Friction is two surfaces, not one

Everything so far in this ladder has pressed a fabric against a rigid flat plate, which is what a thickness gauge, a compression tester and a bearing curve all assume. Almost nothing a fabric actually meets is a rigid flat plate. It meets skin, and it meets other fabric — and another fabric has a bearing curve of its own.

The claim #

The contact between two fabrics is the convolution of their two height distributions, and it is between seven and twenty-five times smaller than either surface against a plate at the same approach.

Three consequences.

A friction coefficient measured against a plate does not transfer. The real area of contact is the quantity friction is proportional to, and it differs by more than an order of magnitude between the two geometries.

The suppression is largest where the touch is lightest. At a fifth of a micrometre of approach the factor is thirteen; at four micrometres it is twenty-five; by twenty micrometres it has fallen to seven. There is a worst case and it sits in the middle of the range a hand explores.

And the registration is an input that nobody sets. Two pieces laid crown into valley and two laid crown on crown are different contacts, and no test specifies which.

The argument #

Two points, one on each surface, are in contact when the sum of their depths below their own surfaces’ maxima is within the approach.

That is the whole of it, and it makes the pair’s bearing curve a convolution. If one surface has a fraction f(z) of its plan at depth z and the other g(z), then the fraction of the pair in contact at an approach δ is the double sum of f(z₁)g(z₂) over all pairs with z₁ + z₂ ≤ δ.

A convolution of two distributions that both start at zero starts much more slowly than either. Each surface has only a per cent or two of its plan within a few micrometres of its own top; requiring both at once multiplies two small numbers, and the product is very small indeed.

What was counted, and how #

The two height distributions are histogrammed off the sampled surfaces — six hundred bins across the cloth’s thickness — and convolved exactly. Nothing is fitted and no distribution is assumed.

The assumption of a shape is precisely what had to be avoided. The standard treatment of rough contact, Greenwood and Williamson’s, takes the asperity heights as Gaussian, which is a reasonable description of a machined metal and is nothing like a woven cloth: a fabric’s height distribution is bimodal, with a peak at each thread system’s crown height and a long tail running down into the interstices. Fitting a Gaussian to it would misstate the top, which is the part that matters.

The measured suppression for a two-and-two twill against itself, as a ratio of single-surface contact to pair contact: 13.5 at 0.7 micrometres of approach, 21.1 at 1.9, 24.7 at 3.7, 12.8 at 7.5, and 6.7 at 18.7. The median gap between the two surfaces is 160 micrometres, which is a large fraction of the cloth’s own thickness — two cloths lying together are mostly not in contact at all.

Why there is a worst case in the middle #

The suppression rises and then falls, and both ends have reasons.

At the very top it is limited by the crowns meeting crowns. However small the approach, there is always some chance that a high point of one surface lands on a high point of the other, and that chance is the product of two small probabilities but it is not zero. So the ratio does not grow without limit as the approach falls; it flattens.

At the bottom it is limited by there being nothing left to suppress. Once the approach is large enough that most of both surfaces is in play, requiring both at once costs less, and the two curves converge.

In between the two effects are both weak and the product of two small numbers is at its smallest relative to either. For this cloth that is at three or four micrometres of approach — which is the depth a light touch produces, and which is exactly the regime in which fabric handle is judged.

What it does to the two coefficients #

This collection has already found that a cloth has two friction coefficients rather than one — that fabric friction refuses to reduce to a single number in the way a textbook coefficient is supposed to.

The pair argument supplies part of the reason. A friction coefficient is a ratio of a force to a load, and the force is proportional to the real area of contact. If the real area depends on what the fabric is sliding against — and by more than an order of magnitude — then the coefficient does too, and there is no property of the fabric alone that it could be.

Three cases, all of them measured under the same name in the literature:

Fabric against a hard smooth plate. The plate’s own bearing curve is a step at zero, so the pair’s curve is the fabric’s own. This is the case every bearing curve in this ladder computes.

Fabric against fabric, at random registration. The convolution above, with a suppression of about twenty at a light touch.

Fabric against fabric, in register. Two pieces of one cloth aligned crown to crown, which is a case the convolution does not describe — it assumes the two surfaces are statistically independent, and two aligned pieces of one cloth are perfectly correlated. Contact would be at the single-surface value, or better.

What it says about a seam under load #

The pair result has an immediate application, and it is one where the geometry is not averaged away.

A seam slips before it breaks: threads are drawn out of the cloth at a stitch line because the friction holding them is finite. Part of that friction is thread against thread inside the fabric — which this collection computes with a capstan argument — and part is cloth against cloth across the seam allowance, which is the pair contact above.

