What friction has to hold in a relaxed knit

A relaxed knit sits on a slope. Its bending energy falls away in both directions, by five millinewtons a stitch across the courses and thirty-nine along the wales, so something is holding it where a tape measure finds it.

The obvious candidate is friction at the interlacings, and this rung does the balance. It comes out differently in the two directions, and one of the two answers is exact.

What friction has available #

A stitch’s own yarn makes two interlacings — its head, where the course above hangs on it, and its sinker, where it hangs on the course below. Each is loaded with the contact force, and each can supply μ times that against sliding.

For an ordinary cotton jersey at a coefficient of nought point three, that is twenty-three millinewtons per stitch. It is an upper bound in the same way the contact force is: a set yarn presses less and therefore grips less.

Across the courses, comfortably #

The force widening a wale is about five millinewtons a stitch. Twenty-three against five is a margin of four and a half, so friction holds that direction easily and there is nothing more to say about it.

That is the direction in which a knit is famously stable in width — a jersey’s wale spacing moves little between relaxation states compared with its course spacing — and it is consistent.

Along the wales, exactly #

The other direction has an identity in it, and the identity is the whole of this rung.

The loop’s height enters the geometry only as the course spacing plus one yarn diameter. So differentiating the energy with respect to the course spacing is the same operation as differentiating it with respect to the loop’s height — and that derivative is what the contact force is defined as.

The force lengthening a knit’s wales is identically the force at its interlacings. Thirty-nine millinewtons appears twice not by coincidence but because it is one number, and the equality holds at every configuration rather than at the relaxed one. It is asserted to a part in a million every time the site is built.

Which makes the balance a pure condition #

Write the balance out. Friction can supply μN at each of two contacts; the force to be held is N. The fabric changing shape has to move the contacts along the yarn, and taking them to slide as far as the fabric moves — the most favourable case, since sliding less dissipates less — the condition is

$$N \;\le\; 2\,\mu\,N$$

and N cancels.

The condition is μ ≥ ½. There is no bending rigidity in it, no loop length, no count, no fibre density, no packing factor and no relaxation state. It is a statement about a friction coefficient and a count of two, and nothing else survives.

And it fails #

Yarn-on-yarn friction is quoted at nought point two to nought point five for the fibres in this collection’s table — cotton, wool, silk, flax, polyester, nylon, viscose. Half is the very top of the range and no fibre sits there routinely.

So friction cannot hold a relaxed knit along its wales, at any construction, in any count, in any fibre. That is not a marginal failure that better numbers might reverse: the margin is twice the coefficient, which is nought point six for an ordinary cotton, and it would need every friction coefficient in the literature to be wrong by a factor of two.

How much sliding a shape change needs #

The one assumption in the balance is the slide distance, and it is worth being explicit because it is where the argument could be attacked.

If the contacts slide less than the fabric moves, friction dissipates less and the condition is harder to meet, not easier. If they slide more — which they can, since the contact point runs along the yarn as the loop reshapes — the condition softens. A slide of twice the fabric’s displacement would bring the requirement down to a quarter, which is inside the range of a real cotton.

So the honest statement is a condition with a stated geometric factor: friction holds if the coefficient times twice the slide ratio exceeds one. At a slide ratio of one, which is the natural reading and the conservative one, it fails for every fibre. Computing the slide ratio properly needs the contact point tracked through the solve, which the machinery could do and does not.

That is the weakest link in this rung and it is named rather than buried.

The two is a count, and it is the only term a knitter can move #

Everything cancelled out of the balance except a friction coefficient and the number two, and the two is worth looking at because it is not a constant of nature. It is a count of interlacings per stitch, and a jersey has the smallest count any weft knit has.

Redo the balance with n contacts instead of two and the condition is

μ ≥ 1 ÷ n.

Against the reported ranges — cotton at two to four tenths, wool as high as five, polyester and nylon at one and a half to three — a jersey is out of reach for every fibre, three contacts is inside cotton’s and wool’s range, four is inside everything’s, and five is comfortable even for polyester.

So the structures that add contacts are the structures that could be frictionally stable, and the trade has names for all of them. A tuck stitch holds an extra loop on the needle, so its yarn meets the fabric more often than a plain loop’s does. A float passes behind several needles and is gripped where it crosses them. And an interlock has two sets of loops interlocking through one another, so each stitch is restrained from both faces rather than one.

