The chatter a stiffer holder removes

Two blocks are dragged across the same surface by the same steadily-moving spring. Everything about the contact is identical — the same materials, the same load, the same speed of pull. One of them slides quietly at the speed it is being pulled. The other lurches, a thousand times a minute, reaching almost a metre a second in each lurch while being pulled at a tenth of a millimetre a second. The only difference between them is the stiffness of the spring.

That is a fact about machinery which anybody who has quieted a squealing brake or a chattering lathe tool already knows in the form of a remedy: shorten the overhang, tighten the mounting, use a stiffer holder. It is not a fact the standard account of stick-slip can produce. The standard account says that static friction exceeds kinetic friction, so a stuck block resists more than a sliding one, so once it breaks free it runs. That is true as far as it goes. It contains no stiffness, so it cannot say when the remedy works.

The account that has no length in it #

Here is the textbook mechanism, drawn as honestly as it can be. A block is held by static friction at μ_s N. The spring stretches until it reaches that value. The block breaks free — the condition that decides sliding rather than tipping having been met — friction drops immediately to μ_k N, and the surplus accelerates it forward until the spring has been unloaded and the block overshoots, decelerates and stops. Then it sticks and the cycle repeats.

The picture is right about what it draws. The difficulty is in what it leaves out, and the omission is easiest to see by asking a dimensional question: what in this model has units of length?

Nothing does. There is a mass, a stiffness, two dimensionless coefficients, a load and a drive speed. From those one can build a time, a force and a distance — but the distance is built out of the spring, not out of the contact. So the model contains no statement whatever about how far the surface has to slide before anything about it changes, and the answer it gives is therefore that friction changes the instant motion begins. That is the assumption doing all the work, and it is what makes the instability unconditional.

Static friction is a response rather than a formula, which is the argument one rung below this one: the contact returns exactly what is asked of it, up to a limit, and the coefficient names the limit rather than the force. That matters here because it is what leaves room for a length. A model with two coefficients and nothing else has no scale in it at all — no distance over which anything changes — and a model with no length cannot say how fast a contact recovers, which is the quantity the whole instability turns on.

What a contact actually does #

Slide two surfaces steadily at a fixed speed and measure the friction. Then change the speed and measure again. Two things happen and they happen on different schedules.

The friction jumps at once, in the same direction as the speed change: going faster is instantaneously harder. That is the direct effect, and it is a statement about the rate at which contacts are being made and broken. Then, over the following slip, the friction drifts back — usually further than it jumped — and settles at a new steady value. That is the evolution effect, and it is a statement about the state of the surface: the population of contacting asperities, which takes time and sliding to renew.

Both effects are logarithmic in speed to very good accuracy, which is why the whole business is written with two dimensionless numbers, conventionally called $a$ and $b$: friction rises by $a$ per e-fold of speed at once, and relaxes by $b$ per e-fold over a sliding distance $D_c$. What is left in steady sliding is the difference:

$$\mu_{ss}(v) = \mu_0 + (a - b)\ln(v/v_0).$$

There is no static coefficient and no kinetic coefficient in that expression. There is one curve, and its slope.

A contact with $b > a$ gives back less resistance the faster it goes — the opposite of a fluid, whose resistance rises with rate by definition, and the reason a lubricant that turns the contact into a film cures a squeal outright. That is the sign that makes trouble, and it is worth being precise about why: a resistance that falls with speed is a negative damping. Anything oscillating in contact with it is being fed rather than drained.

The criterion, which is a stiffness #

Now put the two pieces together. A block sliding steadily is a fixed point of the spring–block system. Perturb it: let the block speed up slightly. The direct effect resists, the state effect eventually gives way, and the spring — this is the part the two-coefficient model cannot see — unloads as the block runs forward, because a block that has moved has taken the tension out of what is pulling it.

So there is a race, and it is decided in the same way an oscillator’s return to rest is decided — by which of two rates wins, not by how large either of them is. The contact sheds friction over a sliding distance $D_c$. The spring sheds force over a distance $\Delta\mu N/k$. Whichever sheds faster wins. If the spring can take the load off the contact at least as fast as the contact loses its grip, nothing accelerates and the block simply creeps forward at the speed it is pulled. If it cannot, there is a surplus at every instant, and the surplus goes into acceleration.

Writing that out gives a critical stiffness:

$$k_c = \frac{(b-a)\,N}{D_c},$$

and a system stiffer than $k_c$ slides steadily while one softer than it does not.

The criterion is a stiffness, and it is easiest to see in the exponents. A damped oscillator grows or decays according to which side of the imaginary axis its exponents sit on, and negative damping puts them on the growing side — so the question is not whether the system has a resonance but whether the effective damping is positive. A contact whose friction falls with speed supplies negative damping, and if the holder is not stiff enough to make up the difference, the oscillation grows.