The second part is much smaller than a plate measurement would suggest, by the factor computed here, and it is smallest at exactly the light clamping loads a sewn seam applies away from the needle line. So a seam’s resistance to slippage is dominated by the thread-in-cloth grip and not by the two plies rubbing, which is the ordering the seam essays already assume and which now has a reason.

The same argument explains a piece of garment behaviour that is otherwise puzzling: two plies of cloth slide over each other far more freely than either slides over a table, which is why a lining works, why a garment with a lining hangs differently from one without, and why two layers of the same cloth feel slippery against one another and grippy against skin.

The registration nobody sets #

Two pieces of the same cloth laid together can be anywhere from perfectly aligned to perfectly out of phase, and it changes the contact by the whole factor computed above.

No test specifies it and no test could. A friction rig clamps two specimens and slides one over the other; the relative phase of their repeats is whatever it happens to be, it changes as the specimens slide, and at a sett of thirty ends to the centimetre the phase runs through a whole cycle every third of a millimetre of travel.

So a fabric-on-fabric friction measurement is an average over registration, taken over however far the rig slides. That is a defensible thing to measure and it is not a property of two surfaces; it is a property of two surfaces and a stroke length.

The prediction that follows is testable and slightly odd: a friction trace against sliding distance should show a periodic component at the repeat’s own pitch, largest for a cloth with a long repeat and a coarse sett. Stick–slip in fabric friction is well documented and is usually attributed to fibre entanglement; a geometric contribution at exactly the thread pitch would be distinguishable from it by its period.

Two different cloths, which is the ordinary case #

Everything above pairs a cloth with itself, which is the extreme case for correlation and the easy case for arithmetic. The ordinary case is two different fabrics, and the convolution handles it without change.

What it predicts is worth stating because it is not symmetric. The pair’s contact is dominated by whichever surface is smoother, in the sense of having more of its plan near its own top. Pair a satin with a plain weave and the plain weave’s scarcity of high points is the limiting factor; the satin’s abundance cannot help, because a contact needs both.

So a smooth fabric against a rough one behaves like a rough pair, and improving the smooth one buys nothing. That is a genuinely useful design statement: in any assembly where two fabrics slide — a lining against a shell, a sheet against a blanket, a belt against a cover — the contact is set by the worse of the two and there is no point finishing the better one.

It also predicts the exception. A plate is the limiting smooth surface: its whole plan is at its own maximum, so the convolution reduces to the other surface’s own curve. Anything with a step-like height distribution behaves the same way, which includes a calendered cloth and a cut pile — both of which present a finite area at zero depth. A cut pile against a woven cloth therefore contacts like the woven cloth alone, and a cut pile against a cut pile contacts on the product of two large fractions rather than two small ones. A carpet against a shoe sole is the one fabric contact in ordinary life where nothing is being suppressed.

The suppression is one over the contact, near enough #

Five measured suppressions at five approaches is a set of numbers. They collapse into one rule, and the rule explains why the factor grows as the touch lightens without needing any of them.

Take each surface’s bearing curve as a power law near its top — φ(δ) = Aδ^p, with p = ½ for line crowns and 1 for points, which is the exponent this ladder is built on. Convolving two such distributions and dividing by the single-surface value gives a suppression of

Γ(2p + 1) ÷ Γ(p + 1)² ÷ φ(δ),

and the leading factor is remarkably insensitive to p: it is 1.27 for line crowns, 1.57 at three quarters, and 2 for points. So across every crown geometry a cloth can have,

suppression ≈ 1.5 ÷ φ, or equivalently pair contact ≈ φ² ÷ 1.5.

Two cloths touch on about the square of what one cloth touches a plate on, and since φ is a few per cent, the square is a few parts in ten thousand.

That single line reproduces the essay’s own falling branch. A suppression of 24.7 implies a single-surface contact of six per cent; 12.8 implies twelve; 6.7 implies twenty-two — all of them plausible values at the approaches quoted, and all of them moving in step. And it explains the shape without any measurement: the suppression grows as the touch lightens because φ shrinks, and it grows at exactly the rate φ shrinks.

It also says where the rule must fail, which is where the essay says the measurement turns over. At very small approaches the two surfaces stop being independent — a crown landing on a crown is not a chance event when the two pieces are cut from one cloth — and a formula built on independence has nothing to say there. The measured turnover is the independence assumption failing, and the closed form marks where.

Which sharpens the design statement, and turns it slightly #

The essay concludes from the two-different-cloths case that the pair is limited by the rougher surface and that improving the smoother one buys nothing. The product form says something adjacent and not quite that.

