

The most important consequence appears when the push is reversed. Load the contact to Q∗Q^*, then reduce the push. The ring that was slipping forwards does not simply slip back. Every point in it stops, because the shear there falls below the limit the moment the push decreases, and for an instant the whole contact is stuck again, with the stiffness of the dashed line. Then a new ring of reverse slip starts at the rim and spreads inward.

The loop a shaken contact traces. The sideways push on a Hertzian contact against its displacement, cycled between plus and minus 0.25, 0.5, 0.9 times the sliding load, in units of μP and of 3μP/16Ga. On each reversal the slipping ring stops, the contact is momentarily stiff again, and a new ring of reverse slip spreads inward — Mindlin and Deresiewicz's rule, which makes each return branch the first loading curve doubled in both directions. The loop encloses the work lost to friction in the slipping ring: 0.0028 at 0.25, 0.0282 at 0.5, 0.3210 at 0.9, in units of μP times 3μP/16Ga. No surface ever slid, and energy is lost on every cycle.
Fig. 4 The push against the displacement for a contact cycled between plus and minus 0.25, 0.5 and 0.9 of the sliding load. Each return branch is the first loading curve doubled in both directions (Mindlin and Deresiewicz). The loops enclose the energy lost to friction in the slipping ring: 0.0028, 0.028 and 0.32 in units of μP\mu P times the sliding displacement.

Mindlin and Deresiewicz showed that the unloading branch has a simple form: it is the first loading curve stretched by a factor of two in both push and displacement, starting from the turning point. The figure draws the resulting loops for three amplitudes. They are closed, because the contact returns to the same state each cycle, and they enclose an area, because the path out and the path back differ. The area is energy, converted to heat in the slipping ring, and it is lost on every cycle although the contact as a whole never slides at all.

The loops grow rapidly with amplitude: 0.0028 at a quarter of the sliding load, 0.028 at half, 0.32 at nine-tenths, in units of μP\mu P times the sliding displacement. A tenfold rise between the first two, for a doubling of amplitude, is the cube law the next figure isolates.

The heat is not the only thing left behind. The slipping ring is where the surfaces rub, and repeated cycles of microscopic slip wear it. The result is fretting, a characteristic damage pattern visible on the contact faces of bolted joints, bearing seats and splines that were never meant to move: an unworn central disc surrounded by a ring of oxidised debris and pitting. Fretting initiates fatigue cracks at the edge of the stuck zone, where the traction changes most sharply, and it is a leading cause of failure in joints that were designed only against slipping.

Damping that grows with the amplitude

The energy lost per cycle has a closed form, and at small amplitudes it is dominated by one power.

Damping that grows with the amplitude. The energy lost per cycle by a Hertzian contact shaken sideways, against the amplitude of the push as a fraction of the sliding load, on logarithmic axes, in units of μP times 3μP/16G*a, with a viscous damper that loses the same energy at a tenth of the sliding load. At small amplitudes the friction loss grows as the cube of the amplitude — the computed slope is 3.002 — against the square for a viscous damper, because both the slipping area and the distance it slips grow with the load. So the fraction of the stored energy lost per cycle is proportional to the amplitude: ten times smaller for a tenfold smaller shake. A contact is nearly lossless for tiny vibrations and strongly damping for large ones, which is why the damping of a bolted structure depends on how hard it is shaken.
Fig. 5 The energy lost per cycle against the amplitude of the push, on logarithmic axes, with a viscous damper that loses the same energy at a tenth of the sliding load (dashed). The frictional loss rises as the cube of the amplitude (computed slope 3.00), the viscous loss as the square.

The figure plots it against amplitude on logarithmic axes. The slope is 3.00: the loss grows as the cube of the amplitude. A linear viscous damper, drawn for comparison through the same point at a tenth of the sliding load, loses energy as the square of the amplitude, because its force and its displacement are both proportional to the amplitude. The frictional contact has one extra factor, because both the width of the slipping ring and the distance it slips grow with the load.

Since the energy stored in the contact’s elastic deformation grows as the square of the amplitude, the fraction of it lost per cycle — which is what an engineer calls the loss factor, and what sets how quickly vibration dies away — is proportional to the amplitude. A contact shaken ten times more gently loses ten times less of its energy per cycle. The three ways of coming to rest treated damping as a fixed coefficient, the same at every amplitude. Here there is no fixed coefficient: small vibrations are barely damped and large ones are heavily damped.

