A bouclé wears from the loops down

A cloth loses its strength before its mass is one of this collection’s firmer results and it has a mechanism rather than a correlation behind it. A rub removes material from a fabric’s crowns; a fabric’s crowns are the parts of its threads that stand highest; and those threads are the ones carrying the tension, so the section lost is section off the load path — and since a thread breaks at its thinnest place, the strength gone is several times the mass gone.

Every step of that argument holds for a woven cloth, for a knit, and for anything else whose surface is made of the threads that hold it together. A bouclé’s surface is not.

A fancy yarn has its crimp in the wrong thread: an effect thread fed eighty per cent faster than its core contains more thread than it is long, the surplus stands off the core in loops, and pulling the yarn stretches the core at once while the loops never straighten. Thirty-nine per cent of the yarn’s mass is on the load path and sixty-one per cent is decoration.

That decoration is also the whole of the cloth’s outside. So the rule inverts, and the inversion has a depth attached to it.

The surface is loop and nothing else #

A bouclé’s jamming bracket runs from the sett at which its loop envelopes touch to the sett at which its count diameters do, and that bracket is twelve and a half times wide — an envelope of 4.13 millimetres against a count diameter of 0.35.

Read as a statement about the surface, that ratio says something simpler than it says about the sett. At every position in the bracket the loops stand clear of the core, because the envelope is the core plus a loop on either side and a loop is never nothing. There is no weight at which a bouclé cloth presents its core to a rubbing surface, and no sett at which the core is the first thing a finger meets.

The depth is the number this account adds. A hundred-gram cloth of the eighty-per-cent yarn sits about a third of the way up its own bracket with its loops pressed to a sixth of their free height — which on a loop that stands 1.95 millimetres free is 0.31 millimetres standing. Below that is the core’s own radius, a further 0.11 millimetres, before a rub is taking section off the thread the cloth’s strength depends on.

So the arithmetic of wear is three tenths of a millimetre of harmless thread, then a tenth of a millimetre to the centre of the load path, and the first of those is three times the second.

The rule inverts, and by most of the cloth #

Put the two accounts side by side and they are the same mechanism with the load path in a different place.

In a woven cloth every thread at the surface is a thread under tension, so the first material a rub removes is load-bearing and the strength falls faster than the mass from the very first rub. In a bouclé the first material a rub removes is effect thread, which carries nothing, so the strength does not fall at all until 61 per cent of the yarn’s mass is gone.

That is not a small correction to the rule; it is the rule with its sign changed, and it has a consequence a specification would care about. A bouclé cloth that has been rubbed until it looks ruined — its loops flattened, broken and pulled away, its whole surface changed — may still break at the load it was woven to break at. And an abrasion test that reports mass loss, which is the ordinary way of reporting one, is reporting almost entirely on the decoration.

More surplus puts more out of harm’s way and does not stand it higher #

The obvious reading of that chart is that a bulkier yarn has more to lose, and it is right. The less obvious one is what happens to the height the mass stands at, which is the quantity that decides how long a rub takes to get through it — and it barely moves.

At a hundred grams a square metre, loops stand 0.363 millimetres at thirty per cent overfeed, 0.310 at eighty, 0.330 at a hundred and sixty and 0.407 at three hundred. A tenfold range of surplus moves the standing height by a fifth, and not monotonically.

The reason is a cancellation and it is worth naming because it looks like an error. A bulkier yarn makes a taller free loop — 1.10 millimetres at thirty per cent against 4.83 at three hundred. It also makes a cloth of a given weight much more openly set, because the yarn is fatter and there is less of it: 6.76 ends a centimetre at thirty per cent against 3.23 at three hundred. An openly set cloth presses its loops less in absolute terms but the loop’s free height has grown faster, so the fraction of free height falls — from a third to a twelfth — and the product is nearly flat.

So overfeeding a bouclé buys mass out of the load path and does not buy depth. A designer choosing a bulkier yarn for durability is choosing more material to lose and the same time before the loss matters, which is a poorer bargain than the yarn’s hand suggests.

A snag is bounded, and the bound is a logarithm #

The other thing that happens to a loop standing three tenths of a millimetre off a cloth is that something catches it, and this is where a bouclé’s reputation is made and lost.

A caught loop is pulled. The thread it needs comes from the loops beside it, and to get there it has to slide past its own binder point and then past the next — and a thread sliding past a wrap is the capstan relation, which multiplies the tension needed by e raised to the friction coefficient times the wrap angle. For half a turn at a friction coefficient of three tenths that factor is 2.57 per binder point.

