What stops a knit extending

The rung below computed a knitted loop’s bending energy across the extension its geometry admits and found something that reads at first like a null result.

It does not change. Not approximately, not to within the model’s noise — exactly, at every extension up to forty per cent, while a woven cloth’s bending energy moves measurably over an eighth of that range. The same model that gives a woven cloth a crimp modulus gives a knit none.

The reason is geometric and is worth having in one sentence. A woven thread’s arcs are held to a radius by the fabric’s own thickness: the arc is D/2 and D is what the two threads must fill between them, so extending the cloth changes the thickness and changes the arc. A knitted loop’s arcs are held to a radius by contact. The loop wraps the loop below and cannot bend tighter than that thread is thick, so its radius is a yarn diameter and stays one however the fabric is pulled.

Something resists a knit being stretched. It is not the loop bending.

What is left is friction #

Extending a knit takes thread out of some loops and puts it into others. It cannot do anything else: the thread is inextensible on this site’s terms, the loops are all one continuous thread, and a fabric that is longer in the wale direction has longer legs and shorter heads.

Thread moving from one loop to the next has to pass through the interlock — the place where the loop’s head is wrapped by the loop above and its foot wraps the loop below — and passing through means sliding against whatever is pressing there. So the resistance is

$$F = \mu \cdot N \cdot (\text{interlocks per unit width})$$

with no yarn stiffness anywhere in it.

Three consequences, and they are what a knit is like #

It is dissipative rather than elastic, so a knit does not spring back. A woven cloth extended and released returns towards a least-energy state and stops in a band around it; the band is friction over stiffness and it is a correction to an elastic story. A knit has no elastic story at all. It stops wherever the driving force falls below the friction, and the “band” is the whole of the behaviour rather than an uncertainty on it.

That is why a knit recovers in the way it does — slowly, incompletely, and better when it is worked — and it is why a knitted garment that has been pulled out of shape stays pulled out of shape until it is washed.

It is proportional to μ, so a softener changes a knit’s dimensional behaviour and a stiffer yarn does not. That is the reverse of the woven case, where the resting band is friction over stiffness and both terms matter. A finisher who softens a knit has changed the whole restoring mechanism by the same factor; one who softens a woven has changed half of it.

And it goes away with agitation. Once the loops have been allowed to slide, the fabric goes to a state set by the loop length and the geometry alone.

The comparison with a woven cloth, which is the sharpest form of it #

Two fabrics, the same yarn, the same friction, both extended.

The woven cloth’s resistance has two parts. The crimp is being redistributed and that costs bending energy, which is stored and returned; and the threads are sliding at their crossings, which costs friction and is not returned. The elastic part is the larger and it is why a woven cloth snaps back when it is released and why its resting band is a band around a minimum rather than an interval with nothing in the middle.

The knit’s resistance has one part. There is no elastic term at all.

That difference is the reason the two fabrics behave nothing alike under exactly the treatments a wearer gives them. A woven cloth pulled at the knee returns most of the way by itself. A knit pulled at the knee stays there, because there is nothing pulling it back, and it recovers only when it is washed — at which point the loops slide again and the fabric returns to the geometric configuration its loop length allows.

Which is why Munden’s constants have no yarn in them #

That last consequence is the one that closes something.

A knit’s dimensions come from its loop sets out the empirical result the whole knitted-fabric trade runs on: in a relaxed plain knit the wales and courses per unit length are each a constant divided by the loop length, and the constants contain no property of the yarn at all. Not its count, not its fibre, not its twist, not its stiffness. About 4.1 and 5.5, for anything.

That is a strange result to meet cold. Every other dimensional property of a fabric on this site depends on the yarn somewhere.

The friction account explains it in a line. If the restoring mechanism were elastic, the relaxed state would be an energy minimum, an energy minimum would depend on the stiffness, and the stiffness would carry the yarn’s properties into the answer. Because the mechanism is dissipative, the relaxed state is not a minimum of anything — it is wherever the fabric stops once the friction has been overcome enough times, and a fully relaxed knit has been agitated until the loops have slid as far as they will go. What is left is a purely geometric configuration: loops packed as the loop length allows.

A dissipative restoring mechanism has no material constant in its endpoint, and that is exactly what Munden measured.

