A light touch never reaches the crowns

The curve that says what a cloth touches with is the sharpest result this collection has produced about a surface. A float of two or more has a plateau whose top is a line, so the contact area opens as a square root of the depth; a float of one has no plateau and its top is a point, so it opens linearly. Two curves with different exponents diverge without limit as the load falls, and an eight-end satin at a thousandth of a cloth’s thickness touches thirty times a plain weave’s area.

The lighter the touch, the larger the difference. That is the argument, and it is correct, and this essay is about the fact that at a light touch the plate is nowhere near the cloth.

The cloth #

How much of a cloth is touching already said this in a sentence and could not do anything about it: “the hair layer takes over below ten micrometres and swallows most of the interesting range.” The bearing curve’s exponent lives at depths between a tenth of a micrometre and thirty. The hair layer reaches hundreds.

What was missing was the arithmetic to say at what pressure the crossing happens, and therefore which of the things anybody does to a fabric are on which side of it. That arithmetic needs a hair population with a length distribution, which now exists.

The claim #

A hair layer carries about two tenths of a kilopascal before a flat plate reaches a woven cloth’s crowns at all. Below that pressure every contact result in this collection describes a surface nothing is touching; above it they apply. A standard thickness test presses at five times the crossover and reads the cloth. Every light-pressure test, and every touch light enough to be called light, reads the hairs.

And for a raised cloth the crossover moves by two orders of magnitude, so nothing anybody does to a flannel by hand ever reaches its surface.

A hair is a column, and that is the whole model #

A protruding fibre meeting a descending plate bends. It is a cantilever, held where it leaves the yarn and loaded at its tip, so the force to deflect its tip by δ is 3EIδ/ℓ³ — and I = πd⁴/64, which for a twelve-micrometre cotton fibre is about 10⁻⁹ mm⁴, a very small number.

It does not bend indefinitely. A column carries load until it buckles, and past its buckling load it lies over and carries no more. For a hair standing free with nothing at its tip that limit is π²EI/4ℓ², about eleven micronewtons for a hair at the population’s own mean length.

So each hair contributes the lesser of a bending force and a buckling force, and the pressure a layer carries at a given plate height is that quantity integrated over the population above the plate. It is a bed of independent springs, which is exactly the admission the height field one level down makes about itself, and it is the same limitation for the same reason.

Where the standard pressures fall, which is on both sides #

The interesting part is not the crossover’s value but what sits around it.

A standard thickness test presses at one kilopascal, five times the crossover. It reads the cloth, which is what it is for, and everything the thickness ladder computed about a foot sinking into a bearing curve is a correct description of what it does.

A light-pressure thickness test presses at a tenth of a kilopascal — the pressure standards specify for compressible fabrics, nonwovens and pile — which is half the crossover. It reads the hair layer.

A fabric brushing against skin exerts of the order of fifty pascals, a quarter of the crossover. It never reaches the cloth.

And a seam under a presser foot is at twenty kilopascals, a hundred times over. The hairs are irrelevant there and always were.

So the standards that exist bracket the crossover, and they were written by people measuring fabrics and noticing that the answer depended on the pressure. The reason it depended on the pressure is in this essay, and the two-figure specification in every thickness standard is a fossil of it.

What it costs a thickness #

The consequence for a number everybody quotes is direct.

An ordinary sheeting measured at the standard pressure reads its own thickness. Measured at a fiftieth of a kilopascal, the foot stops nearly ninety micrometres above the crowns and the reading is nearly half as much again. A third of the light-pressure reading is a population and not a fabric, and nothing has been compressed to make the difference — the foot has simply stopped somewhere else.

For a raised cloth the same comparison spans a factor of three, from six tenths of a millimetre at the standard pressure to one and three quarters at the lightest, and it is why pile and nonwoven standards specify the low pressure. A napped fabric measured at a kilopascal is being measured with its nap crushed flat, which is not the state anybody buys it in.

The result the crossover does not overturn #

It would be easy to read this as demolishing the bearing curve’s exponents and it does not.

The exponent result is about the cloth and it stands. What it needs is a load heavy enough to be on the crown side of the crossover, and every load that does anything to a fabric is. A presser foot, a nip, a squeegee, a knee on a floor, two fabrics rubbing in a Martindale — all of them are one to twenty kilopascals, all of them are above the crossover by a factor of five or more, and for all of them the bearing curve is the operative object.