So the pull needed rises as 2.57 to the number of binder points the thread is dragged through, and the effect thread’s own breaking load caps it.

A snag in a bouclé is a bounded injury, and the bound is six to eight loops depending on the cloth. That is about two centimetres of yarn arriving as one long pulled loop, and then the thread parts rather than running any further.

Two things about that are worth carrying. The bound is logarithmic, so making the binder grip twice as hard does not halve the reach — it divides it by the logarithm of two over the logarithm of 2.57, which is three quarters. And a closer cloth’s snag reaches further, which is the opposite of the expectation: a heavier cloth presses its loops lower, so the loop’s own push is smaller, so the pull starts from a smaller number and takes more doublings to reach the breaking load. Six loops at forty grams a square metre, eight at a hundred and sixty.

The comparison with a knit is the sharp one. A knit runs and a weave frays because a knitted loop released has a neighbour that can release in turn, and the failure propagates without bound until something stops it. A bouclé’s loops are not linked to each other; each is held at its own two binder points, and the capstan puts a ceiling on how far a pull travels. A bouclé snags and does not run, and the reason is a wrap rather than a topology.

What holds a cut loop, and what does not #

A snagged loop that has not broken is still in the cloth, and a snagged loop that has broken is two ends held by the same capstan.

Each end is gripped at every binder point it still passes, so the two ends need 2.57 to the power of however many binder points remain before they can be drawn out — which for anything but the last loop of a length of yarn is a force far beyond what handling supplies. A broken loop stays where it is. It does not shed, it does not work its way out, and it does not leave a hole: it leaves a short pair of ends standing where a loop was.

That is the structural reason a bouclé’s failure mode is a change of texture rather than a loss of material, and it is the same reason with the sign reversed that a raised knit sheds for the whole of its life — there, the anchorage available is seventy-five times less than the fibre needs and nothing is ever close; here, the anchorage is a wrap round a binder and everything is held.

The two also differ in what they are holding. A nap is a population of loose fibres, each anchored only by friction along its buried length. A bouclé loop is a continuous thread that goes on past its binder and into the next loop, so the thing holding it is not friction alone but the fact that pulling it means pulling all of it.

A bouclé pills where a nap sheds #

The wear sequence follows, and it has three stages the arithmetic can name.

First the surface flattens. The loops are pressed past their own maximum force and collapse, which takes a kilopascal and a half — a pressure a hand supplies easily. Nothing is lost; the cloth is simply lower and denser, and it recovers partly when the pressure comes off because the loops are elastic.

Then the loops abrade. Fibre is worn off the apexes, which is where every contact is, and abrasion takes the hairs first applies here with the hairs standing much further out than usual. The fibre removed is effect-thread fibre and it comes off the tops of the loops, so the loops get shorter and thinner and the cloth’s appearance changes continuously.

Then the loops break. An apex worn thin enough parts, and the result is two ends that the capstan holds in place. The cloth now has a short pile where it had loops, which is a different fabric with a different hand — and its strength is unchanged.

A pill is anchored, not made, as this collection’s account of pilling has it, and a bouclé is an unusually good place to anchor one: a broken loop’s two ends are held by a capstan that no amount of rolling will defeat, and the fibre rolling up on them is fibre from the loops around it. So a bouclé pills readily and the pills do not come off, which is a combination the two halves of that account predict together and which is the ordinary complaint about these cloths.

What the loop’s own push has to do with it #

One quantity from the earlier account enters this one and it is easy to miss what it is doing.

The pull that starts a snag is taken as the force the loop is already pushing with at the cloth’s own position — four millinewtons in a hundred-gram cloth. That is the only force in the cloth the arithmetic supplies, and it is the right order: a loop is already loaded against its neighbours, so a pull that is to move thread has to beat what is already there.

It is also the quantity with the fibre in it. A yarn’s stiffness is a bracket five hundred times wide, so the four millinewtons is a lower bound, and a stiffer effect thread starts the snag from a larger number and therefore reaches the breaking load sooner. A bouclé of a coherent effect thread — a filament, a cabled yarn, anything whose fibres cannot slide — would have a snag reach shorter than these by the logarithm of the bracket over the logarithm of the grip, which is about seven binder points: a filament bouclé should snag and break almost at once, and a soft staple one should draw a long loop. That is a prediction about two yarns anybody can pull, and it has not been tested here.

