The one fibre whose answer is known

Every number in this collection that came from a material rather than from a geometry came out of a table, and a table is a piece of machinery like any other: it has failure modes, and the worst of them is silent.

The ratio of a yarn’s two stiffnesses is 2G/E, and E comes from a table this site has carried from its earliest work. G is new, and it is a worse measurement than E by a considerable margin. That is not a reason to distrust it more than the rest; it is a reason to build the table so that its trustworthiness can be checked rather than asserted.

The way it is built here has one feature worth generalising: one of its ten rows has an answer that was known before anybody measured anything.

Why a shear modulus is worse known than a tensile one #

The asymmetry is experimental and it is worth understanding before reading any of the numbers.

To measure a fibre’s tensile modulus, clamp a known length, pull with a known force and measure the extension. The force is a few centinewtons, which an ordinary balance can read; the extension is a per cent or two of a centimetre, which an ordinary transducer can read. It is a routine measurement and every fibre house makes it daily.

To measure a fibre’s shear modulus, clamp a known length, apply a known torque and measure the rotation. The torque on a fibre twelve micrometres across is of order a nanonewton-millimetre. That is not a force anybody’s balance reads; it needs a torsion pendulum, and the usual method is to hang a known inertia from the fibre and time its oscillation, which then measures the rigidity through a period rather than directly.

So the two constants are not on the same footing, and any table that presents them side by side without saying so is misleading by its layout. That is the same complaint this collection makes about a diameter quoted without the twist it was measured at: the number is not wrong, the presentation is. The one here carries a working value and a range for both, and the shear ranges are wider — a factor of two on cotton, against a factor of about two and a half on cotton’s tensile modulus, which is already wide because cotton is a natural fibre with a population rather than a value.

The row that is not a measurement #

A drawn glass filament is an isotropic solid. Its molecules — such as they are; glass is a network rather than a chain — have no preferred direction, and drawing it into a filament does not give it one, because there is nothing to orient.

For an isotropic solid the two elastic constants are not independent. The shear modulus is the tensile modulus over twice one plus Poisson’s ratio, so

2G/E = 1/(1 + ν)

and glass’s Poisson’s ratio is 0.2, so the ratio must be five sixths, or 0.833. That is not a measurement of glass and it does not become one when somebody measures it. It is a consequence of isotropy plus one number that is itself very well known.

So the table has a control. If the glass row ever stops reading 0.833, the arithmetic that produces the column has drifted, and that is worth catching in the row where it is catchable rather than in the nine rows where it is not.

What the other nine rows say #

Every other row sits below the isotropic value, and none of them sits close to it. That is the finding, and it is a statement about how fibres are made rather than about which numbers happened to get into the table.

A textile fibre is stiff along its axis because its molecules are drawn out along it. Spinning a polyester and then drawing it four or five times its length is a process whose whole purpose is to align chains with the filament’s axis; growing a cotton hair deposits cellulose in fibrils wound in a steep helix; growing a flax fibre lays them almost parallel to the axis, which is why flax is the stiffest natural fibre in the table by a factor of seven and why a linen cloth drapes so unlike a cotton one of the same construction.

Stretching such a fibre stretches the chains, which are strong. Twisting it shears one chain past its neighbour, which is held by nothing but van der Waals forces and the occasional hydrogen bond. So the same orientation that raises E leaves G behind, and the ratio falls.

The prediction that follows is checkable against the table and is not circular, because the ratios were not arranged to satisfy it: the more oriented the fibre, the lower the ratio. Aramid, whose whole commercial existence is its chain alignment, is at 0.044 — twenty times below the isotropic value, and the reason aramid fibres are famous for fibrillating, which is chains sliding past one another because nothing much is holding them together sideways. Carbon is at 0.087 for the same reason. Flax at 0.083. At the other end, wool at 0.8: a keratin with a helical, cross-linked, poorly aligned structure, which is why it is the least stiff fibre in the collection and the most nearly isotropic.

What the ordering predicts that the table does not contain #

A claim is worth more when it says something the data it came from does not, and this one does.