This is the dominant damping mechanism in built structures. A steel frame’s material damping is tiny; what dissipates the energy of a vibrating building, bridge or aircraft is mostly the friction in its bolted and riveted joints, and it behaves like this figure rather than like a viscous damper. That is why measured damping ratios depend on how hard a structure is shaken, why a structure tested at small amplitudes looks more lightly damped than it will be in an earthquake, and why turbine blades are sometimes fitted with loose friction dampers designed to slip in exactly this partial way. It also means a vibration that is small enough never decays quickly: at the smallest amplitudes a jointed structure rings as though it had no joints at all.

A stiffness that reads the size of the contact

The initial slope of the compliance curve, 8G∗a8G^*a, has a use of its own. It is proportional to the contact radius and to nothing about friction, because at the very start of the push the whole contact is stuck and the coefficient has not yet been consulted. So a small sideways wiggle, far below the sliding load, measures how big the contact is.

That makes the tangential stiffness an instrument. The same is true of the normal stiffness, 2E∗a2E^*a, which is how nanoindentation measures the area under a sharp tip, and the ratio of the two depends only on the Poisson ratios. Pressing a tip into a surface and wiggling it sideways by a nanometre or two reads off the contact radius, and watching the stiffness fall as the wiggle grows reads off the onset of partial slip, which is one way the friction of single asperities has been measured. The creep before the slide, invisible on the scale of a desk, is the signal.

Stack many such contacts together and the same law sets the stiffness of a granular material. A pile of glass beads under pressure is held together by Hertzian contacts, and a sound wave passing through it shears each contact slightly, in the regime drawn in the third figure. Since each contact’s stiffness grows as its radius, and a Hertzian radius grows as the cube root of its load, the effective-medium theory built on single contacts predicts that the speed of sound rises as the sixth root of the confining pressure. Measurements on bead packs give something closer to the fourth root at low pressure. The difference is not a failure of Mindlin’s solution but of the assumption that every bead touches its neighbours from the start: as the pressure rises, new contacts are recruited, much as the grip that is not a coefficient found new asperities recruited under load, and the packing stiffens faster than any one contact does. The granular column in the heap that becomes a solid is held up by a network of exactly these contacts, most of them partly slipping.

The sound speed also depends on the amplitude, for the reason the last figure gives. A loud wave in a granular medium drives its contacts further into partial slip, softens them and loses energy on every cycle, so granular materials — soils among them — are strongly nonlinear acoustic media. The softening of soil under strong earthquake shaking, which changes how a site responds to the next shock, is partly this effect at the scale of individual grain contacts.

What the elastic contact leaves out

The Cattaneo–Mindlin picture is exact within its assumptions, and the assumptions are specific.

Coulomb’s law holds locally, with one coefficient. Each point of the contact is assumed to stick until its shear reaches μp\mu p and then to slip with shear exactly μp\mu p. Real surfaces have static and kinetic coefficients that differ, coefficients that depend on sliding speed and on how long the contact has been stuck, and interfaces that are themselves made of microscopic asperities. Each of those modifies the loops in detail.

The bodies are smooth and elastically similar. For two bodies of the same material the normal and tangential problems separate. For dissimilar materials a sideways push changes the pressure distribution slightly and the solution becomes coupled. Roughness, which the grip that is not a coefficient found governs how real contact area grows, means that the smooth Hertzian dome is itself an average over many small contacts, each of which has its own partial slip.

The loading is slow and the push does not rotate. A push whose direction changes, or a contact that also twists, has more complicated slip regions that can be crescent-shaped rather than annular. Fast loading brings in the inertia of the bodies and the waves that carry the load. And none of this includes the wear that the slip produces, which changes the contact’s shape as the cycles accumulate.

Still open: how a joint’s damping can be predicted before it is built

The contact of one sphere with one flat is solved. A bolted joint is thousands of such contacts on rough, wavy surfaces under a pressure that is highest near the bolt and falls away from it, with slip occurring first where the pressure is lowest. Predicting its damping and stiffness from its drawings, rather than measuring them on a prototype, is a long-standing goal of structural dynamics and is not yet reliable: models calibrated on one joint often fail on a nominally identical one, because the damping depends on surface details at the scale of micrometres that vary between parts and change as the joint wears. Measurements show the cube law at small amplitudes, as the single contact predicts, and departures from it that depend on the joint; whether the departures can be computed from the surface roughness, or whether each joint must be measured, is an active question in the design of aircraft and turbines, where joint damping decides how long a vibration takes to die away.

