A cloth's Poisson ratio is not a material's

Pull something and it gets longer. It also gets thinner, and the ratio between the two is one of the oldest numbers in mechanics. Rubber is near a half, steel near a third, cork near nothing at all, and the number is quoted on a datasheet next to the density because it is a property of the material and nothing else.

A woven cloth has a number of that shape too. The rung below computes it as the slope of the curve of states a cloth can reach at constant thread length, and it is a perfectly good number: 1.10 for a muslin, 1.13 for a sheeting, 1.58 for a warp-dense poplin.

Every one of those is above a bound that a material is not allowed to exceed. And that is the least of it.

The bound, and what it assumes #

An isotropic linear-elastic continuum has a Poisson ratio of at most one half. That is a published fact and it has a reason: the bulk modulus of such a material has to stay positive — squeeze it from every side and it must get smaller, not larger — and requiring that puts the ratio at or below a half, with the equality reserved for a material that does not change volume at all.

Nothing in the derivation of that bound is derived here, because it is not a fact about cloth and this site does not compute moduli. What is worth being precise about is the list of things it assumes, because a woven fabric meets none of them.

Isotropic, meaning the same in every direction. A cloth has two thread systems at right angles and behaves differently along each, and at forty-five degrees to both it behaves differently again.

Linear, meaning the response is proportional to what is done to it. Peirce’s crimp is not a linear function of anything, and the figures on this page are the demonstration.

Elastic, meaning the response is a material straining. A cloth extending by crimp interchange is a mechanism running, and nothing in it is straining at all.

A continuum, meaning the thing is a homogeneous solid rather than an assembly. A cloth is two families of threads that touch, slide and rotate against one another.

So the bound does not apply, and a cloth exceeding it is not evidence of anything wrong. It is evidence that the number being compared is not the kind of number the bound is about — which is the whole of this essay, said once at the start.

The ratio, computed from the geometry #

The computation has no free parameter in it. At constant thread length the reachable states form a curve; the ratio is minus the transverse strain over the axial one, so it is minus the slope of that curve, scaled by the two reference dimensions. It is read off by central difference at every sampled state.

The values are 1.12, 1.11, 1.07, 1.10, 1.58, 1.09, 1.07 and 1.13, across cloths whose covers run from a fifth to over a half of the surface. Six of the eight sit within a whisker of 1.1, which looks like a law and is not one: the poplin is at 1.58 because its two systems are set very differently, and nothing in the geometry requires the balanced cloths to land where they do.

That much is the first anomaly and the least interesting one. A number above the bound could be shrugged off as a units error or a sign convention. The next three cannot.

It is not a constant #

A material’s ratio is a constant, which is why it can be printed on a datasheet. A cloth’s is not, and it is not close.

Reading the muslin’s ratio at a series of states from three per cent short of its measured construction to three per cent long gives 0.687, 0.797, 0.933, 1.106, 1.333, 1.636, 2.074 — a factor of three across a span a fabric would meet in ordinary use. The voile is worse: 0.545 to 3.237 over the same span, a factor of six. Even the close sheeting, whose whole locus is short, runs 0.753 to 2.009.

Over a band of plus or minus two per cent — which is well inside what a tensile test sweeps through before it reports anything — every cloth in the table roughly doubles.

The ends are worse still, and the reason is exact rather than numerical. The rate at which a spacing changes with its own weave angle is −(l − Dθ) sin θ, and that vanishes twice: when the thread is straight, because sin θ goes to nought, and when it jams, because the straight run l − Dθ does. So whichever system stops the curve is the one that has gone stationary there.

An end set by the warp sends the ratio to infinity — the cloth narrows while it has stopped lengthening. An end set by the weft sends it to zero — the cloth lengthens while it has stopped narrowing. The voile has one of each: its warp goes straight at the extending end and its weft goes straight at the shortening one, so its ratio grows without bound at one end of its curve and falls to nothing at the other. How large or small a number the arithmetic returns there is a fact about how finely the curve was sampled rather than about the cloth, which is why no figure is quoted for it.

The two directions are not reciprocals #

This is the anomaly worth the essay, and it needs a careful distinction first.

The tangent ratios at a single state are exact reciprocals of each other, and that is an identity rather than a finding: a mechanism with one degree of freedom has one slope, and reading a slope the other way up is division. So a cloth’s two tangent ratios multiply to exactly one, every time, on every cloth. An orthotropic elastic sheet is required to keep that product strictly below one — it is the standard admissibility condition, quoted here and not derived — so a cloth sits exactly on a boundary a sheet is not allowed to reach.

