A compression curve is two laws in series

Press a fabric between two plates and record the thickness against the pressure, and the curve that comes out has a shape everybody recognises and nobody derives. It is extremely soft at first — a few pascals move it a long way — then it stiffens sharply, and past that it becomes almost a wall. The literature fits a power law to it and reports an exponent, and the exponents reported range from about a third to about two thirds depending on who fitted and over what range.

The cloth #

The bearing-curve essay got two exponents out of geometry with nothing fitted. Pressure is a stiffness times a strain times a bearing fraction, and the bearing fraction opens as a square root of depth where the crowns are lines and linearly where they are points — so pressure goes as δ^(3/2) and δ², and the approach goes as P^(2/3) and P^(1/2). Measured slopes on the site’s own model came out at 0.67 and 0.50 against those predictions.

And it recorded, in its own summary, that this cannot be the whole of a published exponent: “a published fabric compression exponent is a mixture of that regime and van Wyk’s compaction, fitted over a range that straddles them.” That was correct about there being a mixture and wrong about the second ingredient.

The claim #

A fabric’s compression curve is three regimes in series, not two, and the first of them is not in the fabric. The soft initial region is a hair layer bending; the middle region is the crowns flattening; the wall is van Wyk compaction of the yarn’s own fibres. A power law fitted across the first two produces an exponent that belongs to neither and to their ratio of extents.

The extents are the useful part. On an ordinary woven cloth the hair region occupies the first eighty micrometres and a fifth of a kilopascal; on a raised one it occupies a millimetre and fifteen kilopascals, which is the whole of the range anybody presses a fabric over.

Three laws, each with its own variable #

They are worth setting beside one another because they are functions of different things.

The hairs bend. A protruding fibre is a cantilever with a very small second moment, and a population of them carries a pressure that rises steeply as the plate descends because more hairs engage and each engaged one is pushed further. A hair carries at most its own buckling load, and past that it lies over. The variable is the plate’s height above the crowns.

The crowns flatten. The bearing curve says what fraction of the plan lies within a depth of the highest point, and the pressure to reach that depth is the transverse stiffness times the strain times the fraction. The variable is depth into the cloth.

The fibres compact. Once a crossing cannot get wider it can only get thinner by getting denser, and van Wyk’s law gives pressure as a constant times a modulus times the cube of the fibre volume fraction. The variable is the fraction itself.

Three regimes, three variables, and no reason any power law should fit across a boundary between them.

Where the first boundary is #

The plate reaches the crowns when the hairs can no longer carry it, which for a woven cotton sheeting is at about two tenths of a kilopascal. That is the crossover and it is the first boundary.

Its position is set entirely by the population and not at all by the cloth. Multiply the hairs and it rises; singe them and it falls; change nothing about the weave, the sett or the yarn and it does not move.

Which cloths have a first region at all #

The hair regime is not a universal feature of fabrics and the criterion for having one is already in this ladder.

A hair region needs the hairs to carry a pressure over a distance, and a population that cannot reach across the gaps between its members carries very little over very little. The pure number is n_A λ², the hairs per square millimetre times the square of their own length, and it is under one for every construction in this site’s table.

So a filament fabric has no first region at all, because it has no fibre ends. A singed woven cloth has one so small that a test starting at fifty pascals would miss it. A raised cloth has one that swallows the entire test. Three fabrics, identical in structure, with three qualitatively different compression curves, and the difference is a finishing operation and a fibre choice.

What a fitted exponent is measuring #

Now take a published measurement. A compression test typically runs from a few pascals to a few kilopascals — three orders of magnitude of pressure — and fits one power law to the lot.

The bottom of that range is entirely hair. The middle is crowns. The top is beginning to be compaction. The fitted exponent is a weighted compromise between three laws, and the weights are the fraction of the fitted decades each regime occupies.

That is why the reported exponents scatter. Two laboratories fitting the same fabric over different ranges get different answers, and the difference is not scatter or technique: they have weighted the regimes differently. A laboratory that starts at a kilopascal gets something near the crown exponent; one that starts at ten pascals gets something much lower, because it has included a region where the pressure is carried by fibres standing in the air.

And a laboratory testing a raised fabric gets the hair exponent whatever it does, because on a napped cloth the crossover is above the top of the usual range.

The measurement that would separate them #

The three regimes have different signatures and one of them is cheap to look for.

A hair regime is not repeatable in the way the others are. Bending a hair over is nearly elastic; buckling it is not, and a hair that has been laid flat does not stand back up immediately. So a second compression immediately after a first finds a much smaller hair contribution, and the difference between the first and second cycles is a measurement of the hair layer taken with no extra apparatus. The crown and compaction regimes are far more nearly repeatable — the site’s hysteresis work is about how much less, and it is a matter of tens of per cent rather than of factors.

