A loop has a maximum force in it

A bouclé loop is an elastica, not a semicircle, and the solved shape carries nothing of the fibre: a wool, a mohair and a polyester make the same loop at the same overfeed on the same spacing, because the bending stiffness multiplies every candidate shape’s energy by one factor and leaves the cheapest one cheapest.

That account ended by naming what it could not do. A bouclé cloth sits somewhere in its jamming bracket because its loops have been pressed by their neighbours, to about a sixth of their free height in a hundred-gram cloth of the eighty-per-cent yarn, and how hard a loop resists that was the elastica with a contact on it — a harder problem, and the first place in the whole account where the fibre’s stiffness would matter.

It is not a harder problem. A symmetry the free solution already had makes the pressed loop the same boundary-value problem with one number prescribed instead of found, and once that is seen the solve takes no new apparatus, no penetration test and no contact set.

The apex does the work of a contact condition #

A free loop is antisymmetric about its middle. The thread leaves its first binder point along the core, rises, crosses its own highest point, falls, and arrives at the second binder point along the core again — and at the apex, being the highest point, its tangent is horizontal, which on this yarn means along the core.

So a free loop is already two identical halves, each holding half the thread, each running from a foot to the apex, each with the tangent along the core at both of its ends.

Press the loop down and nothing about that changes. The apex is lower and it is still the highest point, so its tangent is still horizontal; the feet are where the binder holds them; the thread is as long as it was. The pressed loop is two half-loops with the same end conditions and a smaller rise, and a rise is exactly the kind of thing a constrained minimisation takes as a datum.

That is why this account is cheap where the last one expected it to be dear. A contact problem needs the contact set found as part of the answer — which part of the curve touches, where it leaves, and with what pressure. Here the contact is a single point at a known place, so the whole of it is one number.

The free height is where the push is nothing #

The first thing the pressed solve has to do is agree with the free one, and it does so in the strongest available way: the energy’s gradient in the height is nought at the height the free loop takes.

That is not a coincidence to be checked to a tolerance; it is what the free solution is. The free loop is the least-energy shape with its two feet fixed and its length fixed, and among those shapes the height is free — so the energy is stationary in it. A pressed solve that put the minimum anywhere else would be solving a different problem, and the check is run at four length ratios from 1.3 to 4.

It is also the cheapest possible test that the two solves share a basin. The elastica has neighbouring branches — shapes that satisfy both end conditions exactly and are not the minimiser — and a solver walked into one of them does not fail. It agrees, at a higher energy, and the force curve gets a step in it where the branch changed. This solve tries three starts at every height and takes the cheapest converged one, which is what the elastica actually is, and the first version of it did not: continuing from the shape above walked straight into a neighbouring branch at the fourth step and put a jump of sixty per cent into the middle of an otherwise smooth curve.

The push peaks at four fifths and falls away #

With the shape right, the force is the energy’s gradient and the answer has a maximum in it.

A loop is not a spring with a stiffness. A knitted loop pressed by its neighbour was solved the same way and came to tens of millinewtons a stitch, which is the order this one lands at. Over the first fifth of its stroke it behaves like one, resisting more the further it is pressed; past four fifths of its free height the resistance falls, and by a tenth of free height it is pushing with a fifth of its peak.

The reason is what the thread is doing with its surplus. A lightly pressed loop is a bell whose curvature is spread along its whole length; pressing it further does not tighten that bell evenly — it turns the loop into a shape with its curvature concentrated near the feet and a long slack run between them, and a long slack run costs very little. The thread stops being a bent arch and becomes a doubled-back wave, and a wave of a given length across a given gap has an energy that is nearly independent of how high it stands.

A curve with a maximum in it is a softening spring, and a softening spring has a specific consequence: there is no stable intermediate state. A load that exceeds twelve millinewtons a loop does not settle the loop at the corresponding height; it presses it past the peak, where the resistance is lower than the load, and the loop collapses until something else stops it. What stops it is the core, the binder, the neighbouring yarn, or the loop’s own two sides meeting — which happens at the overfeed where a free loop’s neck closes and earlier once the loop is pressed.

That is the mechanical reason a bouclé cloth’s loops are found pressed to a small fraction of their free height rather than to a comfortable half. They did not settle there; they were pushed past their own maximum and fell.

