How much of a cloth is touching

Two surfaces in contact touch on much less of themselves than they appear to. That is the founding observation of the physics of friction, it is why friction is proportional to load rather than to area, and it has been standard for solid bodies since the 1930s. For cloth it has never been computed, because nobody had a surface to compute it from.

The claim #

A woven cloth under an ordinary touch is in contact on a few per cent of its plan, the fraction is decided by the draft far more than by the load, and below about ten micrometres of approach the arithmetic is describing a surface that is not there.

Three statements and they get progressively less comfortable.

The area is small. At five kilopascals — roughly what a hand resting on a table exerts — a two-and-two twill in an ordinary sheeting is touching 4.4 per cent of its plan. At a hundred kilopascals, which is a firm press between finger and thumb, it is touching 17 per cent.

The load barely moves it. Twenty times the pressure buys four times the area, because the bearing curve is a square root twice over: the depth grows slowly with load and the area grows slowly with depth.

And the model runs out before the pressures do. Getting a satin to half a per cent of contact requires an approach of ten nanometres, which is not a distance at which a yarn has a surface. Long before that, what is in contact is the hair layer.

The arithmetic, which is deliberately crude #

Converting a depth into a pressure needs a stiffness, and the model used is the simplest one that is honest: the material inside a contact patch is compressed by the approach over the thread’s own thickness, so the pressure on the patch is G·δ/b, and the pressure on the cloth is that times the bearing fraction.

That is a Winkler foundation — a bed of independent springs — and it is not a contact mechanics. It ignores that the material beside a patch is dragged down with it, it ignores that a thread resting on another thread is supported at a point rather than on a rigid base, and it has no Hertzian pressure distribution in it. Every one of those would change the numbers by tens of per cent and none of them changes an ordering.

What it must not be allowed to conceal is where G comes from. This collection has established that a yarn’s transverse stiffness has no lower bound at all: if the fibres may slide past one another freely, nothing resists a change of the section’s shape at constant area, and a bundle of loose fibres in a sheath is a fluid in cross-section. So the bracket runs from zero to the solid modulus and cannot be narrowed by any care taken over the fibre.

The number used here is therefore the value the compression work fitted to measured fabric thicknesses — four newtons per square millimetre, with published yarn measurements lying between about one and ten. Every pressure in this essay is quoted across that whole range, and the ratios between weaves, which is what the essay is actually about, do not depend on it at all: G appears once, linearly, in every curve.

What was counted, and how #

The bearing area at a depth comes from the closed form rather than from the sampled surface, because every depth in this essay is a few micrometres and a sampling grid cannot resolve a strip a few micrometres wide.

The closed form is exact while the crowns have not merged, which at these depths they have not: the largest contact fraction anywhere in the essay is seventeen per cent, against a cover factor near seventy. Where the two routes overlap they agree to a few per cent, and the residual is the sampling.

The pressure is then found by bisection on a monotone function, which it is: the bearing area only grows with depth and the pressure is the product of two increasing things. No local minimum can hide in it and the answer is exact to the last bit of a double.

Four pressures, and what each of them touches #

The arithmetic is worth reading at pressures a reader can put a name to. All four are for a two-and-two twill in an ordinary sheeting, at the fitted stiffness.

Half a kilopascal, which is a sheet lying on a bed: the plate has sunk 1.1 micrometres and is touching 2.0 per cent of the plan.

One kilopascal, which is what a thickness gauge applies by standard: 1.8 micrometres, 2.6 per cent.

Five kilopascals, a hand resting: 5.4 micrometres, 4.4 per cent.

A hundred kilopascals, a firm pinch between finger and thumb: 27.8 micrometres, 16.8 per cent.

Two hundred times the pressure buys eight times the area. That is the whole character of the answer, and it comes from the two square roots compounding — the bearing curve opens as the square root of depth, and the pressure needed goes as the depth times the area, so the area goes as the two-thirds power of pressure at best. A cloth is not a surface that can be made to touch by pressing harder.

Across the drafts at one pressure the spread is enormous by comparison. At one kilopascal the plain weave is touching 1.2 per cent, the twill 2.6 and the eight-end satin 5.4 — a factor of four and a half between two cloths that differ in nothing but which threads are lifted.

What the number is worth #

Small contact areas explain three things about cloth that are otherwise a matter of assertion.

