Two coefficients, not one

Friction enters this collection in one place and does a great deal of work there. A thread is held at a crossing by μ times the normal force; a cloth frays, a seam slips and a tuft comes away when that product is too small over the length available; a cloth’s resting state is a band whose width is friction over stiffness. Every one of those uses a single coefficient, quoted as a range because it is a measurement, and the range is 0.2 to 0.4 for cotton on cotton.

There is one argument on this site that cannot be made with a single coefficient at all, and it has been made anyway.

A cloth relaxes until its threads stop pushing says that a relaxed cloth does not sit at the bottom of its energy well but anywhere in a band around it where the restoring force is below the friction, and that agitation helps because it lowers the effective friction rather than because it adds energy. That sentence has two coefficients in it. Agitation cannot lower a coefficient; what it does is put the cloth into a sliding state, and a sliding contact resists less than a stuck one — which is a statement about two numbers.

What the second number is #

Kinetic friction runs about three quarters of static for fibre on fibre, and the ratio is remarkably stable: the spread in the reported figures is between measurements rather than between fibres. So it is carried here as a ratio with a range — 0.6 to 0.85, working value 0.75 — rather than as a second table of coefficients, and every result that uses it is reported at both ends.

The site’s existing coefficient is the static one, and saying so settles something that would otherwise be ambiguous everywhere it is used. A pull-out test reports the force at which the thread first moves. That is a static measurement, it is what the trade quotes, and it is what a fraying width and a seam verdict should be computed with — those are all questions about whether something starts to move.

What changes, and what does not #

Almost nothing in the grip ladder changes, and that is worth saying before the thing that does.

A crossover length is the length at which a thread breaks rather than starts to slide, so it is a static question and the static coefficient is the right one. A seam slips or does not according to whether the thread ever begins to come out. A tuft is pulled and either moves or does not. Every verdict in the applied field is a question about the onset of motion, and onset is static.

What changes is everything about a cloth that has already moved, or is being made to move repeatedly, and that is the whole of relaxation.

The resting band, twice #

A cloth left alone is stuck. Its threads are pressed together at every crossing, the restoring force from the crimp is trying to move them, and it succeeds only where that force exceeds μ_s·N. So the states it can rest in are an interval around the least-energy state, and its width is the static friction over the stiffness.

Shake it and the contacts break away. The threads slide at μ_k·N instead, and the cloth keeps moving until the restoring force falls below that — a smaller number, so a narrower band.

And the narrowing is smaller than the ratio #

The tidy version of the argument says the band’s width is friction over stiffness, so lowering the friction in a ratio lowers the width in the same ratio. That would make the narrowing exactly 0.75.

It is 0.87.

The tidy version assumes the restoring force is linear in the extension, and a cloth’s is not. The force rises steeply away from the least-energy state — that is the crimp stiffening as the threads run out of room to interchange — so cutting the friction by a quarter does not move the band’s edges by a quarter of the way in. It moves them by much less, because they were already out where the curve is steep.

At a ratio of 0.6 the narrowing is 0.76; at 0.75 it is 0.87; at 0.85 it is 0.93. The narrowing always lies between the ratio and one, and approaches one as the ratio does, which is what is asserted rather than any of those values.

Which narrows an inference rather than settling one #

The rung below used the band to say something about where a yarn sits in its own bending bracket, and the argument is worth restating because this rung sharpens it.

The free bending bound predicts a resting band of about eleven per cent for a sheeting. A washing test on real cloth finds one to three. That is a gap of about four, and that rung resolved it by concluding that the yarn must be four to twelve times stiffer than the free bound — which is how the bending bracket got narrowed from four hundred to three.

Some of that gap is agitation, because a washing test agitates. How much is now computable: the band narrows to 0.87 of its stuck width, so eleven per cent becomes 9.5, and the gap goes from 3.7 to 3.2. Two coefficients close about a fifth of it. The coefficient ratio alone would have suggested a third, and the curvature takes most of that back.

So that rung’s inference survives and is slightly narrowed. That is the honest result and it is a small one; what makes it worth a rung is that the alternative — treating the ratio as the narrowing — would have overstated the correction by two thirds, and would have looked entirely reasonable.