The secant ratios are what a test reports, and they are not reciprocals at all. Pull the warp two per cent and read the contraction across; pull the weft two per cent and read the contraction along. The two measurements land on different states of the same curve, so there is no reason for them to be related, and they are not.

The products tell the same story. At a two per cent extension the muslin’s two secant ratios multiply to 1.447, the voile’s to 1.822, the sheeting’s to 1.363, the poplin’s to 1.462. Not one of the seven cloths that reach the extension both ways gives a product within two per cent of one, and every one of them is on the wrong side of the boundary an elastic sheet may not cross.

What was counted, and how #

The same eight nominal plain weaves as the two rungs below, with diameters from the counts by conservation of volume at a packing factor of 0.6 and the reference state solved from Peirce’s equations and checked by running them forwards on the answer.

Each cloth’s curve is sampled at 241 states with both thread lengths reconstructed at every one; the worst departure over the whole table is one part in ten thousand million million. The tangent ratio is a central difference along that sampling. The secants are read at the sampled state nearest the extension asked for, and a cloth whose curve does not reach nine tenths of the way past that extension in both directions is declined rather than answered — the cheesecloth drops out of the reciprocity table for exactly that reason, because its whole locus is 1.91 per cent long and a two per cent secant would be a reading off an asymptote.

Two assertions here are the kind that have to be able to fail, and one of them is written to fail on purpose.

A quoted ratio must describe the whole locus. The check takes a single value and the computed series and demands that every state agree with the quote to within a stated tolerance. Fed a real cloth’s own reference value and a tolerance of five per cent, it refuses, and the refusal is the finding. An essay that says the ratio varies and a function that has never rejected a constant are the same sentence written twice.

And the two secants are asserted non-reciprocal, by a function that refuses a pair whose product is within two per cent of one. That one passes, on every cloth, at every extension drawn.

How small an extension the identity survives #

The tangent ratios multiply to exactly one and the secants do not, and the gap between them is reported at three extensions on one cloth. Those three numbers determine a rate, and the rate says how a measurement would have to be run for the identity to be visible at all.

The muslin’s gap runs 0.074 at four tenths of a per cent, 0.416 at two, and 0.619 at two point eight — which is very nearly 0.21 per percentage point of extension, linear across the whole range.

That coefficient is not a new quantity. A secant is the tangent averaged along the path, so the two secants differ by about the tangent’s own rate of change times the extension — and the muslin’s tangent ratio runs from 0.687 to 2.074 across six per cent, which is 0.231 per point. The gap’s slope and the tangent’s slope are the same number, as they must be.

So the reciprocity error is the tangent’s own drift, and it can be inverted into a specification:

to see the two ratios reciprocate to within one per cent, the test must be run at under a twentieth of a per cent of extension.

Five hundred parts per million. No tensile test resolves a transverse contraction at that extension: the specimen has not left the jaws’ own compliance, the crimp has barely begun to move, and the contraction to be measured is a few micrometres across a specimen a few centimetres wide.

The identity is exact and unmeasurable, which is an unusual combination and is worth stating as one. It is not that a measurement would find it approximately; it is that every measurement anybody can make is taken at extensions where it has already failed by tens of per cent.

Which cloths lose it fastest #

The rate differs across the table and it differs in the direction that makes the measurement harder where the cloth is more interesting.

The voile runs 0.545 to 3.237 over the same six per cent, which is 0.449 per point — more than twice the muslin’s. The sheeting, whose locus is short and whose curve is flat, runs 0.753 to 2.009, or 0.209.

An open cloth’s reciprocity fails twice as fast as a close one’s, so its measurement window is half as wide — and an open cloth is exactly where the mechanism is most visible and where somebody would most want to look.

That gives the ordering a physical reading. The rate is the curvature of the locus, and an open cloth has a long locus with a great deal of turning in it while a close one has a short locus that is nearly straight. The cloths whose behaviour is most obviously kinematic are the ones whose kinematic identity is hardest to catch.

And it explains a familiar disagreement #

The rate also accounts for something the trade reports and treats as scatter.

Two laboratories measuring the same fabric’s contraction at two different extensions will disagree, and the size of the disagreement is the rate times the difference in extension. At a fifth of a point per percentage point, a test at one per cent and a test at three per cent differ by 0.4 in a number of order one — forty per cent, from nothing but where the reading was taken.

