The relaxed cloth's contact force

Every force this collection computes at a crossing comes from a tension. A warp end held at some tension arrives at the crossing at the weave angle and leaves at its negative, so the transverse components add and the thread beneath carries 2·T·sin θ. That is exact, it needs nothing fitted, and it is where the site’s first force came from.

It also has a restriction that has been stated in the same breath every time it has been used. It is the contact force in a cloth that is under tension — on the loom, in a seam, in a fabric being pulled. A relaxed cloth is not under tension and its threads still press on one another; what presses them is their own resistance to being bent, and recovering that from Peirce’s geometry is impossible, because Peirce’s path joins arcs of constant curvature to straights and its curvature jumps at every join. The bending energy of such a path is well defined. The contact forces are not recoverable from it. That needs an elastica, this site does not have one, and the rung that computed it recorded the relaxed contact force as unavailable and called it the one number that would close the standing limitation on the site’s own integrity criterion.

This rung supplies it, and not with an elastica.

The inversion, and why it is legitimate #

The chain is three links and each was built for a different reason.

A cloth’s thickness falls monotonically with the load at its crossings. That is the rung below: pressing a crossing flattens both sections, a flattened section is thinner through the cloth by a factor of κ, and the whole geometry re-solves at the smaller thickness. The monotonicity is checked at every step of every sweep and is not assumed.

So the relation is invertible. A measured thickness lying between the relaxed prediction and the thickness at some large load corresponds to exactly one force, found by bisection.

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, it is what a gauge under a light standard pressure reports, and it is quoted in the trade for classes of construction. Nobody has ever measured one in order to recover a contact force.

That last point is what makes this a measurement rather than a fit. The chain has one adjustable constant in it — the yarn’s transverse modulus, which has no lower bound and must be read out of a fabric — and eight cloths are being inverted against it. A single constant reproducing eight thicknesses is a model; eight constants reproducing eight thicknesses would be a table.

What comes out #

Between 0.19 and 0.85 newtons per crossing, across the eight cloths in this site’s table.

The spread is a factor of 4.6, and it is worth being precise about what that is small compared with. The counts in the table run from 10 tex to 60, a factor of six. The area a crossing owns runs from 0.10 mm² to 1.11 mm², a factor of eleven. The predicted thicknesses run from 0.25 mm to 0.60 mm. Eight independent inversions over that range landing inside a factor of five is the model’s only real check, and it passes it.

The ordering is also right in a way nothing arranged. The heaviest cloths carry the largest forces — the duck at 0.70 N and the cheesecloth at 0.85 — and the finest carries the smallest, the batiste at 0.19. A crossing in a coarse open cloth has more thread to hold apart and more area over which to be pressed.

What it is not #

It is smaller than the loom’s, and by a stated factor.

A warp end of 25 tex cotton at one per cent strain carries about 1.13 N, and at a sheeting’s weave angle of 36.8° it presses each pick it crosses with about 1.35 N. The sheeting’s relaxed contact force comes out at 0.42 N. So the finished cloth carries roughly a third of what the loom applied, and the other two thirds went when the cloth came off and relaxed.

That ratio is the honest content of the number and it is worth stating plainly rather than as a footnote. A relaxed cloth is not an unloaded cloth. It is a cloth in which the crimped threads are pushing each other apart and being held together by nothing except their own inability to straighten inside a fabric whose thread lengths are fixed. The force that survives is a residual, it is a substantial fraction of the weaving force, and it is what does all the holding in every piece of cloth that is not currently being pulled.

Which is most of the cloth in the world #

The reason this matters is that almost every question the site asks about grip has been answered with a tension in it.

The crossover length — the gripped length at which a thread breaks rather than slides — is computed from μ times the normal force, and the normal force came from a stated thread tension of half a newton. So the site’s answers about fraying, seam slippage and tuft anchorage have all been answers about cloth under load, quoted at a tension chosen to be reasonable.

