A run is a race between two energies

A knit does not fray; it runs, and the reason is structural: a woven cloth is many threads and a knit is one, so a break in a knit releases a loop which releases the loop below it, and there is nothing to stop the sequence except friction.

The question this rung asks is what drives it. There are two candidates and they suggest opposite remedies, so the answer matters.

The two candidates #

Stored energy. A loop is a bent rod, and unravelling it lets the rod straighten. If the yarn’s natural shape were straight, a stitch would be holding about twenty-five microjoules of bending energy, and releasing it would drive the loop out of the one below.

Applied load. A garment under its own weight, or being pulled on, puts a tension across the fabric, and that tension has to be carried through every interlacing. Where the tension exceeds what friction can hold, the loop slides.

The first would make a run a self-propagating process needing no external force at all. The second makes it a failure under load, which stops when the load does.

Which one it is #

The second, and the first is negligible — for a reason established two rungs back.

A relaxed knit’s yarn has been set into the loop’s own shape, so it holds almost no bending energy at all and has almost nothing to release. The twenty-five microjoules figure is what an unset yarn would hold, and a fabric that has been wet-relaxed and dried is nowhere near it.

That is a strong statement and it is testable against ordinary experience. A run in a stocking laid flat on a table does not travel. It travels when the stocking is on a leg, or is picked up, or is pulled — and it stops the moment the load does.

Which means the lever is friction #

The force holding a loop in place is the coefficient of friction times the contact force times the two interlacings a stitch makes: for an ordinary jersey, about twenty-three millinewtons per loop.

Two and a half grams-force. That is what has to be exceeded before a dropped stitch travels one course, and it is small enough that a stocking’s own weight over a leg can do it while a heavy jersey’s cannot.

Where the twenty-three millinewtons comes from #

Every ingredient is a quantity this ladder computed, and it is worth assembling them so that a reader can disagree with a specific one.

Multiply by a coefficient of friction of nought point three, which is the middle of the reported range for cotton on cotton, and by the two interlacings a stitch’s own yarn makes. That gives twenty-three millinewtons.

Three numbers, of which one is computed, one is measured by other people and one is a count. The count is the least likely to be wrong and the friction coefficient is the most, because friction is two surfaces rather than one and a finished fabric’s coefficient can be half or double the bare fibre’s.

Where the loop is held #

The geometry of the hold is worth looking at, because “two interlacings” is a count anybody can check.

That is very few contacts by any fabric’s standards. A woven thread of the same length meets the fabric four times as often, and each of those contacts is eight times heavier — which is the thirty-fold difference in grip that runs through this whole ladder.

Two light contacts is what a knitted loop has, and it is why the topological possibility of a run is also a mechanical certainty.

The lever a knitter has #

Tightness, and it works on both terms at once.

A shorter loop means the contacts are closer and the contact force is higher, so the hold rises from fifteen millinewtons at a tightness factor of ten to thirty-five at sixteen — a factor of two and a half for a factor of one point six in tightness.

That is why every practical remedy for running is a tightness remedy: a tighter fabric, a finer gauge, a heavier yarn at the same loop length. The trade knows all three work and this is the arithmetic under them.

Why a fabric never breaks its yarn instead #

There is an alternative failure that a knit never takes, and its absence is the sharpest consequence of the numbers.

A thread being pulled through a fabric either slides or breaks, and the gripped length at which the two are equally likely is the crossover length. In a relaxed woven poplin it is about twelve millimetres, so a thread pulled over any ordinary grip breaks rather than travelling — which is why a woven cloth frays for a few millimetres and stops.

In a jersey the crossover length is four hundred and sixty millimetres, longer than any garment. A knitted loop can never be held firmly enough to break rather than slide. So a run is not one of two possible outcomes; it is the only outcome, at every load short of tearing the fabric outright.

What stops a run #

Three things, and only one is a property of the fabric.

The load stopping, which is the usual case and is why most runs are a few courses long.

A structural interruption — a tuck, a float, a change of stitch, a seam — which puts a loop in the path that is held by more than two interlacings. That is what a run-stop course in a stocking is, and it is why the ladder in a stocking always ends at a welt.

Friction rising, which happens if the fabric is locally tighter, wetted, or dirty. A run that stops for no visible reason has usually met one of those.

