Textile Structure
how cloth is put together
Compound and figured cloths

The criterion gets a force

This site's integrity criterion decides exactly whether a draft describes one cloth, and has one standing limitation: it says a tuft bound under one pick and a tuft bound under three are both attached, and it is right, and one of those is a carpet while the other sheds. What separates them needs a normal force in a fabric that is not under tension — the number the rung that computed it recorded as unavailable, and the one a thickness gauge now supplies.
15 min read 6 figures Does it hang togetherMeasured, not claimedA mechanism, not a material

Worth reading first: The criterion cannot see friction · The relaxed cloth's contact force · What holds a tuft in, in newtons.

The criterion this collection is built around is topological and exact. A draft describes one cloth precisely when the above-and-below relation on its threads is strongly connected: no tolerance to choose, no residual to interpret, no material property anywhere in it. It has caught drafts that look entirely reasonable and describe fabric in two independent layers, and it is the reason a figure on this site is a figure rather than a decoration.

It has one limitation and the fancy field is where it bites. The criterion cannot see friction: it says a tuft bound under one pick and a tuft bound under three are both attached, and both of those are true statements about connectivity, and one of those fabrics is a carpet while the other sheds.

That rung named what would be needed to say the difference — a capstan argument, which needs a normal force — and the rung below named why the site could not supply it. The contact force it computes comes from a thread tension, so it is the force in cloth that is being pulled. A carpet on a floor is not being pulled. Recovering the force in a relaxed fabric needs an elastica, this site has none, and the shortfall was recorded as the one number that would close the criterion’s standing limitation.

How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.
Fig. 1 Where the number came from instead. A round thread section is the thickest a thread can be for its area, so a circular geometry predicts the thickest cloth those threads can make — and every cloth measures thinner. The force per crossing that reconciles the two is a relaxed cloth’s own contact force, between 0.19 and 0.85 newtons, read off a thickness gauge rather than out of an elastica.

What the criterion says, and what it cannot

Worth restating precisely, because the distinction is easy to blur and the whole rung sits on it.

The criterion answers whether. Is this fabric one piece or several? That is a yes or no about a graph, it is decidable, and no amount of friction changes the answer: a tuft bound under one pick is topologically attached however slippery the yarn is, and a tuft lying on the surface is topologically detached however sticky.

Nothing on this site has answered how hard. That is a different kind of question — a real number rather than a verdict, with material constants in it, and a range rather than a value. The criterion does not fail to answer it; it is not the kind of statement that could.

The two are complementary and the second has been missing. A fabric that passes the criterion and holds with a tenth of a newton passes and sheds.

The number, and where it came from

Between 0.19 and 0.85 newtons per crossing, across the eight cloths in this site’s table — a range a wetting supplies independently, from a geometry with no measured thickness in it.

The route is a chain of three links and none of them is an elastica. A cloth’s thickness falls monotonically with the load at its crossings, because pressing a crossing flattens both sections and a flattened section is thinner. So the relation inverts. And a fabric’s thickness is measured routinely by people with no interest in yarn mechanics — it is on the specification sheet of every technical cloth, and nobody has ever measured one in order to recover a contact force.

Eight independent inversions land inside a factor of 4.6, across a table whose yarn counts span a factor of six and whose crossings own areas differing by eleven. That spread is the model’s only real check.

What it does to a tuft

What holds a tuft in newtons computed a pile anchorage properly — a capstan series rather than a single factor, because a tuft’s far end is free and each wrap’s friction is dragged around the wraps between it and the pulled end — and got a verdict that was the right kind of verdict and the wrong size.

A V fastening on a duck holds with about a fifth of a newton at ordinary friction; a W fastening with about four fifths. A domestic carpet specification is short by about three and a contract specification by nearly six.

Both of those were computed with a backing tension of one newton, giving a normal force of 1.01 N at the crossing. A carpet on a floor has no backing tension. The duck’s relaxed contact force is 0.70 N, so the whole anchorage falls by a third, and the shortfalls widen from threefold to four and from sixfold to eight.

The compaction law, which is a different function of a different variable. Van Wyk's law for a random fibre assembly: the pressure needed to hold it at a volume fraction rises as the cube of that fraction, because the fibres bend between contacts and the contacts crowd as the assembly densifies. From 0.6 to 0.85 the pressure runs from nothing to 19.1 N/mm². This is not the energy the rest of this family computes: that one conserves the section's area and is quadratic in the log of the aspect ratio, and this one changes the area and is cubic in the packing factor. They meet where the flattened thread is as wide as its own spacing, at an aspect ratio of 4.43 for this cloth. What the plot cannot show is that K is a measurement with a spread of three to one between authors, so every pressure on it is a number known to about one figure.
Fig. 2 What the force does to a tuft, through the law that actually applies at a crossing. Compaction is a different function of a different variable from the thickness curve above — it takes the packing rather than the separation — and the number this rung supplies enters it rather than the other. Getting the right curve is most of what “the criterion gets a force” amounts to.