What the crossover removes is the extrapolation to zero load. The statement “at a thousandth of the cloth’s thickness a satin touches thirty times a plain weave’s area” describes a depth of a third of a micrometre and a pressure far below the crossover, and no plate is ever there. The ordering survives at every real load; the unbounded divergence at vanishing load is a property of a surface nothing reaches.

That is a smaller correction than it sounds and a real one. A limit taken outside the range where the model applies is not a result, and the bearing-curve ladder took one.

Why a raised cloth is a different problem entirely #

Multiply the hair population and the crossover moves with it, because there are more cantilevers sharing the load and each one is engaged sooner.

A cloth raised sixty-fourfold carries nearly fifteen kilopascals before its crowns are reached. That is above every pressure in the list except a presser foot. So on a napped fabric there is no ordinary circumstance in which anything touches the cloth: a hand, a body, another garment, a chair all stop in the nap.

The consequence runs through the whole of this ladder. It is why a napped cloth’s abrasion test measures the nap and never reaches the fabric, why its warmth is the canopy rather than the cloth, and why its hand has nothing to do with its weave. A flannel and a poplin of the same yarn at the same sett feel entirely different and the difference is a population.

The area, which is the number the whole thing was for #

A contact model exists to say how much of a surface is actually touching, and the hair layer’s answer is startlingly small.

At the crossover pressure the plate is resting on every hair in the population, which is 1.4 per square millimetre on a sheeting. Each contributes a contact patch no wider than a fibre — twelve micrometres — so the real contact area is of order a hundredth of a per cent of the plan. The crown model at a kilopascal gives about four per cent.

So passing through the crossover changes the real contact area by more than two orders of magnitude, and it changes it discontinuously in the sense that matters: the two regimes are not a curve with a bend in it, they are two different surfaces being touched.

Everything that scales with real contact area therefore has a step in it at a fifth of a kilopascal. Friction does; heat transfer by conduction across the interface does; the pressure on whatever is being touched does, and it does so upwards — a hundredth of the area carrying the same load is a hundred times the local pressure. A light touch is a low force and a very high pressure, which is not how anybody describes a light touch and is exactly what the arithmetic says.

What was counted, and how #

Four assertions and one of them is about a standard rather than about a fabric.

That the layer carries a positive pressure at zero approach — a model that returned zero would be saying the hairs are infinitely soft, which is what a coverage-only model does say and is why coverage is not the right quantity here.

That the standard test presses above the crossover and the light-pressure test does not. Those are two facts about published standards checked against a computed number, and either could have come out the other way.

And that raising moves the crossover by at least eightfold. It moves it by sixty.

The integration itself has one guard worth naming. The integrand diverges at short hairs: a hair of vanishing length is infinitely stiff in bending, so the bending term goes as 1/h² and the integral over the population does not converge at the origin. The buckling cap fixes the physics and a floor at two fibre diameters fixes the arithmetic, and both are stated at every use. Without the cap the model returns an infinite pressure at zero approach, which is a plausible-looking way of saying that a plate can never reach a cloth at all.

The pressure under one hair does not depend on the touch #

The observation that a light touch is a high pressure deserves a number, and the number turns out to be the same number whatever the touch is — which is a stronger and stranger statement than the one it replaces.

A hair carries at most its buckling load, and it presents at most its own cross-section. So the pressure at a hair’s tip is capped at the buckling load divided by the fibre’s area, which for a cotton fibre at the population’s own mean length is 11 micronewtons over 1.13 × 10⁻¹⁰ square metres:

about 0.1 megapascals, at every hair, at every applied load below the crossover.

Nothing in that depends on how hard the plate is pressing. A lighter touch engages fewer hairs and each of the engaged ones is still at or near its own limit, because a hair either carries very little or is bent over at its buckling load and there is not much range in between. Pressing harder recruits more hairs; it does not press each one harder.

So the layer behaves as a constant-pressure contact with a variable number of contacts, which is an unusual mechanical object and is a fair description of what a soft fabric feels like. Against a nominal fifty pascals from a sleeve brushing skin, the local pressure is two thousand times higher and is the same two thousand times higher whether the sleeve is barely touching or resting.

That has a consequence for the sensation, and it is the one this ladder has been circling. The load a hair delivers to skin is fixed by the fibre and not by the fabric, which is exactly why a prickling fibre’s diameter is the specification and the fabric’s weight is not. A heavier garment presses with more hairs; each of them presses with the same force it always did.

