A cloth compresses along its own bearing curve

Squash a fabric between two plates and record the thickness against the load. The curve that comes out is concave, steep at first and then stiffening, and it has been fitted for seventy years with a power law whose exponent is quoted and never accounted for. The usual explanation is that the fibres are being compacted, which is true at the heavy end and cannot be true at the light end, because at the light end almost none of the cloth is touching anything.

The claim #

The light end of a fabric compression curve is geometry rather than material, and its exponent is decided by whether the weave has a float.

The algebra is one line. The pressure a plate applies is a stiffness times the strain in the material it is squashing times the fraction of the plan doing the squashing:

$$P = G \frac{\delta}{b} A(\delta).$$

For a weave with plateaux, A goes as √δ, so P goes as $\delta^{3/2}$ and the approach goes as $P^{2/3}$. For a plain weave, A goes as δ, so P goes as δ² and the approach goes as $P^{1/2}$.

Nothing in either derivation is fitted. The stiffness G sets the scale of the curve and cancels out of its slope entirely.

Why the exponents survive the crudeness of the model #

The pressure model here is a bed of independent springs, which is not a contact mechanics and would be laughed at by anyone doing Hertzian analysis. It survives because an exponent is more robust than a coefficient.

Replacing the springs with a proper elastic half-space changes the relation between load and approach on a single asperity from linear to a three-halves power, which would move the two exponents to 2/5 and 1/2 respectively. That is a real change and it would be detectable. What it would not do is make them equal: the ratio between the two remains, because it comes from the bearing curve rather than from the contact law, and the bearing curve’s own exponents — one half against one — are pure geometry.

So the prediction that is safe is the ordering and the difference. A floated cloth sinks with a larger exponent than a plain weave of the same yarn, whatever contact law is used, because it is finding contact area faster as it descends.

What was counted, and how #

For each draft, the surface is built and its bearing curve taken in closed form; then for forty pressures spaced logarithmically from 0.05 to 200 kilopascals the depth is found at which the model’s pressure matches, by bisection on a monotone function.

The slope is measured over the light half of the range afterwards, between the fifth point and the twentieth, and compared with the prediction. It comes out at 0.50 for the plain weave and 0.67 for both the twill and the satin — which is 1/2 and 2/3 to the precision the reading allows.

The satin and the twill agreeing exactly is the informative part. They have very different amounts of crown line — 0.70 against 2.10 millimetres per square millimetre — and therefore very different positions on the plot, one being about three times softer than the other at any pressure. They have the same slope, because the slope depends on the exponent and not on the coefficient, and both have plateaux.

The three quantities a compression curve holds #

A pressure–thickness curve is usually read for one number, its slope, and it holds three.

Where it starts is the geometric thickness less whatever the highest crossings contribute, which is a population quantity and belongs to the yarn’s evenness.

How steep it is at the light end is the exponent above, and it belongs to the draft.

Where it sits — how soft the cloth is at any stated pressure — is the crown line times the transverse stiffness, and it belongs to both: the crown line is the draft’s, the stiffness is the yarn’s.

That decomposition is worth having because the three answer to different levers. A finer, more even yarn moves the first. A different weave moves the second and part of the third. A softer or a more loosely spun yarn moves the third alone. A single fitted exponent conflates all of them, which is why published exponents for nominally similar fabrics scatter across a range wider than the difference this essay is claiming to have found.

Where the geometric regime ends #

The curves in the figure bend at the heavy end, and they bend for a reason that has nothing to do with the argument above.

As the depth grows, the crowns widen and eventually merge: the bearing area approaches the cover factor, and after that pressing harder cannot find any more area because there is none. The cloth has stopped being a surface with crowns and has become a slab of fibre with holes in it.

That is where the compaction of a fibre mass takes over, which is van Wyk’s regime and which this collection already computes: the pressure needed to raise a fibre assembly’s packing fraction goes as the cube of that fraction, and the stiffness with it. The two regimes are quite different laws and the transition between them is the merging of the crowns.

For an ordinary sheeting the merge is somewhere near a fifth of the cloth’s thickness of approach, which the model reaches at a few hundred kilopascals — well above anything a hand applies and well below a calender nip. So the whole of the range in which a person handles cloth is in the geometric regime, and the whole of the range in which a machine finishes it is in the compaction one.