What it does to a hysteresis loop #

A band is a static picture and the interesting object is the loop.

Stretch a cloth and let it go. On the way out the threads slide at the kinetic coefficient once they are moving, so the force needed is below what the geometry alone would demand. On the way back they stick first — the restoring force has to reach the static limit before anything moves at all — and then slide at the kinetic one again. So the loop is not a single curve traversed twice at different frictions. It has four parts: stick, slide out, stick, slide back.

The area of the loop is the energy dissipated, and with one coefficient it is 2μ·N times the displacement. With two it is the same expression in the kinetic coefficient, plus a fixed amount at each reversal for breaking away — the difference between the two coefficients times the normal force, twice per cycle, independent of how far the cloth was stretched.

So a cloth cycled through small extensions dissipates disproportionately more per unit of extension than one cycled through large ones, because the break-away cost is paid per cycle rather than per millimetre. That is why a fabric worked back and forth over a short range warms and softens faster than the same fabric stretched once a long way, and it is a statement this site could not have made an hour before it had two coefficients.

None of that is computed here. It is written down because the machinery now permits it and because the shape of it — a per-cycle cost that does not scale with the cycle — is the kind of thing that decides a fatigue behaviour and is recorded as not done.

The loop has an intercept, and the intercept is the difference #

The hysteresis argument above is recorded as not done, and one part of it can be written down without computing anything, because it is a statement about the shape of a measurement rather than about its size.

Dissipation per cycle has two terms. The sliding term is the kinetic coefficient times the normal force times the distance travelled, twice — so it is proportional to the amplitude. The break-away term is the difference between the two coefficients times the normal force, paid once at each reversal against whatever compliance stands between the grip and the load — so it is a constant, independent of the amplitude.

Energy per cycle = A + B·Δ, and neither term is a correction to the other.

Three things follow and all three are checkable on a machine that already exists.

Dissipation per unit of extension diverges at small amplitude. It is A/Δ + B, so a cloth worked back and forth over a tenth of a millimetre dissipates far more per millimetre travelled than the same cloth stretched once through a centimetre. That is the arithmetic behind a seam that warms and softens where a garment flexes, against a panel of the same cloth that is stretched harder and less often and does neither.

There is a crossover amplitude, and it is A/B. Below it the loop is dominated by breaking away and above it by sliding. Its value is the ratio of the coefficients’ difference to the kinetic coefficient, times the break-away compliance — so it is a length set by how far the cloth moves before a contact lets go, which is a small fraction of a thread spacing.

And the intercept is a measurement of the coefficient difference. Plot the loop area against the amplitude and the arithmetic says a straight line with a positive intercept. The slope gives μ_k·N and the intercept gives (μ_s − μ_k)·N times the compliance. That is the difference between the two coefficients, obtained from a fabric test rather than from a yarn-on-yarn friction rig — and the difference is exactly the quantity this rung had to import as a ratio with a range on it.

Whether the intercept is measurable in practice is a question about how much scatter a cyclic fabric test carries, and this collection cannot answer it. What it can say is that the sign and the form are predictions rather than fits: a single-coefficient account predicts a line through the origin, and a two-coefficient account predicts one that does not pass through it. A loop-area sweep at four amplitudes would separate the two accounts, which is a cheaper experiment than either coefficient’s own measurement and is not, as far as this collection has found, run.

The caution is the compliance. The break-away cost is a force times a displacement, and the displacement is whatever elastic travel sits between the two surfaces before one lets go — a property of the thread’s own stiffness at the crossing rather than of the friction. So the intercept is a product of two things and only one of them is the coefficient difference. Reading it as a friction measurement needs the other, and the other is the same transverse stiffness the compression ladder had to fit.

What was counted, and how #

The bands are read off the site’s own load–extension curve at 641 samples, which quantises their width; running at 2,561 samples moves the narrowing by three parts in a thousand, so the sampling is not what produces the result.

The assertions are relations rather than values, at three friction coefficients and three ratios. The narrowing is required to be less than one, which is the direction. It is required to be greater than the ratio, which is the finding. And it is required to rise as the ratio rises, which is the mechanism — a kinetic coefficient nearer the static one narrows the band less.

