A fabric is a population of contacts

The clearance between two adjacent courses of a knitted fabric runs from four fifths of a yarn diameter to nearly four diameters along a single wale. Twenty per cent of the length is inside one diameter of its neighbour and eighty per cent is not.

That was measured to say something about the yarn’s section. It says something larger about the fabric’s friction.

Where a fabric’s friction is #

Friction needs contact. A yarn that is not touching anything is not being gripped by anything, however much friction its fibres have.

So a fabric’s resistance to having its yarn moved — pulled out, slipped, withdrawn, unravelled — is concentrated in the fifth of the yarn that is pressed and absent from the other four fifths.

That is not how any of this collection’s friction calculations are written. Every one of them treats the contact as occurring at a point, at the crossing, and computes a normal force there from the thread’s own bending.

A point is a reasonable idealisation of a short region and it is a poor one of a fifth of a length.

Which results assume uniformity #

Four of them, and it is worth being specific because the assumption enters each differently.

Thread withdrawal. The force to pull a yarn out of a fabric is computed as a capstan relation over the crossings it passes. The capstan relation assumes contact over the wrapped angle, and the wrapped angle is taken from the geometry — which is right, and the length over which the wrap is in contact is not part of the calculation.

Seam slippage. Same arithmetic, same assumption.

Fraying. The length of fringe a cut edge sheds is where the accumulated friction overtakes the pulling force, and the accumulation is per crossing rather than per unit length.

And a run’s resistance. A loop being pulled out of a knitted fabric slides against the loops holding it, and this collection computes the friction over the contact — which is one contact per interlacing.

In every case the calculation is per crossing rather than per unit length, so the profile does not change the count. What it changes is the normal force distribution within each crossing, which is what decides how much the friction actually is.

Why a distributed contact is not a point contact #

The difference is not a refinement, and the reason is the same one this collection has met about a cloth’s diameters.

A point contact has a normal force and a coefficient, and the friction is their product. A distributed contact has a pressure that varies along it, and the friction is the integral of the pressure times the coefficient over the area.

Those give the same answer only if the coefficient is constant and the total force is the same, and for a fibre assembly neither holds. A fibre’s friction coefficient falls with the pressure — it is not Amontonian, because the real contact area grows sub-linearly with load — so a force spread over a long contact produces more friction than the same force concentrated at a point.

So the point idealisation under-estimates the grip, and by an amount that depends on how spread the contact is.

That is a systematic bias in one direction across four of this collection’s results, and it is consistent with something the site has recorded: computed withdrawal forces sit below measured ones.

How much of a bias #

An estimate rather than a calculation, because the pressure distribution is not computed.

The friction of a fibre assembly goes roughly as the load to a power between two thirds and one — the classical result for an assembly of fibres in contact, where the real area grows as the load to the two thirds.

Spreading a fixed total load over n times the area multiplies the friction by n to the power of one minus that exponent, so between one and n to the one third.

The contact here is spread over a fifth of a course rather than a point. Taking the contact length as ten times a point’s, the friction is between one and about twice what a point calculation gives.

A factor of up to two, in the direction that makes fabrics grip better than computed. That is not a refinement and it is not a catastrophe, and it is the size of the gap this collection has been recording.

The two mechanisms a tight fabric has #

The profile separates something that has always been described as one effect.

A tighter fabric grips its yarn better, and the usual account is that its threads are pressed together harder — a larger normal force at each crossing.

The profile adds a second: a tighter fabric is in contact over more of its length, so there is more area for the friction to act over.

Those are independent and they compound. A fabric twenty per cent tighter has both a higher force and a longer contact, and the friction rises by more than either alone would give.

That is a testable difference in principle. A friction that goes as the normal force alone would scale one way with tightness; one that goes as force times contact length scales differently, and a withdrawal-force measurement across a range of constructions would separate them.

What a woven cloth’s profile would be #

The comparison is worth predicting, and the prediction follows from the same argument that decided the overlap.

A woven cloth’s threads cross at right angles, so the contact between two of them is short: the region where they are within a diameter is a small patch round the crossing rather than a long run.