Why a warp knit does not run the same way #

Warp knitting is a different thing entirely, and the difference is exactly the one this arithmetic is about.

A weft knit’s course is one thread, so releasing a loop releases its neighbour along the course. A warp knit’s loops in one course come from many threads, each of which is held by loops in the courses above and below and by its own traverse to the next wale. A released loop has nowhere to release to, because the thread it belongs to continues into a different wale.

So a warp knit’s failure is a hole rather than a ladder, and it is not because its friction is higher. It is because the topology does not supply a next loop.

What the picture cannot show #

The event. A run is a sequence in time and every figure here is a state, so what is drawn is a fabric before and a fabric after, with the force printed rather than shown.

Nor can any figure show the load path. The tension that drives a run arrives from wherever the garment is being pulled, through a fabric that is enormously extensible and therefore distributes it in a way no static drawing carries.

What the energy would have been worth #

The candidate that lost deserves its number, because it is only negligible under a condition that can fail.

The friction work over one course of run is the hold times the distance slipped, which is roughly one loop length: twenty-three millinewtons times three and a half millimetres, or eighty nanojoules. Against twenty-five microjoules stored, that is a factor of three hundred.

So a fabric whose yarn is genuinely unset — a polyester knit that has not been heat-set, most obviously — should run spontaneously, without any load at all, once a stitch is dropped. That is a strong prediction and it matches an unpleasant known behaviour of unset synthetic knits, which are notoriously the worst runners in the trade.

Which makes setting a run remedy #

Read the other way, the arithmetic says heat-setting is not only a dimensional treatment.

A set yarn has almost no stored energy to release, so setting removes the self-propagating driver entirely and leaves only the applied load. That is a large change in kind rather than in degree: a fabric that runs only under load is a fabric a wearer can control, and one that runs on its own is not.

Nobody appears to specify heat-setting as a run remedy, and this says it is one — with the caveat that setting also lowers the contact force and therefore the friction hold, so the two effects act in opposite directions and the net is not computed here.

What is not in the number #

The dynamics. A run travels fast, and a loop being pulled out at speed meets a different friction from one being pulled out slowly — static and kinetic friction are different quantities and the arithmetic here uses one coefficient for both.

The direction is known: kinetic friction is the lower of the two, so a run that has started is held less firmly than one that has not. That is consistent with the observation that runs accelerate, and it means the twenty-three millinewtons is the figure for starting a run rather than for continuing one.

A prediction #

If the hold is friction times a contact force, then anything that changes either should change the run resistance in proportion, and the two are independently adjustable.

A lubricant — a softener, a silicone finish — lowers the coefficient and should make a fabric run more readily at the same construction. A setting treatment lowers the contact force and should do the same. Both are finishes applied for handle, and this says they have a cost in run resistance that is rarely counted.

The experiment is a standard snag-and-run test on one fabric finished three ways, and the prediction is that the softest-handling of the three runs at the lowest load.

The number that is not here #

How far a run travels for a given load, which needs the dynamics and the load path and is not available from a static hold force.

What the hold force gives is a threshold: below it, nothing travels at all. That is the useful half for a designer, because a garment can be specified to keep its working loads under a threshold and cannot easily be specified to limit how far a failure propagates once it starts.

What a hole does that a run does not #

The comparison with the other knitted failure is worth making, because they have different arithmetic and are often described together.

A knit has no hole to lose established that a knit’s openings are not structural in the way a leno’s are — there is no hole whose disappearance destroys the fabric. A hole punched in a knit is a local loss of loops, and whether it grows is exactly the run question asked at its edge.

So a hole in a knit is a run waiting to start, and a hole in a woven cloth is a hole. That is why a small snag in knitwear is a repair job and a small snag in a shirt is not, and it is one more consequence of a crossover length longer than the garment.

The stocking, which is the worked case #

Everything above is easiest to see in the garment the failure is named after, and the numbers there are different in every term.

A stocking is knitted very openly — a low tightness factor, at the slack end of the band or below it — so its contact force is at the bottom of the range and its friction hold is nearer fifteen millinewtons than thirty-five. It is made of a fine continuous filament, whose coefficient of friction against itself is lower than a staple yarn’s. And it is worn at fifty to a hundred per cent extension, where the fabric is carrying a real load rather than lying slack.

