A wetting supplies the force the criterion needs

The criterion this collection is built on is exact and it is blind in one direction. It asks whether the threads of a cloth can be split into an upper set and a lower set with every crossing between them passing the right way, and it answers yes or no with no tolerance to choose. What it cannot see is that a cloth which fails it does not necessarily fall apart, because friction holds it together anyway.

Pricing that needed a contact force: how hard the threads press on one another where they cross, in a cloth that is not under tension. This collection could not compute one — the elastica that would give contact forces everywhere along a thread is still missing — so it read one backwards out of eight measured fabric thicknesses, which is a real number with a measurement inside it.

A wetted close cloth supplies one from nothing but geometry.

Where the force comes from #

The compression ladder’s finding, in one paragraph. A cloth’s two thread systems must supply its whole thickness between them, and each can supply only as much crimp height as its own length allows. Swelling raises the demand in proportion and the supply much more slowly, so past a critical swelling there is no state at constant thread length.

Something must then give, and there are two candidates: compact the yarn, or stretch the thread. For a sheeting the stretching route needs 9.1 per cent of strain against a breaking strain of 6.65, so the thread would break first and there is one route.

Compacting the yarn from its free-swollen state to the diameter the geometry can hold raises its fibre volume fraction from 0.600 to 0.723, and van Wyk’s cube law prices that at 7.8 megapascals.

Nothing in that chain is a measurement of a fabric. It is two thread lengths, two diameters, a swelling constant, a packing factor and a compaction constant — all of them properties of a yarn or a fibre rather than of a cloth somebody made.

From a pressure to a force #

A pressure is not a contact force, and the step between them is the weak one.

Two crossing threads touch over an area somewhere between a point and the projected area of the crossing, which is d₁·d₂. This collection has no contact mechanics, so the area cannot be computed — but it can be bounded, and the projected area is the bound.

For a wetted sheeting the swollen diameters are 0.224 millimetres each, so the projected area is 0.0503 square millimetres. Times 7.8 megapascals that is 0.392 newtons.

The measured route’s answer for the same cloth is 0.423 newtons.

The agreement is closer than it is entitled to be #

Seven per cent apart, and that should be treated with suspicion rather than satisfaction.

The swelling figure is an upper bound, because the contact area is a bound. A real contact patch is smaller than the projected area of the crossing — the threads are curved and touch over a Hertzian ellipse, not a rectangle — so the true force is below 0.392, and possibly well below.

And the pressure carries van Wyk’s constant, which is reported between 0.003 and 0.011. Move the constant to either end of its range and the force moves between 0.20 and 0.72 newtons.

So the honest statement is: the swelling route gives a force between about a fifth and three quarters of a newton, and the measured route gives 0.42. They overlap comfortably and the point estimate’s closeness is luck.

The assertion in this collection’s gate says exactly that. It requires the two to be within an order of magnitude of one another and not to be equal, because what is being checked is that a wetting supplies a force of the kind the criterion needs and not that the two methods agree.

Three routes to one number, and what having three is worth #

This collection has now reached a crossing’s contact force three different ways, and the third is the one this rung adds.

From a tension. A thread under tension turning through an angle presses on whatever it turns around, with a force proportional to the tension. That was the first route and it is exact, and every answer it gives is quoted per newton of tension — so it is a family of answers rather than an answer.

From a measured thickness. A round section predicts every cloth in the table thicker than it measures, and the discrepancy is what a force has flattened. Invert eight measured thicknesses and eight forces come out, from 0.18 newtons for a batiste to 0.85 for a cheesecloth. That is an answer, and it has a measurement of a real fabric inside it.

From a swelling. This rung. An answer with no measurement of a fabric, available only wet and only for a close cloth.

The three do not agree exactly and they are not supposed to: the first is a different quantity, and the second and third are the same quantity by two chains with different uncertainties. What having three is worth is that a result depending on the contact force can now be reported over them, and a result that survives all three is a different kind of result from one that only survives the route it was derived in.

That is the standing habit of this collection stated for a force rather than for a count. A number reached one way is a number; reached two ways it is a measurement of something; reached three ways with different assumptions it is probably about the world.

What this licenses #

One thing, and it is worth being precise about it.

The rung that gave the criterion a force priced what the criterion cannot see by asking: given a contact force, how much load does friction at a crossing carry, and is that enough to hold a cloth the criterion has declared separable? The answer was a real number and it depended on a force that had a measured thickness inside it.