More surplus is a taller loop and a weaker one #

Running the same solve at four overfeeds gives the family, and the two properties a designer would expect to go together go opposite ways.

The peaks are 15.6, 12.2, 7.2 and 3.6 millinewtons at overfeeds of 30, 80, 160 and 300 per cent, and the fraction of free height at which each peak falls is 80, 80, 70 and 50 per cent. So a designer reaching for bulk by feeding the effect thread faster gets a loop that stands higher, gives way sooner in proportion, and resists with a quarter of the force.

Both halves of that follow from the same geometry. A taller loop is a longer thread across the same gap, so its curvature is spread over more length and every curvature is smaller; the energy goes as the curvature squared, so the whole scale of the forces falls. And a longer thread reaches its doubled-back regime earlier in the stroke, because there is more of it to double back.

A bouclé is therefore softer the more it is overfed, and softer by more than its height suggests. That is the mechanical statement behind the bulk these yarns are bought for, and it runs the same way as the crimp a fancy yarn keeps in the wrong thread, and it is the first quantity in this account with the fibre’s stiffness in it — because the force is the rigidity times everything else, while the shape is not.

Five times the stiffness is five times the push and the same picture #

The free loop’s account made a great deal of the stiffness dropping out, and it is worth confirming that it drops back in exactly where expected and nowhere else.

A five-times-stiffer effect thread pressed to the same height takes the identical shape, to six decimal places, and pushes with exactly five times the force. Nothing in between — no partial dependence, no shift of the peak, no change in the fraction of free height at which it falls.

So the whole family of curves above is one curve times a number, and the number is the effect thread’s bending rigidity. That makes the forces here quotable and it makes them a bracket rather than a value: a yarn’s stiffness is a bracket five hundred times wide, running from every fibre free to slide to every fibre locked into a solid rod, and this collection’s standing rule is to quote the lower end and say so. Twelve millinewtons is the floor, and a thirty-tex cotton effect thread whose fibres were fully coherent would push with four hundred and ninety times it.

That bracket is not a defect of the calculation and it is not narrowed by anything here. What the calculation supplies is the shape — where the peak is, how fast it falls, how it moves with the overfeed — and every one of those is a ratio in which the rigidity cancels.

The pressure the whole bracket sits at #

The previous account named exactly what this one was for: a bracket in which a cloth’s position is predicted from its beat-up rather than read from its weight. With the force per loop in hand, that is one multiplication — the force each loop carries at the height the cloth’s weight puts it, times the loops in a square metre.

The promise cannot be kept, and the arithmetic says why.

A heavier cloth is a closer cloth, so it presses its loops further and it has more of them. The force per loop falls as the loop is pressed past its peak; the loops per square metre rise in proportion to the sett. Over a bracket a factor of eight wide in weight the product runs from about 770 to about 1,570 pascals and spends most of its range within ten per cent of 1,500.

So a measurement of the pressure a bouclé cloth carries says almost nothing about where in its bracket the cloth is. A weaver setting the beat-up harder does not move the cloth to a predictable position; the pressure at which it jams is nearly the same at every position, so the beat-up finds whichever position the other settings put it at, and the cloth’s weight remains the only thing that reads its position.

That is a negative result about an instrument and it is worth being precise about what it does and does not say. It does not say the beat-up is irrelevant — a bouclé needs about a kilopascal and a half to jam at all, which is a real specification and is part of what the beat-up’s own force budget has to cover. It says the quantity is flat, so it cannot discriminate. An instrument that reads the same at every value of the thing it is pointed at is not a poor instrument; it is not an instrument.

Why the cancellation is nearly exact #

A near-constant product of two quantities that both move by a factor of several is worth explaining rather than reporting, because the alternative explanation is a mistake in the arithmetic.

The loops in a square metre go as the sett, which goes as the cloth’s weight: a cloth twice as heavy has twice as many yarns in it and therefore twice as many loops. That half is exact and has no mechanics in it.

The force per loop is the part that has to cooperate, and what makes it cooperate is the softening. Below the peak the loop’s force falls roughly in proportion to its height, and the height a cloth presses its loops to falls roughly in proportion to the space between its yarns, which is the same jamming arithmetic every close cloth obeys, which falls roughly as one over the sett. So force per loop goes as one over the sett, loops per area goes as the sett, and the product is flat.