That gives a prediction with a direction and no free parameter: dimensional stability should rise with interlacings per stitch, and a plain jersey should be the least stable weft knit there is. It is, and by a margin large enough that the trade specifies around it — interlock is the fabric chosen when a knitted garment must keep its measurements, and single jersey is the one that famously does not.

The caveat is that this is not the clean cancellation the jersey case was. Adding contacts adds friction only if the extra contacts carry a comparable load, and it changes the driving force too, since a fabric with more interlacings has a different loop geometry and a different energy slope. The count moves both sides of the balance and the arithmetic here moves only one. What survives is the sign: more contacts help, and they help fastest where the coefficient is lowest, which is exactly the fibres a jersey behaves worst in.

How much sliding would rescue a jersey #

The other term with a number in it is the slide ratio, named above as the weakest link. Inverting the condition says precisely how weak it has to be to matter.

Friction holds if 2μr ≥ 1, so the slide ratio needed is r ≥ 1 ÷ 2μ: 1.67 for a cotton at three tenths, 2.5 for a polyester at two, 1.25 for a wool at four.

Those are not absurd numbers, which is what makes the assumption worth attacking rather than dismissing. A ratio above one means the contact point travels further along the yarn than the fabric travels — which is possible, because the contact runs round the curve of the loop as the loop reshapes, and a small change in a tight curve moves the tangency point a long way.

And it is measurable without any of this machinery. Mark a yarn at a contact with a fine line, extend the fabric a known amount, and see how far the mark has moved relative to the crossing. A ratio of one is the natural expectation and the balance’s conservative reading; anything approaching two would rescue friction for cotton and wool and leave the synthetics failing, which would be a different conclusion from the one drawn here and a testable one.

Until that measurement exists, the honest statement is the conditional: friction fails for every fibre at a slide ratio of one, and the number that would change the answer is a single dimensionless ratio nobody has measured.

What is left #

The yarn’s natural shape. If the loop is the shape the yarn was set into, the driving force is not thirty-nine millinewtons but thirty-nine times one minus the set fraction, and there is nothing for friction to hold.

That was previously an attractive explanation. It is now a requirement: no degree of friction resolves the wale direction, so the setting has to be doing it, and the set fraction has to be large enough that whatever is left falls under the friction available. At a coefficient of nought point three that means the yarn is at least forty per cent set.

Which is the first quantitative statement about σ anywhere on this ladder, and it comes not from measuring a fabric but from a balance in which everything else cancelled.

Why the two directions differ #

The asymmetry looks arbitrary and is not. It is a consequence of where the yarn’s diameter sits in the geometry.

The loop’s height is the course spacing plus a diameter, because the loops interlock and the feet rest on the head below. The wale spacing has no such offset: it is just a spacing. So the height is the quantity the contact is directly about, and differentiating with respect to it gives the contact force, while differentiating with respect to the wale spacing gives something else entirely.

One direction is tied to the contact and the other is not, and that is the whole of why one balance is exact and the other is a number.

What the picture cannot show #

A friction coefficient. It is a property of two surfaces in contact, and it is not a property of either of them separately — so no drawing of a fabric contains it and no drawing of a fibre does either.

Nor can any figure show a fabric being held. A held fabric and a fabric in equilibrium look identical from every angle, at every magnification, for ever. The only way to tell them apart is to release one of them.

The same balance across the knittable range #

Since both sides scale together, the failure is not something a knitter can construct their way out of, and it is worth showing rather than asserting.

That flatness is the practical content of the cancellation. Tightening a knit raises the contact force, which raises the friction available and raises the force to be held in exactly the same proportion, so the fabric’s ability to hold its own wale length does not improve at all.

It is the one place on this ladder where tightness is not a lever. Everything else — the modulus, the run resistance, the pilling, the grip on a nap — moves by a factor of two or three across the band, and this moves by nothing.

The experiment that would release it #

Two, and they are cheap.

Lubricate. A knit soaked in a solvent that lowers the yarn-on-yarn coefficient without swelling the fibre should, if any part of the balance is frictional, move — and move in the direction the energy falls. A fabric that does not move at all is a fabric held entirely by set.

Cut a wale. A loaded fabric is one in which every loop is carrying a force; cut a wale and the loops on either side of the cut have lost half their restraint. If they move, the fabric was loaded. The displacement to look for is a fraction of a stitch and it happens once.

Neither has been run here, and the second needs nothing but a fabric, a scalpel and a microscope.