The transition is not a threshold that something has to be pushed past. It is a bifurcation: at $k$ slightly above $k_c$ every disturbance dies away, and at $k$ slightly below it every disturbance grows until it meets the limits of the system. Nothing has to be large to start it.

And the criterion can be checked by measurement rather than argued. Take the same contact — the same $a$, the same $b$, the same $D_c$, the same load — and vary only the stiffness of what is holding it.

Where the length comes from #

$D_c$ is the one quantity in all of this that the two-coefficient account has no room for, and it is not an abstraction. It is the sliding distance over which a contact renews its population of touching points — roughly the size of those points.

For laboratory rock that comes out at a few micrometres, for engineering metals at something similar, and for a fault zone containing gouge, at millimetres to centimetres — which is why the same criterion, with the same two numbers, is used both to design a quiet brake and to ask whether a stretch of fault will creep or break. The scaling is severe: $k_c$ is inversely proportional to $D_c$, so a contact with a coarser memory is stabler, and a fault whose gouge layer thickens can stop producing earthquakes.

That is also why the load appears. Doubling $N$ doubles $k_c$, so a contact pressed harder is more prone to chatter, not less. The instinct to clamp harder is right when the clamping is what raises the stiffness and wrong when it only raises the load, and the two are easy to confuse in a real machine.

What the noise actually is #

A slip event is fast and it is not, by itself, a sound anyone hears. What is heard is the structure ringing afterwards.

Each slip is an impulse delivered to whatever is holding the contact, and a structure struck by an impulse rings at its own frequencies. So the noise is not the friction — it is the holder, excited by the friction, and its pitch says nothing about the sliding and everything about the fixture. That is why the same brake squeals at one frequency on one car and another elsewhere, and why the cure is almost always a change to the mounting rather than to the pad.

The friction supplies the energy and decides whether there is a squeal; the structure decides what it sounds like — and a structure struck repeatedly at its own frequency is being pumped rather than pushed, which is why a squeal grows to a loudness the pedal force never suggests. A violin makes the same trade deliberately: the bow’s rosin is chosen for a strongly velocity-weakening characteristic, the string is soft enough that its stiffness sits well below $k_c$, and the resulting sawtooth — the Helmholtz motion — has a period set by the string’s own allowed modes rather than by the bow. The player controls the amplitude of the events with bow force and their rate with bow speed, and both of those are visible in the model.

The same criterion, on a fault #

The stiffness criterion was derived for a block on a spring, and the spring was a real spring. A fault has no spring attached to it, and the criterion applies to it anyway — which is worth following, because it is the clearest case of a laboratory result crossing eleven orders of magnitude without changing form.

What plays the part of the spring is the rock around the slipping patch. If a circular patch of radius $r$ slips by some amount, the surrounding elastic medium resists, and the resistance per unit slip is of order the shear modulus divided by the radius. So a patch of fault has a stiffness, it is $G/r$, and it is larger for smaller patches. A small patch is a stiff spring.

Put that into the criterion. Stability requires $G/r > (b-a)\sigma/D_c$, which rearranges to a length:

$$h^* \sim \frac{G\,D_c}{(b-a)\,\sigma}.$$

A slipping region smaller than $h^*$ is held by rock stiff enough to bleed off the stress as fast as the contact loses its grip, and it creeps. A region larger than $h^*$ cannot be held, and it accelerates. With a shear modulus of 30 GPa, a memory distance of a centimetre, a velocity-weakening parameter of half a per cent and an effective normal stress of 100 MPa, that length comes out at several hundred metres.

So an earthquake has a minimum size below which it cannot begin, and the process of reaching that size — nucleation — is a slow accelerating creep over a patch growing toward $h^*$. Whether that creep is detectable before the event is the whole of the earthquake-prediction question, and the honest answer is that it has been seen in the laboratory every time and in the field almost never.

The criterion also explains something about depth that has no other explanation. Earthquakes in continental crust nucleate between roughly five and fifteen kilometres and almost never above or below. Above, the fault is unconsolidated gouge, which is velocity-strengthening — $a$ exceeds $b$, the criterion cannot be met at any stiffness, and the fault creeps. Below, the rock is hot enough to deform plastically rather than by brittle slip, and the contact strengthens with rate again. The seismogenic zone is the depth interval over which $a - b$ happens to be negative, and its top and bottom are two different reasons for the same sign change.