The agreement is not exact — 770 pascals at the close end against 1,570 at the middle — and where it fails is where the proportionality fails. At the open end the loops are barely past their peak and the force is nearly flat rather than proportional; at the close end the loop is pressed so far that the elastica is running out of room and the force falls faster than its height. The flatness is a consequence of the softening, and it holds exactly as far as the softening is linear.

What was solved, and how #

A pressed loop is two half-loops. Each is the elastica of half the thread across half the chord, with the tangent along the core at the foot and along the core again at the apex, and the apex’s rise prescribed; the solver is the one built for a thread between two crossings, describing the curve’s angle by sixteen modes and minimising the bending energy under the end conditions. The energy is twice the half-loop’s and the force is its central difference in the rise, with both difference solves started from the converged shape at the height itself. Every solve tries three starts — the shape a step above, a cold start, and a richer basis — and takes the cheapest that converges. A sweep walks downward in steps of two per cent of the free height and stops rather than guessing if the solve fails, which it does at very low heights on the slackest loops, where the thread has to double back on itself and a contact problem begins.

Six things are checked. A loop at its own free height pushes with under two per cent of its peak, at four length ratios — the agreement between the free and pressed solves, stated as the identity it is. It pushes back at every height below that, and pressing it further always costs more energy, at every step, which are the checks on the gradient’s sign. The push peaks between a third and all of the free height and falls to under half its peak by the bottom of the sweep, which is the essay’s claim tested rather than described. A loop with more thread in it peaks lower, at every step of the four ratios. And five times the rigidity is the identical shape and exactly five times the push, to six decimal places, which would fail immediately if any part of the solve had picked up a stiffness it should not have.

The overfeed, the binder spacing, the effect thread’s count and the cloth’s weight are inputs. The bending rigidity is the lower bound of the stiffness bracket and every force is quoted as one.

Where the model stops #

The loop is pressed at a point and a neighbour is not a point. What presses a loop in a cloth is another yarn lying across it, which touches over a length rather than at the apex and may touch off-centre. A distributed contact is a genuine contact problem and this is not one; the point load is an upper bound on how concentrated the pressing can be, so the forces here are the stiffest case of a family whose softer members press over more length.

Nothing stops the loop but the arithmetic. A real loop pressed to a fifth of its free height is close to touching the core, its own two sides, and its neighbours, and this solve knows about none of them. Where the sweep stops is where the elastica stops converging, which is a numerical limit and not a physical one — the physical limit is earlier.

The feet are clamped exactly along the core. A binder wrap has thickness and some give, so a real foot can tilt, and a loop whose feet can turn is a different elastica with a lower stiffness. That would lower every force here and is not computed.

And the pressure is a sum of independent loops. The cloth’s loops are treated as each carrying its own force at its own height, all at the same height, with nothing between them. In a real cloth the loops of neighbouring yarns interleave, some are pressed and some are not, and a fabric is a population of contacts rather than a uniform field — so the pressure here is a mean over a population whose spread is not computed.

Still open: what a loop does when it is pushed sideways #

Every pressing here is straight down on the apex, which is what a yarn lying across the loop does. It is not what abrasion does, what a finger does, or what the next loop along the same yarn does when the cloth is sheared.

A loop pushed sideways is the same elastica with a different condition: the apex displaced along the core rather than towards it, with the rise free. That solve would give a lateral force per loop and, with it, the shear the loops themselves resist — which is a quantity a bouclé cloth has and a plain one does not, because a bouclé’s surface is a forest of elastic arches rather than a set of crossings.

It would also say something the vertical solve cannot. A loop pushed sideways past its own stability turns out of its plane, and a loop out of its plane is a loop lying over its neighbour — which is the beginning of the matting that a bouclé surface does when it is rubbed, which is the population question abrasion takes the hairs first asks of an ordinary cloth, and is where this account’s next question about wear begins.

Who worked it out #

Euler’s elastica of 1744 already contains every shape drawn here, and the elastic-rod literature has treated loops with and without contact many times. What is new here is the observation that a pressed loop needs no contact solve because its apex is a point of horizontal tangency, the maximum in the force–height curve and its movement with the overfeed, and the cancellation that puts the whole jamming bracket at one pressure.