The woven cloth, for contrast #

The same balance on the other side of this collection comes out the other way, and the reason is structural.

A woven cloth’s relaxed state is an energy minimum, so the force friction has to resist at that state is zero, and friction’s whole role is to delay the arrival and to permit states near it. That is why the woven side of this collection has been able to treat friction as a perturbation and get away with it.

A knit’s relaxed state is not a minimum, so friction’s role is to hold a finite force indefinitely, which is a completely different job. The same word covers two mechanisms and the arithmetic separates them.

What this does to the older account #

It sharpens it. The standing explanation for a knit having three sets of dimensions is that friction holds it wherever the last treatment left it, and that treatments which reduce friction — wetting, agitating, tumbling — let it move.

That account is right about how a fabric moves between states and wrong about what holds it in one. Along the wales it is not friction, and the treatments that move a fabric are not only reducing friction: wetting and heating a fibre is exactly the recipe for re-setting it, so the treatment that lets a fabric move is the same treatment that decides where it stops.

Why agitation helps a cloth relax makes the frictional argument on the woven side, where it is correct because a woven cloth genuinely has an energy minimum and friction is only delaying its arrival. A knit has no minimum, so friction is not delaying anything — it is doing the whole job, and in one direction it cannot.

The three states, once more #

The measurement this whole thread is answering to is worth putting beside the balance, because it is what makes the conclusion inescapable rather than merely arithmetic.

A purely frictional account has to explain that ordering, and it cannot: friction has no preferred direction, so releasing a fabric from it should move the fabric downhill, and every published relaxation state is further uphill than the last.

Set explains it in one line, because setting happens during the very treatments that define the states. So the two candidate mechanisms are not competing on equal terms — one of them fails a balance and cannot explain the direction, and the other passes both.

Why nobody hit this before #

Because it needs the contact force, and the contact force needs a thread shape that can be differentiated.

The identity behind it is not deep — the loop’s height is the course spacing plus a diameter, and everybody knows that — but it only becomes visible when both sides are computed in the same units by the same machinery. Two literatures each had one side: knitted-fabric mechanics had loop shapes with imposed dimensions, and the frictional account had a qualitative story about relaxation treatments. The number that connects them is the derivative of one with respect to the other.

The static and the kinetic #

A caution about which coefficient the half applies to.

Friction is two surfaces rather than one, and it also has two values: static, which decides whether something starts to move, and kinetic, which decides what happens once it has. Holding a fabric still is a static question, so the relevant coefficient is the larger of the two, which is the distinction that essay draws — which helps the balance slightly and does not rescue it.

The kinetic value matters for the other question: once a fabric starts to move, does it stop? A structure with a lower kinetic than static coefficient and no energy minimum to arrive at will keep going until something else stops it, which is the geometric edge. That a knit does not do this is more evidence for the set.

What a specification should conclude #

That a knit’s dimensional stability is a setting specification rather than a frictional one, and should be written that way.

A finish that raises friction — a resin, a rougher surface, a lower lubricant loading — buys stability in one direction only and buys none where it is most needed. A treatment that sets the yarn buys it everywhere. That is a different priority from the one a purely frictional account suggests, and it happens to agree with what the trade does: every stability specification in knitting is a relaxation and setting protocol, and none of them is a friction specification.

The trade got there by measuring fabrics. This is the arithmetic saying why the thing it measures is the thing that matters.

What the number means for a fibre choice #

A short practical reading, because the condition names a fibre property directly and few results here do.

The coefficient needed is a half, and the reported ranges are nought point two to nought point four for cotton, nought point two to nought point five for wool, nought point one five to nought point three for polyester and nylon. Wool reaches highest, and wool is also the fibre that sets most completely — so the two mechanisms that could hold a knit are strongest in the same fibre.

That is consistent with wool knitwear’s reputation for dimensional stability once properly relaxed, and with polyester’s for the opposite. It is a weak argument, because so many other things differ between the fibres, and it is the only place on this ladder where a friction coefficient enters a conclusion about a material rather than a structure.

The stronger reading is negative: no fibre’s friction is enough on its own, so a fabric’s stability is decided by its setting rather than by its fibre’s grip, and a fibre chosen for grip alone will disappoint.

Where the ladder goes next #

Nowhere further on this thread; the machinery is spent and what it cannot say is written down. What remains is a joint rather than a fabric — a seam that has to give what the knit gives — and the arithmetic for that has been available on the woven side and not here.