Why the confusion was inevitable #

A published compression curve typically spans one or two decades of pressure, straddles the transition, and is fitted with a single power law. What comes out is an exponent that is a weighted average of two mechanisms, and it moves with the range fitted, the fabric, the weave and the specimen preparation — which is exactly the behaviour of a quantity that is not a property of anything.

Three things follow, and each is checkable.

The exponent should be smaller on a plain weave than on a floated cloth of the same yarn, at the light end. It should rise as the fitted range moves to lighter loads for any cloth with a float, and fall for a plain weave, because both are converging on their own geometric limits. And it should become weave-independent at the heavy end, where the geometry has been erased.

None of those is a measurement this collection has made. They are the shape a measurement would have if this account is right, and the shape it would not have if the compression of a fabric were a material property throughout.

What this does to the softness of cloth #

The practical form of the result is a statement about handle, and it is a mild one because handle has a great deal more in it than compression.

A floated cloth is softer at a light touch and stiffer at a heavy one than a plain weave of the same yarn. Softer at a light touch because it finds contact area quickly and therefore supports a small load with a small approach; stiffer at a heavy one because that same area means the load is spread and the cloth is not being pushed as far into the compaction regime.

Both halves come from the same exponent. A curve with a larger exponent starts flatter and finishes steeper, and the crossing point is where the two cloths of one yarn feel identical — which for the plain weave and the twill above is at about ten kilopascals, right in the middle of the range a finger uses.

That may be why the trade’s language about the softness of weaves is so inconsistent. The comparison genuinely reverses inside the range of pressures a hand applies, so two people pressing at different firmnesses will disagree about which of two cloths is softer, and both will be right.

The crossing, and why the trade cannot agree about softness #

The two curves for a plain weave and a twill of one yarn cross, and where they cross is inside the range a hand explores.

At a tenth of a kilopascal — barely touching — the twill has sunk about a third of the distance the plain weave has, so it is by a wide margin the firmer of the two. At a hundred kilopascals the plain weave has sunk 27.8 micrometres and the twill much less again, so the twill is still firmer. Over the whole range the ordering does not in fact reverse for that pair; what reverses is the rate, and it is the rate a finger reports.

A finger pressing a cloth does not measure a displacement at a pressure. It presses until it feels resistance, which is to say it is sensitive to the slope of the curve rather than to its value, and the slope at a given pressure is the exponent times the depth divided by the pressure. A curve with the larger exponent has the smaller slope at the light end and the larger at the heavy end, so which of two cloths feels softer depends on how hard the person doing the feeling is pressing.

That is a genuinely awkward result for anyone trying to specify handle, and it is not a modelling artefact: it is what having two different exponents means. Two people can press two cloths, disagree completely about which is softer, and both be reporting their measurements correctly.

Why the disagreement about softness is spread over two decades #

The section above says two people pressing at different firmnesses will disagree about which cloth is softer. How wide the region of disagreement is can be worked out, and it turns out to be the reason nobody has ever settled the argument.

What a finger reports is the slope, and for a power law δ = (P/k)^p the slope is pδ/P. So the ratio of two cloths’ slopes at one pressure is

(p₁ ÷ p₂) × (δ₁ ÷ δ₂),

which for a twill against a plain weave is (2/3)/(1/2) = 4/3 times the depth ratio. The twill feels the softer of the two as soon as its depth has reached three quarters of the plain weave’s — a condition on a ratio rather than on either depth.

And that ratio moves as P to the power one sixth, because it is the difference of the two exponents. A sixth power is the slowest dependence anywhere in this collection: to change the depth ratio by a factor of two takes a factor of sixty-four in pressure.

So there is no crossing point in any useful sense. There is a drift, spread over nearly two decades of pressure, in which the two cloths feel more and more alike and then imperceptibly swap — and two assessors pressing within a factor of three of one another are inside that band together, getting different answers, both correctly.

A disagreement that broad cannot be resolved by being more careful, which is the honest reason the trade’s language about which weaves feel soft has never converged. The comparison is not merely pressure-dependent; it is pressure-dependent so weakly that the pressure cannot be pinned down by feel.

What it would take to see the exponents #

The same slowness says why the geometric regime has not been measured, and it sets a specification for the experiment that would settle it.

Distinguishing a half from two thirds means resolving a log–log slope to better than about 0.08. A slope read over n decades of pressure with thickness known to a fractional error ε carries an uncertainty of roughly 1.4ε/(n ln 10) — so

one decade of pressure and thickness to five per cent gives a slope to 0.03, comfortably enough; half a decade gives 0.06, which is marginal; and a quarter of a decade cannot see the difference at all.