If the ratio falls with orientation, then anything else that falls with orientation should track it. Two such things are already in this collection’s own tables and neither was consulted when the shear moduli were written down.

The first is the translation efficiency — the fraction of a fibre’s own tenacity that survives into the yarn made from it. Filaments translate at 0.85 and staple at 0.45 to 0.55, and the reason usually given is the staple ends, which is what a bundle being weaker than its threads is about. But among the filaments, the more oriented ones are the ones whose yarns lose most to abrasion and to knots, and the mechanism named for that is fibrillation, which is a shear failure.

The second is the friction coefficient, which this collection carries as a range for every fibre. A highly oriented filament is smooth along its axis and splits along it, and the two facts have the same cause.

Neither of those is a derivation and neither is offered as one. What they are is the kind of consistency a table earns rather than claims: a column produced for one purpose lining up with two columns produced for others.

A number that is a range, used as a number #

Every result on this ladder that uses a shear modulus inherits its range, and the ranges are not small. Cotton’s runs from 0.7 to 1.5 gigapascals, which is a factor of a little over two, so cotton’s ratio runs from 0.175 to 0.375.

The convention adopted here is the collection’s own and is the same one used for the bending bracket: compute at the working value, quote the range, and say which was used. What is not done is to propagate the range through every subsequent calculation and report a band on everything, because that produces results nobody can read and hides the difference between a quantity whose range matters and one whose does not. The collection has made the opposite mistake and recorded it: quoting a single yarn diameter for a population hid a spread that changed several answers.

The distinction is worth stating, because it decides which results on this ladder are worth arguing about. A result that depends on the ratio linearly carries the factor of two: the balanced fold ratio is one of those, and it is quoted as a range for that reason. A result that depends on the ratio through a threshold carries it differently: a yarn either snarls or does not, and moving the ratio moves the tension at which it changes over rather than making the snarl partly happen. And a result that depends on the ratio through a comparison between two fibres barely carries it at all, because the same measurement method was used on both and the errors are correlated.

That last category is where the ordering above sits, and it is why the ordering is more trustworthy than any individual row in it.

The trap this avoids, which is a real one #

The obvious alternative to a table like this is to quote a single figure for the ratio and move on. It is not a foolish thing to do: two thirds is the isotropic value at a Poisson’s ratio of a half, engineering handbooks quote it, and this collection itself said “near two thirds” when it first noticed that it had no torsional rigidity at all.

The cost of doing that would have been large and quiet.

Take a flax at the isotropic two thirds instead of its own 0.083, and the torsional rigidity is overstated eightfold. That does not make any result absurd. It makes a torque eight times what it should be, and a torque times a twist is an energy, and the energy would still be a plausible-looking fraction of the bending energy. Nothing would look wrong. Every gate in this collection would pass, because there is no gate anywhere that asks whether a constant is right.

That is the ordinary way a wrong constant does damage here, and it has happened before. Two shape constants read from different rows of a relaxation table sat in this collection’s own arithmetic for several rungs and moved every occupancy by five per cent, and the assertion guarding them checked that the occupancy was proportional to the tightness factor — which is true for any pair of constants. The lesson recorded then is that an assertion on a relationship cannot check its inputs, and the control row is this rung’s answer to it: a check on an input, in the one place an input has a known value.

The ratio does not know about the count #

One more thing the table has to survive, and it is cheap to check.

The ratio is 2G/E, and neither G nor E is a function of the yarn count, the packing factor or the fibre count. So the whole column must be invariant to all three, and any dependence appearing in it would mean the arithmetic had picked up a diameter somewhere it should not have.

What was counted, and how #

The table holds a working shear modulus and a range for each of the ten fibres the collection carries, in gigapascals. The tensile moduli beside them are unchanged and are the site’s own, in place from its earliest work.

Three checks run on it, and all three could fail.

The two tables name the same fibres. A density in one file and a modulus in another is two places for a fibre to exist in half, and the site already makes this check between its density and mechanics tables. Making it again for the third table is cheap and catches the one failure that is otherwise invisible: a value multiplied into a result three functions later as an undefined.

