What a knit gives when it is pulled

An early rung of this collection recorded a shortfall in as many words: the extension a knitted loop supplies needs a loop model with bending stiffness in it, which this site does not have. Why a knit recovers quoted a rib’s extension as a lower bound and said it was one.

The model exists now, and the shortfall closes into a curve rather than a number.

What is being computed #

At each imposed wale spacing the fabric’s course spacing is whatever minimises the loop’s energy — chosen by the model rather than assumed — and the shape is re-solved. The force is the rate the energy changes with the wale spacing, converted to a force per unit width of fabric by dividing by the course spacing, because a fabric’s edge of a given length carries one stitch per course spacing across it.

Three quantities go in: a loop length, a yarn diameter and the relaxed spacings. Nothing is fitted and nothing is measured except those.

Where the curve starts #

At zero, and that is not automatic. It starts at zero because the yarn’s natural shape is taken to be the relaxed loop rather than a straight rod — which is the whole content of the rung before this one — so a fabric sitting at its relaxed dimensions holds no energy and needs no force.

An unset model gives a curve that begins at forty-eight newtons a metre and falls, which is not a curve anybody can lay a measurement against.

The soft region #

For the first hundred per cent the fabric takes a few newtons a metre. A metre-wide piece doubled in width needs the weight of three hundred grams.

That is the number every knitted garment is designed around and it has been a qualitative fact for as long as there have been knitted garments. What it comes from is the loop moving: the legs straightening, the heads opening, the course spacing adjusting, all at exactly constant yarn length. The yarn’s own tensile modulus is not in it and does not appear anywhere on the curve.

That is why a knit is a hundred times more extensible than the thread it is made of, and why it is extensible in a fibre with no elastic extension at all. Cotton stretches about seven per cent before it breaks; a cotton jersey stretches a hundred and comes back.

The stiff region #

Past about two hundred per cent the curve turns up hard. The cause is geometric and has nothing elastic in it: the straight line between two interlacings is approaching the yarn available to span it, and the loop is running out of ways to be anything but taut.

The two regions are the reason a knitted fabric’s load–extension curve is the shape it is and a woven cloth’s is not. A woven cloth’s extension comes from crimp interchange and runs out after a few per cent; a knit’s comes from a loop with half its length spare.

Why the computed force is a lower bound #

Three things push the real fabric stiffer than the curve, and all three are absent from the model.

Friction at the interlacings. Every loop that moves slides against the loop it hangs from, against a normal load of tens of millinewtons. That is dissipative, so it adds force on the way out and subtracts it on the way back — which is a hysteresis loop rather than a stiffer curve, and it is why a real measurement does not retrace itself.

Torsion. A loop that is pulled open has a different twist from one that is not, and the twisting energy is not counted.

Courses meeting sideways. The model does not stop adjacent courses crowding one another in the fabric plane, and they do, well before the yarn runs straight.

So the curve is the elastic floor of the real behaviour, and the sentence saying so belongs beside every number read off it.

What the hysteresis is worth #

The frictional part is not computed here and it can be bounded. The work friction dissipates over a cycle is at most the friction force times the distance the contacts slide, and both are available: the friction available at one interlacing is a coefficient times the contact force, and the slide is a fraction of a loop length.

At an ordinary coefficient that comes to a few microjoules a stitch over a large extension, against a stored elastic energy of a few nanojoules at the same point. The dissipated work dominates the stored work by three orders of magnitude, which is a strong statement and an uncomfortable one: most of what a knit does when it is pulled and released is friction, and the elastic curve computed here is the small part.

That is consistent with what is measured. A knitted fabric’s load–extension loop is famously fat, its recovery is famously incomplete, and recovery is measured and nothing predicts it said so on the woven side for the same reason.

A correction to an earlier rung #

A knit is soft because it bends asked the same question with the loop modelled as arcs held to a radius by contact, and got a sharper answer than this one: extending such a loop lengthens its legs and bends nothing further, so the bending cost of extension is exactly zero and the whole of a knit’s first per cent must be friction.

That zero is an artefact of the model rather than a property of the fabric. Arcs of a fixed radius joined by straights have their curvature fixed by construction, so of course moving the straights costs no bending; an elastica has no such construction and its curvature redistributes at every extension. The cost is not zero.

It is, however, tiny — a few nanojoules a stitch against friction’s few microjoules — so the earlier conclusion survives its own reasoning being wrong. The first per cent of a knit’s extension is friction, and the correction is that the elastic part is small rather than absent, which matters because a small quantity can be plotted and an absent one cannot.

What the curve does say despite that #

Its shape, which is geometric rather than energetic, and therefore survives every one of the omissions.