A hair regime is removed by a flame and the others are not. Singeing a specimen and re-testing it removes the first region and leaves the rest where they were, which is about as clean an experiment as this subject offers.

Neither of those is a measurement this collection can make. Both are stated because an argument that a published curve is a mixture ought to say how somebody would check.

Why the wall is in the wrong place too #

There is a second correction and it points the other way.

Van Wyk’s law is a law for a random fibre assembly, and it was derived for one — a wool mass, a batt, a card web. A yarn in a cloth is not random; it is a twisted helical assembly with a packing factor that came from a count, and this site has been treating van Wyk’s constant as a measurement with a threefold spread precisely because the assembly it is being applied to is not the one it was derived on.

The hair layer, by contrast, is a random fibre assembly, and a very dilute one. So van Wyk’s law is a better description of the hair region than of the compaction region, if only the density were high enough for it to apply. It is not: at two parts in ten thousand there are no fibre-to-fibre contacts to speak of, and the pressure is carried by individual cantilevers rather than by a network.

A raised cloth crosses that threshold. A canopy at a per cent or two of fibre is a genuine random assembly, and its compression is van Wyk’s rather than a bed of independent columns. So a napped fabric has four regimes: cantilevers, then a compacting canopy, then crowns, then yarn compaction — and the middle two are the ones that the whole of a hand-feel test lives in.

The energy, which is where the difference is largest #

A compression curve is usually read as a stiffness and it is also an energy: the area under it is the work put into the fabric, and the work not given back is what makes a fabric feel dead or lively.

The hair region contributes almost no energy on a woven cloth, because it is a small force over a small distance — eighty micrometres at under a fifth of a kilopascal. On a raised cloth it contributes nearly all of it, a millimetre at up to fifteen, which is two orders of magnitude more work than the crowns will ever take.

So the sensation of compressing a napped fabric is entirely a sensation of bending fibres in air, and it has no more to do with the cloth underneath than the sound of a carpet has to do with the floor. That is a strong statement and it is a straightforward consequence of where the crossover sits: a hand cannot press hard enough to reach the fabric.

It also explains something the hand-feel literature reports and does not explain, which is that compressional measurements on napped fabrics correlate with almost nothing structural. They are measurements of a population, and the population is set by a raising machine.

What was counted, and how #

The essay’s arithmetic is the crossover and the thickness readings, both of which are asserted on the rung below. What is added here is a comparison rather than a computation, and it is made honestly: the three regimes are computed from three separate pieces of this site’s machinery, and no attempt is made to join them into a single curve.

That refusal is deliberate. Joining them would need the hair layer’s residual stiffness once the plate is on the crowns — hairs are still there, bent flat, and still carrying something — and the crowns’ contribution while the plate is still above them, which is zero. The first of those is not computed anywhere and a smooth interpolation through it would be a picture of a curve rather than a model of one.

What is asserted is the boundary’s existence and its position relative to the standards, which is the claim that does the work.

Where the model stops #

The hair layer’s unloading behaviour is not modelled at all. The essay’s cleanest proposed measurement — the difference between a first and second compression — is a prediction about a quantity the model does not compute.

The three regimes are treated as sequential and they overlap. At any pressure near a boundary both mechanisms are carrying load. The extents quoted are where each stops dominating rather than where each stops.

Van Wyk’s constant is a threefold bracket and every statement about the wall’s position inherits it. That was already recorded and is unchanged.

And the transverse stiffness in the crown regime is fitted, quoted across the range of published yarn compression moduli, and named at every use. This essay does not repair that; it carries it, as the whole ladder does.

What this does to a hand-feel instrument #

The consequence for the one measurement that tries to quantify how a fabric feels is uncomfortable and worth stating.

A hand-feel apparatus compresses a fabric over a low pressure range chosen precisely because it is the range a hand explores, and extracts a compressional energy, a linearity and a resilience from the curve. Every one of those three is, on a napped fabric, a property of the hair layer. On an ordinary woven cloth they straddle the crossover, so all three are mixtures whose proportions depend on the specimen’s finishing history.

That does not make the measurements wrong. A hand also explores that range and also meets the hairs first, so an instrument reporting a mixture is reporting what a hand feels. What it makes them is structurally uninformative: a number that correlates with what people say about a fabric and not with anything in its construction, which is exactly the pattern such measurements show.

The useful move is the one this ladder keeps arriving at. Report the two regimes separately — the same specimen singed and unsinged, or the first cycle against the second — and two numbers appear where there was one, and each of them is about a different object.