It also explains something about how fabrics are judged that otherwise looks like imprecision. Handle assessments are made by people pressing with quite different forces and they agree tolerably well. If the local pressure were proportional to the applied load they could not, because everybody would be sampling a different point on a curve. With a capped local pressure, what varies between assessors is how many contacts they make — a count, which averages — rather than what each contact does.

Which makes the crossover a hairiness reading #

The crossover moved from two tenths of a kilopascal to fifteen when the population was multiplied sixty-fourfold, and the near-proportionality is not a coincidence: at a fixed plate height the pressure is the force per hair times the hairs present, and only the second term moved.

So the crossover pressure is very nearly proportional to the hair density, which makes it a measurement of the hair layer taken with a thickness gauge.

The procedure is two readings and a subtraction. Measure the thickness at the standard kilopascal and again at a fiftieth; the difference is the plate’s stand-off, the stand-off gives the height at which the layer carries a fiftieth of a kilopascal, and the population’s own decay length turns that into a density. No optical instrument, no calibration, and no threshold anybody chose.

And it says which way each finish moves the reading. Singeing lowers the crossover, because it removes the long hairs that were carrying the light loads — so a singed cloth is reached by the plate at a lower pressure and reads closer to its geometric thickness at every pressure. Raising raises it, by nearly two orders of magnitude.

That gives a prediction with the right shape to be tested on a bench that already exists: the gap between a fabric’s standard-pressure and light-pressure thickness readings should collapse after singeing and expand after raising, in proportion to the hair density in each state — and the collapse is a measurement of what the flame actually removed, which is otherwise a difficult thing to establish at all.

Where the model stops #

The cantilevers are independent and they are not. A bent hair leans on its neighbours, which stiffens the layer, so every pressure here is a lower bound and the crossover is a lower bound. Above the canopy threshold the error is large and below it — which is where an ordinary woven cloth sits — the hairs genuinely cannot reach one another and the model is on its home ground.

Only the long population is in it. The short cloud is seven or eight times as much fibre, it is very stiff because it is very short, and it sits exactly where the plate ends up. Including it would raise the pressure near zero approach considerably, which moves the crossover up and strengthens the essay’s claim rather than weakening it.

A hair is straight, elastic and does not slip. A real one is curved, viscoelastic and pulls out of the yarn under load. The first rounds the bending law, the second makes the pressure depend on how fast the foot came down, and the third is the pull-out arithmetic that this site has and did not connect here.

And the emergence angle is a parameter. The pressure goes as the cube of its sine through the free length, which is the strongest single dependence in the calculation and the one with the least behind it.

What a specification would have to say #

The previous rung of this ladder ended by asking what a fabric specification would have to say about contact, and answered it with a transverse stiffness. The answer needs a second line now.

A contact specification needs a pressure and a state. The pressure decides which surface is being described, and the state — as woven, singed, raised — decides where the crossover is. Two cloths identical in every structural respect can be on opposite sides of the crossover at the same pressure, because one of them has been through a flame and the other through a raising machine.

That is unusually actionable for a result in this collection. A thickness, a contact area, a friction coefficient or a compression curve quoted without both is not wrong; it is under-determined, and the two fabrics it could describe differ by a hundredfold in the quantity being reported.

The generalisation #

A quantity computed for a surface applies only above the load at which the surface is reached.

The transferable shape is that a model of contact has a floor, and the floor is set by whatever is in the way rather than by anything in the model. Every asymptotic statement of the form “as the load tends to zero” needs checking against what the load has to get through first, and in a great many real contacts the answer is a layer of something loose that nobody put in the model.

The practical version for this collection: every contact number in the surface ladder should be read with a pressure attached, and the pressure it needs is a fifth of a kilopascal.

Who found it, and when #

That a fabric’s thickness depends on the pressure it is measured at is as old as thickness testing, and every standard says so. Attributing the difference to a hair layer rather than to compressibility is not new either — it is the reason pile fabrics have their own low-pressure specification.

What is new is the number. The crossover has not, as far as this collection can tell, been computed from a hair population and a column formula; it has been handled by specifying pressures that work. And the consequence for the bearing-curve exponents is this site’s own, correcting an essay of its own.

Where the ladder goes next #

Down into the pressures above the crossover, where the two laws meet: a compression curve is two laws in series identifies the soft initial region of every published fabric compression curve as the hair layer and the stiff region as the crowns, which is the mixture the bearing-curve essay named and could not separate.

Sideways, into what the layer is doing while nothing is pressing on it. It holds air, which is most of a fabric’s warmth; it blocks light, which is why a shot cloth has to be filament; and it is worn away and replaced, which is why singeing does not stay done.