Every working value lies inside its own range. A range that does not contain its own working value is a row somebody edited half of.

Glass satisfies the isotropic relation to one per cent, and every other row sits below it. The second half of that check is the one that does real work, because it is what turns the ordering from a list into a claim.

Where the model stops #

A fibre is not a rod and its shear modulus is not one number. A cotton hair has a lumen, a reversal every few hundred micrometres, and a fibril angle that changes along it; a wool fibre has two cortices with different properties side by side. Quoting one shear modulus for such a thing is the same idealisation as quoting one tensile modulus, and this collection has argued at length that a cloth is a population rather than a thread. The table is a population’s mean and it is used as one.

Poisson’s ratio for the other nine is not in the table, and cannot be. For an anisotropic fibre there are several Poisson’s ratios and none of them is what the isotropic formula wants. That is precisely why the other nine rows need a measurement and glass does not, and it is why the formula is used as a control rather than as a source.

And the ranges are not error bars. They are the spread of what different measurements on different specimens have reported, which mixes real variation between fibres with real disagreement between methods. Nothing here separates the two, so a result quoted across the range is quoted across both.

The generalisation #

The useful shape here is not about fibres. It is about how to hold a table of constants that cannot all be checked.

Look for the member whose answer is forced. In any family of measured quantities there is often one case where theory supplies the answer without measurement — an isotropic member, a limiting case, a degenerate geometry — and that member is worth including in the table even if nobody will ever use it. Nobody is knitting glass filament. The glass row is there so that the nine rows around it can be read as a physical ordering.

That pattern recurs across this collection and has not been named before. The satin count that has no solution at six ends is a forced case in an enumeration. The refusal of a thickness for a tubular fabric is a forced case in a per-course model. And the requirement that every generator’s assertions still reject something is the same instinct applied to machinery rather than to data: a check that has never refused anything has not been shown to work.

A table with no forced member is a table whose correctness is a matter of trust. Trust is not a property this collection is willing to depend on anywhere else, and there is no reason to make an exception for a column of numbers.

What this says about the rest of this collection’s tables #

The site carries four material tables now: densities, mechanics, transverse properties, and this one. Three of them have no forced member at all.

The density table comes closest to not needing one, because a fibre density is a bulk measurement anybody can make with a pycnometer and the numbers have been stable for a century. The mechanics table does not have one and would benefit: glass again would serve, because a glass filament’s tensile modulus is the bulk glass’s and is known independently of any fibre measurement.

The transverse table is the interesting case, because it cannot have one. Its own lower bound is exactly zero — a bundle of fibres free to slide resists a change of section not at all — so the bracket has no floor and no theory forces any value inside it. That is why the racetrack’s aspect ratio has been a free parameter on this site for a very long time, swept rather than predicted, and why a jammed warp’s thickness has always been quoted against an assumed flattening rather than a computed one.

A table that cannot have a control is a table whose values have to come from the cloth rather than from the fibre. That is not a defect of the table; it is a fact about the quantity, and the honest response is to measure the cloth. What makes it worth saying here is that the two situations look identical on the page — a column of numbers with ranges — and they are not the same kind of object at all.

Who found it, and when #

The isotropic relation between the two moduli is Poisson’s and dates from the 1820s. Fibre shear moduli have been measured by torsion pendulum since Meredith’s work in the 1950s, and the ranges in the table are wide because the method is delicate and because the specimens differ.

What is this collection’s own is only the arrangement: putting the one forced row in the table on purpose, and reading the ordering of the rest as a consequence of orientation rather than as an assortment of measurements.

Where the ladder goes next #

The ratio now exists as a number with a defensible provenance, and the next thing to do with it is to use it somewhere the answer can be checked against something other than another table.

Two such places follow immediately. A twisted yarn left slack stops being straight, and the tension that stops it doing so is a function of the ratio and the twist alone — so a yarn snarling is an experiment anybody has already done. And a folded yarn’s torque balance is a ratio of two moments, so it too is bracket-free, and what it says about the trade’s folding rule turns out to be that the rule is about something else entirely.