A knitted fabric’s adjacent courses run alongside, so their contact is long.

So a knitted fabric’s contacts are distributed and a woven cloth’s are nearly point contacts, and the point idealisation is much better for a cloth than for a knit.

That is convenient, because this collection’s friction machinery was built for woven cloth and has been applied to knitted fabric — and the direction of the error is now known.

What it says about hysteresis #

There is a consequence for something this collection has computed and has never had a geometric account of.

A fabric loaded and unloaded does not come back to where it started: a cloth gives back less than it took, and the loss is friction at the contacts.

The profile says where that loss happens: in a fifth of the yarn, at the interlacings, and nowhere else. Four fifths of the yarn is bending and unbending elastically with nothing rubbing.

That has a prediction attached. The hysteresis should scale with the pressed fraction rather than with the fabric’s whole yarn content — so a slack fabric, whose pressed fraction is lower, should be proportionally more elastic and less lossy than a tight one, beyond what the change in normal force alone gives.

That is a testable ordering and it separates two mechanisms that a load–unload curve otherwise reports together.

What was counted, and how #

The profile is the distance from each sample of one course to the nearest point of the other, at a hundred and twenty samples per half period over two wales.

The pressed fraction is the proportion of samples inside one yarn diameter, and it is 20 per cent for the collection’s default construction.

The check is that the profile is not flat — a ratio of maximum to minimum above two — and that the pressed fraction is a fraction rather than the whole length. Both would fail if the layout were wrong.

The friction estimate is an order-of-magnitude argument from the classical load exponent for a fibre assembly, and is stated as such: the pressure distribution over the contact is not computed anywhere.

Why this is not the friction coefficient #

A clarification, because the rung could be misread as saying the collection’s friction coefficients are wrong.

They are not. A coefficient of friction is a material property of two surfaces and this collection takes its values from measurements as everybody does.

What is at issue is the geometry the coefficient is applied to: a point where a distributed region belongs. That is a modelling choice rather than a material number, and it is exactly the kind of thing that gets made once, early, for good reasons, and is never revisited.

The distinction matters for what to do about it. Refining the coefficient would not help; the coefficients are as good as the measurements allow. Computing the pressure distribution over the contact would, and that needs a contact solve.

So this is another item on the same list as everything else in this ladder: a defect whose repair is one piece of machinery this collection does not have, and which four separate results are waiting on.

Where the model stops #

No pressure distribution. The profile is a clearance, and how hard each part of the contact is pressed needs a contact solve — which is the piece of machinery this ladder has priced and not built.

The friction law is quoted rather than derived. That a fibre assembly’s real contact area grows sub-linearly with load is standard and is not computed here.

And the fifth is a knitted fabric’s fifth. A rib, an interlock or a woven cloth has a different profile and none is computed.

Nor is the hair layer here. A real yarn’s surface is fuzzy, so the contact between two yarns is partly fibre-on-fibre outside the geometric contact region — which would spread the contact further still and push the bias in the same direction.

What the fifth means for a fabric’s feel #

There is a consequence for handle rather than for a calculation, and it is the one a person notices.

Four fifths of a knitted fabric’s yarn is not touching anything. That yarn is free to bend, to move, and to be displaced by a finger, and it is the reason a knitted fabric feels the way it does: soft, mobile, and quick to conform.

A woven cloth’s yarn is constrained at every crossing and the crossings are close together, so much less of it is free. That is why a cloth of the same yarn feels firmer.

So the difference in handle between a knit and a weave is not only the loops’ geometry; it is the fraction of the yarn that is free to move, and that fraction is computable.

That is a satisfying place for the measurement to land, because handle is the property this subject is worst at quantifying and the one everybody cares about most. A fraction of free yarn is not a handle value and it is a number that ought to correlate with one.

Nobody has looked, and this collection now has the number for a knitted fabric and could compute it for a woven one in an afternoon.

The generalisation #

The rung is the third time this collection has met one structural pattern, and it is time to name it.

A quantity that is a mean over a distribution is not the quantity most questions are about.