A cut edge on a shirt hanging in a wardrobe is not under load. Neither is a carpet before somebody walks on it, nor a seam allowance inside a garment on a shelf. The relaxed contact force is what holds those, and until now this collection could not say what it was.

Substituting the relaxed figure changes those answers by a factor of about three in the direction that matters, and it changes them in a way that is not uniform: the cloths whose relaxed force is furthest below their weaving force are the ones whose grip is most overstated by a tensioned calculation.

And it prices what the criterion could not see #

The site’s integrity criterion is topological. A draft describes one cloth exactly when the above-and-below relation on its threads is strongly connected, and that is a decidable property of a small integer matrix with no tolerance to choose and no residual to interpret. It is the thing this collection is built around.

It also has one standing limitation, stated in the field that most needs it: the criterion cannot see friction. 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 fabrics is a carpet while the other sheds. What separates them is a capstan argument, and a capstan argument needs a normal force.

The normal force it needed was the relaxed one. A carpet on a floor is not under tension.

The tuft argument is the sharpest case because it already has a number attached. A tuft’s anchorage was computed with a stated backing tension of one newton, giving a normal force of 1.01 N at the crossing and a V fastening that holds with a fifth of a newton at ordinary friction — against a domestic carpet specification of several newtons, which it misses by a factor of three, and a contract specification it misses by six.

Those shortfalls were computed for a backing under a newton of tension. A carpet on a floor has none. The duck the tuft argument uses has a relaxed contact force of 0.70 N rather than 1.01, so the whole anchorage falls by a third and the shortfall against specification widens from three-fold to four. The conclusion does not change and its size does, which is the useful kind of correction: the argument that a tuft is not held by friction alone gets stronger, and the gap the backing has to close gets larger.

What was counted, and how #

The bisection runs fourteen halvings over a range of three newtons per crossing, which resolves each answer to about 0.0002 N — far finer than the inputs deserve, and cheap.

The thicknesses are trade figures for cloths of these constructions, not measurements of these eight fabrics, and the same note applies as applies to the cloth table itself: what is being read out of them is an ordering and an order of magnitude. Quoting any single one of these forces to three figures would be quoting the third figure of a thickness nobody measured on that cloth.

Three refusals guard the inversion and each is exercised by the gate. A cloth measuring thicker than a round section predicts returns null with the reason named rather than a force of zero — because that is a cloth the model does not describe, and clamping it to the boundary is exactly the failure this site has now twice shipped and twice found. A thickness no load in the range can reach returns null with its own reason. And a cloth with no measured thickness is refused outright rather than defaulted.

The over-prediction itself is asserted as a relation: every cloth over-predicted, none under, and the smallest over-prediction still substantial. The size of it is a property of the table of measurements and is quoted, not asserted.

And the load’s own weight is not in it. A thickness gauge presses on the cloth to measure it — a light standard pressure, but not nothing — so a measured thickness is a slightly compressed thickness and the force recovered from it is slightly too large. At the pressures a thickness test uses the correction is small compared with the spread between the trade figures themselves, which is why it is noted here rather than applied.

The inversion is worst where the answer is wanted most #

The curve of thickness against contact force is steep at the low end and flat at the high one, and the section above notes what that means for precision in passing. It is worth following through, because it says which of the eight numbers can be leant on and the answer is uncomfortable.

The recovered force’s relative uncertainty is the thickness measurement’s relative uncertainty divided by the local logarithmic slope of the curve. Where the curve is steep, a small change in thickness corresponds to a small change in force and the inversion is sharp; where it is flat, a small change in thickness corresponds to a large change in force and the inversion is nearly blind.

So the well-determined forces are the ones belonging to cloths that have flattened least — the heavy, coarse, firmly set cloths sitting near the top of the curve where a further newton buys very little more flattening. The badly-determined ones belong to the cloths that have flattened most.

That is exactly backwards for what the number is wanted for. The grip arguments this rung was written to correct — fraying, seam slippage, tuft anchorage — matter most in the light, open, finely set cloths, and those are the ones whose recovered force rests on the flattest part of the curve. The duck’s 0.70 N is worth three figures and the batiste’s 0.19 is worth one.