Why fabric friction is not proportional to area. The two coefficients a cloth has are the standing puzzle of textile friction, and part of the reason a cloth’s friction does not scale with its apparent area is that its apparent area is not the area in contact. Doubling a specimen doubles the load as well as the plan, so the real contact area — and with it the friction — stays proportional to load rather than to size.

Why a cloth feels cool and then does not. Heat crosses into a fabric through the material actually touching, and at four per cent the conductance is a twenty-fifth of what a continuous contact would give — which is a quite different route from the still air a fabric holds. The initial coolness of a fabric against skin lasts as long as it takes that contact to grow — which under a resting hand is a matter of the fibres creeping, and is why a fabric feels cool for a second and then does not.

And why a light rub does so much damage. If a load is carried on four per cent of the plan, the pressure on the crowns is twenty-five times the nominal pressure, and it is the crowns that are being rubbed. That is taken up in its own place, where the ratio turns out to be worse than that again.

The pressure on the crowns, which is the number that does damage #

There is a second pressure in every one of these figures and it is the one that is never quoted.

The nominal pressure is the load divided by the plan. The contact pressure is the load divided by the area actually carrying it, and the ratio between them is one over the bearing fraction. At five kilopascals nominal on a twill, the crowns are carrying 5/0.044 — a hundred and thirteen kilopascals, twenty-three times the figure anybody would write down.

That ratio has a floor and it is a long way above one. Even at a hundred kilopascals nominal — pressing hard — the crowns carry six times that. A fabric never distributes a load. It concentrates it, by a factor between six and fifty over any range of loads that a person or a machine is likely to apply, and the concentration is largest exactly where the load is smallest.

Two consequences follow immediately. The threads at a crown are being worked far harder than a nominal pressure suggests, which is why the transverse flattening this collection computes elsewhere begins at loads that look trivial. And the ordering between weaves reverses depending on which pressure is asked about: a satin carries a given load on more area, so its crowns are under less pressure — the same fact that makes a satin touch more makes it press less.

Where the model stops, and this time it stops hard #

The hairs get there first, and they get there by a long way.

The pressure inside a twisted yarn falls to exactly zero at its surface, so the outermost fibres are held by nothing and some of them stand off. A cotton fibre is about fifteen micrometres thick and a protruding hair may stand tens of micrometres clear of the yarn. Every depth in this essay is between a tenth of a micrometre and thirty.

So the honest statement is uncomfortable and worth making plainly. Over most of the range computed here, the surface being described is buried inside the hair layer. What a plate at five kilopascals is actually touching is some number of fibre ends, and the count of those is a spinning property rather than a weaving one. The yarn-surface arithmetic is a lower bound on the contact area — the hairs add to it — and an upper bound on the pressure.

Where it is right is at the other end. Under a heavy load the hairs are crushed flat and irrelevant, the yarn surface is what carries, and the numbers above are the numbers. A calender, a mangle, a nip, a seam under tension — those are the regimes this arithmetic is describing correctly, and they are exactly the regimes in which nobody was in doubt that the cloth was being squashed.

Three smaller limits, for completeness. The springs are independent, so nothing here has a proper contact mechanics in it. The plate is rigid and perfectly flat, which a finger is not and another cloth is emphatically not — that case has two bearing curves rather than one. And the cloth is unrelaxed and unfinished; a cloth that has been washed has crimped further and a cloth that has been calendered has a quite different surface.

The number that resolves the discomfort #

The section above ends with an honest pessimism — that the geometry answers well for a nip and poorly for a hand, because the hairs get there first — and it ends there because the pressure at which the hairs stop mattering was not available. It is now, and it is more favourable than the guess.

A hair layer carries about two tenths of a kilopascal before a flat plate reaches a woven cloth’s crowns at all. Set the four named pressures against it:

All four are on the crown side. The arithmetic in this essay is describing a surface that is genuinely being touched at every pressure it quotes, and the hair layer dominates only below half a kilopascal — which is lighter than a bedsheet.

So the closing summary should be read the other way round. The geometry answers well for a hand, and it answers poorly for a caress: a fabric brushing skin at fifty pascals is a quarter of the crossover, and there the essay’s caution stands in full.

The exception is a raised cloth, where multiplying the hair population moves the crossover to fifteen kilopascals and puts a hand back on the hair side. On a bare woven cloth the crowns are what a hand meets; on a flannel they are not, and the difference is a finishing operation rather than anything in the draft.

That is the sort of correction worth recording rather than quietly applying. The essay’s discomfort was correct as an instinct, was over-stated as an estimate, and is resolved by a number computed later in a different ladder for a different purpose.