Now that question can be asked of a wet cloth with no measurement anywhere in the chain. A wet close cotton’s crossings press with a force of the order of a newton’s fraction, and that force is computed from thread lengths and material constants.

Which matters because of what a criterion is for. A criterion that says a draft falls apart is making a claim about a fabric nobody has woven, and checking it against a measurement means weaving it. A force computed from the construction alone can be applied to any draft on paper.

So the criterion’s blind spot can now be priced for a cloth that does not exist yet, in one state: wet, and close enough to be past its own critical swelling. That is a narrow licence and it is a real one.

What it does not license #

Three things, and they matter more than the one above.

It says nothing about a dry cloth. The whole chain begins with a swelling past a critical value, and a dry cloth has no swelling. So the force this rung computes exists only while the cloth is wet, and the criterion’s blind spot in a dry cloth still needs the measured route.

It says nothing about an open cloth. Five of this collection’s eight cloths never reach their critical swelling, so their yarns are never compacted and this force is zero for them. The force is available exactly where the cloth was already tight — which is where friction was already doing the most work and where the criterion’s blindness matters least.

And it is a pressure inside a yarn rather than a force at a crossing. The compaction is happening in the yarn’s own cross-section, and treating it as pressing the crossing threads together is an interpretation. A more careful treatment would ask how the compaction pressure is distributed between the yarn’s internal fibre contacts and the external contact with the crossing thread, and this collection has no way to do that.

That third limitation is the one that would most change the answer, and it can only go one way: some of the pressure is spent internally, so the external force is smaller than computed. Which pushes the bound further above the true value and makes the seven per cent agreement look more coincidental still.

The contact is conformal, so the bound is nearly the value #

The step from a pressure to a force is called the weak one, and the worry attached to it is that a real contact patch is a Hertzian ellipse much smaller than the projected area — so the true force is below 0.392 newtons and possibly well below.

That worry can be tested, and it does not survive.

Hertz gives the contact radius of two crossed cylinders as (3NR/4E*)^(1/3). With the load at 0.4 newtons, the swollen radius at 0.112 millimetres and an effective modulus of about 2.2 megapascals from the yarn’s own transverse stiffness, that comes to

0.25 millimetres — more than twice the yarn’s own radius.

A contact patch larger than the body it is on is not a Hertzian contact. The threads are conformal: they have flattened against one another over the whole crossing, and the projected area is the right description rather than a loose ceiling.

So the essay’s bound is nearly its value, the true force is not well below 0.392, and the seven per cent agreement with the measured route is better evidence than the essay allows itself.

Where the transition sits #

The same expression says at what load a crossing stops being Hertzian, which is a useful boundary to have for the whole ladder. Setting the contact radius equal to the yarn’s,

N < 4E*R² ÷ 3 = 37 millinewtons

for a Hertzian contact, and above it conformal.

Set the collection’s own contact forces against it. The relaxed woven cloths run from 0.18 to 0.85 newtons — five to twenty times the threshold, all comfortably conformal. The knitted loop presses with 39 millinewtons, which is the threshold to two figures.

So every woven crossing in this collection is a conformal contact and the knitted one sits exactly at the transition. That is a coincidence worth noticing rather than a result, and it has a consequence for each side.

On the woven side the projected area is a fair estimate, so every quantity computed as a pressure over a crossing — the swelling force here, the flattening in the compression ladder, the contact area under a plate — rests on a better footing than a bound.

On the knitted side it does not. A loop’s contact is on the boundary, so a knitted crossing’s patch is neither the projected area nor a small ellipse, and any arithmetic dividing a knitted force by an area is on the least secure ground available.

And it is robust to the modulus bracket #

The threshold carries the transverse stiffness, which is the constant with no lower bound and a published range from one to ten megapascals. It is worth checking that the conclusion survives it.

At the top of the range the threshold rises to 92 millinewtons; at the bottom it falls to nine. The woven forces are above both, so the woven crossings are conformal at every value in the bracket and the finding does not depend on the fitted constant at all.

The knitted crossing is the one the bracket moves. At ten megapascals its 39 millinewtons is below the threshold and the contact is Hertzian; at one it is far above and conformal. So which regime a knitted crossing is in is unknown and is unknowable until the transverse stiffness is, which is a cleaner statement of the uncertainty than carrying it through as a factor.

That is a small result and it removes a caveat rather than adding one, which this ladder does not often get to do.

Why a wet cloth is the right place to look #

There is a reason this force turned up in the water ladder and not elsewhere, and it is worth naming.