With pair contact proportional to φ₁ · φ₂, the two surfaces enter symmetrically. Doubling either one’s contact fraction doubles the pair’s, so improving the better fabric helps exactly as much as improving the worse one — in proportion, though not in absolute terms.

What is true, and is what the essay is reaching for, is that neither can compensate for the other. A product cannot be larger than either factor times one, so a superb surface paired with a poor one is limited by the poor one absolutely: no amount of polishing the satin gets the pair above what the plain weave alone would give against a plate. The scarce surface sets the ceiling; both surfaces set the value.

The practical form of that is more useful than either statement alone. In an assembly where two fabrics slide, the return on improving a surface is proportional and the ceiling is set by the other one — so it is worth improving whichever is cheaper, and worth knowing that the pair will never approach a plate measurement however far either is taken.

And the square explains a familiar disproportion. A finish that doubles a cloth’s contact fraction — a calender, a singe — quadruples its fabric-on-fabric contact while only doubling its contact against a plate. That is why such finishes change how a cloth behaves against another cloth far more than they change how it behaves against an instrument, and why a laboratory measurement of the effect understates what a wearer notices by a whole factor.

Where the model stops #

The hairs, and here they may dominate completely. What actually meets another fabric first is a layer of protruding fibre ends, and two hairy surfaces interpenetrate rather than meeting at a plane. The convolution above is a statement about two yarn surfaces, and on a soft-spun staple cloth it may describe something that never comes into contact at all.

Independence is assumed and is the weakest step. Two surfaces are treated as statistically independent, which is right for two different cloths and wrong for two pieces of one cloth in register. Real fabric contact is somewhere between and nothing here says where.

Nothing deforms. The convolution is geometric: it says what area lies within an approach, not what load is needed. Both surfaces would flatten under contact, which raises the area faster than the geometry alone.

And there is no friction in it. This essay computes an area, and turning an area into a friction force needs a shear strength at the interface — an adhesion, a ploughing term, a fibre-scale entanglement — none of which is geometric and none of which this collection has.

The generalisation #

When two rough bodies meet, the operative distribution is the convolution of their two, and a convolution is smaller near its origin than either factor.

That is the transferable statement and it is the reason contact problems are so much harder than they look. Every intuition about a rough surface is built from pressing it against something flat, and pressing it against something equally rough changes the answer by an order of magnitude in the regime where the pressing is light.

The corollary for measurement is sharp: any quantity proportional to real contact area — friction, thermal contact, adhesion, electrical conduction — is not a property of one surface, and measuring it against a reference flat gives a number that cannot be transferred to a pair. The reference measurement is still worth making; it just does not mean what its units suggest.

The number that would settle the hair question #

The largest doubt in this essay is whether the yarn surfaces ever meet at all, and there is a measurement that would answer it.

Compare a filament cloth and a staple cloth of the same construction. A filament yarn has almost no hair layer; a carded staple yarn has a great deal. If the convolution above is describing what actually touches, the two should show the suppression computed here in about the same ratio, because the geometry is the same. If the hairs dominate, the staple cloth’s contact should be far less suppressed — a brush against a brush interpenetrates and does not have to bring two crowns into coincidence.

And singe one of them. Singeing removes about 0.2 per cent of a cloth’s mass and takes off most of its hair layer, and this collection has already noted that it changes a fabric out of all proportion to what it removes. If a singed staple cloth’s fabric-on-fabric friction moves towards the geometric prediction while its fabric-on-plate friction barely moves, the hair layer is what was carrying the pair contact.

Neither has been done here. They are stated because the essay’s central number is a factor of twenty and the doubt about it is a factor of the same size, and a claim of that shape should carry the experiment that would decide it.

Who found it, and when #

The convolution of two rough surfaces is standard in contact mechanics and dates to the same period as Greenwood and Williamson’s asperity model, which handles it by replacing the pair with an equivalent single rough surface against a flat — legitimate when both distributions are Gaussian, and not available here.

The textile literature on friction is very large and almost entirely empirical: coefficients measured, dependencies on load and speed reported, and the geometry treated as a nuisance. What this collection contributes is a computed height distribution for both surfaces, which makes the convolution something that can be done exactly rather than assumed away.

Where the ladder goes next #

To two surfaces that are not uniform. A crepe is flat in its draft and not in its surface finds that a weave optimised to have no pattern still has a surface with structure in it; and a seam stands proud and wears first is a step in a bearing curve that takes the whole of a garment’s rubbing on two per cent of its area.