That is a strong claim and it is testable in the ordinary way: measure $a$ and $b$ on the rock types found at those depths, at those temperatures and pressures, in a laboratory apparatus a few centimetres across. It has been done, and the sign changes where the seismicity stops.

What a hold is worth #

One more measurement completes the picture, and it is the one that turns the model from a description of sliding into a description of events.

Slide a contact steadily, stop it dead for a while, and start it again. The friction required to restart is larger than the friction that was being sustained, and the excess grows as the logarithm of how long the contact was held — about one per cent of the friction per decade of hold time, over hold times from a second to a week. A contact left alone for a day is a few per cent stronger than one left alone for a minute.

That is what “static friction” is, measured properly: not a constant, but a value that depends on how long the question was left before it was asked. The two-coefficient model’s $\mu_s$ is the answer for whatever hold time the experimenter happened to use, which is why the tabulated values disagree between textbooks by more than their stated precision.

The logarithm has the same origin as the one in the rate law. Junctions creep and grow under load, and a process whose rate falls as the thing it is doing progresses produces a logarithm in time — the same shape as a thermally activated process working against a barrier that rises as it goes.

What this buys, in the fault case, is a recurrence relation. A fault that has just slipped is weak; it then heals logarithmically while the plate loading rebuilds the stress linearly. The next event happens when the rising stress meets the rising strength, so a longer wait gives a stronger fault and therefore a larger stress drop and a larger event. That is the physical content of the observation that a fault segment quiet for a long time is more dangerous than one that ruptured recently — an intuition often stated as though it were mere accounting of accumulated slip, and which is really a statement about healing.

Where the model stops #

The rate law is empirical and its logarithms are not exact. The direct and evolution effects are logarithmic over perhaps five decades of speed and they cannot be logarithmic at zero, where a logarithm has no value. The regularised form used for the figures here — which comes from treating slip as a thermally activated process — agrees with the logarithm everywhere the logarithm is sensible and stays finite where it is not. That regularisation is why the block in these figures creeps instead of sticking, and the creep is real: what it is not is measured, because nobody has measured a friction law at a nanometre a second.

There is only one state variable. A real contact has a distribution of junction ages and a distribution of sizes, and compressing all of that into one number $\theta$ with one memory distance is a strong assumption. It fails most visibly after a long hold, where real surfaces keep strengthening logarithmically in time for as long as anyone is willing to wait, and where a single-state model saturates.

Nothing here is a temperature. Sliding fast enough to matter deposits enough power in a small enough volume to change the surface chemically, and above some speed the contact is no longer the contact that was characterised. Brake fade is that limit, and no rate-and-state law contains it.

And the treatment is one-dimensional. A real brake pad is an extended body that can tilt and can slip on part of its face while sticking on the rest, so the interesting instabilities in machinery are often modes of a system rather than of a point contact. Which of those modes a given design excites is a question about geometry, and it belongs to the collections that own structures and machines rather than to this one.

What the pictures cannot show #

No figure here shows the surface. Every quantity in the model — the state, the memory distance, the two coefficients — is an average over a population of contacting asperities that is never drawn, and the drawing of it would be a drawing of an assumption. What is drawn instead is what a testing machine records, which is a force against time and a displacement against time, and that is exactly what the theory was built to reproduce.

Nor can any figure show the absence the essay is about. A stiff system sliding quietly looks like a horizontal line, and a horizontal line is what a contact with no interesting properties would also produce. The information is in the comparison, which is why every figure here draws two cases that differ in one thing.

Where this ladder goes next #

The rung below this one, friction as a response with no value of its own, establishes what a coefficient of friction means and where it means anything at all — at one instant, at the corner of a response curve. This rung is what happens when that corner is looked at more closely and turns out not to be a corner: there is a curve there, it has a slope, and the sign of the slope is the difference between a machine that works and one that screams.

The pattern generalises past friction, and the generalisation is the part worth carrying away. A system loses stability when something in it gives way faster than the thing holding it can let go. That is the content of the criterion, and it appears wherever a restoring element is in series with a weakening one: in the snap-through of a shallow arch, in the necking of a bar in tension, in the stretch of a van der Waals isotherm where the compressibility turns negative and the smallest fluctuation grows, in a power supply feeding a negative-resistance load, and in a fault whose surrounding rock is either stiff enough to bleed off the stress or not. In every case the stabilising remedy is the same and it is the counterintuitive one — not a weaker weakening, but a stiffer surroundings.

The essay one might expect next is about how the sawtooth is heard rather than how it is made, and that is resonance’s ladder rather than this one. What is left on this ladder is contact as an area rather than as a law — the thing the energy accounting of a slope has to be told about before it can say where the energy went — which is where the coefficient’s independence of area came from in the first place, and where the next rung will start.