And why a knit’s dimensions are a history #

A woven cloth’s relaxed dimensions are a state with an uncertainty on them. A knit’s are a stopping point, and stopping points depend on the path.

So the standard knit relaxation procedures — soak, tumble, repeat, and repeat again — are not making a small correction. They are the whole measurement, and a knit measured before them has no defined size in the sense the trade means. That is why knitted goods are specified after a stated relaxation and woven goods often are not.

It also explains a practical asymmetry. A woven fabric shrinks; a knitted fabric settles. Both are the same word in a specification and they are different mechanisms: the woven’s crimp is redistributing towards a minimum and the knit’s loops are sliding towards wherever friction lets them stop.

What was counted, and how #

The bending result is that rung’s and is re-run here as an assertion rather than quoted: the loop’s bending energy is required to be identical at five extensions to one part in a million million, which is a statement about a closed form rather than about a computation.

The frictional resistance is asserted to be exactly proportional to the friction coefficient — doubling μ doubles it, to twelve figures — because that is the claim that no stiffness has leaked into it. A version of this that computed a force and checked it looked reasonable would have caught nothing.

The interlock count is geometry: two per loop, with the wales per unit length coming from Munden’s own constant and the loop length. It is a count and not an estimate.

The interlock force itself is not available and is not guessed. Everything here is quoted per newton of interlock force, which is a pure number of the geometry and the friction. Getting the force would need for a knit what a thickness inversion gave for a woven — a measured dimension the model over-predicts, read backwards — and the knitted analogue has not been done. That is the largest single thing this rung does not have and it is why no force here is quoted in newtons.

That last figure is worth a sentence, because it is the same mechanism at its limit. Extending a knit slides thread through interlocks that are still closed. A run slides thread through interlocks that have opened. The resistance in the first case is μ·N per interlock and in the second it is nothing, which is why a run is fast and an extension is not — and why a knit’s whole integrity is a statement about interlocks staying shut rather than about anything holding them.

Why an elastane knit is a different fabric entirely #

The argument above says a plain knit has no elastic restoring force at all, and the obvious objection is that knitted garments plainly do spring back. Most of them contain elastane, and that is not a quibble — it is the whole of the difference.

A bare elastomeric thread laid in or plated with the ground yarn puts a genuinely elastic term back into a fabric that had none. It stores energy in the ordinary way, it returns it, and the fabric’s recovery stops being a question about friction. So the two constructions are not the same fabric with a modifier: they are a dissipative mechanism and an elastic one, and everything about how they behave in wear follows from which.

The consequences separate cleanly. A plain knit distorts and stays distorted until it is washed; an elastane knit distorts and returns. A plain knit’s relaxed dimensions depend on how much it has been agitated; an elastane knit’s are set by a force balance and are far more repeatable. A plain knit is softened by a finish that lowers friction; an elastane knit is not, because friction is no longer the restoring term.

And it explains something about specification. Munden’s constants work because a relaxed plain knit’s state is purely geometric — and they should be expected to fail for an elastane fabric, because there the state is an energy minimum and an energy minimum carries the yarn’s properties into the answer. That is a prediction of this rung rather than an observation, and the knitted-fabric literature’s practice of treating elastane constructions separately is consistent with it.

The force is available after all, and it is an order of magnitude #

The interlock force is recorded above as the largest thing this rung does not have. It has since been computed elsewhere in this collection, from a route that has nothing to do with extension, and putting it in converts every ratio here into a number.

The solved loop’s own bending gives the force at an interlock: about forty millinewtons for a 20 tex cotton jersey at a three-and-a-half millimetre loop, at the free end of the yarn’s bending bracket and as an upper bound on an unset yarn.

The interlock count is geometry. At Munden’s fully-relaxed constant the fabric carries 12.3 wales per centimetre, and each loop owns two interlocks, so a centimetre of width presents about twenty-five interlocks to a thread being dragged through.

Multiply by an ordinary friction coefficient:

resistance ≈ 0.3 × 40 mN × 2,500 per metre ≈ 30 newtons per metre of fabric.

Set that beside the elastic force the same fabric pulls back with — between one and three and a half newtons per metre across a garment’s working range — and the account closes.

