What a loop model still cannot say

A model that has just produced a fabric’s contact force, its load–extension curve and its modulus is at its most dangerous, because everything it says sounds equally authoritative. This rung is the list of things it cannot say, each with a number attached where one is available and a measurement that would settle it where one is not.

Five items. Two are approximations that have been priced; three are absences.

One: the curve is planar and a loop is not #

The largest omission and the first to state. The solved centre line lies in the fabric plane; a real loop rides out of that plane at every interlacing, because the two courses have to pass one another.

That ride is not free. It has been priced: an excursion of half a yarn diameter over half a loop length is a curvature of about a fifth of the in-plane peak, and energy goes as the square, so the ride carries about three per cent of the loop’s bending.

Three per cent is small and it is not nothing. Every energy quoted on this ladder is that much low, every force in the same proportion, and every ratio between two fabrics is untouched because both inherit it.

Two: there is no torsion #

A rod that goes round a loop and out the other side has twisted, and twisting costs energy that nothing here counts.

The size of it is not known. A yarn’s torsional rigidity is of the same order as its bending rigidity — for a solid isotropic rod the ratio is fixed by Poisson’s ratio and comes out near two thirds, and a spun yarn is neither solid nor isotropic — and the twist a loop demands is of order one turn per stitch. That puts the torsional energy in the same range as the three per cent above, or an order larger, and there is no honest way to narrow it from here.

What makes it worse than the planar approximation is that torsion is not obviously symmetric between the states being compared. The out-of-plane ride is about the same at every extension; the twist is not, because a loop that has been pulled open has a different writhe from one that has not. So torsion could carry a real part of the load–extension curve’s shape, and nothing here would show it.

What would settle it #

A three-dimensional rod solve with torsion in it, on the same geometry, run at two extensions and compared. That is a larger piece of machinery than this collection has, and it is the obvious next thing to build.

The cheaper partial test is available now and has not been run: measure a knit’s load–extension curve in a yarn with a very low torsional rigidity relative to its bending — a flat tape yarn, or a plied structure with balanced twist — and see whether the computed curve moves closer to the measured one.

Three: adjacent courses can pass through each other #

The model stops a thread reaching further than its own length and stops nothing else. In particular it does not stop the loops of one course from occupying the same place in the fabric plane as the loops of the next.

At the relaxed state that does not matter, because they do not. Stretched a long way it matters enormously, and it is why the geometric ceiling this ladder computes sits at three times the relaxed width while a real jersey jams at about one.

The rung that measures the ceiling treats the gap as the finding rather than as an error, because it says something specific: a jersey does not stop extending because its yarn runs out. Something else stops it first, and the something else is loops meeting sideways.

What would settle that one #

Adding a non-penetration constraint between the sampled points of neighbouring courses, which is a contact problem rather than a boundary-value problem and is a different order of difficulty. It would move the ceiling down and leave the soft part of the curve alone, because the soft part happens where nothing is touching.

The measurement that would confirm it is simpler: photograph a jersey at increasing extension and record the extension at which adjacent courses first touch. If that is where the load–extension curve turns up, the account is right.

Four: the curl radius is still not computed #

Why stockinette curls said it does not predict a radius, and this ladder does not either.

The reason is specific rather than a shortage of effort. What curls a fabric is a moment about an axis in its plane, and the moment a planar loop stores is about the axis perpendicular to it — which bends nothing. The curl comes from the loop’s asymmetry through the thickness: heads on one face, legs on the other, and a stored moment that does not average to zero across the fabric’s own thickness.

That asymmetry is exactly the third dimension the model does not carry. So the curl is not a hard problem that has been deferred; it is the same absence as the first two entries, seen from a different side.

What can be said about curl #

The scaling, and nothing more. A spontaneous curvature is a moment per unit width divided by a bending rigidity, and both halves are now available in the right units: the moment scales as the contact force times the yarn diameter over the wale spacing, and the rigidity is computed from the yarn’s own and the angles it lies at.

That gives the way the curl radius should move with count, loop length and fibre. It does not give the radius, because the constant in front of the moment depends on how the contact loads distribute through the thickness, and a planar model has no thickness for them to distribute through.