Soft, long, and then abruptly stiff, with the stiffening at an extension set by the loop’s own slack. No amount of friction or torsion changes where the geometry runs out, and no fabric with this loop length reaches further whatever its yarn is made of. The elastic floor moves; the wall does not.

That is a stronger statement than it sounds, and it is the reason the curve is worth drawing at all. Everything the model leaves out — friction at the crossings, the yarn’s own torsion, its compression where the loops press — is a force, and a force can only change how hard the fabric is to pull to a given extension. None of them can lengthen the yarn, and the wall is where the yarn has been pulled straight. So the omissions move the curve up and down and cannot move it right, which means a shape argument survives them all and a stiffness argument does not.

And the position of the wall is a specification a maker can act on. It is set by the loop length against the plan dimensions and by nothing else, so a fabric that has to reach a stated extension has to be knitted at a stated loop length, and no choice of fibre or finishing will substitute. That is the one hard constraint in a subject where almost everything else is a trade-off, and it comes out of the geometry rather than out of a measurement.

Where the fabric goes while it does it #

Extension in one direction is not extension in one direction only, and the model chooses the other dimension rather than fixing it.

That sign change is not a curiosity attached to the curve; it is part of what the curve is. The early softness comes partly from the loop opening in the direction being pulled and partly from it being free to grow in the other, and a fabric held to constant length in the wale direction would give a stiffer curve than this one. The rung that pursues the sign change is about why it happens; what matters here is that the transverse condition has to be stated for a load–extension curve to mean anything, and it usually is not.

The energy under the curve #

Integrating a force against an extension gives work, and the work is worth quoting because it is small.

Taking the fabric to a hundred per cent extension stores about two microjoules a stitch, or four joules a square metre. That is a very small quantity — about the energy needed to lift the fabric a few centimetres — and it is a hundredth of what an unset yarn would be holding at rest.

A fabric that is soft, extensible and stores almost nothing is a fabric that will not spring back hard, and that is exactly right. Knitted recovery is not powerful; it is reliable, and reliability comes from the geometry always being available rather than from any force being large.

Reading it as a designer would #

The useful form is not force against extension but the reverse.

To reach fifty per cent extension a garment needs a newton and a half per metre of seam. To reach a hundred, three. To reach two hundred, twenty. A cuff working across the first hundred per cent is working in a region where the force barely changes, which is exactly what a cuff wants — grip that does not tighten as the wearer’s wrist passes through.

What moves the curve #

Two levers and no others.

The loop length moves everything, because it sets the slack. A tighter fabric has less spare yarn per interlacing, so its geometric wall arrives sooner and its soft region is shorter; a slacker one stretches further and more softly. This is the lever a knitter has, and it is the same lever that moves the contact force, the run resistance and the pill supply.

The yarn’s bending rigidity moves the force and not the shape. Every ordinate scales with it and no abscissa does, so a stiffer yarn gives a stiffer fabric with the wall in exactly the same place. And that rigidity is a bracket a hundred and thirty wide, so the vertical scale of this curve is known far less well than its horizontal.

What a measurement would have to control #

If anybody wanted to test this curve, the controls matter more than the instrument, and three of them are usually left loose.

The transverse condition, because a strip held in jaws is not free to contract at the jaws and is free in the middle, so a strip test measures neither the free case computed here nor the fully restrained one. The relaxation state, because the fabric’s own dimensions are a history and a curve measured on a dry-relaxed fabric is a curve about a different fabric from one measured on a fully relaxed one. And the cycle number, because the first pull differs from the tenth by more than the difference this ladder is trying to compute.

None of those is a criticism of anybody’s measurement. They are the reasons a published load–extension curve and a computed one are hard to compare, and stating them is the difference between a comparison that could be made and one that has been.

What recovery itself bounds the sliding to #

The hysteresis is bounded above by the friction force times the distance the contacts slide, and the slide distance is taken as a fraction of a loop length. That assumption can be replaced by a measurement, because the fabric recovers, and recovery is an energy statement.

A knit taken to a hundred per cent extension stores about two microjoules a stitch. For it to come back at all, that stored energy must exceed what friction dissipates on the return — and the friction available at a stitch’s two interlacings is about twenty-three millinewtons. So the slide s must satisfy 23 mN × s ≤ 2 µJ, giving

s ≤ 87 micrometres.

Against a loop length of three and a half millimetres. The contacts can slide at most a fortieth of a loop length over a doubling of the fabric, or the knit would not return, and it does.

That is a genuinely useful bound and it comes from an observation rather than from a model: nothing about the loop’s shape enters it, only the stored energy, the friction and the fact of recovery.