Three factors, all in the same direction, and each worth a factor of about two. The product puts a stocking within reach of its own run threshold in ordinary wear, and it explains why the failure is so characteristic of that one garment and so rare in every other knitted thing.

It also says what the remedies are, and they are the ones the trade uses: a run-stop course every so often, a tighter welt, and a filament with a higher friction or a surface treatment that raises it.

What was known before #

That knits run and wovens fray, structurally, for as long as both have existed. That tightness helps, from the trade. That the mechanism is friction at the interlacings, in the fabric-mechanics literature.

What has not been available is the force, so the two candidate drivers have never been ranked and the standing intuition — that a knit is somehow storing energy that a run releases — has never been checked. It is wrong for a relaxed fabric, and it would be right for an unset one, which is a distinction the arithmetic makes and the intuition does not.

A threshold in wearable units #

Converting the hold into something a garment designer can use takes one more step, and the step is a count.

The hold per loop is twenty-three millinewtons, and a run travels along a wale, so the resisting force per unit width of fabric is the hold divided by the wale spacing: about twenty-eight newtons per metre.

A jersey at fifty per cent extension is carrying about one and a half newtons per metre. So an ordinary garment load is a twentieth of the run threshold, which is why knitwear does not ladder in normal wear, and why a stocking — knitted far more openly, at a fraction of the contact force, and worn at a much higher extension — does.

That comparison is available in one paragraph now and was not available at all before, because it needs both quantities in the same units and one of them did not exist.

What the tightness lever is actually worth #

Tightness was named above as the only lever that works, and the curve gives two points on it: fifteen millinewtons at a tightness factor of ten, thirty-five at sixteen. Two points fix an exponent, and the exponent is worth having because it says how much of a remedy tightness can be.

A ratio of 2.33 in hold across a ratio of 1.60 in tightness makes the hold go as

tightness factor to the power 1.8,

which is close enough to a square to be worth reading as one. And a square is meaningful rather than coincidental here. The tightness factor is the square root of the yarn’s tex divided by the loop length, so its square is tex over loop length squared — the mass of yarn per unit area of the loop’s own footprint, which is the fabric’s areal packing. So the friction hold at a loop is very nearly proportional to how much yarn is crowded into the space the loop occupies, which is what a contact force between two crossing rods ought to depend on and is a small piece of evidence that the contact-force calculation is behaving.

The exponent immediately prices the lever. To double the run threshold takes a tightness factor raised by 2^(1/1.8), which is forty-seven per cent — and the knittable band is ten to sixteen, a range of sixty per cent. So the whole of what a knitter can choose, from the slackest fabric that will hold its shape to the tightest a machine will make, is a factor of 2.3 in run resistance.

That is the number this section exists for, and it is a discouraging one:

tightness cannot make a jersey run-proof; it can make it run at twice the load.

Two things follow. The first is that the trade’s other remedies are not alternatives to tightness but necessities alongside it. A run-stop course does not raise the threshold by a factor of two, it removes the propagation path entirely, and against a lever worth 2.3 across its whole range that is a different order of intervention.

The second is about where in the band a garment should sit, and it argues against the obvious answer. A factor of 2.3 across the band means the marginal return on tightening is largest at the slack end — going from ten to eleven buys seventeen per cent, going from fifteen to sixteen buys twelve — so the cheap improvement is always in tightening a slack fabric rather than in tightening an already tight one. And tightening costs handle, extensibility and yarn, all of which get worse faster at the tight end. The construction that is hardest to justify is the one in the middle, tightened enough to have lost the drape and not enough to have gained a threshold worth naming.

There is one caveat that the exponent cannot carry, and it is the same one the whole ladder carries. The contact force is computed from the loop’s bending, and a set yarn presses less than the calculation says. Setting is not uniform across the tightness range — a tighter loop is bent harder and therefore takes a set more completely — so the real curve is probably flatter than 1.8 at the tight end, and the factor of 2.3 is an upper bound on what tightness buys rather than an estimate of it.

Where the ladder goes next #

To a joint rather than a failure. A seam in a knit has to give what the knit gives, and the arithmetic for how many stitches that takes has been available for woven cloth and not for this.