That is not a demanding experiment. What makes it hard is where the decade has to be: entirely below the crown-merging transition, which for an ordinary sheeting is a few hundred kilopascals, and ideally an order of magnitude below the standard test pressure of one. A decade from 0.02 to 0.2 kilopascals is the measurement, and it means holding a plate’s position to a micrometre while it carries two grams over a square centimetre.

Published curves do the opposite. They span the transition — typically from a fraction of a kilopascal to tens or hundreds — and fit one power law across it, which averages a geometric exponent against a compaction one and returns a number belonging to neither. The instrument is not the obstacle; the fitted range is, and a curve already measured could be re-read over its light end alone by anyone who has the raw points.

That reframes the essay’s three predictions as one request. The measurements very likely exist, in laboratory notebooks and in the low-pressure tails of published curves, and what has never been done is to fit them separately from the rest.

Where the model stops #

The springs are independent. A proper contact mechanics would change both exponents and preserve their difference, and nothing here should be read as a claim about their absolute values.

The stiffness is fitted, and it has no lower bound. A yarn’s resistance to being squashed out of round can be as small as one likes if its fibres are free to slide, so G is a fit to measured thickness rather than a prediction. It sets where the curves sit and not how steep they are.

There is no time in any of it. A cloth under a plate goes on thinning for as long as the plate is there, and a compression measurement is therefore a measurement at a stated dwell. Nothing in this collection’s compression arithmetic has a rate in it — a standing gap, recorded again here.

And the transverse flattening of the threads is left out of the light regime. The threads at a crown are being squashed as well as approached, which is the site’s own pressed-state arithmetic and which changes the section as the pressure rises. Including it would make the crowns widen faster than the geometric bearing curve says, raising the exponent slightly for every weave.

What a knitted fabric does instead #

The argument above is a woven argument, and it is worth saying what changes when the fabric is knitted, because the difference is instructive rather than incidental.

A knitted loop has no float and no plateau. Its highest points are the tops of the legs where the yarn crosses over the loop below, and those are doubly curved — so a plain jersey should sit on the plain weave’s side of the divide, with an exponent near a half at the light end.

But a knit compresses in a way a woven cloth does not, because a knit is soft because it bends rather than because it squashes. Pressing a jersey does not chiefly flatten its yarn; it moves the loops out of plane, and the resistance to that is a bending resistance rather than a transverse one. The bearing curve is still the right description of what is in contact, and the law relating depth to load is not.

That is the boundary of this whole ladder and it is worth naming here. The bearing curve is a statement about a surface, and it is as true of a knit as of a weave. Everything downstream of it that involves a force needs a law for how the material under a crown responds, and a woven cloth’s law and a knit’s are not the same law. Where the essays that follow deal with knitted fabric they deal with the surface and not with the force.

The generalisation #

A contact law between two bodies is the product of a material law and a geometric one, and the exponent belongs to the geometry.

That is the transferable statement, and its useful corollary is that fitting a power law to a compression curve measures a mixture rather than a property. If a system’s contact area grows with approach in a known way, the exponent of the force–displacement law is fixed before any material has been named — and a change of exponent between two specimens is evidence about their shapes rather than about what they are made of.

The special thing about cloth is that the shape is a matrix somebody wrote down. In almost any other rough-contact problem the height distribution is a residue of manufacture and cannot be specified in advance.

Who found it, and when #

The bearing-curve half is Abbott and Firestone’s from 1933 and the contact half is the standard asperity argument of Bowden, Tabor, Greenwood and Williamson. The compaction half is van Wyk’s, 1946, and is the account every textile text gives for fabric compression.

What appears to be missing from the literature is the light-load half: that beneath the compaction regime there is a purely geometric one, that it has its own exponent, and that the exponent is a property of the interlacement. The reason it is missing is probably instrumental — measuring a cloth at a tenth of a kilopascal means measuring a plate’s position to a micrometre while it is barely touching anything.

Where the ladder goes next #

To the same crowns under a rubbing rather than a pressing: a cloth loses its strength before its mass, where the concentration of load on a few per cent of the plan turns into a concentration of damage on one section of every thread.

Sideways, the merging of the crowns at the heavy end is the whole of what a calender does deliberately, and what it buys is not softness but a flat top that reflects.