That is larger than any repeatability either laboratory would admit to, and neither is wrong. A quoted Poisson ratio for a fabric is a reading at an extension, and the extension is almost never in the report — which is the same complaint this collection makes about a thickness quoted without a pressure and a dimension quoted without a state, arriving in the one place where the underlying identity is exact.

Every anomaly has the same cause #

Four separate misbehaviours, and one sentence accounts for all of them: the deformation is a mechanism running at constant thread length, not a material straining.

Above one half, because the bound comes from a volume argument about a continuum and a cloth has no volume constraint of that kind — the space between the threads is free to change. Not a constant, because the mechanism’s geometry changes as it runs, so its slope does too. Diverging at the ends, because a mechanism stops when it runs out of configuration and one of its members goes stationary there. Not reciprocal, because a one-parameter family measured at two different points of itself is being measured at two different states.

That is exactly the argument the bias is a mechanism makes about shear. There, a cloth cut at forty-five degrees extends by a third with nothing in it stretching, because the trellis has a degree of freedom that has nothing to do with the stiffness of its members; and there, too, the extension is bounded by a configuration rather than by a strength, the response is not linear, and calling it elasticity is a category error even though it looks exactly like stretch.

They are one fact seen twice. Shear is the mechanism available at an angle to the threads and crimp interchange is the one available along them, and both are changes of configuration in a structure whose members do not change length.

There is one difference between the two and it matters. Shear is available at almost no cost and crimp interchange is not. A bias strip moves under its own weight; a cloth pulled along the warp resists from the first fraction of a per cent, because straightening a thread means pushing its crossings apart against the cloth. This site cannot say by how much, because saying so needs a force and there is none in any of this.

Where the model stops #

Every claim here needs a cloth with two thread systems and a crimp, and none of it is a statement about materials. The ratio computed on this page is an exchange rate between two geometric quantities at constant thread length, and the reason it breaks every rule a material’s ratio obeys is that it was never one. A fibre has a Poisson ratio in the ordinary sense; the cloth made from it has a number that shares the name and nothing else.

The section is Peirce’s circle. The racetrack would move every figure on this page — a flattened yarn has less crimp for the same cloth and a different jam, so a different curve and a different slope. None of the four anomalies would go away, because none of them depends on the shape of the section.

There is no force, still. The site records this as an open shortfall and this rung does not close it. What is missing is not a refinement: it is why the tangent ratios are exact reciprocals here and are not in any real measurement, because a real cloth’s two directions differ in stiffness as well as in geometry and stiffness is what the reciprocity relation is actually about.

And the model is a plain weave, so a twill’s or a satin’s numbers are not these. The arguments carry over and the arithmetic does not.

Who found it, and when #

That woven fabrics have Poisson ratios above one half is well known in textile mechanics and has been measured many times; values above unity are routine and values above two are reported for closely-set cloths. The measurements came first and they came with the caveat attached, because anybody measuring a fabric this way notices immediately that the number moves.

What the trade does not generally have is the kinematic account of why — that all four departures fall out of one degree of freedom at constant thread length, and that the same argument gives the bias its behaviour. Textile mechanics has the biaxial theory, from the fabric-mechanics work that grew out of Peirce; garment engineering has the practical rule that a fabric’s contraction has to be measured at the strain it will see; and the two are rarely set beside one another.

The strongest evidence that the number is not a material property is a piece of ordinary trade practice. A fabric’s contraction under tension is specified in the state it was finished in and after a stated number of wash cycles, because it changes with both. Nobody quotes a modulus that way, and nobody would accept it.

Where the ladder goes next #

The three rungs of this anchor share one omission and it is the same one every time: no force. The geometry says which states a cloth can reach and says nothing about what it takes to move it between them, so the ladder can compute an exchange rate and cannot compute a stiffness, and cannot say how much cheaper shear is than interchange even though it can say that it is.

Closing that would mean the first quantity on this site that is not a property of the cloth’s geometry, and it would need a yarn’s transverse stiffness and the friction at a crossing — neither of which is anywhere in this site’s machinery, and both of which are measured rather than computed.

Below this rung are the biaxial case and the two bounds on extension; beside it, the four mechanisms a fabric has for getting longer and what each is worth.

What the pictures here cannot show. Every curve on this page is a slope taken from a set of reachable states. A real measurement of a fabric’s contraction is taken from a loading path with friction and hysteresis in it, so it does not return along the line it went out on and it does not fully recover — which is why a shrinkage figure is quoted after a stated number of cycles rather than as a property of the cloth.