Five: the setting cannot be measured from here #

The set fraction brackets every force on the ladder, and it cancels out of the one observation that might have pinned it down: the friction available and the force to be held scale the same way, so a fabric sitting still is consistent with any degree of setting.

Two experiments would separate them and neither has been run here. Cutting a wale and watching whether the fabric moves distinguishes a loaded fabric from an inert one. Boiling an unloaded fabric and watching which way it goes distinguishes friction from set, because their predictions differ in direction.

Until one of those is done, every force in newtons on this ladder is an upper bound and is stated as one.

Six: the contact stops being a point #

Not on the original list, because it is a boundary rather than an absence, but it belongs here.

Where the peak curvature exceeds the wrap’s, the thread would have to bend harder in free space than it does while wrapped, which means the contact is not where the model puts it: it extends, and a little of the free run becomes an arc.

The consequence is small and it is in a known direction. A longer contact means less free run, a slightly stiffer span and a slightly larger force, so the forces at the tight end of the range are underestimates. The correction has not been computed and the figure that would need it is flagged where it is used.

What each omission costs, ranked #

Putting rough sizes on them is more useful than listing them, even where the size is itself uncertain.

The planar approximation costs three per cent of the energy, computed. Torsion costs something between three per cent and a factor of two, uncomputed, and is the one to disbelieve first. The missing course-to-course contact costs a factor of three in the extension ceiling and nothing at all in the soft part of the curve. The curl is a complete absence rather than an error. The setting is a factor of one to zero on every force in newtons and cancels from every ratio.

Ranked by what would most change a published conclusion, torsion is first and the course contact is second. Ranked by what is easiest to fix, the course contact is first.

The four things that are solid #

Setting the absences against what survives them is the point of listing them, and the list on the other side is short and specific.

The geometry — slack, peak curvature, occupancy, the reachable region of the state space — depends on nothing that is missing. The ratios between two fabrics computed the same way are untouched by the rigidity bracket, by the setting and by the three per cent. The direction of the energy gradient does not depend on any of it. And the shape of the load–extension curve, soft then abruptly stiff, comes from the geometry rather than from the energy.

Those four are what the rest of this ladder is built on, and each of them is checked against something the solve did not use.

What is not on the list, and why #

Two things that might be expected and are deliberately absent.

Yarn compression. A yarn pressed at a contact flattens, and this collection has a whole ladder about what that does in a woven cloth. It is not in the loop model because the loads are an order of magnitude smaller and the contacts are points rather than the long arcs a woven crossing has. The flattening at forty millinewtons is a few per cent of a diameter, and it enters the geometry only through the loop height.

Yarn extension. A knit’s whole behaviour up to the ceiling is the loop moving at constant yarn length, and the yarn’s tensile modulus never appears. That is not an omission but a finding: it is why a knit is a hundred times more extensible than the thread it is made of, and the model would be wrong to include it.

What a single-bed model is not #

Everything here is a plain jersey. Rib, interlock, purl and every two-bed structure have their own topology, and what transfers to them is the arithmetic of one loop rather than the fabric-scale results.

The distinction between what transfers and what does not is worth stating exactly, because it is not the obvious one. The arithmetic of one loop transfers: the loop length is still the sum of the same arcs and straights, the free run is still nine tenths of it, the thread still lies in a plane. What does not transfer is everything that came from counting loops in a plan — the stitch density, the areal weight, the extension, the thickness — because a two-bed fabric has loops on two planes and the plan is no longer the fabric.

That is a sharper boundary than “the results are for a jersey”. It says which line of a derivation to start again at, which is one loop up from the end, and it is why this collection’s rib essays reuse the loop solver and none of the jersey’s fabric-scale numbers.

Warp knits are further away still. Warp knitting is a different thing entirely, its loops are held at both ends by neighbouring wales, and none of the single-bed geometry applies without being redone — not even the arithmetic of one loop, because a warp-knitted loop’s two legs run to different wales and the length that closes it is not the length that closes a weft-knitted one.

Where each number in this ladder sits #

A reader arriving at one of these essays alone deserves a way of telling which class its numbers are in, and there are three.

Geometric: slack, occupancy, peak curvature, the reachable states, the extension ceiling, the transverse response. These depend on the loop length, the yarn diameter and two measured spacings, and on nothing else. They are as reliable as Munden’s constants and no more so, since they take those as input.