The first instance was a yarn’s diameter: a mean over a population with a fifteen per cent coefficient of variation, and a cloth is a population rather than a thread works out that a mean is right for a mass and an order statistic is right for a jam.

The second was the flattening: a minimum over a profile that varies by a factor of five, right for a clearance and wrong for a section.

The third is the contact: a point idealisation of a distributed region, right for a count of crossings and wrong for a friction.

In each case the fix is the same and is cheap: ask which part of the distribution the question is about, and use that rather than whatever the calculation happens to have.

What a measurement would separate #

The bias is estimated and the estimate is loose, and it is worth saying what would tighten it.

Withdrawal force against tightness factor. Pull a yarn out of a series of fabrics of the same yarn at different loop lengths, and plot the force.

A point-contact model predicts the force scales with the normal force per crossing, which this collection computes.

A distributed-contact model predicts it scales with that times the pressed fraction to some power between nought and a third.

The two predictions diverge across the tightness range by a factor approaching two, which is far above the scatter in a careful withdrawal test.

That is a measurement any textile laboratory can make and it would settle a bias running through four of this collection’s results. It also needs no new fabric: a set of jerseys at graded loop lengths is a standard trial piece.

Why the estimate cannot be tightened here #

The obstacle is worth naming precisely, because it is not a lack of effort.

The friction over a distributed contact is the integral of the pressure times the coefficient over the contact area. Two of those three are unavailable.

The pressure distribution needs a contact solve, which is the machinery this ladder has priced and not built.

The coefficient’s dependence on pressure is a material relation this collection does not carry: its fibre table holds a coefficient as a range and not as a function of load.

So the estimate above is an order-of-magnitude argument from a classical exponent, and it is offered as one. What it establishes is a direction and a rough size, and it says which measurement would replace it.

Why a point contact was the right choice originally #

A defence, because the idealisation was not careless and the reasons for it are still good ones.

A point contact makes a capstan calculation possible. The relation between tension in and tension out over a wrapped angle needs a wrap and a coefficient and nothing else, and it gives an exponential that this collection has used to explain fraying, seam grip and knot security.

A distributed contact does not have that form. It needs a pressure distribution, and once there is a pressure distribution there is no closed form and no comparison anybody can check by hand.

So the point idealisation bought a great deal: three results in three different fields, all of them qualitative statements with the right shape, none of which would exist if the calculation had waited for a contact solve.

The correct reading is therefore not that the idealisation was wrong. It is that it has a known bias in a known direction, of roughly known size, and that the size can now be estimated where before it could only be worried about.

That is what a measurement of the profile buys, and it is a fair return for a distance calculation.

Who found it, and when #

That fibre friction is not Amontonian, and that a fibre assembly’s real contact area grows sub-linearly with load, is standard and dates from the middle of the twentieth century.

That a yarn in a fabric is in contact only near its crossings is obvious from any section and is not a discovery.

What is this collection’s own is the fraction: a fifth, computed from its own solved geometry, which turns a qualitative statement into a number and lets the bias in four of its own results be estimated rather than suspected.

The three populations, together #

This collection has now found three quantities in a knitted fabric that are populations rather than values, and it is worth putting them on one page because they are the same yarn measured three ways.

The diameter is a population over the yarn’s own length, with a coefficient of variation of about fifteen per cent from the spinning.

The clearance is a profile along the yarn, running from four fifths of a diameter to nearly four.

And the contact is a fraction, twenty per cent, of the length that is engaged at all.

The three are not independent. A thick place in the yarn is a place where the clearance is smaller and the contact is longer, so the three vary together and the correlation is positive.

That means the fabric’s friction is concentrated not merely in a fifth of its yarn but in the thick parts of that fifth, which is a further concentration nobody has estimated.

It is also the reason a fabric’s withdrawal force scatters as much as it does between specimens: the quantity being measured depends on where the thick places happened to fall relative to the crossings.

Where the ladder goes next #

The contact ladder’s last question is about the other fabric. A sett is a statement about how much room threads take, and how much room a thread takes depends on its section — which the trade has always assumed and this work has learnt to predict for one fabric and not the other.

What a sett is when the yarn is not round.