Three quarters of the travel is hair, on a bare cloth #

The fitted exponent is described as a compromise weighted by the decades each regime occupies, and the weighting is worth doing, because it does not come out where the crossover pressure suggests.

What a compression curve records is a thickness change, so the right weight is how many micrometres each regime supplies, not how many decades of pressure it spans.

The hair region supplies ninety micrometres. At a fiftieth of a kilopascal the foot stops that far above the crowns; at the crossover it is on them.

The crown region supplies twenty-eight. From the crowns at two tenths of a kilopascal down to twenty-eight micrometres of sinking at a hundred, which is a firm pinch.

So over the whole of an ordinary test range the reading moves 118 micrometres, and ninety of them — seventy-six per cent — are a fibre population standing in air.

That is a much stronger statement than the essay’s a third of the lightest reading. Three quarters of a compression curve’s travel on a bare cotton cloth is the hair layer, and the fitted exponent is therefore not a mixture weighted towards the fabric — it is very largely the hair layer’s own, with the fabric contributing the last quarter.

Which accounts for the direction of the scatter. The reported exponents run low, from a third upwards, and the geometric prediction for the crowns is a half or two thirds. A fit dominated by a regime that reaches zero thickness at a finite pressure — which a hair layer does, since the plate lands on the crowns — is steeper than any power law and drags a fitted exponent down.

Which makes the singeing experiment quantitative #

The essay proposes singeing a specimen and re-testing as a clean separation, and the arithmetic says what to expect rather than merely that something will change.

Singeing takes the canopy criterion down by two orders of magnitude, so the crossover falls from two tenths of a kilopascal to about two thousandths and the hair region’s extent falls from ninety micrometres to of the order of one.

The total travel therefore falls from 118 micrometres to about 29 — a fourfold reduction in the whole range of a compression curve, from an operation that removes under one per cent of the cloth’s mass.

And the fitted exponent should move to the crowns’ own value. A singed specimen’s compression curve should fit at two thirds for a floated weave and a half for a plain one, where the unsinged specimen fit at a third or thereabouts.

Two predictions, both numerical, both from one specimen measured twice on one instrument, with a gas flame between. A fourfold change in travel and an exponent moving from a third to two thirds is not a subtle effect and would not be mistaken for technique.

And it says what the standards’ two pressures are doing #

The last consequence is about the pair of pressures the thickness standards carry, which the rung below notes sit on opposite sides of the crossover.

They are sampling the two regimes, one each. The light-pressure reading at a tenth of a kilopascal is a measurement of the hair layer with the fabric as a floor; the standard reading at one kilopascal is a measurement of the fabric with the hairs crushed under it.

So the pair is not two attempts at one number with different care. It is two numbers about two objects, and their difference — which nobody reports — is the ninety micrometres this section is about.

The standards already collect the measurement this essay says is missing. What they do not do is subtract.

The generalisation #

When a measured curve is fitted with a single law over a range that contains a boundary, the fitted parameter is a description of the range and not of the material.

The transferable diagnostic is to ask what the extent of each regime is before fitting anything, and to notice that an exponent which moves when the fitting range moves is not an exponent. This site has now met the shape twice: once when a compression exponent turned out to straddle two laws, and once here, when the same exponent turned out to straddle three and to begin outside the object being measured.

The practical rule that follows is short. Fit inside a regime or do not fit. A power law across a boundary is a summary statistic wearing the costume of a physical constant, and it will be quoted as the second.

Who found it, and when #

Van Wyk’s compression law for fibre masses is from 1946 and is the standard treatment. Fabric compression curves and their fitted exponents are a large literature, with de Jong, Postle and Hearle’s work in the 1980s among the more careful, and the sensitivity of the exponent to the fitting range is remarked on there.

The identification of the soft initial region with a hair layer is not new as an idea — it is a standard qualitative explanation and is why compressible-fabric standards specify low pressures. What is new here is the crossover as a computed pressure, and the observation that it sits between two published thickness standards.

Where the ladder goes next #

The compression story continues where the site left it, at the wall, and this essay adds only that the road up to it has one more turning than was thought. What the hair region is worth in its own right — rather than as a nuisance in somebody else’s curve — is the subject of warmth is mostly the hairs, where the same canopy that softens a compression curve is doing the whole of a fabric’s insulation.

And the regime boundary reappears as a measurement problem in abrasion takes the hairs first: the first material off a fabric is the layer that carries the first pascals, and a mass loss quoted at a fixed number of cycles is a mixture in exactly the way a fitted exponent is.