The conclusion does not change, and that is the point

The verdict was already that no fastening reaches a carpet specification by friction alone, and the correct contact force strengthens it rather than reversing it.

That is the useful shape for a correction to have. A number that was wrong by a factor of one and a half, in a direction that widens an already-clear gap, tells the reader that the original conclusion was robust to the error — and tells them what the error was worth, which is a different thing from telling them it did not matter.

What the gap is closed by, in a real carpet, is the backing: a latex or a polymer film applied to the reverse, which locks the tuft’s legs into a continuous solid instead of relying on friction against two yarns. The site records the backing’s contribution as inferred rather than computed, from two directions now, and computing it needs a film modulus and a bond strength that are the coating ladder’s problem rather than this one’s.

Which makes the criterion’s limitation smaller and not smaller in kind

Worth being exact about what has and has not happened.

The criterion still cannot see friction. Nothing about it has changed. It is a statement about a graph and it will always be a statement about a graph.

What has changed is that the complementary statement is now computable. Given a draft, the criterion says whether the fabric is one cloth; given a draft, a construction and a friction range, the machinery now says how hard it holds, in newtons, in a fabric that is not being pulled. The two together are what a specification needs and neither alone is.

So the limitation is no longer a gap in what the site can say. It is a correct statement about the scope of one tool, sitting beside a second tool with a different scope — which is what it should have been all along, and was not, because the second tool did not exist.

Two tools, and the shape of an answer that needs both

Setting the pair out plainly is worth a section, because a specification for a pile fabric is exactly a question that needs both and neither alone.

Is the tuft part of the cloth? The criterion, from the draft, exactly. Yes or no, no constants, no range. A tuft that fails this is not a weak tuft; it is a loose thread lying on a fabric, and no amount of friction would help.

How hard is it held? The capstan, from the draft plus a construction plus a friction range plus a contact force. A number with an interval on it, because two of its four inputs are ranges.

Is that enough? A comparison against a stated specification, which is the applied field’s whole business and is a pair of inequalities.

The three questions are answered by three different kinds of statement and they compose in one direction only. A no to the first makes the second meaningless. A yes to the first says nothing about the second. And the second cannot be turned into a verdict without the third, which comes from outside the fabric entirely.

A V-fastened tuft. A cut pile bound into its ground by V fastening, drawn in section. The pile end passes beneath 3 of the 6 ground picks and wraps 1 half-turn around them in all. The integrity criterion says the tuft is attached; how hard it is held is a different question with a different model behind it.
Fig. 3 The V fastening, which the criterion passes and the specification does not. One pick bound, topologically attached, and holding with about a fifth of a newton at ordinary friction against a domestic carpet requirement several times that. Both statements are true of the same drawing and neither is available from the other.

What was counted, and how

The relaxed contact force is a bisection on the pressing load against a measured thickness, at fourteen halvings over a range of three newtons per crossing, which resolves it far finer than the input deserves.

The thicknesses are trade figures for cloths of these constructions rather than measurements of the eight fabrics named, and the same warning applies as applies to the cloth table itself: what is being read out of them is an ordering and an order of magnitude. Nothing here is quoted to three figures.

The tuft anchorage is that rung’s capstan series, unchanged, with a different normal force in it. Its own assertion — that the ratio between the V and W fastenings is what a series of wraps gives rather than what a single factor gives — is untouched and still runs.

Three refusals guard the inversion and each is exercised. A cloth measuring thicker than a round section predicts returns null with the reason named rather than a force of zero. A thickness no load in the range can reach returns null with its own reason. A cloth with no measured thickness is refused rather than defaulted. Each of those is the failure this site has now shipped twice and found twice — a solver that returns its nearest reachable state as though it were an answer — and they are written as refusals for that reason.

Thickness against pressing force. A duck at a transverse modulus of 4.0 N/mm², with the load at one crossing swept from nothing to 1.60 N. The thickness falls from 0.601 mm to 0.358 mm and never rises. Below 0.00511 N nothing happens at all: the compression energy is quadratic in the log aspect so its slope at a round section is zero, and the bending term's is not, so there is a threshold — and the threshold contains the bending stiffness and the geometry and no transverse modulus whatever. What the plot cannot show is that the small-strain energy it is computed from is being asked to work past an aspect ratio of about two, where a quadratic in the strain is outside its warrant.
Fig. 4 The inversion for the duck a carpet backing is woven from. Thickness against the force at one crossing: monotone, which is what makes the inversion legitimate, and steepest at the low end, which is where the answer sits. The 0.70 N this returns is what replaces a backing tension somebody chose.