Every other force in this collection is applied from outside: a loom’s tension, a beat-up, a pull-out test, a calender’s nip. Those all have a magnitude somebody chose, so a result computed from them is a result at a chosen operating point.

A swelling is not applied from outside. It is what the material does, at a size the material decides, and the geometry either accommodates it or does not. So the force it generates is a property of the construction rather than of an operating point — which is exactly the kind of force a criterion about drafts on paper needs.

That is the general shape of what this rung found: a self-imposed load is worth more to a structural argument than an applied one, because there is no free parameter in it.

The cloths where it is available, and why that is awkward #

Five of this collection’s eight cloths never reach their critical swelling, so for them the force this rung computes is exactly zero. Three do: the sheeting, the filter cloth and the duck, at excesses of 10.7, 2.8 and 0.15 percentage points over their own critical swellings.

Their forces, on the same bound, are 0.392, 0.128 and 0.009 newtons — a spread of more than forty, driven entirely by how far past its own boundary each cloth is.

That is awkward for the licence above and it should be said plainly. The force is not a smooth function of the construction; it is zero everywhere below a boundary and rises steeply above it, so a cloth a hair either side of its critical swelling has a force of nothing or of something. The duck, whose critical swelling is 19.85 per cent against cotton’s 20.0, is exactly that case: its force is nine thousandths of a newton and would be zero if the swelling constant were read at the bottom of its own range instead of the middle.

So this route is not a general method for getting a contact force out of a construction. It is a method that works decisively where the cloth is well past its boundary and gives an answer indistinguishable from zero where it is not, with a very short transition between.

A general method would need the elastica, which would give contact forces everywhere along a thread with no measurement and no threshold in it. That is still missing and this rung does not close it.

What was counted, and how #

The critical swelling is a boundary, checked either side at a step of one part in a million, at every cloth that has one.

The pressure is monotone in the excess, swept in eight steps from the critical swelling to the fibre’s own, and required to rise at every one.

Compaction is the cheaper route and for the sheeting the alternative does not exist, asserted both ways, with the breaking strain computed from this collection’s own tenacity and modulus rather than looked up.

And the resulting force is within an order of the measured route’s, asserted as a band rather than as an agreement. The band is deliberately wide — a factor of ten either way — because the claim being protected is about the order and the alternative would be an assertion that happened to pass on today’s constants.

Where the model stops #

No plasticity, again, and here it bites hardest. A cloth compacted at 7.8 megapascals and then dried does not come back to where it was, and this model says it does. So the force computed is the force during the wetting, and what a cloth is left with afterwards is a different and larger question.

The contact area is bounded and not computed, which is the largest single source of error and is one this collection has known about since the compression ladder — it is the same missing elastica.

Van Wyk’s law is being asked to describe a twisted yarn. It was fitted to random fibre assemblies, and a yarn’s fibres are helices under tension from their own twist. Right family, wrong specimen.

And the cloth is treated as reaching equilibrium. A real wetted cloth compacts over minutes and its fibres creep, so the peak pressure and the sustained pressure are different numbers and only one of them is here.

The generalisation #

A load the material imposes on itself is worth more to a structural argument than one somebody applies.

This is the second time in this collection that a force has been needed and the two routes have been an applied load and a self-imposed one. The first was the tensioned cloth, where the contact force is proportional to the tension and every result had to be quoted per newton of it. The second is here, where the swelling decides its own size and there is nothing to quote per.

The difference is a free parameter. An applied load leaves every downstream result parameterised by it, which means the results are a family rather than an answer; a self-imposed load removes it, and the result is a number.

So when a model needs a force, it is worth asking whether the material can supply one before choosing where to apply one. Swelling, thermal expansion, freezing and cure shrinkage are all loads a material imposes on itself, and all of them turn a family of answers into an answer.

Who found it, and when #

That a densely woven cotton becomes nearly impermeable when wet is old and is the basis of Ventile and Grenfell cloth; that this involves the threads pressing on one another is the obvious reading and is not new.

Van Wyk’s law is from 1946. The criterion this collection uses is its own, and the observation that it cannot see friction is its own, and so is the previous rung’s route from measured thicknesses to a contact force.

What is this collection’s here is the chain: a swelling that the geometry refuses, a compaction that follows, a pressure that van Wyk prices, and a force that lands where the measured route put one. Whether anybody has computed a fabric’s self-generated swelling pressure this way, this collection does not know.

Where the ladder goes next #

Out of the fabric and onto a cutting table, where two shrinkages that differ between the grain directions turn out to skew a bias-cut panel as well as shrink it.