The habit worth carrying away is to ask where a threshold is being crossed locally before the total crosses it. A limit stated for a whole contact — slide at μ times the load — is usually reached first at one edge and then spreads, and everything interesting happens during the spreading. The creep before sliding, the loop without sliding, the damping that grows with amplitude and the fretting ring are all consequences of a law that holds for the total and fails for every part of it.

Part 6 of 6

This essay is one argument about Friction. The others:

Force against slip, and the knee between them. The force a tyre delivers against how much faster its tread is going than the road, for a patch 120 mm long under 4000 N with a friction coefficient of 1. The curve is the integral over the bristles, and the dashed line is the cubic the brush model gives in closed form; they agree to 0.00 per cent of the sliding force. The first slope is 80 kN per unit slip, and it belongs entirely to the elasticity of the rubber — at vanishing slip nothing is sliding, so no friction coefficient can appear in it. Full sliding is reached at 15.0 per cent slip and not before. Everything a driver calls grip lives on the rising part of this curve, at a few per cent of slip, where the patch is partly stuck and partly sliding — and the quantity that decides handling in that region is the slope rather than the friction coefficient at the top. The grip that needs a little slipping Part 5 — A wheel that transmits any force at all is not rolling. Part of its contact patch is stuck to the road and part is already sliding, and the force it delivers is a measure of how much has given up. The useful part of the curve is a few per cent of slip, the peak is not the end of it, and everything past the peak is unstable.

Where this goes next

Force against slip, and the knee between them. The force a tyre delivers against how much faster its tread is going than the road, for a patch 120 mm long under 4000 N with a friction coefficient of 1. The curve is the integral over the bristles, and the dashed line is the cubic the brush model gives in closed form; they agree to 0.00 per cent of the sliding force. The first slope is 80 kN per unit slip, and it belongs entirely to the elasticity of the rubber — at vanishing slip nothing is sliding, so no friction coefficient can appear in it. Full sliding is reached at 15.0 per cent slip and not before. Everything a driver calls grip lives on the rising part of this curve, at a few per cent of slip, where the patch is partly stuck and partly sliding — and the quantity that decides handling in that region is the slope rather than the friction coefficient at the top. The grip that needs a little slipping A wheel that transmits any force at all is not rolling. Part of its contact patch is stuck to the road and part is already sliding, and the force it delivers is a measure of how much has given up. The useful part of the curve is a few per cent of slip, the peak is not the end of it, and everything past the peak is unstable. What friction returns, against what it is asked for. The friction force on a block under a 50 N normal load, against the force applied to it. Below 30.0 N — the static limit μs·N — friction returns exactly what is asked for and nothing moves, so the curve is the 45° line and the coefficient never appears. At that point the surface gives way and the force drops to μk·N = 22.5 N, where it stays however hard the block is pushed. The gap above the flat line is the surplus that accelerates it: 22.5 N at the right-hand edge of the axis. The force that takes what it needs Static friction has no value of its own. It supplies exactly what equilibrium demands and not a newton more, right up to the moment it cannot — which is the only instant in the whole business at which a coefficient of friction means anything at all. Rough contact grows as W^1.000, and one smooth bump as W^0.667. Real contact area against load, both logarithmic, three ways. Plastic flow of the asperities gives A = W/H, a slope of exactly one, which is the account Bowden and Tabor's argument uses. A single elastic sphere gives Hertz's answer, A ∝ W^(2/3), and a friction coefficient falling as W^(−1/3) — Amontons' law would be false. A rough elastic surface, with many asperities spread exponentially in height and integrated here rather than quoted, gives a slope of 1.0000, because pressing harder mostly recruits new contacts rather than enlarging existing ones. The dotted curve repeats that integration with a Gaussian spread of heights and gives 0.9695 — the upper tail of a Gaussian is nearly exponential, so the answer is nearly and not exactly linear. Amontons' law does not need plasticity. It needs roughness, it is exact for one distribution and good to three per cent for another, and it survives whichever way the individual junctions deform. The grip that is not a coefficient A friction coefficient is independent of load and of area, and the reason is usually given as plastic flow of the asperities. It is not — a rough elastic surface gives the same law, integrated here to a slope of 1.0000, because pressing harder recruits new contacts rather than enlarging old ones. What breaks the law is a contact that is smooth, or a material that dissipates in its bulk — where μ passes one and stops being a coefficient at all.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Contact areaDampingFrettingFrictionHertz contactHysteresisPartial slipStatic friction