What was actually closed, and what was not

The shortfall said an elastica would supply the relaxed contact force. It would, and one is still missing, so it is worth being exact about what has replaced it.

What is now available: one contact force per cloth in the table, in newtons, for a fabric that is not under tension, obtained from a measurement anybody can repeat with a thickness gauge.

What is not: the force at any particular point along a thread, the distribution of pressure across a contact patch, the contact force for a cloth outside the table, and the contact force for any fabric whose thickness the compression model does not predict — a knit, a nonwoven, a braid.

So the criterion’s companion is a table rather than a theory. That is enough for the question the fancy field was stuck on, because a pile fabric’s ground is a woven cloth of an ordinary construction and the table covers it, and it is not enough for the general case.

How deep a fastening would have to be

The verdict is that no fastening reaches a carpet specification by friction alone, and the natural question is how deep one would have to go. The capstan gives an answer, and the answer is why the backing exists rather than a deeper weave.

Aspect ratio against pressing force. A duck at a transverse modulus of 4.0 N/mm², with the load at one crossing swept from nothing to 1.60 N. The aspect ratio rises from 1 to 2.22 and never falls. Below 0.00511 N nothing happens at all: the compression energy is quadratic in the log aspect so its slope at a round section is zero, and the bending term's is not, so there is a threshold — and the threshold contains the bending stiffness and the geometry and no transverse modulus whatever. What the plot cannot show is that the small-strain energy it is computed from is being asked to work past an aspect ratio of about two, where a quadratic in the strain is outside its warrant.
Fig. 5 The same force sweep in a duck, which is the ground a tufted carpet is actually made on. Every pressure is larger and the flattening at each is smaller, so a fastening in a heavy ground has more force holding it and less room to be pushed into — both of which the criterion alone could not say.

The two computed points bracket it. A V fastening passes under one pick and holds with 0.14 newtons at the relaxed contact force; a W passes under three and holds with 0.55. Two extra wraps for a factor of 3.9 is about two per wrap, which is what a capstan series gives at this friction and this wrap angle.

Extrapolating from the W: reaching a domestic specification of three newtons needs a further factor of five and a half, which is between two and three more wraps. So a fastening would have to pass under five or six picks to hold by friction alone, and a contract specification would need seven or eight.

That is not a construction anybody could weave, and the reason is worth stating because it is geometric rather than practical. A tuft passing under six picks is a tuft that is mostly in the ground, and the pile it leaves standing is correspondingly short — the yarn has to come from somewhere, and every pick it wraps is pile length spent on anchorage. A carpet whose tufts were anchored by friction would have almost no pile, which is to say it would not be a carpet.

Three things follow that the two computed fastenings do not show on their own.

The exponential is the wrong shape to design against. Each extra wrap doubles the hold and costs a fixed length of yarn, so the yarn spent grows linearly while the hold grows geometrically — which sounds favourable and is not, because the starting point is a factor of twenty short. Doubling from a fifth of a newton takes four or five doublings to matter, and by then the pile is gone.

And it explains the trade’s actual solution completely. A latex backing does not add wraps; it changes the mechanism, locking the tuft’s legs into a continuous solid so that withdrawing one means shearing a film rather than sliding against two yarns. A mechanism change is the right response to a shortfall of twenty, and adding wraps is the right response to a shortfall of two.

The V fastening is not a mistake either. It is short by a factor of twenty and so is the W, near enough, so the extra pick of the W buys very little in a construction that is going to be backed regardless. What it buys is what happens before the backing cures and if the backing fails locally, which is a robustness argument rather than a strength one — and is why the deeper fastening is specified for contract carpet and not for domestic.

The extrapolation’s own caution: two points give a ratio and not a law, and a capstan series’ per-wrap factor is not constant as the wraps accumulate, because each one drags the friction of those before it. The five or six is an order of magnitude and the conclusion — that the gap is a mechanism’s width rather than a wrap’s — is what survives it.

Where the model stops

The force is a woven cloth’s and a carpet backing is a woven cloth, so the substitution is legitimate. It would not be for a tufted carpet on a nonwoven backing, or for a knitted pile, and neither has an equivalent inversion.

The flattening threshold, cloth by cloth. The force at one crossing below which the section stays exactly round, for every cloth in the table. It runs from 0.00162 to 0.00511 N — a few thousandths of a newton, which is a hundred times less than the contact force an ordinary warp tension applies. So every woven cloth is flattened and the interesting question is by how much rather than whether. The threshold is a ratio of two slopes at a round section: how fast the bending energy rises with the aspect ratio, over how fast the thickness falls. The compression energy has zero slope there, so the yarn's transverse modulus — the one constant here that cannot be bounded — does not appear. What the rows cannot show is that this is a threshold in an elastic model with no yield in it anywhere.
Fig. 6 Where the flattening begins, which is where the model stops being able to use a single number. The onset is a threshold and the force this rung supplies is a mean, so a criterion given one force is being given the average of a quantity that has a threshold in it.