Asserting that the narrowing equals the ratio to within some tolerance was the first version, and it failed on its first run at a ratio of 0.6, where the narrowing is 0.755 against a claimed 0.6. That failure is what produced the result, and it is the fourth time in which an assertion written against the numbers in front of it rather than against its claim has caught something. The rule of the house holds: assert the relation or the regime, never the value the defaults happen to produce.

What it does not change, which is most of the site #

Worth ending on the negative result, because a new constant that changed everything would be a suspicious constant.

Every applied answer here is a question about the onset of motion — does the thread start to come out, does the tuft move, does the seam begin to open — and onset is static. So the fraying widths, the seam verdicts, the tuft anchorages and the crossover lengths all keep the coefficient they had, and the site’s friction table is now labelled as static rather than reinterpreted.

Every geometric answer is untouched, because friction was never in it. The matrix, the censuses, the satin theorem, the integrity criterion, the crimp geometry and the cover arithmetic contain no friction coefficient of any kind.

What changes is the small set of results about a cloth that has already moved: the resting band, relaxation, hysteresis, and what a knit does. Those are the places where the history of the fabric matters, and history is exactly where the difference between sticking and sliding lives.

That is a reasonable footprint for a second constant. A distinction between two coefficients ought to matter wherever motion has occurred and nowhere else, and the pattern of what moved and what did not is a check on the account rather than a limitation of it.

Where the model stops #

Both coefficients are quasi-static. What resists an impulsive slide is neither of them, and a beat-up is a blow, a snag is a jerk and a garment in wear is loaded and unloaded thousands of times a day. The beat-up’s frictional half uses a static coefficient for a contact that is being struck.

The ratio is treated as a property of the fibre pair and it is not quite. Kinetic friction depends on sliding speed, and a cloth relaxing in a wash tub slides at speeds nobody here has estimated. The stability of the reported ratio across fibres is what makes a single number defensible; the stability across speeds is not tested.

Nothing here is a real stick-slip cycle. The trace drawn is a sawtooth between two levels, and a real contact has a break-away transient, a velocity-dependent recovery of the static coefficient with time at rest, and — for fibres — a genuine dependence on how long the contact has been sitting. Ageing at rest is why a cloth folded for a year holds its crease.

And the normal force is a tensioned cloth’s in the grip ladder and a relaxed cloth’s in the relaxation ladder, and the two have only just become separate numbers.

The generalisation #

An explanation that names a mechanism the model cannot compute is a placeholder, and it will pass every check until somebody tries to compute it.

“Agitation lowers the effective friction” was written on this site, was correct, was well-motivated, and was not computable with the constants the site carried. Nothing failed. No gate looks for a sentence that requires a quantity the model does not have, and there is no obvious way to make one look.

The second lesson is about proportionality arguments in general. A width divided by a stiffness is proportional to the width only if the stiffness is constant, and the whole reason a resting band is interesting here is that the stiffness is not — the band exists because the restoring force is small near the minimum and grows away from it. So the very feature that makes the band worth computing is the one that breaks the proportionality, and it was invisible until the arithmetic was run.

That pattern is worth carrying past textiles. Wherever a tolerance band comes from a threshold on a nonlinear response, halving the threshold does not halve the band, and the discrepancy is largest exactly where the band is widest and matters most.

Who found it, and when #

The distinction between static and kinetic friction is Coulomb’s and is in every account of the subject. Its measurement for fibre on fibre goes back to the 1940s and the ratio has been stable ever since; the textile literature is unusual mainly in how carefully it separates the two, because yarn friction is anisotropic — against the scales of a wool fibre it differs by direction — and separating the coefficients is forced on anybody measuring it.

That agitation helps a cloth reach its relaxed dimensions is trade practice of the oldest kind and is the entire basis of the standard relaxation procedures a fabric is tested by.

What is this site’s is the arithmetic saying how much of the gap that buys, and the finding that it buys less than the coefficient ratio would suggest.

Where the ladder goes next #

The next rung spends it where it belongs, in the finishing field: why agitation helps a cloth relax and how much of a standard relaxation procedure it explains.

Sideways, the same two coefficients are what a knit’s dimensions are set by — and a knit turns out to have no elastic restoring force at all, so friction is not one term among several there but the whole of the mechanism.