Ratios: the woven-to-knitted contact contrast, the grip per millimetre, the run resistance against tightness, the anisotropy of the bending rigidity. These survive both brackets and are the strongest claims here.

Values in newtons: the contact force, the load–extension curve, the modulus. These are upper bounds at the free end of the rigidity bracket with no setting, and every one of them says so where it appears.

A caution about authority #

The forces on this ladder are computed, checked twice, and quoted with their brackets, and that is a good deal more support than most numbers in textile mechanics carry. It is not the same as a measurement.

Nothing here has been compared with a measured contact force in a knit, because there is not one to compare with — which is why the number was worth computing and also why nobody can yet say it is right. The load–extension curve can be compared with measurements and has not been, and that is the most obviously available next step in the whole ladder.

What the site’s other ladders would want from it #

Three standing shortfalls elsewhere in this collection would be closed by the same missing piece, which is a reason to think the piece is worth building.

What holds a tuft in newtons computed a pile’s anchor force in a woven ground and could not do the knitted case. The strong fibre is the one that pills argued the knitted criterion and could not compute it. A knit gives up its fibres more easily argued the same thing about shedding.

All three are closed by the contact force and are closed in the rungs that follow. What is not closed by it is anything needing a moment about an axis in the fabric’s plane, which is the curl, the rib’s fold, and the way a two-bed fabric balances — and those three stay open at the end of this ladder as they were at the beginning.

Which measurement closes the most of the list #

Six omissions, each with a settling experiment attached, is a research programme rather than a next step. Ranking the experiments by how many items each closes turns it into one, and the ranking is unexpectedly lopsided.

A measured load–extension curve on a real jersey closes three of the six, and it is a test every textile laboratory can run before lunch.

It tests the ceiling directly. The model puts the geometric limit at three times the relaxed width and a jersey jams at about one; the measured curve’s turn-up says where the fabric actually stops, and whether the turn is where adjacent courses meet.

It bounds the torsion. Torsion is the item to disbelieve first because it might carry part of the curve’s shape rather than a constant fraction of its height. A measured curve compared with the computed one separates the two: a discrepancy that is a constant factor is the setting and the rigidity bracket, and one that changes with extension is torsion or the course contact.

And it bounds the setting. Every force here is an upper bound at zero set, so a measured curve lying a stated factor below the computed one puts a number on the set fraction — which is the quantity that brackets every newton on the ladder and which the fabric’s own resting state cannot supply.

The other experiments each close one. Cutting a wale and watching whether the fabric moves distinguishes a loaded fabric from an inert one, and the displacement to look for is a fraction of a loop length — of order a millimetre, visible without an instrument, which makes it the cheapest test in the list and the one with the least in it. Photographing a jersey at increasing extension settles the course contact and nothing else. A three-dimensional rod solve with torsion settles items one and two and is not an experiment at all.

So the ordering is clear and it is not the ordering the list is written in.

The most informative thing available is also the most ordinary. A load–extension curve is the first test anybody runs on a knitted fabric, it exists in every laboratory’s records for every fabric they have ever handled, and this ladder has computed one without ever comparing it with a measurement. That is the single largest gap between what this collection has and what it could have, and it is closed by looking something up rather than by building anything.

The honest summary #

A planar elastica with a natural curvature and point contacts is a good model of a knitted loop’s geometry and a fair model of its forces, with a stated bracket on both. It is not a model of a knitted fabric’s third dimension, and everything that lives in the third dimension — curl, spirality, the interaction between beds, the way a rib folds — is outside it.

The line between those two lists is sharp and it is worth keeping sharp. A model used past its own boundary produces numbers with no warning attached, and this ladder has been careful to put the warning where the number is rather than only here.

Where the ladder goes next #

Out of the machinery and into what it buys. The rest of this ladder spends the forces: a load–extension curve at last, a modulus, a transverse response that changes sign, a criterion for pilling that has been argued and not computed for two ladders, a nap’s anchor in a knit, a run priced as a competition between two forces, and a seam that has to give what the fabric gives.

Every one of those inherits the list above. Each says so once, where the number is.