Two things follow, and the second is a suggestion rather than a caveat.

Quote the ordering rather than the values, which the rung already does, and treat the light cloths’ figures as an order of magnitude. The claim that eight independent inversions land inside a factor of five survives whichever way the light cloths’ uncertainties fall, because their spread is inside that factor already.

And measure the thickness twice. A thickness gauge reports at a specified pressure, and the whole difficulty here is reading a force off one point of a curve. Two readings at two pressures give a local slope, and the slope pins the force far better than the value does — because it is a direct measurement of the very quantity the inversion is short of. Where the absolute reading is nearly blind, the difference between two readings is at its most sensitive, which is the usual arrangement with a flat curve and is why the remedy is available at all.

Nothing in this collection needs a new instrument to do that. Thickness at two pressures is what a compression test already reports, and compressibility is quoted on the specification of every technical cloth as a matter of course — a number gathered for an entirely unrelated purpose, sitting beside the one this rung has been inverting, and containing the information the inversion is missing.

Where the model stops #

This is not an elastica and the shortfall it closes is closed sideways. An elastica would give the contact force from the geometry, at every point along the thread, with no measurement in it. What this rung has is one number per cloth, obtained from one measurement per cloth, through a model with a fitted constant. It answers the question the shortfall asked; it does not do what an elastica would do, and the elastica is still missing.

The force is an average over the crossing. A real contact is a patch with a pressure distribution across it, and the peak pressure at the centre is higher than the average by a factor nobody here has computed. Anything that depends on the peak — fibre damage, the onset of permanent set, whether a filament breaks at a crossing — is out of reach.

Static only. A cloth being worn is loaded and unloaded tens of thousands of times, and the residual force in it after a season is not the residual force in it new. Every force on this site is static, which the rung that computed it recorded and this one does not fix.

And the inversion is one-dimensional where the physics is not. The measured thickness is one number and the state has three coordinates — how the thickness divides, and how flat each system is. What pins the other two is the energy minimum, so they are model outputs and not measurements, and a fabric whose two systems flatten differently from the way the energy says would be reported with the right thickness and the wrong sections.

The generalisation #

A quantity that cannot be computed can sometimes be read off a shape, if the shape has a memory.

The contact force here is unmeasurable directly: nothing can be put between two threads in a fabric without changing what is being measured. What is measurable is a consequence that the material has stored — the flattening — and the storing is what makes the inversion possible. A cloth’s thickness is a strain gauge that was installed at the mill.

The same move works wherever a deformation is retained. A crushed packing tells its compaction pressure. A dented bearing race tells its peak contact load. A compressed gasket tells its bolt torque. A tree ring tells a season. In each case the direct measurement is impossible or destructive and the retained deformation is trivially available, and the whole difficulty is the constitutive law in between — which is exactly where the fitted constant sits here, and exactly why it has to be named every time.

Who found it, and when #

Peirce’s geometry is 1937 and his thickness formula is in it. Kemp’s flattened section is 1958. The observation that measured fabric thicknesses fall well below a circular geometry’s prediction is old, general and usually treated as a reason to prefer the flattened section rather than as a measurement of anything.

Yarn compression has been measured directly since the 1940s, most systematically by van Wyk, and those measurements are what the fitted modulus here is checked against rather than derived from.

What is this site’s is the direction of the arrow. The trade uses a measured flattening to choose a section model; this rung uses a measured thickness to recover a force, which requires the flattening to have been given an energy first — and the energy is the rung two below.

Where the ladder goes next #

Sideways is where this rung is spent. The contact force in a relaxed cloth is what a thread gripped at its interlacings is held by, and putting the two together gives the first account here of how far a cut edge frays that is not an account of a cloth under load.

It is also what the integrity criterion needed to say the thing it structurally cannot: not whether a fabric hangs together, which it decides exactly, but how hard it hangs together.

Further along this ladder, the compression energy still has no compaction regime in it and no plasticity, and both are recorded rather than attempted.