It is an average over the crossing. A real contact is a patch with a pressure distribution, and the peak at the centre is higher than the mean by a factor nobody here has computed. Anything that depends on the peak — whether a fibre is damaged, whether a filament breaks at a crossing — is out of reach.

Static, and a carpet is walked on. Every force on this site is static. A tuft is loosened by many small tugs rather than one pull, the friction that resists a repeated small tug is not the friction that resists a single large one, and the fatigue of a fastening is not modelled anywhere.

And it is not an elastica. The shortfall is closed sideways: one number per cloth from one measurement per cloth through a model with a fitted constant. An elastica would give the contact force from the geometry, everywhere along the thread, with no measurement in it — and it is still missing.

The generalisation

A tool’s limitation is only a gap while nothing else answers the question, and the thing that closes it usually comes from somewhere the tool has no relationship with.

The criterion’s limitation was recorded for a long time as though it were a defect to be repaired, and the natural repair — an elastica, giving contact forces from the same geometry the criterion works on — was correctly identified and correctly judged out of reach. What actually closed it was a thickness gauge, arriving through a compression energy built in the mechanics field for an unrelated reason, and having nothing to do with connectivity at all.

That is worth carrying as a habit rather than a result. When a limitation is recorded, record what would close it and also what kind of thing that is — because the second is what makes a reader notice, much later and in another field, that the thing has turned up.

The narrower lesson is about exact and inexact answers living together. A criterion that answers whether exactly and says nothing about how much is not half a tool. It is one tool, and the temptation to soften it into something that gives a score — to make it answer both questions badly instead of one well — is the temptation this collection has resisted from its first essays. What was needed was a second tool, and it took many essays to build.

Who found it, and when

The strongly-connected criterion for whether a draft describes one cloth is this site’s own formulation of a fact weavers have always known operationally, and it has been here since the collection’s first essays.

The capstan account of a tuft’s anchorage is standard: pile anchorage is measured by a tuft-withdrawal test, the fastening’s wrap count is what a carpet designer chooses between, and the mechanics are well understood.

That a fabric’s thickness over-predicts under a circular thread geometry is old and general, and is usually read as an argument for a flattened section rather than as a measurement of anything.

What is this site’s is the direction of the arrow — a measured thickness read backwards into a force — and the observation that the force it yields is exactly the one a criterion in a different field had been recorded as needing.

Where the ladder goes next

The backing is the gap, from both directions. The pile ladder records the coating’s contribution as inferred, the coating ladder records the same thing from its own side, and closing it needs a film modulus and a bond strength that neither field has.

Sideways, the relaxed contact force is what every grip answer on this site should be computed with when the cloth is not under load, and the fray widths and seam verdicts of the applied field are still computed at a tensioned one.

Where this goes next

A leno and the open plain weave it is not. Two warp ends and the picks they hold. On the left the doup end passes under its partner between picks and comes up the other side; on the right it never crosses, which is an open plain weave at the same sett. The layer count under each panel is computed by the same criterion that decides every other draft, and it returns the same verdict for both. The criterion cannot see friction This site's central check is exact, decidable in linear time, and structurally incapable of distinguishing a carpet from a fabric that sheds. That is not a defect to be repaired — a separation exists or it does not, and there is no margin in it — and the fancy weaves are precisely the constructions that live in the gap. How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering. The relaxed cloth's contact force How hard two threads press on each other in a cloth that is not being pulled is the number this site's own integrity criterion has needed since its first essays, and the route to it was an elastica nobody had. A thickness gauge supplies it instead — because a cloth's thickness is a record of how flat its threads are, and how flat they are is a record of how hard they are pressed. What holds a tuft in, in newtons. The withdrawal force of a V-fastened and a W-fastened tuft over the reported range of yarn-on-yarn friction, on a duck ground with 1.00 N in each pick. Each is the friction at the wraps, with each wrap's share dragged around every wrap between it and the pulled end — a capstan series rather than a single factor. W runs from 0.79 to 6.84 N and V from 0.15 to 0.41. Both are an order of magnitude below what a carpet is specified at, which is a finding about carpets rather than about the model. What the chart cannot show is the backing, which is where the rest of a tufted carpet's anchorage comes from. What holds a tuft in, in newtons The pile ladder computed a tuft's anchorage as a capstan ratio and said, correctly, that a ratio was all it could offer. A ratio multiplies a tension and there was no tension anywhere on this site. There is one now — and the answer, put beside what a carpet is actually specified at, falls short by a factor of three.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

CapstanCloth thicknessContact forceCriterionFrictionIntegrityPileSpecificationStrongly connectedTuft