Which also caps the crown pressure #

The concentration factor — nominal pressure over bearing fraction — is quoted above as running between six and fifty, and the range’s upper end is open in the model: the bearing fraction goes to zero with the load, so the concentration grows without limit as the touch lightens.

The measured behaviour across this essay’s own four points is a bearing fraction rising as about P^0.45, so the concentration falls as P^(−0.55): halving the load raises the crown pressure by forty-seven per cent, and there is nothing in the geometry to stop it.

The hair layer stops it. Below the crossover the crowns are not carrying anything, and what is in contact is a population of fibre tips each pressing at its own buckling stress — which is about a tenth of a megapascal and does not depend on the applied load at all.

So the two ladders bracket the local pressure from both sides:

above 0.2 kilopascals nominal it is the crown pressure, rising as the load falls; below it, it is fixed at the fibre’s own buckling stress.

At the crossover the two are of the same order — a hundred and thirteen kilopascals on the crowns at a hand’s touch, a hundred at a hair tip — which is a coincidence worth noticing and not, on reflection, a surprising one: both are the pressure at which a cellulose structure a few micrometres across gives way.

A cloth’s surface, from a breath to a pinch, delivers a local pressure of the order of a tenth of a megapascal and never anything else. What changes over that whole range is not how hard the contacts press but how many of them there are, and that is as good a one-line description of what a fabric’s surface is for as this ladder has produced.

What a specification would have to say #

None of this is in any fabric specification, and putting it in one would require three numbers rather than the usual zero.

A specification states a thread count, a yarn count and a weave. From those, everything above follows — the surface, the bearing curve, the contact at any stated pressure. So the information is already there and nobody extracts it.

What a specification could usefully add is the pressure at which its numbers are meant. A fabric described as smooth is smooth at a particular touch, in the way that a thread count says nothing without a yarn count beside it; the same fabric under a different touch is being asked a different question of its surface, and two fabrics can trade places between one pressure and another. That is not hypothetical: a plain weave and a satin cross over nowhere in the range computed here, but a heavily calendered plain weave and an as-woven satin cross at about twenty kilopascals, because a flat top bears at once and a round crown has to be crushed first.

The comparison that a hand makes is at about five kilopascals, which is at the very bottom of the range where any of this arithmetic applies. That is an uncomfortable place for a model to be asked to work, and it is exactly where the hair layer dominates — so the honest summary is that the geometry answers well for a nip and poorly for a hand, and the hand is what most specifications are written about.

The generalisation #

A real contact area is a property of the top of a height distribution, and it is small.

The transferable form of that is Bowden and Tabor’s, and it is not this collection’s to claim: whenever two rough bodies meet, the area in contact is a small fraction of the area in view, it grows nearly in proportion to the load rather than to the size of the bodies, and every property that depends on contact — friction, conduction, adhesion, wear — inherits that proportionality.

What a fabric adds is the source of the roughness. In a machined metal the height distribution is a residue of manufacture, near enough Gaussian, and describable by a couple of moments. In a cloth it is designed: the distribution is bimodal, its shape is a matrix somebody chose, and the same yarn can be given contact areas differing by any factor one likes simply by rearranging the lifts.

That is a genuinely unusual situation. Almost nowhere else can the statistics of a surface be specified in advance, exactly, by a discrete choice.

Who found it, and when #

The small-contact-area result is Bowden and Tabor’s, from work through the 1930s and 1940s, and the standard account of friction has rested on it since. Its extension to rough surfaces with a distribution of heights is Greenwood and Williamson’s, 1966, whose model assumes a Gaussian distribution of asperity heights — an assumption a woven cloth flatly violates, since its heights cluster at two crown levels with a gap between them.

The textile literature’s own contact work is nearly all empirical: friction coefficients measured, contact areas inferred from optical or electrical measurements, and the geometry left as a picture. What this collection contributes is the geometry — a bearing curve computed from a draft rather than measured off a specimen — and, with it, the observation that the answer is not a fabric property but a weave property, differing by a factor of fifty between two drafts of one cloth.

Where the ladder goes next #

To the measurement this arithmetic sits inside without anybody noticing: a thickness gauge reads the draft, because a presser foot sinks until the bearing area can carry it, and how far that is depends on the weave.

Sideways, the same curve read as a rate rather than as a state is how a cloth compresses, which turns out to explain the shape of a pressure–thickness curve that has always been fitted with an empirical power law.