The two ends of the bracket are two different cloths to wear #

Everything so far is a hundred-gram cloth, which sits about a third of the way up its own bracket. The bracket is twelve and a half times wide, and its two ends behave differently enough to be worth drawing.

An open bouclé has three times the depth and a third of the material. The loops stand 0.93 millimetres at forty grams a square metre and 0.063 at two hundred and fifty, a range of fifteen, while the decorative share of the mass is 61 per cent at both ends because it is a property of the yarn rather than of the cloth. So the open cloth takes far longer to rub through to its core and has far less core to rub through to; the close cloth is the other way about.

That is a real design choice and the arithmetic states it as a trade rather than a ranking. Depth of protection and quantity of load path move in opposite directions across the bracket, and which of the two a cloth wants is a question about what it is for: an upholstery bouclé wants the load path, a scarf wants the depth.

The snag reach moves the other way from the depth, and by much less: six loops at the open end against eight at the close. It moves at all only because the loop’s own push moves, and it moves little because the reach is a logarithm. A logarithm is what makes the whole bracket behave nearly alike in one respect while differing fifteenfold in another, which is the same flattening that put the whole bracket at one pressure an earlier essay.

What was computed, and how #

The cloth’s position in its own jamming bracket comes from its weight, as the bracket account computes it, and gives the fraction of its free height the loops are pressed to. The free height is the elastica’s, the standing height is the product, and the core’s radius is the count arithmetic’s. The decorative share is the effect thread’s share of the resultant count, which carries the overfeed with it because the effect enters the resultant multiplied by one plus its own surplus. The snag is the capstan on a half turn at each binder, starting from the loop’s own push at the cloth’s position and stopping at the effect thread’s breaking load, with the reach a logarithm of their ratio over the logarithm of the grip.

Five things are checked. The surface is loop at every weight in the bracket, checked as the envelope standing clear of the count diameter at five weights. Most of the mass is on no load path at every one of them. A heavier cloth stands its loops lower, at every step, which is the check on the position being read the right way round. More overfeed puts more of the yarn out of the load path, at every step of four surpluses. And a snag reaches between one and twelve binder points, and reaches further in a closer cloth — a pair of results in which a sign error in the capstan would fail the second immediately.

The friction coefficient, the binder’s half turn, the yarn counts and the cloth weights are inputs. The bending rigidity is the lower bound of the stiffness bracket, as it is wherever that bracket is used.

Where the model stops #

A rub is not a plane coming down. Abrasion is a rubbing surface moving across the cloth, which loads the loops sideways as well as down, and a loop pushed sideways is a different elastica from a loop pressed — it turns out of its plane rather than flattening. Nothing here computes the lateral case, and it is the one that actually matters for how a bouclé mats.

The wear is not rated. How fast fibre comes off an apex depends on the fibre, the abradant, the pressure and the number of cycles, and none of those is structure. What is computed is where the loss falls and how deep the harmless material is, not how long it takes.

The loops are all at one height. A real cloth’s loops are pressed by whatever happens to lie across them, so some stand nearly free and some are flat, and a fabric is a population of contacts rather than a uniform field. The tallest loops meet the abradant first and take all of the early wear, which concentrates the loss further than this account allows for.

And the capstan needs a friction coefficient. Every number about a snag is exponential in it, it is an input rather than a measurement, and a lubricated or a resin-finished binder would give a different reach entirely.

Still open: what a second binder buys #

The whole snag bound rests on one wrap per binder point, which is what a single binder laid in one direction gives. A doubly bound fancy yarn — a binder in each direction, which is ordinary practice for a yarn that is expected to be handled — doubles the wrap at every point and therefore squares the grip: 2.57 becomes 6.6, and the reach falls from seven binder points to about three and a half.

That is a large improvement for a second thread, and the arithmetic says it is bought at the cost the count arithmetic already prices: another fifteen tex on a resultant of eighty-nine, which is a seventh more yarn for half the snag. Whether real doubly bound bouclés snag at half the reach is a measurement anybody with two yarns and a hook could make, and it is the cleanest test this account offers, because the two yarns differ in exactly one thing and the prediction is a factor rather than a direction.

Who worked it out #

That fancy yarns snag, pill and lose their surface before their strength is trade knowledge and is the standard caution against them. The capstan is Euler’s and Eytelwein’s. The inversion of the wear rule above, the depth to the load path read off the cloth’s position in its bracket, and the logarithmic bound on a snag’s reach were computed directly.