Which contradicts what the other balance needs #

The bound has a consequence that runs across ladders, and the contradiction is worth stating rather than leaving to be discovered.

The fabric’s own wale spacing moves 875 micrometres over that same doubling. So the slide ratio — contact travel over fabric travel — is at most 0.099.

The friction balance for a relaxed knit needs the opposite. For friction to hold a jersey along its wales at a coefficient of three tenths, the slide ratio must be at least 1.67 — because friction can only hold what it dissipates against, and a contact that barely moves dissipates barely anything.

The two requirements differ by a factor of seventeen and point opposite ways. Friction cannot both hold a relaxed knit in place and let a stretched one come back. A slide ratio large enough for the first makes the second impossible; one small enough for the second makes the first fail by a wide margin.

Which is a second argument for the set #

Both requirements dissolve at once if the yarn’s natural shape is the loop rather than a straight rod, which is what the rung before this one establishes and what this curve’s starting at zero already assumes.

There is nothing for friction to hold. A set loop at its relaxed dimensions is at its own natural shape, so the driving force the friction balance was trying to resist is not there, and no slide ratio is required.

And there is somewhere for the fabric to return to. Recovery is the yarn going back to the shape it was set into, which needs only enough stored energy to overcome the sliding on the way — the 87-micrometre bound — rather than enough to hold a fabric indefinitely against a permanent force.

So the set is not merely a convenience that makes a curve start at the origin. It is what makes the two frictional requirements compatible, and without it this collection’s knitted ladder carries two energy statements that cannot both be true.

That is the strongest evidence for the set anywhere in this ladder, and it is evidence of an unusual kind: not a measurement, and not a mechanism, but the observation that the alternative leaves two computed quantities contradicting each other by a factor of seventeen.

What the picture cannot show #

The hysteresis. Every figure here is a single-valued curve, and a measured one is a loop with two branches. The elastic curve computed is roughly the mean of those branches minus the friction, and drawing it as one line understates by exactly the thing that is not modelled.

Nor can any of these figures show the fabric failing. The curve stops at the geometric ceiling because there is no state beyond it; a real fabric breaks its yarn or slips its loops long before, and neither event is in the model.

What it settles about an old sentence #

What stops a knit extending argued that a knit’s extension ends when the loop runs out of reconfiguration rather than when the yarn runs out of stretch, and it had no number for either end.

Both ends are now available and one of them turns out to be wrong. The reconfiguration ceiling is real and it is at three hundred per cent — three times where any jersey actually jams. The rung that measures it treats the gap as the finding, and the short version is that a jersey stops extending for a reason that is not in the loop’s own geometry at all.

The comparison the curve makes possible #

A woven cloth of the same yarn, on the same axes, is a nearly vertical line. Its extension is a few per cent, its force is measured in hundreds of newtons a metre, and the two curves do not share a scale.

That is worth drawing rather than saying, and it is the single most legible difference between the two fabrics: two orders of magnitude in extension and two in stiffness, from the same thread, with all of it coming from how much yarn sits between one interlacing and the next.

The same curve at a different construction #

Since the shape is set by the slack and the slack is set by the loop length, changing the loop length moves the wall.

That is the honest content of a tightness factor as a specification. It is quoted as an index of handle and cover, and it is simultaneously an index of extensibility, of contact force, of run resistance and of pilling, because all of them are functions of the same ratio. The rung that pursues that finds it is the model’s only variable.

Where the number came from before #

Nowhere, which is why the shortfall was recorded. The trade measures load–extension curves on a tensile tester and quotes them, and the modelling literature has computed loop shapes since the 1950s without joining the two — because computing a force needs the derivative of an energy with respect to a fabric dimension, and the loop models of that period imposed the dimensions.

That is the same gap the rung on the energy surface found from the other side, and it is the same reason.

Against a woven cloth, on the woven cloth’s own axes #

The comparison is worth making with this collection’s existing woven machinery rather than in words.

A knit’s equivalent figure would need axes running to three hundred per cent. The two fabrics are not near neighbours on one scale; they are objects with different orders of magnitude in the quantity a garment is most often designed around.

What makes it worth saying rather than assuming is that the mechanism is the same in both — thread moving at constant length — and only the budget differs. A woven cloth interchanges crimp between two systems that are both nearly straight. A knit reconfigures a loop with half its length spare. Same physics, two orders of magnitude apart, and the whole difference is how much yarn sits between two interlacings.

Where the ladder goes next #

The slope of this curve at the origin is a modulus, and it is worth its own rung because of what it is made of: a bending rigidity divided by a length to the fourth, which is a fabric property with no fabric material in it.

After that, the transverse behaviour, which changes sign; and the ceiling, which is three times too far out and says so.