The frictional resistance is an order of magnitude above the recovery force, and that single comparison is the quantitative form of everything this rung says qualitatively. A knit distorted and released has a restoring force of a few newtons per metre against a friction of thirty, so it does not move at all until something breaks the contacts loose. It stays where it was put, exactly as observed, and the margin is a factor of ten rather than a near-thing.

Three readings, and the third is a caution.

It explains why agitation is not a refinement but the mechanism. With friction ten times the driving force, a knit left alone never reaches its relaxed dimensions — not slowly, not eventually. It reaches them only when the contacts are made to slide, which is what a tumble does.

And it explains why a softener is the whole lever. Halving μ halves the thirty and leaves the three and a half untouched, so a fabric that could not move at all can move a little. Nothing else on the sheet does that.

And every term is a bracket. The forty millinewtons is at the free bending bound and is an upper bound on an unset yarn; the friction is a range; the loop is a plain jersey’s. So the thirty newtons per metre is an order of magnitude and the ratio of ten is what the argument rests on — which survives comfortably, because closing it would need the friction to be wrong by a factor of ten in one direction or the elastic force by a factor of ten in the other.

Where the model stops #

No interlock force, so no absolute numbers. Stated above. The structure of the answer is complete and its scale is missing.

The loop geometry is a plain single jersey’s. A rib, an interlock fabric or a purl has a different arrangement of the same mechanism, with different interlock counts and different directions of thread movement, and none of them is computed. A second bed changes what a float is and it changes this too.

The thread is inextensible. For cotton at the extensions a knit reaches that is a reasonable approximation and for an elastane-containing knit it is nonsense — an elastic yarn puts a genuine elastic restoring force back into the fabric, which is precisely why it is added, and this rung’s whole finding is about the fabric that does not have one.

The interlocks are counted and their force is not distributed. Every interlock in the fabric is given the same normal force, and a real one has more at the tightly wrapped head than at the loosely held foot. Since the resistance is a sum over interlocks, a distribution with the same mean gives the same total — so the approximation costs nothing for the force and everything for the question of which interlocks slide first, which is what decides where a knit distorts.

And the friction is quasi-static in a fabric that is worked dynamically. A knit in wear is loaded and unloaded continually, at speeds nobody here has estimated, and both coefficients this site carries are slow ones.

The generalisation #

A restoring mechanism that is dissipative has no material constant in its endpoint, and that is a testable signature.

If a system’s rest state depends on a stiffness, the mechanism stores energy. If it does not — if the rest state is purely geometric, with the material’s properties absent from the answer — the mechanism dissipates, and what looks like an equilibrium is a stopping point.

That is a useful thing to be able to read off an empirical result, because the empirical result usually arrives first. Munden’s constants were measured decades before anybody asked what mechanism could produce constants with no material in them, and the absence of a material constant was treated as a convenient simplicity rather than as evidence about the physics. The absence of a constant is evidence, and it points at dissipation.

The converse is worth carrying too. Wherever a rest state is found to be path-dependent — depending on how the system was brought there rather than only on where it is — the same conclusion follows, and the two symptoms travel together: no material constant in the endpoint, and a dependence on history. A knit has both.

Who found it, and when #

Munden’s work on the dimensional properties of plain knitted fabrics is from 1959 and is the foundation of the whole subject: the constants, the loop-length dependence, and the requirement to relax the fabric before measuring it are all there.

That knitted fabrics must be relaxed by agitation, and that their dimensions are otherwise indeterminate, is knitting practice and is written into the standards.

The geometric fact that a knitted loop’s arcs are held by contact rather than by the fabric’s dimensions is that rung’s on this site, and the exact constancy of the bending energy is its result.

What is this rung’s is putting the two together: a mechanism with no elastic term must be dissipative, a dissipative mechanism’s endpoint has no material constant in it, and that is exactly the strange feature of the constants everybody has been using for sixty years.

Where the ladder goes next #

The obvious next thing is the interlock force, by the route that worked for a woven cloth: find a knitted dimension the model over-predicts and read it backwards. A knit’s thickness is the candidate — it is measured routinely and a contact-held loop geometry predicts it — and nothing here has attempted it.

Sideways, the same two coefficients are what make agitation work during relaxation, and the knit is the case where they are not one term among several but the whole of the mechanism.